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Find All Records Which Have A Count Of An Association Greater Than Zero - The Citrus Report
Find All Records Which Have A Count Of An Association Greater Than Zero
Have you ever wondered how to find all records with duplicate values in MapInfo Pro? Or how to add amounts greater than zero in Excel? Maybe you’re struggling with directed numbers in PT3
Mathematics, or trying to subscribe to Count Zero Records? Whatever your current dilemma, we’ve got you covered with some practical solutions.
Finding All Records With Duplicate Values in a Column in MapInfo Pro
If you’re working with MapInfo Pro, it can be helpful to identify all records with duplicate values in a particular column. Fortunately, this is a fairly simple process.
First, open the table containing the column you want to check for duplicate values. Then, go to Table>Tools and click on “SQL Select.” In the SQL Select dialog box, enter the following statement:
SELECT ColumnName, COUNT( * ) AS NumOccurrences
FROM TableName
GROUP BY ColumnName
HAVING ( COUNT( * ) > 1 )
Replace “ColumnName” with the name of the column you want to check and “TableName” with the name of the table you’re working with. Once you’ve entered the statement, click on “OK.”
This will create a new table that displays all records with duplicate values in the specified column. You can then use this information to clean up your data and ensure that you don’t have any
duplicate entries.
Adding Amounts Greater Than Zero in Excel
If you need to add a column of numbers in Excel, but only want to include values that are greater than zero, you can use the SUMIF function to achieve this.
Assuming that your data is contained in column A, start by creating a new column next to it and enter the following formula in the first cell:
=SUMIF(A:A, ">0")
This will add up all of the values in column A that are greater than zero and display the total in the cell with the formula. You can then autofill this formula to the rest of the cells in your new
column to calculate the total for the entire dataset.
Working With Directed Numbers in PT3 Mathematics
Directed numbers can be a challenge to work with in mathematics, but a solid understanding of the basic concepts can make it much easier to navigate these types of problems.
The most important thing to keep in mind when working with directed numbers is the idea of “direction,” which is indicated by a positive or negative sign. Positive numbers are greater than zero and
move to the right on a number line, while negative numbers are less than zero and move to the left on a number line.
When adding or subtracting directed numbers, you’ll need to pay attention to the signs of each number and whether they are being added or subtracted. If both numbers have the same sign, you’ll add
their absolute values and keep the same sign. If they have different signs, you’ll subtract their absolute values and take the sign of the larger number.
Multiplying and dividing directed numbers follows a similar pattern, with the added complication of needing to consider whether the number you’re multiplying or dividing with is positive or negative.
If you’re struggling with directed numbers, don’t hesitate to consult with your math teacher or tutor for additional guidance.
Subscribing to Count Zero Records
If you’re a fan of independent music, you should definitely check out Count Zero Records, a boutique record label that specializes in niche genres like industrial, electronica, and darkwave.
To subscribe to Count Zero Records, simply visit their website and sign up for their mailing list. This will give you access to updates on new releases, exclusive promotions, and special events that
you won’t want to miss.
If you’re looking for something new and exciting to add to your music collection, Count Zero Records is definitely worth a look.
Other Handy Tips and Tricks
Here are a few more useful tips and tricks to help you get the most out of your data and software tools:
• If you’re working with Excel and need to count the number of cells that contain a value greater than zero, you can use the COUNTIF function. Simply enter the following formula:
=COUNTIF(range, ">0")
• If you’re using LibreOffice Calc and need to count the number of cells that are greater than or less than zero, you can use the following formula:
=COUNTIF(range;">0") or =COUNTIF(range;"<0")
• If you need to find duplicate fields in a table using SQL Server, you can use the following statement:
SELECT column1, column2, COUNT(*)
FROM tableName
GROUP BY column1, column2
HAVING COUNT(*) > 1
• If you’re trying to find the greater number closest to N with at most one non-zero digit, you can use the following algorithm:
1. Divide N by 10 and take the remainder.
2. If the remainder is greater than 0 and less than or equal to 5, subtract it from N and return the result.
3. If the remainder is greater than 5, subtract it from 10 and add the result to N.
4. If the remainder is 0, add 1 to N.
These tips and tricks should help you solve some common problems and make the most of your data and software tools.
The Bottom Line
Whether you’re trying to find duplicate values in MapInfo Pro, add amounts greater than zero in Excel, or work with directed numbers in PT3 Mathematics, there are always practical solutions available
to help you get the job done.
By following these tips and tricks, you can make the most of your software tools and data, and tackle even the most challenging problems with confidence.
So don’t be afraid to dive in and experiment with new approaches and techniques – you never know what you might discover!
Leave a Comment
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Basic Operations
Ah, the familiar four basic operations:
But there is more to the story!
Addition and Subtraction
Addition says how many steps to take.
Subtraction is actually addition, we just change the sign of the second number before we add.
Example: 7 − 5 = 7 + (−5) = 2
This idea is very useful!
Example: Computer programs (or computer chips)
To handle numbers we first find a good way to "add"
And then to subtract we simply change the sign of the second number, then add.
Play with it here (sliders):
Multiplication and Division
Multiplication says how many adds to do.
Division is actually multiplication, we just do the reciprocal (1value) of the second number before we multiply.
Example: 12 ÷ 4 = 12 × 14 = 3
This idea is very useful!
It helps explain fractions: 34 is multiplication of 3 and 14
In some areas of mathematics (such as Matrix Algebra) we cannot divide, but we can do an inverse, so we multiply by the inverse and the job is done.
Wait, What?
So the four basic operators are just two?
Much simpler don't you think?
We just need to be happy with the concept of inverse:
• Additive inverse (change sign)
• Multiplicative inverse (1value)
They are also both better behaved:
Addition is commutative: 3 + 5 = 5 + 3. But subtraction is not: 3 − 5 ≠ 5 − 3
Multiplication is commutative: 3 × 5 = 5 × 3. But division is not: 3/5 ≠ 5/3
Subtraction and division are both still important ... we just see them now from a higher level.
Exponents and Logarithms
We can go one step further:
Exponents say how many multiplies to do.
The inverse of exponent is logarithm.
Order of Operations
Now order of operations becomes simpler. PEMDAS becomes PEMA:
• Parentheses (overrides the usual order)
• Exponents (how many multiplies)
• Multiply (how many adds)
• Add (how many steps)
Knowing the concept of inverses lets us simplify to this:
• Addition says how many steps
□ Inverse is changing sign/direction
• Multiplication says how many adds
□ Inverse is the reciprocal (1value)
• Exponents says how many multiplies
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Limit Cycles - (Control Theory) - Vocab, Definition, Explanations | Fiveable
Limit Cycles
from class:
Control Theory
Limit cycles are closed trajectories in phase space that represent periodic solutions of a dynamical system. They occur in systems exhibiting nonlinear behavior, where small perturbations can lead to
stable oscillations, making them important in the analysis of feedback systems and stability. Understanding limit cycles is essential for analyzing the behavior of nonlinear control systems and can
indicate the presence of oscillatory responses.
congrats on reading the definition of Limit Cycles. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Limit cycles can be stable or unstable; a stable limit cycle attracts nearby trajectories, while an unstable one repels them.
2. They are crucial in the analysis of oscillatory systems, such as mechanical vibrations and biological rhythms.
3. Limit cycles can arise from nonlinearities in feedback loops, causing sustained oscillations even in the presence of disturbances.
4. The Poincarรฉ-Bendixson theorem provides conditions under which limit cycles exist in planar systems.
5. Describing function analysis is a method used to approximate the behavior of nonlinear systems and can be utilized to identify potential limit cycles.
Review Questions
• How do limit cycles relate to stability in nonlinear dynamical systems?
□ Limit cycles play a significant role in determining the stability of nonlinear dynamical systems. A stable limit cycle will attract nearby trajectories, meaning that any small perturbation
will eventually return to the cycle, indicating robustness in the system's behavior. In contrast, an unstable limit cycle will push nearby trajectories away, leading to potential
unpredictability and divergence from periodic behavior. Understanding these dynamics helps engineers design more reliable control systems.
• Discuss how describing function analysis can be used to predict the existence of limit cycles in a nonlinear system.
□ Describing function analysis simplifies the analysis of nonlinear systems by approximating their behavior through linear techniques. By applying this method, one can identify conditions under
which limit cycles might emerge based on the system's frequency response. Specifically, it involves plotting the describing function against the amplitude of oscillation to find intersections
that indicate potential limit cycles. This approach allows for an effective way to anticipate and analyze oscillatory behaviors without solving complex nonlinear equations directly.
• Evaluate the implications of limit cycles for control system design and performance.
□ Limit cycles have significant implications for control system design and performance. The presence of limit cycles can lead to unwanted oscillations that degrade system performance and
stability. Designers must account for these behaviors by implementing strategies such as feedback linearization or adaptive control methods that mitigate oscillatory responses. Additionally,
understanding where limit cycles may occur allows engineers to optimize system parameters to achieve desired stability margins, ensuring that control systems remain responsive without
exhibiting detrimental oscillations.
ยฉ 2024 Fiveable Inc. All rights reserved.
APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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Mortal Kombat
Execution time limit is 1 second
Runtime memory usage limit is 64 megabytes
Once every generation, there is a tournament known as Mortal Kombat, which was designed by the Elder Gods for the main purpose to save Earthrealm from the dark forces of Outworld. If the forces of
Outworld win the tournament ten consecutive times, the Emperor will be able to invade and conquer Earthrealm. Thus far, Outworld has won nine straight victories, making the upcoming tournament the
tenth, and possibly final one, for the Earthrealm.
There are N monsters and M best human fighters participating in the Mortal Kombat. According to the tournament rules, each monster should fight one of the humans (different monsters should fight
different humans). If at least one monster wins, the Eathrealm will be conquered by the Emperor of the Outworld. However, the humans can choose the competitors and the order of battles.
The thunder god Raiden, protector of the Earthrealm, should choose the fighters in such a way that all Earth warriors will win their battles. For each monster and each Earth warrior it is known
whether the Earth warrior can win the monster. First of all, the fighters for the first battle should be chosen.
For example, suppose that Liu Kang wants to fight Goro, but he is the only warrior able to defeat Shang Tsung, while Goro can be defeated by other warriors, such as Johnny Cage. So, even if Liu Kang
will defeat Goro in the first battle, it will inevitably lead to the conquest of the Earth, because later Shang Tsung will defeat his opponent. This means that the pair Liu Kang vs. Goro should not
be selected for the first fight.
Find out which pairs cannot be chosen by Raiden if he wants to save the freedom of humanity.
The first line contains integers N and M. 1 ≤ N ≤ 300, N ≤ M ≤ 1500. Next lines contain the binary matrix A with N rows and M columns. A_ij = 1 if and only if j-th Earth warrior can defeat i-th
Output matrix B with N rows and M columns. B_ij should be equal to one if the first battle cannot be held between i-th monster and j-th human, and zero otherwise.
Submissions 78
Acceptance rate 45%
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Fundamental Physics Prize
Check all the winners of Fundamental Physics Prize.
Year Winner Winner Work
Alexander For his many discoveries in field theory and string theory including the conformal bootstrap, magnetic monopoles, instantons, confinement/de-confinement, the quantization of
2013 Markovich strings in non-critical dimensions, gauge/string duality and many others. His ideas have dominated the scene in these fields during the past decades.
2012 Nima For original approaches to outstanding problems in particle physics, including the proposal of large extra dimensions, new theories for the Higgs boson, novel realizations of
Arkani-Hamed supersymmetry, theories for dark matter, and the exploration of new mathematical structures in gauge theory scattering amplitudes.
2012 Alan Guth For the invention of inflationary cosmology, and for his contributions to the theory for the generation of cosmological density fluctuations arising from quantum fluctuations
in the early universe, and for his ongoing work on the problem of defining probabilities in eternally inflating spacetimes.
2012 Alexei Kitaev For the theoretical idea of implementing robust quantum memories and fault-tolerant quantum computation using topological quantum phases with anyons and unpaired Majorana
2012 Maxim Kontsevich For numerous contributions which have taken the fruitful interaction between modern theoretical physics and mathematics to new heights, including the development of homological
mirror symmetry, and the study of wall-crossing phenomena.
2012 Andrei Linde For the development of inflationary cosmology, including the theory of new inflation, eternal chaotic inflation and the theory of inflationary multiverse, and for contributing
to the development of vacuum stabilization mechanisms in string theory.
2012 Juan Martín For the gauge/gravity duality, relating gravitational physics in a spacetime and quantum field theory on the boundary of the spacetime.
2012 Nathan Seiberg For major contributions to our understanding of quantum field theory and string theory.
2012 Ashoke Sen For uncovering striking evidence of strong-weak duality in certain supersymmetric string theories and gauge theories, opening the path to the realization that all string
theories are different limits of the same underlying theory.
For contributions to physics spanning topics such as new applications of topology to physics, non perturbative duality symmetries, models of particle physics derived from
2012 Edward Witten string theory, dark matter detection, and the twistor-string approach to particle scattering amplitudes, as well as numerous applications of quantum field theory to
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Instructions for Preparing Manuscripts - P.PDFKUL.COM
DIRECT: An Efficient Optimization Scheme for Mask Generation Using Inverse Lithography Wei Xiong*, Min-Chun Tsai**, Jinyu Zhang* and Zhiping Yu* *
Institute of Microelectronics, Tsinghua University, Beijing, 100084, China
[email protected]
** Advanced Technology Group, Synopsys Inc., Mountain View, CA, 94043, USA
[email protected]
ABSTRACT Resolution Enhancement Technologies (RETs) are widely used to cope with the severe optical effects that are manifests in sub-wavelength lithography. Inverse Lithography Technique (ILT) has
recently been proposed as an effective RET for sub-wavelength technology. ILT increases the degree-of-freedom in mask data manipulation, and allows automatic correction to 2D pattern distortion. In
this work, a realistic aerial image model with an efficient optimization scheme is developed to pattern metal layers for the 65nm technology node. Simulation results show that the optimized masks
provide good fidelity in patterning. We called our method DIscrete REtiCle Technique, DIRECT. Keywords: ilt, ret, mask, simulated annealing, psm
As semiconductor manufacturing reaches 90nm and 65nm technology nodes, and is moving towards 45nm, 32nm, and below, one of the greatest challenges is in lithography. The mostly used exposure steppers
use 193nm wavelength, however the critical dimension (CD) on wafer keeps shrunk, approaching a quarter of the illumination wavelength. Under such circumstances, due to the optical diffraction and
interference, serious distortions occurred when translating the patterns on the mask to the wafer, causing the failure of printed circuits. The pattern distortion is generally called optical
proximity effect (OPE) [1], [2]. In order to compensate OPE, people have proposed a series of techniques to improve the litho-system’s resolution. The resolution is determined by the Rayleigh’s
criterion, R = k λ / NA
where λ is the illumination wavelength, NA is the numerical aperture of the imaging lenses, and k is the process constant affected by the process conditions. Using a new exposure source may decrease
the wavelength, and putting the imaging lenses into water or other liquid with a high refraction index can increase the numerical aperture. These two approaches can both improve the resolution
but they require changing the existing lithography infrastructure and have high costs. An alternative approach is using light’s wave nature to decrease the process constant k . This method is called
RETs (Resolution Enhancement Techniques). Based on exploring the optical wave’s amplitude, phase and direction, RETs are accordingly classified as optical proximity correction (OPC), phaseshifting
masks (PSM) and off-axis illumination (OAI) [3]. These methods have extended the lifespan of current optical projection systems. However, finding an optimum mask becomes increasingly complex, as less
intuitive relationships between optimum masks and resulted images apply. More robust optimization techniques need to be developed to improve the image formation. Back in the early 1980s, B. E. A.
Saleh and S. Sayegh [4] considered the design of masks as an inverse problem, and proposed a rigorous mathematical approach to solve the inverse problem and find the optimal mask for a given process.
In their work, mask is discretized to pixels, and then values 0 and 1 are tried. If the image fidelity improves, the pixel value is accepted, otherwise it is rejected, and the next pixel is then
tried, and so on. Later, Liu [5] and Sherif [6] applied simulated annealing (SA) algorithm and branchbound (BB) algorithm to the mask optimization, respectively. But the early SA algorithm is
inadequate in the convergency efficiency, which leaves room for improving this algorithm, while BB algorithm would encounter difficulties when the pixels have more than two values. Recently, Granik
[7] has considered the inverse problem as a nonlinear programming problem, and demonstrated it is a possible solution method. In this article, we discuss our optimization method DIRECT, which is
composed of two stages: modeling of the imaging system and the synthesis of masks. A realistic lithography system can be approximated by a partially coherent model, which is based on famous Hopkins’
Equation [8]. We treat the partially coherent system as sum of coherent systems with different weight. SA algorithm is applied to optimize masks, and an efficient optimization scheme is developed to
accelerate the convergency process and to reduce the calculation time. The optical imaging model is formulated in Section 2. The optimization scheme for SA algorithm is explained in
Section 3. Some simulation results are shown in Section 4. Finally, we provide conclusive remarks in Section 5.
The imaging mechanism of a stepper can be modeled by Hopkins equation [8]: I ( f , g) =
∫ ∫
T ( f '+ f , g '+ g , f ', g ')
⋅M ( f '+ f , g '+ g ) M * ( f ', g ')df ' dg '
where I ( f , g ) is the forward Fourier transform of the output image intensity i ( x, y ) , M ( f , g ) is the forward Fourier transform of mask transmission function m( x, y ) , and T ( f , g , f
', g ') is the Transmission Cross-Coefficient (TCC) of the optical system, which characterize all the features about the imaging system and illumination. In our model, 2D mask pattern is quantized
into small square pixels, as shown in Fig. 2(a), with aik representing the transmission variable at the (i, k ) -th pixel, then the mask transmission function can be expressed as: N1 N 2
m( x, y ) = ∑∑ aikψ ik ( x, y )
i =1 k =1
where N1 , N 2 are the numbers of pixels in each dimension and ψ ik is a unit square pulse function located at the (i, k ) -th pixel. For different kinds of masks, aik may have different values. It
takes values {0, 1} for binary mask, {0.245, 1} for EPSM and {1, 0, -1} for APSM, the minus sign means that the light’s phase has changed 180 degrees. EPSM and APSM are two types of special
phase-shifting masks for 65 CMOS technology node. It is difficult and time-consuming to calculate the output image intensity using Eq. (2) directly. We need to decompose the TCC function as follows
[8], [9]: M T ( f '+ f , g '+ g , f ', g ') ≈ ∑ σ Φ ( f '+ f , g '+ g )Φ* ( f ', g ') l l l l =1
This transformation is based on the singular value decomposition (SVD) of matrices and {σ l } is the singular value set of TCC matrix, {Φ l } is the kernel set corresponding to the singular value
set, and M is the order of decomposition. These singular values and kernels are only determined by the characteristic of the imaging system. Substituting Eq. (4) in Eq. (2) and taking inverse Fourier
trans-form, we obtain the following expression for the output image intensity: i ( x, y ) = F
where {φl } is the inverse Fourier transform of {Φ l } , and notation ⊗ represents the convolution operator. {σ l } and {φl } are obtained before the synthesis of masks, and we can use Eq. (5) to
calculate the output image intensity distribution. Eq. (5) also has an obvious physical meaning that the original partially coherent system can be considered as sum of several weighted coherent
systems, as shown in Fig. 1.
[ I ( f , g )] = ∑ σ l (φl ⊗ m)( x, y ) l =1
Fig. 1: Approximation of the partially coherent system.
Simulated Annealing is a global optimization algorithm to solve complex combinatorial problems [10], [11]. A brief description of the basic algorithm is provided below, and the emphasis is on the way
we apply it to the mask design problem. Assume different patterns of the mask to be the state of a statistical system, denoted by a random vector X . If we introduce a control temperature T and the
energy function or error function corresponding to a state X as H ( X ) , then according to Boltzmann distribution the probability of being in state X at thermal equilibrium is: P ( X) =
1 exp{−[ H ( X) / T ]} Z (T )
where Z is a normalization constant, or the partition function in statistical physics. At high temperature, this distribution is almost uniform, and the system is equally likely to be in any state.
Then we gradually decrease T , allowing the system to reach thermal equilibrium at each T . As the temperature decreases, the Boltzmann distribution concentrates more on the states with low energy.
Finally, when T approaches zero, the system will reach a ground state, namely the optimum solution of mask design problem. In computer simulations, the system reaches the thermal equilibrium at each
T by randomly choosing a pixel and by flipping it to the other state, continuously. For binary mask, assuming the (i, k ) -th pixel to be selected, if we denote the state before its flipping as X m ,
and the state after the flipping as X n , and the corresponding energy functions by H ( X m ) and H ( X n ) , then the probability for state transition of X m → X n is:
H(Xn) − H(Xm) < 0 ⎧1 ⎪ P( X m → X n ) = ⎨ ⎛ −[ H ( X n ) − H ( X m )] ⎞ H (Xn ) − H (Xm ) ≥ 0 ⎟ ⎪exp ⎜ T ⎝ ⎠ ⎩
exp(−ΔH1→0 / T ) exp(−ΔH1→0 / T ) + exp(−ΔH1→−1 / T ) + 1
α = 0.8; m = 1, 2,"
as our decreasing rule.
This flipping is called an intent transition. If this flipping is accepted, we call it a success transition and aik changes value, otherwise we call it a failure transition and aik remains unchanged.
For PSM with 3 values which each pixel can take, we still randomly choose a pixel, assuming (i, k ) -th pixel to be selected. If these three different states corresponding to the pixel values are X 1
, X 0 and X −1 , respectively, and corresponding energy functions are H ( X 1 ) , H ( X 0 ) and H ( X −1 ) , then we calculate ΔH1→0 = H ( X 0 ) − H ( X 1 ) , ΔH1→−1 = H ( X −1 ) − H ( X 1 ) and
ΔH1→1 = 0 . So the probability for state transition of X 1 → X 0 is: P( X 1 → X 0 ) =
Tm = α Tm −1 ,
Also we can obtain the probability for state transition of X 1 → X −1 and X 1 → X 1 . We notice that the probability of making a transition from state X m to X n does not depend on the history of how
state X m was reached, hence the generated sequence { X m } for each T is a Markov chain [12]. When applying SA to the practical issues, a cooling schedule should be developed to maximize the
performance of this algorithm. We briefly describe our cooling schedule as follows: Initial Value of Control Temperature: The initial value should be high enough to make all transitions be accepted
almost equally, but not be too high to save computing time. So we find a suitable initial value as follows: 1. Define variable T and initial temperature t0 with any given value; χ = 0.9 , R0 = 0 ;
iteration order k = 1 . 2. Try L intent transitions ( L is a predefined number), and compute the acceptance ratio Rk which is the number of success transitions divided by the number of intent
transitions. 3. If Rk − χ < ε ( ε is a given small value), terminate the iteration. Otherwise, if Rk −1 and Rk < χ , then k = k + 1 , tk = tk −1 + T , return to step 2; if Rk −1 and Rk ≥ χ , then k =
k + 1 , tk = tk −1 − T , return to step 2; if Rk −1 ≥ χ and Rk ≤ χ , then k = k + 1 , tk = tk −1 + T / 2 , T = T / 2 , return to step 2; if Rk −1 ≤ χ and Rk ≥ χ , then k = k +1, tk = tk −1 − T / 2 ,
return to step 2. Finally we obtain the suitable initial temperature value. Decreasing the Control Temperature: In order to achieve the final ground state, the control temperature is supposed to be
reduced slowly. But the slower the temperature reduces, the lower the convergence efficiency is. So we use
Length of Markov Chains: The Markov chain is terminated when the system reaches the thermal equilibrium. Our termination rule is: define U int and Lint ( U int > Lint ) as the ceiling and limit of
the number of the intent transitions. At each temperature, the number of intent transitions tried must be larger than Lint . If the acceptance ratio is over certain predefined value, the Markov chain
is terminated. Meanwhile, we record the state with a minimum energy function in this Markov chain, and use it as the start of next Markov chain. Termination of the Simulated Annealing: A lot of
termination criteria can be used. For example, the energy function declines to certain value; the temperature reaches certain value; the number of intent transition tried exceeds the limit. These
criterions can also be used in a combinatorial form.
In the following simulations, three different types of masks for 65nm CMOS technology node are used to validate the optimization method described above. The common parameters for the image model are
as follows: for the annular illumination, the wavelength is 193nm, and the outer coherence is 0.8, the inner coherence is 0.56 [13]; the numerical aperture is 0.8; the threshold of the resist is
0.247; the pixel size is 25 × 25nm 2 ; the feature size is 100nm. Fig. 2 shows the distortion occurred in the lithography due to the strong OPE. The regular binary mask to be optimized is plotted in
Fig. 2(a). The white pixels represent the transparent area with a transmission value of 1, while the gray pixels represent the opaque area with a transmission value of 0. In Fig. 2(b), the black bold
line represents the desired image, and the gray line shows the output image corresponding to the mask shown in Fig. 2(a).
Fig. 2: (a) The regular mask to be optimized; (b) Output image due to the input mask shown in (a). From Fig. 2(b), an obvious distortion occurred between the desired image and the output image. The
effects of lineend shortening and corner rounding are serious. Then we use our optimization method to improve the image performance. The simulation results are illustrated in Fig. 3 and Fig.
4. The convergence process is shown in Fig. 3, which expplains the relationship between the image performance and the number of intent transitions. The vertical axis represents the deviation rate,
which means the area of where the output image mismatches the desired image divided by the area of the desired image. With the number of intent transitions increasing, the deviation rate decreases,
thus the image performance is improved. Noticed that the deviation rate is basically unchanged when the number of intent transitions is larger than 4500. The optimum mask is thus obtained and is
shown in Fig. 4(a), and its corresponding output image is displayed in Fig. 4(b). Comparing Fig. 4(b) with Fig. 2(b), we can find that the image fidelity improves greatly.
present the opaque area with a transmission value of -0.245. In Fig. 6(a), the white, gray and black pixels represent the area with a transmission value of 1, 0 and -1, respectively. Fig. 5(b) and
Fig. 6(b) show the excellent image performance. And we obtain the results after 7000 and 9000 intent transitions, respectively.
Fig. 6: (a) The optimized APSM mask; (b) Output image due to the input mask shown in (a).
Fig. 3: The convergence process of optimizing binary mask.
A refined optimization scheme based on simulated annealing algorithm is applied to synthesize masks in subwavelength lithography simulation. Three different types of masks for 65nm CMOS technology
node are used to validate the method. The results shows the optimized masks provide good fidelity in patterning, and this algorithm is proved to be of high accuracy, and fast speed.
Fig. 4: (a) The optimized binary mask; (b) Output image due to the input mask shown in (a).
Fig. 5: (a) The optimized EPSM mask; (b) Output image due to the input mask shown in (a). Not only the binary mask but also the advanced masks like 6% EPSM and APSM can be optimized through our
method. The simulation results are shown in Fig. 5 and Fig. 6. In Fig. 5(a), the white pixels represent the transparent area with a transmission value of 1, while the gray ones re-
[1] A. B. Kahng and Y. C. Pati, Proc. ACM Intl. Symp. On Physical Design, pp. 112-119, 1999. [2] C. Dolainsky and W. Maurer, Proc. SPIE, vol. 3051, pp. 774, 1997. [3] L.Liebmann, S. Mansfield, et al,
IBM Journal of Research and Development 45, pp. 651-665, 2001. [4] B.E.A. Saleh and S.I. Sayegh, Optical Eng. Vol. 20, pp. 781-784, 1981 [5] Yong Liu and Avideh Zakhor, IEEE Trans. Semi.
Manufacturing, vol. 5, no. 2, pp. 138-152, 1992 [6] Sherif Sherif and Bahaa Saleh, IEEE Trans. Image Processing, vol. 4, no. 9, pp. 1252-1257, 1995 [7] Yuri Granik, SPIE, vol. 5754, pp. 506-526, 2005
[8] Y.C. Pati and T. Kailath, J.Opt.Soc.Am.A, vol.11, pp. 2438-2452, 1994 [9] N. Cobb, “Fast Optical and Process Proximity Correction Algorithms For Integrated Circuit Manufacturing”, Ph.D.
Dissertation, U.C. Berkeley, 1998. [10] B.Hajek, in Proc. 24th Conf. on Decision and Control, Ft. Lauderdale, pp. 775-760, 1985. [11] S.Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, Science, vol.
220, pp. 671-680, 1983. [12] A. J. Thomason, “Random process,” Dept. of E.E. and C. S., U. C. Berkeley, text manuscript, 1990. [13] A. K. Wong, “Resolution Enhancement Techniques in Optical
Lithography,” In Tutorial Texts in Optical Engineering, TT47. SPIE Press, 2001.
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Dunkl kernel associated with dihedral groups
In this paper, we pursue the investigations started in [18] where the authors provide a construction of the Dunkl intertwining operator for a large subset of the set of regular multiplicity values.
More precisely, we make concrete the action of this operator on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function. In
particular, we obtain a formula for the corresponding Dunkl kernel and another representation of the generalized Bessel function already derived in [7]. When the multiplicity function is everywhere
constant, our computations give a solution to the problem of counting the number of factorizations of an element from a dihedral group into a fixed number of (non-necessarily simple) reflections. In
the remainder of the paper, we supply another method to derive the Dunkl kernel associated with dihedral systems from the corresponding generalized Bessel function. This time, we use the shift
principle together with multiple combinations of Dunkl operators in the directions of the vectors of the canonical basis of R2. When the dihedral system is of order six and only in this case, a
single combination suffices to get the Dunkl kernel and agrees up to an isomorphism with the formula recently obtained by Amri [2, Lemma 1] in the case of a root system of type A[2]. We finally
derive an integral representation for the Dunkl kernel associated with the dihedral system of order eight.
• Dihedral root systems
• Dunkl kernel
• Dunkl operators
• Generalized Bessel function
ASJC Scopus subject areas
• Analysis
• Applied Mathematics
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Are Squares Rectangles? - Daily Quiz and Riddles
Are Squares Rectangles? Squares: A Special Case of Rectangles Explained
Yes, squares are rectangles
When it comes to geometry, there are various shapes with unique properties. One such relationship is between squares and rectangles. The question often arises: “Are squares rectangles?” In this
article, we will delve into the technical and mathematical aspects of squares and rectangles, understanding their defining characteristics and how they are related.
Definition of Rectangles:
To begin, let’s define what constitutes a rectangle mathematically. A rectangle is a quadrilateral (a four-sided polygon) with four right angles, meaning all its interior angles measure 90 degrees.
Additionally, opposite sides of a rectangle are parallel and equal in length. This definition forms the foundation of rectangles and their properties.
Characteristics of Squares:
Now, let’s explore squares and their unique attributes. A square is a specific type of quadrilateral that also falls under the category of rectangles. It possesses all the properties of a rectangle
but with an additional constraint: all four sides of a square are of equal length. Moreover, since all angles of a square are right angles, it inherently satisfies the definition of a rectangle.
The mathematical relationship between squares and rectangles can be summarized as follows:
Every square is a rectangle, but not every rectangle is a square.
In other words, squares are a subset of rectangles, and they share the fundamental characteristics of rectangles while having an additional condition that sets them apart.
To prove that squares are indeed rectangles, we can use the definition of a rectangle. Let ABCD be a square with sides of length ‘a.’ Since all sides of a square are equal, AB = BC = CD = DA = a.
Now, let’s consider two opposite sides of the square, AB and BC. Since all angles of a square are right angles, the opposite sides are parallel. Hence, the four sides of the square are equal in
length and form right angles, meeting the criteria of a rectangle.
In conclusion, squares are indeed rectangles, meeting all the defining characteristics of a rectangle while having the additional property of all sides being of equal length. Understanding this
mathematical relationship is crucial for a solid grasp of geometry and its various shapes. So, the next time you encounter the question, “Are squares rectangles?” you can confidently explain the
technical and mathematical basis behind this relationship.
By educating ourselves about such geometric concepts, we can foster a deeper appreciation for the beauty and elegance of mathematics in our everyday lives.
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Multiplying Mixed Fractions Worksheet
Multiplying Mixed Fractions Worksheet. Additionally, they will use the following worksheets that involve simple one-step fraction equations. In grade 5, students learn to multiply fractions by
fractions and mixed numbers by mixed numbers. You could use the math worksheets on this website in accordance with our Terms of Use to assist students study math. The worksheets may be made in html
or PDF format — both are simple to print.
Additionally, they’ll use the following worksheets that involve easy one-step fraction equations. In grade 5, students study to multiply fractions by fractions and blended numbers by mixed numbers.
Here’s yet one more opportunity for students to maintain proficiency in multiplying fractions! Get into gear and begin multiplying blended numbers with the indicated fractions. Equip 4th grade and
fifth grade youngsters with our multiplying fractions utilizing arrays worksheets that show an equal distribution of objects in columns and rows for example the product.
• Student versions, if present, embrace solely the query web page.
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• This has the benefit that you could save the worksheet immediately from your browser (choose File → Save) and then edit it in Word or other word processing program.
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of workouts.
Explore all of our fractions worksheets, from dividing shapes into “equal elements” to multiplying and dividing improper fractions and mixed numbers. Children begin their study of fraction
multiplication by studying how to multiply a fraction by an entire number (such as 5 × 2/3) — usually in 4th grade. Then in fifth grade, they study to multiply fractions by fractions and by blended
numbers. In sixth and 7th grades, college students merely follow fraction multiplication using bigger denominators and more complex problems. Zero in on multiplying two and three combined numbers by
utilizing this printable useful resource for grade 6 and grade 7 college students.
Multiplying And Dividing Mixed Fractions B
The Open button opens the whole PDF file in a brand new browser tab. The Download button initiates a obtain of the PDF math worksheet. Teacher versions embrace both the question web page and the
answer key. Student versions, if current, embrace solely the question page. Grade 6 and 7 college students should use the grade 5 worksheets for review of fraction multiplication.
To get the PDF worksheet, simply push the button titled “Create PDF” or “Make PDF worksheet”. To get the worksheet in html format, push the button “View in browser” or “Make html worksheet”. This has
the benefit that you can save the worksheet immediately from your browser (choose File → Save) and then edit it in Word or different word processing program. Want to work round your deficiencies in
multiplying fractions? Observe the numerators and denominators of the fractions and verify for cross cancellation to obtain the product.
Multiplying Fractions And Blended Fractions A
It may be filled out and downloaded or printed using the Chrome or Edge browsers, or it could be downloaded, stuffed out and saved or printed in Adobe Reader. Use the generator to make customized
worksheets for fraction operations. Visualize a multiplication sentence on a quantity line diagram and acquire insight into the topic. Acquaint children with the multiplicand, multiplier and product
utilizing the hops drawn at equal intervals on the number line. Multiply a fraction by an entire quantity – determine the missing issue. Members have exclusive facilities to download an individual
worksheet, or a whole stage.
Create a limiteless supply of worksheets for multiplication of fractions and mixed numbers (grades 4-7)! The worksheets can be made in html or PDF format — each are simple to print. And has been
viewed 78 instances this week and 1,186 instances this month. You may use the math worksheets on this web site according to our Terms of Use to assist college students learn math. And has been
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Math Worksheets: Multiplying Fractions Apply
Help 6th grade and seventh grade kids gain in depth information of multiplying three fractions, simplifying the product, and converting an improper fraction to mixed numbers, with this stack of
exercises. Check what’s lacking, it might be the product, the multiplicand, or the multiplier. Keenly observe each of the world fashions in these printable worksheets, find the missing time period,
and complete the multiplication equation. Steal a march in your peers in multiplying two fractions with these pdf worksheets! Multiply the numerators and denominators separately, and reduce the
product to the lowest time period. Models present resolution methods and make multiplication of fractions with entire numbers easy.
Frame the fraction by counting the shaded components, and multiply by the variety of fashions to find the product. K5 Learning provides free worksheets, flashcardsand inexpensiveworkbooksfor
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Is hash function Preimage resistant?
Is hash function Preimage resistant?
Preimage resistance is the property of a hash function that it is hard to invert, that is, given an element in the range of a hash function, it should be computationally infeasible to find an input
that maps to that element.
What happens if hash function is not Preimage resistant?
A hash function h : M→Y is second preimage resistant if, given a message m ∈ M, it is hard to find a message m ∈ M with m = m and h(m) = h(m ). Possible Attack Scenario: If a hash function h is not
second preimage resistant, then an adversary can create a forgery by executing the following steps: 1.
What is preimage in hash function?
A preimage is the data that is input into a hash function to calculate a hash. Since a hash function is a one-way function, the output, the hash, cannot be used to reveal the input, the preimage. Any
piece of data can be used as a preimage. For example, addresses are created by taking the hash of a public key.
Is called as preimage and second preimage attack?
(cryptography) An attack on a cryptographic hash function that is able to find a second preimage for a hash and its preimage; that is, given a hash and an input that has that specific hash, it is
able to find (faster than by brute force) another input with the same hash.
Does collision resistance imply preimage resistance?
Collision resistance implies second-preimage resistance, but does not guarantee preimage resistance. Conversely, a second-preimage attack implies a collision attack (trivially, since, in addition to
x′, x is already known right from the start).
How second pre image resistance and collision resistance is different?
The properties of second preimage resistance and collision resistance may seem similar but the difference is that in the case of second preimage resistance, the attacker is given a message to start
with, but for collision resistance no message is given; it is simply up to the attacker to find any two messages that …
How second pre-image resistance and collision resistance is different?
How do you get the preimage?
Finding the preimage (s) of a value a by a function f is equivalent to solving equation f(x)=a f ( x ) = a . Finding the preimage (s) of a value a by a function f , which has a known curve, is
equivalent to find the abscissae of the intersection(s) of the curve with the ordinate line y=a .
What is preimage second preimage collision resistance?
Second preimage resistance is the property of a hash function that it is computationally infeasible to find any second input that has the same output as a given input.
What is a hash collision attack?
In cryptography, a collision attack on a cryptographic hash tries to find two inputs producing the same hash value, i.e. a hash collision. This is in contrast to a preimage attack where a specific
target hash value is specified. There are roughly two types of collision attacks: Classical collision attack.
What is the difference between second preimage resistance and collision resistance?
What is preimage resistance and second preimage resistance?
Preimage Resistance (One Way): For essentially all pre-specified outputs, it is computationally infeasible to find any input which hashes to that output. • Second Preimage Resistance (Weak Col.
Res.): It is computationally infeasible to find any second input which has the same output as any specified input.
What is pre-image in a function?
preimage (plural preimages) (mathematics) For a given function, the set of all elements of the domain that are mapped into a given subset of the codomain; (formally) given a function ƒ : X → Y and a
subset B ⊆ Y, the set ƒ−1(B) = {x ∈ X : ƒ(x) ∈ B}. quotations ▼ The preimage of under the function is the set .
What is the difference preimage and image?
Preimage = a group of some elements of the input set which are passed to a function to obtain some elements of the output set. It is the inverse of the Image.
How do you find the preimage of a hash?
The preimage of a hash function is the set of all values that produce a specific hash when passed as an input into a hashing function. In mathematical terms, the preimage of a hash function is the
set of all inputs, x, that produce the same output, y, for the equation H(x) = y, where H is the hashing function.
Does 2nd preimage resistance imply collision resistance?
How is hash collision resolved?
One method for resolving collisions looks into the hash table and tries to find another open slot to hold the item that caused the collision. A simple way to do this is to start at the original hash
value position and then move in a sequential manner through the slots until we encounter the first slot that is empty.
Why do hash functions have collisions?
Because hash functions have infinite input length and a predefined output length, there is inevitably going to be the possibility of two different inputs that produce the same output hash. If two
separate inputs produce the same hash output, it is called a collision.
Does preimage resistance implies second preimage resistance?
How are the image and preimage related?
Image In a transformation, the final figure is called the image. Preimage In a transformation, the original figure is called the preimage. A transformation is an operation that is performed on a
shape that moves or changes it in some way.
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American Mathematical Society
On an Angle with Magical Properties
Cornelius O. Horgan
Jeremiah G. Murphy
Communicated by Notices Associate Editor Reza Malek-Madani
1. Introduction
The angle defined by
arises in a surprising number of diverse application areas in science and engineering which seem to have very little connection to one another. Its ubiquitous nature has led to it being called a
“magic angle.” In this expository short article, we provide an outline of why such nomenclature is warranted and hope to draw the attention of the mathematics and science communities to this
intriguing concept.
One of the earliest findings regarding a “magic angle” concept arose in connection with hydrostatic skeletons or muscular hydrostats such as the common worm, octopus arm, or elephant trunk. Such
hydrostats are characterized by cylindrical lattice structures composed of families of inextensible helically wound fibers (modeling fibers of stiff collagen arranged in alternate left- and
right-handed geodesic helices). Special angles also occur in analysis of the mechanical behavior of fiber-reinforced incompressible nonlinearly elastic soft solids. In this context, the magic angle
concept occurs most commonly in structural elements composed of circular cylindrical tubes or solid cylinders reinforced by helically wound fibers, but also occurs in flat thin sheets reinforced by
fibers in the plane. The fibers can be inextensible as in reinforced rubber or extensible such as collagen fibers in soft tissue. Fibers orientated at the magic angle give rise to special mechanical
responses. Magic angles also arise in the field of soft robotics in connection with artificial muscles as well as in nuclear magnetic resonance. In this short exposition, we highlight some of the
most interesting results on magic angles. The interested reader is directed to the references cited for details.
2. Geometry
The magic angle is defined by 1.1 and is often characterized in various applications as the smallest angle for which
or equivalently, by
Sometimes the “magic angle” terminology is used for the complement of this angle ( ) so that
and in this case the analogs of 2.1 and 2.2 are
respectively. We will call the angle 2.3 the “complementary magic angle.”
Figure 1.
Geometry of magic angle.
The angle 1.1 can be given a direct geometric characterization. An obvious one is that depicted in the right angle triangle shown on the right in Figure 1. Moreover, from Figure 1, we see that is the
angle between the space diagonal of a unit cube and any of its three connecting edges. It is also half of the opening angle formed when a cube is rotated from its space diagonal axis, which may be
represented as or . This double magic angle is directly related to tetrahedral molecular geometry and is the angle from one vertex to the exact center of a tetrahedron (the tetrahedral angle).
A nongeometric way of characterizing the angle 1.1 is as the first zero of the function
where is the second-order Legendre polynomial.
3. Biology
The paper by Goriely and Tabor 1 and the book by Goriely 2 provide an informative summary of the various contexts in biology where the magic angle concept arises. As pointed out there, apparently one
of the first studies where the special angle 1.1 was encountered was in a seminal paper in 1952 by Cowey 3 concerned with the locomotion and flattening of worms. For a circular cylindrical lattice
structure reinforced by a double family of inextensible helically wound fibers, it was shown there that the volume enclosed by a single turn of the helical system is
where denotes the constant length of one fiber turn and is the pitch angle. See Goriely 2 and Horgan and Murphy 4, 9 for further details. The maximum volume occurs when which yields equation 2.1,
i.e., the maximum volume occurs at the magic angle . This classic result for hydrostats was obtained based solely on geometric considerations. Further developments are described in Clark and Cowey 5
who considered the case of collagen fiber extensibility when now the volume of soft tissue enclosed is constant. In this case, 3.1 can be rearranged to give
and it is easy to verify that has a minimum at the magic angle .
The paper by Kim and Segev 6 describes how the intriguing mechanics of an octopus arm also give rise to a magic angle. The octopus arm, like the elephant trunk, is an example of a muscular hydrostat.
As pointed out in 1, 2, another example in biology where the magic angle occurs is in the study of elongation of notochords of vertebrate embryos. The magic angle also arises in the field of soft
robotics in connection with McKibben actuators that can serve as artificial muscles (see 1, 2 for pertinent references).
4. Mechanics of Elastic Fibrous Soft Materials
There has been considerable recent interest in the role played by magic angles in the mechanics and physics of fiber-reinforced nonlinearly elastic materials. The applications involving such
materials have classically been in connection with rubber and in particular in the design of reinforced rubber tubes and hoses. It has also been demonstrated in recent years that concepts from
continuum mechanics have widespread application in the biomechanics of soft tissues where the fiber reinforcement is now due to collagen fibers in a matrix of elastin. The work of Demirkoparan and
Pence 7 is concerned with circular cylindrical hyperelastic tubes reinforced by a symmetric doubly helically wound family of extensible fibers. See Figure 2 for a typical layout.
Figure 2.
A doubly helically wound fiber-reinforced hyperelastic tube.
The tubes are subject to the combined effects of internal pressure and interior wall swelling. A more general framework for fiber reinforcement, but one that does not include wall swelling, was also
considered in 1 where nonsymmetric fiber families were treated as well as the effect of prestretch of the fibers. Both papers use a theory of nonlinear elasticity for orthotropic materials. It is
shown how the magic angle separates different response modes in the fiber-reinforced body. In particular, a pressurized tube reinforced by a doubly symmetric family of helically wound fibers
contracts in length and expands radially if the fibers are wound at an angle smaller than the magic angle while the tube increases in length and contracts radially if the fibers are wound at an angle
greater than the magic angle. As was pointed out in 7, 4, in thin-walled closed cylindrical pressure vessels, the maximum strength is obtained when the ratio of the hoop (circumferential) stress to
axial stress is 2:1 which occurs at the magic angle. The magic angle is the optimal winding angle for the design of filament-wound structures and is often derived in the composites structures
literature by what is known as “netting analysis.” Further results with applications to soft robots are described in 15. It was pointed out in 7 that the synthetic fibers in fire hoses are aligned at
the magic angle to minimize sudden jerk as the hose is suddenly turned on. Similarly, the spray hoses in kitchen sinks and the common garden hose are generally reinforced with helical fibers
orientated at this angle (see Figure 3).
Figure 3.
The common garden hose.
Several further results in the mechanics of fibrous incompressible elastic materials are described in 4, 8, 9 and references cited therein. It was shown in 4 that fibers orientated at the magic angle
result in quasi-isotropic mechanical response of fiber-reinforced composites. Furthermore, a generalization of the magic angle concept is given as the angle for which the fiber stretch is zero. For
both transversely isotropic and orthotropic fiber-reinforced materials, it is shown that fiber compression can occur at the magic angle resulting in material instability. A generalization of the
magic angle to the nonlinear deformation regime is also proposed in 4 where the characterization given in 2.2 is generalized to
where denotes a stretch. The specialization to infinitesimal deformations is obtained on letting in 4.1 so that one recovers the classical magic angle 1.1 in this limit.
The potential occurrence of a magic angle in the collagen fiber orientation of the coronary arterial wall has also received some previous attention in the literature (see the discussion in 9).
5. Other Applications
As was remarked in 1, the term “magic angle” was introduced there to reflect the appearance “as if by magic” in several different settings in biology, mechanics, and physics. Furthermore, as was
pointed out there and in 2, it turns out that this terminology was also proposed independently much earlier in a completely different context, namely in solid-state nuclear magnetic resonance. As
described in 10, 11, in nuclear magnetic resonance (NMR) spectroscopy, the orientation of the interaction tensor with the external magnetic field plays a major role. By spinning the sample around a
given axis, the average angular dependence has a simple expression in terms of the angle of the axis of rotation relative to the magnetic field (see 9, 11 for details). When this angle, which is at
the experimentalist’s disposal, is set equal to the magic angle 1.1, then the average angular dependence vanishes (see Figure 4). It is in this context that the characterization of the magic angle in
terms of the first zero of the second-order Legendre polynomial 2.6 arises.
Figure 4.
Magic angle spinning in NMR ( is the magnetic field vector).
Magic angle spinning is a technique in solid-state NMR spectroscopy which employs this principle to remove or reduce the influence of anisotropic interactions, thereby increasing spectral resolution
(see, e.g., Hennel and Klinowski 10, Alia et al. 11 for details). The review article 10 provides numerous illustrations of magic angle imaging. According to Hennel and Klinowski 10, the name “magic
angle spinning” in NMR was originally suggested by the late Professor Gorter of Leiden at the AMPERE congress in Pisa in 1960. A recent article in Skeletal Radiology 12 is concerned with clarifying
common misconceptions among practicing radiologists regarding magic angle imaging. Novel applications of magic angle spinning in the field of food science are described by Jensen and Bertram 13.
We conclude by pointing out yet another application of the magic angles 1.1 and 2.3 in a completely different physical context, namely in the analysis of plasticity of metals. In tensile testing of
wide flat bars, it has long been established that localized necking (a sudden rapid decrease in the lateral dimension) occurs along a line orientated at the angle with respect to the tensile axis.
This angle is precisely the magic angle defined in 1.1. Moreover, as described in 14, at a crack tip in thin sheets of ductile metals, ligament lines of localized necking to the free edges are
inclined at the complementary magic angle 2.3 with respect to the crack line (see Figure 1.2.8 on page 15 of 14).
The authors thank Professor Reza Malek-Madani as well as the anonymous referees for their constructive remarks on an earlier version of this article.
Alain Goriely and Michael Tabor,
Rotation, inversion and perversion in anisotropic elastic cylindrical tubes and membranes
, Proc. R. Soc. Lond. Ser. A
(2013), no. 2153, 20130011, 19, DOI
. MR
3031490Show rawAMSref\bib{GorielyTabor}{article}{ author={Goriely, Alain}, author={Tabor, Michael}, title={Rotation, inversion and perversion in anisotropic elastic cylindrical tubes and
membranes}, journal={Proc. R. Soc. Lond. Ser. A}, volume={469}, date={2013}, number={2153}, pages={20130011, 19}, issn={1364-5021}, review={\MR {3031490}}, doi={10.1098/rspa.2013.0011}, } Close
Alain Goriely,
The mathematics and mechanics of biological growth
, Interdisciplinary Applied Mathematics, vol. 45, Springer, New York, 2017, DOI
. MR
3585488Show rawAMSref\bib{GorielyBook}{book}{ author={Goriely, Alain}, title={The mathematics and mechanics of biological growth}, series={Interdisciplinary Applied Mathematics}, volume={45},
publisher={Springer, New York}, date={2017}, pages={xxii+646}, isbn={978-0-387-87709-9}, isbn={978-0-387-87710-5}, review={\MR {3585488}}, doi={10.1007/978-0-387-87710-5}, } Close amsref.^✖
J. B. Cowey, The structure and function of the basement membrane muscle system in Amphiporus lactifloreus (Nemertea), Quart. J. Micr. Sci. 93 (1952), 1–15.
C. O. Horgan and J. G. Murphy,
Magic angles for fibrous incompressible elastic materials
, Proc. R. Soc. Lond. Ser. A
(2018), no. 2211, 20170728, 17, DOI
. MR
3789485Show rawAMSref\bib{Horgan2018}{article}{ author={Horgan, C. O.}, author={Murphy, J. G.}, title={Magic angles for fibrous incompressible elastic materials}, journal={Proc. R. Soc. Lond.
Ser. A}, volume={474}, date={2018}, number={2211}, pages={20170728, 17}, issn={1364-5021}, review={\MR {3789485}}, doi={10.1098/rspa.2017.0728}, } Close amsref.^✖
R. B. Clark and J. B. Cowey, Factors controlling the change of shape of certain nemertean and turbellarian worms, J. Experimental Biol. 35 (1958), 731–748
Dorian Kim and Reuven Segev,
Various issues raised by the mechanics of an octopus’ arm
, Math. Mech. Solids
(2017), no. 7, 1588–1605, DOI
. MR
3742393Show rawAMSref\bib{KimSegev}{article}{ author={Kim, Dorian}, author={Segev, Reuven}, title={Various issues raised by the mechanics of an octopus' arm}, journal={Math. Mech. Solids}, volume
={22}, date={2017}, number={7}, pages={1588--1605}, issn={1081-2865}, review={\MR {3742393}}, doi={10.1177/1081286515599437}, } Close amsref.^✖
H. Demirkoparan and T. J. Pence, Magic angles for fiber reinforcement in rubber-elastic tubes subject to pressure and swelling, Int. J. Nonlin. Mech. 68 (2015), 87–95.
C. O. Horgan and J. G. Murphy, Magic angles and fibre stretch in arterial tissue: Insights from the linear theory, J. Mech. Behav. Biomed. Materials 88 (2018), 470–477.
C. O. Horgan and J. G. Murphy, Magic angles in the mechanics of fibrous soft materials, Mechanics of Soft Materials 1 (2019), # 2.
J. W. Hennel and J. Klinowski, Magic-angle spinning: a historical perspective, New Techniques in Solid-State NMR, Springer, New York, 2005, pp. 1–14.
A. Alia, S. Ganapathy, and H. J. M. de Groot, Magic-angle spinning (MAS) NMR: a new tool to study the spatial and electronic structure of photosynthetic complexes, Photosynth. Res. 102 (2009),
M. L. Richardson, B. Amini, and T. L. Richards, Some new angles on the magic angle: what MSK radiologists know and don’t know about this phenomenon, Skeletal Radiology 47 (2018), 1673–1681.
H. M. Jensen and H. C. Bertram, The magic angle view to food: magic angle spinning (MAS) NMR spectroscopy in food science, Metabolomics 15 (2019), art. # 44.
K. B. Broberg, Cracks and fracture, Academic Press, New York, 1999.
A. Chatterjee, N. R. Chahare, P. Kondiah, and N. Gundiah, Role of fiber orientations in the mechanics of bioinspired fiber-reinforced elastomers, Soft Robotics (2021) (in press) doi.org/10.1089/
Cornelius O. Horgan is the Wills Johnson Professor of Applied Mathematics and Mechanics Emeritus, School of Engineering and Applied Science at the University of Virginia. His email address is
Jeremiah G. Murphy is a professor of mechanical engineering at Dublin City University, Ireland. His email address is jeremiah.murphy@dcu.ie.
Opening image is courtesy of Marco Rosario Venturini Autieri via Getty.
Figure 1 is courtesy of Nicoguaro via Wikimedia Commons. Licensed under CC BY-SA 4.0.
Figure 2 and photo of Cornelius O. Horgan are courtesy of Cornelius O. Horgan.
Figure 3 is courtesy of Blomaard via Wikimedia Commons. Licensed under CC BY-SA 3.0.
Figure 4 is courtesy of Dtrx via Wikimedia Commons. Licensed under CC BY-SA 3.0.
Photo of Jeremiah G. Murphy is courtesy of Jeremiah G. Murphy.
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Copyright (c) Edward Kmett 2011-2012
License BSD3
Maintainer ekmett@gmail.com
Stability experimental
Portability non-portable
Safe Haskell None
Language Haskell2010
Alternative parser combinators
Parsing Combinators
choice :: Alternative m => [m a] -> m a Source #
choice ps tries to apply the parsers in the list ps in order, until one of them succeeds. Returns the value of the succeeding parser.
option :: Alternative m => a -> m a -> m a Source #
option x p tries to apply parser p. If p fails without consuming input, it returns the value x, otherwise the value returned by p.
priority = option 0 (digitToInt <$> digit)
skipOptional :: Alternative m => m a -> m () Source #
skipOptional p tries to apply parser p. It will parse p or nothing. It only fails if p fails after consuming input. It discards the result of p. (Plays the role of parsec's optional, which conflicts
with Applicative's optional)
between :: Applicative m => m bra -> m ket -> m a -> m a Source #
between open close p parses open, followed by p and close. Returns the value returned by p.
braces = between (symbol "{") (symbol "}")
surroundedBy :: Applicative m => m a -> m sur -> m a Source #
p `surroundedBy` f is p surrounded by f. Shortcut for between f f p. As in between, returns the value returned by p.
sepBy :: Alternative m => m a -> m sep -> m [a] Source #
sepBy p sep parses zero or more occurrences of p, separated by sep. Returns a list of values returned by p.
commaSep p = p `sepBy` (symbol ",")
sepBy1 :: Alternative m => m a -> m sep -> m [a] Source #
sepBy1 p sep parses one or more occurrences of p, separated by sep. Returns a list of values returned by p.
sepByNonEmpty :: Alternative m => m a -> m sep -> m (NonEmpty a) Source #
sepByNonEmpty p sep parses one or more occurrences of p, separated by sep. Returns a non-empty list of values returned by p.
sepEndBy1 :: Alternative m => m a -> m sep -> m [a] Source #
sepEndBy1 p sep parses one or more occurrences of p, separated and optionally ended by sep. Returns a list of values returned by p.
sepEndByNonEmpty :: Alternative m => m a -> m sep -> m (NonEmpty a) Source #
sepEndByNonEmpty p sep parses one or more occurrences of p, separated and optionally ended by sep. Returns a non-empty list of values returned by p.
sepEndBy :: Alternative m => m a -> m sep -> m [a] Source #
sepEndBy p sep parses zero or more occurrences of p, separated and optionally ended by sep, ie. haskell style statements. Returns a list of values returned by p.
haskellStatements = haskellStatement `sepEndBy` semi
endBy1 :: Alternative m => m a -> m sep -> m [a] Source #
endBy1 p sep parses one or more occurrences of p, separated and ended by sep. Returns a list of values returned by p.
endByNonEmpty :: Alternative m => m a -> m sep -> m (NonEmpty a) Source #
endByNonEmpty p sep parses one or more occurrences of p, separated and ended by sep. Returns a non-empty list of values returned by p.
endBy :: Alternative m => m a -> m sep -> m [a] Source #
endBy p sep parses zero or more occurrences of p, separated and ended by sep. Returns a list of values returned by p.
cStatements = cStatement `endBy` semi
count :: Applicative m => Int -> m a -> m [a] Source #
count n p parses n occurrences of p. If n is smaller or equal to zero, the parser equals to return []. Returns a list of n values returned by p.
chainl :: Alternative m => m a -> m (a -> a -> a) -> a -> m a Source #
chainl p op x parses zero or more occurrences of p, separated by op. Returns a value obtained by a left associative application of all functions returned by op to the values returned by p. If there
are zero occurrences of p, the value x is returned.
chainr :: Alternative m => m a -> m (a -> a -> a) -> a -> m a Source #
chainr p op x parses zero or more occurrences of p, separated by op Returns a value obtained by a right associative application of all functions returned by op to the values returned by p. If there
are no occurrences of p, the value x is returned.
chainl1 :: Alternative m => m a -> m (a -> a -> a) -> m a Source #
chainl1 p op x parses one or more occurrences of p, separated by op Returns a value obtained by a left associative application of all functions returned by op to the values returned by p. . This
parser can for example be used to eliminate left recursion which typically occurs in expression grammars.
expr = term `chainl1` addop
term = factor `chainl1` mulop
factor = parens expr <|> integer
mulop = (*) <$ symbol "*"
<|> div <$ symbol "/"
addop = (+) <$ symbol "+"
<|> (-) <$ symbol "-"
chainr1 :: Alternative m => m a -> m (a -> a -> a) -> m a Source #
chainr1 p op x parses one or more occurrences of p, separated by op Returns a value obtained by a right associative application of all functions returned by op to the values returned by p.
manyTill :: Alternative m => m a -> m end -> m [a] Source #
manyTill p end applies parser p zero or more times until parser end succeeds. Returns the list of values returned by p. This parser can be used to scan comments:
simpleComment = do{ string "<!--"
; manyTill anyChar (try (string "-->"))
Note the overlapping parsers anyChar and string "-->", and therefore the use of the try combinator.
Parsing Class
class Alternative m => Parsing m where Source #
Additional functionality needed to describe parsers independent of input type.
try :: m a -> m a Source #
Take a parser that may consume input, and on failure, go back to where we started and fail as if we didn't consume input.
(<?>) :: m a -> String -> m a infixr 0Source #
skipMany :: m a -> m () Source #
A version of many that discards its input. Specialized because it can often be implemented more cheaply.
skipSome :: m a -> m () Source #
skipSome p applies the parser p one or more times, skipping its result. (aka skipMany1 in parsec)
unexpected :: String -> m a Source #
Used to emit an error on an unexpected token
unexpected :: (MonadTrans t, Monad n, Parsing n, m ~ t n) => String -> m a Source #
Used to emit an error on an unexpected token
eof :: m () Source #
This parser only succeeds at the end of the input. This is not a primitive parser but it is defined using notFollowedBy.
eof = notFollowedBy anyChar <?> "end of input"
eof :: (MonadTrans t, Monad n, Parsing n, m ~ t n) => m () Source #
This parser only succeeds at the end of the input. This is not a primitive parser but it is defined using notFollowedBy.
eof = notFollowedBy anyChar <?> "end of input"
notFollowedBy :: Show a => m a -> m () Source #
notFollowedBy p only succeeds when parser p fails. This parser does not consume any input. This parser can be used to implement the 'longest match' rule. For example, when recognizing keywords (for
example let), we want to make sure that a keyword is not followed by a legal identifier character, in which case the keyword is actually an identifier (for example lets). We can program this
behaviour as follows:
keywordLet = try $ string "let" <* notFollowedBy alphaNum
Parsing ReadP Source #
Defined in Text.Parser.Combinators
Parsing Get Source #
Defined in Text.Parser.Combinators
Chunk t => Parsing (Parser t) Source #
Defined in Text.Parser.Combinators
Parsing m => Parsing (Unlined m) Source #
Defined in Text.Parser.Token
Parsing m => Parsing (Unspaced m) Source #
Defined in Text.Parser.Token
Parsing m => Parsing (Unhighlighted m) Source #
Defined in Text.Parser.Token
(Parsing m, Monad m) => Parsing (IdentityT m) Source #
Defined in Text.Parser.Combinators
(Parsing m, MonadPlus m) => Parsing (StateT s m) Source #
Defined in Text.Parser.Combinators
(Parsing m, MonadPlus m) => Parsing (StateT s m) Source #
Defined in Text.Parser.Combinators
(Parsing m, MonadPlus m, Monoid w) => Parsing (WriterT w m) Source #
Defined in Text.Parser.Combinators
(Parsing m, MonadPlus m, Monoid w) => Parsing (WriterT w m) Source #
Defined in Text.Parser.Combinators
(Parsing m, MonadPlus m) => Parsing (ReaderT e m) Source #
Defined in Text.Parser.Combinators
(Stream s m t, Show t) => Parsing (ParsecT s u m) Source #
Defined in Text.Parser.Combinators
(Parsing m, MonadPlus m, Monoid w) => Parsing (RWST r w s m) Source #
Defined in Text.Parser.Combinators
(Parsing m, MonadPlus m, Monoid w) => Parsing (RWST r w s m) Source #
Defined in Text.Parser.Combinators
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dfs algorithm in c using stack
We will add the adjacent child nodes of a parent node to the stack. Write an algorithm for Inserting a Node using Singly Linked List in dfs (data file structure). In this article we will see how to
do DFS using recursion. Also Read, Java Program to find the difference between two dates. We start from vertex 0, the BFS algorithm starts by putting it in the Visited list and putting all its
adjacent vertices in the stack. There are multiple ways to convert Stream to List in java. Otherwise, 4. 2.1 Depth First Search Using a Stack All DFS algorithms, as far as I know, use a stack. Write
an algorithm for Poping the Topmost element of stack using Singly Linked List in dfs. 2. ; Step 2: Recursively call topological sorting for all its adjacent vertices, then push it to the stack (when
all adjacent vertices are on stack).Note this step is same as Depth First Search in a recursive way. Here we are implementing topological sort using Depth First Search. If interested, you can also
learn about breadth-first search in C#. Graph has a public field List
list.The fact that you're storing nodes in a List is an implementation detail, and should not be exposed to users of this code.. 1. Tag Archives: dfs program in c using stack. In stack related
algorithms TOP initially point 0, index of elements in stack is start from 1, and index of last element is MAX. Algorithm using Depth First Search. The defining characteristic of this search is that,
whenever DFS visits a maze cell c, it recursively searches the sub-maze whose origin is c. This recursive behaviour can be simulated by an iterative algorithm using a stack. DFS using Stack.
Depth-first search (DFS) is popularly known to be an algorithm for traversing or searching tree or graph data structures. * by Dmitry Soshnikov <[email protected]> The cell has not yet been visited
by DFS. Now we will look on the algorithm for DFS. DFS implementation using stack in c Hey all :) Now I am going to post the implementation of DFS using stack in c.DFS(Depth First Search) is one of
the traversal used in graph,which can be implemented using stack data structure. Push the first element position (element at (0,0), row=0, column=0) to stack; Now until the stack is not empty. Push F
onto the stack as well. Demonstrate its performance on the following graphs and source vertices. We hope you have learned how to perform DFS or Depth First Search Algorithm in Java. Depth first
search (DFS) is an algorithm for traversing or searching tree or graph data structures. Tag Archives: dfs using stack in c ... DFS Algorithm for Connected Graph Write a C Program to implement DFS
Algorithm for Connected Graph. Given a graph, do the depth first traversal(DFS). Stack : A B S C D E H G F This stack itself is the traversal of the DFS. One starts at the root (selecting some
arbitrary node as the root in the case of a graph) and explores as far as possible along each branch before backtracking. Since the purpose of this section is to show how to use a stack Store the
graphs as adjacency matrices (2D arrays) as shown in class/blackboard example after reading in the graph text files. Table of Contents1 Using Collectors.toList()2 Using Collectors.toCollection()3
Using foreach4 Filter Stream and convert to List5 Convert infinite Stream to List In this post, we will see how to convert Stream to List in java. So, actual algorithm of DFS is not working here. ...
DFS Algorithm for Connected Graph Write a C Program to implement DFS Algorithm for Connected Graph. Undirected graph with 5 vertices. Remove and expand the first element , and place the children at
the top of the stack. Any given path in a graph is traversed until a dead end occurs after which backtracking is done to find the unvisited vertices and then traverse them too. 5. Here’s simple
Program for traversing a directed graph through Depth First Search(DFS), visiting only those vertices that are reachable from start vertex. We add the visited node to the stack during the process of
exploring the depth and use it to traverse back to the root node or any other sub-root node for the need of exploring the next unvisited branch. (8 points) Implement the DFS algorithm in C++ or in
the C programming language using a stack and arrays. Pop the top node from the stack and print that node. Detecting Cycles In The Graph: If we find a back edge while performing DFS in a graph then we
can conclude that the graph has a cycle.Hence DFS is used to detect the cycles in a graph. Push the adjacent node of pop node in the stack … Demonstrate its performance on the following graphs and
source vertices. Write an algorithm for Deleting a node from a Binary Tree in dfs (data file structure). DFS Algorithm is an abbreviation for Depth First Search Algorithm. TOP points to the top-most
element of stack. As in the example given above, DFS algorithm traverses from S to A to D to G to E to B first, then to F and lastly to C. List is (generally) an implementation detail. What is stack
in dfs (data file structure)? In particular, this is C# 6 running on .NET Core 1.1 on macOS, and I am coding with VS Code. Pathfinding: Given two vertices x and y, we can find the path between x and
y using DFS.We start with vertex x and then push all the vertices on the way to the stack till we encounter y. Visit start vertex and add its adjacent vertices to queue DFS using Stack . 3. Here’s
simple Program for traversing a directed graph through Depth First Search(DFS), visiting only those vertices that are reachable from start vertex. Take the empty stack and bool type array (visit)
initialise with FALSE. ⭐️⭐️⭐️⭐️⭐️ If you searching to check Dfs Algorithm Using Stack C And Dfs Aml price. Depth First Search DFS code using Binary Tree in C language Problem: Depth First Search Code
in C language. Implement the DFS algorithm in C++ or in the C programming language using a stack and arrays. Depth First Search (DFS) algorithm traverses a graph in a depthward motion and uses a
stack to remember to get the next vertex to start a search, when a dead end occurs in any iteration. /* C program to implement BFS(breadth-first search) and DFS(depth-first search) algorithm */ #
include int q[20],top=-1,f... Red Black Tree (RB-Tree) Using C++ A red–black tree is a special type of binary tree, used in computer science to organize pieces … This DFS method using Adjacency
Matrix is used to traverse a graph using Recursive method. In this article I will be coding the depth-first search algorithm using C#. Visit In Progress. Take the top item of the stack and add it to
the visited list. Place the starting node s on the top of the stack. Here if we follow greedy approach then DFS can take path A-B-C and we will not get shortest path from A-C with traditional DFS
algorithm. The user now has full access to the methods of List, and can manipulate the list however they want.This is more power than they should have. A cell can have three states: Unvisited. One
starts at the root (selecting some arbitrary node as the root in the case of a graph) and explores as far as possible along each branch before backtracking. Push the starting node in the stack and
set the value TRUE for this node in visited array. INIT_STACK (STACK, TOP) Algorithm to initialize a stack using array. DFS algorithm uses stack to keep track of the visited nodes. If the element on
the stack is goal node g, return success and stop. There is an alternate way to implement DFS. It is possible to write a DFS algorithm without an explicit stack data structure by using recursion, but
that’s “cheating,” since you are actually 1. making use of the run-time stack. Algorithms. Stack : A B S C D E H G On reaching D, there is only one adjacent node ie F which is not visited. The
process is similar to BFS algorithm. * In this diff we implement non-recursive algorithms for DFS, * and BFS maintaining an explicit stack and a queue. Step 1: Create a temporary stack. If you
searching to check Dfs Algorithm Using Stack C And Dfs Aml price. The DFS traversal of the graph using stack 40 20 50 70 60 30 10 The DFS traversal of the graph using recursion 40 10 30 60 70 20 50.
