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Euclidian Geometry
Francis Cuthbertson
From inside the book
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Page 161 ... proportional then any others which have the same ratio as the first and second are proportional to any others which have the same ratio as the third and fourth . For the test expressed
by the definition is satisfied . PROPOSITION IV ...
Page 162 ... proportional . Then rect . ( a , d ) is = rect . ( b , c ) . On a , b construct rectangles , the altitude of each being equal to c , and on c , d construct rectangles , the altitude of
each being equal to a . By hypothesis a b as c : d ...
Page 163 ... proportional . Hence , if three straight lines are proportional the rect- angle contained by the extremes is equal to the square on the mean : and conversely . Hence may be easily
established the following Pro- positions : II - 2 ...
Page 164 ... proportional , then a cas b d . For a b as c : d ; ... rect . ( a , d ) = rect . ( b , c ) ; .. ac as b : d . PROPOSITION VII . If a b as c : d , then ba as d c . For rect . ( a , d ) =
rect . ( b , c ) , and .. ba as d : c ...
Page 167 ... proportional to certain straight lines and angles to certain arcs . Moreover PROPOSITION XIII . If A , B , H be areas and c , d , k straight lines , such that A : B as cd , and B : H as
d : k , then A : H as c : k . For , if be any ...
Popular passages
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The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any
equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth...
... and the principles on which the observations made with these instruments are treated for deduction of the distances and weights of the bo.iits of the Solar System, and of a few stars, omitting
all minutiit of Elementary Class- Books — continued. formulcE, and all troublesome details of calculation.
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What is good Action Space and Hyperparameter for Kumo Torakku track?
• Più recenti
• Maggior numero di voti
• Maggior numero di commenti
Hi Changsoo
Hyperparameters directly impact how the model is updated, they control the settings of the optimization algorithm that is used to "solve" for the model that gives the maximum expected cumulative
return. Changing hyperparameters can improve the convergence of the model, or worsen it. For example, if you increase the learning rate, the weights in your neural network will update with larger
increments. The model may improve (train) faster but the risk is that you miss the optimal solution, or the model never converges as updates are too large. Finding good hyperparameters often required
trying a number of different combinations and then evaluating the performance of the model vs time spent training or some other metric. For example, I am busy training a 3m/s model (with 2 speed
granularity) using a learning race of 0.001 and a low number of epochs. I can see during training at around 90 minutes my model is starting to do a lap now and then. If the learning rate was smaller,
it would probably take longer for my model to complete a lap.
Note that at 3m/s my model will not be as fast as a converged 5m/s (or faster) model, but those will take a long time to converge. We increased the training speed in the console to a max of 8 m/s.
Training at speeds faster than 8m/s tends to send the model spinning off the track.
Kind regards
De Clercq
Hi Changsoo
I did the following tests overnight to show impact of hyperparameters
Trained 4 models on the Kumo Torakku training, each for 180 minutes, using my own reward function that does some center line following, scales reward for driving fast etc.
I alternated
Model 1: 3 m/s 2 speed granularity with learning rate = 0.001 and epochs = 3
Model 2: 3 m/s 2 speed granularity with default hyperparameters
Model 3: 5 m/s 2 speed granularity with learning rate = 0.001 and epochs = 3
Model 4: 5 m/s 2 speed granularity with default hyperparameters
Doing 5 lap evaluation on Kumo Torakku training, showing lap completion percentages
Model 1: 100% 100% 100% 100% 100%
Model 2: 46% 67% 61% 100% 62%
Model 3: 70% 58% 100% 100% 100%
Model 4: 63% 88% 100% 36% 27%
This shows you the impact of playing with the hyperparameters.
Kind regards
De Clercq
Thanks a lot.
It has been a great help.
Hi @DeClercq-AWS,
I'm using your script, but i have a couple of questions:
1- Are Yaw and Steering on Degrees or Radians? It seems to be in Radians.
2- How can we include other parameters?
3- It seems Track Width is not returning the right value. How can we confirm?
4- Is Progress defined from 0-1 or 0-100? It seems it's based on 0-1, but the documentation says 0-100.
Edited by: cladeira on Jun 23, 2019 2:06 PM
The progress in the docs says its a float between 0-100? Or is it 0-1? :)
Progress is a float 0-100.
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The vector(s) which is/are coplanar with vectors \(\hat{i}+\hat{j}+2 \hat{k}\) and \(\hat{i}+2 \hat{j}+\hat{k}\), and perpendicular to the vector \(\hat{i}+\hat{j}+\hat{k}\) is/are (A) \(\hat{j}-\hat
{k}\) (B) \(-\hat{i}+\hat{j}\) (C) \(\hat{i}-\hat{j}\) (D) \(-\hat{j}+\hat{k}\)
Short Answer
Expert verified
None of the options are perpendicular to \((\hat{i}+\hat{j}+\hat{k})\), so none are coplanar with both of the other vectors.
Step by step solution
- Find the Normal Vector
To determine the vectors that are coplanar with the given vectors, first we find a normal vector to the plane that contains them by taking their cross product.
- Calculate the Cross Product
The cross product of \((\hat{i}+\hat{j}+2\hat{k})\) and \((\hat{i}+2\hat{j}+\hat{k})\) can be computed using the determinant method or by components. This vector will be perpendicular to the plane
containing the original two vectors.
- Analyze the Perpendicular Vector
The vector(s) perpendicular to \((\hat{i}+\hat{j}+\hat{k})\) will have a dot product of zero with it. We check each option (A), (B), (C), (D) to see which is perpendicular to this vector.
- Determine the Coplanar and Perpendicular Vector(s)
By calculating the dot product of each option with \((\hat{i}+\hat{j}+\hat{k})\), we find the vector(s) that satisfy the conditions of being coplanar with the given vectors and perpendicular to \((\
- Select the Correct Answer
After calculating, we'll select the option(s) where the dot product with \((\hat{i}+\hat{j}+\hat{k})\) equals to zero, as those will be the correct answers.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Cross Product
Understanding the vector cross product is crucial for solving problems involving three-dimensional vectors, such as finding a vector that is perpendicular to a plane. The cross product, denoted as \(
\times \) between two vectors, results in a third vector that is perpendicular to the plane formed by the original two vectors. The magnitude of this new vector reflects the area of the parallelogram
spanned by the two original vectors.
For example, to calculate the cross product of vectors \( \vec{a} \) and \( \vec{b} \) written in component form as \( \vec{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k} \) and \( \vec{b} = b_1\hat{i}+b_2\
hat{j}+b_3\hat{k} \) respectively, we can arrange their components in a 3x3 matrix, placing the unit vectors in the first row and each vector's components in subsequent rows. The resulting vector has
components calculated by subtracting the products of the diagonal terms moving in opposite directions.
It's important to follow the right-hand rule, which ensures the direction of the cross product vector is correct relative to the original vectors.
Dot Product
The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It's denoted by a dot \(\cdot\) between two
vectors, for example, \(\vec{a} \cdot \vec{b}\). This operation essentially measures how much one vector extends in the direction of the other.
The dot product of two vectors \(\vec{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\) and \(\vec{b} = b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\) would be calculated as \(\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 +
a_3b_3\). If the dot product of two nonzero vectors is zero, they are considered to be orthogonal, or perpendicular to each other. Knowing this property is essential when trying to determine whether
vectors are perpendicular, as required in the exercise provided.
Perpendicular Vectors
Perpendicular vectors in three-dimensional geometry have a unique relationship: they meet at a 90-degree angle, and their dot product equals zero. This property is fundamental when we deal with
orthogonality in vector algebra.
For vectors to be called perpendicular or orthogonal, they don't need to intersect at a specific point in space; they just need to have that 90-degree angle relation with respect to their directions.
As mentioned earlier with the dot product, when we get a result of zero, we can confidently state that the vectors are indeed perpendicular to each other. This concept plays a significant role in the
step-by-step solution provided for the exercise, where checking for the zero dot product helps us identify the correct vector(s) that are perpendicular to a given vector.
Determinant Method
The determinant method is a mathematical approach often applied in vector calculus and linear algebra to compute the cross product of two vectors. It simplifies the calculation by treating the
vectors as rows in a matrix and calculating the determinant. The matrix is usually composed of the unit vectors \(\hat i, \hat j, \hat k\) in the top row and the components of the vectors in the
subsequent rows.
For two vectors \(\vec{a}\) and \(\vec{b}\) given by their components as described previously, we create a matrix with \(\hat i, \hat j, \hat k\) on the top row, the components of \(\vec{a}\) on the
second row, and the components of \(\vec{b}\) on the third row. The determinant of this matrix, with appropriate signs applied to each minor determinant, will give the components of the resulting
vector. This method is very useful for solving vector-related problems, such as finding normal vectors to planes, which is a necessary step addressed in the provided exercise and solution.
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Int'l J. of Communications, Network and System Sciences
Vol.5 No.11(2012), Article ID:24669,7 pages DOI:10.4236/ijcns.2012.511077
Employing Power Allocation to Enhance Zero Forcing Scheme Advantages over Multi-Antenna Multiple Relay Networks
Department of Electrical Engineering (DCCS Lab), Iran University of Science and Technology, Tehran, Iran
Email: afalahati@iust.ac.ir
Received January 21, 2012; revised July 11, 2012; accepted September 16, 2012
Keywords: Cooperative Communication; MIMO; Multiple Antennas Multiple Relay (MAMR) Networks; Zero Forcing; Power Allocation
A multi-antenna multiple relay (MAMR) network is considered and a variation of two-hop zero-forcing amplify-forward relaying method is proposed. Deploying ZF method together with application of
diagonal power allocation matrices at the relays, it is shown that the overall MAMR network is simplified to M independent single antenna multiple relay (SAMR) networks, where M is the number of
source and destination antennas. This enables to incorporate network beamforming proposed for SAMR networks. Accordingly, using the BER as the performance metric, we present simulation results to
show the proposed approach outperforms the common ZF method addressed in the literature.
1. Introduction
It is well established that in most cases relaying techniques provide considerable advantages over direct transmission, provided that the source and relay cooperate efficiently. The choice of relay
function is especially important as it directly affects the potential capacity benefits of node cooperation [1-5]. In this regard, two relaying methods, Amplify-Forward (AF) [6,7] and
estimate-forward (EF) [8,9], are extensively addressed in the literature. As the names imply, the former just amplifies the received signal but the latter estimates the signal with errors and then
forwards it to the destination.
It has been shown that increasing the number of relays has the advantage of increasing the diversity gain and flexibility of the network. However, it renders some new issues to arise [10]. For
instance, the relaying algorithm and power allocation across relays should be addressed in such cases. Relay selection [11,12] and power allocation [13,14] are two well-known methods when the power
management issues are dealt with.
The capacity and reliability of the relay channel can be further improved by using multiple antennas at each node. The use of relays together with multiple antennas has made it a versatile technique
to be used in emerging wireless technologies [15-20]. Relaying strategies for the multi-antenna multiple relay (MAMR) network are more challenging than single antenna network.
AF Multi-Input Multi-Output (MIMO) relay systems have drawn considerable attention in the literature due to their simplicity and ease of implementation. In this regard, a plethora of works are
devoted to finding a proper relaying strategy for AF MAMR networks. In [21], the idea of linear distributed multi-antenna relay beamforming (LDMRB) is introduced where each relay performs a linear
reception and transmission in addition to output power normalization. The linear operations suggested in this paper are Matched Filter (MF), Zero Forcing (ZF) and Minimum Mean Square Error (MMSE).
They are briefly called MF-MF, ZF-ZF and MMSE-MMSE schemes, respectively. In [22], a method based on QR decomposition is suggested which has better performance than the ZF-ZF scheme. Combinations of
various schemes are also considered in [22].
In [23], the so-called incremental cooperative beamforming is introduced and it is shown that it can achieve the network capacity in the asymptotic case for large K with a gap no more than.
In [24], a wireless sensor network composed of a few multi-antenna sensors aimed to transmit a noisy measurement vector parameter to the fusion centre is formulated as a MAMR network.
In [25], it is shown that a MAMR network with single antenna source and destination can be transformed into a single antenna multiple relay (SAMR) network by performing Maximal Ratio Combining (MRC)
at reception and transmission for each relay nodes. This enables the network beamforming introduced in [14] to be readily employed. Indeed, this manuscript is an extension of [25] where, here we
assume the K independent sources send independent data streams to their respected single antenna destinations.
Latest developments are mentioned as the LDMRB to enhance its performance i.e. in [26] MF and MSE are used in reception and transmission of each relay respectively. Although the performance of this
method is better than the proposed method in the current paper, but the current paper is to apply power allocation between relays in ZF-ZF scheme and can be developed further in the future.
Furthermore it is noteworthy that recent papers on this subject use numerical optimization to find the optimized relay matrices, i.e. [27] but these methods are too complicated to implement. Thus we
compare the proposed method with its ancestor (ZF-ZF) method.
In this paper, the idea of LDMRB is used where ZF algorithm is utilized in both reception and transmission. It is shown that using this method the overall MAMR network can be transformed into M
independent SAMR networks. Then the idea of network beamforming that is suggested in [14] for SAMR network can be used to allocate power to any data streams in relays. In other words, in each relay
at first the transmitted vector is estimated using ZF and then based on the network beamforming algorithm [14], the power of each element of the estimated vector is controlled and then it is
forwarded to the destination using ZF precoding.
2. The Relay Network System Model
Figure 1 illustrates a typical MAMR relay network system in which the source and destination have M antennas each. It is assumed that there are K multiple-antenna relays, each having
We consider x as a M × 1 vector whose elements are independent zero mean random variables with covariance matrix
where n[i] is a
Figure 1. A typical MAMR relay system model.
matrix [0] is the noise power associated with each entry[i] are zero mean unit variance drawn from an independent complex Gaussian distribution. But H[i]’s are known at all relays. Moreover, (.)^H is
a Hermitian operation. Assuming the ith relay multiplies its received signal by a weight matrix
where Figure 1:
where [0]. Finally, n[i] for [i] for
3. The ZF-ZF LDMRB Scheme
Suppose a linear MAMR relay in which the relay performs linear operations at both reception and transmission. One can decompose the relay weight matrix
where the superscripts (t), (p) and (r) indicate transmission, power allocation and receiving operations, respectively. Moreover,
Considering the estimated transmitted vector at the ith relay as:
Thus, one can rewrite the transmitted symbol of the ith relay as:
In [21], it is shown that in the absence of power allocation
and similarly, we have:
So the estimated or demodulated vector of the ith relay becomes:
Here it is assumed that the number of relay antennas [i] > M. Moreover, the resulting noise vector at the ith relay is
Note that, the jth entry of the resulting noise vector at the ith relay is
If the receiving matrix is redefined as
The demodulated symbol at the ith relay can now be written as:
Thus the jth^ entry of
where N[0] is the power of
Thus, substituting
As a result, the jth element of the received vector at destination or the received symbol at the jth destination antenna can be represented as:
It can be seen that all interferences are canceled using zero forcing scheme and the symbol received in the jth destination antenna just depends on the symbol transmitted by the jth source antenna.
This resembles to a SAMR network. Therefore, we have M independent SAMR network and hence the so called network beamforming can be applied.
4. The Power Allocation Algorithm
The SNR of the received symbol at the jth destination antenna can be computed as,
It is desired to find
N[0] is a constant term and thus, can be discarded from optimization. We define the following vectors and matrix associated with the jth symbol stream as follows,
The SNR can now be written as:
This relation is similar to the network beamforming problem for single antenna network [14]. Hence, one can apply the optimal power allocation proposed in [14] by Jing et al. If we define
The Jing algorithm [14] is briefly presented here. At first, each relay computes the following parameters:
These are also computed at destination. They are sorted in descending order as follows:
Then the optimal power allocation is obtained as:
where k[0] is the smallest k such that
This procedure is performed for all SAMR networks and
The proposed method can be briefly explained as follows. At first, the receiving
5. Simulation Results
In the simulation of a MAMR network here, the large scale fading experienced by the relays is assumed to be the same. The channel matrices are generated independently during subsequent iterations. It
is also assumed that the first and second hop channels for all relays are known to all nodes. Furthermore, an uncoded QPSK modulation is used and independent symbol sequences is transmitted by each
source antennas.
Figure 2 depicts the overall MAMR system BER performance for a network with 2 relays each with 2 antennas arranged for simulation. As it can be seen for all SNR values, the power allocated ZF method
outperforms the ZF method without power allocation.
Figure 3 shows the BER simulation performance of networks with 4, 6 and 8 relays each with 4 antennas. In this figure, the numbers appeared after the scheme’s name in the legend box determine the
number of nodes and nodes antennas. The first three numbers determines the number of nodes. For instance, 131 means one source, three relays and one destination. The next three numbers determines
number of node’s antennas. For instance, 444 means source, relay and destination each have 4 antennas. In Figure 3 the effect of the number of relays on the performance of the network is evaluated.
It can be seen that Increasing the number of relays, PA’s improvement increases and hence the gap between non power allocated and power allocated ZF increases.
In Figure 4, the number of the source and destination antennas is kept fixed to 2. The number of relays is also fixed to 3 but the relays antennas number varies from 2 to 4. It can be observed that,
by increasing the number of relays antennas, PA’s are improved further. This is indeed for the inevitable increase in diversity to allow more relays to transmit with their full power.
6. Conclusions
A new signaling method for Multi-antenna Multiple Relay networks (MAMR) with the aid of ZF-ZF method at
Figure 2. MAMR network with 2 relay each with 2 antennas.
Figure 3. MAMR network with 4, 6 and 8 relays each with 4 antennas.
Figure 4. Comparison of the designed relay network with the same number of relays and various number of antennas.
the relays is proposed to transform the original network into several single-antenna relay networks. This helps to mitigate the interference term between individual data streams transmitted from the
individual source antennas. Accordingly, the network beamforming which is proved to be the optimal power allocation method for SAMR network [14] is being used.
Simulation results indicate that the proposed method improves the BER comparing with the naive ZF-ZF method in the absence of power allocation.
Future works: the amount of power that is not used for one data stream in a relay can be used by other data stream. This is not considered in this paper and can be the subject for a future work. Also
this method can be generalized to MMSE-MMSE and QR-QR schemes.
7. Acknowledgements
The authors thank ITRC (Iran Telecommunication Research Center) for supporting this project financially.
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23. S. O. Gharan, A. Bayesteh and A. K. Khandani, “Asymptotic Analysis of Amplify and Forward Relaying in a Parallel MIMO Relay Network,” 45th Annual Allerton Conference on Communication, Control,
and Computing, Monticello, 26-28 September 2007.
24. J. Xiao, S. Cui, Z.-Q. Luo and A. J. Goldsmith, “Linear Coherent Decentralized Estimation,” IEEE Transactions on Signal Processing, Vol. 56, No. 2, 2008, pp. 757-770. doi:10.1109/TSP.2007.906762
25. Y. A. Izi and A. Falahati, “On the Cooperation and Power Allocation Schemes for Multiple-Antenna Multiple-Relay Networks,” 5th International Conference on Wireless and Mobile Communications,
Cannes, 23-29 August 2009, pp. 44-48.
26. Y. Zhang, H. W. Luo and W. Chen, “Efficient Relay Beamforming Design With SIC Detection for Dual-Hop MIMO Relay Networks,” IEEE Transactions on Vehicular Technology, Vol. 59, No. 8, 2010, pp.
4192-4197. doi:10.1109/TVT.2010.2065249
27. Y. A. Izi and A. Falahati, “Amplify-Forward Relaying for Multiple-Antenna Multiple Relay Networks under Individual Power Constraint at Each Relay,” EURASIP Journal on Wireless Communications and
Networking, Vol. 2012, No. 1, 2012.
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What is the area of a circle with circumference 51.81 meters?
Here is the answer to questions like: how to find the area of a circle with circumference 51.81 meters?
Circle Calculator
The area of a circle with circumference 51.81 is 213.6
A = πr^2 = π(d2)^2
A = C^24π
π = 3.1415
A = area
C = circumference or perimeter
r = radius, d = diameter
Solution Steps:
Area of a circle in terms of radius:
Area = π·r^2 = 3.14·8.25^2 = 213.6 square meters^(*)
Area of a circle in terms of diameter:
Area = π·(d2)^2 = 3.14·(16.492)^2 = 3.14·(8.25)^2 = 213.6 square meters^(*)
Area of a circle in terms of circumference:
Area = C^24π = 51.81^24π = 2684.28(4·3.14) = 2684.2812.56 = 213.6 square meters^(*)
^(*) 213.60790496922 meters, exactly or limited to the precision of this calculator (13 decimal places).
Note: for simplicity, the operations above were rounded to 2 decimal places and π was rounded to 3.14.
Result in other units of area:
A circle of radius = 8.246 or diameter = 16.49 or circumference = 51.81 meters has an area of:
• 0.0002136 square kilometers (km²)
• 213.6 square meters (m²)
• 2136000 square centimeters (cm²)
• 2.136 × 10^8 square millimeters (mm²)
• 8.24714E-5 square miles (mi²)
• 255.463 square yards (yd²)
• 2299.17 square feet (ft²)
• 331081 square inches (in²)
Site map
Use the this circle area calculator below to find the area of a circle given its circumference, or other parameters. To calculate the area, you just need to enter a positive numeric value in one of
the 3 fields of the calculator. You can also see at the bottom of the calculator, the step-by-step solution.
Formula for area of a circle
Here a three ways to find the area of a circle (formulas):
Circle area formula in terms of radius
A = πr^2
Circle area formula in terms of diameter
A = π(d2)^2
Circle area formula in terms of circumference
A = C^24π
See below some definitions related to the formulas:
Circumference is the linear distance around the circle edge.
The radius of a circle is any of the line segments from its center to its perimeter. The radius is half the diameter or r = d2.
The diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. The diameter is twice the radius or d = 2·r.
The Greek letter π
π represents the number Pi which is defined as the ratio of the circumference of a circle to its diameter or π = Cd . For simplicity, you can use Pi = 3.14 or Pi = 3.1415. Pi is an irrational number.
The first 100 digits of Pi are: 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ...
If you input the radius in centimeters, you will get the answer in square centimeters (cm²), if in inches, will get the answer in square inches (in²) and so on ...
Circumference is often misspelled as circunference.
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After performing Experiment 3, you have determined that your second equivalence point occurred after adding 42.50 mL of 0.1000 MHCl. Assuming that...
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.
After performing Experiment 3, you have determined that your second equivalence point occurred after adding 42.50 mL of 0.1000 MHCl. Assuming that...
After performing Experiment 3, you have determined that your second equivalence point occurred after adding 42.50 mL of 0.1000 MHCl. Assuming that you dissolved your unknown sample in 100.00 mL of
water, calculate the pH of the solution after 17.50 mL of HCl has been added? (Note: Kw = 1.00 x 10-14, Ka1= 4.44 x 10-7 and Ka2= 4.69 x 10-11) Report your pH to two significant figures.
Show more
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Which of the values of x and y makes the following matrices equal:
Hint: In order to solve this problem we need to know if two matrices A and B are said to be equal if A and B have the same order and their corresponding elements are equal. Corresponding elements of
the matrix A and the matrix B are equal, that is the entries of the matrix A and the matrix B in the same position are equal. Knowing this will solve your problem.
Complete step-by-step answer:
We need to find the values of x and y that can make the following matrices equal:
$\left[ \begin{gathered}
3x + 7\,\,\,\,\,\,5 \\
y + 1\,\,\,\,\,2 - 3x \\
\end{gathered} \right] = \left[ \begin{gathered}
0\,\,\,\,\,\,y - 2 \\
8\,\,\,\,\,\,\,\,\,\,4 \\
\end{gathered} \right]$
Two matrices A and B are said to be equal if A and B have the same order and their corresponding elements are equal. Corresponding elements of the matrix A and the matrix B are equal, that is the
entries of the matrix A and the matrix B in the same position are equal.
So, we do
3x + 7 = 0
So, the value of x is $\dfrac{{ - 7}}{3}$.
If we want to check whether the value of x is correct or not then we will check the last element of the matrices that is 2 – 3x = 4
And here the value of x is $\dfrac{{ - 2}}{3}$.
Therefore x cannot have two values in the same matrices. So it is unable to find the values and finding the value of y is also of no use.
Therefore, the correct answer to this problem is B, not possible to find.
Note:When you get to solve such problems you need to know that one variable cannot have different values in the same matrices. Here we have found two different values so either the matrices are not
equal or the matrices given are wrong.
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Reduced-form framework under model uncertainty and generalized Feynman-Kac formula in the G-setting
The thesis dealing with topics under model uncertainty consists of two main parts. In the first part, we introduce a reduced-form framework in the presence of multiple default times under model
uncertainty. In particular, we define a sublinear conditional operator with respect to a family of possibly non-dominated priors for a filtration progressively enlarged by multiple ordered defaults.
Moreover, we analyze the properties of this sublinear conditional expectation as a pricing instrument and consider an application to insurance market modeling with non-linear affine intensities. In
the second part of this thesis, we prove a Feynman-Kac formula under volatility uncertainty which allows to take into account a discounting factor. In the first part, we generalize the results of a
reduced-form framework under model uncertainty for a single default time in order to consider multiple ordered default times. The construction of these default times is based on a generalization of
the Cox model under model uncertainty. Within this setting, we progressively enlarge a reference filtration by N ordered default times and define the sublinear expectation with respect to the
enlarged filtration and a set of possibly non-dominated probability measures. We derive a weak dynamic programming principle for the operator and use it for the valuation of credit portfolio
derivatives under model uncertainty. Moreover, we analyze the properties of the operator as a pricing instrument under model uncertainty. First, we derive some robust superhedging duality results for
payment streams, which allow to interpret the operator as a pricing instrument in the context of superhedging. Second, we use the operator to price a contingent claim such that the extended market is
still arbitrage-free in the sense of “no arbitrage of the first kind”. Moreover, we provide some conditions which guarantee the existence of a modification of the operator which has quasi-sure càdlàg
paths. Finally, we conclude this part by an application to insurance market modeling. For this purpose, we extend the reduced-form framework under model uncertainty for a single default time to
include intensities following a non-linear affine process under parameter uncertainty. This allows to introduce a longevity bond under model uncertainty in a way consistent with the classical case
under a single prior and to compute its valuation numerically. In the second part, we focus on volatility uncertainty and, more specifically on the G-expectation setting. In this setting, we provide
a generalization of a Feynman-Kac formula under volatility uncertainty in presence of a linear term in the PDE due to discounting. We state our result under different hypothesis with respect to the
current result in the literature, where the Lipschitz continuity of some functionals is assumed, which is not necessarily satisfied in our setting. Thus, we establish for the first time a relation
between non-linear PDEs and G-conditional expectation of a discounted payoff. To do so, we introduce a family of fully non-linear PDEs identified by a regularizing parameter with terminal condition φ
at time T > 0, and obtain the G-conditional expectation of a discounted payoff as the limit of the solutions of such a family of PDEs when the regularity parameter goes to zero. Using a stability
result, we can prove that such a limit is a viscosity solution of the limit PDE. Therefore, we are able to show that the G-conditional expectation of the discounted payoff is a solution of the PDE.
In applications, this permits to calculate such a sublinear expectation in a computationally efficient way.
Reduced-form framework under model uncertainty and generalized Feynman-Kac formula in the G-setting / Oberpriller, Katharina. - (2022 May 19).
Reduced-form framework under model uncertainty and generalized Feynman-Kac formula in the G-setting
The thesis dealing with topics under model uncertainty consists of two main parts. In the first part, we introduce a reduced-form framework in the presence of multiple default times under model
uncertainty. In particular, we define a sublinear conditional operator with respect to a family of possibly non-dominated priors for a filtration progressively enlarged by multiple ordered defaults.
Moreover, we analyze the properties of this sublinear conditional expectation as a pricing instrument and consider an application to insurance market modeling with non-linear affine intensities. In
the second part of this thesis, we prove a Feynman-Kac formula under volatility uncertainty which allows to take into account a discounting factor. In the first part, we generalize the results of a
reduced-form framework under model uncertainty for a single default time in order to consider multiple ordered default times. The construction of these default times is based on a generalization of
the Cox model under model uncertainty. Within this setting, we progressively enlarge a reference filtration by N ordered default times and define the sublinear expectation with respect to the
enlarged filtration and a set of possibly non-dominated probability measures. We derive a weak dynamic programming principle for the operator and use it for the valuation of credit portfolio
derivatives under model uncertainty. Moreover, we analyze the properties of the operator as a pricing instrument under model uncertainty. First, we derive some robust superhedging duality results for
payment streams, which allow to interpret the operator as a pricing instrument in the context of superhedging. Second, we use the operator to price a contingent claim such that the extended market is
still arbitrage-free in the sense of “no arbitrage of the first kind”. Moreover, we provide some conditions which guarantee the existence of a modification of the operator which has quasi-sure càdlàg
paths. Finally, we conclude this part by an application to insurance market modeling. For this purpose, we extend the reduced-form framework under model uncertainty for a single default time to
include intensities following a non-linear affine process under parameter uncertainty. This allows to introduce a longevity bond under model uncertainty in a way consistent with the classical case
under a single prior and to compute its valuation numerically. In the second part, we focus on volatility uncertainty and, more specifically on the G-expectation setting. In this setting, we provide
a generalization of a Feynman-Kac formula under volatility uncertainty in presence of a linear term in the PDE due to discounting. We state our result under different hypothesis with respect to the
current result in the literature, where the Lipschitz continuity of some functionals is assumed, which is not necessarily satisfied in our setting. Thus, we establish for the first time a relation
between non-linear PDEs and G-conditional expectation of a discounted payoff. To do so, we introduce a family of fully non-linear PDEs identified by a regularizing parameter with terminal condition φ
at time T > 0, and obtain the G-conditional expectation of a discounted payoff as the limit of the solutions of such a family of PDEs when the regularity parameter goes to zero. Using a stability
result, we can prove that such a limit is a viscosity solution of the limit PDE. Therefore, we are able to show that the G-conditional expectation of the discounted payoff is a solution of the PDE.
In applications, this permits to calculate such a sublinear expectation in a computationally efficient way.
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Chart Data Archives - Peltier Tech
The Problem: Your Y data is in more than a single row or column.
If you try to populate a chart series with 2D data where it isn’t allowed, you’ll encounter this error:
The reference is not valid. References for titles, values, sizes, or data labels must be a single cell, row, or column.
This isn’t strictly true. Not one of the objects listed is restricted to a single cell. Any text element in a chart (chart or axis titles, data labels, or shapes) can link to multiple cells, but the
linked range must be contiguous and in a single row or column; the same is true for the name of a series. The X values of a chart series can be multiple rows or columns, which produce tiered axis
labels such as those shown in LAMBDA Function to Build Three-Tier Year-Quarter-Month Category Axis Labels. The Y values of a chart series must link to data in a single row or column.
The Setup: Y data is in multiple rows or columns.
Here is the problem. The data contains more than one row (below left) or column (below right) but want it to be plotted in a single series. If you select the data and insert a chart, Excel parses the
data into two chart series. The series formulas are shown below the charts, with font colors matching the series colors.
Let’s try to fix this. First, delete the second series of the chart.
Now try to enter the larger range into the series formula.
Excel rejects the changed formula, with the error message described earlier.
There is an exception to the single row or column rule for Y values. You can specify compound (multiple-area) ranges for Y values, as shown below for our multiple row and multiple column data ranges.
The multiple areas in a compound range don’t even all need to be all by row or by column.
This works pretty well, but I think it’s pretty difficult to understand and maintain.
TOROW and TOCOL to the rescue!
Microsoft has released a plethora of new Dynamic Array functions. Among these are TOROW and TOCOL, which are used to arrange values in a 2D range into a new 1D range, shown below under the data
ranges. TOROW and TOCOL produce ranges with the values in the same order, so whether we use one or the other is a matter of preference. There are two series formulas below each chart, showing the
ranges produced by TOROW and TOCOL.
There is a problem, however. The charts don’t look the same for original data in rows vs in columns. This is because both TOROW and TOCOL take all the cells in the first row of the original data and
append all cells in each successive row. This causes the data to be out of order when performing TOROW or TOCOL on columnar data. We can fix this by transposing the data first.
And now all of our charts are consistent.
You could also construct more complicated formulas with other Dynamic Array functions. For example, if I wanted to turn a multiple row range into a single row, I would use:
)(multi-row range)
To convert a multiple-column range into a single column, I would use:
)(multi-column range)
I’m sure people can write more efficient formulas than this, but the TOROW and TOCOL formulas are very concise.
Use Names to keep the worksheet clean
We can implement TOROW and TOCOL in Names rather than in the worksheet, and the Names work just fine in the chart SERIES formulas. Go to Formulas > Define Name; for Name type YrowTOROW; for Scope
select the current sheet (Data); and for Refers to enter =TOROW(), put the cursor between the parentheses, and select C2:F3; then press Enter.
The four relevant names are:
Name: YrowTOROW
Refers To: =TOROW(Data!$C$2:$F$3)
Name: YrowTOCOL
Refers To: =TOCOL(Data!$C$2:$F$3)
Name: YcolTOROWT
Refers To: =TOROW(TRANSPOSE(Data!$C$2:$D$5))
Name: YcolTOCOLT
Refers To: =TOCOL(TRANSPOSE(Data!$C$2:$D$5))
These all produce the same values in the same order in either horizontal or vertical arrays. The chart SERIES formulas do not care. Notice that we applied the lesson from before, of transposing the
columnar data before using TOROW or TOCOL; I’ve appended a T on these Names.
Same result as before. Using Names keeps the worksheet cleaner, but I don’t mind seeing the actual data I’m plotting in my worksheet.
Neat Trick: Double Unary Minus
Last week I learned a new trick. Well, it was new to me, but apparently it has been around for a long time, predating Dynamic Arrays by decades. My colleague Roberto Mensa was showing me some of his
recent charting exercises (check them out at E90E50Charts – Excel Charts Gallery) and he showed me this trick.
You can use a double unary minus, that is, a double minus sign, to force an Excel chart to treat a multiple row or column range as a single array. The double minus is used to convert TRUE and FALSE
to 0 and 1, to convert text into numeric values, and in this case to convert a range into an array. When a 2D array is passed to a chart series, it combines all rows of the array into a 1D array.
The double minus must be used in a Name in order to work with chart data. You can define the following names for row-based or column-based data:
Name: YrowMINUS
Refers To: =--Data!$C$2:$F$3
Name: YcolMINUST
Refers To: =--TRANSPOSE(Data!$C$2:$D$5)
Define them in the scope of the workbook Data, and as before, transpose the columnar data first. Then you can edit the SERIES formulas to use these Names.
This double minus approach doesn’t take precedence over the Names that use TOROW or TOCOL, of course, it’s just another tool for your toolbox.
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9th Class Guess Papers 2024 All Subjects All Punjab Board
Notes: Start practice of these advantageous guess papers without wasting any hour. You will understand the importance of these guess papers in the examination. Hurry up students, its a time to make
the preparation perfect by these guess papers 2024 class 9. You can check 9th class guess paper in pdf here and can also save them along with Class 9 Notes.
9th Class General Math Guess Paper 2024 Punjab Board. To get good marks in general math, students should practice more and consult the guest paper.9th class general maths exam paper is available
online for 10th class students here. Authorities have prepared and uploaded the 10th class general meth gas paper Lahore board for all Urdu medium candidates so that you can better prepare for your
annual exams. These speculation papers have been prepare from previous papers as the same question is repeat every year. For 10th-grade arts students, we’re providing general math estimates here.9th
Class General Math Guess Paper 2024 Punjab Board.
View Online 10th Class Guess Papers 2024
Assessment papers will provide you with frequent and new questions that are expect in this term. In addition to these gas papers, you are also advise to prepare from your book or note so that it will
be 10th class general meth gas paper 2024, not final papers. So don’t rely on speculation and prepare with your books and notes so that this time BISE Lahore authorities change the paper pattern.
Authorities have prepared and uploaded the 10th class general meth gas paper Lahore board for all Urdu medium candidates so that you can better prepare for your annual exams. These speculation papers
have been prepare from previous papers as the same question is repeat every year.
9th Class General Math Guess Paper 2024
9th Class General Math Guess Paper 2024 View-Download
9th Class Pairing Scheme 2024 All Subjects View Online
9th Class Notes 2024 All Subjects View Online
9th Class Date Sheet 2024 All BISE Boards View Online
9th Class Roll Number Slip 2024 All BISE Boards View Online
9th Class Guess Paper 2024 All Subjects View Online
Math Guess Paper Punjab Board
General math is an important subject for 10th-grade arts students. Students should practice these Gus papers as much as possible for better scores in the exam. These speculation papers are perfect
for better preparation. Tenth graders can get these valuable guest maths from the website Taleem24.com and also download them in PDF. All students can download guest papers in PDF and prepare
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General Math Guess Paper 2024 Punjab Board
By following this advice you can avoid any kind of insult about your paper because if you deliberately prepare your paper in any way you will get good marks. Mathematics is usually a difficult
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you will pass in your paper. Scroll down this page and prepare for your exam with this BISE Lahore Bard Urdu Medium School General Mathematics Estimation Paper.
UN,IT NO.1 PERCENTAGE, RATIO AND PROPORTION.
1. EXERCISE NO. 1.1
1. Q.1 (i)
2. Q.2 (i) ,(v)
3. Q.4 (vii)
2. EXERCISE NO.1.2 Q.1, 2,3,4,5,7, 9
1. EXAMPLE -2 1.1.3
2. EXERCISE NO. 1.4 Q.1 , 2, 3, 7, 8
3. EXERCISE -1.5 Q.1,2,3,5,7,8,9,10
Chap NO.2 ZAKAT, USHR AND INHERITANCE.
1. EXERCISE NO. 2.1
1. Q.1
2. Q.2
3. Q.3
4. Q.4
5. Q.5
6. Q.6
7. Q.7
8. Q.10
9. Q.11
10. Q. 12
1. EXERCISE NO. 3.1
1. Q.1 (i) , (vi)
2. Q.4
3. Q.5
4. Q.3 (i) , (iv) , (v)
2. EXERCISE NO. 3.2 Q.1 (i)
1. Q.5
2. Q.6
3. Q.2 (ii)
3. EXERCISE NO. 3.3
1. Q.1
2. Q.2
3. Q.3
1. EXERCISE NO. 4.1
1. Q.1
2. Q.2
2. EXERCIS NO. 4.2
1. Q.1
2. Q.2
3. Q.3
4. Q.6
3. EXERCISE NO. 4.3
1. Q.6
2. Q.7
3. Q.8
4. Q.9
1. EXERCISE NO. 5.1
1. Q. 1
2. Q.5
3. Q.6
4. Q.7
2. EXERCISE NO. 5.3
1. Q. 2
2. Q. 4
3. Q.5
4. Q. 7
U,NIT NO.6 EXPENSENTS AND LOGARITHMS.
1. EXERCISE NO. 6.1
1. Q.3 (iii)
2. Q.4 (x)
3. Q.5 (iii)
2. EXERCISE NO.6.2
1. Q.28 (v)
2. Q.28
3. Q.29
3. EXERCISE NO.6.3
1. Q.1 1 , 3
2. Q.11
3. Q.12
4. Q.13
5. Q.14
6. Q.15
4. EXERCISE NO. 6.4
1. Q.4 (ii) Q.3
2. Q.4
3. Q.5
4. Q.6
5. EXERCISE NO.6.5
1. Q.1 (iii)
2. Q.2 (i)
3. Q.3
4. Q.4
5. Q.7
6. Q.9
7. Q.10
8. Q.11
U,NIT, NO.7 ARITHMETICS AND GEOMETRIC SEQUENCE.
1. EXERCISE NO. 7.1
1. Q.1 (i) , (iii), (iv) , (vii)
2. EXERCISE NO. 7.2
1. Q.1 (i),(iv)
2. Q.2 (i), (v)
3. Q.4
4. Q.10
3. EXERCISE NO. 7.3
1. Q.1 (i) , (ii), (iii) , (iv)
2. Q.2
3. Q.3
4. Q.4
5. Q.5
6. Q.9
4. EXERCISE NO.7.5
1. Q.1 (i) , (ii)
2. Q.2 (i) , (ii)
UNIT“ NO.8 SET AND FUNCTIONS
1. EXPERCISE NO. 8.1
1. Q.1
2. Q.2
3. Q.3
4. Q.4
5. Q.5
6. Q.6
7. Q.7
2. EXERCISE NO. 8.2 Q.1
1. Q.2
2. Q.3
3. Q.4
4. Q.5
.UNI,T NO.9 LINEAR GRAPHS
1. EXERCISE NO. 9.1
1. Q.1 (vi)
2. Q.3 (i) , (ii), (iv)
2. EXERCISE NO. 9.3
1. Q.1 (b)
2. Q.3
UNI,T NO.10 BASIC STATISTICS
1. REVIEW EXERCISE. Q.3
1. Q.5
2. Q.6
• Express 95% as fractions in lowest form.
• Express the decimal 0.065 as percentage.
• If there are 800 cars in a car parking and 80% of them are Pakistan made. Find the number of Pakistani cars.
• Simplify the ration 24:12 in the simplest form
• Express 12/10 : 28/10 in its simple form
• What is meant by cost price and sale price?
• Define direct proportion.
• Find the ratio of Rs. 160 per meter to Rs.150 per meter.
• Express ratio 2/3,3/5 in its simplest form.
• Find the unknown ‘x’ in the proportion x : 3 :: 60 :15
• Calculate Zakat on Gold amounting to Rs. 11, 1,000/-
• Calculate the amount of Zakat on an amount of Rs. 5000000.
• Find CP when SP = Rs.572/- and profit is 5%
Find the SP when CP= Rs. 1540/- Loss = 5%
• Find the marked Price when SP = Rs. 2400, Discount = 4%
• Find CP when SP = Rs.851 and loss = 5%
• Convert 700 Saudi Riyal into Pakistan rupees. When the rate of Saudi Riyal = Rs.22, 400.
• Convert 250 US dollars into Sterling pound.
• Define excise duty
• The annual income of a flat is Rs. 14, 00,000. Find the tax payable at the rate of 16%
• Define Sales tax.
• A computer price is Rs. 34800 inclusive of 16% sales tax what is the original price of the computer.
• Simplify (x^2y^3)^1/6
• Write 0.00018 in scientific notation.
• Solve the equation Log, (x+1) = 2
• Write down the value of log 52.13
• Express in exponential form:
• F.ind A.M between 4 and 8
• F.ind G.M. between 4 and 9
• Find the 5^th term of a G.P. 4, -12, 36………..
• Convert 20 ^oC into ^oF.
• Find the range of the given data 10,15,9,5, 22
• Find the standard division of the values 2,3,6,8,11
• Amina scored 45 out of 50 in a math test, 64 out of 75 in chemistry test and 72 out of 80 in a physics test. In which subject did she perform best?
• A shop keeper plans to produce 200 articles with the help for 5 persons working 8 hours daily. How many articles can be made by 8 persons if they work 5 hours daily.
• The price of 20 pens is Rs.2000. What will be the price of 40 such pens?
• Calculate zakat on gold of worth Rs. 8, 00,000, cash of amount Rs. 4, 00,000 and silver of weight 50 tola (Rs.5000 per tola).
• If 15% discount on MP of a heater is allowed and still makes a profit of 2%. If it is sold on MP, what is profit percentage?
• Find compound profit on Rs.800 for 4 years @ 6 percent per annum.
• Evaluate. 8.67 x 3.94/1.78
• Draw the graph of y = 3x
• Distribute Rs. 33,000 as a profit in a business regarding three persons if their shares are in the ratio 3: 5:3
• Insert two G.M. between 4 and 1/2
General Maths Matriculation Guess Papers
These days just focus on your studies and avoid wasting all your time as you have very few days left to start your exams. Stay tuned to get a general match to review the annual Matriculation Exam.
Therefore, as soon as the 10th class estimates for SSC Part 2 General Match are prepare, they will be upload on this page. You can leave a comment in the comments section below to get the latest
updates on this test. Guess the pamphlets are going to be upload here soon, they are not ready yet and we are going to do it. As soon as we have prepared these estimate sheets, we will upload them
here and then you will be able to download the online guess paper for the 10th class General Meth Gas Paper 2020 Lahore Board.
9th Class General Math Guess Paper 2024
We work to provide our students with the easiest way to prepare for exams so that our students can prepare easily. These Guess Papers are well-design guest papers and have been develop by our
experience instructors. Tenth-grade papers are also give to the students here. Download 10th Class General Mathematics Estimation Sheet.Take advantage of this amazing facility. Visit our website for
more important updates and study materials.
Notes: Start practice of these advantageous guess papers without wasting any hour. You will understand the importance of these guess papers in the examination. Hurry up students, its a time to make
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Ninth Grade Islamiat Paper Scheme includes all the topics from which the question paper will be . In addition, the Paper Scheme provides students with complete details that must be follow in the
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Guess Paper Punjab Board 9th Class Elective Islamiat
In addition, the Paper Scheme provides students with complete details that must be follow in the 9th-grade annual examinations. 9th Class Elective Islamiat Guess Paper Punjab Board.9th Class Elective
Islamiat Guess Paper Punjab Board.How can you consider taking an exam if a student does not know how to fill out the form? Therefore, for this reason, you should learn how to format your paper
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9th Class Elective Islamiat Guess Paper 2024 All Chapter
1 Quran Majeed View-Download
2 Allah aur Us kay Rasool ki Muhabbat View-Download
3 Ilm ki Farziyat-o-Fazeelat View-Download
4 Zakat View-Download
5 Complete Book MCQs View-Download
6 Important Short Questions View-Download
9th Class Pairing Scheme 2024 All Subjects View Online
9th Class Notes 2024 All Subjects View Online
9th Class Date Sheet 2024 All BISE Boards View Online
9th Class Roll Number Slip 2024 All BISE Boards View Online
9th Class Guess Paper 2024 All Subjects View Online
9th Class Elective Islamiat Guess Paper 2024
How can you consider taking an exam if a student does not know how to fill out the form? Therefore, for this reason, you should learn how to format your paper through the 9th class paper scheme.
Ninth Grade Islamiat Paper Scheme includes all the topics from which the question paper will be . How can you consider taking an exam if a student does not know how to fill out the form? Therefore,
for this reason, you should learn how to format your paper through the 9th class paper scheme.
View Online 9th Class Islamic Paper Pattern
How can you consider taking an exam if a student does not know how to fill out the form? Therefore, for this reason, you should learn how to format your paper through the 9th class paper scheme. This
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Punjab Board 9th Class Elective Islamiat Guess Paper
Now you can get Paper Scheme for all Class 9 subjects along with Class 9th Paper Patterns. 9th Class Elective Islamiat Guess Paper Punjab Board.9th Class Elective Islamiat Guess Paper Punjab Board..
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Notes: Start practice of these advantageous guess papers without wasting any hour. You will understand the importance of these guess papers in the examination. Hurry up students, its a time to make
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9th Class Computer Science Guess Paper 2024. 9th Class Guess Paper tells you how to format the paper to perform in the examination hall. How can you consider taking an exam if a student does not know
how to fill out the form? Therefore, for this reason, you should learn how to format your paper through the 9th class paper scheme. To help students, we uploaded the latest pair of schemes/paper
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9th Class Computer Science Full Guess Paper
9th Class Computer Science Guess Paper 2024 View-Download
All Chapter Punjab Board 9th Class Computer Science Guess Paper 2024
9th Class Computer Science PDF
Chapters Chapter Name Medium
1 Problem Solving Urdu Medium
2 Binary System Urdu Medium
3 Networks Urdu Medium
4 Data and Privacy Urdu Medium
5 Designing Website Urdu Medium
9th Class Pairing Scheme 2024 All Subjects View Online
9th Class Notes 2024 All Subjects View Online
9th Class Date Sheet 2024 All BISE Boards View Online
9th Class Roll Number Slip 2024 All BISE Boards View Online
9th Class Guess Paper 2024 All Subjects View Online
Computer Science Guess Paper 9th Class
Candidates are advise not to worry. For the convenience of the students, the authorities provide a pairing scheme to the students so that they can prepare for the annual examinations accordingly.
Students looking for 9th-grade computer science Guess Paper can view the curriculum from this platform. During the preparation for the annual exams, students are concern about which chapters should
be prepare and which areas need more time to prepare.SSC Science Group or Arts Group all subject pair scheme is available here. 9th Class Guess Paper tells you how to format the paper to perform in
the examination hall. How can you consider taking an exam if a student does not know how to fill out the form? Therefore, for this reason, you should learn how to format your paper through the 9th
class paper scheme.
Computer Guess Paper 2024 Punjab Board
In addition, a list of practical sections related to computer science is also provided and candidates will have to prepare theory as well as practical sections. Candidates should consider a paper
scheme before preparing a preparation plan as it will guide them and help them to cover the most important and key topics of the article. Candidates have to go through this scheme which includes
various chapters like problem-solving, binary system, networks, data and privacy, website design. The division of questions in each chapter is state for the purpose and thematic part.
Download Matric 9th 10th Class Guess Paper
Dear students, if you have any problem downloading the smart syllabus or Guess Paper, let us know in the comment box below. If you are worrie and looking for a peer scheme for all matriculation
subjects or are reviewing the 9th and 10th-grade assessment scheme, check out this page. All the boards of Punjab like Lahore, Gujranwala, Multan, Sahiwal, Bahawalpur, Sargodha, Rawalpindi,
Faisalabad, and DG Khan, 9th / 10th class paired schemes for each subject are available here.
9th Class Computer Guess Paper
9th Class Guess Paper tells you how to format the paper to perform in the examination hall. How can you consider taking an exam if a student does not know how to fill out the form? Therefore, for
this reason, you should learn how to format your paper through the 9th class paper scheme. 9th Class Guess Paper tells you how to format the paper to perform in the examination hall. How can you
consider taking an exam if a student does not know how to fill out the form? Therefore, for this reason, you should learn how to format your paper through the 9th class paper scheme.
9th Class Computer Science Guess Paper
MCQs: ALL MCQs FROM EXERCISE OF COMPUTER SCIENCE TEXT BOOK.
• Define Problem solving.
• What do you mean by well –defined problem
• What do you know understanding the problem or analyzing a problem?
• What do you mean by planning a solution? Mention strategies for problem solving.
• Define Algorithm and Flowchart.
• What is the purpose of Oval shape/ Terminal shape symbol in flowchart.
• What is the purpose of Input/output symbol of flowchart?
• What is the purpose Rectangle symbol in flow chart.
• Draw a flowchart to convert Celsius to Fahrenheit temperature.
• Draw a flowchart determine whether a given number is odd or even.
• What is role algorithm is problem solving.
• What is Test Data? Give example.
• Compare verification and validation.
• What are disadvantages of Flowchart?
• What are advantages and disadvantages of Algorithm.
• Convert (69610)10 to Hexadecimal.
• Difference between Volatile and non –volatile memory.
• Difference between Temporary Memory and Permanent Storage.
• What is Number System?
• Difference between Decimal, Binary and Hexadecimal number system.
• Convert 0B9 (16) into decimal.
• What is computer memory? Also write its types.
• What is storage Device?
• What is difference between internal storage and external storage?
• What is ASCII?
• What is Boolean proposition? Give some examples.
• What is a truth table?
• What is AND Operation.
• What is OR Operation.
• What is NOT Operation.
• What is truth table for complex Boolean expression.
• Describe computer Net Work.
• What is Net Work and Net Works?
• What is Client?
• What is a Server?
• What is Network Topology? Mention its types.
• What is Data Communication? Mention its components.
• What is sender (transmitter and receiver) (sink).
• What is the difference between static and dynamic IP.
• Describe the working of Web Browser.
• What is advantages of user communication on networks?
• What is Data Transmission?
• What is TCP/IP model?
• Define HTTP and SMTP.
• What is IP Address?
• What is Router?
• Write advantages and disadvantages of Star topology and bus topology.
• Define Communication Channel.
• Give a reason to add CAPTCHA websites.
• What do mean by Privacy of Data.
• Define Piracy.
• What do you mean by confidentiality of data? Give example.
• What do you mean by software piracy?
• What do you know about Copyright Law?
• What is cracking key?
• What is computer virus?
• What is secret code OR Key?
• Who is hacker?
• What is the difference between Cryptographic keys and Password?
• Write any two characteristics of good password.
• What is Cybercrime? Describe its Activities.
• What is Hacking?
• What is the use of cookies?
• What do you mean by computerized system?
• Difference between ordered and unordered list.
• What is the difference between hyperlink and anchor?
• What do you know about HTML?
• Define Hypertext.
• What is the role of HTML? M.I.
• Name the types of Lists in HTML.
• What is the use of hyperlinks?
• How we create web page using HTML.
• Difference between paired tags and singular tags.
• What are attributes in HTML?
• Define HTML and body tags.
• What is text formatting? And how we bold the text.
• Describe Nested list.
• Which tag is used for adding image?
• What is metadata?
• Explain Step involved to create a HTML page.
• Define problem analysis your answer along with an example.
• How do you determine require of flow chart.
• Find LCM of two numbers.
• Input a year and determine whether n1 and divided n2 or not.
• Conversion : (69610)10 to hexadecimal
• Conversion : (ABCD)16 to binary.
• Difference between volatile and non- volatile memory.
• Difference between temporary and permanent storage.
• What is TCP/IP model? Describe its fiver layers with their function.
• What is the sizes of IPv4 and IPv6? Explain the method the calculate the size of these both standard.
• What do we need an installation key where can be protected with a password.
• Give the reason add captcha on website.
• What is patent, and why do we need to register it.
• Write down the procedure to create a web page and display it.
• How would you add image and set back ground to a web page.
• What is difference between hyperlink and anchor.
• Write down the procedure to create table.
Notes: Start practice of these advantageous guess papers without wasting any hour. You will understand the importance of these guess papers in the examination. Hurry up students, its a time to make
the preparation perfect by these guess papers 2024 class 9. You can check 9th class guess paper in pdf here and can also save them along with Class 9 Notes.
9th Class Home Economics Guess Paper 2024. SSC Science Group or Arts Group all subject Guess Paper is available here. 9th Class Guess Paper tells you how to format the paper to perform in the
examination hall. How can you consider taking an exam if a student does not know how to fill out the form? Therefore, for this reason, you should learn how to format your paper through the 9th class
paper scheme.Like this page for the latest updates regarding annual matric exam preparation. To help students, we uploaded the latest Guess Paper of schemes/paper schemes for each subject in all
classes.9th Class Home Economics Guess Paper 2024.
9th Class Home Economics Guess Paper
9th Class Home Economics Guess Paper View-Download
9th Class Pairing Scheme 2024 All Subjects View Online
9th Class Notes 2024 All Subjects View Online
9th Class Date Sheet 2024 All BISE Boards View Online
9th Class Roll Number Slip 2024 All BISE Boards View Online
9th Class Guess Paper 2024 All Subjects View Online
Home Economics Paper Scheme 2024 9th Class
Candidates are advise not to worry. For the convenience of the students, the authorities provide a Guess Paper to the students so that they can prepare for the annual examinations accordingly.
Students looking for 9th-grade Economics science Guess Paper can view the curriculum from this platform. During the preparation for the annual exams, students are concern about which chapters should
be prepare and which areas need more time to prepare.9th Class Home Economics Guess Paper.
9th Class Home Economics Guess Paper
9th Class Guess Paper tells you how to format the paper to perform in the examination hall. How can you consider taking an exam if a student does not know how to fill out the form? Therefore, for
this reason, you should learn how to format your paper through the 9th class paper scheme. The division of questions in each chapter is state for the purpose and thematic part.
Home Economics Guess Paper Punjab Board
9th Class Guess Paper tells you how to format the paper to perform in the examination hall. How can you consider taking an exam if a student does not know how to fill out the form? Therefore, for
this reason, you should learn how to format your paper through the 9th class paper scheme. To this end, you can collaborate with your teachers and come up with different ideas using past papers and
sample papers. Ninth-grade past papers and sample papers are design in such a way that they reflect the complete paper pattern issue by the Board of Education. You can check out the 9th class smart
syllabus as it will be easier for you to prepare for the final exam.
Notes: Start practice of these advantageous guess papers without wasting any hour. You will understand the importance of these guess papers in the examination. Hurry up students, its a time to make
the preparation perfect by these guess papers 2024 class 9. You can check 9th class guess paper in pdf here and can also save them along with Class 9 Notes.
9th Class Education Guess Paper 2024 Punjab Board. For better practice and better performance in board exams, we have created the best Education Guess Paper for our hardworking students which is
provided to you online here. You can view the 9th Class Education Assessment Paper here. These speculative papers help you a lot in the board exam as the questions of these papers are by our highly
experience teachers. Ninth-grade students can get ninth-grade Education exam papers from here.9th Class Education Guess Paper 2024 Punjab Board.
9th Class Education Guess Paper 2024 All Chapter
9th Class Education Guess Paper 2024 View-Download
Evaluate 9th grade Education Guess papers
In addition, the education Ninth Grade Evaluation Paper is free to download and you can save it in PDF format. There are a lot of guest papers that are give for the 9th Class and are very important
from a paper point of view. Education science assessment papers for 9th-grade students are available here. These guest papers are very important for ninth-grade arts students. Students can get very
good marks in board exams if they prepare these assessment papers before the exams. These Guess Papers give you an idea of the types of questions that can be ask in an Education Board exam.
9th Class Education MCQ 2024
9th Class Education Guess Paper 2024 View-Download
9th Class Pairing Scheme 2024 All Subjects View Online
9th Class Notes 2024 All Subjects View Online
9th Class Date Sheet 2024 All BISE Boards View Online
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9th Class Education Guess Paper
If you are looking for FBISE Federal Board Matric Model Papers, you are on the right page. If you are looking for different study materials for the annual matriculation exam, then the model papers
will help you. The best thing about this board is that it issues exam syllabi and model papers before the exams so that the students can guess what is going to happen in the exam. This is an easy way
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scroll down and get the best result from this page.
FBISE Federal Board Matriculation Model Papers
The board released the ninth and tenth-grade model papers. You can download in PDF format which is essential for all of you where you can prepare online federal board model papers for 9th class as
well as federal board model papers for 10th class and 9th class. , Which the authorities are providing. Of this board. You can click on any of the following articles where you can find past Federal
Board papers in PDF.
9th Class Guess Paper Punjab Board
These include subjects such as physics, computer, chemistry, biology, and mathematics. For better preparation for exams, 9th class guess papers are available here. These assessment papers are
designed specifically for ninth-grade students. Now, 9th-grade students can get these important guest papers online from here and practice easily. These ninth-grade speculation papers are definitely
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9th Class Education short Question Guess
1. What is mean by non formal education?
2. What is Formal eduction?
3. Define informal education
4. function of education
5. difference of knowledge and education
6. why is knowledge of education consider as science?
7. Culture Heritage of a nation?
8. Terms: developemnt , adolescence, growth
9. what is natural and physical change?
10. mean of “Home Environment”
11. two example of physical difference?
12. define terms: Education, society. community
Education Long Question Guess 2024
So students best of luck for tomorrow paper and get education guess paper 2024 of 9th class. Happy Exams.
Notes: Start practice of these advantageous guess papers without wasting any hour. You will understand the importance of these guess papers in the examination. Hurry up students, its a time to make
the preparation perfect by these guess papers 2024 class 9. You can check 9th class guess paper in pdf here and can also save them along with Class 9 Notes.
Although the PDF file guess paper also contains important long questions, I have put it here. A picture of long questions to download in pictorial form as well. You can see a list of all the
important long questions given in the picture here. This is also for FBISE and the Federal Board. Students can also visit my other website which is Urdu to download many other things for FBISE.
Edcateweb.com is the educational portal of students. 9th Class Urdu Guess Papers 2024 Punjab Board.
9th Class Urdu Short Guess Papers 2024
9th Class Urdu Guess Paper 2024 View / Download
9th Class Pairing Scheme 2024 All Subjects View Online
9th Class Notes 2024 All Subjects View Online
9th Class Date Sheet 2024 All BISE Boards View Online
9th Class Roll Number Slip 2024 All BISE Boards View Online
9th Class Guess Paper 2024 All Subjects View Online
9th Class Urdu Guess Papers Punjab Board
In 2024, we assembled and compiled class assessment test papers for 9th-grade students, and tried to get the best assessment paper for 2024 with the least amount of effort. 9th Grade Urdu Assessment
Paper Punjab Board. Here I have given estimates for all the subjects of the 9th class. If you want to get an estimate for all articles in Category 9 of 2024, you can get it. The following is the
Pakistan Study Guest Paper of the Ninth Lahore Board. Now, these guess papers can also be downloaded in PDF format. Although the PDF file guest paper also contains important long questions, I have
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9th Class Urdu Guess Papers
Here are another 9th grade Pak study speculation papers and I would suggest looking at them too. Therefore, you should also download the guest paper of Study 9th Class of Bihar-ul-Alam. The link is
This is 9th grade Pak Study Urdu Medium Estimates Paper for all Punjab Boards for Board Examinations. The assessment paper is general and not specific to any particular board. The appraisal paper is
in PDF format and you can easily download the file to your computer or smartphone. The link to download the file is given below the image. Guessing paper always evaluates important and common issues
and saves you energy and time. As we know, the lesson of lesson 9 has already started, so get good marks with minimum effort.
View Online Guess Papers Punjab Board
A quick and clever way to prepare for the exam with the assessment paper will help you get good marks. Every student wants to cover the whole curriculum, but it is difficult for all students.
Therefore, we recommend that you prepare your next paper with the help of estimation papers. One way for smart students is to build all the smart buses, then guess and revisit all the important
questions.For the lazy students who can’t cover all the courses in class 9 and want a smart and easy way to prepare the test paper, the second is easy. Trust me, you will get good marks, but only if
you have at least all the problems. That’s why we got 9th estimate papers for all subjects in 2024.
Urdu Guess Papers Punjab Board
We were advised by the professor and the staff on board for these estimates. So I hope you can cover and read 60% of the papers, remembering all the lessons of Lesson 9. Collect all 9 level
speculations and leaked papers. Download Level 9 Urdu Intermediate Assessment Test Paper for Pakistan Studies for Board. Director Examinations on this page.Good luck, dear students! 9th-grade exam
papers for all subjects are online. Grade 9 candidates can get guest papers for any subject from this page.
9th Class Urdu Guess Papers
Estimation papers will be uploaded soon, and everyone who visits this website can get free online estimation papers on any topic from this page. There are very few boards, you can get these boards
from this page just to evaluate the paper. If you are a student of another board of directors, you should follow the homepage of this website, from there you will definitely get the board evaluation
article. Important short questions of ninth grade Urdu are given in the guest paper.
Notes: Start practice of these advantageous guess papers without wasting any hour. You will understand the importance of these guess papers in the examination. Hurry up students, its a time to make
the preparation perfect by these guess papers 2024 class 9. You can check 9th class guess paper in pdf here and can also save them along with Class 9 Notes.
9th Class Pak Studies Guess Paper 2024 Punjab Board. 9th Class Pak Studies Guess Paper – Notice PDF Guess Paper is for Lahore Board, Gujranwala Board, Multan Board, Sahiwal Board, DG Khan Board,
Rawalpindi Board, and Sargodha Board. This is also for FBISE and the Federal Board. Students can also visit my other website which is a free study to download many other things for FBISE. In 2024,
we’ve compiled and compiled class estimating test papers for 9th Class, and strive to achieve the best results with the least amount of effort.9th Class Pak Studies Guess Papers 2024 Punjab Board.
Guess Papers Punjab Board 9th Class Pak Studies
Guessing paper always evaluates important and common issues and saves you energy and time. As we know, the lesson of lesson 9 has already started, so get good marks with minimum effort. A quick and
clever way to prepare for the exam with the assessment paper will help you get good marks. Every student wants to cover the whole curriculum, but it is difficult for all students. Therefore, we
recommend that you prepare your next paper with the help of estimation papers. One way for smart students is to build all the smart buses, then guess and revisit all the important questions.
9th Class Pak Studies Guess Papers 2024 All Chapter
9th Class Pak Studies Guess Paper 2024 English Medium View / Download
9th Class Pak Studies Guess Paper 2024 Urdu Medium View / Download
9th Class Pairing Scheme 2024 All Subjects View Online
9th Class Notes 2024 All Subjects View Online
9th Class Date Sheet 2024 All BISE Boards View Online
9th Class Roll Number Slip 2024 All BISE Boards View Online
9th Class Guess Paper 2024 All Subjects View Online
Pak Studies Guess Papers
For the lazy students who can’t cover all the courses in class 9 and want a smart and easy way to prepare the test paper, the second is easy. Trust me, you will get good marks, but only if you have
at least all the problems. That’s why we got the 9th estimate papers for all subjects in 2024. We were advised by the professor and the staff on board for these estimates. So I hope you can cover and
read 60% of the papers, remembering all the lessons of Lesson 9. Collect all 9 level speculations and leaked papers in 2024. Download Level 9 Urdu Intermediate Assessment Test Paper for Pakistan
Studies for Board. Director Examinations on this page.
9th Class Pak Studies Guess Papers
Here I have given estimates for all the subjects of the 9th class. If you want to get an estimate for all articles in Category 9 of 2024, you can get it. The following is the Pakistan Study Guess
Paper of the Ninth Lahore Board in 2024. Now these guest papers can also be downloaded in PDF format. Although the PDF file guest paper also contains important long questions, I have put it here. A
picture of long questions to download in pictorial form as well. You can see a list of all the important long questions given in the picture here.
9th Class Pak Studies Guess Papers
Here are another 9th grade Pak study speculation papers and I would suggest looking at them too. Therefore, you should also download the guest paper of Study 9th Class of Bihar-ul-Alam. The link is
This is the 9th grade Pak Study Urdu Medium Estimates Paper for all Punjab Boards for Board Examinations. The assessment paper is general and not specific to any particular board. The appraisal paper
is in PDF format and you can easily download the file to your computer or smartphone. The link to download the file is given below the image. Good luck, dear students! 9th-grade exam papers for all
subjects are online.
9th Class Pak Studies Guess Paper 2024
Short Questions :Unit #1
1. Toheed?
2. Aqeedah e Risalat?
3. Nazrya Pakistan?
4. Allama Iqbal ne Muslim millet K bare kya frmaya??
5. Do Qomi nazrya?
6. Aqliyyeton se mutalliq Quaid ka frman?
7. Lafz Pakistan kub tujweez kya gya??
Long Questions
1. Islami iqdar jo nazrya Pakistan ki asaas hain?
2. Allama Iqbal k irshadat ki roshni Main nazrya Pakistan?
3. Hindustan Main muslmano ki muashi halet??
Short Questions: Unit #2
1. Crips mission ki Teen tjaaweez?
2. Jinnah Gandhi muzakrat 1944 Quaid ka jwab?
3. Kabeena mission main plan main sobai group ki tushkeel??
4. Qrar Dad e Pakistan ka mutn??
5. Aboori hkoomut mai shamil wuzra K names??
6. Rollet Act 1919 pr Quaid ka moaqquf??
7. Quaid ne Safeer e amn ka khitab kaise paya???
Long Questions
1. Quaid ka Pakistan ki tushkeel Main kirdar?
2. Hindustan main no abadyati nizam??
3. Cabina mission plan 1946 ???
Short Questions: Unit #3
1. Junglat ki kmi ki wjoohat???
2. Pakistan ka muhalle wqoo??
3. Zmeeni aloodgi Main kmi k lye iqdamat??
4. Drra tochi OR Gomal kis pharri silsile pr waqe hai??
5. 5 glaciers K names??
6. Pakistan K 5 qudrti khitton K names??
7. Toba kakerr ka pharri silsila khaan waqe hai??
8. Durend line kise kehte hain? Mahol se kya murad?
9. Pakistan K 2 berajon K names?
Long Questions
1. Aabo hwa ka insani zindgi pr asr?
2. Dryaon ka nizam? Junglaat ki ahmyet?
3. Mahol ko drpesh khutraat??
Short Questions: Unit #4
1. 1956 k aain ki 5 islami dufaat?
2. 1965 ki jung Main behrya ka kirdar?
3. 1965 ki jung K 2 asbaab?
4. Malakund division kaise bna?
5. Muashi traqqi??
6. Wahid shehryet?
7. Radcliffe ki ghair munsfana tuqseem??
8. Union council OR Union committee ?
9. Pakistan ki ibtdai mushklaat?
10. Qrar daad e Mqaasid?
11. Mushrqi Pakistan ki alehdgi ki wjoohat? 1962 ka aain??
Long Question : Unit #5
Islam main ortooon ka muqaam OR huqooq?
English Guess Paper
Notes: Start practice of these advantageous guess papers without wasting any hour. You will understand the importance of these guess papers in the examination. Hurry up students, its a time to make
the preparation perfect by these guess papers 2024 class 9. You can check 9th class guess paper in pdf here and can also save them along with Class 9 Notes.
9th Class English Guess Paper 2024 Punjab Boards. These include subjects such as physics, computer, chemistry, biology and mathematics. For better preparation of exams, 9th class guess papers are
available here. These assessment sheets are designed specifically for 9th Class students. Now, 9th Class can get these important guess papers online from here and practice easily. These 9th Class
speculation papers are definitely very useful and important for your best preparation for board exam.Good luck, dear students!
