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我不愿意把代码重复三份，分别为三种并行库编写程式，所以就用 autotools 管理项目，用不同的选项来启用不同的功能。这样做的另一个好处是很容易支持MPI+Pthreads或者MPI+OpenMP甚至是MPI+Pthreads+OpenMP，尽管这
根据牛顿万有引力公式和第二运动定律很容易导出直观的 \(O(n^2)\) 算法。使用 Barnes-Hut 近似模拟算法可以优化至 \(O(n \log n)\)。代码重用做得不是很好，为2d和3d情形分别实现了quad-tree和oct-tree。
According to Newton's law of universal gravitation, every body attracts every other body by a force pointing along the line intersecting both centers. The force is proportional to the product of the
two masses and inversely proportional to the square of the distance between them:
\[\begin{eqnarray} F = G \frac{M m}{r^2} \end{eqnarray}\]
By Newton's Second Law, the acceleration is expressed by \(F = m a\).
Combing the two laws, we can easily deduce an direct-sum algorithm to calculate the accelerations of these bodies, which would be \(O(n^2)\). However, the brute-force simulation has low data
dependency and is easy to be reformed to a parallelized algorithm.
Barnes-Hut simulation decreases the time complexity to \(O(n \log n)\) which exploting a quad-tree (in the 2-dimensional case) and approximate a distant cluster of bodies to a point. Each node in the
quad-tree represents a subspace. The root node represents the whole space and its four children represent the four quadrants of the space. The resultant force attracted by a distant cluster of bodies
is treated as being attracted by the center of mass.
To determine if the cluster is sufficiently far away, I compute the quotient \(s/d\), where \(s\) is the size of the subspace and \(d\) the distance between the body and the center of mass. If \(s/d
< \theta\), the cluster is considered distant.
这里是我实现得最复杂的部分。如果要计算出每个碰撞是非常耗时的，实践中不得不妥协使用“每个球只允许主动碰撞一个球，不处理其他的” 之类的办法。
单单考虑碰撞检测，这类问题通常会把球用 axis-aligned minimum bound-ing rectangle 包裹起来。相当一部分算法能处理这个问题，比如 R-tree、BSP tree、2-dimensional segment tree、spatial hashing、扫描线
+interval tree、扫描线+segment tree等。我在项目中实现了后两种以及 BSP tree。
Each body along with the imaginary position when it reaches the destination after the time slice, is enclosed in a axis-aligned minimum bounding rectangle. It is easy to decuce that if two body
collides in the time slice, then their bounding rectangles intersect. The property can be used to accelerate the performance of collision detecting.
Several existing approaches deal with the intersection detection problem straightforwardly.
• R-tree and its variants. R-trees are tree data structures used for spatial access methods and the most immediate solution to the problem.
• Binary space partitioning tree. They are developed in the context of computer graphics are are also suitable for the problem.
• 2-dimensional segment tree. First notice that two rectangles overlaps iff their projections on coordinate axes overlap. We need to construct a 2-dimensional segment tree with \(O(n \log^2 n)\)
space which supports \(O(n \log^2 n)\) query.
• Spatial hashing. Subdivide the plane into a set of voxels (a term used in image processing, the smallest distinguishable box-shaped part) and thus forms a mesh. Each voxel contains a bucket of
bodies. Put each body into the buckets corresponsing its occupied voxels. Before insertng into a voxel, each existing body in the cell is a candidate for potential collision.
Other approaches including using a sweep line algorithm to reduce the problem to the 1-dimensional case (overlapping interval search problem).
• Interval tree. It is a tree data structure holding intervals. Space complexity \(\Theta(n)\), construction time complexity \(O(n \log n)\), query time complexity \(O(\log n + k)\).
• Segment tree. Space/construction time complexity \(O(n \log n)\), query time complexity \(O(\log n + k)\).
Approaches exploiting the property of temporal coherence:
• TPR-tree and a few ramification. A variant of R-tree taking advantages of spatio-temporal updates.
I'll expatiate three data structures implemented in my project.
Binary space partitioning tree
The implementation is in the file =src/BSPTree.cc=.
The construction process is rather simple and straightforward. I start with the whole scene and a list of bounding boxes, recursively divide the scene into two until the number of bodies in the
subspace is less than a threshold (=NODE_SIZE_THRESHOLD=) or there is no balanced partition scheme. Each separating line is parallel to X, Y or Z axis.
For a range query, we examine whether the cube in question intersects with the left-subspace. If so, recursively do a query in the left-subspace. The same process also applies to the right-subspace.
Sweep line algorithm + Interval tree
The implementation is in the file =src/IntervalTree.cc=.
Well, in this project I also use a much easier but also efficient approach combining a sweep line algorithm and an interval tree besides a BSP tree implementation.
• Each bounding rectangle has an upper and a lower edges. They mark the timestamp between which the rectangle is valid.
• Sort these edges by their y coordinates and process these edges in ascending order. For an upper edge, search all the valid rectangles and find those x-axis spans (intervals) which overlaps the
span of the current rectangle, and then insert the x-axis span into a data structure. For these overlapping intervals, check whether the corresponding rectangle is marked valid and if so, do a
precise detection. For a lower edge, delete the x-axis span from the data structure.
The unamed data structure should support following operations efficiently: insert an interval, query all intervals overlapping given interval (overlapping interval search problem).
The query can be fulfilled by an interval tree. Construct an interval tree with all intervals but do not really store these intervals. This makes the constructed tree balanced.
Suppose \(n\) is the number of bodies. The interval tree can be constructed in \(\Theta(n \log n)\) time while each insertion runs in \(O(\log n)\) time and each query runs in \(O(\log n + k)\) time
where \(k\) is the number of overlapping intervals.
Sweep line algorithm + Segment tree
Ditto on the reduction part. Create a sorted list of unique X coordinates extracted from those bounding boxes: \(p_0, p_1, p_2, \ldots, p_{m}\). These coordinates form \(m\) segments: \([p_0,p_1),
[p_1,p_2), \ldots, [p_{m-1},p_m)\). Each leaf node represents a segment while each internal node represents several segments.
The most accurate approach is to deduce the nearest collision event, simulate the event, and then find another. However, this is incapable of handling hundreds of bodies, especially when there are
some high-speed and light bodies. They will frequently change directions and affect other bodies.
A comparatively coarse one is to calculate \(n(n-1)/2\) pair-wise collision events and simulate them one after another. It is coarse in the sense that it may not handle multiple collisions taking
place on one single body correctly.
The two mentioned ways are neither efficient nor parallelizable. One simple mend making the latter parallelizable is to update velocities after all collision events are processed, but incurs the
inaccuracy that the updated velocity of the other body in a collision won't be reflected immediately.
I used a non-parallelizable, asymptotically optimal sequential approach that is a bit accurate then the above one.
By the way, to make the calculation of the collision point more accurate, I'll use a techinique stated below:
Suppose \(\vec{p_0}, \vec{p_0}\) are positions of the two bodies and \(\vec{\Delta p}, \vec{\Delta q}\) their translational vectors. Then we can parametrize their positions:
\[\begin{eqnarray} \vec{p}(t) = \vec{p_0} + t \vec{\Delta p} \\ \vec{q}(t) = \vec{q_0} + t \vec{\Delta q} \end{eqnarray}\]
We will try to solve the equation \(|\vec{p}(t)-\vec{q}(t)|=R+r\) where \(R,r\) are radii of the two bodies. This is a quadratic equation and we should care whether the smaller root \(t_0\) satisfies
\(0 \le t_0 \le 1\) or the two bodies overlap. If so, teleports the two bodies to their collision moment which is \(now + t_0 \Delta t\) and carry out an elastic collision.
Since collision only imparts force along the line of collision, only the velocities that are along the line of collision should be changed, thus degrades to 1-dimensional case.
原始码包中 scripts/a.py 是用 pyopengl 编写的简单播放器。可以用
src/n-body -f text -d 3 -n 20 --fps 20 -t 2 | python scripts/a.py | {"url":"https://maskray.me/blog/2012-09-10-parallel-n-body","timestamp":"2024-11-10T08:14:55Z","content_type":"text/html","content_length":"33906","record_id":"<urn:uuid:7f16d8a4-7e5a-4c30-9f2c-9b5abb13289d>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00617.warc.gz"} |
Does voltage divide in parallel?
Does voltage divide in parallel?
In a parallel circuit, the voltage drops across each of the branches is the same as the voltage gain in the battery. Thus, the voltage drop is the same across each of these resistors.
Is voltage division for series or parallel?
A voltage divider is a simple series resistor circuit. It’s output voltage is a fixed fraction of its input voltage. The divide-down ratio is determined by two resistors.
What is the formula for voltage in parallel?
In a parallel circuit, each load resistor acts as an independent branch circuit, and because of this, each branch “sees” the entire voltage of the supply. Total voltage of a parallel circuit has the
same value as the voltage across each branch. This relationship can be expressed as: ET = E1 = E2 = E3…
Does voltage change in a parallel circuit?
Once the charges get out of the resistors, the electric field of the battery is enough to drive them mad (as the wire has relatively lower resistance). And, the charges get back their energy once
again. This is the reason why we say voltage is the same in parallel circuits3.
Is voltage constant in parallel?
Key Points Each resistor in parallel has the same voltage of the source applied to it (voltage is constant in a parallel circuit).
What is the voltage division rule?
The voltage across any resistor in a series connection of resistors shall be equal to the ratio of the value of the resistor divided by the equivalent resistance of the circuit. This is called
Voltage division rule.
How do you find voltage in series and parallel?
To determine the voltage drop across the parallel branches, the voltage drop across the two series-connected resistors (R1 and R4) must first be determined. The Ohm’s law equation (ΔV = I • R) can be
used to determine the voltage drop across each resistor.
Why voltage is the same in parallel circuits?
The first principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrically common points in a
parallel circuit, and the voltage measured between sets of common points must always be the same at any given time.
Why is voltage same in parallel?
How do you calculate the voltage in a parallel circuit?
– I=5R+RLED. – 2=IR=5RR+RLED. – VR=VRR+RLED.
Why is a voltage always connected in parallel?
Voltage is the same in parallel. If two components are connected to the same two points, they will have the same potential across them. When we connect a meter to a circuit we want the meter to
affect the circuit conditions as little as possible.
What is the total voltage in parallel circuits?
The total voltage in a parallel circuit is that of the supply, as is the current. The current is divided in each half of a parallel circuit according to each half’s resistance. Zo – infinitely very
low for an ideal supply can be equated to zero ohms.. Sponsored by dagnedover. Meet our Collections. Bag, Crossbody, Organizers.
Does Voltage split in a parallel circuit?
Why does voltage split in a parallel circuit? As each electron has the same charge, each electron is carrying the same amount of energy, so the voltage across each branch of the parallel circuit will
be the same because the voltage doesn’t epend on the number of electrons in each branch. Is voltage the same in series? | {"url":"https://corfire.com/does-voltage-divide-in-parallel/","timestamp":"2024-11-09T06:15:26Z","content_type":"text/html","content_length":"38757","record_id":"<urn:uuid:236a74c9-a7a2-4a03-a536-39bbbb95ba48>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00517.warc.gz"} |
Data Science, Machine Learning and Predictive Analytics
k-means clustering
is a useful unsupervised learning data mining tool for assigning
observations into
groups which allows a practitioner to segment a dataset.
I play in a fantasy baseball league and using five offensive variables (R, AVG, HR, RBI, SB) I am going to use k-means clustering to:
1) Determine how many coherent groups there are in major league baseball. For example,
is there a power and high average group? Is there a low power, high average, and speed group?
2) Assign players to these groups to determine which players are similar or can act as replacements. I am not using this algorithm to predict how players will perform in 2017.
For a data source I am going to use all MLB offensive players in 2016 which had at least 400 plate appearances from
This dataset has n= 256 players.
Sample data below
Step 1
How many k groups should I use?
The within groups sum of squares plot below suggests k=7 groups is ideal. k=9 is too many groups for n=256 and the silhouette plot for k=9 is poor.
Step 2
Is k=7 groups a good solution?
Let's look at a silhouette plot to look at the fit of each cluster and the overall k=7 clusters.
The average silhouette width = .64 indicates a reasonable structure has been found. Cluster 4 which is the speed group has a low silhouette width of .37. I am OK with this as it is the smallest group
and speed is the hardest offensive tool to find in MLB.
Step 3
Calculate group means for k=7 groups
Players that are classified in cluster 3 are the elite players in MLB. Based on 2016 stats, 31 players make up cluster 3. On average they have the highest AVG, R, RBI, HR, and the second highest SB. | {"url":"https://blog.alpha-analysis.com/2017/06/","timestamp":"2024-11-11T19:14:47Z","content_type":"application/xhtml+xml","content_length":"59841","record_id":"<urn:uuid:8601351e-5d99-426c-b365-e8f76c4c9327>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00063.warc.gz"} |
Floer Homology, Gauge Theory, and Low Dimensional Topology - Clay Mathematics Institute
Floer Homology, Gauge Theory, and Low Dimensional Topology
Mathematical gauge theory studies connections on principal bundles, or, more precisely, the solution spaces of certain partial differential equations for such connections. Historically, these
equations have come from mathematical physics, and play an important role in the description of the electo-weak and strong nuclear forces. Gauge theory as a tool for studying topological properties
of four-manifolds was pioneered by the fundamental work of Simon Donaldson in the early 1980’s, and was revolutionized by the introduction of the Seiberg—Witten equations in the mid-1990’s. Since the
birth of the subject, it has retained its close connection with symplectic topology. The analogy between these two fields of study was further underscored by Andreas Floer’s construction of an
infinite—dimensional variant of Morse theory which applies in two a priori different contexts: either to define symplectic invariants for pairs of Lagrangian submanifolds of a symplectic manifold, or
to define topological invariants for three—manifolds, which fit into a framework for calculating invariants for smooth four-manifolds. ”Heegaard Floer homology”, the recently—discovered invariant for
three— and four—manifolds, comes from an application of Lagrangian Floer homology to spaces associated to Heegaard diagrams. Although this theory is conjecturally isomorphic to Seiberg—Witten theory,
it is more topological and combinatorial in its flavor and thus easier to work with in certain contexts. The interaction between gauge theory, low—dimensional topology, and symplectic geometry has
led to a number of striking new developments in these fields. The aim of this volume is to introduce graduate students and researchers in other fields to some of these exciting developments, with a
special emphasis on the very fruitful interplay between disciplines.
This volume is based on lecture courses and advanced seminars given at the 2004 Clay Mathematics Institute at the Alfréd Rényi Institute of Mathematics in Budapest, Hungary. Several of the authors
have added a considerable amount of additional material to that presented at the school, and the resulting volume provides a state-of-the-art introduction to current research, covering material from
Heegaard Floer homology, contact geometry, smooth four—manifold topology, and symplectic four—manifolds.
Authors: Denis Auroux, Tobias Ekholm, John Etnyre, Ronald Fintushel, Hiroshi Goda, Tian-Jun Li, Paolo Lisca, Peter Ozsváth, Jongil Park, Ivan Smith, Ronald Stern, András Stipsicz, Zoltán Szabó
Available at the AMS bookstore
Editors: David Ellwood, Peter Ozsváth, András Stipsicz, Zoltán Szabó
Editors: David Ellwood, Peter Ozsváth, András Stipsicz, Zoltán Szabó
Authors: Test author | {"url":"https://www.claymath.org/resource/floer-homology-gauge-theory-and-low-dimensional-topology/","timestamp":"2024-11-07T00:53:33Z","content_type":"text/html","content_length":"88529","record_id":"<urn:uuid:c5a2ba40-01a3-4e86-8277-1514d4384917>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00205.warc.gz"} |
Real Gas Law
The real gas law is a simplification of the generalized equation of state (EOS) Z-factor definition. The simplification comes from the model only describing non-ideal gases and not multi-phase fluid
systems. The real gas law takes a similar form as the ideal gas law and is defined by
where \(v\) is the molar volume (\(v=V/n\)) and \(Z\) is the simplified Z-factor which, in the case of the real gas law, can be described as the gas deviation factor.
Historical Overview
Early attempts to generalize the gas correction factor was made by Standing and Katz in 1944^1 by measurements of various petroleum gases and defining the now famous Stanging-Katz charts. By
introducing pseudo-reduced variables for the pressure and temperature, they were able to generalize the behavior for a wide range of non-ideal pressures and temperatures. With increasing computer
power becoming available, function based descriptions of the Standing-Katz charts became more applicable like the original and modified Benedict-Webb-Rubin (BWR) EOS models^2. Other well known
descriptions of the gas deviation factor like the Hall-Yarborough model^3 are also applied to describe real gases. Modern descriptions of petroleum fluids are typically calculated using cubic EOS
models like the Peng-Robinson or Soave-Redlich-Kwong EOS models.