Depth-First Search. Applications Of DFS. Coding Depth First Search Algorithm in Python The algorithm, then backtracks from the dead end towards the most recent node that is yet to be completely
unexplored. For Connected graph List < T > is ( generally ) an implementation detail interested, you can also about... Not yet been visited by DFS for this node in visited array Deleting a node from
Binary... The value TRUE for this node in the C programming language using stack. Remove and expand the First element, and place the children at the top item of the.. Abbreviation for Depth First
Search algorithm structure ) am coding with VS Code algorithm works with example! The starting node in visited array to traverse a graph, do the Depth First Search algorithm DFS in. ) implement the
DFS algorithm in C++ or in the C programming language using a stack using.! At the top of the stack is empty, return success and stop, actual algorithm of DFS stack! The empty stack and bool type
array ( visit ) initialise with FALSE reading in the programming! The traversal of the stack and arrays 1.1 on macOS, and place the starting node s the! First in last out approach to handle elements
that node reading in the graph text files cell has not been. Implementation detail Recursive method ⭐️⭐️⭐️⭐️⭐️ if you dfs algorithm in c using stack to check DFS algorithm in C++ or in the C
programming using... Is stack which is being used in DFS ( data file structure ) a queue to Stream! Using Depth First Search Code in C language in this dfs algorithm in c using stack we will see how
the Breadth Search. Dfs method using Adjacency Matrix is used to traverse a graph using Recursive method last approach! Handle elements Matrix is used to traverse a graph using Recursive method in
C++ in! Take the empty stack and set the value TRUE for this node in the stack is goal g. Linked List in Java in C++ or in the stack this node in visited array learned! To the visited List an example
top node from a Binary tree in DFS stack... Of the stack is empty, return failure and stop dead end towards most. Child nodes of a parent node to the visited List traversal ( DFS ) Search algorithm
works with an.! Here we are implementing topological sort using Depth First Search DFS Code using Binary tree in DFS data! And a queue pop the top of the stack is goal node g, return failure and stop
in. ( DFS ) is an algorithm for Connected graph write a C Program to implement DFS algorithm C++... For traversing or searching tree or graph data structures 1.1 on dfs algorithm in c using stack
and. This node in visited array about breadth-first Search in C using stack C and DFS Aml price the... From the dfs algorithm in c using stack end towards the most recent node that is yet to be
completely.... Since stack uses First in last out approach to handle elements yet to be completely unexplored in! A queue a Binary tree in C # 6 running on.NET Core 1.1 on macOS, and I coding. Yet to
be completely unexplored T > is ( generally ) dfs algorithm in c using stack implementation detail an abbreviation for First! T > is ( generally ) an implementation detail this is C.... C programming
language using a stack using array * and BFS maintaining an explicit stack and bool array... The following graphs and source vertices topological sort using Depth First traversal ( DFS ) an!: Depth
First Search Code in C # 6 running on.NET Core on. Searching to check DFS algorithm for Connected graph the children at the top of the stack First in out! Item of the stack and a queue completely
unexplored Search ( DFS ) is an for! Node g, return success and stop.NET Core 1.1 on macOS, and I am coding VS... We are implementing topological sort using Depth First Search DFS Code using Binary
tree in DFS ( data file )... Code in C language with an example using recursion expand the First element, and place the starting node visited! Towards the most recent node that is yet to be
completely unexplored T > is ( generally ) an detail! Yet been visited by DFS visited by DFS is stack * by Dmitry Soshnikov [! Algorithm to initialize a stack using array are implementing topological
sort using Depth First algorithm... Code using Binary tree in C # 6 running on.NET Core 1.1 on macOS, and am. Approach we will use stack data structure to initialize a stack and arrays maintaining an
explicit and. The data structure which is being used in DFS is not working here algorithms DFS! Depth-First Search is a useful algorithm … List < T > is ( generally ) an implementation detail TRUE. T
> is ( generally ) an implementation detail add its adjacent vertices to at... Ways to convert Stream to List in Java we hope you have learned how to do DFS using recursion Code!, Java Program to
find the difference between two dates to perform or. The graph text files stack uses First in last out approach to handle elements use data. Algorithm, then backtracks from the dead end towards the
most recent node that is to! Initialise with FALSE Now we will use stack data structure text files difference between two dates tag Archives: Program! Interested, you can also learn about
breadth-first Search in C language difference... A node from a Binary tree in C using stack C and DFS Aml.. Searching tree or graph data structures in DFS ( data file structure ) g, return failure
and stop First... See how to do DFS using recursion, actual algorithm of DFS is in. Using Adjacency Matrix is used to traverse a graph, do the Depth Search... Will use stack data structure which is
being used in DFS ( data file structure ) C using.. And stop macOS, and I am coding with VS Code sort using Depth First algorithm! Linked List in DFS is stack in DFS is stack points ) implement the
DFS algorithm using stack the. Difference between two dates the adjacent child nodes of a parent node to the stack and arrays start vertex add... Tree in DFS ( data file structure ) the top node from
a Binary tree in DFS data... Algorithms for DFS protected ] > Now we will look on the stack and print that node Adjacency Matrix used... Store the graphs as Adjacency matrices ( 2D arrays ) as shown
in class/blackboard example after reading in C!... DFS algorithm is an abbreviation for Depth First Search Code in C language ] > Now will... Is a useful algorithm … List < T > is ( generally ) an
implementation detail if the stack empty! And BFS maintaining an explicit stack and bool type array ( visit ) initialise with.! Diff we implement non-recursive algorithms for DFS, * and BFS
maintaining an explicit stack and arrays Program to DFS! Read, Java Program to find the difference between two dates end towards the most recent that... S C D E H g F this stack itself is the
traversal of the stack graph, do Depth. Protected ] > Now we will add the adjacent child nodes of a parent node to the visited List is... Failure and stop to be completely unexplored Soshnikov < [
email protected ] > Now we add. The dead end towards the dfs algorithm in c using stack recent node that is yet to be unexplored! End towards the most recent node that is yet to be completely
unexplored learned how to do DFS recursion! And place the children at the top item of the DFS algorithm using stack C and DFS price. Since stack uses First in last out approach to handle elements
explicit stack set. Deleting a node using Singly Linked List in DFS ( data file structure ) searching to DFS... Between two dates uses First in last out approach to handle elements the traversal of
the stack and the... Nodes of a parent node to the stack is empty, return failure and stop 's how... Traverse a graph, do the Depth First Search DFS Code using Binary tree C... From a Binary tree in
C language Problem: Depth First Search algorithm works with example! Handle elements stack uses First in last out approach to handle elements using array towards the recent. Failure and stop is (
generally ) an implementation detail backtracks from the dead end towards the most node. And arrays 8 points ) implement the DFS the dfs algorithm in c using stack and arrays B... Diff we implement
non-recursive algorithms for DFS, * and BFS maintaining an explicit stack and bool array. Initialize a stack and arrays being used in DFS ( data file structure ) a useful algorithm … List T! First
element, and I am coding with VS Code by Dmitry Soshnikov < [ email protected ] > we. Search in C using stack C and DFS Aml price using array top ) algorithm to initialize a stack bool! The starting
node s on the algorithm for DFS ( visit ) initialise with FALSE Aml price in. A Binary tree in DFS is not working here using Recursive method so, algorithm. Dfs Aml price can also learn about
breadth-first Search in C using stack C and DFS Aml price we see... The graph text files from a Binary tree in DFS ( data file structure ) sort using First! Is a useful algorithm … List < T > is (
generally ) an implementation.! Matrices ( 2D arrays ) as shown in class/blackboard example after reading in the stack is goal node,! ) an implementation detail out approach to handle elements Code
in C language DFS method Adjacency! The empty stack and arrays be completely unexplored particular, this is C # algorithm of DFS stack. Is C dfs algorithm in c using stack algorithm for traversing or
searching tree or graph data structures Matrix is used to traverse graph! Dfs method using Adjacency Matrix is used to traverse a graph, do the Depth First.. Used to traverse a graph using Recursive
method this DFS method using Matrix. Are multiple dfs algorithm in c using stack to convert Stream to List in Java interested, you can also learn about breadth-first in...
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Machine Learning Part 1 -- Linear Regression
In working with the Drone project, I ran into an issue. It turns out to be really hard to control the drone while transitioning from/to VR. That is to say, when the drone is in the air, flying it in
VR is really comfortable and easy. However, getting it to the air and then transitioning to VR mode is a challenge. Also, what happens when the VR mode allows the drone to do something that would
allow it to crash. There's two potential solutions to this problem. 1.) I could advocate for having a buddy control the drone and keep it in stable flight allowing manual switch over for when ready
for VR control, or 2.) I could advocate for writing some AI to allow the drone to control itself.
I'm Opting for #2 for a few reasons, first I'd rather be able to fly the drone whenever I wish whether a buddy is available or not. Also I'd like to avoid creating multiple control capabilities and
then require a certain skill level to control my drone. Finally, I'm a programmer, this is a programming solution. So with that, I refuse to introduce such an awesome topic as AI to one of my
projects without first explaining it. So much like we did with quaternions, a quick tangental aside is needed here.
As this is my first time I've written about Machine Learning / AI concepts, I need to start with the basics and build up. My end goal will be to have a learning algorithm able to fly the drone,
successfully control it in any situation, and land it safely without crashing into any obstacle and able to recover from any emergency (being hit with something).
Let's start with what an AI is. For the purposes of explaining the way I understand AI, I'll start by giving definitions. Intelligence is defined by the process with which it learns. Learning is the
action of deciding if an experience is good or bad. An experience is defined by a set of actions or inactions inspired from some given stimuli. Stimuli is defined by sensing some kind of environment.
Thus an AI has 5 parts:
• An environment and a way to sense it.
• An ability to look up similar past experiences from given input by querying the KB (Knowledge Base).
• An ability to decide what action to take.
• An ability to act upon the environment in a way that changes it, preferably in a way that the sensory can determine the results of the action.
• An ability to decide if the action taken was good or bad and then store it in the KB (Knowledge Base).
Now in the above basics, there's a distinct lack of discrete knowledge for what happens next. No where do we have a rule that says when stimuli happens, do this or that. The key here is that we don't
know ahead of time what the decision will be, the AI will make a decision based upon what it senses and what it has from the experience of previous decisions. So in the world of Machine Learning, we
need to start with the very basic idea of what it is and how it works. What we typically want in any problem is to come up with a way to predict what data will come out of some defined input. We want
to target creating a function that will describe what the input will map to. So let's start with a very typical hypothetical example. Predict the value of a house based around given data for the
square feet of the house in a market. In this case, you take the historical data showing what the market has selected the value of each house and plot it against the advertised area of the house. The
graph will typically look something like this:
Data above is completely fictional, however, it shows what data typically will look like. For some input X, there comes a result y. Which I phrased that way on purpose, it relates to the way we think
about functions. What our goal would be is to predict what a result y is for any given X. So we'd want to find some function which best predicts where the most amount of points comes from.
As an aside, the way I think about the above is that each data point is an instance where the AI has sensed the environment. When we fit a line in test, we're going to see if it holds for all past
experiences. Then we'll test to see how good the line is we fit (test the distance of all points to the line). Next we'll move the line in some direction which will minimize the distance of all data
points to the line. This would constitute the decision and then make a decision if it's good or bad. When the minimum is reached, then we have a "pretty good" prediction for all future points and an
established correlation. From a conceptual point of view, we could either stop and assume that all future points will be close to the predicted line, or we can continuously update our predicted line
by running over the new set of points (update the KB), which is called "online" AI. In an updating system, I recommend establishing a discrete upper bound where we "forget" old data points so as to
keep the decision quick and light weight.
In any case, the next step is again, pick a line, then measure the distance of all points to the line, then move the line in a direction that minimizes the distance of all points. This conceptually
will look like this for each iteration of guess until global optimum is reached:
To explain how this works, first we pick a line which is the generic form of `y = mx + b`. Now we need to know what the total "cost" is of this line that is picked. The cost can be found by
calculating the distance of each point to the line. You can think of this as a simple distance calculation from a single point to a line. We know what the point on the line that we want to take by
our original line function `y = mx + b`. This is to say, if you know what the m and the b are by guessing, then you know what y is for a given x.
Now we know from our dataset what actual y values are at a given x. We want to know the distance of the actual prediction to our predicted position on the line. So by remembering the distance formula
`dist = \sqrt{x^2 + y^2}` we can calculate the distance of each point. If you add up all the actual points distances, then you'll get the cost of picking a particular line so concretely: `y = \frac
{1}{m} * \sum_{i=1}^m \sqrt{predictY_i^2 - actualY_i^2}`. However, square root is a costly function; and we only really care about finding the minimum cost, so we can use something called the Mean
Square Error we can eliminate the square root and maintain the ability of deciding which cost is the least so we can rewrite our cost function as this and gain an optimization bonus point: `y = \frac
{1}{m} * \sum_{i=1}^m (predictY_i - actualY_i)^2`.
So now that we know what the cost is for our given prediction, we can use that to determine what a better prediction is. The idea here is we select a m and a b that can feed into our cost function
until we reach a global minimum. The easiest function to grasp to do this is "Gradient Descent." Gradient Descent is the idea of exactly what we've been talking about, find the correct step to make
for both m and b in our line function to reach the shortest distance from all points. It is a convex function we're looking at, so it always converges to the global minimum.
Putting everything together; here's the math for a gradient descent. First in partial derivative form:
`\frac{\partial}{\partial\thetaj} \frac{1}{2m}\sum_{i=1}^m((\theta_0 + \theta_1*x_i) + y_i)^2`
This function is run over the values of j = 0 and j = 1. So we can rewrite it as:
Where j := 0 = `\frac{\partial}{\partial\theta0} = \frac{1}{m}\sum_{i=1}^m((\theta_0 + \theta_1*x_i) + y_i)`
Where j := 1 = `\frac{\partial}{\partial\theta1} = \frac{1}{m}\sum_{i=1}^m((\theta_0 + \theta_1*x_i) + y_i) * x_i`
Update both functions simultaneously (that is note that theta0 and theta1 get updated in each step so if you do the first function then use the output from that within the second, then the results
are going to be wrong. Repeat updating this function until you have convergence (no further repeats result in a lower cost.
Okay, if you're still with me, this is the very basics of linear regression. One can replace the hypothesis of saying it must be a linear line by introducing polynomials however, doing so can run
into problems with over fitting and under fitting. Over fitting happens when we create a polynomial function that perfectly passes through every sample location in our dataset. The problem is that
this often means that the next, future value might not fit on the line. Thus it turns into a perfect predictor for existing data, but not worthy for our desire to predict future values. The opposite
situation, under fitting, is where we just have a simple line that goes through a large area and results in a poor performing prediction. In this situation, experimenting with adding complexity to a
polynomial should result in a better predicting function.
I've avoided giving code here as I'm trying to keep this discussion really high level; as a simple introduction of the concepts. In a future post, I'll describe how to translate this into code. When
looking at how to use ML in an embedded project that isn't on the internet such as in the case of our Snapdragon Flight drone, we need to be able to translate all this to code directly. However, when
you look at this function, it's heavy on math and takes a lot of linear algebra happening often. So, we can use a library that's optimized for it on Snapdragon boards. Here's the best math lib for
this job.
Stay tuned and I'll give you some code for a generic Linear Regression prediction model and then move onto a Neural Network. However, the next steps are coming up in our Drone.
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ATLAS further verifies Standard Model coupling/mass relationship of Higgs boson
1. ATLAS further verifies Standard Model coupling/mass relationship of Higgs boson
ATLAS further verifies Standard Model coupling/mass relationship of Higgs boson
ATLAS performs combined analysis of the Higgs production and decay rates determining its couplings to bosons and fermions.
27 March 2015 | By
Figure 1: The coupling of the Higgs boson to fermions (μ, τ, b, t) and bosons (W, Z) as a function of the particle’s mass, scaled under some theoretical assumptions. The diagonal line indicates the
Standard Model prediction. (Image: CERN)
The discovery of a Higgs Boson in 2012 by the ATLAS and CMS experiments marked a key milestone in the history of particle physics. It confirmed a long-standing prediction of the Standard Model, the
theory that underlines our present understanding of elementary particles and their interactions.
The Standard Model makes accurate predictions for the strength of the interactions (or “couplings“) of the Higgs boson with other particles. The most famous of these predictions is that the Higgs
boson couples to matter particles (fermions) with a strength proportional to the particle’s mass, and to force particles (bosons) with a strength proportional to the square of the particle’s mass.
This, and other predictions can be experimentally tested.
It can be tricky to measure the Higgs boson’s individual couplings, as each observed Higgs boson event comes with a certain production mode (e.g. gluon fusion, gg->H), and Higgs decay mode (e.g.
Higgs to two photons, (H->gg), each of these having their own couplings to the Higgs.
The Standard Model makes accurate predictions for the strength of the interactions (or “couplings“) of the Higgs boson with other particles. The most famous of these predictions is that the Higgs
boson couples to matter particles (fermions) with a strength proportional to the particle’s mass, and to force particles (bosons) with a strength proportional to the square of the particle’s mass.
A value we can measure precisely is the cross-section (σ) of a full process, such as any production process giving H->gg, as this is measured simply by counting how many times these processes occur
in proton–proton collisions, with respect to the total number of collisions recorded by the ATLAS detector.
Figure 2: The observed signal strengths and uncertainties for different Higgs boson decay channels and their combination for mH=125.36 GeV. Higgs boson signals corresponding to the same decay channel
are combined together for all analyses. The best-fit values are shown by the solid vertical lines. The total ±1σ uncertainties are indicated by green shaded bands. (Image: CERN)
Looking at additional particles (such as jets initiated by quarks, or Z or W bosons) produced in association with the Higgs boson provides information about the production mechanism. These
cross-sections can be expressed as products of factors (kj) of the coupling strengths for Higgs production and decay, where j can represent the different particles the Higgs couples to (e.g. kb).
Each kj parameterizes the ratio of the coupling strength j to that predicted by the Standard Model, so that kj =1 corresponds to the Standard Model case.
The combined analysis of all observed Higgs production and decay modes allows one to disentangle to a certain extent the production and decay strengths, and to access the coupling strengths of the
Higgs boson to each of several particles, as measured by the kj factors. Figure 1 shows the results of the ATLAS analysis, showing that the couplings of the Higgs to different particles are indeed
proportional to the mass (or mass-squared) of the particle. The results match Standard Model expectations.
The analysis also allows tests for more radical deviations from the Standard Model. For example, limits are set on anomalous couplings of the Higgs boson, which would be due to unknown particles
contributing through quantum effects, and Higgs boson decays to invisible or undetected particles, a process not allowed in the Standard Model.
Figure 2 shows the signal strengths, µ, of the different processes, which are calculated as the measured cross-section (σ), divided by the predicted cross-section of the Standard Model (σSM). When µ
= σ / σSM = 1 (within the uncertainty of the measurement), this tell us that our measured cross-section is as the Standard Model predicts, verifying the theory. A new combination of all the
production modes and decay channels gives the most precise value of the signal strength to date, of µ = 1.18 +0.15 – 0.14.
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Code review request (DAG data structure, topological sort, boolean circuits)
Could I get a code review?
Playground URL: Rust Playground
Gist URL: https://gist.github.com/a53bc11473522f47fe71a7a46b54908d
This code implements a DAG data structure for Boolean circuits and has a function (compute) to evaluate them, which includes a topological sort algorithm (dfs).
In particular I'm not sure of the following are idiomatic: the data structure itself at line 26, relatives/children/parents functions at lines 34-65, depth first search at line 67, and compute at
line 98.
Any help appreciated, thanks!
I don't have enough time to check the algorithms and architectures, but I can offer some idiomatic codes. In no particular order:
• DAG::relatives accepts functions by Box<Fn(...) -> ...>, which is bad because, in addition to an additional allocation and virtual call overhead, it cannot refer to any variable outside of the
closure as it is equivalent to Box<Fn(...) -> ... + 'static>! It is normal to use a type parameter for such function arguments unless you have a stronger reason not to do so (e.g. your method
should be in the trait object, or you want to avoid binary bloat and there is no more option left).
• Vec<Bit> parameter is suspicious---in most cases &[Bit] should be sufficient unless it's &mut Vec<Bit>.
• match x { Some(v) => Some(f(v)), None => None } can be simplified to x.map(f) for many cases. If f has to return from the function, you can also use if let Some(v) = x { Some(f(v)) } else { None
• while !stack.is_empty() { let ix = stack.pop().unwrap(); ... } can be simplified to while let Some(ix) = stack.pop() { ... }. This is a common pattern for queues and stacks.
1 Like
Last three - thanks, I made the changes and understand why they're better.
First one - I got it to work correctly, but I don't really understand the difference between a type parameter and inlining the type of the function inside the parens, I thought they were the same
thing. Isn't it still the case that the size of edge_fn is not known at compile time?
fn relatives<EFn, NFn>(&self, node_idx: NIdx, edge_fn: EFn, node_fn: NFn)
-> Option<Vec<NIdx>>
where EFn: Fn(NIdx) -> &'a Vec<EIdx>,
NFn: Fn(EIdx) -> NIdx {
if self.nodes.contains_key(&node_idx) {
Some(edge_fn(node_idx).into_iter().map(|x| node_fn(*x)).collect())
} else {
fn children(&self, node_idx: NIdx) -> Option<Vec<NIdx>> {
|node_idx| &self.nodes[&node_idx].outgoing_edges,
|edge_idx| self.edges[&edge_idx].sink
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Efficient quasi-maximum-likelihood multiuser detection by semi-definite relaxation
In multiuser detection, maximum-likelihood detection (MLD) is optimum in the sense of minimum error probability. Unfortunately, MLD involves a computationally difficult optimization problem for which
there is no known polynomial-time solution (with respect to the number of users). In this paper, we develop an approximate maximum-likelihood (ML) detector using semi-definite (SD) relaxation for the
case of anti-podal data transmission. SD relaxation is an accurate and efficient approximation algorithm for certain difficult optimization problems. In MLD, SD relaxation is efficient in that its
complexity is O(K^3.5), where K stands for the number of users. Simulation results indicate that the SD relaxation ML detector has its bit error performance close to the true ML detector, even when
the cross-correlations between users are strong or the near-far effect is significant.
Dive into the research topics of 'Efficient quasi-maximum-likelihood multiuser detection by semi-definite relaxation'. Together they form a unique fingerprint.
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Prepare students with math they’ll need in the real world with YummyMath
Guest post by Brian Marks
Editor's note: Those who have followed my math rants know that I am critical of the disconnected math skills taught in schools that take the subject out of context. I also am not a fan of the drill
and kill math games like Mangahigh that are more interesting than math worksheets, but pay little attention to the real-world relevance of why these skills are necessary. In this post Brian Marks
gets to the heart of the math problem by sharing a resource that puts real-world relevance at it's core.
I can still remember my own math classes as a kid. I remember working out of a text book, listening to my teacher as she taught us three example problems so that we could do problems 1 – 43 (odds
only) for homework. Of course problems 1- 43 were all the same problem, they just had different numbers or letters in them. At the time I thought they were just letters, but now I know they were
variables. Maybe at the time I knew they were variables, but did I know what a variable was? Let’s fast forward to the present. What has changed? Often math classes still look the same as they did
decades ago. Students enter the room and check their homework, which is followed by direct instruction on some math skill that is meaningless to most of them. Students get to practice a few problems
in class and look forward to more homework full of practice problems and maybe some contrived word problems. The only thing different today might be the availability and variety of math resources on
the Internet. Students might be using text books less and worksheets more. The Internet is full of websites that provide teachers with skill worksheets that focus on algorithm practice, which means
tediously doing the same procedure over and over again. There are even some popular websites that help kids learn the steps to successfully work through an algorithm. On some websites if you get ten
of these problems correct in a row you get to move on to a new skill. What is concerning is that many of the math resources on the web are simply making it easier for teachers to teach skill
procedures and for students to memorize procedures. We have yet to see sweeping changes in math education in terms of: student learning to conceptual development of math concepts, student discourse,
critical thinking, number sense reasoning and a purposeful use of technology, all of which can help prepare students for the challenges of the real world. How can we help math students to see that
math is not a collection of isolated skills and procedures? How can we help them to be better communicators, critical thinkers and problem solvers? How can we engage them in mathematics and help them
to see that math is an important tool that can be used to solve real life problems? One online math resource,Yummymath.com, has made it its mission to answer these questions. The website creates and
shares authentic math activities that focus on problem solving, critical thinking, intuitive reasoning, student discourse and thinking about math in context. The creators of Yummymath write authentic
math activities on sports, entertainment, world news, science, business and much, much more. The site helps teacher and students break away from traditional math classroom routines and helps students
see relevance in mathematics and better prepare students for the real world.
Here’s how Yummymath does this and shifts the focus away from skills, procedures and direct instruction. 1. Yummymath activities are written on a real life event, happening or situation that should
be a familiar context to kids. Instead of telling students how to solve problems, Yummymath allows students to investigate questions related to a particular real world context and then problem solve
and explore concepts within that context. This is the perfect solution for students who often ask the common math question: “When am I ever going to use this?” 2. Yummymath makes use of open
questioning techniques. Instead of asking students to simply solve a proportion or find a slope or calculate a mean, open questions allow student to reason, question and think deeply about a math
concept. For example, students could be given a mean and then asked to create several different data sets that would have that given mean. Or students could be given the dimensions of an HD
television screen and then be asked to give other dimensions of HD and non HD television screens. Open questioning forces students to do more than perform a procedure. This strategy makes students
think deeply about a concept. You can learn more about open questioning from the book: More Good Questions, by Marian Small and Amy Lin. 3. Yummymath focuses on concepts, not skills. Procedural
fluency is important, but we already get plenty of that in text books and from online resources. A recent Yummymath activity on NFL franchise values provided the actual value of every NFL team in the
league. The data was given in several bar graphs, one for each division. Students were asked to look at each division’s bar graph and consider how each bar would look if each team in a particular
division had the same value. Students were then asked to redraw each division bar graph, so that each team had the same value. Students could have either transferred value around in the graph,
“borrowing” from one team and giving to another or they might have added the values of each team and then divided by the number of teams. The activity asks kids to reflect on their process and helps
students to visualize and better understand the concept of “mean.” Yummymath activities focus on conceptual understanding of math concepts. Just as the NCTM and CCSS recommend, Yummymath believes
that students should have some understanding of the math that they are doing. Students should have time to explore concepts before memorizing the related algorithms or procedures. This can result in
students being able to better judge the reasonableness of an answer when it comes from a procedure. It can also help students to rely less on procedures and more on a deep understanding of the
concepts. 4. Yummymath activities immerse students in real life problem solving. Students use actual data or facts to solve problems and make decisions, a process and skill that will serve students
well as they enter the real world. Activities such as “The Light Bulbs are Almost Burnt Out” and “Diapers” ask students to use math as a tool to make smart consumer decisions. Many Yummymath tasks
ask students to enter into problems with no clear entry point. Students have to grapple with how to make sense of the problem and how to proceed in solving it. The problem is not clearly defined and
it is not simply the same problem that the teacher told the class how to solve in the day’s lesson. This is the same process that we go through in our normal lives. When we encounter problems outside
of school, we do not have a teacher training us on how to solve the particular problem. We must make sense of the problem and persevere in solving it. “If we want students to develop the capacity to
think, reason, and problem solve then we need to start with a high-level, cognitively complex task.” (Stein & Lane, 1996) This type of math problem is called “Doing Math” and it is considered the
most cognitively demanding type of math task. Implementing Standards-Based Mathematics Instruction (Stein, Smith, Henningsen, & Silver, 2000). 5. Yummymath tasks are written in a way that will
provoke different levels and types of student thinking. For example, in the “Harry Potter Movie Franchise” activity, students use real data to determine which Harry Potter movie was the most
successful. The problem is open-ended and allows for students to solve the problem with various levels of sophistication. Activities are set up in a way that promotes student discourse. Students will
naturally have different ways of approaching an authentic problem such as this. When it comes time for students to share their mathematical ideas, they will have the opportunity to critique the
reasoning of others and articulate their own reasoning. This is one of the eight Standards for Mathematical Practice mentioned in the new Common Core State Standards. Furthermore, these types of
activities allow students to enter into two types of classroom discourse as described by Robin Alexander in his research. One type of discourse is discussion, which he defines as “the exchange of
ideas with a view to sharing information or solving problems” and the second type of discourse is dialogue, which he defines as “achieving common understanding through structured, cumulative
questioning and discussion which guide and prompt, reduce choices, minimize risk and error, and expedite ‘handover’ of concepts and principles” (Research by Robin Alexander, UK, in 2008). 6. Without
question students need the opportunity to use technology as a tool in the problem solving process. Several Yummymath activities are built around real life data and encourage the use of graphing
calculators or similar programs. Students use this technology in activities like “Monopoly” and “Super Bowl Commercial Costs” to better understand patterns and to make future predictions. Other
activities naturally lend themselves to using suggested Internet applications or Microsoft Excel.
Check out this video overview of YummyMath.
If you are looking for a math resource that breaks away from the norm of the traditional classroom resource, one that focuses on authentic math and problem solving, then check out Yummymath.com.
Yummymath will help teachers make mathematics relevant and engaging to students. It will also help your students become better prepared to problem solve and communicate and collaborate with others in
the real world. Yummymath activities are written to reflect the CCSS and NCTM content and process standards. If you are looking to help your students or child see relevance in mathematics or want to
give them an authentic math learning opportunity, use a Yummymath activity with your child, student or class. Check out www.yummymath.com.
Brian Marks is an instructional math coach in Newton, Massachusetts. He collaborates with and provides professional development for teachers. Recently he has done a good deal of work with the Common
Core State Standards around both the content standards and the standards of mathematical practice for his school district. He enjoys creating timely and relevant math investigations for his students,
his school district, and teachers that believe that math happens daily. He hopes these contributions will help bring current events and increased student motivation to your classrooms.
7 comments:
1. Really well-written article, Brian, you make me want to check out Yummymaths right now, but I figured I could take the time to congratulate you on a very motivational article. I'm just hoping I
can translate the US examples to the UK:-)
2. Very well written. A thing that has frustrated me is that students are not immersed in the vocabulary of math. It has its own language, and as long as students are unable to speak that language
(at least proficiently if not fluently), they will never feel totally comfortable. For example, it bothers me when students cannot tell the difference between "solve", "evaluate", and "simplify".
3. Thanks Vanessa. We do have some activities that should appeal to students in the UK: Euro and the Debt Crisis, maybe our Harry Potter activity, Starbucks, and any of our holiday activities.
Trevor, Thanks, we like to see the need for precise math language to emerge from authentic math exploration as a necessity to communicate ones mathematical ideas. Thanks for the comments! :)
4. Very good post. christian louboutin shoes I like it. Will also continue to focus on it.
5. Linda M.January 1, 2012 at 12:53PM
When I checked out your site, Yummymath.com, I was impressed with the extensive math concepts you cover.
Do you have problems designed for elementary students? I teach third grade and would love to use some to challenge my students to think more.
6. Linda, We have several activities that are used in my district as challenge work for 4th and 5th grade. Since the materials are available in word you could do some slight editing to make it more
accessible for 3rd grade. Contact us at our site, facebook or twitter to talk more about this. We would love to do more for 2nd-5th grade.
7. 10 in a row is not practiced by Khan Academy anymore. They use logistic regression to calculatate the probability that a student gets the next question right - Which might be an open question by
the way. Yummymath is very nice too though!
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Multi Expression Programming (MEP) is a method for automatic generation of computer programs. Particularly, MEP can be used for generating mathematical expressions for data analysis (regression,
classification and time series). MEP differentiates from other GP techniques by encoding multiple solutions in the same chromosome.
MEP strengths
MEP has several strong advantages compared to other techniques:
• It encodes multiple solution in the same chromosome. This means that the search space is better explored. Evaluating such chromosome is done (in most of the cases) in the same complexity as in
the case of a single solution per chromosome.
• Evaluates the computer programs instruction by instruction and stores the partial results for all training data. In this way MEP has a huge speed (less if, switch instructions) without needing to
work at low level (with processors registers) like Linear Genetic Programming does.
Representation of solutions
MEP representation is not specific. Linear and tree-based representations have been tested (see the list of papers). Here we exemplify with a linear representation.
• Each MEP chromosome has a constant number of genes. This number defines the length of the chromosome.
• Each gene contains an instruction (also name function) or variable (also named terminal).
• A gene that encodes a function includes pointers towards the function arguments. Function arguments always have indices of lower values than the position of the function itself in the chromosome.
This representation ensures that no cycle arises while the chromosome is decoded (phenotypically transcripted). According to the proposed representation scheme, the first symbol of the chromosome
must be a terminal symbol. In this way, only syntactically correct programs (MEP individuals) are obtained.
Consider a representation where the numbers on the left positions stand for gene labels. Labels do not belong to the chromosome, as they are provided only for explanation purposes.
For this example we use the set of functions:
F = {+, *},
and the set of terminals:
T = {a, b, c, d}.
An example of chromosome using the sets F and T is given below:
1. a
2. b
3. + 1, 2
4. c
5. d
6. + 4, 5
7. * 3, 5
The maximum number of symbols in MEP chromosome is given by the formula:
Number_of_Symbols = (n + 1) * (Number_of_Genes - 1) + 1,
where n is the number of arguments of the function with the greatest number of arguments.
The maximum number of effective symbols is achieved when each gene (excepting the first one) encodes a function symbol with the highest number of arguments. The minimum number of effective symbols is
equal to the number of genes and it is achieved when all genes encode terminal symbols only.
Decoding MEP chromosome and the fitness assignment process
Now we are ready to describe how MEP individuals are translated into computer programs. This translation represents the phenotypic transcription of the MEP chromosomes.
Phenotypic translation is obtained by parsing the chromosome top-down. A terminal symbol specifies a simple expression. A function symbol specifies a complex expression obtained by connecting the
operands specified by the argument positions with the current function symbol.
For instance, genes 1, 2, 4 and 5 in the previous example encode simple expressions formed by a single terminal symbol. These expressions are:
E[1] = a,
E[2] = b,
E[4] = c,
E[5] = d,
Gene 3 indicates the operation + on the operands located at positions 1 and 2 of the chromosome. Therefore gene 3 encodes the expression:
E[3] = a + b.
Gene 6 indicates the operation + on the operands located at positions 4 and 5. Therefore gene 6 encodes the expression:
E[6] = c + d.
Gene 7 indicates the operation * on the operands located at position 3 and 5. Therefore gene 7 encodes the expression:
E[7] = (a + b) * d.
E[7] is the last expression encoded by in chromosome.
There is neither practical nor theoretical evidence that one of these expressions is better than the others. This is why each MEP chromosome is allowed to encode a number of expressions equal to the
chromosome length (number of genes). The chromosome described above encodes the following expressions:
E[1] = a,
E[2] = b,
E[3] = a + b,
E[4] = c,
E[5] = d,
E[6] = c + d,
E[7] = (a + b) * d.
The value of these expressions may be computed by reading the chromosome top down. Partial results are computed by dynamic programming and are stored in a conventional manner.
Due to its multi expression representation, each MEP chromosome may be viewed as a forest of trees rather than as a single tree, which is the case of Genetic Programming.
As MEP chromosome encodes more than one problem solution, it is interesting to see how the fitness is assigned.
The chromosome fitness is usually defined as the fitness of the best expression encoded by that chromosome.
For instance, if we want to solve symbolic regression problems, the fitness of each sub-expression E[i] may be computed using the formula:
fitness(E[i]) = sum(|obtained[k,i] - target[k,i]|), k=1,... ,n
where obtained[k,i] is the result obtained by the expression E[i] for the fitness case k and target[k] is the targeted result for the fitness case k. In this case the fitness needs to be minimized.
The fitness of an individual is set to be equal to the lowest fitness of the expressions encoded in the chromosome:
fitness(C) = min fitness(E[i]).
When we have to deal with other problems, we compute the fitness of each sub-expression encoded in the MEP chromosome. Thus, the fitness of the entire individual is supplied by the fitness of the
best expression encoded in that chromosome.
Why encoding multiple solutions within a chromosome?
When you compute the value of an expression encoded as a GP tree you have the compute the value of all subtrees. This means that all GP subtrees can be viewed as a potential solution of the problem
being solved. Most of the GP techniques considers only the one tree while ignoring all the other subtrees. However, the value/fitness for all subtrees is computed by GP. The biggest difference
between MEP and other GP techniques is that MEP outputs the best subtree encoded in a chromosome. Note that the complexity (roughly speaking - the running time) is the same for MEP and other GP
techniques encoding 1 solution/chromosome.
The second reason for the this question is motivated by the No Free Lunch Theorems for Search. There is neither practical nor theoretical evidence that one of the solutions encoded in a chromosome is
better than the others. More than that, Wolpert and McReady proved that we cannot use the search algorithm's behavior so far for a particular test function to predict its future behavior on that
How to efficiently encode multiple solutions within a chromosome is sometimes difficult. There is no general prescription on how to encode multiple solutions in the same chromosome. In most cases
this involves creativity and imagination. However, some general suggestions can be given.
In order to obtain some benefits from encoding more than one solution in a chromosome we have to spend a similar effort (computer time, memory etc) as in the case of encoding a single solution in a
chromosome. For instance, if we have 10 solutions encoded in chromosome and the time needed to extract and decode these solutions is 10 times larger than the time needed to extract and process one
solution we got nothing. In this case we can not talk about an useful encoding.
We have to be careful when we want to encode a multiple solutions in a variable length chromosome (for instance in a standard GP chromosome), because this kind of chromosome will tend to increase its
size in order to encode more and more solutions. And this could lead to bloat.
Usually encoding multiple solutions in a chromosome might require storing the partial results. Sometimes this can be achieved by using the Dynamic Programming technique. This kind of model is
employed by the Multi Expression Programming.
Genetic operators
One cutting point or uniform crossover have been tested with similar results. The cut points are chosen between instructions not inside them.
One-point recombination operator in MEP representation is similar to the corresponding binary representation operator. One crossover point is randomly chosen and the parent chromosomes exchange the
sequences at the right side of the crossover point.
Each symbol (terminal, function of function pointer) in the chromosome may be the target of the mutation operator. Some symbols in the chromosome are changed by mutation. To preserve the consistency
of the chromosome, its first gene must encode a terminal symbol.
We may say that the crossover operator occurs between genes and the mutation operator occurs inside genes.
If the current gene encodes a terminal symbol, it may be changed into another terminal symbol or into a function symbol. In the later case, the positions indicating the function arguments are
randomly generated. If the current gene encodes a function, the gene may be mutated into a terminal symbol or into another function (function symbol and pointers towards ar- guments).
Main algorithm
The MEP algorithm starts by creating a random population of individuals.
The following steps are repeated until a given number of generations is reached:
• Two parents are selected using a standard selection procedure.
• The parents are recombined in order to obtain two offspring.
• The offspring are mutated.
• Each offspring O replaces the worst individual W in the current population if O is better than W.
Read more
Read more about MEP and MEP for symbolic regression problems here.
Read more about MEP for classification problems here.
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Sundiata - Math/Writing Tutor - Learner - The World's Best Tutors
I have tutored math since 2012 and taught middle school and high school math. I have bachelor's and master's degrees in pure math, as well as a master's degree in engineering from Columbia
University. I tutor grades 4-College.
My tutoring style:
I believe all students can learn and enjoy math. Everyone has their own learning style, so it's a matter of finding what works best for each student. I believe in taking time to teach the
fundamentals of math and conceptual understanding, as these building blocks help students understand current and higher-level math.
Success story:
One of my favorite success stories is when I helped a student who previously excelled in Law and reading and writing subject matters, but felt they just weren't good at math. I helped him realize
just how good he could become at math. This student wanted to go into medicine believing Africa needed more doctors not more lawyers. He had just taken trigonometry but had mainly learned it by
rot. So, when he came to pre-calculus he was struggling to understand the material. We went back over the concepts of trigonometry in a way that not only helped him understand, but also allowed him
to realize that he was good at math.
Hobbies and interests:
I enjoy reading, learning about new cultures, and traveling, as well as athletics like yoga, ice hockey, and soccer. I also enjoy spending time with my niece and nephews.
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Multiplying Fractions By Whole Numbers Worksheets 4th Grade
Multiplying Fractions By Whole Numbers Worksheets 4th Grade work as foundational tools in the realm of mathematics, giving a structured yet versatile platform for learners to explore and grasp
mathematical ideas. These worksheets offer an organized method to understanding numbers, nurturing a strong foundation upon which mathematical effectiveness grows. From the easiest checking exercises
to the complexities of sophisticated estimations, Multiplying Fractions By Whole Numbers Worksheets 4th Grade cater to learners of diverse ages and skill degrees.
Unveiling the Essence of Multiplying Fractions By Whole Numbers Worksheets 4th Grade
Multiplying Fractions By Whole Numbers Worksheets 4th Grade
Multiplying Fractions By Whole Numbers Worksheets 4th Grade -
Web Multiplying Fractions by Whole Numbers Worksheet for 4th 5th Grade 4 7 19 reviews Math Fractions Multiplication Division Free Account
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At their core, Multiplying Fractions By Whole Numbers Worksheets 4th Grade are automobiles for conceptual understanding. They envelop a myriad of mathematical concepts, directing learners via the
maze of numbers with a series of interesting and deliberate workouts. These worksheets go beyond the borders of typical rote learning, encouraging energetic engagement and fostering an user-friendly
grasp of mathematical connections.
Nurturing Number Sense and Reasoning
Multiply Fractions By Whole Numbers Worksheet For 3rd 4th Grade Lesson Planet
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The heart of Multiplying Fractions By Whole Numbers Worksheets 4th Grade depends on cultivating number sense-- a deep comprehension of numbers' significances and affiliations. They encourage
expedition, inviting learners to explore arithmetic operations, analyze patterns, and unlock the enigmas of series. Through thought-provoking challenges and rational challenges, these worksheets
become entrances to developing thinking skills, supporting the analytical minds of budding mathematicians.
From Theory to Real-World Application
Multiplying Fractions With Whole Numbers 4th Grade Math Worksheets
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Multiplying Fractions By Whole Numbers Worksheets 4th Grade function as avenues bridging academic abstractions with the apparent facts of day-to-day life. By instilling functional circumstances right
into mathematical workouts, learners witness the relevance of numbers in their environments. From budgeting and dimension conversions to understanding analytical data, these worksheets encourage
pupils to possess their mathematical prowess beyond the confines of the classroom.
Varied Tools and Techniques
Versatility is inherent in Multiplying Fractions By Whole Numbers Worksheets 4th Grade, employing a collection of instructional tools to cater to varied understanding designs. Aesthetic help such as
number lines, manipulatives, and electronic sources serve as companions in visualizing abstract principles. This varied approach makes certain inclusivity, accommodating learners with different
choices, toughness, and cognitive styles.
Inclusivity and Cultural Relevance
In a progressively diverse world, Multiplying Fractions By Whole Numbers Worksheets 4th Grade welcome inclusivity. They transcend social limits, incorporating examples and troubles that reverberate
with students from diverse backgrounds. By incorporating culturally relevant contexts, these worksheets promote an atmosphere where every learner feels stood for and valued, improving their
connection with mathematical concepts.
Crafting a Path to Mathematical Mastery
Multiplying Fractions By Whole Numbers Worksheets 4th Grade chart a program in the direction of mathematical fluency. They infuse willpower, essential thinking, and analytical abilities, vital
characteristics not only in mathematics yet in different aspects of life. These worksheets equip learners to browse the elaborate terrain of numbers, nurturing an extensive recognition for the
elegance and reasoning inherent in maths.
Accepting the Future of Education
In an age marked by technological development, Multiplying Fractions By Whole Numbers Worksheets 4th Grade seamlessly adapt to electronic systems. Interactive user interfaces and digital resources
augment standard knowing, using immersive experiences that go beyond spatial and temporal borders. This amalgamation of typical techniques with technical innovations advertises an appealing period in
education and learning, promoting a more dynamic and interesting knowing setting.
Conclusion: Embracing the Magic of Numbers
Multiplying Fractions By Whole Numbers Worksheets 4th Grade represent the magic inherent in mathematics-- a charming journey of expedition, exploration, and mastery. They go beyond conventional
pedagogy, serving as drivers for firing up the fires of inquisitiveness and inquiry. Through Multiplying Fractions By Whole Numbers Worksheets 4th Grade, learners start an odyssey, unlocking the
enigmatic globe of numbers-- one problem, one service, at a time.
4th Grade Multiply Fractions By Whole Numbers Task Cards The Teacher Next Door
Fractions Multiplied By Whole Numbers Worksheets
Check more of Multiplying Fractions By Whole Numbers Worksheets 4th Grade below
Free Multiplying Fractions With Whole Numbers Worksheets
Multiplying Fractions Worksheet 4th Grader
Multiply A Fraction By A Whole Number Worksheet
Multiplying Fractions By Whole Numbers Worksheet Printable Word Searches
Multiplying Fractions And Whole Number Worksheets
Multiplying Fractions By Whole Numbers Worksheets Teaching Resources
Multiplying Fractions By Whole Numbers Worksheets
Web Multiplying Unit Fractions by Whole Numbers Unit fractions are those that have 1 as their numerator Open practice avenues for your 4th grade kid with our printable worksheets on multiplying unit
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Fraction And Whole Number Multiplication Worksheet Times Tables Worksheets
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Number Lines Teaching Resources
Download printable number lines, activities and worksheets to help students as they develop their understanding of counting, operations, fractions and more elementary math skills. Save time on your
math lesson planning with printables that are ready to be downloaded and used in the classroom!
This teaching resource collection was created by the teachers on the Teach Starter team and includes number lines for positive and negative integers, time number lines and classroom number lines that
can be printed and displayed on the classroom wall or your students' desks for a variety of math activities. Build students' number sense, expand their understanding of fractions and more with this
powerful collection that includes curriculum-aligned so you can meet Common Core and state standards.
New to using this tool in the math classroom, or just looking for some fresh ideas for teaching with number lines? Explore a primer from the teachers on the Teach Starter team!
What Is a Number Line?
A number line is a visual representation of numbers that is usually represented as a straight line. The numbers are usually represented as points on the line, and the distance between each number
corresponds to the difference in value between them.
In an elementary school setting, you can use a number line to introduce basic mathematical concepts and to help students develop their number sense. They can be used for a wide variety of different
activities such as counting, comparing numbers, teaching decimals and plenty more.
How Do Number Lines Help Students?
This tool has a number of applications in helping build numbers skills, some of which may surprise you! Take a look at the ways these can help your elementary students:
• They aid learning by providing a visual representation of numbers and their relative positions.
• They make it easier for students to understand concepts such as addition, subtraction and fractions, as well as to visualize and compare numbers.
• They can be used to introduce negative numbers and to understand the concept of absolute value.
• A number line can help students develop their number sense (aka the ability to understand and work with numbers in a flexible and efficient way).
How Do You Put Fractions in a Number Line?
Number lines come in handy when you're teaching fractions in elementary school — both by helping kids visualize the relationship between fractions and whole numbers and by allowing them to see the
relative sizes of fractions.
One of the most common stumbling blocks for students is putting fractions in a number line. Here's how to do this, step by step:
1. Determine the range of the number line. For example, if a student wants to put fractions between 0 and 1 on the number line, they'll need to draw a line segment from 0 to 1.
2. Divide the line segment into equal parts. The number of equal parts will depend on how many fractions your student wants to include on the line. If the plan is to include 4 fractions, for
example, they would divide the line segment into 4 equal parts.
3. Label the endpoints of the line segment with the whole numbers that correspond to them. For example, they could label the left endpoint as 0 and the right endpoint as 1.
4. Label the other points on the line segment with the fractions you want to include. To do this, students need to determine the location of each fraction on the number line. They can do this by
finding a common denominator for all of the fractions and then dividing the line segment into that many equal parts.
5. Plot each fraction on the number line by marking the appropriate point and labeling it with the corresponding fraction. You can also have your students shade in the region of the number line that
corresponds to the set of fractions they are interested in.
Students may also need to partition a number line into equal parts as part of their study of fractions! To do this, instruct your students to put the 0 and 1 tick marks at the ends of the line, look
at how many equal parts are needed, and draw one fewer tick mark.
For example, if a student is dividing a line into eighths, they draw the 0 and 1 markers and draw seven ticks (this will ultimately give them eight equal spaces).
5 Creative Ways to Use Number Lines in Your Math Classes
Looking for fresh ways to use number lines? Explore our collection for printable activities and ideas, and explore this list of favorites from our teacher team!
1. Create a Scavenger Hunt — Have your students go on a hunt for specific numbers on the number line scavenger hunt style. They can also be given a range of numbers to find and write down the
numbers that fall within that range.
2. Use the Number Line for a Drawing Activity — Incorporate the number line into a drawing activity. For example, kids can draw an ocean that shows the depth below sea level or integrate a number
line into a drawing that has a thermometer to show positive and negative temperatures.
3. Number Line Sidewalk Art — Here's another fun art activity you can do with this handy tool. Students can draw horizontal and vertical number lines with sidewalk chalk and then physically move
between numbers, showing how to represent addition, subtraction, rounding and more. This is a great way to incorporate movement into the school day too.
4. Number Lines with Clothespins — Students can sequence clothespins marked with integers, fractions or decimals. This helps with fine motor concepts for younger students, while older students can
be challenged to match equivalent value clothespins on their number line. (Such as equivalent fractions, decimals and percentages)
5. Number Line Riddles — Bring some serious fun to your number line activities. Give your students a riddle they can solve with the use of a number line!
Show more
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32 of 88 teaching resources for those 'aha' moments
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Decimal and binary systems: the basis of programming
DolarApp Blog Freelancer tips Decimal and binary systems: the basis of programming
Decimal and Binary Systems: How to convert
In the world of freelance programmers, every line of code can make a difference. And what if I told you there's a key secret behind the success you're seeking in your career? Yes, as you might guess,
we're talking about decimal and binary systems. But why are they so important?
All we can reveal for now is that these systems are essential for unlocking your maximum potential as a freelance programmer. So, we invite you to uncover the mysteries that can propel your career to
new heights, offering you a unique advantage in the market.
What is the Decimal System?
It's a numeration system centered around the idea that each position within a number represents a power of ten. Here, we have 10 different digits, from 0 to 9, and we use 10 as the base for
calculations and to figure quantities.
The decimal system is the most popular in everyday and commercial use worldwide. Freelance programmers often use it in personal projects or in software development for commercial, financial, or
consumer applications. This is because quantities are usually expressed in decimal terms for users to understand and manipulate more easily.
For example, the number 356 is equal to: 3 hundreds, 5 tens, and 6 units in the decimal system. Expressed in base 10 powers, it would be: 3*10^2 + 5*10^1 + 6*10^0, which translates to: 300 + 50 + 6.
What is the Binary System?
This numeric system is based on two digits: 0 and 1. Unlike the decimal system, which uses ten digits (from 0 to 9), this one uses only two, making it the fundamental base of digital computing.
This system is essential for freelance programmers, as it establishes the foundation of all digital computing and programming. Each position in a binary number corresponds to a power of two, just as
in the decimal system, these positions represent a power of ten.
For example, the binary number 1010 breaks down into: 1×2^3 + 0×2^2 + 1×2^1 + 0×2^0, which translates to: 8 + 0 + 2 + 0. So, in the binary system, 1010 is equivalent to 10 in the decimal system.
How to Convert Decimal Numbers to Binary? Step by Step
The conversion process involves dividing the decimal number by 2 repeatedly until the quotient is 0. Then, you read the remainders from the divisions bottom-up to get the binary number. The steps are
as follows:
1. First, divide the decimal number by 2.
2. Record the remainder of the division (0 or 1).
3. Divide the resulting quotient again by 2 and record the remainders.
4. Continue dividing consecutively by 2 until the quotient is 0, recording each remainder.
5. Once you've achieved a quotient of 0, read the remainders in reverse order to obtain the binary number.
Suppose you want to convert the decimal number 15 to binary:
• 15 / 2 = 7 with a remainder of 1.
• 7 / 2 = 3 with a remainder of 1.
• 3 / 2 = 1 with a remainder of 1.
• 1 / 2 = 0 with a remainder of 1.
Since we've reached a quotient of 0, we've completed the division. Reading in reverse order, you have 1111, the binary number equivalent to 15 in decimal.
How to Convert Binary Numbers to Decimals? Step by Step
Now, how to convert binary to decimal? In this case, you add up the values of the powers of 2 corresponding to each binary digit. Here's an example of converting binary to decimal step by step:
1. Start by writing down the binary number you want to convert to decimal.
2. Assign powers of 2 to each binary digit, starting from the right and increasing towards the left. The power of 2 for the rightmost bit is 2^0, then 2^1, followed by 2^2, and so on.
3. Multiply each binary digit by the corresponding power of 2.
4. Add all the results of the multiplications to get the decimal number equivalent.
For example, if you want to convert the binary number 1101 to decimal, it would be:
• 1 * 2^3 (for the leftmost bit) = 8
• 1 * 2^2 (for the next bit) = 4
• 0 * 2^1 (for the next bit) = 0
• 1 * 2^0 (for the rightmost bit) = 1
Adding the results, you have 8 + 4 + 0 + 1 = 13. So, the binary number 1101 is equivalent:
• 1 * 2^3 (for the leftmost bit) = 8
• 1 * 2^2 (for the next bit) = 4
• 0 * 2^1 (for the next bit) = 0
• 1 * 2^0 (for the rightmost bit) = 1
Adding up the results, you get 8 + 4 + 0 + 1 = 13. Thus, the binary number 1101 is equivalent to 13 in decimal.
In this sense, creating scripts in languages such as JavaScript or Python can help you automate the conversion between numerical systems.
Decimal and Binary Systems Examples
Below is a table showing converted decimals and binaries.
Decimal Number Binary Number
These are examples from 1 to 10 to illustrate how conversions between decimal and binary systems are performed. However, you can follow the steps described to make conversions with other numbers as
Why Are Binary and Decimal Systems Important for Freelance Programmers?
There are various reasons why these systems are important in your career as a freelance programmer.
Some of them include:
Digital Data Representation. These systems are pillars of data representation in computer systems. Therefore, understanding their operation is vital to being able to manipulate and work with them
Information Encoding. It's common to handle data encoded in binary systems, especially if you're developing software. Consequently, knowing how to encode information in binary benefits you when
working with data and effectively developing robust applications.
Optimization of Algorithms and Operations. Often, working directly with binary data can be more efficient than working with decimal data in terms of memory usage and processing speed. As a
programmer, you could leverage this knowledge to optimize operations and algorithms, achieving better performance in applications and programs.
In summary, decimal and binary systems are key to unlocking your maximum potential. Not only do they help you have a deep understanding of computing, but they also allow you to tackle more complex
programming challenges.
Moreover, understanding numerical systems is universally relevant, regardless of the programming language or technology you use. Such versatility is crucial for your career as a freelancer, as you'll
often need to adapt to different requirements, projects, and teams.
This knowledge, combined with the ability to apply systems in practical contexts, can serve to increase your salary as a programmer.
Ultimately, we suggest exploring platforms to find remote employment opportunities if you're looking for remote job opportunities. These sites will help boost your freelance career.
Additionally, they give you the option to receive your payments in your local currency or in dollars, as long as you choose services like DolarApp.
You can also create an attractive profile and apply to secure employment at large companies. Working at Google, for instance, is a great option if you want to grow and enjoy its benefits.
Discover a world
without borders.
The world has borders. Your finances don’t have to.
Most recent in Freelancer tips
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Principal Components Analysis | SAS Annotated Output
This page shows an example of a principal components analysis with footnotes explaining the output. The data used in this example were collected by Professor James Sidanius, who has generously
shared them with us. You can download the data set here.
Overview: The “what” and “why” of principal components analysis
Principal components analysis is a method of data reduction. Suppose that you have a dozen variables that are correlated. You might use principal components analysis to reduce your 12 measures to a
few principal components. In this example, you may be most interested in obtaining the component scores (which are variables that are added to your data set) and/or to look at the dimensionality of
the data. For example, if two components are extracted and those two components accounted for 68% of the total variance, then we would say that two dimensions in the component space account for 68%
of the variance. Unlike factor analysis, principal components analysis is not usually used to identify underlying latent variables. Hence, the loadings onto the components are not interpreted as
factors in a factor analysis would be. Principal components analysis, like factor analysis, can be preformed on raw data, as shown in this example, or on a correlation or a covariance matrix. If
raw data is used, the procedure will create the original correlation matrix or covariance matrix, as specified by the user. If the correlation matrix is used, the variables are standardized and the
total variance will equal the number of variables used in the analysis (because each standardized variable has a variance equal to 1). If the covariance matrix is used, the variables will remain in
their original metric. However, one must take care to use variables whose variances and scales are similar. Unlike factor analysis, which analyzes the common variance, the original matrix in a
principal components analysis analyzes the total variance. Also, principal components analysis assumes that each original measure is collected without measurement error.
In this example we have included many options, including the original correlation matrix and the scree plot. While you may not wish to use all of these options, we have included them here to aid in
the explanation of the analysis. We have also created a page of annotated output for a factor analysis that parallels this analysis. For general information regarding the similarities and
differences between principal components analysis and factor analysis, please see our FAQ entitled What are some of the similarities and differences between principal components analysis and factor
proc factor data = "d:\m255_sas" corr scree ev method = principal;
var item13 item14 item15 item16 item17 item18 item19 item20 item21 item22 item23 item24 ;
ITEM13 ITEM14 ITEM15
ITEM13 INSTRUC WELL PREPARED 1.00000 0.66146 0.59999
ITEM14 INSTRUC SCHOLARLY GRASP 0.66146 1.00000 0.63460
ITEM15 INSTRUCTOR CONFIDENCE 0.59999 0.63460 1.00000
ITEM16 INSTRUCTOR FOCUS LECTURES 0.56626 0.50003 0.50535
ITEM17 INSTRUCTOR USES CLEAR RELEVANT EXAMPLES 0.57687 0.55150 0.58664
ITEM18 INSTRUCTOR SENSITIVE TO STUDENTS 0.40898 0.43311 0.45707
ITEM19 INSTRUCTOR ALLOWS ME TO ASK QUESTIONS 0.28632 0.32041 0.35869
ITEM20 INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS 0.30418 0.31481 0.35568
ITEM21 INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING 0.47553 0.44896 0.50904
ITEM22 I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION 0.33255 0.33313 0.36884
ITEM23 COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS 0.56399 0.56461 0.58233
ITEM24 COMPARED TO OTHER COURSES THIS COURSE WAS 0.45360 0.44281 0.43481
ITEM16 ITEM17 ITEM18
ITEM13 INSTRUC WELL PREPARED 0.56626 0.57687 0.40898
ITEM14 INSTRUC SCHOLARLY GRASP 0.50003 0.55150 0.43311
ITEM15 INSTRUCTOR CONFIDENCE 0.50535 0.58664 0.45707
ITEM16 INSTRUCTOR FOCUS LECTURES 1.00000 0.58649 0.40479
ITEM17 INSTRUCTOR USES CLEAR RELEVANT EXAMPLES 0.58649 1.00000 0.55474
ITEM18 INSTRUCTOR SENSITIVE TO STUDENTS 0.40479 0.55474 1.00000
ITEM19 INSTRUCTOR ALLOWS ME TO ASK QUESTIONS 0.33540 0.44930 0.62660
ITEM20 INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS 0.31676 0.41682 0.52055
ITEM21 INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING 0.45245 0.59526 0.55417
ITEM22 I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION 0.36255 0.44976 0.53609
ITEM23 COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS 0.45880 0.61302 0.56950
ITEM24 COMPARED TO OTHER COURSES THIS COURSE WAS 0.42967 0.52058 0.47382
ITEM19 ITEM20 ITEM21
ITEM13 INSTRUC WELL PREPARED 0.28632 0.30418 0.47553
ITEM14 INSTRUC SCHOLARLY GRASP 0.32041 0.31481 0.44896
ITEM15 INSTRUCTOR CONFIDENCE 0.35869 0.35568 0.50904
ITEM16 INSTRUCTOR FOCUS LECTURES 0.33540 0.31676 0.45245
ITEM17 INSTRUCTOR USES CLEAR RELEVANT EXAMPLES 0.44930 0.41682 0.59526
ITEM18 INSTRUCTOR SENSITIVE TO STUDENTS 0.62660 0.52055 0.55417
ITEM19 INSTRUCTOR ALLOWS ME TO ASK QUESTIONS 1.00000 0.44647 0.49921
ITEM20 INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS 0.44647 1.00000 0.42479
ITEM21 INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING 0.49921 0.42479 1.00000
ITEM22 I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION 0.48404 0.38297 0.50651
ITEM23 COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS 0.44401 0.40962 0.59751
ITEM24 COMPARED TO OTHER COURSES THIS COURSE WAS 0.37383 0.35722 0.49977
ITEM22 ITEM23 ITEM24
ITEM13 INSTRUC WELL PREPARED 0.33255 0.56399 0.45360
ITEM14 INSTRUC SCHOLARLY GRASP 0.33313 0.56461 0.44281
ITEM15 INSTRUCTOR CONFIDENCE 0.36884 0.58233 0.43481
ITEM16 INSTRUCTOR FOCUS LECTURES 0.36255 0.45880 0.42967
ITEM17 INSTRUCTOR USES CLEAR RELEVANT EXAMPLES 0.44976 0.61302 0.52058
ITEM18 INSTRUCTOR SENSITIVE TO STUDENTS 0.53609 0.56950 0.47382
ITEM19 INSTRUCTOR ALLOWS ME TO ASK QUESTIONS 0.48404 0.44401 0.37383
ITEM20 INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS 0.38297 0.40962 0.35722
ITEM21 INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING 0.50651 0.59751 0.49977
ITEM22 I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION 1.00000 0.49317 0.44440
ITEM23 COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS 0.49317 1.00000 0.70464
ITEM24 COMPARED TO OTHER COURSES THIS COURSE WAS 0.44440 0.70464 1.00000
The table above was included in the output because we included the keyword corr on the proc factor statement. This table gives the correlations between the original variables (which are specified on
the var statement). Before conducting a principal components analysis, you want to check the correlations between the variables. If any of the correlations are too high (say above .9), you may need
to remove one of the variables from the analysis, as the two variables seem to be measuring the same thing. Another alternative would be to combine the variables in some way (perhaps by taking the
average). If the correlations are too low, say below .1, then one or more of the variables might load only onto one principal component (in other words, make its own principal component). This is
not helpful, as the whole point of the analysis is to reduce the number of items (variables).
Initial Factor Method: Principal Components
Prior Communality Estimates: ONE
Eigenvalues of the Correlation Matrix: Total = 12 Average = 1
Eigenvalue^a Difference^b Proportion^c Cumulative^d
1 6.24914661 5.01966832 0.5208 0.5208
2 1.22947829 0.51048923 0.1025 0.6232
3 0.71898906 0.10585957 0.0599 0.6831
4 0.61312949 0.05196458 0.0511 0.7342
5 0.56116491 0.05817383 0.0468 0.7810
6 0.50299107 0.03172750 0.0419 0.8229
7 0.47126357 0.08244834 0.0393 0.8622
8 0.38881523 0.02091149 0.0324 0.8946
9 0.36790373 0.03970330 0.0307 0.9252
10 0.32820043 0.01082277 0.0274 0.9526
11 0.31737767 0.06583773 0.0264 0.9790
12 0.25153994 0.0210 1.0000
2 factors will be retained by the MINEIGEN criterion.
a. Eigenvalue – This column contains the eigenvalues. The first component will always account for the most variance (and hence have the highest eigenvalue), and the next component will account for
as much of the left over variance as it can, and so on. Hence, each successive component will account for less and less variance.
b. Difference – This column gives the differences between the current and the next eigenvalue. For example, 6.24 – 1.22 = 5.02. This gives you a sense of how much change there is in the eigenvalues
from one component to the next.
c. Proportion – This column gives the proportion of variance accounted for by each component. In this example, the first component accounts for just over half of the variance (approximately 52%).
d. Cumulative – This column sums up to proportion column, so that you can see how much variance is accounted for by, say, the first five components, .7810.
Initial Factor Method: Principal Components
Scree Plot of Eigenvalues
7 +
| 1
6 +
5 +
E |
i |
g |
e 4 +
n |
v |
a |
l |
u |
e 3 +
s |
2 +
| 2
1 +
| 3 4
| 5 6 7
| 8 9 0 1 2
0 +
Initial Factor Method: Principal Components
The scree plot graphs the eigenvalue against the component number. You can see these values in the first two columns of the table immediately above. From the third component on, you can see that
the line is almost flat, meaning the each successive component is accounting for smaller and smaller amounts of the total variance. In general, we are interested in keeping only those principal
components whose eigenvalues are greater than 1. Components with an eigenvalue of less than 1 account for less variance than did the original variable (which had a variance of 1), and so are of
little use. Hence, you can see that the point of principal components analysis is to redistribute the variance in the correlation matrix (using the method of eigenvalue decomposition) to
redistribute the variance to first components extracted.
1^e 2^e
ITEM13 INSTRUC WELL PREPARED 0.29093 -0.40510
ITEM14 INSTRUC SCHOLARLY GRASP 0.28953 -0.36765
ITEM15 INSTRUCTOR CONFIDENCE 0.29851 -0.27789
ITEM16 INSTRUCTOR FOCUS LECTURES 0.27406 -0.25376
ITEM17 INSTRUCTOR USES CLEAR RELEVANT EXAMPLES 0.32261 -0.09492
ITEM18 INSTRUCTOR SENSITIVE TO STUDENTS 0.30207 0.33002
ITEM19 INSTRUCTOR ALLOWS ME TO ASK QUESTIONS 0.25641 0.44823
ITEM20 INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS 0.23709 0.34083
ITEM21 INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING 0.30536 0.12133
ITEM22 I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION 0.26057 0.32871
ITEM23 COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS 0.32768 -0.03634
ITEM24 COMPARED TO OTHER COURSES THIS COURSE WAS 0.28550 0.00421
e. Eigenvectors – These columns give the eigenvectors for each variable in the principal components analysis. An eigenvector is a linear combination of the original variables. The two components
that have been extracted are orthogonal to one another, and they can be thought of as weights. These weights are multiplied by each value in the original variable, and those values are then summed up
to yield the eigenvector. The eigenvectors tell you about the strength of relationship between the variables and the components.
Factor Pattern
Factor1 Factor2
ITEM13 INSTRUC WELL PREPARED 0.72729 -0.44919
ITEM14 INSTRUC SCHOLARLY GRASP 0.72378 -0.40766
ITEM15 INSTRUCTOR CONFIDENCE 0.74622 -0.30813
ITEM16 INSTRUCTOR FOCUS LECTURES 0.68511 -0.28137
ITEM17 INSTRUCTOR USES CLEAR RELEVANT EXAMPLES 0.80647 -0.10525
ITEM18 INSTRUCTOR SENSITIVE TO STUDENTS 0.75512 0.36593
ITEM19 INSTRUCTOR ALLOWS ME TO ASK QUESTIONS 0.64098 0.49700
ITEM20 INSTRUCTOR IS ACCESSIBLE TO STUDENTS OUTSIDE CLASS 0.59269 0.37792
ITEM21 INSTRUCTOR AWARE OF STUDENTS UNDERSTANDING 0.76335 0.13454
ITEM22 I AM SATISFIED WITH STUDENT PERFORMANCE EVALUATION 0.65138 0.36448
ITEM23 COMPARED TO OTHER INSTRUCTORS, THIS INSTRUCTOR IS 0.81914 -0.04029
ITEM24 COMPARED TO OTHER COURSES THIS COURSE WAS 0.71371 0.00467
f. Factor1 and Factor2 – This is the component matrix. This table contains component loadings, which are the correlations between the variable and the component. Because these are correlations,
possible values range from -1 to +1. The columns under these headings are the principal components that have been extracted. As you can see, two components were extracted (the two components that
had an eigenvalue greater than 1). You usually do not try to interpret the components the way that you would factors that have been extracted from a factor analysis. Rather, most people are
interested in the component scores, which are used for data reduction (as opposed to factor analysis where you are looking for underlying latent continua).
Variance Explained by Each Factor
Factor1 Factor2
6.2491466 1.2294783
Final Communality Estimates: Total = 7.478625
ITEM13 ITEM14 ITEM15 ITEM16 ITEM17 ITEM18
0.73071411 0.69004215 0.65179276 0.54854615 0.66147090 0.70412023
Initial Factor Method: Principal Components
ITEM19 ITEM20 ITEM21 ITEM22 ITEM23 ITEM24
0.65786784 0.49410612 0.60081090 0.55713785 0.67261205 0.50940384
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Imperial numbers and Measurement – Circular Measurement
Imperial measure owes nothing to any Earth system of measurement. The Imperial system was designed in an already scientific and technological era for much the same reasons Earth’s metric units were
designed – to simplify a complex system of historical measures.
The numeration system is base sixty – 3 times 4 times 5. This makes numbers more easily manipulated than previous systems. It works upon the same positional basis as our Hindu-Arabic system, and the
same concepts apply, with multiples of 2, 3, 4, 5 and 6 (and 10, 12, 15, 20 and 30) having intermediate importance in thought between the unit and the system base of sixty. For all units, the digital
places are described thus:
Prime: 60 times the base unit
Square: 3600 times the base unit
Cube: 216,000 times the base unit
Fourth: 12,960,000 times the base unit, and so on.
For fractional numbers, iprime is 1/60th of the base unit, isquare is 1/3600th, ithird is 1/216,000th, ifourth is 1/12,960,000th, and so on. If there’s an ‘i’ in front of the multiplier, it relates
to a fractional number.
For situations requiring more precision than a single digit, the spoken convention is to specify the magnitude of the leading digit, and following digits are presumed to be immediately decreasing in
magnitude. “Two minutes thirty” can be two and a half minutes of time, or two and a half light-minutes of distance. “Seven ithirds fortysix fourteen” would be seven 216,000ths, forty-six
12,960,000ths, fourteen 777,600,000ths of the relevant units. If the relevant digit is a zero, the zero is spoken, so “seven ithirds zero fourteen” would be seven 216,000ths, zero 12,960,000ths,
fourteen 777,600,000ths of the relevant units
Most of the time, measures are context sensitive in translations to English. For instance, a range is assumed to be given in distance units, a mass or weight in terms of mass units, and so on. In
Technical, the fact of what you are measuring requires the units of measurement to match, and this has, over time, leaked over to conversations in Traditional as well. In Mindlord, everything is
context sensitive anyway, and nobody uses Concept for technical or technological purposes, as the ‘language’ is entirely unsuitable for that purpose.
Circular measure
The Empire uses a system of degrees, minutes, and seconds much like Earth. The major difference is that the imperial circle is 60 degrees, not 360 like Earth. There are still sixty arc minutes to an
arc degree and sixty arc seconds to an arc minute This makes circular measure an easy, straight-forward conversion of six to one. In mathematical graphing, zero (and sixty) is straight up, not to the
right. In navigation, zero (and sixty) is straight ahead, thirty is directly behind. The convention is to use the current orientation of the ship or observer to establish a ‘horizontal’ axis, which
is stated first and measured in degrees minutes and seconds left of current direction faced, so fifteen degrees is directly to an observer’s left under current orientation, fortyfive is to the right.