9th Class English Guess Paper 2024
9th Class English Short Guess Paper 2024 View-Download
9th Class English Guess Paper For Weak Student 2024 View-Download
9th Class Pairing Scheme 2024 All Subjects View Online
9th Class Notes 2024 All Subjects View Online
9th Class Date Sheet 2024 All BISE Boards View Online
9th Class Roll Number Slip 2024 All BISE Boards View Online
9th Class Guess Paper 2024 All Subjects View Online
Guess Paper Punjab Boards 9th Class English
There are very few boards, you can get these boards from this page just to evaluate the paper. If you are a student of another board of directors, you should follow the homepage of this website, from
there you will definitely get the board evaluation article. You can also download ninth-grade ALP NOTES based on ALP Smart Curriculum. You can also strengthen your test score with a large number of
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Punjab Boards 9th Class English Guess Paper 2024
Guess the papers are for reference only, and candidates should be fully prepared for the exam. The Board of Intermediate and Secondary Education is a commanding body functioning under the Ministry of
Education. There are many boards that are currently working with Lahore Board Faisalabad, Sargodha, Gujranwala, and many others under the supervision of BISE. The main purpose of the board is to
provide quality education to every student. The main objective of the board is to improve the conduct of crystal clear examinations in the country. Since regular and private students have registered
with the board and thousands of students have obtained their degrees from the board. All students are looking forward to ninth grade.
9th Class English Guess Paper Punjab Boards
Date Sheet BISE Lahore should be linked to Edcateweb.com to get the latest updates of the Date Sheet. A quick and clever way to prepare for the exam by guessing the questions will help you get good
marks. Every student wants to cover the whole curriculum, but it is difficult for all students. Therefore, we recommend that you prepare your next paper with the help of guest papers. 9th-grade
students can prepare all subjects in two ways. One way for smart students is to build all the smart buses, then guess and revisit all the important questions. For the lazy students who can’t cover
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English Guess Paper Punjab Boards 9th Class
The guest paper included all the important questions, paragraphs, abstracts, articles, etc. of the English 10th class. Gus Paper is in PDF and you can download it from Hafeezullah Notes. While I have
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Guess papers for 9th graders, and strive to get the best results with the least effort. Evaluating always helps you figure out the most important and common issues, which can save you energy and
time. As we know, Lesson 9 exam paper has started, so you can get good marks with minimum effort.
9th Class English Guess Paper 2024
Sr# Paragraphs Lesson
1 The period of waiting had…… 1
2 When Hazrat Muhammad (SAW) was thirty… 1
3 In the fifth and sixth…… 1
4 The revelation of the divine……. 1
5 Since this belief was ……… 1
6 Patriotism means love for……… 2
7 Patriotism gives people strength…… 2
8 Quaid-e-Azam Muhammad Ali……. 2
9 Teacher: OK, as we have decided……. 3
10 Student 1: Media helps……… 3
11 The preparation for this journey…… 4
12 Her grandfather, Hazrat Abu Quhafa…. 4
13 Hazrat Asma will always remain…… 4
14 During the early and difficult……. 6
15 The whole journey of the great……. 6
16 The ideology of Pakistan was…… 6
17 Quaid-e-Azam was a man of strong……. 6
18 Today the Quaid’s Pakistan is……… 7
19 Blue Mosque reflects the architecture…… 7
20 The upper level of the interior is…… 7
21 The most important element in……. 7
22 The Masjid has a specious………. 7
23 In the evening, a large number……. 7
24 I was upset. The advice to leave……. 9
25 She and her family had entered……. 9
25 I continued to work on Hira…… 9
26 Drug addiction is really a very…… 10
27 Drug addiction is caused by……… 10
28 Noise pollution is defined as any…. 11
29 Another source of noise pollution…. 11
30 Noise pollution not only causes…… 11
31 In stories, the doomed hero is usually…. 12
32 This day I should devote…….. 12
33 How was it possible, I asked…….. 12
34 Now and then, I have…… 12
Questions: Answer the following Questions.
1. What type of land Arabia is?
2. For which ability were the Arabs famous?
3. What was the first revelation?
4. Where is Makkah situated?
5. Why was Quran sent in Arabic?
6. What is the highest military award of Pakistan?
7. How will you define patriotism?
8. What are the qualities of a patriot?
9. Who offers for the country?
10. As a citizen of Pakistan what are your duties towards your country?
11. What are the two major means of communication?
12. How does media provide entertainment?
13. What happens when media is allowed to play its role unchecked?
14. What type of information does media provide?
15. What is the most important function that media performs?
16. Why was Abu Jehil furious?
17. Why was Abu Quhafa worried?
18. How did Hazrat Asma console her grandfather?
19. Who was Hazrat Abdullah bin Zubair (R.A.)?
20. What message do you get from the life of the whole nation?
21. Why did Quaid want the oneness of whole nation?
22. How much confidence did Quaid-e-Azam have in his nation?
23. How can we become a strong nation?
24. What was the Quaid’s concept of our nation?
25. What can be the possible solution to our present problems?
26. Who was appointed as architect of the Masjid?
27. Who constructed Masjid Sophia?
28. How does the interior of Masjid look?
29. Why Sultan Ahmad Masjid is also known as Blue Masjid?
30. Where is the royal room situated?
31. What is an ICU in a hospital?
32. Why did the nurse ask Hira’s sister to come and talk to her?
33. Describe some qualities of the nurse.
34. Why did the nurse disagree with the doctor’s point of view?
35. What are the effects of drug addiction?
36. What are the causes of drug addiction?
37. What is the role of counseling in preventing drug addiction?
38. Are drug addicts aware of the dangers of drugs?
39. How do you define noise pollution?
40. How is transport a source of noise pollution?
41. Why noise hazardous for human health?
42. Why has she no time to waste in longings?
43. Who was Helen Keller?
44. Why makes you that the author is sad and depressed?
Question: Words/Phrases/Idioms.
Century, Nationalism, Companion, Specious, Conquest, Supreme, Refuge, Humality, Delegation, Invasion, Quietly, Embolish, Urge, Geared up, Gave away, Flamboyant, Ignorance, Global village, Solitude,
Monument, Responsible, Constructive role, Bits and pieces, Gradually, Influential, To keep an eye, Man in the street, State. Determines, Entertain, Raising spirit, Miserable, Verge, Impact, all a
prey, Care, Sacrifice, MAd with anger, Pass through, Commendable, Fit of fury, Impressive, Perilous, Properity, Reveal the secret, Dexterously, Aptitude, Motherland, Migration, Specious, Damage.
Question: Write a letter to your
1. mother who is worried about your health.
2. father asking about the health of your mother.
3. mother about the test you have just taken.
4. brother about the importance of science subjects.
5. father requesting him to send you some extra funds.
6. sister thanking her for a gift.
7. friend condoling her for a gift
Question: Important stories.
1. A friend in need is a friend indeed
2. Greed is a curse
3. Haste makes waste
4. Never tell a lie
5. The boy who cried wolf
Question: Important Dialogues
1. Dialogue between teacher and student.
2. Dialogue between two students regarding prayers.
3. Dialogue between a brother and a sister.
Question: Important comprehension
1. Once a stag was………….
2. King Robert Bruce……………
3. For three years……………….
4. Newspapers keep us…………
5. Early rising is a……………
6. One day, a girl found……….
7. A tailor ran a shop……….
8. Musa was in chief………….
Question: Change of voice
1. The mother loves the children.
2. They are buying this house.
3. She gave me five films.
4. Why did she write such a letter?
5. They had gained nothing.
6. He will write a letter.
7. We shall have killed the snake.
8. She likes apples.
9. The boy is climbing the wall.
10. We did not hear a sound.
11. The board has given me a gold medal.
12. We use milk for making cheese.
13. Why is he mending the hair?
14. The driver opened the door of the car.
15. She has not beaten the dog.
16. She bought five video films.
17. She was teaching the students.
18. They have bought a house.
19. The teacher was helping the students.
20. Why were they beating the boys.
21. They have not done their job.
22. A car ran over an old man.
23. He will give you a box of chocolates.
24. We shall have finished our work.
25. He took away my books.
26. They caught the thief.
27. The boy makes the picture.
28. They had not done their homework.
9th Class English Guess Paper Punjab Boards
These ninth class guest papers of the Punjab Board have been provided to the students in the form of best representation here. Now, you don’t need to buy the 9th class assessment paper Punjab Board
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Punjab Boards 9th Class English Guess Paper
9th Class English Guess Paper Punjab Boards. For the Board’s annual examination, these Guest Papers Class 9 Punjab Board and Guest Papers Class 9 Lahore Board are the best. We provide you with
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9th Class Islamiat Guess Papers 2024 Punjab Board.9th Class Islamiat Guess Paper – Notice PDF 2024 Guess Paper is for Lahore Board, Gujranwala Board, Multan Board, Sahiwal Board, DG Khan Board,
Rawalpindi Board, and Sargodha Board.9th Class Islamiat Guess Papers 2024 Punjab Board. Students can also visit my other website which is a free study to download many other things for FBISE. In 2024
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Guessing paper always evaluates important and common issues and saves you energy and time. As we know, the lesson of lesson 9 has already started, so get good marks with minimum effort. A quick and
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9th Class Islamiat Guess Papers 2024 All Chapter
1 Quran Majeed View-Download
2 Allah aur Us kay Rasool ki Muhabbat View-Download
3 Ilm ki Farziyat-o-Fazeelat View-Download
4 Zakat View-Download
5 Complete Book MCQs View-Download
6 Important Short Questions View-Download
9th Class Pairing Scheme 2024 All Subjects View Online
9th Class Notes 2024 All Subjects View Online
9th Class Date Sheet 2024 All BISE Boards View Online
9th Class Roll Number Slip 2024 All BISE Boards View Online
9th Class Guess Paper 2024 All Subjects View Online
9th Class Islamiat Guess Papers
For the lazy students who can’t cover all the courses in class 9 and want a smart and easy way to prepare the test paper, the second is easy. Trust me, you will get good marks, but only if you have
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Guess Papers 2024 9th Class Islamiat
Here are another 9th grade Pak study speculation papers and I would suggest looking at them too. Therefore, you should also download the guest paper of Study 9th Class 2024 of Bihar-ul-Alam. The link
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This is the 9th grade Pak Study Urdu Medium Estimates Paper for all Punjab Boards for 2024 Board Examinations. The assessment paper is general and not specific to any particular board. The appraisal
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Grade 9 candidates can get guest papers for any subject from this page. Estimation papers will be uploaded soon, and everyone who visits this website can get free online estimation papers on any
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Notes: Start practice of these advantageous guess papers without wasting any hour. You will understand the importance of these guess papers in the examination. Hurry up students, its a time to make
the preparation perfect by these guess papers 2024 class 9. You can check 9th class guess paper in pdf here and can also save them along with Class 9 Notes.
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9th Class Math Guess Papers 2024
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9th Class Math Guess Papers 2024 All Chapters
9th Class Math PDF Chapter Wise
Chapters Chapter Name Medium
1 Matrices and determinants English Medium
2 Real and Complex Numbers. English Medium
3 Logarithms English Medium
4 Algebraic Expressions and Algebraic Formulas English Medium
5 Factorization. English Medium
6 Algebraic Manipulation English Medium
7 Linear Equations and Inequalities English Medium
8 Linear Graphs & Their Application English Medium
9 Introduction to coordinate geometry English Medium
10 Congruent Triangles English Medium
11 Parallelograms and Triangles English Medium
12 Line Bisectors and angle Bisectors English Medium
13 Sides and Angles of A Triangle English Medium
14 Ratio and Proportion English Medium
15 Pythagoras Theorem English Medium
16 Theorems Related with Area English Medium
17 Practical Geometry-Triangles English Medium
9th Class Pairing Scheme 2024 All Subjects View Online
9th Class Notes 2024 All Subjects View Online
9th Class Date Sheet 2024 All BISE Boards View Online
9th Class Roll Number Slip 2024 All BISE Boards View Online
9th Class Guess Paper 2024 All Subjects View Online
Guess Papers Punjab Board 9th Class Math
9th Class Math Guess Papers Punjab Board. Our assessment paper gives students a complete direction on which they should study the questions in a short time to get good scores. Our assessment test
paper is also very useful for ordinary students as they can prepare it in less time. The important mathematical estimation papers are given here. Edcateweb.com is a student education portal. In 2024,
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save you energy and time. Every student wants to cover the whole curriculum, but it is difficult for all students.9th Class Math Guess Papers Punjab Board.
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Confidence intervals with current status data
Piet Groeneboom and Kim Hendrickx
In the current status model, a positive variable of interest \(X\) with distribution function \(F_0\) is not observed directly. A censoring variable \(T \sim G\) is observed instead together with the
indicator \(\Delta=(X \le T)\). curstatCI provides functions to estimate the distribution function \(F_0\) and to construct pointswise confidence intervals around \(F_0(t)\) based on an observed
sample \((T_1, \Delta_1),\ldots, (T_n, \Delta_n)\) of size \(n\) from the observable random vector \((T, \Delta)\). The methods used in this package are described in Groeneboom and Hendrickx (2017).
More details on the current status model can be found in Groeneboom and Jongbloed (2014).
To illustrate the usage of the functions provided in the curstatCI package, we consider a sample \((X_1,\ldots,X_n)\) of size \(n=1000\) from the truncated exponential distribution on \([0,2]\) with
distribution function \(F_0(t)=(1-\exp(-t))/(1-\exp(-2)\) if \(t \in [0,2]\). The observation points \((T_1,\ldots,T_n)\) are sampled from a uniform distribution on \([0,2]\). This data setting is
also considered in the simulation section of Groeneboom and Hendrickx (2017). The R-code to generate the observable random vector \((t_1,\delta_1), \ldots, (t_n, \delta_n)\) is given below.
t<-rep(NA, n)
delta<-rep(NA, n)
for(i in (1:n) ){
if(y<=t[i]){ delta[i]<-1}
The nonparametric maximum likelihood estimator (MLE)
The MLE \(F_n\) of \(F_0\) is defined as the maximizer of the log likelihood given by (up to a constant not depending on \(F\)) \[ \sum_{i=1}^n \Delta_i\log F(T_i) + (1-\Delta_i)\log(1-F(T_i)),\]
over all possible distribution functions \(F\). As can be seen from its structure, the log likelihood only depends on the value that \(F\) takes at the observed time points \(T_i\). The values in
between are irrelevant as long as \(F\) is increasing. The MLE \(F_n\) is a step function which can be characterized by the left derivative of the greatest convex minorant of the cumulative
sumdiagran consisting of the points \(P_0=(0,0)\) and \[P_i=\left(\sum_{j=1}^i w_j,\sum_{j=1}^i f_{1j}\right),\,i=1,\dots,m,\]
where the \(w_j\) are weights, given by the number of observations at point \(T_{(j)}\), assuming that \(T_{(1)}<\dots<T_{(m)}\) (\(m\) being the number of different observations in the sample) are
the order statistics of the sample \((T_1,\Delta_1),\dots,(T_n,\Delta_n)\) and where \(f_{1j}\) is the number of \(\Delta_k\) equal to one at the \(j\)th order statistic of the sample. When no ties
are present in the data, \(w_j=1, m=n\) and \(f_{1j}=\Delta_{(j)}\), where \(\Delta_{(j)}\) corresponds to \(T_{(j)}\).
The function ComputeMLE in the package curstatCI computes the values of the MLE at the distinct jump points of the stepwise MLE. The current status data needs to be formatted as follows. The first
column contains the observations \(t_{(1)}<\dots<t_{(m)}\) in ascending order. The second and third columns contain the variables \(f_{1j}\) and \(w_j\) corresponding to \(t_{(j)}\).
A<-cbind(t[order(t)], delta[order(t)], rep(1,n))
## [,1] [,2] [,3]
## [1,] 0.0007901406 0 1
## [2,] 0.0064286250 0 1
## [3,] 0.0075551583 0 1
## [4,] 0.0121238651 0 1
## [5,] 0.0150197768 0 1
## [6,] 0.0179643347 0 1
The function ComputeMLE returns the jump points and corresponding values of the MLE.
## x mle
## 1 0.00000000 0.00000000
## 2 0.04461921 0.05405405
## 3 0.12463964 0.15000000
## 4 0.18832966 0.18181818
## 5 0.21623244 0.21428571
## 6 0.24208302 0.25000000
## 7 0.25054101 0.30555556
## 8 0.32536475 0.38461538
## 9 0.34771374 0.38888889
## 10 0.39735814 0.45454545
## 11 0.55754520 0.50000000
## 12 0.63327266 0.51515152
## 13 0.70567068 0.54838710
## 14 0.74357757 0.61363636
## 15 0.82979603 0.66666667
## 16 0.83427289 0.69230769
## 17 0.90577147 0.69902913
## 18 1.13136189 0.75000000
## 19 1.14422337 0.81967213
## 20 1.38870142 0.85714286
## 21 1.40659256 0.90000000
## 22 1.48428796 0.92307692
## 23 1.55291452 0.97297297
## 24 1.77735579 0.98666667
## 25 1.91620524 1.00000000
The number of jump points in the simulated data example equals 24. The MLE is zero at all points smaller than the first jump point 0.04461921 and one at all points larger than or equal to the last
jump point 1.91620524. A picture of the step function \(F_n\) is given below.
plot(mle$x, mle$mle,type='s', ylim=c(0,1),xlim=c(0,2), main="",ylab="",xlab="",las=1)
The smoothed maximum likelihood estimator (SMLE)
Starting from the nonparametric MLE \(F_n\), a smooth estimator \(\tilde{F}_{nh}\) of the distribution function \(F_0\) can be obtained by smoothing the MLE \(F_n\) using a kernel function \(K\) and
bandwidth \(h>0\). We use the triweight kernel defined by \[ K(t)=\frac{35}{32}\left(1-t^2\right)^31_{[-1,1]}(t).\] The SMLE is next defined by \[ \tilde F_{nh}(t)=\int\mathbb K\left(\frac{t-x}{h}\
right)\,d F_n(x), \] where \[\mathbb K(t)=\int_{-\infty}^t K(x)\,dx.\] The function ComputeSMLE computes the values of the SMLE in the points x based on a pre-specified bandwidth choice \(h\). A
user-specified bandwidth vector bw of size length(x) is used for each point in the vetor x. A data-driven bandwidth choice for each point in x is returned by the function ComputeBW which is described
For our simulated data set, a picture of the SMLE \(\tilde F_{nh}(t)\) using a bandwidth \(h=2n^{-1/5}\), evaluated in the points \(t=0.02,0.04,\ldots,1.98\) together with the true distribution
function \(F_0\) is obtained as follows:
grid<-seq(0.02,1.98, 0.02)
smle<-ComputeSMLE(data=A, x=grid, bw=bw)
plot(grid, smle,type='l', ylim=c(0,1), main="",ylab="",xlab="",las=1)
lines(grid, (1-exp(-grid))/(1-exp(-2.0)), col=2, lty=2)
Pointwise confidence intervals around the SMLE using a data-driven pointwise bandwidth vector
The nonparametric bootstrap procedure, which consists of resampling (with replacement) the \((T_i,\Delta_i)\) from the original sample is consistent for generating the limiting distribution of the
SMLE \(\tilde F_{nh}(t)\) under current status data (see Groeneboom and Hendrickx (2017)). As a consequence, valid pointwise confidence intervals for the distribution function \(F_0(t)\) can be
constructed using a bootstrap sample \((T_1^*,\Delta_1^*),\ldots (T_n^*,\Delta_n^*)\). The \(1-\alpha\) bootstrap confidence intervals generated by the function ComputeConfIntervals are given by \[ \
left[\tilde F_{nh}(t)-Q_{1-\alpha/2}^*(t)\sqrt{S_{nh}(t)}, \tilde F_{nh}(t)-Q_{\alpha/2}^*(t)\sqrt{S_{nh}(t)}\right],\] where \(Q_{\alpha}^*(t)\) is the \(\alpha\)th quantile of \(B\) values of \(W_
{nh}^*(t)\) defined by, \[W_{nh}^*(t)=\left\{\tilde F_{nh}^*(t)-\tilde F_{nh}(t)\right\}/\sqrt{S_{nh}^*(t)},\] where \(\tilde F_{nh}^*(t)\) is the SMLE in the bootstrap sample and \(S_{nh}(t)\) is
given by: \[S_{nh}(t)=\frac1{(nh)^{2}}\sum_{i=1}^n K\left(\frac{t-T_i}h\right)^2\left(\Delta_i-F_n(T_i)\right)^2.\] \(S_{nh}^*(t)\) is the bootstrap analogue of \(S_{nh}(t)\) obtained by replacing \
((T_i,\Delta_i)\) in the expression above by \((T_i^*,\Delta_i^*)\). The number of bootstrap samples \(B\) used by the function ComputeSMLE equals \(B=1000\).
The bandwidth for estimating the SMLE \(\tilde F_{nh}\) at point \(t\), obtained by minimizing the pointwise Mean Squared Error (MSE) using the subsampling principle in combination with
undersmoothing is given by the function ComputeBW. To obtain an approximation of the optimal bandwidth minimizing the pointwise MSE, \(1000\) bootstrap subsamples of size \(m=o(n)\) are generated
from the original sample using the subsampling principle and \(c_{t, opt}\) is selected as the minimizer of \[ \sum_{b=1}^{1000}\left\{\tilde F_{m,cm^{-1/5}}^b(t) - \tilde F_{n,c_0n^{-1/5}}(t) \right
\}^2,\] where \(\tilde F_{n,c_0n^{-1/5}}\) is the SMLE in the original sample of size \(n\) using an initial bandwidth \(c_0n^{-1/5}\). The bandwidth used for estimating the SMLE is next given by \(h
When the bandwidth \(h\) is small, it can happen that no time points are observed within the interval \([t-h, t+h]\). As a consequence the estimate of the variance \(S_{nh}(t)\) is zero and the
Studentized Confidence intervals are unsatisfactory. If this happens, the function ComputeConfIntervals returns the classical \(1-\alpha\) confidence intervals given by: \[ \left[\tilde F_{nh}(t)-Z_
{1-\alpha/2}^*(t), \tilde F_{nh}(t)-Z_{\alpha/2}^*(t)\right],\] where \(Z_{\alpha}^*(t)\) is the \(\alpha\)th quantile of \(B\) values of \(V_{nh}^*(t)\) defined by, \[V_{nh}^*(t)=\tilde F_{nh}^*(t)-
\tilde F_{nh}(t).\]
Besides the upper and lower bounds of the \(1-\alpha\) bootstrap confidence intervals, the function ComputeConfIntervals also returns the output of the functions ComputeMLE and ComputeSMLE. The
theory for the construction of pointwise confidence intervals is limited to points within the observation interval. It is not recomended to construct intervals in points smaller resp. larger than the
smallst resp. largest observed time point.
The general approach for constructing the confidence intervals is the following: First decide upon the points where the confidence intervals need to be computed. If no particular interest in certain
points is requested, it is useful to consider a grid of points within the interval starting from the smallest observed time point \(t_i\) until the largest observed time point \(t_j\). The function
ComputeConfIntervals can also deal with positive values outside this interval but some numerical instability is to be expected and the results are no longer trustworthy.
## [1] 0.0007901406 1.9997072918
grid<-seq(0.01,1.99 ,by=0.01)
Next, select the bandwidth vector for estimating the SMLE \(\tilde F_{nh}(t)\) for each point in the grid. If no pre-specified bandwidth value is preferred, a data-driven bandwidth vector can be
obtained by the function ComputeBW.
bw<-ComputeBW(data=A, x=grid)
## The computations took 4.025 seconds
plot(grid, bw, main="",ylim=c(0.5,0.7),ylab="",xlab="",las=1)
The bandwidth vector obatined by the function ComputeBW is used as input bandwidth for the function ComputeConfIntervals.
out<-ComputeConfIntervals(data=A,x=grid,alpha=0.05, bw=bw)
## The program produces the Studentized nonparametric bootstrap confidence intervals for the cdf, using the SMLE.
## Number of unique observations: 1000
## Sample size n = 1000
## Number of Studentized Intervals = 199
## Number of Non-Studentized Intervals = 0
## The computations took 4.075 seconds
The function ComputeConfIntervals informs about the number of times the Studentized resp. classical bootstrap confidence intervals are calculated. The default method is the Studentized bootstrap
confidence interval. In this simulated data example, the variance estimate for the Studentized confidence intervals is available for each point in the grid and out$Studentized equals grid.
## $names
## [1] "MLE" "SMLE" "CI" "Studentized"
## [5] "NonStudentized"
## numeric(0)
A picture of the the SMLE together with the pointwise confidence intervals in the gridpoints and the true distribution function \(F_0\) is given below:
plot(grid, out$SMLE,type='l', ylim=c(0,1), main="",ylab="",xlab="",las=1)
lines(grid, left, col=4)
lines(grid, right, col=4)
segments(grid,left, grid, right)
lines(grid, (1-exp(-grid))/(1-exp(-2.0)), col=2)
Data applications
Hepatitis A data
Niels Keiding (1991) considered a cross-sectional study on the Hepatitis A virus from Bulgaria. In 1964 samples were collected from school children and blood donors on the presence or absence of
Hepatitis A immunity. In total \(n=850\) individuals ranging from 1 to 86 years old were tested for immunization. It is assumed that, once infected with Hepatitis A, lifelong immunity is achieved. To
estimate the sero-prevalence for Hepatitis A in Bulgaria, 95% confidence intervals around the distribution function for the time to infection are computed using the ComputeConfIntervals function in
the package curstatCI. Since only 22 out of the 850 individuals were older than 75 years, who, moreover, all had antibodies for Hepatitis A, it seems sensible to restrict the range to [1,75]. The
resulting confidence intervals are obtained as follows:
## t freq1 freq2
## 1 1 3 16
## 2 2 3 15
## 3 3 3 16
## 4 4 4 13
## 5 5 7 12
## 6 6 4 15
out<-ComputeConfIntervals(data=hepatitisA,x=grid,alpha=0.05, bw=bw)
The estimated prevalence of Hepatitis A at the age of 18 is 0.51, about half of the infections in Bulgaria happen during childhood.
## [1] 0.5109369
plot(grid, out$SMLE,type='l', ylim=c(0,1), main="",ylab="",xlab="",las=1)
lines(grid, left, col=4)
lines(grid, right, col=4)
segments(grid,left, grid, right)
\(\tilde F_{nh}(1)\) is computed using the classical confidence interval instead of the Studentized confidence interval.
## [1] 1
N. Keiding et al. (1996) considered a current status data set on the prevalence of rubella in 230 Austrian males with ages ranging from three months up to 80 years. Rubella is a highly contagious
childhood disease spread by airborne and droplet transmission. The symptoms (such as rash, sore throat, mild fever and swollen glands) are less severe in children than in adults. Since the Austrian
vaccination policy against rubella only vaccinated girls, the male individuals included in the data set represent an unvaccinated population and (lifelong) immunity could only be acquired if the
individual got the disease. Pointwise confidence intervals are useful to investigate the time to immunization (i.e. the time to infection) against rubella.
## t freq1 freq2
## 1 0.2740 0 1
## 2 0.3781 0 1
## 3 0.5288 0 1
## 4 0.5342 0 1
## 5 0.9452 1 1
## 6 0.9479 0 1
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.274 8.868 25.595 28.970 44.888 80.118
out<-ComputeConfIntervals(data=rubella,x=grid,alpha=0.05, bw=bw)
The SMLE increases steeply in the ages before adulthood which is in line with the fact that rubella is considered as a childhood disease.
plot(grid, out$SMLE,type='l', ylim=c(0,1), main="",ylab="",xlab="",las=1)
lines(grid, left, col=4)
lines(grid, right, col=4)
segments(grid,left, grid, right)
Groeneboom, P., and K. Hendrickx. 2017. “The Nonparametric Bootstrap for the Current Status Model.” Electron. J. Statist. 11 (2): 3446–84. doi:10.1214/17-EJS1345.
Groeneboom, P., and G. Jongbloed. 2014. Nonparametric Estimation Under Shape Constraints. Cambridge: Cambridge Univ. Press.
Keiding, N., K. Begtrup, T.H. Scheike, and G. Hasibeder. 1996. “Estimation from Current Status Data in Continuous Time.” Lifetime Data Anal. 2: 119–29.
Keiding, Niels. 1991. “Age-Specific Incidence and Prevalence: A Statistical Perspective.” J. Roy. Statist. Soc. Ser. A 154 (3): 371–412. doi:10.2307/2983150.
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Diffusion coefficient - (Statistical Mechanics) - Vocab, Definition, Explanations | Fiveable
Diffusion coefficient
from class:
Statistical Mechanics
The diffusion coefficient is a numerical value that quantifies how easily particles spread out or diffuse through a medium over time. It plays a crucial role in understanding the dynamics of Brownian
motion, the process of diffusion itself, and various transport phenomena, indicating how fast particles move from areas of high concentration to low concentration.
congrats on reading the definition of diffusion coefficient. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The diffusion coefficient is typically denoted by the symbol 'D' and has units of area per time (e.g., m²/s).
2. It depends on factors such as temperature, viscosity of the medium, and the size of the diffusing particles.
3. In Brownian motion, the diffusion coefficient can be related to the mean squared displacement of particles, helping to describe their random paths.
4. The diffusion coefficient can vary widely between different materials and conditions; for example, it is generally larger in gases than in liquids.
5. Understanding the diffusion coefficient is crucial for fields like chemistry, biology, and materials science, as it affects reaction rates and transport processes.
Review Questions
• How does the diffusion coefficient relate to Brownian motion and what implications does it have on particle movement?
□ The diffusion coefficient directly impacts Brownian motion by quantifying how quickly particles spread out in a fluid due to random collisions with other molecules. A higher diffusion
coefficient indicates that particles move more freely and spread faster from areas of high concentration. This relationship helps us understand particle behavior in various contexts, from
microscopic systems to larger-scale phenomena.
• Discuss Fick's laws of diffusion and explain how they incorporate the concept of the diffusion coefficient in predicting concentration changes over time.
□ Fick's first law states that the flux of diffusing particles is proportional to the concentration gradient, which involves the diffusion coefficient as a constant of proportionality. Fick's
second law builds on this by describing how this flux leads to changes in concentration over time. Together, these laws illustrate how the diffusion coefficient is essential for calculating
how substances move through different media based on their concentration differences.
• Evaluate how changes in temperature and medium viscosity can affect the diffusion coefficient and impact practical applications such as drug delivery systems.
□ An increase in temperature typically enhances particle kinetic energy, leading to a higher diffusion coefficient, while increased viscosity in a medium slows down particle movement and
reduces it. In practical applications like drug delivery systems, understanding these variations is crucial; for instance, a drug's effectiveness can be influenced by its diffusion rate
through bodily tissues. Thus, adjusting these factors can optimize drug release profiles and improve therapeutic outcomes.
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ECCC - Alexander Smal
All reports by Author Alexander Smal:
TR24-037 | 26th February 2024
Yaroslav Alekseev, Yuval Filmus, Alexander Smal
Lifting dichotomies
Revisions: 2
Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure $A$ for some function $f$, we compose $f$ with a
carefully chosen gadget function $g$ and get essentially the same lower bound on a complexity measure $B$ for the ... more >>>
TR23-016 | 22nd February 2023
Yuval Filmus, Edward Hirsch, Artur Riazanov, Alexander Smal, Marc Vinyals
Proving Unsatisfiability with Hitting Formulas
Revisions: 1
Hitting formulas have been studied in many different contexts at least since [Iwama 1989]. A hitting formula is a set of Boolean clauses such that any two of the clauses cannot be simultaneously
falsified. [Peitl and Szeider 2022] conjectured that the family of unsatisfiable hitting formulas should contain the hardest ... more >>>
TR22-016 | 15th February 2022
Artur Ignatiev, Ivan Mihajlin, Alexander Smal
Super-cubic lower bound for generalized Karchmer-Wigderson games
Revisions: 1
In this paper, we prove a super-cubic lower bound on the size of a communication protocol for generalized Karchmer-Wigderson game for some explicit function $f: \{0,1\}^n\to \{0,1\}^{\log n}$. Lower
bounds for original Karchmer-Wigderson games correspond to De Morgan formula lower bounds, thus the best known size lower bound is cubic. ... more >>>
TR20-117 | 4th August 2020
Yuriy Dementiev, Artur Ignatiev, Vyacheslav Sidelnik, Alexander Smal, Mikhail Ushakov
New bounds on the half-duplex communication complexity
Revisions: 3
In this work, we continue the research started in [HIMS18], where the authors suggested to study the half-duplex communication complexity. Unlike the classical model of communication complexity
introduced by Yao, in the half-duplex model, Alice and Bob can speak or listen simultaneously, as if they were talking using a walkie-talkie. ... more >>>
TR20-116 | 1st August 2020
Ivan Mihajlin, Alexander Smal
Toward better depth lower bounds: the XOR-KRW conjecture
Revisions: 2
In this paper, we propose a new conjecture, the XOR-KRW conjecture, which is a relaxation of the Karchmer-Raz-Wigderson conjecture [KRW95]. This relaxation is still strong enough to imply $\mathbf{P}
\not\subseteq \mathbf{NC}^1$ if proven. We also present a weaker version of this conjecture that might be used for breaking $n^3$ lower ... more >>>
TR18-089 | 27th April 2018
Kenneth Hoover, Russell Impagliazzo, Ivan Mihajlin, Alexander Smal
Half-duplex communication complexity
Revisions: 6
Suppose Alice and Bob are communicating bits to each other in order to compute some function $f$, but instead of a classical communication channel they have a pair of walkie-talkie devices. They can
use some classical communication protocol for $f$ where each round one player sends bit and the other ... more >>>
TR17-191 | 15th December 2017
Alexander Smal, Navid Talebanfard
Prediction from Partial Information and Hindsight, an Alternative Proof
Revisions: 2
Let $X$ be a random variable distributed over $n$-bit strings with $H(X) \ge n - k$, where $k \ll n$. Using subadditivity we know that a random coordinate looks random. Meir and Wigderson [TR17-149]
showed a random coordinate looks random to an adversary who is allowed to query around $n/k$ ... more >>>
TR16-022 | 22nd February 2016
Alexander Golovnev, Alexander Kulikov, Alexander Smal, Suguru Tamaki
Circuit size lower bounds and #SAT upper bounds through a general framework
Revisions: 2
Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary
Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the ... more >>>
TR11-091 | 20th May 2011
Edward Hirsch, Dmitry Itsykson, Valeria Nikolaenko, Alexander Smal
Optimal heuristic algorithms for the image of an injective function
The existence of optimal algorithms is not known for any decision problem in NP$\setminus$P. We consider the problem of testing the membership in the image of an injective function. We construct
optimal heuristic algorithms for this problem in both randomized and deterministic settings (a heuristic algorithm can err on a ... more >>>
TR10-193 | 5th December 2010
Edward Hirsch, Dmitry Itsykson, Ivan Monakhov, Alexander Smal
On optimal heuristic randomized semidecision procedures, with applications to proof complexity and cryptography
The existence of an optimal propositional proof system is a major open question in proof complexity; many people conjecture that such systems do not exist. Krajicek and Pudlak (1989) show that this
question is equivalent to the existence of an algorithm that is optimal on all propositional tautologies. Monroe (2009) ... more >>>
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Bias and consistency
Suppose you have two ways to estimate something you’re interested in. One is biased and one is unbiased. Surely the unbiased method is better, right? Not necessarily. Statistical bias is not as bad
as it sounds.
Under ideal conditions, an unbiased estimator gives the correct answer on average, but each particular estimate may be ridiculous. Suppose you ask me to estimate how many dwarfs were in Snow White
and the Seven Dwarfs. If I alternately guess 100 and −272, each guess will be wildly wrong. But if 75% of the time I guess 100 and 25% of the time guess −272, my average guess will be 7 and so my
estimates will be unbiased. But if half the time I guess 8 and half the time I guess 7, my average guess will be 7.5 and my process will be biased. However, each estimate will be more accurate.
Consistency is a weaker condition than unbiasedness. Consistency says that if you feed your method enough data generated from your assumed model, your estimates will converge to the correct value.
But if your model is not exactly correct (and it never is) will you get a reasonably good result? It’s possible for an inconsistent method to provide good results in practice and it’s possible that a
consistent method may not.
In his blog post on cross validation, Rob Hyndman mentions a paper that shows one validation method is consistent and another is not. Rob concludes
Frankly, I don’t consider this is a very important result as there is never a true model. In reality, every model is wrong, so consistency is not really an interesting property.
In the context of his post, Rob argues that the most important test of a statistical method is how well it predicts future data. Some people have commented that this comes down too hard on
consistency. But we’re talking about a blog post, and blogs don’t use the same kind of carefully qualified language that formal papers do. Perhaps in a more formal setting Rob might argue that a
gross failure of consistency gives one reason to suspect a method won’t predict well, but a lack of complete consistency shouldn’t remove a method from consideration. Such language may be
inoffensive, but it lacks the verve of his original statement.
Too often bias and consistency are seen as all-or-nothing properties. In theoretical statistics, one typically asks whether a method is biased, not how biased it is. The same is true of consistency.
Bias and consistency are only two criteria by which methods can be evaluated. A small amount of bias or inconsistency may be an acceptable trade-off in exchange for better performance by other
criteria such as efficiency or robustness.
Related posts
3 thoughts on “Bias and consistency”
1. Couple more examples — James-Stein estimator for mean of 3d Gaussian is biased but has uniformly lower squared error. Best unbiased estimator of 1/p for a Binomial(n,p) distributed variable has
infinite variance.
In machine learning, a well known maxim is that learning is impossible without bias. It’s statistics, estimation seems unbiased because the statistician restricts learning to a single model but
the bias is still there, it just happens before automatic inference starts.
2. Regarding bias and learning, many people prefer implicit bias to explicit bias. As long as the bias is implicit in the choice of model, we can pretend it doesn’t exist. :)
3. Good points and I wish you would go a little further. It seems to me that one reason the public has skepticism about findings which really are pretty consistent, e.g. , global warming, is that
statisticians and other researchers exaggerate the impact of such factors as a slightly biased estimate.
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How To Calculate Square Footage - Easy to Calculate
When you are looking for a place to live it is very common to ask about its square footage. This refers to the place’s area measured in units of square feet. Everywhere in the world with the
exception of just a few countries, this would be measured in square meters. But no worries! It is pretty easy to convert one into the other. Keep reading to learn how.
Usually, the bigger a house is, the more expensive it will be in a given area of a city. On the other hand, if you want to sell your house it is essential that you report its square footage to
potential buyers, so they can compare it to other options in the market.
This is why learning how to calculate the square footage of any space, whether you plan to sell it, rent it or even renovate it, is a very useful skill to have. Let’s discover how to do it for
commonly- and irregularly-shaped rooms.
How to calculate square footage
To calculate a flat space’s square footage follow these steps:
1. Divide the entire area into as many adjacent common shapes (rectangles, triangles, circles, etc.) as you need without leaving any uncovered space.
2. Calculate the area of every shape independently in square feet, using the following equations. To do so, make sure the lengths and radii are in feet.
3. Add all of the resulting areas together to calculate the room’s total square footage:
What is square footage
An area is the amount of two-dimensional space covered by any surface. You can also think of it as the space enclosed by any flat closed curve, meaning one which starts and ends at the same place.
Think about a rope. If you tie both its ends together and lay it on the floor you can produce an infinite amount of shapes. The following image shows three possible configurations. The rope
represents the perimeter of the shape, and the space enclosed by it is its area. The former is fixed in this case, because the rope does not change its total length. On the other hand, the total area
depends on the shape that you create with it.
Different flat shapes have different areas, or cover different amounts of space, as long as the perimeter stays the same. If the perimeter of a shape increases, it will be able to enclose a bigger
space, meaning its area will increase, while if it decreases, the total area enclosed by it will be less.
How an area is measured depends on the specific shape enclosing it. Since it is essentially flat, it spreads over bidimensional space, which implies it has a width and a length. In principle, the
area is determined by multiplying these dimensions. Since both are measured in meters, the area has units of square meters.
In the United States, Canada and the United Kingdom, feet (ft) are widely used as a unit for length. In the rest of the world meters are used instead. The relation between one and the other is the
This means that an area can be measured both in square meters and square feet. To convert one into the other fractions can be used. Since 1 m equals 3.3 ft, then their ratio must be equal to 1, and
since multiplying any number by 1 yields the same number, we can write the following equation:
So, in order to convert square meters to square feet, you just need to multiply the former by 10.8.
In these examples we have been dealing with flat shapes. But what happens when a shape is not flat but bumpy and uneven? What is its area then? This happens, for example, with land terrains which
have hills or slopes.
Imagine you have a piece of cloth big enough to cover the entire uneven terrain. Since fabric is flexible it will be able to take the three-dimensional shape of the land. Once entirely covered, you
cut the cloth following the terrain’s boundaries, so that its entire area matches that of the land.
If you then extend the fabric on a flat surface you will be able to measure a flat area as big as the surface of your terrain. In this scenario, the area is called surface area. Although it might
sound redundant, using the word “surface” just implies that the area being measured is distributed on top of a 3-dimensional object.
Now, most homes have flat floors, even if distributed in different spaces with different heights. This makes measuring the total area of a room or an entire apartment or house, a lot easier.
Since square feet is the most common unit to measure areas in countries which use the imperial system, the square footage of any space simply refers to its area measured in said units.
How to measure the square footage of a rectangular room
Most rooms or living spaces are rectangular. This means their square footage matches that of a rectangle with the same width and length as the room. In this case, its area is calculated by
multiplying both dimensions. Look at the following image. The width of the room is 12 feet, and its length is 16 feet.
The total square footage of the room is then:
Remember there is no real difference between the width and the length of a rectangle. We only use these words to differentiate both dimensions from our perspective.
How to calculate the square footage of an irregular room
Sometimes, architects get a little more creative than usual and design spaces that no longer have simple rectangular shapes. How can we measure the square footage of such a space then?
In this case, a very useful option is to fit common shapes, which areas are known, into the space you want to measure. Take a look at the following image. Although the room is not a simple rectangle,
it can be thought of as the sum of two adjacent rectangles:
The area of the left rectangle is:
And the area of the right rectangle is:
The total area of the room is the sum of both rectangles:
The same strategy can be implemented with many other room shapes, even those with round perimeters. Take a look at the following example. The room can be divided into two shapes: a rectangle and a
semicircle (half a circle).
The area of the rectangle is:
The area of the semicircle is simply half of the area of an entire circle with the same radius. In this case, the radius is 6 ft, which can be extracted by looking at the room’s length (12 ft), which
equals the circle’s diameter, and dividing it by two. The area of this section is:
The total area of the room is then:
Since many rooms with irregular shapes can be fitted with common shapes as rectangles, triangles or circles, or parts of them, here is a summary of their areas in terms of their main dimensions:
Other helpful sources
If you want to measure the area of a room with an irregular shape, you can use this helpful calculator by HomeAdvisor. Add as many individual shapes as you need to describe the entire space and input
their dimensions. Then click “calculate” and see the final result for the room’s area.
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Getting Started with dsims
L Marshall
Distance Sampling Simulations
Additional vignettes can be found in the distance sampling examples page of our website: http://examples.distancesampling.org
This vignette introduces the basic procedure for setting up and running a distance sampling simulation using ‘dsims’ (Marshall 2021). The ‘dsims’ package uses the distance sampling survey design
package ‘dssd’ (Marshall 2020a) to define the design and generate the surveys (sets of transects). For further details on defining designs please refer to the ‘dssd’ vignettes. ‘dsims’ was designed
to be largely similar to the ‘DSsim’ package (Marshall 2020b) in terms of work flow, functions and arguments. The main differences in terms of its use lie in the definition of the designs which can
now be generated in R using the ‘dssd’ package (these packages are automatically linked) and the definition of analyses. Analyses are now defined using terminology based on the ‘Distance’ package
(Miller 2020). In addition, the underlying functionality now makes use of the ‘sf’ package (Pebesma and Baston 2021).
Distance Sampling techniques provide design based estimates of density and abundance for populations. The accuracy of these estimates relies on valid survey design. While general rules of thumb can
help guide our design choices, simulations emulating a specific set of survey characteristics can often help us achieve more efficient and robust designs for individual studies. For example,
simulations can help us investigate how effort allocation can affect our estimates or the effects of a more efficient design which has less uniform coverage probability. Due to the individual nature
of each study, each with their specific set of characteristics, simulation can be a powerful tool in evaluating survey design.
Setting up the Region
We will use the St Andrews bay area as an example study region for these simulations. This is a single strata study region which has been projected into metres. We will first load the ‘dsims’
package, this will also automatically load the ‘dssd’ package. As this shapefile does not have a projection recorded (in an associated .prj file) we tell ‘dsims’ that the units are metres.
## Loading required package: dssd
# Find the file path to the example shapefile in dssd
shapefile.name <- system.file("extdata", "StAndrew.shp", package = "dssd")
# Create the survey region object
region <- make.region(region.name = "St Andrews bay",
shape = shapefile.name,
units = "m")
Defining the study population
To define a study population we require a number of intermediate steps. We describe these in turn below.
Population Density Grid
The first step in defining your study population is to set up the density grid. One way to do this is to first create a flat surface and then add hot and low spots to represent where you think you
might have areas of higher and lower density of animals.
If we were to assume that there were 300 groups in the St Andrews bay study area (which is a fairly large number!) this would only give us an average density of 3.04-07 groups per square metre. For
this simulation, as we will use a fixed population size, we do not need to worry about the absolute values of the density surface. Instead, it can be simpler to work with larger values and be aware
that we are defining a relative density surface. So where we create a surface to have a density of twice that in another area that relationship will be maintained (be it at much smaller absolute
values) when we later generate the population.
For the purposes of simulation you will likely want to test over a range of plausible animal distributions (if you knew exactly how many you were going to find at any given location you probably
wouldn’t be doing the study!). When testing non-uniform coverage designs it is advisable to try out worst case scenarios, i.e. set density in the area of higher or lower coverage to differ from the
majority of the survey region. This will give an idea of the degree of potential bias which could be introduced.
In this example, for the equal spaced zigzag design, as it is generated in a convex hull the areas with differing coverage are likely to be at the very top and very bottom of the survey region. In
the density grid below these areas are shown to have lower animal density than the rest of the survey region, a likely scenario when a study region has been constructed in order to catch the range of
a population of interest.
# We first create a flat density grid
density <- make.density(region = region,
x.space = 500,
constant = 1)
# Now we can add some high and low points to give some spatial variability
density <- add.hotspot(object = density,
centre = c(-170000, 6255000),
sigma = 8000,
amplitude = 4)
density <- add.hotspot(object = density,
centre = c(-160000, 6275000),
sigma = 6000,
amplitude = 4)
density <- add.hotspot(object = density,
centre = c(-155000, 6260000),
sigma = 3000,
amplitude = 2)
density <- add.hotspot(object = density,
centre = c(-150000, 6240000),
sigma = 10000,
amplitude = -0.9)
density <- add.hotspot(object = density,
centre = c(-155000, 6285000),
sigma = 10000,
amplitude = -1)
# I will choose to plot in km rather than m (scale = 0.001)
plot(density, region, scale = 0.001)
In some situations you may not need to rely on constructing a density distribution from scratch. Now we will demonstrate how to use a gam to construct the density surface. As I do not have data for
this area I will use the density grid I created above as an example dataset. I will fit a gam to this data and then use this to create a new density object. As I need to restrict the predicted values
to be greater than zero, I will use a log link with the gaussian error distribution. This can also be a useful trick if you want to turn something created using the above method, which can look a bit
lumpy and bumpy, into a smoother distribution surface. The gam fitted must only use a smooth over x and y to fit the model as no other predictor covariates will be present in the density surface.
# First extract the data above - this is simple in this case as we only have a single strata
# Multi-strata regions will involve combining the density grids for each strata into a
# single dataset.
density.data <- density@density.surface[[1]]
## Simple feature collection with 6 features and 4 fields
## Geometry type: POLYGON
## Dimension: XY
## Bounding box: xmin: -157572.4 ymin: 6241463 xmax: -154890.4 ymax: 6241543
## CRS: NA
## strata density x y geometry
## 34 St Andrews bay 0.6128054 -157640.4 6241293 POLYGON ((-157390.4 6241543...
## 35 St Andrews bay 0.5614958 -157140.4 6241293 POLYGON ((-157390.4 6241543...
## 36 St Andrews bay 0.5125986 -156640.4 6241293 POLYGON ((-156890.4 6241543...
## 37 St Andrews bay 0.4662975 -156140.4 6241293 POLYGON ((-156390.4 6241543...
## 38 St Andrews bay 0.4227525 -155640.4 6241293 POLYGON ((-155890.4 6241543...
## 39 St Andrews bay 0.3821010 -155140.4 6241293 POLYGON ((-155390.4 6241543...
## Loading required package: nlme
## This is mgcv 1.9-0. For overview type 'help("mgcv-package")'.
fit.gam <- gam(density ~ s(x,y), data = density.data, family = gaussian(link="log"))
# Use the gam object to create a density object
gam.density <- make.density(region = region,
x.space = 500,
fitted.model = fit.gam)
plot(gam.density, region, scale = 0.001)
Other Population Parameters
Once we have created a plausible animal density distribution we can go on to define other population parameters. We do this by constructing a population description.
We will assume animals occur in small clusters so we will first create a covariate list and define the distribution for cluster size (which must be named “size”) as a zero-truncated Poisson
distribution with mean equal to 3. For those of you familiar with ‘DSsim’ please note the simplified format for defining population covariates.
The other population value we have to define is the population size. As we have clusters in our population, N will refer to the number of clusters rather than individuals. We will set the number of
clusters to be 100. We then leave the fixed.N argument as the default TRUE to say we would like to generate the population based on the population size rather than the density surface.
# Create a covariate list describing the distribution of cluster sizes
covariates <- list(size = list(distribution = "ztruncpois", mean = 3))
# Define the population description
pop.desc <- make.population.description(region = region,
density = gam.density,
covariates = covariates,
N = 300,
fixed.N = TRUE)
Coverage Grid
It is good practice to create a coverage grid over your study area to assess how coverage probability varies spatially across your study area for any specified designs. For designs where there may be
non-uniform coverage, we advise coverage probability is assessed prior to running any simulations. However, as this step is not essential for running simulations we will omit it here and refer you to
the ‘dssd’ vignettes for further details.
Defining the Design
‘dsims’ working together with ‘dssd’ provides a number of point and line transect designs. Further details on defining designs can be found in the ‘dssd’ help and vignettes. We also provide examples
online at .
For these simulations we will compare two line transect designs, systematically spaced parallel lines and equal spaced zigzag lines. The zigzag design will be generated within a convex hull to try to
minimise the off-effort transit time between the ends of transects.
The design angles for each design were selected so that the transects run roughly perpendicular to the coast. The way the two designs are defined means that this is 90 degrees for the parallel line
design and 0 for the zigzag design. Both designs assumed a minus sampling protocol and the truncation distance was set at 750m from the transect. The spacings for each design were selected to give
the same trackline lengths of around 450 km (this was assessed by running the coverage simulations for these designs using ‘run.coverage’, see help in ‘dssd’). The trackline lengths can be thought of
as an indicator of the cost of the survey as they give the total travel time (both on and off effort) from the beginning of the first transect to the end of the last transect.
parallel.design <- make.design(region = region,
design = "systematic",
spacing = 2500,
edge.protocol = "minus",
design.angle = 90,
truncation = 750)
zigzag.design <- make.design(region = region,
design = "eszigzag",
spacing = 2233,
edge.protocol = "minus",
design.angle = 0,
bounding.shape = "convex.hull",
truncation = 750)
Defining Detectability
Once we have defined both the population of interest and the design which we will use to survey our population we now need to provide information about how detectable the individuals or clusters are.
For this example we will assume that larger clusters are more detectable. Take care when defining covariate parameters that the covariate names match those in the population description.
When setting the basic scale parameter along side covariate parameters values we need be aware of how the covariate parameter values are incorporated. The covariate parameter values provided adjust
the value of the scale parameter on the log scale. The scale parameter for any individual (\(\sigma_j\)) can be calculated as:
\[\sigma_j = exp(log(\sigma_0)+\sum_{i=1}^{k}\beta_ix_{ij})\] where \(j\) is the individual, \(\sigma_0\) is the base line scale parameter (passed in as argument ‘scale.param’ on the natural scale),
the \(\beta_i\)’s are the covariate parameters passed in on the log scale for each covariate \(i\) and the \(x_{ij}\) values are the covariate values for covariate \(i\) and individual \(j\).
We will assume a half normal detection function with a scale parameter of 300. We will set the truncation distance to be the same as the design at 750 m. and set the covariate slope coefficient on
the log scale to log(1.08) = 0.077. We can check what our detection functions will look like for the different covariate values by plotting them. To plot the example detection functions we need to
provide the population description as well as detectability.
# Define the covariate parameters on the log scale
cov.param <- list(size = log(1.08))
# Create the detectability description
detect <- make.detectability(key.function = "hn",
scale.param = 300,
cov.param = cov.param,
truncation = 750)
# Plot the simulation detection functions
plot(detect, pop.desc)
We can also calculate the average detection function for our mean cluster size of 3 as defined in our population description:
\[\sigma_{size = 3} = exp(log(300)+log(1.05)*3) = 347.3 \]
Defining Analyses
The final component to a simulation is the analysis or set of analyses you wish to fit to the simulated data. We will define a number of models and allow automatic model selection based on the
minimum AIC value. The models included below are a half-normal with no covariates, a hazard rate with no covariates and a half-normal with cluster size as a covariate. We will leave the truncation
value at 750 as previously defined (it must be <= to the truncation values used previously). We will use the default error variance estimator “R2”. See ‘?mrds::varn’ for descriptions of the various
empirical variance estimators for encounter rate.
Putting the Simulation Together
Now we have all the simulation components defined we can create our simulation objects. We will create one for the systematic parallel line design and one for the equal spaced zigzag design.
sim.parallel <- make.simulation(reps = 999,
design = parallel.design,
population.description = pop.desc,
detectability = detect,
ds.analysis = analyses)
sim.zigzag <- make.simulation(reps = 999,
design = zigzag.design,
population.description = pop.desc,
detectability = detect,
ds.analysis = analyses)
Once you have created a simulation we recommend you check to see what a simulated survey might look like.
# Generate a single instance of a survey: a population, set of transects
# and the resulting distance data
eg.parallel.survey <- run.survey(sim.parallel)
# Plot it to view a summary
plot(eg.parallel.survey, region)
# Generate a single instance of a survey: a population, set of transects
# and the resulting distance data
eg.zigzag.survey <- run.survey(sim.zigzag)
# Plot it to view a summary
plot(eg.zigzag.survey, region)
Running the Simulation
The simulations can be run as follows. Note that these will take some time to run!
Simulation Results
Once the simulations have run we can view a summary of the results. Viewing a summary of a simulation will first summarise the simulation setup and then if the simulation has been run provide a
summary of the results. A glossary is also provided to aid interpretation of the results. Note that each run will produce slightly different results due to the random component of the generation of
both the populations and the sets of survey transects.
Firstly, for the systematic parallel lines design we can see that there is very low bias 1.85% for the estimated abundance / density of individuals. The bias is even lower at only 0.16% for the
estimated abundance / density of clusters. Also we can see that the analyses have done a good job at estimating the mean cluster size, there is only 1.72% bias.
We can also see that the 95% confidence intervals calculated for the abundance / density estimates are in fact capturing the true value around 97% of the time (CI.coverage.prob). We can also note
that the observed standard deviation of the estimates of the mean is a bit lower than the mean se, meaning we are realising a lower variance than we would estimate. This is often seen with systematic
designs as the default variance estimator assumes a completely random allocation of transect locations, systematic designs usually have lower variance.
Reassuringly, these results are as expected for the systematic parallel line design. We expect low bias, as by definition, parallel line designs produce a very uniform coverage probability. The only
areas where this design might not produce uniform coverage is around the boundary where there could be minor edge effects due to the minus sampling.
## GLOSSARY
## --------
## Summary of Simulation Output
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Region : the region name.
## No. Repetitions : the number of times the simulation was repeated.
## No. Excluded Repetitions : the number of times the simulation failed
## (too few sightings, model fitting failure etc.)
## Summary for Individuals
## ~~~~~~~~~~~~~~~~~~~~~~~
## Summary Statistics:
## mean.Cover.Area : mean covered across simulation.
## mean.Effort : mean effort across simulation.
## mean.n : mean number of observed objects across
## simulation.
## mean.n.miss.dist: mean number of observed objects where no distance
## was recorded (only displayed if value > 0).
## no.zero.n : number of surveys in simulation where
## nothing was detected (only displayed if value > 0).
## mean.ER : mean encounter rate across simulation.
## mean.se.ER : mean standard error of the encounter rates
## across simulation.
## sd.mean.ER : standard deviation of the encounter rates
## across simulation.
## Estimates of Abundance:
## Truth : true population size, (or mean of true
## population sizes across simulation for Poisson N.
## mean.Estimate : mean estimate of abundance across simulation.
## percent.bias : the percentage of bias in the estimates.
## RMSE : root mean squared error/no. successful reps
## CI.coverage.prob : proportion of times the 95% confidence interval
## contained the true value.
## mean.se : the mean standard error of the estimates of
## abundance
## sd.of.means : the standard deviation of the estimates
## Estimates of Density:
## Truth : true average density.
## mean.Estimate : mean estimate of density across simulation.
## percent.bias : the percentage of bias in the estimates.
## RMSE : root mean squared error/no. successful reps
## CI.coverage.prob : proportion of times the 95% confidence interval
## contained the true value.
## mean.se : the mean standard error of the estimates.
## sd.of.means : the standard deviation of the estimates.
## Detection Function Values
## ~~~~~~~~~~~~~~~~~~~~~~~~~
## mean.observed.Pa : mean proportion of individuals/clusters observed in
## the covered region.
## mean.estimte.Pa : mean estimate of the proportion of individuals/
## clusters observed in the covered region.
## sd.estimate.Pa : standard deviation of the mean estimates of the
## proportion of individuals/clusters observed in the
## covered region.
## mean.ESW : mean estimated strip width.
## sd.ESW : standard deviation of the mean estimated strip widths.
## Region: St Andrews bay
## No. Repetitions: 999
## No. Excluded Repetitions: 0
## Using only repetitions where all models converged.
## Design: Systematic parallel line design
## design.type : Systematic parallel line design
## bounding.shape : rectangle
## spacing : 2500
## design.angle : 90
## edge.protocol : minus
## Individual Level Covariate Summary:
## size:ztruncpois , mean = 3
## Population Detectability Summary:
## key.function = hn
## scale.param = 300
## truncation = 750
## Covariate Detectability Summary (params on log scale):
## size parameters:
## Strata St Andrews bay
## 0.07696104
## Analysis Summary:
## Candidate Models:
## Model 1: key function 'hn', formula '~1', was selected 474 time(s).
## Model 2: key function 'hr', formula '~1', was selected 201 time(s).
## Model 3: key function 'hn', formula '~size', was selected 324 time(s).
## criteria = AIC
## variance.estimator = R2
## truncation = 750
## Summary for Individuals
## Estimates of Abundance (N)
## Truth mean.Estimate percent.bias RMSE CI.coverage.prob mean.se sd.of.means
## 1 900 916.67 1.85 149.89 0.97 155.99 149.04
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Estimates of Density (D)
## Truth mean.Estimate percent.bias RMSE CI.coverage.prob
## 1 9.113923e-07 9.282781e-07 1.852743 1.517923e-07 0.968969
## mean.se sd.of.means
## 1 1.579674e-07 1.509258e-07
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Summary for Clusters
## Summary Statistics
## mean.Cover.Area mean.Effort mean.n mean.k mean.ER mean.se.ER
## 1 592153913 394769.3 106.7317 15.82883 0.0002704223 3.569565e-05
## sd.mean.ER
## 1 2.137828e-05
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Estimates of Abundance (N)
## Truth mean.Estimate percent.bias RMSE CI.coverage.prob mean.se sd.of.means
## 1 300 300.49 0.16 45.02 0.97 49.01 45.04
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Estimates of Density (D)
## Truth mean.Estimate percent.bias RMSE CI.coverage.prob
## 1 3.037974e-07 3.042914e-07 0.1626056 4.55915e-08 0.970971
## mean.se sd.of.means
## 1 4.962823e-08 4.561166e-08
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Estimates of Expected Cluster Size
## Truth mean.Expected.S percent.bias mean.se.ExpS sd.mean.ExpS
## 1 3 3.05 1.72 0.16 0.2
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Detection Function Values
## mean.observed.Pa mean.estimate.Pa sd.estimate.Pa mean.ESW sd.ESW
## 1 0.6 0.6 0.07 451.01 54.08
We can now check the results for the zigzag design. While zigzag designs generated inside a convex hull can be much more efficient than parallel line designs (less off-effort transit) there is the
possibility of non-uniform coverage. The coverage can be assessed by running ‘run.coverage’ but by itself this does not give much of an indication of the likely effects on the survey results. The
degree to which non-uniform coverage may affect survey results is determined not only by the variability in coverage but also in how that combines with the density of animals in the region. Note that
while we have run only one density scenario here, if you have non-uniform coverage probability it is advisable to test the effects under a range of plausible animal distributions.
Under this assumed distribution of animals, it looks like any effects of non-uniform coverage are going to have minimal effects on the estimates of abundance / density. For individuals the bias is
around 2.5% and for clusters it is 0.65%. Similar to the parallel line design, the confidence intervals are also giving a coverage of 97%.
What we can note is that the improved efficiency of this design has increased our on effort line length and corresponding covered area and is thus giving us a bit better precision than the systematic
parallel line design.
## GLOSSARY
## --------
## Summary of Simulation Output
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Region : the region name.
## No. Repetitions : the number of times the simulation was repeated.
## No. Excluded Repetitions : the number of times the simulation failed
## (too few sightings, model fitting failure etc.)
## Summary for Individuals
## ~~~~~~~~~~~~~~~~~~~~~~~
## Summary Statistics:
## mean.Cover.Area : mean covered across simulation.
## mean.Effort : mean effort across simulation.
## mean.n : mean number of observed objects across
## simulation.
## mean.n.miss.dist: mean number of observed objects where no distance
## was recorded (only displayed if value > 0).
## no.zero.n : number of surveys in simulation where
## nothing was detected (only displayed if value > 0).
## mean.ER : mean encounter rate across simulation.
## mean.se.ER : mean standard error of the encounter rates
## across simulation.
## sd.mean.ER : standard deviation of the encounter rates
## across simulation.
## Estimates of Abundance:
## Truth : true population size, (or mean of true
## population sizes across simulation for Poisson N.
## mean.Estimate : mean estimate of abundance across simulation.
## percent.bias : the percentage of bias in the estimates.
## RMSE : root mean squared error/no. successful reps
## CI.coverage.prob : proportion of times the 95% confidence interval
## contained the true value.
## mean.se : the mean standard error of the estimates of
## abundance
## sd.of.means : the standard deviation of the estimates
## Estimates of Density:
## Truth : true average density.
## mean.Estimate : mean estimate of density across simulation.
## percent.bias : the percentage of bias in the estimates.
## RMSE : root mean squared error/no. successful reps
## CI.coverage.prob : proportion of times the 95% confidence interval
## contained the true value.
## mean.se : the mean standard error of the estimates.
## sd.of.means : the standard deviation of the estimates.
## Detection Function Values
## ~~~~~~~~~~~~~~~~~~~~~~~~~
## mean.observed.Pa : mean proportion of individuals/clusters observed in
## the covered region.
## mean.estimte.Pa : mean estimate of the proportion of individuals/
## clusters observed in the covered region.
## sd.estimate.Pa : standard deviation of the mean estimates of the
## proportion of individuals/clusters observed in the
## covered region.
## mean.ESW : mean estimated strip width.
## sd.ESW : standard deviation of the mean estimated strip widths.
## Region: St Andrews bay
## No. Repetitions: 999
## No. Excluded Repetitions: 0
## Using only repetitions where all models converged.
## Design: Equal spaced zigzag line design
## design.type : Equal spaced zigzag line design
## bounding.shape : convex.hull
## spacing : 2233
## design.angle : 0
## edge.protocol : minus
## Individual Level Covariate Summary:
## size:ztruncpois , mean = 3
## Population Detectability Summary:
## key.function = hn
## scale.param = 300
## truncation = 750
## Covariate Detectability Summary (params on log scale):
## size parameters:
## Strata St Andrews bay
## 0.07696104
## Analysis Summary:
## Candidate Models:
## Model 1: key function 'hn', formula '~1', was selected 476 time(s).
## Model 2: key function 'hr', formula '~1', was selected 178 time(s).
## Model 3: key function 'hn', formula '~size', was selected 345 time(s).
## criteria = AIC
## variance.estimator = R2
## truncation = 750
## Summary for Individuals
## Estimates of Abundance (N)
## Truth mean.Estimate percent.bias RMSE CI.coverage.prob mean.se sd.of.means
## 1 900 922.42 2.49 134.29 0.97 145.28 132.47
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Estimates of Density (D)
## Truth mean.Estimate percent.bias RMSE CI.coverage.prob
## 1 9.113923e-07 9.340926e-07 2.490729 1.359901e-07 0.971972
## mean.se sd.of.means
## 1 1.471139e-07 1.341493e-07
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Summary for Clusters
## Summary Statistics
## mean.Cover.Area mean.Effort mean.n mean.k mean.ER mean.se.ER
## 1 663209990 442140 120.3654 18.47948 0.0002722232 3.346453e-05
## sd.mean.ER
## 1 2.143639e-05
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Estimates of Abundance (N)
## Truth mean.Estimate percent.bias RMSE CI.coverage.prob mean.se sd.of.means
## 1 300 301.94 0.65 41.32 0.97 45.58 41.29
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Estimates of Density (D)
## Truth mean.Estimate percent.bias RMSE CI.coverage.prob
## 1 3.037974e-07 3.057631e-07 0.6470286 4.18412e-08 0.973974
## mean.se sd.of.means
## 1 4.616181e-08 4.181593e-08
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Estimates of Expected Cluster Size
## Truth mean.Expected.S percent.bias mean.se.ExpS sd.mean.ExpS
## 1 3 3.06 1.97 0.15 0.2
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
## Detection Function Values
## mean.observed.Pa mean.estimate.Pa sd.estimate.Pa mean.ESW sd.ESW
## 1 0.6 0.6 0.07 450.54 49.27
Histograms of the estimates of abundance from each of the simulation replicates can also be viewed to check for the possible effects of extreme values or skewed distributions.
We can see in Figure 9 that there were a couple of high estimates generated >500 for both the parallel line and zigzag designs. These probably represent data sets that were difficult to fit a model
too (perhaps a chance spiked data set). Most of the estimates are centered around truth but these occasional high estimates may have increased the mean value slightly and could be associated with the
small amount of positive bias.
Simulation Conclusions
Under these simulation assumptions it appears that the zigzag design will cost us a little in accuracy but allow us to gain some precision. It should be noted that the cost in accuracy will vary
depending on the distribution of animals in the survey region.
|
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Category talk:Monads - Rosetta CodeCategory talk:Monads
Lead Section
A suggestion - perhaps the list monad ? It's a useful route to cartesian products, and enables filtering and list comprehension expressions.
See, for example: https://en.wikibooks.org/wiki/Haskell/Understanding_monads/List
For the Maybe monad, the task description probably needs to add:
1. A brief motivation - for example safe versions of partial numeric functions 2. A couple of specific 'safe' functions to implement and compose
Perhaps, for example, avoiding division by zero, or square roots (or logs?) of negative numbers ?
I made the lead section more explicit to allow editors to implement a List or a Maybe Monad, and in the Clojure example I went ahead and implemented both to show that anyone can do both. As
for a section on motivation, I'm all for it, and I'm new to Rosetta Code so I'm wondering if that's a common practice. Also, I figure if someone is coming to the page, they're past the point
where they need a motivation to learn about monads ;). But like I said, I like the idea, so if you want to add a blurb in the lead about the benefits of monads and their common use cases,
feel free to update! Domgetter (talk) 12:01, 30 January 2016 (UTC)
I think you have assumed too much about the relevance and implementations of monads. I've left some quips in the J implementation to illustrate this issue. But I'll update the description
with a bit about motivation. --Rdm (talk) 18:51, 30 January 2016 (UTC)
Contributors may not need much motivation, but helpful, I think, to frame things more for readers. In that context good to see some introductory and motivational text developing. On
the specifics, I might personally tend to use the term 'pattern' in preference to 'mechanism'. The pattern itself has an algebraic or category theoretic definition, but the key point
for a programmer is really that it provides a well modelled and reliable approach to allowing composition of functions whose arguments consist of modalised or wrapped data types. The
nature of the wrapping and its value/relevance varies (protection against illegal values for partial functions, binding of names to multiple possible values etc etc), but the
possibility of composition is the core value and relevance of all monadic patterns.Hout (talk) 22:12, 30 January 2016 (UTC)
Hmm... "pattern", like "mechanism" is a somewhat ambiguous term with multiple plausibly relevant definitions for something like this (and also wit a good variety of
not-so-relevant definitions). I don't really have any strong preference for one term over the other, but if the distinction matters it would probably be good to provide some
exposition and/or links to suggest something about the relevant aspects of that distinction - our audience here is going to include programmers working with languages or in
environments where these sorts of distinctions are meaningless or, worse: misleading. --Rdm (talk) 23:15, 30 January 2016 (UTC)
PS not sure, FWIW, that I am convinced by the necessity (or empirical reality) of a connection between monadic patterns and multiple dispatch :-) The point about the bind function is
that it applies a type-specific function to whatever is inside the monadic wrapping. There is no need for run-time branching on a detected type, or for functions which can operate on
multiple types Hout (talk) 23:03, 30 January 2016 (UTC)
Correct - the whole point of the monad definition is to hide that implementation detail. But there is no need to hide something which does not exist. --Rdm (talk) 23:17, 30
January 2016 (UTC)
Rereading your explanation in terms of unfied handling of multiple types I do wonder whether a misundertanding might have arisen ? No polymorphism is entailed, and none of the
functions involved, either on the surface, or in the implementation of a particular type of monad, either necessarily can or pretend to apply to more than one type. Or do you
simply mean that composition of wrapped data can be achieved using the same technique with various types of wrapping ? Hout (talk) 01:08, 31 January 2016 (UTC)
Yes, I mean that that these monads are about working with wrapped types (and the point is to be able to work with more than one of these wrapped types). That said, the
word "pattern" also can carry a connotation of "pattern matching" which in turn can carry an implication of polymorphism - not a part of this task, certainly, but for some
readers the use of the term "pattern" might imply something different. --Rdm (talk) 01:36, 31 January 2016 (UTC)
OK, I think I might disagree with you that "the point is to be able to work with more than one of these wrapped types". The fact that Kliesli composition is useful
with various wrapping types does enable a unified treatment of various monads in some coding interfaces, but the point or value of the Maybe monad is not, I think,
that there are other monads - it is that we can get the type-safety of a wrapper without losing the structuring value of functional composition. Similarly, the value
and point of the list monad is not that there are other monads - it is that we get a simple mechanism (by combining composition with the binding of a name to multiple
values) for obtaining cartesian products, expressing set comprehensions, filtering etc. In other words, your formulation appears to be locating the essential value of
monadic patterns in as aspect that seems a little more peripheral or incidental to me. Hout (talk) 02:28, 31 January 2016 (UTC)
I do not know if your point here is that the "bottom type" is not a distinct type from number or if your point here is that you do not "work with" the bottom type.