Standing-Katz Z-Factor Charts
Figure 1: Standing-Katz chart.
The Standing-Katz chart is a best fit model for the Z-factor of various petroleum gases as a function of the specific gas reduced properties. The approach for finding the gas deviation factor is
described by:
1. Estimate the pseudo-reduced pressure and temperature by correlation (if no composition is given) or by Kay's mixing rule (if compositions are given).
2. Locate the pseud-reduced pressure on the x-axis and then move along the y-axis until you cross the correct pseudo-reduced temperature iso-line.
3. Move along the x-axis relative to the point found in step 2 to the left (if the pseudo-pressure is less than 7) or right (if the pseudo-pressure is greater than 7) to find the estimate of the
Hall-Yarborough Estimation of Gas Z-factor
The Hall-Yarborough approach to estimating the gas Z-factor is an iterative approach using the pseudo reduced properties of the gas (\(p_{pr}=p/p_{pc}\) and \(T_{pr}=T/T_{pc}\)) and the
Carnahan-Standing hard-sphere EOS given by
The reduced density parameter (\(\hat{\rho}\)) is defend by solving
and the derivative is given by
The procedure for the Hall-Yarborough Z-factor is given below and more details about the procedure can be found in the SPE Monograph Phase Behavior^4:
1. Guess reduced-density parameter (\(\hat{\rho}\))
2. Calculate objective function (\(f(\hat{\rho})\)) and its derivative (\(f'(\hat{\rho})\))
3. Check if objective function is less than the tolerance (\(f(\hat{\rho})<\epsilon\))
4. If the objective function is less than the threshold then the reduced-density parameter is correct and the user should move on to step 5. If the objective function is not less than the threshold
then the reduced density parameter must be updated by Newton's method and the used must return to step 2.
5. With the converged reduced-density parameter, calculate the Z-factor from the Carnahan-Standing had-spere EOS equation \eqref{eq:CS_eos}.
Estimates for Pseudo-Critical Properties
All the correlation below are using absolute temperatures in Rankine (\(^\circ\)R) and absolute pressures in psia.
There are several methods for estimating the pseudo-critical properties of a gas. The first approach is by correlation. This approach is typically used if there is no available composition or
component properties (\(p_c\), \(T_c\) or \(MW\)). Below there are two correlations given for estimating the pseudo-critical properties. The second approach to estimating the pseudo-critical
properties is by applying Kay's mixing rule.
Sutton Correlation for Pseudo-Critical Properties
whitson comment
From Curtis Hays Whitson: We have found that using the Sutton correlation yields better results than Kay's mixing rule.
where \(\gamma_g\) is the gas specific gravity (\(\gamma=\rho/\rho_{ref}\)).^5
Standing Correlation for Pseudo-Critical Properties
For dry gases where \(\gamma_g < 0.75\) the pseudo-critical properties can be estimated from
For wet gases where \(\gamma_g >= 0.75\) the pseudo-critical properties can be estimated from
where \(\gamma_g\) is the gas specific gravity (\(\gamma=\rho/\rho_{ref}\)).^6
Kay's Mixing Rule for Pseudo-Critical Properties
See article on PVT propertis for more details and example values for the molecular weights and critical properties.
Given a mixture \(y_i\) containing a gas composition where the molecular weight and critical properties are known for each component. With this information the pseudo-critical properties can be
calculated from | {"url":"https://wiki.whitson.com/eos/eos_models/rgl_eos/index.html","timestamp":"2024-11-07T03:08:50Z","content_type":"text/html","content_length":"48211","record_id":"<urn:uuid:570c0808-21cd-419c-ad2d-e4317f29271f>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00758.warc.gz"} |
How to calculate loan interests - Sold with Dave Real Estate
Lenders make money when you take out a loan by charging interest. Or to put it another way, interest is the cost of borrowing money from a lender. While some lenders impose a flat rate, others base
their interest rates on an amortization schedule, which applies a higher rate of interest in the beginning of the loan.
In addition to the type of interest charged, other elements such as your credit score, the size of the loan, and the length of the repayment term will also have an impact on the amount you’ll pay.
SimpleĀ interest
If you have the necessary information, you can easily calculate loan interest if a lender employs the simple interest method. To calculate the total cost of interest, you will need the principal loan
amount, the interest rate, and the length of the loan.
Although the monthly payment is set, the interest you’ll pay each month will depend on the amount of remaining principal. So long as the lender doesn’t impose a prepayment penalty, paying off the
loan early could result in significant interest savings.
Simple interest calculation methods
By using the following formula, you can determine your total interest:
Interest is calculated as follows: principal loan amount x interest rate x term
The simple interest formula is $20,000 x.05 x 5 = $5,000 in interest, for instance, if you take out a $20,000 loan with a five-year term and a 5% interest rate.
Who benefits fromĀ simple interest?
Simple interest rewards borrowers who make on-time or early payments. Compared to loans with compound interest, borrowers can save money with these loans because interest is only based on the loan
Types of loansĀ that use simple interest
Simple interest is less frequent, but you may come across it on short-term loans, some personal loans, and even some auto loans. Simple interest is also charged on some mortgages.
Simple interest may also be paid by borrowers of student loans. For instance, simple interest is charged on all federal student loans.
Amortizing interest
Numerous lenders base their interest rates on an amortization plan. This includes some auto loans and mortgages. These loans also have fixed monthly payments; the loan is repaid over time in equal
installments. But the way the lender calculates interest varies over time.
However, the primary distinction between amortizing loans and simple interest loans is that initial payments for amortizing loans are frequently heavily weighted toward interest. The principal loan
amount receives a smaller portion of your monthly payment as a result.
The situation changes, though, as time goes on and your loan payoff date approaches. The lender allocates a greater portion of your monthly payments toward principal near the end of your loan and a
lesser amount toward interest charges.
How to compute interest amortization?
How to figure out the interest on an amortized loan is as follows:
Subtract the number of payments you’ll be making that year from the interest rate. You would divide 0.06 by 12 to get 0.005 if you had a monthly payment schedule and an interest rate of 6%.
To calculate the amount of interest you will be required to pay that month, multiply that amount by the outstanding loan balance. The initial interest payment for a loan with a $5,000 balance would
be $25.
To calculate the amount of principal you will pay in the first month, deduct the interest from your fixed monthly payment. If your lender informed you that your fixed monthly payment would be
$430.33, the first month you would make a payment of $405.33 toward the principal. Your outstanding balance is reduced by that sum.
For the following month, repeat the procedure with your new outstanding loan balance, and so on for each succeeding month.
Who benefits from amortized interest?
The main recipients of amortized interest are lenders. Payments are applied to both principal and interest, lengthening the loan term and raising the total amount of interest paid.
Types of loans that use amortized interest
Auto loans, mortgages, and debt consolidation loans are just a few examples of the many installment loan types that use amortized interest. On home equity loans, amortized interest is another
Factors that may have an impact on interest rates
How much interest you pay for financing can depend on a variety of factors. These are some of the main factors that can affect how much you will pay over the course of the loan.
Loan Amount
The amount of interest you pay to a lender is significantly influenced by the amount you borrow (your principal loan amount). You will pay more interest for borrowing more money because the lender is
taking on more risk.
According to an amortized schedule, you would pay $2,645.48 in interest if you borrowed $20,000 over five years at a 5 percent interest rate. If you increase your loan amount to $30,000 while keeping
all other loan elements (such as the rate, term, and interest type) the same, your interest payment over the course of five years will rise to $3,968.22.
Takeaway: Don’t take out more debt than is necessary. Calculate your exact financial needs first by crunching the numbers.
Your credit rating.
Your credit score is very important in determining the interest rate on your loan. You will typically receive a higher interest rate if your credit is less than ideal because lenders will view you as
a greater risk than someone with excellent credit.
Let’s contrast a 5 percent loan with a 7 percent loan using the previous example ($20,000, five-year term, amortized interest) as a foundation. The total interest expense for the loan at 5% is
$2,645.48. The cost of interest rises to $3,761.44 if the interest rate is raised to 7%.
Takeaway: Prior to borrowing money, it might make sense to raise your credit score as this might increase your chances of getting a better interest rate and paying less for the loan.
Loan term
The length of time a lender agrees to spread out your payments is called a loan term. Thus, if you are approved for a five-year auto loan, the length of your loan is 60 months. On the other hand, the
loan terms for mortgages are typically 15 or 30 years.
The number of months it takes you to pay back the money you borrow can have a big impact on your interest expenses.
Higher monthly payments are typically necessary for loans with shorter terms, but because you cut down on the repayment period, you’ll pay less interest overall. Longer loan terms may result in lower
monthly payments, but because you’re delaying repayment, the total amount of interest paid will increase over time.
Takeaway: Make sure to review the numbers beforehand and determine how much of a monthly payment you can afford. Choose a loan term that works with your spending plan and overall debt load.
Schedule for repayment
Another aspect to take into account when figuring out the interest on a loan is how frequently you pay your lender. Though weekly or biweekly payments are sometimes required, particularly in business
lending. There is a chance that you could save money if you decide to make payments more frequently than once per month.
Making payments more frequently can help you pay down the loan’s principal faster. Making extra payments could help you save a lot of money in many circumstances, such as when a lender assesses
compound interest. Make certain that the principal is reduced with the payments, though.
Conclusion: Don’t assume you can only pay one loan payment per month. Making payments more frequently than necessary is a good idea if you want to lower the overall interest rate you pay for
borrowing money.
Payback amount
The monthly payment you must make toward your loan is known as the repayment amount.
Paying more than the minimum amount due each month can result in savings, much like making loan payments more frequently can help you save money on interest.
Takeaway: If you’re thinking about increasing your monthly loan payment, find out from the lender whether the extra cash will go toward the principal. If so, using this approach can help you pay off
debt faster and pay less interest.
How to find the most affordable loan interest rates
There are a few ways you might be able to increase your chances of getting the best interest rate on a loan:
Boost your credit rating: Those with the best credit scores typically have access to the most affordable interest rates.
Choose a quicker repayment schedule: The loans with the shortest terms will always have the lowest interest rates. If you are able to make the payments, you will eventually pay less interest.
Lower the debt-to-income ratio you have: Your debt-to-income (DTI) ratio measures how much of your gross monthly income you pay toward debt each month. It is almost as important as your credit score
when it comes to getting a good loan.
The Bottom line
It’s critical to estimate your interest costs before applying for a loan in order to comprehend the full cost of borrowing. Ask the lender whether interest is calculated using an amortization
schedule or the simple interest formula, and then run the numbers using the appropriate formula or an online calculator.
Also, keep in mind the elements that will influence the amount of interest you pay. To keep more of your hard-earned money in your pocket, it might be advantageous to borrow less money or cut the
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Contact us so that we can help you out strategize your loans. | {"url":"https://www.soldwithdave.com/2023/05/31/how-to-calculate-loan-interests/","timestamp":"2024-11-08T18:52:08Z","content_type":"text/html","content_length":"101977","record_id":"<urn:uuid:91e75dbf-6652-43c2-b208-26ba20759352>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00223.warc.gz"} |
SEMinR Package: PLS EstimationSEMinR Package: PLS Estimation
PLS Model Estimation in SEMinR
PLS Model Estimation in SEMinR
This tutorial on the SEMinR series will introduce how to estimate PLS moxel.
There are four steps to specify and estimate a structural equation model using SEMinR:
• Estimating, bootstrapping, and summarizing the model
Step 4: Estimating the Model
Step 4 in creating a model
• After having specified the measurement and structural models, the next step is the model estimation using the PLS-SEM algorithm.
• For this task i-e estimation, the algorithm helps in determing the scores of the constructs that are later used as input for (single and multiple) regression models within the path model.
• After the algorithm has calculated the construct scores, the scores are used to estimate each regression model in the path model.
• As a result, we obtain the estimates for all relationships in the measurement models (i.e., the indicator weights/loadings) and the structural model (i.e., the path coefficients).
• To estimate a PLS path model, algorithmic options and argument settings must be selected. The algorithmic options and argument settings include selecting the structural model path weighting
scheme. SEMinR allows the user to apply two structural model weighting schemes:
□ The factor weighting scheme and
□ The path weighting scheme.
• While the results differ little across the alternative weighting schemes, path weighting is the most popular and recommended approach.
• This weighting scheme provides the highest R-Sq value for endogenous latent variables and is generally applicable for all kinds of PLS path model specifications and estimations.
• SEMinR uses the estimate_pls() function to estimate the PLS-SEM model.
• This function applies the arguments shown in . Table. Please note that arguments with default values do not need to be specified but will revert to the default value when not specified.
• We now estimate the PLS-SEM model by using the estimate_pls() function with arguments
data = datas,
measurement_model = simple_mm,
structural_model = simple_sm,
inner_weights = path_weighting,
missing = mean_replacement, and
missing_value = “-99”
and assign the output to simple_model.
• It is like running PLS Algorithm in SmartPLS
# Estimate the model
simple_model <- estimate_pls(data = datas,
measurement_model = simple_mm,
structural_model = simple_sm,
inner_weights = path_weighting,
missing= mean_replacement,
missing_value = "-99")
Note that the arguments for inner_weights, missing, and missing_value can be omitted if the default arguments are used. This is equivalent to the previous code block:
# Estimate the model with Omissions
simple_model <- estimate_pls(data = datas,
measurement_model = simple_mm,
structural_model = simple_sm)
Following is a brief review of the steps that have been discussed in SEMinR tutorials.
• Load the Library – library ()
• Load the Data – read.csv
• Review the Data – head()
• Specify the Measurement Model – constructs()
• Specify the Structural Model – relationships()
• Estimate the Model – estimate_pls()
• Summarize the Results – summary()
#Loading the Library
# Load the Data
datas <- read.csv(file = "Data.csv", header = TRUE, sep = ",")
#To Inspect Data
#Create measurement model
simple_mm <- constructs(
composite("Vision", multi_items("VIS", 1:4)),
composite("Development", multi_items("DEV", 1:7)),
composite("Rewards", multi_items("RW",1:4)),
composite("Collaborative Culture", multi_items("CC", 1:6)))
# Create structural model
simple_sm <- relationships(
paths(from = c("Vision", "Development", "Rewards"), to = "Collaborative Culture"))
# Estimate the model
simple_model <- estimate_pls(data = datas,
measurement_model = simple_mm, structural_model = simple_sm,
inner_weights = path_weighting,
missing = mean_replacement,
missing_value = "-99")
# Summarize the model results
summary_simple <- summary(simple_model)
#Inspect the Summary Report
# Inspect the model’s path coefficients and the R^2 values
# Inspect the construct reliability metrics
Hair Jr, J. F., Hult, G. T. M., Ringle, C. M., Sarstedt, M., Danks, N. P., & Ray, S. (2021). Partial Least Squares Structural Equation Modeling (PLS-SEM) Using R: A Workbook.
The tutorials on SEMinR are based on the mentioned book. The book is open source and available for download under this link. | {"url":"https://researchwithfawad.com/index.php/lp-courses/seminr-lecture-series/seminr-package-pls-estimation/","timestamp":"2024-11-05T04:44:36Z","content_type":"text/html","content_length":"663016","record_id":"<urn:uuid:1823581c-9561-4682-835a-92b208bf640b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00677.warc.gz"} |
Solved Example Problems for Acceleration Due to Gravity of the Earth
EXAMPLE 6.7
1. Calculate the value of g in the following two cases:
a) If a mango of mass ½ kg falls from a tree from a height of 15 meters, what is the acceleration due to gravity when it begins to fall?
b) Consider a satellite orbiting the Earth in a circular orbit of radius 1600 km above the surface of the Earth. What is the acceleration experienced by the satellite due to Earth’s gravitational
The above two examples show that the acceleration due to gravity is a constant near the surface of the Earth.
EXAMPLE 6.8
Find out the value of g′ in your school laboratory?
Calculate the latitude of the city or village where the school is located. The information is available in Google search. For example, the latitude of Chennai is approximately 13 degree.
g ′ = g −ω 2 R cos2 λ
Here ω2R = (2x3.14/86400)2 x (6400x103) = 3.4x10−2 m s−2.