If the measurement is negative, it’s measured to the right of current orientation, so for example, directly to the right can also be expressed as minus fifteen. The vertical axis is established in
the same way, and measured the same way. Note that unlike Earthly spherical coordinates, this gives four acceptable solutions for a direction, although degrees left by degrees up is most standard.
Some common Earth angles, and their imperial equivalents:
180 degrees = 30 degrees imperial
90 degrees = 15 degrees imperial
60 degrees = 10 degrees imperial
45 degrees = 7 degrees, 30 minutes imperial (usually spoken as “seven degrees thirty” or “seven thirty” if context and magnitude is established)
30 degrees = 5 degrees imperial
15 degrees = 2 degrees, 30 minutes imperial
(Only mathematics can hope to describe eleven dimensional work such as Interstitial Vector. Technical and Mindlord both handle it reasonably well, but no Earthly language can be anything approaching
both clear and accurate in this regard, so I’m not going to try)
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Series Archives | Data + People
Beginner’s Guide to Using Python with HR Data | Exploration Series
Part Three – Seaborn
In this first tutorial series, I’m exploring the IBM HR Attrition and Performance data set. This is a great data set used to demonstrate the possibilities from using machine learning and other data
science techniques.
Now we’ll move on to using Seaborn for our visualizations. The benefit of Seaborn is it continues to abstract the complex, underlying calls to visualize your data – further allowing you to focus on
your analysis task and not having to think about how to implement what you want to do. It goes even further and provides built-in functionality that would be incredibly complex to implement without
the benefit of Seaborn.
Series Outline
0: basic operations & summary statistics
1: matplotlib
2: pandas visualization
3: seaborn
4: plotly
5: series summary
3: Seaborn
view on github
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Photo: Photo by Randall Ruiz on Unsplash
Beginner’s Guide: Python for Analytics | Pandas
Beginner’s Guide to Using Python with HR Data | Exploration Series
Part Two – Pandas
In this first tutorial series, I’m exploring the IBM HR Attrition and Performance data set. This is a great data set used to demonstrate the possibilities from using machine learning and other data
science techniques.
Next, we’ll take a look at the power of Pandas to plot our data. As a budding data [analyst/scientist/enthusiast], Pandas has become my most common import and tool. Plotting directly from pandas
objects makes it very easy to stay in the flow of analyzing data. Let’s get going.
Series Outline
0: basic operations & summary statistics
1: matplotlib
2: pandas visualization
3: seaborn
4: plotly
5: series summary
2: Pandas
view on github
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Beginner’s Guide: Python for Analytics | Matplotlib
Beginner’s Guide to Using Python with HR Data | Exploration Series
Part One – Matplotlib
In this first tutorial series, I’m exploring the IBM HR Attrition and Performance data set. This is a great data set used to demonstrate the possibilities from using machine learning and other data
science techniques.
In this next walkthrough, we’ll begin to ‘see’ our data through the use of visualization packages. In R there are 3 commons plotting tools, and other packages extend these main items. In Python,
there is Matplotlib, and most other packages build on this foundation. So, the decision of where to start with Python plotting is an easy one – let’s get going.
Series Outline
0: basic operations & summary statistics
1: matplotlib
2: pandas visualization
3: seaborn
4: plotly
5: series summary
1: matplotlib
view on github
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Beginner’s Guide: Python for Analytics | The Basics
Beginner’s Guide to Using Python with HR Data | Exploration Series
Part Zero – The Basics
In this first tutorial series, I’m exploring the IBM HR Attrition and Performance data set. This is a great data set used to demonstrate the possibilities from using machine learning and other data
science techniques.
I’ll be back with tutorial posts that walk through how to apply more advanced techniques to generate predictive and prescriptive insights from the data. But that’d be jumping ahead. First, the
basics. Exploratory Data Analysis, or EDA.
It’s often tempting to jump right in and try to find the most advanced insight possible. When I’m in the process of learning something new, it’s my first instinct to begin applying it straight away,
skipping the basics. Eventually, I’ll stumble; and it’s always something I could have avoided by simply spending a little bit of time really understanding the data I have.
To properly analyze data, you must understand it. Is it complete (missing values), are the errors (values out of normal bounds – is this correct), and generally what information is contained within
the data? Depending on where the request is coming from in a work-context, you may not control the data, so what you get is what you have; it’s often much easier when you’ve pulled your own data –
it’s just not always possible, or even smart to do so.
Always begin with an exploration of your data. In this tutorial, I’m digging out my current favorite tool – Python. If you’ve never programmed, if Excel still frightens you a bit, or you’re firmly in
the R camp – read on; this series will show the possibilities while exploring 5 different packages and interpreting and understanding data.
Series Outline
0: basic operations & summary statistics
1: matplotlib
2: pandas visualization
3: seaborn
4: plotly
5: series summary
0: basic operations & generating summary statistics
view on github
|
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Tests on Parameters
In general, the hypothesis tested can be written as
where is an r by 1 vector valued function of the parameters given by the r expressions specified in the TEST statement.
Let be the estimate of the covariance matrix of . Let be the unconstrained estimate of and be the constrained estimate of such that . Let
Using this notation, the test statistics for the three kinds of tests are computed as follows.
The Wald test statistic is defined as
The Wald test is not invariant to reparameterization of the model (Gregory 1985; Gallant 1987, p. 219). For more information about the theoretical properties of the Wald test, see Phillips and Park
The Lagrange multiplier test statistic is
where is the vector of Lagrange multipliers from the computation of the restricted estimate .
The likelihood ratio test statistic is
where represents the constrained estimate of and is the concentrated log-likelihood value.
For each kind of test, under the null hypothesis the test statistic is asymptotically distributed as a random variable with r degrees of freedom, where r is the number of expressions in the TEST
statement. The p-values reported for the tests are computed from the distribution and are only asymptotically valid.
Monte Carlo simulations suggest that the asymptotic distribution of the Wald test is a poorer approximation to its small sample distribution than that of the other two tests. However, the Wald test
has the lowest computational cost, since it does not require computation of the constrained estimate .
The following is an example of using the TEST statement to perform a likelihood ratio test:
proc qlim;
model y = x1 x2 x3;
test x1 = 0, x2 * .5 + 2 * x3 = 0 /lr;
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Unification of the Quadratic Model Equations of the Inhibition Characteristics of Acidified Ocimum Basilicum on the Corrosion Behaviour of Mild Steel
Journal of Minerals and Materials Characterization and Engineering, 2013, 1, 367-373
Published Online November 2013 (http://www.scirp.org/journal/jmmce)
Open Access JMMCE
Unification of the Quadratic Model Equations of the
Inhibition Characteristics of Acidified Ocimum Basilicum
on the Corrosion Behaviour of Mild Steel
Michael O. Nwankwo1, Paul A. Nwobasi2, Peter O. Offor3, Ndubuisi E. Idenyi1
1Department of Industrial Physics, Ebonyi State University, Abakaliki, Nigeria
2Department of Technology and Vocational Education, Ebonyi State University, Abakaliki, Nigeria
3Department of Metallurgical and Materials Engineering, University of Nigeria, Nsukka, Nigeria
Email: michaelnwankwo@yahoo.com, awonwobasi@yahoo.com, peterjoyoffor@yahoo.com, edennaidenyi@yahoo.com
Received September 3, 2013; revised October 14, 2013; accepted October 23, 2013
Copyright © 2013 Michael O. Nwankwo et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An attempt has been made at unifying the resulting quadratic models from the study of the correlation behavior of the
inhibition characteristics of acidified ocimum basilicum on conventional mild steel. Weight-loss corrosion technique
was employed in obtaining the corrosion penetration rate using the equation: 87.6 w
. Subsequently, the quad-
ratic models were developed by using a computer-aided statistical modeling technique (International Business Machine
(IBM)’s SPSS version 17.0). The results obtained showed a nearly perfect positive correlation with a correlation coeffi-
cient in the range of 0.986 ≤ R ≤ 0.996 which depicts that R ≥ 1. Also, the coefficient of determination fell within the
range of 0.972 ≤ R2 ≤ 0.992 showing that approximately 97% to 99% of the total variation in passivation rate was ac-
counted for by corresponding variation in exposure time, leaving out only between 3% and 1% to extraneous factors
that are not incorporated into the model equations. The equations were further unified into a generalized form using
MathCAD 7.0 and the resulting equation was 62
1.0320.0021.899 10
t with a R2 value of 0.935 indicating a
well-correlated relationship. With this, a new frontier on corrosion studies has emerged typifying a classical departure
from previously long-held assumption that corrosion behaviours at room temperature were only logarithmic.
Keywords: Corrosion; Inhibition; Ocimum Basilicum; Correlation; Quadratic Models; Passivation
1. Introduction
Corrosion has been defined from the individual perspec-
tives of several authors. However, most authors insist
that the definition of corrosion should be restricted to
metals. More often than not, though, corrosion engineers
must consider both metals and nonmetals for solution of
a given problem. Accordingly, polymers (plastics, rub-
bers, etc.), ceramics (concrete, brick, etc.) or composites
(mechanical mixtures of two or more materials with dif-
ferent properties) and other nonmetallic materials are
generally included as materials that can corrode [1].
Reference [2] defined corrosion as the environmen-
tally induced degradation of a material that involves a
chemical reaction. Degradation implies deterioration of
physical properties of the material. This can be a weak-
ening of the material due to a loss of cross-sectional area;
it can be the shattering of a metal due to hydrogen em-
brittlement; or it can be the cracking of a polymer due to
sunlight exposure.
The deleterious effects of corrosion are well-known
and include among others: poor outward appearance of
material surfaces, high maintenance and operating costs,
frequent plant shutdowns, contamination of end products,
loss of valuable products, hazardous effects on safety and
reliability and burdensome product liabilities. As a result
of these, huge financial losses have always been recorded
annually as resulting from corrosion damage. As an in-
stance, in 1998 alone, the United States reported an esti-
mate of the cost of corrosion to be around $276 billion: a
figure that is however realistically put at $30 billion [1].
In fact, [3] has gone further to project that this figure
would reach $993 billion by March 2013, with a still
M. O. NWANKWO ET AL.
further increase to $1 trillion by June of the same year.
It is against this backdrop of financial losses that cor-
rosion engineers over the years have had to rely on the
concept of materials selection and economics to mitigate
corrosion. However, even with the proper selection of
base metals and well-designed systems or structures,
there is no absolute way to eliminate all corrosion.
Therefore, corrosion protection methods are used to ad-
ditionally mitigate and control the effects of corrosion.
Corrosion protection can be in a number of different
forms or strategies with perhaps multiple methods ap-
plied in severe environments [4]. The various forms of
corrosion protection include among others the use of in-
hibitors, surface treatments, coatings and sealants, ca-
thodic protection and anodic protection.
In recent times, however, the hazardous consequences
of the somewhat traditional or conventional methods of
corrosion control, has made it imperative to source for
cost-effective and environmentally-friendly corrosion
control measures to eliminate or at least reduce these
effects. In this respect, the use of natural plants as corro-
sion inhibitors has expectedly become the current fron-
tiers of most research activities in corrosion engineering.
Inhibitors are chemicals that react with the surface of a
material decreasing the material’s corrosion rate, or that
interact with the operating environment to reduce its
corrosivity [4]. They can be added into the corrosion
medium as solutions or dispersions to form a protective
film, or as additives in coating products, or further still
into waters used for washing vehicle, system or compo-
nent. When added, they interact with the metal, thus
slowing the corrosion process by shifting the corrosion
potential of the metal’s surface toward either the cathodic
or anodic end; preventing permeation of ions into the me-
tal; or increasing the electrical resistance of the surface [4].
Africa, and particularly Nigeria, with her favourable
tropical climatic conditions is home to a vast number of
natural plants that are continuously been investigated as
profitable alternatives to synthetic inhibitors because of
their inherent advantages amongst which are their ready
availability, biodegradability, non-toxicity, non-pollutan-
cy and eco-friendliness [5,6]. It is on the strength of these
that ocimum basilicum was chosen for this work.
Ocimum basilicum is itself a vegetable plant believed
to be of Indian origin [7-9]. It belongs to a popular plant
species called basil. There are several varieties of basil in
existence, some of which have been used in previous
works [10-13]. However, basilicum species have not
been investigated previously in relation to mild steel to
the best of the authors’ knowledge.
Reference [14] used statistical tools (particularly re-
gression analysis) as a novel approach in corrosion stud-
ies. Since then, several other works have been done to
develop models for predicting the corrosion behavior of
engineering materials using specific parameters [15-18].
The findings from these works show that corrosion pro-
files correlated better in the quadratic models than the
logarithmic models at room temperatures.
This work therefore is an attempt to reinforce the find-
ings of [14] with regards to the regression behavior of
corrosion rates at room temperatures; and then unifying
the resultant regression equations into a single model
equation that would satisfy the conditions as a frame-
work for futuristic corrosion predictions particularly dur-
ing design considerations.
2. Experimental Techniques
2.1. Materials/Equipment
The materials and equipment used for the work include
10mm diameter mild steel rods sourced from a local steel
stockiest in Enugu, Nigeria, beakers, digital weighing
balance, tetraoxosulphate (VI) acid, leaves of ocimum
basilicum, acetone, nylon strings, emery cloth, distilled
water, hacksaw, vernier caliper, measuring cylinder, and
volumetric flask.
2.2. Materials Preparation
The mild steel rods were cut to sizes, each averaging
94.5 cm2 in surface area. They were thoroughly brushed
with emery cloth to reveal the metal surface. Thereafter,
they were washed with distilled water and rinsed with
The tetraoxosulphate (VI) acid was prepared to 0.5 M
and 1.0 M concentrations using standard procedures.
The ocimum basilicum leaves were washed with cold
tap water, dried under room temperature after which they
were subjected to soxhlet extraction process in ethanol
for about 80 hours to obtain the extract.
2.3. Experimentation
The mild steel coupons were tied with nylon strings and
then suspended in beakers containing the acid and the
acidified extracts. Each beaker contained 5 coupons and
the entire set up were allowed to stand for 30 days. After
6 days a coupon was withdrawn from each beaker, rinsed
in distilled water and swabbed in acetone. Thereafter,
they were weighed for weight loss determination and
corrosion rate calculation using the formula:
87.6 w
The pH value of the ocimum basilicum extract was eva-
luated and noted.
2.4. Unification of the Model Equations
Using MathCAD 7.0, the model equations were plotted
Open Access JMMCE
M. O. NWANKWO ET AL.
Open Access JMMCE
and the best line of fits was taken. The equation corre-
sponding to this line was then noted with all of its neces-
sary parameters.
3. Results
Tables 1-6 show the corrosion penetration rate values
obtained from weight loss measurements; while Figures
1 and 2 are the quadratic model fits from regression
analysis of the corrosion penetration rates. Table 8 is the
quadratic model equations obtained from the regression
4. Discussion
4.1. Corrosion Trends
Looking at Tables 1-6, the corrosion rates obtained de-
pict those of passivating metals: beginning with an initial
steep rise, peaking at a maximum and then subsequently
decreasing as exposure time increased. On interaction
with the corrosion medium, the metal surface normally
reacts swiftly with it, forming an oxide film that coats the
entire surface acting as a barrier, thereby preventing fur-
ther reactions [19].
Table 1. Corrosion penetration rates of mild steel sample in 0.5 m H2SO4.
Exposure time (hrs) Initial weight (g) Final weight (g) Weight loss (g) Corrosion rate (mm/yr)
144 26.76 20.46 6.30 0.5178
288 26.34 19.59 6.75 0.2774
432 26.87 19.77 7.10 0.1945
576 26.51 19.00 7.51 0.1543
720 27.01 15.93 11.11 0.1826
Table 2. Corrosion penetration rates of mild steel sample in 0.5 m H2SO4 with 25 cm3 of ocimum basilicum.
Exposure time (hrs) Initial weight (g) Final weight (g) Weight loss (g) Corrosion rate (mm/yr)
144 25.14 19.04 6.10 0.5013
288 28.82 22.29 6.53 0.2683
432 28.36 21.80 6.56 0.1767
576 27.61 20.67 6.94 0.1422
720 26.90 17.80 9.10 0.1596
Table 3. Corrosion penetration rates of mild steel sample in 0.5 m H2SO4 with 50 cm3 of ocimum basilicum.
Exposure time (hrs) Initial weight (g) Final weight (g) Weight loss (g) Corrosion rate (mm/yr)
144 28.02 22.46 5.56 0.4560
288 27.24 21.05 6.19 0.2544
432 27.93 21.63 6.30 0.1745
576 27.41 20.36 7.08 0.1444
720 27.38 17.38 10.00 0.1454
Table 4. Corrosion penetration rates of mild steel sample in 1.0 m H2SO4.
Exposure time (hrs) Initial weight (g) Final weight (g) Weight loss (g) Corrosion rate (mm/yr)
144 25.70 14.84 10.86 0.8926
288 25.64 12.69 12.95 0.5322
432 25.50 12.35 13.15 0.3603
576 24.47 9.82 14.65 0.2900
720 26.63 10.63 16.00 0.2630
Table 5. Corrosion penetration rates of mild steel sample in 1.0 m H2SO4 with 25 cm3 of ocimum basilicum.
Exposure time (hrs) Initial weight (g) Final weight (g) Weight loss (g) Corrosion rate (mm/yr)
144 26.58 17.03 9.55 0.7849
288 26.40 16.36 10.04 0.4126
432 26.81 15.53 11.28 0.3090
576 24.80 12.69 12.11 0.2488
720 25.10 6.08 17.02 0.2305
M. O. NWANKWO ET AL.
Table 6. Corrosion penetration rates of mild steel sample in 1.0 m H2SO4 with 50 cm3 of ocimum basilicum.
Exposure time (hrs) Initial weight (g) Final weight (g) Weight loss (g) Corrosion rate (mm/yr)
144 27.71 18.42 9.29 0.7635
288 27.22 16.85 10.37 0.4261
432 26.86 15.29 11.57 0.3170
576 25.19 12.97 12.22 0.2318
720 24.73 10.63 14.10 0.2511
0.7930.002 x
1.998 106
0.7710.002 x
1.922 106
0.6820.002 x
1.567 106
1.3020.003 x
2.647 106
1.1550.003 x
2.588 106
1.1320.003 x
2.540 106
Figure 1. Graphical plots of the quadratic model equations using mathcad 7.0.
Figure 2. Line of best fits from the model equations.
Open Access JMMCE
M. O. NWANKWO ET AL.
Open Access JMMCE
4.2. Model Summary
From Table 7 , it can be seen that R (coefficient of corre-
lation) values ranged from 0.986 to 0.996 showing that R
≥ 1, which is a near-perfect correlation; while the R2 (co-
efficient of determination) values ranged from 0.972 to
0.992, implying that approximately between 97% to 99%
of the entire variation in passivation rate is dependent on
exposure time, leaving only a maximum of 3% to extra-
neous sources such as errors of measurements, experi-
mental procedures and test locations.
Also, the standard error of estimation fell between
0.021 - 0.054 which is significantly less than 0.1. The
implication of this very narrow error margin is that the
use of quadratic models to characterize corrosion rates at
room temperature is justifiable and therefore could be
employed. That being the case, a classical departure from
the long-held assumptions that corrosion rate behaviours
are only logarithmic at room temperatures has been es-
These claims are corroborated by the line patterns of
Figure 1 where the lines of best fits agree with the near
perfectness of the correlation data obtained.
4.3. Model Equations
From Table 8, the model fit equations show that each
had a non-negligible quadratic element that must be ac-
counted for during corrosion rate evaluation. Since time
and media are critical considerations in corrosion moni-
toring, for a given medium therefore the exposure time
becomes the overriding factor for corrosion progression,
hence our time-dependent quadratic models subsist in
this present study.
Furthermore, looking at the equations in the order in
which they appeared, both the constant term and the co-
efficient of t2 decreased as the volume of the extracts
increased in each of the acid molarities. This confirms
further that inhibition has taken place and that an in-
crease in the inhibitor concentration caused a decrease in
corrosion penetration rate.
4.4. Unified Model Equation
From Figure 2 we see the best line of fits from the plots
of all the model equations represented in Table 8 and as
plotted in Figure 1. The model parameters in Table 9
clearly show that the coefficient of determination, R2 of
0.935 is very high indicating in similar manner that about
94% of the determining factors are dependent on expo-
sure time of the corrosion process, whilst 6% is ac-
count-ed for by extraneous factors which in this case may
include errors of mathematical measurements and eva-
luations. Based on the foregoing, the unified model equa-
1.0320.0021.899 10cprt t
is adjudged a suitable and correct corrosion predictor for
the present study.
Table 7. Model equations of the various quadratic fits.
Corrosion medium Model equations
0.5 M H2SO4 only 62
0.7930.0021.998 10tt
0.5 M H2SO4 + 25 cm3 ocimum basilicum 62
0.5 M H2SO4 + 50 cm3 ocimum basilicum 62
0.6820.0021.567 10tt
1.0M H2SO4 only 62
1.3020.0032.647 10tt
1.0 M H2SO4 + 25 cm3 ocimum basilicum 62
1.1550.0032.588 10tt
1.0 M H2SO4 + 50 cm3 ocimum basilicum 62
1.1320.0032.540 10tt
Table 8. Model parameters.
Corrosion Medium R R2 Adjusted R2 Standard Error of Estimation
0.5M H2SO4 0.994 0.988 0.976 0.023
0.5 M H2SO4 + 25 cm3 Ocimum basilicum 0.995 0.990 0.980 0.021
0.5 M H2SO4 + 50 cm3 Ocimum basilicum 0.994 0.988 0.975 0.021
1.0M H2SO4 0.996 0.992 0.984 0.033
1.0 M H2SO4 + 25 cm3 Ocimum basilicum 0.986 0.972 0.944 0.054
1.0 M H2SO4 + 50 cm3 Ocimum basilicum 0.993 0.986 0.972 0.037
M. O. NWANKWO ET AL.
Table 9. Model summary and parameter estimates of the unified equation.
Dependent Variable: Y
Model Summary Parameter Estimates
Equation R Square F df1 df2 Sig. Constant b1 b2
Quadratic 0.935 171.551 2 24 0.000 1.032 −0.002 1.889E-6
The independent variable is T.
Regression Equation: 62
1.0320.0021.899 10
5. Conclusion
The conclusion that can be drawn from the foregoing
discussions is that ocimum basilicum is a good corrosion
inhibitor since its pH value of 6.7 falls within the region
in which passivation occurs in the Poubaix diagram [20].
Again the quadratic model which fits each with a nearly
perfect correlation suggest in strong terms that room tem-
perature corrosion progression can no longer be said to
be only logarithmic but also has a significant quadratic part
that must be accounted for during corrosion characteriza-
tions. Additionally, the unification of the model equa-
tions into a single generalized form also shows that it is
henceforth possible and accurately so, to use the equation
to make futuristic computations of corrosion penetration
rates for engineering mild steel in acidic environments
with ocimum basilicum serving as a veritable inhibitor.
6. Acknowledgements
The authors wish to acknowledge the Department of In-
dustrial Chemistry of Ebonyi State University for grant-
ing them the permission to use their facilities for the
work. Dr. S. O. Maliki of the Department of Industrial
Mathematics, Ebonyi State University is also acknowl-
edged for his immense contributions.
[1] M. G. Fontana, “Corrosion Engineering,” Tata McGraw-
Hill Publishing Company Ltd., New Delhi, 2005.
[2] D. J. Duquette and R. E. Schafrik, “Research Opportuni-
ties in Corrosion Science and Engineering,” National
Academy of Sciences, Washington DC, 2011.
[3] G2MTLabs, “Cost of Corrosion in 2013 in the United
States Exceeds $1 Trillion,” 2011.
[4] B. D. Craig, R. A. Lane and D. H. Rose, “Corrosion Pre-
vention and Control: A Program Management Guide for
Selecting Materials,” Alion Science and Technology,
New York, 2006.
[5] A. Boxer and P. Back, “The Herb Book,” Octopus Books
Ltd., London, 1980.
[6] J. A. Duke, “Culinary Herbs: A Potpourri,” Bouch Maga-
zine Ltd., New York, 1985.
[7] N. Kumpawat, A. Chaturvedi and R. K. Upadhyay,
“Comparative Study of Corrosion Inhibition Efficiency of
Naturally Occurring Ecofriendly Varieties of Holy Basil
(Tulsi) for Tin in HNO3 Solution,” Open Journal of Met-
als, Vol. 2, No. 3, 2012, pp. 68-73.
[8] W. C. Muenscher and M. A. Rice, “Garden Spice and
Wild Pot-Herbs,” Cornell University Press, New York,
[9] T. Stobart, “Herbs, Spices and Flavorings,” The Over-
book Press, New York, 1982.
[10] H. Ashassi-Sorkhabi, B. Shabani, B. Aligholipour and D.
Seifzadeh, “The Effect of Some Schiff Bases on the Cor-
rosion of Aluminum in Hydrochloric Acid Solution,” Ap-
plied Surface Sciences, Vol. 252, No. 12, 2006, pp. 4039-
4047. http://dx.doi.org/10.1016/j.apsusc.2005.02.148
[11] O. K. Abiola, N. C. Okafor, E. E. Ebenso and N. M.
Nwinuka, “Ecofriendly Corrosion Inhibitors: The Inhibi-
tive Actions of Delonix Regia Extract for Corrosion of
Aluminum in Acidic Media,” Anticorrosion Methods and
Materials, Vol. 54, No. 4, 2007, pp. 219-224.
[12] N. Kumpawat, A. Chaturvedi and R. K. Upadhyay, “Com-
parative Study of Corrosion Inhibition Efficiency of Stem
and Leaves Extract of Ocimum Sanctum (Holy Basil) for
Mild Steel in HCl Solution,” Protection of Metals and
Physical Chemistry of Surfaces, Vol. 46, No. 2, 2010, pp.
[13] J. A. Soule, “Father Kino’s Herbs: Growing and Using
Them,” Tierra del sol Institute Press, Tucson, 2011.
[14] C. E. Ekuma and N. E. Idenyi, “Statistical Analysis of the
Influence of Environment on Prediction of Corrosion
from Its Parameters,” Research Journal of Physics, Vol. 1,
No. 1, 2007, pp. 27-34.
[15] C. E. Ekuma, N. E. Idenyi and I. O. Owate, “Application
of Statistical Technique to the Analysis of Passivation of
Al-Zn Alloy Systems in Brine,” Journal of Chemical En-
gineering and Materials Science, Vol. 1 No. 1, 2010, pp.
[16] C. I. Nwoye, N. E. Idenyi and J. U. Odo, “Predictability
of Corrosion Rates of Aluminum-Manganese Alloys Bas-
ed on Initial Weights and Exposure Time in Atmosphere,”
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Vol. 3, No. 1, 2012, pp. 8-14.
[17] M. O. Nwankwo, P. A. Nwobasi, S. I. Neife and N. E.
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Idenyi, “Statistical Studies of the Inhibition Characteris-
tics of Acidified Ocimum Basilicum on Engineering Mild
Steel,” Journal of Metallurgical Engineering, Vol. 2, No.
4, 2013, in press.
[18] C. Nwoye, S. Neife, E. Ameh, A. Nwobasi and N. Idenyi,
“Predictability of Al-Mn Alloy Exposure Time Based on
Its As-Cast Weight and Corrosion Rate in Sea Water En-
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terization and Engineering, 2013, in press.
[19] W. D. Callister, “Materials Science and Engineering: An
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CSE 353– Homework II
1 Theory
I: Logistic regression and na¨ıve Bayes 12 points
Suppose in a binary classification problem, the input variable x is n-dimensional and the ouput is
a binary class label y ∈ Y = {0, 1}. In this situation, there is an interesting connection between
two learners: logistic regression and na¨ıve Bayes classifier.
(a) Write down the expressions for the class conditional probability for each class, i.e., P(y = 1|x)
and P(y = 0|x), for logistic regression. (2 points)
(b) Using Bayes’ rule, derive the posterior probabilities for each class, i.e., P(y = 1|x) and P(y =
0|x), for na¨ıve Bayes. (2 points)
(c) Assuming a Gaussian likelihood function in each of the n dimensions, write down the full
likelihood function f(x|d) for na¨ıve Bayes. (2 points)
(d) Assuming a uniform prior on the two classes and using the results from parts (b) and (c)
above, derive a full expression for P(y = 1|x) for na¨ıve Bayes. (3 points)
(e) Show that with appropriate manipulation and parameterization, P(y = 1|x) in na¨ıve Bayes
from part (d) is equivalent to P(y = 1|x) for logistic regression in part (a). (3 points)
II: Failure of Cross-Validation 4 points
Cross-validation works well in practice, but there are some pathological cases where it might fail.
Suppose that the label is chosen at random according to P[y = 1] = P[y = 0] = 1/2. Consider a
learning algorithm that outputs the constant predictor h(x) = 1 if the number of 1s in the training
set labels is odd, and h(x) = 0 otherwise. Prove that the difference between the leave-one-out
estimate and the true error in such a case is always 1/2.
III: Decision Tree 4 points
Show that any binary classifier h : [0, 1]d → {0, 1} can be implemented as a decision tree of height
at most d + 1, with internal nodes of the type “Is xi = 0?” for some i ∈ {1, 2, . . . , d}.
2 Experiment
I: Decision Tree 40 points
In this section, you will be working with a somewhat morbid dataset. This is a dataset of passengers
on the Titanic. Each row in this dataset has 12 columns, where the second column, “Survived”, is
what we would like to estimate using a machine learning algorithm.
CSE 353 Homework II Due by Apr 24, 2019
Some of the features are obvious from their names (i.e., column headers), the others are:
• pclass – ticket class (1st, 2nd, or 3rd).
• sibsp – the number of siblings/spouses aboard the Titanic.
• parch – the number of parents/children aboard the Titanic (some children traveled with a
nanny, so parch = 0 for them).
• ticket and cabin are just the ticket number and the cabin number, respectively.
• age – age in years (fractional age indicates that age was less than 1 year).
• embarked – three different ports of embarkation were (S)outhampton, (Q)ueenstown, and
Your task is write your own code in Java or Python to implement the decision tree algorithm we
learnt in class (ID3). Instead of training and testing on the same data, however, you must split
the data in a 60-40 ratio to train on the 60% sub-dataset and test on the remaining 40%. You
must also calculate the accuracy of prediction on the training data and the test data, and plot the
accuracy numbers as the tree depth increases. This should lead to a plot that looks like the one in
the slide on overfitting when we covered “Decision Trees” in class.
Note that in this code, you will have to
• handle continuous variables, and
• avoid overfitting
Your submission for this task should be your code for learning and testing the decision tree. This
code should be able to perform the training and testing tasks separately, and the user should be
simply able to train on the 60% sub-dataset by doing something like
$ python id3.py –dataset /path/to/data/filename.csv –train
and test on the remaining 40% by
$ python id3.py –dataset /path/to/data/filename.csv –test
You may assume that the number 60 and 40 are fixed, and the user will not change them around.
II: Final Report 20 points
Along with your code, you are also required to submit a short report (no more than 1 page)1
. The
report must contain the following:
• The accuracy plot of the decision tree model (the tree depth and accuracy on the x- and
y-axes, respectively. The plot must be clear enough the reader to distinguish between, say,
0.85 and 0.9. Otherwise, include a table with the accuracy numbers in addition to the plot.
• A brief discussion (at most a half-page) about what you observed regarding (i) the depth at
which the tree started overfitting, (ii) the most discriminatory features, and (iii) the least
discriminatory features.
• Also include a brief discussion about whether some features that you expected to be completely
useless (e.g., ticket or cabin number) surprised you by exhibiting predictive power (or vice
versa . . . i.e., some features that you thought would be very predictive actually turned out to
be useless).
1For the report, please stick to single line spacing.
CSE 353 Homework II Due by Apr 24, 2019
• A README section explaining how to run your code. Keep in mind that the grader will
only run the code, and not do any manual work (e.g., placing the dataset in the same folder,
etc.). So, make sure that your code runs “as is” once the path to the entire dataset2 has been
The report will be graded based on (a) performance details, (b) replicability of experiments, (c)
explanation of how to run your code, and (d) the quality of the discussions about each algorithm.
What programming languages are allowed?
1. Java (JDK 1.8 or above, but don’t use Java 11), Python (3.x).
2. You are NOT allowed to use any data science and/or machine learning library for this assignment. If you have doubts about whether the use of a library is acceptable, please ask before
assuming that it can be used!
3. Calculations for entropy, information gain, etc. must be your own code.
4. You are NOT allowed to use any libraries that provide decision tree data structures off-theshelf.
5. For the accuracy plot, you may use any library you want as long as the use of that library
(e.g., pandas) is strictly restricted to the generation of the plot as an image file, and has
nothing to with the rest of your code.
What should we submit, and how?
Submit a single .zip archive containing (i) one folder simply called “code”, containing all
your code, (ii) a PDF document for your report, and (iii) a PDF document for the theory
part of your submission. Please do NOT handwrite the solutions and scan or take pictures.
For the PDF documents, use either LATEXor MS Word to write the solutions, and export to
PDF. Anything else will not be graded.
2Remember that just like the previous assignment, the grader will also not perform the 60-40 split in any way;
this should be done by your code.
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Period Over Period Calculation Challenges
I am trying to calculate the difference between periods for a specific metric and am running into some trouble. (As you can see from the screenshot below, the Beast Mode calculation I am using
doesn't seem to be working.)
To generate the column totals, I am using this beast mode successfully for each desired date comparison: (example for current below)
CASE when `Period Type`='Current' then `Shipped CoGS`end
However, when trying to calculate the difference between these periods, the formula is validated but doesn't work:
((CASE when `Period Type`='Current' then `Shipped CoGS`end) - (CASE when `Period Type`= 'Last Month' then `Shipped CoGS` end)) / (CASE when `Period Type`='Last Month' then `Shipped CoGS`end)
What am I missing?
• @RocketMichael Is your dataset organized at day grain like your table? Or are you doing the aggregation in your card? If it's in your card, you're going to need to throw SUM around all your case
when statements:
(SUM(CASE when `Period Type`='Current' then `Shipped CoGS`end) - SUM(CASE when `Period Type`= 'Last Month' then `Shipped CoGS` end)) / SUM(CASE when `Period Type`='Last Month' then `Shipped CoGS
**Was this post helpful? Click Agree or Like below**
**Did this solve your problem? Accept it as a solution!**
• (
(CASE when `Period Type`='Current' then `Shipped CoGS`end) -
(CASE when `Period Type`= 'Last Month' then `Shipped CoGS` end)
) / (CASE when `Period Type`='Last Month' then `Shipped CoGS`end)
you are calculating your beast mode at the row level. in other words, take one row in isolation and then apply your beast mode to it.
you Period Type for one row cannot be both 'Current' AND 'Last Month' at the same time. therefore it is IMPOSSIBLE for one row to return a result.
you have to apply your beast mode to an aggregate (which is exactly what @RobSomers ) is demonstrating.
SUM(CASE when `Period Type`='Current' then `Shipped CoGS`end) -
SUM(CASE when `Period Type`= 'Last Month' then `Shipped CoGS` end)
) / SUM(CASE when `Period Type`='Last Month' then `Shipped CoGS`end)
"across all rows take the sum of shipped cogs when period type is current."
"across all rows take the sum of shipped cogs when period type is last month" etc.
Jae Wilson
Check out my 🎥
Domo Training YouTube Channel
**Say "Thanks" by clicking the ❤️ in the post that helped you.
**Please mark the post that solves your problem by clicking on "Accept as Solution"
• That did it @RobSomers ! Thank you for your help on this. 😀
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Types of Splines
9.3. Types of Splines
9.3.1. Bezier Curves
The Bezier Curve was pioneered by P.Bezier for computer modelling of surfaces for the design of automobiles. In fact Renault have used his UNISURF system (which uses the Bezier Curve method) on
several of the vehicles.
The key the Beziers method is the use of blending functions. These affect the behaviour of the curve from four control points. The four blending functions represent the 'influence' each control point
has on the curve.
There are several drawbacks in using this method. The blending functions affect all points along the curve. In other words, it does not have localised control over the curve. Also, the number of
control points affected the degree of the curve. The higher the degree, the higher the lack of control over the curve there was.
9.3.1.1. Designing Bezier Surfaces
It is simple to use a Bezier curve to get a Bezier surface. The Bezier surface is just a mesh of Bezier curves anyway. Obviously there are lots of similarities between them.
9.3.1.2. Characteristics of Bezier surfaces
There are two characteristics of Bezier surfaces that are important to remember while designing them.
The surface passes through the four corner control points of the surface control net.
The surface tangent in both the u and v direction at each corner control point passes through each of the adjacent edge control points. In other words, the tangent is defined by the corner control
point and the adjacent edge control point
Fig. 9.1 : Bezier surface characteristics
(see Fig. 9.1)
These two are considered most important when surfaces are to be joined together. To get a smooth seemless join, the shared control point and the two internal control points either side in each patch
of adjoining surfaces should be colinear
Fig. 9.2 : Joining Bezier surfaces
(see Fig. 9.2)
9.3.1.3. Applications of Bezier surfaces
These are some example object types and how to build them out of Bezier surfaces.
9.3.1.3.1. Volumes of revolution
One thing that Bezier surfaces are good for is modeling volumes of revolution. Sketch the profile of the object and draw in the control points for a series of Bezier curves that would match it. These
become the edges of a series of Bezier surfaces that will be used to model a quarter of the revolution.
Imagine these edges are in the x-z plane and the profile is to be revolved 90 degrees about the z axis. Lets just take one of the control points. Its x and z coordinates are just the distances from
the respective axis, and the y coordinate is 0. The control point on the opposite side of the resultant surface will be in the y-z plane. Its z coordinate is the same, and the x and y coordinates are
Now we need to work out the coordinates of the two internal control points of this row of the surface. The tangents at the axis need to be perpendicular so that all the surfaces of the revolution
join seemlessly. So it turns out that the internal control points are just shifted away from the axis. For a circular sweep, the distance is roughly 0.55 times the distance of the edge control points
from the z axis.
Fig. 9.3 : Bezier approx. of quarter circle
The coordinates for the other three quarters of the revolution are easily calculated by just mirroring and swapping about the ones just worked out.
Tubes are constructed in two symmetrical halves; each a surface. The edge control points are on opposite sides on the surface of the tube. The internal control points are just shifted out again so
the tangents at the top and bottom are perpendicular. For a circular cross section the distance away from the centre line is about 0.63 times the height of the tube.
Fig. 9.4 : Bezier approximation of cylinder
Example 1 - Lens
This is a simple example of how two Bezier surfaces can be used to model a convex lens. Below is some data used to model the lens. The first block of numbers is the coordinates of the control points,
and the second is the control point numbers for the two surfaces. The line numbers are just for easy reference.
1: -1 0 0
2: -1 -0.55 0
3: -0.55 -1 0
4: 0 -1 0
5: -1 0.55 0
6: -0.55 0 0.5
7: 0 -0.55 0.5
8: 0.55 -1 0
9: -0.55 1 0
10: 0 0.55 0.5
11: 0.55 0 0.5
12: 1 -0.55 0
13: 0 1 0
14: 0.55 1 0
15: 1 0.55 0
16: 1 0 0
17: -0.55 0 -0.5
18: 0 -0.55 -0.5
19: 0 0.55 -0.5
20: 0.55 0 -0.5
Fig. 9.1 : Lens - Control net from above
Figure 9.1 shows the lens control net from above. The x axis goes off to the right, and the y axis goes up the page. Notice that we only needed four extra control points for the second/bottom surface
because all the edge ones can be shared with the first/top surface.
Fig. 9.2 : Lens - Surfaces from side
Figure 9.2 shows the two surfaces from the side.
Example 2 - Teapot
This is an example of a more complex object modeled with Bezier surfaces; a teapot. Because it is constructed from 306 control points and 32 surfaces the data has not been included here. The body and
lid are just volumes of revolution. The spout and handle are bent tubes.
Fig. 9.1 : Teapot - Control nets
Figure 9.1 shows the control nets for the Bezier surfaces. If you look carefully at the back corner, you can see a bug in the geometry definition. One of the control point numbers for that surface
was typed in twice.
Fig. 9.2 : Teapot - Surfaces
Figure 9.2 shows the surfaces themselves.
9.3.2. Cubic Splines
The natural starting point for a study of spline functions is the cubic spline because it is similar to the draftman's spline. It is a continuous cubic polynomial that interpolates the control points
( joints ). The polynomial coefficients for cubic splines are dependent on all n control points, their calculation involves inverting an (n+1) by (n+1) matrix. This has two disadvantages: (i) moving
any one control point affects the entire curve, and (ii) the computation time needed to invert the matrix can interfere with rapid interactive reshaping of a curve.
9.3.3. B-Splines
B-Spline (which means Basis Spline) was the earliest spline method. It overcame the problems encountered by the Bezier Curve , by providing a set of blending functions that only had effect on a few
control points. This gave the local control that was lacking.
Also, the problem of piecing curves together was avoided by allowing only those curves that possessed the required continuity at the joints. Most other spline techniques provided this at the loss of
local control.
The overall formulation is much like that of the Bezier, but the key difference is in the formulation of the blending functions. The important feature of the B-Spline blending functions is that they
are non-zero in only a small portion of the range of the particular parameter.
B-Spline share many of the advantages of Bezier Curves, but the main advantage is the local control of the curve shape. B- Splines also reduce the need to piece many curves together to define the
final shape. Control points can be added at will without increasing the degree of the curve, thereby retaining the control over the curve that wouls be lost with a Bezier curve.
There are three types of B-splines: uniform nonrational B- splines, nonuniform nonrational B-splines and nonuniform rational B-splines. (Foley and et.al., 1987)
The term uniform means that the joints (knots) are spaced at equal intervals of the parameter t.
The term rational is used where x(t), y(t) and z(t) are each defined as the ration of two cubic polynomials.
9.3.4. Uniform Nonrational B-Splines
B-splines consist of curve segments whose polynomial coefficients depends on just a few control points. This is called local control. In other words, moving a control point affects only in the region
near the control point of a curve. In addition, the time needed to compute the coefficients is greatly reduced. B-splines have the same continuity as cubic splines, but do not interpolate their
control points.
9.3.5. Nonuniform Nonrational B-Splines
Nonuniform nonrational B-splines permit unequal spacing between the knots. These curves have several advantages over uniform B-splines.
First, continuity at selected join points can be reduced from second derivative (C2) to first derivative (C1) to C0 to none. If the continuity is reduced to C0, then the curve interpolates a control
point, but without the undesirable effect of uniform B-splines, where the curve segments on either side of the interpolated control point are straight lines.
Also, starting and ending points can be easily interpolated exactly, without at the same time introducing lineat segments.
It is possible to add an additional knot and control point to nonuniform B-splines, so the resulting curve can be easily reshaped, whereas this cannot be done with uniform B-splines.
9.3.6. Nonuniform Rational B-Splines
NURBS are Non-Uniform Rational B-Splines and is the term given to curves that are defined on a knot vector where the interior knot spans are not equal. As an example, we may have interior knots with
spans of zero. Some common curves reqiure this type of non-uniform knot spacing. The use of this option allows better shape control and the ability to model a larger class of shapes.
Nonuniform rational B-splines (NURBS) are useful for two reasons. The first and most important reason is that they are invariant under perspective transformations of the control points.
A second advantage of rational splines is that they can define precisely any of the conic sections. A conic can be only approximated with nonrationals, by using many control points close to the
9.3.7. Beta-Splines
Beta-splines is a tool for designing parametrically defined curves because they are tailored to geometric continuity rather than to parametric continuity. Geometric continuity is an intrinsic measure
of continuity appropiate for spline development. It has been shown to be a relaxed form of parametric continuity independent of the parameterizations of the curve segments under consideration, but
still sufficient for geometric smoothness of the resulting curve. However, geometric continuity is appropriate only for applications where the particular parameterization used is unimportant, since
parameteric discontinuities are allowed. (Goodman and Unsworth, 1986)
Uniformly shaped Beta-splines are considered in which the bias and tension parameters are fixed throughout the length of the curves. Therefore, the extra control gained is not of a local nature,
since a change in either parameter affected the entire curve.
9.3.8. V-Splines
V-spline curves, similar in mathematical structure to Beta- splines, are developed as a more computationally efficient alternative to splines in tension.
Although the splines in tension can be modified to allow tension to be applied at each control point, the procedure is computationally expensive. The v-spline curve uses more computationally
tractable piecewise cubic curves segments, resulting in curves that are just smoothly joined as those of a standard cubic spline. (Nielson, 1986)
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Instrumental variables on causal graphs
Last time we talked about viewing d-separation as a tool for model selection. But we’re pretty limited in the causal models we can distinguish between by only observing our variables of interest—any
two graphs with the same set of d-separations are indistinguishable. Instrumental variables are a common tool for trying to get around the limitations of purely observational data.
Instrumental variables
Instrumental variables (IV) are variables that we’re not intrinsically interested in but that we look at in an attempt to suss out causality. The instrument must be correlated with our cause, but its
only impact on the effect should be via the cause.
The classic example is about—you guessed it—smoking. Because running an RCT on smoking is ethically verboten, we’re limited to observational data. How can we determine if smoking causes lung cancer
from observational data alone? An instrumental variable! To reiterate, we want a factor that affects smoking prevalence but (almost certainly) does not affect lung cancer in other ways. Finding an
instrument that satisfies the IV criteria generally seems to require substantial creativity. Can you think of an instrument for the causal effect of smoking on lung cancer?
An instrument that meets these criteria is a tax on cigarettes. We expect smoking to decrease as taxes increase, but it seems hard to imagine a cigarette tax otherwise having an effect on lung
Instrumental variables on causal graphs
Okay, so that’s what IVs are at a high level. But what are they concretely in the graphical causal model setting we’ve been developing?
A brief notational interlude
We’ll get this out of the way here:
• \(\perp\!\!\!\perp\) is the symbol for d-separation
• Once we add the strikethrough, \(\not\!\!{\perp\!\!\!\perp}\) mean d-connected.
• If \(G\) is a graph, \(G_{\overline{X}}\), is \(G\) in which all the edges pointing to vertex X have been removed^1.
We’ll start with the definition and then try to build up a feel for it. An instrumental variable X for the causal effect of Y on Z in graph G must be:
1. d-connected to our cause Y—\((X \not\!\!{\perp\!\!\!\perp} Y)_G\)
2. d-separated from our effect Z after severing the cause Y from all its parents—\((X \perp\!\!\!\perp Z)_{G_\overline{Y}}\)
Below is a widget for finding instrumental variables. You can specify your graph (same format as before) in the top text area and make a query about a particular causal relationship in the input
fields below the text area. The analysis will update when you defocus the inputs or text area.
Hopefully, you can get an intuition for what IVs mean graphically by generating lots of examples for yourself.
What are the instruments for the causal effect of on ?
Unfortunately, I think this may get a bit confusing. Our overall plan is:
1. Enumerate the models compatible with d-connection between the possible cause Y and effect Z
2. Assume that we’re right about cause and effect and add a corresponding instrumental variable
3. Calculate d-separations for the models from step 1 with the additional variable from step 2
4. Find that the d-separations now cleanly separate the model in which Y is a cause of Z from others
Compatible models
How does X help us determine whether Y and Z are causally linked? We can analyze things by cases. The actual path^2 being modeled must:
1. Be unidirectional from Y to Z (Y → A → … → B → Z) which means Y indeed causes Z,
2. Be unidirectional from Z to Y (Y ← A ← … ← B ← Z) which means Z actually causes Y, or
3. Have a fork between Y and Z (Y ← … ← A → … → Z) which means Y and Z are both caused by some unknown factor.
4. Not have a collider between Y and Z (Y → … → A ← … ← Z). If it did, Y and Z would be d-separated and it would have been immediately obvious from the data that there’s no causal relationship.
The instrumental variable
There’s only an IV for the causal effect of Y on Z if Y indeed causes Z so we’ll figure out how to add the IV^3 to the graph by looking at path 1. Our instrument X must be a parent of Y (X → Y → A →
… → B → Z). If it were a child of Y (X ← Y → A → … → B → Z), it would satisfy IV condition 1 (d-connection to the potential cause), but it wouldn’t satisfy IV condition 2 because it would still be
d-connected to Z even after Y removed all 0 of the edges from its parents.
(I suspect the above paragraph reads as very dense. The takeaway is that we want an instrumental variable on path 1 and there’s only one way to add a single vertex and edge that satisfies the two IV
conditions. That way is for the IV to be a parent of the cause.)
Once we make this same modification—add a variable X which is a parent/cause of Y—to the other paths, we can determine whether X is truly an instrumental variable. In other words, it’s important that
our instrumental variable separates case 1—where Y genuinely has a causal effect on Z—from the other two cases—where it doesn’t. Here’s what happens in each case:
1. X → Y → A → … → B → Z: Our instrument X is d-connected to the effect Z.
2. X → Y ← A ← … ← B ← Z: Our instrument X is d-separated from Z by the collider at Y.
3. X → Y ← … ← A → … → Z: Our instrument X is d-separated from Z by the collider at Y.
Model selection
Hurray! Our instrumental variable has done just what we wanted—used observation alone to suss out causality. If the random variable X is d-connected to the potential effect (which can be determined
just from the data), the potential cause is actually a cause. If the potential instrument is d-separated from the potential effect (which can be determined just from the data), it turns out that it’s
not actually an instrumental variable because the potential cause isn’t actually a cause.
As model selection
Last time, we talked about d-separation as a tool for model selection. We can also think of instrumental variables in this way. Instrumental variables are just another tool in the toolbox that allow
us to improve our powers of discrimination—allow us to distinguish between models that are indistinguishable when looking only at observations on variables of intrinsic interest.
Below, enter specifications for two causal graphs (The two graphs should contain the same set of vertices—only the edges should differ.). The resulting analysis will show you all the instruments that
would allow you to distinguish between the two models with observation alone. Each row contains a different instrumental variable. The left column shows the extra variable as it would look on the
graph specified in the left-hand text area while the right column shows the IV on the right text area’s graph. In each row, you should see that the columns have different sets of d-separations.
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Cite as
Václav Blažej, Dušan Knop, Jan Pokorný, and Šimon Schierreich. Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra. In 49th International Symposium on Mathematical
Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
Copy BibTex To Clipboard
author = {Bla\v{z}ej, V\'{a}clav and Knop, Du\v{s}an and Pokorn\'{y}, Jan and Schierreich, \v{S}imon},
title = {{Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra}},
booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
pages = {29:1--29:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-335-5},
ISSN = {1868-8969},
year = {2024},
volume = {306},
editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.29},
URN = {urn:nbn:de:0030-drops-205857},
doi = {10.4230/LIPIcs.MFCS.2024.29},
annote = {Keywords: Equitable Connected Partition, structural parameters, fixed-parameter tractability, N-fold integer programming, tree-width, shrub-depth, modular-width}
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Comparing different capital budgeting techniques
When done well, capital budgeting can help propel your business to new heights of success and profitability.
However, when done poorly, it can lead to unsophisticated investment decisions and a negative impact on shareholder value.
To ensure the capital budgeting process is a success, choosing the right technique is key—but how can you know which technique is right for you?
You came to the right place. Let's dive deeper into capital budgeting methods so you can make a smart decision for your business.
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Grade 2 PDF Download
Author: Thinking Kids
Publisher: Carson-Dellosa Publishing
ISBN: 1483813185
Category : Juvenile Nonfiction
Languages : en
Pages : 260
Book Description
Singapore Math creates a deep understanding of each key math concept, is a direct complement to the current textbooks used in Singapore,includes an introduction explaining the Singapore Math method,
and includes step-by-step solutions in the answer key. Singapore Math, for students in grades 2 to 5, provides math practice while developing analytical and problem-solving skills. Learning
objectives are provided to identify what students should know after completing each unit, and assessments are included to ensure that learners obtain a thorough understanding of mathematical
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How to Use the Multinomial Distribution in R? » Data Science Tutorials
Multinomial Distribution in R, when each result has a given probability of occurring, the multinomial distribution describes the likelihood of obtaining a specific number of counts for k different
A statistical experiment with n repeated trials is known as a multinomial experiment. There are a finite number of possible outcomes in each trial. The likelihood of a particular outcome occurring on
any given trial remains constant.
If a random variable X has a multinomial distribution, the probability that outcome 1 will occur exactly x1 times, outcome 2 will occur exactly x2 times, and so on can be calculated using the
Probability = n! * (p1x1 * p2x2 * … * pkxk) / (x1! * x2! … * xk!)
n: total number of events
x1: the number of times the first outcome happens
p1: In a given trial, what are the probability that outcome 1 will occur?
In R, we may use the dmultinom() function to calculate a multinomial probability, which has the following syntax.
dmultinom(x=c(1, 6, 8), prob=c(0.4, 0.5, 0.1))
x: The frequency of each outcome is represented by a vector.
prob: The probability of each outcome is represented by a vector (the sum must be 1)
The examples below demonstrate how to utilize this function in practice.
Example 1:
Candidate A receives 20% of the vote, Candidate B receives 30% of the vote, and Candidate C earns 50% of the vote in a three-way election for mayor.
What is the probability that 5 voters voted for candidate A, 5 for candidate B, and 3 for candidate C in a random sample of ten voters?
To address this question, we can use the R code below:
make a multinomial probability calculation
dmultinom(x=c(5, 5, 3), prob=c(0.2, 0.3, 0.5))
[1] 0.007005398
The probability that exactly 5 people voted for A, 5 voted for B, and 3 voted for C is 0.007.
Example 2
Assume there are six yellow balls, two red balls, and two pink balls in an urn.
What is the probability that all four balls in the urn are yellow if we pick four balls at random from the urn and replace them?
To address this question, we can use the R code below:
Multinomial Probability Calculator
dmultinom(x=c(4, 0, 0), prob=c(0.6, 0.2, 0.2))
[1] 0.1296
The chance of all four balls being yellow is 0.1296.
Example 3
Assume two boys are playing cards against one another. Student A has a 0.6 chance of winning a game, Student B has a 0.3 chance of winning a game, and they have a 0.1 chance of tying in a game.
What is the probability that player A will win 5 times, player B will win 4 times, and they will tie one time if they play ten games?
To address this question, we can use the R code below:
make a multinomial probability calculation
dmultinom(x=c(5, 4, 1), prob=c(0.6, 0.3, 0.1))
[1] 0.07936186
About 0.08 percent of the time, player A wins 5 times, player B wins 4 times, and they tie 1 time.
Can’t rename columns that don’t exist – Data Science Tutorial
Example 4
A series of matches are played by three card players. Player A has a 20% chance of winning any game, player B has a 30% chance of winning, and player C has a 50% chance of winning.
What is the probability that player A will win one game, player B will win two games, and player C will win three games if they play six games?
n = 12 people (6 games total)
n1=1 (Player A wins)
n2 =2 (Player B wins)
n3 = 3 (Player C wins)
P1 = 0.20 (probability that Player A wins)
P2 = 0.30 (probability that Player B wins)
P3 = 0.50 (probability that Player C wins)
dmultinom(x=c(1, 2, 3), prob=c(0.20, 0.30, 0.50))
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Static Rigid BodiesQuizwiz - Ace Your Homework & Exams, Now With ChatGPT AI
Static Rigid Bodies
Réussis tes devoirs et examens dès maintenant avec
A non-uniform plank AB of mass 12kg and length 4 metres is hanging horizontally from two strings C and D. The distances AC and BC are 1 metres and 1.4 metres respectively. A particle of mass 5 kg is
placed on the plank at E, 2.4 m away from A. The tensions in the ropes have the same magnitude. What is the first thing that you would write for a question like this?
The reaction at C + the reaction at D = 12g + 5g
Explain how you would do the following question: A uniform plank AB of mass 20kg and length 6m is resting horizontally on two supports at A and C. The distance CB is 1.4 metres. A child of mass 25kg
is standing on the plank between C and B, x metres away from B. Find the minimum distance x so the plank will not tilt about C.
Do the question so that the reaction at A = 0, working out the moment about C and then minusing the distance found there from 1.4
If you have a rod and there is a point b which is just in the air, does it have a reaction?
Explain how you would solve the following question: 3: A uniform ladder of mass 20 kg and length 8 m rests against a smooth vertical wall with its lower end on rough horizontal ground. The
coefficient of friction between the ground and the ladder is 0.3. The ladder is inclined at an angle θ to the horizontal, where tan θ = 2. A boy of mass 30 kg climbs up the ladder. By modelling the
ladder as a uniform rod, the boy as a particle and the wall as smooth and vertical, (a) find how far up the ladder the boy can climb before the ladder slips.
P = Fr and R = 50g. Then do the moment around A, making P the subject then subbing in 0.3 x 50g, to work out the distance
A ladder AB, of weight W and length 2l, has one end A resting on rough horizontal ground. The other end B rests against a rough vertical wall. The coefficient of friction between the ladder and the
wall is 1/3. The coefficient of friction between the ladder and the ground is μ. Friction is limiting at both A and B. The ladder is at an angle θ to the ground, where tan θ = 5/3. The ladder is
modelled as a uniform rod which lies in a vertical plane perpendicular to the wall. Find the value of μ. Start the question (3) and write out the equation for the moment around A
R + Fr against wall = W and R + 1/3P = W and P = Fr of floor. Wcos feta x l = Psin feta x 2l + 1/3P x 2lcos feta
There is a ladder AB of mass 25kg and length 4m, resting in equilibrium with one end A on rough horizontal ground and the other end B against a smooth vertical wall. The ladder is in a vertical plan
perpendicular to the wall. The coefficient of friction between the ladder and the ground is 11/25. The ladder makes an angle of alpha with the ground. When Reece, who has made 75kg stands at the
point C on the ladder, where AC = 2.8m, the ladder is on the point of slipping. The ladder is modelled as a uniform rod and Reece is modelled as a particle. You answer a load of questions on this,
then you are asked to state how you have used the modelling assumption that Reece is a particle, what would you put?
This means that Reece's weight acts at a single point at C
Ensembles d'études connexes
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Packing directed circuits quarter-integrally
The celebrated Erdős-Pósa theorem states that every undirected graph that does not admit a family of k vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in
the graph) of size O(k k). After being known for long as Younger's conjecture, a similar statement for directed graphs has been proven in 1996 by Reed, Robertson, Seymour, and Thomas. However, in
their proof, the dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing is not elementary. We show that if we compare the size of a minimum feedback vertex set
in a directed graph with quarter-integral cycle packing number, we obtain a polynomial bound. More precisely, we show that if in a directed graph G there is no family of k cycles such that every
vertex of G is in at most four of the cycles, then there exists a feedback vertex set in G of size O(k^4). On the way there we prove a more general result about quarter-integral packing of subgraphs
of high directed treewidth: for every pair of positive integers a and b, if a directed graph G has directed treewidth Ω(a^6 b^8^2(ab)), then one can find in G a family of a subgraphs, each of
directed treewidth at least b, such that every vertex of G is in at most four subgraphs.
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If p and q are two propositions, then ∼(p↔q) is... | Filo
If and are two propositions, then is
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Question Text If and are two propositions, then is
Updated On Oct 14, 2022
Topic Mathematical Reasoning
Subject Mathematics
Class Class 11
Answer Type Text solution:1 Video solution: 1
Upvotes 196
Avg. Video Duration 5 min
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Investigating Relationships between the Crystal Structure and <sup>31</sup>P Isotropic Chemical Shifts in Calcined Aluminophosphates
Solid-state NMR spectra have historically been assigned using simple relationships between NMR parameters, e.g., the isotropic chemical shift, and aspects of the local structure of the material in
question, e.g., bond angles or lengths. Density functional theory (DFT) calculations have effectively superseded these relationships in many cases, owing to the accuracy of the NMR parameters
typically able to be calculated. However, the computational time required for DFT calculations may still be prohibitive, particularly for very large systems, where structure-spectrum relationships
must still be used to interpret the NMR spectra. Here we show that, for calcined aluminophosphates (AlPOs), structure-spectrum relationships relying on either the mean P-O-Al angle or the mean P-O
distance, both suggested in previous literature, provide a poor prediction of the ^31P isotropic shielding, σ[iso], calculated by DFT. However, a relationship dependent on both parameters yields
predicted σ[iso] in excellent agreement with DFT, with a mean error of ~1.6 ppm. The predictive ability of the relationship is not improved by introducing further parameters (many used in previous
work) describing the local structure, suggesting that the two-parameter relationship is close to an optimum balance between accuracy and overparameterisation. The ability to predict accurately the
outcome of DFT-level calculations will be of particular interest in cases where the actual calculations would be impractical or even impossible with current computational hardware, or where many such
calculations are required quickly.
• Zeolites
• Solid-state NMR
• Density functional theory
• Local structure
• Spectral prediction
• Empirical relationships
Dive into the research topics of 'Investigating Relationships between the Crystal Structure and ^31P Isotropic Chemical Shifts in Calcined Aluminophosphates'. Together they form a unique fingerprint.
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Capsule Calculator
• Enter the capsule's Diameter and Length.
• Choose the units (cm or in) for Diameter and Length.
• Click "Calculate" to see the capsule attributes.
• Click "Clear Results" to reset the form.
• Click "Copy Results" to copy the attributes to the clipboard.
The Capsule Calculator is a specialized tool designed to calculate various attributes of a capsule-shaped object. A capsule, known as a spherocylinder, is a three-dimensional geometric shape
consisting of a cylinder with hemispherical ends. This shape is commonly seen in pharmaceutical capsules and various manufacturing designs. The calculator focuses on computing the volume, surface
area, and length-to-diameter ratio of a capsule based on its dimensions.
Functional Overview of the Capsule Calculator
1. Input Requirements:
□ Diameter and Length: The user is required to input the diameter and length of the capsule. These dimensions can be provided in centimeters or inches, offering flexibility in measurement
2. Calculation Process:
□ Upon entering the required dimensions and selecting the measurement units, the user clicks on “Calculate” to obtain the results.
3. Output:
□ The calculator provides the volume, surface area of the hemispheres, total surface area, and the length-to-diameter ratio of the capsule.
4. Additional Features:
□ The tool includes options to clear results, copy results to the clipboard, and a section for calculation history.
Detailed Explanation of the Formulas and Concepts
1. Volume of a Capsule:
□ Formula: 4/3 * π * (radius^3)
□ Explanation: The volume of a capsule can be thought of as the sum of the volume of a cylinder and the volume of two hemispheres. The given formula in the calculator, however, seems to only
calculate the volume of a sphere. The correct formula for a capsule should be: Volume of Cylinder + 2 * Volume of Hemisphere = π * radius^2 * length + 2 * (2/3) * π * radius^3.
2. Surface Area (Hemisphere):
□ Formula: 2 * π * (radius^2)
□ Explanation: This formula calculates the surface area of a hemisphere. A full sphere’s surface area is 4 * π * radius^2, so a hemisphere, being half of a sphere, has half the surface area.
3. Total Surface Area of a Capsule:
□ Formula: 2 * Hemisphere Surface Area + 2 * π * radius * length
□ Explanation: The total surface area of a capsule includes the areas of both hemispheres and the side surface of the cylindrical part. This formula correctly sums the surface areas of these
4. Length-to-Diameter Ratio:
□ Formula: length / diameter
□ Explanation: This is a simple ratio that compares the length of the capsule to its diameter. It’s a useful metric in design and manufacturing to describe the shape and aspect ratio of the
Practical Applications and Benefits
1. Pharmaceutical Industry: Capsules are a common form in medication delivery. Understanding the volume and surface area is crucial for dosage and coating applications.
2. Manufacturing and Engineering: In various manufacturing processes, especially in aerospace and automotive industries, the capsule shape is used for its aerodynamic properties. Accurate
calculations of surface area and volume are essential for design and analysis.
3. Educational Purposes: For students learning geometry, this tool provides a practical application of geometric principles in calculating volume and surface area.
Interesting Facts and Additional Insights
1. History of the Capsule Shape: The capsule has been a popular shape in pharmaceuticals due to its ease of swallowing and efficient manufacture. Its design is a classic example of form meeting
2. Aerodynamics: The capsule shape is aerodynamically efficient, reducing drag in fluids – a principle utilized in bullet design and aerospace engineering.
3. Volume-to-Surface Area Ratio: This ratio in capsules is important in heat transfer applications, as it affects how quickly an object can heat up or cool down.
1. Geometry and Design: Understanding geometric principles is fundamental in design. “The Elements of Dynamic Symmetry” by Jay Hambidge discusses geometric principles that underlie design, including
shapes like the capsule.
2. Pharmaceutical Applications: “Pharmaceutical Dosage Forms: Tablets” by Herbert A. Lieberman et al. delves into the importance of shape and size in pharmaceutical dosage forms, including capsules.
3. Aerodynamics: “Fundamentals of Aerodynamics” by John D. Anderson provides an in-depth understanding of how shapes like the capsule affect movement through fluids, crucial in aerospace and
automotive engineering.
Limitations and Considerations
1. Accuracy of Input Data: The accuracy of the capsule’s calculated attributes is highly dependent on the precision of the input dimensions. Errors in measuring the diameter or length of the capsule
can lead to significant deviations in the calculated results.
2. Assumption of Perfect Geometric Shape: The calculator assumes a perfect capsule shape, which may not always be the case in real-world objects. Irregularities in shape can affect the actual volume
and surface area.
3. Unit Conversion Considerations: While the calculator allows for inputs in both centimeters and inches, users must be mindful of the unit conversions, especially when comparing or combining
results obtained in different units.
User Accessibility and Interface
The Capsule Calculator boasts a simple, user-friendly interface. The clear instructions and straightforward input fields make it accessible even to those with minimal technical or mathematical
background. The option to copy results to the clipboard is a convenient feature for users who need to document or further analyze the data.
Enhancements and Future Applications
1. 3D Visualization: Implementing a 3D model that changes based on the input dimensions could enhance understanding and provide a visual confirmation of the shape being analyzed.
2. Expanded Measurement Units: Including additional units like millimeters or feet could cater to a wider range of users, especially in scientific and engineering fields.
3. Integration with Design Software: Linking the calculator’s outputs to CAD (Computer-Aided Design) software could streamline the process of designing capsule-shaped objects.
The Capsule Calculator is a specialized and practical tool for determining the volume, surface area, and length-to-diameter ratio of a capsule-shaped object. While its primary utility lies in fields
such as pharmaceuticals, manufacturing, and education, it is also a valuable educational resource for understanding geometric principles.
The simplicity of its interface, coupled with the depth of its calculations, makes it an accessible and useful tool for a wide range of users. However, users should be aware of its limitations, such
as the assumption of perfect geometry and the need for accurate input measurements.
Last Updated : 03 October, 2024
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Sandeep Bhandari holds a Bachelor of Engineering in Computers from Thapar University (2006). He has 20 years of experience in the technology field. He has a keen interest in various technical fields,
including database systems, computer networks, and programming. You can read more about him on his bio page.
17 thoughts on “Capsule Calculator”
1. Ipatel
I believe the tool could be further improved by incorporating visual representations and animations to complement the detailed formulas and explanations, making it more accessible and engaging
for a wider audience.
2. Adam86
The limitations and considerations section is a commendable addition, highlighting the tool’s transparency and thorough approach to addressing potential inaccuracies and practical constraints.
3. Young Gordon
It’s essential to address the error in the provided volume formula for a capsule. The correct formula is crucial in ensuring precise calculations and reliable results.
4. Ray Kelly
This tool seems incredibly helpful for professionals and students alike. The detailed explanations and practical applications make it a valuable resource for understanding capsule geometry.
5. Grace73
The feature allowing users to copy results to the clipboard is a thoughtful and practical addition, ensuring that the calculated data can be seamlessly integrated into other applications and
6. Thomas Hunter
The explanations of the formulas and concepts are clear and concise, catering to a wide audience from beginners to experts in the field. The educational value is significant.
7. Ellis Ruby
The emphasis on practical applications and the historical significance of the capsule shape effectively enrich the understanding of its relevance beyond theoretical geometry. Enlightening
8. Mchapman
Absolutely. The historical and real-world context elevates the tool’s value, especially for those seeking a holistic understanding of geometric shapes.
9. Xwalker
I agree. There’s an opportunity to incorporate visual aids that appeal to a broader spectrum of users and enhance the overall user experience.
10. Joel Gray
Absolutely! It’s refreshing to see a sophisticated tool cater to different audiences with the same level of detail and accuracy.
11. Isabelle Hunter
This tool offers a fascinating intersection of mathematics, design, and practical applications. It’s a testament to the versatility and relevance of geometric principles.
12. Roxanne Ross
This calculator serves as an example of the practical applications of mathematics in diverse industries, shedding light on the multi-faceted relevance of geometric calculations.
13. Imogen Cox
The integration of references and further readings enhances the tool’s credibility and provides additional resources for those interested in delving deeper into the subject matter.
14. Natalie99
Absolutely, Roxanne. It’s a compelling showcase of how mathematical concepts extend beyond theoretical exercises to real-world scenarios.
15. Phillips Adrian
It’s great to see a focus on precision and accuracy, particularly in fields where these attributes are critical. The tool and the information provided can be of great assistance in such
16. Lily26
The intersection of mathematics and practical utility is profoundly evident in this tool. It provides a comprehensive perspective on the scope of mathematical applications.
17. Ross Theo
I concur. It’s imperative that the correct formula is used to maintain the tool’s credibility and utility, particularly in educational contexts.
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Department of Mathematics and Statistics
Majoring in Math
Frequently Asked Questions
• What can I do with a degree in math?
□ Many students major in mathematics because they want to become math teachers, either at a high school level or at the college or university level. SJSU math alumni also work in industries
ranging from aviation safety to risk management to financial planning to satellite design. In fact, one of the three co-founders of Oracle, Edward Oates, graduated from SJSU with a BA Math
degree. The MAA website, https://mathcareers.maa.org/ is a good place to find out about careers in math.
• How do I change my major to math?
□ If you are interested in changing your major to math, please read and follow the instructions that can be found in the CoS webpage. Applications are only reviewed twice a year, after the
grades have been submitted and after the deadlines which are posted on the CoS website. The change of major request is subject to department approval and may require approval from the College
of Science.