If you could clarify that point maybe I could understand what you are getting at? --Rdm (talk) 05:03, 31 January 2016 (UTC),
My point is that I think your formulation might mislead – the monadic pattern is not a technique for constructing types (it's a way of setting up a category
with a particular set of morphisms, encoded by specific functions), and it neither entails nor conceals multiple despatch. To answer your question, which
concerns (and is perhaps over-determined by a focus on ?) one particular monad ( the Maybe monad ), none of the top level or called functions which constitute
the morphisms of the category are polymorphically applicable to two different types (e.g. to a choice between bottom and a non-bottom value), and none branch
on whether they are presented with bottom or not bottom.
I think it might be more helpful to readers to simply 1. introduce monads as a way of working with wrapped/modalised data which preserves the possibility of
composition 2. Emphasize that although each monad involves (in fact is defined by) a 'bind' function (for piercing the wrapping and applying a function
directly to its contents) and a 'return' (or wrapping) function, the behaviour and value of particular monads is very variable and quite unpredictable. The
value and nature of the list monad really has nothing more in common with the Maybe monad than those two defining functions. To characterise it as a way of
constructing lists, or as involving some kind of multiple despatch, might tend, I think, to lead astray or simply confuse Hout (talk) 08:43, 31 January 2016
In many typical implementations, a monad is implemented as a type. So I do not know what you mean by "the monadic pattern is not a technique for constructing
types". Maybe, though, you have a specific idea about what techniques are, and that is important? Then again, people use the monadic pattern is to deal with
other types (plural) and of course with the values which those types can take on. so I don't know what you mean by "it neither entails nor conceals multiple
despatch". I understand that there will be examples which do not require multiple dispatch, but I would ask you to find a more generally relevant
(cross-language) phrase if you want to fix that issue.
As for your "none of the top level or called functions which constitute the morphisms of the category are polymorphically applicable to two different types"
statement, that is exactly what I meant when I said it hides multiple dispatch. That said, I am not wedded to the term "multiple dispatch" -it's really no
concern of mine whether the handling of types is at "compile time" at "run time" at "parse time" or at "lunch time". And if you can come up with a description
of multiple type handling which uses less jargon, I'd be all in favor of that.
That said, we are talking about language features, here, and we are also in a context where we have to discuss what's relevant in contexts involving many
languages. I think your points would be relevant if we could assume a language which has a native implementation of monads. Your points could also be relevant
in a few other languages. However, that is not sufficient to convey the ideas to users of other languages.
If you try to narrow the discussion to be specifically tailored to readers who already know what you are talking about, you are not going to convey any useful
information to the reader.
If you are going to address the rosettacode audience, you are going to need to break out of your jargon focus enough to convey to programmers and users of
languages which do not have a native monad implementation enough about how it *works* (as opposed to how it's *implemented*) for them to implement something.
(As an aside, I think the relationship between "list" and "maybe" might be closer than you think. Yes, the implementations will usually be distinct. But if you
constrain a list to have a maximum of one element, the zero length list corresponds to the bottom type and the 1 length list to the other type. That's only
"nothing to do with" if you do not appreciate what analogies have to do with reasoning - and that hypothetical lack of appreciation, if someone were inflicted
with it, could be a significant handicap in many situations.)
Anyways, I think I understand where you are coming from. And, if you can come up with a better phrasing which clearly communicates what needs to be implemented
to users of arbitrary languages - something which focuses on what happens in the implementation rather than on jargon or on what monads are not - that would be
great. However, when I look out at what's been done for various popular languages, I find that a number of them seem to have a only specific monad
implementation which looks an awful lot like the maybe implementation - in other words, not at all the sort of generality which I think you are trying reach
for. So ... perhaps what we are talking about here probably has more to do with proper use of english grammar than it has to do with how to make a working
monad implementation? --Rdm (talk) 09:43, 31 January 2016 (UTC)
Perhaps the solution is to break this task out into several specific monads, and present the nature and value of each individually, avoiding entanglement with
too much algebraic abstraction. This might enable something like, in one case: "The Maybe monad is a way of safely composing partial functions, which avoids
the need for exception handling" and in another "The List monad is a way of composing functions whose argument is bound to a range of possible values, rather
than to one unique value. It enables simple encoding of cartesian products and set comprehension expressions." ("A way of ..." can then be concretised in terms
of implementing two functions - bind/inject/chain (whatever you want to call it) and return/wrap).
I do not think that is necessary, but it sounds doable. --Rdm (talk) 09:43, 31 January 2016 (UTC)
PS I think perhaps your formulation offers a description of algebraic data types rather than of monads. Hout (talk) 09:31, 31 January 2016 (UTC)
Where are you implementing monads without any such types? Would you object to a discussion of types which happens to also mention values that can be
represented using those types? --Rdm (talk) 09:43, 31 January 2016 (UTC)
Useful monads are likely to involve algebraic datatypes, just as elephants are likely to involve lungs, but monads are not defined by their algebraic datatypes
any more than elephants are defined by their lungs. Not all algebraic datatypes are monadic, and not everything that has a lung is an elephant. The list monad
just involves ordinary lists, and no polymorphism. What makes it monadic (and useful) is the bind and return functions. Categories are defined by their
morphisms - not by the nature or contents of their objects in general, or by the presence of algebraic datatypes in particular. I suggest we adopt the solution
below, and make sure that we clarify what functions have to be implemented to ensure that composition works, and that composed functions are not presented with
arguments of the wrong type. Hout (talk) 16:51, 31 January 2016 (UTC)
(1) While the general case algebraic data type is not monadic, I've yet to see an algebraic data type which cannot be used in a monad.
(2) Change of list length is roughly equivalent to a type change. --Rdm (talk) 21:29, 31 January 2016 (UTC)
I think these issues may now be resolved by the edits, but FWIW, the problem with characterising monads by foregrounding algebraic datatypes was mainly one of
emphasis. A monad is a category – a set of related morphisms or functions, if you like – it's not a datatype – and the simplest monad, in which the Return
function is equivalent to the ID function (see under Identity Monad), involves no 'wrapper' or modal context at all. From the practical coding perspective, a
monad is a pattern of function nesting, and the practical steps involved in creating that pattern consist of writing two functions. As you point out, a useful
monad is very likely to involve some kind of modal wrapper (though even the identity monad is used in programming), but while that may be an empirically
(though not formally) *almost* necessary condition, it is neither practically sufficient, nor conceptually central enough to provide an adequate substitute for
foregrounding the monadic functions themselves, and the value of the function nesting which they enable. (In my personal view :-) Hout (talk) 17:42, 2 February
2016 (UTC)
Ok, so why do you distinguish between list monad and maybe monad? More specifically, if I am working in a language where there's no enforced distinction
between a list and a maybe (where the distinction comes from the supporting code - how the data gets used), it seems that I'd use identical implementations of
return and <bind>. Do you disagree with this? If so, why do you say there's a problem with characterising monads by foregrounding algebraic datatypes? (If not,
just let me know and I'll be happy.) --Rdm (talk) 08:07, 3 February 2016 (UTC)
Structure of Monads on Rosetta Code
Perhaps we should use the overall structure used for Sorting Algorithms. That is, make Monads redirect to Category:Monads, which lists all Monads, and then, e.g. the List Monad, make List_monad
redirect to Monads/List_monad in the same way that Merge_sort redirects to Sorting_algorithms/Merge_sort. This way it fits an existing structure on the site, and it makes it clear that there are many
different kinds of Monads. Plus, each kind of Monad can have a task tailored to its semantics. Any thoughts? Domgetter (talk) 09:25, 31 January 2016 (UTC)
Yes, I agree – I think that would work better Hout (talk) 09:37, 31 January 2016 (UTC)
Looks like that turned out nicely. It feels much more organized and uncluttered. Domgetter (talk) 08:52, 1 February 2016 (UTC)
Good. A useful category to introduce, I think. Perhaps a 3rd entry ? The Writer monad is useful, and quite easily understood and written - maybe worth adding, for further comparison
across languages Hout (talk) 19:58, 1 February 2016 (UTC)
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Re: Hep with the formula
04-16-2024 11:57 PM
I have a formula below for calculating the sum of gross risk potential in the table. I want the sum calculation based on the risk name instead of summing up the entire column. I tried the below
formula but it is not giving the correct value.
I can give an example
The total sum of gross risk potential is 100. But the gross risk potential for risk A is 20. I want to get this separate value for all the risks i have in a single column without creating multiple
TotalRiskPotentialPerRisk =
Top_Risk_Controls[Riskname] = EARLIER(Top_Risk_Controls[Riskname])
In the above formula, im getting the same values as the gross risk potential column instead of the sum of them based on the risk name. How to get the sum of the values based on the risk name.
Please note that the data source is the sharepoint list.
04-17-2024 06:05 PM
04-21-2024 01:39 AM
04-21-2024 12:59 PM
04-24-2024 02:12 AM
04-25-2024 06:05 PM
05-01-2024 06:53 AM
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$K$th order statistic in $O(N)$¶
Given an array $A$ of size $N$ and a number $K$. The problem is to find $K$-th largest number in the array, i.e., $K$-th order statistic.
The basic idea - to use the idea of quick sort algorithm. Actually, the algorithm is simple, it is more difficult to prove that it runs in an average of $O(N)$, in contrast to the quick sort.
Implementation (not recursive)¶
template <class T>
T order_statistics (std::vector<T> a, unsigned n, unsigned k)
using std::swap;
for (unsigned l=1, r=n; ; )
if (r <= l+1)
// the current part size is either 1 or 2, so it is easy to find the answer
if (r == l+1 && a[r] < a[l])
swap (a[l], a[r]);
return a[k];
// ordering a[l], a[l+1], a[r]
unsigned mid = (l + r) >> 1;
swap (a[mid], a[l+1]);
if (a[l] > a[r])
swap (a[l], a[r]);
if (a[l+1] > a[r])
swap (a[l+1], a[r]);
if (a[l] > a[l+1])
swap (a[l], a[l+1]);
// performing division
// barrier is a[l + 1], i.e. median among a[l], a[l + 1], a[r]
i = l+1,
j = r;
const T
cur = a[l+1];
for (;;)
while (a[++i] < cur) ;
while (a[--j] > cur) ;
if (i > j)
swap (a[i], a[j]);
// inserting the barrier
a[l+1] = a[j];
a[j] = cur;
// we continue to work in that part, which must contain the required element
if (j >= k)
r = j-1;
if (j <= k)
l = i;
• The randomized algorithm above is named quickselect. You should do random shuffle on $A$ before calling it or use a random element as a barrier for it to run properly. There are also
deterministic algorithms that solve the specified problem in linear time, such as median of medians.
• A deterministic linear solution is implemented in C++ standard library as std::nth_element.
• Finding $K$ smallest elements can be reduced to finding $K$-th element with a linear overhead, as they're exactly the elements that are smaller than $K$-th.
Practice Problems¶
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How do you find the critical numbers of y = sin^2 x? | HIX Tutor
How do you find the critical numbers of #y = sin^2 x#?
Answer 1
We know that #cos2x=cos^2x-sin^2x=>cos2x=1-2*sin^2x=> sin^2x=1/2[1-cos2x]#
Hence #y# becomes #y=1/2*[1-cos2x]#
Now the critical values #c# of #f# are those if and only if
i. #f'(c)=0# ii. #f'(c)# is undefined
So the critical values are those for which #dy/(dc)=0# hence
The critical points are at #x=(n*pi)/2#
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Answer 2
To find the critical numbers of ( y = \sin^2(x) ), you first need to find the derivative of the function with respect to ( x ), and then solve for values of ( x ) where the derivative is equal to
zero or undefined.
First, find the derivative of ( \sin^2(x) ) using the chain rule. The derivative of ( \sin^2(x) ) with respect to ( x ) is ( 2\sin(x)\cos(x) ).
Next, set the derivative equal to zero and solve for ( x ).
( 2\sin(x)\cos(x) = 0 )
This equation is satisfied when either ( \sin(x) = 0 ) or ( \cos(x) = 0 ).
The solutions for ( \sin(x) = 0 ) occur at ( x = k\pi ), where ( k ) is an integer.
The solutions for ( \cos(x) = 0 ) occur at ( x = \frac{\pi}{2} + k\pi ) and ( x = \frac{3\pi}{2} + k\pi ), where ( k ) is an integer.
These values of ( x ) are the critical numbers of the function ( y = \sin^2(x) ).
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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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How to Measure a 3 Phase Power Current - Precision Motor Repair
When Your Business needs to know what type of Power you have.
Here's how to measure a 3 phase power current.
Perhaps the idea of measuring a 3 phase power current sounds intimidating to some of you. After all, a reliance on 3 phase power sources doesn't necessarily indicate an understanding of (or even
interest in) the math which lies behind them.
Even so, some of you might be a tad bit curious, on how power is rated.
So how do you calculate 3 phase power?
While you're certainly welcome to take a peek at Wikipedia's technical entry, we think that you'll find our simple approach to the this task a bit more...suitable for beginners.
That being said, let's hop right into things. Our first order of business is to establish the variables.
Putting the Variables on the Table
Every good lesson should clearly define the variables at its start, and, needless to say, we want this to be a good lesson. Consequently, we'll be taking this time to briefly touch on watts, apparent
power, and power factors.
A watt (W) is a measurement of power. This unit is used to measure the power a circuit takes in. Kilowatts (kW) can also be used to measure this power; one kilowatt is the equivalent of 1000 watts.
Apparent power (VA or volt-ampere) is calculated by finding the product of the voltage and the current; apparent power can also be measured in kilovolt-amperes (kVA). A kVA is equal to 1000
The power factor (pf) is the relationship between kilo-volt amperes and kilowatts. It can be represented as:
kW = kVA x pf
Notice that this formula can be algebraically rearranged in order to calculate each component. The power factor, for instance, can be represented as:
pf = kW/kVA
The kilo-volt amperes, on the other hand, can be represented as:
kVA = kW / pf
Calculating a Single Phase Power Current
Although our ultimate goal is to teach you how to calculate a 3 phase power current, we (and most other people) presume that teaching you how to calculate a single phase power current will lay some
important groundwork for what you have at your business now and
what you may be needing in the future.
There are two reasons for our presumption here, the first of which is that calculating a single phase power current is much simpler than calculating a multiphase or three
phase power current.
The second, more important reason has everything to do with the fact that you can actually use the logic and formula for calculating single phase power currents when calculating multiphase power
But enough of that talk. Let's get down to business.
Generally speaking, you won't be responsible for calculating all of the values of the variables; some, such as the voltage or power factor, will be supplied. You presumably don't, after all, have
access to a voltmeter or any other instrument of that nature.
Having said that, you can easily use the variables that you know the value of to find any unknown values. If, for example, you are given the power factor and wattage, you can make quick work of
finding the apparent power.
Remember that the power factor is the relationship between the kilo-volt amperes and kilowatts. This relationship has previously been expressed as:
kW = kVA x pf
If we algebraically rearrange this equation in order to solve for the apparent power (kVA), we get:
kVA = kW/pf
Thus, we can divide our wattage by the power factor in order to find our apparent power.
What, however, do we make of this apparent power?
At this point, we must introduce a new formula that will allow us to calculate the current. Luckily for us, there is a simple one:
Current = kVA (or VA)/voltage
Using this formula, we simply divide the kVA we've calculated by the voltage (which should be given) in order to calculate the current.
Calculating a 3 Phase Power Current
Now that we've calculated a single phase power current, we can move on to doing the same for 3 phase power currents. Although there is a formula for calculating 3 phase power currents, we'll be
teaching you a more intuitive way of completing this task.
Before we get into the math, however, you must understand exactly how a 3 phase system differs from a single phase system.
To put things simply, the crucial difference between the two systems is the voltage; three phase systems have line to line voltage (VLL) and phase voltage (VLN).
The relationship between the line to line voltage and the phase voltage can be written as:
VLN = VLL/sqrt(3)
For our purposes, you don't need to have an in-depth understanding of these two variables. You only need to keep the relationship between them in mind.
You also shouldn't worry about calculating both of them; at least one of them will be given to you.
Using the method we'll be teaching you, the general idea is to convert a 3 phase system into a single phase one.
In order to make this conversion, however, you need to understand that, for our purposes here, a 3 phase system is essentially producing 3 times as many kilowatts as a single phase system; this
difference in power produced actually makes it pretty easy to see why some people upgrade to 3 phase power.
The apparent power is also tripled in a 3 phase system.
That said, in order to calculate a 3 phase power current using this method, you'll need to divide your wattage by 3 before plugging the value into this formula:
kVA = kW/pf
You should notice that this is the exact same formula that was used above for single phase systems.
You should then follow this formula up by dividing the kVA by the voltage (your VLN in the case of a 3 phase system) in order to calculate the current.
In this case, however, there is an added step.
Remember that you divided by 3 in order to set up an equation for a single phase system. Because of this division, then, your answer only reflects the output of a single phase.
In order to find the output of the 3 phase system you started with, you need only multiply the current you calculated by 3.
Simple, right?
Well, only if the system is balanced.
Although our calculations assume that a 3 phase system will be balanced, the reality of things is that most systems are not so conveniently balanced. That is to say, each phase doesn't always produce
an equal amount of power.
In such cases, you'll need to rely on much more complicated math to get an accurate answer. That math, however, is a bit too complex (polar coordinates and all) to go into detail about here.
So what do you do?
How about more POWER?
As it turns out, some sources say that you can take the average of the 3 phases and use that value in your equations. Still, it should be noted that this method is not going to get an exact answer.
Even if you can't calculate the exact answer when dealing with an unbalanced system, though, you've at least figured out (numerically, of course) just what makes 3 phase power that popular kid on the
playground that many businesses want on their side.
And who knows? You might even want it on your side one day. Want to inquire more
about the power at your business or facility? Call Precision Motor Repair for Dyna-Phase
units and get the 3 phase power current you need to run more efficiently.
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Understanding Mathematical Functions: Which Of The Following Is A True
When it comes to understanding mathematical functions, it's important to grasp the definition of a function and the importance of understanding how they work in mathematics. A mathematical function
is a relationship between a set of inputs and a set of possible outputs, where each input is related to exactly one output. Functions are a fundamental concept in mathematics and are used to describe
many real-world phenomena. Understanding functions is crucial for solving equations, modeling data, and making predictions in various fields such as engineering, physics, economics, and more.
Key Takeaways
• Mathematical functions are a fundamental concept in mathematics and are used to describe many real-world phenomena.
• Understanding functions is crucial for solving equations, modeling data, and making predictions in various fields such as engineering, physics, economics, and more.
• Key characteristics of functions include input and output, one-to-one correspondence, and domain and range.
• There are various types of functions, including linear, quadratic, exponential, and logarithmic functions.
• Functions have real-life applications in fields such as economics, physics, and biology, and are essential for understanding and solving problems in these areas.
Key Characteristics of Functions
Mathematical functions are crucial components of various mathematical and scientific calculations. Understanding the key characteristics of functions is essential for comprehending their behavior and
application in various fields.
A. Input and Output
At the core of a mathematical function lies the concept of input and output. A function takes an input (or independent variable) and produces an output (or dependent variable). The relationship
between the input and output is defined by the function itself.
B. One-to-one correspondence
A fundamental characteristic of a function is the concept of one-to-one correspondence, which means that each input value corresponds to exactly one output value. In other words, for every x-value in
the domain of the function, there is only one corresponding y-value in the range of the function.
C. Domain and Range
The domain of a function consists of all possible input values for the function, while the range consists of all possible output values. The domain and range are critical in understanding the
behavior and limitations of a function. For example, some functions may have restrictions on certain input values that result in undefined or imaginary outputs.
Types of Functions
Understanding mathematical functions is essential for anyone studying mathematics or related fields. Functions are a fundamental concept in mathematics and form the basis for various mathematical
models and analyses. There are several types of functions, each with its own unique characteristics and applications. In this chapter, we will explore the different types of functions, including
linear functions, quadratic functions, exponential functions, and logarithmic functions.
Linear functions
Linear functions are some of the most basic and widely used functions in mathematics. They are represented by the equation y = mx + b, where m is the slope and b is the y-intercept. The graph of a
linear function is a straight line, and the rate of change is constant. Linear functions have a wide range of applications in fields such as physics, engineering, economics, and finance. They are
often used to model and analyze relationships between two variables.
Quadratic functions
Quadratic functions are second-degree functions, meaning that the highest exponent of the variable is 2. The general form of a quadratic function is y = ax^2 + bx + c, where a, b, and c are constants
and a ≠ 0. The graph of a quadratic function is a parabola, which can open upward or downward depending on the value of a. Quadratic functions are commonly used to model various phenomena, such as
the motion of projectiles, the shape of certain curves, and the behavior of certain physical systems.
Exponential functions
Exponential functions are functions in which the variable appears in the exponent. The general form of an exponential function is y = ab^x, where a and b are constants and b is the base of the
exponential. The graph of an exponential function is a curve that increases or decreases rapidly, depending on the value of b. Exponential functions are used to model processes that exhibit
exponential growth or decay, such as population growth, radioactive decay, and compound interest.
Logarithmic functions
Logarithmic functions are the inverse of exponential functions. The general form of a logarithmic function is y = log_b(x), where b is the base of the logarithm. The graph of a logarithmic function
is a curve that increases or decreases slowly, depending on the base of the logarithm. Logarithmic functions are used to model various phenomena, such as the measurement of sound intensity, the
response of certain physical systems, and the analysis of algorithms and computational complexity.
Common Misconceptions about Functions
When it comes to mathematical functions, there are several misconceptions that are commonly held. Let’s address some of them:
A. Functions must be expressed as a formula
One common misconception about functions is that they must be expressed as a specific formula. While many functions can be represented by a formula, it is not a requirement. Functions can be defined
in a variety of ways, including through verbal descriptions, tables, or graphs. In fact, there are some functions that do not have an algebraic expression at all. Therefore, it is important to
understand that functions can be defined in various ways and are not limited to being represented by a formula.
B. Functions can only have numerical inputs
Another misconception is that functions can only have numerical inputs. In reality, functions can have a wide range of inputs, including numerical, algebraic, or even geometric inputs. For example, a
function can take a set of points in a coordinate plane as input, rather than just numerical values. This misconception stems from the idea that functions are solely mathematical concepts, but they
can actually be applied to a variety of contexts beyond just numerical inputs.
C. Functions must have a specific shape on a graph
There is a common belief that functions must have a specific shape on a graph, such as a straight line or a parabola. While many functions do have recognizable graph shapes, this is not a requirement
for a function. In fact, functions can have a wide range of graph shapes, including curves, step functions, and even irregular shapes. It is important to understand that the graph of a function can
vary widely based on its specific properties and behaviors, and it does not have to conform to any specific shape.
Testing for Functions
When dealing with mathematical functions, it is important to be able to test whether a given relationship is a true function. There are several methods that can be used to determine this, including
the vertical line test, horizontal line test, and algebraic methods.
A. Vertical line test
The vertical line test is a simple graphical method used to determine if a given relationship is a function. To perform the vertical line test, simply draw vertical lines through the graph of the
relationship. If at any point a vertical line intersects the graph at more than one point, then the relationship is not a function. If the vertical line only intersects the graph at one point for
every possible input value, then the relationship is indeed a function.
B. Horizontal line test
The horizontal line test is another graphical method used to test for functions. Similar to the vertical line test, the horizontal line test involves drawing horizontal lines through the graph of the
relationship. If a horizontal line intersects the graph at more than one point, then the relationship is not a function. On the other hand, if the horizontal line only intersects the graph at one
point for every possible input value, then the relationship is a function.
C. Using algebraic methods to determine if a relationship is a function
In addition to graphical methods, algebraic methods can also be used to test whether a given relationship is a function. One such method involves examining the input-output pairs of the relationship.
If each input value corresponds to only one output value, then the relationship is a function. However, if a single input value corresponds to multiple output values, then the relationship is not a
Real-life Applications of Functions
One of the most fascinating aspects of mathematical functions is their wide range of applications in real-life scenarios. Functions are used to model and analyze various phenomena in fields such as
economics, physics, and biology.
A. Economics - supply and demand functions
In economics, functions play a crucial role in understanding the relationship between supply and demand. The supply and demand functions help economists and businesses to analyze market trends, make
pricing decisions, and forecast future demand for goods and services. By using mathematical functions, economists can quantify the impact of various factors such as price changes, consumer
preferences, and production costs on the supply and demand equilibrium.
B. Physics - motion and force functions
Functions are extensively used in physics to describe the motion and forces acting on objects. Motion functions, such as position, velocity, and acceleration functions, provide a mathematical
representation of an object's movement through space and time. Force functions, on the other hand, help physicists analyze the impact of different forces on an object's motion, allowing them to
predict trajectories and design systems that utilize these principles.
C. Biology - population growth functions
In biology, functions are used to model and study population dynamics. Population growth functions, such as exponential and logistic growth functions, are used to analyze the changes in population
size over time, taking into account factors such as birth rates, death rates, and environmental limitations. These functions are essential for understanding the dynamics of ecosystems, predicting
species extinction risks, and developing strategies for sustainable resource management.
Understanding functions is crucial in various fields such as science, engineering, economics, and more. It provides a framework for solving problems and making predictions based on data. I encourage
everyone to continue exploring and learning about mathematical functions, as it opens up a world of possibilities for understanding the world around us.
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1. Unit 1 Topic 3 Counting and probability
2. Topic: Review of the fundamentals of probability
Sub-topics: ACMMM052 review probability as a measure of ‘the likelihood of occurrence’ of an event (1) ACMMM053 review the probability scale: 0≤P(A)≤10≤P(A)≤1 for each event A,A, with P(A)=0PA=0 if…
(1) ACMMM054 review the rules: P(A⎯⎯⎯)=1−P(A) PA¯=1-PA and P(A∪B)=P(A)+P(B)−P(A∩B)PA∪B=PA+PB-PA∩B (0) ACMMM055 use relative frequencies obtained from data as point estimates of probabilities. (0)
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Quantum Computing Technology and Roadmap
Future of Quantum Computing: Unlocking the Possibilities
- Overview
A quantum computing roadmap is a plan that outlines a company's goals for quantum computing, such as building an error-corrected quantum computer or making commercial quantum computing a reality.
Some companies that have quantum computing roadmaps include:
• IBM: IBM's 10-year plan includes developing new quantum processors, software, and services, as well as creating a software error-correction code. The roadmap also focuses on quantum-centric
supercomputing, which uses a heterogeneous computing architecture that combines quantum and classical computation.
• Infleqtion: Infleqtion's five-year roadmap, called Scorpius, aims to make commercial quantum computing a reality within five years. The roadmap focuses on making commercial scale computation
accessible in areas like material science, energy, and machine learning.
• QuEra: QuEra's roadmap focuses on developing large-scale, fault-tolerant quantum computers that can solve problems that are difficult for classical computers. The roadmap includes goals like
reaching 100 logical error-corrected qubits by 2026.
The overall purpose of this roadmap is to help facilitate progress in quantum computing research towards the era of quantum computer science. This is a living document and will be updated as needed.
- The Future of Quantum Computing
The future of quantum computing is here. As quantum computing progresses rapidly, it will have a major impact on the future of computing. Quantum computers could change the way we think about
computing, exponentially increasing processing speeds and allowing access to data that was previously inaccessible.
Quantum computing is both the present and the future. Unlike classical computing, which uses bits to represent data and perform operations, quantum computing uses quantum bits (qubits), which can
exist in multiple states with a certain probability (called superposition). This would allow quantum computers to perform certain types of calculations faster than classical computers.
While it's still a nascent technology, significant progress has been made in the field in recent years. Quantum computers have already been built and are used by researchers and companies for a
variety of tasks, such as optimization problems and simulations of quantum systems.
Overall, the future of quantum computing is bright, with the potential to revolutionize fields from medicine to finance to cybersecurity. Even so, quantum computing may not be widely available and
practical in the real world for several years.
- Why Do We Need Quantum Computing?
Quantum computers, sometimes called probabilistic or nondeterministic computers, are considered the most important computing technology of this century. It is a computing marvel that harnesses the
natural world to produce machines with powerful processing potential. Our world and reality itself is quantum. Real-world quantum systems cannot be modeled on classical computers.
Today's digital technologies are basically arithmetic devices that perform mathematical operations. We benefit greatly from computing in all its forms. Computers are very important in our life.
Hardware and software are what keep each object functioning properly. However, they have some limitations, which is why we need quantum computers.
Although the name sounds complicated, it is not difficult to define. It is a machine that uses the properties of quantum physics to store data and perform computations. They perform calculations,
just like the processors you find everywhere from your smartphone to your smartwatch. The difference, however, is that quantum computers are much more powerful than classical computers.
Classical computers encode information in binary bits. Computers use binary signals to process data. Data is represented as 1 or 0. A bit is a relatively simple state that represents one result or
another, for example, a switch can be on or off. Sequences of 0s and 1s give us a lot of computing power.
The longer the processing time, the more computing power is required. However, despite all the processing advances, traditional computing devices still face challenging tasks. Our current machines
are inaccurate because electrons orbiting atoms are in superposition in the real world.
Our current computers cannot calculate probabilities because electrons exist in more than one state at the same time. Quantum computers can take advantage of the fact that they operate using
superposition. Superposition is a characteristic of subatomic particles such as electrons and photons that can exist in two different states at the same time.
- The Challenge of Quantum Computing
One of the main problems facing quantum computers today is that entangled qubits quickly become incoherent relative to other qubits. Therefore, algorithms need to do their work quickly before the
qubits become incoherent.
Currently, most quantum computers can only keep a few dozen qubits coherent. A recent study showed that cosmic rays introduce a series of decoherence errors that are difficult to correct using
standard error correction techniques. This results in our inability to represent meaningful real-life problems on quantum computers.
Also, there is no uniformity in the underlying quantum computing hardware. Currently, companies are looking at different ways to build quantum computers -- for example, Quantum Annealer, Analog
Quantum Computer, and Universal Quantum Computer. This is very similar to our multiple transistor designs in the early days of computing. Therefore, only certain problems can be efficiently mapped
onto specific types of underlying quantum computing hardware.
Research is underway to solve the decoherence problem and design a universal quantum computer, and we are still about few years away from solving meaningful problems on a quantum computer. At the
same time, we anticipate deploying quantum computers and classical computers in a hybrid fashion to provide computational efficiency.
- How Quantum Computers are Deployed
The promise of quantum computing is that it will help us solve certain types of problems that today's classical computers cannot solve in a reasonable amount of time.
Quantum computing has captured the imagination of many business executives. By promising to solve problems that classical computers cannot reasonably solve, the right combination of quantum hardware
and software can lead to competitive advantages, new revenue streams, cost reductions, and other bottom-line benefits.
Quantum computers require custom hardware; today, only large hyperscalers and a handful of hardware companies offer quantum computer simulators and quantum computers of limited size as cloud
Quantum computers are currently targeting problems that are computationally intensive and latency-insensitive. Furthermore, today's quantum computer architectures are not mature enough to handle
large amounts of data. Therefore, in many cases, quantum computers are often deployed in a hybrid fashion with classical computers.
Although a quantum computer itself doesn't consume much power during computations, it requires specialized cryogenic refrigerators to keep superconducting temperatures low.
- Moving Quantum Computers to Production
As with other frontier technologies, companies are approaching quantum computing in different ways. Some companies have taken a wait-and-see approach, accepting the calculated risk that they may have
to play catch-up at high speed in a year or two.
In other cases, interested individuals have explored quantum computing as amateur skunkworks work, then successfully persuaded their managers to turn their work into an official project rather than a
secret one.
Still others have embraced quantum computing from the top, setting up exploratory teams and tasking them with building internal quantum capabilities and identifying relevant use cases. Once
identified, the companies selected several use cases for proof-of-concept projects.
Now that some of these proof-of-concepts have been successfully completed, companies are starting to consider possible production deployments of quantum solutions.
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Angles of a Parallelogram- Theorems, Proofs, Properties
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Angles of a Parallelogram
There are four interior angles in a parallelogram and the sum of the interior angles of a parallelogram is always 360°. The opposite angles of a parallelogram are equal and the consecutive angles of
a parallelogram are supplementary. Let us read more about the properties of the angles of a parallelogram in detail.
1. Properties of Angles of a Parallelogram
2. Theorems Related to Angles of a Parallelogram
3. FAQs on Angles of a Parallelogram
Properties of Angles of a Parallelogram
A parallelogram is a quadrilateral with equal and parallel opposite sides. There are some special properties of a parallelogram that make it different from the other quadrilaterals. Observe the
following parallelogram to relate to its properties given below:
• The opposite angles of a parallelogram are congruent (equal). Here, ∠A = ∠C; ∠D = ∠B.
• All the angles of a parallelogram add up to 360°. Here,∠A + ∠B + ∠C + ∠D = 360°.
• All the respective consecutive angles are supplementary. Here, ∠A + ∠B = 180°; ∠B + ∠C = 180°; ∠C + ∠D = 180°; ∠D + ∠A = 180°
Theorems Related to Angles of a Parallelogram
The theorems related to the angles of a parallelogram are helpful to solve the problems related to a parallelogram. Two of the important theorems are given below:
• The opposite angles of a parallelogram are equal.
• Consecutive angles of a parallelogram are supplementary.
Let us learn about these two special theorems of a parallelogram in detail.
Opposite Angles of a Parallelogram are Equal
Theorem: In a parallelogram, the opposite angles are equal.
Given: ABCD is a parallelogram, with four angles ∠A, ∠B, ∠C, ∠D respectively.
To Prove: ∠A =∠C and ∠B=∠D
Proof: In the parallelogram ABCD, diagonal AC is dividing the parallelogram into two triangles. On comparing triangles ABC, and ADC. Here we have:
AC = AC (common sides)
∠1 = ∠4 (alternate interior angles)
∠2 = ∠3 (alternate interior angles)
Thus, the two triangles are congruent, △ABC ≅ △ADC
This gives ∠B = ∠D by CPCT (corresponding parts of congruent triangles).
Similarly, we can show that ∠A =∠C.
Hence proved, that opposite angles in any parallelogram are equal.
The converse of the above theorem says if the opposite angles of a quadrilateral are equal, then it is a parallelogram. Let us prove the same.
Given: ∠A =∠C and ∠B=∠D in the quadrilateral ABCD.
To Prove: ABCD is a parallelogram.
The sum of all the four angles of this quadrilateral is equal to 360°.
= [∠A + ∠B + ∠C + ∠D = 360º]
= 2(∠A + ∠B) = 360º (We can substitute ∠C with ∠A and ∠D with ∠B since it is given that ∠A =∠C and ∠B =∠D)
= ∠A + ∠B = 180º . This shows that the consecutive angles are supplementary. Hence, it means that AD || BC. Similarly, we can show that AB || CD.
Hence, AD || BC, and AB || CD.
Therefore ABCD is a parallelogram.
Consecutive Angles of a Parallelogram are Supplementary
The consecutive angles of a parallelogram are supplementary. Let us prove this property considering the following given fact and using the same figure.
Given: ABCD is a parallelogram, with four angles ∠A, ∠B, ∠C, ∠D respectively.
To prove: ∠A + ∠B = 180°, ∠C + ∠D = 180°.
Proof: If AD is considered to be a transversal and AB || CD.
According to the property of transversal, we know that the interior angles on the same side of a transversal are supplementary.
Therefore, ∠A + ∠D = 180°.
∠B + ∠C = 180°
∠C + ∠D = 180°
∠A + ∠B = 180°
Therefore, the sum of the respective two adjacent angles of a parallelogram is equal to 180°.
Hence, it is proved that the consecutive angles of a parallelogram are supplementary.
Related Articles on Angles of a Parallelogram
Check out the interesting articles given below that are related to the angles of a parallelogram.
Solved Examples on Angles of a Parallelogram
1. Example 1: One angle of a parallelogram measures 75°. Find the measure of its adjacent angle and the measure of all the remaining angles of the parallelogram.
Given that one angle of a parallelogram = 75°
Let the adjacent angle be x
We know that the consecutive (adjacent) angles of a parallelogram are supplementary.
Therefore, 75° + x° = 180°
x = 180° - 75° = 105°
To find the measure of all the four angles of a parallelogram we know that the opposite angles of a parallelogram are congruent.
Hence, ∠1 = 75°, ∠2 = 105°, ∠3 = 75°, ∠4 = 105°
2. Example 2: The values of the opposite angles of a parallelogram are given as follows: ∠1 = 75°, ∠3 = (x + 30)°, find the value of x.
Given: ∠1 and ∠3 are opposite angles of a parallelogram.
Given: ∠1 = 75° and ∠3 = (x + 30)°
We know that the opposite angles of a parallelogram are equal.
(x + 30)° = 75°
x = 75° - 30°
x = 45°
Hence, the value of x is 45°.
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Practice Questions on Angles of a Parallelogram
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FAQs on Angles of a Parallelogram
Do Angles in a Parallelogram add up to 360°?
Yes, all the interior angles of a parallelogram add up to 360°. For example, in a parallelogram ABCD, ∠A + ∠B + ∠C + ∠D = 360°. According to the angle sum property of polygons, the sum of the
interior angles in a polygon can be calculated with the help of the number of triangles that can be formed inside it. In this case, a parallelogram consists of 2 triangles, so, the sum of the
interior angles is 360°. This can also be calculated by the formula, S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. Here, 'n' = 4. Therefore, the sum of the interior
angles of a parallelogram = S = (4 − 2) × 180° = (4 − 2) × 180° = 2 × 180° = 360°.
What is the Relationship Between the Adjacent Angles of a Parallelogram?
The adjacent angles of a parallelogram are also known as consecutive angles and they are always supplementary (180°).
How are the Opposite Angles of a Parallelogram Related?
The opposite angles of a parallelogram are always equal, whereas, the adjacent angles of a parallelogram are always supplementary.
How to Find the Missing Angles of a Parallelogram?
We can easily find the missing angles of a parallelogram with the help of three special properties:
• The opposite angles of a parallelogram are congruent.
• The consecutive angles of a parallelogram are supplementary.
• The sum of all the angles of a parallelogram is equal to 360°.
What are the Interior Angles of a Parallelogram?
The angles made on the inside of a parallelogram and formed by each pair of adjacent sides are its interior angles. The interior angles of a parallelogram sum up to 360° and any two adjacent
(consecutive) angles of a parallelogram are supplementary.
Are all Angles in a Parallelogram Equal?
No, all the angles of a parallelogram are not equal. There are two basic theorems related to the angles of a parallelogram which state that the opposite angles of a parallelogram are equal and the
consecutive (adjacent) angles are supplementary.
What is the Sum of the Interior Angles of a Parallelogram?
The sum of the interior angles of a parallelogram is always 360°. According to the angle sum property of polygons, the sum of the interior angles of a polygon can be found by the formula, S = (n − 2)
× 180°, where 'n' shows the number of sides in the polygon. In this case, 'n' = 4. Therefore, the sum of the interior angles of a parallelogram = S = (4 − 2) × 180° = (4 − 2) × 180° = 2 × 180° =
Are the Angles of a Parallelogram 90 Degrees?
In some parallelograms like rectangles and squares, all the angles measure 90°. However, the angles in the other parallelograms may not necessarily be 90°.
Are the Opposite Angles of a Parallelogram Congruent?
Yes, the opposite angles of a parallelogram are congruent. However, the adjacent angles of a parallelogram are always supplementary.
Are Consecutive Angles of a Parallelogram Congruent?
No, the consecutive (adjacent) angles of a parallelogram are not congruent, they are supplementary.
Are the Opposite Angles of a Parallelogram Supplementary?
No, according to the theorems based on the angles of a parallelogram, the opposite angles are not supplementary, they are equal.
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Overview of the ACT Mathematics Test
The ACT is a standardized test that assesses your university readiness and considers your grades during the admissions process. This is one aspect of measuring your eligibility for college or
university admission. The test has five sections:
• English (45 minutes- 75 questions)
• Math (60 minutes- 60 questions)
• Reading (35 minutes- 40 questions)
• Science (35 minutes- 40 questions)
• Writing (40 minutes- Optional)
You have 2 hours and 55 minutes, plus an optional 40-minute Writing test to answer 215 questions.
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How is the structure of the ACT Mathematics Test?
The math test is one of the longest parts of the test. You will be given a full hour to complete 60 questions. All of these questions are multiple-choice, and each relies on your understanding of
some mathematical concepts. Most of what you will find in the ACT math section is very similar to what you learned in high school. Like other sections of the ACT, the math test is divided into
several categories of knowledge, each question of the test is placed in a specific category. These categories involve Elementary Algebra, Pre-Algebra, Intermediate Algebra, Coordinate Geometry, Plane
Geometry, and Trigonometry.
Pre-Algebra (20-25%):
Pre-Algebra is typically taught in middle school as a warm-up class for regular algebra. It is designed to introduce concepts such as integers, factoring, and orders of operation. The topics covered
• Basic operations
• How to use place value
• Factoring
• Square roots and exponents
• Scientific notation
• Ratios, proportion, and percent
• Data collection and interpretation
Elementary Algebra (15-20%):
Elementary algebra extends to the topics taught in pre-algebra. This includes the use of variables, how algebraic equations work. The topics covered include:
• More exponents and square roots
• Using substitution to solve Algebraic expressions
• Using variables
• Understanding how algebraic operations work
• Using factoring to solve quadratic equations
Intermediate Algebra (15-20%):
Intermediate Algebra involves quadratic equations, getting deeper into functions, and, relations. More advanced topics such as matrices and complex numbers are also introduced. The topics covered
• Quadratic formula and inequalities
• Radical and rational expressions
• Absolute value equations and inequalities
• Systems of equations
• Functions and modeling
• Matrices
• Polynomial roots
• Complex numbers
Coordinate Geometry (15-20%):
Coordinate Geometry includes basic level concepts using points and lines using 2-digit coordinates. Graphing is a huge component because it shows how you turn algebraic equations into a photo. The
topics covered include:
• Relationships between equations and graphs
• Parallel and perpendicular lines
• Slope line intercept
• Distance formula
• Midpoint formula
• Conics
• Graph inequalities
Plane Geometry (20-25%):
Plane geometry is made of coordinate geometry. The focus now moves from the coordinates and lines to the shapes on the flat screen. The topics covered include:
• Angles and relations for parallel lines and perpendicular
• Properties of triangles, circles, rectangles, parallelograms, and trapezoids
• Geometric proofs
• Transformations
• Volume
• Applying geometric principles to 3 dimensions
Trigonometry (5-10%):
Trigonometry is the study of angle functions and how they are used in mathematical calculations. Triangles are widely studied to help explain angular relationships.
• Trigonometric relations in right triangles
• Trigonometric function values and properties
• Graph trigonometric functions
• How to use trigonometric identities
• How to solve trigonometric equations
• Trigonometric function modeling
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Do you give a formula sheet on the ACT Mathematics Test?
You do not have the right to have a formula in ACT, so memorize geometry, algebra, and trigonometry formulas before the ACT.
Is the ACT Mathematics Test hard?
In ACT Math, you have 60 minutes to answer 60 math questions. This is usually pretty hard for most students to get through – the answer to each question is just 60 seconds, and some of these
questions take a long time.
Can you use a calculator on the ACT Mathematics Test?
The use of certified calculators in the ACT exam is allowed, but test designers note that all questions are technically solved without a calculator.
How is the ACT Test scored?
In the ACT, each subject area is given a scaled score between 1 and 36.
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Need Math worksheets to help you measure your exam readiness for your upcoming ACT test?
Have a look at our comprehensive ACT Math Worksheets to help you practice and prepare for the ACT Math test.
Want to review the most common ACT Math formula?
Here is our complete list of ACT Math formulas.
Looking for FREE ACT Math websites to find free online resources?
Here is our complete list of Top 10 Free Websites for ACT Math Preparation.
Need a practice test to help you improve your ACT Math score?
Have a look at our Full-Length ACT Math Practice Test and Free ACT Math Practice Test.
Do you know the key differences between SAT Math and ACT Math?
Find your answer here: SAT Math vs. ACT Math: the key differences.
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Future Value of $1 Per Period
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Define Future Value of $1 Per Period in Real Estate
Future Value of $1 Per Period:
The "future value of $1 per period" is a term used to describe how much a series of payments of one dollar (or any specified amount) will grow over time. It's like saving a little bit of money every
week or month, and watching it grow into a larger amount over time.
FV = PMT x (((1 + i)^n - 1) / i)
For example, let's say you make a payment of $1 at the end of every month for 10 years, and the interest rate is 5% per year. The future value of that stream of payments would be $147.26. This means
that by saving $1 a month for 10 years and earning interest, your money grew to $147.26.
"A Deep Dive for Real Estate Appraisers"
To calculate the future value of $1 per period using an HP12c calculator, you would use the "PMT" key, which stands for "payment", along with the "FV" key for future value. Here's an example of how
to do it:
Let's say you want to know how much a stream of payments of $1 per month will be worth in 5 years if you invest it at an annual interest rate of 6%. Here are the steps:
Enter the number of years: Press "5" and then the "N" key.
Enter the interest rate: Press "6" and then the "%i" key.
Enter the payment amount: Press "1" and then the "CHS" key (to make it negative), and then the "PMT" key.
Enter the present value: Press "0" and then the "PV" key (since there is no initial investment).
Calculate the future value: Press "FV" and the calculator will display the answer.
The answer should be $79.61, which means that by saving $1 per month for 5 years at 6% interest, your money will grow to $79.61. You can also use this formula to calculate the future value of any
FV = PMT x (((1 + i)^n - 1) / i)
where FV is the future value, PMT is the payment amount, i is the interest rate per period, and n is the number of periods.
"Wit & Whimsy with the Dumb Ox: Unlocking Knowledge with Rhyme:"
The future value of $1 per period, it's quite a feat!
It's like planting a little seed and watching it grow to a treat.
You make a payment of $1 every period you choose,
And watch your money grow, it's quite a ruse!
Let's say you save $1 every month for 10 years,
And the interest rate is 5%, there are no fears.
Your payments grow and grow, just like a tree,
And the future value is $147.26, oh what a spree!
So remember, the future value of $1 per period,
Is like saving a little bit, and watching your money grow as a reward.
It's one of the six functions of a dollar, so don't be deterred,
Save your money wisely, and watch it grow, oh what a word!
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Transformation of time domain TDR to its frequency domain S11 (Return Loss) using FFT - Gquipment
Transformation of time domain TDR to its frequency domain S11 (Return Loss) using FFT
In this article, we explain how to transform a time domain TDR (Time Domain Reflectometry) measurement into a frequency domain return loss graph. Since the TDR trace is in the time domain and the
return loss (S[11]) is in the frequency domain, an FFT is used to perform the transformation. The steps are implemented using a Python script. The article concludes with some examples of different
1. Why transform a TDR trace to return loss (S[11])?
Before we answer this question, it's a good idea to summarize what the different purposes of the TDR and return loss measurements are. We start with the TDR which shows the reflection of a DUT
(Device Under Test) as a function of time. Most TDRs use a step function as excitation because it is easier to implement and the reflection created matches the impedance of the DUT. It's a very
intuitive way to look at a DUT because you can easily track how the impedance changes through the DUT. When the propagation speed (Vp) is known, it is also easy to convert time into distance. It is
therefore a powerful technique to precisely determine the impedance at a specific location within the DUT. The resolution is limited by the step rise time and the bandwidth of the front end. Figure
1a shows an example of the reflection created by a TDR. The DUT is a Beatty, which consists of a 25 ohm trace centered in a 50 ohm microstrip (Figure 1b).
Figure 1a: TDR trace of Beatty, reflection coefficient
Figure 1b: TDR trace of Beatty, impedance
The return loss plot on the other hand, shows S[11] (return loss) over frequency. Which is nothing else then the reflection coefficient (Gamma or S[11]) as a function of frequency, plotted in a
convenient scale (dB). Of course, frequency is the domain that most of the RF engineering work takes place in, so, the return loss plot fits in naturally. And while the return loss plot also
corresponds with the impedance of the DUT, the link with the physical position is not apparent anymore.
The return loss graph, on the other hand, shows S[11] versus frequency. That is nothing more than the reflection coefficient (Gamma or S[11]) as a function of the frequency, plotted in a handy scale
(dB). Of course, frequency is the domain in which most RF engineering takes place, so the return loss diagram fits naturally into this. And although the return loss also corresponds to the impedance
of the DUT, the link with the physical position can no longer be found.
So why transform a TDR reflection trace into its frequency domain return loss? The first reason can be very practical: because your laboratory only has a decent TDR and no VNA. Or the bandwidth of
your TDR is significantly higher than that of your VNA.
The second use case is that you want to know the return loss of your DUT, without the reflection of nearby disturbances, such as connectors or other nearby disturbances. Keep in mind that reflections
interfere with each other and the end result of this interference is what you see in the return loss graph. And while it is very difficult to distinguish between these disturbances and the DUT
response in the frequency domain, you can easily do this in the time domain. Using a TDR trace you can easily locate the reflection you are interested in. As long as they are sufficiently far apart
from the disturbance and/or the resolution of your TDR is high enough.
Figure 2: TDR trace of a GCPW, including SMA connectors
This brings us to the powerful use case of isolating the reflection of a disturbance (e.g. from a connector) and transforming it into the frequency domain to obtain the S[11] parameter. This can then
be used as input for a de-embedding proces to eliminate the effects of connectors on the measurements of the S-parameters of a DUT. Figure 2 shows an interesting candidate for such a procedure. It is
the TDR trace of a well-matched GCPW (about 49 ohms, starting at about 70 ps) where the SMA connector (-80 ps) and the SMA landing area (between 0ps and 70 ps) create reflections that are much
stronger than that of the GCPW itself.
2. The transformation process
The Fourier transform and its inverse can be used to transform between the impulse response in the time domain and the frequency response in the frequency domain and vice versa. To transform a TDR
time series (trace), we must use the impulse response and not the step response generated by a TDR. Therefore, the first step is to calculate the first derivative of the TDR trace, which gives us the
time-domain impulse response of the reflection coefficient. Figure 3 shows the result of this operation using the same Beatty as from the previous section.
Figure 3: Beatty Impulse Response of the reflection coefficient
This impulse response is then transformed into the frequency domain by applying the FFT operation. This generates a frequency response with N frequency points, where N equals the number of samples of
the time series (TDR trace).
Each frequency point consists of a complex number, otherwise all phase information would be lost. The spectrum also has a negative and positive frequency range. This is a result of the way each
frequency point is described mathematically. Both spectra contain the same information because the positive spectrum is the complex conjugate of the negative spectrum. Long story short, you can
safely discard one of the two spectra.
The magnitude and phase information can be easily retrieved from the spectrum by calculating the magnitude and phase angle for each complex frequency point. The return loss is calculated using RL =
2.1 Scaling
In FFT and inverse FFT operations it is common to scale the response to maintain a certain quantity, such as power or amplitude. In our case we want to preserve the magnitude of the reflection
coefficient. This is achieved by multiplying the FFT response by the sampling time. The calculation of the first derivative of the reflection coefficient does the opposite: it calculates the d(Gamma)
/dt. Both operations therefore cancel each other out. Therefore, you will not find these scaling operations in the pseudocode presented in section 2.3.
2.2 The frequency dimension
The frequency axis of the return loss graph depends on several aspects such as the number of samples taken, the sampling rate and the bandwidth of the TDR measurement to name the most important ones.
In this section we will explore how each affects the frequency dimension of the return loss diagram.
The most important parameters are the time steps dt (or 1/Fs) and the number of samples N. The first observation we make is that the FFT creates a spectrum of N frequency points. The time step dt
defines the frequency range as (1/dt) Hz. It is easy to see that the frequency steps (the so-called bins) are distributed over the number of frequency points N. Each frequency bin therefore has a
frequency resolution of 1/dt/N or Fs/N.
This relationship shows that the frequency resolution is increased by using more samples and/or increasing the time step (decreasing the sample rate). Remember that the spectrum has negative and
positive frequencies, which carry the same information. Since we are throwing away half the spectrum, we only have N/2 frequency points to work with in a return loss graph. The number of relevant
frequency points is also limited by the bandwidth B of the TDR measuring equipment, as there is no point in displaying frequency content outside this bandwidth. So we also have the restriction that
each frequency point must satisfy f[i] < B.
Finally, the sampling frequency Fs is limited by the Nyquist frequency. If we assume a TDR bandwidth of B , then Fs > 2*B.
It is instructive to see how all these parameters relate to each other by putting them in a table. We used N=512 and B=20GHz.
Number of points
Time step (ps) Fs/2 (GHz) Frequency resolution (GHz) (f[i] < B)
1 500 1.95 10
2 250 0.98 20
4 125 0.49 40
10 50 0.20 100
20 25 0.10 200
40 12.5 0.05 400
Table 1 : Frequency related parameters of the FFT transformation
The table shows that the best results are obtained with a time step of 20 ps. The Nyquist frequency in this case is still twice the bandwidth and the FFT spectrum has 200 frequency points that fall
within the bandwidth of the TDR measuring equipment. The last row in the table violates the Nyquist criterion and would also require more points than the available 256 frequency points.
Figure 4: Return loss graph (frequency response), 10 ps time step, N=512
The return loss plot for a 10 ps time step is shown in Figure 4. The 4 ps version is shown in Figure 5, showing the lack of resolution and irregular pattern of peaks and valleys.
Figure 5: Return loss graph (frequency response), 4 ps time step, N=512
2.3 Python pseudo code
The following pseudocode summarizes the steps using Python with the NumPy library. The truncation of the spectrum to the maximum bandwidth is not shown in this code snippet.
# Pseudo code, laying out the essential steps only
# Trace data is in x_values
sampling_interval = np.mean(np.diff(x_values))
# First derivative
diff_gamma_values = np.gradient(gamma_values)
# FFT
fourier_data = fft(diff_gama_values)
# Get the frequencies corresponding to the FFT result
frequencies = np.fft.fftfreq(len(diff_gama_values), d=sampling_interval)
# Calculate the magnitude of the complex Fourier transform data
magnitude = np.abs(fourier_data)
# Return loss
magnitude = 20 * np.log10(magnitude)
# Plot (frequency, magnitude)
3. Some more examples
We also performed two other experiments. One with a 50 ohm transmission line because it has a relatively low reflection coefficient. And another using one of the output ports of a two-resistor
splitter. It has a reflection coefficient of 0.25, which is flat over the frequency range.
3.1 A transmission line
In picture 1 you can see the microstrip (GitHub PCB008003042) and a MINI-ENCL-MINI-EXT-FX used as fixture. Figure 6 and 7 show the TDR trace (impedance and reflection) and the generated return loss
plot is shown in figure 8.
Picture 1: Transmission line in enclosure
The TDR trace clearly shows that the microstrip with its 53 ohms remains within the tolerance limit of 10%. The SMA mating and connector landing areas have similar amplitude in reflections. The SMA
landing area also seems to have a bit too much capacitance. This example also shows again that you can easily locate impedance discontinuities with a TDR trace.
Figure 6: TDR trace, impedance, 2 ps time step, N=512
Figure 7: TDR trace, reflection, 2 ps time step, N=512
This transmission line, with both connectors, creates a return loss graph as shown below. Up to 10 GHz the return loss is 25 dB or better, while up to 18 GHz it hovers around 20 dB. This trace has
100 frequency data points and the TDR is sampled with a time step of 10 ps.
Figure 8: Return loss plot, B=20 GHz, number of point=100
3.2 The output port of a 2R splitter
The final experiment was performed using a SPLT-2R1-0-6000-B, a power splitter that uses a two-resistor configuration. One of its features is that the output ports are not matched to 50 ohm. They
exhibit an impedance of 83 ohm. This is not the place to discuss why this is the case, but here we use it as a convenient way to create a flat impedance that is not equal to 50 ohm. If you'd like to
learn more about these kind of power splitters, you can check out out our overview article on power splitters and couplers.
Picture 2: Measuring the output port of a two resistor splitter
The DTR trace clearly shows that the output port indeed has an impedance of 83 ohms. The remaining ports are terminated at 50 ohms.
Figure 9: TDR trace, 2 ps time step, N=512
After running the Python script, we get the return loss graph as shown in Figure 10. This shows that the return loss is indeed more or less flat and centered around 12 dB, which corresponds to 83
ohms (or a reflection coefficient of 0.25). This trace was also sampled with a time step of 10 ps. The frequency axis is truncated at 6 GHz, which is the operating range of the power splitter. The
chart uses 31 data points.
Figure 10: Return loss plot, B=6 GHz, number of point=31
4. Next steps
In this article we looked at how we can use a TDR measurement to derive the return loss (S[11] parameter) of a DUT. One of the issues we haven't discussed yet is that any discontinuity at the
beginning and end of the time series (TDR trace) will introduce an error in the return loss. This problem can be solved by using a technique called windowing. The applied window determines which part
of the trace is included in the TDR trace used for the transformation process and to what extent. You could say that in this article we used a rectangular window that started at the beginning and
ended at the end of the TDR trace. It turns out that if you use a window with a shape more similar to the shape shown in Figure 11, you can minimize the errors in the return loss graph introduced by
the discontinuities in the TRD trace, because they are suppressed by the applied window.
Figure 11: Example of a window
Another interesting topic is to use this windowing technique to select part of the TDR trace, in order to determine the S[11] parameter of only the connectors. These can then be used as input for a
de-embedding process.
Posted by
Peter Goossens Applications
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Srinivasa Ramanujan – 16×16 Biography Magic Square
This Biography Magic Square summarizes the important events that in the life of Sri Srinivasa Ramanujan.
How it was constructed:
Important dates in the life of Ramanujan were taken, two digits at a time, representing either the date or the month or the first or second part of the four-digit year. As an example, Ramanujan’s
birth-day 22-12-1887 is taken in four separate entries as 22 12 18 87. These were then laid out in the top of the Magic Square, in the first column. Then, a complete Magic Square was built on top of
these numbers, with the following additional feature : each square indicated by a separate color (in this case, there are 4 such 4×4 sub-sqaures), which are magic squares themselves!
Srinivasa Ramanujan 16×16 biography Magic Square
(Download Ramanujan_16x16_biography_Magic_Square in excel format).
This is a smaller version of the 100-by-100 and 125-by-125 biography magic squares that we have constructed.
This was earlier published in an article “A Unique Novel Homage to the Great Indian Mathematician” in the March 2013 (Volume 23, Pg 146-147) Mathematics Newsletter published by the Ramanujan
Mathematics Society. (download free).
If you find this interesting, you could construct your own! If you want some help, drop a mail to me at contact[at]jollymaths[dot]com.
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LSAT Tip of the Week: Analytical Reasoning Practice Problem #2
analytical reasoning (logic games) question. Our practice question will be from the June 2007 LSAT.
There are exactly three recycling centers in Rivertown: Center 1, Center 2, and Center 3. Exactly five kinds of material are recycled at these recycling centers: glass, newsprint, plastic, tin, and
wood. Each recycling center recycles at least two but no more than three of these kinds of material. The following conditions must hold:
Every kind of material that Center 2 recycles is also recycled at Center 1.
Only one of the recycling centers recycles plastic, and that recycling center does not recycle glass.
First, we know that there are 3 sections and that each recycles at least 2 but no more than 3.
Rule 1: If it recycles W it also recycles N, so:
W -> N & N -> W (The latter holds to be true since it is called a contrapositive).
Rule 2: If in Center 2 then also in Center 1, so:
2 -> 1 & 1->2 (Again, this is the contrapositive)
This rule has a hidden point as well, it states that if Center 2 recycles three materials, Center 1 will recycle those three as well. If Center 2 recycles 2 materials, Center 1 will recycle those 2
as well. Anything 2 has, 1 does as well.
Rule 3: Only one P, and that center does not recycle G. This allows us to infer that P cannot be in Center 2 because of Rule 2.
Which one of the following could be an accurate account of all the kinds of material recycled at each recycling center in Rivertown?
(A) Center 1: newsprint, plastic, wood; Center 2: newsprint, wood; Center 3: glass, tin, wood
(B) Center 1: glass, newsprint, tin; Center 2: glass, newsprint, tin; Center 3: newsprint, plastic, wood
(C) Center 1: glass, newsprint, wood; Center 2: glass, newsprint, tin; Center 3: plastic, tin
(D) Center 1: glass, plastic, tin; Center 2: glass, tin; Center 3: newsprint, wood
(E) Center 1: newsprint, plastic, wood; Center 2: newsprint, plastic, wood; Center 3: glass, newsprint, tin
We aren’t given any new rules here, so we will just use the rules we have already pulled out.
Rule #1: If they recycle W they must have N too. Option (a) does not have both, we eliminate this. Going through the rest we know that no other options violate rule 1, so let’s move on to rule 2.
Rule #2: If it’s in center 2, it must also be in center 1. Option (c) is missing tin in center 1, so we eliminate this. Going through the rest we do not see any other violations of this rule, so
let’s move on to rule 3.
Rule #3: Only one center can recycle plastic and that one cannot have glass. We see that in option (d) both glass and plastic is being recycled in the same center, so we can cross this off. We also
see that option (e) violates this as two centers are recycling plastic. We are now left with (b), which is our correct answer.
This week, we will focus on an example of how to setup an analytical reasoning (logic games) question. Our practice question will be from the June 2007 LSAT.
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Area of a Triangle
Area: Learn
Area is the space within a shape. The units of area are the squares of the units of length (that is, if a shape's sides are measured in meters, then the shape's area is measured in square meters).
Triangles have an interesting relationship with rectangles. You can see in the diagram below how a triangle can be made by cutting a rectangle in half along a diagonal.
So, finding the area of a triangle uses the same base times height multiplication as a rectangle, but then divide by 2 to find the area of just the triangle. Here is an example: Find the area of a
triangle with a base of 4m and a height of 5m.
• Multiply the base by the height: 4*5 = 20
• Divide the result in half: 20÷2 = 10m^2
Any side of the triangle can be used as the base, but then the height must be the line which is drawn from the base side to the opposite angle (vertex). This is shown in the diagram below.
As a shortcut to keep the numbers small, you can divide either the base or height by two before multiplying. Here is an example using the same triangle as above (4m base and 5m height):
• Instead of dividing by 2 at the end, let's divide the 4 first: 4÷2=2
• Now multiply the base and height: 2*5 = 10m^2
Area: Practice
What is the area of a triangle whose
base length is cm
height length is cm?
Press the Start Button To Begin
You have 0 correct and 0 incorrect.
This is 0 percent correct.
Game Name Description Best Score
How many correct answers can you get in 60 seconds? 0
Extra time is awarded for each correct answer. 0
Play longer by getting more correct.
How fast can you get 20 more correct answers than wrong answers? 999
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Lab00: Simple Python exercises
Lab00: Simple Python exercises¶
1. Make a list of the numbers 1 to 10.
2. Make a list of the first 10 square numbers.
3. Make a list of the first 10 square numbers in reversed order.
4. Repeat 3 but only keep the square numbers divisible by 4.
5. A Pythagoran triple is a tuple of 3 positive integers (a, b, c) such that \(a^2 + b^2 =c ^2\). Find the unique Pythagorean triples where c is less than 20.
dna = """
6. Remove any blank space characters including newlines in dna.
7. Find the unique bases in dna.
8. Find the position of the first occrurence of ‘C’ in dna.
9. Find the position of the second occurrence of ‘G’ in dna.
10. What is the sequence of the complementary strand of DNA? Recall from grade school biology that A is complementary to T and C is complementary to G.
11. Can you print the complemntary DNA strand with only 40 characters to a line
Using regular Python
Using bash magic
12. A 1-D random walk experiment starts from value 0, then either adds or subtracts 1 at each step. Run \(n\) such random walks, each time recording the final value after \(k\) steps. Show the counts
of eac final value. What is the mean and standard deviaiton of the final values?
Let \(n\) = 10000 and \(k=100\). Use the standard libary package random to generate random steps.
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nForum - Search Results Feed (Tag: arithmetic-geometry)Riemann zeta functionGalois representationspecial values of L-functionsL-functionring of integersmultiple zeta valuesfracture theoremgenus of a number fieldSpec(Z)Fermat curvearithmetic jet spacemoduli space of Calabi-Yau spacesautomorphic L-functionLanglands program and membrane instantons?Borger's absolute geometryfinite fields as prime geodesics, not as knotsprime geodesicDiophantine equationDirichlet theta functionvacuum amplitudeArtin L-functions analogous to zeta of flat-connection-twisted Laplace operatorArtin L-function and Alexander polynomial ?Langlands correspondence & equivariant elliptic cohomology & modular functor & WZW model & CS theoryTheta-functions and L-functions ?Langlands correspondence
We have had our share of the debate of whether $Spec(\mathbb{Z})$ is really usefully analogous to a 3-manifold, and of how the $Spec(\mathbb{F}_p)$-s inside it then are analogous to knots in a
Here is a thought (maybe this was voiced before and I am just being really slow, please bear with me):
things would seem to fall into place much better if we thought of the $Spec(\mathbb{F}_p) \hookrightarrow Spec(\mathbb{Z})$ not as analogous to knots, but as analogous to the prime geodesics inside a
hyperbolic 3-manifold.
With this and its generalization to function fields, then the analogy between the Selberg zeta function for 3-manifolds and the Artin L-function (pointed out here) would become even better: in both
cases we’d have the infinite product over all prime geodesics of, essentially, the determinant of the monodromy of the given flat connection over that geodesic.
Also, thinking of the $Spec(\mathbb{F}_p)$ not as knots but as prime geodesics removes all the awkward aspects of the former interpretation, such as why on earth one would be required to consider all
these knots at once (which does not fit the analogy with knots in CS theory). Of course the prime geodesics would also be knots, technically, but I am talking here about the difference between
thinking of them playing the conceptual role of the knots in CS theory (which are things we choose at will to build observables) and the prime geodesics, which are given to us by the gods as a means
to compute the perturbative CS path integral.
Finally, there is of course much support from other directions of an analogy between prime geodesics and prime numbers (asymptotics etc.).
So it would seem to make much sense.
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An Object Is Moving To The Right At A Constant Velocity. What Will Happen If Aforce Of 20 N Starts Acting
B. Is correct answer
The angular acceleration of the washing machine spin is 53.5 x 10⁻³.
What is meant by angular acceleration ?
Angular acceleration is defined as the time rate of change of angular velocity in a circular motion.
Initial angular velocity, ω₁ = 20 rad/s
Final angular velocity, ω₂ = 8 rad/s
Angular displacement, θ = 2[tex]\pi[/tex] x no. of revolutions
θ = 2[tex]\pi[/tex] x 500 = 1000[tex]\pi[/tex]
a) Using the equation of circular motion,
ω₂² - ω₁² = 2αθ
Therefore, the angular acceleration,
α = (ω₂² - ω₁²)/2θ
α = (20² - 8²)/2 x 1000[tex]\pi[/tex]
α = 53.5 x 10⁻³ rad/s²
b) Time taken by the machine spin to slow down,
t = (ω₂ - ω₁)/α
t = (20 - 8)/53.5 x 10⁻³
t = 224.2 s
a) The angular acceleration of the washing machine spin is 53.5 x 10⁻³.
b) Time taken by the washing machine spin to slow down is 224.2 s.
To learn more about angular acceleration, click:
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The Persistence of 360
Imperials vs Metrics: Asked and Answered
Somebody asked questions about why America.. whoa here we go.. was too stupid to adapt the metric system like Europe. Well the obvious conventional wisdom is that we're stupid and have bad habits.
But let's think about that for a moment.
Let's start with a yard. Not your yard, a yardstick. But your yard could be measured with a yardstick just as easily as it could be measured with a meter stick. Why? Because we know what a yard is
and we know what a meter is and we know arithmetic.
Let's say your yard is rather large, like mine. It would then be 50 feet by 40 feet = 2000 square feet. Now there's something very nice about feet that we know, because feet divide into 12 inches. 12
is a special number because you can easily divide it in halves, thirds and quarters, and that is what you generally do with land. Yards are conveniently 3 feet. 36 can be divided very nicely too. But
when you look at 360, then you get something that's rather strikingly brilliant.
360 can be divided by 2,3,4,5,6,8,9,10,12,15,18 and 20. That's rather clever don't you think? You can't do that with decimals so easily, and after all the point of any system is not that things can
be measured. We could invent meters and re-invent them to be any arbitrary length and do the arithmetic. The strength of the metric system lies in its ability to scale by orders of magnitude. But
scaling by orders of magnitude is not something that happens very often in the human scale. In engineering, sure, but not in home construction, food recipes, human body measurement or any other of a
dozen activities (hmm dozen) that we are regularly involved in. Now we could go into all of the ways we measure all of the things we do from miles per hour to paper sizes, but that long long
discussion actually gets to the heart of Metric vs Imperial. In short, however, this is all about fractions. Or if you like to be mathematical about it, it's about superior highly composite numbers.