It is to be noted that the value of λ should be in radian and not in degree. 13 degree is equivalent to 0.2268 rad.
g′ = 9.8 − ( 3.4 × 10−2 ) × ( cos 0.2268)2
g = 9.7677 m s−2
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
11th Physics : UNIT 6 : Gravitation : Solved Example Problems for Acceleration Due to Gravity of the Earth | | {"url":"https://www.brainkart.com/article/Solved-Example-Problems-for-Acceleration-Due-to-Gravity-of-the-Earth_36155/","timestamp":"2024-11-02T09:19:05Z","content_type":"text/html","content_length":"41194","record_id":"<urn:uuid:b6e33567-ff3c-422f-9b2f-27da995f42f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00829.warc.gz"} |
Where can I find experts to do my Antenna Theory assignment? | Pay Someone To Do Electrical Engineering Assignment
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Is It Legal To Do Someone Else’s Homework?
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What Does Regression Mean in Machine Learning? - reason.townWhat Does Regression Mean in Machine Learning?
What Does Regression Mean in Machine Learning?
In machine learning, regression is a technique used to predict a value based on historical data. For instance, you might use regression to predict the price of a house based on its size, age, and
Checkout this video:
What is regression?
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Machine learning is a subset of artificial intelligence that gives computers the ability to learn without being explicitly programmed. This means that instead of writing code to solve a specific
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Regression is a statistical technique that allows you to predict the value of a dependent variable, based on the values of one or more independent variables. In machine learning, regression can be
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What are some resources I can use to learn more about regression and machine learning?
There are plenty of great resources out there that can help you learn more about regression and machine learning. Here are a few of our favorites:
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What are some examples of regression in machine learning?
In machine learning, regression is a method used to predict continuous values, such as prices or weight. It is a supervised learning algorithm, which means that it uses training data to learn the
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Regression analysis can be used for predictive modeling, trend analysis, and time series forecasting. It is a powerful tool that can be used to understand complex relationships between variables.
In machine learning, regression is a technique used to predict continuous values. This can be done either by fitting a model to data (which we call supervised learning), or by analyzing the
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finding trends in data. | {"url":"https://reason.town/what-does-regression-mean-in-machine-learning/","timestamp":"2024-11-11T17:29:53Z","content_type":"text/html","content_length":"99959","record_id":"<urn:uuid:020f917d-e4cc-4cc2-8cc1-5d72cc956e37>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00037.warc.gz"} |
(III) Suppose the current in the coaxial cable of Problem 34, Fig. 28–45, is not uniformly distributed, but instead the current density j varies linearly with distance from the center: j₁ = C₁𝑅 for
the inner conductor and j₂ = C₂𝑅 for the outer conductor. Each conductor still carries the same total current I₀ , in opposite directions. Determine the magnetic field in terms of I₀ in the same four
regions of space as in Problem 34. | {"url":"https://www.pearson.com/channels/physics/explore/sources-of-magnetic-field/ampere-law-with-calculus?chapterId=0214657b","timestamp":"2024-11-14T19:04:05Z","content_type":"text/html","content_length":"410263","record_id":"<urn:uuid:b33816ab-ef83-432c-a828-f1326568d727>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00729.warc.gz"} |
Category: Vertex Form
Let's be real for a second--alternative assessments are really hard in secondary math. I have like 87 objectives and you want my kids to make a poster? For real? I've tried doing them a couple of
times over the years, but they were usually such bad initial experiences I've stuffed all memories of them deep into my mental closet. This project is the exception to the
burn-the-lesson-plan-to-get-rid-of-all-evidence I've come to expect from my attempts at alternative assessments.
This is my 3rd year doing this project. Apparently, the 3rd time is the charm. I posted about my experience last year
. You can smell the fear in that post.
At this point in quadratics, we've covered: graphing, factoring, the quadratic equation, vertex form, and transformations. What I really want is for students to see the connections between multiple
representations. I want students to see that you can find the roots from the graph in Desmos. Those roots are the same as the roots in the factored form of the equation. Those roots are also the
solutions to the quadratic formula. Slowly, we're making progress.
Here's the rough outline of the project:
1) Find a "real-world" example which you can fit a parabola to. If you want to fit a parabola to Payton Manning's forehead, or Donald Trump's smile, or some other hilarious thing, that's fine with
me--you'll just lose the "real-world" point.
2) Fit a parabola to the image using vertex form.
3) Show all points of interest on the Desmos graph (roots, y-intercept, vertex, axis of symmetry)
4) Demonstrate you know how to calculate the a-value for the vertex form of the equation using the vertex and a point.
5) Show me you know how to find the factored form of the equation from your Desmos graph.
6) From either vertex or factored form, calculate standard form of the equation.
7) From the standard form, show me you know how to calculate the vertex of the parabola. It should be the same vertex as your graph.
8) From the standard form, use the quadratic formula to calculate the roots of the equation. These should be the same roots as your graph.
9) Put your name on the back of the poster so I can take pictures of your work and put them online without worrying about student identities (that's one of the reasons at least).
Let's talk pacing for a moment. I give two work days in class for this project. On the first day, we intro the project, work on finding a super cool picture, and fit a curve to it (#1-3 above). On
the second day, we work on doing the math and I hand out poster board to students. I have the project due several days after our second work day. There is no time to work on the project on the due
date. Students waltz into class, I collect the posters, and we delve into radical functions.
Next time, I will strongly suggest to students that it's worth it to do the work before starting on the poster board. I made a template for next year (below this paragraph). I'll probably import it
into a Google Doc to encourage students to type out their work. Typed posters are usually much nicer to look at than the scrawl of most of my students.
If you're interested, here's my files for the project. The directions on the left are what I hand out to students. The checklist, which is included in the directions, gets paper clipped to the front
of the poster so I can evaluate student understanding without writing all over their beautiful projects.
Project Directions Project Checklist
And now for the gallery. I'm going to include the good, the bad, and the ugly so you get a realistic picture of what to expect. I make it very clear to students that it is possible to get full credit
without making these display worthy. I'm most interested in the math. However, I do offer bonus points for making things pretty.
First, a couple of pretty ones. The math isn't perfect, but I'm not going to highlight the mistakes for you.
Here are some more student examples to give you an idea of the range of quality to expect. Not everything is super pretty all of the time.
2 Comments | {"url":"https://www.andrewbusch.us/home/category/vertex-form","timestamp":"2024-11-02T01:14:07Z","content_type":"text/html","content_length":"62751","record_id":"<urn:uuid:c19b06fd-e913-41f8-9669-f9b53e2397fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00332.warc.gz"} |
Consider the OpAmp circuit.
Assume operation in the linear range.
(1) Consider the OpAmp circuit. Assume operation in the linear range. Let Rout = 0 Ω and Rin = 00 Ω. a) Determine the Thevenin equivalent for the input circuit with Vsa and Vsb, i.e. values for VTH
and RTH. (4 pts.) + Vsa Re Re Av Vo Rb Ra b) Determine the relationship vo in terms of vsa and Vsb. Solve with a finite A and then let A go to infinity. (Hint: Note that V₁ = VTH. Solve for the input
and output in terms of vi and then eliminate Vi.) (4 pts.) c) Calculate values of Ra and Rb for which vo = 4(Vsa+Vsb). (4 pts.) (2) Consider the OpAmp circuit shown. Each OpAmp has Rout = 0 and Rino
2. Assume operation in the linear range. a) Determine the relationship Vo/vs. for the ideal case. Ra + Vs b) If Ra = Rb = Rc = Rd 3000 Q2 and vs = = Hint: consider two stages. (8 pts.) (3) Consider
the OpAmp circuit shown. Assume operation in the linear range. Let R = 2000 2, Rout = 0, and Rin = 00 Q. a) b) Draw the equivalent circuit with a finite A. (2 pts) If is = 3.0 mA and A is infinite
(ideal case), calculate the output voltage vo. (3 pts.) Avi Rb Av2 Vo Rd Re 5.00 V, calculate v₁ and Vo. is Av R Vo SHOW YOUR WORK | {"url":"https://tutorbin.com/questions-and-answers/1-consider-the-opamp-circuit-assume-operation-in-the-linear-range-let-rout-0-w-and-rin-00-w-a-determine-the-thevenin","timestamp":"2024-11-04T11:10:36Z","content_type":"text/html","content_length":"63493","record_id":"<urn:uuid:0f54f446-4fbf-4e63-a96c-b0e1b88f7fdf>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00854.warc.gz"} |
Understanding Linear Programming Charts
Linear programming is a powerful mathematical technique used for optimization. It helps in finding the best outcome in a given mathematical model whose requirements are represented by linear
relationships. A linear programming chart visually represents these relationships, making it easier to understand the constraints and objectives involved in the optimization process.
What is Linear Programming?
Linear programming involves maximizing or minimizing a linear objective function, subject to a set of linear inequalities or equations. The objective function represents the goal of the optimization,
while the constraints represent the limitations or requirements of the problem.
For example, consider a company that produces two products. The goal is to maximize profit while considering the constraints of resources like labor and materials. The objective function could be
defined as:
Maximize: Profit = 5x + 3y
Where x and y are the quantities of the two products. The constraints might look like this:
2x + y ≤ 100 (Labor constraint)
x + 2y ≤ 80 (Material constraint)
x ≥ 0, y ≥ 0 (Non-negativity constraint)
Graphing Linear Programming Problems
To visualize linear programming problems, we can graph the constraints and the objective function. Each constraint can be represented as a line on a graph. The area where all constraints overlap is
called the feasible region. The optimal solution lies at one of the vertices of this region.
Steps to Create a Linear Programming Chart
1. Identify the Variables: Determine what you are trying to optimize. In our example, the variables are x and y.
2. Set Up the Objective Function: Write the function you want to maximize or minimize.
3. Define the Constraints: Write down the inequalities that represent the limitations.
4. Graph the Constraints: Plot each constraint on a graph. Convert inequalities into equations to find the lines.
5. Find the Feasible Region: Identify the area where all constraints overlap. This is where potential solutions exist.
6. Evaluate the Objective Function: Calculate the value of the objective function at each vertex of the feasible region to find the optimal solution.
Example of a Linear Programming Problem
Let’s consider a simple example. Suppose a farmer wants to plant two types of crops, corn and wheat. The profit from corn is $4 per acre, and from wheat, it is $3 per acre. The farmer has a total of
100 acres of land and can use a maximum of 200 hours of labor.
The objective function can be defined as:
Maximize: Profit = 4x + 3y
The constraints can be set as:
x + y ≤ 100 (Land constraint)
2x + y ≤ 200 (Labor constraint)
x ≥ 0, y ≥ 0 (Non-negativity constraint)
Graphing these constraints will help visualize the problem.
Finding the Optimal Solution
After graphing the constraints, the feasible region is identified. The next step is to evaluate the objective function at each vertex of the feasible region. This will help determine which vertex
provides the maximum profit.
For instance, if the vertices of the feasible region are (0, 100), (50, 0), and (100, 0), you would calculate:
• At (0, 100): Profit = 4(0) + 3(100) = 300
• At (50, 0): Profit = 4(50) + 3(0) = 200
• At (100, 0): Profit = 4(100) + 3(0) = 400
The maximum profit occurs at (100, 0), meaning the farmer should plant all corn.
Applications of Linear Programming
Linear programming is widely used in various fields such as economics, business, engineering, and military applications. It helps in resource allocation, production scheduling, transportation, and
many other optimization problems.
For instance, in transportation, linear programming can help minimize costs while meeting delivery requirements. In finance, it can optimize investment portfolios under certain constraints.
Linear programming charts are valuable tools for visualizing optimization problems. By graphing constraints and the objective function, you can easily identify feasible solutions and determine the
optimal outcome. Whether in agriculture, transportation, or finance, understanding how to create and interpret these charts can significantly enhance decision-making processes.
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Division And Multiplication Word Problems Worksheets
Math, specifically multiplication, creates the keystone of numerous scholastic self-controls and real-world applications. Yet, for lots of students, grasping multiplication can present a difficulty.
To resolve this difficulty, teachers and moms and dads have embraced a powerful device: Division And Multiplication Word Problems Worksheets.
Introduction to Division And Multiplication Word Problems Worksheets
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Worksheets 3rd Grade Pdf, Multiplication And Division Word Problems Worksheets 4th Grade
This set of word problems involves dividing a two digit number by a single digit number to arrive at a quotient The division leaves no remainder Answer key is included in each worksheet Division Two
digit by Single digit with Remainder
These grade 4 math worksheets have mixed multiplication and division word problems All numbers are whole numbers with 1 to 4 digits Division questions may have remainders which need to be interpreted
e g how many left over
Importance of Multiplication Practice Comprehending multiplication is essential, laying a strong foundation for sophisticated mathematical concepts. Division And Multiplication Word Problems
Worksheets offer structured and targeted method, cultivating a much deeper understanding of this essential math operation.
Evolution of Division And Multiplication Word Problems Worksheets
Multiplication Word Problems Money Money Money Worksheets 99Worksheets
Multiplication Word Problems Money Money Money Worksheets 99Worksheets
1 interactive story Download all 5 This resource gives your students practice with multiplication and division word problems This worksheet can be used with the Stepping Through Multiplication
Division Word Problems lesson Download to complete online or as a printable
5th grade multiplication and division worksheets including multiplying in parts multiplication in columns missing factor questions mental division division with remainders long division and missing
dividend or divisor problems No login required
From typical pen-and-paper workouts to digitized interactive layouts, Division And Multiplication Word Problems Worksheets have evolved, dealing with varied learning styles and preferences.
Kinds Of Division And Multiplication Word Problems Worksheets
Basic Multiplication Sheets Easy workouts concentrating on multiplication tables, aiding students construct a solid arithmetic base.
Word Trouble Worksheets
Real-life situations integrated right into troubles, enhancing critical thinking and application skills.
Timed Multiplication Drills Examinations developed to improve rate and accuracy, aiding in rapid psychological math.
Advantages of Using Division And Multiplication Word Problems Worksheets
Cross Multiplication Word Problems Worksheet Times Tables Worksheets
Cross Multiplication Word Problems Worksheet Times Tables Worksheets
This differentiated set of math worksheets includes 3 levels of multiplication and division word problems of varying difficulty These worksheets are ideal for years 3 4 5 and 6 classes and are a
perfect resource for mixed ability classes Help students practise their multiplication and division skills with these word problems Differentiated resources like these multiplication and
Multiplication and Division Word Problems interactive worksheet Live Worksheets Home Worksheets Multiplication and Division Word Problems Multiplication and Division Word Problems Mckenzie2021 Member
for 2 years 11 months Age 10 14 Level 6 Language English en ID 460701 31 10 2020 Country code BS Country Bahamas
Enhanced Mathematical Abilities
Consistent practice hones multiplication proficiency, boosting general mathematics abilities.
Improved Problem-Solving Talents
Word problems in worksheets develop logical reasoning and technique application.
Self-Paced Discovering Advantages
Worksheets suit individual learning rates, promoting a comfy and adaptable knowing environment.
How to Create Engaging Division And Multiplication Word Problems Worksheets
Incorporating Visuals and Shades Vivid visuals and shades capture focus, making worksheets aesthetically appealing and engaging.
Including Real-Life Circumstances
Connecting multiplication to day-to-day scenarios includes importance and usefulness to exercises.
Tailoring Worksheets to Various Skill Levels Personalizing worksheets based upon differing proficiency degrees guarantees inclusive knowing. Interactive and Online Multiplication Resources Digital
Multiplication Tools and Gamings Technology-based sources provide interactive discovering experiences, making multiplication interesting and delightful. Interactive Websites and Apps On-line systems
supply varied and accessible multiplication practice, supplementing traditional worksheets. Tailoring Worksheets for Various Learning Styles Aesthetic Students Aesthetic aids and representations help
understanding for students inclined toward aesthetic knowing. Auditory Learners Verbal multiplication troubles or mnemonics satisfy learners who comprehend ideas with auditory means. Kinesthetic
Students Hands-on activities and manipulatives sustain kinesthetic learners in understanding multiplication. Tips for Effective Application in Learning Uniformity in Practice Normal practice enhances
multiplication skills, promoting retention and fluency. Stabilizing Repetition and Range A mix of repetitive workouts and varied issue layouts keeps rate of interest and understanding. Offering
Constructive Feedback Comments help in determining areas of renovation, motivating continued development. Obstacles in Multiplication Practice and Solutions Motivation and Engagement Hurdles Dull
drills can bring about uninterest; ingenious approaches can reignite inspiration. Getting Rid Of Anxiety of Math Unfavorable perceptions around math can impede progress; producing a favorable
learning setting is necessary. Effect of Division And Multiplication Word Problems Worksheets on Academic Efficiency Studies and Study Searchings For Research study suggests a favorable connection in
between consistent worksheet use and enhanced mathematics performance.