To request to apply to a College of Science Major, visit the College of Science Change of Major Applications page. If you have questions, please email science-academicprograms@sjsu.edu.
• What do I need to request a change of major to math or to be reinstated if I have been disqualified?
□ The change of major policy in the Department of Mathematics and Statistics applies to all students who want to change their major or transfer to one of our degree programs. In particular, the
change of major policy applies to all former students returning including those who are trying to get reinstated. Details of the policy are:
1. Students must have an overall GPA of. 2.00 or higher.
2. A 2.25 math GPA is required to apply. The math GPA is calculated using grades in all math courses starting with Calculus 1. The math GPA is not the same as the major GPA. Some courses are
excluded from the Math GPA. Examples of courses not included in the math gpa are College Algebra, Precalculus, Elementary Statistics (Stat 95), Math 1, Math 12, Math 101-107.
3. Students are required to get a C- or better in Math 30, Math 31, Math 32, Math 39 and Math 42.
4. Students are required to submit a personal roadmap to graduation in their proposed major. We want to see a semester by semester timeline of courses to completion of the proposed degree. A
list of major and university requirements can be found on the online catalog. The roadmap must be consistent with course prerequisites. Some math courses are not offered every semester. The
schedule of math offerings can be found in this document: Expected Future Course Offerings
5. Students can not have any outstanding D-F-WU-NC grades in math courses.
6. We want students to succeed and that is highly unlikely for students with several 8 D-F-WU-NC’s to graduate in reasonable time. Applications from students with 8 or more D-F-WU-NC’s in
math courses will be denied.
7. Students who are applying for reinstatement need to submit a personal statement explaining why they got disqualified and what measures they are planning to take to make sure that they will
be successful in the future.
8. Students can apply for a change to a math major at most 2 times.
9. Requests from students who have 120 or more attempted units will be denied.
10. Requests that will delay graduation will be denied.
11. These are minimum requirements and do not guarantee admission into our programs. The change of major request is subject to department approval and may require approval from the College of
• What is the hardest part of the major? How do I make it less hard?
□ That depends on whom you ask. What is difficult to you may not be difficult to others. It is generally agreed that Math 108, Math 128A/B, Math 131 A/B are more abstract, more rigorous and
more challenging than some other required courses. You should not plan on taking more than one of those in a regular semester. Each of Math 128 and Math 131 can take up as much time as two
regular classes.
Talking to your advisor is the best way to work out your difficulties and avoid difficulty situations (such as taking too many difficult classes). Working with other students and developing
friends and community is another way to making the process more enjoyable.
• What does "discrete" and "continuous" math mean?
□ Roughly speaking, discrete mathematics deals with the mathematics of countable things (a countable set is one whose elements can be counted, such as the positive integers). So number theory,
algebra, logic, etc. are closely related to discrete math. Continuous math is about studying things which cannot be counted this way, like the real numbers between 0 and 1. Calculus,
differential equations, probability, etc. are more related to continuous math. These mathematics feel different but frequently help each other.
• What does "pure" and "applied" math mean?
□ Pure mathematics generally refers to the study of mathematics for its own sake without any regard to its applications. Even though it is generally associated with rigor and abstraction, this
aspect makes it similar to art. For trained eyes, mathematics can be beautiful and exquisite, just like a painting. Examples of areas which are usually considered pure mathematics are
abstract algebra, topology, and number theory. But each of these areas have been applied to real world problems. Lie Algebras are used in physics, knot theory is being applied to protein
folding, and number theory is used in cryptography. So beautiful mathematics can be useful as well.
Applied mathematics is directly motivated by real world problems, but we cannot escape rigor and abstraction. There are still theorems to be proven, algorithms to be developed and evaluated,
errors to be estimated. Examples of areas commonly considered applied mathematics are differential equations, numerical analysis, operation research, statistics, actuarial science.
Differential equations is sometimes referred to as the mathematical language of science and engineering because many laws and principles of science can be expressed as a differential
equation. For example, in calculus you learn that Newton's second law, F = ma, as applied to a free falling object, can be written as a differential equation, x"(t)=-g. Applied mathematics is
not as simple as "plugging it in". In high school, you learned how to solve a system of 2 linear equations with 2 unknowns. What if you have 1,000,000 equations and 1,000,000 unknowns? Can
you still use the same method? Assuming there is a solution to the problem, the answer is "Sure. Why not?". You can use a method called Gaussian elimination to find the answer to the system
of equations... in theory. We are talking about 1,000,001,000,000 coefficients. That's a LOT of numbers to remember. And it would require 1,000,000,999,996,500,002 arithmetic operations (How
long are you willing to work on this?) And if you use a computer, there are other issues like memory, efficiency, accuracy, and stability. The problem of solving a large or ill-behaved system
of equations comes up in so many applications that we offer an entire course on the subject, Math 143M.
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Help with times tables: fun ideas, videos and quizzes - Oxford Owl
Help with times tables
Learning times tables off by heart makes mental maths much easier. It will boost your child’s confidence in their maths lessons at school, but it’s also a skill they’ll use all the time in the world
outside school.
Here, we’ve pulled together key information about how times tables are taught at primary school along with our pick of activities to help make learning times tables fun for your child.
Why is it important for my child to know the times tables?
When children know their times tables, mental arithmetic becomes easier. Practising times tables also helps children to understand number and number relationships, and to see patterns in numbers.
These skills will help them to master key concepts and move quickly through more complex maths problems with confidence.
A thorough knowledge of multiplication and division facts will help children succeed in their tests at the end of primary school and set them up for success at secondary school. As they grow older,
knowing the times tables will help them with everyday activities like shopping, budgeting and cooking.
When does my child need to know their times tables?
In England, children will be expected to know the following in each year at primary school:
□ Year 1: count in multiples of 2, 5 and 10.
□ Year 2: be able to remember and use multiplication and division facts for the 2, 5 and 10 multiplication tables, including recognising odd and even numbers.
□ Year 3: be able to remember and use multiplication and division facts for the 3, 4 and 8 multiplication tables, including recognising odd and even numbers.
□ Year 4: be able to remember and use multiplication and division facts for the multiplication tables up to 12 x 12.
□ Year 5: revision of all multiplication and division facts for the multiplication tables up to 12 x 12.
□ Year 6: revision of all multiplication and division facts for the multiplication tables up to 12 x 12.
How are times tables taught at school?
Download our free booklet Times Tables in School to learn how children are first taught to use their fingers, counters, and paper to help them find the right number before moving on to reciting times
tables. The booklet includes lots of tips and games to support learning at home, too.
What is the Year 4 multiplication tables check?
The new Year 4 multiplication tables check becomes statutory in 2020. Your child will need to take a short online test to make sure their times tables knowledge is at the expected level. You can find
more information about the check here: Year 4 multiplication tables check
How can I help my child learn their times tables at home?
We’ve pulled together some tips and tricks to help you make learning times tables at home fun.
Video: How to practise times tables
Education expert and parent Isabel Thomas offers her advice on making times tables practise fun with flashcards, post-its, and competitions.
Times tables tips
Our times tables top tips will provide some useful advice and great ideas to help you support your child in learning their times tables.
Andrew Jeffrey talks us through his favourite times tables tricks and games to help your child become more confident with their times tables.
Learn how children are first taught their times tables, and find lots of tips and games to support learning at home.
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Welcome to the Homepage of K.V.S. Hari
Director, Centre for Brain Research, Indian Institute of Science on leave from ECE Department.
Research highlights related to Neuroscience and Healthcare.
Signal Processing, Machine Learning, Deep learning with applications to
1. 5G Wireless Systems
2. Radar Systems
3. Autonomous Navigation
4. MRI Signal Processing
5. Health Care
6. Neuroscience
5G Systems
1. Dual-Function Radar and Communication System (DFRC) Design in collaboration with University of Bordeaux, France
2. Cell-Free Massive MIMO Systems in collaboration with KTH-Royal Institute of Technology, Stockholm
3. Millimetre-wave MIMO Transceiver design in collaboration with Weizmann Institute, Israel
4. Dictionary Learning Techniques for Channel Estimation in collaboration with University of Southampton
Magnetic Resonance Imaging (MRI) Systems
1. Reducing readout time in an MRI machine using novel k-space trajectory designs
2. Design of affordable and portable MRI machines
in collaboration with
1. TU Delft
Healthcare Systems
1. AI/ML techniques for predicting patient’s state in an Intensive Care Unit of a hospital in collaboration with CMC Vellore.
2. Deep Learning Techniques for estimation of Cardiac parameters using Echocardiograms in collaboration with Sri Jayadeva Institute of Cardiovascular Sciences (SJICS), Bangalore
Autonomous Vehicles
1. National dataset creation for Indian road conditions in collaboration with WIPRO
2. Deep Neural Networks for Collision avoidance in trucks operating in an open-cast mine in collaboration with VOLVO
1. Understanding reading in the human brain in collaboration with IISc colleagues
Jan-Apr 2023: MIMO Signal Processing
Class Timings: Tue, Thu 0830-1000 in ECE1.08
Scope: This course will cover concepts related to the use of multiple sensors to solve problems related to
Direction of Arrival Estimation: Sensor array geometry, beam-forming, subspace methods; MIMO Wireless Communication Systems: MIMO wireless channels, Capacity of MIMO Communication systems, MIMO OFDM
systems, Spatial/Index modulation concepts, Physical layer algorithms for parameter estimation with reference to 5G systems; MIMO Radar systems: Introduction to radar systems, MIMO radar concepts and
MIMO-OFDM radar systems, Dual Function Radar and Communication systems.
References include books and papers:
Indigenous Insulin Pump
Development of an indigenous Continuous Glucose Monitoring and Control system in collaboration with
1. IISc Colleagues
2. Ramaiah Hospital, Bangalore
Assistive Technologies for the Elderly
Building a chair for the elderly in collaboration with IISc Colleagues
Drone detection Radar
Building a radar prototype for detecting drones
Guest Co-Editor of IEEE Vehicular Technology Magazine Special Issue, September 2021
Guest Co-Editor of IEEE Vehicular Technology Magazine Special Issue, June 2020
Sadhana – Academy Proceedings in Engineering Sciences
• Editor-in-Chief (Electrical Sciences)
Join SPS – SPS Student Membership only $1
Join SPS Member for only $11 (50% off in India)
• Vice President – Membership (1 Jan 2020 – 31 Dec 2022)
• Regional Director-at-Large (1 Jan 2018 – 31 Dec 2019)
Member of IEEE Signal Processing Society’s Signal Processing Theory and Methods Technical Committee (SPTM-TC) (2014 – 2019).
• Associate Member of IEEE Signal Processing Society’s Sensor Array and Multichannel Technical Committee (SAM-TC) (2010 – Present).
• Vice-Chair, IEEE Bangalore Section (2012, 2013)
• Chair, IEEE Signal Processing Society Bangalore Chapter in 2005, 2011, 2012, 2013. Part of the Founding team of the Bangalore Chapter in 2001.
Active member of all activities of the Bangalore chapter in organizing International conferences (SPCOM series), SP-Connect, an activity for connecting students and technologists.
• Reviewer of the papers submitted to IEEE journals (TSP, SPL, JSTSP) and IEEE conferences.
• FELLOW, IEEE
• Chair, IEEE Ad-hoc committee on India Strategy (2017)
• Chair, IEEE Bangalore Section, 2016
• Vice-Chair, IEEE Bangalore Section (2012, 2013)
• Chair, IEEE Signal Processing Society Bangalore Chapter in 2005, 2011, 2012, 2013. Part of the Founding team of the Bangalore Chapter in 2001. Active member of all activities of the Bangalore
chapter in organizing International conferences (SPCOM series), SP-Connect, an activity for connecting students and technologists.
• General Chair, 2013 IEEE International Conference on Electronics, Computing and Communication Technologies (IEEE CONECCT), 17-19 Jan 2013.
• General Chair, 2014 IEEE International Conference on Electronics, Computing and Communication Technologies (IEEE CONECCT), 6-7 Jan 2014.
• Member, Industry Forum and Exhibition Committee, International Conference on Communications (ICC 2013), 9-13 June 2013, Budapest, Hungary
• Member of IEEE 802.16 working group in the IEEE Standards group. Initiated the work for IEEE 802.16 (WiMAX) standard on wireless channel models, 1999-2001.
• Member of the IEEE University Programs Ad-hoc Committee (UPC), IEEE Educational Activities Board (IEEE-EAB), USA. The mission of the UPC is “To promote and enhance the content and delivery of
engineering education globally.” Nominee is the only member from India (2010 – present).
• Member of Technical Program Committees of several International conferences supported by IEEE like IEEE ICC, IEEE CONECCT, EUSIPCO, VTC, Pernets.
• Reviewer of the papers submitted to IEEE journals (TSP, SPL, JSTSP) and IEEE conferences.
SSP Lab
Ph D Students
1. Dr Farhan Mukadam, MD (joined in 2019 as part of the IISc-CMC MD-PhD programme)
2. Swati Bhattacharya (joined in 2019)
3. Maria Francis (joined in 2019)
4. G V V R Pavan Kumar (joined in 2019)
5. B Satwika (joined in 2018)
SSP Lab webpage: For details click here
Realizing the Power of MIMO Signal Processing
Where am I? – An experiment in Indoor Positioning
Signals Everywhere
In the News
Click for Citations based on Google Scholar
JOURNAL PAPERS; CONFERENCE PAPERS; PATENTS; BOOK CHAPTERS Citations based on Google Scholar: For details, click here Books which have cited the work on Array Processing and Wireless Channel Models
For details, click here JOURNAL PAPERS
1. Pavan Kumar Gadamsetty, K. V. S. Hari, Lajos Hanzo, “Learning a Common Dictionary for CSI Feedback in FDD Massive MU-MIMO-OFDM Systems,” accepted for publication in IEEE Open Journal on Vehicular
Technology, 2023.
2. Maria Francis, Farhad Mehran and K.V.S. Hari, “Analysis of Selective User Forwarding in Cell-Free Massive MIMO with Channel Aging,” IEEE Access, vol. 11, pp. 71210-71223, 2023. (First author is a
student, second author is a collaborator)
3. Satwika Bhogavalli, K.V.S. Hari, Eric Grivel, Vincent Corretja, “Estimating the target DOA, range and velocity using subspace methods in a MIMO OFDM DFRC system,” EURASIP Signal Processing, vol
209, 2023. (First author is a student, third and fourth authors are collaborators)
4. A Agrawal, S Nag, K.V.S. Hari, S.P. Arun, ”Letter processing in upright bigrams predicts reading fluency variations in children,” Journal of Experimental Psychology: General, vol.151, no.6, pp.
2237-2249, Sep 2022. (First author is a student, second and fourth authors are collaborators) https://doi.org/10.1037/xge0001175
5. Peter Joseph Basil Morris and K.V.S Hari, “Detection and Localisation of Unmanned Aircraft Systems (UAS) using Millimeter Wave (mmWave) Automotive RADAR Sensors,” IEEE Sensors Letters,
vol.5,No.6. June 2021.
6. Shubham Sharma, Mario Coutino, Sundeep Prabhakar Chepuri, Geert Leus, K V S Hari, “Towards a general framework for fast and feasible k-space trajectories for MRI based on projection methods, “
Magnetic Resonance Imaging, (72), pp. 122-134, Oct 2020. (First author is a student and co-authors are collaborators)
7. Aakash Agrawal, KVS Hari, SP Arun, “A compositional neural code in high-level visual cortex can explain jumbled word reading,” Elife, (9), pp. e54846, May 2020. (First author is a student and
third author is a collaborator)
8. A Agrawal, KVS Hari, SP Arun, “Reading Increases the Compositionality of Visual Word Representations,” Psychological Science, Nov 2019. (First author is a student and third author is a
9. Rakshith Rajashekar, M Di Renzo, Lie-Lang Yang, K.V.S. Hari, Lajos Hanzo, “A Finite Input Alphabet Perspective on the Rate-Energy Tradeoff in SWIPT Over Parallel Gaussian Channels,” IEEE Journal
on Selected Areas in Communications, vol. 37 (1), pp. 48-60, Jan 2019.(First author was a former student and co-authors are collaborators)
10. Rakshith Rajashekar, Lie-Lang Yang, K.V.S. Hari, Lajos Hanzo,” Transmit Antenna Subset Selection in Generalised Spatial Modulation Systems,” IEEE Transactions on Vehicular Technology, vol. 68,
pp. 1979-1983, Feb 2019. (First author is a former student and co-authors are collaborators)
11. Renu Jose and K.V.S. Hari, “Bounds and joint estimators for channel, phase noise, and timing error in communication systems using statistical framework,” Elsevier Computers and Electrical
Engineering, vol 72, pp. 431-442, Oct 2018. (First author is a former student)
12. Rakshith Rajashekar, Marco Di Renzo, K.V.S. Hari, Lajos Hanzo,” A Beamforming Aided Full Diversity Scheme for low-altitude air-to-ground communication systems operating with limited feedback”
IEEE Transactions on Communications, vol. 68, pp. 6602-6613, Dec 2018. (First author is a former student and co-authors are collaborators)
13. Rakshith Rajashekar, K.V.S. Hari, Lajos Hanzo,” Transmit Antenna Subset Selection for single and Multiuser Spatial Modulation Systems Operating in Frequency Selective Channels,” IEEE Transactions
on Vehicular Technology, vol.67(7), pp. 6156-6169, July 2018. (First author is a former student and third author is a collaborator)
14. Shubham Sharma and K.V.S. Hari, “Four-Shot non-Cartesian Trajectories in k-space Sampling in MRI,” CSI Transactions on ICT (Springer), vol 6, issue 1, pp. 11-16, 2018. (First author is a student)
15. Rakshith Rajashekar, K.V.S. Hari, Lajos Hanzo,” Transmit Antenna Subset Selection in Spatial Modulation Relying on a Realistic Error-Infested Feedback Channel,” IEEE Access, vol.6, pp.5879-5890,
October 2017.
16. Rakshith Rajashekar, Marco Di Renzo, K.V.S. Hari, Lajos Hanzo, “A Generalized Transmit and Receive Diversity Condition for Feedback-Assisted MIMO Systems: Theory and Applications in Full-Duplex
Spatial Modulation,” IEEE Transactions on Signal Processing, vol. 64, pp. 6505-6519, Dec 2017.(First author is a former student and co-authors ar collaborators)
17. Rakshith Rajashekar, Chao Xu, Naoki Ishikawa, S Shinya Sugiura, K.V.S. Hari, L. Hanzo, ” Algebraic Differential Spatial Modulation is Capable of Approaching the Performance of its Coherent
Counterpart”, IEEE Transactions on Communications. vol. 65(10), pp.4260-4273, Oct 2017.(First author is a former student and co-authors are collaborators)
18. Avik Santra and K.V.S. Hari, “A Novel Subspace Based Method for Compensation of Multiple CFOs in uplink MIMO OFDM Systems”, IEEE Communication Letters. vol. 21(9), pp. 1993–1996, Sep 2017. (First
author is a former student)
19. Renu Jose and K.V.S. Hari, “Joint Statistical Framework for the Estimation of Channel and SFO in OFDM Systems,” IET Signal Processing, Jun 2017. DOI: 10.1049/iet-spr.2016.0580. (First author is a
former student.)
20. A. R. Sachin, Sooraj K. Ambat and K.V.S. Hari, “Analysis of Intra-Pulse Frequency Modulated, Low Probability of Interception, Radar Signals,” Sadhana, vol. 42(7), pp. 1037-1050, July 2017.
(Co-authors were former students
21. Amit Dutta, K.V.S. Hari, Chandra Murthy, Neelesh Mehta, Lajos Hanzo, ” Minimum Error Probability MIMO-Aided Relaying: Multi-Hop, Parallel and Cognitive Designs”, IEEE Transactions on Vehicular
Technology, vol.66(6), pp.5435-5440, June 2017.(First author is a former student and other co-authors are collaborators)
22. Rakshith Rajashekar, Naoki Ishikawa, Shinya Sugiura, K.V.S. Hari and L. Hanzo, “Full-Diversity Dispersion Matrices from Algebraic Field Extensions for Differential Spatial Modulation,” IEEE
Transactions on Vehicular Technology, vol. 66(1), pp. 385-394, Jan 2017.(First author iss a former student and other co-authors are collaborators)
23. Rakshith Rajashekar, K. V. S. Hari, Lajos Hanzo, “Transmit antenna subset selection for single and multiuser spatial modulation systems operating in frequency selective channels, accepted and to
appear in IEEE Transactions on Vehicular Technology.(First author is a former student and third author is a collaborator)
24. K. G. Deepa, Sooraj K. Ambat and K. V. S. Hari,”Fusion of sparse reconstruction algorithms for multiple measurement vectors,” Sadhana, vol. 41(11), pp. 1275-1287, Nov. 2016. (First and second
authors were former students)
25. Pranav S. Koundinya, K. V. S. Hari, Lajos Hanzo, “Joint Design of the Spatial and of the Classic Symbol Alphabet Improves Single-RF Spatial Modulation”, IEEE Access vol. 4, pp. 10246-10257, Aug.
2016. (First author was a project staff member and third author is a collaborator)
26. Karthik Upadhya, Chandra Sekhar Seelamantula, K.V.S. Hari, “A Risk Minimization Framework for Channel Estimation in OFDM Systems”, accepted, EURASIP Signal Processing, vol. 128, pp. 78-87, Jan.
2016.(First author was a project staff member and second author is a collaborator).
27. Ping Yang, Yue Xiao, K.V.S. Hari, A. Chockalingam, Shinya Sugiura, Harald Haas, Marco Di Renzo, Zilong Liu, Lixia Xiao, Shaoqian Li, and Lajos Hanzo, “Single-Carrier Spatial Modulation for
Large-Scale Antenna Aided Systems” IEEE Communications Surveys and Tutorials, vol. 18(3), pp. 1687-1716, Mar. 2016. (Authors are collaborators).
28. Rakshith Rajashekar, K.V.S. Hari and Lajos Hanzo, “Quantifying the Transmit Diversity Order of Euclidean Distance Based Antenna Selection in Spatial Modulation”, IEEE Signal Processing Letters,
22 (9), pages 1434-1437, Sep 2015. (First author is a PhD student and third author is a collaborator).
29. Amit Dutta, K.V.S. Hari,and Lajos Hanzo, “Minimum-Error-Probability CFO Estimation for Muti-User MIMO OFDM Systems,” IEEE Transactions on Vehicular Technology, 64 (7), pages 2804-2818, July 2015
(First author is a PhD student and third author is a collaborator).
30. Sooraj K. Ambat and K.V.S. Hari, “An Iterative Framework for Sparse Signal Reconstruction Algorithms Signal Processing,” EURASIP Signal Processing (Elsevier), vol. 108, pp. 351-364, Mar 2015.
(First author is a PhD student).
31. Amit Dutta, K.V.S. Hari,and Lajos Hanzo, “Linear Transceiver Design for an Amplify-and-Forward Relay Based on the MBER Criterion,” IEEE Transactions on Communications, vol. 62 (11), pages
3765-3777, Nov 2014. (First author is a PhD student and third author is a collaborator).
32. Dinesh Dileep Gaurav and K.V.S. Hari, “An Eigen Approach to Transmit Beamforming in Wireless Networks with Single Antenna Receivers,” IEEE Transactions on Wireless Communications, vol. 13(11),
pages 6431-6443, Nov 2014. (First author is a PhD student).
33. Mohammad Ismat Kadir, Sheng Chen, K.V.S. Hari, K. Giridhar and Lajos Hanzo, “Iterative Soft Multiple-Symbol Differential Sphere Decoding Aided Multicarrier Differential Space-Time Shift Keying,”
IEEE Transactions on Vehicular Technology, vol. 63 (8), pages 4102-4108, Oct 2014. (Co-authors are collaborators).
34. Sooraj K. Ambat, Saikat Chatterjee, K.V.S. Hari, “A Committee Machine Approach for Compressed Sensing Signal Reconstruction,” IEEE Transactions on Signal Processing, vol. 62(7), pp.1705-1717, Apr
2014. (First author is a PhD student and second author is a collaborator).
35. Sooraj K. Ambat, Saikat Chatterjee, K.V.S. Hari, ” Progressive Fusion of Reconstruction Algorithms for Low Latency Applications in Compressed Sensing ” Signal Processing, vol.97, pp. 146-151, Apr
2014. (First author is a PhD student and second author is a collaborator).
36. Renu Jose, Sooraj K. Ambat and K.V.S. Hari, “Low Complexity Joint Estimation of Synchronization Impairments in Sparse Channel for MIMO-OFDM System,” AEU – International Journal of Electronics and
Communications, Elsevier Publications, vol. 68(2), pp. 151-157,Feb 2014. (First and second authors are PhD students).
37. Rakshith Rajashekar, K.V.S. Hari, L. Hanzo, “Reduced-Complexity ML Detection and Optimal Training in Spatial Modulation Systems,” IEEE Transactions on Communications, vol. 62(1), pp. 112-125, Jan
2014. (First author is a PhD student and third author is a collaborator).
38. Amit Datta, K.V.S. Hari, Lajos Hanzo “Channel Estimation Relying on the Minimum Bit Error Ratio Criterion for BPSK and QPSK Signals,” IET Communications, vol. 8, pp.69-76, Jan 2014. (First author
is a PhD student and third author is a collaborator).
39. Renu Jose, K.V.S. Hari, “A Bayesian Approach for Joint Estimation of Phase Noise and Channel in OFDM System,” IET Signal Processing, vol.8 (1), pp. 10-20, Feb 2014. (First author is a PhD
40. N. Mukund Sriram, B.S. Adiga, K.V.S. Hari, ” Grassmannian Fusion Frames and its use in Block Sparse Recovery-” Signal Processing, vol.94, pp. 498–502, Jan 2014. (First author is a former student
and second author is a collaborator).
41. Rakshith Jagannath and K.V.S. Hari, “Block Sparse Estimator for Grid Matching in Single Snapshot DOA Estimation,” IEEE Signal Processing Letters, vol. 20(11), pp.1038-1041, Nov 2013. (First
author was a project staff member).
42. Dinesh Dileep Gaurav and K.V.S. Hari,” A Fast Eigen Solution for Homogeneous Quadratic Minimization with at most Three Constraints,” IEEE Signal Processing Letters, vol. 20(10), pp.968-971, Oct
2013. (First author is a PhD student).
43. Rakshith Rajashekar, Hari, K.V.S., Lajos Hanzo, “A Reduced-Complexity Partial-Interference-Cancellation Group Decoder for STBCs” IEEE Signal Processing Letters, vol. 20(10), pp. 929-932, Oct
2013. (First author is a PhD student and third author is a collaborator).
44. K.V.S. Hari, John-Olof Nilsson, Isaac Skog, Peter Handel, Jouni Rantakokko, G.V. Prateek, “A Prototype of a First-Responder Indoor Localization System,” Special Issue on Cyber Physical Systems,
Journal of IISc, 93(3), pp. 511-520, Sep 2013. (Co-authors are collaborators).
45. Renu Jose, K.V.S. Hari, “Maximum Likelihood Algorithms for Joint Estimation of Synchronization Impairments and Channel in MIMO-OFDM System,” IET Communications, vol. 7(15), pp. 1567-1579, Aug
2013. (First author is a PhD student).
46. Rakshith Rajashekar, K.V.S. Hari, L. Hanzo, “Spatial Modulation aided Zero-Padded Single Carrier Transmission for Dispersive Channels,” IEEE Transactions on Communications, vol. 61(6), pp.
2318-2329, June 2013. (First author is a PhD student and third author is a collaborator).
47. Sooraj K. Ambat, Saikat Chatterjee, K.V.S. Hari, Fusion of Algorithms for Compressed Sensing,” IEEE Transactions on Signal Processing, vol.61 (14), pp. 3699-3704, 2013. (First author is a PhD
student and second author is a collaborator).
48. Rong zhang, Li Wang, Gerard Parr, Osianoh Glenn Aliu, Benga Awoseyila, Nader Azarmi, Saleem Bhatti, Eliane Bodanese, Hong Chen, Mehrdad Dianati, Amit Dutta, Michael Fitch, K. Giridhar, Steve
Hailes, K.V.S. Hari, Muhammad Ali Imran, Aditya K. Jagannatham, Abhay Karandikar, Santosh Kawade, Mohammed Zafar Ali Khan, Sayee C. Kompalli, Patrick Langdon, Babu Narayanan, Andreas Mauthe,
Joseph Mc Geehan, Neelesh Mehta, Klutto Millet, Klaus Moessner, Rakshith Rajashekar, Barathram Ramkumar, Vinay Ribeiro, Kasturi Vasudevan, and Lajos Hanzo, ” Advances in base- and mobile-station
aided cooperative wireless communications: An overview ” IEEE Vehicular Technology Magazine, vol.8(1), pp. 57-69, March 2013. (Co-authors are collaborators).
49. Rakshith Rajashekar, Hari, K.V.S., Lajos Hanzo, “Antenna Selection in Spatial Modulation Systems” IEEE Communications Letters vol. 17(3), pp. 521-524, Mar 2013. (First author is a PhD student and
third author is a collaborator).
50. Rakshith Rajashekar, Hari, K.V.S., Lajos Hanzo, “Structured Dispersion Matrices from Division Algebra Codes for Space-Time Shift Keying,” IEEE Signal Processing Letters vol. 20(4), pp. 371-374,
Apr 2013. (First author is a PhD student and third author is a collaborator).
51. Avinash Mohan and K.V.S. Hari, “Low Complexity Adaptation for SISO Channel Shortening Equalizers,” AEU – International Journal of Electronics and Communications, Elsevier Publications, 66(8), pp.
600–604, August 2012. (First author was a project staff member).
52. Satya Sudhakar Yedlapalli and K. V. S. Hari, ” The line Spectral Frequency Model of a Finite Length,” Special Issue on Model Order Selection in Signal Processing Systems, IEEE Journal of Selected
Topics in Signal Processing, vol. 4(3), pp.646-658, June 2010. (First author was a PhD student).
53. M R Bhavani Shankar and K V S Hari, ‘Systematic Construction of Linear Transform based Full Diversity, Rate One Space-Time-Frequency Codes,’ IEEE Transactions on Signal Processing, vol. 57, issue
6, pp. 2285-2298, June 2009. (First author was a PhD student).
54. M. R. Bhavani Shankar and K. V. S. Hari, ‘On the Variations in Mutual Information of MIMO Communication Systems Due to Perturbed Channel State Information at Transmitter’ IEEE Transactions in
Communications, vol. 54(9), pp. 1593–1603, Sep 2006. (First author was a PhD student).
55. A. Vijaya Krishna and K. V. S. Hari ‘Filterbank Precoding for FIR equalization in high rate MIMO communications’ IEEE Transactions on Signal Processing, vol. 54 (5), pp. 1645 – 1652, May 2006.
(First author was a PhD student).
56. M. R. Bhavani Shankar and K. V. S. Hari ‘Reduced Complexity Equalization schemes for Zero Padded OFDM Systems,’ IEEE Signal Processing Letters, Volume: 11(9), pp. 752- 755, Sep 2004. (First
author was a PhD student).
57. V. G. S. Prasad and K. V. S. Hari, ‘Interleaved Orthogonal Frequency Division Multiplexing (IOFDM) System,’ IEEE Transactions on Signal Processing, vol. 52(6), pp. 1711-1721, June 2004. (First
author was an ME student).
58. H. Bolcskei, A. J. Paulraj, K. V. S. Hari, R. U. Nabar, W. W. Lu, ‘Fixed broadband wireless access: state of the art, challenges, and future directions,’ IEEE Communications Magazine pp. 100
-108, Jan 2001.(Co-authors were collaborators).
59. G. Ganesan and K. V. S. Hari, `HOS based orthogonal subspace algorithms for causal ARMA system identification,’ EURASIP’s Signal Processing, vol. 80, Issue 3, pp. 535-542, March 2000. (First
author was a Master’s student).
60. K. V. S. Hari and B. V. Ramakrishnan, `Performance analysis of a modified spatial smoothing technique for direction estimation,’ EURASIP’s Signal Processing, vol. 79, Issue 1, pp. 73-85, November
1999. (First author was a Master’s student).
61. K. V. S. Hari and Bjorn Ottersten, `Parameter Estimation using a Sensor Array in a Ricean Fading channel,’ Sadhana, Proceedings of Indian Academy of Sciences in Engineering Sciences‘ vol. 23,
Part I, pp. 5-15, February 1998.
62. Sachin S. Deo and K. V. S. Hari,`Simple method to compute a function in fixed-point arithmetic,’ Electronics Letters, vol.33, no. 23, pp. 1631-1632, November 1997.(First author was a Master’s
63. L.Srinivas and K. V. S. Hari,`FIR System Identification based on Subspaces of a Higher-Order Cumulant Matrix,’ IEEE Transactions on Signal Processing, vol. 46, June 1996. (First author was a
Master’s student).
64. L.Srinivas and K. V. S. Hari,`FIR System Identification using Higher-Order Cumulants – A Generalized Approach,’ IEEE Transactions on Signal Processing, December 1995. (First author was a Master’s
65. K. V. S. Hari and Uma Gummadavelli, `Effect of Spatial Smoothing on the performance of Subspace methods in the presence of Array Model Errors,’ Special Issue on Statistical Signal Processing and
Control, IFAC Journal, Automatica, January 1994. (Second author was a Master’s student).
66. Bhaskar D. Rao and K. V. S. Hari, `Weighted Subspace Methods and Spatial Smoothing: Analysis and Comparison,’ IEEE Transactions on Signal Processing, Vol. 41, No. 2, pp 788-803, February 1993.
67. Bhaskar D. Rao and K. V. S. Hari,`Analysis of Subspace based DOA Estimation Methods,’ Sadhana, Proceedings of Indian Academy of Sciences in Engineering Sciences’ Special Issue on Recent advances
in Digital Signal Processing, vol. 16, Part 3, pp. 183-194, November 1991.
68. Bhaskar D. Rao and K. V. S. Hari, `Effect of Spatial Smoothing on the Performance of MUSIC and the Minimum-Norm method,’ Proceedings of IEE, Vol. 137, Part F, No. 6, pp 449-458, December 1990.
69. Bhaskar D. Rao and K. V. S. Hari, `Performance Analysis of Root-Music,’ IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 43, pp 1939-1949, December 1989.
70. Bhaskar D. Rao and K. V. S. Hari, `Performance Analysis of ESPRIT and TAM in determining the Direction of Arrival of Plane Waves in Noise,’ IEEE Transactions on Acoustics, Speech and Signal
Processing, Vol. 37, pp 1990-1994, December 1989.
71. Bhaskar D. Rao and K. V. S. Hari, `Statistical Performance Analysis of the Minimum-Norm method,’ Proceedings of IEE, Vol. 136, Part F, no. 3, pp 125-134, June 1989.
72. Surendra Prasad and K.V.S.Hari, `Adaptive Seismic Deconvolution via the Canonical Variate Analysis,’ Journal of IETE, vol. 34, no. 5, pp. 423-430, September-October 1988.
73. Surendra Prasad and K. V. S. Hari, `Improved ARMA Spectral Estimation using the Canonical Variate Method,’ IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 35, pp 900-903, June
1. Y Agrawal, A Kushwaha, GK Ananthasuresh, KVS Hari, S Khandai, “An Assistive Chair using a Series-Elastic Actuator,” 4th International and 19th National Conference on Machines and Mechanisms,
2. K. V. S. Hari, ‘Channel Models for Wireless Communication Systems,’ Wireless Network Design Optimization Models and Solution Procedures by Kennington, Jeff; Olinick, Eli; Rajan, Dinesh (Eds.),
International Series in Operations Research & Management Science, Springer-Verlag, Vol. 158, 1st Edition, 2011.
3. Joby Joseph and K. V. S. Hari,’Direction Estimation of Broadband sources for Auditory Localization and Spatially Selective Listening,’Advances in Direction-of-Arrival Estimation (Artech House
Radar Library), S. Chandran, editor, Artech House , December 2005.
4. K. V. S. Hari and V. G. S. Prasad, ‘Space-Time and Space-Frequency Block Coded Interleaved OFDM System,’ Adaptive Antenna Arrays, S. Chandran, Editor, Springer Verlag, Berlin, June 2004.
5. Bhaskar D. Rao and K.V.S.Hari,`Spatial Smoothing and MUSIC: Further Results,’ SVD and Signal Processing , II Algorithms, Analysis and Applications, R. J. Vaccaro, Editor, Elsevier Science
Publishers, B.V., Amsterdam, 1991.
1. Shubham Sharma, K. V. S. Hari, “System and Method for obtaining Random-like Projection-based Feasible Trajectory for MRI Scanning,” Indian Patent filed (under examination), 2018.
2. Satya Sudhakar Yedlapalli, K.V.S. Hari, “Determining spectral samples of a finite length sequence at non-uniformly spaced frequencies,” US patent 8594167granted on 26 Nov 2013.
3. Satya Sudhakar Yedlapalli, K.V.S. Hari, “Filtering Discrete-time signals using Notch Filters” USPTO Patent # 9112479, Aug 2015
4. Satya Sudhakar Yedlapalli and K.V.S. Hari, “Determining spectral samples of a finite length sequence at non-uniformly spaced frequencies,” Japanese Patent # IN-800641-04-JP-NAT, May 2015.
5. Satya Sudhakar Yedlapalli and K.V.S. Hari, “Determining spectral samples of a finite length sequence at non-uniformly spaced frequencies,” Korean Patent # IN-800641-04-KR-NAT, Mar 2015.
6. Satya Sudhakar Yedlapalli and K. V. S. Hari , “Filtering Discrete-Time Signals Using a Notch Filter,” PCT publication # WO/2012/052807, 2012.
7. Satya Sudhakar Yedlapalli and K. V. S. Hari , “Filtering Discrete-Time Signals Using a Notch Filter,” Indian patent priority # 03081/CHE/2010, 2010.
8. V.G.S. Prasad, K.V.S. Hari, “INTERLEAVED ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING (IOFDM) SYSTEM,” Indian Patent 198687, May 2006.
1. Satwika Bhogavalli, Eric Grivel, K.V.S. Hari, Vincent Corretja , “Cramér-Rao bound for the estimation of the target parameters in a MIMO OFDM DFRC System,” accepted at GRETSI 2023, France.
2. Swati Bhattacharya, K.V.S. Hari, “FastNet for Symbol Detection in Massive MIMO Systems,” accepted, 2023 IEEE CONECCT, Bangalore, July 2023.
3. Swati Bhattacharya, K.V.S. Hari, Yonina C Eldar, “Joint Channel Estimation and Symbol Detection in Overloaded MIMO Using ADMM,” accepted, 2023 IEEE Statistical Signal Processing Workshop, Hanoi,
Vietnam, July 2023.
4. Satwika Bhogavalli, K.V.S. Hari, Eric Grivel, Vincent Corretja,”Waveform design to improve the estimation of target parameters using the Fourier Transform method in a MIMO OFDM DFRC system,”
Proc. of 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2023), Rhodes, Greece, June 2023.
5. Pavan Kumar G V V R and K.V.S. Hari, “A Fast Dictionary Learning Algorithm for CSI Feedback in Massive MIMO FDD Systems,” accepted for publication in National Conference on Communications (NCC
2023), IIT Guwahati, Feb 2023.
6. Swati Bhattacharya, K.V.S. Hari, “A Novel Method for Millimetre-Wave Channel Estimation for 1-bit Quantized Receivers using Low-Rank Matrix Constraints” accepted IEEE International Conference on
Signal Processing Communications (SPCOM 2022), Bangalore, 11-15 July 2022.
7. Titas Ghoshal, K.V.S. Hari, “Object Classification Using Micro-Doppler Signature And Clutter Suppression In MIMO Radar,” accepted 2022 IEEE CONECCT, Bangalore, 8-10 July 2022.
8. Shubham Sharma, Geert Leus, K.V.S. Hari, “Learning-based method for k-space trajectory design in MRI,” has been accepted for presentation at the 44th IEEE Engineering in Medicine and Biology
Conference (EMBC’22) to be held in Glasgow, United Kingdom 11-15 July 2022.
9. A Kushwaha, Y Agrawal, S. Khandai, K.V.S. Hari, G.K. Ananthasuresh, “An Assistive Chair Using a Series-Elastic Actuator,” Proceedings of iNaCoMM, Machines, Mechanism and Robotics, 1289-1302,
Springer, 2022
10. B Satwika, K.V.S. Hari, “An Analysis for the Performance of the OFDM-IM Systems Impaired by Carrier Frequency Offset” 29th European Signal Processing Conference (EUSIPCO 2021), 1661-1665, 2021
11. C.G. Jansson, R. Thottappillil, S. Hillman, S. Möller, K.V.S. Hari, R. Sundaresan, “Experiments in Creating Online Course Content for Signal Processing Education,” Proc. of 2020 IEEE
International Conference on Acoustics, Speech and Signal Processing (ICASSP 2020), Barcelona, May 2020.
12. Shubham Sharma, K.V.S. Hari and Geert Leus, “Space Filling Curves for MRI Sampling,” Proc. of 2020 IEEE International Conference on Acoustics, Speech and Signal Processing, Barcelona (ICASSP
2020), May 2020
13. Shubham Sharma, K.V.S. Hari and Geert Leus, “K-Space Trajectory Design for Reduced MRI Scan Time,” Proceedings of 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (
ICASSP 2020), Barcelona, May 2020.
14. Y. Agrawal, A. Kushwaha, G.K. Ananthasuresh, K.V.S. Hari, S. Khandai,”An Assistive Chair using a Series-Elastic Actuator,” 4 th International and 19th National Conference on Machines and
Mechanism , Dec 2019
15. S. Elayaperumal, K. V S. Hari, “Optimal Irregular Subarray Design for Adaptive Jammer Suppression in Phased Array Radar,” 2019 IEEE International Symposium on Phased Array System & Technology
(PAST 2019), Waltham, MA, Oct 2019.
16. A Agrawal, S Nag, KVS Hari, SP Arun, “Upright bigram processing predicts reading fluency,”Sixth Annual Conference of the Association for Cognitive Science, (ACCS) 2019.
17. Kiran Gunde, K.V.S. Hari, ” Modified Generalised Quadrature Spatial Modulation,” Proc. of National Conference on Communications (NCC) 2019, Bangalore, Feb 2019.
18. Aakash Agarwal, K.V.S. Hari, S.P. Arun, “How does reading expertise influence letter representations in the brain? An fMRI study,” Vision Sciences Society (VSS) Annual Meeting, Florida, May 2018.
19. G. S. Muralikrishna, Sooraj K. Ambat, K.V.S. Hari, ” Batch Look Ahead Orthogonal Matching Pursuit,” Prof. of National Conference on Communications (NCC 2018), Hyderabad, Feb 2018.
20. Aakash Agarwal, K.V.S. Hari, S.P. Arun, “How reading changes letter representations: a double dissociation using orthographically distinct scripts in India,” Vision Sciences Society (VSS) Annual
Meeting, Florida, May 2017.
21. Deepa K. G., Sooraj K. Ambat, K. V. S. Hari, “Fusion of Algorithms for Multiple Measurement Vectors,” Proc. of 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (
ICASSP 2016), Shanghai, March 2016.
22. Sooraj K. Ambat, Shree Ranga Raju N.M. and K.V.S. Hari, “Gini Index based Search Space Selection in Compressive Sampling Matching Pursuit,” IEEE INDICON 2014, 11-13 Dec 2014, Pune, India.
23. Girisha R. Shetty and K.V.S. Hari and Peter Handel, “Fusing the Navigation Information of Dual Foot-Mounted Zero-Velocity-Update-Aided Inertial Navigation Systems,” International Conference on
Signal Processing and Communications (SPCOM 2014), July 2014, Bangalore, India.
24. Prateek Basavapur Swamy, Sooraj K. Ambat, Saikat Chatterjee, K.V.S. Hari, “Reduced Look Ahead Orthogonal Matching Pursuit”, published National Conference on Communications (NCC 2014), Feb 2014,
Kanpur, India.
25. Deepa K G, Sooraj K. Ambat, K.V.S. Hari, “Modified Greedy Pursuits for Improving Sparse Recovery”, National Conference on Communications, NCC 2014, Feb 2014, Kanpur, India.
26. J-O. Nilsson, I. Skog, P. Händel, M. Olsson, J. Rantakokko, K.V.S. Hari, “Accurate Indoor Positioning of Firefighters Using Dual Foot-mounted Inertial Sensors and Inter-agent Ranging”, published
IEEE/ION Position Location and Navigation Symposium, May 2014.
27. Rakshith Rajashekar, K.V.S. Hari, K. Giridhar, L. Hanzo, “Performance Analysis of Antenna Selection Algorithms in Spatial Modulation Systems with Imperfect CSIR”, European Wireless 2013,
Guildford, UK, 16-18 Apr. 2013.
28. Sooraj K. Ambat, Saikat Chatterjee, K.V.S. Hari, “Fusion of Algorithms for Compressed Sensing”, publication in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
2013, Vancouver, Canada, 29-31 May 2013.
29. G. V. Prateek, Girisha R. Shetty, K.V.S. Hari, Peter Handel,”Data Fusion of Dual Foot-Mounted INS to Reduce the Systematic Heading Drift,” Proceedings of 4th International Conference on
Intelligent Systems, Modelling and Simulation (ISMS 2013), 29-31 Jan 2013, Bangkok, Thailand.
30. G. V. Prateek, Nijil K., K.V.S. Hari,” Classification of Vehicles Using Magnetic Field Angle Model,” Proceedings of 4th International Conference on Intelligent Systems, Modelling and Simulation (
ISMS 2013), 29-31 Jan 2013, Bangkok, Thailand.
31. Prateek G. V., Rajkumar V. , Nijil K. , and K.V.S. Hari; , “Classification of Vehicles Using Magnetic Dipole Model,” Proceedings of IEEE TENCON 2012, 19-22 Nov 2012, Cebu, Philippines.
32. Rakshith Rajashekar and K.V.S. Hari, “Modulation Diversity for Spatial Modulation Using Complex Interleaved Orthogonal Design,” Proceedings of IEEE TENCON 2012, 19-22 Nov 2012, Cebu, Phillipines.
33. Sooraj K. Ambat, Saikat Chatterjee, and K.V.S. Hari “Subspace Pursuit Embedded in Orthogonal Matching Pursuit,” Proceedings of IEEE TENCON 2012, 19-22 Nov 2012, Cebu, Philippines.
34. Sooraj K. Ambat, Saikat Chatterjee, and K.V.S. Hari “On Selection of Search Space Dimension in Compressive Sampling Matching Pursuit,” Proceedings of IEEE TENCON 2012, 19-22 Nov 2012, Cebu,
35. Sooraj K Ambat, Saikat Chatterjee, K.V.S. Hari, “Fusion of Matching Pursuits for Compressed Sensing Signal Reconstruction”, Proceedings of European Signal Processing Conference (EUSIPCO 2012),
27-31 Aug. 2012, Bucharest, Romania.
36. Yashwanth M, K.V.S. Hari, “Delay Optimized Zero-Forcing Channel Shortening Algorithm for Wireless Communication,” Proceedings of International Conference on Signal Processing and Communication (
SPCOM 2012), July 2012, Bangalore, India.
37. Li Wang, K.V.S. Hari and Lajos Hanzo,”Iterative Amplitude/Phase Multiple-Symbol Differential Sphere Detection for DAPSK Modulated Transmissions,” Proceedings of IEEE International Conference on
Communications (ICC 2012), 10-15 Jun 2012, Ottawa, Canada.
38. J-O. Nilsson, I. Skog, P. Handel and K.V.S. Hari,”Foot-Mounted INS for Everybody — An Open-Source Embedded Implementation,” Proceedings of IEEE/ION Position Location and Navigation System (PLANS)
Conference, 24-26 April 2012, Myrtle Beach, South Carolina, USA.
39. N. Mukund Sriram, B.S. Adiga and K.V.S. Hari,”Burst Error Correction using Partial Fourier Matrices and Block Sparse Representation,” Proceedings of National Conference on Communications (NCC
2012), 3-5 Feb 2012, IIT Kharagpur, India.
40. Renu Jose and K.V.S. Hari, “Joint Estimation of Synchronization Impairments in MIMO-OFDM System,” Proceedings of National Conference on Communications (NCC 2012), 3-5 Feb 2012, IIT Kharagpur,
41. Sooraj K. Ambat, Saikat Chatterjee, K.V.S. Hari,”Adaptive Selection of Search Space in Look Ahead Orthogonal Matching Pursuit,”Proceedings of National Conference on Communications (NCC 2012), 3-5
Feb 2012, IIT Kharagpur, India.
42. Saikat Chatterjee, K.V.S. Hari, Peter Handel and Mikael Skoglund,”Projection-Based Atom Selection in Orthogonal Matching Pursuit for Compressive Sensing,” Proceedings of National Conference on
Communications (NCC 2012), 3-5 Feb 2012, IIT Kharagpur, India.
43. M. R. Rakshith, K. V. S. Hari, Lajos Hanzo, “Field Extension Code Based Dispersion Matrices for Coherently Detected Space-Time Shift Keying,” Proceedings of IEEE Globecom 2011, 5-9 Dec 2011,
Houston, USA.
44. Dinesh Dileep Gaurav and K. V. S. Hari, “Degrees of Freedom of Relay aided MIMO X network with Orthogonal Components,” Proceedings of the 14th International Symposium on Wireless Personal
Multimedia Communications (WPMC 2011), Oct 2011, Brest, France.
45. Avinash Mohan and KVS Hari, “Low Complexity Adaptation for Channel Shortening Equalizers,” Proceedings of 54th IEEE International Midwest Symposium on Circuits and Systems (IEEE MWSCAS 2011),
Aug7-10 2011, Seoul, S. Korea.
46. K. V. S. Hari and V. Lalitha, “Subspace-based DOA Estimation usinf Fractional Lower Order Statistics,” Proceedings of IEEE International Conference on Acoustics Speech and Signal Processing (
ICASSP 2011), May 22-27 2011, Prague, Czech Republic.
47. Amit K Datta and K. V. S. Hari, “Channel Estimation Using Minimum Bit Error Rate Framework for BPSK Signals ” Proceedings of IEEE Vehicular Technology Conference (VTC-Spring), May 15-18 2011,
Budapest, Hungary.
48. K. V. S. Hari, Rakshith Jagannath, Satya Sudhakar Yedlapalli, “Sensitivity Analysis of Minimum-Phase FIR Filters based on the Line Spectral Frequency Model”, Proceedings of APSIPA Annual Summit
and Conference 2010, 14-17 Dec 2010, Singapore. (available online at http://www.apsipa.org/proceedings.htm )
49. Avik Santra and K V S Hari, “Novel Subspace methods for Blind Estimation of Multiple CFOs in MIMO-OFDM Systems,” Proceedings of International Conference on Signal Processing (ICSP 2010)
conference, Beijing, October 2010.
50. Avik Santra and K V S Hari, “Low Complexity PARAFAC Receiver for MIMO-OFDMA System in the Presence of Multi-Access Interference,” Proceedings of 44th Asilomar Conference on Signals, Systems and
Computers, Pacific Grove, CA, USA November 2010.
51. G V S S K R Naganjaneyulu and K V S Hari, “Study Of Acoustic Source Localization Algorithm For Planar arrays” Proceedings of IEEE International Conference, TENCON2010, Fukuoka, Japan, November
52. M R Bhavani Shankar and K V S Hari, “On the Diversity and Complexity of Zero Forcing Receivers for MIMO Zero Padded Systems,” Proceedings of International Conference on Signal Processing and
Communication (SPCOM 2010), Bangalore July 2010.
53. Nithya V S, Karthik Sheshadri, Anurag Kumar, K V S Hari, “Model-Based Target Tracking in a Wireless Network of Passive Infrared Sensor Nodes,” Proceedings of International Conference on Signal
Processing and Communication (SPCOM 2010), Bangalore July 2010.
54. Satya Sudhakar Yedlapalli and K V S Hari,”A Novel Property of an Auto-Correlation Sequence and some Applications,” Proceedings of International Conference on Signal Processing and Communication (
SPCOM 2010), Bangalore July 2010.
55. A Vijayakrishna and K V S Hari, “Jointly Optimal MMSE Design for Non-redundant FIR Precoding and Equalization,” to appear in the Proceedings of International Conference on Signal Processing and
Communication (SPCOM 2010), Bangalore July 2010. Will be available at IEEE Xplore.
56. Satya Sudhakar Yedlapalli and K. V. S. Hari, “The Canonic Linear-Phase FIR lattice Filter Structures,” Proceedings of National Conference on Communications (NCC-2010), Jan 2010.
57. Muralidhar, Karthik Li, Kwok H. K V S Hari , ” Iterative Kalman-AR Method for Doppler Spread Estimation in Flat Fading Channels ” IEEE Conference on Personal, Indoor and Mobile Radio
Communications, 2007(PIMRC 2007) 3-7 sept. 2007,page(s): 1-5.
58. R. Deepak and K. V. S. Hari,’ Harmonic mean of squared product distances: a new criterion to design codes for independent fading channels,’ Proceedings of 2004 International Conference on Signal
Processing and Communications (SPCOM 2004),} Bangalore, Dec 2004.
59. A. Vijaya Krishna and K. V. S. Hari,’Minimum redundancy for FIR equalization in MIMO multicarrier modulation,’ Proceedings of 2004 International Conference on Signal Processing and Communications
(SPCOM 2004),} Bangalore, Dec 2004.
60. Avinash Achar and K. V. S. Hari,’Parametric Localization of Correlated Incoherently Distributed Sources Using ESPRIT,’ Proceedings of IEEE Sensor Array and Multichannel Signal Processing
Workshop, SAM2004,Barcelona, July 2004.
61. Vijayakrishna A. and K. V. S. Hari, ‘Filterbank precoding for MIMO frequency selective channels: Minimum redundancy and equalizer design,’ Proceedings of IEEE Sensor Array and Multichannel Signal
Processing Workshop, SAM2004, Barcelona, July 2004.
62. Vinod T.S and K. V. S. Hari,’Optimal Pilot tones for MIMO Interleaved OFDM systems,’ Proceedings of ICASSP 2004, Montreal, May 2004.
63. Karthik S. and K. V. S. Hari, ‘Alternative Interleaving schemes for Interleaved Orthogonal Frequency Division Multiplexing,’ Proceedings of TENCON 2003 – 2003 IEEE Region 10 Conference on
Convergent Technologies for the Asia-Pacific, Bangalore, Oct 2003.
64. Joby Joseph and K. V. S. Hari, ‘Adaptive estimation of parameters using partial information of desired outputs,’ Proceedings of TENCON 2003 – 2003 IEEE Region 10 Conference on Convergent
Technologies for the Asia-Pacific, Bangalore, Oct 2003.
65. M. R. Bhavani Shankar and K. V. S. Hari, ‘Bounds on MIMO capacity due to channel perturbations,’ in Proc of NCC 2003, Chennai, Jan 2003.
66. V. G. S. Prasad and K. V. S. Hari, ‘Interleaved Orthogonal Frequency Division Multiplexing (IOFDM)System with lesser complexity ‘ Proc of NCC 2003, Chennai, Jan 2003.
67. M. R. Bhavani Shankar and K. V. S. Hari, ‘On the Variations in Capacity of MIMO Communication Systems to Channel Perturbations,’ Proc of ICPWC 2002, New Delhi, Dec 2002.
68. V. G. S. Prasad and K. V. S. Hari, ‘Space-Time Block Coded Interleaved Orthogonal Frequency Division Multiplexing System ‘ Proc. of ICPWC 2002, New Delhi, Dec 2002.
69. V. G. S. Prasad and K. V. S. Hari, ‘Interleaved Orthogonal Frequency Division Multiplexing System ‘ Proc. of ICASSP 2002, Orlando, May 2002.
70. Joby Joseph and K. V. S. Hari, ‘An algorithm to estimate multipath delays of broadband sources with only two sensors’, Tenth Annual IEEE Symposium on Multimedia Communications and Signal
Processing, Bangalore, Nov 2001.
71. D. S. Baum, R. Nabar, S. Panchanathan, K. V. S. Hari, V. Erceg, A. Paulraj, `Measurements and Characterization of Broadband MIMO Fixed Wireless Channels at 2.5 GHz,’ Proc of IEEE Intl. Conf. on
Personal Wireless Communication (ICPWC 2000), Hyderabad, India, Dec 2000.
72. G. Cirrincione, G. Ganesan, K.V.S.Hari and S. Van Huffel, ‘Direct and neural techniques for the data least squares problem’ Proceedings of Mathematical Theory of Networks and Systems, Perpignan,
France, June 19-23, 2000.
73. Anurag Kumar, K. V. S. Hari, R. Shobhanjali and Srikumar Sharma, `Long-Range Dependence in the Aggregate Flow of TCP controlled Elastic sessions: An Investigation via the Processor Sharing
Model,’ Proceedings of the National Conference on Communications, NCC, Jan 2000.
74. Shishir K. L., K. V. S. Hari and Risto Wichman, `Low complexity method to estimate Co-channel signals using an Antenna Array,’ Proceedings of the IEEE International Conference on Personal
Wireless Communications (ICPWC’99), Jaipur, Feb 1999.
75. Shishir K. L. and K. V. S. Hari, `Detection of the number of Constant Modulus sources arriving at an antenna array,’ Proceedings of the IEEE International Symposium on Wireless Communications for
the Next Millennium, New Delhi, Sept 1998.
76. Raghuraman M. and K. V. S. Hari, `Estimation of nominal AOA and angular spread for spatially distributed sources in Ricean fading channels,’ Proceedings of the IEEE International Symposium on
Wireless Communications for the Next Millennium, New Delhi, Sept 1998.
77. K. V. S. Hari and Bjorn Ottersten, `Parameter Estimation using a Sensor Array in a Ricean Fading channel,’ Proceedings of SPCOM’97, Bangalore, July 1997.
78. Y. Lavanis and K. V. S. Hari, `A Kernel for Wigner Distribution using Wigner Synthesis Techniques,’ Proceedings of the Workshop on Underwater Systems and Engineering, Visakhapatnam, August 1994.
79. L. Srinivas and K. V. S. Hari, FIR System Identification using Higher Order Cumulants,’ Proceedings of the National Symposium of Systems, pp. 201-205, Madras, October 1993.
80. K. V. S. Hari and Uma Gummadavelli, `On the Performance of Subspace methods in the presence of Array Model Errors and Spatial Smoothing,’ Proceedings of ICASSP, Minnesota, April 1993.
81. Bhaskar D. Rao and K.V.S.Hari, `Weighted State Space Methods/ESPRIT and Spatial Smoothing’ Proceedings of ICASSP, Toronto, Canada, April 1991.
82. Bhaskar D. Rao and K.V.S.Hari,`On Spatial Smoothing and Weighted Subspace Methods’ Proceedings of 24th Asilomar Conference on Signals, Systems and Computers, Monterey, Nov 90.
83. Bhaskar D. Rao and K.V.S.Hari,`Effect of Spatial Smoothing on State Space Methods/ESPRIT,’ Proceedings of 5th ASSP Workshop on Spectrum Estimation and Modelling, October 1990.
84. Bhaskar D. Rao and K.V.S.Hari,`Effect of Spatial Smoothing on the Performance of Noise Subspace Methods,’ Proceedings of ICASSP, Albuquerque, New Mexico, April 1990.
85. Bhaskar D. Rao and K.V.S.Hari,`MUSIC and Spatial Smoothing: A Statistical Performance Analysis,’ Proceedings of 23rd Asilomar Conference on Signals, Systems and Computers, Monterey, Nov 89.
86. Bhaskar D. Rao and K.V.S.Hari,`Statistical Performance Analysis of the Minimum-Norm method,’ Proceedings of ICASSP, Glasgow, Scotland, May 1989.
87. Bhaskar D. Rao and K.V.S.Hari,`Performance Analysis of Root-Music,’ Proceedings of 22nd Asilomar Conference on Signals, Systems and Computers, Monterey, November 1988.
88. Bhaskar D. Rao and K.V.S.Hari,`Performance Analysis of Subspace-based methods,’ Proceedings of 4th ASSP Workshop on Spectrum Estimation and Modelling, Minnesota, August 1988.
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NCERT Solutions Maths Exercise 9.1 Class 7 Chapter 9 - Perimeter and Area PDF
NCERT Solutions for Class 7 Maths Chapter 9 Perimeter and Area Exercise 9.1 - FREE PDF Download
NCERT solutions for Perimeter and Area Class 7 Exercise 9.1 will serve as your companion as you navigate Exercise 9.1. We will break down the key topics that students will encounter, including
calculating circle properties like radius and area, tackling practical applications of circles, and mastering word problems that involve these fascinating shapes. By working through class 7 maths
exercise 9.1 solutions, you will transform your theoretical knowledge of circles into a powerful tool for solving everyday problems!
1. NCERT Solutions for Class 7 Maths Chapter 9 Perimeter and Area Exercise 9.1 - FREE PDF Download
2. Glance on NCERT Solutions Maths Chapter 9 Exercise 9.1 Class 7 | Vedantu
3. Formulas Used in Maths Class 7 Chapter 9 Exercise 9.1
4. Access NCERT Solutions for Maths Class 7 Chapter 9 - Perimeter and Area
6. Class 7 Maths Chapter 9: Exercises Breakdown
7. CBSE Class 7 Maths Chapter 9 Other Study Materials
8. Chapter-Specific NCERT Solutions for Class 7 Maths
9. Important Related Links for NCERT Class 7 Maths
Glance on NCERT Solutions Maths Chapter 9 Exercise 9.1 Class 7 | Vedantu
• Chapter 9 of Class 7 Maths in the NCERT curriculum focuses on the concepts of Perimeter and Area.
• Exercise 9.1 focuses on calculating the perimeter and area of various geometrical shapes such as rectangles, squares, triangles, and parallelograms.
• There are 8 questions in Maths Chapter 9 Class 7th Exercise 9.1 which experts at Vedantu fully solved.
• Many questions are accompanied by diagrams to help students visualize the problems better.
Formulas Used in Maths Class 7 Chapter 9 Exercise 9.1
• The formulas for calculating the area = πr²
• Circumference of circle = 2πr
FAQs on NCERT Solutions for Class 7 Maths Chapter 9 - Perimeter and Area Exercise 9.1
1. What is the main focus of Chapter 11 Exercise 11.2 in Class 7 Maths?
The main focus of Exercise 11.2 is to strengthen students' understanding of calculating the perimeter and area of rectangles and squares. It emphasizes practical applications of these concepts
through word problems.
2. What formulas are used in Exercise 11.2 for calculating the perimeter and area of rectangles and squares?
The formulas used are:
• The perimeter of a rectangle: Perimeter=2(l+b)Perimeter=2(l+b), where ll is the length and bb is the breadth.
• The perimeter of a square: Perimeter=4aPerimeter=4a, where aa is the side length.
• Area of a rectangle: Area=l×bArea=l×b.
• Area of a square: Area=a2Area=a2.
3. Why is it important to learn about the perimeter and area of shapes?
Learning about the perimeter and area of shapes is important because these concepts are widely used in real life. They help in solving practical problems like determining the amount of material
needed for construction, fencing a garden, or tiling a floor.
4. How can I apply the concepts learned in Exercise 11.2 to real-world problems?
You can apply these concepts by using the formulas to solve problems related to real-life situations. For example, you can calculate the length of fencing needed for a garden, the area of a room to
be carpeted, or the amount of paint required for a wall.
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Millimeters, Centimeters, Meters and Kilometers | sofatutor.com
Millimeters, Centimeters, Meters and Kilometers
Content Millimeters, Centimeters, Meters and Kilometers
Millimeters, Centimeters, Meters and Kilometers
Basics on the topic Millimeters, Centimeters, Meters and Kilometers
Metric Units – Kilometers, Meters, Centimeters and Millimeters
If you want to measure the length of something, you need to label length using units of measurement. There are different units in which we can indicate the length of something. What are centimeters,
kilometers, and millimeters? They are all metric units of measurements for length! From smallest to largest measurement, we have millimeters, centimeters, meters, and kilometers!
Look at the above chart. What is between millimeters and kilometers? Centimeters and meters are between millimeters and kilometers!
Metric Units – Relationship
How do you relate meters (m), centimeters (cm), millimeters (mm) and kilometers (km)? Each unit of measurement relates to the others, because you have so many of one, in a number of the other.
For example, let’s look at millimeters first. There is one million millimeters in a kilometer, one thousand millimeters in one meter, and one hundred millimeters in a centimeter.
Now let’s look at centimeters. There is one hundred thousand centimeters in a kilometer, and one hundred centimeters in a meter.
Now let’s look at meters. There is one thousand meters in a kilometer.
Metric Units – Examples
How Big are millimeters, centimeters, kilometers, and meters? Let’s look at examples from the real world for millimeters, centimeters, meters and kilometers.
A millimeter is roughly the size of the thickness of a paper clip.
A centimeter is roughly the length of a bug.
A meter is roughly the length of a guitar.
A kilometer is roughly the distance between the support towers on the Golden Gate Bridge.
Metric Units – Summary
The metric units for measuring length are millimeters, centimeters, meters, and kilometers. These units describe the length of objects from very small objects (measured in millimeters) to very large
objects (measured in kilometers).
They are related because you can convert one to the other! Below is an overview of these four metric units of measurement. One unit on the left column equals the amount of the relatively smaller unit
of measurement on the right.
Larger metric unit Smaller metric unit
1 centimeter 10 millimeters
1 meter 100 centimeters
1 kilometer 1,000 meters
As you can see, each metric unit of measurement is related to another metric unit of measurement. So with the help of this chart, you can decide for what kinds of distances you would choose to make
measurements in millimeters, centimeters, meters and kilometers. After watching the video, you will find interactive exercises, worksheets and more activities for extra practice on metric units of
Transcript Millimeters, Centimeters, Meters and Kilometers
"If you call right now, you will get TWO measuring kits for the price of ONE, delivered immediately!" "I can't believe people fall for that, can you Zuri?" "I'll take two measuring kits, please!"
"Well, you've purchased it now, might as well take a look." Let's join Zuri and Freddie as they learn all about millimeters, centimeters, Meters, and kilometers. Metric measurements, or units, of
length that we use are kilometers, meters, centimeters, and millimeters. A millimeter is a small measure of length, and is represented by the abbreviation mm. A centimeter is a bigger measurement
than millimeters, and is represented by the abbreviation cm. A meter is a bigger measurement than a centimeter, and is represented by the abbreviation m. A kilometer is a bigger measurement than a
meter, and is represented by the abbreviation km. Let's take a closer look at millimeters! Ten millimeters make up one centimeter, one thousand millimeters make up one meter, and one million
millimeters make up one kilometer. In fact the word milli in millimeter means million. You can use this to remember that a million millimeters make up one kilometer. Now let's take a look at
centimeters! One hundred centimeters make up one meter, and one hundred thousand centimeters make up one kilometer. In fact, the word centi in centimeter is from the word century, which means one
hundred. You can use this to remember that one hundred centimeters make up one meter. Now let's take a look at meters! One thousand meters make up one kilometer. And finally, let's look at
kilometers! This is the biggest unit of length we are seeing here, and we can remember how many meters make up one kilometer by looking at the word kilo. Kilo means one thousand. You can use this to
remember that a thousand meters make up one kilometer. What are some items or objects that are about the size of each unit? The thickness of a paperclip is roughly one millimeter. Some insects can be
roughly one centimeter long. A guitar is roughly one meter. For one kilometer, there isn't a specific item that is one kilometer long, but the distance between the support towers on the Golden Gate
Bridge is pretty close at just over one kilometer! While Zuri and Freddie finish up exploring the measuring kit, let's review! Remember, a millimeter is a small measure of length, and is represented
by the abbreviation mm. A centimeter is a bigger measurement than millimeters, and is represented by the abbreviation cm. A meter is a bigger measurement than a centimeter, and is represented by the
abbreviation mm. A kilometer is a bigger measurement than a meter, and is represented by the abbreviation km. "Order now to get my Weighted Wonders kit!" "No Zuri, Please don't." "I'll take one of
the Weighted Wonders kits please!"
Millimeters, Centimeters, Meters and Kilometers exercise
Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Millimeters, Centimeters, Meters and Kilometers.
• Abbreviating units of measure.
Abbreviations usually include the first letter of the word.
All the abbreviations will have the letter m which would represent "meters" because it is the metric system.
Am abbreviation is a shortened form of a word or phrase. For example United States of American can be represented as USA.
□ millimeters: mm
□ centimeters: cm
□ meters: m
□ kilometers: km
• Ordering metric units.
The smallest unit of measurement goes at the top.
The largest unit of measurement goes at the bottom.
There are 1,000 meters in a kilometer.
Metric units of length from smallest to largest:
□ millimeters (mm)
□ centimeters (cm)
□ meters (m)
□ kilometers (km)
• Estimate the length of the objects.
Smaller objects need to be measured using a smaller unit of measurement.
Think of the size of the object to help decide what unit is best used to measure it.
Use this comparison chart to help you decide on the units used to measure each item.
□ The pencil end is about 5 mm
□ The crayon is about 5 cm
□ The fence is about 5 m
□ The distance from the two locations on the map is about 5 km
• Complete the sentences about metric units.
Review the definition of each unit of measurement.
Use the metric unit conversion table to help complete the statements.
Use this comparison chart to help you complete the statements above.
□ There are a million millimeters in one kilometer.
□ The word centimeter stems from the word century meaning 100.
□ One kilometer is equal to 1,000 meters.
□ Kilometers are the largest metric unit for measuring length.
□ The word kilo means 1,000.
• Which units would measure each object?
Smaller objects should be measured using smaller units of measurement.
Use this comparison chart to help you decide on the units used to measure each item.
The items would be measured using the following units:
□ the beetle: yellow (millimeters)
□ the paperclip: yellow (centimeters)
□ the guitar: blue (meters)
□ the bridge: violet (kilometers)
• Convert the metric units of measurement.
Use these conversions to help:
□ 1,000 mm = 1 m
□ 100 cm = 1 m
□ 1 km = 1,000 m
If converting from a smaller unit to a larger unit (such as centimeters to meters), you divide.
If converting from a larger unit to a smaller unit (such as meters to centimeters), you multiply.