Now a friend told me that there are two kinds of engineers in the world. The kind that have put a man on the moon, and the kind that use the metric system. (ooh burn!) The picture above is taken from
his machine shop. It sounds like braggadocio until you listen for another minute. Then the man says 'SAE'. Now anybody who's turned a wrench in their lives knows something about SAE sockets vs Metric
sockets. But if you look at what SAE actually was and is, things get very interesting.
Check out this dude Kettering. I don't know about you, but to me that's a career full of wowsers. 186 patents is no accident. He helped build GM into the world's largest company. But what's
fascinating, as when I listen enough, is to learn how many parts in the world's market for parts are of SAE measure rather than of metric. I often think in terms of meters nowadays, but only for
things that are linear. I will always think of pies when I have to divide, and I will think of fractions. But isn't it interesting, just looking at the picture, which is a better, simpler, faster way
to measure?
Maybe the metric system is why we haven't put another man on the moon.
Now I'm going to show the other side here, because what we've done in America which is metric to the bone before just about every other nation is in our currency. More than just about anything,
currency lies in the realm of Extremistan. Orders of magnitude are quite proper here. But at the small level of the ordinary consumer, we don't do much with pricing that accords to common 360 sense.
For example. In old money there were 12 pence to the shilling. That would be quite nice if you purchased many things by the dozen. If a dozen eggs cost a shilling, a half dozen eggs would be
sixpence. A smart shopkeeper would try to price such things that way for the convenience of his customers. Then again, one might be a bit too clever (by half?) using the same system to obfuscate, as
bakers may have throwing in for their extra sized dozen. We certainly don't price things logically at the farmer's market level, but clearly decimal pricing lets us see unit prices reflect economies
of scale in bulk pricing. So for things that scale orders of magnitude, decimal is better.
By the way, what time is it?
No matter how you cut it, measurements are irregular and none can conform to a single rule...much like languages or certainly parts thereof. There are 365.25 days in a rotation of the sun by the
earth. not a number easily divisible by 10 which is the decimal system in a nutshell. Good luck with arranging a calendar of 100 days, or weeks of 10 days.
I work in my engine building hobby hobby with both decimal systems and imperial. There is also decimal imperial as opposed to fractional. Both make sense. Imperial is more organic.. surely all the
rage these days.
Expand full comment
However you are forgetting one area that metric does quite nicely.
1 cubic meter of fresh water at sea level is 1000 KG or 1 metric ton (or more properly, a tonne). Volume to Mass.
Yeah, I know the old phrase "a pint's a pound the world around", but doing engineering, the type where three digits is enough (and four too many, and five is WAY out), the mass to volume above
makes for easy rules of thumb. Yeah, you do the math afterwards, but the metric equivalence above is very useful. I deal... well used to... deal in sea ships, coal, cars, and containers. The "pint
is a pound" really isn't helpful there. Mind you, everything is still in TEU, or Twenty-foot Equivalent Units, but they do have metric standards.
And don't get me started on baking. Metric and by weight, thank you very much.
So I must cordially disagree. I remain in favor, outside of time and navigation issues, the metric system.
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American Mathematical Society
Solutions of conservation laws satisfy the monotonicity property: the number of local extrema is a non-increasing function of time, and local maximum/minimum values decrease/increase monotonically in
time. This paper investigates this property from a numerical standpoint. We introduce a class of fully discrete in space and time, high order accurate, difference schemes, called generalized monotone
schemes. Convergence toward the entropy solution is proven via a new technique of proof, assuming that the initial data has a finite number of extremum values only, and the flux-function is strictly
convex. We define discrete paths of extrema by tracking local extremum values in the approximate solution. In the course of the analysis we establish the pointwise convergence of the trace of the
solution along a path of extremum. As a corollary, we obtain a proof of convergence for a MUSCL-type scheme that is second order accurate away from sonic points and extrema. References
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English and French summaries). MR 1300961
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• Haim Nessyahu, Eitan Tadmor, and Tamir Tassa, The convergence rate of Godunov type schemes, SIAM J. Numer. Anal. 31 (1994), no. 1, 1–16. MR 1259963, DOI 10.1137/0731001
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• Chi-Wang Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Statist. Comput. 9 (1988), no. 6, 1073–1084. MR 963855, DOI 10.1137/0909073
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• Huanan Yang, On wavewise entropy inequalities for high-resolution schemes. I. The semidiscrete case, Math. Comp. 65 (1996), no. 213, 45–67, S1–S13. MR 1320900, DOI 10.1090/S0025-5718-96-00668-0
• H. Yang, On wavewise entropy inequalities for high-resolution schemes II : fully-discrete MUSCL schemes with exact evolution in small time, SIAM J. Numer. Anal., to appear.
Similar Articles
• Retrieve articles in Mathematics of Computation with MSC (1991): 35L65, 65M12
• Retrieve articles in all journals with MSC (1991): 35L65, 65M12
Additional Information
• Philippe G. LeFloch
• Affiliation: Centre de Mathématiques Appliquées and Centre National de la Recherche Scientifique, URA 756, Ecole Polytechnique, 91128 Palaiseau, France
• Email: lefloch@cmapx.polytechnique.fr
• Jian-Guo Liu
• Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
• MR Author ID: 233036
• ORCID: 0000-0002-9911-4045
• Email: jliu@math.temple.edu
• Received by editor(s): May 5, 1997
• Received by editor(s) in revised form: November 10, 1997
• Published electronically: February 13, 1999
• Additional Notes: The first author was supported in parts by the Centre National de la Recherche Scientifique, and by the National Science Foundation under grants DMS-88-06731, DMS 94-01003 and
DMS 95-02766, and a Faculty Early Career Development award (CAREER) from NSF. The second author was partially supported by DOE grant DE-FG02 88ER-25053.
• © Copyright 1999 American Mathematical Society
• Journal: Math. Comp. 68 (1999), 1025-1055
• MSC (1991): Primary 35L65, 65M12
• DOI: https://doi.org/10.1090/S0025-5718-99-01062-5
• MathSciNet review: 1627801
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Python - (Systems Biology) - Vocab, Definition, Explanations | Fiveable
from class:
Systems Biology
Python is a high-level programming language known for its readability and versatility, making it widely used in various fields, including network visualization and analysis tools. Its simplicity
allows researchers to focus on solving problems rather than getting bogged down by complex syntax. This has made Python a popular choice among scientists and analysts who work with large data sets
and need effective tools for data visualization and analysis.
congrats on reading the definition of Python. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Python supports multiple programming paradigms, including procedural, object-oriented, and functional programming, which allows users to choose the best approach for their network visualization
2. The extensive library ecosystem in Python means users can easily integrate various tools for specific tasks, such as data manipulation with Pandas or plotting with Matplotlib.
3. Python’s community is large and active, providing extensive documentation and support through forums, tutorials, and shared code repositories.
4. Using Python for network visualization can greatly enhance the ability to analyze complex data structures, as it provides various libraries that simplify the process of creating visual
5. The use of Python in scientific computing is growing, with many researchers adopting it for its ability to handle large datasets and create sophisticated visualizations quickly.
Review Questions
• How does Python's versatility contribute to its effectiveness in network visualization and analysis?
□ Python's versatility stems from its support for multiple programming paradigms and a rich ecosystem of libraries. This allows users to tackle different aspects of network visualization and
analysis using the most suitable methods. For example, users can manipulate data with Pandas, visualize it with Matplotlib, or analyze networks with NetworkX. This combination empowers
researchers to create comprehensive visualizations tailored to their specific needs.
• Discuss the role of libraries like NetworkX and Matplotlib in enhancing Python's capabilities for network analysis.
□ Libraries like NetworkX and Matplotlib play crucial roles in expanding Python's capabilities for network analysis. NetworkX provides specialized tools for creating and analyzing complex
networks, enabling users to study relationships between data points effectively. Meanwhile, Matplotlib allows users to visualize these networks clearly through graphs and plots. Together,
these libraries streamline the workflow from data manipulation to visualization, making Python an essential tool in systems biology.
• Evaluate the impact of Python's community support on its adoption for scientific research and data analysis.
□ Python's strong community support has significantly impacted its adoption in scientific research and data analysis. With extensive documentation, forums, and collaborative platforms
available, researchers can quickly find solutions to common problems or learn new techniques. This collaborative environment encourages knowledge sharing and innovation among scientists,
fostering the development of new tools and methodologies. As a result, Python continues to grow as a preferred language in fields requiring rigorous data analysis and visualization.
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2003 OASDI Trustees Report
Short-Range Actuarial Estimates
For the short range (2003-2012), the Trustees measure trust fund adequacy by comparing assets at the beginning of each year to projected expenditures for that year under the intermediate set of
assumptions. Having a trust fund ratio of 100 percent or more--that is, assets at the beginning of each year at least equal to projected outgo during the year--is considered a good indication of a
trust fund's ability to cover most short-term contingencies. Both the OASI and the DI trust fund ratios under the intermediate assumptions exceed 100 percent throughout the short-range period and
therefore satisfy the Trustees' short-term test for financial adequacy. Figure II.D1 below shows the trust fund ratios for the combined OASI and DI Trust Funds for the next 10 years.
OASDI Trust Fund
[Assets as a
percentage of
Long-Range Actuarial Estimates
The financial status of the trust funds over the next 75 years is measured in terms of costs and income as a percentage of taxable payroll, trust fund ratios, the actuarial balance (also as a
percentage of taxable payroll), and the open group unfunded obligation (expressed in present-value dollars). Considering Social Security's cost as a percentage of the total U.S. economy (gross
domestic product or GDP) provides an additional perspective.
The year-by-year relationship of the income and cost rates shown in figure II.D2 illustrates the expected pattern of cash flow for the OASDI program over the full 75-year period. Under the
intermediate assumptions, the OASDI cost rate is projected to decline slightly and then remain flat for the next several years. It then begins to increase rapidly and first exceeds the income rate in
2018, producing cash-flow deficits thereafter. Despite these cash-flow deficits, trust fund interest earnings and assets will allow continuation of full benefit payments until 2042, when the trust
funds will be exhausted. Pressures on the Federal Budget will thus emerge well before 2042. Even if a trust fund's assets are exhausted, however, tax income will continue to flow into the fund.
Present tax rates would be sufficient to pay 73 percent of scheduled benefits after trust fund exhaustion in 2042 and 65 percent of scheduled benefits in 2077.
Income and
Cost Rates
Social Security's cost rate generally will continue rising through about 2030 as the baby-boom generation reaches retirement age. Thereafter, the cost rate is estimated to rise at a slower rate for
about 15 years as the baby boom ages and begins to decrease in size. Continued reductions in death rates and relatively low birth rates will cause a significant upward shift in the average age of the
population and will push the cost rate to nearly 20 percent of taxable payroll by 2077 under the intermediate assumptions. In a pay-as-you-go system such as OASDI, this 20-percent cost rate means the
combination of the payroll tax (now totaling 12.4 percent) and proceeds from income taxes on benefits (expected to be 1.0 percent of taxable payroll in 2077) would have to equal 20 percent to pay all
currently scheduled benefits. Although the annual projections do not extend beyond 2077, the upward shift in the average age of the population is likely to continue and to increase the gap between
OASDI costs and income.
The primary reason that the OASDI cost rate will increase rapidly between 2010 and 2030 is that, as the large baby-boom generation born in the years 1946 through 1964 retires, the number of
beneficiaries will increase much more rapidly than the number of workers. The estimated number of workers per beneficiary is shown in figure II.D3. In 2002, there were about 3.3 workers for every
OASDI beneficiary. The baby-boom generation will have largely retired by 2030, and the projected ratio of workers to beneficiaries will be only 2.2 at that time. Thereafter, the number of workers per
beneficiary will slowly decline, and the OASDI cost rate will continue to increase.
of Covered
Workers Per
The maximum projected trust fund ratios for the OASI, DI, and combined funds appear in table II.D1. The year in which the maximum projected trust fund ratio is attained and the year in which the
assets are projected to be exhausted are shown as well.
Table II.D1.--Projected Maximum Trust Fund Ratios
Achieved and
Trust Fund Exhaustion Dates Under the Intermediate
│ │OASI│ DI │OASDI │
│ Maximum trust fund ratio (percent)│ 526│ 226│ 471│
││ Year attained│2016│2007│ 2016│
│ Year of exhaustion│2044│2028│ 2042│
The actuarial balance is a measure of the program's financial status for the 75-year valuation period as a whole. It is essentially the difference between income and cost of the program expressed as
a percentage of taxable payroll over the valuation period. This single number summarizes the adequacy of program financing for the period. When the actuarial balance is negative, the actuarial
deficit can be interpreted as the percentage that would have to be added to the current law income rate in each of the next 75 years, or subtracted from the cost rate in each year, to bring the funds
into actuarial balance. In this report, the actuarial balance under the intermediate assumptions is a deficit of 1.92 percent of taxable payroll for the combined OASI and DI Trust Funds. The
actuarial deficit was1.87 percent in the 2002 report and has been in the range of 1.46 percent to 2.23 percent for the last ten reports.
Another way to illustrate the financial shortfall of the OASDI system is to examine the cumulative value of taxes less costs, in present value. Figure II.D4 shows the present value of cumulative
OASDI taxes less costs over the next 75 years. The balance of the combined trust funds peaks at $2.3 trillion in 2017 (in present value) and then turns downward. Through the end of 2077, the combined
funds have a present-value unfunded obligation of $3.5 trillion.
OASDI Income Less
Cost, Based on
Present Law Tax
Rates and
Scheduled Benefits
[Present value as
of 1-1-2003, in
Still another important way to look at Social Security's future is to view its cost as a share of the U.S. economy. Figure II.D5 shows that Social Security's cost as a percentage of GDP will grow 1.6
times from 4.4 percent in 2002 to 7.0 percent in 2077. Over the same period, the cost of Social Security expressed as a percentage of taxable payroll will grow from 10.95 percent to 19.92 percent.
Cost as a
Percentage of
Even a 75-year period is not long enough to provide a complete picture of Social Security's financial condition. Figures II.D6 and II.D7 show that the program's financial condition continues to
worsen at the end of the period. Some experts have noted that overemphasis on summary measures for a 75-year period can lead to incorrect perceptions and to policy prescriptions that do not move
towards a sustainable system. In order to provide a more complete description of Social Security's very long-run financial condition this year the Trustees present actuarial estimates over a time
period that extends to the infinite horizon. These calculations show that extending the horizon indeed increases the unfunded obligation, indicating that much larger changes would be required to
achieve solvency over the infinite future as compared to changes needed according to 75-year period measures.
Changes From Last Year's Report
This year's projections are little changed from those in last year's report, as shown in figure II.D6. The long-range actuarial deficit has increased from 1.87 percent to 1.92 percent of taxable
payroll--primarily because the valuation period now extends through 2077. On balance, changes in assumptions, methods, and data have slightly lessened the negative impact of changing the valuation
period (see table II.D2). The open group unfunded obligation over the 75-year projection period has increased from $3.3 trillion to $3.5 trillion.
Table II.D2.--Reasons for Change in the 75-Year Actuarial
Under Intermediate Assumptions
[As a percentage of taxable payroll]
│ Item │OASI │ DI │Combined │
│Shown in last year's report: │
││ Income rate│11.79│1.92│ 13.72│
││ Cost rate│13.33│2.26│ 15.59│
││ Actuarial balance│-1.54│-.34│ -1.87│
│ Changes in actuarial balance due to changes in:│
│││ Legislation / Regulation│ .00│ .00│ .00│
│││ Valuation period│ -.06│-.01│ -.07│
│││ │ │ │ │
│││ │ │ │ │
│││ Demographic data and assumptions│ -.03│-.01│ -.04│
│││ Economic data and assumptions│ +.01│-.01│ .00│
│││ Disability data and assumptions│ .00│ .00│ .00│
│││ Projection methods and data│ +.05│+.01│ +.06│
││ Total change in actuarial balance│ -.03│-.02│ -.04│
│ Shown in this report:│
││ Actuarial balance│-1.56│-.35│ -1.92│
││ Income rate│11.85│1.93│ 13.78│
││ Cost rate│13.41│2.29│ 15.70│
Note: Totals do not necessarily equal the sums of rounded components.
A higher assumed rate of immigration improves the projected financial situation of the trust funds in the early years, however, and delays the start of cash-flow deficits from 2017 to 2018. The year
in which the combined trust funds will be exhausted also slips one year--from 2041 to 2042.
Cost as
2002 and
Uncertainty of the Projections
Significant uncertainty surrounds the intermediate assumptions. The Trustees have traditionally used low cost (alternative I) and high cost (alternative III) assumptions to indicate this uncertainty.
Figure II.D7 shows the projected trust fund ratios for the combined OASI and DI Trust Funds under the intermediate, low cost, and high cost assumptions. The low cost alternative is characterized by
assumptions that improve the financial condition of the trust funds, including a higher fertility rate, slower improvement in mortality, a higher real-wage differential, and lower unemployment. The
high cost alternative, in contrast, features a lower fertility rate, more rapid declines in mortality, a lower real-wage differential, and higher unemployment.
OASDI Trust Fund
Ratios Under
[Assets as a
percentage of
These three alternatives have traditionally been constructed to provide a reasonable range of possible future experience. However, these alternatives do not address the probability that actual
experience will be within or outside the range. As an additional way of illustrating uncertainty, this Trustees Report for the first time uses a model of the trust funds that provides an explicit
probability distribution of possible future outcomes (see appendix E). The results of this model suggest that outcomes better than the traditional low cost alternative and outcomes worse than the
high cost alternative have very low probabilities of occurring.
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Learning in Games
040152 UK Learning in Games (2012S)
Continuous assessment of course work
Note: The time of your registration within the registration period has no effect on the allocation of places (no first come, first served).
• Registration is open from Th 09.02.2012 09:00 to Mo 20.02.2012 17:00
• Registration is open from Mo 27.02.2012 09:00 to Tu 28.02.2012 17:00
• Deregistration possible until We 14.03.2012 23:59
max. 50 participants
Language: English
Classes (iCal) - next class is marked with N
• Thursday 01.03. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 08.03. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 15.03. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Friday 23.03. 10:45 - 12:15 (ehem. Hörsaal 23 Hauptgebäude, 1.Stock, Stiege 5)
• Thursday 29.03. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 19.04. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 26.04. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 03.05. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 10.05. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 24.05. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 31.05. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 14.06. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 21.06. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
• Thursday 28.06. 12:00 - 13:30 (ehem. Seminarraum 10 Hauptgebäude, Tiefparterre Stiege 5 Hof 3)
Aims, contents and method of the course
Game theory is methodology for understanding how to make choices when facing others that are also making choices. It is an analysis of strategic decision making. Game theory is the central and most
important tool in Economics.
The classical approach is to assume rationality. Players correctly anticipate what others are doing and in equilibrium none of the players has an incentive to deviate when assuming that none of the
others deviate.
But do we really believe that players have the same beliefs and follow this equilibrium concept? In this lecture we relax these assumptions, do not make heroic assumptions about players think about
each other. We assume instead that they learn over time. We investigate what happens in the long run. What will they learn? Will they all do the same thing? Will they learn to play an equilibrium as
defined above? Will they sometimes never learn anything as they constantly change their actions?
This is the topic of this course.
In very simple games we will investigate what players learn to play. This will of course depend on how they learn. The models of learning highlighted below will be considered.Methodology:
There is some mathematics involved as one has to understand how play in the population changes over time. However, emphasis will be on understanding intuition in simple settings, and analysis of
examples, not proving theorems nor proving general results. Notes are supplied as often the original papers too mathematical.Specific Models
Evolutionary Learning:
This is a simplified biological story in which there are only two types of players, incumbants and mutants. Those that are more successful reproduce. Search is for an evolutionarily stable strategy.
In an extended version there are many types and one considers change of play according to the replicator dynamics.
Assessment and permitted materials
Minimum requirements and assessment criteria
The objective is to receive an overview of the literature and to understand, under which
circumstances we can trust classic game theory and when not.
Examination topics
Reading list
- Weibull (1995), Evolutionary Game Theory, MIT Press
This is a simplified cultural learning story in which individuals observe what others do and imitate those that are more successful. The specific type of imitation matters so we first need to
investigate which type of imitation or learning rule has good properties. Then we can move on to make predictions. It turns out that the underlying dynamics is the replicator dynamics encountered
- K.H. Schlag, Why Imitate, and if so, How? A Boundedly Rational Approach to Multi-Armed
Bandits, Journal of Economic Theory 78(1) (1998), 130-156.- Alos-Ferrer, C. and K.H. Schlag (2007), Imitation and Learning, Handbook of Rational and Social Choice, Chapter 11,
papers/Imitation.pdfReenforcement Learning:
In this model, players experiment between the different strategies, choosing those that performed well more likely.Reference:
- Börgers, T. and R. Sarin (1997) Learning Through Reinforcement and Replicator Dynamics, Journal of Economic Theory 77, 1-14Best Response Learning:
Next we consider more sophisticated players who know which game they are playing and who have information about what others did in the last round. A variant is fictitious play where players look at
the entire history of play by others. They do not anticipate what others will do.Reference:
- Fudenberg and Levine (1998), The theory of learning in games, MIT Press.
Anticipatory Learning:
Finally we briefly consider learning when players anticipate that others will also change their strategy.Reference:
- R. Selten, Anticipatory learning in 2-person games, in R. Selten (Ed.), Game Equilibrium
Models. vol. I, Springer-Verlag, Berlin (1991), pp. 98-254
Last modified: Tu 01.10.2024 00:08
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Simon Donaldson extras
We give three autobiographies of Simon Donaldson. The first we have constructed from the Heidelberg Laureate Forum interview he gave in
. The other two are autobiographies, one from the book
, the other written at the time he won the Shaw Prize in
. This is followed by an essay on the Shaw Prize
. We then give the citation for Donaldson winning the
Breakthrough Prize in Mathematics, and finally the relevant part of the citation for the
Wolf Prize.
Click on a link below to go to that section
1. The HLF Portraits: Simon Donaldson.
I was born into a scientific household. My mother did a degree in natural sciences in Cambridge. She didn't pursue it - her generation didn't after they got married. My father was an engineer. He
started his career in the navy but he left the navy about the time he got married. Then he worked in the physiology laboratory in Cambridge. He was building the apparatus for experiments. He was
working with well-known people such as Andrew Huxley trying to understand the nervous system. I was one of four children. There were plentiful supplies of books but my parents allowed all the
children to develop their own interests. What tended to be even more conspicuous than books, there were various projects. My father always had projects he was doing in the house - he built model
airplanes, he would mend things in the house - there was a lot of activity and an interest in doing things.
I'm the third of four, three boys and a girl. My sister is older then me. There were underlying expectations but these were very general. I went to a prep school in Cambridge, a feeder school for an
independent secondary school. I think most of the things I know I learnt there. I was most interested in history at that time - the history school masters were keen on me. I didn't do extraordinarily
well nor was I particularly hard working.
When I was twelve the family moved because my father got a different job. Actually it was working with some of the same people but rather than working for Cambridge University they set up a research
unit funded by the Medical Research Council. At that time my father was working on artificial vision and they had some success in doing that but maybe it was a bit before its time. I have the
impression that now everything has gone much further. Maybe some of the technology wasn't quite there at that time. They were also working on some less ambitious things, basically when you stimulate
the nervous system artificially. He shared the struggles of his work with his children.
We moved to Kent, near to a town called Sevenoaks which is a commuter town twenty miles south of London. I went to Sevenoaks School which is one of the slightly anomalous schools. It was a private
school but since there was no local grammar school so half of the pupils were actually supported by the state and half were private. It had a broader social mix than one might think for an
independent school. It had a reputation of being quite innovative - practised lots of trendy theories of education - this was about 1970. I didn't completely like it to begin with. It took me some
years to settle in to this school. I had grown up thinking I had a path to the schools of Cambridge that I was going to follow and then this was different. But actually the disruption was probably
quite good for me. For a time I was not especially happy there.
Part of my problem was my passion for boats all through this time. At that point in my life I wanted to be a designer of yachts. Going back to the school it was fortunate that sailing was a big thing
at the school so through the sailing I became happy in the school. I would design yachts, not constructing them of course, but I was quite serious about it. I had books about the calculations you
must make and all the things you must think of. There was more mathematics involved with these things than I had learnt at school. My father showed me some mathematical techniques that I was using
describing curves by the superimposition of sine waves.
So I was intrigued by all that - it was about this time that I became interested in mathematics - partly from the yacht design and partly because I found it very interesting and beautiful. Another
thing was that it fitted in with what I was capable of doing in the sense that most of my family would be doing more projects where you actually built something - they would make a boat or make an
airplane or something. But I hadn't got on very well with physically making things. I could make things, but not physical things, I could make things that existed as ideas. They were like projects
but they were not practical projects. Projects of the mind - somehow I found that very congenial. I wanted to learn calculus when I heard about it; some years before it would be covered in school, I
would study it.
I should have spoken about my grandfather - my mother's father - he wasn't a mathematician but he was a retired schoolmaster. He was extraordinarily keen on my development; in fact he took a great
interest so he would give me many many books of all kinds but, in particular, at the stage we are talking about now he would go down to the book shops in Cambridge and buy me some suitable maths
books. Some of them are still on these shelves. I might have had some interest that stayed dormant if there hadn't been a book to develop it. He took a special interest in all his grandchildren, but
probably particularly in me. There were some excellent mathematics teachers in the school - they did take a special interest in me - but I wouldn't say they were the seed - they were nurturing.
From the time I was about fourteen, I gave up the idea of being a yacht designer - I decided I wanted to be a mathematician. That was presented to me like someone might want to be a concert pianist -
many may want to do it but few will succeed. But in reality its not quite like that. But that was how I thought about it at the time. This was some very desirable thing to try to do but I had a plan
B - to go and be an accountant or something like that. Most of my family went to Cambridge and, particularly in mathematics, it had the long tradition of being the main centre. Quite a big element at
that time is the competitive tradition - there were scholarships to Oxford and Cambridge and financially they weren't worth very much but they but in the world of people at schools applying to Oxford
and Cambridge they were a big thing. In the end I got a scholarship - the point I'm trying to make is that there is an element of competition in mathematics of being the smartest quickest person
which, at this period, was a significant thing. Also in Cambridge there is a competition to be the top. I never was the top - it was a competition I would enter and I would do reasonably but I was
never outstanding.
I became intrigued by the branch of mathematics called geometry which is a bit distant from what most people will think of as geometry. It does involve working with ideas in a visual way even if they
are very abstract ideas. In the Cambridge course at that time there was almost nothing about geometry but there was one small course and there were applied mathematics courses like relativity and
such things. The fact that it was not really taught added to the intrigue - a secret subject. As I soon found out, just down the road in Oxford, there were lots of people who did exactly that. In
Cambridge there were people more around Stephen Hawking who were in a sense doing the same kind of thing but from an applied direction. Part of this is that at Cambridge mathematics is divided into
pure and applied but I had decided I was going to be pure but a lot of activity was in applied. At Oxford they didn't make the same distinction - it is more unified.
I applied to Oxford for a Ph.D. I did alright in my exams in Cambridge. There was someone in Cambridge who was very important to me. There were several but the person I'm thinking of is Frank Adams.
He was a topologist and I somehow managed to impress him. He was one of my referees for Oxford. He was very kind. After all my exams at Cambridge, Adams wrote to me saying, "I was one of your
examiners and I want to congratulate you on what you wrote in these exams," This was kind of him. I didn't answer the questions to get the maximum marks in the least time - it wasn't my way of doing
things. I tend to know a little about many things rather than knowing a lot about one particular thing. I bring together different ideas.
At Oxford it was two people who were important to me, Atiyah and Penrose. I went to Oxford and it was the activity and culture that developed around them that was important. One thing is that they
are both remarkable people and particularly Atiyah is a very charismatic person who would have been at the top of any area he wanted. But also there was a wider movement at that time in mathematics
partly in less division into compartments also specifically in extending the connection between geometry and physics. That was a much more general thing that was going on - a general intellectual
movement; in particular Oxford was at the centre of that. It was exactly the right thing for me. I had then, and I have now, very little knowledge of advanced physics. I had a good background in
classical physics but I have no, or essentially very little, technical knowledge of the front line of physics. But certain ideas and questions arose in physics which could be understood in
mathematical terms. The lack of technical knowledge of physics was not a major barrier. I remember someone telling me at that time, "Don't try to learn quantum field theory until you have a tenured
Hitchin was my formal director. He suggested the direction for my dissertation, particularly work he'd done a few years before with Atiyah. But I don't think he meant me to work on the conjecture in
general. What he suggested was that I checked some constructions for a special case. He never really said, "This is your project." Nevertheless, I started to try to solve the conjecture. I should say
that I didn't meet Atiyah in the first year I was in Oxford. I worked on this conjecture. The thing there was that all these geometers in Oxford didn't do analysis - weren't experts on partial
differential equations. I'm exaggerating, of course, but there was a kind of a gap there intellectually, particularly with non-linear partial differential equations. But partly because I had a lot of
analysis in Cambridge and partly because it is something I find congenial, I embarked on solving this problem using techniques from analysis, particularly partial differential equations.
I was struggling to understand the techniques from partial differential equations that I had to learn for myself. I'm trying to understand something, then the way is to do thought experiments - if
this is true what would be the consequences. So doing that, I had this idea about how the solutions to this equation could behave. That gave me this picture which gave a space for parameterising all
the solutions. I looked at the properties of this space. On the other direction, on the side of topology, if you have such a space there are basic consequences so I put these consequences together
and it transpired that this would be some new information about the manifolds - the 3-dimensional manifolds. I was not starting off thinking it was special - I didn't even know this information was
new. At one time I thought it would be an interesting new approach to something well-known.
I had a colleague whom I shared an office with, Mike Hopkins, who is now a very distinguished topologist. He filled me in on some crucial information. I realised that this information would be
important - new information. It depended on both knowing about the partial differential equations and about the topology, so maybe the Cambridge background was rather good there because we had not
only learnt the geometry in a formal way, but had learnt other things. So with those two things anyone would see you got information but there were not too many people who had that combination of
expertise. Once I'd made this observation it was obvious that I should go full steam ahead to develop it into a much bigger theory of how these spaces could give information about 4-dimensional
manifolds. It took me a month or so to find out what it was good for. At this stage it was all thought experiments. If this is true then this follows, then this, etc. There was a lot of hard work
filling in all the proofs and details.
For my first year at Oxford I was working with Hitchin but then he suggested I move to Atiyah. It was about a month after I started working with Atiyah that I came along and said, "I've got this
formula," He was excited. It would be reasonable to say, if it is very broadly interpreted, that I'm still working somewhere in between partial differential equation approaches and with topological
ideas. The precise mix of these things have evolved over the years. Another way to look at it is to say that from the early 1980s we suddenly discovered a lot more about these 4-dimensional manifolds
- at least for 10 years after that there was an explosion of knowledge but after that the pace of discovery slowed down. There are still fundamental questions we have no way of attacking. So I still
really like to feel I'm attacking these questions but because they are not really accessible I work round them. But they are still the centre of my interest. If you asked what I really want to
understand, then it is these questions about 4-dimensional manifolds and not just the answers to the questions but why are these connections coming from physics and things like that. Interaction
between these fields had thrived. It is for younger people who have a deeper understanding of quantum field theory.
2. The Mathematicians autobiography of Simon Donaldson.
My father had a large influence on my development into a mathematician, at least in a general way. I have an early memory of him saying with relish : "... and then I shall be able to get back to
research" (presumably, after completing some chores which he had described to me). I had no idea what "research" might be, but from that time the word was tinged with glamour and romance.
My father was an engineer (my two brothers followed him) and was often sceptical of overly theoretical work. "All they produce is paper" - I hear him fulminating - "I bet they haven't touched a
screwdriver in years". (Although this was not meant completely seriously; he had a deep interest in all kinds of science). Our house was always full of projects of a creative, practical kind:
building model aircraft and so on. My own attempts in this direction were usually less successful; my vision of what I wanted to achieve outran my patience and ability to bring it about. So that was
partly how I moved towards mathematics, where vision was not trammelled by irksome practical difficulties. Another thing that was very important was that I was fascinated by sailing, sailing boats,
and anything nautical. So, when I was about thirteen, I decided that my career was to be a yacht designer, and began to design some. (I had no intention of actually building these yachts - that could
wait until I had wealthy clients - so the enterprise was not limited by practicalities.) I went into this deeply. To design a yacht you need to calculate volumes, areas, moments, and so on from your
plans. So it was quite natural for me to learn more mathematics. Gradually, the mathematics became the centre of my interests and the yacht designing fell into the background.
I was also lucky to have excellent mathematics teachers. It was important for me to be able to do well in mathematics and physics at school. A large influence in that direction came from my
grandfather. He was a teacher of modern languages and, at a younger age, encouraged my interest in history and academic things generally.
By the time I was about sixteen, I had a fairly definite idea that what I wanted to be was a mathematician and some notion of what that was. I would investigate various questions that occurred to me,
almost never making any definite progress. So in a way I was precocious (although definitely not in the sense of being particularly good at tests of the Mathematical Olympiad kind or, later, the
undergraduate exams in Cambridge). This made the transition to life as a proper research mathematician, as a doctoral student in Oxford with Nigel Hitchin and Michael Atiyah, comparatively easy for
I mostly work by drawing pictures (vestiges of my yacht plans?) so I was naturally attracted to geometry. When I was an undergraduate, it was not so easy to learn about differential geometry, since
it did not really come into the standard course, but this added to the allure of the subject. Holding fast to my metaphorical paternal screwdriver, I prefer problems that are quite concrete and
specific, where one feels one is actually producing and working with some definite mathematical object.
I was blessed with good fortune at the start of my research career. At that time (1980) the Yang-Mills equations, arising in particle physics, were making a big impact in pure mathematics,
particularly in connection with geometry and Roger Penrose's twistor theory. The project Hitchin suggested to me involved a rather different kind of question, connecting differential and algebraic
geometry but veering more towards analysis and partial differential equations. Happily - with rather different purposes in mind, Karen Uhlenbeck and Cliff Taubes had, in the few years before, gone a
long way in developing the relevant analytical techniques. Of course this was before the Internet made it so easy to find papers, and I remember the exciting day when I received Uhlenbeck's preprints
by mail from the USA. This was when I was a first-year doctoral student. A good approach in research, I find, is to imagine what should be true, i.e. a picture of what properties some mathematical
objects have, and then explore the consequences. If the consequences lead to a contradiction, that shows that the picture needs to be modified; on the other hand, if the consequences fit in with what
one knows otherwise and lead to some interesting further predictions, that is good evidence for the correctness of the picture. Following this strategy (although certainly not consciously) and
exploring the properties of Yang-Mills instantons, I stumbled on, at the beginning of my second year, an entirely unexpected application of these to the topology of four-dimensional manifolds. The
two main themes of my research in the twenty-seven years since then have been extending this and, in a different direction, developing the links between algebraic geometry, differential geometry, and
partial differential equations.
3. The Shaw Prize autobiography of Simon K Donaldson.
I was born in 1957 in Cambridge, England, the third of four children. At that time, my father worked as an electrical engineer in the Physiology Department of the University. My mother had been
brought up in Cambridge and had taken a Science degree there. When I was 12 we moved to a village in Kent, following my father's appointment to lead a team in London developing neurological implants.
The passion of my youth was sailing. Through this, I became interested in the design of boats, and in turn in mathematics. From the age of about 16, I spent much time studying books, puzzling over
problems and trying to explore. I did well in mathematics and physics at school, but not outstandingly so.
In 1976 I returned to Cambridge for my first degree. The subject I liked best was geometry, although there was rather little of this in the Cambridge course at that time and my main training was in
analysis, topology and traditional Mathematical Physics. The word "geometry" may convey a misleading impression. The modern subject is very far from Euclid's, and it is perhaps better to think of
vector calculus and, for example, the geometrical notion of the "flux" of a vector field.
In 1980 I moved to Oxford to work for a doctorate, supervised by Nigel Hitchin. This was an exciting time in Oxford. Penrose's "twistor theory" was dominant, an early example of the now-pervasive
interaction between geometry and fundamental physics. Sir Michael Atiyah, who supervised my work later, was a driving force in this and a few years before he, Hitchin, Drinfeld and Manin had done
renowned work on Yang-Mills instantons, using twistor theory and geometry of complex variables. These instantons solve generalisations of Maxwell's equations.
In my thesis I studied two different, but related, topics which have developed into the two themes of most of my subsequent research. The first theme is the interaction between differential and
algebraic geometry. The problem that Hitchin proposed to me was to relate the instantons over complex spaces to "bundles" studied by algebraic geometers. What was unusual, in terms of the Oxford
environment, was that I tried to tackle this problem using analytical techniques. This kind of approach was by no means new in other problems and in other parts of the world, but was not a strong
tradition in the UK. I learnt the trade by studying preprints of Cliff Taubes and Karen Uhlenbeck, which opened up the analytical approach to Yang-Mills theory.
My original focus was on the case of a complex space, but certain central questions made sense on any 4-dimensional manifold. Thinking about these, and combining with an existence result proved by
Taubes, led to the other topic of my thesis: the application of Yang-Mills theory to 4-manifold topology. This was quite unexpected. While the basic argument "stared me in the face", thanks to my
training in topology, hard work was required to carry it through in detail.
Now I turn to the decade 1983-93. I spent a year in Princeton, and met my wife, Nora, during a visit to the University of Maryland. Our three children, Andres, Jane and Nicholas were born, joining my
step-daughter Adriana. (Andres and Jane have both now gained degrees from Cambridge and Nicholas is nearing the end of high school. Adriana is a mathematics teacher.) In research, my focus was on
developing the topological applications into a general theory. This was as part of a large endeavour by many mathematicians around the world. I wrote one monograph, with Kronheimer, about these ideas
and another dealing with Floer's theory, which extends them to 3-dimensional manifolds. I was made a Professor in Oxford in 1985 and was fortunate to have many research students.
From about 1994 I developed a different research strand, introducing techniques into symplectic topology. The ideas meshed in well with the results obtained by Taubes around that time. After two
decades of dramatic development, 4-manifold theory has reached a much more steady state and, while a great deal is known, there are huge areas where we are entirely ignorant. This strand was an
example of an attempt to find some new approach.
We spent the year 1997-8 in Stanford, and I moved back to my current position at Imperial College, London. My wife runs a Medical Statistics Unit at King's College. There was little geometry in
Imperial then, but now, thanks largely to the drive of my colleague Richard Thomas, we have one of the main centres for research in this area. My work over the past decade has in a sense returned to
my thesis problem, but extended into Riemannian geometry. This is an area with a longer history and the problems are much harder, but the theory is developing in an exciting way. Many excellent young
mathematicians around the world are entering the field and contributing to these developments.
7 October 2009, Hong Kong
4. The Shaw Prize essay 2009.
Over the past 30 years, geometry in 3 and 4 dimensions has been totally revolutionised by new ideas emerging from theoretical physics. Old problems have been solved but, more importantly, new vistas
have been opened up which will keep mathematicians busy for decades to come.
While the initial spark has come from physics (where it was extensively pursued by Edward Witten), the detailed mathematical development has required the full armoury of non-linear analysis, where
deep technical arguments have to be carefully guided by geometric insight and topological considerations.
The two main pioneers who both initiated and developed key aspects of this new field are Simon K Donaldson and Clifford H Taubes. Together with their students, they have established an active school
of research which is both wide-ranging, original and deep. Most of the results, including some very recent ones, are due to them.
To set the scene, it is helpful to look back over the previous two centuries. The 19th century was dominated by the geometry of 2-dimensional surfaces, starting with the work of Abel on algebraic
functions, and developing into the theory of complex Riemann surfaces. By the beginning of the 20th century, Poincaré had introduced topological ideas which were to prove so fruitful, notably in the
work of Hodge on higher dimensional algebraic geometry and also in the global analysis of dynamical systems.
In the latter half of the 20th century there was spectacular progress in understanding the topology of higher dimensional manifolds and fairly complete results were obtained in dimensions 5 or
greater. The two "low dimensions" of 3 and 4, arguably the most important for the real physical world, presented serious difficulties but these were expected to be surmounted, along established
lines, in the near future.
In the 1980's this complacent view was shattered by the impact of new ideas coming from physics. The first breakthrough was made by Simon K Donaldson in his PhD thesis where he used the Yang-Mills
equations of $SU(2)$-gauge theory to study 4-dimensional smooth (differentiable) manifolds. Specifically, Donaldson studied the moduli (or parameter) space of all $SU(2)$-instantons, solutions of the
self-dual $SU(2)$ Yang-Mills equations (which minimise the Yang-Mills functional), and used it as a tool to derive results about the 4-manifold. This instanton moduli space depends on a choice of
Riemannian metric on the 4-manifold but Donaldson was able to produce results which were independent of the metric.
There are serious analytical difficulties in carrying out this programme and Donaldson had to rely on the earlier work of Karen Uhlenbeck and Clifford H Taubes. As these new ideas were developed and
expanded by Donaldson, Taubes and others, spectacular results came tumbling out. Here is an abbreviated list, which shows the wide and unexpected gulf between topological 4-manifold (where the
problems had just been solved by Michael Freedman) and smooth 4-manifold:
(1) Many compact topological 4-manifold which have no smooth structure.
(2) Many inequivalent smooth structures on compact 4-manifold.
(3) Uncountably many inequivalent smooth structures on Euclidean 4-space.
(4) New invariants of smooth structures.
The invariants in (4) were first introduced by Donaldson using his instanton moduli space. Subsequently, an alternative and somewhat simpler approach emerged, again from physics, in the form of
Seiberg-Witten theory. Here, one just counted the finite number of solutions of the Seiberg-Witten equations (i.e. the moduli space was now zero dimensional).
One of Taubes' great achievements was to relate Seiberg-Witten invariants to those introduced earlier by Gromov for symplectic manifolds. Such manifolds occur both as phase spaces in classical
mechanics and in complex algebraic geometry, through the Kahler metrics inherited from projective space and exploited by Hodge. Although symplectic manifolds need not carry a complex structure, they
always carry an almost (i.e. non-integrable) complex structure. Gromov introduced the idea of "pseudo-holomorphic curves" on symplectic manifolds and obtained invariants by suitably counting such
curves. Taubes, in a series of long and difficult papers, proved that, for a symplectic 4-manifold, the Seiberg-Witten invariants essentially coincide with the Gromov-Witten invariants (an extension
of the Gromov invariants). The key step in the work of Taubes is the construction of a pseudo-holomorphic curve from a solution of the Seiberg-Witten equations. This is fundamental since it connects
gauge theory (a theory of potentials and fields) to sub-varieties (curves). Roughly, it represents a kind of non-linear duality.
In fact, extending complex algebraic geometry to symplectic manifolds (of any even dimension) was again pioneered by Donaldson who proved various existence theorems such as the existence of
symplectic submanifolds. In the apparently large gap between algebraic geometry and theoretical physics, symplectic manifolds form a natural bridge and the recent results of Donaldson, Taubes and
others provide, so to speak, a handrail across the bridge.
All this work in 4 dimensions has an impact on 3 dimensions, especially through the work of Andreas Floer, and Taubes has made many contributions in this direction. His most outstanding result is his
very recent proof, in 3 dimensions, of a long-standing conjecture of Alan Weinstein. This asserts the existence of a closed orbit for a Reeb vector field on a contact 3-manifold. Contact 3-manifolds
arise naturally as level sets of Hamiltonian functions (energy) on a symplectic 4-manifold, and the Weinstein conjecture now asserts the existence of a closed orbit of the Hamiltonian vector field.
This latest tour de force of Taubes exhibits his real power as a geometric analyst.
In recent years Donaldson has turned his attention to the hard problem of finding Hermitian metrics of constant scalar curvature on compact complex manifolds. The famous solution by Yau of the Calabi
conjecture is an example of such problems. Donaldson has recast the constant scalar curvature problem in terms of moment maps, an idea derived from symplectic geometry which played a key role in
gauge theory. This construction of metrics is a much deeper problem, being extremely non-linear but Donaldson has already made incisive progress on the analytical questions involved. This new work of
Donaldson represents an exciting new advance which is currently attracting much attention.
This quick summary of the contributions of both Donaldson and Taubes shows how they have transformed our understanding of 3 and 4 dimensions. New ideas from physics, together with deep and delicate
analysis in a topological framework, have been the hallmark of their work. They are fully deserving of the Shaw Prize in Mathematical Sciences for 2009.
Mathematical Sciences Selection Committee
The Shaw Prize
7 October 2009, Hong Kong
5. Simon Donaldson: 2015 Breakthrough Prize in Mathematics.
Simon Donaldson was awarded the 2015 Breakthrough Prize in Mathematics: "For the new revolutionary invariants of four-dimensional manifolds and for the study of the relation between stability in
algebraic geometry and in global differential geometry, both for bundles and for Fano varieties."
The Science
We experience the world in three dimensions: up and down, left and right, forward and back. But according to Einstein, there are actually four: his theory of general relativity integrates time with
the three spatial dimensions. And while we can't visualise four dimensions, we can analyse them with mathematics. Simon Donaldson has transformed our understanding of four-dimensional shapes, showing
which ones can be "tamed" with the kind of equations that mathematicians can solve, and which can't. In the process, he both provided powerful new tools for physicists and incorporated new ideas from
physics into mathematics.
Comments by Simon Donaldson
It has been my great good fortune that my career has spanned a period of exceptionally exciting developments in my field. Ideas and techniques from different areas - topology, physics, differential
equations, geometry - have become interwoven in ways that no one would have predicted half a century ago. It is a privilege to have been able to witness this and take some part in it. Mathematics has
a long time scale. One of the pleasantest things is, looking back in time, to contemplate how the developments we see in our lifetimes fit into the longer term. And, looking forward in time, we have
confidence that the problems that seem to us intractable will yield to future advances, invisible to us now. I owe enormous thanks to my advisors, Michael Atiyah and Nigel Hitchin, and to all the
mathematicians who made Oxford in the 1980s such a special place. What I learned then has underpinned my whole career. Looking forward, rather than back, it has been my great good fortune to have
been able to watch the development of many extraordinary research students. I would like to thank my wife, Nora, my parents, and all my family for their support.
6. Sir Simon Kirwan Donaldson: Wolf Prize Laureate in Mathematics 2020.
Simon Donaldson and Yakov Eliashberg were jointly awarded the Wolf Prize: "for their contributions to differential geometry and topology."
Sir Simon Kirwan Donaldson (born 1957, Cambridge, U.K.) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson-Thomas
Donaldson's passion of youth was sailing. Through this, he became interested in the design of boats, and in turn in mathematics. Donaldson gained a BA degree in mathematics from Pembroke College,
Cambridge in 1979, and in 1980 began postgraduate work at Worcester College, Oxford.
As a graduate student, Donaldson made a spectacular discovery on the nature or 4-dimensional geometry and topology which is considered one of the great events of 20th century mathematics. He showed
there are phenomena in 4-dimensions which have no counterpart in any other dimension. This was totally unexpected, running against the perceived wisdom of the time.
Not only did Donaldson make this discovery but he also produced new tools with which to study it, involving deep new ideas in global nonlinear analysis, topology, and algebraic geometry.
After gaining his DPhil degree from Oxford University in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford, he spent the academic year 1983-84 at the Institute for
Advanced Study in Princeton, and returned to Oxford as Wallis Professor of Mathematics in 1985. After spending one year visiting Stanford University, he moved to Imperial College London in 1998.
Donaldson is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University and a Professor in Pure Mathematics at Imperial College London.
Donaldson's work is remarkable in its reversal of the usual direction of ideas from mathematics being applied to solve problems in physics.
A trademark of Donaldson's work is to use geometric ideas in infinite dimensions, and deep non-linear analysis, to give new ways to solve partial differential equations (PDE). In this way he used the
Yang-Mills equations, which has its origin in quantum field theory, to solve problems in pure mathematics (Kähler manifolds) and changed our understanding of symplectic manifolds. These are the phase
spaces of classical mechanics, and he has shown that large parts of the powerful theory of algebraic geometry can be extended to them.
Applying physics to problems or pure mathematics was a stunning reversal of the usual interaction between the subjects and has helped develop a new unification of the subjects over the last 20 years,
resulting in great progress in both. His use of moduli (or parameter) spaces of solutions of physical equations - and the interpretation of this technique as a form of quantum field theory - is now
pervasive throughout many branches of modem mathematics and physics as a way to produce "Donaldson-type Invariants" of geometries of all types. In the last 5 years he has been making great progress
with special geometries crucial to string theory in dimensions six ("Donaldson-Thomas theory"), seven and eight.
Professor Simon Donaldson is awarded the Wolf Prize for his leadership in geometry in the last 35 years. His work has been a unique combination of novel ideas in global non-linear analysis, topology,
algebraic geometry, and theoretical physics, following his fundamental work on 4-manifolds and gauge theory. Especially remarkable is his recent work on symplectic and Kähler geometry.
Last Updated December 2023
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Best Practices For Displaying Math Equations In Latex Documents - Latexum
Best Practices For Displaying Math Equations In Latex Documents
Properly Displaying Equations
When writing a LaTeX document that contains mathematical equations, it is important to display the equations properly so they render clearly and correctly. Here are some best practices for basic
equation formatting:
• Use the equation environment for single-line equations. This will place the equation on a separate line and center it. For example:
$$E = mc^2$$
• Use the align environment for multi-line and aligned equations. This allows you to align elements across multiple lines. For example:
a+b &= c \\
d+e &= f
• Use proper math mode symbols like \$ $ or \( \) to surround math expressions in a sentence. For example:
Euler's identity $e^{i\pi} + 1 = 0$ is a famous equation.
Using the correct environments and delimiters for equations ensures they will display properly when you compile your LaTeX document.
Formatting Equations Nicely
In addition to technically correct displays, you also want your equations to have good visual formatting to make them easier to read. Here are some formatting best practices:
• Add some spacing around display equations using the \\[ ] constructs. This adds a bit of visual separation between the equations and surrounding text. For example:
Text above equation\\[2ex]
$$x = 3$$\\[2ex]
Text below equation
• Break lines appropriately in long, multi-line equations to improve readability. Use alignments and line breaks at logical points, often operators or relations:
x =& 3 + 4 + 5 + 6 + \\
& 7 + 8 + 9 + 10
Use text mode next to equations to explain the meaning of variables and key components. This helps readers understand your mathematical notation. For example:
$$A = \pi r^2 \text{ where r is the radius}$$
Following these basic formatting guidelines will make your equations easier to quickly parse and understand.
Numbering Equations
When you have multiple equations in a document, it is helpful to number them so you can refer back to specific equations in the text. Here are recommendations for numbering equations:
• Automatically number a display equation by using the \begin{equation} and \end{equation} environment. Equations will be numbered sequentially.
$$x^2 + y^2 = z^2$$
• Manually specify an equation number with the \tag{} command inside an equation. This overrides the automatic numbering:
$$x + y = z \tag{1.3'}$$
• Refer to a numbered equation in text using the \eqref{} command. It will print the appropriate number:
As seen in equation \eqref{eqn:emc2}...
Numbering and referencing equations can make discussions significantly clearer by grounding concepts to specific locations in the document.
Common Math Functions
LaTeX provides a wide selection of mathematical symbols, functions, and formatting capabilities. Some common examples include:
• Greek letters - Frequently used for variables and mathematical constants like angles. Access them using names like \alpha, \beta, \theta:
$$\alpha + \beta = \gamma$$
• Fractions - Created using the \frac{} command:
\[ z = \frac{x+y}{k} \]
• Square roots - Done with \sqrt{}:
\[ x = \sqrt{b^2 - 4ac} \]
• Summation and integrals - Implemented with \sum and \int:
\[ \sum_{i=1}^n x_i
\, , \,
\int_{a}^{b} x^2 dx
Leveraging these common math functions fluently can drastically reduce equation authoring time and make documents very readable.
Troubleshooting Errors
Even seasoned LaTeX users run into issues with equations on occasions. Some tips for catching errors:
• Check for any missing \$ delimiters around inline math expressions. Equations inside sentences should be wrapped in these.
• Check for any missing { or } brackets around equation components. This can cause cascading errors.
• Use the showkeys LaTeX package to see debugging markers next to all equation tags and labels. This makes missing labels obvious.
• Carefully inspect compiler/build tool logs for line numbers associated with errors. This locates issue areas quickly.
Catching any syntax issues early on saves future headaches when trying to style and reference equations. Having a consistent way to check grammar helps significantly.
Equations are key building blocks in technical LaTeX documents across science and engineering. By following best practices like:
• Properly styling equations
• Formatting equations for readability
• Numbering equations appropriately
• Leveraging common LaTeX math functionality
• Carefully troubleshooting issues
You can author beautiful documents rich in mathematical expressions. Taken together, these recommendations help make the equation authoring process easier and improve comprehension for readers. So
incorporate them into your workflows to take your LaTeX skills to the next level.
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Testing for image symmetries – with application to confocal microscopy
Statistical tests are introduced for checking whether an image function f(x, y) defined on the unit disc D = {(x, y) : x2 + y2 ≤ 1} is invariant under certain symmetry transformations of D, given
that discrete and noisy data are observed. We consider invariance under reflections or under rotations by rational angles, as well as joint invariance under both reflections and rotations.
Furthermore, we propose a test for rotational invariance of f(x, y), i.e., for checking whether f(x, y), after transformation to polar coordinates, only depends on the radius and not on the angle.
These symmetry relations can be naturally expressed as restrictions for the Zernike moments of the image function f(x, y), i.e., the Fourier coefficients with respect to the Zernike orthogonal basis.
Therefore, our test statistics are based on checking whether the estimated Zernike coefficients approximately satisfy those restrictions. This is carried out by forming the L2 distance between the
image function and its transformed version obtained by some symmetry transformation. We derive the asymptotic distribution of the test statistics under both the hypothesis of symmetry as well as
under fixed alternatives. Furthermore, we investigate the quality of the asymptotic approximations via simulation studies. The usefulness our theory is verified by examining an important problem in
confocal microscopy, i.e., we investigate possible imprecise alignments in the optical path of the microscope. For optical systems with rotational symmetry, the theoretical point-spread-function
(PSF) is reflection symmetric with respect to two orthogonal axes, and rotationally invariant if the detector plane matches the optical plane of the microscope. We use our tests to investigate
whether the required symmetries can indeed be detected in the empirical PSF.
Image symmetry, Nanoscale bioimaging, Nonparametric estimation, Point-spread-function, Symmetry detection, Zernike moment
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What is 16 Celsius to Fahrenheit? - ConvertTemperatureintoCelsius.info
16 degrees Celsius is equal to 60.8 degrees Fahrenheit.
Celsius and Fahrenheit are two common temperature scales used around the world, with Celsius being the standard in most countries and Fahrenheit used primarily in the United States. The two scales
use different fixed points: the freezing point of water is 0 degrees Celsius and 32 degrees Fahrenheit, while the boiling point of water is 100 degrees Celsius and 212 degrees Fahrenheit.
To convert Celsius to Fahrenheit, you can use the following formula:
°F = (°C × 9/5) + 32
So, to convert 16 degrees Celsius to Fahrenheit, you would use the formula:
°F = (16 × 9/5) + 32
°F = 28.8 + 32
°F = 60.8
Therefore, 16 degrees Celsius is equal to 60.8 degrees Fahrenheit.
Understanding the relationship between Celsius and Fahrenheit can be helpful in various situations, such as when traveling to a country that uses a different temperature scale or when working on
scientific calculations that involve temperature conversions.
It’s important to note that while the Celsius scale is based on the freezing and boiling points of water, the Fahrenheit scale was originally defined by the coldest temperature reached in winter in a
specific location and the average human body temperature. This is why the two scales have different numerical values for the same temperature.
In conclusion, 16 degrees Celsius is equivalent to 60.8 degrees Fahrenheit. Understanding the relationship between these two temperature scales can be beneficial in a variety of contexts and can help
individuals navigate different measurement systems.
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Interesting Possibilities (response To Harv) - an Astronomy Net God & Science Forum Message
Hi Harv,
All along this current dialog I have expressed my optimism that we were closing the gap of misunderstanding between us. With your latest post, however, you have clearly shown me that my optimism was
wholly unwarranted. It is as if we don't understand what each other is saying in the slightest.
***P.:Even though Dick has not formally studied the foundations of mathematics, he has nonetheless realized this same distinction, and he recognizes the profound impact it can have on the
possibilities to answer a question posed by Einstein. I can't quote it exactly, but it is the question of what, if anything, can we know about our universe as a result of pure thought. Mathematics
has paved the way showing what kind of logical structures can be developed by pure thought. Dick has gone beyond that to show that this strictly logical structure implies some necessary constraints
on any possible communicable universe. This without any appeal whatsoever to any data or information supposedly coming from any real universe. H.:I have a problem with this paragraph. Let me reword
it the way that I hear it:***
If that is the way you heard it, Harv, let me apologize for being so unclear in what I wrote.
***"A game invented by primitive lifeforms called Monopoly has paved the way showing what kind of pictures in Pictionary (another game) can be drawn by these primitive lifeform's from their pure
primitive thoughts. Dick has gone beyond that to show that the pictures in Pictionary implies necessary constraints on any words that come from the mouth of these primitive lifeforms. This without
any appeal whatsoever to any data or information supposedly coming from any real universe experienced by the primitive lifeforms."***
Your comparison of what I wrote with your description of Monopoly and Pictionary is so far off base that it would be an utter waste of time to comment on it. Instead, let me try to rewrite my
paragraph in such a way that it won't come across to you distorted beyond recognition or comprehension.
You didn't comment on the first half of my paragraph in which I merely posed the question of, "What, if anything, can we know about our universe as a result of pure thought?" I hope it was clear that
that question is the beginning idea of the paragraph, and it is that question for which the remainder of the paragraph attempts to sketch out an answer.
Let me proceed more slowly and carefully this time. In fact, I'll number the steps in order to make it easier for us to identify exactly where you are unable to follow my explanation.
1. We wish to examine, and attempt to answer, the question "What, if anything, can we know about our universe as a result of pure thought?"
2. To even consider this question, we must first have a basic understanding of the English language and a working knowledge of each of the separate words in the question which is, at some level,
consistent with the working knowledge of all the people included in the term 'we'.
3. Beyond that basic level, we must be clear in our mutual understanding of the connotation, in the context of the question, of each of the key terms in the question. To that end, each of these terms
will be defined in turn in a sequence which progressively builds on terms earlier defined.
4. Thought is defined to be the type of activity involving the manipulation of ideas as experienced by the author of this post, hereinafter referred to as "I" or "me" depending on the grammatical
5. I assume that there are other thinkers who experience thought much as I do, and these others are identified as separate and distinct live people, one of whom is identified as Harv, hereinafter
referred to as "you". The people who happen to read this post will comprise the set of people hereinafter referred to as "we".
6. To say "we know something" means that we are able to describe or explain that something in English language sentences that are sufficiently clear in their meaning that we are all able to
understand the description or explanation and have reasonable confidence that we all understand in the same way.
7. To say "we can know something" means that it is possible in principle for us to come to know it in the future even though we may not know it now.
8. To ask "What can we know about something?" is to pose a question which is to be answered by English sentences describing or explaining facts, features, or constraints, which when understood,
increase the totality of what we know about the something.
9. To ask "What if anything can we know about something?" is to admit the possibility that it may not be possible to know anything about it even in principle.
10. To say "pure thought", we mean thought in which, among the ideas being manipulated, no knowledge of our universe is present or admitted (meaning "allowed in" -- not "acknowledged".) In other
words, pure thought means the manipulation of ideas that have no tangible relationship to what was referred to in the original question as "our universe". (Yes, I'm aware that I haven't defined 'our
universe' yet. Be patient.)
11. Even though we each may have a belief that we know something about our universe, the definition of 'pure thought' in 10 forces us to accept the term 'our universe' as undefined. It must be
considered in the same way as we consider an unknown variable, say x, in an algebraic expression.
12. To help clarify this, consider the question, "What can we know about a green apple without examining it?" Can we know that it is green? That it is an apple? Well, yes. The question itself gives
that information to us.
13. So, if we ask "What can we know about our universe?", and then proceed to define 'our universe' in any way whatsoever, then we can say that we know our universe to be whatever we defined it to
be. For example, if we defined our universe to be 'whatever exists', then at the outset, we know that our universe exists. In this case, I think you would agree, it would be meaningless to assert
that we have thereby gained some new knowledge of our universe.
14. This fact, that our universe exists, cannot be allowed into the set of ideas we are to manipulate as defined by 'pure thought' . So even though we may be able to define 'our universe' in other
contexts, it must be left undefined for the purposes of this discussion.
15. Finally, to say "to know as a result of pure thought" means that the ideas are manipulated strictly according to the formal rules of logic.
16. Having made these definitions, we should now have a mutually clear understanding of the question, "What, if anything, can we know about our universe as a result of pure thought?"
17. It means that we are asking what, if anything, may be discovered about a completely unknown thing or entity that we refer to with the symbol "our universe" by only manipulating ideas according to
the rules of logic and by making no assumptions about, or appeal to anything about, "our universe" whatsoever.
18. At this point, we need to point out that, in spite of any history of discovery, development, or codification of the rules of logic that might have taken place, the application of them to ideas
that are devoid of any tangible relationship to anything in "our universe" does not violate our definition of 'pure thought'.
19. I am now ready to explain what I meant when I wrote "Mathematics has paved the way showing what kind of logical structures can be developed by pure thought."
20. Mathematics, as it was developed on Earth, has left an "epistemological trail", as you have pointed out. The concepts incorporated in the body of mathematics came from concepts held by people
about "our universe". Some of these seem to be warranted and others have been shown to be unwarranted.
21. In the last century, however, mathematicians have systematically removed from the body of mathematics all tangible relationships between mathematical concepts and any concepts having anything to
do with anything thought to be part of "our universe".
22. This fact is largely unknown and completely uninteresting to most people, including scientists who use the most sophisticated mathematical results in their work. Nonetheless, it is a fact.
23. At this point in history, the body of mathematics is a logical structure which can be developed by pure thought alone. Thus I used the road-construction metaphor that "Mathematics has paved the
way" toward the answer to the question of "What, if anything, can we know about our universe as a result of pure thought?" by providing the body of mathematics as a basis on which we may build
further logical structures strictly using pure thought.
24. I will now explain what I meant when I wrote, "Dick has gone beyond that to show that this strictly logical structure implies some necessary constraints on any possible communicable universe."
25. By "this strictly logical structure", I am referring to the body of mathematics described in 21 above.
26. When I said, "Dick has gone beyond that", I meant that he started with the body of mathematics, assumed absolutely nothing about any putative "universe", defined some arbitrary (albeit
controversial) terms, and deduced what I claim to be a theorem which states a specific set of constraints which apply to arbitrary subsets of arbitrary sets of numbers.
27. The theorem, which naturally falls within the discipline of Statistical Analysis under Probability Theory, describes necessary constraints on any functions which describe the probability of
sampling a particular subset of a given set of numbers (BTW the "given set of numbers" is typically referred to as "the Universe" in conventional Probability Theory.)
28. For the record, Dick does not agree with my classification of his result as a theorem. A long-standing debate on this issue is still under way. Without dragging that debate into this thread, I
would still appreciate Dick's correction to my description of the constraints in 27 above if it is in error.
29. So, with the exception of a possible mis-statement of the constraints, I hope I have explained that, "Dick has gone beyond [conventional mathematics] to show that this strictly logical structure
implies some necessary constraints on any possible [set of numbers]."
30. Now, referring back to number 6, we see that in order to know anything, it is necessary and sufficient that we be able to produce English language statements containing understandable
descriptions or explanations.
31. The purpose of these statements, and indeed of language itself, is to facilitate communication among people.
32. Thus we can say that in order to communicate anything we must encode the description or explanation in language statements. (At least that is the present state of human affairs. It may be
possible in the future, or maybe even now for some people, that telepathic or other non-language communication will be possible. But at the present time, we may restrict our definition of
'communication' to that of the transfer of ideas using language.)
33. Let us now consider the undefined variable, "our universe". It is unfortunate that the term includes the word 'our' rather than the word 'any' or 'some', but this is only a trivial arbitrary
choice of a symbolic tag. Since the term "our universe" is completely undefined, it may just as well represent any so-called universe which we may not choose to call "our own".
34. Returning to our basic question, "What, if anything, can we know about our universe as a result of pure thought?", it is clear by definition (6) that anything that would be possible to know would
also be communicable.
35. Therefore, anything we can in principle know about "our universe" must be communicable.
36. So we may partially answer our basic question at this point by saying that the only things we can know about our universe, as the result of pure thought alone, would be aspects or features of it
that are communicable.
37. To the extent that "our universe" or indeed "any universe" has communicable features or aspects, it would be reasonable to call them 'communicable universes'. Some may be more communicable than
38. From 30 and 32, we may infer that descriptions or explanations of the communicable aspects or features of any communicable universe may be encoded in language.
39. It is well known that all language descriptions and explanations can be encoded in sets of numbers, just as this post was so encoded on its way from my keyboard to your screen.
40. So combining 39 and 29 we conclude that, "Dick has gone beyond [conventional mathematics] to show that this strictly logical structure implies some necessary constraints on any possible
communicable universe."
41. Which brings us to the final sentence of my mis-understood paragraph: "This without any appeal whatsoever to any data or information supposedly coming from any real universe."
42. If you can forgive and overlook my grammatical error, I think that by reviewing 1 through 41 above, you can convince yourself that no appeal was made in this argument to any data or information
from any real universe whatsoever.
***I have a problem with this paragraph.***
I hope this has cleared it up for you, Harv.
***As you see, logic and math are too different games (e.g., Pictionary and Monopoly). Pictionary doesn't restrict Monopoly (or vice versa), but they may have rules in common. For example, in
Pictionary and Monopoly there should be multiple players and they should take turns, etc.***
Logic and math are indeed two different games, but I can find almost no part of your analogy that applies. Math has no rules -- the rules are all supplied by logic.
Math consists solely of a body of definitions, axioms, and theorems which have been shown to be consistent according to the rules of logic. Except for some possible inspiration from some unknown
source, which if it happens goes unacknowledged, the entire body of mathematics, as represented by the formal mathematical literature, is an invention of human minds.
Logic, on the other hand, was not invented by human minds. Instead, logic seems to come as part of the "original equipment" of a human mind. The actual origin and explanation for the rules of logic,
as far as I can tell, are a complete mystery.
Except for the fact that both the rules of logic, and the propositions of mathematics, can be expressed in human-devised symbolism, there is almost no similarity between mathematics and logic.
As for "restriction", logic severely "restricts" mathematics, but not at all vice-versa. Propositions can only be added to the body of a particular mathematical structure if they can be inferred from
the previous body using only the rules of logic. Thus logic imposes a severe restriction on what propositions may make up a body of mathematics. There is no restriction of the sort whatsoever imposed
on logic by the body of mathematics.
I will concede, however, that both mathematics and logic may be considered by multiple people.
I am somewhat dismayed by our divergence, Harv, and I hesitate to express any optimism that you will understand this post. But I have done my best to make it clear, and if I have failed, I apologize.
I don't know what else to do.