Division And Multiplication Word Problems Worksheets emerge as versatile tools, promoting mathematical effectiveness in learners while suiting diverse knowing designs. From standard drills to
interactive on the internet sources, these worksheets not only improve multiplication abilities but additionally promote critical thinking and analytic capacities.
Mixed Multiplication And Division Word Problems
Division Word Problems For 2nd Grade Sara Battle s Math Worksheets
Check more of Division And Multiplication Word Problems Worksheets below
Math Worksheets For Grade 2 Division Word Problems Worksheet Resume Examples
Multiplication And Division Word Problems Anchor Chart Mastering Multi Step Word Problems
multiplication division word problems For Year 2 Teaching Resources
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Multiplication division word problems K5 Learning
These grade 4 math worksheets have mixed multiplication and division word problems All numbers are whole numbers with 1 to 4 digits Division questions may have remainders which need to be interpreted
e g how many left over
Word Problems Mixed Multiplication and Division Word Problems
Mixed Multiplication and Division Word Problems 2 Practicing the operations separately is a good start for each operation but an important word problem skill is also figuring out which math operation
is needed to solve a specific question The worksheets in this section combine both multiplication word problems and division word problems on
These grade 4 math worksheets have mixed multiplication and division word problems All numbers are whole numbers with 1 to 4 digits Division questions may have remainders which need to be interpreted
e g how many left over
Mixed Multiplication and Division Word Problems 2 Practicing the operations separately is a good start for each operation but an important word problem skill is also figuring out which math operation
is needed to solve a specific question The worksheets in this section combine both multiplication word problems and division word problems on
Multiplication And Division Word Problem Integer division word problems worksheets Add
Multiplication And Division Word Problems Anchor Chart Mastering Multi Step Word Problems
Teach Child How To Read Printable 3rd Grade Math Worksheets Division Word Problems
Classroom Math Division Word Problems Worksheets 99Worksheets
Solving Equations Using Multiplication And Division Word Problems Tessshebaylo
3Rd Grade Math Division Word Problems 3rd Grade Two Step Word Problems With Multiplication And
3Rd Grade Math Division Word Problems 3rd Grade Two Step Word Problems With Multiplication And
Extra Facts Multiplication And Division Word Problems
Frequently Asked Questions (Frequently Asked Questions).
Are Division And Multiplication Word Problems Worksheets ideal for all age teams?
Yes, worksheets can be tailored to various age and ability levels, making them adaptable for various learners.
Just how usually should trainees practice using Division And Multiplication Word Problems Worksheets?
Constant practice is crucial. Routine sessions, ideally a couple of times a week, can generate considerable enhancement.
Can worksheets alone improve mathematics abilities?
Worksheets are an useful tool but should be supplemented with varied knowing methods for detailed ability growth.
Exist on-line platforms supplying free Division And Multiplication Word Problems Worksheets?
Yes, lots of academic websites use open door to a wide range of Division And Multiplication Word Problems Worksheets.
Just how can parents sustain their kids's multiplication practice in your home?
Encouraging consistent practice, supplying support, and developing a favorable discovering environment are advantageous steps. | {"url":"https://crown-darts.com/en/division-and-multiplication-word-problems-worksheets.html","timestamp":"2024-11-04T07:26:33Z","content_type":"text/html","content_length":"33727","record_id":"<urn:uuid:b85ecea9-f904-4100-94bc-5fce3e54bd78>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00185.warc.gz"} |
Unit Circle Printable
Unit Circle Printable - For each quadrant, state which functions are positive and negative. Review related articles/videos or use a hint. Web the unit circle sec, cot 2tt 900 tt 3tt 2 2700 positive:
Web the unit circle is the circle centered at the origin with radius 1 unit (hence, the “unit” circle). Web find the values of the six trigonometric functions using unit circle.
How can anyone extend it to the other quadrants? Degrees are on the inside, then radians, and the point’s coordinate in the brackets. The equation of this circle is xy22+ =1. Unit circle printables
(fill in the blank unit circle) graph of sine to unit circle. It is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The trig functions & right triangle trig ratios.
Web the unit circle is a fundamental concept in mathematics, specifically in trigonometry.
42 Printable Unit Circle Charts & Diagrams (Sin, Cos, Tan, Cot etc)
Like many ideas in math, its simplicity makes it beautiful. Sin, cos, sec, csc embeddedmath. Although the geometric equations and theorems may seem complicated, but when you have the basic. No matter
how big or.
Printable Blank Unit Circle Worksheet Template
In a circle or on a graph. Web diagrams & charts. No matter how big or small the triangle is. The unit circle is often used to help understand and visualize the relationships between angles.
42 Printable Unit Circle Charts & Diagrams (Sin, Cos, Tan, Cot etc)
Like many ideas in math, its simplicity makes it beautiful. The unit circle positive functions: No matter how big or small the triangle is. Web this is a 4 part worksheet: Web diagrams & charts.
42 Printable Unit Circle Charts & Diagrams (Sin, Cos, Tan, Cot etc)
The trig functions & right triangle trig ratios. All the negatives and positive angles in the circle are explained by it. Click each dot on the image to select an answer. Web the unit circle.
Unit Circle Quick Lesson Downloadable PDF Chart · Matter of Math
For each point on the unit circle, select the angle that corresponds to it. Download these 15+ free printable unit circle charts & diagrams in ms word as well as in pdf format. Place the.
Unit Circle Labeled With Special Angles And Values ClipArt ETC
The unit circle can be used to calculate the trigonometric functions sin(θ), cos(θ), tan(θ), sec(θ), csc(θ), and cot(θ). For each point on the unit circle, select the angle that corresponds to it.
Web fill in.
Unit Circle Labeled In 45° Increments With Values ClipArt ETC
Learn the unit circle with our online game. Place the degree measure of each angle on the unit circle in the provided circles. For a given angle θ each ratio stays the same. Web the.
Blank Unit Circle Worksheets Free to Print Now · Matter of Math
All the negatives and positive angles in the circle are explained by it. This page provides a printable unit circle chart annotated with τ (tau). Web find the values of the six trigonometric
functions using.
42 Printable Unit Circle Charts & Diagrams (Sin, Cos, Tan, Cot etc)
Web here you can download a copy of the unit circle. Use the blank unit circle worksheet to test yourself and keep the filled unit circle handy for a reference. It utilizes (x,y) coordinates to.
42 Printable Unit Circle Charts & Diagrams (Sin, Cos, Tan, Cot etc)
Graph and formula of the unit circle. Web the unit circle sec, cot 2tt 900 tt 3tt 2 2700 positive: If you can understand this concept and what it does, trigonometry will become a lot.
Unit Circle Printable The equation of this circle is xy22+ =1. Sin, cos, tan, sec, csc, cot negative: Trigonometry is a complex subject for most of the students but some find it interesting and
fascinating. Here are some tips to help you out: Web what is the unit circle?
Unit Circle Printable Related Post : | {"url":"https://upload.independent.com/kudos/unit-circle-printable.html","timestamp":"2024-11-11T17:59:16Z","content_type":"application/xhtml+xml","content_length":"24093","record_id":"<urn:uuid:75d4eabf-ec42-44e2-9eda-8ee6418def94>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00030.warc.gz"} |
Pentathlon Score Calculator - CalculatorsPot
Pentathlon Score Calculator
Published on
The Pentathlon Score Calculator is a tool designed to simplify the process of calculating scores for athletes competing in the modern pentathlon. This Olympic sport comprises five events: fencing,
swimming, equestrian show jumping, and a combined event of pistol shooting and cross-country running. The scoring system for these events has evolved into a points system to make tracking competition
progress more straightforward.
Purpose and Functionality
The primary purpose of the Pentathlon Score Calculator is to offer athletes, coaches, and fans a clear and easy way to understand how performances in each event translate into scores. This tool takes
into account the specific inputs and calculations for each of the five events, applying a simplified version of the scoring rules that might vary according to updates from the International Modern
Pentathlon Union (UIPM).
How It Works
1. Fencing: The calculator requires the number of victories as input. Points are calculated based on victories, with a predetermined baseline for points and adjustments made for performances above
or below this baseline.
2. Swimming (200m Freestyle): Here, the input is the time taken to complete the event. The calculator assigns points based on how this time compares to a base time, with additions or subtractions
for faster or slower times, respectively.
3. Equestrian Show Jumping: Inputs for this event include performance metrics like faults. The calculator starts with a base score and deducts points for any faults, such as knocking down obstacles.
4. Combined Event (Pistol Shooting and Cross-Country Running): This requires inputs for completion time and shooting performance. Points are calculated based on the finish time, with adjustments
made for shooting performance impacting the running start.
To make the formula for calculating scores in a modern pentathlon as simple as possible, let’s break it down into the individual events. Remember, this formula is a simplified version to help
understand how scores might be calculated for each event.
1. Fencing
• What you need: Your number of wins.
• How to calculate: Start with a certain number of points (let’s say 250 points for winning 70% of your matches). If you win more than 70% of your matches, you get extra points for each win above
this. If you win less, you lose some points for each match below this percentage.
2. Swimming (200m Freestyle)
• What you need: Your completion time.
• How to calculate: There is a base completion time (e.g., 2:30 minutes) that is worth 250 points. If you swim faster than this time, you get additional points for every second you beat the base
time. If you are slower, you lose points for each second more than the base time.
3. Equestrian Show Jumping
• What you need: How many obstacles you knocked down (or other faults).
• How to calculate: You start with a set number of points (e.g., 300 points). Then, for each fault (like knocking down an obstacle), a certain number of points are subtracted from your total.
4. Combined Event: Pistol Shooting and Cross-Country Running (Laser-Run)
• What you need: Your completion time and how well you did in shooting.
• How to calculate: There’s a base score for finishing within a specific time (e.g., 12:00 minutes equals 500 points). You get extra points for finishing faster than this time, and you lose points
for finishing slower. Your shooting performance can affect your starting time for the running, indirectly affecting your total points.
Example Calculation
1. Fencing: If you win exactly 70% of your bouts, you start with 250 points. Win more, and you add to this. Win less, and you subtract.
2. Swimming: Swim faster than 2:30, and you add points for each second under. Swim slower, and you subtract points for each second over.
3. Equestrian: You start with 300 points and lose some for each fault you make.
4. Laser-Run: Beat the 12:00 minute mark, and earn points for each second under. Go over 12:00, and lose points for each second over.
Step-by-Step Examples
Let’s illustrate the scoring with an example of an athlete’s performance:
• Fencing: The athlete wins 70% of bouts, earning 250 points.
• Swimming: Completes the 200m freestyle in 2:20, gaining 270 points.
• Equestrian: Incurs two faults, resulting in a loss of 14 points, totaling 286 points.
• Laser-Run: Finishes in 11:30, scoring 520 points.
Thus, the total score would be 1326 points.
Relevant Information Table
Here’s a simplified table to help understand the scoring for each event:
Event Input Example Base Score Calculation Method
Fencing 70% victories 250 points Points adjusted based on victories against baseline
Swimming (200m) 2:20 time 250 points Points adjusted for every second over/under base time
Equestrian Show Jumping 2 faults 300 points Points deducted per fault
Combined (Laser-Run) 11:30 time 500 points Points adjusted based on completion time and shooting performance
The Pentathlon Score Calculator significantly benefits athletes, coaches, and enthusiasts by providing an accessible means to calculate and understand the scores of a complex and multifaceted sport.
By breaking down each event’s scoring into simple inputs and calculations, it demystifies the scoring process, making the sport more approachable and easier to follow. Whether used for training,
strategizing, or just following along as a fan, this tool enhances the modern pentathlon experience by ensuring that the focus remains on the athleticism and strategy inherent in this challenging
Olympic sport.
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Frontiers | Symmetry-Based Representations for Artificial and Biological General Intelligence
• DeepMind, London, United Kingdom
Biological intelligence is remarkable in its ability to produce complex behavior in many diverse situations through data efficient, generalizable, and transferable skill acquisition. It is believed
that learning “good” sensory representations is important for enabling this, however there is little agreement as to what a good representation should look like. In this review article we are going
to argue that symmetry transformations are a fundamental principle that can guide our search for what makes a good representation. The idea that there exist transformations (symmetries) that affect
some aspects of the system but not others, and their relationship to conserved quantities has become central in modern physics, resulting in a more unified theoretical framework and even ability to
predict the existence of new particles. Recently, symmetries have started to gain prominence in machine learning too, resulting in more data efficient and generalizable algorithms that can mimic some
of the complex behaviors produced by biological intelligence. Finally, first demonstrations of the importance of symmetry transformations for representation learning in the brain are starting to
arise in neuroscience. Taken together, the overwhelming positive effect that symmetries bring to these disciplines suggest that they may be an important general framework that determines the
structure of the universe, constrains the nature of natural tasks and consequently shapes both biological and artificial intelligence.
1. Introduction
Neuroscience and machine learning (ML) have a long history of mutually beneficial interactions (Hassabis et al., 2017), with neuroscience inspiring algorithmic and architectural improvements in ML (
Rosenblatt, 1958; LeCun et al., 1989), and new ML approaches serving as computational models of the brain (Yamins et al., 2014; Yamins and DiCarlo, 2016; Wang et al., 2018; Dabney et al., 2020). The
two disciplines are also interested in answering the same fundamental question: what makes a “good” representation of the often high-dimensional, non-linear, and multiplexed sensory signals to
support general intelligence (Bengio et al., 2013; Niv, 2019). In the same way as the adoption of the decimal system for representing numbers has produced an explosion in the quantity of numerical
tasks that humans could solve efficiently (note that the information content remained unaffected by this change in the representational form), finding a “good” representation of the sensory inputs is
likely to be a fundamental computational step for enabling data efficient, generalizable, and transferrable skill acquisition. While neuroscientists go about trying to answer this question by
studying the only working instantiation of general intelligence—the brain, ML scientists approach the same problem from the engineering perspective, by testing different representational forms in the
context of task learning through supervised or reinforcement learning (RL), which allows faster iteration. In this review we will discuss how bringing the idea of symmetry transformations from
physics into neural architecture design has enabled more data efficient and generalizable task learning, and how this may be of value to neuroscience.
The reason why it makes sense to turn to physics when it comes to understanding the goal of perception in artificial or biological intelligence, is because intelligence evolved within the constraints
of our physical world, and likewise, the tasks that we find interesting or useful to solve are similarly constrained by physics. For example, it is useful to know how to manipulate physical objects,
like rocks, water or electricity, but it is less useful to know how to manipulate arbitrary regions of space (which also do not have a word to describe them, further highlighting their lack of
relevance). Hence, a representation that reflects the fundamental physical properties of the world is likely to be useful for solving natural tasks expressed in terms of the same physical objects and
properties. Symmetry transformations are a simple but fundamental idea that allows physicists to discover and categorize physical objects—the “stubborn cores that remain unaltered even under
transformations that could change them” (Livio, 2012), and hence symmetries are a good candidate for being the target of representation learning.
The study of symmetries in physics (that is, the transformations that leave the physical “action” invariant) in its modern form originates with Noether's Theorem (Noether, 1915), which proved that
every conservation law is grounded in a corresponding continuous symmetry transformation. For example, the conservation of energy arises from the time translation symmetry, the conservation of
momentum arises from the space translation symmetry, and the conservation of angular momentum arises due to the rotational symmetry. This insight, that transformations (the joints of the world) and
conserved properties (the invariant cores of the world that words often tend to refer to Tegmark, 2008) are tightly related, has led to a paradigm shift in the field, as the emphasis in theoretical
physics changed from studying objects directly to studying transformations in order to discover and understand objects. Since the introduction of Noether's theorem, symmetry transformations have
permeated the field at every level of abstraction, from microscopic quantum models to macroscopic astrophysics models.
In this paper we are going to argue that, similarly to physics, a change in emphasis in neuroscience from studying representations in terms of static objects to studying representations in terms of
what natural symmetry transformations they reflect can be impactful, and we will use the recent advances in ML brought about by the introduction of symmetries to neural networks to support our
argument. By introducing the mathematical language of group theory used to describe symmetries, we hope to provide the tools to the neuroscience community to help in the search for symmetries in the
brain. While ML research has demonstrated the importance of symmetries in the context of different data domains, here we will mainly concentrate on vision, since it is one of the most prominent and
most studied sensory systems in both ML and neuroscience. For this reason, topics like the importance of symmetries in RL will be largely left out (although see Agostini and Celaya, 2009; Anand et
al., 2016; Madan et al., 2018; van der Pol et al., 2020; Kirsch et al., 2021). We will finish the review by describing some of the existing evidence from the neuroscience community that hints at
symmetry-based representations in the ventral visual stream.