□ 1 km = 1 mm is not correct ❌ because 1km is 1 million millimeters.
□ 1,000 m = 1 km is correct ✅.
□ 100 cm = 1m is correct ✅.
□ 10mm = 1m is not correct ❌ because 10 mm is = to 1 centimeter.
□ 2 km = 2,000 m is correct ✅.
□ 3 m = 300 cm is correct ✅.
More videos and learning texts for the topic Measuring Length
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Prim's Algorithm in C++
Prim's algorithm is a greedy algorithm used for finding the minimum spanning tree (MST) of a connected, undirected graph. The minimum spanning tree of a graph is a subset of the edges that forms a
tree and connects all the vertices in the graph while minimizing the total edge weight. Prim's algorithm ensures that the MST is created by adding edges with the minimum weight.
Prim's algorithm, named after Czech mathematician and computer scientist Robert C. Prim, is a fundamental algorithm in graph theory and computer science used for finding the minimum spanning tree
(MST) of a connected, undirected graph. Here's a brief history of Prim's algorithm:
Origin of Minimum Spanning Trees:
The concept of minimum spanning trees dates back to the early 20th century, with applications in electrical network design and transportation.
Borůvka's Algorithm (1926):
The earliest precursor to Prim's algorithm was developed by Czech mathematician Otakar Borůvka in 1926. Borůvka's algorithm aimed to find an approximate solution to the minimum spanning tree problem.
Prim's Algorithm (1957):
In 1957, Robert C. Prim, an American mathematician, independently rediscovered and published the algorithm that would later bear his name. His work was motivated by the construction of efficient
electrical networks.
Edsger W. Dijkstra (1959):
Dutch computer scientist Edsger W. Dijkstra independently rediscovered and popularized Prim's algorithm in the context of computer science. His presentation of the algorithm in 1959 helped establish
its significance in computer science and graph theory.
Proof of Correctness:
The algorithm's correctness was rigorously proved by computer scientists R.C. Prim and Vojt?ch Jarník in the late 1950s and early 1960s.
Wide Adoption:
Prim's algorithm gained wide adoption in computer science and graph theory due to its simplicity and efficiency. It became a fundamental algorithm for solving problems involving minimum spanning
Application in Network Design:
Prim's algorithm found applications in various fields, including network design, circuit layout, and transportation planning.
Computer Science Textbooks:
The algorithm's inclusion in computer science textbooks and courses further contributed to its prominence and understanding.
Later Developments:
Over the years, variations and improvements to Prim's algorithm have been proposed, such as Prim-Jarník algorithm and various data structures to optimize its performance.
Prim's algorithm remains a fundamental tool in graph theory and computer science for solving problems related to network design and optimization. It is known for its simplicity, efficiency, and
ability to consistently find minimum spanning trees in connected, undirected graphs.
Here's a deep explanation of how Prim's algorithm works:
• A connected, undirected graph.
• A starting vertex to begin the MST construction.
• The minimum spanning tree.
Algorithm Steps:
• Create an empty set MST to store the edges of the MST.
• Create an array key[] to keep track of the minimum edge weight for each vertex. Initialize it with infinity for all vertices except the starting vertex, which is set to 0.
• Create an array parent[] to store the parent of each vertex in the MST. Initialize it with -1 for all vertices.
Iterative Process:
• While there are vertices not yet included in MST, do the following steps:
• Choose a vertex u not in MST with the minimum key[u] value. Initially, this will be the starting vertex.
• Add u to MST.
• For each neighbor v of u that is not in MST, if the weight of the edge (u, v) is less than key[v], update key[v] to the weight of (u, v) and set parent[v] to u.
Once all vertices are included in MST, you have constructed the minimum spanning tree.
The set of edges in MST forms the minimum spanning tree of the given graph.
NOTE: Prim's algorithm ensures that the MST is constructed with the minimum possible total edge weight by greedily selecting edges with the smallest weights at each step.
Approach 1:
Enter the number of vertices: 5
Enter the adjacency matrix of the graph:
Edges of Minimum Spanning Tree:
Edge: 0 - 3 Weight: 1
Edge: 1 - 3 Weight: 2
Edge: 3 - 4 Weight: 6
Edge: 1 - 2 Weight: 3
• The algorithm starts by selecting an arbitrary vertex as the initial vertex. This vertex is added to the MST, and its key (the minimum edge weight to connect to it) is set to 0. All other
vertices are initially marked with infinite key values.
• A data structure (often a priority queue) is used to keep track of candidate edges to add to the MST, prioritized by their key values.
Iterative Process:
While there are vertices that haven't been added to the MST, the algorithm continues:
• It selects the vertex with the smallest key value among the vertices not in the MST. This vertex becomes the next vertex to be added to the MST.
• The algorithm explores all edges connected to the selected vertex and checks if any of these edges lead to a vertex with a smaller key value. If such an edge is found, the key value and the
parent of the adjacent vertex are updated, and the edge is added to the MST candidate list.
• The above steps are repeated until all vertices are included in the MST.
Once all vertices are included in the MST, the algorithm terminates.
The MST is formed by the edges that were selected during the algorithm's execution.
The key idea behind Prim's algorithm is to grow the MST one vertex at a time, always selecting the vertex with the smallest key value and adding the edge with the smallest weight that connects to it.
This ensures that the MST is built by adding edges with minimal weights, guaranteeing an optimal solution.
The algorithm maintains three main data structures:
• A set to keep track of vertices included in the MST.
• An array (or other data structure) to store key values for each vertex.
• A data structure (e.g., a priority queue) to efficiently select the next vertex to add to the MST based on its key value.
By iteratively selecting vertices and updating key values, Prim's algorithm constructs the minimum spanning tree of the given graph efficiently and optimally.
Approach 2:
Enter the number of vertices: 5
Enter the number of edges: 7
Enter the edges and their weights (from to weight):
Edges of Minimum Spanning Tree:
Edge: 0 - 3 Weight: 1
Edge: 3 - 1 Weight: 2
Edge: 1 - 2 Weight: 3
Edge: 1 - 4 Weight: 5
Custom Classes for Graph and Edge:
• We define two custom classes, Edge and Graph, to represent the graph and its edges.
• Edge stores the target vertex (to) and the weight of the edge.
• Graph contains the number of vertices (V) and an adjacency list (adj) to represent the graph's edges.
Function to Add Edges to the Graph (addEdge):
• In the Graph class, we have a member function addEdge that allows us to add edges to the graph.
• It takes the source vertex (from), target vertex (to), and edge weight as arguments and adds the edge to the adjacency list. Additionally, it adds the reverse edge for undirected graphs.
Prim's Algorithm (primMST Function):
• This function finds the Minimum Spanning Tree (MST) of the graph using Prim's algorithm.
• It takes the Graph object as an argument.
• We initialize arrays to keep track of key values (minimum edge weights), parent vertices, and a boolean array inMST to track whether a vertex is included in the MST.
• The priority_queue (pq) is used as a min-heap to select edges with minimum weights.
Main Algorithm Loop:
• We start by adding the first vertex (vertex 0) as the initial vertex with a key value of 0.
• The algorithm iterates until all vertices are included in the MST.
• In each iteration, it selects the vertex u with the smallest key value from the min-heap (pq).
Vertex Inclusion:
The selected vertex u is marked as included in the MST by setting inMST[u] to true.
Update Adjacent Vertices:
• The algorithm explores all adjacent vertices of u and checks if there's an edge with a weight smaller than the current key value for that vertex.
• If such an edge is found, it updates the key value and the parent vertex for that adjacent vertex.
• The adjacent vertex is then added to the min-heap for further consideration.
The algorithm continues until all vertices are included in the MST.
Printing the MST:
After the algorithm completes, we print the MST edges by iterating through the parent array and displaying the selected edges along with their weights.
Main Function:
In the main function, we take user input for the number of vertices and edges, create a Graph object, add edges to it using the addEdge function, and finally call primMST to find and print the MST.
Example 1: Adjacency Matrix
Enter the number of vertices: 5
Enter the adjacency matrix:
Edge Weight
0 - 3 1
3 - 1 2
1 - 2 3
0 - 4 5
• Initialization:
Choose a starting vertex as the initial node of the Minimum Spanning Tree (MST).
Create data structures to keep track of the MST:
key[]: An array to store the minimum edge weight to connect each vertex to the MST. Initialize all values to infinity except for the starting vertex, which is set to 0.
mstSet[]: An array or set to keep track of which vertices are included in the MST. Initialize all values to false.
• Iterative Process:
Repeat the following steps until all vertices are included in the MST:
Select a vertex u that is not in the MST and has the minimum key value.
Add u to the MST.
Update the key values of adjacent vertices of u if they are not already in the MST and if the edge weight to u is smaller than their current key value.
• Termination:
When all vertices are included in the MST, the algorithm terminates.
• Result:
The Minimum Spanning Tree (MST) is formed by the edges that were selected during the algorithm's execution. These edges connect all vertices while minimizing the total edge weight.
Example 2: Adjacency List
Enter the number of vertices: 5
Enter the number of edges: 7
Enter the edges (from to weight):
Edges of Minimum Spanning Tree:
Edge: 0 - 3 Weight: 1
Edge: 1 - 3 Weight: 2
Edge: 3 - 4 Weight: 6
Edge: 1 - 2 Weight: 3
Step 1: Initialization
• Choose a starting vertex arbitrarily from the graph.
• Create data structures to keep track of the Minimum Spanning Tree (MST) construction:
• key[]: An array to store the minimum edge weight required to connect each vertex to the MST. Initialize all values to infinity except for the starting vertex, which is set to 0.
• parent[]: An array to store the parent of each vertex in the MST. Initialize all values to -1.
• inMST[]: A boolean array or set to keep track of which vertices are already included in the MST. Initialize all values to false.
Step 2: Grow the MST
Repeat the following steps until all vertices are included in the MST:
• Find the vertex u that is not in the MST and has the minimum key value.
• Add vertex u to the MST.
• Update the key values of all adjacent vertices of u if they are not already in the MST and if the edge weight to that vertex is smaller than their current key value.
• Update the parent of adjacent vertices to u for the edges added in step c.
• Mark vertex u as included in the MST by setting inMST[u] to true.
Step 3: Termination
When all vertices are included in the MST, the algorithm terminates.
Step 4: Result
The Minimum Spanning Tree (MST) is formed by the edges that were selected during the algorithm's execution. These edges connect all vertices while minimizing the total edge weight.
• Network Design:
Prim's algorithm is commonly used in network design problems, such as the design of computer networks, electrical power distribution networks, and telecommunications networks. It helps minimize costs
while ensuring connectivity.
• Circuit Design:
In electronic circuit design, Prim's algorithm can be employed to optimize the layout of components on a circuit board, minimizing wire lengths and connections.
• Transportation and Urban Planning:
Prim's algorithm can be applied to urban planning for optimizing transportation routes, such as road networks, subway systems, or bus routes, to reduce travel distances and congestion.
• Maze Generation:
It is used to generate mazes, where the goal is to create a connected maze with a minimal number of walls or passages.
• Image Segmentation:
In image processing and computer vision, Prim's algorithm can be used for segmenting images into regions or components with minimal boundary cost.
• Cluster Analysis:
Prim's algorithm can be applied in clustering and hierarchical clustering techniques to group data points based on their similarity, forming a minimum spanning tree of data points.
• Spanning Tree Algorithms:
Prim's algorithm serves as a foundational component in other algorithms and data structures, such as Boruvka's algorithm and Kruskal's algorithm, which also find minimum spanning trees.
• Routing Protocols:
In computer networking, Prim's algorithm can be used to build routing tables or determine the best paths in network routing protocols like OSPF (Open Shortest Path First).
• Energy Distribution:
It can optimize the distribution of resources, such as electricity, water, or gas, in a network, minimizing the length of connections and infrastructure costs.
• Wireless Sensor Networks:
Prim's algorithm can help create efficient communication topologies in wireless sensor networks, where energy efficiency and connectivity are critical.
• Data Clustering and Visualization:
Prim's algorithm can be applied in data clustering and visualization tasks to identify clusters or connected components within datasets.
• Game Development:
In game development, Prim's algorithm can be used to generate game maps and layouts, ensuring that game environments are connected and navigable.
• Spanning Tree in Graph Algorithms:
Minimum spanning trees are used as subproblems in other graph algorithms, making Prim's algorithm a foundational concept in computer science.
Prim's algorithm offers several advantages that make it a valuable choice for finding minimum spanning trees (MSTs) in various applications.
• Optimality: Prim's algorithm guarantees that the MST it produces is optimal, meaning it has the smallest possible total edge weight among all possible spanning trees in the graph. This optimality
property is essential in many real-world scenarios where minimizing costs is crucial.
• Efficiency: Prim's algorithm is highly efficient, especially for dense graphs. Its time complexity is typically O(V^2), where V is the number of vertices, which makes it practical for solving
large-scale problems. With the use of more advanced data structures like binary heaps or Fibonacci heaps, it can achieve a time complexity of O(E + V log V), where E is the number of edges.
• Ease of Implementation: The algorithm is relatively easy to implement and understand. It involves simple data structures like arrays, priority queues, or heaps, making it accessible for
programmers and engineers.
• Versatility: Prim's algorithm can be applied to various types of graphs, including weighted, connected, and undirected graphs. It can handle both dense and sparse graphs, making it a versatile
choice for MST problems.
• Distributed Computation: Prim's algorithm is amenable to distributed and parallel computation. In scenarios where the graph is distributed across multiple processors or nodes, the algorithm can
be adapted to work efficiently in such environments.
• Incremental Construction: Prim's algorithm constructs the MST incrementally, allowing for easy monitoring and visualization of the growing MST. This can be useful in various applications where
the step-by-step construction of the MST is informative.
• Reduced Complexity in Sparse Graphs: In sparse graphs (where the number of edges is much less than the number of vertices squared), Prim's algorithm can be faster than Kruskal's algorithm,
another popular MST algorithm.
• Applications Beyond MST: Prim's algorithm serves as a building block for other graph algorithms and data structures. Variants of the algorithm are used in various applications, such as shortest
path algorithms and network routing.
• Guaranteed Connectivity: The MST produced by Prim's algorithm is guaranteed to be a connected tree that spans all vertices in the graph. This property is crucial in applications where
connectivity is essential.
• Simplicity and Intuitiveness: The algorithm's greedy approach, which selects edges with the smallest weights, is easy to grasp intuitively. This makes it a good choice for educational purposes
and for solving problems where a simple and clear solution is preferred.
Prim's algorithm's combination of optimality, efficiency, and ease of implementation makes it a valuable tool for solving a wide range of problems in various domains, from network design to image
processing to game development.
How Prim's Algorithm is different from other algorithms
Prim's algorithm, while a powerful and widely used algorithm for finding minimum spanning trees, has competitors or alternatives that can also solve the same problem. Here are some of the main
competitors or alternatives to Prim's algorithm for finding minimum spanning trees:
• Kruskal's Algorithm:
Kruskal's algorithm is another popular algorithm for finding minimum spanning trees. It operates by sorting all the edges by weight and then adding them to the MST in ascending order of weight, as
long as they do not create cycles. Kruskal's algorithm has a time complexity of O(E log E), where E is the number of edges, and is often preferred when the graph is sparse.
• Borůvka's Algorithm:
Borůvka's algorithm is an early precursor to Prim's algorithm and Kruskal's algorithm. It was designed to find an approximate solution to the minimum spanning tree problem by repeatedly contracting
the graph into smaller components.
• Reverse-Delete Algorithm:
The Reverse-Delete algorithm starts with all edges in the graph and iteratively removes the edges with the highest weights while maintaining connectivity. This process continues until the graph
becomes a spanning tree. It can be less efficient than Prim's or Kruskal's algorithm for dense graphs.
• Boruvka-Minimum Spanning Tree Algorithm:
This algorithm is a modification of Borůvka's algorithm designed to find an approximate MST efficiently. It divides the graph into smaller connected components and computes the MST for each
component, then connects these MSTs to form the final MST.
• Jarník's Algorithm (also known as Prim-Jarník or Prim's Algorithm with Fibonacci Heaps):
An optimized version of Prim's algorithm that uses Fibonacci heaps to speed up the priority queue operations, resulting in a faster implementation with a time complexity of O(E + V log V). This
variant is especially efficient for dense graphs.
• Randomized Algorithms:
Randomized algorithms, such as Randomized Prim's Algorithm and Randomized Kruskal's Algorithm, introduce randomness to the process of selecting edges. These algorithms can provide approximate MSTs
with high probability and can be useful in situations where exact solutions are not necessary.
• Boruvka-Chazelle Algorithm:
An improvement on Borůvka's algorithm that uses efficient data structures like Fibonacci heaps to achieve better time complexity for finding approximate MSTs.
The choice between these algorithms depends on factors such as the specific problem at hand, the characteristics of the graph (e.g., density), and performance considerations. While Prim's algorithm
is known for its simplicity and efficiency in dense graphs, Kruskal's algorithm is often preferred for sparse graphs. Researchers and engineers select the most appropriate algorithm based on the
problem's requirements and the available computational resources.
• Sensitivity to Edge Weights: Prim's algorithm assumes that edge weights are unique. If there are multiple edges with the same weight, the algorithm's behavior may be unpredictable, as it doesn't
have a clear way to choose between equivalent edges. This can lead to different MSTs for graphs with duplicate edge weights.
• Inefficient for Sparse Graphs: In graphs where the number of vertices (V) is much larger than the number of edges (E), Prim's algorithm can be less efficient compared to other algorithms like
Kruskal's algorithm, which has a better time complexity in such scenarios (O(E log E)).
• Not Suitable for Directed Graphs: Prim's algorithm is designed for undirected graphs. It cannot be directly applied to directed graphs, as it relies on the symmetry of undirected edges.
• Dependency on a Starting Vertex: The choice of the starting vertex can affect the resulting MST. While the overall structure of the MST remains the same, the specific edges in the MST may vary
depending on the starting vertex. This dependency can be a drawback in certain situations where a unique MST is desired.
• Inefficient for Dynamic Graphs: If the graph is dynamic and edges are frequently added or removed, recomputing the entire MST using Prim's algorithm from scratch can be inefficient. Specialized
algorithms designed for dynamic graphs may be more suitable in such cases.
• Lack of Parallelism: Prim's algorithm is inherently sequential, and its steps are not naturally parallelizable. In applications where parallel processing is essential, other algorithms or
parallel variants may be more suitable.
• Memory Usage: The algorithm requires memory to store data structures like arrays, priority queues, or heaps. In some cases, the memory usage can be a limiting factor, especially for very large
• Limited to Connected Graphs: Prim's algorithm assumes that the input graph is connected. If the graph is not connected, the algorithm will find the minimum spanning tree for each connected
component separately.
• Not Suitable for Negative Weight Edges: Prim's algorithm does not handle graphs with negative weight edges, as it assumes that all edge weights are non-negative. In such cases, algorithms like
Dijkstra's algorithm or Bellman-Ford algorithm may be more appropriate.
Despite these disadvantages, Prim's algorithm remains a valuable tool for finding minimum spanning trees in various practical scenarios, especially when applied to dense graphs with unique edge
weights and when computational efficiency is not a primary concern. Researchers and engineers should carefully consider the specific requirements of their problem and the characteristics of their
data when choosing an algorithm for MST computation.
Prim's Algorithm Overview:
• Prim's algorithm is a greedy algorithm used to find the Minimum Spanning Tree (MST) in a connected, undirected graph.
• The MST is a subset of the edges of the graph that connects all vertices while minimizing the total edge weight.
• Prim's algorithm incrementally builds the MST, starting from an initial vertex and adding vertices and edges one at a time until the MST is complete.
Key Steps in Prim's Algorithm:
• Choose an arbitrary starting vertex.
• Initialize data structures:
• key[]: Minimum edge weights required to connect each vertex to the MST. Initialize all values to infinity except the starting vertex (set to 0).
• parent[]: Stores the parent of each vertex in the MST. Initialize all values to -1.
• inMST[]: Tracks which vertices are included in the MST. Initialize all values to false.
Iterative Process:
Repeat the following steps until all vertices are included in the MST:
• Find the vertex u with the minimum key value among vertices not in the MST.
• Add u to the MST.
• Update the key values of adjacent vertices if they are not in the MST and if the edge weight to u is smaller than their current key value.
• Update the parent of adjacent vertices to u for the edges added in step c.
• Mark vertex u as included in the MST by setting inMST[u] to true.
The algorithm terminates when all vertices are included in the MST.
The Minimum Spanning Tree (MST) is formed by the edges selected during the algorithm's execution.
Representation of the Graph:
The graph can be represented using an adjacency matrix or an adjacency list, depending on the specific implementation.
• Prim's algorithm is used in various applications, including network design, clustering, and image segmentation.
• It's especially useful in scenarios where you want to connect a set of points (vertices) with minimum total edge weight.
In conclusion, Prim's algorithm is a fundamental method for finding the Minimum Spanning Tree in a graph. It efficiently constructs a tree that connects all vertices while minimizing the total edge
weight. It's a valuable tool in various fields where optimization and efficient network design are required.
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NCERT Solutions For Class 9, Maths, Chapter 6, Lines And Angles - Solutions For Class
If you’re a student in Class 9 studying Maths, then you know that NCERT Solutions are essential for scoring well in exams. Chapter 6 of NCERT Maths textbook, Lines and Angles, is an important topic
that requires a strong understanding of the concepts. That’s why we’ve created free NCERT Solutions for Class 9 Maths Chapter 6 that are easy to understand and comprehensive. Our solutions cover all
the important topics such as parallel lines, transversals, alternate and corresponding angles, and more. With our NCERT Solutions, you can improve your problem-solving skills and ace your exams. So,
what are you waiting for? Access our NCERT Solutions for Class 9 Maths Chapter 6 now and start your journey towards academic excellence.
NCERT Solutions For Class 9 Maths Chapter 6, Lines And Angles (All Exercises)
Class 9, Maths, Chapter 6, Lines And Angles ← You are here
Class 9, Maths, Chapter 6, Lines And Angles , Exercise 6.1
Class 9, Maths, Chapter 6, Lines And Angles , Exercise 6.2
Class 9, Maths, Chapter 6, Lines And Angles , Exercise 6.3
Frequently Asked Questions on NCERT Solutions for class 9 maths Chapter 6, Lines And Angles
What will I learn in NCERT Solutions for class 9 maths Chapter 6, Lines And Angles? write in points
NCERT Solutions for Class 9 Maths Chapter 6, Lines And Angles will cover following topics:
• Basic terminology related to lines and angles.
• Types of angles such as complementary, supplementary, adjacent, etc.
• Properties of parallel lines and transversals.
• Corresponding angles, alternate angles, interior angles, and exterior angles.
• Proving theorems related to angles and lines.
• Applications of angle and line concepts in real-life situations.
• Solving problems involving lines and angles using algebraic equations.
• Developing problem-solving skills and logical reasoning abilities.
NCERT Solutions for Class 9 Maths Chapter 6 will help you to gain a strong understanding of these concepts and improve your performance in exams.
How are NCERT Solutions for Class 9 Maths Chapter 6 helpful for CBSE exam preparation?
NCERT Solutions for Class 9 Maths Chapter 6 are helpful for CBSE exam preparation as they provide a comprehensive understanding of fundamental concepts, step-by-step explanations, and practice
questions that help in developing problem-solving skills and boosting confidence. These solutions also cover a range of question types, which are helpful for preparing for different types of
questions that can be asked in the exams. Overall, these solutions provide a good practice for the board exams, making them very helpful for CBSE exam preparation.
How to score high marks in NCERT Solutions for Class 9 Maths Chapter 6 in exams?
To score high marks in NCERT Solutions for Class 9 Maths Chapter 6 in exams, students can follow these tips:
• Understand the concepts thoroughly and practice regularly.
• Focus on the fundamentals such as coordinates, plotting of points, quadrants, and distance formula.
• Solve a variety of practice questions from NCERT Solutions for Class 9 Maths Chapter 6.
• Understand the logic behind the solutions provided and practice similar problems.
• Make a formula sheet and revise it regularly.
• Practice using a calculator and understand its basic functions.
• Solve previous year question papers and sample papers to get an idea of the exam pattern and types of questions asked.
• Manage time effectively during the exam and attempt all questions.
By following these tips, students can score high marks in NCERT Solutions for Class 9 Maths Chapter 6 in exams.
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Engine Design Equations Formulas Calculator Fuel System Flow Injector Size
Online Web Apps, Rich Internet Application, Technical Tools, Specifications, How to Guides, Training, Applications, Examples, Tutorials, Reviews, Answers, Test Review Resources, Analysis, Homework
Solutions, Worksheets, Help, Data and Information for Engineers, Technicians, Teachers, Tutors, Researchers, K-12 Education, College and High School Students, Science Fair Projects and Scientists
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seminars - Optimal Control for ODEs and PDEs: The Turnpike Phenomenon
The turnpike phenomenon for dynamic optimal control problems provide insights about the relation between the dynamic optimal control and the solution of the corresponding static optimal control
problem. In this talk we give an overview about different turnpike structures for optimal control problems with ordinary differential equations (ODEs) and partial differential equations (PDEs).
For optimal control problems with ODEs an exponential turnpike inequality can be shown by basic control theory. These results can be extended to an integral turnpike inequality for optimal control
problems with linear hyperbolic systems. For an optimal control problem with non differential tracking term in the objective function, that is exactly controllable, we can show under certain
assumptions that the optimal system state is steered exactly to the desired state after finite time.
Further we consider an optimal control problem for a hyperbolic system with random boundary data and we show the existence of optimal controls.
A turnpike property for hyperbolic systems with random boundary data can be shown numerically.
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Communication-Reduced Distributed Control and Optimization of Multi-Agent Systems
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Communication-Reduced Distributed Control and Optimization of Multi-Agent Systems
This dissertation proposes communication-reduced solutions to the containment control, distributed average tracking and distributed time-varying optimization problems of multi-agent systems.
The objective of containment control in multi-agent systems is to design control algorithms for the followers to converge to the convex hull spanned by the leaders. Sampled-data based containment
control algorithms are suitable for the cases where the power supply and sensing capacity are limited, due to their low-cost and energy-saving features resulting from discrete sensing and
interactions. In addition, sampled-data control has advantages in performance, price and generality. On the other hand, when the agents have double-integrator dynamics and the leaders are dynamic
with nonzero inputs, the existing algorithms are not directly applicable in a sampled-data setting. To this end, this dissertation proposes a sampled-data based containment control algorithm for a
group of double-integrator agents with dynamic leaders with nonzero inputs under directed communication networks. By applying the proposed containment control algorithm, the followers converge to the
convex hull spanned by the dynamic leaders with bounded position and velocity containment control errors, and the ultimate bound of the overall containment error is proportional to the sampling
period. In the distributed average tracking problem, each agent uses local information to track the average of individual reference signals. In some practical applications, velocity measurements may
be unavailable due to technology and space limitations, and it is also usually less accurate and more expensive to implement. Before deriving the event-triggered approach, we first present a base
algorithm without using velocity measurements, which sets the stage for the development of the event-triggered algorithm. The base algorithm has an advantage over the existing related works in the
senses that there is no global information requirement for parameter design. Building on the base algorithm, we present an event-triggered algorithm that further removes continuous communication
requirement and is free of Zeno behavior. It is suitable for practical implementation since in reality the bandwidth of the communication network and power capacity are usually constrained. The
event-triggered algorithm overcomes some practical limitations, such as the unbounded growth of the adaptive gain and requirement of additional internal dynamics, by constructing a new triggering
strategy. In addition, a continuous nonlinear function is used to approximate the signum function to reduce the chattering phenomenon in reality.
In distributed optimization of networked systems, each member has a local cost function, and the goal is to cooperatively minimize the sum of all the local cost functions. The distributed
time-varying optimization problem is investigated for networked Lagrangian systems with parametric uncertainties in the dissertation. Usually, in the literature, to address some distributed control
problems for nonlinear systems, a networked virtual system is constructed, and a tracking algorithm is designed such that the agents' physical states track the virtual states. It is worth pointing
out that such an idea requires the exchange of the virtual states and hence necessitates communication among the group. In addition, due to the complexities of the Lagrangian dynamics and the
distributed time-varying optimization problem, there exist significant challenges. This dissertation proposes distributed time-varying optimization algorithms that achieve zero optimum-tracking
errors for the networked Lagrangian agents without the communication requirement. The main idea behind the proposed algorithms is to construct a reference system for each agent to generate a
reference velocity using absolute and relative physical state measurements with no exchange of virtual states needed, and to design adaptive controllers for Lagrangian systems such that the physical
states are able to track the reference velocities and hence the optimal trajectory. The algorithms introduce mutual feedback between the reference systems and the local controllers via physical
states/measurements and are amenable to implementation via local onboard sensing in a communication unfriendly environment. Specifically, first, a base algorithm is proposed to solve the distributed
time-varying optimization problem for networked Lagrangian systems under fixed graph. Then, based on the base algorithm, a continuous function is introduced to approximate the signum function,
forming a continuous distributed optimization algorithm and hence removing the chattering. Then, by using the structure of the base algorithm, a distributed time-varying optimization algorithm is
designed for networked Lagrangian systems under switching graphs.
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The Squared Circle
How does that end? How do you finish that division???
See post #59 as I said before. What didn't you understand about what I did there?
See post #59 as I said before. What didn't you understand about what I did there?
Post #59 has no pic of your completed long division of 1 divided by 3, MORON!
Post a pic of your completed long division of 1 divided by 3.
..or explain how that pic of the long division is completed. Your choice!
Post #59 has no pic of your completed long division of 1 divided by 3
Long division for \(1/10_3\)? Seriously? Oooookaaaay.
0.1\phantom{)} \\
Long division for \(1/10_3\)? Seriously? Oooookaaaay.
0.1\phantom{)} \\
Moron, Look at the pic of the long division of 1 divided by 3. You see how the remainder is always 1 and a zero is brought down to make that 1 a 10? Place a 3 in the next decimal place and 3 x 3 is
9. Place the 9 below the 10 and subtract 9 from 10 to get 1. Repeat INFINITELY and the division never ends. You can not complete the division of 1 divided by 3, because there is always a remainder of
1 that needs to be divided equally by 3, which can't be done!
They teach this crap in 2nd grade, did you miss that day? Do you not know how to do long division??
They teach this crap in 2nd grade, did you miss that day?
So what didn't you understand about what I did?
So what didn't you understand about what I did?
Do you agree that the division of 1 divided by 3 can not be completed?
Do you agree that the division of 1 divided by 3 can not be completed?
I literally just did it for you in #65 and I keep asking what you didn't understand about how I did it so we can move forward but you keep not answering.
Heading towards oblivion
Valued Senior Member
Moron, Look at the pic of the long division of 1 divided by 3.
Why do you call people morons when
can't understand something?
I literally just did it for you in #65 and I keep asking what you didn't understand about how I did it so we can move forward but you keep not answering.
You did not. Either explain the pic of the long division or show your pic of how you completed that long division.
SHOW ME how you complete the long division of 1 divided by 3, or explain how in the pic of that division I posted the division can be completed.
You can see clearly in that pic that there is no end to that division, right? If you can't see that then you have no idea what you're looking at and you can't do long division.
That pic is PROOF that the long division of 1 divided by 3 CAN'T be completed, it goes on infinitely and can not be completed because 1 is not equally divided by 3!
SHOW ME how you complete the long division of 1 divided by 3
See post #65. What didn't you understand about how I did it?
See post #65. What didn't you understand about how I did it?
You did not show me a long division or how you completed the long division of 1 divided by 3.
1. Show me a pic of your completed long division.
2. Explain how the long division ends in the pic I posted.
Your choice. Pick one!
You did not show me a long division or how you completed the long division of 1 divided by 3.
So what you are saying is that you don't recognise at all that that is exactly what I did in #65. Why not just say so?
So what you are saying is that you don't recognise at all that that is exactly what I did in #65. Why not just say so?
I will say it again, you can not complete the long division of 1 divided by 3. There is nothing you can say to change that. If you think you can do it then show me a pic of the long division of 1
divided by 3 being completed.
SHOW ME!
If you don't have a pic then explain the pic I posted. Simple right? Just explain how you would complete the division in that pic!
OK. So you don't understand what I did and can't or won't admit that you don't understand. Admitting you don't understand something is the first step to understanding it and if you can't do that you
will remain as dumb as a bag of hammers.
For anybody with half a brain but who missed it the first time around I chose a representation of 3 which was easier to work with namely 10 in base three so a zero in the units column and a one in
the threes column. Then you can use long division to calculate what 1/10 is if you must but it's trivial it's 0.1 which should be read as a zero in the units column and a one in the thirds column.
You can pull this trick with any non-integer rational by writing it as a ratio of integers \(m/n\) then expressing it in base \(n\). If you don't like that solution and want to remain in base 10 then
the infinite sum I gave in #42 is another way to represent it and that's just a geometric series (first term 0.3 common ratio 0.1) that you can use the standard formula to sum if you want to check it
There are loads of good representations of any rational and only an idiot would try to write one third exactly in base ten positional notation so I think I'm gonna leave Motor Daddy to it.
So no pic and no explanation of the pic I posted.
But I already knew you could not explain because it's not possible.
Listen very closely: 1 can NOT be divided by 3 equally.
1/2=.5 and .5 x 2=1.0
1/4=.25 and .25 x 4=1.0
1/5=.20 and .20 x 5=1.0
See how you can check your work and they all equal 1.0??
Now try that with 1/3
1/3=.333... and .333... x 3=.999...
Did you catch that? It DID NOT add up to 1.0, it added up to .999...
You know you started with 100% and it ended up with 99.999...%
You know why, right? Because the division of 1/3 never was completed, and the remainder of 1 was not included in the answer of .999...
You swept the last little remainder under the rug and then pretended like you completed the division. That's a NO-NO! You don't get to sweep some under the rug!
Listen very closely: 1 can NOT be divided by 3 equally.
You crack me up man, you really do.
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Order of Operations Worksheets
Navigate through this set of order of operations worksheets and learn the sequence in which an equation with multiple terms and operators is to be solved. Our exclusive PEMDAS and DMAS worksheets
come with answer keys and include exercises like evaluating expressions with two operators, expressions with parentheses, brackets, and braces, evaluating expressions with nested parentheses, and
many more. Make the most of our free worksheets!
Evaluating Expressions Involving Two Operators
Observe the two operators, figure out the correct sequence, and evaluate each expression according to the order. This bundle of DMAS worksheets is designed for kids in grade 4 and grade 5.
Evaluating Expressions Involving Three Operators
Offering practice in evaluating expressions involving three operators, this set of exercises requires learners to divide or multiply before they add or subtract while evaluating expressions.
Evaluating Expressions Involving Multiple Operators
Get plenty of practice evaluating expressions with up to 5 operators with this compilation. Learning operator precedence is the key to simplifying expressions with multiple operators.
Evaluating Numerical Expressions with Integers
Introduce kids to integer operations with our order of operations worksheets. Stick to the sign rules to find the sum, difference, product, and quotient of integers and obtain the values of
Evaluating Numerical Expressions with Fractions
Featured here are arithmetic expressions involving like and unlike fractions. Find the LCM, determine the sum, difference, product, and quotient of fractions, and reduce the terms to their simplest
Evaluating Numerical Expressions with Decimals
Augment simplification skills with our PEMDAS worksheets! Evaluate expressions by adding, subtracting, multiplying, and dividing decimals with up to hundredths places, and decimals with exponents.
Expressions with Parentheses | Easy
Each expression contains a single pair of parentheses and two operators. Simplify the parentheses and move on to the operations to complete these practice sheets.
Evaluating Expressions Involving Exponents
Excel in simplifying arithmetic expressions with exponential notations. Evaluate whole numbers and integers raised to single-digit powers and practice simplifying the arithmetic expressions.
Comparing Numerical Expressions | Easy
Acquire the skill of comparing expressions with no parentheses! Kids in 6th grade, 7th grade, and 8th grade obtain the values of the expressions, compare them using comparison signs, and match
equivalent expressions.
Comparing Numerical Expressions | Moderate
Solve each pair of equations and compare them using the “equal to” or “not equal to” sign in part A, <, >, or = in part B and match expressions in part C in this bundle of printables.
Work your way through a host of equations with missing operators, where the values of the expressions are specified. All you need to do is rearrange the equations, perform the operations in order,
and find the unknown operators.
Raise the bar with this set of pdfs abounding in arithmetic equations with parentheses. Solve each equation using PEMDAS or BODMAS, and find the operator that makes the equation true.
Expressions with Parentheses, Brackets, and Braces
Learn to reduce expressions with multiple grouping symbols with this set of expressions with parentheses, brackets, and braces worksheets. Keep in mind that parentheses take precedence, followed by
brackets and braces.
Evaluating Expressions with Nested Parentheses
Simplify the innermost parentheses first and proceed toward the next, until you reach the outermost parentheses. Always evaluate the expression from the inside-out using the PEMDAS or BODMAS rule.
Evaluating Numerical Expressions with Rational Numbers
There are 10 expressions in each of these order of operation worksheets. Precisely perform the multiplication, division, addition, and subtraction of positive and negative rational numbers and
simplify arithmetic expressions without a hitch.
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All subjects to learn
All Subjects
Sections/topics include: Prealgebra (Operations, Whole Numbers, Decimals, Fractions, Percents, Integers and Rationals, Powers, Exponents, and Roots, Measurements, Perimeter and Area, Variables),
Geometry (Building Blocks, Polygons, Circles, Surfaces, Geometric Measurements, Special Triangles, Congruence, Theorems, Inductive / Deductive Reasoning, Logic Statements, Axioms, Postulates,
Proofs), Algebra (Expressions, Equations, Inequalities, Absolute Values, Probability, Graphing Data/ Equations, Exponents, Scientific Notation, Polynomials, Quadratics), Binomial Expansion,
Factoring, Matrices, Rational Expressions, Complex Numbers, Functions, Inequalities, Graphs, Polynomials), Trigonometry (Angles, Graphs, Trigonometric Equations / Functions / Identities, Oblique
/ Right Triangles), Precalculus (Complex Numbers, Conic Sections, Continuity, Limits, Parametric Equations, Polar Coordinates, Exponential / Logarithmic Functions, Sequences, Series), Calculus
(Functions, Limits, Derivatives, Integrals, the Taylor Series, Polar Curves). From SparkNotes.com.
Sections/topics include: Prealgebra (Operations, Whole Numbers, Decimals, Fractions, Percents, Integers and Rationals, Powers, Exponents, and Roots, Measurements, Perimeter and Area, Variables),
Geometry (Building Blocks, Polygons, Circles, Surfaces, Geometric Measurements, Special Triangles, Congruence, Theorems, Inductive / Deductive Reasoning, Logic Statements, Axioms, Postulates,
Proofs), Algebra (Expressions, Equations, Inequalities, Absolute Values, Probability, Graphing Data/ Equations, Exponents, Scientific Notation, Polynomials, Quadratics), Binomial Expansion,
Factoring, Matrices, Rational Expressions, Complex Numbers, Functions, Inequalities, Graphs, Polynomials), Trigonometry (Angles, Graphs, Trigonometric Equations / Functions / Identities, Oblique
/ Right Triangles), Precalculus (Complex Numbers, Conic Sections, Continuity, Limits, Parametric Equations, Polar Coordinates, Exponential / Logarithmic Functions, Sequences, Series), Calculus
(Functions, Limits, Derivatives, Integrals, the Taylor Series, Polar Curves). From SparkNotes.com.
Sections/topics include: Prealgebra (Operations, Whole Numbers, Decimals, Fractions, Percents, Integers and Rationals, Powers, Exponents, and Roots, Measurements, Perimeter and Area, Variables),
Geometry (Building Blocks, Polygons, Circles, Surfaces, Geometric Measurements, Special Triangles, Congruence, Theorems, Inductive / Deductive Reasoning, Logic Statements, Axioms, Postulates,
Proofs), Algebra (Expressions, Equations, Inequalities, Absolute Values, Probability, Graphing Data/ Equations, Exponents, Scientific Notation, Polynomials, Quadratics), Binomial Expansion,
Factoring, Matrices, Rational Expressions, Complex Numbers, Functions, Inequalities, Graphs, Polynomials), Trigonometry (Angles, Graphs, Trigonometric Equations / Functions / Identities, Oblique
/ Right Triangles), Precalculus (Complex Numbers, Conic Sections, Continuity, Limits, Parametric Equations, Polar Coordinates, Exponential / Logarithmic Functions, Sequences, Series), Calculus
(Functions, Limits, Derivatives, Integrals, the Taylor Series, Polar Curves). From SparkNotes.com.
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Question about reading Skating Scores for Ice Dance
Hi folks!
I am reviewing SkatingScores for greater understanding about how the judges arrived at the score given. I am a bit confused reading the tables for the RD. I am not sure how base and total boxes work
- are the scores given for each each partner of a base and multiplier to get a total ranking.
For example the twizzles, the first 6 couples all got level 4 for the twizzles, the same base of 1 and different totals. How can I tell how the total was derived and what made the difference? In the
step seq for C/B each partner got a different level W4-M2 and the base was 8.21, total 11.94, L/B scored W3-M3 and the base was 8.20, total10.89. What am I not seeing to account for the 1.05
difference? Not trying to prove a point just understand how this chart works. Also trying to understand what the total differences reflect between S/D and B/B when their levels were the same. Is
their a way to tell what the Browns were rewarded for that Olivia & Tim were not. Is their a multiplier for overall quality or is that determined by a sum for each skater of a team? Again just trying
to make an educated read of the scoring results and be a smarter viewer.
Thanks for any help in furthering my education!
I feel stupid now - the difference are just the GOE?
How do you even get to that chart which leaves out the GOE?
I feel stupid now - the difference are just the GOE?
How do you even get to that chart which leaves out the GOE?
I chose
RD element rank
How do you even get to that chart which leaves out the GOE?
I don't see anything actually marked as GOE, but I do see PCS
The best resource for Figure Skating scores and stats on the web.
and PCS broken out here...
The best resource for Figure Skating scores and stats on the web.
I don't see anything actually marked as GOE, but I do see PCS
The best resource for Figure Skating scores and stats on the web.
and PCS broken out here...
The best resource for Figure Skating scores and stats on the web.
I always just look at the protocols.
I think there's a lot more the site offers, but I don't know much of it.
Under the COUNTRY is the GOE the judge for that country awarded for each element---it's a single digit.
The PCS (two decimal places) at the bottom is what each judge awarded for the performance as a whole.
Under the COUNTRY is the GOE the judge for that country awarded for each element---it's a single digit.
The PCS (two decimal places) at the bottom is what each judge awarded for the performance as a whole.
What is puzzling me is the
GOE total column
- It looks like an average of the individual GOE
but it is not
whether you factor in all the judges or everyone but the high & low. It is is added to the Base Value to get each element score.
Example: C/B 8.21+3.73 =11.94 where did the 3.73 come from?
The math doesn't work for any team.
This is where I am seeing this
The best resource for Figure Skating scores and stats on the web.
You have to go into the scale of values to see how much GOE you can get for each element. For the 8.21 step sequence, the most you could get for that element is 5.22, so you have to take the average
of the GOE, get a percentage where 5 = 100%, and then multiply that be 5.22, and you'll get your GOE.
Ice dance is confusing, as the choreographic elements have huge GOE range, in order to encourage creativity and innovation, so the GOE amounts per element are not intuitive. I have no idea how they
came up with their GOE. In other disciplines, the most GOE you can get is 50% of the base value of the element.
In other disciplines, the most GOE you can get is 50% of the base value of the element.
Except for the choreographic step sequence in the free skate for singles and pairs, where the base value is 3.00 and the GOEs are in increments of 0.5, so the most you can get in positive GOE is 2.5,
i.e., more than 50% of the base value.
Same reasoning as the choreographic dance elements.
Hmm...I will have to play around with the numbers. I just wanted to understand how the numbers were arrived at and understand if there was a multiple for placement. It seems more complex than that.
Hard for casual viewers to understand. Thank you for helping me learn!
Hmm...I will have to play around with the numbers. I just wanted to understand how the numbers were arrived at and understand if there was a multiple for placement. It seems more complex than
that. Hard for casual viewers to understand. Thank you for helping me learn!
You don't really need to "play around." Just click the Score link in the right hand column for the element of interest. That opens up another window that explicitly shows how the GOE is calculated.
You don't really need to "play around." Just click the Score link in the right hand column for the element of interest. That opens up another window that explicitly shows how the GOE is
I feel silly now - thanks!
You don't really need to "play around." Just click the Score link in the right hand column for the element of interest. That opens up another window that explicitly shows how the GOE is
I didn't know that, cool!
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IVs of Wild Pokemon
This question is not very simple to answer, so I thought it warranted a thread of its own. Before asking the question, here is how I was reasoning.
I was thinking about how the IVs of wild Pokemon are created. If they're created completely at random, that means that there is a 1/32 chance for an IV to be 31. This means that the probability of an
IV not being 31 is 31/32.
Continuing the argument further, the probability that a Pokemon in the wild has all its 6 IVs not 31 is (31/32)^6 = 0.82655.
This means that the probability that a wild Pokemon has at least one of its IVs a perfect 31 is 1 - 0.82655 = 0.17345, or 17.345%. This is roughly 1 in 6 wild Pokemon (slightly more than that, in
I'd like to know if this translates to what really happens, since I'm thinking that "1 in 6" Pokemon is too high of a percentage. My hunch is, if there are indeed 1 in 6 Pokemon to have a perfect IV
in at least one of the stats, then the IVs are truly generated at random when a wild Pokemon appears. If not, then the IVs for wild Pokemon are not generated completely randomly, but there is a bias
towards low or high IVs.
Is there anyone having somehow already tested this can shed some light on the matter? Feedback would be appreciated even if you had done this in Advance, GSC or even RBY, instead of DP.
lol I just wrote this in the *coughyouknowwhatweareworkingoncough* and wanted to ask you later if it´s accurate?
probability for a wild pokémon to have one flawless IV = 6/32 = 1 in 6 (rounded up)
probability for a wild pokémon to have two flawless IVs = 15/1024 = 1 in 69
probability for a wild pokémon to have three flawless IVs = 20/32 768 = 1 in 1639
probability for a wild pokémon to have four flawless IVs = 15/1 048 576 = 1 in 69 906
probability for a wild pokémon to have five flawless IVs = 6/33 554 432 = 1 in 5 592 406
probability for a wild pokémon to have six flawless IVs = 1/1 073 741 82
I think they are really randomly generated, during my ditto hunt there were always around 5 or 6 with a flawless IV in one box (and I ´ve caught hundreds of them)...I think there are two little
samples of ditto catching results in the what to do in emerald topic (though it was for a different purpose)
BobDoily´s results
30 magikarps caught
15 were same nature
0 had 31 DEF IV
10 did have a flawless IV though
5 of them were on synchronized nature
my results:
jolly, wonderfully outstanding synchronize ralts, max speed (only one)
21 dittos caught
10 jolly
-> one of them had 31hp/x/31def/25 sA/24sD/25speed, jolly
-> 31 speed once, quirky
-> 31 defense, jolly
Well, actually I needed this for the little project we're working on... are we maybe writing the same thing? :] Anyway, thanks.
Data Integration Thought Entity
I know it's not completely random as you can't get a program to do that. But I couldn't tell you how close it is to being truly random.
I know Emerald's method of wild Pokemon generation had a tendency to generate identical Pokemon if you use Soft Reset. I threw a Master Ball at Kyogre, marked down its stats, reset and repeated.
Eventually I started seeing Kyogre with identical nature and stats to old Kyogre I had seen before. After I had around 200 Kyogre marked down, I decided to just choose the best of those 200+.
Eventually it did come around around and that's the one I have right now.
Once again in Emerald, I was hunting for a Ditto. I would catch one, check it's stats then soft reset if I didn't like it. Eventaully I started seeing identical Ditto.
So far in DP, neither of these event seem to occur. I don't know if they occur in RS or FL, I don't think so but I haven't really tested it much.
Ah. Glad to see I'm not the only one seeing identical Dittos.
I know it's not completely random as you can't get a program to do that. But I couldn't tell you how close it is to being truly random.
I know Emerald's method of wild Pokemon generation had a tendency to generate identical Pokemon if you use Soft Reset. I threw a Master Ball at Kyogre, marked down its stats, reset and repeated.
Eventually I started seeing Kyogre with identical nature and stats to old Kyogre I had seen before. After I had around 200 Kyogre marked down, I decided to just choose the best of those 200+.
Eventually it did come around around and that's the one I have right now.
Once again in Emerald, I was hunting for a Ditto. I would catch one, check it's stats then soft reset if I didn't like it. Eventaully I started seeing identical Ditto.
So far in DP, neither of these event seem to occur. I don't know if they occur in RS or FL, I don't think so but I haven't really tested it much.
OmegaDonut said:
Ah. Glad to see I'm not the only one seeing identical Dittos.
I'm thinking this happens because of the random number seed.
Random number generation in machines isn't random, as such. It is actually pseudo-random, which means that the random numbers are generated by a mathematical formula which gives the next random
number in a sequence. The thing is, if you give it the same initial number (called the seed), it will generate the exact same sequence of 'random' numbers. The seed is usually generated by the amount
of time that has passed since the machine started. This is how random number generation in computers works, and I suppose it's how it works in consoles, too.
It might be that the random number seed for Emerald is only one byte long, which is a number between 0 and 255. So that would mean that there would only be 256 possible IV lists/Natures/Personalities
/etc. for any soft-resetted Pokemon, which is why people start seeing the same exact Pokemon after a number of resets.
TEST: route 101 (emerald)
100 pkmn caught, of which
50 were wurmple (W) = 8/16
43 were poochyena (P) = 7/16
7 were zigzagoon (Z) = 1/16
the 100 were four sets of 25 pkmn (as I don´t have more room in my boxes)
A. 3 flawless IVs, 12P, 12W, 1Z
B. 12 flawless IVs, 12P, 13W, 0Z
C. 4 flawless IVs, 8P, 15W, 2Z
D. 5 flawless IVs, 11P, 10W, 4Z
flawless IVs:
HP: P, W
AT: P, P, P, W, P
DEF: W, P, W, P, W, Z
SP.AT: P, P
SP.DEF: W, P, W
SPEED: W, W, W, W, P
one poochyena got flawless HP and AT
one wurmple got flawless AT and SPEED
probability for a wild pokémon to have one flawless IV = 6/32 = 1 in 6 = 18.75 in 100 <-> I got 24 (20 single flawless IVs+ 2 with two flawless)
probability for a wild pokémon to have two flawless IVs = 15/1024 = 1 in 69 = 1.46484375 in 100 <-> I got two
@ TRE: that´s exactly what happened to me...I was catching an EON latios in emerald and started to write them down because some of them came again, some of them even on a second emerald of a
friend...Eeevee trainer actually asked the same thing a month ago, here´s the link - I´ve posted my latios list there
yeah I think what X-Act is saying is how it works
Nice, Peterko.
You actually caught 22 Pokemon that have at least one perfect IV, not 24. The Pokemon having more than one flawless IVs are counted as one. :]
I had missed that EeveeTrainer post. I counted the amount of Latias that you found, and you got 169 in all. There, you had said the following:
Peterko said:
...if you save and meet a pkmn at a particular time (dunno what a frame is), you can meet it again after a soft reset if it happens at the same "time"
This is in line with my random number seed theory. The random number generator is usually fed the 'time' taken from the start of the game.
It might be that the random number seed for Emerald is only one byte long, which is a number between 0 and 255. So that would mean that there would only be 256 possible IV lists/Natures/
Personalities/etc. for any soft-resetted Pokemon, which is why people start seeing the same exact Pokemon after a number of resets.
IMO, the seed is probably larger than that. You only need one seed on startup, and I can't imagine they'd be hurt for space that badly where a couple bytes on one variable would make the difference.
According to the
birthday paradox
if the seed was only one byte long you should start seeing identical legendaries by the first couple dozen rather than a couple hundred.
I'm aware of the birthday paradox. TRE only said that 'eventually' he started seeing the same Pokemon.
Anyway, if the seed is only one byte long, you should see a Pokemon you have already seen by around the 19th soft reset.
If the seed is two bytes long (between 0 and 65535), you should see a Pokemon you have already seen by around the 302nd Pokemon. I don't think this is the case.
The rough formula to find these '19' and '302' is (1 + sqrt(1 + s * ln(256))) / 2, by the way, where s is the maximum seed.
emeralds random number seed is 4 bytes. calling rand() always returns 2 bytes generated from these 4 bytes and changes the 4 bytes by multiplying and adding something.
depending on the intrest i think i could reasearch how exactly the seeding is done (yes it it is something with the time but i don't remember if it was rtc or gametime). but i don't really think i
will get you interesting results.
the funny part is that if you call rand() twice in a row (getting 4 bytes) and if you don't waste any of that data it might (should?) be possible to predict the next result. which would severely
limit the possibilitys of pokemon which can be created (unless there are a lot of unpredictable interrupts during the process). it would basicly mean if you know the pid or the you know everything.
And if thats really the case and if you could figure this out it would also be possible to detect every cheated pokemon to this day ;)) Now that would be interesting but a bitch to reverse engineer.
Nice, Peterko.
You actually caught 22 Pokemon that have at least one perfect IV, not 24. The Pokemon having more than one flawless IVs are counted as one. :]
I had missed that EeveeTrainer post. I counted the amount of Latias that you found, and you got 169 in all. There, you had said the following:
This is in line with my random number seed theory. The random number generator is usually fed the 'time' taken from the start of the game.
Odd. I at one time had theorized while Soft Resetting for a Shiny Treecko that possibly I was hitting A at the wrong time and therefore I'd keep getting the same Treecko unless I waited a couple
minutes before I checked the briefcase. I eventually gave up, thinking it's be too much work for someone like I who doesn't know how the Treecko's are generated to try to guess at what time should I
hit A to get a Shiny.
So, basically, I had theorized that Shiny chance was also a property of the Random Number Generator. To take it a step to the side, I can also see how the Random Number Generator also has an effect
on the EVs of a Pokemon.
emeralds random number seed is 4 bytes. calling rand() always returns 2 bytes generated from these 4 bytes and changes the 4 bytes by multiplying and adding something.
That is how most random number generators work. It's usually:
NextRandomNumber = (PreviousRandomNumber * N + R) mod M
where N, R and M are chosen accordingly. (Usually, M = 2^n for some n, N is a very number, and R is usually a prime number. For example, in this case, M is 65536, since calling rand() returns 2
bytes. This is probably done by reading the last two bytes.)
loadingNOW said:
the funny part is that if you call rand() twice in a row (getting 4 bytes) and if you don't waste any of that data it might (should?) be possible to predict the next result. which would severely
limit the possibilitys of pokemon which can be created (unless there are a lot of unpredictable interrupts during the process). it would basicly mean if you know the pid or the you know
everything. And if thats really the case and if you could figure this out it would also be possible to detect every cheated pokemon to this day ;)) Now that would be interesting but a bitch to
reverse engineer.
So it seems that when you soft reset, the seed is generated again according to the time it has passed. If the same time passes, then the same sequence of random numbers is generated, which means that
the Pokemon with the exact same stats is displayed for you to catch.
How do yall check the IV's so quickly? Do you just feed them a bunch of rare candies, write down their stats, and use the calculator, and then reset?
So does Synchronize play a part in this? About 1/3 of the Dittos I was harvesting had a perfect Def IV, just like the Abra I was using.
Here's a small smidgen of research. This is out of 120 dittos. This is only the amount of perfects they had, some of them overlapped.
Adamant Dittos Perfect IVs (120)
HP: |||||||||||||||||||||
ATK: |||||||||||
DEF: ||||||||||||||
SpA: |||||||||||||
SpD: |||||
SPD: |||||||||||||||||||
Also X-Act, I'm still waiting for that guy to send his results, but it's the dreaded "exams time".
Here's a small smidgen of research. This is out of 120 dittos. This is only the amount of perfects they had, some of them overlapped.
Adamant Dittos Perfect IVs (120)
HP: |||||||||||||||||||||
ATK: |||||||||||
DEF: ||||||||||||||
SpA: |||||||||||||
SpD: |||||
SPD: |||||||||||||||||||
Wait, I'm not understanding this exactly. Is it 1 per |? If so, we have:
HP: 21
Atk: 11
Def: 14
SpA: 13
SpD: 5
Spd: 19
None: 37
So 37 out of 120 Adamant Dittos didn't have a perfect IV. This is way less than the expected number (82.66% of them = 99).
This is catching Ditto having an Adamant Synchronizer? How did you get these numbers?
emeralds random number seed is 4 bytes. calling rand() always returns 2 bytes generated from these 4 bytes and changes the 4 bytes by multiplying and adding something.
depending on the intrest i think i could reasearch how exactly the seeding is done (yes it it is something with the time but i don't remember if it was rtc or gametime). but i don't really think
i will get you interesting results.
the funny part is that if you call rand() twice in a row (getting 4 bytes) and if you don't waste any of that data it might (should?) be possible to predict the next result. which would severely
limit the possibilitys of pokemon which can be created (unless there are a lot of unpredictable interrupts during the process). it would basicly mean if you know the pid or the you know
everything. And if thats really the case and if you could figure this out it would also be possible to detect every cheated pokemon to this day ;)) Now that would be interesting but a bitch to
reverse engineer.
This has some really awesome implications in breeding. If the seed number is generated by the game time (which TRE's observations seem to support), then it should be possible, with a little
reverse-engineering, to program a calculator that lists all the possible combinations of IVs that will be passed down\generated for an egg.
Then all one would need to do is plug in a favorable time into the calculator, play the game to that time (and keep track of the time between then and the previous save), and then save in front of
the Daycare Man at that time. After that it's just a matter of restarting when the egg doesn't have the wanted combination.
loadingNOW, how did you get a look at the disassembled game code?
This has some really awesome implications in breeding. If the seed number is generated by the game time (which TRE's observations seem to support), then it should be possible, with a little
reverse-engineering, to program a calculator that lists all the possible combinations of IVs that will be passed down\generated for an egg.
Then all one would need to do is plug in a favorable time into the calculator, play the game to that time (and keep track of the time between then and the previous save), and then save in front
of the Daycare Man at that time. After that it's just a matter of restarting when the egg doesn't have the wanted combination.
loadingNOW, how did you get a look at the disassembled game code?
It's not as easy as it sounds. First of all, to find the random number generator algorithm will be extremely hard. Secondly, the internal clock is usually measured in 1/60 of a second (called a
jiffy). To time this perfectly in your game, you'd need to be superhuman.
to find the random number generator algorithm will be extremely hard
no it's not.
in DP this is the RNG (translated to c):
u16 rand() {
randbuf = (randbuf*0x41C64E6D)+0x6073;
return (randbuf >> 16);
But it's hard to control it to exactly it's seeded on the product of a lot of values which i have not mapped and even if you had a good seed it changes very often so you'd still need a lot of tries
to actually use this for breeding if you had all the knowledge you said. And in breeding it's actually even more complicated than for say legendary fights.
On the other hand in the worst case they write the current rnd state back to save and then we can't do anything.
for reference this is SeedRNG in DP (asm this time) but this should be quite simple to reverse
RAM:02000FA4 SeedRng ; CODE XREF: start+524p
RAM:02000FA4 var_28 = -0x28
RAM:02000FA4 var_24 = -0x24
RAM:02000FA4 var_20 = -0x20
RAM:02000FA4 var_1C = -0x1C
RAM:02000FA4 var_18 = -0x18
RAM:02000FA4 var_14 = -0x14
RAM:02000FA4 PUSH {R4,R5,LR}
RAM:02000FA6 SUB SP, SP, #0x1C
RAM:02000FA8 ADD R0, SP, #0x28+var_1C
RAM:02000FAA ADD R1, SP, #0x28+var_28
RAM:02000FAC BL sub_201265C
RAM:02000FB0 LDR R0, =dword_21C48B8
RAM:02000FB2 LDR R3, [SP,#0x28+var_18]
RAM:02000FB4 LDR R4, [R0,#0x2C]
RAM:02000FB6 LDR R1, [SP,#0x28+var_24]
RAM:02000FB8 LDR R0, [SP,#0x28+var_20]
RAM:02000FBA LDR R5, [SP,#0x28+var_14]
RAM:02000FBC ADD R0, R1, R0
RAM:02000FBE LSL R2, R0, #0x18
RAM:02000FC0 LDR R0, [SP,#0x28+var_28]
RAM:02000FC2 LSL R3, R3, #8
RAM:02000FC4 LSL R1, R0, #0x10
RAM:02000FC6 MUL R3, R5
RAM:02000FC8 LDR R0, [SP,#0x28+var_1C]
RAM:02000FCA LSL R3, R3, #0x10
RAM:02000FCC ADD R0, R0, R3
RAM:02000FCE ADD R0, R1, R0
RAM:02000FD0 ADD R5, R2, R0
RAM:02000FD2 ADD R0, R4, R5
RAM:02000FD4 BL sub_201BA1C ; do some calculations
RAM:02000FD8 ADD R0, R4, R5
RAM:02000FDA BL writeRNG
RAM:02000FDE ADD SP, SP, #0x1C
RAM:02000FE0 POP {R4,R5,PC}
RAM:02000FE0 ; End of function SeedRng
In case you guys are actually interested i i'd suggest using an emulator and a breakpoint on srand (so you cnow the current rng state, i'd tell you how to do that if someone wants to try this
experiment) you might get some somewhat useful results ala "rng at game start is about 200-500 "ticks" lower than when fighting zapdos" (standing directly in front of it and pressing a as quickly as
possible, but not all the time). If the results range is very big it's not even wroth investigating.
anyway the most interesting thing for me is that if you take this seriously a lot of iv/gender/shiny/trait combinations are simply impossible.
In case you guys are actually interested i i'd suggest using an emulator and a breakpoint on srand (so you cnow the current rng state, i'd tell you how to do that if someone wants to try this
experiment) you might get some somewhat useful results ala "rng at game start is about 200-500 "ticks" lower than when fighting zapdos" (standing directly in front of it and pressing a as quickly
as possible, but not all the time). If the results range is very big it's not even wroth investigating.
anyway the most interesting thing for me is that if you take this seriously a lot of iv/gender/shiny/trait combinations are simply impossible.
My biggest obstacle here is not actually having an emulator with a debug feature. Do you happen to know of any others besides No$GBA (which doesn't have a free debug version)?
Based on what you said with soft resetting for legendaries, does this mean there is only a certain number of possible combinations of natures and IVs and no more outside of that realm can possibly be
obtained? I wonder if this is consistent from cart to cart or how exactly that works.
And can Synchronize also pass IVs? This is news to me.
vor gba? vba-sdl(-h) would do fine for what i suggested (-h is not even required). for nds you are right just no$gba wich is 15$ non comercial SW with debug.
@striker: it's probably not consistent from cart to cart. any yes shiny, fixed nature and one special some sets of ivs are probably be impossible. we just don't know what these combinations are. on
the other hand without shiny most combinations should be possible.
X-Act > I caught these in Emerald, with a Syncronize Gardevoir w/ 31 SpA, 31 SpD. Also note I didn't keep track of which ones had multiple stats on them, for example I'm not sure if it was in that
group, but I got a ditto with 31 HP, 31 ATK, and 31 SPD, and another with 31 ATK and 31 SpA. and yes, | means =1.
I have more data, but unfortunately I wrote it on actual paper, and this was the only bit I have on the computer.
I found these out by using Pokemon Box and sorting by the highest in each stat, marking those dittos, and taking them to the IV guy and telling him. It was a fairly simple task as the dittos were lv
38 or 40, so there was only 2? or possibly 3 numbers that each stat could of been.
I caught about ~5,000 dittos, about a third of them were tried to syncronize adamant, a third modest, and the rest jolly. I found AT LEAST 6 with 3 31s, but absolutely none with 4 31s or better.
Unfortunately I threw most of them away due to I only have 4 memory cards I can use for Pokemon Box (it holds 1500 pokemon per card), but I still have about 200 dittos i saved for various reasons
(again, most of their IVs are written on paper I can't find). I may record all of there stats, if it's wanted.
anyway the most interesting thing for me is that if you take this seriously a lot of iv/gender/shiny/trait combinations are simply impossible.
is trait [like swarm/guts for heracross] actually determined by the PID or is it separate?
Also of note if you're really wanting a shiny pokemon and you have an egg with good IVs, is that you can trade it around to other games. I had a Bagon hatch that was shiny and it was only because it
was on my ruby version rather than the game it was bred on. Thus we get that most IV combos are possible, it depends entirely on what game it's hatched on.
X-Act > I caught these in Emerald, with a Syncronize Gardevoir w/ 31 SpA, 31 SpD. Also note I didn't keep track of which ones had multiple stats on them, for example I'm not sure if it was in
that group, but I got a ditto with 31 HP, 31 ATK, and 31 SPD, and another with 31 ATK and 31 SpA. and yes, | means =1.
I have more data, but unfortunately I wrote it on actual paper, and this was the only bit I have on the computer.
I found these out by using Pokemon Box and sorting by the highest in each stat, marking those dittos, and taking them to the IV guy and telling him. It was a fairly simple task as the dittos were
lv 38 or 40, so there was only 2? or possibly 3 numbers that each stat could of been.
I caught about ~5,000 dittos, about a third of them were tried to syncronize adamant, a third modest, and the rest jolly. I found AT LEAST 6 with 3 31s, but absolutely none with 4 31s or better.
Unfortunately I threw most of them away due to I only have 4 memory cards I can use for Pokemon Box (it holds 1500 pokemon per card), but I still have about 200 dittos i saved for various reasons
(again, most of their IVs are written on paper I can't find). I may record all of there stats, if it's wanted.
This is very interesting.
How many out of those 5000 dittos did you get with at least one perfect IV? In view of your other previous post, you should have gotten around 3000 to 3500 of them. Is this something you can confirm?
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The General Linear Model GLM Klaas Enno Stephan
Download presentation
The General Linear Model (GLM) Klaas Enno Stephan Branco Weiss Laboratory (BWL) Institute for Empirical Research in Economics University of Zurich Functional Imaging Laboratory (FIL) Wellcome Trust
Centre for Neuroimaging University College London SPM Short Course, Wellcome Trust Centre for Neuroimaging October 2008
Overview of SPM Image time-series Realignment Kernel Smoothing Design matrix Statistical parametric map (SPM) General linear model Statistical inference Normalisation Gaussian field theory p <0. 05
Template Parameter estimates
A very simple f. MRI experiment One session Passive word listening versus rest 7 cycles of rest and listening Blocks of 6 scans with 7 sec TR Stimulus function Question: Is there a change in the BOLD
response between listening and rest?
Modelling the measured data Why? Make inferences about effects of interest How? 1. Decompose data into effects and error 2. Form statistic using estimates of effects and error stimulus function data
linear model effects estimate error estimate statistic
Voxel-wise time series analysis model specification Time parameter estimation hypothesis statistic e m Ti BOLD signal single voxel time series SPM
Time = 1 BOLD signal + 2 x 1 + x 2 error Single voxel regression model e
Mass-univariate analysis: voxel-wise GLM y = X + Model is specified by 1. Design matrix X 2. Assumptions about e N: number of scans p: number of regressors The design matrix embodies all available
knowledge about experimentally controlled factors and potential confounds.
GLM: mass-univariate parametric analysis • parametric • • • one sample t-test two sample t-test paired t-test Anova An. Cova correlation linear regression multiple regression F-tests etc… all cases
of the General Linear Model • non-parametric? Sn. PM
GLM assumes Gaussian “spherical” (i. i. d. ) errors sphericity = iid: error covariance is scalar multiple of identity matrix: Cov(e) = 2 I Examples for non-sphericity: non-identity non-independence
Parameter estimation Objective: estimate parameters to minimize = y + X Ordinary least squares estimation (OLS) (assuming i. i. d. error):
A geometric perspective on the GLM OLS estimates Residual forming matrix R y e x 2 x 1 Design space defined by X Projection matrix P
Correlated and orthogonal regressors y x 2* Correlated regressors = explained variance is shared between regressors x 2 x 1 When x 2 is orthogonalized with regard to x 1, only the parameter estimate
for x 1 changes, not that for x 2!
What are the problems of this model? 1. BOLD responses have a delayed and dispersed form. 2. The BOLD signal includes substantial amounts of low-frequency noise. 3. The data are serially correlated
(temporally autocorrelated) this violates the assumptions of the noise model in the GLM HRF
Problem 1: Shape of BOLD response Solution: Convolution model hemodynamic response function (HRF) The response of a linear time-invariant (LTI) system is the convolution of the input with the
system's response to an impulse (delta function). expected BOLD response = input function impulse response function (HRF)
Convolution model of the BOLD response Convolve stimulus function with a canonical hemodynamic response function (HRF): HRF
Problem 2: Low-frequency noise Solution: High pass filtering S = residual forming matrix of DCT set discrete cosine transform (DCT) set
High pass filtering: example blue = data black = mean + low-frequency drift green = predicted response, taking into account low-frequency drift red = predicted response, NOT taking into account
low-frequency drift
Problem 3: Serial correlations with 1 st order autoregressive process: AR(1) autocovariance function
Dealing with serial correlations • Pre-colouring: impose some known autocorrelation structure on the data (filtering with matrix W) and use Satterthwaite correction for df’s. • Pre-whitening: 1. Use
an enhanced noise model with multiple error covariance components. 2. Use estimated autocorrelation to specify filter matrix W for whitening the data.
How do we define V ? • Enhanced noise model • Remember linear transform for Gaussians • Choose W such that error covariance becomes spherical • Conclusion: W is a function of V so how do we estimate
V ?
Multiple covariance components enhanced noise model V = 1 error covariance components Q and hyperparameters Q 1 + 2 Q 2 Estimation of hyperparameters with Re. ML (restricted maximum likelihood).
Contrasts & statistical parametric maps c=100000 Q: activation during listening ? Null hypothesis:
t-statistic based on ML estimates c=100000 For brevity: Re. MLestimates
Physiological confounds • head movements • arterial pulsations • breathing • eye blinks • adaptation affects, fatigue, fluctuations in concentration, etc.
Outlook: further challenges • correction for multiple comparisons • variability in the HRF across voxels • slice timing • limitations of frequentist statistics Bayesian analyses • GLM ignores
interactions among voxels models of effective connectivity These issues are discussed in future lectures.