Warm regards,
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List of registered participants
• Dmitri Alekseevsky (Moscow , Russia), Hopf Bundle, Listing's Law and Saccades;
• David Blair (East Lansing, USA)
• Victor Buchstaber (Moscow, Russia)
• Bang-Yen Chen (Michigan State University, East Lansing, USA), Great antipodal sets and recent applications;
• Ryszard Deszcz (Wroclaw, Poland), Curvature properties of pseudosymmetry type of some 2-quasi Einstein manifolds;
• Branko Dragovich (Belgrade, Serbia), Nonlocal de Sitter Gravity $\sqrt{dS}$ and the Dark Side of the Universe;
• Graham Hall (Aberdeen, UK)
• Stefan Ivanov (Sofia, Bulgaria), The Riemannian curvature identities on Almost Complex Calabi-Yau with torsion 6-manifold and generalized Ricci solitons;
• Louis Kauffman (University of Ilinois at Chicago, Chicago, USA), Knot Dynamics and Vortex Reconnection;
• Miodrag Mateljević (SANU, Belgrade, Serbia), Lorentz Transformation and time dilatation;
• Josef Mikeš (Palacky University Olomouc, Olomouc, Czech Republic), Geodesic mappings and their generalizations;
• Dmitry Millionschikov (Moscow , Russia), Minimal models of nilmanifold and complex structures;
• Svetislav Minčić (Niš, Serbia)
• Andrey Mironov (Novosibirsk, Russia)
• Alexandr Mishchenko (Lomonosov Moscow State University, Moscow, Russia), Maslov index and infinitesimal Lagrangian manifolds;
• Yuri Nikolayevsky ( Melbourne, Australia), Killing tensors on symmetric spaces;
• Masafumi Okumura (Saitama, Japan)
• Leopold Verstraelen (Leuven, Belgium)
• Andrei Vesnin (Sobolev Institute of Mathematics and Tomsk State University, Novosibirsk, Russia), On Yamada polynomial of spatial graphs and Jones polynomial of related links;
• Iskander Taimanov (Novosibirsk, Russia)
• Alexey Tuzhilin (Moscow, Russia), Gromov-Hausdorff Distance and Geometric Optimization Problems;
• Luc Vrancken (Valenciennes, France)
• Vitaly Balashchenko (Faculty of Mathematics and Mechanics Minsk, Belarus), Left-invariant $f$-structures on low-dimensional solvable Lie groups;
• Qing Li (Shijiazhuang traditional Chinese hospital, Shijiazhuang, China), A continuum space is the infinitely great;
• Miroslava Antić (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia)
• Nadezda Guseva (Moscow Pedagogical State University, Moscow, Russia), Geodesic mappings and their generalizations;
• Nenad Lazarov (Vinča Institute of Nuclear Sciences - National Institute of the Republic of Serbia, University of Belgrade, Belgrade, Serbia), The velocity of one dimension cosmos;
• Ariana Pitea (National University of Science and Technology Politehnica Bucharest, Bucharest, Romania), Numerical algorithms on Hadamard manifolds;
• Vladimir Rovenski (Department of Mathematics, University of Haifa, Haifa, Israel), Weak nearly Sasakian and weak nearly cosymplectic manifolds;
• Pooja Rani (IIT BHU, Varanasi, India), The Brylinski beta function of a double layer;
• Milica Cvetković (The Academy of Applied Technical and Preschool Studies in Niš, Niš, Serbia), The Willmore energy variations and energy efficient architecture;
• Nenad Vesić (Mathematical Institute of Serbian Academy of Sciences and Arts, Niš, Serbia), New invariants for conformal mappings of Riemannian spaces;
• Milan Zlatanović (Niš, Serbia)
• Mića Stanković (Niš, Serbia), On some generalizations of Kahlerian spaces;
• Vladica Andrejić (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia)
• Vsevolod Sakbaev (FSI Federal Research Centre Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences, Moscow, Russia), Singularity formation for solution of NSE and its
• Lyudmila Efremova (Nizhny Novgorod State University, Nizhny Novgorod, Russia), On maps obtained by small perturbations of skew products;
• Vladislava Milenković (Faculty of Technology, University of Niš, Leskovac, Serbia), New types of mappings of generalized Riemannian spaces;
• Andronick Arutyunov (Moscow Institute of Physcics and Technolohies, Moscow, Russia), On a coarse geometric approach to operator algebras;
• Igor Uljarević (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia), Size in contact geometry;
• Paweł Walczak (Uniwersity of Lodz, Faculty of Mathematics and Computer Science, Lodz, Poland), Einstein Foliations;
• Muhittin Evren Aydin (Firat University, Elazig, Turkey), Pointwise rectifying submanifolds and anti-torqued vector fields;
• Srđan Vukmirović (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia)
• Zoran Rakić (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia)
• Miloš Petrović (University of Niš, Faculty of Agriculture in Kruševac, Kruševac, Serbia), Composition of conformal and projective mappings of generalized Riemannian spaces in Eisenhart's sense
preserving certain tensors;
• Ljubica Velimirović (Faculty of Sciences and Mathematics, University of Niš, Niš, Serbia)
• Ana Velimirović (Metropolitan, Belgrade, Serbia)
• Marija Najdanović (University of Priština in Kosovska Mitrovica, Faculty of Sciences and Mathematics, Kosovska Mitrovica, Serbia), On deformations preserving dual arc length in dual 3-space;
• Emilija Nešović (Faculty of Science, University of Kragujevac, Kragujevac, Serbia)
• Mancho Manev (Paisii Hilendarski University of Plovdiv & Medical University, Plovdiv, Bulgaria)
• Ivan Dimitrijević (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia), The Schwarzschild-de Sitter Metric of Nonlocal $\sqrt{dS}$ Gravity;
• Miroslav Maksimović (University of Priština in Kosovska Mitrovica, Faculty of Sciences and Mathematics, Kosovska Mitrovica, Serbia), Some curvature properties of quarter-symmetric metric
• Đorđe Kocić (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia), The shape operator of real hypersurfaces in S^6(1);
• Jovana Ormanović (Faculty of Mathematics, University of Belgrade, Belgrade, Serbia)
• İrem Küpeli Erken (Bursa Technical University, Faculty of Engineering and Natural Scıence, Bursa, Turkey), Some Remarks on Quasi-para-Sasakian Manifolds;
• Mustafa Özkan (Bursa Technical University, Faculty of Engineering and Natural Scıence, Bursa, Turkey), Fischer-Marsden Equation on Paracontact Geometry;
• Mansi Mishra (Indian Institute of Technology, B.H.U., Varanasi, India), Weyl transform of a measure;
• Siraj Udin (Department of Mathematics, Jamia Millia Islamia, New Delhi, India), Sequential and iterated warped products;
• Milica Stojanović (University of Belgrade, Faculty of Organizational Sciences, Belgrade, Serbia), 3-Triangulation of polyhedra and their connection graphs;
• Tijana Šukilović (Faculty of Mathematics, University of Belgrade, Belgrade, Serbia), Polynomial integrability of sub-Riemannian geodesic flows on compact Lie groups;
• Katarina Lukić (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia), The Jacobi-orthogonality and Osserman tensors;
• Murali Vemuri(Indian Institute of Technology, Varanasi, India), A generalization of a result of Minakshisundaram and Pleijel;
• Jelena Stojanov (University of Novi Sad, technical faculty "Mihajlo Pupin" Zrenjanin, Serbia), Geometrical eigenproblem of various types higher order tensors in Riemannian space;
• Andrijana Dekić (Mathematical Institute of Serbian Academy of Sciences and Arts, Belgrade, Serbia), Almost Kahler structures on complex hyperbolic space;
• Milan Pavlović (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia), Integrability of the sub-Riemannian geodesic flow of the left-invariant metric on the Heisenberg group;
• Velichka Milousheva (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria), Timelike Surfaces with Parallel Normalized Mean Curvature Vector Field and their
Canonical Parameters;
• Miloš Đorić (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia), Yamabe CR solitons on Kahler manifolds;
• Anton Vikhrov (Lomonosov Moscow State University, Moscow, Russia), Geodesics in Gromov–Hausdorff class;
• Ekaterina Zhikhareva(Lomonosov Moscow State University, Moscow, Russia), Semisimple algebraic Nijenhuis operators on small dimentional Lie algebras;
• Daniil Ilyukhin(Lomonosov Moscow State University, Moscow, Russia), The Fermat-Torricelli problem in normed spaces;
• Craig van Coevering (Boğazici University, İstanbul, Türkiye), Extremal K\"{a}hler metrics and the moment map;
• Handan Yıldırım (İstanbul University, Science Faculty, İstanbul, Türkiye), Legendrian dual surfaces, lying in the $3$-dimensional de Sitter space, of a spacelike curve in the $3$-dimensional
• Nikolay Antonov (Sofia, Bulgaria)
• Victor Aguilar (Casa Grande, USA), Geometry Notation;
• Teresa Arias-Marco (Universidad de Extremadura, Badajoz, Spain), Asymptotics of the mixed-Steklov spectrum;
• Dhriti Sundar Patra (Department of Mathematics, IIT Hyderabad Kandi, Sangareddy), Characterizations of Ricci-Bourguignon almost solitons;
• Dragan Đokić (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia)
• Alexander Petkov (Sofia University, Faculty of Mathematics and Informatics, Sofia, Bulgaria), Li-Yau sub-gradient estimates and Perelman-type entropy formulas for the heat equation in
quaternionic contact geometry;
• HAZAL YÜRÜK (ISTANBUL TECHNICAL UNIVERSITY, İstanbul, Türkiye)
• Ljiljana Radović (Facuilty of Mechanical Engineering, University of Niš, Niš, Serbia), The geometrical personalization of human organs 3D models by using the Characteristic Product Features
• Zdenek Dusek (Institute of Technology and Business in Ceske Budejovice, Ceske Budejovice, Czech Republic), Finsler geodesic orbit metrics;
• Mirjana Đorić (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia)
• Theodore Popelensky (Moscow State University, Moscow, Russia)
• Gleb Nosovskiy (Moscow State University, Moscow, Russia), Implementation of Postprocessing Procedure of a Rapid Algorithm of Geometric Coding of Digital Images Using CUDA Architecture;
• Yurii Nikonorov (Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz, Russia), On geodesic orbit nilmanifolds;
• Cornelia-Livia Bejan ("Gh. Asachi" Technical University of Iasi, Romania), Almost complex manifolds with Norden metrics;
• Ilja Gogić (University of Zagreb, Department of Mathematics, Zagreb, Croatia), Applications of algebraic topology to operator algebras: Homogeneous C*-algebras;
• Miroslava Petrović-Torgašev (State University of Novi Pazar, Department of Sciences and Mathematics, Novi Pazar, Serbia)
• Miloš Arsenović (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia), Convergence theorems for sequences of operator valued functions;
• Miguel Brozos Vasquez (Ferrol, Spain)
• Anastasiia Shubert (Moscow State University, Moscow, Russia), Geometric Properties of the Inertia Tensor of a Solid Body;
• Dušan Simjanović (Metropolitan University, Niš, Serbia), Invariants for geometric mappings;
• Eduardo Garcia-Rio (University of Santiago de Compostela, Santiago de Compostela, Spain), Critical metrics for quadratic curvature functionals;
• Khalifa Al Shaqsi (University of Nizwa, Oman), Neutrosophic Touchard Polynomials for Subclass of Analytic Functions;
• Aleksandar Lipkovski (University of Belgrade, Faculty of Mathematics, Belgrade, Serbia), Calculation of quiver representations of finite ring digraphs;
• Jelena Matović (Academy of Technical and Art Applied Studies - School of Electrical and Computer Engineering, Belgrade, Serbia), Calculation of quiver representations of finite ring digraphs;
• FAN Huijun (School of Mathematical Sciences, Peking University), Spectrum problem of Schrodinger operators related to Landau-Ginzburg models;
• Lidija Rančić (University of Niš, Faculty of Electronic Engineering, Niš, Serbia), Basins of attractions of a new iterative method for finding simple zeros;
• Svetozar Rančić (University of Niš, Faculty of Sciences and Mathematics, Niš, Serbia), Basins of attractions of a new iterative method for finding simple zeros;
• Anton Savin (RUDN University, Moscow, Russia), On the Index Problem On Manifolds with Group Actions. Contributions of Fixed Points;
• Ivan Limonchenko (Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia), Persistent homology theory and toric topology;
• Nikola Velimirović (State University of Novi Pazar, Serbia)
• Kaori Yamaguchi (Ritsumeikan University, Kusatsu, Shiga, Japan), On statistics which are almost sufficient from the viewpoint of the Fisher metrics;
• Đorđe Baralić (Mathematical Institute SANU, Belgrade, Serbia), The mod $p$ Buchstaber invariant;
• Elena Makhrova (Lobachevsky State University of Nizhni Novgorod, Nizhny Novgorod, Russia), The structure of dendrites and dynamics of continuous maps on them;
• Lyudmila Efremova (Nizhny Novgorod State University, Nizhny Novgorod, Russia), On maps obtained by small perturbations of skew products;
• Miguel Brozos-Vázquez (CITMAga-Universidade da Coruña, Spain), Remarks on weighted Einstein manifolds;
• Sergei Agapov (Sobolev Institute of Mathematics SB RAS, Novosibirsk Russia), Integrable geodesic flows with rational first integrals;
• Denis Ilyutko (Lomonosov Moscow State University, Moscow, Russia), Weight systems of framed chord diagrams: the Circuit-Nullity formula and Lie algebras;
• Elena Kudryavtseva (Lomonosov Moscow State University, Moscow, Russia), Generic bifurcations of tori in 3D integrable Hamiltonian systems;
• Sergey Stepanov (Financial University under the Government of the Russian Federation, Moscow, Russia), New vanishing theorem of conformal Killing forms;
• Filip Živanović (Simons Center for Geometry and Physics, NY, United States), (Non) existence of Lagrangians in hyperkähler manifolds;
• Gabriel-Eduard Vîlcu (University of Bucharest & National University of Science and Technology Politehnica Bucharest & "Gheorghe Mihoc - Caius Iacob" Institute of Mathematical Statistics and
Applied Mathematics of the Romanian Academy, Romania), On the stability of the identity map of compact space forms;
• Crina-Daniela Neacsu (National University of Science and Technology Politehnica Bucharest, Romania), On the stability of the identity map of compact space forms;
• Vladimir Dragović (University of Texas at Dallas, MISANU), How to connect two most beautiful geometric theorems?;
• Admir Huseini(University of St. Cyrill and Methodius, Skopje, North Macedonia), Shape morphing using interpolation between implicit functions;
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There are 14 legs in this group. There are 6 heads in this group.
There are 14 legs in this group.
There are 6 heads in this group.
There are 12 ears in this group.
There are 50 fingers in this group.
There is 1 tail in this group.
Who could be in this group?
How would these figures change if the dog ran away? :)
thats 4 girbils, 1 duck, and a man with 10 arms
the answer is 5 people and a dog or a cat or anything else with four legs a tail and two ears! very simple really
25 gerbils and one with a tail and the rest have none
1 answer
(click or scroll down)
To determine who could be in this group, we can analyze the given information about legs, heads, ears, fingers, and tails. Let's break it down:
- There are 14 legs in this group. Since each person has 2 legs, the number of people is 14 divided by 2, which equals 7 people.
- There are 6 heads in this group. This means there are 6 individuals in total, as every person has one head. We already know there are 7 people, so this number does not match.
- There are 12 ears in this group. Each person has 2 ears, so the number of people should be 12 divided by 2, which equals 6 people. Again, this number does not match the previous calculation.
From these calculations, we see that the given information is inconsistent. There is no way to determine exactly who could be in this group based on the provided information.
Now, let's consider what would happen if the dog ran away:
- If the dog ran away, we would lose 4 legs (since dogs have 4 legs).
- This means there would now be 10 legs in the group (14 legs - 4 legs).
- Since each person has 2 legs, the number of people would be 10 divided by 2, which equals 5 people.
- As for the other animals mentioned (4 gerbils, 1 duck, and a man with 10 arms), they are not accounted for in the given information, so it is unclear how they would change.
In conclusion, the figures would change to 5 people if the dog ran away, but the impact on the other mentioned animals cannot be determined based on the given information.
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Copyright (c) The FFI Task Force 2000-2002
License see libraries/base/LICENSE
Maintainer ffi@haskell.org
Stability internal
Portability non-portable (GHC Extensions)
Safe Haskell None
Language Haskell2010
data Ptr a Source #
A value of type Ptr a represents a pointer to an object, or an array of objects, which may be marshalled to or from Haskell values of type a.
The type a will often be an instance of class Storable which provides the marshalling operations. However this is not essential, and you can provide your own operations to access the pointer. For
example you might write small foreign functions to get or set the fields of a C struct.
Generic1 (URec (Ptr ()) :: k -> Type) Source #
Defined in GHC.Internal.Generics
type Rep1 (URec (Ptr ()) :: k -> Type) Since: base-4.9.0.0
Defined in GHC.Internal.Generics
Eq1 (UAddr :: Type -> Type) Source # Since: base-4.21.0.0
Defined in Data.Functor.Classes
Ord1 (UAddr :: Type -> Type) Source # Since: base-4.21.0.0
Defined in Data.Functor.Classes
Show1 (UAddr :: Type -> Type) Source # Since: base-4.21.0.0
Defined in Data.Functor.Classes
Data a => Data (Ptr a) Source # Since: base-4.8.0.0
Defined in GHC.Internal.Data.Data
Foldable (UAddr :: Type -> Type) Source # Since: base-4.9.0.0
Defined in GHC.Internal.Data.Foldable
Traversable (UAddr :: Type -> Type) Source # Since: base-4.9.0.0
Defined in GHC.Internal.Data.Traversable
Storable (Ptr a) Source # Since: base-2.1
Defined in GHC.Internal.Foreign.Storable
Show (Ptr a) Source # Since: base-2.1
Defined in GHC.Internal.Ptr
Eq (Ptr a) Source # Since: base-2.1
Defined in GHC.Internal.Ptr
Ord (Ptr a) Source # Since: base-2.1
Defined in GHC.Internal.Ptr
Functor (URec (Ptr ()) :: Type -> Type) Source # Since: base-4.9.0.0
Defined in GHC.Internal.Generics
Show (UAddr p) Source # Since: base-4.21.0.0
Defined in GHC.Internal.Generics
Generic (URec (Ptr ()) p) Source #
Defined in GHC.Internal.Generics
type Rep (URec (Ptr ()) p) Since: base-4.9.0.0
Defined in GHC.Internal.Generics
Eq (URec (Ptr ()) p) Source # Since: base-4.9.0.0
Defined in GHC.Internal.Generics
Ord (URec (Ptr ()) p) Source # Since: base-4.9.0.0
Defined in GHC.Internal.Generics
Used for marking occurrences of Addr#
data URec (Ptr ()) (p :: k) Source #
Since: base-4.9.0.0
Defined in GHC.Internal.Generics
type Rep1 (URec (Ptr ()) :: k -> Type) Source # Since: base-4.9.0.0
Defined in GHC.Internal.Generics
type Rep (URec (Ptr ()) p) Source # Since: base-4.9.0.0
Defined in GHC.Internal.Generics
data FunPtr a Source #
A value of type FunPtr a is a pointer to a function callable from foreign code. The type a will normally be a foreign type, a function type with zero or more arguments where
• the argument types are marshallable foreign types, i.e. Char, Int, Double, Float, Bool, Int8, Int16, Int32, Int64, Word8, Word16, Word32, Word64, Ptr a, FunPtr a, StablePtr a or a renaming of any
of these using newtype.
• the return type is either a marshallable foreign type or has the form IO t where t is a marshallable foreign type or ().
A value of type FunPtr a may be a pointer to a foreign function, either returned by another foreign function or imported with a a static address import like
foreign import ccall "stdlib.h &free"
p_free :: FunPtr (Ptr a -> IO ())
or a pointer to a Haskell function created using a wrapper stub declared to produce a FunPtr of the correct type. For example:
type Compare = Int -> Int -> Bool
foreign import ccall "wrapper"
mkCompare :: Compare -> IO (FunPtr Compare)
Calls to wrapper stubs like mkCompare allocate storage, which should be released with freeHaskellFunPtr when no longer required.
To convert FunPtr values to corresponding Haskell functions, one can define a dynamic stub for the specific foreign type, e.g.
type IntFunction = CInt -> IO ()
foreign import ccall "dynamic"
mkFun :: FunPtr IntFunction -> IntFunction
Storable (FunPtr a) Source # Since: base-2.1
Defined in GHC.Internal.Foreign.Storable
Show (FunPtr a) Source # Since: base-2.1
Defined in GHC.Internal.Ptr
Eq (FunPtr a) Source #
Defined in GHC.Internal.Ptr
Ord (FunPtr a) Source #
Defined in GHC.Internal.Ptr
nullPtr :: Ptr a Source #
The constant nullPtr contains a distinguished value of Ptr that is not associated with a valid memory location.
alignPtr :: Ptr a -> Int -> Ptr a Source #
Given an arbitrary address and an alignment constraint, alignPtr yields the next higher address that fulfills the alignment constraint. An alignment constraint x is fulfilled by any address divisible
by x. This operation is idempotent.
minusPtr :: Ptr a -> Ptr b -> Int Source #
Computes the offset required to get from the second to the first argument. We have
p2 == p1 `plusPtr` (p2 `minusPtr` p1)
Unsafe functions
castFunPtrToPtr :: FunPtr a -> Ptr b Source #
Casts a FunPtr to a Ptr.
Note: this is valid only on architectures where data and function pointers range over the same set of addresses, and should only be used for bindings to external libraries whose interface already
relies on this assumption.
castPtrToFunPtr :: Ptr a -> FunPtr b Source #
Casts a Ptr to a FunPtr.
Note: this is valid only on architectures where data and function pointers range over the same set of addresses, and should only be used for bindings to external libraries whose interface already
relies on this assumption.
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Addition of Radius-Vectors
Fix a point O in the plane. Point O is called the origin. The directed segment OA from the origin to an arbitrary point A in the plane is known as the A's radius-vector. Radius-vectors of two points
can be added according to the rule of parallelogram. Sometimes we forget to mention the origin and talk of the sum A + B of two points, which usually happens in affine geometry. The reason for this
laxity is that the sums OA + OB and O'A + O'B are related in a very simple manner. What is the relationship?
|Activities| |Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny
The sum A + B is translated in the direction opposite to that of OO', but by the same distance.
The point is to prove that the quadrilateral OS'SO' is a parallelogram. To this end, consider two parallelograms, OASB and O'AS'B. The two share the diagonal AB. As is well known, the parallelogram
is characterized by the property that its diagonals are divided in half by the point of their intersection. It thus follows that the diagonals OS and O'S' of the parallelograms OASB and O'AS'B both
path through the midpoint of AB and are divided in half by that point.
In other words, in the quadrilateral OS'SO', the diagonals are divided into half by their point of intersection. Therefore, the quadrilateral is a parallelogram.
There's a less formal explanation. Since it's all about vectors, it seems intuitively clear that when one of the points A or B is translated by a vector v, the sum A + B undergoes the same
transformation: (A + v) + B = v + (A + B). If both A and B are shifted by v, the sum is translated twice as far, by 2v. However, when three points A, B and O are translated by v, the sum is obviously
translated in the same manner, i.e. only by v. Comparing the last two cases we concluded that shifting the origin has a "detrimental" effect on the translation of the sum: instead of moving by 2v,
the sum only moves v. It also clear that effects of translation of any of the three points is independent of effects caused by translations of the other two points. Therefore, v - 2v is the effect on
the sum A + B of translating the origin by the vector v.
|Activities| |Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny
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Items where Subject is "44 Integral transforms, operational calculus"
Jump to:
Number of items at this level: 12.
Assier, Raphael and Peake, Nigel (2012) On the diffraction of acoustic waves by a quarter-plane. Wave Motion, 49 (1). pp. 64-82.
Assier, Raphael and Peake, Nigel (2012) Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane. IMA Journal of Applied
Mathematics, 77 (5). pp. 605-625.
Fusai, Gianluca and Abrahams, David and Sgarra, Carlo (2006) An exact analytical solution for discrete barrier options. Finance and Stochastics, 10 (1). pp. 1-26. ISSN 1432-1122
Lionheart, William and Sharafutdinov, Vladimir (2008) Reconstruction algorithm for the linearized polarization tomography problem with incomplete data. [MIMS Preprint]
Lionheart, William and Sharafutdinov, Vladimir (2008) Reconstruction algorithm for the polarization tomography problem with incomplete data. [MIMS Preprint]
Lionheart, William and Sharafutdinov, Vladimir (2008) Reconstruction algorithm for the polarization tomography problem with incomplete data. [MIMS Preprint]
Lionheart, William R.B. and Adler, Andy (2021) The SVD of the linearized EIT problem on a disk. In: 21 st International Conference on Biomedical Applications of ELECTRICAL IMPEDANCE TOMOGRAPHY, 14–16
June 2021, National University of Ireland, Galway.
Lionheart, William R.B. and Graham, Oliver Bistatic two dimensional synthetic aperture radar as a tensor tomography problem. In: 3rd IMA Conference on Inverse Problems from Theory to Application, May
3, 2022 - May 5, 2022, International Centre for Mathematical Sciences (ICMS)Edinburgh, , UK. (Unpublished)
Lionheart, William R.B. and Withers, Philip J. (2014) Diffraction tomography of strain. [MIMS Preprint]
Lionheart, William R.B. and Withers, Philip J. (2014) Diffraction tomography of strain. [MIMS Preprint]
Tregidgo, Henry F.J. (2013) Implementation and Analysis of Katsevich Reconstruction for Helical Scan CT. Masters thesis, The University of Manchester.
Watson, Francis Maurice (2016) BETTER IMAGING FOR LANDMINE DETECTION: AN EXPLORATION OF 3D FULL-WAVE INVERSION FOR GROUND-PENETRATING RADAR. Doctoral thesis, The University of Manchester.
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Diamond Problem Solver
This diamond problem solver will let you do various mathematical problems in diamond layout. Want to know how? If yes, then it's time to move by! Stay focused!
What Is Diamond Problem In Mathematics?
In mathematics: “A particular method to resolve mathematical queries by filling up tetra fields is known as the diamond method”
Pattern of Diamond Problem:
In this pattern:
• B = factor 2
• Product = the number got by multiplying both the factors
• Sum= addition result of both the factors
Our best diamond calculator math will instantly resolve for each and every element involved in the diamond figure problem.
How To Do Diamond Problems?
When you try to resolve various diamond queries, you will come across three major sub categories. No doubt our diamond problem solver will allow you to simplify and generate answers without any
hurdle. But it is also crucial to consider the manual calculations. So let’s go through these one by one:
Method # 01:
When Two Factors Are Provided:
Look at the figure above. What do you think you are being played with? Let us tell you. Here you are provided with a couple of factors with product and sum being unknown. So there are a couple
methods to find these terms:
Sum = Factor 1 + Factor 2 Product = Factor 1 * Factor 2
• By using this online diamond calculator that displays immediate outputs of the the inputs being provided
Method # 02:
When product or sum plus one factor is provided:
Here arise two formulas which are enlisted as follows:
When sum and one factor is given:
Sum = Factor 1 + Factor 2 Factor 1 = Sum - Factor 2
When product and one factor is given:
Product = Factor 1 * Factor 2 Factor 1 = Product/Factor 2
Method # 03:
When only product and sum are given:
This is a little bit complicated. But do not worry as the free diamond problem calculator will assist you to resolve such problems in a good way. Anyways, coming to the point, consider the following
scenario: Look at the above figure. You are only provided with a product. To determine the remaining parameters, you need to follow up the key points mentioned below:
• Get going for determining the pairs that will result in the product number
• Also, remember to take in consideration the negative number pairs
• At last, add each pair to check which pair will give you the number equivalent to the sum given
How To Solve Diamond Problems?
Let’s help you people in getting a firm grip over the concept in depth by resolving the example below. What you need to do here is just to understand the calculations by being focused!
Example # 01:
Resolve the following diamond problem:
Now here we have: Possible multiple pairs of 9:
3*3 9*1 -3*-3 -9*-1
Finding the sum of all pairs one by one:
3+3 = 6 9+1 = 10 -3+(-3) = -3-3 = -6 -9+(-1) = -9-1 = -10
Only the first pair gives you the sum that is given, so we have:
Factor 1 = 3
Factor 2 = 3
Is a diamond a square?
No doubt the definition is that if the diamond and square do match a lot, we can say square is actually a diamond as well.
What is diamond problem in C#?
The diamond problem is considered a phenomenon that arises when two classes originate from a single parent class. The fourth class is then obtained by the combinations of the previous two generated.
What is the Product and Sum if the factors are 3 and 4?
Sum = 3+4
Sum = 7
Product = 3*4
Product = 12
You can also cross check related diamond math problems by using this diamond math calculator.
From the source of wikipedia: Diamond problem, Approaches From the source of khan academy: Multiple Inheritance in C++, Illustration
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How to Write a Linear Function - Simple Steps for Beginners
To write a linear function, I typically start by determining the slope and the y-intercept. This form, known as slope-intercept form, is written as $y = mx + b$, where (m) represents the slope or the
rate of change, and (b) signifies the y-intercept, the point where the function crosses the y-axis.
Plotting a linear function on a graph reveals a straight line where every point confirms the consistent rate of change of the function.
A common scenario is finding the equation of a linear function from two points. In this case, I calculate the slope as $m = \frac{{y_2 – y_1}}{{x_2 – x_1}}$.
Subsequently, I use one of the points to solve for (b) in the slope-intercept equation. Through this process, establishing relationships and patterns becomes straightforward, as linear functions
offer a dependable way to understand how one variable responds to changes in another.
Graphing linear equations helps to visualize how the slope of the line represents the rate of change, allowing me to predict and compare values with ease. Curious about translating your scenarios
into linear models?
Stick around, and I’ll demonstrate how simple it can be.
Writing Linear Functions
When I approach linear functions, I focus on the simplicity and utility they offer. These functions graph as straight lines and are foundational in algebra. The general formula of a linear function
is $$ y = mx + b $$, where ‘m’ stands for the slope and ‘b’ is the y-intercept.
To write a linear function, I first identify two important components: the slope and the y-intercept.
• The slope quantifies the steepness of a line and is calculated by the slope formula $$ m = \frac{{\text{rise}}}{{\text{run}}} $$, which is the change in y over the change in x between any two
points $$ x_1, y_1 $$ and $$ x_2, y_2 $$ on the line: $$ m = \frac{{y_2 – y_1}}{{x_2 – x_1}} $$
• The y-intercept is the point where the line crosses the y-axis, at $$ x = 0 $$
Given two points $$ x_1, y_1 $$ and $$ x_2, y_2 $$, I can form the equation of a line:
1. Calculate the slope (m).
2. Use the slope and one coordinate pair in the point-slope form: $$ y – y_1 = m(x – x_1) $$
3. Solve for y to get the slope-intercept form: $$ y = mx + b $$
For graphing linear functions, I plot the y-intercept and use the slope to find another point. Drawing a line through these points gives me the graph of the function.
Form Equation
Point-Slope $$ y – y_1 = m(x – x_1) $$
Slope-Intercept $$ y = mx + b $$
Standard $$ Ax + By = C $$
Special cases include horizontal and vertical lines. A horizontal line has a slope of 0 and is written as $$ y = b $$. A vertical line has an undefined slope and is represented by $$ x = a $$, where
‘a’ is the x-intercept.
In function notation, we might write $$ f(x) = mx + b $$, which emphasizes the output value (f(x)) for a given input (x).
Remember, when writing equations for vertical and horizontal lines, the standard forms are simply $$ x = a $$ (vertical) and $$ y = b $$ (horizontal), where the line crosses the x-axis and y-axis,
Applications and Variations
When I work with linear functions, I find their applications to be quite extensive. They often model real-life situations effectively, especially when it comes to capturing a relationship between two
variables where one is a constant multiple of the other.
In these instances, the equation of the linear function is typically written in slope-intercept form, which looks like $f(x) = mx + b$, where ( m ) represents the slope and ( b ) the y-intercept.
For example, if I want to model population change over time, I may look at the year-over-year increase as a consistent rate, which translates into a straight-line graph.
This also applies when considering the relationship between pressure and depth in fluid mechanics, where pressure increases linearly with depth.
In graphing these functions, it’s essential to understand basic math transformations that make data interpretation more straightforward.
Let’s say I have a maglev train characterized by a linear function, and I want to compare it with another function to understand its intersections or potential parallelism. I can apply
transformations like vertical shifts vertical stretches or compressions to reframe the data without altering the integrity of the model.
Moreover, recognizing when lines are parallel or perpendicular is crucial, especially in fields like engineering or architecture.
I remember setting up two functions and then evaluating their slopes: if the slopes were negative reciprocals of each other, the lines were perpendicular, and if they were equal, the lines were
When adjusting a function to suit a particular domain or range, I might perform vertical shifts or compressions. A vertical shift involves adding or subtracting a constant to the function, written as
( f(x) = mx + b + k ), where ( k ) is the constant.
On the other hand, a vertical stretch or compression involves multiplying the function by a constant, written as ( f(x) = cmx + b ) for some constant ( c ).
Below is a table reflecting these transformations on a basic linear function ( f(x) = x ):
Transformation Function form Graphical change
Vertical Shift ( f(x) = x + k ) Shifts graph up or down by ( k ) units
Stretch ( f(x) = cx ) Stretches graph vertically by a factor of ( c )
Compression $f(x) = \frac{1}{c}x $ Compresses graph vertically by a factor of ( c )
In my experience, understanding and applying these principles of linear functions fosters a more profound comprehension of their real-world applications and empowers me to model complex relationships
with ease.
In mastering the art of writing linear functions, I’ve emphasized the importance of identifying two key components: the slope and the y-intercept.
By grasping the slope, represented as m in the linear function formula $y = mx + b$, you’ve learned how it dictates the steepness or incline of the line. The y-intercept, denoted by b, reveals where
the line crosses the y-axis.
I’ve shown how to use two points on a graph to determine the slope with the formula $m = \frac{y_2 – y_1}{x_2 – x_1}$.
Once the slope is known, and with the y-intercept in hand, constructing the equation of a line is straightforward. This linear equation becomes a powerful tool for representing real-world situations
and solving problems that involve constant rates of change.
Remember, linear functions are the cornerstone of algebra and serve as a foundation for understanding more complex mathematical concepts.
By ensuring that you follow the slope-intercept form, you’ve got a reliable method to write and interpret linear functions with confidence.
Keep practicing, and you’ll find that these concepts become second nature as you continue your journey in mathematics.
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Hypothesis Testing of a Proportion Calculator
100 randomly selected items were tested. It was found that 40 of the items tested positive.
Test the hypothesis that exactly 50% of the items tested positive at α = 0.05
State the null and alternative hypothesis:
: p = 0.5
: p ≠ 0.5
p^ = 0.4
Calculate our test statistic z:
z = 0.4 - 0.5
√0.5(1 - 0.5)/100
z = -2
Checking our table of z-scores for α = 0.05%, we get:
Z = 1.6449
Our rejection region is Z > 1.6449
Since our test statistic of -2 is less than our Z-value of 1.6449, it is not in the rejection region, so we accept H[0]
What is the Answer?
Since our test statistic of -2 is less than our Z-value of 1.6449, it is not in the rejection region, so we accept H[0]
How does the Hypothesis Testing for a proportion Calculator work?
Free Hypothesis Testing for a proportion Calculator - Performs hypothesis testing using a test statistic for a proportion value.
This calculator has 4 inputs.
What 2 formulas are used for the Hypothesis Testing for a proportion Calculator?
p^ = x/n
z = (p^ - p)/sqrt(p(1 - p)/n)
For more math formulas, check out our
Formula Dossier
What 6 concepts are covered in the Hypothesis Testing for a proportion Calculator?
alternative hypothesis
opposite of null hypothesis. One of the proposed proposition in the hypothesis test.
hypothesis testing
statistical test using a statement of a possible explanation for some conclusions
hypothesis testing for a proportion
an act in statistics whereby an analyst tests an assumption regarding a population proportion
null hypothesis
in a statistical test, the hypothesis that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error.
sample size
measures the number of individual samples measured or observations used in a survey or experiment.
test statistic
a number calculated by a statistical test
Add This Calculator To Your Website
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PK model library
The PK library is a library of standard PK models. Instead of writing the structural model yourself, you can select among simple PK models already written for you. When you select the PK library, a
full list of all available models appear, which you can filter by selecting options for administration, distribution and elimination.
Here we provide general guidelines to guide you towards the standard PK model that is the most suited to your dataset.
If you open any of the .txt files in this library, corresponding to the model written in mlxtran-formatted code, you see that the pkmodel macro is always used. This macro enables to define all
standard PK models in a compact manner (except for https://monolixsuite.slp-software.com/monolix/?contextKey=multiple-administration-routes ). It uses the analytical solution of the ODE system to
simulate it, if it is available (which is the case for all models with linear elimination and without transit compartments), and otherwise the ODE system itself. A complete list of the analytical
solutions for standard PK and PKPD models is available in this document: PKPDlibrary.pdf
If none of the standard PK models seems to fit your needs, consider using a model from our other PK libraries (PK double absorption, TMDD). To define your own custom PK model, jump directly to data
and models.
All PK models in the library correspond to a system of equations with the following structure:
• an input rate which depends on the type of administration
• an ODE system based on a number of compartments
• an elimination rate which depends on the type of elimination selected.
The equations express the concentration Cc(t) in the central compartment at a time t after the last drug administration:
• Single dose: at time t after dose D given at time
• Multiples doses: at time t after n doses n given at time
• Steady state: at a time t after dose D given at time D given at interval
Routes of administration
You should know the type of administration to use based on the study that generated the dataset. The route of administration will determine the dynamics of the input rate In(t) for the system
described in Distribution.
To explore the differences in administration routes, we will see how they impact the input rate of a model with 1 compartment and linear elimination, parameterized with the clearance.
The ODE system for this model is:
Intravenous bolus
The intravenous bolus is an injection which quickly raises the concentration of the substance in the blood to an effective level. When bolus is selected, if D is the dose administered at time
The response in the central compartment to a bolus in case of a model with 1 compartment with linear elimination is a decreasing exponential:
This section focuses on intravenous infusion. For subcutaneous infusions, check the dedicated subsection in the oral/extravascular section.
To have the input modeled as an infusion instead of a bolus, you need to have a column tagged as ‘infusion rate’ or ‘infusion duration’ in your dataset in Monolix (the duration will be deduced from
the amount and rate columns if rate is specified, and vice versa). In Simulx, you need to create a treatment element that is an infusion.
The intravenous infusion of duration
The response in the central compartment in case of a model with 1 compartment with linear elimination is increasing as soon as the infusion starts, and decreasing as soon as it stops.
Note that bolus and infusion models are encoded exactly the same way with the pkmodel macro. It is the pkmodel macro that will check if a column tagged as INFUSION RATE or INFUSION DURATION appears
in the dataset, and if this is the case, use the analytical solution of the infusion equations instead of the bolus equations.
Oral/extravascular administration
In the case of an oral or extravascular administration, you have additional options for delay and for the type of absorption.
Absorption: 0 or 1st order?
The type of absorption will determine if the peak in the response is sharp or smooth.
A zero-order absorption is modeled with the same input rate as for an infusion, i.e. with a square pulse input, with 2 differences:
• the duration of the input Tk0 is not known, it is thus part of the parameters to optimize, and it will influence the height and the duration of the response,
• a delay can be added between the dosing time and the start of the pulse, and this delay can also be optimized.
A first-order absorption will instead correspond to a sharper input pulse with a slow exponential decay which depends on the absorption rate. Because of this, the response peak is smoother. This
input rate is the analytical solution of a system with a depot compartment where the dose would be added at time ka:
We compare here zero and 1st order absorption in the case of a lag time, but if there is no lag in the response, the no delay option can be selected and the parameter Tlag will disappear.
Delay: lag time or transit compartments?
In the case of a 1st order absorption, it is possible to select between two types of delay: a lag time or transit compartments. The type of delay selected will change the rising phase of the pulse
input. A lag time corresponds to an instantaneous increase of the input rate at time in a specific page. Transit compartments are sometimes more accurate than a lag time but the models take longer to
simulate because there is no analytical solution available.
Instead of adding the dose to a depot compartment which would be also the absorption compartment at ktr between the compartments are added between the depot and the absorption compartment.
The input rate is the solution of the following system:
This absorption model gives more flexibility to fit the absorption phase with the 3 following parameters: transit rate ktr, the absorption rate ka and the mean transit time
Subcutaneous infusion
To have the input modeled as a subcutaneous infusion, you need to
• select a model with an oral/extravascular administration
• have a column tagged as ‘infusion rate’ or ‘infusion duration’ in your dataset in Monolix (the duration will be deduced from the amount and rate columns if rate is specified, and vice versa), or
create a treatment element that is an infusion in Simulx.
Only 1st-order absorption is available in this case, and the input rate in case of Tlag (or no delay, i.e. Tlag = 0) is the solution of the following subsystem, combining the equations described
above for infusion and for first-order absorption:
Multiple administration routes
It is possible to use a combination of an oral or extravascular administration, and a bolus or an infusion. For this, a column should be tagged as administration ID in the dataset with 1 for all
doses administered intravenously and 2 for orally or extravascularly administered doses. In this case, the input rate is the sum of:
• an infusion rate if columns tagged as INFUSION RATE or INFUSION DURATION appear in the data, or a bolus rate otherwise,
• a rate for oral administration defined above.
In practice, most of the time the two administrations are not simultaneous and the rate takes in turn the value of a rate for an infusion/bolus and for an oral administration, according to the dosing
times in the data set.
For each of the combined models, there is a possibility to incorporate the bioavailability by selecting the model with parameter F. This will multiply the oral/extravascular administration input rate
by the number F. We give this possibility only in case of multiple administration routes because in a single route setting, F and V would only appear in the ratio V/F which makes them
The input rate defined by the administration route impacts the amount of drug in the central compartment. You can select to have additional peripheral compartments leading to a system with 1, 2 or 3
compartments in total:
The ODE system for standard PK models involves one species per compartment. The amount of drug in the central compartment is called A[c], and for the 2nd and 3rd compartment it is called A[2] and A
[3. ]The output of the model is the concentration in the central compartment C[c].
The ODE system for each case is given here, parameterized by the volumes in each compartment V1, V2, V3 and the intercompartmental clearance Q.
Instead of using the intercompartmental clearance Q[i] between the central compartment and a peripheral compartment i, and a volume V[i] for each compartment, it is also possible to use the volume V
in the central compartment only and transfer rates for each peripheral compartmental compartment i:
In and El rates are defined by the selected administration route and the elimination rate.
To show the impact of the number of compartments, we will see how they impact the response of a model with a bolus and linear elimination, parameterized with the clearance.
Impact of parameters in the case of 1 compartment
In the case of 1 compartment, the ODE system for a bolus with linear elimination parameterized with the clearance for
The response is given by a decreasing exponential Cl/V is kept constant (to focus on the effect of the distribution parameters), increasing the volume V will translate the whole response down:
Impact of parameters in the case of 2 compartments
In the case of 2 compartments, the ODE system for a bolus with linear elimination leads to a response involving a sum of decreasing exponentials such that log(Cc shows 2 different slopes. Increasing
V1 makes the response start lower, increasing V2 decreases the value at which the slope changes, and increasing Q moves this point earlier, making the slope change also sharper.
Impact of parameters in the case of 3 compartments
The case of 3 compartments is similar to the case of 2 compartments. log(Cc) shows 3 different slopes. V1, V2 and Q2 influence the early dynamics by playing on the initial value, and on the time and
height of the first point of slope change. Q3 and V3 influence the later dynamics by playing on the second point of slope change.
The elimination rate denoted in the previous section as El can be either a linear rate involving the clearance Cl, or a Michaelis Menten rate involving the constants Km and Vm. Another possible
parameterization for the linear rate is using the elimination rate k=Cl/V. To visualize the difference in the response, we show below the response of a 1-compartment model to a bolus in the two
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Balance model complexity and cross-validated score
Balance model complexity and cross-validated score#
This example balances model complexity and cross-validated score by finding a decent accuracy within 1 standard deviation of the best accuracy score while minimising the number of PCA components [1].
The figure shows the trade-off between cross-validated score and the number of PCA components. The balanced case is when n_components=10 and accuracy=0.88, which falls into the range within 1
standard deviation of the best accuracy score.
[1] Hastie, T., Tibshirani, R.,, Friedman, J. (2001). Model Assessment and Selection. The Elements of Statistical Learning (pp. 219-260). New York, NY, USA: Springer New York Inc..
The best_index_ is 2
The n_components selected is 10
The corresponding accuracy score is 0.88
# Author: Wenhao Zhang <wenhaoz@ucla.edu>
import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import load_digits
from sklearn.decomposition import PCA
from sklearn.model_selection import GridSearchCV
from sklearn.pipeline import Pipeline
from sklearn.svm import LinearSVC
def lower_bound(cv_results):
Calculate the lower bound within 1 standard deviation
of the best `mean_test_scores`.
cv_results : dict of numpy(masked) ndarrays
See attribute cv_results_ of `GridSearchCV`
Lower bound within 1 standard deviation of the
best `mean_test_score`.
best_score_idx = np.argmax(cv_results["mean_test_score"])
return (
- cv_results["std_test_score"][best_score_idx]
def best_low_complexity(cv_results):
Balance model complexity with cross-validated score.
cv_results : dict of numpy(masked) ndarrays
See attribute cv_results_ of `GridSearchCV`.
Index of a model that has the fewest PCA components
while has its test score within 1 standard deviation of the best
threshold = lower_bound(cv_results)
candidate_idx = np.flatnonzero(cv_results["mean_test_score"] >= threshold)
best_idx = candidate_idx[
return best_idx
pipe = Pipeline(
("reduce_dim", PCA(random_state=42)),
("classify", LinearSVC(random_state=42, C=0.01)),
param_grid = {"reduce_dim__n_components": [6, 8, 10, 12, 14]}
grid = GridSearchCV(
X, y = load_digits(return_X_y=True)
grid.fit(X, y)
n_components = grid.cv_results_["param_reduce_dim__n_components"]
test_scores = grid.cv_results_["mean_test_score"]
plt.bar(n_components, test_scores, width=1.3, color="b")
lower = lower_bound(grid.cv_results_)
plt.axhline(np.max(test_scores), linestyle="--", color="y", label="Best score")
plt.axhline(lower, linestyle="--", color=".5", label="Best score - 1 std")
plt.title("Balance model complexity and cross-validated score")
plt.xlabel("Number of PCA components used")
plt.ylabel("Digit classification accuracy")
plt.ylim((0, 1.0))
plt.legend(loc="upper left")
best_index_ = grid.best_index_
print("The best_index_ is %d" % best_index_)
print("The n_components selected is %d" % n_components[best_index_])
"The corresponding accuracy score is %.2f"
% grid.cv_results_["mean_test_score"][best_index_]
Total running time of the script: (0 minutes 1.005 seconds)
Related examples
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cutting planes
We analyze an inner approximation scheme for probability maximization. The approach was proposed in Fabian, Csizmas, Drenyovszki, Van Ackooij, Vajnai, Kovacs, Szantai (2018) Probability maximization
by inner approximation, Acta Polytechnica Hungarica 15:105-125, as an analogue of a classic dual approach in the handling of probabilistic constraints. Even a basic implementation of the maximization
scheme proved … Read more
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Price and Nominal Wage Phillips Curves and the Dynamics of Distribution in Japan
The first term on the right-hand side of equation (2) shows how fluctuations in the capacity utilization rate affect the nominal wage level. Nevertheless, we accept the rate of capacity utilization
as an explanatory variable for the nominal wage Phillips curve. The second term on the right-hand side of equation (2) shows that a change in the price of a good affects the rate of change in the
level of the nominal wage through bargaining between workers and management.
We specify the demand regime that represents how a change in wage share affects the rate of capacity utilization. The second term on the right-hand side of equation (11) shows how wage share affects
the rate of capacity utilization, and the sign of φ2 corresponds to demand regimes.
3 Estimation
However, if φ1Ω2 > Ω1φ2, even under these combinations, the steady-state equilibrium is stable because the state of. The source of value-added nominal output is the corporate statistics, and the GDP
deflator is the National Accounts Report. Real capital stock is the average of real fixed capital stock at the beginning and end of the period.
The source of this is the preliminary quarterly estimates of the gross capital stock of private enterprises published by the Bureau of Statistics. Labor input is obtained by multiplying the number of
employees by working hours per employee. The source of the number of employees is the Corporation Statistics, and that of working hours per capita is the Monthly Labor Survey published by the
Ministry of Health, Labor and Welfare.
Since none of the data series mentioned above are seasonally adjusted, we use them after seasonal adjustment based on X-12-ARIMA programs. The rate of change in each variable is obtained by
calculating the rate in each quarterly data series year-on-year. The expected inflation rate pe is obtained by taking a linearly decreasing moving average of the price inflation rates with linearly
decreasing weights over the past twelve quarters.8.
Estimation results
If there are nonsignificant explanatory variables in the estimated outcomes, we retain them as long as the adjusted R-squared value of the estimated equation that includes them is higher than the
value of the estimated equation that does not include them. However, if the adjusted R-squared value of the estimated equation increases using lagged data, we accept lagged explanatory variables. In
all estimated equations from the p-value of the J-statistic, the null hypothesis that the model specification is correct is not rejected, even at the 10% significance level.
Therefore, the real wage turns out to be procyclical with the capacity utilization rate. Since the sign of Ω1 is negative, the wage share is shown to be countercyclical with the capacity utilization
rate. In other words, the Japanese distribution regime was defined as a countercyclical wage share during the study period due to the rigidity of the domestic labor market.
As a result, even though real wages were procyclical to the rate of capacity utilization, wage shares were countercyclical. Since all the stability conditions are satisfied, we can consider that the
dynamic system of the rate of capacity utilization and wage share was stable in Japan during the study period. Therefore, wage share was seen to be regulated within a constant range and the dynamic
system was stable.
We therefore conclude that the stability of the Japanese economy depended on cooperative relations between the worker and the management.
4 Structural changes
Structural change in the nominal Phillips curve
First, the capacity utilization rate coefficient ut is not statistically significant before 1991, even at the 10% level, while it is statistically significant at the 1% level thereafter. However, for
a similar period, Kurosaka and Goto (1987) argue that the Japanese nominal wage was rigid to a change in output because the unemployment rate was stable and the Okun coefficient very large. In other
words, in Japan until the 1980s, because most workers were classified as regular, whose employment was strongly secured, the nominal wage did not reflect the change in output, explaining the lack of
a correlation between wage change nominal and capacity utilization rate. in 1977Q3–1991Q1.
By contrast, after the 1990s, due to the proportional increase in non-regular workers, nominal wages became sensitive to the change in output. Therefore, there is clear correlation between the change
in nominal wage and rate of capacity utilization in 1991Q1-2007Q3. Second, the coefficient of the rate of change in price Pˆt−2 is statistically significant at the 1% level before 1991, while after
1991 it is not statistically significant even at the 10%.
For most of the 1990s and 2000s, inflation was very low or even negative, and it was difficult for firms to allow nominal wages to reflect the change in commodity prices due to the downward rigidity
of the former. Therefore, there was no correlation between the change in the nominal wage and the change in the commodity price in Japan after the 1990s.12. However, after the 1990s, the proportion
of non-regular workers whose nominal wages were not affected by collective bargaining increased, weakening the impact of labor-management compromise on the nominal wage.
In summary, in Japan after 1991, the effect of the rate of capacity utilization on the nominal wage strengthened, while the effects of the rate of price change and the wage share weakened.
Structural change in the rate of change in labour productivity
Third, the wage share coefficient ψt−3 is statistically significant at the 1% level before 1991, but not statistically significant even at the 10% level after that. As noted above, the nominal wage
until the 1980s was heavily influenced by labor-management cooperation because workers were typically employed on a regular basis with long-term employment contracts. Furthermore, Nakata (2007)
demonstrates that after 2000, many large Japanese firms adjusted their employment levels of regular workers, who were heavily insured until the late 1990s.
As before, we check whether this breakpoint is appropriate using Wald and the likelihood ratio tests and find it statistically significant at the 1% level. First, the coefficient of the rate of
capacity utilization ut is statistically significant at the 1% level before 2000, while it is not statistically significant after, even at the 10%. Before 2000, labor productivity and output were
correlated because firms could not flexibly adjust their labor input in line with a change in output.
However, after 2000 there was no longer a correlation because companies flexibly adjusted their labor input and the labor hoarding effect was lost. Second, the coefficient of the wage share ψt−4 is
statistically significant at the 1% level before 2000, while afterward it is not statistically significant, even at the 10% level. This change means that the effect of the creation of the reserve
army was lost in the wage share.
In summary, in Japan after 2000, the effects of the capacity utilization rate and the wage share on the rate of change of labor productivity disappeared.
The stability of dynamics in each period
On the contrary, the wage share was still countercyclical during this period because the employment adjustment remained rigid and the capacity utilization rate had positive effects on labor
productivity. During this period there was a combination of profit-led demand and countercyclical wage share and the absolute value of Ω2 fell below that seen in 1977Q3-1991Q1. As a result, the sign
of it changed from positive to negative, and therefore the dynamics were unstable.
J The absolute value of Ω2 was small because the mechanism for regulating the wage share through collective bargaining was weakened by the proportional increase in non-regular workers excluded from
unions. 1991Q1-2000Q3 corresponds to the 'lost decade' in Japan, the prolonged recession after the bursting of the bubble economy. During this period, Japanese companies suffered from a decline in
capacity utilization and a profit squeeze with an increase in the wage share.
During this period, the necessary condition was that the sign of Ω2 was negative and the absolute value of Ω2 large enough due to the combination of a profit-driven demand regime and a
countercyclical wage share. In other words, the distributional regime shifted from a countercyclical wage distribution regime to a procyclical wage distribution regime as firms accelerated employment
adjustment and with the change. As a result, the combination of demand and distribution regimes resulted in a profit-led demand regime and a procyclical wage share.
Therefore, despite the combination of a profitable demand regime and a countercyclical wage share, the dynamics were stable. During the period 1991Q1–2000Q3, the Japanese economy switched to a labor
market-led regime (ie, real wages became procyclical) as the nominal wage reflected the change in output due to the increase in the share of nonregular workers. This proportional increase in the
number of non-regular employees also weakened the mechanism of wage share regulation, as collective bargaining had little effect on the wages of non-regular employees.
As a result, dynamics became unstable during this period, leading to the long recession in Japan in the 1990s. In the third quarter of 2000 and the third quarter of 2007, the distributive regime
switched from a countercyclical wage share regime to a procyclical wage share regime, as Japanese firms accelerated their employment adjustment and the labor hoarding effect was lost. As a result,
the dynamics recovered despite the lack of wage share regulation, as the combination of the profit-oriented demand regime and the procyclical wage share unambiguously stabilized the dynamics.
Some studies of these two types of Phillips curves focus on the effect of monetary policy on macroeconomic stability.13. Second, we did not focus on the employment rate to avoid the problem of
overdetermination in our model. Finally, we analyzed structural changes only in terms of the nominal wage Phillips curve and the rate of change in labor productivity.
The Institutional Structure of the Modern Japanese Economy and the Protracted Recession.” Kyoto University, 2006 (Chapter 5 of the doctoral dissertation in Japanese). An Empirical Analysis of the
Income Distribution and Demand Formation Pattern of the Japanese Economy.” Political Economy Quarterly 47, no. Endogenous Technological Change, Income Distribution, and Unemployment Through Class
Conflict.” Structural Change and Economic Dynamics 21, no.
Estimated Nonlinearities and Multiple Equilibria in a Model of Distributive-Demand Cycles.” International Review of Applied Economics 25, iss.
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Degrees and choice numbers
The choice number ch(G) of a graph G = (V, E) is the minimum number k such that for every assignment of a list S(ν) of at least k colors to each vertex ν ∈ V, there is a proper vertex coloring of G
assigning to each vertex ν a color from its list S(ν). We prove that if the minimum degree of G is d, then its choice number is at least (1/2 - 0(1)) log[2] d, where the 0(1)-term tends to zero as d
tends to infinity. This is tight up to a constant factor of 2 + 0(1), improves an estimate established by the author, and settles a problem raised by him and Krivelevich.
Dive into the research topics of 'Degrees and choice numbers'. Together they form a unique fingerprint.
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Types of Diffraction with Examples
Types of Diffraction of light with examples
In this post, You’ll Learn about Diffraction of light in a comprehensive way.
So, If you want to get benefits from this post, then you’ll love this post.
• Diffraction Definition
• Diffraction Types
• Diffraction Examples
• Much more
Let’s dive right in:
What is Diffraction of light?
The bending of light waves around the corners of an obstacle and spreading of light waves into geometrical shadow is called diffraction. Fraunhofer Diffraction and Fresnel Diffraction are two Types
of Diffraction of Light. Bending of Light around the corners of Window is an example of Diffraction.
Diffraction effect depends upon the size of the obstacle. Diffraction of light takes place if the size of the obstacle is comparable to the wavelength of light. Light waves are very small in
wavelength,i.e, from 4×10^-7 m to 7 × 10 ^-7 m. If the size of opening or obstacle is near to this limit, only then we can observe the phenomenon of diffraction.
Types of diffraction in physics
Diffraction of light can be divided into two types:
• Fraunhofer Diffraction
• Fresnel Diffraction
Fraunhofer Diffraction
In Fraunhofer diffraction:
• Source and the screen are far away from each other.
• Incident wavefronts on the diffracting obstacle are plane.
• Diffraction obstacle gives rise to wavefronts which are also plane.
• Plane diffracting wavefronts are converged by means of a convex lens to produce a diffraction pattern.
Fresnel Diffraction
In Fresnel diffraction:
• Source and screen are not far away from each other.
• Incident wavefronts are spherical.
• Wavefronts leaving the obstacles are also spherical.
• The convex lens is not needed to converge the spherical wavefronts.
See Also: Refraction of light
Diffraction of light
In Young’s double-slit experiment for the interference of light, the central region of the fringe system is bright. If light travels in a straight path, the central region should appear dark i.e.,
the shadow of the screen between the two slits. Another simple experiment can be performed by exhibiting the same effect.
Consider that a small and smooth ball of about 3 mm in diameter is illuminated by a point source of light. The shadow of the object is received on a screen as shown in the figure. The shadow of the
spherical object is not completely dark but has a bright spot at its centre. According to Huygens’s principle, each point on the rim of the sphere behaves as a source of secondary wavelets which
illuminate the central region of the shadow.
These two experiments clearly show that when light travels past an obstacle, it does not proceed exactly along a straight path, but bends around the obstacle. The phenomenon is found to be prominent
when the wavelength of light is compared with the size of the obstacle or aperture of the slit. The diffraction of light occurs, in effect, due to the interference between rays coming from different
parts of the same wavefront.
See Also : Interference of light
Diffraction due to Narrow slit
The figure shows the experimental arrangement for studying diffraction of light due to the narrow slit. The slit AB of width d is illuminated by a parallel beam of monochromatic light of wavelength
λ. The screen S is placed parallel to the slit for observing the effects of the diffraction of light. A small portion of the incident wavefront passes through the narrow slit. Each point of this
section of the wavefront sends out secondary wavelets to the screen. These wavelets then interfere to produce the diffraction pattern. It becomes simple to deal with rays instead of wavefronts as
shown in the figure.
In this figure, only nine rays have been drawn whereas actually there are a large number of them. Let us consider rays 1 and 5 which are in phase on in the wavefront AB. When these reach the
wavefront AC, ray 5 would have a path difference ab say equal to λ/2. Thus, when these two rays reach point p on the screen; they will interfere destructively. Similarly, each pair 2 and 6,3 and 7,4
and 8 differ in the path by λ/2 and will do the same. But the path difference ab=d/2 sinθ.
The equation for the first minimum is, then
d/2 sinθ =λ/2
or d sinθ=λ
In general, the conditions for different orders of minima on either side of the centre are given by:
d sinθ =mλ
where m=± (1,2,3,….)
The region between any two consecutive minima both above and below O will be bright. A narrow slit, therefore, produces a series of bright and dark regions with the first bright region at the centre
of the pattern.
Diffraction of X-rays by Crystals
The wavelength of an electromagnetic wave can be determined if a grating of the proper spacing i.e. of the order of wavelength λ of the wave, is available. X-rays are electromagnetic waves of very
short wavelengths ( of the order of 0.1 nm). It would be impossible to construct a grating having such a small spacing by the cutting process. However: the atomic spacing in a solid is known to be
about 0.1 nm. In 1913, Max von Laue suggested that the regular array of atoms in a crystal solid could act as a three-dimensional diffraction pattern are complex because of the three-dimensional
diffraction grating for X-rays. The subsequent experiment confirmed this prediction. The diffraction patterns are complex because of the three-dimensional nature of the crystal. Nevertheless, x-ray
diffraction has proved to be an invaluable technique for studying crystalline structures and for understanding the structure of matter.
A collimated beam of x-rays is incident on a crystal. The diffracted beams are very intense in certain directions, corresponding to constructive interference from waves reflected from layers of atoms
in the crystal. The diffracted beams can be detected by a photographic film, and they form array of spots known as a Laue pattern. One can deduce the crystalline structure by analyzing the positions
and intensities of the various spots in the pattern.
Bragg’s Equation
Suppose that an x-ray beam is incident at an angle θ on one of the planes. The beam can be reflected from both the upper plane and the lower plane travels farther than the beam reflected from the
upper plane. It is clear that beam 2 travels a greater distance than beam 1 after reflection from atoms of the plane. Thus the distance BC+CD is the effective path difference between the two
reflected beams 1 and 2.
Watch also:
Related Searches of Diffraction of Light:
Related Searches In Physics are:
External sources
• https://en.wikipedia.org/wiki/Diffraction
• https://www.quora.com/What-is-diffraction-of-light
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ummaries for
Lecture Summaries for Differential Equations
Lecture 01: An introduction to the very basic definitions and terminology of differential equations,
as well as a discussion of central issues and objectives for the course.
Lecture 02: Solving first order linear differential equations and initial value problems using
integrating factors .
Lecture 03: Solving separable equations.
Lecture 04: The Existence and Uniqueness Theorem for solving general first order linear equations.
Lecture 05: Applications of first order ODEs involving continuous compounding, and population
dynamics using the logistic equation .
Lecture 06: Solving the logistic equation, and an application of first order ODEs to a problem
of physics.
Lecture 07: Solving exact equations.
Lecture 08: Sketching a proof of the Existence and Uniqueness Theorem for first order ODEs.
Lecture 09: An introduction to difference equations and their solutions, focusing on first order
linear difference equations.
Lecture 10: An application of first order linear difference equations, as well as a brief discussion
of non- linear difference equations , their solutions, and stairstep diagrams.
Lecture 11: An introduction to second order ODEs and initial value problems, and a discussion
of solutions to second order homogeneous constant coefficient equations.
Lecture 12: A discussion of existence and uniqueness results for second order linear ODEs, and
of fundamental sets of solutions and the importance of the Wronskian of solutions.
Lecture 13: A discussion of the structure of the set of solutions to a linear homogeneous ODE
from a linear algebra perspective ; concepts such as linear independence, span, and basis are used
to better understand fundamental sets of solutions.
Lecture 14: Solving ODEs with characteristic equation having non-real complex roots.
Lecture 15: Solving ODEs with characteristic equation having repeated roots.
Lecture 16: Solving second order linear non-homogeneous equations using the method of undetermined
coefficients .
Lecture 17: Solving second order linear non-homogeneous equations using the method of variation
of parameters.
Lecture 18: A discussion of the structure of solution sets to higher order linear equations, the
basic Existence and Uniqueness Theorem, and a generalization of the Wronskian.
Lecture 19: Solving higher order constant coefficient homogeneous equations.
Lecture 20: Solving higher order non-homogeneous equations using the method of undetermined
coefficients .
Lecture 21: Solving higher order non-homogeneous equations using the method of variation of
Lecture 22: A review of the most fundamental properties of power series.
Lecture 23: Solving differential equations and initial value problems using power series.
Lecture 24: An example of how to use power series to solve non-constant coefficient ODEs,
and a discussion of the basic theorem underlying the use of power series to solve ODEs.
Lecture 25: A review of improper integration and an introduction to the Laplace transform.
Lecture 26: A discussion of the main properties of the Laplace transform which make it useful
for solving initial value problems .
Lecture 27: A discussion of how the Laplace transform and its inverse act on unit step functions,
exponentials, and products of these functions with others.
Lecture 28: An introduction to the convolution of two functions, and an examination of how
the Laplace transform acts on such a convolution.
Lecture 29: An introduction to systems of equations and the basic existence and uniqueness
result for the corresponding initial value problems .
Lecture 30: An introduction to vector function notation, and a discussion of the structure of
solution sets to homogeneous systems and the importance of the Wronskian.
Lecture 31: Solving constant coefficient linear homogeneous systems using eigenvalues and
Lecture 32: Solving constant coefficient linear homogeneous systems in the case where an
eigenvalue is complex .
Lecture 33: Solving constant coefficient linear homogeneous systems in the case where there is
a repeated eigenvalue.
Lecture 34: Viewing solutions to linear homogeneous systems in terms of fundamental matrices
and the exponential of a matrix .
Lecture 35: Solving non-homogeneous systems using diagonalization and variation of parameters.
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Motor Constant Calculator - Calculator Wow
Motor Constant Calculator
The Motor Constant Calculator is an essential tool for engineers, technicians, and hobbyists involved in the design and analysis of electric motors. This calculator helps determine the motor
constant, a critical parameter that reflects the performance and efficiency of a motor. By inputting the torque and speed values, users can compute the motor constant, which aids in assessing how
effectively a motor converts electrical power into mechanical power.
Understanding the motor constant is vital for several reasons:
1. Motor Efficiency: The motor constant provides insight into how efficiently a motor converts electrical power into mechanical power. A higher motor constant generally indicates better performance
and efficiency.
2. Design Optimization: Engineers use the motor constant to optimize motor designs. By analyzing different configurations and parameters, they can enhance motor performance for specific
3. Selection of Motors: When choosing a motor for a particular application, the motor constant helps in comparing different motors’ performance. It ensures that the selected motor meets the required
performance criteria.
4. Energy Consumption: Calculating the motor constant allows for better understanding and management of energy consumption. Efficient motors reduce operational costs and energy usage.
5. Reliability and Durability: Knowing the motor constant helps in predicting how a motor will perform under various conditions, which is crucial for ensuring long-term reliability and durability.
How to Use
Using the Motor Constant Calculator is straightforward:
1. Input Torque: Enter the torque value in Newton-meters (Nm). Torque represents the rotational force produced by the motor.
2. Input Speed: Enter the speed in revolutions per minute (RPM). Speed indicates how quickly the motor rotates.
3. Calculate: Click the “Calculate” button. The calculator will use the formula to compute the motor constant.
4. View Result: The motor constant will be displayed, showing how effectively the motor converts electrical power into mechanical power.
Km = Torque / sqrt(Speed in radians/second)
To convert RPM to radians per second, use the formula: Speed in radians/second = Speed in RPM × (2 × π / 60).
FAQs and Answers
1. What is the Motor Constant Calculator?
The Motor Constant Calculator determines the motor constant, which measures a motor’s efficiency based on its torque and speed.
2. How does the calculator work?
It uses the formula Km = Torque / sqrt(Speed in radians/second) to calculate the motor constant.
3. What units are used in the calculation?
Torque is entered in Newton-meters (Nm), and speed is entered in revolutions per minute (RPM). The result is in units of the motor constant.
4. Why is the motor constant important?
It helps in assessing motor efficiency, optimizing designs, selecting appropriate motors, and managing energy consumption.
5. How do I convert RPM to radians per second?
Use the formula Speed in radians/second = Speed in RPM × (2 × π / 60) to convert RPM to radians per second.
6. Can I use this calculator for different types of motors?
Yes, it applies to various types of electric motors, including DC and AC motors.
7. What if I enter incorrect values?
Ensure accurate input values for precise calculations. Incorrect values may lead to inaccurate results.
8. Is the calculator easy to use?
Yes, the calculator is user-friendly and designed for quick and straightforward calculations.
9. Can this calculator help with motor design?
Yes, it is useful for optimizing motor designs and comparing different motor configurations.
10. How often should I use this calculator?
Use it whenever you need to assess or compare motor performance, particularly during design and selection processes.
The Motor Constant Calculator is a powerful tool for evaluating and optimizing electric motors’ performance. By understanding how to use this calculator effectively, engineers and technicians can
make informed decisions about motor efficiency, design, and selection. Accurate calculations ensure better performance, energy management, and overall reliability, making this tool invaluable for
anyone working with electric motors.
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Two pulse mid point converter | 2 Pulse Mid Point Converter
Two pulse mid point converter:
Two pulse mid point converter are, in general, single phase converters. The pulse number of the converter indicates the frequency of the ripple voltage superimposing the average dc voltage at the
terminals. The schematic of a Two pulse mid point converter is shown in Fig. 3.15. In the figure Ï…[s1]Â and Ï…[s2]Â are midpoint to line voltages on the secondary side, which are out of phase by
The branch which has a higher voltage with respect to the midpoint can be made to conduct by giving a firing pulse to the thyristor in that branch. The midpoint of the secondary serves as the return
for the current. The thyristor T[1] can be fired when Ï…[s1]Â is positive and T[2] can be fired when Ï…[s2]Â is positive with respect to the midpoint. The thyristor conducts the load current when its
voltage is positive, once it is turned ON. The output voltage and curÂrent waveforms, assuming a highly inductive load are given in Fig. 3.16. If a thyristor, say T[1]Â is switched ON when the
voltage is positive the current through the load builds up. T[1] maintains conduction, depending upon the nature of the load, even in the period when Ï…[s1]Â is negative. At this stage Ï…[s2]
 becomes positive and thyristor T[2] takes over if a firing pulse is given. AssumÂing instantaneous commutation, performance, equations of the converter can be derived. Assuming a turns ratio of
unity for the transformer and neglecting the commutation reactances, the average voltage at the dc terminals of a Two pulse mid point converter can be derived as
where α is the firing angle of the converter. For firing delay angles in the range 0 to 90° the converter operates as a rectifier providing dc voltage at the load. Power transfer takes place from
ac to dc. For angles in the range 90° to 180° (theoretically) the converter operates in the inverting mode. If there is a dc source on the dc side, dc power can be converted to ac. However, in
practice a firing angle of 180° cannot be reached due to overlap and the finite amount of time taken by the thyristor to go to a blocking state.
The average dc voltage is maximum at a firing angle of 0°, It decreases as the firing angle changes from 0 to 90°, and finally becomes zero when the firing angle is 90°. The voltage reverses its
polarity for firing angles greater than 90° and increases with reversed polarity as a is increased beyond 90°. It reaches its maximum negative value at α = 180° (Fig. 3.17).
The average value of the thyristor current
The rms value of the thyristor current
Using these values of voltages and currents the ratings of the converter trans-former can be obtained. The rating of the secondary winding is
The design rating of the transformer is
The design rating of transformer is
The increased rating of the transformers is due to the dc component of the current.
The peak forward or reverse voltages applied to the thyristors are
The premagnetisation of the transformer existing in the circuit of Fig. 3.15 can be avoided by the ring connection of the transformer shown in Fig. 3.18. Overlap the current through the load builds
up when a thyristor is turned ON and the voltage across it is positive. Say, for example, T[1] is fired when V[s1]Â is positive. T[1] maintains its conduction till the next thyristor is fired at any
instant during its positive voltage. V[s2]Â becomes positive when V[s1]Â becomes negative. Thyristor T[2] is fired. The current transfer takes place from T[1] to T[2]Â The preceding discussion has
assumed instantaneous commutation. But in practice, due to the leakage reactance of the transformer, and the line inductances and additional inductances of the circuit (to protect the thyristors from
di/dt), the transfer of current is never instantaneous, but takes a definite amount of time. During commutation both the thyristors conduct. The current of the outgoing thyristor decreases and that
of the incoming one increases. The process is complete when all the current has been transferred to the incoming thyristor. The angle of overlap is denoted by u. The voltage and current wave forms of
the converter, taking overlap into consideration, are shown in Fig. 3.19.