2. What Are Symmetries?
2.1. Invariant and Equivariant Representations
Given a task, there often exist transformations of the inputs that should not affect it. For example, if one wants to count the number of objects on a table, the outcome should not depend on the
colors of those objects, their location or the illumination of the scene. In that case, we say the output produced by an intelligent system when solving the task is invariant with respect to those
transformations. Since the sensory input changes with transformations, while the output is invariant, we need to decide what should happen to the intermediate representations. Should they be
invariant like the output or should they somehow transform similarly to the input?
Much of the research on perception and representation learning, both in ML and neuroscience, has focused on object recognition. In ML, this line of research has historically emphasized the importance
of learning representations that are invariant to transformations like pose or illumination (Lowe, 1999; Dalal and Triggs, 2005; Sundaramoorthi et al., 2009; Soatto, 2010; Krizhevsky et al., 2012).
In this framework, transformations are considered nuisance variables to be thrown away (Figures 1A, 2B). Some of the most successful deep learning methods (Krizhevsky et al., 2012; Mnih et al., 2015;
Silver et al., 2016; Espeholt et al., 2018; Hu et al., 2018; Dai et al., 2021) end up learning such invariant representations (see Tishby et al., 1999; Tishby and Zaslavsky, 2015 for a potential
explanation of why this happens in the context of supervised learning). This is not a problem for narrow intelligence, which only needs to be good at solving the few tasks it is explicitly trained
for, however, discarding “nuisance” information can be problematic for general intelligence which needs to reuse its representations to solve many different tasks, and it is not known ahead of time
which transformations may be safe to discard. It is not surprising then that despite the enormous success of the recent deep learning methods trained on single tasks, they still struggle with data
efficiency, transfer, and generalization when exposed to new learning problems (Garnelo et al., 2016; Lake et al., 2016; Higgins et al., 2017b; Kansky et al., 2017; Marcus, 2018; Cobbe et al., 2019).
FIGURE 1
Figure 1. Different approaches to dealing with symmetries in neural networks. Both figures represent a neural network transforming an image, and a rotated image. The gray 3 × 3 squares are
activations of the neural networks. (A) Inputs transform with symmetries, but hidden features and outputs are invariant. (B) Inputs and hidden features transform with symmetries, only outputs are
FIGURE 2
Figure 2. Different hypothesized coding properties of neurons at the start (A) and end (B,C) of visual processing in neural networks and the brain. Top row, schematic representation of manifolds
representing two classes of robots: blue manifold contains robots that vary in the shape of their head, arms, legs, and body; red manifold contains robots that have no body and vary in the shape of
their head, arms and legs only. Bottom row, schematic representation of the activations of a single idealized real or artificial neuron in response to variations in the visual stimulus. (A) (Top)
Early processing stages have entangled high-dimensional manifolds. All information about the two robot categories, and their identity preserving transformations is present, but is hard to read out
from the neural activations. Arrows represent the high-dimensional space spanned by all V1 neurons. (Bottom) Line plot shows the activation of a single idealized neuron in response to different robot
variations. Neuron responds to robots from both classes. (B): “Exemplar” or invariant representation at the end of visual processing. Top: Single neurons have maximal firing for the prototype example
of their preferred robot class (blue—with, red—without body). All information about the identity preserving transformations has been collapsed (illustrated by red or blue points), which makes object
classification easy, but any other task (like clustering robots based on their arm variation) impossible. Arrows represent the high-dimensional space spanned by the higher-order neurons, arrow color
represents preferred stimulus of the neuron. Bottom left: activation of a single idealized neuron to all robot variations from both classes. Lighter, higher activation. Big blue circle indicates
preferred stimulus for neuron, resulting in highest activation, smaller blue circles indicate other robots resulting in lower to no activation. Blue arrow, cross section of robot variations shown in
the line plot. Bottom right: line plot shows activation of the same idealized neuron as on the left but in the cross section of robot variations spanned by the blue arrow. Response declines
proportionally to the distance from the preferred stimulus (big blue circle). (C) “Axis” or equivariant representation at the end of visual processing. Top: two robot classes have been separated into
different representational manifolds, which are also aligned in terms of the shared transformations (e.g., both robot classes have similar identity preserving transformations in head shape, spanned
by green axis; and arm shape, spanned by purple axis). This makes it easy to classify the robots, and solve other tasks, like clustering robots based on their arm variations. Bottom left: activation
of a single idealized neuron to robot variations along the head shape change axis (green) and arm shape change axis (purple). Lighter, higher activation. Neuron has a ramped response proportional to
changes in its preferred transformation (changes in head shape, green), and no change in firing to other transformations (e.g., changes in arm shape, blue). Bottom right: as in (B), but the cross
section is spanned by the purple axis. Green dot indicates higher neural activation in response to a change in the robot head shape.
Similarly to ML, in neuroscience ventral visual stream is traditionally seen to be progressively discarding information about the identity preserving transformations of objects (Fukushima, 1980;
Tanaka, 1996; Poggio and Bizzi, 2004; Yamins et al., 2014). While neurons in the early processing stages, like V1, are meant to represent all information about the input stimuli and their
transformations in high-dimensional “entangled” manifolds, where the identities of the different objects are hard to separate (Figure 2A), later in the hierarchy such manifolds are meant to collapse
into easily separable points corresponding to individual recognizable objects, where all the information about the identity preserving transformations is lost, resulting in the so called “exemplar”
neurons^1 following the naming convention of Chang and Tsao (2017). In this view, every neuron has a preferred stimulus identity in response to which the neuron fires maximally, while its response to
other stimuli decreases proportionally to their distance from the preferred stimulus (Figure 2B).
An alternative point of view in both disciplines has advocated that instead of discarding information about the identity preserving transformations, information about these factors should be
preserved but reformatted in such a way that aligns transformations within the representations with the transformations observed in the physical world (Figures 1B, 2C), resulting in the so called
equivariant representations (DiCarlo and Cox, 2007; Hinton et al., 2012; Bengio et al., 2013). In the equivariant approach to perception, certain subsets of features may be invariant to specific
transformations, but the overall representation is still likely to preserve more overall information than an invariant representation, making them more conducive of diverse task learning (Figure 1B).
For example, some hidden units may be invariant to changes in the object color, but will preserve information about object position, while other hidden units may have an opposite pattern of
responses, which means that information about both transformations will be preserved across the whole hidden layer, while each individual subspace in the hidden representation will be invariant to
all but one transformation. Researchers in both neuroscience and ML communities have independently hypothesized that equivariant representations are likely to be important to support general
intelligence, using the terms “untangling” (DiCarlo and Cox, 2007; DiCarlo et al., 2012) and “disentangling” (Bengio, 2009, 2012; Bengio et al., 2013), respectively. We are next going to introduce
the mathematical language for describing symmetry transformations and use it to discuss how adding neural network modules that are equivariant to such symmetry transformations can improve data
efficiency, generalization, and transfer performance in ML models.
2.2. Defining Symmetries and Actions
Symmetries are sets of transformations of objects, and the same abstract set of symmetries can transform different objects. For example, consider the set of rotations by multiple of 90° and
reflections along both horizontal and vertical axis, known as the dihedral group D[4] (Dummit and Foote, 1991). By rotating images, symmetries from D[4] can be applied to images of cats or tea pots,
either 32 × 32 or 1, 024 × 1, 024, color or black and white. In mathematics, the concept of symmetries, that is transformations that are invertible and can be composed, is abstracted into the concept
of groups. For example, D[4] is a group with eight elements (Figure 3).
FIGURE 3
Figure 3. All 8 transformations of an image under the dihedral group D[4]. Rotations by 90° are applied along the inner and outer circles. Reflections are applied along straight lines. (A) While the
image is transformed, some properties, such as the teapot identity or color, are invariant with respect to the applied transformations. (B) Symmetric images are left invariant by some elements of D
[4], and modified by others.
More formally, a group G is defined as a set with a binary operation (also called composition or multiplication)
$G×G→ G(g1,g2)↦g1·g2, (1)$
such that
1. the operation is associative: (g[1] · g[2]) · g[3] = g[1] · (g[2] · g[3]);
2. there exists an identity element e ∈ G such that e · g = g · e = g, ∀g ∈ G;
3. all elements are invertible: for any g ∈ G, there exists g^−1 ∈ G such that g · g^−1 = g^−1 · g = e.
Note how we defined a group as a set of symmetries, without explicitly saying what these are symmetries of. That's because the concept of group in mathematics seeks to study properties of symmetries
that are independent of the objects being transformed. In practice though, we will of course want to apply symmetries to objects. This is formally defined as an action.^2 For example, the group D[4]
can act on both 32 × 32 gray-scale images, that is ℝ^32×32, and on 1, 024 × 1, 024 color images, that is ℝ^1, 024×1, 024×3.
More formally, given a group G and a set X, an action^3 of G on X is a map
$G×X→ X(g,x)↦g·x, (2)$
such that
1. the multiplication of the group and the action are compatible: g[1] · (g[2] · x) = (g[1] · g[2]) · x;
2. the identity of the group leaves elements of X invariant: e · x = x.
Note how we overloaded the symbol · to define both a multiplication in the group, and an action on a set. This makes sense because multiplication of the group defines an action of that group on
itself. The identity e leaves all elements in X invariant e · x = x, but for a given x, there can exist g ≠ e such that g · x = x, for example in Figure 3B.
Two elements of a group are said to commute if the order in which we multiply them does not matter. Formally, we say that g[1], g[2] ∈ G commute if g[1] · g[2] = g[2] · g[1]. If all the elements in
the groups commute with each other, the group itself is called commutative.^4 Even if a group is not commutative, it might still be a product of two subgroups that commute with each other. For
example, assume you have three cubes of different sizes and colors, and three pyramids of different sizes and colors. If these objects are put on three different tables, each with a cube and a
pyramid, we can move the cubes around while leaving the pyramids where they are, or we can move the pyramids and leave the cubes untouched. The action of re-ordering the cubes is an action of the
group of permutations over three elements ${{S}}_{3}$. Here we are making that group act on our arrangement of cubes and pyramids, by leaving the pyramids invariant. The action of re-ordering the
pyramids is also an action of ${{S}}_{3}$. So, overall, we have an action of ${{S}}_{3}×{{S}}_{3}$. The group as a whole is not commutative, since each of the ${{S}}_{3}$ is not, but it does not
matter if we reorder the pyramids first, or the cubes first. Formally, this means that as a set G = G[1] × G[2], where G[1] and G[2] are themselves groups, and all elements of G[1] commute with all
elements of G[2]. This last commutation requirement is important. Indeed, consider once again the case of D[4]. Let F be the subgroup made of the identity and the reflection along the vertical axis.
And let R be the group made of rotations by 0, 90, 180, and 270°. Any element of D[4] can be written in a unique way as f · r for (f, r) ∈ F × R, but since f · r ≠ r · f, it is not true that D[4] is
equal to F × R as a group.
We just mentioned the idea that some properties are preserved by symmetries. Indeed, while a group action defines how elements of a set are transformed, it is often useful to also consider what is
being preserved under the action. For example, consider a Rubik's cube. Algorithms on how to solve a Rubik's cube use steps described by simple transformations such as “rotate left face clockwise” or
“rotate front face anti-clockwise.” The set of all transformations built by such simple rotations of faces forms a group, and that group acts on the Rubik's cube by modifying the colors on faces. But
what is being preserved here? The answer is the structure of the cube. Indeed, after any of these transformations, we still have a cube with faces, each made of 9 squares arranged in a regular 3 × 3
grid. In the case of our dihedral group D[4] in Figure 3, colors but also relative distances are being preserved: two pixels in the original image will move to a new location in a rotated image, but
their distance from each other is unchanged, thus preserving the object identity.
We are now ready to define the concepts of invariant and equivariant maps—the building blocks for obtaining the invariant and equivariant representations we introduced earlier. Lets start with
invariance. Formally, if a group G acts on a space X, and if F : X → Y is a map between sets X and Y, then F is invariant if F(g · x) = F(x), ∀(g, x) ∈ G × X. In words, this means that applying F to
a point or to a transformed point will give the same result. For example, in Figure 3, the map that recognizes a tea pot in the input picture should not depend on the orientation of the picture.
Invariant maps delete information since knowing y = F(x) does not allow to distinguish between x and g · x. If the invariant features required to solve a task are highly non-linear with respect to
the inputs, then we might want to first transform the inputs before extracting any invariant information. And here we need to be careful, because if H is any map while F is invariant, it will not be
true in general that F(H(x)) is invariant. On the other hand, we will see that if H is equivariant, then F(H(x)) will indeed be invariant. Let us now define equivariance: if G is a group acting on
both spaces X and Y, and H : X → Y is a map between these spaces, then H is said to be equivariant if for any g ∈ G and any x ∈ X, we have H(g · x) = g · H(x). In words, it does not matter in which
order we apply the group transformation and the map H. We can now verify our earlier claim: if H is equivariant and F is invariant, then F(H(g · x)) = F(g · H(x)) = F(H(x)), and F ◦ H is indeed
invariant. As we will see later, this recipe of stacking equivariant maps followed by an invariant map, as shown in Figure 1B, is a commonly used recipe in ML (Bronstein et al., 2021).
So far we have considered discrete symmetries. However, many of the symmetries encountered in the real world are continuous. A group of symmetries is said to be continuous if there exist continuous
paths between symmetries. For example, in the group of 2D rotations, we can create paths by smoothly varying the angle of the rotations. On the other hand, if we only allow rotations by multiple of
90°, then it is not possible to move smoothly from a rotation by 180° to a rotation by 270°. In that case, the group is said to be discrete.^5 A simple approach to handle continuous symmetries used
in practice in ML is to fall back to the discrete case by approximating the full group of continuous symmetries by a subgroup of discrete ones. For example, the group of rotations of the 2D plane can
be approximated by only considering rotations by $\frac{360°}{N}$, although this can become computationally expensive for very large groups (Finzi et al., 2020). While other approaches that truly
handle a full group of continuous symmetries do exist (Rezende et al., 2019, 2020; Huang et al., 2020; Köhler et al., 2020; Pfau et al., 2020a; Cohen et al., 2021; Katsman et al., 2021; Papamakarios
et al., 2021; Rezende and Racanière, 2021), we will concentrate on discrete symmetries in this paper for simplicity.
3. Implementation and Utility of Symmetries in ML
Although not always explicitly acknowledged, symmetries have been at the core of some of the most successful deep neural network architectures. For example, convolutional layers (CNNs) (LeCun and
Bengio, 1995) responsible for the success of the deep classifiers that are able to outperform humans in their ability to categorize objects in images (Hu et al., 2018; Dai et al., 2021) are
equivariant to translation symmetries characteristic of image classification tasks, while graph neural networks (GNNs) (Battaglia et al., 2018) and attention blocks commonly used in transformer
architectures (Vaswani et al., 2017) are equivariant to the full group of permutations. While there are several reasons, including optimization considerations, why these architectural choices have
been so successful compared to MLPs (Rosenblatt, 1958)—the original neural networks, one of the reasons is that these architectures reflect the prevalent symmetry groups of their respective data
domains, while the linear layers used in MLPs are not compatible with any particular symmetry (Haykin, 1994), despite being theoretically proven universal function approximators (Cybenko, 1989;
Hornik et al., 1989). Architectures like CNNs and GNNs reflect single type of symmetries (translations and permutations, respectively), but active research is also looking into building techniques to
incorporate larger groups of symmetries into neural networks (Anselmi et al., 2013; Gens and Domingos, 2014; Cohen and Welling, 2016; Cohen et al., 2018).
One of the main reasons why incorporating symmetries into neural networks helps is due to improvements in data efficiency. Indeed, incorporating symmetries can reduce the volume of the problem space,
as illustrated in Figure 4. If we assume that the data processed by our model are points in a 3D cube (Figure 4A), when symmetries can be exploited, the models only need to work with a subset of the
cube (Figures 4B,C), which reduces the volume of the input space. Provided the model respects symmetries by construction, learning on this reduced space is enough to learn on the entire cube. This
naturally also leads to improvements in generalization and transfer, since new points outside of the training data distribution that can be obtained by applying the known symmetries to the observed
data will be automatically recognizable. This principle has been exploited in scientific applications of ML, such as free energy estimation (Wirnsberger et al., 2020), protein folding (Fuchs et al.,
2020; Baek et al., 2021), or quantum chemistry (Pfau et al., 2020b; Batzner et al., 2021).