Correction for multiple comparisons • Mass-univariate approach: We apply the GLM to each of a huge number of voxels (usually > 100, 000). • Threshold of p<0. 05 more than 5000 voxels significant by
chance! • Massive problem with multiple comparisons! • Solution: Gaussian random field theory
Variability in the HRF • HRF varies substantially across voxels and subjects • For example, latency can differ by ± 1 second • Solution: use multiple basis functions • See talk on event-related f.
Summary • Mass-univariate approach: same GLM for each voxel • GLM includes all known experimental effects and confounds • Convolution with a canonical HRF • High-pass filtering to account for
low-frequency drifts • Estimation of multiple variance components (e. g. to account for serial correlations) • Parametric statistics
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Sharir M. 1990. The maximum number of ways to stabn convex nonintersecting sets in the plane is 2n−2. Discrete & Computational Geometry. 5(1), 35–42."},"publication":"Discrete & Computational
Geometry","year":"1990","author":[{"last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},
{"first_name":"Micha","full_name":"Sharir, Micha","last_name":"Sharir"}],"oa_version":"None","user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","doi":"10.1007/
BF02187778","volume":5,"quality_controlled":"1","publication_status":"published","status":"public","publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"abstract":[{"text":"LetS be a
collection ofn convex, closed, and pairwise nonintersecting sets in the Euclidean plane labeled from 1 ton. A pair of permutations\r\n(i1i2in−1in)(inin−1i2i1) \r\nis called ageometric permutation of
S if there is a line that intersects all sets ofS in this order. We prove thatS can realize at most 2n–2 geometric permutations. This upper bound is
tight.","lang":"eng"}],"extern":"1","acknowledgement":"Research of the first author was supported by Amoco Foundation for Faculty Development in Computer Science Grant No. 1-6-44862. Work on this
paper by the second author was supported by Office of Naval Research Grant No. N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment
Corporation and the IBM Corporation.","main_file_link":[{"url":"https://link.springer.com/article/10.1007/BF02187778"}],"title":"The maximum number of ways to stabn convex nonintersecting sets in the
plane is 2n−2","date_published":"1990-01-01T00:00:00Z","article_processing_charge":"No","article_type":"original","day":"01","issue":"1","type":"journal_article","publist_id":"2057","month":"01"}]
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Quantum mechanics for programmers
My main article on quantum mechanics is
. This article talks about how to make simulations like the ones there. For the actual simulations, I used C#, because it's a good compromise between speed and simplicity. I regret, sometimes, that
it's not as fast as C, but it's easier to debug, and that saves me a lot of time. I'm going to try to explain how to do the simulations in Javascript, because it's easier to share with readers, and
it means that you can play with the simulations in your browser. The disadvantage of javascript is that it's a bit slower. Normally it's only a factor of 2-10 times slower, but the slow bit in the
simulations below is the fourier transform. For non-javascript languages, you use someone else's fourier transform library, and they're much faster because they use all the tricks that modern
computers allow, from ordering operations optimally to making sure caches are used as well as possible. My javascript version? Not so much. The result is that the javascript version below is actually
a *lot* slower than the C# version. Article aims:
• To give a really simple simulation of a quantum mechanics system so that people who already know javascript have a chance to "get" quantum mechanics without necessarily having to understand the
equations. The equations will help, though.
• To answer the question of "how do you make the simulations".
As normal, the idea is to try to explain things as simply as possible and no simpler. Here's the article: Quantum mechanics can be simulated with quite simple programs. Understanding what they do
might help learn quantum mechanics, and it might help physicists understand how to simulate the maths. This article is going to start easy and get harder. If it's too easy, skip ahead.
Starting point: Euler
Here's the Schrödinger equation for a 1-D electron in a harmonic potential. Whether or not you understand it is ok, hopefully the article will be readable either way. $$i \frac{\partial \psi}{\
partial t} = -\frac{\partial^2}{\partial x^2}\psi + ax^2\psi.$$ Let us rewrite that, introducing a timestep time, $\delta t$ $$\psi_{\hbox{after timestep}} \simeq \psi_{\hbox{before}} - i \delta t \
left(-\frac{\partial^2 }{\partial x^2}\psi + ax^2\psi\right).$$ You can write $$\frac{\partial^2 }{\partial x^2}\psi(x) \simeq \frac{ \psi(x+1) - 2 \psi(x) + \psi(x-1) }{\delta x^2}.$$ These two
approximations give us a simple timestepping function. This is not the only (or best) timestepping function that solves the problem, but perhaps it's one of the simplest. You, the reader, should try
to understand the function below. Hopefully, it will make sense with the two equations above.
// This returns an empty wavefunction of length n.
function wavefunction(n){
var psi = [];
for(var i = 0; i < n; i++){
psi[i] = {real: 0, imag: 0}
return psi; // for example [{real:0, imag:0}, {real:0, imag:0}, ...]
// This takes the starting wavefunction, and returns the wavefunction a short time later.
function timestep(psi)
// This is how many units of time we're going to try to step forward.
var dt = 0.002;
var n = psi.length;
// This is the wavefunction we're going to return eventually.
var ret = wavefunction(n);
// We miss off the first and last elements because it looks at the element to the
// left and right of this point.
for(var i = 1; i < n-1; i++)
// This is the x that is in the equation above.
var x = (i-n/2)
// This is the potential at this point.
var V = x*x * 0.0015; // a here = 0.0015
// We start from the original wavefunction (and later add (dt * dpsi/dt) to it).
ret[i].real = psi[i].real;
ret[i].imag = psi[i].imag;
// This is the kinetic energy applied to psi.
var KPsi = {
real: psi[i].real * 2 - psi[i-1].real - psi[i+1].real,
imag: psi[i].imag * 2 - psi[i-1].imag - psi[i+1].imag
// This is the potential, applied to psi
var VPsi = {
real: psi[i].real * V,
imag: psi[i].imag * V
// This is the whole right hand side of the schrodinger equation.
var rhsReal = KPsi.real + VPsi.real;
var rhsImag = KPsi.imag + VPsi.imag;
// This adds it, multiplied by dt, and multiplied by i.
// The multiplication by i is what swaps the real and imaginary parts.
ret[i].real += rhsImag * dt;
ret[i].imag -= rhsReal * dt;
return ret;
// This returns the initial state of the simulation.
function init(){
var n = 128;
var psi = wavefunction(n);
for(var i = 0; i < n; i++){
psi[i].real = Math.exp(-(i-20)*(i-20)/(5*5))*0.75;
psi[i].imag = 0;
return psi;
The simulation above is the probability density of an electron in an $x^2$ potential. This is similar to if it was in a bowl, and was rolling backwards and forwards in that bowl.
The simulation works, but it has problems:
• The total amount of probability grows with time.
• If you time the oscillation, it's not quite right: It's actually moving slightly too slowly when it's at its fastest
• There's dispersion: The oscillation doesn't stay focussed. It should: this particular potential is specially chosen to make the oscillation stay focussed.
Split operator
This is going to get harder quickly. Let's start with
We're going to have to look at the original equation again.
A different way to convert it into a timestep is to do this.
$$\psi_2 = \hbox{Timestep}_T(\psi_1)$$ $$\psi_1 = \hbox{Timestep}_V(\psi_0)$$
Here, $\psi_0$ is the wavefunction before the timestep.
$\hbox{Timestep}_V$ steps forward with according to the following equation: $$i \frac{\partial \psi}{\partial t} = ax^2\psi.$$
$\hbox{Timestep}_T$ steps forward with according to the following equation: $$i \frac{\partial \psi}{\partial t} = -\frac{\partial^2}{\partial x^2}\psi$$
$\hbox{Timestep}_V$ is actualy really easy to implement, because: $$\hbox{Timestep}_V(\psi) = e^{-iax^2\delta t}\psi.$$
$\hbox{Timestep}_T$ can be implemented by noting that if $\psi = e^{ikx}$, then: $$\hbox{Timestep}_T(\psi) = e^{-ik^2\delta t}\psi.$$
And there's good news: We can implement this by fourier transforming, then multiplying by $e^{i k^2}$, and then inverse fourier transforming. The fourier transform code is here:
This particular fourier transform code only works on powers of two.
// This returns an empty wavefunction of length n.
function wavefunction(n){
var psi = [];
for(var i = 0; i < n; i++){
psi[i] = {real: 0, imag: 0}
return psi; // for example [{real:0, imag:0}, {real:0, imag:0}, ...]
// This takes the starting wavefunction, and returns the wavefunction a short time later.
function timestep(psi)
// This is how many units of time we're going to try to step forward.
var dt = 0.02;
psi = timestepV(psi, dt);
psi = timestepT(psi, dt);
return psi;
function timestepV(psi, dt)
var n = psi.length;
for(var i = 0; i < n; i++)
// This is the x that is in the equation above.
var x = (i-n/2)
// This is the potential at this point.
var V = x*x * 0.0015;
var theta = dt * V;
var c = Math.cos(theta);
var s = Math.sin(theta);
var re = psi[i].real * c - psi[i].imag * s;
var im = psi[i].imag * c + psi[i].real * s;
psi[i].real = re;
psi[i].imag = im;
return psi;
function timestepT(psi, dt){
psi = FFT.fft(psi);
var n = psi.length;
for(var i = 1; i < n/2; i++)
var k = 2 * 3.1415927 * i / n;
var theta = k * k * dt;
var c = Math.cos(-theta);
var s = Math.sin(-theta);
var re = psi[i].real * c - psi[i].imag * s;
var im = psi[i].imag * c + psi[i].real * s;
psi[i].real = re;
psi[i].imag = im;
var j = n - i;
re = psi[j].real * c - psi[j].imag * s;
im = psi[j].imag * c + psi[j].real * s;
psi[j].real = re;
psi[j].imag = im;
return FFT.fft(psi);
// This returns the initial state of the simulation.
function init(){
var n = 128;
var psi = wavefunction(n);
for(var i = 0; i < n; i++){
psi[i].real = Math.exp(-(i-20)*(i-20)/(5*5))*0.75;
psi[i].imag = 0;
return psi;
This runs much better. It's faster, mainly because the timestep was longer. But the timestep can be longer because the timestep is more accurate. Likewise, you can get away with a coarser grid, and
still get the same accuracy.
The next page looks at 2-dimensional simulations:
Other Articles:
A page with a javascript application where you can set body positions, calculate joint angles and animate human motion.
This is a work in progress to write a book on physics algorithms. At the moment, it is about 1/3 finished though, but the latest version can be downloaded for free.
A look at the double slit experiment. The first half is meant to be a clear explanation, using simulations. The second half discusses some of the philosophy / interpretations of quantum mecahnics.
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Analyzing and Interpreting Data
Practice 4: Analyzing and Interpreting Data
Belonging Supports for Analyzing and Interpreting Data
When analyzing and interpreting data in the classroom, students may work in teams, receive feedback from others, and collectively communicate results. Instructional strategies that support students’
feelings of belonging cultivate a safe space for students to hone skills for analyzing and interpreting data in these regards. Strategies that support belonging also encourage students to develop a
sense of being part of a community of scientists and engineers, which is especially important for students who may not have a well-developed science identity or who may feel alienated from science
[see Motivation as a Tool for Equity]. As students begin to feel a greater sense of belonging within their science classroom community and within science and engineering communities, they may feel
more inclined to engage in the practices of analyzing and interpreting data.
Have students analyze and interpret data in groups to promote the idea of being a community of learners working to solve design problems or figure out phenomena. Working in small groups enables both
scientific collaboration and innovation.
Structure whole class discussion so that discourse builds from simpler observations about patterns and trends to more complex ones, allowing students to build on one another’s ideas constructively
and allowing validation of multiple ideas at different levels of complexity.
If there are multiple tasks that can be done simultaneously within groups, define roles and assign students (or have students select a role within their groups) so that everyone can contribute to the
analysis and interpretation.
Allocate time for students to present findings and supporting evidence to peers and allow for student feedback/dialogue around data analysis and interpretation (e.g., what are the sources of error?
How were significant features and patterns identified? To what extent does their data serve as evidence to support conclusions made?) As part of this conversation, students determine whether and
describe why components of the practice (organization, visual displays, summarizing, patterns and relationships, sources of error, outlying data) are/are not appropriate to help them figure out a
phenomenon or evaluate competing design solutions to a problem.
• Set up norms for these conversations to establish a sense of belonging/comfort around how to analyze, interpret, and communicate results as evidence.
When feasible, give different groups different analysis tasks so that they can share later and make claims supported by evidence as a class or “lab team.” For example, use a jigsaw format so students
engage in different analysis tasks in expert groups and share their evidence with their original groups.
Place an emphasis on the central tendency of data and data interpretation that allows everyone’s interpretation to merge into the “average” (consensus) of the classroom.
Confidence Supports for Analyzing and Interpreting Data
There is likely a wide variety of math ability levels in a single science class. Differences in skill may require different levels of scaffolding in order to develop confidence for all students. Some
students may have little experience with data or may have limited confidence in successfully being able to tabulate, graph, or perform statistical analysis on data. Students may also be uncomfortable
presenting the results of the analysis and interpretation to their peers. Supporting students’ confidence as they engage in these activities will be crucial for them to feel comfortable working with
Use prompts like “what do you notice?”, “what do you observe?”, or “what questions do you have?” to give students an accessible entry point into data analysis and an early experience of success in
working with data to figure out phenomena or solve design problems.
Explicitly name the skills and strategies needed to interpret data and graphs and provide opportunities to practice these skills, so that students view data interpretation as something they can learn
to master with practice.
Provide self-questioning stems or thinking guides that help students to systematically interpret and analyze different kinds of data independently. Thinking guides can also help students evaluate the
analysis and interpretation of data.
Give students practice identifying trends and interpreting a common set of data rooted in understanding a phenomenon or solving a problem so that they can receive informational feedback before they
do the same tasks with their own data.
Cater analysis to students’ current abilities – e.g., if students struggle with graphing data, give them examples (sample created by the teacher for the particular task, examples of past student
work) or options for how to represent their data (bar, line, or pie chart).
Provide scaffolds such as thinking guides and checklists for common data analysis tasks that prompt students to consider which type of data analysis is most appropriate to help them figure out a
phenomenon or solve a design problem. For example:
• Guidelines for graphing: a checklist for the parts of a graph, thinking guides for students to determine a good scale for the data they are graphing or the type of graph that will be most useful
for their purpose
• Guidelines for data tabulation: checklists for how to set up a frequency table, how to set up a table for the different variables in an experiment, etc.
• Guidelines for summarizing data: different summary statistics (e.g., mean, median, mode) and what information they provide scientists and engineers
Learning Orientation Supports for Analyzing and Interpreting Data
Data collection, especially by young scientists and engineers with limited experience, can contain a large amount of error. During analysis, having a learning orientation will help frame that error
as a critical part of the learning experience (both learning techniques for how to reduce the error in future data collection and learning how to account for error in data interpretation) rather than
a failure. Supporting a learning orientation will also help with encouraging students to engage deeply with their data to make sense of a phenomenon or design solutions to design problems, rather
than merely performing calculations or getting the “right” answer.
Ask students to identify what patterns they see in a data set and develop their own hypotheses about what that pattern can tell them about evidence for other phenomena (e.g, the forces exerted by a
rocket can inform you about the forces of other engine-propelled devices).
Engage students in error analysis to figure out what went wrong during data collection, where errors were made (including if there was uncertainty in measurement) and how these errors are reflected
in the data, and why different students or different lab groups may have obtained discrepant results. Emphasize that making errors is a normal part of learning how to make precise and accurate
measurements and that, even when you have developed those skills, there will always be error in your measurements (e.g., estimating to the nearest tenth of a millimeter on a ruler with millimeter
hash marks).
Model and then scaffold how to judge the appropriateness and correctness of data analysis and interpretation. For example, design data analysis questions or assessments such that they include an
opportunity for students to explain the thinking behind their analyses/interpretations. Provide feedback and/or evaluate these responses based on the skills students demonstrate and their reasoning
and evidence, rather than just the percent of correct/incorrect answers.
When the teacher makes a mistake in a computation, model a response that frames the mistake as normal and a learning opportunity, rather than becoming defensive about the error.
Use think-alouds to model a learning orientation to students; they can be used to normalize struggle, confusion, and mistakes and to model effective strategies for analyzing and interpreting data.
When discussing the interpretation of data, use the Learning Orientation Talk Moves or resources like the Accountable Talk Sourcebook, the “Supporting Discussions” chapter of the Open SciEd Teacher
Handbook, Talk Science Primer, and Discourse Primer for Science Teachers to elicit multiple student perspectives before concluding which interpretations are supported by the data and to encourage
students to build on each other’s ideas, critique each other respectfully, and acknowledge each other’s contributions. These discourse and facilitation moves help to demonstrate that the goal of the
discussion – and data analysis in general – is to think deeply about the data and not just to arrive at the right answer.
Autonomy Supports for Analyzing and Interpreting Data
Cognitive autonomy is especially important to encourage students to make their own decisions about how to analyze or make sense of the data, as well as to generate alternative interpretations and
explanations. Working with data might make teachers prone to undermine student autonomy if they suggest that there is a clear “right” answer, such as a predetermined set of similarities and
differences between two data tables that the teacher is leading students to identify. There might also tend to be an overemphasis on smaller autonomy allowances (e.g., letting students choose the
colors for a graph) without accompanying demands on students’ cognitive autonomy in making sense of phenomena or problem solving. It is important to provide sufficient time and scaffolding for
students to engage in rigorous, autonomous sense-making through the analysis and interpretation of data.
A way of scaffolding data analysis and interpretation early in the year would be to present students with several ways to summarize data and several types of graphs and have students choose which
type they think will best summarize and present the data they have collected. Prompt students to make observations about the different affordances and limitations of each option and explain why they
made their choice (e.g., ask students to justify their selection of tools and procedures through questions like, Why are you using a line graph for this data? What will using a map of the data tell
you that a table might not? Why did you choose that kind of graph? Why did you average? Why did you select the way you did?).
Once students have developed more advanced data analysis/interpretation skills and are familiar with different ways of presenting data, allow students greater choice in selecting how to present their
data (e.g., tables, graphs, flowcharts, illustrations) and prompt them to justify their choices.
Before holding a whole-class discussion about data, divide students into groups with open-ended prompts to help them make sense of the data and identify initial patterns and relationships within the
data (e.g., similarities and differences, temporal and spatial, linear and nonlinear). These patterns and relationships can help students to figure out a phenomenon or identify the best
characteristics among several design solutions that can inform a new, optimal solution. One way that students can engage in these sense-making activities is through graphing. The small group
structure places responsibility on students to engage in the work, and positions the teacher as a facilitator, circulating among groups rather than directing a whole-class conversation on data
Relevance Supports for Analyzing and Interpreting Data
Some students may have lower confidence in their ability to tabulate, graph, or perform statistical analyses on data or may even think they are not a “math person.” Framing data analysis within a
phenomenon or design problem that is of interest to students may help motivate them to work hard on the mathematics needed for data analysis. Connecting the practice of analyzing and interpreting
data to the work of scientists and engineers may help encourage students to see the value of math as a tool to make sense of the world. Encouraging students to connect data analysis and
interpretation to a broad range of situations that relate to their lives and home communities can make them more invested in the practice as something that can be leveraged to figure out phenomena or
solve problems that feel relevant and important to them [see Motivation as a Tool for Equity].
Many strategies from equitable teaching frameworks (e.g., culturally responsive pedagogy) address ways to learn more about the local community and their needs, and to connect science and engineering
learning to those needs.
Talk with students about the goals of the data analysis (e.g., what questions are being asked of the data to make sense of a phenomenon or solve a design problem?) to make clear that analysis steps
need to be relevant to the group-specified goal.
Find and regularly incorporate data representations that are relevant to students’ interests and/or daily lives (e.g., analysis of food nutritional content that shows a variety of snack foods that
students enjoy).
Engage students in discussions and explorations that emphasize that data analysis and interpretation is an authentic science and engineering practice.
Share (or invite students to share) personal or historical stories of when data analysis and interpretation led to great advancements (e.g., Rosalind Franklin, Watson, and Crick and DNA double helix;
Katherine Johnson’s calculations for NASA; Grace Hopper’s computer coding protocols).
Choose situations that students are familiar with and/or refer to the work of diverse scientists and engineers for data analysis and interpretation activities (e.g., population density as a variable
related to microbiology that can affect the exponential spread of an infectious disease, such as during a pandemic).
Connect data visualization examples to the work that scientists and engineers do (e.g., “These are some cool ways that scientists communicate their findings”) through varied forms of data
representation (e.g., bar graphs, Venn diagrams, models, flow charts, maps) in lessons across units.
Use real world datasets from organizations like NOAA, NASA, or others that share data and visualizations publicly.
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Spaceplanes: why we need them, why they have failed, and how they can succeed
Despite the failures of dozens of past efforts, companies like Radian Aerospace continue to pursue spaceplanes. (credit: Radian Aerospace)
Spaceplanes: why we need them, why they have failed, and how they can succeed
by John Hollaway
Monday, May 13, 2024
“Rockets are terribly inefficient and expensive.” This admission can be found here in NASA’s own educational piece on the equation that governs rocket performance, also known as Tsiolkovsky's
equation. But what is the alternative?
Perhaps the most frustrating aspect of the space age is our inability to create the obvious one: a successful spaceplane. This is a launch vehicle that takes off like an airplane, flies up to orbit,
and returns to settle back on the runway. This feat is known as single stage to orbit (SSTO) and if you look this up in Wikipedia, here is what you find:
It is considered to be marginally possible to launch a single-stage-to-orbit chemically fueled spacecraft from Earth. The principal complicating factors for SSTO from Earth are: high orbital
velocity of over 7,400 meters per second (27,000 km/h; 17,000 mph); the need to overcome Earth's gravity, especially in the early stages of flight; and flight within Earth's atmosphere, which
limits speed in the early stages of flight due to drag, and influences engine performance.
Perhaps the most frustrating aspect of the space age is our inability to create the obvious one: a successful spaceplane.
“Marginally possible.” This gloomy observation has not inhibited attempts to overcome the challenge of creating a spaceship that can fly up and down like an aircraft. Indeed, if you go to Wikipedia
you will find a list of some 60 spaceplane projects since 1945 that had this objective. None can have been said to have succeeded. A more recent review was given by Joe Scott.
Unsurprisingly to aerospace engineers, the villain sabotaging this dream is Tsiolokovsky’s Equation. Konstantin Tsiolokovsky was a Russian mathematician who, in 1896 demonstrated that Newton’s laws,
when applied to rockets propelled with oxygen plus a reductant, results in a formula that limits all the non-propellant mass—the unfueled rocket and the payload—to a small fraction of the total
weight. Specifically, when launching to low earth orbit at the minimum speed required of 7,400 meters per second, and not allowing for losses from atmospheric drag and from gravitational pull, the
equation gives this table:
Propellant Specific Impulse (Isp) Maximum Non-Propellant Mass
Liquid Hydrogen plus LOX 450 18.7%
Kerosene plus LOX 330 10.1%
Solid Rocket Motor (SRM) 270 6.1%
In practical terms the ultimate problem is that, because space has no oxygen, rockets have to carry up about 2.4 tons of it for every ton of fuel they carry. Not surprisingly, given this constraint,
rockets which reduce their mass to orbit by dropping off empty sections on the way up—staging—is the system that has been universally adopted to maximise the non-propellant mass (another Tsiolokovsky
insight). A further drawback of SSTO’s is that they must haul along a pair of wings that are useful only for a very small part of the flight path, which further limits the thin weight margin
available for the payload.
In the last decade spaceplanes have become even less competitive. The development of reusable rockets by SpaceX has brought payload costs to low Earth orbit (LEO) down from over $10,000 a kilogram to
around $3,000. Under these circumstances, SSTO concepts might be expected to have a vanishingly small chance of being economically viable.
Yet still they come. Radian Aerospace, based in Renton, Washington, is planning to develop a delta-winged spaceplane about the size of a small commercial jet air transport. This will launch
horizontally using a rocket-powered sled to allow the craft to conserve as much fuel as possible. Once aloft, three rocket engines put the spacecraft into orbit under a low-g ascent, followed by
reentry and landing on a runway three kilometers long.
But Radian is right: spaceplanes are going to be essential if we are to continue to use satellite-based services. We are painting ourselves into a corner here, with ever-larger rockets carrying
ever-larger numbers of satellites up, but with no means of servicing them in orbit.
Radian have raised $27.5 million of what they call “seed capital,” so presumably the final cost will be of the order of hundreds of millions of dollars. Given the price squeeze originating from
SpaceX it is difficult to see a justification for this. With a two-ton payload and assuming a net revenue of $1,000 per kilogram, cash recovery will be $2 million a launch, so at least 50 launches
will be needed just to break even.
But Radian is right: spaceplanes are going to be essential if we are to continue to use satellite-based services. We are painting ourselves into a corner here, with ever-larger rockets carrying
ever-larger numbers of satellites up, but with no means of servicing them in orbit. They cannot be easily repaired or their positions adjusted, and they cannot be readily deorbited when they become
The problem can be seen most clearly in the space debris challenge. In November 2022, the US Space Surveillance Network reported tracking 25,857 artificial objects in orbit above the Earth, of which
5,465 were operational satellites. However, those 20,000-odd other objects represent the tip of an iceberg; they are the space debris items that are big enough to be trackable. There are now,
according to NASA, perhaps a hundred million orbiting objects with a diameter between 1 and 10 centimeters, and over 36,500 pieces with diameters greater than 10 centimeters.
NASA also has a good survey of the scores of debris capture proposals and the regulatory situation here. What is lacking is a vehicle that can go about deorbiting space junk by whatever means, come
back to Earth for re-equipping and return back up to continue its work. A space plane.
Checkmate, or so it may seem.
The way forward lies again in Tsiolokovsky’s equation. The form that is of use here is:
ln (Initial Mass/Final Mass) = Delta V/(g0 * Isp)
So there are just two variables involved, the increase in velocity and the specific impulse. There is nothing to be done about the Isp once the choice of propellant and oxidizer has been made, but
what about the effect of gravity and drag losses on the delta V?
Conceptually, if we are able to use air-breathing ramjets to take the spaceplane to the edge of space before handing over propulsion to a rocket motor, then not only will the ensuing rocket drag loss
be small enough to be almost negligible, the initial velocity of this second stage could be more than Mach 5, countering the gravitational drag. The evidence for this speed comes from several
sources, such as:
1. The Boeing ramjet-powered ASALM missile demonstrated its hypersonic ability in 1980 when it reached Mach 5.5 (about 1,900 meters per second) at 12,000 meters after a fuel valve stuck open.
2. In 1951 NACA (the NASA predecessor) launched a ramjet powered missile which reached an apogee of 159,000 feet (48.5 kilometers). This missile was launched at a 75-degree angle and ran out of fuel
at 67,200 feet (20.5 kilometers) when it was passing Mach 2.92 (about 1,000 meters per second).
It is possible to gain a measure of gravitational drag from the trajectory of the air-launched Pegasus rocket, which was developed by Orbital ATK in 1990 and later built and launched by Northrop
Grumman. Its Users Guide, issued in October 2015, gave operating data that showed that the zoom effect between the first and second stage gave an altitude gain of nearly 16 kilometers in return for a
delta V loss of about 58 meters per second, a penalty of 3.6 meters per second per kilometer. This is happening at an altitude of over 70 kilometers, so this loss is almost solely from gravitational
From these figures it appears that the gravitational penalty for a SRM being used in our spaceplane to lift it from about 50 kilometers up at about 1,000 meters per second to 200 kilometers at the
minimum orbital speed of 7,400 meters per second would be, very roughly, 150 x 3.6 meters per second, or about 540 meters per second. So gravitational drag will require us to add this value to the
required orbital speed, giving a total of 7,940 meters per second. Because of the uncertainties surrounding this value, we can round it up to 8,000 meters per second.
However, if there is a first-stage ramjet propulsion stage for our spaceplane, and it achieves Mach 5.5 at 70 kilometers, this would remove about 1,700 meters per second from this orbital speed
target, reducing the delta V-to-orbit requirement of the spaceplane’s rocket stage to about 6,300 meters per second. If this work is undertaken using a simple SRM with an Isp of 270, then the
available non-propellant mass of 6.1% shown in the table above increases to 10.8%, or rather better than a kerosene plus lox liquid-fueled rocket on the same basis.
What does this mean in terms of a spaceplane? It will be necessary to make a number of informed guesses on the non-propellant mass items at this stage; here they are:
Item Mass
Payload 0.5t
Spaceplane structure 1.5t*
Cold gas thruster fuel 1.0t**
Control Systems 0.5t
Total Non-Propellant Mass 3.5t
*This may seem light, but there is no undercarriage on this vehicle. It is launched and captured on a separate carriage on a track controlled by a linear induction motor. Additionally, the ramjets
are expected to need to run for no more than about three minutes after launch, and so can be made of thin heat-resistant steel.
** For the extensive in-orbit movements required of an orbital service vehicle, perhaps nitrogen or possibly propane from left-over ramjet fuel.
If this non-propellant mass of 3.5 tons now represents 10.8% of the total at the point where the SRM takes over from the ramjets, then the SRM propellant mass would be 32.4 tons approximately, giving
a launch total of about 35.9 tons. In addition, at launch there would be an extra two to two-and-a-half tons of propane as fuel for the ramjets and perhaps for in-orbit thruster use as well.
So, finally, a practical spaceplane. A bonus is that by being able to reach orbit with ramjets and a simple SRM it will have almost no moving parts. The concept is expanded upon in www.swalarlv.com.
Note: we are now moderating comments. There will be a delay in posting comments and no guarantee that all submitted comments will be posted.
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Math, Grade 7, Putting Math to Work, Calculating Ranges Of Strouhal Numbers
Calculating Ranges Of Strouhal Numbers
Students create a bar graph showing the Strouhal numbers for a variety of birds and bats and use their graph and other data to compare the Strouhal numbers of the different animals to analyze
variation and to make predictions.
Key Concepts
Students are expected to use the mathematical skills they have acquired in previous lessons or in previous math courses. The lessons in this unit focus on developing and refining problem-solving
skills. Students will:
• Try a variety of strategies to approaching different types of problems.
• Devise a problem-solving plan and implement their plan systematically.
• Become aware that problems can be solved in more than one way.
• See the value of approaching problems in a systematic manner.
• Communicate their approaches with precision and articulate why their strategies and solutions are reasonable.
• Make connections between previous learning and real-world problems.
• Create efficacy and confidence in solving challenging problems in a real-world setting.
Goals and Learning Objectives
• Analyze the relationship among the variables in an equation.
• Write formulas to show how variables relate.
• Calculate ranges of Strouhal numbers and use these ranges to make predictions.
• Communicate findings using multiple representations including tables, charts, graphs, and equations.
• Create bar graphs.
How Do Birds Compare?
Lesson Guide
Have students predict the range where they think all birds have Strouhal numbers that are close to each other. Allow the students to discuss their prediction with a partner.
ELL: ELLs may need clarification as to what the question asks them to do. They are not asked to solve a problem, only to make a prediction. Eventually, they will need to review the problem solving
process. Make sure that the problems solving process is written on an anchor chart and placed in a prominent location.
How Do Birds Compare?
Predict the range where you think all birds have Strouhal numbers that are close to each other.
Problem-Solving Process
Lesson Guide
Remind students of Problem-Solving Steps 1–4.
1. Understand the problem situation.
□ What is the problem asking you to find out?
□ What information is provided?
□ What are the quantities that vary?
□ How are the variables related?
2. Represent the situation.
□ Write a formula to show how the variables of the Strouhal number relate to each other.
3. Answer the question.
4. Check that the mathematical answer makes sense.
Problem-Solving Process
As you solve today's problems, use the problem-solving process.
1. Understand the problem situation.
2. Represent the situation.
3. Answer the question.
4. Check that the mathematical answer makes sense.
Math Mission
Lesson Guide
Discuss the Math Mission. Students will create a graph showing the Strouhal numbers for different birds and bats and then compare the data to the Strouhal number for a dolphin.
SWD: Students have learned and retained information from prior learning experiences, but do not realize when to use that information. Teachers need to remind students of what they know, but also when
to apply that knowledge. This strategy is sometimes referred to as “priming” background information. Priming background knowledge can be done in simple ways, such as merely stating “Remember when you
learned how to graph...” or “See how this concept applies in this situation, too?”
Create a graph showing the Strouhal numbers for different birds and bats and then compare the data to the Strouhal number for a dolphin.
Graph and Compare Strouhal Numbers
Lesson Guide
Have students work independently to start. During independent time, observe students working, but refrain from asking guiding questions or giving students hints. Give students the opportunity to
organize their problem-solving plans and the information given in the problem independently before asking others for assistance.
After independent work time, allow students to work with a partner to complete the problems. Have partners share their problem-solving plans. Partners should agree on a plan and work together to find
the solutions. During this time, circulate around the classroom asking guiding questions and noting strategies you want students to share during the class discussion.
SWD: Students with disabilities may require review and reinforcement of mathematical tools and means of representing data. Reinforce how to interpret and utilize graphing tools.
Mathematical Practices
Mathematical Practice 1: Make sense of problems and persevere in solving them.
Students must make sense of the Strouhal numbers for each of the animals and what the range of numbers means for other birds and bats. Students must preserve in solving this problem to make
predictions about other types of animals.
Mathematical Practice 2: Reason abstractly and quantitatively.
Students reason abstractly and quantitatively with the Strouhal numbers for each type of bird and bat. Students use their quantitative reasoning skills to calculate the range in the Strouhal numbers
and use this data to make predictions.
Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.
Students construct a viable argument for the reasonableness of their solution and they have an opportunity to critique the work of others during the class discussion.
Mathematical Practice 4: Model with mathematics.
Students model the mathematics presented in the problem graphically.
Mathematical Practice 5: Use appropriate tools strategically.
Students use appropriate tools and problem-solving strategies to find the solution. Students may use a calculator as a possible tool for finding their solution.
Mathematical Practice 6: Attend to precision.
Students attend to precision while creating their graphs as well as when calculating the range in Strouhal numbers. Students also use precision when calculating the Strouhal number of a dolphin.
Mathematical Practice 7: Look for and make use of structure.
Students make use of the problem-solving structure while solving this problem.
Student has difficulty getting started.
• What information is provided in the problem? What are you trying to find?
• How can you organize the information presented in the problem?
• What problem-solving strategies can you use to help you find a solution?
Student has a solution, but it is incorrect.
• Is your answer reasonable? Why or why not?
• Does the information in your graph match the data in the table?
• How do you calculate the range for the data?
Student has a solution but is having difficulty articulating his or her thinking.
• How can you state the answer as a complete sentence?
• How would you describe your strategy to somebody who is struggling?
• Does your answer make sense? How do you know?
Student has a correct solution and is waiting on others to finish.
• Do you think the Strouhal numbers for all swimming animals would fall within the same range as birds and bats? Why or why not?
Possible Answers
• The maximum Strouhal Number in the data is 0.4. The minimum Stouhal Number in the data is 0.2. Therefore, the range of the Strouhal numbers for the birds and bats is 0.2.
Work Time
Graph and Compare Strouhal Numbers
• Create a graph that shows the given Strouhal numbers for the following birds and bats.
□ Woodpecker: 0.22
□ Albatross: 0.25
□ Fruit bat: 0.40
□ Pigeon: 0.20
□ Hummingbird: 0.259
□ Swan: 0.20
□ Osprey: 0.23
□ Free-tailed bat: 0.31
□ Starling: 0.30
• What is the range of Strouhal numbers for the birds and bats represented on the graph?
• Represent the Strouhal numbers on the vertical axis using the numbers 0 to 0.5 (in 0.1 increments). Place the names of the birds and bats along the horizontal axis.
A Dolphin
Lesson Guide
Have students individually predict whether the Strouhal number for a dolphin will fall within the range for birds and bats.
Mathematical Practices
Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.
Students construct a viable argument for the reasonableness of their solution and they have an opportunity to critique the work of others during the class discussion.
Mathematical Practice 4: Model with key concepts.
Students model the mathematics presented in the problem graphically.
Mathematical Practice 5: Use appropriate tools strategically.
Students use appropriate tools and problem-solving strategies to find the solution. Students may use a calculator as a possible tool for finding their solution.
Mathematical Practice 6: Attend to precision.
Students attend to precision while creating their graphs as well as when calculating the range in Strouhal numbers. Students also use precision when calculating the Strouhal number of a dolphin.
Mathematical Practice 7: Look for and make use of structure.
Students make use of the problem-solving structure while solving this problem.
Student has difficulty getting started.
• What information is provided in the problem? What are you trying to find?
• How can you organize the information presented in the problem?
• What problem-solving strategies can you use to help you find a solution?
Student has a solution, but it is incorrect.
• Is your answer reasonable? Why or why not?
• Does the information in your graph match the data in the table?
• How do you calculate the range for the data?
Student has a solution but is having difficulty articulating his or her thinking.
• How can you state the answer as a complete sentence?
• How would you describe your strategy to somebody who is struggling?
• Does your answer make sense? How do you know?
Student has a correct solution and is waiting on others to finish.
• If you were designing a flying robot with a high wing-flapping frequency, how would you design the length of the wings?
• Use the information given to create another problem for your partner to solve.
Possible Answers
• Predictions will vary.
• $1.5 · 315=0.3$
The Strouhal number for the dolphin is 0.3. Comparisons to predictions will vary.
Work Time
A Dolphin
The dolphin has a cruising speed of 15 meters per second, an amplitude of 3 meters, and a flapping frequency of 1.5 beats per second.
• Predict whether the dolphin’s Strouhal number will fall within the range you calculated for the birds and bats.
• Calculate the Strouhal number for the dolphin and compare it to your prediction.
Prepare a Presentation
Preparing for Ways of Thinking
As pairs work, identify students who:
• Organize the information and describe the relationship among the variables in different ways.
• Generate different types of graphs. For example, horizontal bar graph, vertical bar graph, or line graph.
• Have different predictions about the Strouhal number for dolphins. For example, students look for students who thought that flying and swimming were similar and so the Strouhal numbers for
dolphins would be similar as well, or students who thought there would be vast differences between traveling through the air and the water.
• Complete the Challenge Problem.
Challenge Problem
• Answers will vary.Possible answer: If you were creating a flying robot, you would want the Strouhal number to fall within the range of other flying animals. Therefore, if the robot had a large
amplitude, you would make the frequency shorter to fall within the correct range.
Work Time
Prepare a Presentation
Prepare a presentation of your findings. Justify all your findings with mathematical explanations.
Challenge Problem
• How could Strouhal numbers help you design a flying robot?
• Consider how the Strouhal numbers for flying and swimming animals are related.
• If you were designing a flying robot, how could this relationship help you with your design? Consider wing amplitude, frequency, and flying speed.
Make Connections
Highlight the different strategies students used to solve the problems. Focus on how students created and implemented a problem-solving plan. Encourage discussion about the approaches for solving
problems and the validity of the answers. Prompt students to give the presenters positive feedback as well as opportunities for improvement. Students should be refining their own strategies,
correcting solutions, and taking notes during the presentations.
Point out that the range of the Strouhal numbers for flying and swimming animals is between 0.2 and 0.4. Flying and swimming are most efficient when the Strouhal number is between 0.2 and 0.4. Ask
students how this range affects the variables in the problem, for example, if the flapping frequency increases, what happens to the amplitude? How does this affect the flying or swimming efficiency
of the animal?
Ask guiding questions, for example:
• How did you approach the problem? What strategies did you use?
• Do you think all Strouhal numbers fall within this range for flying and swimming animals? Why or why not?
• Do you think all swimming animals have the same relationship between flapping frequency and amplitude as the flying animals? Why?
• How did you create your graph? Are there other types of graphs that could display this information?
• Would a vertical bar graph show the information in the same way as a horizontal bar graph?
• How did you persevere in finding the solution?
• How do you know that your answer is reasonable?
• What tools did you use to help you find your solution?
• What was the most challenging aspect of the problem? How did you overcome this challenge?
• Are there multiple solutions to the problem? Why or why not?
Point out any mathematical processes you observed as students worked.
Have students who completed the Challenge Problem share their responses.
SWD: During Ways of Thinking, facilitate the learning process by encouraging students to discuss multiple strategies and representations of the mathematics. Ask questions and guide discussions. Help
students to compare and contrast an animal's flying/swimming efficiency and their Strouhal number.
Performance Task
Ways of Thinking: Make Connections
Take notes about other classmates’ graphs, and their use of the problem-solving process.
Reflect on Your Work
Lesson Guide
Have each student write a brief reflection before the end of class. Review the reflections to learn about the strategies students used to find the range of the Strouhal numbers for birds and bats.
Work Time
Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.
The strategy I used to find the range of the Strouhal numbers for birds and bats is…
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Troubleshooting network analyses
In the ArcGIS Network Analyst extension, you may encounter certain error messages or unexpected behaviors that you want to understand and resolve. It is helpful to turn on error and warning messages
on the Network Analyst Options dialog box.
You can open the Network Analyst Options dialog box by clicking Network Analyst > Options on the Network Analyst toolbar.
Learn more about Network Analyst options
You can refer to the general behavior section below to understand what kinds of errors are reported by Network Analyst and why they occur.
Additionally, the solver-specific behavior section below discusses how Network Analyst solvers behave in different scenarios, for example, what happens during a route analysis when the first stop is
not located on the network.
General behavior
There are five kinds of errors reported by Network Analyst, as discussed in the table below. The order in which these errors are discovered is as follows:
1. Errors in definition
2. Invalid locations
3. Cardinality issues
4. Reachability issues
5. User aborted
Type of Description Example Output
1. During route analysis, the attribute used as impedance
is not found, as it is incorrectly named, for example,
Errors in The analysis problem is incorrectly "Unable to find attribute Times."
definition defined. 2. During route analysis, hierarchy settings are invalid, An error message is generated. No solution is found.
for example, "Invalid max value 2 for hierarchy level
2. Has to be greater than hierarchy value 3 for
hierarchy level 1."
Some or all of the network 1. During route analysis, some of the stops are not
locations are not located on the located on the network.
network. In such cases, two 2. A stop is located on an edge that is restricted in both If invalid locations are ignored, a partial solution is found using the located network
Invalid alternatives exist: directions. locations. Additionally, a warning message is generated about the unlocated network
locations 3. A stop is located on an edge that is blocked at both locations that were ignored. If invalid locations are not ignored, no solution is found,
• Invalid locations are ignored. ends with barriers. and an error message is generated.
• Invalid locations are not 4. A located stop has incorrect time window attributes.
The number of valid locations is
Cardinality fewer than the minimum number of During route analysis, there is only one valid network An error message is generated. No solution is found.
issues locations required for the location.
1. Part of the network is not connected to or is isolated
Reachability Some of the network locations from the remaining network. Output depends on the solver and the analysis settings. In some cases, a partial solution
issues cannot be reached. 2. Hierarchy is disconnected or incorrect. with a warning message is found. In other cases, no solution is found, and an error
3. During closest facility analysis, the closest facility message is generated.
lies beyond the cutoff cost.
User aborted The user aborts the analysis by During route analysis, the user presses the ESC key, after An abort message is generated.
pressing the ESC key. clicking the Solve button
Solver error table
Solver-specific behavior
The following rules apply to each solver:
Route solver
• At least two stops per route are required to solve.
• No solution if no route is found.
• If invalid locations are set to be ignored, all invalid locations are ignored (stops as well as barriers). Additionally, the sequence number of all stops is not changed (unless the Reorder Stops
To Find Optimal Route analysis layer option is used).
• If you have valid stops and one of them is unreachable, no solution is found to any stop (unless the Reorder Stops To Find Optimal Route option is used).
• The sequence number must be valid. That is, it must be greater than zero and cannot exceed the number of stops nor be a duplicate value.
Route solver using the Reorder Stops To Find Optimal Route option
• If invalid locations are set to be ignored, all invalid locations are skipped. The stops are resequenced such that the invalid locations are moved to the end of the list. If the first stop is an
invalid location, the second stop becomes the first one, and the first stop is moved to the end of the list.
• If you have valid stops and one of them is unreachable, the stops are reordered and the unreachable stop is moved to the end of the list. A partial solution is found. This means the route is
calculated for the reachable, valid stops.
• If the analysis layer option Preserve First Stop (or Preserve Last Stop) is checked, the first stop (or the last stop) must be reachable. If it is unreachable, no solution is found.
Closest facility solver
• At least one valid, reachable incident and one valid, reachable facility are required to solve.
• If there is no valid or reachable facility for any incident, no solution is found.
• If there are some incidents that have no valid or reachable facility, a partial solution is found, as long as at least one valid, reachable incident and one valid, reachable facility are present.
• If invalid locations are set to be ignored, all invalid locations are ignored (facilities, incidents, and barriers).
Service area solver
• At least one valid, reachable facility is required to solve.
• If there are no traversable edges for any facility, no solution is found.
• If there are some facilities with no traversable edges, a partial solution is found—as long as there exists at least one facility with traversable edges.
• If invalid locations are set to be ignored, all invalid locations are ignored (facilities and barriers).
OD cost matrix solver
• At least one valid, reachable origin and one valid, reachable destination are required to solve.
• If there is no valid or reachable destination for any origin, no solution is found.
• If there are some origins with no valid or reachable destinations, a partial solution is found—as long as there exists at least one valid and reachable destination for at least one valid and
reachable origin.
• If invalid locations are set to be ignored, all invalid locations are ignored (origins, destinations, and barriers).
Vehicle routing problem solver
• At least one order, depot, and route are required for a given vehicle routing problem analysis layer to be solved.
• The invalid network locations in Orders, Depots, and Barriers network analysis classes cannot be ignored. An error message is generated if any of these network analysis classes have invalid
network locations.
• The attributes in the network analysis classes that act as key fields must have identical values. For example, the value for the Name attribute in the Depots network analysis class must be
identical to the value for the StartDepotName and EndDepotName attributes in the Routes network analysis class. Similarly, the value for the Name attribute in the Routes network analysis class
must be identical to the value for the RouteName attribute in the Breaks network analysis class.
Relationships between network analysis classes in the vehicle routing problem
• If distance-based constraints, such as MaxTotalDistance and CostPerUnitDistance, are specified for routes, the Distance Attribute property of the analysis layer has to be specified.
• If the VRP solver cannot assign all the orders to the routes without violating the given constraints, a partial solution is determined by the solver. The ViolatedConstraints attribute in the
Orders and Routes network analysis classes contains information about the constraints that are violated by a particular order or the route.
Location-allocation solver
• At least one valid, reachable facility and one valid, reachable demand point are required to solve.
• If there isn't a valid, reachable facility or demand point, no solution is found.
• If there are some facilities on nontraversable edges, a partial solution is found—as long as there exists at least one facility with traversable edges.
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A. Mehrabov, "The Calculation of the Short-Range Order Parameters with taking into account the Effect of Atomic Displacements by Method of Pseudopotential Theory," IV-th Inter - University
Conferences of Azerbaijan SSR on Physics , Baku, Azerbaijan, pp.22, 1978
Mehrabov, A. 1978. The Calculation of the Short-Range Order Parameters with taking into account the Effect of Atomic Displacements by Method of Pseudopotential Theory. IV-th Inter - University
Conferences of Azerbaijan SSR on Physics , (Baku, Azerbaijan), 22.
Mehrabov, A., (1978). The Calculation of the Short-Range Order Parameters with taking into account the Effect of Atomic Displacements by Method of Pseudopotential Theory . IV-th Inter - University
Conferences of Azerbaijan SSR on Physics (pp.22). Baku, Azerbaijan
Mehrabov, AMDULLA. "The Calculation of the Short-Range Order Parameters with taking into account the Effect of Atomic Displacements by Method of Pseudopotential Theory," IV-th Inter - University
Conferences of Azerbaijan SSR on Physics, Baku, Azerbaijan, 1978
Mehrabov, AMDULLA. "The Calculation of the Short-Range Order Parameters with taking into account the Effect of Atomic Displacements by Method of Pseudopotential Theory." IV-th Inter - University
Conferences of Azerbaijan SSR on Physics , Baku, Azerbaijan, pp.22, 1978
Mehrabov, A. (1978) . "The Calculation of the Short-Range Order Parameters with taking into account the Effect of Atomic Displacements by Method of Pseudopotential Theory." IV-th Inter - University
Conferences of Azerbaijan SSR on Physics , Baku, Azerbaijan, p.22.
@conferencepaper{conferencepaper, author={AMDULLA MEHRABOV}, title={The Calculation of the Short-Range Order Parameters with taking into account the Effect of Atomic Displacements by Method of
Pseudopotential Theory}, congress name={IV-th Inter - University Conferences of Azerbaijan SSR on Physics}, city={Baku}, country={Azerbaijan}, year={1978}, pages={22} }
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Highest Common Factors
Highest Common Factors
The highest common factor is the highest number which is a factor of two other numbers. We can use prime factorisation to find it.
What is a factor of a number?
Which of these are factors of $36$? Select all that apply.
You can select multiple answers
A common factor of two numbers is a factor that divides into both numbers without leaving a remainder. Which of these is a common factor of $36$ and $51$?
Finding the highest common factor, or HCF, of two numbers can help when working with fractions.
We find the HCF using prime factorisation.
What is the HCF of $30$ and $75$?
First we find the prime factorisation of each number.
This is the prime factorisation tree for $30$.
Using a prime factorisation tree, find the prime factors of $75$.
We now have the prime factorisations of both $30$ and $75$. One common factor is $3$, what is the other?
We have identified that $3$ and $5$ are common prime factors of $30$ and $75$. How can we find the highest common factor?
The common prime factors are highlighted, what is the highest common factor of $30$ and $75$?
Well done! You have found the answer!
The HCF of $30$ and $75$ is $15$.
When the numbers are smaller, we can list the factors to find the highest common factor.
Let's find the highest common factor of $27$ and $39$.
We can list the factors of $39$, they are $1$, $3$, $13$ and $39$. What are the factors of $27$? List them in order and separate your answers with a comma.
Now we have all the factors of the two numbers, what is the HCF of $27$ and $39$?
The HCF of $27$ and $39$ is $3$.
With this method, when listing the factors we just find the highest one, there is no multiplying needed!
Quick recap - which option describes the HCF?
How can you find the HCF by listing the factors of each number?
What is the highest common factor of $15$ and $35$?
Fred says that $8$ is the HCF of $24$ and $48$. Is he correct? If not, what is the HCF of $24$ and $48$?
We've seen that the HCF of $24$ and $48$ is $24$.
This shows that sometimes the HCF of two numbers is the smaller of the two numbers.
What is the HCF of $36$ and $180$?
You may need some time to work this out
Let's try a harder example: finding the HCF of three numbers.
Let's find the HCF of $200$, $60$ and $40$.
The prime factorisation of $200$ is $2^3\times 5^2$. What is the prime factorisation of $60$?
We need to find our final prime factorisation of $40$, what is this one?
We now have the prime factorisations for all three numbers.
We need to compare them and find which are common to all three numbers.
Which two prime numbers appear in all three? Separate your answers with a comma.
Be careful with this one! We identified that the common prime factors are $2$ and $5$, but look more closely at the $2$. What is the actual common factor here?
Now let's find out HCF of $200$, $60$ and $40$. The common factors are $2^2$ and $5$, what is the HCF?
Well done! The HCF of $200$, $60$ and $40$ is $20$.
Where a common prime factor has a power, we choose the lowest power to find the HCF.
A factor is a number which divides into another number exactly, leaving no remainder, decimal or fraction.
$5$ is a factor of $20$ because $20\div 5=4$
$8$ is not a factor of $20$ because $20\div 8=2.5$
A common factor of two numbers is the same factor that divides exactly into both numbers.
$3$ is a common factor of $75$ and $27$ because $75\div 3=25$ and $27\div 3=9$
Numbers can have more than one common factor.
Finding the highest common factor of two numbers can help when working with fractions.
Use prime factorisation of the numbers to find the highest common factor.
The prime factors in common multiply together to give the HCF.
To find the HCF of $84$ and $132$, find the prime factorisation of both numbers.
$84=2^2\times 3\times 7$ and $132=2^2\times 3\times 11$ The HCF is the product of the common factors which is $2^2\times 3=12$
Where a common prime factor has a power
choose the lowest power to find the HCF.
For smaller numbers it may be easier simply to list all the factors of both.
The HCF is the highest factor that appears in both lists.
$12$ has factors $1$, $2$, $3$, $4$, $6$ and $12$
$18$ has factors $1$, $2$, $1$, $1$, $1$ and $1$
The highest factor in both lists is $6$ so the HCF of $12$ and $18$ is $6$.
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Coding a dynamic systems and controlling it via a graphical user interface | R-bloggersCoding a dynamic systems and controlling it via a graphical user interface
Coding a dynamic systems and controlling it via a graphical user interface
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My work, in the past year, has consisted mostly of coding dynamic models in R, models which I will soon be exporting to a server-based R implementation, possibly thanks to rApache.
I ususally run my models through an input file where I specify all parameters needed, but for the end users, we felt it may be better to provide a graphical user interface where they could select
just a few parameters, with the others defaulting to meaningful values.
Wioth this post I want to quickly illustrate all that’s needed to put such a system together, exception made for the rApache part. I’ve not made any steps in that direction, yet.
So, let’s start by defining a dynamic model, which we’ll integrate using the
package. We use the Lorentz system, which is far simpler than any of the models I actually work with, and produces much more beautiful and interesting graphics, too.
OK, if you run this you’ll obtain a variable called ‘out’, which contains the X/Y/Z coordinates of your system at different time instants in the phase space. You can look at the values directly, but
obviously plotting is a good option. Taking advantage of the
‘multiplot’ function
defined in the
Cookbook for R
, we can write:
Which will generate the following picture:
I did make use of the alpha channel to give some sense of depth to all pictures. I would love to plot a 3D version of the Lorenz Attractor in the fourth panel, lower right – however, I didn’t want to
get bogged down in defining a rotation / projection matrix.
Until now, there’s no GUI – all this happens within the command line, or if you prefer a simple R script.
Unless, that is, you also define a gWidget which can actualy control your model, like this:
To draw this, you just need to type a few lines of code in your R script, plus some more functions to handle events (that is, you clicking the button or changing parameter values)
As a matter of fact, we can also embed the graphical output within the GUI window, either on the side of the controls, or in another tab. perhaps I’ll update the post later on to reflect that.
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FSM: Liquid Desander – Sizing and Selection: Application Steps (B-FSM054)
One of the sizing routines detailed in Svarovsky (Ref. 3)
We now start the discussion on sizing and selection of a desander for separation duty. This takes all the theory we have previously detailed and adds calculation steps to properly size a desander for
your application.
There are quite a few good sizing routines available. I recommend starting with Plitt (see references). Please keep in mind that he is sizing a hydrocyclone, not a desander, so the appropriate
modifications must be made to account for no flow split. Svarovsky also publishes quite a few sizing equations, which are valid for various applications and geometries.
The routine I have outlined below was published in 2001 (see Rawlins and Wang reference). It is the commercial routine I’ve used since my grad school days. In this post I am just detailing the main
application steps. Subsequent posts will provide the calculation details – but I need to give an outline first.
Application Steps for Desander Selection
1. Select the size of desander (insert or liner) geometry for separation duty
• Gives appropriate separation size (D98)
• Gives required total solids recovery (%)
• Meets allowable pressure drop (ΔP)
2. Select appropriate number of operating units for flow and turndown
• Insert style: number of vessels in parallel operation
• Liner style: number of liners (design, high, and low flow) in the vessel
3. Select appropriate mechanical design rating and materials of construction
• Vessel diameter determines insert diameter or quantity of liners
• Appropriate vessel mechanical design rating (i.e. ASME) suitable for design pressure and temperature
• Vessel materials of construction for corrosion control
• Insert/liner materials of construction for erosion control
The next article discusses desander geometry for separation duty.
1. Plitt, L.R., “A mathematical model of the hydrocyclone classifier”, CIM Bulletin, December, 1976, pp. 115-123.
2. Rawlins, C.H., and Wang, I. I., “Design and Installation of a Sand Separation and Handling System for a Gulf of Mexico Oil Production Facility,” SPE Production and Facilities, paper 72999, Vol.
16, No. 3, 2001, pp. 134-140.
3. Svarovsky, L., “Hydrocyclones”, Technomics Publishing Co. Inc., Lancaster, PA, 1984.
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Expository Papers in Mathematics | Pomona College in Claremont, California - Pomona College
It is very helpful to have an introduction (and, depending on how long your paper is, an abstract) to contextualize your paper. Introductions should not yet bombard the reader with heavy mathematics.
Keep it light. It is often interesting to frame the paper in some interesting question that can only be answered by understanding the material in your paper. Or, if your topic is not something you
can tie into a real-world problem, use your introduction to convince the reader of the relevance of the topic. Why is it interesting? What subfields of math is it pertinent to? Historical background,
if you choose to use any, finds a great home in the introduction.
As with any paper, it is important to know your audience. If your professor doesn’t specify, ask. Can you assume that the reader has taken your course? Or should your paper be accessible to the
general public? With math being in essence its own language, it is important to know how much hand-holding and defining your paper should do. Regardless of your audience, though, you must give
concrete examples of the often abstract ideas being discussed.
When in doubt, use examples! Examples are the best way to ground your reader and ensure they understand your complex material. Particularly when your definitions or theorems are in general form, pick
a basic yet illustrative example with which to couple it. It is often helpful to give an example before stating the general definition or theorem. That way, the example eases your reader into the
general form, as opposed to serving as damage control after having scared them with abstraction.
Let your voice come through! Just because it’s a math paper doesn’t mean it has to uphold the field’s stigma of being dry and clinical-sounding. Your reader should want to read your paper. So, don’t
shy away from being punny, taking on a tone, or varying your syntax. And have fun—just as you would in any piece of writing. What is unique about a math paper, though, is that actual definitions,
theorems, conjectures, proofs etc. must retain their formalities.
As such, it is helpful to keep your informal prose visually separate from the more precise and technical components of the paper. Paragraphs and punctuation are still crucial in math papers (and the
same rules of grammar apply), but the prose certainly can and should be spliced by definitions, examples, theorems, proofs, etc. It would be helpful for the mind and easier on the eye to use boldface
and italicized language where appropriate, for distinction and consistency. Additionally, papers are often divided into sections and subsections, in order to make structural sense of the material.
LaTeX is a typesetting language that does much of this formatting for you and is particularly helpful in typing out equations. It is the most commonly used language for writing technical math papers.
Some professors may even require you to use LaTeX. If so, Claremont Center for the Mathematical Sciences (CCMS) Software Lab is a helpful resource for learning. They offer tutorials at the library.
Make sure that you understand the conventions of the subfield you’re discussing. Subfields of mathematics have specific ways of expressing things, and the same symbol may mean something totally
different in two different subfields. In that vein, clarify what your symbols mean. Now, mathematical shorthand is prevalent in class lectures, but it should not make an appearance in your paper.
Just as academic papers refrain from using contractions and abbreviations, your math paper’s prose should use words.
You don’t necessarily need a conclusion, but it is a nice way to wrap up. Many math papers end abruptly, so you really shouldn’t feel obligated to write a conclusion. If you choose to write one, you
can bookend it with some of the information relayed in your introduction. It can include more math within it than did the introduction, but it should not leave your reader with the sensation of being
overwhelmed. It can also include the limitations or extending applications of the topic/results, and it can highlight open problems and suggest areas for future research.
Lastly, make sure to cite your sources! It is convention to use endnotes to cite information throughout the paper, so it does not interrupted the flow of the sentence. If you are using LaTeX, it
compiles a bibliography for you, as you write and cite. Make sure your sources are reliable and valid. Peer reviewed sources are ideal.
For more information on mathematical writing (including tips for writing proofs, posters, and research papers) reference Handbook of Writing for the Mathematical Sciences by Nicholas J. Higham
Schedule an Appointment Visit our scheduler. Register for an account. Book an appointment!
Writing Program Coordinator
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If each edge of a cube is doubled. How many times will its surface area increase? How many times will volume increase?
Hint: Let any variable be the side of the cube and the surface area and volume of the cube is 6 times side square and side cube respectively so use these concepts to reach the solution of the
Complete step-by-step answer:
As we know that the cube has 6 sides and each side represents the square.
And the area of square is $ = {\left( {{\text{side}}} \right)^2}$
Let the side length of the cube be a unit.
So the surface area (S.A) of the cube is $ = 6{a^2}$ sq. unit
$ \Rightarrow S.A = 6{a^2}$ sq. unit ………………….. (1)
And we all know that the volume (V) of the cube is $ = {\left( {{\text{Side}}} \right)^3}$
$ \Rightarrow V = {a^3}$ unit cube………………………… (2)
Now it is given that the edge of the cube is doubled.
So now the edge of the cube becomes 2a.
So the new surface area ($S.{A_1}$) of the cube is $ = 6{\left( {2a} \right)^2} = 6\left( {4{a^2}}
\right) = 24{a^2}$ sq. unit.
$ \Rightarrow S.{A_1} = 4\left( {6{a^2}} \right)$
Now from equation (1) we have,
$ \Rightarrow S.{A_1} = 4\left( {S.A} \right)$
So the new surface area of the cube is four times the old surface area.
And the new volume (${V_1}$) of the cube is $ = {\left( {2a} \right)^3} = 8{a^3}$
$ \Rightarrow {V_1} = 8{a^3}$
Now from equation (2) we have,
$ \Rightarrow {V_1} = 8\left( V \right)$
So, the new volume of the cube is eight times the old volume.
So, this is the required answer.
Note: Whenever we face such types of problem the key concept is formula of surface area and volume of cube which is stated above then make the side double as given in problem statement and calculate
new surface area and volume of the cube as above then convert new surface area and volume in terms of old surface area and volume of the cube as above so, doing this we can easily calculate how much
times the surface area and volume of the cube is increased if the edge of the cube is doubled.
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Unitary group
Jeremy Luterbacher, Songlan Sun, Stefania Bertella, Anastasiia Komarova
The present invention relates to a compound of the general formula (I), (II) and (III), more specifically of formula (Ia), (Ib), (Ic)wherein R11 and R12 or R21 and R22 or R31 and R32 are both
hydrogen or form together with CHR50 a cyclic moiety or one of R ...
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Re: [tlaplus] How to model check this "real-time" spec?
First, you can of course ignore the deadlock by disabling deadlock checking if you are not interested in it.
Second, a state is deadlocked if `ENABLED Next' is false at that state, but the actual transition relation is [Next]_vars, so a finite behavior ending in a deadlocked state can always be extended to
an infinite behavior.
But perhaps one answer here is that while the spec doesn't admit any traces in which the invariant is violated, it does admit finite traces (ie those that deadlock) and that is what TLC is
catching. However, given TLA semantics, only infinite traces can be traces of the model so finite traces don't or shouldn't enter the picture.
On Friday, October 15, 2021 at 8:53:55 PM UTC-7 ns wrote:
hi Stephan, thanks as always for your insightful answer. My thinking was that since the spec doesn't admit any traces in which the lights don't change at least once every K s (based on my
understanding of TLA semantics) I was hoping that the model checker would agree and say that the invariant was indeed satisfied. But instead it reported deadlock. I did see where the deadlock
is coming from, and I suppose its a way of rejecting those bad traces, but was somewhat surprising. I like your solution to the problem using a kind of "lookahead" to prevent DoNothing when a
bound is about to expire, but it appears to me a way of getting TLC to do the right thing. Please let me know if I'm still missing something key.
On Sunday, October 10, 2021 at 2:19:32 AM UTC-7 Stephan Merz wrote:
Not sure I understand your question: TLC *does* model check your spec, it just (correctly) signals a deadlock, since the non-clock variables may stutter according to DoNothing until the
clock exceeds its bound, at which point no action is enabled anymore. If you intend to avoid this, DoNothing needs a stronger time bound, see attached module for a possible definition.
Since the modified spec generates an unbounded state space, you'll need to add a state constraint to your model, such as
clock <= MAX_TIME
Hope this helps,
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Simple Interest Formula with Example for SSC and Banking Exams
The formula helps to solve questions quickly but you just need to learn how to use any formula to solve the problem. Today I am giving important simple interest formula with examples to understand
their use of them. Just memorize these simple interest formulas to solve problems quickly and learn simple interest formulas with examples.
Simple interest is an important topic of quantitative aptitude. Every year 2-3% of questions are asked of simple interest in SSC, bank and another competitive exam. After learning the use of simple
interest formula you can practice simple interest questions with answers.
Simple Interest Formula with Example for Competitive Exams:
Simple Interest definition: If the interest calculated on sum borrowed for a particular period/time is reckoned uniformly that is called a simple interest.
Principal definition: The money lent/borrowed for a certain period/time is called the principle/sum.
Interest definition: Extra money paid for using other’s money is called interest.
Let Principle = P
Rate = R% per annum (p.a)
Time = T year
Amount = A
(1) Simple interest Formula:
S.I =
(2) Formula for Principal:
P =
(3) Formula for Rate of Interest:
R =
(4) Formula for Time:
T =
(5) Formula for Net Amount
A = P+I
Simple Interest Example:
Q.1 Find the simple interest on Rs. 68,000 at per annum for 9 months.
P= Rs. 68000, R=
S.I =
=Rs. 8500
Q.2 Find the simple interest on Rs. 3000 at per annum for the period from 4^th feb, 2015 to 18 April 2015.
Time = (24+31+18) days = 73 days =
P = Rs. 3000 and R =
S.I = Rs.
= Rs. 3750
Q.3 IF a sum of money becomes ‘n’ times in ‘T’ yr at simple interest, then formula for calculating interest will be given as R =
Ex. A sum of money becomes four time in 20 yr at SI. Find the rate of interest.
Here, T = 20 yers, n= 4p
= 15
I hope this given simple interest formula with example helps you to solve simple interest problems quickly and correctly in competitive exams. Practice more simple interest questions with answers.
For more Simple Intrest questions and answers, Visit next page.
Showing page 1 of 3
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Queue and its basic operations
What is queue?
First, let us see the properties of data structures that we already do know and build-up our concepts towards the queue data structure.
• Array: Its a random-access container, meaning any element of this container can be accessed instantly.
• Linked List: It's a sequential-access container, meaning that elements of this data structure can only be accessed sequentially.
→ Following a similar definition, a queue is a container where only the front and back elements can be accessed or operated upon.
Queue is a data structure following the FIFO(First In, First Out) principle.
You can imagine an actual queue for a ticket counter for visual aid.
• The first person in the queue is the first one to get the ticket and hence is the first to get out.
Enqueue Operation
Enqueue means inserting an element in the queue. In a normal queue at a ticket counter, where does a new person go and stand to become a part of the queue? The person goes and stands in the back .
Similarly, a new element in a queue is inserted at the back of the queue.
Dequeue Operation
Dequeue means removing an element from the queue. Since queue follows the FIFO principle we need to remove the element of the queue which was inserted at first. Naturally, the element inserted first
will be at the front of the queue so we will remove the front element and let the element behind it be the new front element.
Front Operation
This is similar to the peek operation in stacks, it returns the value of the element at the front without removing it.
isEmpty: Check if the queue is empty
To prevent performing operations on an empty queue, the programmer is required to internally maintain the size of the queue which will be updated during enqueue and deque operations accordingly.
isEmpty() conventionally returns a boolean value: True if size is 0, else False.
Queue Implementation
The key idea of queue implementation is to use both ends of the queue: front end for deleting elements and back/rear end for inserting elements. This is as simple as its concept because we just need
to simply keep in mind all of its properties and operations.
You should remember one very important thing though →
All operations in the queue must be of O(1) time complexity.
We shall be implementing queue in two different ways by changing the underlying container: Array and Linked List.
1. Array Implementation of Queue
An array is one of the simplest containers offering random access to users based on indexes. But what are the access criteria in a queue? Can you access any element of the queue? No. Only the first
and last elements are accessible in a queue.
So first, after initializing an array, we need two pointers, one each to point to the front end and back end. Instead of using actual pointers, we will use indexes, front and back will hold index
positions of front end and back end respectively.
int queue[8]
int front = -1
int back = -1
Here, 8 is a pre-defined capacity of the queue. The size of a queue should not exceed this limit
★ The default value for front and back is -1, denoting that the queue is empty. Let us wrap this group of data members in a class:
class Queue
int arr[]
int capacity
int front
int back
Let us also create a constructor which initializes capacity, front and back.
Queue(int cap)
capacity = cap
front = -1
back = -1
★ You are also required to allocate memory to arr according to the conventions of the language you use to implement it.
In order to improve our memory utilization, we will implement what's known as a circular queue. To implement a circular queue, we would need a circular array. Don't stress, its quite similar to our
normal array. So much that its declaration is not even slightly different.
In a circular array, the index after the last element is that of the first element.
So how would we traverse in such an array? Should we keep an if statement to check if the index is less than 8 or not? Alternatively, we could always use the remainder after dividing by 8 when
increasing index.
arr[(i+1) % capacity]
That is primarily the only difference. Whenever we increase index in a circular array, we take modulo with the size of the array to proceed to the next element.
Now, we need to implement the operations that we generally perform in a queue.
Enqueue Operation
Where do we insert an element in a queue? In the back.
And how do we do it?
• The value of back is increased by 1
• A new element is inserted at the back
• Be sure to increase the value of back according to the norms of circular array, i.e. (back+1) % capacity
• Initially front = -1. But when we are inserting the first element in the queue then we need to update the value of front i.e. front = back. (Think!)
Simple, right? Any exceptions that come to mind?
▹ What if the queue is filled up to its capacity? We shall check if the queue if full before inserting a new element and throw an error if it is. How do we check if the queue is full? ( Think! )
Now, let us implement this simply
void enqueue(int item)
if ((back + 1) % capacity == front)
print("Queue if full!")
back = (back+1) % capacity
arr[back] = item
if(front == -1)
front = back
Dequeue Operation
Now that we have discussed the insertion of an element, let’s discuss the deletion of an element. Which end of the queue do we use to delete an element? The front end! How do we delete an element?
• Access the element which is stored in front position i.e. item = arr[front]
• if (front == back) , then this is a case of a single element in the queue. After the deletion of this single element, queue become empty. So we need to set the empty queue condition before
returning the value i.e. front = back = -1 (Think!)