The effect of overlap is to cause a kind of voltage drop at the output terminals. The average value of the dc voltage at the dc terminals of the converter is
Therefore, as the converter is loaded there is a reduction in the terminal voltage. This reduction is called voltage regulation. Besides overlap, the drops in the thyristors and circuit resistances
contribute to voltage regulation. During overlap the rate of change of current causes a drop in the inductive reactances in series with the thyristors, which is the main cause of voltage regulation.
The mean dc voltage of a Two pulse mid point converter is superimposed by a ripple voltage of twice the supply frequency. The ripple content is minimum at α = 0° and increases to a maximum at α =
90°. When α is increased further the ripple content decreases and falls to minimum when α = 180°.
When the load is purely resistive, the current in it becomes discontinuous. To explain this, note that the current is in phase with the voltage. When the load voltage falls to zero and the thyristor
is reverse biased, conduction ceases. When the load has sufficient inductance one thyristor conducts for 180° and the other thyristor takes over before the load current falls to zero. Thus,
conduction is made continuous. The load current becomes pure dc if the load has infinite inductance.
The mean value of the dc output voltage would be different for the cases of continuous and discontinuous conduction for the same firing angle being less for the former case. This is mainly because
negative excursions of the voltage are possible across the load in the case of continuous conduction, as the load current is maintained even after the voltage has become negative. The back emf load
on a converter is prone to discontinuous conduction. A resistive load inherently has discontinuous conduction. These cases are depicted in Fig. 3.20.
The performance of a converter is characterised by the superimposed ripple content of ac voltage on the mean dc voltage. The effective value of the rth harmonic referred to V[di] is neglecting the
overlap. When the effect of the overlap is taken into considerÂation, the effective value of the rth harmonic referred to V[di]Â would be
The ripple content can be easily calculated as the ratio of the effective value of superimposed ac voltage to ideal dc voltage
For a Two pulse mid point converter the ripple content is 48.2% for α = 0° and 111.1% for α = 90°.
A smoothing inductance is necessary in the load circuit. This inductance serves two purposes:
• to smooth the ripple content of output current
• to make conduction continuous in the load or to minimise the possiÂbility of discontinuous conduction.
The value of L[d] is normally determined such as to avoid discontinuous conÂduction rather than to smoothen out the ripple content. The layout of the smoothing inductance is rather large. The
smoothing inductance required in the load circuit is
V[di] average value of dc voltage
α firing angle at which the smoothing is required
I[d] is the dc current at which the conduction must be continuous
The performance of a converter is also characterised by harmonic currents on the ac side. The harmonic components present on the ac side do not contribute to any power transfer. On the other hand,
they cause undesirable effects in converter operation and also reduce the power factor markedly. They may cause resonance effects due to the line inductance and capacitance. When overlap is
neglected, the rms value of the harmonic current referred to the fundamental is
The effective value of the line current expressed as the ratio of the fundamental
For a Two pulse mid point converter this ratio is 111.1%. This distortion of the input ac current is also seen by examining the fundamental content of the input current. The ratio of the fundamental
component to the total rms current, g = I[1L]/I[L]. For a Two pulse mid point converter
When the overlap μ is taken into consideration
The effect of overlap is to reduce the distortion on the ac side and decrease the rms value of a harmonic.
The reactive power required by a converter is also a significant factor in evaluating its performance. The fundamental displacement factor is the phase difference between the voltage and the
fundamental of input current. From the waveforms of Fig. 3.16 the displacement factor is cos α. The total power factor on the input side is somewhat less than the displacement factor. It can be
shown that the total power factor is given by
For a Two pulse mid point converter, g = 0.9. The harmonics therefore effectively deÂcrease the pf even though they do not contribute to power transfer.
The reactive power required by a converter is due to the phase control emÂployed, as well as commutation. Unlike the active power which is decided by the fundamental only, the control reactive power
is decided by the harmonics also. The fundamental displacement factor is the cosine of the control anÂgle. When commutation is considered there is a certain overlap angle, because of which the
current waveform shifts further to the right, increasing the angle of lag of the current. The overlap angle depends upon the firing angle. The reactive power required because of phase control is V
[di]V[d] sin α and it increases as the firing angle increases, or the pf becomes poor. The reactive power due to commutation overlap is
where u[o] is the overlap angle at α = 0°. It can be shown that the reactive power required by the converter at a given firing angle α is
The fundamental displacement factor, taking overlap into consideration is approximately cos(α + u/2) or cos(α + 2u/3) depending upon 60 < α < 90° or 0 < α < 30°.
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Configure Simulation Conditions
Select solver, set initial conditions, pick input data set, set step size
After you build a model in Simulink^®, you can configure the simulation to run quickly and accurately without making structural changes to the model.
The first step in configuring your simulation is to select a solver. By default, Simulink automatically selects a variable-step solver. You can fine tune the solver options or select a different
solver in the Solver Pane of the Configuration Parameters dialog box.
Sometimes, a simulation can slow down or stall. Use the Solver Profiler to identify bottlenecks in the simulation and get recommendations to improve the solver performance.
Simulink.BlockDiagram.getAlgebraicLoops Identify and analyze algebraic loops in a model
solverprofiler.profileModel Programmatically analyze solver performance for model using Solver Profiler
Model Settings
Related Information
Featured Examples
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Give one example for the fact that a number which is rational but not an integer.
Hint: First, before proceeding for this, we must know the condition that the rational number is always an integer is not necessary but an integer can be written as a rational number always. Then,
rational number is the number of the form $ \dfrac{p}{q} $ where $ q\ne 0 $ and an integer is the number that has an integer value and always comes in total range of number from $ -\infty $ to $ \
infty $ .Then, the example considered by us as $ \dfrac{3}{4} $ is not an integer but still a rational number.
Complete step-by-step answer:
In this question, we are supposed to find a number which is rational but not an integer.
So, before proceeding for this, we must know the condition that the rational number is always an integer is not necessary but an integer can be written as a rational number always.
Here, we are said to give an example of the rational number which is not an integer.
But, before that, we all should know what is a rational number.
So, the rational number is the number of the form $ \dfrac{p}{q} $ where $ q\ne 0 $ .
However, an integer is the number that has an integer value and always comes in the total range of numbers from $ -\infty $ to $ \infty $ .
Now, to take an example of any rational number which is not an integer is:
$ \dfrac{3}{4} $
Here, the example considered by us as $ \dfrac{3}{4} $ is not an integer but still a rational number.
Now, to prove that it is not an integer, find the decimal value of the number considered as:
$ \dfrac{3}{4}=0.75 $
So, it gives the value 0.75 and by the definition of integers, it is not an integer value.
But when we go for the definition of rational number, $ \dfrac{3}{4} $ is of form $ \dfrac{p}{q} $ and also its denominator is not zero which states that it is a valid rational number.
Similarly, we can take many numbers which are rational numbers but not integers.
Hence, it is proved that $ \dfrac{3}{4} $ is a rational number but not an integer.
Note: Now, to solve these types of questions, it was not necessary to take this fixed example as $ \dfrac{3}{4} $ to prove that a rational number is not necessarily an integer. So, in mathematics of
the numbers, we have a number of rational numbers like $ \dfrac{1}{2},\dfrac{6}{8},\dfrac{1}{5} $ and much more that are not integers. So we can take any rational number whose fraction doesn’t give
an integer value to prove this statement.
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On the symbol error rate of M-ary MPSK over generalized fading channels with additive Laplacian noise
This paper considers the symbol error rate of M-ary phase shift keying (MPSK) constellations over extended Generalized-K fading with Laplacian noise and using a minimum distance detector. A generic
closed form expression of the conditional and the average probability of error is obtained and simplified in terms of the Fox's H function. More simplifications to well known functions for some
special cases of fading are also presented. Finally, the mathematical formalism is validated with some numerical results examples done by computer based simulations. © 2014 IEEE.
Dive into the research topics of 'On the symbol error rate of M-ary MPSK over generalized fading channels with additive Laplacian noise'. Together they form a unique fingerprint.
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Mechanical Behavior of a Virus - Problem Set 3 | Mechanical Behavior of Materials | Materials Science and Engineering | MIT OpenCourseWare
(b) Considering Ivanovska, et al.’s Fig. 1E [1] rendering of a bacteriophage prohead “shell”, compute the stress required to initiate plastic deformation if the prohead were considered to be a short
cylindrical shell under uniaxial compression.
Ivanovska, et al. [1] states on p. 7604 of the manuscript that proheads responded linearly to indentation forces up to approximately 2.8 nN. Define this as F[yield]. From this, we can calculate the
stress required to initiate plastic deformation, σ[y], based on the cross-sectional area A. In other words, we’re given F[yield] = 2.8 nN; and σ[y] = F[yield]/A.
Finding cross-sectional area A: The manuscript states that the cantilever used was OMCL-RC800PSA from Olympus. Based on the manufacturer specifications [2], the probe is a sharpened pyramidal tip,
but we’ll approximate it as a cone here with a tip radius r = 15 nm, tip angle θ = 45°, tip height h = 2.9 μm (Fig. 1). Because the prohead and cantilever are of comparable sizes, we cannot model the
prohead surface area being acted upon as a flat plane. Thus we cannot simply take the tip area to be our cross sectional area. Our actual cross-sectional area is the projected contact area (shaded
region in Fig. 1). Therefore, using cone geometry, we can find A via A = π(r[1])^2, where r[1] is as indicated in Fig. 3. To calculate this, we need to know the strain at yield point, which is
provided by the manuscript (pg. 7604): strain[yield] = maximum linear displacement = 12 nm. Using trigonometry, we’re able to find the following value: r[1] = 19.97 nm. Therefore, A = 1252.2 nm^2.
Fig. 1. Cantilever Probe Tip. Dimensions based on manufacturer specifications of OMCL-RC800PSA from Olympus. Tip is approximated as a cone with tip radius r, tip height h and tip angle theta. Shaded
region indicates cross sectional area used for yield stress calculation.
Finding σ[y]: σ[y] = 2.8 nN / (1252.2 nm^2) = 2.2 MPa
(c) These authors [1] state that imaging of the proheads in contact mode atomic force microscopy (AFM) allows for the study of prohead deformation under “uniaxial pressure.” This is incorrect at
several levels. Develop a brief, justified objection to this claim, considering the design of the experiment in detail.
The authors used scanning force microscopy (SFM) to image the proheads under different maximal loading forces and assumed this to be “uniaxial pressure”. However, given the imaging modality rasters
an AFM tip, with a contact region of approximately 5 nm, across the virus prohead, which itself has an inherent curvature, the applied force is clearly not uniaxial, see figure below. Along the
curvature of the virus prohead the applied force is resolved into various components. Also, the size of AFM tips (< 20 nm) are on the order of the size of a prohead and thus cannot be considered as a
point load when forces are applied to the prohead.
(d) These authors also claim that the mechanical testing of such proheads enables prediction of the elastic properties of the bacteriophage, by recourse to nonlinear elastic bucking of shells. Based
on the data they present in this paper and known continuum mechanical analysis of shell elasticity, is this claim justified? Timoshenko [5] is an excellent resource on mechanical analysis of shells.
The authors do nanoindentaion of the bacteriophage φ 29 shells and then model the shells using a continuum approach, assuming it is homogeneous and a thin shell. However, as we know the shell’s
microstructure is inhomogeneous and they even admitted there is a bimodal distribution of elastic moduli. Also, the authors considered the shells as spherical thin shells, assuming h/R = 0.1. In
order to be regraded as flat plates (thin shells), the shells may have the thickness (h) which is less than one hundredth of the least radius (R) of curvature [3]. This means that the nonlinearity of
the shells can be reduced to the solvable linear shells [4]. If the ratio of the thickness to the radius is comparable to the one tenth like this case, the shells may be considered as the moderately
thick shells. In thick shells, the interaction due to bending term and stretching term cannot be evaded. Therefore, their claim would not be justified.
[1] Ivanovska, I. L., P. J. Pablo, B. Ibarra, G. Sgalari, F. C. MacKintosh, J. L. Carrascosa, C. F. Schmidt, and G. J. L. Wuite. “Bacteriophage Capsids: Tough Nanoshells with Complex Elastic
Properties.” PNAS 101 (2004): 7600-7605.
[2] Micro Cantilever.
[3] Cox, H. L. The Buckling of Cylindrical Plates and Shells. New York, NY: Pergamon, 1963, pp. 71-72.
[4] Gould, Phillip L. Analysis of Shells and Plates. Upper Saddle River, NJ: Prentice Hall, 1998, pp. 461-463. ISBN: 9780133749502.
[5] Timoshenko, S., and S. Woinowsky-Krieger. Theory of Plates and Shells . New York, NY: McGraw-Hill, 1959.
Plasticity and fracture of microelectronic thin films/lines
Effects of multidimensional defects on III-V semiconductor mechanics
Defect nucleation in crystalline metals
Roleof water in accelerated fracture of fiber optic glass
Carbon nanotube mechanics
Superelastic and superplastic alloys
Mechanical behavior of a virus | Problem Set 2 | Problem Set 3 | Problem Set 5
Effects of radiation on mechanical behavior of crystalline materials
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Russell's Paradox
Russell's Paradox is a paradox in set theory. One can classify sets into one of two categories. The first category contains sets that are not members of themselves. This contains most of the sets
that we run into in everyday life. For example, the set of all penguins falls in this category, because the set of all penguins is a set, not a penguin. The second category contains sets that are
members of themselves. The set of all non-penguins, for example, is not a penguin and thus is a member of itself. So is the set of all sets.
Now, take the set of all sets that are not members of themselves. In which category does it belong? If this set is not a member of itself, then it is a member of itself. If it is, then it isn't. So,
this set is a member of itself if and only if it is not a member of itself, which is the paradox.
This paradox was very significant when it was discovered by Bertrand Russell in 1902. Around this time, attempts were being made to put mathematics on a foundation of set theory. However, if this
foundation contains a contradiction, this is a very big problem, since, according to the laws of logic, one could then prove any mathematical statement to be true.
Obviously this paradox needed to be resolved. The "naive set theory" of the time allowed for a lot of leeway in how a set could be defined. The way that Russell and others resolved the paradox was to
disallow predicates such as "the set of all sets that do not contain themselves" to be valid definitions of a set.
This paradox is similar to the Cretan Liar paradox.
View An article about Russell's Paradox at the Stanford Encyclopedia of Philosophy.
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Linear Recurrences
This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.
Linear recurrences with constant coefficients are an interesting class of recurrence equations that can be solved explicitly. The most famous example are certainly the Fibonacci numbers with the
equation f(n) = f(n-1) + f(n - 2) and the quite non-obvious closed form (φ^n - (-φ)^-n) / √5 where φ is the golden ratio.
In this work, I build on existing tools in Isabelle – such as formal power series and polynomial factorisation algorithms – to develop a theory of these recurrences and derive a fully executable
solver for them that can be exported to programming languages like Haskell.
Session Linear_Recurrences
Session Linear_Recurrences_Solver
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planar density calculator
Calculate the planar density of the (110) plane of a BCC unit cell 2.Calculate the planar density of the (110) plane of a FCC unit cell 3.Calculate the planar density of the (111) plane of a SC unit
cell 4.Calculate the planar density of the (111) plane of a BCC unit cell Expert Answer 100% (6 ratings) GO The planar density of a face centered cubic unit cell can be calculated with a few simple
steps. Download "Animol" app from Apple App Store or Google Play Store and watch these videos on Mobile! We can use 4 other way(s) to calculate the same, which is/are as follows -, Planar Density for
FCC 111 plane Calculator. How to calculate Planar Density using this online calculator? The Planar Density for FCC 111 plane formula is defined a number of atoms per unit area that are centered on a
particular crystallographic plane and is represented as. x z y Green ones touches each other. We can use 4 other way(s) to calculate the same, which is/are as follows -, Planar Density for FCC 100
Plane Calculator. Planar Density for FCC 100 Plane calculator uses Planar Density = 0.25/(Radius of Constituent Particle^2) to calculate the Planar Density, The Planar Density for FCC 100 plane
formula is defined as number of atoms per unit area that are centered on a particular crystallographic plane. The Planar density is the ratio of number of atoms lying on the atomic plane per unit
area of the atomic plane. Substitute the value calculated in step 1 for the numerator and the value calculated in step 2 for the denominator. P 3.53 (a): Linear Density for BCC Calculate the linear
density for the following directions in terms of R: [100] [110] [111] * Planar Density of (100) Iron Solution: At T < 912C iron has the BCC structure. How to Calculate Planar Density for FCC 111
plane? The Planar density is the ratio of number of atoms lying on the atomic plane per unit area of the atomic plane. Here, _A+B is the plane-averaged density of the combined A+B system, and _A and
_B are the plane-averaged. If we consider the anions (O^{2-} ions) to be located at the FCC positions of a cubic unit cell as shown for the Cl^-ions of Figure 11.5 and Figure EP11.6, then the (111)
plane contains the equivalent of two anions. Planar Density for BCC 100 Plane calculator uses Planar Density = 0.19/(Radius of Constituent Particle^2) to calculate the Planar Density, The Planar
Density for BCC 100 plane formula is defined as the number of atoms per unit area of the BCC (100) plane. Planar Density for FCC 111 plane calculator uses Planar Density = 0.29/(Radius of Constituent
Particle^2) to calculate the Planar Density, The Planar Density for FCC 111 plane formula is defined a number of atoms per unit area that are centered on a particular crystallographic plane. How to
Calculate Planar Density for BCC 100 Plane? Planar density (PD ) is taken as the number of atoms perunit area that are centered on a particular crystallographic plane, The Area of Plane is the
geometrical area of plane in the given direction. Planar Density for FCC 100 Plane calculator uses Planar Density = 0.25/ (Radius of Constituent Particle^2) to calculate the Planar Density, The
Planar Density for FCC 100 plane formula is defined as number of atoms per unit area that are centered on a particular crystallographic plane. Planar Density calculator uses Planar Density = Number
of Atoms Centered on Plane/Area of Plane to calculate the Planar Density, The Planar Density formula is defined as the number of atoms centered on plane per unit area of the plane. 6071.42857142857
--> No Conversion Required, The Planar Density formula is defined as the number of atoms centered on plane per unit area of the plane and is represented as, The Planar Density formula is defined as
the number of atoms centered on plane per unit area of the plane is calculated using. Calculate the linear density for the [111] direction in a FCC structure.c. For (111): From the sketch, we can
determine that the area of the (111) plane is (v2a./2) (va/V2) = 0.866a.. Planar Density for BCC 100 Plane Solution. 10+ Density of different Cubic Cell Calculators, Number of Atoms Centered on
Direction Vector. Planar Density is denoted by P.D symbol. . Find Your Next Great Science Fair Project! Here is how the Planar Density for FCC 100 Plane calculation can be explained with given input
values -> 0.694444 = 0.25/(0.6^2). Whether you need help solving quadratic equations, inspiration for the upcoming science fair or the latest update on a major storm, Sciencing is here to help.
Pragati Jaju has created this Calculator and 50+ more calculators! As an example, the area of a (1 1 0) plane of an FCC crystal is 8_sqrt(2)_R^2 where "R" is the radius of an atom within the plane.
Planar Density for BCC 100 Plane calculator uses. Paige Turner started writing professionally in 2009. (a) (100) plane (FCC) planar density= Enter your answer for (100) plane (FCC) in accordance to
the question statement /R2 (110) plane (FCC) planar density= Enter your answer for (110) plane This problem has been solved! Planar density (PD ) is taken as the number of atoms perunit area that are
centered on a particular crystallographic plane. National Institute of Information Technology. #GATEWaytoESE #adityathakur #material_scienceFor news of different exams , doubt discussion , quiz ,
quality question solving , notes join our telegram grou. Drive the expression between atomic radius " \ ( R \) " and unit cell length " \ ( a \) " for \ ( F C C \) and calculate planar density for
the (101) plane in terms of atomic radius \ ( R \). or To use this online calculator for Planar Density for BCC 100 Plane, enter Radius of Constituent Particle (R) and hit the calculate button.
Calculate the number of atoms centered on a given plane. 10+ Density of different Cubic Cell Calculators, Number of Atoms Centered on Direction Vector. Planar Density is defined as the number of
atoms per unit area that are centered on a particular crystallographic plane. The Planar Density for FCC 100 plane formula is defined as number of atoms per unit area that are centered on a
particular crystallographic plane is calculated using. Planar Density Calculate the planer density of the (110) plane for the FCC crystal Compute planar area Compute number of atoms For an atom to be
counted, it has to be centered on that plane. She holds a Master of Science in engineering from New York University. Planar density = Area of atoms/ Area of plane = 2*pi r2/ 8*r2 = pi / 4 Wiki User
2011-11-10 04:57:31 This answer is: Study guides Stu's Guide 4 cards Test- Nicole Test Proton number of. The Number of Atoms Centered on Plane is the total number of atoms lying on the given plane.
Express your final results in terms of the atomic radius, R. a. How to calculate Planar Density using this online calculator? Akshada Kulkarni has verified this Calculator and 900+ more calculators!
How to calculate Planar Density for FCC 100 Plane? To use this online calculator for Planar Density for FCC 100 Plane, enter Radius of Constituent Particle (R) and hit the calculate button. Atomic
radius is 4.05A. Sample calculations are performed for. Population density is an example of areal numberDec 14 2016 B. National Institute of Information Technology. For the example string that weighs
0.0025 kg and is 0.43 m long, perform this operation as follows: 0.0025/0.43 = 0.00582 kg/m. k = 510 N/m %D k : May 19 2022 Air enters the 1-m2 inlet of an aircraft engine at 100 kPa and 20C with a
velocity of 180 m/s. Calculate the linear density for the [111] direction in a BCC structure.b. . The Radius of Constituent Particle is the radius of the atom present in the unit cell. Mechanical
Engineering questions and answers. Planar Density calculator uses Planar Density = Number of Atoms Centered on Plane/Area of Plane to calculate the Planar Density, The Planar Density formula is
defined as the number of atoms centered on plane per unit area of the plane. Substitute the value calculated in step 1 for the numerator and the value calculated in step 2 for the denominator. Planar
density is a measure of packing density in crystals. How to Calculate Planar Density for FCC 100 Plane? Pragati Jaju has created this Calculator and 50+ more calculators! Planar Density is denoted by
P.D symbol. The Planar Density for FCC 111 plane formula is defined a number of atoms per unit area that are centered on a particular crystallographic plane is calculated using. Planar Density is
defined as the number of atoms per unit area that are centered on a particular crystallographic plane. The units of planar density are mm^-1, cm^-1. Which, if any, of these planes is close packed?
The plane-averaged charge density difference can be written as =_A+B _A _B . 10+ Density of different Cubic Cell Calculators, Number of Atoms Centered on Direction Vector, Planar Density is defined
as number of atoms per unit area that are centered on a particular. The contact surface is smooth. crystallographic plane. Planar Density - Number of atoms . 8.05555555555556E+15 --> No Conversion
Required. Akshada Kulkarni has verified this Calculator and 900+ more calculators! Planar Density is defined as number of atoms per unit area that are centered on a particular crystallographic plane.
In the present paper, a new position-duplication-number method is developed to calculate the planar density of all Bravais lattices and all crystal structure types using the formula ( hkl ) ( m .
Here is how the Planar Density for BCC 100 Plane calculation can be explained with given input values -> 0.527778 = 0.19/(0.6^2). Here is how the Planar Density calculation can be explained with
given input values -> 6071.429 = 34/0.0056. As described above the maximum energy density of a Gaussian beam . Articles published in Journal of Applied Crystallography focus on these methods and
their use in identifying structural and diffusion-controlled phase transformations, structure . It is the reciprocal of area. Question: 6. What is Planar Density for FCC 111 plane? Here is how the
Planar Density for FCC 111 plane calculation can be explained with given input values -> 0.805556 = 0.29/(0.6^2). x z y R R R R a a a How many ways are there to calculate Planar Density? If the block
is subjected to a force of F = 570 N, determine its velocity in m/s when s = 0.5 m. When s = 0, the block is at rest and the spring is uncompressed. It's unit is reciprocal of area. Find the area of
the plane. In this formula, Planar Density uses Radius of Constituent Particle. Planar Density is defined as number of atoms per unit area that are centered on a particular A large collection of
element-wise planar densities for compounds obtained from the Materials Project is calculated using brute force computational geometry methods, where the planar density is . In this formula, Planar
Density uses Number of Atoms Centered on Plane & Area of Plane. 10+ Density of different Cubic Cell Calculators, Number of Atoms Centered on Direction Vector. Calculate planar density with the
formula: PD = Number of atoms centered on a given plane / Area of the plane. Planar Density is denoted by P.D symbol. Planar Density is denoted by P.D symbol. How many ways are there to calculate
Planar Density? =number of atoms centered on plane/area of plane . What is the atomic density of 111 FCC? Planar Density for BCC 100 Plane calculator uses Planar Density = 0.19/ (Radius of
Constituent Particle^2) to calculate the Planar Density, The Planar Density for BCC 100 plane formula is defined as the number of atoms per unit area of the BCC (100) plane. (a) Calculate the planar
density of atoms on (111) and (110) planes in BCC and FCC unit Cells. Our goal is to make science relevant and fun for everyone. Engineering. The Planar Density for FCC 100 plane formula is defined
as number of atoms per unit area that are centered on a particular crystallographic plane and is represented as. Planar Density The basic component of a crystal structure is a unit cell. - Calculate
Linear or Planar Density . 6.94444444444445E+15 --> No Conversion Required. How to calculate Planar Density for FCC 100 Plane using this online calculator? Planar Density for FCC 111 plane Solution.
The Planar Density for FCC 111 plane formula is defined a number of atoms per unit area that are centered on a particular crystallographic plane and is represented as P.D = 0.29/ (R^2) or Planar
Density = 0.29/ (Radius of Constituent Particle^2). In crystalline materials such as metals, atoms are packed on periodic, three-dimensional arrays. Planar Density for FCC 100 Plane calculator uses.
Her articles on business, health, technology and travel have been published on various websites ever since. The density of a material, typically denoted using the Greek symbol , is defined as its
mass per unit volume. As an example, there are 2 atoms on a (1 1 0) plane of an FCC crystal. 2022 Leaf Group Ltd. / Leaf Group Media, All Rights Reserved. However, it is important to pay special
attention to the units used for density calculations. The Dimensional Formula of Linear Mass Density = M1L-1T0. National Institute of Information Technology. The Planar density is the ratio of number
of atoms lying on the atomic plane per unit area of the atomic plane. We can use 4 other way(s) to calculate the same, which is/are as follows -. Calculate planar density with the formula: PD =
Number of atoms centered on a given plane / Area of the plane. In non-crystalline materials such as silicon oxide, atoms are not subject to periodic packing. Mechanical Engineering. LINEAR AND PLANAR
DENSITIES Linear density (LD ) is defined as the number of atoms per unit length whose centers lie on the direction vector for a specific crystallographic direction; that is, = number of atoms
centered on direction vector length of direction vector 5.27777777777778E+15 --> No Conversion Required. How many ways are there to calculate Planar Density? Materials Science problem deriving the
planar density of a Face Centered Cubic unit cell in the (100) and (110) planes. Pragati Jaju has created this Calculator and 50+ more calculators! How many ways are there to calculate Planar
Density? Planar Density is denoted by P.D symbol. The units of planar density are mm^-2, cm^-2. This coplanar orbit is a remnant of how the solar system formed. Along with the calculated linear mass
density, two conversion scales will show a range of mass with a fixed length, and a range of length with a fixed mass, converted to linear density, relating to each calculated result. Pragati Jaju
has created this Calculator and 50+ more calculators! What is Planar Density for BCC 100 Plane? The Radius of Constituent Particle is the radius of the atom present in the unit cell. 18.4K
subscribers 11K views 2 years ago Materials science relies on calculations of linear and planar density frequently when determining things like slip systems. (b) Calculate the linear density of atoms
on [111] and [110] planes in BCC and FCC unit cells. = m V where: is the density m is the mass V is the volume The calculation of density is quite straightforward. How to calculate Planar Density for
FCC 111 plane? Up to 256 cash back fCalculate planar density of the plane answered in e. Planar Density calculator uses Planar Density Number of Atoms Centered on PlaneArea of Plane to calculate the
Planar Density The Planar Density formula is defined as the number of. User Guide. The unit of planar density are mm^-2, mm^-1. 7E4 2 points planar density 0.1144 x points/crn2 cm) Atomic radius is
4.05A (c) Draw your own conclusions from parts 4 (a) and 4 (b) 7. What is Planar Density for FCC 100 Plane? The Radius of Constituent Particle is the radius of the atom present in the unit cell. To
use this online calculator for Planar Density, enter Number of Atoms Centered on Plane (N) & Area of Plane (A) and hit the calculate button. Planar Density for FCC 100 Plane Solution. (a) Indicate
the melting points or . Akshada Kulkarni has verified this Calculator and 900+ more calculators! To use this online calculator for Planar Density for FCC 111 plane, enter Radius of Constituent
Particle (R) and hit the calculate button. (100) Radius of iron R = 0.1241 nm R 3 . Calculate planar density with the formula: PD = Number of atoms centered on a given plane / Area of the plane.
Atoms can be packed together densely or loosely. National Institute of Information Technology. How to calculate for the linear and planar density in crystals.0:00 Start0:10 Linear Density2:13 Planar
Density What is the planar density for the (110) plane in a FCC structure The Planar Density for BCC 100 plane formula is defined as the number of atoms per unit area of the BCC (100) plane and is
represented as. The Planar Density for BCC 100 plane formula is defined as the number of atoms per unit area of the BCC (100) plane is calculated using. 332 Determine the planar density and packing
fraction for FCC nickel the (110), and (111) planes. The Planar density is the ratio of number of atoms lying on the atomic plane per unit area of the atomic plane. Many research topics in condensed
matter research, materials science and the life sciences make use of crystallographic methods to study crystalline and non-crystalline matter with neutrons, X-rays and electrons. What is the formula
for linear density? The units for planar density are reciprocal area (e.g., nm2,m2). We can use 4 other way(s) to calculate the same, which is/are as follows -, Planar Density for BCC 100 Plane
Calculator. n = 0.5 atoms L L = a line length . 1. Therefore, 110 mg of water (density = 1.00) will equal 11. How to calculate Planar Density for BCC 100 Plane using this online calculator? Planar
density is the fraction of total crystallographic plane area that is occupied by atoms. How to calculate Planar Density for BCC 100 Plane? How to calculate Planar Density for FCC 111 plane using this
online calculator? Temperature and Pressure - Online calculator figures and table showing density and specific weight of pentane C 5 H 12 at temperatures ranging from -130 to 325 C -200 to 620 F at
atmospheric and higher pressure -. Planar Density is denoted by P.D symbol. See the answer a) Calculate planar densities for the (100), (110), and (111) planes for FCC. Cite this Article Did you find
this page helpful? Akshada Kulkarni has verified this Calculator and 900+ more calculators! This calculator is used to determine the linear mass density from the total measured mass and length of an
item. In this formula, Planar Density uses Radius of Constituent Particle. Body-centered Cubic Crystal Structure (BCC) First, we should find the lattice parameter(a) in terms of atomic radius(R). It
depends on the density of the liquid. The Radius of Constituent Particle is the radius of the atom present in the unit cell. Planar Density is denoted by P.D symbol. In this formula, Planar Density
uses Radius of Constituent Particle. You can easily calculate the volume by using the formula: density=mass (g)/volume (mL). on = 3.5167 (100): planar density (3.5167 packing fraction = (4r/v)2
0.1527 x 10-16 - O. [3 \times 60 = 180 = \frac{1}{2} anion + (3\times \frac{1}{2} ) anions at each midpoint of the sides of the (111) planar triangle of Figure EP11.6 = a total of 2 anions within the
. What is the planar density for the (110) plane in a BCC structure?d. Then, we can find linear density or planar density. Here, we show how to calculalte. Planar Density is defined as number of
atoms per unit area that are centered on a particular crystallographic plane. Divide the mass of the string by its length to get linear density in kilograms per meter. Planar Density for FCC 111
plane calculator uses.
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Math Colloquia - On the resolution of the Gibbs phenomenon
Since Fourier introduced the Fourier series to solve the heat equation, the Fourier or polynomial approximation has served as a useful tool in solving various problems arising in industrial
applications. If the function to approximate with the finite Fourier series is smooth enough, the error between the function and the approximation decays uniformly. If, however, the function is
nonperiodic or has a jump discontinuity, the approximation becomes oscillatory near the jump discontinuity and the error does not decay uniformly anymore. This is known as the Gibbs-Wilbraham
phenomenon. The Gibbs phenomenon is a theoretically well-understood simple phenomenon, but its resolution is not and thus has continuously inspired researchers to develop theories on its resolution.
Resolving the Gibbs phenomenon involves recovering the uniform convergence of the error while the Gibbs oscillations are well suppressed. This talk explains recent progresses on the resolution of the
Gibbs phenomenon focusing on the discussion of how to recover the uniform convergence from the Fourier partial sum and its numerical implementation. There is no best methodology on the resolution of
the Gibbs phenomenon and each methodology has its own merits with differences demonstrated when implemented. This talk also explains possible issues when the methodology is implemented numerically.
The talk is intended for a general audience.
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Printable algebra test
printable algebra test Related topics: adding-subtracting-fractions
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TNF Posted: Friday 14th of Sep 09:10
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AllejHat Posted: Sunday 16th of Sep 08:58
Don’t fear, Algebrator is here ! I was in a same situation sometime back, when my friend suggested that I should try Algebrator. And I didn’t just pass my test; I went on to score really
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It’s right here: https://softmath.com/about-algebra-help.html. Buy it and try it, if you don’t like it (which I think can’t be true) then they even have an unquestionable money back
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A Guide Towards Sheet Metal Bending Calculations - richcontentdaily.com
A Guide Towards Sheet Metal Bending Calculations
by Jeremy Wall
written by Jeremy Wall
Every aspect of sheet metal bending is crucial. If one aspect is not followed, then the outcome may not be as desired. The process involved in metal bending requires plasticizing sheet metals in
order to change their shape. Therefore, every specification, especially in pattern dimensions, must be followed. Thus, to obtain the proper sheet metal bending, some calculations need to take place.
This article aims to offer a guide towards sheet metal bending calculations.
A guide towards sheet metal bending calculations
· K-factor calculation
In sheet metal bending, K refers to a constant used in the calculations of sheet-metal flat length. K-factor is the ratio of material’s neutral line location to its thickness. Generally, it is
described by t/T, where T represents thickness and t represents the material’s neutral line location. The critical point to note while calculating the K-factor is that its formula does not consider
the forming stress but the neutral line location geometric calculation. The K-factor calculations depend on various aspects. The various aspects include; bending operation type that is air bending,
bottoming, coining, and so many others and the type of material used.
· Bend allowance calculation
Bend allowance refers to a bend’s arc length that is measured from the material’s neutral axis. Generally, bend allowance can be defined as the material required to include the part’s leg length to
obtain the right size for the cut’s flat pattern. Additionally, it calculates the material required to get the sheet-metal flat length. The Bend allowance calculation formula is given by;
Bend allowance can be calculated by; A
A refers to 90 degrees angles, R refers to a radius, K refers to the K factor, and T refers to thickness.
· Bend deduction calculation
Bend deduction is the difference between the total flange length after bending and the beginning flat length. The total flange length sum after bending is complete should be higher than the flat
length before bending. The sheet metal bending process ensures that the bend’s inner surface is compressed while the bend’s outer surface is stretched. The bend deduction calculation is given by;
Bend deduction= 2(R+T) tan- bend allowance
Benefits of sheet metal bending
· Cost-effective
Sheet metal bending is cost-effective as it does not allow for material waste during its processes. Thus, the material saved can be bent into more useful shapes.
· Bending accuracy
Sheet metal bending machine ensures that there is bending accuracy. Thus, parts are produced according to one’s exact specifications. Even though the processes require sheet metal deformation, the
deformation results are usually accurate.
· Low labor costs
The low labor costs are because the latest sheet metal bending machine allows for automation processes to happen. The automation process ensures that the sheet bending machine requires only one or
two operators.
· Efficiency
Sheet metal bending machines are usually high-performing. The high-performing machines guarantee machine efficiency. Thus, it encourages time-saving.
Bend deduction, bend allowance, and K factor are essential factors in sheet metal bending calculations. Any mistake done during the calculations can ensure that the results obtained are not what was
expected. Therefore, to get the specified part, it is essential to get correct calculations.
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Jeremy Wall
Jeremy Wall who graduated from Yale University and works for national television. He is very interested in manufacturing related topics and often gives his own opinions and opinions.
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The Stacks project
Lemma 60.9.5. In Situation 60.7.5. Let $X' \subset X$ and $S' \subset S$ be open subschemes such that $X'$ maps into $S'$. Then there is a fully faithful functor $\text{Cris}(X'/S') \to \text{Cris}(X
/S)$ which gives rise to a morphism of topoi fitting into the commutative diagram
\[ \xymatrix{ (X'/S')_{\text{cris}} \ar[r] \ar[d]_{u_{X'/S'}} & (X/S)_{\text{cris}} \ar[d]^{u_{X/S}} \\ \mathop{\mathit{Sh}}\nolimits (X'_{Zar}) \ar[r] & \mathop{\mathit{Sh}}\nolimits (X_{Zar}) } \]
Moreover, this diagram is an example of localization of morphisms of topoi as in Sites, Lemma 7.31.1.
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Maths Isn’t Only Used in the OfficeMaths Isn't Only Used in the Office
Maths Isn’t Only Used in the Office
Math is a game of certainty, not guesswork, and wherever you go in the world, one language will always stay the same: the language of numbers. Whether you were a genius in school or not, we simply
can’t get away from the fact that, even if it’s fundamental, is all around us every day. It’s in our budgets, in the amount of flour that goes into a cake, in calculating how much gas money we need
for a road trip. It’s also in casinos, everywhere you look. While many people would naturally associate math in casinos with a frowned-upon activity like counting cards, that’s not the only place you
see it. Of course, the element of chance and random results in gambling means that it can only take you so far in working out the probability of a win or loss and what you need next to get a little
closer to winning, but it’s still there behind the scenes. You don’t have to be the one who went to university three years early on a scholarship to understand the basic ways that math is used in
gambling. Join us as we take a look at the connection between the two!
The Link
To start to understand the link between math and gambling (which is really the link between quantifiable probability and luck), we must travel back in time to the 16th century and meet a man called
Gerolamo Cardano. Cardano was the author of one of the very first gambling manuals, which outlined ways for gamblers to make their way through something called “sample space.” Let’s look at an
Two dice are able to make a total of 36 combinations when rolled together, but only one of those combinations would produce a result of two sixes.
Some mathematicians cite this as the beginning of the probability theory, which is what all strategic gambling is based upon. Whether it was or wasn’t, this work helped set the course for a whole new
area of mathematical study. Back in the present, players who understand this theory are able to utilize it to minimize their level of risk. Whatever the game, the exact same rules, and laws of
probability will always apply. This might sound a little complicated at first but don’t worry, it’ll get simpler as we explain! A player uses probability theory to ASSESS the risk of the bet that
they might place by taking stock of the PROBABILITY of a win, what the VALUE of that win might be, the DURATION of the game they’re playing, what the VOLATILITY INDEX is for this particular bet and
the BET they are placing.
Most of those terms probably seem quite clear and easy to understand, so we’ll explain what the weirdest term means: volatility index. This is the term used to describe deviation in a mathematical
situation. This volatility index tells a player if they might win more or less than what the “expected” value is. This index helps to essentially quantify luck by telling players what their odds are.
While casino games are all about the odds, there are mathematical strategies that you can apply to help yourself get a clearer idea of what’s going on. Some players just walk into a casino to have a
little fun and see if luck favors them that night, but others play casino games a little more seriously. For example, there are lots of professional poker games in which you would be lost without
strategy! There’s an element of the unknown in gambling that we cannot ever fully quantify, but because there are a finite amount of possible outcomes where numbered dice or cards or random number
generators are involved, math can help us whittle it down and understand it a little bit better. It might seem like complicated stuff at first, but the longer you look at the systems that are applied
here, the more clear they become.
Does Math Really Help?
How much of the potential immediate future can you really “predict” by figuring out what the chances are of various scenarios coming to pass? How much can math really help you in playing casino
games? The answer is a fair amount if it is applied correctly. It’s much more challenging to use it to your advantage in land-based casinos but it is still definitely an option. There are many
tutorials and videos out there that can help you understand probability math and use it to your best advantage in every sphere of life, including casino games.
Wrap Up
Your brain is a wonderful thing, and it’s the best, fastest computer on the earth today. The link between math and gambling is an exciting one to explore with this living computer at your disposal;
will you be exploring it more?
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Electrical engineering - CavIndia
GATE Electrical Engineering
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GATE Electrical Engg Online Test Series
Check Scholarship Coupon Details and Course Duration before purchase. Best GATE Electrical Engg Test Series in India. 60 Electrical Engg, 50 Aptitude, 30 Maths Tests. 5 Mocks held in Dec and Jan.
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You can enrol through this website or through our app using upi, wallet payments, net-banking, debit and credit cards of most banks. If you still have a difficulty, please call or whatsapp on
9930406349 and we will assist you. After enrolment, please fill enrollment form here:
Each of us requires a different kind of study program based upon our style/preference of studying. Normally, all students take our study material and test series. Many also take video lectures as it
helps them clear concepts. A lot depends upon time available to prepare, current stage of preparation, etc. If you are still unsure, please contact us.
Yes, there may be few scholarships available for students from top colleges, students with good grades, students from EWS and for students whose parents are from teaching or defence services. Pls
contact us on 9930406349 via whatsapp with details of course you wish to join and scholarship category needed, along with relevant documents.
As a registered Career Avenues student, you can ask your doubts here and our faculty will get back to you.
Typically 5-6 months are required, but some students need a longer time frame based on other commitments. College students start preparation 12-18 months before GATE to have more time to practise
questions as they may have semester exams as well.
We suggest about 800 to 1200 hours of preparation time overall. This can be divided into 3-4 months or 12-18 months, based on your schedule.
A good Score for GATE Electrical (EE) is Considered To Be: 55
Steps And Strategy To Prepare For GATE Electrical (EE) Exam
1. Take a diagnostic test – best diagnostic test is a GATE paper of any of the previous 3 years.
2. Note down what you have scored and what was the actual GATE qualifying score cut-off. Note that qualification does not help you much. What you need is a good score. So note the good score
mentioned above and measure the gap between your score and a good score.
3. Note the GATE syllabus and mark your topics that you are good at. First try to master subjects that you are good at.
4. However, some subjects like Analog and Digital Electronics, Conrol System and Power Systems have a high weightage. So you should definitely prepare these.
5. General Aptitude does not require preparation. It requires practice. So just practice solving Aptitude questions every day for 30 minutes.
6. Mathematics may have a very high weightage. But note that to get these 6-10 marks, what you have to study and practice is typically more than a core subject. So if you wish to eliminate some
topics in Maths, it is fine. Master topics that you are good at.
7. Take lots of section tests and Mocks. Career Avenues provides an excellent test series for GATE Electrical (EE).Â
8. In case you require focused GATE study material and books, you should take Career Avenues GATE Electrical (EE) study material which has been made by IIT alumni and is focused towards GATE.Â
• Being a GATE aspirant, it is very important that you first know what is the syllabus for GATE Electrical (EE) Examination before you start preparation.
• Keep handy the updated copy of GATE Electrical (EE) Examination syllabus.
• Go through the complete and updated syllabus, highlight important subjects and topics based on Past GATE Electrical (EE) Papers and Weightage plus your understanding of particular subject or
• Keep tracking and prioritizing your preparation-to-do list and the syllabus for the GATE Electrical (EE) examination.
Section I: Engineering Mathematics
Matrix Algebra, Systems of linear equations, Eigenvalues, Eigenvectors.
Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial Derivatives, Maxima and minima, Multiple integrals, Fourier series, Vector identities,
Directional derivatives, Line integral, Surface integral, Volume integral, Stokes theorem, Gauss theorem, Greens theorem.
First order equations (linear and nonlinear), Higher order linear differential equations with constant coefficients, Method of variation of parameters, Cauchys equation, Eulers equation, Initial and
boundary value problems, Partial Differential Equations, Method of separation of variables.
Analytic functions, Cauchys integral theorem, Cauchys integral formula, Taylor series, Laurent series, Residue theorem, Solution integrals.
• Probability and Statistics:
Sampling theorems, Conditional probability, Mean, Median, Mode, Standard Deviation, Random variables, Discrete and Continuous distributions, Poisson distribution, Normal distribution, Binomial
distribution, Correlation analysis, Regression analysis.
Solutions of nonlinear algebraic equations, Single and Multi-step methods for differential equations.
Fourier Transform, Laplace Transform, z-Transform.
Section II: Electric Circuits
Network graph, KCL, KVL, Node and Mesh analysis, Transient response of dc and ac networks, Sinusoidal steady-state analysis, Resonance, Passive filters, Ideal current and voltage sources, Thevenins
theorem, Nortons theorem, Superposition theorem, Maximum power transfer theorem, Two-port networks, Three phase circuits, Power and power factor in ac circuits.
Section III: Electromagnetic Fields
Coulomb’s Law, Electric Field Intensity, Electric Flux Density, Gauss’s Law, Divergence, Electric field and potential due to point, line, plane and spherical charge distributions, Effect of
dielectric medium, Capacitance of simple configurations, Biot-Savarts law, Amperes law, Curl, Faradays law, Lorentz force, Inductance, Magnetomotive force, Reluctance, Magnetic circuits, Self and
Mutual inductance of simple configurations.
Section IV: Signals and Systems
Representation of continuous and discrete-time signals, Shifting and scaling operations, Linear Time Invariant and Causal systems, Fourier series representation of continuous periodic signals,
Sampling theorem, Applications of Fourier Transform, Laplace Transform and z-Transform.
Section V: Electrical Machines
Single phase transformer: equivalent circuit, phasor diagram, open circuit and short circuit tests, regulation and efficiency; Three phase transformers: connections, parallel operation;
Autotransformer, Electromechanical energy conversion principles, DC machines: separately excited, series and shunt, motoring and generating mode of operation and their characteristics, starting and
speed control of dc motors; Three phase induction motors: principle of operation, types, performance, torque-speed characteristics, no-load and blocked rotor tests, equivalent circuit, starting and
speed control; Operating principle of single phase induction motors; Synchronous machines: cylindrical and salient pole machines, performance, regulation and parallel operation of generators,
starting of synchronous motor, characteristics; Types of losses and efficiency calculations of electric machines.
Section VI: Power Systems
Power generation concepts, ac and dc transmission concepts, Models and performance of transmission lines and cables, Series and shunt compensation, Electric field distribution and insulators,
Distribution systems, Per-unit quantities, Bus admittance matrix, Gauss-Seidel and Newton-Raphson load flow methods, Voltage and Frequency control, Power factor correction, Symmetrical components,
Symmetrical and unsymmetrical fault analysis, Principles of over-current, differential and distance protection; Circuit breakers, System stability concepts, Equal area criterion.
Section VII: Control Systems
Mathematical modeling and representation of systems, Feedback principle, transfer function, Block diagrams and Signal flow graphs, Transient and Steady-state analysis of linear time invariant
systems, Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci, Stability analysis, Lag, Lead and Lead-Lag compensators; P, PI and PID controllers; State space model, State transition matrix.
Section VIII: Electrical and Electronic Measurements
Bridges and Potentiometers, Measurement of voltage, current, power, energy and power factor; Instrument transformers, Digital voltmeters and multimeters, Phase, Time and Frequency measurement;
Oscilloscopes, Error analysis.
Section IX: Analog and Digital Electronics
Characteristics of diodes, BJT, MOSFET; Simple diode circuits: clipping, clamping, rectifiers; Amplifiers: Biasing, Equivalent circuit and Frequency response; Oscillators and Feedback amplifiers;
Operational amplifiers: Characteristics and applications; Simple active filters, VCOs and Timers, Combinational and Sequential logic circuits, Multiplexer, DE multiplexer, Schmitt trigger, Sample and
hold circuits, A/D and D/A converters, 8085Microprocessor: Architecture, Programming and Interfacing.
Section X: Power Electronics
Characteristics of semiconductor power devices: Diode, Thyristor, Triac, GTO, MOSFET, IGBT; DC to DC conversion: Buck, Boost and Buck-Boost converters; Single and three phase configuration of
uncontrolled rectifiers, Line commutated thyristor based converters, Bidirectional ac to dc voltage source converters, Issues of line current harmonics, Power factor, Distortion factor of ac to dc
converters, Single phase and three phase inverters, Sinusoidal pulse width modulation.
Here are some recommended books for GATE Electrical Engineering (EE) preparation:
1. Electric Circuits:
□ “Fundamentals of Electric Circuits” by Charles K. Alexander, Matthew N.O. Sadiku.
□ “Engineering Circuit Analysis” by William H. Hayt, Jack E. Kemmerly.
2. Signals & Systems:
□ “Signals & Systems” by Alan V. Oppenheim.
□ “Signals & Systems” by Tarun Kumar Rawat.
3. Analog Circuits:
□ “Microelectronic Circuits” by Adel S. Sedra, Kenneth C. Smith.
□ “Integrated Electronics” by Millman & Halkias.
4. Electrical Machines:
□ “Electrical Machinery” by P.S. Bhimra.
□ “Electrical Machines” by J.B. Gupta.
5. Measurements:
□ “A Course in Electrical and Electronic Measurements and Instrumentation” by A. K. Sawhney, Puneet Sawhney.
□ “Electronic Instrumentation and Measurement Techniques” by William David Cooper.
6. Digital Circuits:
□ “Digital Systems: Principles and Applications” by Ronald J. Tocci, Neal S. Widmer, Greg Moss.
□ “Digital Logic and Computer Design” by M. Morris Mano.
7. Control Systems:
□ “Control Systems Engineering” by I.J. Nagrath, M. Gopal.
□ “Automatic Control Systems” by Benjamin C. Kuo.
8. Electromagnetic Theory:
□ “Engineering Electromagnetics” by William H. Hayt Jr., John A. Buck.
□ “Electromagnetic Waves” by R.K. Shevgaonkar.
9. Power Electronics:
□ “Power Electronics” by P.S. Bimbhra.
□ “Power Electronics: Circuits, Devices, and Applications” by Muhammad H. Rashid.
10. Power Systems:
□ “Electrical Power Systems” by C.L. Wadhwa.
□ “Principles of Power Systems” by V.K. Mehta, Rohit Mehta.
Please note that while these books are recommended, it’s important to refer to the official GATE syllabus and previous years’ question papers for a better understanding of the exam pattern and focus
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Mathematics for the Liberal Arts
Learning Outcomes
• Determine whether a graph has an Euler path and/ or circuit
• Use Fleury’s algorithm to find an Euler circuit
• Add edges to a graph to create an Euler circuit if one doesn’t exist
• Identify whether a graph has a Hamiltonian circuit or path
• Find the optimal Hamiltonian circuit for a graph using the brute force algorithm, the nearest neighbor algorithm, and the sorted edges algorithm
• Identify a connected graph that is a spanning tree
• Use Kruskal’s algorithm to form a spanning tree, and a minimum cost spanning tree
Hamiltonian Circuits and the Traveling Salesman Problem
In the last section, we considered optimizing a walking route for a postal carrier. How is this different than the requirements of a package delivery driver? While the postal carrier needed to walk
down every street (edge) to deliver the mail, the package delivery driver instead needs to visit every one of a set of delivery locations. Instead of looking for a circuit that covers every edge
once, the package deliverer is interested in a circuit that visits every vertex once.
Hamiltonian Circuits and Paths
A Hamiltonian circuit is a circuit that visits every vertex once with no repeats. Being a circuit, it must start and end at the same vertex. A Hamiltonian path also visits every vertex once with no
repeats, but does not have to start and end at the same vertex.
Hamiltonian circuits are named for William Rowan Hamilton who studied them in the 1800’s.
One Hamiltonian circuit is shown on the graph below. There are several other Hamiltonian circuits possible on this graph. Notice that the circuit only has to visit every vertex once; it does not need
to use every edge.
This circuit could be notated by the sequence of vertices visited, starting and ending at the same vertex: ABFGCDHMLKJEA. Notice that the same circuit could be written in reverse order, or starting
and ending at a different vertex.
Unlike with Euler circuits, there is no nice theorem that allows us to instantly determine whether or not a Hamiltonian circuit exists for all graphs.[1]
Does a Hamiltonian path or circuit exist on the graph below?
We can see that once we travel to vertex E there is no way to leave without returning to C, so there is no possibility of a Hamiltonian circuit. If we start at vertex E we can find several
Hamiltonian paths, such as ECDAB and ECABD
Try It
With Hamiltonian circuits, our focus will not be on existence, but on the question of optimization; given a graph where the edges have weights, can we find the optimal Hamiltonian circuit; the one
with lowest total weight.
Watch this video to see the examples above worked out.
This problem is called the Traveling salesman problem (TSP) because the question can be framed like this: Suppose a salesman needs to give sales pitches in four cities. He looks up the airfares
between each city, and puts the costs in a graph. In what order should he travel to visit each city once then return home with the lowest cost?
To answer this question of how to find the lowest cost Hamiltonian circuit, we will consider some possible approaches. The first option that might come to mind is to just try all different possible
question can be framed like this: Suppose a salesman needs to give sales pitches in four cities. He looks up the airfares between each city, and puts the costs in a graph. In what order should he
travel to visit each city once then return home with the lowest cost?
To answer this question of how to find the lowest cost Hamiltonian circuit, we will consider some possible approaches. The first option that might come to mind is to just try all different possible
Brute Force Algorithm (a.k.a. exhaustive search)
1. List all possible Hamiltonian circuits
2. Find the length of each circuit by adding the edge weights
3. Select the circuit with minimal total weight.
Apply the Brute force algorithm to find the minimum cost Hamiltonian circuit on the graph below.
To apply the Brute force algorithm, we list all possible Hamiltonian circuits and calculate their weight:
Circuit Weight
ABCDA 4+13+8+1 = 26
ABDCA 4+9+8+2 = 23
ACBDA 2+13+9+1 = 25
Note: These are the unique circuits on this graph. All other possible circuits are the reverse of the listed ones or start at a different vertex, but result in the same weights.
From this we can see that the second circuit, ABDCA, is the optimal circuit.
Watch these examples worked again in the following video.
Try It
The Brute force algorithm is optimal; it will always produce the Hamiltonian circuit with minimum weight. Is it efficient? To answer that question, we need to consider how many Hamiltonian circuits a
graph could have. For simplicity, let’s look at the worst-case possibility, where every vertex is connected to every other vertex. This is called a complete graph.
Suppose we had a complete graph with five vertices like the air travel graph above. From Seattle there are four cities we can visit first. From each of those, there are three choices. From each of
those cities, there are two possible cities to visit next. There is then only one choice for the last city before returning home.
This can be shown visually:
Counting the number of routes, we can see thereare [latex]4\cdot{3}\cdot{2}\cdot{1}[/latex] routes. For six cities there would be [latex]5\cdot{4}\cdot{3}\cdot{2}\cdot{1}[/latex] routes.
Number of Possible Circuits
For N vertices in a complete graph, there will be [latex](n-1)!=(n-1)(n-2)(n-3)\dots{3}\cdot{2}\cdot{1}[/latex] routes. Half of these are duplicates in reverse order, so there are [latex]\frac
{(n-1)!}{2}[/latex] unique circuits.
The exclamation symbol, !, is read “factorial” and is shorthand for the product shown.
How many circuits would a complete graph with 8 vertices have?
A complete graph with 8 vertices would have = 5040 possible Hamiltonian circuits. Half of the circuits are duplicates of other circuits but in reverse order, leaving 2520 unique routes.
While this is a lot, it doesn’t seem unreasonably huge. But consider what happens as the number of cities increase:
Cities Unique Hamiltonian Circuits
9 8!/2 = 20,160
10 9!/2 = 181,440
11 10!/2 = 1,814,400
15 14!/2 = 43,589,145,600
20 19!/2 = 60,822,550,204,416,000
Watch these examples worked again in the following video.
As you can see the number of circuits is growing extremely quickly. If a computer looked at one billion circuits a second, it would still take almost two years to examine all the possible circuits
with only 20 cities! Certainly Brute Force is not an efficient algorithm.
Nearest Neighbor Algorithm (NNA)
1. Select a starting point.
2. Move to the nearest unvisited vertex (the edge with smallest weight).
3. Repeat until the circuit is complete.
Unfortunately, no one has yet found an efficient and optimal algorithm to solve the TSP, and it is very unlikely anyone ever will. Since it is not practical to use brute force to solve the problem,
we turn instead to heuristic algorithms; efficient algorithms that give approximate solutions. In other words, heuristic algorithms are fast, but may or may not produce the optimal circuit.
Consider our earlier graph, shown to the right.
Starting at vertex A, the nearest neighbor is vertex D with a weight of 1.
From D, the nearest neighbor is C, with a weight of 8.
From C, our only option is to move to vertex B, the only unvisited vertex, with a cost of 13.
From B we return to A with a weight of 4.
The resulting circuit is ADCBA with a total weight of [latex]1+8+13+4 = 26[/latex].
Watch the example worked out in the following video.
We ended up finding the worst circuit in the graph! What happened? Unfortunately, while it is very easy to implement, the NNA is a greedy algorithm, meaning it only looks at the immediate decision
without considering the consequences in the future. In this case, following the edge AD forced us to use the very expensive edge BC later.
Consider again our salesman. Starting in Seattle, the nearest neighbor (cheapest flight) is to LA, at a cost of $70. From there:
LA to Chicago: $100
Chicago to Atlanta: $75
Atlanta to Dallas: $85
Dallas to Seattle: $120
Total cost: $450
In this case, nearest neighbor did find the optimal circuit.
Watch this example worked out again in this video.
Going back to our first example, how could we improve the outcome? One option would be to redo the nearest neighbor algorithm with a different starting point to see if the result changed. Since
nearest neighbor is so fast, doing it several times isn’t a big deal.
We will revisit the graph from Example 17.
Starting at vertex A resulted in a circuit with weight 26.
Starting at vertex B, the nearest neighbor circuit is BADCB with a weight of 4+1+8+13 = 26. This is the same circuit we found starting at vertex A. No better.
Starting at vertex C, the nearest neighbor circuit is CADBC with a weight of 2+1+9+13 = 25. Better!
Starting at vertex D, the nearest neighbor circuit is DACBA. Notice that this is actually the same circuit we found starting at C, just written with a different starting vertex.
The RNNA was able to produce a slightly better circuit with a weight of 25, but still not the optimal circuit in this case. Notice that even though we found the circuit by starting at vertex C, we
could still write the circuit starting at A: ADBCA or ACBDA.
Try It
The table below shows the time, in milliseconds, it takes to send a packet of data between computers on a network. If data needed to be sent in sequence to each computer, then notification needed to
come back to the original computer, we would be solving the TSP. The computers are labeled A-F for convenience.
A B C D E F
A — 44 34 12 40 41
B 44 — 31 43 24 50
C 34 31 — 20 39 27
D 12 43 20 — 11 17
E 40 24 39 11 — 42
F 41 50 27 17 42 —
a. Find the circuit generated by the NNA starting at vertex B.
b. Find the circuit generated by the RNNA.
While certainly better than the basic NNA, unfortunately, the RNNA is still greedy and will produce very bad results for some graphs. As an alternative, our next approach will step back and look at
the “big picture” – it will select first the edges that are shortest, and then fill in the gaps.
Using the four vertex graph from earlier, we can use the Sorted Edges algorithm.
The cheapest edge is AD, with a cost of 1. We highlight that edge to mark it selected.
The next shortest edge is AC, with a weight of 2, so we highlight that edge.
For the third edge, we’d like to add AB, but that would give vertex A degree 3, which is not allowed in a Hamiltonian circuit. The next shortest edge is CD, but that edge would create a circuit ACDA
that does not include vertex B, so we reject that edge. The next shortest edge is BD, so we add that edge to the graph.
We then add the last edge to complete the circuit: ACBDA with weight 25.
Notice that the algorithm did not produce the optimal circuit in this case; the optimal circuit is ACDBA with weight 23.
While the Sorted Edge algorithm overcomes some of the shortcomings of NNA, it is still only a heuristic algorithm, and does not guarantee the optimal circuit.
Your teacher’s band, Derivative Work, is doing a bar tour in Oregon. The driving distances are shown below. Plan an efficient route for your teacher to visit all the cities and return to the starting
location. Use NNA starting at Portland, and then use Sorted Edges.
Ashland Astoria Bend Corvallis Crater Lake Eugene Newport Portland Salem Seaside
Ashland – 374 200 223 108 178 252 285 240 356
Astoria 374 – 255 166 433 199 135 95 136 17
Bend 200 255 – 128 277 128 180 160 131 247
Corvallis 223 166 128 – 430 47 52 84 40 155
Crater Lake 108 433 277 430 – 453 478 344 389 423
Eugene 178 199 128 47 453 – 91 110 64 181
Newport 252 135 180 52 478 91 – 114 83 117
Portland 285 95 160 84 344 110 114 – 47 78
Salem 240 136 131 40 389 64 83 47 – 118
Seaside 356 17 247 155 423 181 117 78 118 –
To see the entire table, scroll to the right
Using NNA with a large number of cities, you might find it helpful to mark off the cities as they’re visited to keep from accidently visiting them again. Looking in the row for Portland, the smallest
distance is 47, to Salem. Following that idea, our circuit will be:
Portland to Salem 47
Salem to Corvallis 40
Corvallis to Eugene 47
Eugene to Newport 91
Newport to Seaside 117
Seaside to Astoria 17
Astoria to Bend 255
Bend to Ashland 200
Ashland to Crater Lake 108
Crater Lake to Portland 344
Total trip length: 1266 miles
Using Sorted Edges, you might find it helpful to draw an empty graph, perhaps by drawing vertices in a circular pattern. Adding edges to the graph as you select them will help you visualize any
circuits or vertices with degree 3.
We start adding the shortest edges:
Seaside to Astoria 17 miles
Corvallis to Salem 40 miles
Portland to Salem 47 miles
Corvallis to Eugene 47 miles
The graph after adding these edges is shown to the right. The next shortest edge is from Corvallis to Newport at 52 miles, but adding that edge would give Corvallis degree 3.
Continuing on, we can skip over any edge pair that contains Salem or Corvallis, since they both already have degree 2.
Portland to Seaside 78 miles
Eugene to Newport 91 miles
Portland to Astoria (reject – closes circuit)
Ashland to Crater Lk 108 miles
The graph after adding these edges is shown to the right. At this point, we can skip over any edge pair that contains Salem, Seaside, Eugene, Portland, or Corvallis since they already have degree 2.
Newport to Astoria (reject – closes circuit)
Newport to Bend 180 miles
Bend to Ashland 200 miles
At this point the only way to complete the circuit is to add:
Crater Lk to Astoria 433 miles. The final circuit, written to start at Portland, is:
Portland, Salem, Corvallis, Eugene, Newport, Bend, Ashland, Crater Lake, Astoria, Seaside, Portland. Total trip length: 1241 miles.
While better than the NNA route, neither algorithm produced the optimal route. The following route can make the tour in 1069 miles:
Portland, Astoria, Seaside, Newport, Corvallis, Eugene, Ashland, Crater Lake, Bend, Salem, Portland
Watch the example of nearest neighbor algorithm for traveling from city to city using a table worked out in the video below.
In the next video we use the same table, but use sorted edges to plan the trip.
Try It
Find the circuit produced by the Sorted Edges algorithm using the graph below.
Spanning Trees
A company requires reliable internet and phone connectivity between their five offices (named A, B, C, D, and E for simplicity) in New York, so they decide to lease dedicated lines from the phone
company. The phone company will charge for each link made. The costs, in thousands of dollars per year, are shown in the graph.
In this case, we don’t need to find a circuit, or even a specific path; all we need to do is make sure we can make a call from any office to any other. In other words, we need to be sure there is a
path from any vertex to any other vertex.
Spanning Tree
A spanning tree is a connected graph using all vertices in which there are no circuits.
In other words, there is a path from any vertex to any other vertex, but no circuits.
Some examples of spanning trees are shown below. Notice there are no circuits in the trees, and it is fine to have vertices with degree higher than two.
Usually we have a starting graph to work from, like in the phone example above. In this case, we form our spanning tree by finding a subgraph – a new graph formed using all the vertices but only some
of the edges from the original graph. No edges will be created where they didn’t already exist.
Of course, any random spanning tree isn’t really what we want. We want the minimum cost spanning tree (MCST).
Minimum Cost Spanning Tree (MCST)
The minimum cost spanning tree is the spanning tree with the smallest total edge weight.
A nearest neighbor style approach doesn’t make as much sense here since we don’t need a circuit, so instead we will take an approach similar to sorted edges.
Kruskal’s Algorithm
1. Select the cheapest unused edge in the graph.
2. Repeat step 1, adding the cheapest unused edge, unless:
3. adding the edge would create a circuit
Repeat until a spanning tree is formed
Using our phone line graph from above, begin adding edges:
AB $4 OK
AE $5 OK
BE $6 reject – closes circuit ABEA
DC $7 OK
AC $8 OK
At this point we stop – every vertex is now connected, so we have formed a spanning tree with cost $24 thousand a year.
Remarkably, Kruskal’s algorithm is both optimal and efficient; we are guaranteed to always produce the optimal MCST.
The power company needs to lay updated distribution lines connecting the ten Oregon cities below to the power grid. How can they minimize the amount of new line to lay?
Ashland Astoria Bend Corvallis Crater Lake Eugene Newport Portland Salem Seaside
Ashland – 374 200 223 108 178 252 285 240 356
Astoria 374 – 255 166 433 199 135 95 136 17
Bend 200 255 – 128 277 128 180 160 131 247
Corvallis 223 166 128 – 430 47 52 84 40 155
Crater Lake 108 433 277 430 – 453 478 344 389 423
Eugene 178 199 128 47 453 – 91 110 64 181
Newport 252 135 180 52 478 91 – 114 83 117
Portland 285 95 160 84 344 110 114 – 47 78
Salem 240 136 131 40 389 64 83 47 – 118
Seaside 356 17 247 155 423 181 117 78 118 –
To see the entire table, scroll to the right
Using Kruskal’s algorithm, we add edges from cheapest to most expensive, rejecting any that close a circuit. We stop when the graph is connected.
Seaside to Astoria 17 milesCorvallis to Salem 40 miles
Portland to Salem 47 miles
Corvallis to Eugene 47 miles
Corvallis to Newport 52 miles
Salem to Eugene reject – closes circuit
Portland to Seaside 78 miles
The graph up to this point is shown below.
Newport to Salem reject
Corvallis to Portland reject
Eugene to Newport reject
Portland to Astoria reject
Ashland to Crater Lk 108 miles
Eugene to Portland reject
Newport to Portland reject
Newport to Seaside reject
Salem to Seaside reject
Bend to Eugene 128 miles
Bend to Salem reject
Astoria to Newport reject
Salem to Astoria reject
Corvallis to Seaside reject
Portland to Bend reject
Astoria to Corvallis reject
Eugene to Ashland 178 miles
This connects the graph. The total length of cable to lay would be 695 miles.
Watch the example above worked out in the following video, without a table.
Now we present the same example, with a table in the following video.
[1] There are some theorems that can be used in specific circumstances, such as Dirac’s theorem, which says that a Hamiltonian circuit must exist on a graph with n vertices if each vertex has degree
n/2 or greater.
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tensor_product(G, H)[source]¶
Return the tensor product of G and H.
The tensor product P of the graphs G and H has a node set that is the Cartesian product of the node sets,
Tensor product is sometimes also referred to as the categorical product, direct product, cardinal product or conjunction.
Parameters: H (G,) – Networkx graphs.
Returns: P – The tensor product of G and H. P will be a multi-graph if either G or H is a multi-graph, will be a directed if G and H are directed, and undirected if G and H are undirected.
Return type: NetworkX graph
Raises: NetworkXError – If G and H are not both directed or both undirected.
Node attributes in P are two-tuple of the G and H node attributes. Missing attributes are assigned None.
>>> G = nx.Graph()
>>> H = nx.Graph()
>>> G.add_node(0,a1=True)
>>> H.add_node('a',a2='Spam')
>>> P = nx.tensor_product(G,H)
>>> P.nodes()
[(0, 'a')]
Edge attributes and edge keys (for multigraphs) are also copied to the new product graph
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Strategies for Scoring High in Parametric Inference
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Key Topics
• Understanding Key Concepts
□ 1. Sufficiency and Sufficiency Theorems
□ 2. Completeness and Completeness Theorems
□ 3. Exponential Families of Distributions
• Point Estimation
□ Criteria for Evaluating Estimators
□ Advanced Topics in Point Estimation
• Bayesian Techniques
• Tests of Hypotheses
• Strategies for Scoring High in Parametric Inference Assignments
• Conclusion
Parametric inference, a cornerstone of statistics, involves making inferences about population parameters based on sample data. Point estimation, a crucial aspect of this field, entails estimating a
single value for an unknown parameter from the available data. In this blog, we will delve into various strategies and concepts to help you understand and excel in point estimation and related topics
in parametric inference. If you're seeking assistance with your Inference assignment, this blog will provide comprehensive insights and strategies to enhance your understanding and proficiency in
these statistical concepts.