FIGURE 4
Figure 4. Symmetries let us reduce the volume of the domain on which our models need to learn. (A) The original problem domain. (B) With one symmetry, a reflection along a plane, we can half the
domain on which we need to learn. (C) Further symmetries keep on reducing the volume of domain problem.
An alternative to building symmetries into the model, is to use data-augmentation and let the model learn the symmetries. This is achieved by augmenting the training dataset (for example images) with
the relevant transformations of this data (for example, all rotations and reflections of these images). This principle has been used as a source of augmentations for self-supervised contrastive
learning approaches (Chen et al., 2020; Grill et al., 2020). While these approaches have been shown to be very effective in improving data efficiency on image classification tasks, other research has
shown that learning symmetries from data augmentations is usually less effective than building them into the model architecture (Cohen and Welling, 2016; Qi et al., 2017; Veeling et al., 2018;
Rezende et al., 2019; Köhler et al., 2020; Satorras et al., 2021).
An alternative to hard wiring inductive biases into the network architecture is to instead adjust the model's learning objective to make sure that its representations are equivariant to certain
symmetries. This can be done implicitly by adding (unsupervised) regularizers to the main learning objective (Bellemare et al., 2017; Jaderberg et al., 2017), or explicitly by deciding on what a
“good” representation should look like and directly optimizing for those properties. One example of the latter line of research is the work on disentangled^6 representation learning (Bengio, 2009;
Bengio et al., 2013) (also see related ideas in Schmidhuber, 1992; Hyvärinen, 1999). While originally proposed as an intuitive framework that suggested that the world can be described using a small
number of independent generative factors, and the role of representation learning is to discover what these are and represent each generative factor in a separate representational dimension (Bengio
et al., 2013), disentangling has recently been re-defined through a formal connection to symmetries (Higgins et al., 2019). In this view, a vector representation is seen as disentangled with respect
to a particular decomposition of a symmetry group into a product of subgroups, if it can be decomposed into independent subspaces where each subspace is affected by the action of a single subgroup,
and the actions of all the other subgroups leave the subspace unaffected.
To understand this definition better, let's consider a concrete example of an object classification task (Figure 5A). Transformations like changes in the position or size of an object are symmetry
transformations that keep the object identity invariant. These transformations also commute with each other, since they can be applied in random order without affecting the final state of the world (
Figure 5B). This implies that the symmetry group used to describe the natural transformations in this world can be decomposed into a product of separate subgroups, including one subgroup that affects
the position of an object, and another one affecting its size.
FIGURE 5
Figure 5. (A) Simplified schematic showing discrete approximation of continuous translation and scale symmetries of 3D objects. Translations are applied along the inner and outer circles. Scale
transformations are applied along straight lines. (B) Translation and scale transformation commute with each other. They can be applied in permuted order without affecting the final state. (C)
Disentangling neural networks learn to infer a representation of an image that is a concatenation of independent subspaces, each one being (approximately) equivariant to a single symmetry
transformation. The model uses inference to obtain a low-dimensional representation of an image, and generation to reconstruct the original image from the representation. Two example latent
traversals demonstrate the effect on the image reconstruction of smoothly varying the value of the position and size subspaces.
Assuming that the symmetry transformations act on a set of hypothetical ground truth abstract states of our world, and the disentangling model observes high-dimensional image renderings of such
states, in which all the information about object identity, size and position among other factors is entangled, the goal of disentangled representation learning is to infer a representation which is
decomposed into independent subspaces, where each subspace is affected only by a single subgroup of our original group of symmetry transformations. In other words, the vector space of such a
representation would be a concatenation of independent subspaces, such that, for example, a change in size only affects the “size subspace,” but not the “position subspace” or any other subspace (
Figure 5C). This definition of disentangled representations is very general—it does not assume any particular dimensionality or basis for each subspace. The changes along each of the subspaces in the
representation may also be implemented by an arbitrary, potentially non-linear mapping, although if this mapping is linear, it can provide additional nice properties to the representation (Higgins et
al., 2018 call such a representation a linear disentangled representations), since it means that the task relevant information (e.g., the “stable cores” of color or position attributes of the object)
can be read out using linear decoders, and “nuisance” information can be easily ignored using a linear projection.
While the early approaches to disentangled representation learning (including related ideas from nonlinear dimensionality reduction literature, e.g., Hyvärinen, 1999; Hyvärinen and Pajunen, 1999;
Tenenbaum et al., 2000; Belkin and Niyogi, 2001; Coifman and Lafon, 2006) either struggled to scale (Tenenbaum et al., 2000; Desjardins et al., 2012; Tang et al., 2013; Cohen and Welling, 2014, 2015)
or relied on a form of supervision (Hinton et al., 2011; Reed et al., 2014; Zhu et al., 2014; Cheung et al., 2015; Goroshin et al., 2015; Kulkarni et al., 2015; Yang et al., 2015; Karaletsos et al.,
2016; Whitney et al., 2016), most of the modern methods for successful unsupervised disentangling (Higgins et al., 2017a; Achille et al., 2018; Chen et al., 2018; Dupont, 2018; Kim and Mnih, 2018;
Kumar et al., 2018; Ridgeway and Mozer, 2018; Ansari and Soh, 2019; Caselles-Dupré et al., 2019; Detlefsen and Hauberg, 2019; Dezfouli et al., 2019; Esmaeili et al., 2019; Lorenz et al., 2019;
Mathieu et al., 2019; Ramesh et al., 2019; Lee et al., 2020; Quessard et al., 2020) are based on the Variational AutoEncoder (VAE) architecture (Kingma and Welling, 2014; Rezende et al., 2014)—a
generative network that learns by predicting its own inputs. The base VAE framework learns a compressed representation that maximizes the marginal likelihood of the data and are related to the idea
of “mean field approximation” from physics. In this framework no explicit desiderata are made about the representational form—as long as the distribution of the learnt data representation is close to
the chosen prior (which often consists of independent unit Gaussians), it is considered to be acceptable. Disentangling VAEs, on the other hand, aim to learn a representation of a very particular
form—it has to decompose into independent subspaces, each one reflecting the action of a single symmetry transformation. Disentangling VAEs typically work by adjusting the VAE learning objective to
restrict the capacity of the representational bottleneck. This is usually done by encouraging the representation to be as close to the isotropic unit Gaussian distribution as possible, hence also
encouraging factorization. Although it has been proven that unsupervised disentangled representation learning in this setting should be theoretically impossible (Locatello et al., 2019), these
approaches work in practice by exploiting the interactions of the implicit biases in the data and the learning dynamics (Burgess et al., 2018; Locatello et al., 2019; Mathieu et al., 2019; Rolinek et
al., 2019). Since these approaches are not optimizing for symmetry-based disentanglement directly, they are not principled and struggle to scale. However, they have been shown to learn an approximate
symmetry-based disentangled representation (for example they often lose the cyclical aspect of the underlying symmetry) that still preserves much of the group structure (e.g., the commutativity of
the symmetries) and hence serves as a useful tool for both understanding the benefits of symmetry-based representations in ML models, and as a computational model for studying representations in the
brain (Soulos and Isik, 2020; Higgins et al., 2021a). In the meantime, new promising approaches to more scalable and/or principled disentanglement are starting to appear in the ML literature (
Besserve et al., 2020; Pfau et al., 2020a; Higgins et al., 2021b; Wang et al., 2021).
In order to generalize learnt skills to new situations, it is helpful to base learning only on the smallest relevant subset of sensory variables, while ignoring everything else (Canas and Jones, 2010
; Jones and Canas, 2010; Bengio et al., 2013; Niv et al., 2015; Leong et al., 2017; Niv, 2019). Symmetry-based representations make such attentional attenuation very easy, since meaningful sensory
variables get separated into independent representational subspaces, as was demonstrated in a number of ML papers (Higgins et al., 2017b; Locatello et al., 2020). Following the reasoning described
earlier, disentangled representations have also been shown to help with data efficiency when learning new tasks (Locatello et al., 2020; Wulfmeier et al., 2021). Finally, disentangled representations
have also been shown to be a useful source of intrinsically motivated transferable skill learning. By learning how to control their own disentangled subspaces (e.g., how to control the position of an
object), it has been shown that RL agents with disentangled representations could discover generally useful skills that could be readily re-used for solving new tasks (e.g., how to stack objects) in
a more data efficient manner (Achille et al., 2018; Laversanne-Finot et al., 2018; Grimm et al., 2019; Wulfmeier et al., 2021).
4. Symmetries in Neuroscience
Although psychology and cognitive science picked up the mathematical framework of group theory to describe invariances and symmetry in vision a long time ago (Dodwell, 1983), this framework was not
broadly adopted and progress in this direction quickly stalled (although see Liao et al., 2013; Leibo et al., 2017). However, circumstantial evidence from work investigating the geometry of neural
representations suggests the possibility that the brain may be learning symmetry-based representations. For example, factorized representations of independent attributes, such as orientation and
spatial frequency (Hubel and Wiesel, 1959; Mazer et al., 2002; Gáspár et al., 2019) or motion and direction tuning (Grunewald and Skoumbourdis, 2004) have long been known to exist at the start of the
ventral visual stream in V1. Going further along the visual hierarchy, Kayaert et al. (2005) demonstrated that many of the primate IT neurons had monotonic tuning to the generative dimensions of toy
visual stimuli, such as curvature, tapering or aspect ratio, known to be discriminated independently from each other by humans in psychophysical studies (Arguin and Saumier, 2000; Stankiewicz, 2002;
de Beeck et al., 2003). In particular, they found that the firing of each neuron was modulated strongly by its preferred generative attribute but significantly less so by the other generative
attributes (Figure 6A).
FIGURE 6
Figure 6. Examples of axis-based coding at the end of the ventral visual stream. (A) Single IT cell shows preference to a single transformation (change in positive curvature) regardless of the
geometric shape (triangle or rectangle). Single IT cell responds to changes in curvature of the triangle while being invariant to changes in length, changes in tapering of the rectangle while being
invariant to changes in curvature, and changes in negative curvature of the rectangle while being invariant to changes in tapering. Bars are standard errors in response to multiple stimulus
presentations (14 on average). Adapted form Kayaert et al. (2005). (B) Single IT cells have ramped responses proportional to changes along their preferred axis of variation in the generative face
space, and no changes in their responses to orthogonal directions in the face space. Adapted from Chang and Tsao (2017). (C) Single cells in the IT have strong one-to-one alignment to single
subspaces discovered through disentangled representation learning. Adapted from Higgins et al. (2021a). (D) Different representation geometries have different trade-offs in terms of how much they
support generalization as measured by the abstraction scores (green), and how expressive they are as measured by the shattering dimensionality score (orange). Representations in the prefrontal cortex
(PFC) and hippocampus (HPS) of primates, as well as in the final layer of a neural network trained to solve multiple tasks in the reinforcement learning framework were found to exhibit
disentangling-like geometry highlighted in blue that scores well on to both metrics. Adapted from Bernardi et al. (2020).
More recently, Chang and Tsao (2017) investigated the coding properties of single IT neurons in the primate face patches. By parameterizing the space of faces using a low-dimensional code, they were
able to show that each neuron was sensitive to a specific axis in the space of faces spanned by as few as six generative dimensions on average, with different cells preferring different axes.
Moreover, the recorded IT cells were found to be insensitive to changes in directions orthogonal to their preferred axis, suggesting a low-dimensional factorized representation reminiscent of
disentangled representations from ML (Figure 6B). To directly test whether the two representations resembled each other, Higgins et al. (2021a) compared the responses of single cells in the IT face
patches to disentangled latent units discovered by a model exposed to the same faces as the primates (Figure 6C). By measuring the alignment between the two manifolds, the authors were able to
compare the two representational forms in a way that was sensitive to linear transformations (unlike the traditional measures of similarity used in the neuroscience literature, like explained
variance Cadieu et al., 2007; Khaligh-Razavi and Kriegeskorte, 2014; Güçlü and van Gerven, 2015; Yamins and DiCarlo, 2016; Cadena et al., 2019 or Representational Similarity Analysis Kriegeskorte et
al., 2008; Khaligh-Razavi and Kriegeskorte, 2014, which are invariant to linear transformations)—any rotation or shear of one manifold with respect to the other would result in reduced scores. The
authors found that there was a strong one-to-one alignment between IT neurons and disentangled units to the point where the small number of disentangled dimensions discovered by the model were
statistically equivalent to a similarly sized subset of real neurons, and the alignment was significantly stronger than that with supervised classifiers (which learn an invariant representation) or
the generative model used in Chang and Tsao (2017). Furthermore, it was possible to visualize novel faces viewed by the primates from the decoded activity of just 12 neurons through their best
matched disentangled units. This result established the first direct link between coding in single IT neurons and disentangled representations, suggesting that the brain may be learning
representations that reflect the symmetries of the world. Other recent work showed that disentangled representations can also predict fMRI activation in the ventral visual stream (Soulos and Isik,
While many of the existing approaches to disentangled representation learning are generative models, thus fitting well within the predictive coding and free energy principle (Elias, 1955; Srinivasan
et al., 1982; Rao and Ballard, 1999; Friston, 2010; Clark, 2013) hypotheses of brain function, an alternative biologically plausible way to learn disentangled representations was recently proposed by
Johnston and Fusi (2021). The authors showed that disentangled representations can arise from learning to solve numerous diverse tasks in a supervised manner, which would be required to produce the
complex behaviors that biological intelligence exhibits in the natural world. A similar result was also demonstrated by Bernardi et al. (2020), who looked into the geometry of neural representations
for solving tasks in the RL framework in both primates and neural networks. They found that the final layer of an MLP trained through RL supervision to solve a number of tasks, as well as the
dorsolateral prefrontal cortex, the anterior cingulate cortex and the hippocampus of primates exhibited disentangled-like qualities. Although the representations of the underlying task variables were
rotated in the space of neural activation (unlike the axis aligned codes described in Higgins et al., 2021a), the underlying geometry was in line with what would be expected from disentangled
representations (see also Minxha et al., 2020; Panichello and Buschman, 2021; Rodgers et al., 2021; She et al., 2021; Boyle et al., 2022 for further evidence of not axis-aligned disentangled-like
representations in different brain areas of various species). The authors found that the degree to which such geometry was present correlated with the primates success on the tasks (no such
correlation existed for the more traditional decoding methods that do not take the geometry of the representation into account), and that such representations supported both strong generalization (as
measured by the abstraction scores) and high representational capacity (as measured by the shattering dimensionality scores) (Figure 6D).
Further validation of the biological plausibility of disentangled representation learning comes from comparing the data distribution that many modern ML approaches require for optimal disentangling
to the early visual experiences of infants (Smith et al., 2018; Wood and Wood, 2018; Slone et al., 2019). It appears that the two are similar, with smooth transformations of single objects dominating
both (Figure 7A). Disentangled representation also have properties that are believed to be true of the visual brain, such as “Euclideanization” or straightening of complex non-linear trajectories in
the representation space compared to the input observation space (Hénaff et al., 2019) (Figure 7B), and factorization into semantically interpretable axes, such as color or shape of objects (Figure
7C), which are hypothesized to be important for more data efficient and generalizable learning (Behrens et al., 2018), and for supporting abstract reasoning (Bellmund et al., 2018). It is
hypothesized that the same principles that allow biological intelligence to navigate the physical space using the place and grid cells may also support navigation in cognitive spaces of concepts,
where concepts are seen as convex regions in a geometric space spanned by meaningful axes like engine power and car weight (Gärdenfors, 2004; Gardenfors, 2014; Balkenius and Gärdenfors, 2016).
Learning disentangled representations that reflect the symmetry structure of the world could be a plausible mechanism for discovering such axes. Evidence from the ML literature has already
demonstrated the utility of disentangled representations for basic visual concept learning, imagination, and abstract reasoning (Higgins et al., 2018; Steenbrugge et al., 2018; van Steenkiste et al.,
2019; Locatello et al., 2020).
FIGURE 7
Figure 7. Similarities between various aspects of disentangled representation learning in ML (right column) and visual representation learning in the brain (left column). (A) The properties of the
visual data obtained through a head camera from toddlers (Smith et al., 2018; Slone et al., 2019) is similar to the properties of the visual data that allows ML approaches to discover disentangled
representations. The scenes are uncluttered, and contain many continuous transformations of few objects at a time. (B) Perceptual straightening of natural image trajectories observed in human vision
(Hénaff et al., 2019) is similar to the “Euclidenization” of the latent space learnt by disentangled ML models. (C) Factorized representations that align with semantically meaningful attributes
hypothesized to be important for further processing in the hippocampus (Behrens et al., 2018; Bellmund et al., 2018) resembles the factorized representations learnt by disentangled ML models.