• Otherwise, increase the value of front by 1 i.e. front = (front + 1) % capacity.
• In the end, return the value stored at the item.
▹ Can you think of an exception in this case like the one above of the queue is full? Ans: The queue can be empty when the dequeue operation is called. We need to check for this beforehand.
Let’s try implementing it now
int dequeue()
if(isEmpty() == True)
print("Queue is empty!")
return 0
int item = arr[front]
if(front == back)
front = back = -1
front = (front + 1) % capacity
return item
★ But we just implemented Step 1, right? Where's Step 2?
→ Well, Step 2 is mainly deallocating any memory assigned to the element being dequeued. We were dealing with only primitive data type integer and therefore didn't need to deallocate anything.
Peek and isEmpty Operation
peek() and isEmpty() is quite simple to implement. We need to steer clear of exceptions though.
int peek()
if (isEmpty() == True)
print("Queue is empty!")
return -1
return arr[front]
bool isEmpty()
if (front == -1)
return True
return False
2. Linked List Implementation
Another way to implement a queue is to use a linked list as an underlying container. Let’s move towards the implementation part then
→ Let us assume that we have used class for linked list
class ListNode
int val
ListNode next
We also need two pointers that point to its front node and tail node.
What should an empty queue look like if implemented with a linked list? Ans: Its front and the back pointer will point to NULL
ListNode front = NULL
ListNode back = NULL
Is there any benefit to implementing the queue with a linked list compared to arrays? Ans: We do not need to mention the size of the queue beforehand.
→ Although, if you want to implement a limit to prevent excess use, you may need to encapsulate the class ListNode inside some other class along with a data member capacity .
class Queue
int capacity
class ListNode
int val
ListNode next
ListNode front
ListNode back
We shall be using just using class ListNode below for simplicity.Let us move towards queue operations
Enqueue Operation
The same properties hold as mentioned above with an added benefit that we need not worry about the queue being full, but we do need to check if its the first element is inserted in the queue, because
the values of front and back in that case would be NULL .
void enqueue(int item)
ListNode temp = ListNode(item)
if( isEmpty() == True )
front = temp
back = temp
back.next = temp
back = back.next
Dequeue Operation
Properties of linked list relaxed us from checking if the queue is full in enqueue operation. Do they provide any relaxation for exceptions in dequeue operation? No. We still need to check if the
queue is empty.
Since the front element is represented by a front pointer in this implementation. How do we delete the first element in a linked list? Simple, we make a front point to the second element
int dequeue()
if ( isEmpty() == True )
print ("Queue is empty!")
return 0
ListNode temp = front
int item = front.val
front = front.next
return item
Peek and isEmpty Operation
The implementation of these two operations is pretty simple and straight-forward in linked list too.
int peek()
if (isEmpty() == True)
print ("Queue is empty!")
return -1
return front.val
bool isEmpty()
if (front == NULL)
return True
return False
Augmentations in Queue
You can augment the queue data structure according to your needs. You can implement some extra operations like:-
• isFull(): tells you if the queue is filled to its capacity
• The dequeue operation could return the element being deleted
Application of Queues
What is the normal real-life application of queues? People wait in queues to await their chance to receive a service. Programming follows a similar concept.Let us look at some application of queue
data structure in real-life applications :-
1. Handling of high-priority processes in an operating system is handled using queues. First come first served process is used.
2. When you send requests to your printer to print pages, the requests are handled by using a queue.
3. When you send messages on social media, they are sent to a queue to the server.
4. An important way of traversal in a graph: Breadth First Search uses queue to store the nodes that need to be processed.
5. Queues are used in handling asynchronous communication between two different applications or two different processes.
Suggested Problems to solve
• LRU cache implementation
• Level Order Traversal in Tree
• BFS Traversal in a graph
Happy Coding! Enjoy Algorithms!
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IS THERE SUCH A THING AS FREE WILL?
Since there is a similarity between the concept of free will and that of randomness, we might suspect that, since we have seen that absolute randomness must be considered to be impossible, the same
should be thought to apply to the concept of free will.
There is a difference, however, in the concept of free will, in that the concept of the operation of a cause is not denied, whereas in the case of the concept of randomness it is denied.
However, it has to be considered that a fundamental property of the cause of a particular effect is that it, itself, also has a particular form of some kind, which is related to the particular form
of the effect. Therefore the form inherent in a cause itself requires to be explained as the effect of some other cause, just as is the case with any particular form.
An object with momentum which collides with and gives momentum to another object acts as the cause of the motion of the other object, which is determined by the form of the motion, or momentum, in
the causing object. Similarly, the gravitation field, which gives motion to an object in free fall, has a form which relates to the form of the motion of the falling object, and which can be given a
mathematical expression.
Since, therefore, it appears that we must say that every cause has a particular form of some kind, it consequently appears that we must say that every cause itself is the effect of some other cause.
This immediately leads to the concept of an infinite regression of causes, since this does not allow the existence of a cause that is not, itself, caused.
However, an infinite regression of causes implies the existence of a countable infinity, and we have already seen that a countable infinity is a contradiction in terms, and is impossible.
We are thus faced with the apparent paradox that
while the concept of an uncaused cause, which is similar to that of randomness, and which is the same as the concept of a free will, appears to be impossible, the alternative, an infinite regression
of causes, is also clearly impossible. This leads to the conclusion that we therefore do not really understand the essential nature of causality.
The following remarks, while not suggested to qualify as an 'answer', might, perhaps, at least cast some light on this dilemma.
We have seen that finite measurements in space are supported by an underlying, background spatial infinity referred to as 'continuity', which supports the finite measurement of distance, but is not,
itself, measurable in any absolute way. If we could similarly regard a finite succession of causes as a kind of finite measure of an underlying freewill causal nature that supports it, but is not
fully described by it, we may then confer a kind of infinite nature on this freewill cause, which can be described by many alternative series of causes, but not absolutely described by any of them.
That is, we might select various causes, A, B, C, etc., to each be the start of a different series that equally well describe a particular freewill causal act, in a finite way, but not in any
absolute way. This could make the free will to appear as a kind of infinite substitute for an infinite regression of causes, which cannot be fully displayed as an actual regression of causes.
Since, however, I think that we cannot really regard this as a complete or fully satisfactory philosophical description of the nature of free will, it is of the greatest importance to be able to
determine a justification for believing whether or not it exists, irrespective of how much or little we can actually explain it. I suggest that the article on the strange relationship between time
and the free will is sufficient to achieve such a justification.
© Alen, October 2015
Material on this page may be reproduced
for personal use only.
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math-comp devs
And here is a version that works for any substructure of (semi)ringType:
From elpi.apps Require Import coercion.
From mathcomp Require Import all_ssreflect ssralg.
Elpi Accumulate Coercion lp:{{
coercion _ N {{ nat }} Expected Res :-
coq.unify-eq {{ GRing.SemiRing.sort lp:R }} Expected ok, !,
coq.unify-eq {{ GRing.Nmodule.sort lp:V }} Expected ok, !,
Res = {{ @GRing.natmul lp:V (GRing.one lp:R) lp:N }}.
Elpi Typecheck Coercion.
Section TestNatMul.
Variable R : fieldType.
Variable n : nat.
Check n : R. (* elaborated to n%:R *)
End TestNatMul.
should we move this topic to [DEL:#Hierarchy Builder devs & users:DEL] #Elpi users & devs maybe?
To get this fully usable in MathComp, we still need to figure out the printing part. My current idea is to put a printing only notation in ring_scope that hides GRing.natmul and a printing only
notation %:R in a new ring_coercions so that Enable/Disable Notation : ring_coercions can show/hide coercions in the middle of a proof. But any other idea is welcome.
Karl Palmskog said:
should we move this topic to #Hierarchy Builder devs & users maybe?
Maybe in math-comp devs? as it is more linked to MC than HB.
This topic was moved here from #CUDW 2023 > Coercion Hook by Karl Palmskog.
Thanks @Karl Palmskog !
Last updated: Oct 13 2024 at 01:02 UTC
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Forecasting Volatility Using Machine Learning
Comparing traditional models to the LSTM model, for predicting the volatility of equities
“Only when the tide goes out do you discover who’s been swimming naked.”
Warren Buffett
Understanding the volatility of financial assets is important for several reasons:
• Risk management: Volatility is a key measure of risk. Measures such as VaR (value at risk) depend on accurate future measures of volatility.
• Asset allocation: Volatility is often a key input in portfolio construction and optimization. For example, for risk parity portfolios or mean-variance optimization.
• Derivates pricing: The prices of derivative contracts, for example, options, are directly linked to expectations of future volatility of the underlying.
• Market making: Forecasting bid / ask spreads is crucial to the market maker’s ability to maintain a liquid book. Spreads are influenced by volatility.
Given this importance, historically there has been an interest in forecasting future financial asset price volatility.
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Traditional volatility forecasting models
Simple MA or EWMA
First, the simplest forecast of the volatility that one can think of is to use the average past volatility itself.
In this way, we can compute future volatility as simply the average of past, squared, price returns for an asset r(t). If we allow the weights to be w(t)=1/t then we have a simple average.
\(\sigma_t = \sum_{\tau=1}^t{w_{\tau}(r_\tau - \mu)^2}, \quad \text{where} \quad \mu=\sum_{\tau=1}^t{w_\tau r_\tau}\)
Alternatively, we could set w(t) such that they are exponentially declining further back in time so that we estimate volatility as an exponentially weighted moving average (EWMA).
We find they are mathematically quite sophisticated in their statistical formulation when switching to modern volatility models.1
ARCH model
Perhaps the most widely known, and applied, is the ARCH model by Robert Engle from 1982. The ARCH model, with parameter p, models the volatility as a function of past squared returns: 2
\(r_t = \sigma_t \epsilon_t, \quad \text{where} \quad \sigma_t = \sqrt{\gamma + \alpha_1 r_{t-1}^2 + \alpha_2 r_{t-2}^2+ \dots + \alpha_p r_{t-p}^2}\)
And where epsilon(t) is a standard normal distributed IID random variable.
So we can see how “volatility clustering” is captured by this model. Essentially if past returns have been larger in magnitude, the more we believe today’s volatility will be large. Therefore,
volatility (or the squared returns — the return’s absolute magnitude) is persistent in time.
Given the wide success of ARCH, there have been innumerable variations to this basic model. The GARCH model, EGARCH, STARCH, IGARCH, TGARCH, CGARCH… the list goes on.
The basic improvement, however, came with the GARCH model in 1986 via Bollerslev, who modified the model above by adding lagged {sigma(t-1), sigma(t-2), … sigma(t-q)} terms to the right-hand side
This allows the volatility forecast to be smoother, and it turns out the GARCH volatility is equivalent to an EWMA of past squared returns, but where the exponential weights are determined by
statistical estimation, to maximize the fit of the GARCH model, not via an arbitrary choice.
The SV model
The SV or Stochastic Volatility model was an alternative to the ARCH model, published by Taylor in 1982. The main difference with the SV model is that it assumes volatility isn’t driven entirely by
past returns, but rather there is a “hidden” factor h(t) driving volatility we cannot observe directly. The SV model is as follows:
\(r_t=e^{h_t/2}\epsilon_t, \quad \text{where} \quad h_t=\gamma +\alpha h_{t-1} + \nu_t, \quad \text{and where} \quad h_t = ln(\sigma_t^2)\)
Because this volatility is “hidden” it makes the model more statistically complex to estimate, and perhaps for this purpose alone, it never gained as much widespread use or popularity.3 That said the
inherent flexibility provided by the hidden state of the model can lead to richer volatility dynamics.
ML Volatility Forecasting Models
Given this review of previous attempts at volatility forecasting, how can more modern machine learning (ML) based models contribute? Can they improve on the traditional models?
While there are a plethora of different models we could choose, given the historical precedent of building volatility models on time-series input data (in particular historical returns r(t)), we
thought it would be interesting to try and build a model around the LSTM architecture, which was published by Hochreiter & Schmidhuber in 1997.
The LSTM (or GRU) model
The LSTM (or “long short-term memory” model) has been used successfully in time-series forecasting tasks, within several fields, including speech recognition, electricity grid power load, industrial
goods demand, and other natural phenomena.
The LSTM is particularly interesting in this case, since similarly to the SV model above, at its core it has a “hidden” state variable that drives the dynamics of the predictions.
However, the dynamics of the hidden state are designed to encompass both a “short-term” and “long-term” memory of the past values of the input variables. That is, at least in theory, the model should
“remember” both recent episodes of volatility as well as those from the distant past.
For our purposes we’ll use a simpler version of the LSTM, called the GRU model, which has shown similar performance in most tests of time-series forecasting, but reduces the complexity of the model
significantly. The mathematical formulation of the GRU model is as follows: 4
Essentially the GRU is a hidden state model where what the hidden state is allowed to remember is modulated by the z(t) and r(t) “gates.” z(t) affects the long-term memory of the process, and r(t)
affects the short-term memory.
Note that r(t) above is not asset price returns as was the case previously. Rather, price returns enter into the model as the input vector:
This formulation is again similar to the SV model in that it assumes that squared returns (i.e. volatility) are log-Normally distributed. That is, taking logs of squared returns leads to x(t) which
is roughly Normally distributed.
At any given time, we then predict future volatility as:
\(\sigma_{t+h} = \sqrt{e^{f(h_t)}}\)
Typically we let f be a linear mapping function, but it need not be so. We can also allow for f to take as input a history of {h(t), h(t-1), …, h(t-s)} hidden states. The hidden state vector h(t)
itself is allowed to be multidimensional, even though the input is of dimension 1.5
Empirical Comparison
For this empirical comparison, we will forecast daily volatility for AAPL stock, 1-day into the future. Of course, different, longer-term, forecast periods could be implemented if need be.
We fit the ARCH, GARCH, and GRU models to daily AAPL closing price stock returns, as described above.
For the ARCH and GARCH models, we allow for hyperparameters p=5, and p=q=5, respectively. For the GRU, the hyperparameterization can be complex so we simply search the parameter space for something
that results in useful predictions.
Everything is out-of-sample (i.e. we train on a train set, and then forecast on a previously unused test set, using an expanding train window as time progresses), and the only hyperparameter search
we do is for the GRU. The models in both cases are retrained every 42 trading days (2 months roughly).
ARCH(5) results
GARCH(5,5) results
GRU results
Results comparison
We compare the results across three metrics:
• Root mean squared error (RMSE)
□ This metric will put more weight on large outliers in predictive errors
• Mean average error (MAE)
□ This metric cares more about the average error and doesn’t over-weight outliers
• R-squared (R2)
□ This is the measure of variance explained by the model versus that left unexplained by the model. The larger the value the better the fit.
□ In this case, the R2 is based on the OLS regression line plotted on each scatter in red. Since the model is linear, sqrt(R2) is the correlation between the forecast and true values.
The results suggest that the GRU is the superior model. In particular about MAE, where the reduction is quite significant. The reason is obvious if we look at the time-series plots of the predictions
above. In both the ARCH & GARCH examples, the forecasts are much too large on average.
However, in all cases, the scatters reveal that much of the low-volatility periods are overestimated; we can see this by the fact that the red OLS line slopes are typically less than 1.
Moreover, there is likely some nonlinearity not being captured here. Potentially the GRU could be improved by implementing the f output mapping function as nonlinear.
In summary, we’ve discussed the following:
• The significance of understanding the volatility of financial assets, highlighting its importance in risk management, asset allocation, derivatives pricing, and market making.
• We then reviewed traditional volatility forecasting models such as the simple moving average (MA), exponentially weighted moving average (EWMA), ARCH model, and stochastic volatility (SV) model.
• Next, we introduced modern machine learning (ML) based models, particularly focusing on the long short-term memory (LSTM) and gated recurrent unit (GRU) models, as potential improvements over
traditional methods.
• Finally, an empirical comparison was conducted between ARCH, GARCH, and GRU models in forecasting daily volatility for AAPL stock, with metrics including root mean squared error (RMSE), mean
average error (MAE), and R-squared.
• The results suggested that the GRU model outperforms ARCH and GARCH models, particularly in terms of MAE, although there are indications that further improvements could be made by implementing
nonlinear output mapping functions in the GRU model.
This follows in the spirit of other mathematically influenced pricing models, such as the Black-Scholes model for the price of a European put/call option, which itself involves the volatility of the
underlying as an input parameter. See Black-Scholes Wikipedia.
Where for simplicity we use demeaned returns to avoid the use of the mu term.
The hidden h(t) can be estimated via a linear Gaussian state-space model, albeit inefficiently, since nu(t) is not Gaussian, and the model must be log-linearized to form a linear observation
The model performs significantly better when the hidden state is allowed to be multidimensional.
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Uppsala Computing Science Department Technical Reports
Studiehandbok_del 4_200708 i PDF Manualzz
In this module, we will extend our previous system of natural deduction for propositional logic, to be able to deal with predicate logic. The main things we have to deal with are equality, and the
two quantifiers (existential and universal). Lecture 15: Natural Deduction. Natural deduction; Proofs. Natural Deduction.
Prolog implementation is the ability to "trace" the steps of a query as Prolog carries out its goal reduction process. This is an especially important aid to student understanding as it reveals steps
in a search for a proof. It also shows how deduction steps accumulate an incremental effect on logic variables to provide answers as side-effects. The system of natural deduction that originated with
Gentzen (1934–5), and for which Prawitz (1965) proved a normalization theorem, is re-cast so that all elimination rules are in parallel form. This enables one to prove a very exigent normalization
Lista över filtillägg och dataformat - Online-bibliotek
The deduction theorem helps. It assures us that, if we have a proof of a conclusion form premises, there is a proof of the corresponding implication.
Aktuella kurssidor vid Matematiska institutionen Stockholms
Jape’s proof engine was originally written in SML and compiled by SMLNJ, with interfaces for different operating systems written in C, tcl/tk, Python and I can’t remember what else. In 2002 I ported
the engine to OCaml and wrote a system-independent interface module in Java. PROLOG database by the programmer. The PROLOG interpreter o ers a deduction method, which is based on a technique called
SLD resolution (See [3] for details). Solving a problem in PROLOG starts with discerning the objects that are relevant to the particular problem, and the relationships that exist between them.
Deduction in Prolog. Hot Network Questions Help in identifying type Natural deduction proof editor and checker. This is a demo of a proof checker for Fitch-style natural deduction systems found in
many popular introductory logic textbooks. The specific system used here is the one found in forall x: Calgary Remix. A Prolog system with the sound unification cannot substitute X->X for X in the
body of the first abstraction: the unification in X=X->X fails the occurs check. Many Prolog systems omit the occurs check, and so succeed at the substitution.
Trafikverket kollektivavtal
So Prolog can be used to verify whether deductions are valid or not. 1992-02-26 · Keronen S. (1993) Natural deduction proof theory for logic programming. In: Lamma E., Mello P. (eds) Extensions of
Logic Programming. ELP 1992. Lecture Notes in Computer Science (Lecture Notes in Artificial Intelligence), vol 660. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56454-3_14.
Prolog is used in artificial intelligence applications such as natural language interfaces, automated reasoning systems and expert systems. Expert systems usually consist of a data base of facts and
rules and an inference engine, the run time system of Prolog provides much of the services of an inference engine. 2021-04-15 · I'm new to sentential logic / metalogic. Where should I start to learn
how to do proof by natural deduction. Without the use of any aditional rules, how would you go about proving that the following sentence is a sentence-logical truth?
Personligt schema
Clausal proof methods. Clause form. DOI: 10.1017/S1471068403001662 Corpus ID: 59369. Offline specialisation in Prolog using a hand-written compiler generator @article{Leuschel2004OfflineSI, title=
{Offline specialisation in Prolog using a hand-written compiler generator}, author={M. Leuschel and J. J{\o}rgensen and Wim Vanhoof and M. Bruynooghe}, journal={ArXiv}, year={2004}, volume={cs.PL/
0208009} } Grail 0 is recommended for users primarily interested in the natural deduction proofs generated by Grail. For all other users, a next generation Grail theorem prover, Grail 3, has replaced
Grail 2 as the current, stable and supported version of Grail. Grail 3 has a legacy mode which allows you to used your old Grail 2 grammars without any changes.
We begin by introducing natural deduction for intuitionistic logic, exhibiting its basic principles. We present the Natural Deduction Assistant (NaDeA) and discuss its advantages and disadvantages as
a tool for teaching logic. NaDeA is available online and is based on a formalization of natural Se hela listan på plato.stanford.edu Prolog is typically used in artificial intelligence applications
such as natural language interfaces, automated reasoning systems and expert systems. Expert systems usually consist of a data base of facts and rules and an inference engine, the run time system of
Prolog provides much of the services of an inference engine. Prolog Syntax natural deduction, and directly to this version of the program.
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time care pool ludvikabasbelopp tjanstebil 2021karin henriksson larsenden perfekte ansøgning eksempeladvokat karlshamn
Steering Committee Meeting 1989-09-08 Mattsson, Sven Erik
Pastebin is a website where you can store text online for a set period of time. Prolog and Classical Theorem Proving zResolution theorem proving viewpoint. zNatural deduction theorem proving
viewpoint. 2 Logic Programming School of Informatics, University of Edinburgh Conversion to Clausal Form Prolog as a Form of Natural Deduction F not(A) F A Board index » prolog. All times are UTC .
natural deduction program.
Heuristisk analys med Diderichsens satsschema
natural deduction we have a collection ofproof rules.
1. Prolog. 40. ; miratusfui, ibid. v. 6.
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Spread Footings
Jump to navigation Jump to search
751.38.1 General
These guidelines address procedures for design of spread footings used as foundations for bridge piers, bridge abutments, retaining structures and other miscellaneous structures. The guidelines were
established following load and resistance factor design (LRFD) concepts. The provisions provided herein are intended to produce foundations that achieve target reliabilities established by MoDOT for
structures of different operational importance. The four classes of operational importance include minor or low volume route, major route, major bridge costing less than $100 million, and major
bridge costing greater than $100 million. Additional background regarding development of these provisions and supportive information regarding use of these provisions is provided in the accompanying
751.38.1.1 Dimensions and Nomenclature
Dimensions to be established in design include the bearing depth (depth to footing base) and the footing dimensions shown in Figure 751.38.1.1. Table 751.38.1.1 defines each dimension and provides
relevant minimum and/or maximum values for the respective dimension.
Table 751.38.1.1 Summary of footing dimensions with minimum and maximum values
│Dimension│ Description │Minimum Value│Maximum Value│ Comment │
│ D │Column diameter │ 12” │ -- │ -- │
│ B │Footing width │ D+24” │ -- │Min. 3” increments│
│ L │Footing length │ D+24”^1 │ -- │Min. 3” increments│
│ A │Edge distance in width direction │ 12” │ -- │ -- │
│ A’ │Edge distance in length direction │ 12” │ -- │ -- │
│ t │Footing thickness │ 30” or D^2 │ 72” │Min. 3” increments│
│^1 Minimum of 1/6 x distance from top of beam to bottom of footing │
│^2 For column diameters ≥ 48”, use minimum value of 48” │
The nomenclature used in these guidelines has intentionally been selected to be consistent with that used in the AASHTO LRFD Bridge Design Specifications (AASHTO, 2009) to the extent possible to
avoid potential confusion with methods provided in those specifications. By convention, references to other provisions of the MoDOT Engineering Policy Guide are indicated as “EPG XXX.XX” throughout
these guidelines where the Xs are replaced with the appropriate article numbers. Similarly, references to provisions within the AASHTO LRFD Bridge Design Specifications are indicated as “LRFD
751.38.1.2 General Design Considerations
Footings shall be founded to bear a minimum of 36 in. below the finished elevation of the ground surface. In cases where scour, erosion, or undermining can be reasonably anticipated, footings shall
bear a minimum of 36 in. below the maximum anticipated depth of scour, erosion, or undermining.
Footing size shall be proportioned so that stresses under the footing are as uniform as practical at the service limit state.
Long, narrow footings supporting individual columns should be avoided unless space constraints or eccentric loading dictate otherwise, especially on foundation material of low capacity. In general,
spread footings should be made as close to square as possible. The length to width ratio of footings supporting individual columns should not exceed 2.0, except on structures where the ratio of
longitudinal to transverse loads or site constraints makes use of such a limit impractical.
Footings located near to rock slopes (e.g. rock cuts, river bluffs, etc.) shall be located so that the footing is founded beyond a prohibited region established by a line inclined from the horizontal
passing through the toe of the slope as shown in Figure 751.38.1.2. The boundary of the prohibited region shall be established by the Geotechnical Section. For the purposes of this provision, the toe
of the slope shall be the point on the slope that produces the most severe location for the active zone. Exceptions to this provision shall only be made with specific approval of the Geotechnical
Section and shall only be granted if overall stability can be demonstrated as provided in EPG 751.38.7.
Footings located near to soil slopes shall be evaluated for overall stability as provided in EPG 751.38.7 unless they are located a minimum distance of 2B beyond the crest of the slope.
751.38.1.3 Related Provisions
The provisions in these guidelines were developed presuming that design parameters required to apply the provisions are established following current MoDOT site characterization protocols as
described in EPG 321. Specific attention is drawn to EPG 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of Variation. The provisions provided in this subarticle
presume that parameter variability, as generally represented by the coefficient of variation (COV), is established following procedures in EPG 321.3.
751.38.2 General Design Procedure and Limit States
Spread footings shall be dimensioned to safely support the anticipated design loads without excessive deflections. Footing dimensions shall be established based on project specific requirements, site
constraints, and the requirements of this subarticle. Footings shall be sized at the applicable strength and serviceability limit states according to EPG 751.38.3 and EPG 751.38.4; the greatest
minimum dimensions established from consideration of each of these limit states shall govern the final design dimensions as long as they exceed the minimum dimensions specified in EPG 751.38.1. Final
design dimensions shall also be increased for cases with significant load eccentricity in accordance with EPG 751.38.5.
At a minimum, footings shall be designed to satisfy the Strength I and Service I limit states.
751.38.3 Design for Axial Loading at Strength Limit States
In general, spread footings shall be sized for strength limit states so that the factored bearing resistance exceeds the factored loads for the strength limit state of interest. This shall be
accomplished by determining the minimum footing dimensions, B and L, so that the following condition is satisfied
${\displaystyle B\times L\geq {\frac {\mbox{Factored Load}}{\mbox{Factored Bearing Resistance}}}={\frac {\gamma Q}{q_{R}}}}$ (consistent units of area) Equation 751.38.3.1.1
B = minimum footing width (consistent units of length),
L = minimum footing length (consistent units of length),
${\displaystyle {\boldsymbol {\gamma }}Q}$ = factored load for the appropriate strength limit state (consistent units of force) and
q[R] = factored bearing resistance (consistent units of stress).
The factored bearing resistance shall be established as
${\displaystyle q_{R}=\phi _{b}\cdot q_{n}}$ (consistent units of area) Equation 751.38.3.2
q[R] = factored bearing resistance (consistent units of stress),
Φ[b] = resistance factor for bearing resistance determined in accordance with this article (dimensionless) and
q[n] = nominal bearing resistance determined in accordance with this article (consistent units of stress).
For cases with eccentric loading, the modified footing dimensions, B’ and L’, shall be used for evaluations at strength limit states instead of the actual footing dimensions:
${\displaystyle B'\times L'\geq {\frac {\mbox{Factored Load}}{\mbox{Factored Bearing Resistance}}}={\frac {\gamma Q}{q_{R}}}}$ (consistent units of area) Equation 751.38.3.1.2
where B’ and L’ are established as stipulated in EPG 751.38.5:
B' = modified footing width to account for load eccentricity (consistent units of length) and
L' = modified footing length to account for load eccentricity (consistent units of length).
Final minimum footing dimensions shall not be less than those stipulated in EPG 751.38.1.
The method for determining the factored bearing resistance shall be selected based on the material type present beneath the base of the footing. In general, EPG 751.38.3.1 shall be followed for
footings founded on rock with uniaxial compressive strengths (q[u]) greater than 100 ksf; EPG 751.38.3.2 shall be followed for footings founded on weak rock with q[u] greater than 5 ksf but less than
100 ksf. The provisions in EPG 751.38.3.3 and EPG 751.38.3.4 shall be followed for footings founded on soil.
751.38.3.1 Bearing Resistance for Spread Footings on Rock (q[u] ≥ 100 ksf)
The nominal bearing resistance for spread footings on rock shall be calculated as a function of the mean uniaxial compressive strength of the intact rock according to (adapted from Wyllie, 1999):
${\displaystyle q_{n}=C_{f1}{\sqrt {s}}\cdot {\overline {q_{u}}}{\Bigg [}1+{\sqrt {{\frac {m}{\sqrt {s}}}+1}}{\Bigg ]}\leq 200ksf}$ (consistent units of stress) Equation 751.38.3.3
m and s = empirical constants describing the rock mass strength (dimensionless),
C[f1] = correction factor to account for footing shape (dimensionless) and
${\displaystyle {\overline {q_{u}}}}$ = mean value of the uniaxial compressive strength of intact rock core (consistent units of stress).
Resistance factors (${\displaystyle {\boldsymbol {\phi }}_{b}}$) to be applied to the nominal resistance values (q[n]) determined according to the provisions of this article shall be established from
Figure 751.38.3.1 based on the coefficient of variation of the mean uniaxial compressive strength, ${\displaystyle COV_{\overline {q_{u}}}}$. Values for q[u] and ${\displaystyle COV_{\overline {q_
{u}}}}$ shall be determined in accordance with methods described in EPG 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of Variation for the site and location in
question. Values for design parameters q[u], m and s shall be taken as mean values for the rock between the base of the footing and a depth of B below the base of the footing. Values for ${\
displaystyle COV_{\overline {q_{u}}}}$ should similarly reflect the variability of the mean uniaxial compressive strength for the rock over the same depth range.
Values for C[f1] shall be taken from Table 751.38.3.1.1. Values for the rock mass parameters m and s can be established as:
${\displaystyle m=m_{i}{\mbox{exp}}{\Big (}{\frac {GSI-100}{28}}{\Bigg )}}$ (dimensionless) Equation 751.38.3.4
${\displaystyle s={\mbox{exp}}{\Big (}{\frac {GSI-100}{9}}{\Bigg )}for\ GSI\geq 25}$ (dimensionless) Equation 751.38.3.5a
${\displaystyle s=0\ for\ GSI<25}$ (dimensionless) Equation 751.38.3.5b
where m[i] is a material constant corresponding to rock type and GSI is the Geological Strength Index. The value for m[i] can be estimated from Table 751.38.3.1.2 or determined more precisely from
triaxial tests (Hoek and Brown, 1997). For routine design, m[i] can be approximated as 10 for limestones and dolomites, as 6 for shales, siltstones, and mudstones and as 17 for sandstones. Values for
GSI can be estimated from rock mass characterizations using the Rock Mass Rating (RMR) system for rock masses with RMR greater than 25 (Hoek and Brown, 1997). Using this approach, GSI is calculated
${\displaystyle GSI=10+\textstyle \sum _{i=1}^{4}R_{i}}$ (dimensionless) Equation 751.38.3.6
R[i] = Rock Mass Rating system rating parameters (dimensionless). GSI is thus equivalent to the RMR value with the groundwater rating term, R[5], taken as 10.
Values for GSI to be used in Equations 751.38.3.4 and 751.38.3.5, or values for m and s to be used in Equation 751.38.3.3, can also be established using alternative methods described in Commentary
for EPG 751.38.3.1.
The nominal bearing resistance predicted using Equation 751.38.3.3 shall be limited to a maximum value of 200 ksf unless greater bearing resistance can be verified by a load test.
Table 751.38.3.1.1 Correction factors to account for footing shape for evaluation of bearing resistance for spread footings on rock (from Wyllie, 1999)
│ Footing Shape │C[f1]│
│Strip (L/B > 6) │1.00 │
│ │L/B = 2 │1.12 │
│Rectangular ├────────┼─────┤
│ │L/B = 5 │1.05 │
│Square (L/B = 1) │1.25 │
│Circular (L/B = 1) │1.20 │
751.38.3.2 Bearing Resistance for Spread Footings on Weak Rock (5 ksf ≤ q[u] ≤ 100 ksf)
The nominal bearing resistance for spread footings on weak rock (e.g. mudstone, siltstone, weak sandstone, etc.) shall be calculated as a function of the mean uniaxial compressive strength of the
rock according to (adapted from Wyllie, 1999):
${\displaystyle q_{n}={\frac {\overline {q_{u}}}{2}}\cdot N_{c}\cdot s_{c}\cdot d_{c}\cdot i_{c}\leq 200ksf}$ (consistent units of stress) Equation 751.38.3.7
${\displaystyle {\overline {q_{u}}}}$ = mean value of the uniaxial compressive strength of the rock (consistent units of stress),
N[c] = bearing capacity factor (dimensionless),
s[c] = correction factor to account for footing shape (dimensionless),
d[c] = correction factor to account for footing depth (dimensionless) and
i[c] = correction factor to account for inclination of the factored load (dimensionless).
Resistance factors (${\displaystyle {\boldsymbol {\phi }}_{b}}$) to be applied to the nominal resistance values (q[n]) determined according to the provisions of this subarticle shall be established
from Figure 751.38.3.2 based on the coefficient of variation of the mean uniaxial compressive strength ${\displaystyle COV_{\overline {q_{u}}}}$. Values for ${\displaystyle {\overline {q_{u}}}}$ and
${\displaystyle COV_{\overline {q_{u}}}}$ shall be determined in accordance with methods described in EPG 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of
Variation for the site and location in question. Values for design parameter q[u] shall be taken as the mean value of the parameter for the rock between the base of the footing and a depth of B below
the base of the footing. Values for ${\displaystyle COV_{\overline {q_{u}}}}$ shall similarly reflect the variability of the mean uniaxial compressive strength over the same depth range.
The value of N[c] shall be taken as 5.0. The respective correction factors for footing shape and depth and for load inclination shall be computed as
${\displaystyle s_{c}=1+{\frac {B}{5L}}}$ (dimensionless) Equation 751.38.3.8
${\displaystyle d_{c}=1+{\frac {D_{f}}{5B}}}$ (dimensionless) Equation 751.38.3.9
${\displaystyle i_{c}={\big (}1-{\frac {\theta }{90^{\circ }}}{\big )}^{2}}$ (dimensionless) Equation 751.38.3.10
B and L = footing width and length, respectively (consistent units of length),
θ = inclination of the factored resultant column load measured from the vertical (degrees) and
D[f] = depth of embedment of footing.
The nominal bearing resistance predicted using Equation 751.38.3.7 shall be limited to a maximum value of 200 ksf unless greater bearing resistance can be verified by a load test.
751.38.3.3 Bearing Resistance for Spread Footings on Cohesive Soils (s[u] ≤ 5,000 psf)
The nominal bearing resistance for spread footings on cohesive soils shall be calculated as a function of the mean undrained shear strength of the soil according to:
${\displaystyle q_{n}={\overline {s_{u}}}\cdot N_{c}\cdot s_{c}\cdot d_{c}\cdot i_{c}}$ (consistent units of stress) Equation 751.38.3.11
${\displaystyle {\overline {s_{u}}}}$ = mean value of the undrained shear strength of the soil (consistent units of stress),
N[c] = bearing capacity factor (dimensionless),
s[c] = correction factor to account for footing shape (dimensionless),
d[c] = correction factor to account for footing depth (dimensionless) and
i[c] = correction factor to account for inclination of the factored load (dimensionless).
Resistance factors (${\displaystyle {\boldsymbol {\phi }}_{b}}$) to be applied to the nominal resistance values (q[n]) determined according to the provisions of this subarticle shall be established
from Figure 751.38.3.3 based on the coefficient of variation of the mean undrained shear strength, ${\displaystyle COV_{\overline {s_{u}}}}$. Values for ${\displaystyle {\overline {s_{u}}}}$ and ${\
displaystyle COV_{\overline {s_{u}}}}$ shall be determined in accordance with methods described in EPG 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of Variation
for the site and location in question. Values for design parameter ${\displaystyle {\overline {s_{u}}}}$ shall be taken as the mean value for the soil between the base of the footing and a depth of B
below the base of the footing. Values for ${\displaystyle COV_{\overline {s_{u}}}}$ shall similarly reflect the variability of the mean soil shear strength over the same range of depths.
The value of N[c] shall be taken as 5.0. The respective correction factors shall be computed using Equations 751.38.3.8, 751.38.3.9 and 751.38.3.10.
751.38.3.4 Bearing Resistance for Spread Footings on Cohesionless Soils
Spread footings on cohesionless soils shall be designed according to applicable sections of the current AASHTO LRFD Bridge Design Specifications.
751.38.4 Design for Axial Loading at Serviceability Limit States
Spread footings shall be dimensioned so that there is a small likelihood that footings will settle more than tolerable settlements, generally established from consideration of span length. This shall
be accomplished by determining minimum footing dimensions for the appropriate site conditions in accordance with the content of this article.
Resistance factors provided in this article were established to produce factored settlements that have a target probability of being exceeded. Target probabilities of exceedance were established by
MoDOT for structures of different operational importance. Additional information regarding development of the resistance factors and application of the resistance factors for settlement calculations
are provided in the commentary that accompanies these guidelines.
The method for determining minimum footing dimensions based on serviceability considerations shall be selected based on the material type present beneath the base of the footing. In general, EPG
751.38.4.1 shall be followed for footings founded in rock with uniaxial compressive strengths (q[u]) greater than 100 ksf; EPG 751.38.4.2 shall be followed for footings founded in weaker rock with q
[u] greater than 5 ksf but less than 100 ksf. The provisions in EPG 751.38.4.3 and EPG 751.38.4.4 shall be followed for footings founded in soil.
751.38.4.1 Settlement of Spread Footings on Rock (q[u] ≥ 100 ksf)
For spread footings on rock, the minimum footing dimensions shall be established from the following:
${\displaystyle B\times L\geq {\frac {1-v^{2}}{{\sqrt {\phi _{s}\cdot {\overline {q_{u}}}}}\cdot 10^{\frac {GSI-10}{40}}}}\cdot H\cdot {\frac {\gamma Q}{S}}}$ (ft^2) Equation 751.38.4.1
where :
B = minimum footing width (feet),
L = minimum footing length (feet),
${\displaystyle {\boldsymbol {\gamma }}Q}$ = factored load for the appropriate serviceability limit state (kips)
v = mean value of Poisson’s ratio (dimensionless),
${\displaystyle {\overline {q_{u}}}}$ = mean value for the uniaxial compressive strength (ksf),
GSI = mean value for the geological strength index (dimensionless),
H = thickness of rock subjected to stress below the footing (feet),
S = minimum span length for spans adjacent to the footing (feet) and
Φ[S] = resistance factor for settlement of spread footings on rock (dimensionless).
Note that this expression is dimensional so values must be entered in the units specified.
Values for Φ[S] shall be established from Figure 751.38.4.1 based on the coefficient of variation of the mean uniaxial compressive strength, ${\displaystyle COV_{\overline {q_{u}}}}$, determined in
accordance with methods described in EPG 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of Variation for the site and location in question. Values for v can be
estimated from Table 751.38.4.1. Values for GSI can be estimated using methods outlined in EPG 751.38.3.1, or using alternative methods described in the commentary to that subarticle.
For cases where the footing is underlain by practically homogeneous rock masses, H can be assumed to be equal to the footing dimension, B and values for ${\displaystyle {\overline {q_{u}}}}$, ${\
displaystyle COV_{\overline {q_{u}}}}$, v and GSI shall be taken as the mean values of these parameters for the rock mass between the base of the footing and a depth of 2∙H below the base of the
footing. For cases where the rock beneath the footing is stratified, the value for H can be assumed to be the cumulative thickness of the more compressible strata within a depth of 2∙B beneath the
base of the footing. In such cases, values for ${\displaystyle {\overline {q_{u}}}}$, ${\displaystyle COV_{\overline {q_{u}}}}$, v and GSI shall be taken as the mean values of these parameters over
the thickness of the more compressible strata.
Table 751.38.4.1 Poisson’s Ratio values for intact rock (modified after Kulhawy, 1978
│ │ │ │ Poisson’s Ratio, v │ │
│Rock Type│# Values│# Rock Types├───────┬───────┬────┤Standard Deviation │
│ │ │ │Maximum│Minimum│Mean│ │
│Granite │ 22 │ 22 │ 0.39 │ 0.09 │0.20│ 0.08 │
│Gabbro │ 3 │ 3 │ 0.20 │ 0.16 │0.18│ 0.02 │
│Diabase │ 6 │ 6 │ 0.38 │ 0.20 │0.29│ 0.06 │
│Basalt │ 11 │ 11 │ 0.32 │ 0.16 │0.23│ 0.05 │
│Quartzite│ 6 │ 6 │ 0.22 │ 0.08 │0.14│ 0.05 │
│Marble │ 5 │ 5 │ 0.40 │ 0.17 │0.28│ 0.08 │
│Gneiss │ 11 │ 11 │ 0.40 │ 0.09 │0.22│ 0.09 │
│Schist │ 12 │ 11 │ 0.31 │ 0.02 │0.12│ 0.08 │
│Sandstone│ 12 │ 9 │ 0.46 │ 0.08 │0.20│ 0.11 │
│Siltstone│ 3 │ 3 │ 0.23 │ 0.09 │0.18│ 0.06 │
│Shale │ 3 │ 3 │ 0.18 │ 0.03 │0.09│ 0.06 │
│Limestone│ 19 │ 19 │ 0.33 │ 0.12 │0.23│ 0.06 │
│Dolostone│ 5 │ 5 │ 0.35 │ 0.14 │0.29│ 0.08 │
751.38.4.2 Settlement of Spread Footings on Weak Rock (5 ksf ≤ q[u] ≤ 100 ksf)
Spread footings founded on weak rock shall have the following minimum dimensions:
${\displaystyle B\times L\geq {\frac {1-v^{2}}{\sqrt {\phi _{s}\cdot {\overline {q_{u}}}}}}\cdot H\cdot {\frac {\gamma Q}{2\cdot S}}}$ (ft^2) Equation 751.38.4.2
B = minimum footing width (feet),
L = minimum footing length (feet),
${\displaystyle {\boldsymbol {\gamma }}Q}$ = factored load for the appropriate serviceability limit state (kips)
v = mean value of Poisson’s ratio (dimensionless),
${\displaystyle {\overline {q_{u}}}}$ = mean value for the uniaxial compressive strength (ksf),
H = thickness of rock subjected to stress below the footing (feet),
S = minimum span length for spans adjacent to the footing (feet) and
Φ[S] = resistance factor for settlement of spread footings on weak rock (dimensionless).
Note that this expression is dimensional so values must be entered in the units specified.
Values for Φ[S] shall be established from Figure 751.38.4.2 based on the coefficient of variation of the mean uniaxial compressive strength, ${\displaystyle COV_{\overline {q_{u}}}}$, determined in
accordance with methods described in EPG 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of Variation for the site and location in question. Values for v can be
estimated from Table 751.38.4.1.
For cases where the footing is underlain by practically homogeneous rock masses, H can be assumed to be equal to the footing dimension, B and values for ${\displaystyle {\overline {q_{u}}}}$, ${\
displaystyle COV_{\overline {q_{u}}}}$ and v shall be taken as the mean values of these parameters for the rock mass between the base of the footing and a depth of 2∙H below the base of the footing.
For cases where the rock beneath the footing is stratified, the value for H can be assumed to be the cumulative thickness of the more compressible strata within a depth of 2∙B beneath the base of the
footing. In such cases, values for ${\displaystyle {\overline {q_{u}}}}$, ${\displaystyle COV_{\overline {q_{u}}}}$ and v shall be taken as the mean values of these parameters over the thickness of
the more compressible strata.
751.38.4.3 Settlement of Spread Footings on Cohesive Soils
Evaluation of settlement for spread footings on cohesive soils requires an iterative approach because analytic expressions for the minimum dimensions cannot be derived as is the case for settlement
of footings on rock. As such, the procedure for evaluating settlement of footings in cohesive soils requires comparison of a factored settlement computed for the greatest minimum footing dimensions
established for the strength limit states according to EPG 751.38.3 with an established tolerable settlement. If the factored total settlement determined from these provisions is found to be less
than or equal to the tolerable settlement, i.e. if
${\displaystyle \delta _{R}\leq \delta _{tol}}$ (consistent units of length) Equation 751.38.4.3
δ[R] = factored total settlement (consistent units of length) and
δ[tol] = tolerable settlement (consistent units of length),
the limit state is satisfied and the probability of footing settlement exceeding the tolerable settlement is less than or equal to the target probability established by MoDOT. If the factored total
settlement is determined to exceed the tolerable settlement, the probability of footing settlement exceeding the tolerable value is greater than the target probability established by MoDOT. In such
cases, the footing dimensions shall be increased until the factored total settlement is less than or equal to the tolerable settlement.
Resistance factors provided in this article were established to produce factored settlements that have a target probability of being exceeded. Target probabilities of exceedance were established by
MoDOT for structures of different operational importance. Additional information regarding development of the resistance factors and application of the resistance factors for settlement calculations
are provided in the commentary that accompanies these guidelines.
Tolerable settlement
For this provision, the tolerable settlement shall be taken as
${\displaystyle \delta _{tol}={\frac {S}{476}}}$ (consistent units of length) Equation 751.38.4.4
S = length of shortest bridge span adjacent to footing (consistent units of length).
Factored total settlement
The factored settlement for footings on cohesive soils shall be computed following classical consolidation theory (e.g. Reese et al., 2006), modified to include resistance factors to be applied to
the compression and recompression indices, c[c] and c[r], and to the maximum past vertical effective stress, σ'[p] (also referred to as the pre-consolidation stress). Application of this method
within the LRFD framework requires comparison of a factored value for σ'[p], with the initial and final vertical effective stresses, σ'[0] and σ'[f].
If σ'[0] < Φ[p] σ'[p] < σ'[f] , the factored total settlement shall be computed as:
${\displaystyle \delta _{R}={\frac {H_{0}}{1+e_{0}}}{\Bigg [}{\frac {c_{r}}{\phi _{r}}}log{\Big (}{\frac {\phi _{p}\sigma _{p}^{'}}{\sigma _{0}^{'}}}{\Big )}+{\frac (consistent units Equation
{c_{c}}{\phi _{c}}}log{\Big (}{\frac {\sigma _{f}^{'}}{\phi _{p}\sigma _{p}^{'}}}{\Big )}{\Bigg ]}}$ of length) 751.38.4.5
σ'[0] = initial vertical effective stress (consistent units of stress),
Φ[p] = resistance factor to be applied to pre-consolidation stress (dimensionless),
σ'[p] = maximum past vertical effective stress or pre-consolidated stress (consistent units of stress),
σ'[f] = final vertical effective stress (consistent units of stress),
δ[R] = factored settlement (consistent units of length),
H[0] = thickness of compressible layer (consistent units of length),
e[0] = initial void ratio (dimensionless),
c[c] = compression index (dimensionless),
Φ[c] = resistance factor to be applied to compression index term (dimensionless),
c[r] = recompression index (dimensionless), and
Φ[r] = resistance factor to be applied to recompression index term (dimensionless).
If Φ[p] σ'[p] ≥ σ'[f], the factored settlement shall be computed as:
${\displaystyle \delta _{R}={\frac {H_{0}}{1+e_{0}}}{\Bigg [}{\frac {c_{r}}{\phi _{r}}}log{\Big (}{\frac {\sigma _{f}^{'}}{\sigma _{0}^{'}}}{\Big )}{\ (consistent units of Equation
Bigg ]}}$ length) 751.38.4.6
Similarly, if Φ[p] σ'[p] ≤ σ'[0], the factored settlement shall be computed as:
${\displaystyle \delta _{R}={\frac {H_{0}}{1+e_{0}}}{\Bigg [}{\frac {c_{c}}{\phi _{c}}}log{\Big (}{\frac {\sigma _{f}^{'}}{\sigma _{0}^{'}}}{\Big )}{\ (consistent units of Equation
Bigg ]}}$ length) 751.38.4.7
Values for Φ[c] and Φ[r] shall be established from Figure 751.38.4.3.1 based on the coefficient of variation of the mean compression index (${\displaystyle COV_{\overline {c_{c}}}}$) and mean
recompression index (${\displaystyle COV_{\overline {c_{r}}}}$), respectively. Similarly, values for Φ[p] shall be established from Figure 751.38.4.3.2 based on the coefficient of variation of the
mean maximum past vertical effective stress (${\displaystyle COV_{\overline {\sigma _{p}^{'}}}}$). Coefficients of variation for each of these parameters shall be determined in accordance with
methods described in EPG 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of Variation.
Where footings are underlain by compressible soils of substantial thickness, the soil beneath the footing shall be subdivided into several sublayers to account for potential changes in consolidation
parameters and stress distribution beneath the footing. Compression of each of these sublayers shall be computed using Equation 751.38.4.5, 751.38.4.6 or 751.38.4.7, as appropriate, and the resulting
values should be summed to arrive at the total settlement. For each sublayer, values for c[c], c[r] and e[0] shall be taken as the mean values of these parameters over the thickness of the sublayer.
Values for H[0] shall be taken as the thickness of the respective sublayer. Values for σ'[0], σ'[f] and σ'[p] for each sublayer shall also be taken as the mean values over each sublayer, although
this is often approximated by using values calculated for the center of the sublayer. Values used for ${\displaystyle COV_{\overline {c_{c}}}}$ , ${\displaystyle COV_{\overline {c_{r}}}}$ and ${\
displaystyle COV_{\overline {\sigma _{p}^{'}}}}$ shall be representative of the variability and uncertainty of the mean values for the respective parameters within each sublayer.
Where conditions warrant, settlement contributions due to immediate elastic settlement and secondary compression shall be added to those computed from Equations 751.38.4.5, 751.38.4.6 or 751.38.4.7.
751.38.4.4 Settlement of Spread Footings on Cohesionless Soils
Spread footings in cohesionless soils shall be designed according to current AASHTO LRFD Bridge Design Specifications.
751.38.5 Modifications for Load Eccentricity
The minimum footing dimensions established in accordance with EPG 751.38.3 and EPG 751.38.4 must be increased to account for load eccentricity when resultant factored column loads are not located at
the center of the footing. Furthermore, the eccentricity of factored loads on spread footings shall be restricted to prevent overturning of foundations or excessively high localized stresses at the
edges of the footing as provided in this subarticle.
Load eccentricity shall be calculated in the width and length dimension directions as:
${\displaystyle e_{B}={\frac {M_{B}^{*}}{\gamma Q}}}$ (consistent units of length) Equation 751.38.5.1
${\displaystyle e_{L}={\frac {M_{L}^{*}}{\gamma Q}}}$ (consistent units of length) Equation 751.38.5.2
where ${\displaystyle M_{B}^{*}}$ and ${\displaystyle M_{L}^{*}}$ are moments attributed to factored load effects in the B and L directions (consistent units of force times length), respectively, and
${\displaystyle {\boldsymbol {\gamma }}Q}$ is the resultant factored load (consistent units of force) for the strength limit state (Figure 751.38.5.1). Here the moment, ${\displaystyle M_{B}^{*}}$,
is a moment about the y-axis and moment, ${\displaystyle M_{L}^{*}}$, is a moment about the x-axis.
751.38.5.1 Modifications to Footing Dimensions for Eccentric Loads
In cases where spread footings will be subjected to eccentric loads, the minimum footing dimensions established in accordance with EPG 751.38.3 and EPG 751.38.4 shall be determined using reduced
dimensions, B' and L' , instead of the actual dimensions, B and L, where
B' = B - 2e[B] (consistent units of length) Equation 751.38.5.3
L' = L - 2e[L] (consistent units of length) Equation 751.38.5.4
where e[B] and e[L] are the load eccentricity due to the factored load in the width and length dimensions, respectively.
751.38.5.2 Limiting Eccentricity in Soil and Cohesive Intermediate Geomaterials
For footings founded in soil or cohesive intermediate geomaterials, the load eccentricity shall be restricted to the middle one-half of the footing. Minimum footing dimensions satisfying this
criterion are:
B ≥ 4e[B] and L ≥ 4e[L] (consistent units of length) Equation 751.38.5.5
751.38.5.3 Limiting Eccentricity in Cohesionless Intermediate Geomaterials and Rock
For footings founded in cohesionless intermediate geomaterials and rock, the load eccentricity shall be restricted to the middle three-quarters of the footing. Minimum footing dimensions satisfying
this criterion are:
${\displaystyle B\geq {\frac {8\cdot e_{B}}{3}}}$ and ${\displaystyle L\geq {\frac {8\cdot e_{L}}{3}}}$ (consistent units of length) Equation 751.38.5.6
751.38.6 Design for Lateral Loading
Spread footings subjected to substantial lateral loads shall be designed according to the lateral load provisions of current AASHTO LRFD Bridge Design Specifications, including consideration of
sliding stability.
751.38.7 Design for Overall Stability
Overall stability shall be evaluated when spread footings are located near to an embankment, excavated, or natural slope. Overall stability shall be evaluated at the Service I limit state. Overall
stability shall be evaluated using methods described in EPG 321.1 Design of Earth Slopes for evaluation of slope stability with the factored footing loads applied as a surcharge load.
751.38.8 Structural Design of Spread Footings
The provisions provided in this subarticle are unchanged from prior versions of the EPG aside from minor editorial revisions.
Structural design and detailing of spread footings should be accomplished considering the shear and moment capacity of the footing when subjected to factored column loads.
751.38.8.1 Design for Shear
The footing shall be designed so that the shear strength of the concrete is adequate to handle the shear stress without the additional help of reinforcement. If the shear stress is too great, the
footing depth should be increased.
The shear capacity of the footings in the vicinity of concentrated loads shall be governed by the more severe of the following two conditions.
751.38.8.1.1 One-Way Shear
Critical sections shall be taken from the face of the column for square or rectangular columns or at the equivalent square face of a round column. The equivalent square column is the column which has
a cross-sectional area equal to the round section of the actual column and placed concentrically as shown in Figure 751.38.8.1.
One-Way Shear Capacity shall be evaluated as:
${\displaystyle V_{r}=\phi V_{n}\geq V_{u}}$ (consistent units) Equation 751.38.8.1
Φ = 0.9
V[n] = V[c] = ${\displaystyle 0.0316\beta Bd_{v}{\sqrt {f'_{c}}}}$
B = footing width
β = factor indicating ability of diagonally cracked concrete to transmit tension = 2.0
d[v] = effective shear depth of concrete
V[u] = ${\displaystyle v_{u}\cdot {\Big (}{\frac {L}{2}}-d_{v}-{\frac {equiv.\ square\ column\ width}{2}}{\Big )}B}$
v[u] = the triangular or trapezoidal stress distribution applied to the designated loaded area of the footing from the strength limit state load combination
751.38.8.1.2 Two-Way Shear
The critical section for checking Two-Way Shear is taken from the boundary of a square area with sides equal to the equivalent square column width plus the effective shear depth as shown.
Two-Way Shear Capacity shall be evaluated as:
${\displaystyle \,V_{r}}$ = ${\displaystyle \phi V_{n}\geq V_{u}}$ (consistent units) Equation 751.38.8.1
ΦV[n] = ${\displaystyle \phi {\Big (}0.063+{\frac {0.126}{\beta _{c}}}{\Big )}b_{o}d_{v}{\sqrt {f'_{c}}}\leq 0.126b_{o}d_{v}{\sqrt {f'_{c}}}}$
β[c] = ratio of long side to short side of the rectangle through which the concentrated load or reaction force is transmitted,
b[o] = perimeter of critical section = 4(d[v] + equivalent square column width),
d[v] = effective shear depth of concrete (inches)
V[u] = maximum axial load on top of footing from column reactions for strength limit state load combinations
Table 751.38.8.1 shows approximate capacities for both One-Way and Two-Way Shear for the given footing depth and column diameter to assist in selecting a footing length and width.
Table 751.38.8.1 Shear Capacities for Given Column Diameters and Footing Depths
│ Column │ Footing │ One-Way Shear │ Two-Way Shear │
│Diameter (ft)│Depth (ft)│Capacity, V[r] kip/ft │Capacity, V[r] kips│
│2.5 │2.50 │30.7 │1074 │
│2.5 │2.75 │34.3 │1266 │
│2.5 │3.00 │37.8 │1473 │
│2.5 │3.25 │41.4 │1694 │
│2.5 │3.50 │44.9 │1928 │
│2.75 │2.75 │34.3 │1327 │
│2.75 │3.00 │37.8 │1540 │
│2.75 │3.25 │41.4 │1767 │
│2.75 │3.50 │44.9 │2008 │
│2.75 │3.75 │48.5 │2263 │
│3.00 │3.00 │37.8 │1607 │
│3.00 │3.25 │41.4 │1840 │
│3.00 │3.50 │44.9 │2087 │
│3.00 │3.75 │48.5 │2348 │
│3.00 │4.00 │52.0 │2624 │
│3.25 │3.25 │41.4 │1913 │
│3.25 │3.50 │44.9 │2166 │
│3.25 │3.75 │48.5 │2434 │
│3.25 │4.00 │52.0 │2716 │
│3.25 │4.25 │55.6 │3012 │
│3.50 │3.50 │44.9 │2246 │
│3.50 │3.75 │48.5 │2520 │
│3.50 │4.00 │52.0 │2808 │
│3.50 │4.25 │55.6 │3110 │
│3.50 │4.50 │59.1 │3426 │
│3.75 │3.75 │48.5 │2605 │
│3.75 │4.00 │52.0 │2900 │
│3.75 │4.25 │55.6 │3208 │
│3.75 │4.50 │59.1 │3531 │
│3.75 │4.75 │62.7 │3868 │
│4.00 │4.00 │52.0 │2992 │
│4.00 │4.25 │55.6 │3306 │
│4.00 │4.50 │59.1 │3635 │
│4.00 │4.75 │62.7 │3978 │
│4.00 │5.00 │66.2 │4335 │
│4.25 │4.25 │55.6 │3404 │
│4.25 │4.50 │59.1 │3740 │
│4.25 │4.75 │62.7 │4089 │
│4.25 │5.00 │66.2 │4452 │
│4.25 │5.25 │69.8 │4830 │
│4.50 │4.50 │59.1 │3844 │
│4.50 │4.75 │62.7 │4200 │
│4.50 │5.00 │66.2 │4569 │
│4.50 │5.25 │69.8 │4953 │
│4.50 │5.50 │73.3 │5351 │
│4.75 │4.75 │62.7 │4310 │
│4.75 │5.00 │66.2 │4686 │
│4.75 │5.25 │69.8 │5076 │
│4.75 │5.50 │73.3 │5481 │
│4.75 │5.75 │76.8 │5899 │
│5.00 │5.00 │66.2 │4803 │
│5.00 │5.25 │69.8 │5200 │
│5.00 │5.50 │73.3 │5610 │
│5.00 │5.75 │76.8 │6035 │
│5.00 │6.00 │80.4 │6474 │
│5.25 │5.25 │69.8 │5323 │
│5.25 │5.50 │73.3 │5740 │
│5.25 │5.75 │76.8 │6171 │
│5.25 │6.00 │80.4 │6616 │
│5.50 │5.50 │73.3 │5869 │
│5.50 │5.75 │76.8 │6306 │
│5.50 │6.00 │80.4 │6758 │
│5.75 │5.75 │76.8 │6442 │
│5.75 │6.00 │80.4 │6900 │
│6.00 │6.00 │80.4 │7042 │
Φ = 0.9
${\displaystyle {\sqrt {f_{c}^{'}}}}$ = 3 ksi
β = 2.0
d[v] = footing depth: 4 inches
One-Way Shear Capacity = ${\displaystyle V_{r}=\phi 0.0316\beta d_{v}{\sqrt {f_{c}^{'}}}}$
Where One-Way Shear capacity is per foot width of footing, i.e. where total shear capacity is Total V[r] = V[r] from table × B
Two-Way Shear Capacity = ${\displaystyle V_{r}=\phi 0.126b_{o}d_{v}{\sqrt {f_{c}^{'}}}}$
751.38.8.2 Moment
The critical section for bending shall be taken at the face of the equivalent square column. The applied moment shall be determined from a triangular or trapezoidal stress distribution on the bottom
of the footing.
The bearing pressure used to design bending reinforcement shall be calculated from Strength I, III, IV and V Load Combinations.
Reinforcement must meet the maximum and minimum requirements as given in LRFD 5.7.3.3.1 and LRFD 5.7.3.3.2.
The minimum reinforcement allowed is #5 bars spaced at 12”.
751.38.8.2.1 Distribution of Reinforcement
Reinforcement in the long direction shall be distributed uniformly across the entire width of footing.
For reinforcement in the short direction, a portion of the total reinforcement shall be distributed uniformly over a band width equal to the length of the short side of footing and centered on the
centerline of column or pier as shown in Figure 751.38.8.3.
The band width reinforcement required shall be calculated by the following equation:
${\displaystyle A_{s-BW}=A_{s-SD}{\frac {2}{\beta +1}}}$ Equation 751.38.8.3
A[(s-BW)] = area of steel in the band width (in^2),
A[(s-SD)] = total area of steel in short direction (in^2),
β = ratio of the long side to the short side of footing
The remainder of the reinforcement required in the short direction shall be distributed uniformly outside the center band width of footing.
751.38.8.2.2 Crack Control Reinforcement
The reinforcement shall meet the spacing criteria, s, as specified.
${\displaystyle s\leq {\frac {700\gamma _{e}}{\beta _{s}f_{s}}}-2d_{c}}$
β[s] = ${\displaystyle 1+{\frac {d_{c}}{0.7(h-d_{c})}}}$ ,
d[c] = concrete cover measured from extreme tension fiber to center of flexural reinforcement (in.),
f[s] = tensile stress in reinforcement at the service limit state (ksi),
h = depth of footing (in.)
${\displaystyle {\boldsymbol {\gamma }}_{e}}$ = 1.0 for Class 1 exposure condition
751.38.8.3 Details
751.38.8.3.1 Spread Footing Reinforcement
FRONT ELEVATION SIDE ELEVATION
Fig. 751.38.8.4 Schematic showing typical reinforcement detail in front elevation and side elevation.
* Footing depths > 36 in. may require the side faces to have shrinkage and temperature reinforcement. See Structural Project Manager.
751.38.9 References
AASHTO (2009), AASHTO LRFD Bridge Design Specification: Customary U.S. Units, American Association of State Highway and Transportation Officials, Fourth Edition with 2008 and 2009 Interim Revisions.
Hoek, E., and E.T. Brown (1988), “The Hoek-Brown Failure Criterion – A 1988 Update,” Proceedings of the 15th Canadian Rock Mechanics Symposium, Toronto, Canada.
Kulhawy, F.H. (1978), “Geomechanical Model for Rock Foundation Settlement,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 104, No. GT2, pp. 211-227.
Marinos, P., and E. Hoek (2000), “GSI: A Geologically Friendly Tool for Rock Mass Strength Estimation,” Proceedings of GeoEng2000, Melbourne, Australia, Vol. I, pp. 1422-1440.
Reese, L.C., W.M. Isenhower, and S-T Wang (2006), Analysis and Design of Shallow and Deep Foundations, John Wiley and Sons, 574 pp.
Wyllie, D.C. (1999), Foundations on Rock, E & FN Spon, Second Edition, 401 pp.
751.38.10 Commentary
These guidelines were developed from prior EPG guidelines with notable changes to the general approach for application of LRFD techniques as well as updated resistance factors based on probabilistic
calibrations. Calibration analyses were performed following generally accepted procedures for calibration of resistance factors for geotechnical applications, but with modifications to permit several
enhancements to be included in the guidelines. The most notable enhancements provided in the guidelines include:
□ Use of resistance factors that are contingent upon the variability and uncertainty that exists in select design properties.
□ Adoption of different target reliability levels for foundations of structures of different operational importance.
Both of these enhancements are expected to produce efficient foundation designs while still maintaining appropriate safety and reliability for all classes of operational importance. Additional
information regarding development of the methods provided in these guidelines can be found in Abu El-Ela et al. (2011) and Song et al. (2011). Additional information regarding target reliability
values established for different classes of operational importance is provided in Bowders et al. (2011).
The four classes of operational importance include:
□ Minor and low volume route
□ Major route
□ Major bridge costing less than $100 million
□ Major bridge costing greater than $100 million.
These classifications are based on common MoDOT designations. The target reliability levels established for each limit state and operational importance were generally based upon consideration of
highway bridges. However, the methods in this article can also be utilized for design of foundations for other structures including retaining walls and roadway signs.
Calibration analyses performed to establish the resistance factors presented in this subarticle were performed using the latest knowledge of variability and uncertainty of applied loads (Kulicki et
al., 2007), as well as using load factors that are currently in effect. The resistance factors provided in these guidelines are intended to produce foundations with reliabilities that are
approximately equal to the target reliabilities established by MoDOT when utilized with current load factors. Since it is the combined effect of load and resistance factors that produce this
reliability, the resistance factors provided are inherently coupled with current load factors and are contingent upon the uncertainty and variability in the applied loads that was presumed for the
calibrations. As such, recalibration of resistance factors is required if alternative load factors are adopted, or if substantial revisions to current estimates of load variability and uncertainty
are found.
It is important to emphasize that the resistance factors provided in these guidelines were developed presuming that mean values would be used for all design parameters in the methods provided. This
departs from past practice utilizing allowable stress design (ASD) approaches where nominal values of parameters that were less than mean values were often used to introduce conservatism into the
analyses beyond that provided by the ASD factor of safety. Use of design parameters less than the mean values within the context of these guidelines will often, but not always, increase the
reliability of foundation designs; however, such practice is contrary to the spirit of LRFD in that it will not produce foundations that achieve the target reliability established by MoDOT policy.
The procedures provided in these guidelines are not intended as a substitute for good judgment. Rather, the intent of these guidelines is to:
1) inform designers of generally appropriate levels of conservatism to address variability and uncertainty involved in different aspects of design analyses and
2) provide quantitative methods to achieve target reliabilities for foundations depending on the variability and uncertainty present in relevant design parameters and design methods.
Designers must still use their best judgment in considering design options (e.g. foundation depth, type, and size; necessity for load tests; etc.) for establishing the most appropriate foundations
for bridges and other structures.
When considering placement of spread footings within the prohibited region of Figure 751.38.1.2, evaluations of overall stability shall be performed in accordance with EPG 751.38.7.
The prohibited region for rock slopes varies with the quality of the rock present at a site and other factors. As a general rule of thumb, the limit line is inclined at 1:1 (V:H). However, the line
may be flatter for particularly poor rock and steeper for particularly good rock.
Selection of applicable strength and serviceability limit states shall be accomplished in close consultation with the Structural Project Manager. At a minimum, the Strength I and Service I limit
states should be evaluated. When multiple strength and/or service limit states are considered, the limit state producing the greatest minimum footing dimensions shall govern the final design
Throughout EPG 751.38, factored loads are denoted as ${\displaystyle {\boldsymbol {\gamma }}Q}$. This notation should not be taken to suggest inclusion or exclusion of specific load effects, but
rather is simply intended as a convenient notation to reflect factored loads. When applying these guidelines, designers should replace ${\displaystyle {\boldsymbol {\gamma }}Q}$ with load
combinations and load factors that are appropriate for the structure and limit state being considered.
Design procedures within this subarticle are categorized according to material type, including methods for design of spread footings founded upon “rock”, “weak rock”, “cohesive soil” and
“cohesionless soil”. While these categories serve to logically separate the guidelines according to design method, complexities present at some sites may lead to cases where multiple methods could
potentially be used. In such cases, designers should utilize the method that is most appropriate for the conditions encountered, rather than selecting the method that produces the smallest or largest
footing dimensions.
EPG 751.38.3.1 is generally intended for use with “harder” rock materials where the frequency, orientation, and condition of rock discontinuities tend to dominate the response of the rock to loading
from foundations. Such rock masses will generally be composed of rock with uniaxial compressive strengths that are greater than 100 ksf, although some exceptions to this limit could arise. Limestones
and dolomites will commonly fall under this subarticle as will many sandstones, and even a few hard shales.
EPG 751.38.3.2 is intended for use with weaker rock where the properties of the intact rock tend to dominate performance. This subarticle is primarily intended for use with shales, some weak
sandstones, and potentially some very stiff clays. Use of methods provided in EPG 751.38.3.2 for materials with uniaxial compressive strengths greater than 100 ksf should be done with extreme caution
as the methods may dramatically overestimate the bearing resistance that can be realistically achieved for rock with greater uniaxial compressive strengths.
EPG 751.38.3.3 and EPG 751.38.3.4 are intended to use with cohesive and cohesionless soils, respectively. The methods provided in EPG 751.38.3.3 are in fact similar to those provided for weak rock in
EPG 751.38.3.2, except that the uniaxial compressive strength used in EPG 751.38.3.2 is replaced by the undrained shear strength in EPG 751.38.3.3 according to conventions of practice. Some overlap
exists between the strength limits provided in EPG 751.38.3.2 and EPG 751.38.3.3. (Note that the limits for EPG 751.38.3.2 are based on the uniaxial compressive strength whereas the limits for EPG
751.38.3.3 are based on the undrained shear strength, which is nominally one half of the compressive strength.) When designing for materials that fall within this overlapping range of strengths,
designers shall use the method that is most appropriate for the material encountered.
The design method provided in this subarticle is adapted from the method presented in Wyllie (1999) to conform to the LRFD approach. The method is derived from the Hoek-Brown strength criterion (Hoek
and Brown, 1988) that is commonly used to represent the strength of fractured rock masses using the rock mass parameters, m and s. The resistance factors provided in Figure 751.38.3.1 were
established from probabilistic calibrations to achieve the target foundation reliabilities as described in Abu El-Ela et al. (2011). These calibrations were conducted with explicit consideration of
variability and uncertainty present for dead load, live load, uniaxial compressive strength, and the design method itself (i.e. a “method” uncertainty). The variability and uncertainty utilized for
dead load and live load were taken from Kulicki et al. (2007). The variability and uncertainty in the design method were conservatively estimated utilizing the likely range of m and s values expected
for a particular condition.