Mastering point estimation requires a deep understanding of sufficiency, completeness, exponential families of distributions, and advanced criteria like mean square error, unbiasedness, and
consistency. Additionally, Bayesian techniques, hypothesis testing methods, and strategic study approaches play a vital role in scoring well in assignments. If you're looking for help with your
statistics assignment, mastering these concepts and techniques will be essential for achieving proficiency and success in your studies.
By focusing on these fundamental concepts, practicing their applications through examples, and employing effective study techniques, you can enhance your comprehension and proficiency in point
estimation, enabling you to excel in your parametric inference assignments and examinations.
Understanding Key Concepts
1. Sufficiency and Sufficiency Theorems
Sufficiency is a fundamental concept in statistics that determines if a statistic contains all the information in the sample relevant to the parameter being estimated. A statistic is sufficient if no
other statistic that can be calculated from the same sample provides any additional information about the parameter. This is crucial because it allows us to reduce the amount of data needed for
estimation without losing information.
The Factorization Theorem, a key result in sufficiency theory, provides a formal criterion for identifying sufficient statistics. It states that a statistic is sufficient if the joint probability
distribution of the sample data can be factored into two functions: one that depends on the sample data only through the sufficient statistic and another that depends on the parameter only.
Minimal sufficiency is an extension of sufficiency that indicates a statistic is minimal if it cannot be reduced further to another sufficient statistic. Understanding these concepts will help you
identify the most efficient statistics for your estimations.
2. Completeness and Completeness Theorems
Completeness is another critical property that a family of probability distributions can possess. A family of distributions is complete if the only function of the parameter that has expected value
zero for all distributions in the family is the function that is zero everywhere. Completeness ensures that the best unbiased estimator for the parameter exists and is unique.
Lehmann-Scheffe theorem is a fundamental result that relates sufficiency and completeness. It states that under certain conditions, the best unbiased estimator of a parameter is a function of a
complete, sufficient statistic. This theorem highlights the importance of using complete, sufficient statistics in parametric inference.
Basu's theorem connects sufficiency and independence of a statistic and an ancillary statistic, another vital concept in the theory of estimation. An ancillary statistic is one whose distribution
does not depend on the parameter being estimated, which can simplify the process of estimation.
3. Exponential Families of Distributions
Exponential families of distributions are a class of probability distributions that have a specific form, making them mathematically tractable. They include many commonly used distributions such as
the normal, exponential, and gamma distributions.
These distributions are often characterized by canonical parameters and canonical sufficient statistics, which simplify the process of estimating their parameters. Canonical parameters are natural
parameters that characterize the distribution in the exponential family, while canonical sufficient statistics are sufficient statistics that are natural to the exponential family and often used in
deriving estimators.
Point Estimation
Point estimation is a fundamental concept in statistics that involves the process of using sample data to estimate an unknown parameter of interest with a single value, referred to as a point
estimate. This estimate is crucial in making decisions and drawing conclusions about populations based on sample data. The goal of point estimation is to find the most likely value of the parameter
given the observed data. Point estimates can be evaluated using several criteria, such as mean square error, unbiasedness, relative efficiency, consistency, and more. Understanding these criteria is
essential for determining the reliability and accuracy of point estimates. Moreover, point estimation plays a vital role in various fields, including economics, biology, engineering, and social
sciences, where understanding population parameters is critical for decision-making and policy development.
Criteria for Evaluating Estimators
There are several criteria for evaluating the quality of point estimators:
• Mean Square Error (MSE):MSE is the expected value of the squared difference between the estimator and the parameter.
• Unbiasedness: An estimator is unbiased if its expected value equals the true parameter value.
• Relative Efficiency: Relative efficiency compares the performance of two estimators in terms of their variances.
• Cramer-Rao Inequality: The Cramer-Rao inequality provides a lower bound on the variance of any unbiased estimator.
• Consistency:An estimator is consistent if it converges to the true parameter value as the sample size increases.
Advanced Topics in Point Estimation
Advanced topics in point estimation include:
• UMVUE (Uniformly Minimum Variance Unbiased Estimator): The UMVUE is the estimator that has the smallest variance among all unbiased estimators.
• Rao-Blackwell Theorem: This theorem states that a function of a sufficient statistic that is an unbiased estimator is also the best unbiased estimator (minimum variance estimator) in the class of
unbiased estimators.
• Bayesian Estimation:Bayesian estimation involves using prior beliefs about the parameter to update and obtain a posterior distribution, from which estimates of the parameter can be made.
Bayesian Techniques
Bayesian estimation offers a distinctive approach to parametric inference, integrating prior beliefs about the parameter with observed data to refine those beliefs.
• Priors: These encapsulate existing knowledge or assumptions about the parameter before data collection, serving as the foundation for Bayesian analysis.
• Posteriors: As data is observed, priors are updated to form posteriors, representing the revised beliefs about the parameter based on both prior knowledge and observed data.
• Bayes' Estimators: These estimators are formulated using Bayesian principles, often by minimizing the expected loss based on the posterior distribution. They provide a coherent framework for
decision-making in uncertain environments.
• Bayesian Credible Regions: These regions in parameter space signify intervals with a specified probability of encompassing the true parameter value, offering a measure of uncertainty in Bayesian
Tests of Hypotheses
Statistical hypothesis testing is a fundamental part of parametric inference, involving several key components:
• Hypotheses:In hypothesis testing, we state a null hypothesis (typically denoted as 𝐻0) and an alternative hypothesis (𝐻1or 𝐻𝑎). The null hypothesis represents the status quo or no effect, while
the alternative hypothesis represents what we are testing for.
• Critical Regions: These are defined regions of the sample space that, if the test statistic falls within them, lead to the rejection of the null hypothesis. The size and shape of these regions
are determined by the significance level of the test.
• Neyman-Pearson Lemma:This lemma provides a method for constructing the most powerful tests, which maximize the probability of correctly rejecting the null hypothesis when it is false, subject to
a constraint on the probability of rejecting it when it is true.
• UMP (Uniformly Most Powerful), UMPU (Uniformly Most Powerful Unbiased), and LMP (Locally Most Powerful) tests: These are types of hypothesis tests that are optimal in various senses. UMP tests
are the most powerful tests for a given significance level, UMPU tests are the most powerful unbiased tests, and LMP tests are the most powerful tests in a local region of the parameter space.
• Monotone Likelihood Ratio Family: This refers to a class of distributions that satisfy the monotone likelihood ratio property. This property ensures that the likelihood ratio test is monotonic in
the sense that the test statistic either always increases or always decreases as the sample size increases, simplifying hypothesis testing.
Understanding these components is essential for designing effective hypothesis tests and interpreting their results accurately in parametric inference.
Strategies for Scoring High in Parametric Inference Assignments
Mastering point estimation and related topics in parametric inference requires a combination of theoretical understanding and practical application. Here are some strategies to help you excel in your
• Understand Theoretical Concepts: Focus on understanding the definitions, theorems, and their implications. Practice deriving sufficient statistics, applying the factorization theorem, and
checking the conditions for completeness.
• Practice Calculations:Work through examples to calculate UMVUEs, apply the Cramer-Rao inequality, and understand how to derive and apply Bayesian estimators.
• Review Assumptions:Understand the assumptions under which the theorems and inequalities hold. This will help in applying them correctly and avoiding common mistakes.
• Work with Real-World Data: Apply these concepts to real data sets to gain practical experience and see how the theory applies in practice.
• Review Numerical Methods: Understand computational methods for estimation and hypothesis testing, including numerical optimizations and simulation-based techniques.
• Utilize Resources:Use textbooks, lecture notes, online resources, and practice problems to reinforce your understanding.
Mastering point estimation and related topics in parametric inference is an intricate journey that demands a comprehensive grasp of theoretical underpinnings, adept computational abilities, and the
acumen to apply concepts in real-world scenarios. By immersing yourself in understanding key concepts such as sufficiency, completeness, and Bayesian techniques, and honing your skills through
diligent practice, you can navigate the complexities of parametric inference with confidence. Remember, consistent practice coupled with a resilient attitude towards challenges is paramount for
achieving excellence in this domain. Embrace the learning process, seek out resources for reinforcement, and never shy away from seeking clarification when needed. With dedication and perseverance,
you can unlock the door to success in this challenging yet immensely rewarding field.
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Data collection mechanisms
• Generalizing the censoring problem to other data collection mechanisms.
• Examples of other data collection mechanisms.
Rather than memorizing several data collection mechanisms, it is more important to recognize that it is simply a special (but important) example of probabilistic modelling and the first step of our
Bayesian recipe.
• In censoring, we knew how many \(H_i\)’s were above the detection limit.
• In truncation, a different setup, we now have even less information:
□ we only observe the \(H_i\)’s that are below the limit…
□ …we don’t know how many were above the limit.
• Mathematically, when the \(H_i\) have a continuous distribution this can be modelled as:
\[\begin{align*} X &\sim \text{prior}() \\ N &\sim \text{DiscreteDistribution}() \\ H_1, \dots, H_N &\sim \text{likelihood}(X) \\ I_i &= \mathbb{1}[H_i \le L] \\ Y &= \{ H_i : I_i = 1 \}. \end
• Here \(I_i\) is an “inclusion indicator”.
• Bayesian analysis will be based on \(X | Y\)
Example: CRISPR-Cas9 unique molecular identifier (UMI) family size. “Families of cells” that left zero progenies are not observed!
Non-ignorable missingness
Truncation can be generalized as follows:
• Instead of a deterministic criterion based on \(H_i\) to decide if to include in the set of observations or not,
• make that decision based on some probability model \(p\) that could depend on \(h_i\) and \(x\), \(p(x, h_i) \in [0, 1]\):
\[\begin{align*} X &\sim \text{prior}() \\ N &\sim \text{DiscreteDistribution}() \\ H_1, \dots, H_N &\sim \text{likelihood}(X) \\ I_i &\sim {\mathrm{Bern}}(p(X, H_i)) \\ Y &= \{ H_i : I_i = 1 \}. \
Question: how would you set \(p(x, h)\) to recover truncation as a special case of non-ignorable missingness?
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Jonas Lindstrøm
I recently had to write a simple algorithm that computes the bit-length of an integer (the number of digits in the binary expansion of the integer) given only bitwise shifts and comparison operators.
It is simple to compute the bit-length in linear time in the number of digits of the integer by computing the binary expansion and counting the number of digits, but it is also possible to do it in
logarithmic running time in an upper bound for the bit-length of the integer. However, I wasn’t able to find such an algorithm described anywhere online so I share my solution here in case anyone
else run into the same problem.
The idea behind the algorithm is to find the bit-length of an integer n \geq 0 using binary search with the following criterion: Find the unique m such that n \gg m = 0 but n \gg (m - 1) = 1 where \
gg denotes a bitwise right shift. Note that m is the bit-length of n. Since the algorithm is a binary search, the running time is logarithmic in the maximal length of n.
Below are both a recursive and an iterative solution written in Java. They should be easy to translate to other languages.
Recursive solution
public static int bitLength(int n, int maxBitLength) {
if (n <= 1) {
return n;
int m = maxBitLength >> 1;
int nPrime = n >> m;
if (nPrime > 0) {
return m + bitLength(nPrime, maxBitLength - m);
return bitLength(n, m);
Iterative solution
public static int bitLength(int n, int maxBitLength) {
if (n <= 1) {
return n;
int length = 1;
while (maxBitLength > 1) {
int m = maxBitLength >> 1;
int nPrime = n >> m;
if (nPrime > 0) {
length += m;
n = nPrime;
maxBitLength = maxBitLength - m;
} else {
maxBitLength = m;
return length;
High resolution fractal flames
Fractal flames are a type of iterated function systems invented by Scott Draves in 1992. The fixed sets of fractal flames may be computed using the chaos game (as described in an earlier post), and
the resulting histogram may be visualised as beautiful fractal-like images. If the histogram also has a dimension storing what function from the function system was used to get to a particular point,
it may even be coloured.
There are a lot of software available to generate fractal flames, and I have built yet another one focussed on generating very high resolution images for printing. The image below has resolution 7087
x 7087 and been generated after about 4 hours of computation on a laptop. It is free to use under a Creative Commons BY-NC 4.0 license.
Scott Draves & Eric Reckase (2003), The Fractal Flame Algorithm, https://flam3.com/flame_draves.pdf
On the creation of “The Nørgård Palindrome”
The Nørgård Palindrome is an ambient electronic music track released recently by Morten Bach and me. It is composed algorithmically and recorded live in studio using a lot of synthesizers. It is the
second track of the album, the first being “Lorenz-6674089274190705457 (Seltsamer Attraktor)” which was described in another post.
The arpeggio-like tracks in The Nørgård Palindrome is created from an integer sequence first studied by the danish composer Per Nørgård in 1959 who called it an “infinite series”. It may be defined
a_0 &= 0, \\
a_{2n} &= -a_n, \\
a_{2n + 1} &= a_n + 1.
The first terms of the sequence are
0, 1, -1, 2, 1, 0, -2, 3, -1, 2, 0, 1, 2, -1, -3, 4, 1, 0, \ldots
The sequence is interesting from a purely mathematical view point, which has been studied by several authors, for example by Au, Drexler-Lemire & Shallit (2017). Considering only the parity of the
sequence yields the Thue-Morse sequence, which is a famous and well-studied sequence.
However, we will, as Per Nørgård, use the sequence to compose music. The sequence is most famously used in the symphony “Voyage into the Golden Screen”, where Per Nørgård mapped the first terms of
the sequence to notes by picking a base note corresponding to 0 and then map an integer k to the note k semitones above the base note.
In The Nørgård Palindrome, we do the same, although we use a diatonic scale instead of a chromatic scale, and get the following notes when using a C-minor scale with 0 mapping to C:
It turns out that certain patterns are repeated throughout the sequence, although sometimes transposed, which makes the sequence very usable in music.
In the video below we play the first 144 notes slowly along while showing the progression of the corresponding sequence.
The first 144 notes of Nørgårds’ infinite series mapped to notes in a diatonic scale.
In The Nørgård Palindrome, we compute a large number of terms, allowing us to play the sequence very fast for a long time, and when done, we play the sequence backwards. This voice is played as a
canon in two, and the places where the voices are in harmony or aligned emphasises the structure of the sequence.
The recurring theme is also composed from the sequence using a pentatonic scale and played slower.
The code use to generate the sequence and the MIDI-files used on the track is available on GitHub. The track is released as part of the album pieces of infinity 01 which is available on most
streaming services, including Spotify and iTunes.
On the creation of “Lorenz-6674089274190705457 (Seltsamer Attraktor)”
Lorenz-6674089274190705457 (Seltsamer Attraktor) is an ambient music track released by Morten Bach and me. It was composed algorithmically and recorded live in studio using a number of synthesizers.
This post will describe how the track was composed.
The Lorenz system is a system of ordinary differential equations
\frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma(y - x), \\
\frac{\mathrm{d}y}{\mathrm{d}t} &= x(\rho - z) - y, \\
\frac{\mathrm{d}z}{\mathrm{d}t} &= xy - \beta z.
where \sigma, \rho and \beta are positive real numbers. The system was first studied by Edward Lorenz and Helen Fetter as a simulation of atmospheric convection. It is known to behave chaotically for
certain parameters since small changes in the starting point changes the future of a solution radically, an example of the so-called butterfly effect.
The differential equations above gives a formula for what direction a curve should move after it reaches a point (x,y,z) \in \mathbb{R}^3. As an example, for (1,1,1) we get the direction (0, \rho -
1, 1 - \beta).
In the composition of Lorenz-6674089274190705457 (Seltsamer Attraktor), we chose \sigma = 10, \rho = 28 and \beta = 2 and consider three curves with randomly chosen starting points. The number
6674089274190705457 is the seed of the pseudo-random number generator used to pick the starting points, so another seed would give other starting points and hence a different track.
The curves are computed numerically. Above we show an example of a curve for t \in [0, 5]. The points corresponding to a discrete subset of the three curves we get from the given seed are mapped to
notes. More precisely, we pick the points where t = 0.07k for k \in \mathbb{N}.
We consider the projection of curves to the (x,z)-plane. The part of this plane where the curve resides is divided into a grid as illustrated above. If the point to be mapped to a note is in the
(i,j)‘th square, the note is chosen as the j‘th note in a predefined diatonic scale (in this case C-minor) with duration 2^{-i} time-units. The resulting track is saved as a MIDI-file.
The composition of the track is visualised in a video available on YouTube. Here, all three voices are shown as separate curves along with the actual track.
The Lorenz system and this mapping into musical notes was chosen to give an interesting, and somewhat linear (musically speaking) and continuously evolving dynamic. Using this mapping, the voices
composed moves both fast and slow at different times. The continuity of the curves also ensures that the movement of each voice is linear (going either up or down continuously).
The track is available on most streaming services and music platforms, eg. Spotify or iTunes. The code used to generate the tracks is available on GitHub.
Visualizing fractals with the Chaos Game
Many fractals may be described as the fixed set of an iterated function set (IFS). Perhaps most famously, the Sierpiński Triangle is such a fractal. Formally, an IFS is a set of maps on a metric
space, eg. \mathbb{R}^n, which map points closer to each other.
Hutchinson proved in 1981 that an IFS has a unique compact fixed set S – a set where all points are mapped back into the set. Now, for some choices of IFS on the plane, the set S is very interesting
and shows fractal properties. The Sierpiński Triangle is for example the fixed set of the following IFS:
(x,y) \mapsto \frac{1}{2}(x,y),\\ (x,y) \mapsto \frac{1}{2}(x-2,y), \\ (x,y) \mapsto \frac{1}{2}(x – 1, y – \sqrt{3})
A common way to visualise the fixed set of an IFS is by using the so-called Chaos game. Here, a point in the plane is picked at random. Then we apply one of the functions of the IFS, chosen at
random, to the point. The result is another point in the plane which we again apply one of the function chosen at random on. At each step we plot the point, and we may continue for as long as we like
and with as many initial points as we want.
The Sierpiński Triangle.
Another possible fractal which may be constructed as the fixed set of an IFS is the Barnsley Fern. Here the functions are (with points written as column vectors):
\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 0 & 0 \\ 0 & 0.16 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix},
\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 0.85 & 0.04 \\ -0.04 & 0.85 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix},
\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 0.20 & -0.26 \\ 0.23 & 0.22 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix},
\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} -0.15 & 0.28 \\ 0.26 & 0.24 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}.
Here, the the probability to pick the first map should be 1%, the second should be 85% and the remaining two should be 7% each. This will yield the picture below:
The Barnsley Fern.
A more complicated family of fractals representable by an IFS are the so-called fractal flames. For these fractals, the functions in their corresponding IFS’s are of the form P \circ V \circ T where
P and T are affine linear transformations and V is a non-linear functions, a so-called variation.
A fractal flame.
Slowly transforming the parameters in the transformations of a fractal flame can be used to create movies.
Colouring the fractals may be done in different ways, the simplest being simply plotting each point while iterating in the chaos game. A slightly better way, which is used here, is the log-density
method. Here the image to be rendered is divided into pixels, and the number of times each pixel is hit in the chaos game is saved. Now, the colour of a pixel is determined as the ratio \log n / \log
m where n is the number of times the pixel was hit and m is the maximum number of times a pixel in the image has been hit.
The software used to generate the images in this post is available on GitHub.
On the distribution of expected goals
I have recently read The Expected Goals Philosophy by James Tippett, where the idea behind “Expected goals” is explained very well. In The Expected Goals Philosophy, the probability that a given team
will win based on their expected goals is computed by simulating the game many times, but it can actually be computed analytically which I will do here.
Also in The Expected Goals Philosophy, a phenomenon where a team creating a few good chances will win against a team creating many smaller chances even though the expected number of goals is exactly
the same, is introduced and explained. Here I will give a different, and perhaps more rigorous explanation than the one given in the book to this curious phenomenon.
The distribution of xG
First, lets sum up the concept of expected goals: Given the shots a football team has during a match, each with some probability of ending up as a goal, the expected goals (xG) is the expected value
of the total number of goals which equals the sum of the probabilities of each shot ending up in goal.
A shot with probability p of ending up in goal can be considered to be a Bernoulli random variable, so the expected goals of a team is the sum of many Bernoulli random variables, one for each shot.
It follows that the expected goals of a team has a Poisson binomial distribution.
Consider the following example from The Expected Goals Philosophy. In 2019 Arsenal played Manchester United. The shots taken by each team and their estimated probabilities of ending up in goal are
listed below:
Arsenal shots: (0.02, 0.02, 0.03, 0.04, 0.04, 0.05, 0.06, 0.07, 0.09, 0.10, 0.12, 0.13, 0.76)
Manchester United shots: (0.01, 0.02, 0.02, 0.02, 0.03, 0.05, 0.05, 0.05, 0.06, 0.22, 0.30, 0.43, 0.48, 0.63)
The expected value of a Poisson binomial distribution is the sum of the probabilities of each experiment (shot in this case), so calculating the expected goals for each team is simple: Arsenal has xG
= 1.53 and Manchester United has xG = 2.37. But to consider the distribution of who will win the game, we need to consider the probability mass function of the expected goals which, as we saw, has a
Poisson binomial distrbution.
The pmf of a Poisson binomial distributed random variable X with n parameters p_1, \ldots, p_n (shots with estimated xG’s in this case) may be calculated as follows: The probability that exactly k
shots succeeds is equal to the sum of all possible combinations of k shots succeeding and the remaining n-k shots missed, e.g.
P(X = k) = \sum_{A \in F_{k,n}} \prod_{i \in A} p_i \prod_{j \in F_{k,n} \setminus A} (1-p_j).
Here F_{k,n} is the set of all subsets of size k of \{1,\ldots,n\}. The pmf in this form is cumbersome to compute when the number of parameters (in this case the number of shots) increases. But
luckily there are smarter ways to compute them, eg. a recursive method which is used in the code used to compute the actual distribution:
Now computing the probability of the possible outcomes of the game is straight-forward: For each possible number of goals Arsenal could have scored, we consider the probability that Manchester United
has scored fewer, more or the same amount of goals. And since the event that Arsenal scores for example one goal and the event that Arsenal scores two goals are disjoint, the probabilities may be
summed. Also, the expected goals of the two teams are assumed to be independent, so if we let A denote Arsenals xG and M denote Manchester Uniteds xG we for have:
P(\text{Arsenal wins}) = \sum_{i = 1}^\infty P(A = i) \sum_{j = 0}^{i-1} P(M = j)
Say we consider the event that Arsenal has scored two goals. Then the probability that they will win in this case is equal to the probability that Manchester United scored either a single goal or no
goals. These probabilities are read from the above chart and added: 0.04 + 0.19 = 0.23.
This computation gives us the following probabilities for a win, draw or loose for Arsenal resp.: 0.18, 0.23, 0.59. These numbers are very close to the probabilities given in the The Expected Goals
Philosophy where they were computed running 100.000 simulations.
Skewness of xG
In The Expected Goals Philosophy, a curious phenomenon is presented, namely that a team creating many small chances is more likely to loose to a team creating few large chances, even though the two
teams’ expected number of goals are equal. In the book, the phenomenon is explained by the larger variance of the former teams xG, which is correct, but it is perhaps more precise to say, that it is
due to the skewness of the distribution.
The example from the book is the case where one team, Team Coin, has four shots each with probability 1/2 of ending up in goal, and another team, Team Die Shots, has 12 shots each with probability 1/
6 of ending up in goal. Since the probabilities for each shot ending up in goal are the same in the two cases, the xG for both teams are binomial distributed, which is somewhat simpler than the
Poisson binomial distribution. A plot similar to the one above looks like this:
Note that Team Die’s is skewed to the right. In general, for binomial distributions, the distribution is symmetric if p = 0.5. But if p > 0.5, the distribution is skewed to the left (because the
skewness is negative) and if p < 0.5, the distribution is skewed to the right (because the skewness is positive). In this case, Team Die’s distribution is skewed to the right so it has more of its
mass to the left of the mean, meaning that the probability of scoring few goals is bigger than the probability of scoring more. Team Coin’s distribution, on the other hand, is completely symmetric
(because the skewness is 0), meaning that the probability of scoring fewer goals than the mean is exactly the same as scoring more. Since the mean of the two are the same, the result is that Team
Coin has a higher probability of ending up the winner.
The code for computing the distribution of the outcome of a football game based on the expected goals is available here.
Algorithmic composition with Langton’s Ant
Langton’s Ant is a simulation in two dimensions which has been proven to be a universal Turing machine – so it can in principal be used to compute anything computable by a computer.
The simulation consists of an infinite board of squares which can be either white or black. Now, an ant walks around the board. If the ant lands on a white square, it turns right, flips the color of
the square and moves forward. one square If the square is black, the ant turns left, flips the color of the square and moves forward one square.
When visualised, the behaviour of this system changes over time from structured and simple to more chaotic. However, the system is completely deterministic, determined only by the starting state.
In the video above, a simulation with two ants runs over 500 steps and every time a square flips from black to white a note is played. The note to be played is determined as follows:
• The board is divided into 7×7 sub-boards.
• These squares are enumerated from the bottom left from 0 to 48.
• When a square is flipped from black to white, the number assigned to the square determines the note as the number of semitones above A1.
Seven is chosen as the width of the sub-squares because it is the number of semitones in a fifth, so the ants moves either chromatically (horizontally) or in fifths (vertically). In the beginning,
they are moving independently and very structured, but when their paths meet, a more complex, chaotic behaviour emerges.
Ruffini – abstract algebra in Java
Class inheritance and generics/templates makes it possible to write code with nice abstractions very similar to the abstractions used in math, particularly in abstract algebra, where a computation
can be described for an entire class of algebraic structures sharing some similar properties (eg. groups and rings). This has been described nicely in the book “From Mathematics to Generic
Programming” by Alexander A. Stepanov and Daniel E. Rose.
Inspired by this book, I have implemented a library for computations on abstract algebraic structures such as groups, rings and fields. The library is called Ruffini (named after the italian
mathematician Paolo Ruffini) and is developed in Java using generics to achieve the same kind of abstraction as in abstract algebra, eg. that you do not specify what specific algebraic structure and
elements are used, but only what abstract structure it has, eg. that it is a group or a ring.
Abstract algebraic structures are defined in Ruffini by a number of interfaces extending each other, each describing operations on elements of some set represented by a generic class E. The simplest
such interface is a semigroup:
public interface Semigroup<E> {
* Return the result of the product <i>ab</i> in this
* semigroup.
* @param a An element <i>a</i> in this algebraic structure.
* @param b Another element <i>b</i> in this algebraic
* structure.
* @return The product of the two elements, <i>ab</i>.
public E multiply(E a, E b);
The semigroup is extended by a monoid interface which adds a getIdentity() method and by a group interface which adds an invert(E) method. Continuing like this, we end up with interfaces for more
complicated structures like rings, fields and vector spaces.
Now, each of these interfaces describes functions that can be applied to a generic type E (eg. that given two instances of E, the multiply method above returns a new instance of type E). As in
abstract algebra, these functions can be used to describe complicated computations without specifying exactly what algebraic structure is being used.
Ruffini currently implements a number of algorithms that is defined and described for the abstract structures, including the discrete Fourier transform, the Euclidean algorithm, computing Gröbner
bases, the Gram-Schmidt process and Gaussian elimination. The abstraction makes it possible to write the algorithms only once in a clear, while still being usable on any of the algebraic structures
defined in the library, such as integers, rational numbers, integers modulo n, finite fields, polynomial rings and matrix rings. Below is the code for the Gram-Schmidt process:
public List<V> apply(List<V> vectors) {
List<V> U = new ArrayList<>();
Projection<V, S> projection =
new Projection<>(vectorSpace, innerProduct);
for (V v : vectors) {
for (V u : U) {
V p = projection.apply(v, u);
v = vectorSpace.subtract(v, p);
return U;
Here V and S are the generic types for resp. the vectors and scalars of the underlying vector space, and Projection is a class defining projection given a specific vector space and an inner product.
The library is can be downloaded from GitHub: https://github.com/jonas-lj/Ruffini.
Solving the 0-1 Knapsack problem with Excel
Given a list of items each with a value and a weight, the Knapsack problem seeks to find the set of items with the largest combined value within a given weight limit. There are a nice dynamic
programming solution which I decided to implement in a spread sheet. I used Google Sheets but the solution is exported as an excel-sheet.
The solution builds a Knapsack table for round 0,1,…,limit. In each round r the solution for the problem with limit r is constructed as a column in the table, so the table has to be as wide as the
maximum limit. Once the table is built, the solution can be found using backtracking. This is all described pretty well on Wikipedia, https://en.wikipedia.org/wiki/Knapsack_problem#0/
The main challenge was to translate this algorithm from procedural pseudocode to a spreadsheet. Building the table is simple enough (once you learn the OFFSET command in Excel which allows you to add
or subtract a variable number of rows and columns from a given position), but the backtracking was a bit more tricky.
Assuming that the weights are stored in column A from row 3 and the corresponding values are stored in column B, the table starts in column D. The round numbers are stored in row 1 from columnD. Row
2 are just 0’s and all other entries are (the one below is from D3):
=IF($A3>D$1; D2; MAX(D2; OFFSET(INDIRECT(ADDRESS(ROW(); COLUMN()));-1;-$A3) + $B3))
The table stops after round 40, so we find the solution with weight at most 40.
The backtracking to find the actual solution after building the table is done in column AT and AU. In the first column, the number of weight spent after 40 rounds are calculated from bottom to top
row using the formula (from AT3):
=IF(OFFSET(INDIRECT(ADDRESS(ROW(); COLUMN()));0;-AT23-2) <> OFFSET(INDIRECT(ADDRESS(ROW(); COLUMN()));-1;-AT23-2); A22 + AT23; AT23)
The solution is shown in column AU where for each row we simply check if the accumulated weight increased with this item.
Gram-Schmidt process for integers
Given a set of linearly independent vectors, the Gram-Schmidt process produces a new set of vectors spanning the same subset, but the new set of vectors are now mutually orthogonal. The algorithm is
usually stated as working over a vector space, but will in this post be altered to work over a module which is a vector space where the scalars are elements in a ring instead of a field. This could
for instance be the case if we are given a set of integer vectors we wish to orthogonalise.
The original version can be described as follows: Given a set of vectors v_1, \ldots, v_n we define u_1 = v_1 and
u_k = v_k - \text{proj}_{u_1}(v_k) - \text{proj}_{u_2}(v_k) - \cdots - \text{proj}_{u_{k-1}}(v_k)
for k = 2,3,\ldots,n. The projection operator is defined as
\text{proj}_u(v) = \frac{\langle u, v \rangle}{\langle u, u \rangle} u.
If we say that all v_i have integer entries, we see that u_i must have rational entries, and simply scaling each vector with a multiple of all the denominators of the entries will give a vector
parallel to the original vector but with integer entries. But what if we are interested in an algorithm that can only represent integers?
The algorithm is presented below. Note that the algorithm works over any module with inner product (aka a Hilbert Module).
# Given: Set of vectors V.
# Returns: A set of mutually orthogonal vectors spanning
# the same subspace as the vectors in V.
U := Ø
k := 1
while V ≠ Ø do
poll vector v from V
w := kv
for (u,m) in U do
w -= m〈v,u〉u
# Optional: Divide all entries in w by gcd of entries
if V ≠ Ø do
n :=〈w,w〉
for (u,m) in U do
m *= n
Put (w,k) in U
k *= n
else do
# No more iterations after this one
Put (w,k) in U
return first coordinates of elements in U
When used on integers, the entries in the vectors grow very fast, but this may be avoided by dividing w by the greatest common divisor of the entries.
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Asimov’s Biographical Encyclopedia of Science and Technology
I am taking a class on linear algebra at the moment, which means that I am breathing in matrices and Gaussian elimination on a daily basis. There is a footnote in our textbook where row reduction
algorithms are discussed: "The algorithm here is a variant of what is commonly called Gaussian elimination. A similar elimination method for linear systems was used by Chinese mathematicians in about
250 B.C. The process was unknown in Western culture until the nineteenth century, when a famous German mathematician, Carl Friedrich Gauss, discovered it. A German engineer, Wilhelm Jordan,
popularized the algorithm in an 1888 text on geodesy" (Linear Algebra and its Applications, 4th edition, by David C. Lay - 2012 - p. 12). Geodesy is mathematically determining the size and shape of
the Earth. I remembered Gauss's name coming up in my statistics class; it turns out his contributions to both linear algebra and statistics were secondary byproducts of his interest in geodesy. I
decided to look him up in my reference works and discovered that his contributions to mathematics and science, in general, were so great that Gaussian elimination isn't even mentioned.
Even though it was just published 32 years after Gauss's death, Johnson's Universal Cyclopaedia (1887) features a compact summary of his life and accomplishments: Gauss (Karl Friedrich), b. in
Brunswick, Germany, Apr. 30, 1777; was educated at the expense of the duke of Brunswick, who had heard of his precocious mathematical talents; solved when eighteen years old the problem of the
division of the circle into seventeen equal parts, and afterwards became famous for skill in the indeterminate analysis and in curious numerical questions; demonstrated Fermat's theorem; became in
1807 professor of astronomy at Göttingen and director of the observatory; received in 1810 the Lalande medal for calculating by a new method the orbits of Ceres and Pallas; was made in 1816 a court
councillor, and in 1845 a privy councillor of Hanover; made after 1821 important improvements in geodetic methods and instruments; after 1831 devoted much attention to terrestrial magnetism. D. at
Göttingen Feb 23, 1855. Gauss is regarded as one of the first mathematicians of this century (vol. 3, p. 403).
The 11th edition Encyclopaedia Britannica (1910) is a bit more thorough. The article on Gauss can be found in volume 11, beginning with the fact that he was "born of humble parents" (535). Some fun
extracts (pp. 535-536):
In 1807 he was appointed director of the Göttingen observatory, an office which he retained to his death: it is said that he never slept away from under the roof of his observatory, except on one
occasion, when he accepted an invitation from Baron von Humboldt to attend a meeting of natural philosophers in Berlin. [...] With [Wilhelm] Weber's assistance he erected in 1833 in Göttingen a
magnetic observatory free from iron (as Humboldt and F. J. D. Arago had previously done on a smaller scale), where he made magnetic observations, and from this same observatory he sent telegraphic
signals to the neighboring town, thus showing the practicality of an electromagnetic telegraph. [...] Running through these volumes in order, we have in the second the memoir, Summatio quarundam
serierum singularium, the memoirs on the theory of biquadratic residues, in which the notion of complex numbers of the form a + bi was first introduced into the theory of numbers; and included in the
Nachlass are some valuable tables. That for the conversion of a fraction into decimals (giving the complete period for all the prime numbers up to 997) is a specimen of the extraordinary love which
Gauss had for long arithmetical calculations, and the amount of work gone through in the construction of the table of the number of the classes of binary quadratic forms must also have been
A much longer entry on Gauss appears in the Macropaedia section of the 15th-edition Encyclopaedia Britannica (here the 1997 printing, volume 19, pp. 697-698). This article comments a little bit on
his personal life outside of his mathematical discoveries, and also mentions his contributions to statistics (couched in the context of his interest in geodesy); here is a fraction of the details
found within:
His own dictum, "Mathematics, queen of the sciences, and arithmetic, the queen of mathematics," aptly conveys his perception of the pivotal role of mathematics in science. [...] His first wife died
in 1809, after a marriage of four years and soon after the birth of their third child. From his second marriage (1810-31) were born two sons and a daughter. [...] By introducing what is now known as
the Gaussian error curve, he showed how probability could be represented by a bell-shaped curve, commonly called the normal curve of variation, which is basic to descriptions of statistically
distributed data. [...] The most important result of their [Weber + Gauss's] work in electromagnetism was the development, by other workers, of electric telegraphy. Because their finances were
limited, their experiments were on a small scale; Gauss was rather frightened at the thought of worldwide communication. [...] Teaching was his only aversion, and, thus, he had only a few students.
Instead he effected the development of mathematics through his publications, about 155 titles, to which he devoted the greatest care. Three principles guided his work: "Pauca, sed matura" ("Few, but
ripe"), his favourite saying; the motto "Ut nihil amplius desiderandum relictum sit" ("That nothing further remains to be done"); and his requirement of utmost rigour. It is evident from his
posthumous works that there are extensive and important papers that he never published because, in his opinion, they did not satisfy one of these principles. He pursued a research topic in
mathematics only when he might anticipate meaningful relationships of ideas and results that were commendable because of their elegance or generality.
A fun text in general for looking at the lives of great scientists and mathematicians comes from another prolific writer, Isaac Asimov, in Asimov's Biographical Encyclopedia of Science and
Technology: The Lives & Achievements of 1195 Great Scientists from Ancient Times to the Present; I have the revised version from 1972. The entries in this book are arranged chronologically, but there
is a handy alphabetical index at the front of the book, which helped me quickly locate the biography of Gauss on pages 249-251. My favorite quips:
At the age of three, he was already correcting his father's sums, and all his life he kept all sorts of numerical records, even useless ones such as the length of the lives of famous men, in days. He
was virtually mad over numbers. [...] All of this was not without a price, for his intense concentration on the great work that poured form him withdrew him sometimes from contact with humanity.
There is a story that when he was told, in 1807, that his wife was dying, he looked up from the problem that engaged him and muttered, "Tell her to wait a moment till I'm through." [...] His agile
mind never seemed to cease. At the age of sixty-two he taught himself Russian. [...] Each of his two wives died young and only one of his six children survived him. His life was filled with personal
tragedy, and though he died wealthy, he also died embittered.
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Upcoming Events | Mathematics
Main content start
prove theorem 1.5, following sections 4.3–4.5
Clifford Taubes (Harvard University)
The Vafa Witten equations on 4-manifolds are the variational equations of a functional that generalizes one of the Chern-Simons functionals for SL(2;C) connections on 3-manifolds (and it reduces to
that on products of a 3-manifold with the circle). Being that the moduli space of solutions…
Cheng-Chiang Tsai (Academia Sinica)
Moy-Prasad filtration subgroups are generalization of congruence subgroups for $GL_n(Q_p)$ to a general $p$-adic reductive group $G(F)$. Moy-Prasad proved that any irreducible smooth representation
of $G(F)$ has its restriction to a Moy-Prasad subgroup given by an irreducible representation (…
Jonathan Tidor (Stanford)
Semialgebraic graphs are a convenient way to encode many problems in discrete geometry. These include the Erdős unit distance problem and many of its variants, the point-line incidence problems
studied by Szemerédi–Trotter and by Guth–Katz, more general problems about incidences of…
Let W be a complete finite type Liouville manifold. One can associate to each closed subset K of W that is conical at infinity an invariant SH_W(K). I will first explain the construction of SH_W(K)
and note how it recovers known invariants through special choices of K. Then, I will prove a big…
Josef Greilhuber (Stanford)
The theme for Student Analysis in the second half of fall quarter is geometric wave equations and wave maps. This will be the third talk on this theme.
Deding Yang (Peking University)
Sebastian Haney (Columbia)
One of the earliest achievements of mirror symmetry was the prediction of genus zero Gromov-Witten invariants for the quintic threefold in terms of period integrals on the mirror. Analogous
predictions for open Gromov-Witten invariants in closed Calabi-Yau threefolds can be …
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231.Scientific proof
I have contacted some kabbala students who consider themselves to be knowers to the point that they have set up the Sanhedrin, a name given in the Mishna to a counsel of seventy-one Jewish sages,
members are not elected, nor are their positions permanent. Any scholar, at any time, may gain a place on the legislature by proving a greater level of scholarship in Jewish law than a current
I did so after an article which mentioned re-establishing the temple and it showed a vital mistake. In order to rebuild the temple, they would need to know how to build it in accordance with God’s
laws, meaning that it has to be in harmony with his ratio’s and measurements. When I pointed it out they were rude and unwilling to listen because they are the authority, sadly this is not the case.
They have a representation of the tree of knowledge and the tree of life which is close to the right structure but as it was passed on by those who were not knowledgeable some of the vital things
were lost and building on an incorrect structure literally brings the house down.
Anyone who has seriously studied Stan Tenen of the Meru foundation on the Hebrew letters will also know of the court cases against Dan Winter and material presented by Drunvalo Melchizadek. Words
such as unscientific, masquerades as a scientist and obsessed, falsified mathematics etc. are being used, so I contacted both Stan Tenen and Dan Winter on the one hand pointing out that none of them
have any rights of ownership and pointing out that they only have rudimentary bits of a greater whole, a structure, the only structure that contains all, but not based on letters but on numbers. This
was Dan Winter’s reply: I would suggest it is essential to graduate from numerology (appropriately not respected by science) to what is measurable. Namely converting the geometric ratios to the
physics of electric field symmetry, that is key to bio-activity. When you know how to identify a produced electric field which causes a seed to germinate by its frequency signature, then there is
something to share.
According to him just about everything is self-organized electric phase conjugation, it shows not only did he judge without seriously looking into it but as I have explained before looked with
conditioned eyes, while openness caused them to find/see something ego/personality near to directly closes their eyes again for truth and think that their finding and only their finding, holds the
answer. And again in his article today he mentions the frequency I have mentioned before 1356. That will release hydrogen from h20 linking it to the golden ratio and the Planck length but without
knowing where these numbers really come from it does not mean a lot.
Someone who realises that is J.Luliano:
There is a most profound and beautiful question associated with the observed coupling constant, e the amplitude for a real electron to emit or absorb a real photon.
It is a simple number that has been experimentally determined to be close to -0.08542455. My physicist friends won’t recognize this number, because they like to remember it as the inverse of its
square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical
physicists put this number up on their wall and worry about it. Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural
logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we
don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this
number come out, without putting it in secretly!
Now let’s look at what he has found and the number he based it on and where it comes from and after that I will show you how it is interlocked, where it comes from and while it is shown to play a
role in the micro cosmos. I I would also link it to the macro cosmos.
For this we need to go to the kabbalah (it will help you to have the picture of the star of Bethlehem, tree of life and knowledge next to it when reading the following).
The vessels of the three Sephiroth Kether . Hokmah and Binah at first performed well in the task of holding the light, but when the light poured down through the lower vessels, from Hesed through
Yesod, these six lower vessels shattered and were dispersed in to the chaotic void of the Tehiru. This was the Shebirat ha kelim, the breaking of the vessels, the original vessels were in what is now
the world of Atziluth, but when the light from above penetrated the Sephira Malkuth, this shattered into 288 sparks which failed to return to the primordial source but instead fell through the worlds
and became attached and trapped in the broken fragments of the vessels which formed the Kelipoth, the shells or husks. These husks became the evil forces of the Sitra Ahra, the other or under world,
preventing the return of the sparks of divine light to its source, thus the light or energy of creation fell into matter.
To help you a little, the top star of David/hexagram has two pyramids, one pointing up and one down. The one pointing up, the top , receives the light and spreads out to the pyramid its lower two
points and reflects back but when it breaks through into the lower part (the pentagram) it is scattered /divided onto the 288 degrees. And as you know the top has 288 degrees and the bottom has 288
degrees that make up 576 degrees or the shape of the eternity 8 this 288 multiplied by 288 gives you 82944.
More on this later, but we now move to show how this number is connected to the scientific constants.SEPHIRA MALKUTH: “288 sparks from broken vessels” sqrt 82944 = 288
288^2 = 82944
DG Leahy: 82944 and the four forces of nature
The Feigenbaum number is derived directly out of the 82944 formula:
[tan^-1[1/(.367872976^Pi)]] + Pi = 4.669204313…..tan^-1 in radians
actual = 4.669201609…
4.669201609/4.669204313 = 99.999942%
The golden section is derived out of fine-structure:
[[[COS^-1 ((1.618033989-1)/2)]*4] ^ (2/Pi)] / 100 = .367872976
COS^-1 in degrees
The deepest meaning of the 1/Pi (1/3.141592654) phenomena is as the
supreme exponential link to the “hard” constants of Nature:
FEIGENBAUM CONSTANT = 4.669201609…rule of order in chaotic systems
F= 4.6692043132…..tan in radians
100/[(tan(F-Pi)…………………….^ (1/Pi)] = 36.7872976…
GOLDEN MEAN = 1.618033989…rule of order in living forms systems
Phi = 1.618033989 = (1+sqrt5)/2…COS^-1 in degrees
[COS^-1[((sqrt 5 -1)/4)*1152]……..^ (1/Pi) = 36.7872976…
FINE-STRUCTURE CONSTANT = a(em) =amplitude for an electron-photon exchange, rule of proton-electron interactions.
a(em) = 137.036000986…cos in radians
(cos 1/(a(em))*100……………………………= 36.7872976…
COLLECTIVE UNCONSCIOUS = 82944 = (288^2)…rule of brain/nervous/systems/symbolic
Prof. Leahy =(82944 logic) and Sephira Malkuth=(288 sparks)
82944…………………………….. ^ (1/Pi) = 36.7872976…
COLLECTIVE UNCONSCIOUS: Egyptian form, Cheops pyramid, Beta=.37
height = 486.256005976 feet = ht
base leg = 763.81 feet…cos in radians = bl
[cos [(10^(2*ht/bl))/(.37^2)]] * 100……..= 36.7872976…
COLLECTIVE UNCONSCIOUS : Pi form…in radians
[cos[(10^((2/Pi)+(2/Pi))]/(.370000606^2)]*100 = 36.7872976…
COLLECTIVE UNCONSCIOUS: Christian(666), Hebrew(288) form
cos in radians…
cos[(10^(287.999975988/37))/(666^2)]……..= 36.7872976…
NATURES CHARGED MASSES: proton, positive; electron, negative
pmev = proton million electron volts = 938.271998 (NIST 1998)
emev = electron million electron volts = .5109994691
[(10*pmev)^[(1/Pi)*(emev^-(1/Pi))]]……………………..= 36.7872976…
NATURES CHARGED MASSES: and the collective unconscious as the electrons carrier
pmev = 938.271998
emev = .51100091734
pmev / (emev^2) / 6.66…………………………= 36.7872976…
COLLECTIVE UNCONSCIOUS ROOT = 144/37and 1/Pi
[10^((1/Pi)+(1/Pi))] *1800 = 144.000014266/37
NATURES STRONG FORCE: a(s) = 14 and the fermat form: e^(Pi+8)
a(s) = 13.99995242
[(10^((2/Pi)+(2/Pi))] * a(s) = sqrt (e^(Pi+8)..e=2.7182818…
NATURES CHARGED/UNCHARGED PAIR: neutron-proton, harmonic mean of 57 and 37.
nmev = neutron = 939.5653313 (NIST 1998 = 939.56533mev
pmev = proton
(2109/940) ^ (1/Pi) = nmev – pmev = [[(1/57+1/37)^-1]/10] ^ (1/Pi)
NATURES CHARGED/UNCHARGED PAIR: fine-structure constant and Fermat form
[[sqrt[(e^(Pi+8)) * a(em)]]/10] ^ (1/Pi) = nmev- pmev
NATURES CHARGED/UNCHARGED PAIR: strong , fine-structure forces
a(s) = 13.99990799
[[[10^((2/Pi)+(2/Pi))] * a(s) * sqrt a(em) ]/10] ^ (1/Pi) = nmev – pmev
NATURES SECOND-ORDER PHASE TRANSITION, BETA (.37) and a(em)
[10^((2/Pi)+(2/Pi))] * a(em) = .370000605064 ^ 2
NATURES DENSEST OBJECT:PLANCK MASS = Planck mass with Planck length as radius
a(s) = 14.00051693 =strong nuclear force
M(p) = Planck mass = 2.17656..* (10^-8) kilograms (1998 NIST)
1/[[10^((4/Pi)+(4/Pi)))] * (a(s)^2) * 666] = 2.1765 * (10^-8)
GRAVITATIONAL FORCE: G(n) newtons = 6.6739*(10^-11) mks
a(s) = 14.00051693
h =plancks constant = 6.626068758*(10^-34) joules (1998 NIST)
c = speed of light = 299792458 m/s (1998 NIST)
h * c *(10^(16/Pi)) * (a(s)^4) * (666^2) / 2 / Pi = 6.6739* (10^-11)
NUCLEAR WEAK FORCE: fermi coupling-charge =G(w) =.0000116639 F(m)
through the Feigenbaum constant:
emev = .510998902
pmev = 938.271998
F = Feigenbaum constant = 4.669201609
G(w) = fermi-coupling charge = (1998 NIST = .0000116639 F(m))
emev / pmev / F / 10 = .0000116640288484
J.Iuliano please look at http://dgleahy.com/dgl/p22.html
Having shown where this number comes from I will also show you how it relates to the zodiacal cycle, the zodiac and its 12 signs are 25920 years, it is also a circle of 360 degrees of 52,36 cubits
which makes 188496. Within this circle you have the eight of 576 degrees and the famous 18 times makes 82944. But it is also known that the tree has 32 paths so when we divide 188596 by 32 we get
58905 which is half the centre of the ark of Noah 11781. If we divide 25920 by 81 we get 320, and if we multiply 58905 by 320 we get 188496 and if we multiply 320 by 81 we get 25920 again. I have
shown you that the Torah provides us with both 288 and 3125, now if we divide 25820 by 31,25 we get 82944, if we multiply 188596 by 3125 we get 58905. Having explained that the clay tablet of the
Sumerians tells us: you do not know the value of 18, then let us multiply the Torah number 31,25 times 18 and the answer is 5625 which is the ARK OF COVENANT.
If we divide 82944 by the cubit 5236 we get 1584.11 which is half 3168.22 which is the value of lord Jesus Christ. If multiplied by 7 we get 11088.77 which is the two trees and the two sevens as
explained in other articles. If we divide 11088 by the 77 we get our 144.
There is the fine structure constant (e^2)/2/E/h/c=1.3703596. A number that every physicist has written on his blackboard or kept in the back of his mind, they find it but do not know where this
number comes from, in other words they see it to be a constant but do not know the cause of it. A little tip here, the full number is 2 leaves 62964(0.4) times 14 is 881496 (188496) and 81 times is
51——84 twice 5184 is 10368 times 8 is 82944.
The 3-4-5 right triangle reflects an angle of 52 degrees 36 minutes.
The cubit 52.36 52×36 is 1872 18×72 is 1296 the difference between the two is 576. The eight. 12X96 is 1152 which is two times 576, the tree of good and evil. And 5+7+6 is of course the Sumerian 18.
576 x 5625 is 3240000.
Matter, cycles, the uni-verse a uni-ting verse/song.
Moshiya van den Broek
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s & Reflection
Solutions to Lab 8 exercises & Reflection¶
Written solutions¶
The written solutions for this week can be download as PDF below.
Video explanations of some solutions¶
Videos for questions 2--4 and 6 are shown in the hints page.
If you want a refresher on what logarithms are then please watch this explanation:
Please let me know if you want clarification on any other exercises.
Why do we use big-O notation instead of actual CPU time on actual machines?
In fact, if you were comparing a few different algorithms then it makes perfect sense to compare actual code implementations of the algorithms at hand. These time measurements would be very useful
practically because they include real-world factors such as the CPU registers sizes, the cache sizes, etc.
So why do we still use big-O notation then?
This practical approach suffers from being "local", in the sense that it is very accurate for your local machines in space and time, and in the historical timeline.
Imagine you find a very good paper from 50 years ago but where the code was run on a machine from those days -- how useful would the time measurements shown there be to you nowadays?
The paper is also quite likely to have tested the algorithms on relatively small parameter sizes. Your machine is probably able to handle larger sizes, but this "practical paper" gives you little
insight into what to expect.
This is why we prefer the use of big-O notation. The theoretical analysis of the algorithm should still apply 50 years on, as long as the general architecture of the machine is fairly similar.
The theoretical big-O analysis also identifies the important parameters affecting the complexity. It shows how these parameters affect the cost (polynomially or super-polynomially).
You don't have to hand-in your reflection -- this is not an assessment. Keep your notes and go over them as you understand the material more. Some of the above ideas will become clear in one week,
while others will be met again towards the end of the module!
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Diagnostic Medicine/Pathobiology
If no dates are listed directly under a course, then the course meets Tuesday, January 16, 2024 through Friday, May 3, 2024.
DMP313Introduction to Epidemiology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 14914 3 T U 10:00-11:15 a.m. LSP 123 books Cernicchiaro, Natalia None
DMP314Environmental and Public Health
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 14888 3 T U 4:30-6:00 p.m. LSP 127 books Kastner, Justin Jon None
DMP611Cow Calf Health Systems
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 11877 2 M W 9:30-10:20 a.m. CL 205 books Hanzlicek, Gregg Alan None
DMP680Prb/Pathobiology - Top/Vary by Student
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A IND 12004 1-5 Appointment books Nagaraja, T.G. None
• Enrollment restrictions: Instructor consent
DMP802Environmental Health
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 14179 3 M 5:30-8:00 p.m. VCS N202 books Kincaid, Margaret Mercedes None
DMP814Veterinary Bacteriology & Mycology Lecture
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 15759 3 books None
Meets January 16 - May 3, 2024: M W 8:00-8:50 a.m. VMT 301 Chengappa, M M
Nagaraja, T.G.
Meets January 16 - May 3, 2024: U 1:00-1:50 p.m. VMT 301 Chengappa, M M
Nagaraja, T.G.
DMP815Multidisciplinary Thought and Presentation
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
ZA REC 14259 3 Distance books Kastner, Justin Jon None
• Enrollment restrictions: Instructor consent
DMP816Trade and Agricultural Health
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
ZA LEC 13986 2 Distance books Kastner, Justin Jon None
DMP817Principles of Veterinary Immunology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 10575 3 M W F 11:00-11:50 a.m. VMT 301 books Mwangi, Waithaka None
DMP818Veterinary Epidemiology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 10576 2 M W 9:00-9:50 a.m. VMT 301 books Renter, David Gregory None
Hanthorn, Christy J
DMP831Veterinary Virology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
B LEC 16007 3 books None
DMP840Public Health Practice
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A FLD 11934 3-6 Appointment books Mulcahy, Ellyn None
• Enrollment restrictions: Instructor consent
DMP841Veterinary Public Health
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 10579 2 books None
Meets January 3 - April 21, 2024: T 10:00-10:50 a.m. VCS A101 Mulcahy, Ellyn
Meets January 3 - April 21, 2024: F 11:00-11:50 a.m. VCS A101 Mulcahy, Ellyn
• Section meets from January 3 through April 21, 2024.
DMP854Intermediate Epidemiology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 11988 3 W 3:30-5:00 p.m. VMS 343 books Sanderson, Michael W None
DMP857Systemic Pathology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
01B LAB 10577 M W F 9:00-10:50 a.m. VMT 204 books Mosier, Derek A None
A LEC 10578 5 M W F 8:00-8:50 a.m. VMT 201 books Mosier, Derek A None
DMP870Pathobiology Seminar
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A SEM 10584 1 F 8:30-9:20 a.m. VCS N202 books Larson, Haley Elizabeth None
Dhakal, Santosh
DMP880Prb/Pathobiology - Top/Vary By Student
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A IND 10585 1-6 Appointment books Nagaraja, T.G. None
• Enrollment restrictions: Instructor consent
DMP880Prb/Pathobiology - Top/Pathobiology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
ZA IND 15062 1 Distance books Mulcahy, Ellyn None
• Enrollment restrictions: Instructor consent
DMP888Globalization, Cooperation, and the Food Trade
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
ZA LEC 13988 1 Distance books Kastner, Justin Jon None
DMP895Top/Pathobiology - Top/Vary By Student
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A IND 10586 2 Appointment books Nagaraja, T.G. None
• Enrollment restrictions: Instructor consent
DMP898MS Report in Pathobiology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A IND 14901 2 CNC Appointment books Nagaraja, T.G. None
DMP899Ms Research in Pathobiology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A RSH 10587 0-18 Appointment books Mosier, Derek A None
DMP910Pathogenic Mechanisms of Viruses
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 16706 3 Appointment books Vanlandingham, Dana L None
Miller, Laura Caldwell
DMP970Pathobiology Seminar
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A SEM 10588 1 F 8:30-9:20 a.m. VCS N202 books Larson, Haley Elizabeth None
Dhakal, Santosh
DMP980Prb/Pathobiology - Top/Vary By Student
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A IND 10589 1-6 Appointment books Nagaraja, T.G. None
• Enrollment restrictions: Instructor consent
DMP999Phd Research in Pathobiology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A RSH 10590 1-18 Appointment books Mosier, Derek A None
VDMP811Clinical Pathology I
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 15811 2 T 1:00-4:50 p.m. VMT 301 books Pohlman, Lisa M None
VDMP814Veterinary Bacteriology & Mycology Lecture
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 15101 3 books None
Meets January 16 - May 3, 2024: M W 8:00-8:50 a.m. VMT 301 Chengappa, M M
Nagaraja, T.G.
Meets January 16 - May 3, 2024: U 1:00-1:50 p.m. VMT 301 Chengappa, M M
Nagaraja, T.G.
VDMP817Principles of Veterinary Immunology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 14893 3 M W F 11:00-11:50 a.m. VMT 301 books Mwangi, Waithaka None
VDMP818Veterinary Epidemiology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 14892 2 M W 9:00-9:50 a.m. VMT 301 books Renter, David Gregory None
Hanthorn, Christy J
VDMP841Veterinary Public Health
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A LEC 14910 2 books None
Meets January 16 - May 3, 2024: T 3:00-3:50 p.m. VCS A101 Mulcahy, Ellyn
Meets January 16 - May 3, 2024: F 11:00-11:50 a.m. VCS A101 Mulcahy, Ellyn
B LEC 16834 2 books None
Meets January 2 - April 19, 2024: T 10:00-10:50 a.m. VCS A101 Mulcahy, Ellyn
Meets January 2 - April 19, 2024: F 11:00-11:50 a.m. VCS A101 Mulcahy, Ellyn
• Section meets from January 2 through April 19, 2024.
VDMP857Systemic Pathology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
01B LAB 14904 M W F 9:00-10:50 a.m. VMT 204 books Mosier, Derek A None
A LEC 14905 5 M W F 8:00-8:50 a.m. VMT 201 books Mosier, Derek A None
VDMP888Globalization, Cooperation, and the Food Trade
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
B LEC 15844 1 U 3:00-3:50 p.m. VCS N202 books Kastner, Justin Jon None
VDMP891DMP Vet Med Elective - Top/Lab Animal Sciences
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
D LEC 14992 1-3 T F 1:00-1:50 p.m. VCS N202 books Olson, Sally Ann None
VDMP891DMP Vet Med Elective - Top/Ecotoxicology
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
E LEC 15842 1 T 4:00-4:50 p.m. VMT 201 books Mosier, Derek A None
VDMP902Diagnostic Medicine
Section Type Number Units Basis Days Hours Facility Books Instructor K-State 8
A REC 10581 3 Appointment books Plattner, Brandon Lee None
• Section meets from December 4, 2023 through May 5, 2024.
For more information, visit the Diagnostic Medicine/Pathobiology home page.
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Odds | Definition & Meaning
Odds|Definition & Meaning
Chances and probability also have a straightforward relationship: the odds of an occurrence are the ratio of the odds that the result will occur to the odds that it won’t. In typical notation, s is
the outcome’s probability, whereas 1-s is the probability that the outcome does not occur. By looking at the results of rolling a six-sided die, odds can be shown.
Figure 1 – Probability and odds
Figure 1 illustrates the difference between probability and odds.
There is a 1:5 chance of rolling a 6. This is since 1 event (rolling a 6) results in the desired outcome of “rolling a 6,” whereas 5 other events do not (rolling a 1,2,3,4, or 5). The likelihood of
rolling a 4 or 5 is 2:4. This is since only 2 occurrences (rolling either a 4 or 5) result in the desired outcome of “rolling either a 4 or 5,” whereas 4 events do not (rolling a 1, 2, 3 or 6).
If neither a 4 nor a 5 is rolled, the chances are 4:2. This is because there are four events—rolling a 1, 2, 3, or 6—that result in the desired result of “not rolling a 4 or 5,” and only two that do
not (rolling a 4 or 5).
Even though the likelihood of an event is distinct from the odds, they are connected and may be estimated from one another. The percentage of total occurrences over the number of events, or 2/(2+4),
which is 1/3, 0.33, or 33%, determines the likelihood of rolling a 4 or 5.
Difference Between Probability and Odds
The percentage of times you anticipate seeing a certain occurrence across many trials is the chance that it will happen. Probabilities are always in the 0 to 1 range. Odds are calculated by dividing
the probability that an event will occur by the probability that it will not occur. Figure 2 illustrates the graphical representation of odds and probability.
Figure 2 – Graphical representation of probability and odds
Odds in Statistics
In statistics, odds are a way to convey relative probabilities; they are sometimes referred to as “the odds are on your side.” The ratio of the likelihood that an event will occur to the likelihood
that it won’t occur is known as the odds (in favour) of an event or proposition.
Since there are only two outcomes, this is a Bernoulli trial, mathematically speaking. This is the ratio of the number of outcomes with a desired event to the number of outcomes without it. For a
finite sample space with equal probabilities, we can write them as S and F (success and failure) or W and L (wins and losses).
The probability that a randomly selected day of the week falls on a weekend, for instance, is two to five (2:5). This is because the days of the week form a sampling space of seven possible outcomes,
and the event only happens on Saturday and Sunday, as opposed to the other five.
In contrast, a probability space with a finite number of equally likely events may be used to describe odds as a ratio of integers. The prepositions and it can be used to express odds and probability
in prose.
For example, ”Too many odds for (or against) and too many odds for [event]” refers to odds that are the ratio of the number of (equally likely) outcomes for or against (or vice versa). point. “So
many [outcomes] odds, so many [outcomes]” refers to probability, which is the number of (equally likely) outcomes.
For instance, “chances of a weekend are 2 in 7,” but “odds of a weekend are 2 to 5″.
Application of Odds
Odds and comparable ratios may be more intuitive or practical in probability theory and statistics than probabilities. The log odds is sometimes employed in these situations.
The most straightforward way to multiply or divide odds is to use a log, which transforms addition from multiplication and subtraction from division. This is crucial in the logistic model because the
target variable’s log odds are created by linearly combining the observed variables.
Similar ratios are employed in other areas of statistics; the likelihood ratio in likelihood statistics, which is utilized as the Bayes factor in Bayesian statistics, is of essential significance.
Odds are very helpful in sequential decision-making issues, such as how to halt (online) on last specific event problems that are resolved by the odds algorithm.
An odds ratio is a ratio of odds or a ratio of probability ratios, and odds are a ratio of probabilities. Odds ratios are commonly used in clinical trial analysis. Although they possess useful
mathematical properties, they can also lead to results that defy common sense.
For example, an event with an 80% chance of occurring is 4 times more likely to occur than an event with a 20% chance of 16 times larger than the likely event. (1-4 or 4-1 or 0.25). Figure 3 shows
the odds formula.
Some Examples of Odds
Example 1
Find odds in favor of throwing a die to get five.
The total number of outcomes when a die is rolled is Six. The number of times the outcome is five is equal to one, and the number of unfavorable outcomes is (6-1) =5. So odds in favor of throwing a
die to get five is 1:5.
Example 2
Find odds in throwing a die and getting a five.
The total number of outcomes when a die is rolled is Six. The number of times the outcome is five is equal to one, and the number of unfavorable outcomes is (6-1) =5. So the odds in against throwing
a die to get five is 5:1.
All images were created using GeoGebra.
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A parallel-plate capacitor has plates of area 0.16 m2 and a
separation of 1.20 cm. A...
A parallel-plate capacitor has plates of area 0.16 m2 and a separation of 1.20 cm. A...
A parallel-plate capacitor has plates of area 0.16 m2 and a separation of 1.20 cm. A
battery charges the plates to a potential difference of 180 V and is then
disconnected. A dielectric slab of thickness 0.4 cm and dielectric constant K=3 is
then placed symmetrically between the plates.
a) What is the capacitance before the slab is inserted?
b) What is the capacitance with the slab in place?
c) What is the free charge q before the slab is inserted?
d) What is the free charge q after the slab is inserted?
e) What is the magnitude of the electric field in the space between the plates
and dielectric?
f) What is the magnitude of the electric field in the dielectric itself?
g) With the slab in place, what is the potential difference across the plates?
h) How much external work is involved in the process of inserting the slab?
i) What is the minimum dielectric strength that the dielectric must withstand
so that it will not break when it is placed in the capacitor?
j) Assume that I repeat the process (charge my initial capacitor to the same
battery, then remove the battery) but then instead of introducing a slab of
dielectric I introduce an identically shaped slab made out of solid copper.
How will any of the answers to questions b), d), e), f), g) and h) ?
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Count & Sum Distinct Values based on Criteria | COUNT(DISTINCT(COLLECT(...) | Returning 1 ???
In a ROLLUP sheet, referencing a source sheet, I'm trying to count (and sum) DISTINCT values that meet multiple criteria. My formula keeps returning "1", when in the shown scenario below has "80"
distinct values. Hoping I'm just missing something simple.
Rollup sheet purpose = count the distinct values rented within individual machine classes.
[Example (shown in the screenshot): YTD there were 270 invoices for machine class = "TELESCOPIC FORKLIFT". Within those 270 invoices, there were only 80 distinct serialized machines. So, I want my
formula to produce the "80" distinct serialized machines, but the result of my formula is 1. Some serial#'s found within this machine class "TELESCOPIC FORKLIFT" are numeric, and some are
[Example (shown in the screenshot): YTD there were 446 invoices for machine class = "SCISSOR LIFT". Within those 446 invoices, there were only 119 distinct serialized machines. The formula produced
the correct result. ALL serial#'s found within this machine class "SCISSOR LIFT" are numeric only.]
Issue = I have some machine classes where my current formula returns only "1" when I know there are more than 1 distinct serialized machines rented in that machine class.
Notable =
1.) For some of the machine classes my formula produces the correct result, and some do not.
2.) None of the columns in the source sheet are dropdowns.
3.) Within machine classes where the formula works, all machine serial #'s are of consistent format (either the serial # is ALL numbers or ALL serial numbers are alpha-numeric). Whereas the machine
classes where the formula does NOT work, the serial #'s are NOT consistently formatted (some serial #'s are all numbers AND some serial #'s are alpha-numeric).
In the subsequent columns ("1", "2", .....), I use the same formula, adding a second criteria to return the count of distinct serialized machines within that month (Jan =1). The formula in those
columns present the same issue in the same rows.
Thank you for your help!
Best Answer
• The problem is the mix of data types in the serial number column. You will need to add a helper column (can be hidden after setting up) that converts every row within that column into text and
then reference this in your formula.
=[Serial #]@row + ""
=[Column Name]@row plus quote quote
• The problem is the mix of data types in the serial number column. You will need to add a helper column (can be hidden after setting up) that converts every row within that column into text and
then reference this in your formula.
=[Serial #]@row + ""
=[Column Name]@row plus quote quote
• @Paul Newcome to the rescue again and FAST! Thank you very much, Paul!
Help Article Resources
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Declination and maximum altitude of the Sun above the horizon on a given date
This online calculator calculates the declination of the Sun on a given date and the maximum altitude above the horizon on that day for a given latitude
A short text about the declination and altitude of the Sun above the horizon as a function of latitude can be found below the calculator.
The path of the Sun at different latitudes
Declination means declination in the second equatorial coordinate system, which, unlike declination in the first equatorial coordinate system, does not change due to the daily motion of the Earth.
This is due to the fact that the second coordinate in the second equatorial coordinate system - direct ascent, is counted from the point of the vernal equinox, which is stationary. Accordingly, the
Sun reaches its declination, and highest position, once a day, at true noon.
The maximum altitude of the Sun is related to the latitude at which the observer is located and the Sun's declination by the following relationship: h[max] = 𝜎 + (90° - 𝜑), where 𝜎 is the declination
of the Sun, 𝜑 is the latitude of the place.
The maximum value of the Sun's declination 𝜎, on the day of the summer solstice, is equal to the angle of inclination of the Earth's axis - 23°26′14″ (approximately, since the value of the angle of
inclination is constantly changing slightly due to various effects). Accordingly, it can be seen that for latitudes smaller than 23°26′14″ (south of the Tropic of Cancer and north of the Tropic of
Capricorn), the value in the formula above can sometimes exceed the maximum height of the Sun above the horizon of 90° degrees (zenith). In practice, this means that, in the case of the northern
hemisphere, the Sun culminates not south of the zenith (when at noon the shadow points north), but north, and consequently the shadow at noon points south. The calculator always shows a value less
than 90° by subtracting the resulting value from 180°.
On the Tropic of Cancer itself, on the day of the summer solstice, the sun passes through the zenith, that is, at noon it is directly overhead - the day of zero shadow. South of the Tropic of Cancer,
the day of zero shadow happens twice a year - when the declination of the Sun is equal to the latitude of the place of observation. But for latitudes above 66°33′46″ (90° - 23°26′14″) (north of the
Arctic Circle and south of the Arctic Circle), the altitude can be negative, corresponding to a polar night.
To summarize all this, I will cite the astronomy characteristics of temperature belts from Bakulin's textbook^1, Chapter 1, § 17. The daily path of the Sun at different latitudes:
1. In frigid zones (from 𝜑 = ± 66° 34' to 𝜑 = ± 90°), the Sun can be a non-sunsetting and non-sunrise luminary. Polar day and polar night can last from 24 hours to half a year.
2. In temperate zones (from 𝜑 = ± 23° 26' to 𝜑 = ± 66° 34') the Sun rises and sets every day, but is never at zenith. Polar days and nights never occur here. The duration of day and night is shorter
than 24 hours. In summer, the day is longer than the night and vice versa in winter.
3. In the torrid zone (from 𝜑 = + 23° 26' to 𝜑 = - 23° 26') the Sun is also always a rising and setting luminary and is at its zenith twice a year (once in the tropics) at noon. The days it happens
depends on the latitude, and at the equator the Sun at zenith happens on the day of the vernal and the day of the autumnal equinoxes.