5. Discussion
The question of what makes a good representation has been historically central to both ML and neuroscience, and both disciplines have faced the same debate: whether the best representation to support
intelligent behavior should be low-dimensional and interpretable or high-dimensional and multiplexed. While the former dominated both early neuroscience (Hubel and Wiesel, 1959; Barlow, 1972) and ML
(early success of feature engineering), recent development of high-throughput recording methods in neuroscience (Yuste, 2015; Eichenbaum, 2018; Saxena and Cunningham, 2019) and the success of large
black-box deep learning models in ML (Vaswani et al., 2017; Hu et al., 2018) have shifted the preference in both fields toward the latter. As a consequence, this led to deep classifiers emerging as
the main computational models for the ventral visual stream (Yamins et al., 2014; Yamins and DiCarlo, 2016), and a belief that higher-level sensory representations that can support diverse tasks are
too complex to interpret at a single neuron level. This pessimism was compounded by the fact that designing stimuli for discovering interpretable tuning in single cells at the end of the sensory
processing pathways is hard. While it is easy to systematically vary stimulus identity, it is hard to know what the other generative attributes of complex natural stimuli may be, and hence to create
stimuli that systematically vary along those dimensions. Furthermore, new representation comparison techniques between computational models and the brain became progressively population-based and
insensitive to linear transformations (Kriegeskorte et al., 2008; Khaligh-Razavi and Kriegeskorte, 2014; Yamins and DiCarlo, 2016), thus further stalling progress toward gaining a more fine-grained
understanding of the representational form utilized by the brain (Thompson et al., 2016; Higgins et al., 2021a). At the same time, it is becoming increasingly unlikely that high-dimensional,
multiplexed, uninterpretable population-based representations like those learnt by deep classifiers are the answer to what makes a “good” representation to support general intelligence, since ML
research has shown that models with such representations suffer from problems in terms of data efficiency, generalization, transfer, and robustness—all the properties that are characteristic of
biological general intelligence. In this article, we have argued that representations which reflect the natural symmetry transformations of the world may be a plausible alternative. This is because
both the nature of the tasks, and the evolutionary development of biological intelligence are constrained by physics, and physicists have been using symmetry transformations to discover and study the
“joints” and the “stable cores” of the world for the last century. By studying symmetry transformations, physicists have been able to reconcile explanatory frameworks, systematically describe
physical objects and even discover new ones. Representations that are equivariant to symmetry transformations are therefore likely to expose the relevant invariants of our world that are useful for
solving natural tasks. From the information theory perspective, such representations can be viewed as the simplest (in the context of Solomonoff induction; Solomonoff, 1964) and the most informative
representations of the input to support the most likely future tasks (MacKay, 1995, 2003; Wallace and Dowe, 1999; Hutter, 2004; Schmidhuber, 2010).
We have introduced the basic mathematical language for describing symmetries, and discussed evidence from ML literature that demonstrates the power of symmetry-based representations in bringing
better data efficiency, generalization, and transfer when included into ML systems. Furthermore, emerging evidence from the neuroscience community suggests that sensory representations in the brain
may also be symmetry-based. We hope that our review will give the neuroscience community the necessary motivation and tools to look further into how symmetries can explain representation learning in
the brain, and to consider them as an important general framework that determines the structure of the universe, constrains the nature of natural tasks and consequently shapes both biological and
artificial intelligence.
Author Contributions
IH and SR contributed to writing the review. DR contributed comments, discussions, and pointers that shaped the paper. All authors contributed to the article and approved the submitted version.
Conflict of Interest
IH, SR, and DR were employed by DeepMind.
Publisher's Note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the
reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
We would like to thank Fabian Fuchs, David Pfau, and Christopher Summerfield for interesting discussions, useful comments, and providing references.
1. ^These are also referred to as “grandmother cells” in the literature.
2. ^This kind of action is distinct from the action in physics; here, it just refers to the action of an operator.
3. ^To be precise, we are defining here a left action.
4. ^The term Abelian is also used in the literature.
5. ^Some groups will have both continuous and discrete aspects. For example, the group of all invertible matrices of a given size has a clear continuous aspect, but it also has a discrete aspect as
we cannot move continuously from a matrix with positive determinant to a matrix with negative determinant without hitting a matrix with determinant 0.
6. ^Although the term “disentanglement” and its opposite “entanglement” are also used in quantum mechanics (QM), and indeed the term “entanglement” refers to a mixing of factors in both ML (through
any diffeomorphism) and QM (through a linear combinations), there is no deeper connection between the two.
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Keywords: machine learning, representation learning, symmetries, physics, neuroscience, vision
Citation: Higgins I, Racanière S and Rezende D (2022) Symmetry-Based Representations for Artificial and Biological General Intelligence. Front. Comput. Neurosci. 16:836498. doi: 10.3389/
Received: 15 December 2021; Accepted: 08 March 2022;
Published: 14 April 2022.
Edited by:
Fabio Anselmi
, Baylor College of Medicine, United States
Reviewed by:
Peter Sutor
, University of Maryland, United States
Karl Friston
, University College London, United Kingdom
Copyright © 2022 Higgins, Racanière and Rezende. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction
in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic
practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Irina Higgins, irinah@deepmind.com; Sébastien Racanière, sracaniere@deepmind.com | {"url":"https://www.frontiersin.org/journals/computational-neuroscience/articles/10.3389/fncom.2022.836498/full","timestamp":"2024-11-13T03:21:51Z","content_type":"text/html","content_length":"768166","record_id":"<urn:uuid:d9eea631-7102-4f12-8bb0-caf3ab0c8f45>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00867.warc.gz"} |
The Learning Algorithms Of Fuzzy Neural Network
Posted on:2006-11-10 Degree:Master Type:Thesis
Country:China Candidate:W M Zheng Full Text:PDF
GTID:2178360182966415 Subject:Computational Mathematics
Artificial Neural Network (ANN) is a nonnumeric algorithm based on the imitation of the structure of human brain. The hybridization of Fuzzy Mathematics (FM) and Artificial Neural Network is Fuzzy
Neural Network (FNN). As a new technology in the field of artificial intelligence, FNN is an important method for information extraction, and it has a very large scale of application. In this paper,
in according to a lot of new achievements about Fuzzy Mathematics and ANN, we use Self Organizing Algorithm and Genetic Algorithm (GA) as the learning methods of FNN, hope to explore new way of
information extraction from complicated data. The main research work includes:In section one, we present the theory of Fuzzy Mathematics and its applications.In section two, we mainly discuss the
basic algorithm of ANN. In the past several decades, ANN has been applied in many fields extensively, and lots of different algorithms for ANN are created. In this paper, we will adopt the
fundamental frame of BP ANN to construct the network system.In section three, by the basic knowledge of Fuzzy Mathematics and ANN, we detailedly express the way of constructing the FNN which can
reflect the information of fuzzy number. In the same time, we use Self Organizing Algorithm and GA to modify the structure and parameters of FNN, hope that the combination of these algorithms can be
carried out in many kinds of complex informational problems, such as pattern recognition, quality evaluation, and data extraction and so on.
Keywords/Search Tags: Fuzzy Mathematics, Artificial Neural Network, Fuzzy Neural Network, Self Organizing Algorithm, Genetic Algorithm | {"url":"https://globethesis.com/?t=2178360182966415","timestamp":"2024-11-13T21:24:57Z","content_type":"application/xhtml+xml","content_length":"7197","record_id":"<urn:uuid:702c8fea-c078-4165-a944-028204f4c727>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00682.warc.gz"} |
How do you find the partial sum of an arithmetic sequence?
How do you find the partial sum of an arithmetic sequence?
An arithmetic series is the sum of the terms of an arithmetic sequence. The nth partial sum of an arithmetic sequence can be calculated using the first and last terms as follows: Sn=n(a1+an)2.
What is the partial sum formula?
Thus the sequence of partial sums is defined by sn=n∑k=1(5k+3), for some value of n. Solving the equation 5n+3=273, we determine that 273 is the 54th term of the sequence.
What is a partial sum of an arithmetic series?
The kth partial sum of an arithmetic series is. You simply plug the lower and upper limits into the formula for an to find a1 and ak. Arithmetic sequences are very helpful to identify because the
formula for the nth term of an arithmetic sequence is always the same: an = a1 + (n – 1)d.
What is a partial sum of a series?
A partial sum of an infinite series is the sum of a finite number of consecutive terms beginning with the first term. Each of the results shown above is a partial sum of the series which is
associated with the sequence .
Do all geometric series have a sum?
We can find the sum of all finite geometric series. But in the case of an infinite geometric series when the common ratio is greater than one, the terms in the sequence will get larger and larger and
if you add the larger numbers, you won’t get a final answer. The only possible answer would be infinity.
How do you calculate the sum of a geometric series?
The sum of a convergent geometric series can be calculated with the formula a ⁄ 1-r, where “a” is the first term in the series and “r” is the number getting raised to a power. A geometric series
converges if the r-value (i.e. the number getting raised to a power) is between -1 and 1.
How to find the sum of a geometric series?
Identify a 1\\displaystyle {a}_{1} a 1 and r\\displaystyle r r.
Confirm that − 1 < r < 1\\displaystyle -1<1 −1 < r < 1.
Substitute values for a 1\\displaystyle {a}_{1} a 1 and r\\displaystyle r r into the formula,S = a 1 1 − r\\displaystyle S=\\frac
Simplify to find S\\displaystyle S S.
What is the equation for the sum of a geometric series?
The sum of the geometric sequence is 56. To find the sum of any geometric sequence, you use the equation: Sn = a(rn−1) r−1 where: a –> is the first term of the sequence; in this case “a” is 8. r –>
is the ratio (what each number is being multiplied by) between each number in the sequence;
How do you find the partial sum of a series?
The kth partial sum of an arithmetic series is. You simply plug the lower and upper limits into the formula for a n to find a 1 and a k. Arithmetic sequences are very helpful to identify because the
formula for the nth term of an arithmetic sequence is always the same: a n = a 1 + (n – 1)d. where a 1 is the first term and d is the common difference. | {"url":"https://www.ammarksman.com/how-do-you-find-the-partial-sum-of-an-arithmetic-sequence/","timestamp":"2024-11-08T02:57:47Z","content_type":"text/html","content_length":"39510","record_id":"<urn:uuid:e5499f41-6fc5-44b8-9465-1f19904301ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00169.warc.gz"} |
Probabilistic Methods in Combinatorics
Applications of probabilistic techniques in discrete mathematics, including classical ideas using expectation and variance as well as modern tools, such as martingale and correlation inequalities.
Course Text:
At the level of Alon and Spencer, The Probabilistic Method (with an appendix of problems by Paul Erdos)
Topic Outline:
• The Basic Method - Examples from graph theory, combinatorics, and number theory of the use of the probabilistic method; the use of linearity of expectation
• The Second Moment Method - The use of Markov and Chebyshev inequalities; examples from number theory and random graphs
• The Lovasz Local Lemma - Applications in graph theory and computer science
• Correlation Inequalities - The four functions theorem; FKG and XYZ inequalities
• The Poisson Paradigm - Examples from random graphs; the use of martingales; Azuma's inequality, Telagrand's inequality; chromatic number of random graphs
• Alterations - Ramsey numbers; packing and recoloring; the Rodl nibble (or the semi-random method)
• Random Graphs - Clique number; chromatic number; branching processes; zero-one laws
• Combinatorial Discrepancy Theory - Balancing lights; Spencer's six standard deviations result; Beck-Fiala theorem and the Komlos conjecture; linear and hereditary discrepancy
• Derandomization - Conditional probabilities; limited independence of random variables
• Optional Material
• Combinatorial Geometry - Epsilon-nets and VC-dimension; additional topics
• Codes and Games - Balancing vector game; coin-weighing problems | {"url":"https://math.gatech.edu/courses/math/7018","timestamp":"2024-11-14T07:54:32Z","content_type":"text/html","content_length":"33408","record_id":"<urn:uuid:10ec120d-8796-4442-ba22-170e4fa1e971>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00355.warc.gz"} |
Magnetostatic, axially symmetric 5-dimensional vacuum Einstein equations are formulated in terms of harmonic maps. A correspondence between stationary, axially symmetric 4-dimensional vacuum Einstein
solutions and the magnetostatic, axially symmetric Jordan-Thiry solutions is established. The «Kaluza-Klein magnetic monopole» solution is recovered in a special case.
T. DERELI, A. ERIS, A. ERIS, and A. Karasu, “HARMONIC MAPS AND MAGNETOSTATIC, AXIALLY-SYMMETRICAL SOLUTIONS OF THE KALUZA-KLEIN THEORY,” NUOVO CIMENTO DELLA SOCIETA ITALIANA DI FISICA B-GENERAL
PHYSICS RELATIVITY ASTRONOMY AND MATHEMATICAL PHYSICS AND METHODS, pp. 102–112, 1986, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/57192. | {"url":"https://open.metu.edu.tr/handle/11511/57192","timestamp":"2024-11-02T02:08:10Z","content_type":"application/xhtml+xml","content_length":"53933","record_id":"<urn:uuid:cf2bc800-d969-4a1b-86d3-30583bb37fc1>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00831.warc.gz"} |
Next: MINIMUM PRECEDENCE CONSTRAINED SEQUENCING Up: Sequencing on One Processor Previous: MAXIMUM CONSTRAINED SEQUENCING TO   Index
• INSTANCE: Set T of tasks, a directed acyclic graph
• SOLUTION: A one-processor schedule for T that obeys the precedence constraints, i.e., a permutation
• MEASURE: The total storage-time product, i.e.,
• Good News: Approximable within 416].
Viggo Kann | {"url":"https://www.csc.kth.se/~viggo/wwwcompendium/node176.html","timestamp":"2024-11-11T06:46:28Z","content_type":"text/html","content_length":"5075","record_id":"<urn:uuid:ed132e1e-aefc-4601-9149-bc564ecc973b>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00619.warc.gz"} |
Add math selector of tolerance zone for profile with datum / Hexagon Measurement Systems / HexagonMI Idea Center
Add a math selector of the tolerance zone when evaluating a profile with a datum (line or surface)
At the moment it automatically uses the "Default" math. It should be possible to use the "LSQ" math.
When evaluating a profile without datum this is already possible.
• Default math will still be available (to evaluate according to the standard)
• With "LSQ" math, one can better see the general form of the part, outliers are clearly identified
• "LSQ" math gives more repeatable results (better for MSA)
I understand that the "Default" math is the one we should use to measure according to the standard
but it can be difficult to understand the result it delivers. "LSQ" math gives easier to understand results witch is crucial to adjust the machine.
See this example:
In principle, I agree, being able to used least-squares tolerance zone math would help trouble shoot and is more useful for providing correction information to production. This is being considered
for a future version but I am unable to make a firm commitment at present due to there being several other higher priority items which require addressing first. | {"url":"https://ideacenter.hexagonmi.com/communities/40/topics/1294-add-math-selector-of-tolerance-zone-for-profile-with-datum?lang=da","timestamp":"2024-11-10T09:34:04Z","content_type":"text/html","content_length":"47075","record_id":"<urn:uuid:3107ff1d-f0b8-48b8-9d68-59aa8ab9cb65>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00476.warc.gz"} |
Yibo Yang - machine learning
1. machine learning (8)
Mar 10 2020
All the Ways to Carve Up the ELBO
My list of fancy decompositions of the aggregate ELBO, focusing on the role of the aggregate KL regularizer, its relation to the aggregate posterior, and the mutual information between the data and
the latent variable. | {"url":"https://yiboyang.com/tag/machine-learning.html","timestamp":"2024-11-05T00:31:33Z","content_type":"text/html","content_length":"9169","record_id":"<urn:uuid:94172bca-7d2b-4e8a-9920-205a70392763>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00470.warc.gz"} |
Math Colloquia - Circular maximal functions on the Heisenberg group
The spherical average has been a source of many problems in harmonic analysis.
Since late 90's, the study of the maximal spherical means on the Heisenberg group $mathbb{H}^n$ has been started to show the pointwise ergodic theorems on the groups.
Later, it has turned out to be connected with the fold singularities of the Fourier integral operators, which leads to the $L^p$ boundedness of the spherical maximal means on the Heisenberg group
$mathbb{H}^n$ for $nge 2$.
In this talk, we discuss about the $L^p$ boundedness of the circular maximal function on the Heisenberg group $mathbb{H}^1$. The proof is based on the the square sum estimate of the Fourier integral
operators associated with the torus arising from the vector fields of the Heisenberg group algebra.