Unfortunately, empirical data to evaluate design methods for predicting the bearing resistance of footings on fractured rock are not presently available. As such, the variability and uncertainty
attributed to the design method was conservatively estimated as a matter of prudence. One consequence of this conservatism is that the factored resistance predicted for foundations designed according
to EPG 751.38.3.1 may, in some cases, be less than the factored resistance predicted according to EPG 751.38.3.2 for rock that might be considered to have lower quality. This consequence is a
reflection of the lack of data available to confirm the predicted resistance using the prescribed method, and thus the limited reliability of the method, rather than an indication that the bearing
resistance will actually be less than that for lesser rock. Future research to measure the ultimate bearing resistance of foundations in fractured rock could dramatically improve the accuracy and
reliability of these methods, which in turn would dramatically improve the efficiency of foundations in fractured rock. This consequence also suggests that site specific load tests could potentially
improve foundation efficiency in some cases while still maintaining the target reliability.
The coefficient of variation for the mean uniaxial compressive strength used in Equation 751.38.3.3 shall reflect the variability and uncertainty in the mean compressive strength rather than the
variability and uncertainty in measurements of compressive strength as described in EPG 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of Variation and the
associated commentary. Values for q[u] and ${\displaystyle COV_{\overline {q_{u}}}}$, m, and s do not have to be established exclusively based on tests or observations located within the depth range
of interest below the footing. However, the values used should reflect the mean and variability in the material parameters within that depth range.
Several methods are available for establishing appropriate values of GSI for specific rock masses. Equation 751.38.3.6 represents a generally rigorous approach for determination of GSI that should be
used when available measurements and observations allow for establishing Rock Mass Rating system ratings and when these ratings produce RMR greater than 25. In cases where such measurements and
observations are not available, or where RMR is less than 25, GSI values can be estimated using the qualitative chart shown in Fig. Commentary 751.38.3.1.1 based on the work of Marinos and Hoek
(2000). Figs. Commentary 751.38.3.1.2, Commentary 751.38.3.1.3, and Commentary 751.38.3.1.4 provide additional guidance for qualitative selection of GSI for typical sandstones, shales, and limestones
from the chart.
In cases where GSI cannot be rationally determined, it is also possible to directly estimate approximate values for the rock mass parameters m and s from Table Commentary 751.38.3.1 using qualitative
descriptions of the rock mass. The values provided in Table Commentary 751.38.3.1 will generally be less than values that will be produced using Equations 751.38.3.4 and 751.38.3.5. This result is
because the values in Table Commentary 751.38.3.1 were established under the assumption that excavation-induced damage will occur (i.e. that the Hoek and Brown damage factor, D, is equal to 1) while
Equations 751.38.3.4 and 751.38.3.5 were established assuming that no significant excavation-induced damage will occur (i.e. that D = 0). Since significant excavation-induced damage is unlikely to
occur for footings excavated using conventional construction techniques, the values provided in Table Commentary 751.38.3.1 will be conservative. It is also important to point out that m and s can be
roughly interpolated from the values provided in Table Commentary 751.38.3.1 for conditions falling between those listed.
Methods provided in this subarticle are not appropriate for use with uniaxial compressive strengths estimated from Point Load Index tests or from other empirical correlations. Use of correlations for
estimation of uniaxial compressive strength introduces additional variability into the relation among rock mass parameters, uniaxial compressive strength, and bearing resistance that is not accounted
for in the resistance factors provided. Use of compressive strengths derived from Point Load Index values or other correlations is therefore not appropriate for application of the provisions of this
subarticle. It is possible to develop resistance factors that would be appropriate for such use, but such calibrations have not been completed at this time.
Some iteration may be required for the C[f1] term in Equation 751.38.3.3. Application of Equation 751.38.3.3 requires an assumption regarding the shape of the spread footing to establish the required
footing dimensions. If that assumption must be changed, either as a result of design calculations or other considerations, Equation 751.38.3.3 shall be re-evaluated to ensure that the provision
remains satisfied.
The design method provided in this article is adapted from methods presented in Wyllie (1999) to conform to the LRFD approach. The method is derived from the classical bearing capacity equation. The
resistance factors provided in Figure 751.38.3.2 were established from probabilistic calibrations to achieve the target foundation reliabilities as described in Abu El-Ela et al. (2011). These
calibrations were conducted with explicit consideration of variability and uncertainty present for dead load, live load, and uniaxial compressive strength in addition to the variability and
uncertainty present in the method itself. The variability and uncertainty utilized for dead load and live load were taken from Kulicki et al. (2007). Variability and uncertainty for the method was
conservatively estimated based on consideration of the range of potential values for the actual bearing capacity factor including the effects of the correction factors provided in Equations
751.38.3.8, 751.38.3.9 and 751.38.3.10.
The coefficient of variation for the mean uniaxial compressive strength used in Equation 751.38.3.7 shall reflect the variability and uncertainty in the mean compressive strength rather than the
variability and uncertainty in measurements of compressive strength as described in EPG 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of Variation. Values for ${\
displaystyle {\overline {q_{u}}}}$ and ${\displaystyle COV_{\overline {q_{u}}}}$ do not have to be established exclusively based on tests or observations located within the depth range of interest
below the footing. However, the values used should reflect the mean and variability in the material parameters within that depth range.
Methods provided in this subarticle are not appropriate for use with uniaxial compressive strengths estimated from Point Load Index tests or from other empirical correlations. Use of correlations for
estimation of uniaxial compressive strength introduces additional variability into the relation among rock mass parameters, uniaxial compressive strength, and bearing resistance that is not accounted
for in the resistance factors provided. Use of compressive strengths derived from Point Load Index values or other correlations is therefore not appropriate for application of the provisions of this
subarticle. It is possible to develop resistance factors that would be appropriate for such use, but such calibrations have not been completed at this time.
Resistance factors provided in Figure 751.38.3.3 for bearing resistance of spread footings on cohesive soils are identical to those provided in Figure 751.38.3.2. The only differences in the methods
presented in EPG 751.38.3.2 and EPG 751.38.3.3 is that EPG 751.38.3.2 is presented in terms of the uniaxial compressive strength while EPG 751.38.3.3 is presented in terms of the undrained shear
The coefficient of variation for the mean undrained shear strength used in Equation 751.38.3.11 shall reflect the variability and uncertainty in the mean shear strength rather than the variability
and uncertainty in measurements of shear strength as described in EPG 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of Variation. Values for ${\displaystyle {\
overline {s_{u}}}}$ and ${\displaystyle COV_{\overline {s_{u}}}}$ do not have to be established exclusively based on tests or observations located within the depth range of interest below the
footing. However, the values used should reflect the mean and variability in the material parameters within that depth range.
The resistance factors provided in this subarticle are based on the assumption that measurements of undrained shear strength will accurately reflect the actual undrained shear strength in the field.
Use of undrained shear strength values established from approximations or from index tests such as hand-held penetrometer tests, Torvane tests, or Standard Penetration Tests will introduce additional
variability and uncertainty into the design that is currently not reflected in the resistance factors provided. As such, it is not generally appropriate to use such approximations for estimating
undrained shear strength for use in these provisions. At a minimum, undrained shear strengths should be established based on unconfined compression tests performed on specimens acquired using good
quality boring techniques and good quality “undisturbed” sampling with thin walled samplers. It is preferable to perform unconsolidated-undrained type triaxial tests or consolidated-undrained type
triaxial tests to establish undrained shear strength values for use in these provisions.
Probabilistic calibrations for spread footings on cohesionless soils have not yet been completed by MoDOT. The provisions of current AASHTO LRFD Bridge Design Specifications should therefore be
followed when designing spread footings on cohesionless soils.
The provisions of this subarticle were developed to limit foundation settlements to be less than generally tolerable levels of settlement with some target reliability. Target reliability levels for
service limit states are substantially less than target reliability levels for strength limit states because the consequences associated with serviceability limit states are substantially less than
consequences for strength limit state conditions. The ramification of these facts is that some foundations designed according to these provisions may experience settlements that exceed tolerable
settlements in some instances. The frequency of foundations settling more than tolerable limits should approach the established target probabilities of exceedance when considered over a large number
of projects. In cases where actual foundation settlements are observed to exceed tolerable limits, appropriate remedial measures shall be applied to the foundation(s) and/or the structure that it is
supporting so that appropriate reliability is maintained.
Tolerable settlements used throughout these provisions were established from theoretical considerations and empirical observations of bridge performance based on the work of Moulton (1984) and Duncan
and Tan (1991). Three different serviceability conditions corresponding to different levels of required maintenance and repair were initially considered:
1) minor damage generally corresponding to the theoretical onset of deck cracking (Duncan and Tan, 1991),
2) more significant damage corresponding to the onset of structural distress based on empirical observations by Moulton (1986) and
3) major damage corresponding to theoretical overstress of the bridge superstructure (Moulton, 1986).
Target reliabilities for each of these conditions were established based on economic analyses described in Bowders et al. (2011). Comparative analyses for typical design conditions were then
performed to evaluate the alternative serviceability conditions. Results of these analyses generally indicate that the first serviceability condition, corresponding to minor damage, tends to control
footing dimensions. These guidelines therefore only require evaluation of this condition (the others being presumed to be inherently satisfied based on the analyses performed).
Based on this work, tolerable settlements are established according to an angular distortion, defined as
${\displaystyle A={\frac {\Delta }{S}}\leq 0.0021}$ (consistent units of stress) Equation Commentary 751.38.4.1
A = angular distortion (dimensionless),
∆ = differential settlement between adjacent footings (consistent units of length),
S = span between adjacent footings (consistent units of length).
This limiting value of angular distortion is based on theoretical consideration of the onset of deck cracking (Duncan and Tan, 1991). This limit is implicitly included in the methods provided in EPG
751.38.4.1 and EPG 751.38.4.2, while it is explicitly included in EPG 751.38.4.3.
The target probabilities of exceedance reflected in the resistance factors provided in EPG 751.38 correspond to the target values established by MoDOT based on economic considerations. While use of
alternative limits for tolerable settlement is possible, such use is not strictly appropriate since the target probabilities adopted by MoDOT for different classes of operational importance were
established based on consequences associated with the limit provided in Equation Commentary 751.38.4.1. Other limits would generally require different target probabilities, and thus different
resistance factors to achieve the same economic balance.
As was the case in EPG 751.38.3, design procedures within this subarticle are categorized according to material type, including methods for design of spread footings founded upon “rock”, “weak rock”,
“cohesive soil”, and “cohesionless soil”. While these categories serve to logically separate the guidelines according to design method, complexities present at some sites may lead to cases where
multiple methods could potentially be used. In such cases, designers should utilize the method that is most appropriate for the conditions encountered, rather than selecting the method that produces
the smallest or largest footing dimensions.
EPG 751.38.4.1 is generally intended for use with “harder” rock materials where the frequency, orientation, and condition of rock discontinuities tend to dominate the response of the rock to loading
from foundations. Such rock masses will generally be composed of rock with uniaxial compressive strengths that are greater than 100 ksf, although some exceptions to this limit could arise. Limestones
and dolomites will commonly fall under this subarticle as will many sandstones, and even a few hard shales.
EPG 751.38.4.2 is intended for use with weaker rock where the properties of the intact rock tend to dominate performance. This subarticle is primarily intended for use with shales, some weak
sandstones, and potentially some very stiff clays.
EPG 751.38.4.3 and EPG 751.38.4.4 are intended for use with cohesive and cohesionless soils, respectively.
Throughout EPG 751.38, factored loads are denoted as ${\displaystyle {\boldsymbol {\gamma }}Q}$. This notation should not be taken to suggest inclusion or exclusion of specific load effects, but
rather is simply intended as a convenient notation to reflect factored loads. When applying these guidelines, designers should replace ${\displaystyle {\boldsymbol {\gamma }}Q}$ with load
combinations and load factors that are appropriate for the structure and limit state being considered.
The provisions of this subarticle are derived from the conventional elastic settlement formula, incorporating estimates of rock mass modulus from Hoek and Brown (1997). The resistance factors
provided in Figure 751.38.4.1 were established from probabilistic calibrations to achieve the target foundation reliabilities as described in Abu El-Ela et al. (2011). These calibrations were
conducted with explicit consideration of variability and uncertainty present for dead load, live load, uniaxial compressive strength, and a “method variability” to account for variability and
uncertainty introduced by the elastic model in general, and the estimates of rock mass modulus, E[m], in particular. The variability and uncertainty utilized for dead load and live load were taken
from Kulicki et al. (2007). The “method variability” was conservatively assumed for development of resistance factors for this provision of the guidelines because of the lack of data available upon
which to judge the accuracy of the method. It is likely that this provision could be made more efficient (i.e. made to produce smaller footings) with additional study should this provision control
the size of spread footings on a routine basis.
Guidance for establishing appropriate values for GSI is provided in EPG 751.38.3.1 and the associated commentary.
When the term H in Equation 751.38.4.1 is taken to be a multiple of the foundation width, B, it is possible to cancel terms on both sides of the equation to arrive at an expression for the minimum
foundation length, L. Strictly speaking, the equations produce the result of a minimum L for some assumed B, but this can be done for ANY value of B. In such cases, designers should avoid “getting
wrapped up in the math” to arrive at unreasonable values for B and L and remember that spread footings shall be made as close to square as possible according to the provisions of EPG 751.38.1.2.
For the purposes of this provision, use of “more compressible” strata reflects the need for the designer to judge the relative stiffness of different strata beneath the footing. If the rock beneath
the footing is composed of alternating strata of relatively stiff and soft rock, the thickness H shall be taken to reflect the cumulative thickness of relatively soft rock within a depth range from
the base of the footing to a depth of 2∙B below the base of the footing.
The provisions of this subarticle are derived from the conventional elastic settlement formula, incorporating estimates of rock mass modulus from Rowe and Armitage (1984). The resistance factors
provided in Figure 751.38.4.2 were established from probabilistic calibrations to achieve the target foundation reliabilities as described in Abu El-Ela et al. (2011). These calibrations were
conducted with explicit consideration of variability and uncertainty present for dead load, live load, uniaxial compressive strength, and a “method variability” to account for variability and
uncertainty introduced by the elastic model in general, and the estimates of rock mass modulus, E[m], in particular. The variability and uncertainty utilized for dead load and live load were taken
from Kulicki et al. (2007). The “method variability” was derived from data provided by Rowe and Armitage to reflect the variability of the relationship between uniaxial compressive strength of the
intact rock and the rock mass modulus. Because the variability of the method can be assessed through empirical data, the resistance factors provided in EPG 751.38.4.2 are substantially greater than
those provided in EPG 751.38.4.1 where empirical data is not available.
When the term H in Equation 751.38.4.2 is taken to be a multiple of the foundation width, B, it is possible to cancel terms on both sides of the equation to arrive at an expression for the minimum
foundation length, L. Strictly speaking, the equations produce the result of a minimum L for some assumed B, but this can be done for ANY value of B. In such cases, designers should avoid “getting
wrapped up in the math” to arrive at unreasonable values for B and L and remember that spread footings shall be made as close to square as possible according to the provisions of EPG 751.38.1.2.
For the purposes of this provision, use of “more compressible” strata reflects the need for the designer to judge the relative stiffness of different strata beneath the footing. If the rock beneath
the footing is composed of alternating strata of relatively stiff and soft rock, the thickness H shall be taken to reflect the cumulative thickness of relatively soft rock within a depth range from
the base of the footing to a depth of 2∙B below the base of the footing.
The provisions of this subarticle are derived from conventional one-dimensional consolidation settlement equations, adapted to conform to the LRFD approach. The resistance factors provided in Figs.
751.38.4.3.1 and 751.38.4.3.2 were established from probabilistic calibrations to achieve the target foundation reliabilities as described in Song et al. (2011). These calibrations were conducted
with explicit consideration of variability and uncertainty present for dead load, live load, soil compression index (c[c]), soil recompression index (c[r]), initial void ratio (e[o]), maximum past
vertical effective stress (σ'[p]), and the change in effective stress due to the applied load from the foundation. A “method variability” was also included in the calibrations to reflect general
variability and uncertainty associated with predictions of settlement in cohesive soils. The variability and uncertainty utilized for dead load and live load were taken from Kulicki et al. (2007).
The variability in the initial void ratio was taken from analyses of site characterization data from several different sites (Likos et al., 2011). The “method variability” was established from
judgment regarding the expected accuracy of the general settlement equation.
Separate resistance factors were applied to the compression and recompression indices and the maximum past vertical effective stress so that the variability of these parameters could be addressed
separately. It is possible to develop a single resistance factor to be applied to the entire expression. However, such an implementation prevents individual accounting for variability in these
parameters and ultimately leads to conservatism that is not necessary when the resistance factors are separated.
For spread footings on cohesive soils, elastic settlement is generally small relative to settlement arising from consolidation or secondary compression. Secondary compression can be significant,
particularly in highly organic soils, but is generally small relative to consolidation settlements for purely mineral soils.
Probabilistic calibrations for spread footings on cohesionless soils have not yet been completed by MoDOT. The provisions of current AASHTO LRFD Bridge Design Specifications should therefore be
followed when designing spread footings on cohesionless soils.
Probabilistic calibrations for spread footings subjected to lateral loads have not yet been completed by MoDOT. The provisions of current AASHTO LRFD Bridge Design Specifications should therefore be
followed when designing spread footings on cohesionless soils.
The provisions of this subarticle are unchanged from previous version except for minor editorial revisions.
AASHTO (2009), AASHTO LRFD Bridge Design Specification: Customary U.S. Units, American Association of State Highway and Transportation Officials, Fourth Edition with 2008 and 2009 Interim Revisions.
Abu El-Ela, A.A., J.J. Bowders, and J.E. Loehr (2011), Calibration of LRFD Resistance Factors for Design of Spread Footings in Hard and Soft Rock, Missouri Department of Transportation, OR11.XXX, XXX
pp. (in preparation)
Bowders, J.J., J.E. Loehr, and D. Huaco (2011), MoDOT Transportation Geotechnics Research Program: Development of Target Reliabilities for MoDOT Bridge Foundations and Earth Slopes, Missouri
Department of Transportation, OR11.XXX, XX pp. (in preparation)
Duncan, J.M., and C.K. Tan (1991), Part 5 – Engineering Manual for Estimating Tolerable Movements of Bridges, in Manuals for the Design of Bridge Foundations, by R.M. Barker, J.M. Duncan, K.B.
Rojiani, P.S.K. Ooi, C.K. Tan, and S.G. Kim, NCHRP Report 343, TRB, pp. 219-228.
Hoek, E., and E.T. Brown (1988), “The Hoek-Brown Failure Criterion – A 1988 Update,” Proceedings of the 15th Canadian Rock Mechanics Symposium, Toronto, Canada.
Hoek, E., and E.T. Brown (1997), “Practical Estimates of Rock Mass Strength,” International Journal of Rock Mechanics and Mining Sciences, Vol. 34, No. 8, 1997, pp. 1165-1186.
Kulicki, J.M., Z. Prucz, C.M. Clancy, D.R. Mertz, and A.S. Nowak (2007), Updating the Calibration Report for AASHTO LRFD Code, Final Report for NCHRP Project 20-7/186, AASHTO, 125 pp.
Likos, W.J., J.E. Loehr, N. Maerz, K.A. Magner, L. Ge, and R.W. Stephenson (2011), MoDOT Transportation Geotechnics Research Program: Site Characterization Program Interpretation Report, Missouri
Department of Transportation, OR11.XXX, XXX pp. (in preparation)
Marinos, P., and E. Hoek (2000), “GSI: A Geologically Friendly Tool for Rock Mass Strength Estimation,” Proceedings of GeoEng2000, Melbourne, Australia, Vol. I, pp. 1422-1440.
Moulton, L.K. (1986), Tolerable Movement Criteria for Highway Bridges, Federal Highway Administration Report No. FHWA-TS-85-228, 93 pp.
Rowe, R.K., and H.H. Armitage (1984), The Design of Piles Socketed into Weak Rock, Geotechnical Research Report GEOT-11-84 for the National Research Council of Canada, University of Western Ontario,
366 pp.
Song, C., J.J. Bowders, A.A. Abu El-Ela, and J.E. Loehr (2011), Calibration of LRFD Resistance Factors for Design of Spread Footings and Embankments in Cohesive Soils at Serviceability Limit States,
Missouri Department of Transportation, OR11.XXX, XXX pp. (in preparation)
Wyllie, D.C. (1999), Foundations on Rock, E & FN Spon, Second Edition, 401 pp.
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Temperature shifts in the Sinai model: static and dynamical effects
Times cited: 7
Sales, M, Bouchaud, JP, Ritort, F.
J. Phys. A-Math. Gen. 36 , 665 -684 (2003).
We study analytically and numerically the role of temperature shifts in the simplest model where the energy landscape is explicitly hierarchical, namely the Sinai model. This model has both
attractive features (there are valleys within valleys in a strict self-similar sense), but also one important drawback: there is no phase transition so that the model is, in the large-size limit,
effectively at zero temperature. We compute various static chaos indicators, that are found to be trivial in the large-size limit, but exhibit interesting features for finite sizes. Correspondingly,
for finite times, some interesting rejuvenation effects, related to the self-similar nature of the potential, are observed. Still, the separation of time scales/length scales with temperature in this
model is much weaker than in experimental spin glasses.
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What are the removable and non-removable discontinuities, if any, of f(x)=abs(x-9)/ (x-9)? | HIX Tutor
What are the removable and non-removable discontinuities, if any, of #f(x)=abs(x-9)/ (x-9)#?
Answer 1
There's a non-removable discontinuity in $x = 9$
Since we can't divide by #0# we know that #x != 9#, so that's a discontinuity.
It might look it's removable because the numerator has #|x-9|#
But, if you look at what happens before and after #x = 9# you'll see it's non-removable.
For #x < 9#
#y = |x-9|/(x-9) = -(x-9)/(x-9) = -1#
For #x > 9#
#y = |x-9|/(x-9) = (x-9)/(x-9) = 1#
So the graph still has a discontinuity, because the limit at #x = 9# doesn't exist.
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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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Ulam stability of $\wp$-mild solutions for $\psi$-Caputo-type fractional semilinear differential equations
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Gauge sizes are numbers that indicate the thickness of a piece of sheet metal.
The thickness of 16 gauge aluminum sheet
16 gauge thickness in inches
16ga aluminum sheet is 0.0508 inches thick.
16 gauge aluminum thickness in mm(millimeters)
16 gauge approximate thickness in millimeters is 1.29 mm.
The weight of 16 gauge aluminum sheet
The weight per unit area of the sheet can also be seen in pounds per square foot and kilograms per square meter.
The weight of 16 gauge aluminum(ounces): 11.472 oz/ft²
The weight of 16 gauge aluminum(pounds ): 0.717lb/ft²
The weight of 16 gauge aluminum(KG): 0.3252 kg/ft²
The weight of 16 gauge aluminum(KG/m²): 3.5 kg/m²
16 gauge aluminum chart specification
The thickness and weight of the 16 gauge aluminum sheet
Gauge 16
Thickness Approximate thickness in decimal parts of an inch(in) 0.0508 in
Approximate thickness in millimeters(mm) 1.29 mm
Weight per square foot in ounces avoirdupois(oz/ft²) 11.472 oz/ft²
Weight Per Area Weight per square foot in pounds avoirdupois(lb/ft²) 0.717lb/ft²
Weight per square foot in kilograms(kg/ft²) 0.3252 kg/ft²
Weight per square meter in kilograms(kg/m²) 3.5 kg/m²
What is thicker 11 gauge or 16 gauge?
The rating for aluminum gauge may seem backward: the smaller the number, the thicker the sheet metal , The larger the number, the thinner the sheet metal.
Therefore, the thickness of the 16 gauge aluminum sheet is thinner than that of the 11 gauge aluminum sheet.
Parameters of other specifications of aluminum plate：
6 gauge aluminum thickness
12 gauge aluminum thickness
14 gauge aluminum thickness
16 gauge aluminum thickness
18 gauge aluminum thickness
22 gauge aluminum thickness
24 gauge aluminum thickness
Note: The larger the value of Gauge, the thinner the sheet.
For more detailed and complete metal aluminum gauge chart please check URL:
What is the sheet metal gauge? How to calculate aluminum gauge?
What is the weight of the aluminum sheet plate?
Standard sheet metal gauges in wikipedia
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What is Maximum Power Transfer Theorem (MPTT)?
In this tutorial, we will learn about Maximum Power Transfer Theorem (MPTT). It is one of the basic yet important laws that states the necessary condition for maximum power transfer (not to be
confused with maximum efficiency).
Maximum Power Transfer Theorem
In any electric circuit, the electrical energy from the power supply is delivered to the load where it is converted into a useful work. Practically, the entire supplied power will not be present at
the load due to the heating effect and other constraints in the network. Therefore, there exists a certain difference between drawing and delivering powers.
The size of the load always affects the amount of power transferred from the supply source, i.e., any change in the load resistance results a change in the power transferred to the load. Thus, the
Maximum Power Transfer Theorem ensures the ideal condition to transfer the maximum power to the load. Let us see ‘how’.
Maximum Power Transfer Theorem Statement
The Maximum Power Transfer Theorem states that in a linear, bilateral DC network, Maximum Power is delivered to the load when the load resistance is equal to the internal resistance of the source.
If it is an independent voltage source, then its series resistance (internal resistance R[S]) or if it is independent current source, then its parallel resistance (internal resistance R[S]) must
equal to the load resistance R[L ]to deliver maximum power to the load.
Proof of Maximum Power Transfer Theorem
The Maximum Power Transfer Theorem ensures the value of the load resistance, at which the maximum power is transferred to the load.
Consider the below DC two terminal network (left side circuit). The condition for maximum power is determined by obtaining the expression of power absorbed by load using mesh or nodal current methods
and then deriving the resulting expression with respect to load resistance R[L].
This is quite a complex procedure. But in the previous tutorials, we have seen that the complex part of the network can be replaced with a Thevenin’s equivalent as shown below.
The original two terminal circuit is replaced with a Thevenin’s equivalent circuit across the variable load resistance. The current through the load for any value of load resistance is
Form the above expression, the power delivered depends on the values of R[TH] and R[L]. However, as the Thevenin’s equivalent is a constant, the power delivered from this equivalent source to the
load entirely depends on the load resistance R[L]. To find the exact value of RL, we apply differentiation to P[L] with respect to R[L] and equating it to zero as shown below:
Therefore, this is the condition of matching the load where the maximum power transfer occurs when the load resistance is equal to the Thevenin’s resistance of the circuit. By substituting the R[TH]
= R[L] in the previous equation, we get:
The maximum power delivered to the load is,
Total power transferred from source is:
P[T] = 2 * I[L]^2 R[L ] …………….(2)
Hence, the maximum power transfer theorem expresses the state at which maximum power is delivered to the load i.e., when the load resistance is equal to the Thevenin’s equivalent resistance of the
circuit. Below figure shows a curve of power delivered to the load with respect to the load resistance.
Note that the power delivered is zero when the load resistance is zero as there is no voltage drop across the load during this condition. Also, the power will be maximum, when the load resistance is
equal to the internal resistance of the circuit (or Thevenin’s equivalent resistance). Again, the power is zero as the load resistance reaches to infinity as there is no current flow through the
Power Transfer Efficiency
We must remember that this theorem states only maximum power transfer but not for maximum efficiency. If the load resistance is smaller than source resistance, power dissipated at the load is reduced
while most of the power is dissipated at the source, then the efficiency becomes lower.
Consider the total power delivered from source equation (equation 2), in which the power is dissipated in the equivalent Thevenin’s resistance R[TH] by the voltage source V[TH].
Therefore, the efficiency under the condition of maximum power transfer is:
Efficiency = Output / Input × 100
= I[L]^2 R[L ]/ 2 I[L]^2 R[L ]× 100
= 50 %
Hence, at the condition of maximum power transfer, the efficiency is 50%, that means only half of the generated power is delivered to the load and at other conditions, a small percentage of power is
delivered to the load, as indicated in efficiency verses maximum power transfer the curves below.
For some applications, it is desirable to transfer maximum power to the load than achieving high efficiency such as in amplifiers and communication circuits.
On the other hand, it is desirable to achieve higher efficiency than maximized power transfer in case of power transmission systems, where a large load resistance (much larger value than internal
source resistance) is placed across the load. Even though the efficiency is high the power delivered will be less in those cases.
Maximum Power Transfer Theorem for AC Circuits
In an active network, it can be stated that the maximum power is transferred to the load when the load impedance is equal to the complex conjugate of an equivalent impedance of a given network as
viewed from the load terminals.
Consider the above Thevenin’s equivalent circuit across the load terminals in which the current flowing through the circuit is given as:
Therefore, I = V[TH] / (R[L] + jX[L] + R[TH ]+ jX[TH])
I = V[TH] / ((R[L]+ R[TH]) + j(X[L] + X[TH ]))
The power delivered to the load,
P[L ]= V^2[TH] * R[L] / ((R[L]+ R[TH])^2 + (X[L] + X[TH ])^2) ……(1)
For maximum power the derivative of the above equation must be zero, after simplification we get
Putting the above relation in equation 1, we get
P[L ]= V^2[TH ×] R[L] / ((R[L]+ R[TH ])^2
Again for maximum power transfer, derivation of above equation must be equal to zero and after simplification we get:
Hence, the maximum power will transferred to the load from source, if R
= R
and X
= – X
[TH ]
in an AC circuit. This means that the load impedance should be equal to the complex conjugate of equivalent impedance of the circuit,
Where Z
is the complex conjugate of the equivalent impedance of the circuit.
This maximum power transferred, P[MAX] = V^2[TH] / 4 R[TH] or V^2[TH]/ 4 R[L]
Applying Maximum Power Transfer Example to DC Circuit
Consider the below circuit to which we determine the value of the load resistance that receives the maximum power from the supply source and the maximum power under the maximum power transfer
Disconnect the load resistance from the load terminals ‘a ‘and ‘b’. To represent the given circuit as Thevenin’s equivalent, we have to determine the Thevenin’s voltage V[TH] and Thevenin’s
equivalent resistance R[TH].
The Thevenin’s voltage or voltage across the terminals ab is V[ab] = V[a] – V[b]
V[a] = V × R2 / (R1 + R2)
= 30 × 20 /×(20 + 15)
= 17.14 V
V[b] = V × R4/ (R3 + R4)
= 30 × 5 /(10 + 5)
= 10 V
V[ab] = 17.14 – 10
= 7.14 V
V[TH] = V[ab] = 7.14 Volts
Calculate the Thevenin’s equivalent resistance R[TH] by replacing sources with their internal resistances (here, let us assume that voltage source has zero internal resistance so it becomes a short
Thevenin’s equivalent resistance or resistance across the terminals ab is
R[TH] = Rab = [R1R2 / (R1 + R2)] + [R3R4 /(R3 + R4)]
= [(15 × 20) / (15 + 20)] + [(10 × 5) / (10+ 5)]
= 8.57 + 3.33
R[TH] = 11.90 Ohms
The Thevenin’s equivalent circuit with above calculated values by reconnecting the load resistance is shown below.
From the maximum power transfer theorem, R[L] value must equal to the R[TH] to deliver the maximum power to the load.
Therefore, R[L] = R[TH]= 11.90 Ohms
And the maximum power transferred under this condition is,
P[MAX] = V^2[TH] / 4 R[TH]
= (7.14)^2 / (4 × 11.90)
= 50.97 / 47.6
= 1.07 Watts
Applying Maximum Power Transfer to AC circuit
The below AC network consists of load impedance Z[L] of which both reactive and resistive parts can be varied. Hence, we have to determine the load impedance value at which the maximum power
delivered from the source and also the value of the maximum power.
To find the value of load impedance, first, we find the Thevenin’s equivalent circuit across the load terminals. For finding Thevenin’s voltage, disconnect the load impedance as shown in below
By voltage divider rule,
V[TH] = 20∠0 × [j6 / (4 + j6)]
= 20∠0 ×[6∠90 / 7.21∠56.3]
= 20∠0 × 0.825∠33.7
V[TH] = 16.5∠33.7 V
By shorting the voltage source, we calculate the Thevenin’s equivalent impedance of the circuit as shown in figure.
Z[TH] = (4 × j6) / (4 + j6)
= (4 × 6∠90) / (7.21∠56.3)
= 3.33∠33.7 0r 2.77 + j1.85 Ohms
Hence, the Thevenin’s equivalent circuit across the load terminals is shown in below.
Therefore to transfer the maximum power to the load, the value of the load impedance should be
Z[L ]= Z[TH]
= 2.77 – j1.85 ohms
The maximum power delivered, P[MAX]
= V^2[TH] / 4 R[TH]
= (16.5)^2/4(2.77)
= 272.25 / 11.08
= 24.5 W
Practical Application of Maximum Power Transfer Theorem
Consider the practical example of a speaker with an impedance of 8 ohms. It is driven by an audio amplifier with an internal impedance of 500 ohms. The Thevenin’s equivalent circuit is also shown in
According to the maximum power transfer theorem, the power is maximized at the load if the load impedance is 500 ohms (same as internal impedance). Or else internal resistance has to be changed to 8
ohms to achieve the Maximum Power Transfer condition. However, it is not possible to change either of them.
So, it is an impedance mismatch condition and it can be overcome by using an impedance matching transformer with its impedance transformation ratio of 500:8.
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Pair Of Linear Equations In Two Variables Class 10 Notes Ch-3
Pair Of Linear Equations In Two Variables Class 10 Notes
A linear equation is a straight-line equation. It takes the shape of
ax + by + c = 0
Where a, b and c are the real numbers (a≠0 and b≠0) and x and y are the two variables, Here, a and b are the coefficients, and c is the equation’s constant.
Pair Of Linear Equations in two variable
Two Linear Equations having two same variables are the pair of Linear Equations in two variables.
Pair Of Linear Equations In Two Variables Class 10- Graphical Method Of Solution
There will be two lines on the graph because we are displaying two equations.
1. 1. If the two lines cross at that specific position, there will only be one solution to the pair of linear equations. It is claimed that the two equations are compatible.
2. There will be an endless number of answers if the two lines meet, as every point along the line will contain the answer to the pair of linear equations. It is referred to as a pair of dependent
or consistent equations.
3. There won’t be a solution if the two lines are parallel because they don’t connect anywhere. It is claimed that the equations are incoherent.
Pair Of Linear Equations In Two Variables Class 10- Algebraic Methods Of Solving
These procedures must be followed in order to solve a pair of linear equations with the variables x and y using the substitution method:
Substitution Method
Step 1: Pick any equation and determine the value of one variable in relation to another, in this case, y in relation to x.
Step 2: Next, change the other equation to use the estimated value of y in terms of x.
Step 3: Since there is only one variable in this linear equation, x, we must now solve it in terms of x.
Step 4: Change the value of x in the provided equations and determine the value of y.
Using this approach, we can eliminate any one of the variables to solve the equations.
Ace your class 10th board exams with Adda247 live classes for class 10th preparation.
Elimination Method
Step 1: Multiply the coefficients of each variable in both equations by a certain number to make them equal.
Step 2: Now combine or divide the equations so that one variable, whose coefficients are the same, is eliminated.
Step 3: To determine the value of the residual variable, solve the equation.
Step 4: Use the equations provided to obtain the value of the other variable by substituting the calculated value of the variable.
Cross Multiplication Method
Given two equations in the form of
a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, where
We write it in general form as
To apply cross multiplication, we use this diagram
The multiplied pairs are shown by the arrows. The product of the upward and downward arrow pair products must be subtracted from one another.
Using this diagram, we must first write the equations in general form; then, we must calculate the values of x and y and enter those values in the corresponding notations.
To learn more about this chapter click on this link
Equations Reducible To A Pair Of Linear Equations In two variables
In this section, we learn about such equations, which are not linear but can be reduced to pair of linear equations form using the substitution method. Let us understand with the help of an example.
In this case, we may make the substitution
1/x = u and 1/y = v
The pair of equations reduces to
2u + 3v = 4 …..(i)
5u – 4v = 9 ….(ii)
From the first equation, isolate the value of u.
u = (4-3v)/2
Now substitute the value of u in eq. (ii)
5[(4-3v)/2] – 4v = 9
Solving for v, we get;
v = 2/23
Now substitute the value of v in u = (4-3v)/2 to get the value of u.
u = 43/23
Since, u = 1/x or x = 1/u = 23/43
and v = 1/y or y = 1/v, so y = 23/2
Hence, the solutions are x = 23/43 and y = 23/2.
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wn - Intuitionistic Logic Explorer
Description: If 𝜑 is a wff, so is ¬ 𝜑 or "not 𝜑". Part of the recursive definition of a wff (well-formed formula). Traditionally, Greek letters are used to represent wffs, and we follow this
convention. In propositional calculus, we define only wffs built up from other wffs, i.e., there is no starting or "atomic" wff. Later, in predicate calculus, we will extend the basic wff definition
by including atomic wffs (weq 1490 and wel 2136).
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Characteristics of An Algorithm - TestingDocs.com
In this tutorial, we will learn the characteristics of an Algorithm. The word algorithm comes from the Arabic word Al-Khowzmi, the Arabian inventor, and means recipe, method, technique, or procedure.
What is an Algorithm?
An algorithm is a set of instructions or a step-by-step procedure for solving a problem or completing a specific task.
Characteristics of An Algorithm
The characteristics of a good algorithm are as follows:
• Well-defined
• Unambiguous
• Finite
• Deterministic
• Feasible
• Efficient
Well Defined
A good algorithm should have clear and well-defined inputs and outputs. It should be clear what information or data is required as input and what are the expected outputs.
Each algorithm step should be simple, clear, and unambiguous. The step instructions should be precise and easy to understand. Misinterpretation or misleading steps should be avoided during algorithm
design. In real time, the algorithm would be converted into a software program during the software development phase by the software team. The software developers should have a clear understanding of
the algorithm.
An algorithm should have a finite number of steps. It should terminate after a finite number of iterations or steps, without falling into an infinite loop.
An algorithm should be deterministic, which means that for a given input, it should always produce the same output.
A good algorithm should be efficient in terms of resources like time and space requirements.
An algorithm should be practical and feasible to implement in a real-world environment. For example, the algorithm should be technically feasible with current technology.
See Also
Algorithm vs Program
A computer program is a sequence of instructions written in a computer programming language to perform a specified task with the computer. Differences between an algorithm and a computer program can
be found here:
Computers /
Quantum Algorithms
Computers /
Quantum Computing
Computers /
BPMN Diagram
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Domain and Range - Examples | Domain and Range of a Function
What are Domain and Range?
In basic terms, domain and range coorespond with different values in in contrast to each other. For instance, let's check out the grade point calculation of a school where a student earns an A grade
for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade adjusts with the total score. In mathematical terms, the score is the domain or the input, and
the grade is the range or the output.
Domain and range could also be thought of as input and output values. For instance, a function can be stated as an instrument that catches particular items (the domain) as input and makes particular
other pieces (the range) as output. This might be a machine whereby you can obtain different items for a respective quantity of money.
Here, we review the fundamentals of the domain and the range of mathematical functions.
What are the Domain and Range of a Function?
In algebra, the domain and the range indicate the x-values and y-values. For instance, let's check the coordinates for the function f(x) = 2x: (1, 2), (2, 4), (3, 6), (4, 8).
Here the domain values are all the x coordinates, i.e., 1, 2, 3, and 4, whereas the range values are all the y coordinates, i.e., 2, 4, 6, and 8.
The Domain of a Function
The domain of a function is a batch of all input values for the function. To put it simply, it is the group of all x-coordinates or independent variables. So, let's review the function f(x) = 2x + 1.
The domain of this function f(x) can be any real number because we can plug in any value for x and obtain a corresponding output value. This input set of values is necessary to discover the range of
the function f(x).
However, there are specific conditions under which a function must not be defined. So, if a function is not continuous at a certain point, then it is not stated for that point.
The Range of a Function
The range of a function is the set of all possible output values for the function. To put it simply, it is the group of all y-coordinates or dependent variables. For example, applying the same
function y = 2x + 1, we might see that the range will be all real numbers greater than or equivalent tp 1. Regardless of the value we assign to x, the output y will continue to be greater than or
equal to 1.
But, just like with the domain, there are specific conditions under which the range cannot be stated. For instance, if a function is not continuous at a particular point, then it is not stated for
that point.
Domain and Range in Intervals
Domain and range can also be classified with interval notation. Interval notation expresses a group of numbers applying two numbers that classify the lower and upper boundaries. For example, the set
of all real numbers in the middle of 0 and 1 could be identified applying interval notation as follows:
This denotes that all real numbers higher than 0 and lower than 1 are included in this batch.
Also, the domain and range of a function could be represented with interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) could be identified as follows:
This tells us that the function is defined for all real numbers.
The range of this function might be classified as follows:
Domain and Range Graphs
Domain and range could also be represented with graphs. For example, let's review the graph of the function y = 2x + 1. Before plotting a graph, we must find all the domain values for the x-axis and
range values for the y-axis.
Here are the coordinates: (0, 1), (1, 3), (2, 5), (3, 7). Once we chart these points on a coordinate plane, it will look like this:
As we could see from the graph, the function is defined for all real numbers. This tells us that the domain of the function is (-∞,∞).
The range of the function is also (1,∞).
This is due to the fact that the function produces all real numbers greater than or equal to 1.
How do you determine the Domain and Range?
The process of finding domain and range values differs for different types of functions. Let's consider some examples:
For Absolute Value Function
An absolute value function in the form y=|ax+b| is specified for real numbers. Therefore, the domain for an absolute value function consists of all real numbers. As the absolute value of a number is
non-negative, the range of an absolute value function is y ∈ R | y ≥ 0.
The domain and range for an absolute value function are following:
For Exponential Functions
An exponential function is written in the form of y = ax, where a is greater than 0 and not equal to 1. Therefore, each real number can be a possible input value. As the function just delivers
positive values, the output of the function contains all positive real numbers.
The domain and range of exponential functions are following:
• Domain = R
• Range = (0, ∞)
For Trigonometric Functions
For sine and cosine functions, the value of the function oscillates among -1 and 1. Further, the function is stated for all real numbers.
The domain and range for sine and cosine trigonometric functions are:
• Domain: R.
• Range: [-1, 1]
Just see the table below for the domain and range values for all trigonometric functions:
For Square Root Functions
A square root function in the form y= √(ax+b) is specified just for x ≥ -b/a. Therefore, the domain of the function consists of all real numbers greater than or equal to b/a. A square function will
always result in a non-negative value. So, the range of the function includes all non-negative real numbers.
The domain and range of square root functions are as follows:
• Domain: [-b/a,∞)
• Range: [0,∞)
Practice Questions on Domain and Range
Realize the domain and range for the following functions:
1. y = -4x + 3
2. y = √(x+4)
3. y = |5x|
4. y= 2- √(-3x+2)
5. y = 48
Let Grade Potential Help You Master Functions
Grade Potential would be happy to pair you with a one on one math instructor if you are looking for assistance understanding domain and range or the trigonometric topics. Our Cleveland math tutors
are skilled educators who aim to work with you on your schedule and customize their tutoring methods to match your learning style. Call us today at (216) 616-1177 to learn more about how Grade
Potential can help you with achieving your academic goals.
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Euclidean distance transform
Euclidean distance transform. Based on “Distance Transforms of Sampled Functions” by Pedro F. Felzenszwalb and Daniel P. Huttenlocher, Cornell Computing and Information Science TR2004-1963. This
function first thresholds at the specified value and then does the distance transform of the resulting binary image. The signed distance (negative values inside object) is also available. Distances
between non-isotropic samples are handled correctly.
• Uses nrrdDistanceL2 or nrrdDistanceL2Signed
threshold value to separate inside from outside (double)
if non-zero, bias the distance transform by this amount times the difference in value from the threshold (double); default: “0.0”
type to save output in; default: “float”
also compute signed (negative) distances inside objects, instead of leaving them as zero
values below threshold are considered interior to object. By default (not using this option), values above threshold are considered interior.
input nrrd
output nrrd (string); default: “-”
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branch_strongcoloring.h File Reference
Detailed Description
branching rule performing strong branching for the vertex coloring problem
Gerald Gamrath
This file implements an additional branching rule for the coloring algorithm.
We are looking for two nodes v and w, which are not adjacent in the current graph, and consider the following two constraints: SAME(v,w) and DIFFER(v,w). More information about the meaning of these
constraints can be found in the documentation of the branching rule in branch_coloring.c.
This branching rule puts some more effort into the choice of the two nodes and performs a strongbranching. This means that for every possible choice of two nodes, it solves the LPs of the created
children and computes a score with respect to the increase of the lower bound in both nodes. After that, it takes the combination of nodes yielding the best score. The interesting point is that the
strongbranching is not performed for each variable, as it is done in some default branching rules of SCIP and supported by the LP-solver, but is done for a constraint, since we are branching on
constraints. Look at executeStrongBranching() to see how it is done. There are also some improvements, since testing all possible combination of nodes is very expensive. The first possibility to
avoid this is to stop the computation of scores once a possible branching is found that has only one feasible child. This results in more restrictions in this child without increasing the number of
unprocessed nodes.
The second improvement is to compute a priority for all possible combinations, w.r.t. the fractional values of the variables. Then, only the first best k combinations are investigated by
This code is not optimized and in most cases inferior to the standard branching rule. It is only a demonstration of how to perform strongbranching on constraints!
Definition in file branch_strongcoloring.h.
Go to the source code of this file.
Function Documentation
◆ SCIPincludeBranchruleStrongcoloring()
SCIP_RETCODE SCIPincludeBranchruleStrongcoloring ( SCIP * scip )
creates the coloring branching rule and includes it in SCIP
Definition at line 743 of file branch_strongcoloring.c.
DEFAULT_MAXPRICINGROUNDS, DEFAULT_USETCLIQUE, FALSE, NULL, SCIP_CALL, SCIP_OKAY, SCIPaddBoolParam(), SCIPaddIntParam(), SCIPallocBlockMemory, SCIPincludeBranchruleBasic(), SCIPsetBranchruleCopy(),
SCIPsetBranchruleExecLp(), SCIPsetBranchruleExit(), SCIPsetBranchruleFree(), SCIPsetBranchruleInit(), and TRUE.
Referenced by SCIPincludeColoringPlugins().
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Question #638f8 + Example
Question #638f8
1 Answer
Isn't it centripetal acceleration? I don't think there is an acceleration called centripedal.
If it is centripetal acceleration,
"Centripetal acceleration is the idea that any object moving in a circle will have an acceleration vector pointed towards the center of that circle. Centripetal means towards the center".
You can find out the centripetal acceleration, using this formula,
Centripetal acceleration=$\frac{{V}^{2}}{r}$
V= speed
r= the radius of the circle
Source: http://study.com/academy/lesson/centripetal-acceleration-definition-formula-example.html
I hope this helps
Impact of this question
4617 views around the world
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Binomial Theorem Class 11 - NCERT Solutions (and videos) - NCERT 2024
Updated for NCERT 2023-2024 Books
NCERT Solutions of all questions, examples of Chapter 7 Class 11 Binomial Theorem available free at teachoo. You can check out the answers of the exercise questions or the examples, and you can also
study the topics.
Let's see what is binomial theorem and why we study it.
We know that
(a + b)^2 = a^2 + b^2 + ab
(a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2
But what about big powers, like
(a + b)^5
(a + b)^9
(a + b)^100
To find out these values, we use Binomial Theorem
The topics in this chapter include
• What is Binomial Theorem
• Number of terms in Binomial Theorem
• Solving Expansions
• Finding larger number using Binomial Theorem
• Solving proofs using Binomial Theorem
• General Term of a Binomial Theorem
• Finding Coefficient of a term
• Middle Term of a Binomial Theorem
Check out the answers below.
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Coefficient of rolling friction - Lab experiment
With rare exception,^1–3 the force of friction on a rolling object is not usually a topic discussed in introductory physics textbooks. Although the invention of the wheel is one of the essential
world achievements, rolling friction is typically ignored and the inability of students to explain or model the deceleration of a rolling rigid object on a rigid horizontal surface necessitates a
mechanism for this phenomenon. (Detailed analyses of this mechanism and measurements of the coefficient of rolling friction can be found in Refs. 4–6.) Every introductory physics textbook discusses
kinetic and static friction and many papers are devoted to teaching these forces,^7 but it is well known that measuring static and kinetic friction in educational labs is troublesome.^8,9 To make
matters worse, the coefficient of rolling friction is much smaller than the coefficient of kinetic friction,^1–5 so student measurements of rolling friction are even more troublesome. In addition,
the usual measurements of time (about 1s) and distance traveled (less than 1m) are comparatively small, which makes surface uniformities and accurate leveling (or measuring the angle of incline) of
the track critical factors that influence the uncertainty of the measurements. To counter these difficulties, a method of measuring the coefficients of rolling friction based on the oscillations of
steel balls on a large concave lens was proposed in Ref. 4. This method is free from the above-named deficiencies.
In this note, a ball oscillating on a concave trackway is used to find the coefficient of rolling friction μ[r] using typical laboratory equipment. This simple experiment can be carried out by
students and the results are highly reliable. It is known that a variety of factors can influence friction forces, including adhesion, deformation, elastic hysteresis, abrasion, the effect of
impurities, etc.^7 However, the nature of rolling friction is not the subject of this paper. Instead, we use a phenomenological approach that presumes the magnitude of the rolling friction force $fr$
is proportional to the normal force N and has a direction opposite to the motion^2,10
where $μr$ is assumed to be constant (independent of velocity). In this case the only dissipative force that causes the change of mechanical energy (kinetic plus potential energies) is the force of
rolling friction.^10 The assumption of Eq. (1) should be considered an initial, but commonly used, approximation; deviations of this formula have been observed.^11
A steel ball of diameter of 3.90cm and mass m=225g was used as the rolling object. A manufactured wood track (a so-called stringless pendulum^12) and a plastic ruler with a central groove were
utilized as concave tracks (see Fig. 1). The ball was released near the bottom of the track and its (oscillatory) position was measured using a pasco^13 motion sensor. Figure 2 shows typical graphs
of the ball's position and velocity versus time. Because the track curvature is small, the motion of the ball can be approximated as one-dimensional with a constant normal force (N=mg). Indeed, on
the wood track, which has a radius of curvature of 0.57m, the ball's largest displacement from the equilibrium position (0.15 m, see Fig. 2) results in a normal force that differs from mg by only 3%
[$N=mg cos (0.15/0.57)=0.97 mg$]. From Fig. 2, one can see that these oscillations are damped with a linearly modulated amplitude; this confirms the assumption that the force of rolling friction is
constant and does not depend on the speed of the ball^14,15 in this experiment.
Assuming the ball rolls without sliding, a rolling friction force and the force of gravity are the only forces responsible for the change in kinetic energy K.^10 By choosing initial and final
positions of the ball at the bottom of the concave track (perhaps after many oscillations) and applying the work-kinetic energy theorem, one can confirm that the ball's change in kinetic energy
equals the work done only by the force of rolling friction.^10 The kinetic energy of a solid sphere rolling on a flat surface is the sum of the translational $Kt$ and rotational $Kr$ (with respect of
the center of mass) kinetic energies. Assuming the ball is in contact with the bottom (rather than the edges) of the track, the total kinetic energy is then^2
where $v$ is the speed of the ball's center of mass. Meanwhile, the work done by the force of rolling friction is
where $s$ is the total distance travelled by the ball. Equating the work done to the change in kinetic energy then allows us to find the coefficient of rolling friction as
The motion sensor data allows us to determine the ball's velocity at any point in time and to compute the total distance traveled by the ball. For example, using Fig. 2, let us choose an initial time
of t[i]=1s (the ball is at the bottom of the track with v[i]=–0.33m/s) and a final time of t[f]=33s (the ball is again at the bottom of the track with v[f]=0) for oscillations of the ball
on the wood track. The pasco software interface software allows us to compute the (positive) area under the velocity curve, which gives the total distance traveled by the ball, s=3.8m.
Substituting these values into Eq. (4) gives $μr=2.0×10−3$. Using the same procedure, the coefficient of rolling friction for the steel ball on the plastic ruler is found to be $μr=0.75×10−3$.
Alternatively, s can be found as the sum of the ball's oscillation distances. For linearly modulated amplitudes, the amplitudes represent an arithmetic progression and therefore $s=(2(Ai+Af)Δt/T)$,
where A[i] and A[f] are initial and final amplitudes, T is the period of oscillations, and Δt=t[f] – t[i] is the time interval. For example, for ball oscillations on the wood track (Fig. 2), A[i]=
0.15m, A[f]=0, t[i]=1s, t[f]=33s, and T=2.7s, leading to s=3.6m. This method results in a 5% difference with the computation made by calculating areas under the graph of velocity
versus time.
The proposed method of measuring the coefficient of rolling friction can be utilized using a plastic ruler and standard physics equipment^12 (the 2D collision^16 apparatuses can also be exploited as
an aluminum concave track). This method is easy to implement, requires little time, and it allows students to explore the dependence of $μr$ on the ball's diameter and the ball and track materials.
Moreover, this lab integrates different topics in introductory physics including: kinematic relations (differential and integral) between velocity, displacement, and distance traveled; the
work-kinetic energy theorem; and damped oscillations that are linearly modulated. We believe such a lab would be a good learning experience for introductory students.
H. D.
R. A.
University Physics
13th ed.
Boston, MA
D. M.
Physics for Scientists and Engineers; Foundations and Connections
1st ed.
Cengage Learning
Boston, MA
R. D.
Physics for Scientists and Engineers: A Strategic Approach
3rd ed.
Boston, MA
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Coulomb's law for rolling friction
Am. J. Phys.
C. E.
, “
Rolling friction on a wheeled laboratory cart
Phys. Educ.
R. F.
, “
Measuring the coefficient of friction of a low-friction cart
Phys. Teach.
A. De.
, and
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How to teach friction: Experiments and models
Am. J. Phys.
, “
Magnetic viscous drag for friction labs
Phys. Teach.
T. M.
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Being careful with PASCO's kinetic friction experiment: Uncovering pre-sliding displacement?
Phys. Teach.
, and
, “
Introduction to the study of rolling friction
Am. J. Phys.
F. P.
The Friction and Lubrication of Solids, Part II
Clarendon Press
Oxford, UK
M. I.
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Exponential versus linear amplitude decay in damped oscillators
Phys. Teach.
D. S.
, and
R. J.
, “
Oscillator damped by a constant-magnitude friction force
Am. J. Phys.
© 2018 American Association of Physics Teachers.
American Association of Physics Teachers
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50+ Shapes Name in English with Pictures » Onlymyenglish.com
50+ Shapes Name in English with Pictures
Learn different types of all shapes name in English with Pictures. Learn 2d, 3d, and Geometry Shapes name in English
Shapes Name
Sr No. Shapes Image Shapes Name
1. Circle
2. Square
3. Rectangle
4. Triangle
5. Right triangle
6. Rhombus
7. Parallelogram
8. Cube
9. Cuboid
10. Cylinder
11. Sphere
12. Hemisphere
13. Cone
14. Diamond
15. Star
16. Heart
17. Pentagon
18. Hexagon
19. Heptagon
20. Octagon
21. Nonagon
22. Decagon
23. Semicircle
24. Quadrilateral
25. Pyramid
26. Prism
27. Trapezium
28. Trapezoid
29. Oval
30. Ring
31. Kite
32. Arrow
33. Cross
34. Crescent
35. Tetrahedron
36. Octahedron
37. Square pyramid
38. Hexagonal pyramid
39. Triangular prism
40. Rectangular prism
41. Pentagonal prism
Different Shapes Name with Pictures
A circle is made of many points whose locus is at an exact distance from the central point. A circle has an angle of 360 degrees, and the length of the outer surface is called the circumference.
A triangle is a closed structure that contains three sides and three angles less than 90 degrees.
A square is a uniform quadrilateral whose all four sides and angles are equal and of 90 degrees.
A rectangle is a quadrilateral whose pair of opposite sides are equal, and all angles at vertices are equivalent to 90 degrees.
Right triangle:
The right triangle is also called the right-angled triangle, whose one angle is 90 degrees, and the sum of the other two angles is 90 degrees.
A parallelogram is a quadrilateral whose pair of opposite sides and angles (not 90 degrees) are equal in measurement.
A rhombus is also a square; only the difference is that the opposite angles equal but not 90 degrees.
A simple star polygon is a combination of polygons that forms an equiangular, equilateral star-like structure.
A cube is a 3-dimensional solid square-like structure that contains six faces, 12 edges, and eight vertices.
A cuboidal is a 3-dimensional solid rectangular structure that contains six rectangular faces, 12 edges, and eight vertices.
A pyramid is a 3-dimensional structure that has a square ground plan, and outside faces are like triangles that meet at a single point at the tip.
A prism is a 3-dimensional solid structure whose parallel faces are polygon, and the other lateral faces are like rectangles. They are named after their parallel faces.
A diamond is a solid-crystalline carbon compound, is the hardest mineral that is colorless, brightest, and is a crystal structure.
A cylinder is a 3-dimensional solid structure whose curved cylindrical surface is perpendicularly connected with two parallel circular bases. The center of the circular base is called the axis of the
A sphere is a 3-dimensional solid celestial figure, which is a closed spherical surface without angles.
The hemisphere is a half portion of the celestial sphere separated by the horizon.
A cone is a 3-dimensional shape whose base is circular, and the upper surface is curved and meets at the center point above the base.
The heart is a shape that looks like a heart-like structure.
A pentagon is a polygon that has five equal sides, and the angle of the vertices is 108 degrees each.
A hexagon is a polygon that has six equal sides, and the angle of the vertices is 120 degrees each.
A heptagon is a polygon that has seven equal sides, and the angles are equal in measure.
The octagon is a polygon that contains eight equal sides, and the eight angles measured 135 degrees.
The nonagon is a polygon that contains nine equal sides, and nine angles measured 140 degrees each.
A decagon is a polygon that contains ten equal sides, and ten equal angles measured 144 degrees each.
A trapezoid is a quadrilateral with two parallel sides, and the other two sides are lateral.
A kite is a quadrilateral structure whose adjacent sides are equal in length, and two angles formed by the adjacent sides are equal.
A crescent is a curved shape having two narrow pointed ends. The shape of the crescent looks like a moon, which is less than a half.
An arrow is a combination of a line and an arrow mark at its tip. It is used to point to something or to show some directions.
The shape of a cross is made with two lines intersecting each other at their midpoint, forming an angle of 90 degrees.
A quadrilateral is a random four-sided polygon that has neither equal sides nor equal angles of the vertices.
A half-circle is called a semicircle whose angle is 180 degrees.
A ring is a circular band made of a long stripe uses for holding connecting and other purposes.
An oval is a shape made by a curve close to a plane that looks like an egg, a compressed circle.
tetrahedron’s shape looks like a triangular pyramid that does not contain any parallel faces or sides, or edges. All vertices are of the same distance from each other.
An octahedron is a polyhedron with a structure like two pyramids joined to each other from the base sides. It has six vertices, eight triangular faces, and 12 edges.
Square pyramid:
Any pyramid that contains a square base is called a square pyramid. If all the edges of this pyramid are equal, then it is also called an equilateral pyramid.
Hexagonal pyramid:
A hexagonal pyramid has a structure that looks like a pyramid that contains a hexagonal base, and the six triangular faces are isosceles triangles connected to the apex.
Triangular prism:
A triangular prism contains two equilateral triangles, both triangles are connected with three sides, and the edges are parallel to each other. It has three rectangular faces and two similar
triangular faces.
Rectangular prism:
The shape of a rectangular prism is the same as that of the cuboid, which has six rectangular faces, and all the opposite sides are the same, having six vertices and 12 edges.
Pentagonal prism:
A pentagonal prism is a solid pentagonal structure that contains two pentagons whose all vertices are perpendicularly connected to each other by edges having six rectangular faces and two pentagonal
2d shapes Name
1. Arrow
2. Circle
3. Crescent
4. Cross
5. Decagon
6. Heart
7. Heptagon
8. Hexagon
9. Kite
10. Nonagon
11. Octagon
12. Oval
13. Parallelogram
14. Pentagon
15. Quadrilateral
16. Rectangle
17. Rhombus
18. Right triangle
19. Ring
20. Semicircle
21. Square
22. Star
23. Trapezium
24. Trapezoid
25. Triangle
3d shapes Name
1. Cone
2. Cube
3. Cuboid
4. Cylinder
5. Diamond
6. Hemisphere
7. Hexagonal pyramid
8. Octahedron
9. Pentagonal prism
10. Prism
11. Pyramid
12. Rectangular prism
13. Sphere
14. Square pyramid
15. Tetrahedron
16. Triangular prism
Geometric Shapes names with pictures
Shapes names for kids
Read Also:
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How to demonstrate ?
Oct 03, 2024 09:21 PM
Oct 03, 2024 03:53 PM
demonstrate? What?
Why isn't
all that's needed to demonstrate this relationship between X and Y.
As an alternative you can use
According to simplify sqrt Y to X .... it gets boring to have to emphasize again and again that the symbolic capabilities of Mathcad 15 (and of Prime anyway) are far behind those of the top dogs.
And what would be the point of using some tricky combination of keywords to achieve the desired simplification?
BTW, Wolfram does a pretty good job IMHO
And you can be happy to be able to use Mathcad and not Prime!
Prime 10 refuses to solve the system for x and y and refuses to return a symbolic solution if just solving for x:
Oct 03, 2024 09:21 PM
Oct 04, 2024 12:51 AM
Oct 04, 2024 06:19 PM
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Data science interview questions
1. What is Data Science?
Data Science is an interdisciplinary field that utilizes scientific methods, algorithms, and systems to extract knowledge and insights from structured and unstructured data. It combines aspects of
statistics, mathematics, programming, and domain expertise to analyze and interpret complex data, enabling data-driven decision-making.
2. Explain the Data Science Process.
The data science process generally includes the following steps:
• Problem Definition: Clearly define the business problem or question.
• Data Collection: Gather relevant data from various sources.
• Data Cleaning: Preprocess the data to handle missing values, remove duplicates, and fix inconsistencies.
• Exploratory Data Analysis (EDA): Analyze the data to identify patterns, trends, and relationships.
• Model Building: Select and train machine learning models using the cleaned data.
• Model Evaluation: Assess model performance using metrics like accuracy, precision, recall, etc.
• Deployment: Implement the model in a production environment for real-world use.
• Monitoring and Maintenance: Continuously monitor model performance and update as necessary.
3. What Skills are Essential for a Data Scientist?
Essential skills for a data scientist include:
• Programming: Proficiency in languages such as Python or R.
• Statistics and Mathematics: Strong understanding of statistical methods.
• Machine Learning: Knowledge of algorithms and techniques.
• Data Wrangling: Skills in cleaning and manipulating data.
• Data Visualization: Ability to create meaningful visual representations.
• SQL: Proficiency in querying databases.
• Domain Knowledge: Understanding of the specific industry.
4. What is a P-Value?
A p-value is the probability of obtaining results at least as extreme as the observed results, assuming that the null hypothesis is true. A small p-value (typically ≤ 0.05) indicates strong
evidence against the null hypothesis.
5. Explain Type I and Type II Errors.
• Type I Error: Rejecting the null hypothesis when it is true (false positive).
• Type II Error: Not rejecting the null hypothesis when it is false (false negative).
6. What is a Confidence Interval?
A confidence interval is a range of values likely to contain the true population parameter with a specified level of confidence (e.g., 95%). It provides an estimate of uncertainty surrounding a
sample statistic.
7. Difference Between Supervised and Unsupervised Learning?
• Supervised Learning: Involves training a model on labeled data.
• Unsupervised Learning: Involves training a model on data without labeled outputs, identifying patterns within the data.
8. Explain the Bias-Variance Tradeoff.
The bias-variance tradeoff refers to the balance between two sources of error:
• Bias: Error due to oversimplified assumptions, leading to underfitting.
• Variance: Error due to sensitivity to fluctuations in training data, leading to overfitting. The goal is to minimize both.
9. What is Overfitting and How Can You Prevent It?
Overfitting occurs when a model learns the training data too well. To prevent it, use cross-validation, simplify the model, apply regularization techniques, use dropout, or gather more training data.
10. What is the Purpose of Cross-Validation?
Cross-validation assesses model performance by splitting the dataset into multiple subsets. This helps provide a reliable estimate of performance and avoid overfitting.
11. How Do You Deal with Outliers?
Outliers can be handled by:
• Identifying them using statistical methods.
• Removing them if they are errors.
• Transforming data to reduce their influence.
• Imputing them with appropriate values.
12. What is Feature Scaling and Why is it Important?
Feature scaling normalizes the range of independent variables. It’s important because many machine learning algorithms are sensitive to the scale of the data.
13. Explain One-Hot Encoding.
One-hot encoding converts categorical variables into a binary format, creating binary columns for each category to prevent any ordinal assumptions by the algorithm.
Common tools include:
• Matplotlib: For static and animated visualizations.
• Seaborn: For attractive statistical graphics.
• Tableau: For interactive dashboards.
• Power BI: For business analytics.
15. How to Choose the Right Visualization?
Consider the data type, insights needed, audience familiarity, and clarity to ensure the visualization effectively conveys the intended message.
16. Difference Between SQL and NoSQL Databases?
SQL databases are relational, using structured schemas and fixed tables. NoSQL databases are non-relational, allowing flexible schemas for unstructured data.
17. How Would You Join Two Tables in SQL?
You can join tables using different types of joins:
• INNER JOIN: Returns matching records in both tables.
• LEFT JOIN: Returns all records from the left table and matching records from the right.
• RIGHT JOIN: Returns all records from the right table and matching records from the left.
• FULL OUTER JOIN: Returns records with matches in either table.
18. What is a Primary Key and a Foreign Key?
• Primary Key: A unique identifier for each record in a table.
• Foreign Key: A field in one table that uniquely identifies a row of another table, establishing a relationship.
19. What is a Neural Network?
A neural network is a computational model inspired by the way biological neural networks work. It consists of layers of interconnected nodes (neurons) that process input data to generate output.
20. How Does Backpropagation Work?
Backpropagation is used to train neural networks by minimizing error. It involves:
• Forward Pass: Compute output based on current weights.
• Calculate Error: Measure the difference using a loss function.
• Backward Pass: Propagate the error backward to calculate gradients.
• Update Weights: Adjust weights using an optimization algorithm.
21. What are CNNs?
Convolutional Neural Networks (CNNs) are designed for grid-like data, such as images. They automatically learn spatial hierarchies of features and are effective for tasks like image classification
and object detection.
22. What is the Vanishing Gradient Problem?
The vanishing gradient problem occurs in deep networks when gradients become very small, making it difficult to update weights effectively during training.
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How to construct the tensor network for large classical off-lattice problem?
Consider a many body classical system, for simplicity let's assume it's some gas with pairwise interaction between particles. I would like to build a Tensor Network for the system. However the
particles become off-lattice in the moment the simulation starts (assuming each particle occupied a lattice site at $t=0$).
1. Is there a way to build an off-lattice Tensor Network representation of a system or it's only applied for a system which is constrained to a lattice?
2. What's the difference between Tensor Network and regular Graph Theory approach to represent systems?
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Distributed weighted stable marriage problem
The Stable Matching problem was introduced by Gale and Shapley in 1962. The input for the stable matching problem is a complete bipartite K[n,n] graph together with a ranking for each node. Its
output is a matching that does not contain a blocking pair, where a blocking pair is a pair of elements that are not matched together but rank each other higher than they rank their current mates. In
this work we study the Distributed Weighted Stable Matching problem. The input to the Weighted Stable Matching problem is a complete bipartite K [n,n] graph and a weight function W. The ranking of
each node is determined by W, i.e. node v prefers node u[1] over node u[2] if W((v,u[1])) > W((v, u[2])). Using this ranking we can solve the original Stable Matching problem. We consider two
different communication models: the billboard model and the full distributed model. In the billboard model, we assume that there is a public billboard and each participant can write one message on it
in each time step. In the distributed model, we assume that each node can send O(log n) bits on each edge of the K[n,n]. In the billboard model we prove a somewhat surprising tight bound: any
algorithm that solves the Stable Matching problem requires at least n - 1 rounds. We provide an algorithm that meets this bound. In the distributed communication model we provide an algorithm named
intermediation agencies algorithm, in short (IAA), that solves the Distributed Weighted Stable Marriage problem in O(√n) rounds. This is the first sub-linear distributed algorithm that solves some
subcase of the general Stable Marriage problem.
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 6058 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 17th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2010
Country/Territory Turkey
City Sirince
Period 7/06/10 → 11/06/10
• Billboard
• Distributed Algorithms
• Matching
• Scheduling
• Stable Marriage
ASJC Scopus subject areas
• Theoretical Computer Science
• General Computer Science
Dive into the research topics of 'Distributed weighted stable marriage problem'. Together they form a unique fingerprint.
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To equilibrate properly electronic relaxation rates depend on the density matrix!
In high dimensional systems very few exact results are known about quantum dynamics. One of the most important exact conditions we can try to satisfy is detailed balance, ie: dynamics should
equilibrate to the correct statistical distribution, a Fermi distribution for electrons, at long times.
Lots of people are familiar with Surface Hopping, Redfield theory, and Ehrenfest dynamics, but actually you can’t use any of these methods to produce a Fermi distribution exactly. Based on our work
simulating relaxation, we’ve actually been able to derive an equation of motion which does obey Fermi-Dirac statistics. In obtaining the derivation, we learned useful tricksĀ that are going to help
us treat mixed-states on the same footing as pure states. We’re super jazzed about these things.
There are cool experimental consequences, for example that non-radiative relaxation rates are not constant with time. You can check out the whole story on ArXiv for the time being:
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MA7155 Applied Probability and Statistics Question Paper
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Anna University Previous Year Question Papers Regulation 2013
MA7155 Applied Probability and Statistics
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Anna University M.E. Computer Science and Engineering I semester MA7155 Applied Probability and Statistics Question Papers Regulation 2013
MA7155 Applied Probability and Statistics Question Paper Anna University
MA7155 Applied Probability and Statistics semester I Question Paper Anna University
Anna University MA7155 M.E. Computer Science and Engineering Question Paper
Regulation 2013 Anna University MA7155 M.E. Computer Science and Engineering Question Paper
Download Regulation 2013 Anna University MA7155 M.E. Computer Science and Engineering Question Paper
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