1. A course of general astronomy / P. I. Bakulin, E. V. Kononovich, V. I. Moroz. - Moscow: Nauka, 1976. ↩
Similar calculators
PLANETCALC, Declination and maximum altitude of the Sun above the horizon on a given date
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[Complex plane] arg[(z-1)/(z+1)] = pi/3
• Thread starter tusher
• Start date
In summary: Notice that the denominator is 0 when x= -1. This is because the original equation represents a circle of radius 1, centred at (-1,0), which becomes a point when z = -1. So the equation
shows that there are no values of z for which the argument of (z-1)/(z+1) is defined, so it does not represent a circle.In summary, the argument of (z-1)/(z+1) does not represent a circle as there
are no values of z for which the argument is defined.
Homework Statement
Show that arg[(z-1)/(z+1)] represents a circle. Find it's radius and centre.
Homework Equations
The Attempt at a Solution
using z = (x+iy) i narrowed down to (z-1)/(z+1) = (iy)/(1+x) , assuming it was a circle.
What next?
Is this correct approach??
using z = (x+iy) i narrowed down to (z-1)/(z+1) = (iy)/(1+x)
... how did you get that? Please show your working.
The next step is to take the argument and show that it is the same as the equation for a circle.
I miscalculated. :(
The steps are..
[itex]\frac{z-1}{z+1}[/itex] = [itex]\frac{(x+iy-1)}{(x+iy+1)}[/itex]
= [itex]\frac{(x+iy-1)(x-iy-1)}{(x+iy+1)(x-iy-1)}[/itex]
= [itex]\frac{(x)^{2}-2x+1+y^{2}}{x^{2}-1-2iy-(iy)^{2}}[/itex]
Supposing,x[itex]^{2}[/itex] + y[itex]^{2}[/itex] = 1
we get ,
I have no idea what to do next.
Here, let me tidy that up a bit: $$\begin{align}
C &=\frac{z-1}{z+1} & & \text{(1)}\\
&= \frac{(x+iy-1)}{(x+iy+1)} & & \text(2)\\
&= \frac{(x+iy-1)(x-iy-1)}{(x+iy+1)(x-iy-1)} & & \text(3)\\
&=\frac{(x-1)^{2}-(iy)^{2}}{(x^{2})-(iy)^{2}} & & \text(4)\\
&= \frac{(x)^{2}-2x+1+y^{2}}{x^{2}-1-2iy-(iy)^{2}} & & \text(5)\\
& =\frac{x^{2}+y^{2}-2x+1}{x^{2}+y^{2}-2iy-1} & & \text(6)
(... I've called the original function C and numbered the lines to make it easier to talk about.)
What was the idea behind step 3? Please show your reasoning.
Note: at some stage you'll have to find arg[C] and show that it is a circle.
Supposing,x[itex]^{2}[/itex] + y[itex]^{2}[/itex] = 1
we get ,
... how does that follow? Again, I don't see your reasoning.
(Assume that I know the maths but I'm in another country on the other side of the World and we may have different conventions in how we approach math problems here. I won't be offended.)
...lets see: $$c=\frac{x-1}{iy}=-i\frac{x-1}{y}$$... is purely imaginary so the argument is: ##\text{Arg}[c]=\pm\frac{\pi}{2}## ... i.e. it is a couple of points, not a circle.
Perhaps you should end up with something like: ##\text{Arg}[c] = x^2+y^2-k## ... where ##r=\text{Arg}[c]+k## is the (real) radius?
It's kinda hard to see how that would work.
How can an argument come to a circle - it takes 2D and turns it into 1D?
So there is something missing from the problem statement.Generally: $$\frac{z-z_1}{z-z_2}=c$$... is a circle in the complex plane if c ≠ 1 and is real.
... overall you need to think how the argument comes into this.
Note: C=r is the equation of a circle, radius r, with it's center at (x,y)=(1,0).
If A=arg[C] then the radius of the circle is r=√tan(A/2).
... depending on how the argument is defined in your course.
Aside: good use of the equation editor.
If you hot "quote" under this post you get to see how I tidied that up for you ;)
So if we consider A(-1,0) and B(1,0) to be 2 points on the argand plane and z be any arbitrary point then line segment zA will be z+1 and zB will be z-1. The arguments will be arg(z+1) and arg(z-1)
respectively. So the angle between the 2 lines will be arg((z-1)/z+1)). We have been given that this angle is constant(π/2). We know that any angle subtended by the end points of a chord on a circle
is always equal. So we know that locus of z is a circle.
The center can be found out be taking the incenter of an equilateral triangle(one of the many possibilities of triangle zAB)whose base is AB and radius will be the distance from this point to any of
the 2 points(A or B)
Hope this was helpfull
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
tusher said:
I miscalculated. :(
The steps are..
[itex]\frac{z-1}{z+1}[/itex] = [itex]\frac{(x+iy-1)}{(x+iy+1)}[/itex]
= [itex]\frac{(x+iy-1)(x-iy-1)}{(x+iy+1)(x-iy-1)}[/itex]
= [itex]\frac{(x)^{2}-2x+1+y^{2}}{x^{2}-1-2iy-(iy)^{2}}[/itex]
Supposing,x[itex]^{2}[/itex] + y[itex]^{2}[/itex] = 1
we get ,
I have no idea what to do next.
I cannot see why you "simplify" the way you do. To convert(z-1)/(z+1) to X + iY form you need to multiply and divide by (x
1-iy), not the (x-1 - iy) that you used. I get
$$\frac{z-1}{z+1} = \frac{x^2+y^2-1 + i 2y}{x^2+y^2+2x+1}.$$
FAQ: [Complex plane] arg[(z-1)/(z+1)] = pi/3
1. What is the complex plane?
The complex plane is a geometric representation of the complex numbers, where the horizontal axis represents the real numbers and the vertical axis represents the imaginary numbers.
2. What does "arg" mean in the context of this equation?
"Arg" is short for argument and in this context, it refers to the angle or direction of a complex number on the complex plane. It is measured counterclockwise from the positive real axis.
3. How is the complex number (z-1)/(z+1) related to the complex plane?
The complex number (z-1)/(z+1) represents a point on the complex plane, with the numerator being the distance from the origin and the denominator being the direction or angle from the positive real
4. What does the equation arg[(z-1)/(z+1)] = pi/3 represent on the complex plane?
This equation represents a line on the complex plane that makes an angle of pi/3 with the positive real axis, passing through the point (1,0) and (0,1).
5. How many solutions are there to this equation on the complex plane?
There are infinitely many solutions to this equation, as the complex plane is continuous and any angle can be represented by multiple points on the plane.
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Linear Congruence Equation
Linear Congruence Equation¶
This equation is of the form:
$$a \cdot x \equiv b \pmod n,$$
where $a$, $b$ and $n$ are given integers and $x$ is an unknown integer.
It is required to find the value $x$ from the interval $[0, n-1]$ (clearly, on the entire number line there can be infinitely many solutions that will differ from each other in $n \cdot k$ , where
$k$ is any integer). If the solution is not unique, then we will consider how to get all the solutions.
Solution by finding the inverse element¶
Let us first consider a simpler case where $a$ and $n$ are coprime ($\gcd(a, n) = 1$). Then one can find the inverse of $a$, and multiplying both sides of the equation with the inverse, and we can
get a unique solution.
$$x \equiv b \cdot a ^ {- 1} \pmod n$$
Now consider the case where $a$ and $n$ are not coprime ($\gcd(a, n) \ne 1$). Then the solution will not always exist (for example $2 \cdot x \equiv 1 \pmod 4$ has no solution).
Let $g = \gcd(a, n)$, i.e. the greatest common divisor of $a$ and $n$ (which in this case is greater than one).
Then, if $b$ is not divisible by $g$, there is no solution. In fact, for any $x$ the left side of the equation $a \cdot x \pmod n$ , is always divisible by $g$, while the right-hand side is not
divisible by it, hence it follows that there are no solutions.
If $g$ divides $b$, then by dividing both sides of the equation by $g$ (i.e. dividing $a$, $b$ and $n$ by $g$), we receive a new equation:
$$a^\prime \cdot x \equiv b^\prime \pmod{n^\prime}$$
in which $a^\prime$ and $n^\prime$ are already relatively prime, and we have already learned how to handle such an equation. We get $x^\prime$ as solution for $x$.
It is clear that this $x^\prime$ will also be a solution of the original equation. However it will not be the only solution. It can be shown that the original equation has exactly $g$ solutions, and
they will look like this:
$$x_i \equiv (x^\prime + i\cdot n^\prime) \pmod n \quad \text{for } i = 0 \ldots g-1$$
Summarizing, we can say that the number of solutions of the linear congruence equation is equal to either $g = \gcd(a, n)$ or to zero.
Solution with the Extended Euclidean Algorithm¶
We can rewrite the linear congruence to the following Diophantine equation:
$$a \cdot x + n \cdot k = b,$$
where $x$ and $k$ are unknown integers.
The method of solving this equation is described in the corresponding article Linear Diophantine equations and it consists of applying the Extended Euclidean Algorithm.
It also describes the method of obtaining all solutions of this equation from one found solution, and incidentally this method, when carefully considered, is absolutely equivalent to the method
described in the previous section.
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Max Daniels
Multi-layer State Evolution Under Random Convolutional Design.
Max Daniels*, Cédric Gerbelot*, Florent Krzakala, Lenka Zdeborová. Published in NeurIPS 2022.
We study signal recovery in a multi-layer model with convolutional matrices, which is a simple model for a convolutional neural network. We prove state evolution equations for Approximate Message
Passing, an algorithm that can be used to compute posterior statistics in high dimensional Bayesian models. These equations provide a tractable method to predict signal recovery error in a large-size
Score-based Generative Neural Networks for Large-Scale Optimal Transport.
Max Daniels, Tyler Maunu, Paul Hand. Published in NeurIPS 2021.
We propose a new method to solve a regularized form of the Optimal Transport problem. The goal is to learn a transportation plan between a given source and target probability distribution so that the
cost to execute that plan is minimized. We prove global optimization guarantees for a fast, large-scale learning algorithm to solve this problem and we demonstrate strong empirical performance.
Generator Surgery for Compressed Sensing
Jung Yeon Park*, Niklas Smedemark-Marguilies*, Max Daniels, Rose Yu, Jan-Willem van de Meent, Paul Hand. Presented at NeurIPS 2020 Deep Inverse Workshop.
Generative priors for imaging inverse problems allow one to model images from a dataset by generating samples from the dataset. We demonstrate a simple method to improve recovery performance by
modifying these priors after training, but this performance boost comes at the cost of no longer being able to generate samples using the prior.
Invertible generative models for inverse problems: mitigating representation error and dataset bias.
Muhammad Asim*, Max Daniels*, Oscar Leong, Paul Hand, and Ali Ahmed. Published in ICML 2020.
In an imaging inverse problem, one must recover missing information about a target image using prior assumptions on the image structure. We show that Invertible Neural Networks can be used to vastly
outperform classical approaches when one has access to a dataset of known images.
Statistical Distances and Their Implications to GAN Training.
Max Daniels. Presented at VISxAI workshop at IEEE VIS 2019. Honorable mention for best submission.
This is an interactive article about the role of statistical distances like Kullback Leibler Divergence and Earth Mover's Distance in training Generative Adversarial Networks (GANs).
An Overview of Graph Spectral Clustering and Partial Differential Equations.
Max Daniels*, Catherine Huang*, Chloe Makdad*, Shubham Makharia*. Product of a 2020 summer undergraduate research program run by the Institute for Computational and Experimental Research in
Clustering is a useful tool in data analysis. We explain the connection between the graph spectral clustering algorithm and the physical process of heat diffusion.
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Causality Archives • Statisticelle
Directed acyclic graphs (DAGs), and causal graphs in general, provide a framework for making assumptions explicit and identifying confounders or mediators of the relationship between the exposure of
interest and outcome that need to be adjusted for in analysis. Recently, I ran into the need to generate data from a DAG for a paper I am writing with my peers Kevin McIntyre and Joshua Wiener. After
a quick Google search, I was pleasantly surprised to see there were several options to do so. In particular, the dagR library provides “functions to draw, manipulate, [and] evaluate directed acyclic
graphs and simulate corresponding data”.
Besides dagR‘s reference manual, a short letter published in Epidemiology, and a limited collection of examples, I couldn’t find too many resources regarding how to use the functionality provided by
dagR. The goal of this blog post is to provide an expository example of how to create a DAG and generate data from it using the dagR library.
To simulate data from a DAG with dagR, we need to:
1. Create the DAG of interest using the dag.init function by specifying its nodes (exposure, outcome, and covariates) and their directed arcs (directed arrows to/from nodes).
2. Pass the DAG from (1) to the dag.sim function and specify the number of observations to be generated, arc coefficients, node types (binary or continuous), and parameters of the node distributions
(Normal or Bernoulli).
For this tutorial, we are going to try to replicate the simple confounding/common cause DAG presented in Figure 1b as well as the more complex DAG in Figure 2a of Shier and Platt’s (2008) paper,
Reducing bias through directed acyclic graphs.
Continue reading Using a DAG to simulate data with the dagR library
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Course Descriptions
West Lafayette Campus
Mathematics Courses
Click on name of course to view prerequisites and additional information.
MA 10800 - Mathematics As A Profession And A Discipline
A seminar course for undergraduate students interested in majoring in an area of mathematics at Purdue. The purpose is to build prospective mathematics majors' awareness of opportunities to enhance
their experiences at Purdue and of career paths available for graduates with a good mathematical background. The format of most classes is a presentation and discussion with an invited speaker/guest,
including experts on a different aspect of mathematics in our world today. This course is recommended for undergraduates in their first or second year at Purdue. 1 credit hour
MA 13700 - Mathematics For Elementary Teachers I
Designed for prospective elementary school teachers. Problem solving. Numerical reasoning including self-generated and conventional algorithms. Whole and fractional number systems, elementary number
theory. (Not available for credit toward graduation in the College of Science.). 3 credit hours
MA 13800 - Mathematics For Elementary Teachers II
Elementary school teachers must understand how multiplication gives rise to exponents and how to represent, interpret, and compute exponents from problem situations. They must also understand how to
represent practical situations using algebraic and fractional expressions, and verbally interpret graphs of functions. They have to know basic concepts of probability theory. This course covers
conceptual and practical notions of exponents and radicals; algebraic and rational functions, algebraic equations and inequalities, systems of linear equations, polynomial, exponential, and
logarithmic functions. Notions of probability. 3 credit hours
MA 13900 - Mathematics For Elementary Teachers III
Geometric, measurement and spatial reasoning in one, two and three dimensions as the basis for elementary school geometry. Metric and non-metric geometry, transformation geometry. (Not available for
credit toward graduation in the College of Science.) 3 credit hours
Exponents and radicals; algebraic and fractional expressions. Equations and inequalities, systems of linear equations. Polynomial, exponential, and logarithmic functions. Not open to students with
credit in MA 15900. Not available for credit toward graduation in the School of Science. CTL:IMA 1601 College Algebra 0 OR 3 credit hours
MA 15555 - Quantitative Reasoning
This course will cover important mathematical ideas, including proportion, weighted averages, linear models, exponential models, basic probability and statistics, and some algebra, by using concrete
real-world problems. It will not be a prerequisite for any other mathematics course. CTL: Quantitative Reasoning 0 OR 3 credit hours
MA 15800 - Precalculus - Functions And Trigonometry
Functions, Trigonometry, and Algebra of calculus topics designed to fully prepare students for all first semester calculus courses. Functions topics include Quadratic, Higher Order Polynomials,
Rational, Exponential, Logarithmic, and Trigonometric. Other focuses include graphing of functions and solving application problems. Not Available for credit toward graduation in the College of
Science. Students may not receive credit for both MA 15400 and MA 15800. Students may not receive credit for both MA 15900 and MA 15800. 0 OR 3 credit hours
Topics include trigonometric and exponential functions; limits and differentiation, rules of differentiation, maxima, minima and optimization; curve sketching, integration, anti-derivatives,
fundamental theorem of calculus. Properties of definite integrals and numerical methods. Applications to life, managerial and social sciences. CTL:IMA 1604 Calculus - Short I 0 OR 3 credit hours
MA 16020 - Applied Calculus II
This course covers techniques of integration; infinite series, convergence tests; differentiation and integration of functions of several variables; maxima and minima, optimization; differential
equations and initial value problems; matrices, determinants, eigenvalues and eigenvectors. Applications. CTL:IMA 1605 Calculus - Short II 0 OR 3 credit hours
MA 16100 - Plane Analytic Geometry And Calculus I
Introduction to differential and integral calculus of one variable, with applications. Some schools or departments may allow only 4 credit hours toward graduation for this course. Designed for
students who have not had at least a one-semester calculus course in high school, with a grade of "A" or "B". Not open to students with credit in MA 16500. Demonstrated competence in college algebra
and trigonometry. 0 OR 5 credit hours
MA 16200 - Plane Analytic Geometry And Calculus II
Continuation of MA 16100. Vectors in two and three dimensions, techniques of integration, infinite series, conic sections, polar coordinates, surfaces in three dimensions. Some schools or departments
may allow only 4 credit hours toward graduation for this course. 0 OR 5 credit hours
MA 16290 - Data Science Labs: Calculus
This course consists of weekly computer laboratories which apply concepts learned in Calculus I and II to data science problems. Main topics covered include function sampling and approximation,
numerical differentiation, numerical integration, Jupyter notebooks, introductory Python programming, object oriented programming, and data acquisition with microcontrollers and sensors. 1 credit
MA 16500 - Analytic Geometry And Calculus I
Introduction to differential and integral calculus of one variable, with applications. Conic sections. Designed for students who have had at least a one-semester calculus course in high school, with
a grade of "A" or "B", but are not qualified to enter MA 16200 or 16600, or the advanced placement courses MA 27100. Demonstrated competence in college algebra and trigonometry. CTL:IMA 1602 Calculus
- Long I 0 OR 4 credit hours
MA 16600 - Analytic Geometry And Calculus II
Continuation of MA 16500. Vectors in two and three dimensions. Techniques of integration, infinite series, polar coordinates, surfaces in three dimensions. Not open to students with credit in MA
16200. CTL:IMA 1603 Calculus - Long II 0 OR 4 credit hours
MA 17000 - Introduction To Actuarial Science
(STAT 17000) An introduction to actuarial science from the point of view of practicing actuaries from life insurance, casualty insurance and consulting; introduction to insurance and the mathematical
theory of interest; application of spreadsheets to problems related to actuarial science. 0 OR 2 credit hours
MA 18300 - Professional Practicum I
Professional Practicum. For Cooperative Education students only; must be accepted for the program by the cooperative program coordinator. Permission of department required. 0 credit hours
MA 19000 - Topics In Mathematics For Undergraduates
Supervised reading courses as well as special topics courses for undergraduates are given under this number. Permission of instructor required. 0 to 5 credit hours
MA 25000 - Problem Solving In Probability
(STAT 25000) This course is designed to teach techniques for solving problems in probability theory which are relevant to the actuarial sciences. It is intended to help actuarial students prepare for
the Society of Actuaries and Casualty Actuarial Society Exam P/1. Credit of Examination is not available for this course. 2 credit hours
MA 26100 - Multivariate Calculus
Planes, lines, and curves in three dimensions. Differential calculus of several variables; multiple integrals. Introduction to vector calculus. Not open to students with credit in MA 27100. 0 OR 4
credit hours
MA 26190 - Data Science Labs: Multivariate Calculus
This course consists of weekly computer laboratories which apply concepts learned in Multivariate Calculus (Calculus III) to data science problems. The students will also practice programming in
Python and use sensors and microprocessors to acquire data. Topics covered include representation and perception of color, motion detection in videos, and construction of a planimeter based on
Green's theorem. 1 credit hour
MA 26200 - Linear Algebra And Differential Equations
Linear algebra, elements of differential equations. Not open to students with credit in MA 26500 or 26600. 0 OR 4 credit hours
Introduction to linear algebra. Systems of linear equations, matrix algebra, vector spaces, determinants, eigenvalues and eigenvectors, diagonalization of matrices, applications. Not open to students
with credit in MA 26200, 27200, 35000 or 35100. 3 credit hours
MA 26600 - Ordinary Differential Equations
First order equations, second and nth order linear equations, series solutions, solution by Laplace transform, systems of linear equations. It is preferable but not required to take MA 26500 either
first or concurrently. Not open to students with credit in MA 26200, 27200, 36000, 36100, or 36600. 3 credit hours
MA 27101 - Honors Multivariate Calculus
This course is the Honors version of MA 26100, Multivariate Calculus; it will also include a review of infinite series. The course is intended for first-year students who have credit for Calculus I
and II. There will be a significant emphasis on conceptual explanation, but not on formal proof. Permission of department is required. 0 OR 5 credit hours
MA 27900 - Modern Mathematics In Science And Society
The course covers topics in combinatorics and probability applied to real life situations such as the paradoxes of democracy, weighted voting, fair division, apportionment, traveling salesmen, the
mathematics of networks, Fibonacci numbers, golden ratio, growth patterns in nature, mathematics of money, symmetry, fractals, censuses and surveys, random sampling, sample spaces, permutations and
uniform probability spaces. 3 credit hours
MA 29000 - Topics In Mathematics For Undergraduates
Supervised reading courses as well as special topics courses for undergraduates are given under this number. Permission of instructor required. 1 to 5 credit hours
MA 29199 - Cooperative Experience I
Professional experience in mathematics. Program coordinated by school with cooperation of participating employers. Students submit summary report and company evaluation. Professional Practice
students only. Permission of department required. 0 credit hours
MA 29299 - Cooperative Experience II
Professional experience in mathematics. Program coordinated by school with cooperation of participating employers. Students submit summary report and company evaluation. Professional Practice
students only. Permission of department required. 0 credit hours
MA 30100 - An Introduction To Proof Through Real Analysis
An introduction to abstract reasoning in the context of real analysis. Topics may include axioms for the real numbers, mathematical induction, formal definition of limits, density, decimal
representations, convergence of sequences and series, continuity, differentiability, the extreme value, mean value and intermediate value theorems, and cardinality. The emphasis, however, is more on
the concept of proof than on any one given topic. 3 credit hours
MA 30300 - Differential Equations And Partial Differential Equations For Engineering And The Sciences
This is a methods course for juniors in any branch of engineering and science, designed to follow MA 26200 or MA 26600. Materials to be covered are: linear systems of ordinary differential equations,
nonlinear systems, Fourier series, separation of variables for partial differential equations, and Sturm-Liouville theory. 3 credit hours
MA 34100 - Foundations Of Analysis
An introductory course in rigorous analysis, covering real numbers, sequences, series, continuous functions, differentiation, and Riemann integration. MA 30100 is helpful but not required. 3 credit
MA 34900 - Signals And Systems For Mathematicians
This course introduces the mathematical framework for the description, analysis and processing of signals such as music, speech and images. Main topics covered include signal representations in
different bases; continuous-time signal sampling; and signal processing by linear and time-invariant systems. 3 credit hours
MA 35100 - Elementary Linear Algebra
Systems of linear equations, finite dimensional vector spaces, matrices, determinants, eigenvalues and eigenvector applications to analytical geometry. Not open to students with credit in MA 26500. 3
credit hours
Theoretical background for methods and results that appear in MA 35100. Inner products, orthogonality, and applications including least squares. 3 credit hours
MA 36200 - Topics In Vector Calculus
Multivariate calculus; partial differentiation; implicit function theorems and transformations; line and surface integrals; vector fields; theorems of Gauss, Green, and Stokes. Credit granted for
only one of MA 36200 and 51000. 3 credit hours
MA 36600 - Ordinary Differential Equations
An introduction to ordinary differential equations with emphasis on problem solving and applications. The one-hour computer lab will give students an opportunity for hands-on experience with both the
theory and applications of the subject. 0 OR 4 credit hours
MA 37300 - Financial Mathematics
A mathematical treatment of some fundamental concepts of financial mathematics and their application to real world business situations and basic risk management. Includes discussions of valuing
investments, capital budgeting, valuing contingent cash flows, yield curves, spot rates, forward rates, short sales, Macaulay duration, modified duration, convexity, and immunization. Provides
preparation for the SOA/CAS Actuarial Exam FM/2. 3 credit hours
MA 37400 - Mathematical Foundations For Machine Learning
This course combines data, computation, and inferential thinking to solve challenging problems. In this class, we explore key areas of machine learning including question formulation, statistical
inference, predictive modeling, and decision making. Through a strong emphasis on data-centric computing, and quantitative critical thinking, this class covers key principles and techniques of
machine learning. These include algorithms for machine learning methods including regression, classification, and clustering; and statistical concepts of measurement error and prediction. 3 credit
MA 37500 - Introduction To Discrete Mathematics
Induction, permutations, combinations, finite probability, relations, graphs, trees, graph algorithms, recurrence relations, generating functions. Problem solving in all these areas. Credit granted
for only one of MA 27600 and 37500. 3 credit hours
MA 38500 - Introduction To Logic
Propositional calculus and predicate calculus with applications to mathematical proofs, valid arguments, switching theory, and formal languages. 3 credit hours
MA 38600 - Professional Practicum IV
Professional Practicum. Permission of department required. 0 credit hours
MA 39000 - Topics In Mathematics For Undergraduates
Supervised reading courses as well as special topics courses for undergraduates are given under this number. Permission of instructor required. 1 to 5 credit hours
MA 39399 - Cooperative Experience III
Professional experience in mathematics. Program coordinated by school with cooperation of participating employers. Students submit summary report and company evaluation. Professional Practice
students only. Permission of department required. 0 credit hours
MA 39499 - Extensive Cooperative Experience IV
Professional experience in mathematics. Program coordinated by school with cooperation of participating employers. Students submit summary report and company evaluation. Professional Practice
students only. Permission of department required. 0 credit hours
MA 39599 - Extensive Cooperative Experience V
Professional experience in mathematics. Program coordinated by school with cooperation of participating employers. Students submit summary report and company evaluation. Professional Practice
students only. Permission of department required. 0 credit hours
(STAT 41600) An introduction to mathematical probability suitable as a preparation for actuarial science, statistical theory, and mathematical modeling. General probability rules, conditional
probability and Bayes theorem, discrete and continuous random variables, moments and moment generating functions, joint and conditional distributions, standard discrete and continuous distributions
and their properties, law of large numbers and central limit theorem. 3 credit hours
MA 42100 - Linear Programming And Optimization Techniques
Solution of linear programming problems by the simplex method, duality theory, transportation problems, assignment problems, network analysis, dynamic programming. 3 credit hours
MA 42500 - Elements Of Complex Analysis
Complex numbers and complex-valued functions; differentiation of complex functions; power series, uniform convergence; integration, contour integrals; elementary conformal mapping. 3 credit hours
MA 42800 - Introduction To Fourier Analysis
Topics include: Fourier series, convolutions, kernels, summation methods, Fourier transforms, and applications to the wave, heat, and Laplace equations. 3 credit hours
MA 43200 - Elementary Stochastic Processes
An introduction to some classes of stochastic processes that arise in probabilistic models of time-dependent random processes. The main stochastic processes studied will be discrete time Markov
chains and Poisson processes. Other possible topics covered may include continuous time Markov chains, renewal processes, queueing networks, and martingales. 3 credit hours
MA 44000 - Honors Real Analysis I
Real analysis in one and n-dimensional Euclidean spaces. Topics include the completeness property of real numbers, topology of Euclidean spaces, Heine-Borel theorem, convergence of sequences and
series in Euclidean spaces, limit superior and limit inferior, Bolzano-Weierstrass theorem, continuity, uniform continuity, limits and uniform convergence of functions, Riemann or Riemann-Stieltjes
integrals. 3 credit hours
MA 44200 - Honors Real Analysis II
Real analysis in one and n-dimensional Euclidean spaces--continued from MA 44000. Topics include mappings of Euclidean spaces and their derivatives, multivariable chain rule, inverse function theorem
and implicit function theorem, sets with content and integration in n dimensions, the integrability theorem, Jacobian and change of variables theorem, related topics. 3 credit hours
This course, which is essentially the first half of MA 55300, is recommended for students wanting a more substantial background in algebra than is afforded by MA 45300, in particular students
intending to do graduate work in science or engineering. Topics include the elements of number theory and group theory; unique factorization in polynomial rings and in principal ideal domains. 3
credit hours
MA 45300 - Elements Of Algebra I
Fundamental properties of integers, polynomials, groups, rings, and fields, with emphasis on problem solving and applications. Not open to students with credit in MA 45000. 3 credit hours
MA 45401 - Galois Theory Honors
This course will give a thorough introduction to Galois theory. Galois theory is a fundamental tool in many areas of mathematics, including number theory and algebraic geometry. This course will
increase students' mathematical maturity and prepare them for graduate school. Topics include finite extension fields and their symmetries, ruler and compass constructions, complex roots of unity,
solvable groups, and the solvability of polynomial equations by arithmetic and radical operations. This course is intended for third- or fourth-year students who have taken MA 45000 (Algebra Honors)
or MA 45300 (Elements of Algebra I). 3 credit hours
This course begins at the high-school level and then moves quickly to intermediate and advanced topics including an introduction to non-Euclidean geometry. Emphasis on proofs. 3 credit hours
MA 46200 - Elementary Differential Geometry
The geometry of curves and surfaces based on familiar parts of calculus and linear algebra. An introduction to the study of differentiable manifolds and Riemannian geometry. 3 credit hours
MA 47201 - Actuarial Models-Life Contingencies
Mathematical foundation of actuarial science, emphasizing probability models for life contingencies as the basis for analyzing life insurance and life annuities and determining premiums and reserves.
This course provides the background for Course MLC of the Society of Actuaries and Course 3L of the Casualty Actuarial Society. 0 OR 4 credit hours
MA 48100 - Advanced Problem-Solving Seminar
Seminar intended to prepare students for the national Putnam examination in mathematics. 3 credit hours
MA 48400 - Seminar On Teaching College Algebra And Trigonometry
This course is a seminar on the teaching of mathematics for our best undergraduate mathematics education students. It provides supervised teaching experience along with a chance for the students to
perfect their knowledge of algebra before going on to be high school teachers. Students who take this class will also teach a section of MA 15300. Permission of instructor required. 3 credit hours
MA 48700 - Professional Practicum V
r 1.00. Professional Practicum. Permission of department required. 0 OR 1 credit hour
MA 49000 - Topics In Mathematics For Undergraduates
Supervised reading courses as well as special topics courses for undergraduates are given under this number. Permission of instructor required. 1 to 6 credit hours
MA 49500 - Advanced Topics In Mathematics For Undergraduates
Advanced topics courses in mathematics for undergraduates are given under this number. Permission of instructor required. 1 to 5 credit hours
Group theory: definitions, examples, subgroups, quotient groups, homomorphisms, and isomorphism theorems. Ring theory: definitions, examples, homomorphisms, ideals, quotient rings, fraction fields,
polynomial rings, Euclidean domains, and unique factorization domains. Field theory: algebraic field extensions, straightedge and compass constructions. 3 credit hours
Completeness of the real number system, basic topological properties, compactness, sequences and series, absolute convergence of series, rearrangement of series, properties of continuous functions,
the Riemann-Stieltjes integral, sequences and series of functions, uniform convergence, the Stone-Weierstrass theorem, equicontinuity, and the Arzela-Ascoli theorem. 3 credit hours
Calculus of functions of several variables and of vector fields in orthogonal coordinate systems. Optimization problems, implicit function theorem, Green's theorem, Stokes' theorem, divergence
theorems. Applications to engineering and the physical sciences. Not open to students with credit in MA 36200 or 41000. 3 credit hours
MA 51100 - Linear Algebra With Applications
Real and complex vector spaces; linear transformations; Gram-Schmidt process and projections; least squares; QR and LU factorization; diagonalization, real and complex spectral theorem; Schur
triangular form; Jordan canonical form; quadratic forms. 3 credit hours
(CS 51400) Iterative methods for solving nonlinear; linear difference equations, applications to solution of polynomial equations; differentiation and integration formulas; numerical solution of
ordinary differential equations; roundoff error bounds. 3 credit hours
MA 51800 - Advanced Discrete Mathematics
The course covers mathematics useful in analyzing computer algorithms. Topics include recurrence relations, evaluation of sums, integer functions, elementary number theory, binomial coefficients,
generating functions, discrete probability, and asymptotic methods. 3 credit hours
MA 51900 - Introduction To Probability
(STAT 51900) Algebra of sets, sample spaces, combinatorial problems, independence, random variables, distribution functions, moment generating functions, special continuous and discrete
distributions, distribution of a function of a random variable, limit theorems. 3 credit hours
MA 52000 - Boundary Value Problems Of Differential Equations
Separation of variables; Fourier series; boundary value problems; Fourier transforms; Bessel functions; Legendre polynomials. 3 credit hours
MA 52100 - Introduction To Optimization Problems
Necessary and sufficient conditions for local extrema in programming problems and in the calculus of variations. Control problems; statement of maximum principles and applications. Discrete control
problems. 3 credit hours
MA 52300 - Introduction To Partial Differential Equations
First order quasi-linear equations and their applications to physical and social sciences; the Cauchy-Kovalevsky theorem; characteristics, classification and canonical forms of linear equations;
equations of mathematical physics; study of Laplace, wave and heat equations; methods of solution. 3 credit hours
MA 52500 - Introduction To Complex Analysis
Complex numbers and complex-valued functions of one complex variable; differentiation and contour integration; Cauchy's theorem; Taylor and Laurent series; residues; conformal mapping; applications.
Not open to students with credit in MA 42500. 3 credit hours
MA 52700 - Advanced Mathematics For Engineers And Physicists I
MA 52700 is not a prerequisite for MA 52800; these courses can be taken independently. Topics in MA 52700 include linear algebra, systems of ordinary differential equations, Laplace transforms,
Fourier series and transforms, and partial differential equations. MA 51100 is recommended. 3 credit hours
MA 52800 - Advanced Mathematics For Engineers And Physicists II
MA 52700 is not a prerequisite for MA 52800; these courses can be taken independently. Topics in MA 52800 include divergence theorem, Stokes theorem, complex variables, contour integration, calculus
of residues and applications, conformal mapping, and potential theory. MA 51000 is recommended. 3 credit hours
MA 53000 - Functions Of A Complex Variable I
Complex numbers and complex-valued functions of one complex variable; differentiation and contour integration; Cauchy's theorem; Taylor and Laurent series; residues; conformal mapping; special
topics. More mathematically rigorous than MA 52500. 3 credit hours
MA 53100 - Functions Of A Complex Variable II
Advanced topics. 3 credit hours
MA 53200 - Elements Of Stochastic Processes
(STAT 53200) A basic course in stochastic models, including discrete and continuous time Markov chains and Brownian motion, as well as an introduction to topics such as Gaussian processes, queues,
epidemic models, branching processes, renewal processes, replacement, and reliability problems. 3 credit hours
MA 53800 - Probability Theory I
(STAT 53800) Mathematically rigorous, measure-theoretic introduction to probability spaces, random variables, independence, weak and strong laws of large numbers, conditional expectations, and
martingales. 3 credit hours
MA 53900 - Probability Theory II.
(STAT 53900) Convergence of probability laws; characteristic functions; convergence to the normal law; infinitely divisible and stable laws; Brownian motion and the invariance principle. 3 credit
MA 54200 - Theory Of Distributions And Applications
Definition and basic properties of distributions; convolution and Fourier transforms; applications to partial differential equations; Sobolev spaces. 3 credit hours
MA 54300 - Ordinary Differential Equations And Dynamical Systems
This course focuses on the theory of ordinary differential equations and methods of proof for developing this theory. Topics include basic results for linear systems, the local theory for nonlinear
systems (existence and uniqueness, dependence on parameters, flows and linearization, stable manifold theorem) and the global theory for nonlinear systems (global existence, limit sets and periodic
orbits, Poincare maps). Permission of instructor required. 3 credit hours
MA 54400 - Real Analysis And Measure Theory
Metric space topology; continuity, convergence; equicontinuity; compactness; bounded variation, Helly selection theorem; Riemann-Stieltjes integral; Lebesgue measure; abstract measure spaces;
LP-spaces; Holder and Minkowski inequalities; Riesz-Fischer theorem. 3 credit hours
MA 54500 - Functions Of Several Variables And Related Topics
Differentiation of functions; Besicovitch covering theorem; differentiation of one measure with respect to another; Hardy-Littlewood maximal function; functions of several variables; Sobolev spaces.
3 credit hours
MA 54600 - Introduction To Functional Analysis
Fundamentals of functional analysis. Banach spaces, Hahn-Banach theorem. Principle of uniform boundedness. Closed graph and open mapping theorems. Applications. Hilbert spaces. Orthonormal sets.
Spectral theorem for Hermitian operators and compact operators. 3 credit hours
MA 55300 - Introduction To Abstract Algebra
Group theory: Sylow theorems, Jordan-Holder theorem, solvable groups. Ring theory: unique factorization in polynomial rings and principal ideal domains. Field theory: ruler and compass constructions,
roots of unity, finite fields, Galois theory, solvability of equations by radicals. 3 credit hours
Review of basics: vector spaces, dimension, linear maps, matrices determinants, linear equations. Bilinear forms; inner product spaces; spectral theory; eigenvalues. Modules over a principal ideal
domain; finitely generated abelian groups; Jordan and rational canonical forms for a linear transformation. 3 credit hours
MA 55600 - Introduction To The Theory Of Numbers
Divisibility, congruences, quadratic residues, Diophantine equations, the sequence of primes. 3 credit hours
Review of fundamental structures of algebra (groups, rings, fields, modules, algebras); Jordan-Holder and Sylow theorems; Galois theory; bilinear forms; modules over principal ideal domains; Artinian
rings and semisimple modules. Polynomial and power series rings; Noetherian rings and modules; localization; integral dependence; rudiments of algebraic geometry and algebraic number theory;
ramification theory. 3 credit hours
MA 55800 - Abstract Algebra II
A continuation of MA 55700. 3 credit hours
MA 56000 - Fundamental Concepts Of Geometry
Foundations of Euclidean geometry, including a critique of Euclid's "Elements" and a detailed study of an axiom system such as that of Hilbert. Independence of the parallel axiom and introduction to
non-Euclidean geometry. 3 credit hours
MA 56200 - Introduction To Differential Geometry And Topology
Smooth manifolds; tangent vectors; inverse and implicit function theorems; submanifolds; vector fields; integral curves; differential forms; the exterior derivative; DeRham cohomology groups;
surfaces in E3., Gaussian curvature; two dimensional Riemannian geometry; Gauss-Bonnet and Poincare theorems on vector fields. 3 credit hours
MA 57100 - Elementary Topology
Fundamentals of point set topology with a brief introduction to the fundamental group and related topics, topological and metric spaces, compactness, connectedness, separation properties, local
compactness, introduction to function spaces, basic notions involving deformations of continuous paths. 3 credit hours
MA 57200 - Introduction In Algebraic Topology
Singular homology theory; Eilenberg-Steenrod axioms; simplicial and cell complexes; elementary homotopy theory; Lefschetz fixed point theorem. 3 credit hours
MA 57300 - Numerical Solution Of Ordinary Differential Equations
Numerical solution of initial-value problems by Runge-Kutta methods, general one-step methods, and multistep methods; analysis of truncation error, discretization error, and rounding error; stability
of multistep methods; numerical solution of boundary- and eigen-value problems by initial-value techniques and finite difference methods. 3 credit hours
MA 57400 - Numerical Optimization
Convex optimization algorithms using modern large-scale algorithms for convex optimization, with a heavy emphasis on analysis including monotone operator, fixed point iteration and duality in
splitting methods. The course will cover and focus on the following three parts: smooth optimization algorithms, nonsmooth convex optimization algorithms, and stochastic and randomized algorithms.
Permission of department required. Prerequisites: MA 51100 and MA 50400. 3 credit hours
Introduction to graph theory with applications. 3 credit hours
MA 58400 - Algebraic Number Theory
Dedekind domains, norm, discriminant, different, finiteness of class number, Dirichlet unit theorem, quadratic and cyclotomic extensions, quadratic reciprocity, decomposition and inertia groups,
completions and local fields. 3 credit hours
MA 58500 - Mathematical Logic I
Propositional and predicate calculus; the Godel completeness and compactness theorem, primitive recursive and recursive functions; the Godel incompleteness theorem; Tarski's theorem; Church's
theorem; recursive undecidability; special topics such as nonstandard analysis. 3 credit hours
MA 59000 - Topics In Mathematics
Supervised reading courses as well as dual-level special topics courses are given under this number. Permission of instructor required. 0 to 5 credit hours
MA 59500 - Topics In Mathematics
Special topics courses including dual-level special topics. Permission of instructor required. 1 to 5 credit hours
MA 59800 - Topics In Mathematics
Supervised reading courses as well as dual-level special topics courses are given under this number. Permission of instructor required. 1 to 5 credit hours
MA 61100 - Methods Of Applied Mathematics I
Banach and Hilbert spaces; linear operators; spectral theory of compact linear operators; applications to linear integral equations and to regular Sturm-Liouville problems for ordinary differential
equations. Prerequisite: MA 51100, 54400. 3 credit hours
MA 61500 - Numerical Methods For Partial Differential Equations I
(CS 615) Finite element method for elliptic partial differential equations; weak formulation; finite-dimensional approximations; error bounds; algorithmic issues; solving sparse linear systems;
finite element method for parabolic partial differential equations; backward difference and Crank-Nicholson time-stepping; introduction to finite difference methods for elliptic, parabolic, and
hyperbolic equations; stability, consistency, and convergence; discrete maximum principles. Prerequisite: MA 51400, 52300. 3 credit hours
MA 62000 - Mathematical Theory Of Optimal Control
Existence theorems; the maximum principle; relationship to the calculus of variations; linear systems with quadratic criteria; applications. Offered in alternate years. Prerequisite: MA 54400. 3
credit hours
MA 63100 - Several Complex Variables
Power series, holomorphic functions, representation by integrals, extension of functions, holomorphically convex domains. Local theory of analytic sets (Weierstrass preparation theorem and
consequences). Functions and sets in the projective space Pn (theorems of Weierstrass and Chow and their extensions). Prerequisite: MA 53000. 3 credit hours
MA 63800 - Stochastic Processes I
(STAT 638) Advanced topics in probability theory which may include stationary processes, independent increment processes, Gaussian processes; martingales, Markov processes, ergodic theory.
Prerequisite: MA 53900. 3 credit hours
MA 63900 - Stochastic Process II.
(STAT 63900) Continuation of MA 63800. 3 credit hours
MA 64200 - Methods Of Linear And Nonlinear Partial Differential Equations I
Second order elliptic equations including maximum principles, Harnack inequality, Schauder estimates, and Sobolev estimates. Applications of linear theory to nonlinear equations. Prerequisite: MA
52300. 3 credit hours
MA 64300 - Methods Of Partial Differential Equations II
Continuation of MA 642. Topics to be covered are Lp theory for solutions of elliptic equations, including Moser's estimates, Aleksandrov maximum principle, and the Calderon-Zygmund theory.
Introduction to evolution problems for parabolic and hyperbolic equations, including Galerkin approximation and semigroup methods. Applications to nonlinear problems. Prerequisite: MA 64200. 3 credit
MA 64400 - Calculus Of Variations
Direct methods; necessary and sufficient conditions for lower semicontinuity of multiple integrals; existence theorems and connections with optimal control theory. Prerequisite: MA 54400. 3 credit
MA 65000 - Commutative Algebra
The study of those rings of importance in algebraic and analytic geometry and algebraic number theory. Prerequisite: MA 55800. 3 credit hours
MA 66100 - Modern Differential Geometry
Topics chosen by the instructor. Prerequisite: MA 54400, 55400. 3 credit hours
MA 66300 - Algebraic Curves And Functions I
Algebraic functions of one variable from the geometric, algebraic, or function-theoretic points of view. Riemann-Roch theorem, differentials. Prerequisite: MA 55800. 3 credit hours
MA 66400 - Algebraic Curves And Functions II
Continuation of MA 663. Topics chosen by the instructor. Prerequisite: MA 66300. 3 credit hours
Topics of current interest will be chosen by the instructor. Prerequisite: MA 65000 or 66300. 3 credit hours
Ideles, adeles, L-functions, Artin symbol, reciprocity, local and global class fields, Kronecker-Weber Theorem. Prerequisite: MA 58400. 3 credit hours
Topics vary. Permission of instructor required. 1 to 3 credit hours
MA 69200 - Topics Applied Math
Topics in applied math. Permission of instructor required. 1 to 3 credit hours
Topics in analysis. Permission of instructor required. 1 to 3 credit hours
MA 69400 - Topics In Differential Equations
Topics In Differential Equations. Permission of instructor required. 1 to 3 credit hours
Topics in geometry. Permission of instructor required. 1 to 3 credit hours
Topics in topology. Permission of instructor required. 1 to 3 credit hours
MA 69900 - Research PhD Thesis
Research PhD Thesis. Permission of instructor required. 1 to 18 credit hours
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How to Add Fractions with Different Denominators.
In an effort to understand how to add fractions with different denominators, we must understand what a fraction is and how to represent it.Basically, a fraction is a number that represents a part of
a whole and it is written with one number on top, of the other, separated by a line.
We can be define a denominator as the bottom number in a fraction that shows the number of equal parts an item is divided into.
While adding fractions with different denominators may seem hard, however, all you need to do is convert the given fractions to like fractions to get common denominators so that it becomes easier to
add them.
How to Add Fractions with Different Denominators.
Method 1:
There are three ways in which you can add fractions with unlike denominators. The first one is where you find we find the Least common multiple of the denominators and then sum up the numerators to
get our answer. This is the most used method.
Example 1:
We start by finding the L.C.M of the denominators.
Method 2:
This next method is the easy way as all you need to do is cross multiply the two fractions and add the results together to get the numerator of the answer. Then multiply the two denominators together
to get the denominator of the answer. You must always make sure that your fraction is in its simplest form.
Example 2:
Method 3:
This method applies when adding more than two fractions. It is almost the same as the second method with a one tweak. Lets consider the fraction below.
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Do you Know Top Deep Learning Algorithms Explained
Let’s discuss the endless possibilities of deep learning and the top deep learning algorithms behind the popular deep learning applications like language recognition, autonomous vehicles, deep
learning robots, etc.
Deep learning has achieved massive popularity in scientific computing, and its algorithms are helpful in various industries that solve complex problems. Each deep learning algorithm uses different
types of neural networks to perform specific tasks.
While firing Siri or Alexa with questions, people usually wonder how machines deliver super-human accuracy. It is possible with deep learning – the amazingly intimidating area of data science.
Top deep learning algorithms explained
What is Deep Learning?
Deep learning makes use of artificial neural networks to deliver sophisticated computations on vast amounts of data. Therefore, it is a type of machine learning that acts based on the structure and
function of the human brain.
Thus, deep learning algorithms train machines by learning from examples. Also, industries such as health care, eCommerce, entertainment, and advertising usually use deep learning.
Thus, it is a subset of artificial intelligence with networks competent of unsupervised learning from unstructured or unlabeled data.
Deep learning has risen hand-in-hand with the digital era. Therefore, this has caused a revolution in data extraction in all forms and from every region of the world. This data, renowned as big data,
is drawn from sensational sources like social media, internet search engines, e-commerce platforms, and online cinemas.
How do deep learning algorithms work?
While deep learning algorithms highlight self-learning representations, they depend upon ANNs that reflect how the brain computes information.
Therefore, during the training process, algorithms use unknown elements in the input distribution to obtain features, group objects, and discover functional data patterns.
Deep learning models use several algorithms. As no one network is considerably whole, some algorithms are highly suitable to perform specific tasks. Thus, it’s good to gain a solid understanding of
all primary algorithms to choose the right ones.
What is a Neural network?
A neural network is a web-like human brain consisting of artificial neurons, also known as nodes. Hence, these nodes are piled next to each other in three layers:
• The input layer
• The hidden layer(s)
• The output layer
Therefore, data provides each node with information in the form of inputs. Thus, the node doubles the inputs with random weights, calculates them, and adds a bias. Lastly, nonlinear functions, also
known as activation functions, determine which neuron to fire.
Top deep learning algorithms
Deep learning algorithms operate with almost all kinds of data and need large amounts of computing power and information to solve complicated problems. Now, let us deep-dive into the list of deep
learning models.
Convolutional Neural Networks (CNNs)
CNNs, also recognized as ConvNets, consist of multiple layers mainly for image processing and object detection. Also, it can be called a deep learning algorithm for image processing.
Yann LeCun produced the first CNN in 1988 and was named LeNet. Hence, it is for recognizing characters like ZIP codes and digits.
Thus, CNNs are broadly used to identify satellite images, process medical images, forecast time series, and detect anomalies.
How Do CNNs Work?
CNN’s have various layers that process and extract features from data:
• CNN has a convolution layer that has different filters to perform the convolution operation.
• It also has a ReLU layer to execute operations on elements. The output is a revised feature map.
• The revised feature map next feeds into a pooling layer. Pooling is a down-sampling operation that lessens the dimensions of the feature map.
• The pooling layer then transforms the resulting two-dimensional arrays from the pooled feature map into a single, long, continuous, linear vector by flattening it.
• A fully connected layer forms when the flattened matrix from the pooling layer is served as an input, classifying and identifying the images.
Long Short Term Memory Networks (LSTMs)
LSTMs are a set of Recurrent Neural Networks (RNN) specialized in learning and memorizing long-term dependencies. Recalling past information for a long duration is the default behaviour.
LSTMs preserve information over time. Â Thus, they are helpful in time-series prediction as they remember previous inputs.
Therefore, LSTMs have a chain-like structure where four interacting layers communicate uniquely. Besides time-series predictions, LSTMs are typically for speech recognition, music composition, and
pharmaceutical development.
How Do LSTMs Work?
• Firstly, they ignore irrelevant parts of the previous state
• Next, they selectively renew the cell-state values
• Ultimately, the output of certain parts of the cell state
Recurrent Neural Networks (RNNs)
RNNs have associations that form directed cycles, allowing the LSTM to be the inputs to the current phase. RNNs are usually for image captioning, time-series analysis, natural-language processing,
handwriting recognition, and machine translation.
How Do RNNs work?
• The output of the LSTM becomes an input to the current phase, enabling the memory of past inputs due to its efficient internal memory.
• RNNs can process inputs of different lengths. The more the computation, the more are the chances of gathering information, and the model size does not grow with the input size.
Generative Adversarial Networks (GANs)
GANs are generative deep learning algorithms that produce new data instances that relate to the training data. Therefore, GAN has two components: a generator, which generates fake data, and a
discriminator, which learns from that phony information.
How Do GANs work?
• The discriminator gets the difference between the generator’s fake data and the actual sample data.
• Hence, during the initial training, the generator creates fake data, and the discriminator quickly learns to tell that it’s false.
• The GAN sends the output to the generator and the discriminator to renew the model.
Radial Basis Function Networks (RBFNs)
RBFNs are unique types of feedforward neural networks that utilize radial basis functions as activation functions. Therefore, they have input, hidden, and output layers used for classification,
regression, and time-series prediction.
How Do RBFNs Work?
RBFN utilizes trial and error to define the structure of the network. It has two steps as follows:
• Firstly, the centres of the hidden layer using an unsupervised learning algorithm are determined.
• Lastly, the weights with linear regression are determined.
Multilayer Perceptrons (MLPs)
MLPs is a great place to start learning about deep learning technology.
It belongs to the family of feedforward neural networks with various layers of perceptrons that have activation functions. Therefore, MLPs consist of an input layer and an output layer that is
entirely connected.
They have an equal number of input and output layers but may have various hidden layers and can be used to build speech recognition, image recognition, and machine-translation software.
How Do MLPs Work?
• MLPs serve the data to the input layer of the network. The layers of neurons join in a graph so that the signal passes in one direction.
• It computes the input with the weights that exist between the input layer and the hidden layers.
• MLPs use activation functions to decide which nodes to fire.
• It also trains the model to know the correlation and learn the dependencies within the independent and the target variables of a training data set.
Self Organizing Maps (SOMs)
Professor Teuvo Kohonen invented SOMs, enabling data visualization to decrease data dimensions through self-organizing artificial neural networks.
Data visualization tries to solve the problem that humans cannot easily visualize high-dimensional data. Thus, SOMs help users know this high-dimensional information.
How Do SOMs Work?
• SOMs initialize weights for each node and pick a vector at random from the training data.
• They monitor each node to find which weights are the most suitable input vector. Therefore, the best node is knowingly the Best Matching Unit (BMU).
• SOMs find the BMU’s neighborhood, and the amount of neighbors decreases over time.
• Thus, the closer a node is to a BMU, the more its weight changes. The farther the neighbor is from the BMU, the less it learns.
Deep Belief Networks (DBNs)
DBNs are generative models that consist of various layers of stochastic, latent variables. Therefore, the latent variables have binary values and are usually called hidden units.
Deep Belief Networks (DBNs) are for image recognition, video recognition, and motion-capture data.
How Do DBNs Work?
• Greedy learning algorithms guide DBNs. The greedy learning algorithm practices a layer-by-layer approach for getting the top-down, generative weights.
• DBNs run the steps of Gibbs sampling over the top two hidden layers.
• Therefore, they draw a sample from the noticeable units using a single pass of ancestral sampling through the remaining model.
• DBNs discover that a single, bottom-up pass can infer the values of the latent variables in every layer.
Restricted Boltzmann Machines (RBMs)
RBMs are stochastic neural networks that can acquire from a probability distribution over a set of inputs.
Thus, this deep learning algorithm is for dimensionality reduction, classification, regression, collaborative filtering, and feature learning. RBMs constitute the building blocks of DBNs.
How Do RBMs Work?
RBMs consist of two layers:
• Visible units
• Hidden units
Every visible unit is symmetrically related to all hidden units. RBMs consists of a bias unit attached to all visible and hidden units but lacks output nodes.
Autoencoder is an unsupervised ANN that recognizes how to compress and encode data efficiently.
Then determines how to reconstruct the data back from the encoded compression to a representation that is as close to the initial input provided at first.
Hence, autoencoders should encode the image and then reduce the input size into a smaller entity. Lastly, the autoencoder decodes the image to generate the reconstructed image.
Deep learning has emerged over the past years, and deep learning algorithms have become broadly popular in many industries. Hence, it has made computers work smart and make them work according to
one’s needs.
With the ever-growing data, these algorithms would only grow more efficient with time and would indeed be able to replicate the human brain.
Also Read:
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Height is computed on the x-coordinates and it is the difference of minimum and maximum x-values after eliminating the outer points on the either ends by using the 95% approximation of the x-values
of the given input neuron.
Function Output Type : Numeric
Function Output Type : Real
Calculated : At each tracing point
Returns a value : For whole arbor
│Metric │Total_Sum│#Compartments │#Compartments │Minimum│Average│Maximum│S.D.│
│ │ │(considered) │(discarded) │ │ │ │ │
│Soma_Surface│159.722 │1 │(1101) │159.722│159.722│159.722│0 │
*All units are in microns
Values to consider : Min/ Avg/ Max
Output Interpretation :
If the user believes that the input neuron is not oriented properly, one can do principal component analysis (PCA) first by selecting PCA in the function panel. In this case, width is computed using
the same 95% approximation on the new x-axis after shifting and rotating the cell to a different axis based on component analysis.
Total_Sum = 159.722 gives Total_Sum, Minimum = 159.722 gives minimum, Maximum = 159.722 gives maximum for given input neuron.
References :
- Although the Y-value at each tracing point is considered, the height is computed for the entire neuron. So, one should only use the Min/Avg/Max values for this metric. NOT Total_Sum.
-One can choose the rule "type=3" in the specificity panel to limit the height analysis to dendrites only.
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Lab Week 14
Learning Objectives
• Implement a recursive mathematical function to draw a set of circles on the canvas.
• Draw a recursive tree.
• Draw a Sierpinski triangle.
In this lab/recitation, you will write some p5.js programs that use the techniques you’ve learned in class so far. The goal here is not just to get the programs done as quickly as possible, but also
to help your peers if they get stuck and to discuss alternate ways to solve the same problems. You will be put into breakout rooms in Zoom to work on your code and then discuss your answers with each
other, or to help each other if you get stuck. Then we will return to discuss the results together as a group.
For each problem, you will start with a copy of the uncompressed template-p5only.zip in a folder named lab-14. Rename the folder as andrewID-14-A, andrewID-14-B, etc. as appropriate.
A. Fibonacci Numbers
The Fibonacci sequence is the following sequence of numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
These numbers occur in nature, architecture, music, and art. We will implement a version of the program completed in class that will display a set of circles where the number of circles is given by a
Fibonacci number. The program will advance through the numbers, drawing more circles each time.
If you look at the sequence carefully, you will see that any Fibonacci number in the sequence is the sum of the previous two Fibonacci numbers. (Does that sound recursive to you?) Of course, two of
the numbers don’t follow this rule; which ones?
We say the 0th Fibonacci number is 1, the 1st Fibonacci number is 1, the 2nd Fibonacci number is 2, the 3rd Fibonacci number is 3, the 4th Fibonacci number is 5, the 5th Fibonacci number is 8, etc.
n 0 1 2 3 4 5 6 7 8 9 ...
fibonacci(n) 1 1 2 3 5 8 13 21 34 55 ...
Complete the fibonacci function below so that the program computes and returns the n^th Fibonacci number recursively.
var n = 0;
function setup() {
createCanvas(400, 400);
function draw() {
var numCircles = fibonacci(n);
text(n.toString(), 10, 10);
text(numCircles.toString(), 10, 30);
n += 1;
function fibonacci(n) {
// replace the question marks with the required expressions:
if ( ??????????????? ) return 1;
else return ( ??????????????? );
function drawCircles(numCircles) {
for (var i = 0; i < numCircles; i++) {
fill(color(random(0,256), random(0, 256), random(0, 256)));
circle(random(0, width), random(0, height), random(10, 30));
B. Recursive Tree
We wish to draw the tree above. If you look at it recursively, a tree consists of a trunk, along with two trees on top of it, one at an angle 10 degrees to the left and one at an angle 10 degrees to
the right. Each pair of trees is one level shorter than the whole tree.
Complete the recursive function drawTree so that it draws this recursive tree. Look at the comments to guide you along.
var numLevels = 8;
var branchLength = 40;
function setup() {
createCanvas(400, 400);
function draw() {
translate(200, 350); // location of base of tree
drawTree(numLevels, branchLength);
function drawTree(levels, length) {
// base case: if there are no more levels, just return:
line(0, 0, 0, -length); // draw the trunk of the current tree
// move the origin to the top of the trunk of the current tree:
// rotate to the left 10 degrees of the initial trunk:
// draw a tree with one less level with the same trunk length:
// rotate to the right 10 degrees from the initial trunk:
// draw a tree with one less level with the same trunk length:
C. Sierpinski Triangle
This is an example of a fractal, an image that is self-similar. When you look at parts of the image, you see the original image.
One of the most famous fractals is the Sierpinski Triangle which is a triangle that is divided up into three triangles which are divided up into three triangles each, which are divided up into… you
get the idea.
To draw this sketch, we start with a large triangle of blue. We then find the midpoints of each of the three sides and draw a triangle between these points in the background color. Finally, we repeat
the midpoint process with each of the three remaining blue triangles that are formed when we draw the triangle in the background color. Each of these triangles is “split” into three smaller
triangles. This repeats until we reach the number of levels of repetition.
Complete the function splitIntoThree based on the comments to guide you.
var numLevels = 4;
function setup() {
createCanvas(400, 400);
function draw() {
fill(0, 0, 255);
triangle(200, 50, 350, 350, 50, 350);
splitIntoThree(numLevels, 200, 50, 350, 350, 50, 350);
function splitIntoThree(levels, x0, y0, x1, y1, x2, y2) {
// base case: if there are no more levels left to draw, we're done:
fill(220); // background color
var x01 = midpt(x0, x1); // midpoint of x between x0 and x1
var y01 = midpt(y0, y1); // etc.
var x12 = midpt(x1, x2);
var y12 = midpt(y1, y2);
var x20 = midpt(x2, x0);
var y20 = midpt(y2, y0);
// draw a triangle using the midpoints:
// split each of the remaining blue triangles with one less level
// (hint: you should have three recursive calls here,
// each call will have a list of x and y points for one of the
// remaining blue triangles):
function midpt(a, b) {
return (a + b)/2;
At the end of the lab, zip the lab-14 folder (whatever you got done) and submit it to Autolab. Do not worry if you did not complete all of the programming problems but you should have made it through
problems A and B, and you should have some attempt at problem C.
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Top Notch Algebra 2 Tutoring in Pittsburgh | Grade Potential
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Finite element analysis model of rotary forging for assembling wheel hub bearing assembly – Post 2
Finite element analysis model of rotary forging for assembling wheel hub
bearing assembly – Post 2/2
This post is in continuation with last week’s post on the finite element analysis model of rotary forging process of a wheel hub bearing assembly. Adding on to that introductory post, the details of
the analysis model and the predictions will be discussed in this post.
This kind of rotary forging processes requires higher computational times using a full domain analysis model because they have pretty larger strokes compared to conventional forging. It should be
noted that the plastic deformation occurs due to a localized contact of the forming die with material in the early stage, implying that the plastic deformation has a restricted effect on the
neighborhood of the local contact region. This fact can be observed from Figure 1, indicating that the contact region during rotary forging for assembling a wheel hub bearing assembly is quite small.
Figure 1: Contact area during rotary forging
Therefore, as shown in Figure 2, a partial analysis model defined by two artificial planes of symmetry is proposed for the engineering analysis model, which was successfully applied for simulating a
flow forming process (Cho et al., 2011). Of course, it is noted that the analysis results of this model may be more or less different from the real phenomena, especially at the planes of symmetry.
However, it can be expected that quite reliable predictions can be obtained for the present rotary forging process because the plastic deformation occurring at the local contact area has little
influence on that at the opposite side as shown in Figure 1, emphasizing that the effective strain rate distribution is concentrated around local contact area.
Figure 2: Finite element analysis model for the 60° analysis model
To check validity of the proposed approach, a rotary forging process of Figure 2(a) was analyzed, which was previously studied using a hexahedral element (Moon et al., 2007). The shape of a preform
and its dimensions are shown in Figure 4(b). The lower part of the preform is far away from being plastically deformed and its displacement was constrained by a constraint box in which all nodal
degrees of freedom of the nodes are constrained.
The flow stress was defined by
The upper die revolves without any power exerted and the friction between the upper die and material was thus neglected. If the friction is considered, the revolving velocity will become an unknown
variable, which may enhance a negligible part of solution accuracy at a greater expense of computational time.
Figure 2 is a 60° analysis model, which is composed of two artificial planes of symmetry, a part of material defined by them, a constraint box and tools or dies. First, to reveal the size effect of
the analysis domain defined by two planes of symmetry, 30°, 60° and 90° analysis models were investigated.
Figure 3 shows the predicted configurations of the deformed material together with the inner race of hub bearing unit at the selected planes for the 60° analysis model. As shown in the figure, there
exists a non-negligible difference in deformed shape of material around the bent region between the mid-plane and symmetric plane. It can be seen that change of the shape is stationary round the
mid-plane between 20° and 40° planes, implying that the mid-plane is quite far away from the effect of the assumed artificial plane.
Figure 3: Predictions for the 60° analysis model
Figure 4: Deformation history for the 60° analysis model
Figure 4 shows the history of deformation of sections A and B for the 60° analysis model. It can be seen at a glance that the deformation history of section A is nearly the same with that of section
B as a whole. However, around the final stroke a distinct difference in contact region between the two planes can be observed. It is noteworthy that the size of cavity formed between the hub and
inner race of hub bearing has a strong influence on the forming load when the process is controlled in terms of displacement (Shim et al., 2012) because the free surface around the major deforming
region becomes very small at the final stroke.
Figure 5. Comparison of mid-planes of the 30°, 60° and 90° analysis models and experiments
Figures 5(a)-(c) compare the deformed shapes at the mid-plane for the 30°, 60° and 90° analysis models, indicating that all the predictions are nearly the same. Comparison of the predictions in
Figures 5(a), (b) and (c) with the experiments in Figure 5(d) shows that they are acceptable. Less than one hour of computational time was taken for the 60° analysis model. Figure 6 shows the
predictions obtained by the 60° analysis model at the final stroke with emphasis on finite element mesh system used.
Figure 6. Predictions by the 60 ° final shape analysis model
Concluding remarks:
A computationally efficient finite element analysis model was proposed for analyzing a rotary forging process. The model is composed of one or two artificial planes of symmetry and a part of material
defined by them. The model assumes that plastic deformation is concentrated on relatively small contact area and was employed for simulating a cold rotary forging process of a wheel hub bearing
assembly, after local contact area was found to be very small, which is a typical application example of the proposed analysis model. The simulation was conducted using a rigid-plastic finite element
method assisted by an intelligent remeshing technique.
Three cases of 30°, 60° and 90° analysis models were studied to validate the present finite element analysis model. The predictions at their planes of symmetry and mid-planes were investigated and
compared with the experiments, revealing that the predictions at the mid-planes are in good agreement with the experiments for all the cases while those at the planes of symmetry are more or less
different from the actual phenomena. Based on the discussion about the predictions, the 60° analysis model is recommended for both computational efficiency and solution reliability. With the present
finite element analysis model, computational time could be reduced drastically.
[1] Cho, J. M., Jung, Y. D., Lee, M. C., Joun, M. S., 2011.Finite element model of simulating a chipless forming process based on flow forming. Proceedings of the Korean Society for Technology of
Plasticity, 143-146
[2] Choi, S., Na, K.H., Kim, J. H., 1997. Upper-bound analysis of the rotary forging of a cylindrical billet. Journal of Material Processing Technology, 67(1-3), 78-82.
[3] Guangchun, W., Guoqun, Z., 2002. Simulation and analysis of rotary forging of a ring workpiece using finite element method. Finite Elements in Analysis and Design, 38(12), 1151-1164.
[4] Han, X., Hua, L., 2013. 3D FE modelling of contact pressure response in cold rotary forging. Tribology International, 57, 115-123.
[5] Hawkyard, J. B., Gurnani, C. K. S., Johnson, W., 1977. Pressure distribution measurements in rotary forging. Journal of Mechanical Engineering Science, 19(4), 135-137.
[6] Liu, G., Yuan, S. J., Wang, Z. R., Zhou, D. C., 2004. Explanation of the mushroom effect in the rotary forging of a cylinder. Journal of Material Processing Technology, 151(1-3), 178–82.
[7] Moon, H. K., Lee, M. C., Joun, M. S., 2007. An approximate efficient finite element approach to simulating a rotary forming process and its application to a wheel-bearing assembly. Finite
Elements in Analysis and Design, 44(1-2), 17-23.
[8] Munshi, M., Shah, K., Cho, H., Altan, T., 2005. Finite element analysis of orbital forming used in spindle/inner ring assembly. 8th ICTP, Verona,Italy.
[9] Shim, G. H., Kim, D. K., Choi, M. H., Kim, E. Z., Joun, M. S., 2012. Proposal of an optimized forging process for assembling hub bearing unit. Proceedings of the Korean Society for Technology of
Plasticity, 346-349.
[10] Toda, K., Ishii, T., Kashiwagi, S., Mitarai, T., 2001. Development of hub units with shaft clinching for automotive wheel bearing. KOYO Engineering Journal English Edition, 158, 26-30.
[11] Wang, G. C., Guan, J., Zhao, G. Q., 2005. A photo-plastic experimental study on deformation of rotary forging a ring workpiece. Journal of Material Processing Technology, 169(1), 108–114.
[12] Yuan, S. J., Wang, X. H., Liu, G., Zhou, D. C., 1998. The precision forming of pin parts by cold-drawing and rotary-forging. Journal of Material Processing Technology, 86(1), 252–256.
[13] Zhou, D. C., Yuan, S. J., Wang, Z. R., Xiao, Z. R., 1992. Defects caused in forming process of rotary forged parts and their preventive methods. Journal of Material Processing Technology, 32
(1-2), 471–479.
Do follow us on LinkedIn to stay updated and know more interesting simulation examples from a wide variety of metal forming processes.
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Semiconductors/Op-Amps - Wikibooks, open books for an open world
Op-Amps stands for Operational Amplifier, a device usually manufactured as an Integrated Circuit (IC) . User don't need to know about the Itegrated Circuit inside in order to work with Op-Amps . All
you need to know is that Op-Amps acts as an amplifier, it amplifies the difference of the 2 different voltages at input by a gain or amplification factor A
Output of an operational amplifier:
${\displaystyle V_{O}=A(V^{+}-V^{-})}$
Ideally ${\displaystyle A}$ is assumed to be equal to infinity. However, in practical op-amps, it has a high value. Furthermore, the gain ${\displaystyle A}$ is a function of frequency.
Below is the most commonly used configuration of Op-Amps
Inverting amplifier
${\displaystyle V_{\mathrm {out} }=-V_{\mathrm {in} }{\frac {R_{f}}{R_{\mathrm {in} }}}}$
• The output voltage is a negative voltage equal to the input voltage amplified by a factor ${\displaystyle {\frac {R_{f}}{R_{\mathrm {in} }}}}$
Positive amplifer.
${\displaystyle V_{\mathrm {out} }=V_{\mathrm {in} }\left(1+{R_{2} \over R_{1}}\right)}$
• The output voltage is a positive voltage equal to the input voltage amplified by a factor ${\displaystyle \left(1+{R_{2} \over R_{1}}\right)}$
• From the circuit of Positive Amplifier, If R[2] = 0 thì V[o] = V[i]
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