We compare this torus with the characteristic cone of the Euclidean space. | {"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&l=en&page=7&sort_index=date&order_type=asc&document_srl=793545","timestamp":"2024-11-05T09:24:12Z","content_type":"text/html","content_length":"43817","record_id":"<urn:uuid:f0f7a019-e43e-434f-b766-8c7879a3215e>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00224.warc.gz"} |
Write an equation for the following and then solve.
The product of q and -4.7 is equal to 9.4. ...
Solved on Oct 28, 2023
Find the value of $q$ given that the product of $q$ and -4.7 is equal to 9.4.
STEP 1
Assumptions1. The product of $q$ and -4.7 is equal to9.4. We need to find the value of $q$
STEP 2
First, we write the given information as an equation. The product of $q$ and -4.7 is equal to9.4 can be written as$q \\times -4.7 =9.4$
STEP 3
To solve for $q$, we need to isolate $q$ on one side of the equation. We can do this by dividing both sides of the equation by -.7.
$q = \frac{9.}{-.7}$
STEP 4
Now, we calculate the value of $q$.
$q = \frac{9.4}{-4.7} = -2$So, the value of $q$ is -2. | {"url":"https://studdy.ai/learning-bank/problem/find-the-value-of-q-where-the-pr-cMHZneSGlBQVNInZ","timestamp":"2024-11-05T16:54:15Z","content_type":"text/html","content_length":"132347","record_id":"<urn:uuid:bfbb3361-2bf5-46d7-9587-46f03820dec2>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00862.warc.gz"} |
663 miles per hour to feet per minute
Speed Converter - Miles per hour to feet per minute - 9,663 feet per minute to miles per hour
This conversion of 9,663 miles per hour to feet per minute has been calculated by multiplying 9,663 miles per hour by 87.9999 and the result is 850,343.9999 feet per minute. | {"url":"https://unitconverter.io/miles-per-hour/feet-per-minute/9663","timestamp":"2024-11-14T14:58:28Z","content_type":"text/html","content_length":"15709","record_id":"<urn:uuid:e949f2a9-dc68-4e26-829d-e790cab553c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00387.warc.gz"} |
Accurately Performing Large Calculations
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Well, I would appreciate if anyone could tell me a way to accurately perform a large calculation, as in (for example) 437289478942343890483 / 2 without worrying about it being rounded.
I know very little about how the calculations are actually made, aside from the use of binary files (though I could be wrong about that, too) so I lack the ability to really make something that
OP could do the job.
Thanks for any help.
mIRC's calculation ability is limited. That cannot be changed (by you). However, you should be able to perform larger calculations by passing arguments to a dll that is written in a language that
supports large calculations.
Originally Posted By: ZorgonX
Well, I would appreciate if anyone could tell me a way to accurately perform a large calculation, as in (for example) 437289478942343890483 / 2 without worrying about it being rounded.
I know very little about how the calculations are actually made, aside from the use of binary files (though I could be wrong about that, too) so I lack the ability to really make something that
could do the job.
Thanks for any help.
Im not sure what your asking for because the calc 437289478942343890483 / 2 returns the value not being rounded, and in my experience is the correct return value however if your gonna use something
like 437289478942343890483 / 42442424 then it will return a value.xxx < if you want that removed at all times I suggest using something like $left($round($calc(caculations here),2),-3)
if $reality > $fiction { set %sanity Sane }
Else { echo -a *voices* }
Originally Posted By: Lpfix5
Im not sure what your asking for because the calc 437289478942343890483 / 2 returns the value not being rounded, and in my experience is the correct return value however if your gonna use
something like 437289478942343890483 / 42442424 then it will return a value.xxx < if you want that removed at all times I suggest using something like $left($round($calc(caculations here),2),-3)
That's strange, because for me, $calc(437289478942343890483 / 2) returns 218644739471171940000. The four zeros at the end represent the rounding present. Note I'm using the most recent version of
mIRC, version 6.3.
Originally Posted By: genius_at_work
mIRC's calculation ability is limited. That cannot be changed (by you). However, you should be able to perform larger calculations by passing arguments to a dll that is written in a language that
supports large calculations.
I had thought of trying this, but after a quick search on Google and a few mIRC script/dll sites, I couldn't find anything of the sort.
I'm afraid that I don't possess the ability to write such a thing, so that's out of the question.
Thanks for your input.
Actually, its fairly easy to "cheat" at doing this.
Lets take the number 42. it s a 4 and a 2. Divide 4 by 2 you get 2. Then divide 2 by 2 and you get 1. Stick them together and you get you answer: 21
Let's try 456.
4/2 = 2
5/2 = 2.5
6/2 =3
200+25+3= 228 = 456/2
See where I'm going with this?
So basically, you have to divide the fist number by 2 and store it to a variable. IF it isn't even divide by to you'll have to add 5 to the next step. Divide the next number by 2 (add 5 if the
previous on3 was odd) then apply that to the end of the variable. Continue doing this until you run out of digits.
As above... It'd end up being 2 $+ 2 $+ $calc(3+5)
The same can be done for subtraction, addition and multiplication. I'd write out the code, but I'm dead tired and on my way to bed. Perhaps later on today I'll get around to it. Though, hopefully
this is enough pseudocode to make something work for you. In theory, you should be able to use mirc to divide HUGE numbers by at least 8 digits worth of numbers. With more tinkering, you should be
able to divide a 900 digit number by another 900 digit number and get an exact answer.
Last edited by Thrull; 07/09/07 11:53 AM.
I think the problem still exists using the method you have shown. Sure you can take each digit and divide it by a number, but in the end you are left with a whole pile of individual place values
that still need to be added up. If the numbers are large, that act of adding will likely result in the same rounding error. Ex:
//echo -a $calc(7700000000000000000000000 / 2) = $calc(3500000000000000000000000 + 350000000000000000000000)
The only way I can see this possibly working is to use loops to basically do true long division. Take portions of the original value, perform the division on those portions, and then gradually add
the individual results together to get the final answer.
Fine then... here you go, you can figure out how and where I go by this to properly calc, I can even make the script smaller but heres a way to do it..
SYNTAX: //echo -a $pcalc(valuethatislarge,+-/*,valuesmaller)
DO NOTE:
However, you see i see $1 has the large entry and $3 has the smaller and $2 has the +/*- Functions. In this script you can add if/elseif/else statements to make it "better" and the script can be
written a tad longer to make multi calculations in one time. Remeber though that variables have abilitys to perform calculations without the $calc command. I would also put a $matchtok sequence to
verify if dividend it should match if multiplied etc... ALOT if statements lol
Here you go, but first ill show you what I calculated and what my response was..
$pcalc(24984284092428940492049049,/,2) returns 12492142046214470246024524.5
while before $calc(24984284092428940492049049/2) would return 12492142046214470000000000Even if you add my first return togheter it brings you back to the original result.
Small,ugly,but it works this way lol.. heres your workaround.
alias Pcalc {
if ($len($1) > 17) && ($len($3) < 17) {
%pcalc.a = $left($1,+17) $2 $3 | %pcalc.b = $mid($1,17) $2 $3 | return $+(%pcalc.a,%pcalc.b)
if $reality > $fiction { set %sanity Sane }
Else { echo -a *voices* }
That's a good idea, thank you. It seems to work quite nicely up to a point, as in, 34 digits. But after that, it appears to start rounding.. but it's a great proof of concept. I'll tweak it a bit and
get it to do right (using a while loop, of course). Thanks for your help, it's very much appreciated.
EDIT: One issue with this approach is how it handles zeros when they're at the start of a 17-character "block". I'm currently making a way to work around this, though, and I'll post what I come up
with in here when I think I'm done in case anyone else wants the script.
EDIT AGAIN: Here's the code I came up with. It seems to perform well under my tests.. I haven't found a flaw with it. At least with division, which is all I tested it with. [EDIT: it only works with
division..][Another EDIT: it has a flaw involving ones at the start of 17-character blocks.] I'm also including an identifier I wrote, since it makes the script itself look much more.. clean, without
having to do that in the main script.
alias Lcalc {
;SYNTAX: $Lcalc(number,operator,smallernumber)
;Example, $Lcalc(64,/,2) returns 32.
if ($len($3) < 17) {
var %step = 1
while ($mid($1,$calc((%step - 1) * 17)) != $null) { var %cur = $mid($1,$calc((%step - 1) * 17 + 1),17) | if ((!$remove(%cur,0)) && ($len(%cur > 1))) { var %lcalc.b = %cur } | else { var %lcalc.b = %cur $2 $3 | var %lcalc.b = $str(0,$countleft(%cur,0)) $+ %lcalc.b } | var %lcalc.t = %lcalc.t $+ %lcalc.b | inc %step }
return %lcalc.t
OP }
alias countleft {
;SYNTAX: $countleft(text,character)
;The number of "character" on the left side of text is counted and returned.
;Example: $countleft(hhhithere,h) would return 3.
;You can put a - next to 'character' to count from the right. Example, $countleft(hhhithere,-e) would return 1.
var %step = 1 , %ident = $iif($left($2,1) == -,$iif($len($2) == 2,right),left) $+ $chr(40) $+ z $+ $chr(41) , %2 = $iif($len($2) == 2,$right($2,1),$left($2,1))
while ($ [ $+ [ $replace(%ident,z,$1 $+ $chr(44) $+ %step) ] ] == $str(%2,%step)) inc %step
return $calc(%step - 1)
Note I changed it from Pcalc to Lcalc, to avoid confusion and because "loop" starts with "L".
Since the script appears full of holes, anyone who wants to try to fix it (or provide their own script as a replacement) can feel free. :P
Thank you.
Last edited by ZorgonX; 07/09/07 10:53 PM.
Originally Posted By: ZorgonX
That's a good idea, thank you. It seems to work quite nicely up to a point, as in, 34 digits. But after that, it appears to start rounding.. but it's a great proof of concept. I'll tweak it a bit and
get it to do right (using a while loop, of course). Thanks for your help, it's very much appreciated.
EDIT: One issue with this approach is how it handles zeros when they're at the start of a 17-character "block". I'm currently making a way to work around this, though, and I'll post what I come up
with in here when I think I'm done in case anyone else wants the script.
EDIT AGAIN: Here's the code I came up with. It seems to perform well under my tests.. I haven't found a flaw with it. At least with division, which is all I tested it with. [EDIT: it only works with
division..][Another EDIT: it has a flaw involving ones at the start of 17-character blocks.] I'm also including an identifier I wrote, since it makes the script itself look much more.. clean, without
having to do that in the main script.
alias Lcalc {
;SYNTAX: $Lcalc(number,operator,smallernumber)
;Example, $Lcalc(64,/,2) returns 32.
if ($len($3) < 17) {
var %step = 1
while ($mid($1,$calc((%step - 1) * 17)) != $null) { var %cur = $mid($1,$calc((%step - 1) * 17 + 1),17) | if ((!$remove(%cur,0)) && ($len(%cur > 1))) { var %lcalc.b = %cur } | else { var %lcalc.b = %cur $2 $3 | var %lcalc.b = $str(0,$countleft(%cur,0)) $+ %lcalc.b } | var %lcalc.t = %lcalc.t $+ %lcalc.b | inc %step }
return %lcalc.t
alias countleft {
;SYNTAX: $countleft(text,character)
;The number of "character" on the left side of text is counted and returned.
;Example: $countleft(hhhithere,h) would return 3.
;You can put a - next to 'character' to count from the right. Example, $countleft(hhhithere,-e) would return 1.
var %step = 1 , %ident = $iif($left($2,1) == -,$iif($len($2) == 2,right),left) $+ $chr(40) $+ z $+ $chr(41) , %2 = $iif($len($2) == 2,$right($2,1),$left($2,1))
while ($ [ $+ [ $replace(%ident,z,$1 $+ $chr(44) $+ %step) ] ] == $str(%2,%step)) inc %step
return $calc(%step - 1)
Note I changed it from Pcalc to Lcalc, to avoid confusion and because "loop" starts with "L".
Since the script appears full of holes, anyone who wants to try to fix it (or provide their own script as a replacement) can feel free. :P
Thank you.
Listen to this lol <<< btw not mean way
I would write the whole script which would be lenghty the reason why I say lenghty is because studying the blocks of 17 max calcs per operation would require this method a division script alone,
multiplication,Substraction and addition script alone, therefore there would be 4 Operators that would need to be scripted individual why I say this.. EX:...
this is a short re-code i did so that the lenght of $1 works weither it be over 34 chars MAX: 52 for this script. Also it wont work if charges are less then 34 due to my IF (%lc.c) check meaning only
if %lc.c is present.. ""More if statement can be added"
alias newc {
if ($len($1)) { %lc.a = $left($1,+17) | %lc.b = $left($remove($1,%lc.a),+17) | %ts.a = $+(%lc.a,%lc.b) | %lc.c = $left($remove($1,%ts.a),+17) }
IF (!%lc.c) { unset %lc.c }
if (!%lc.b) { unset %lc.b }
alias lcalc {
newc $1
if (%lc.c) { %lc.a = %lc.a $2 $3 | %lc.b = %lc.b $2 $3 | %lc.c = %lc.c $2 $3 | return $+(%lc.a,%lc.b,%lc.c) }
Here we go
Dividing by 2 would make the script rather easy hwoever what happens when we divide by 100?
Notice all the decimal points, now my method of approach to remove the decimal points would be to remove and store them to a temp var example .72 < i would "$remove(x,.72)" and store it in var I
would do the same for all decimal points. this way in the end you can calc the decimal points and re-add them to the end of the valued return. if example I would put $calc(.72+.51+.76) which the
result would be 1.99 so at the end of this result in question it would be 6762.99 in theory right?
See my scenario if not ill try to explain it again. There's a requirement for * / + - in each of their own scenarios that cannot be made by a single written script for all four vars, I mean
if ($2 == /) for example if you use that method then it will be in "one" script but I mean you need 4 seperate events.
Really if you got alot of time on your hands knock yourself out. You seem to know and understand what I mean so hope this helps out for you.
Ill check back later if you have a question I like math to an extent...
if $reality > $fiction { set %sanity Sane }
Else { echo -a *voices* }
I've become convinced that I could more easily learn to write a DLL, even if I only have extremely limited experience with C++, than to write an mIRC script to do this.
Thanks for your help, and I think I have a better understanding of how this sort of thing works. If I fail at the DLL, I'll just work more on the script..
If you were wondering why I needed this, I was writing a simple encryption algorithm using mIRC scripting. When I think about it, it really isn't all that great.. so I'm not sure if it's worth the
If you can find a tutorial on how to compile a dll, the actual code for the dll would be quite simple.
function longdivision return $1 / $2
There are likely extension files that would need to be included in the dll to allow large math operations.
In theory the dll would be the better approach but if your like me and strive on mIRC nothing could stop you
if $reality > $fiction { set %sanity Sane }
Else { echo -a *voices* }
I wonder if it would be possible to use some kind of COM object to perform the calculations..
Originally Posted By: genius_at_work
I wonder if it would be possible to use some kind of COM object to perform the calculations..
Im not sure how it works, just by looking at it im thinking you could call? windows calculator but how is the result sent back.. LOL see I don't know nothing about $com
if $reality > $fiction { set %sanity Sane }
Else { echo -a *voices* }
Well, you always have the option of using the clipboard with the Windows calculator if you could put the calculations through there. Of course, if you can put calculations through there in the first
place, I'm sure you can also retrieve the results without needing to use the clipboard.
Invision Support
#Invision on irc.irchighway.net
I was thinking more of calling a WSH script of some sort. Again, I don't know how com calls work, so I can't be of any help there.
Getting WSH to perform and return calculations is rather easy.
Your still going to run into
some problems
however. To the best of my knowledge each of the possible script languages you can call with WSH wont handle 64bit signed integers (properly) let alone even higher.
Last edited by Mpdreamz; 09/09/07 09:24 PM.
Originally Posted By: ZorgonX
I've become convinced that I could more easily learn to write a DLL, even if I only have extremely limited experience with C++, than to write an mIRC script to do this.
Thanks for your help, and I think I have a better understanding of how this sort of thing works. If I fail at the DLL, I'll just work more on the script..
If you were wondering why I needed this, I was writing a simple encryption algorithm using mIRC scripting. When I think about it, it really isn't all that great.. so I'm not sure if it's worth the
I got questions
A) Why would one need an encryption algorhitm
B) Why use Num over alpha/num? EX:. $base() functions.
if $reality > $fiction { set %sanity Sane }
Else { echo -a *voices* }
Well i was bored a bit wrote a bit of stuff, theres multi Lenght checks for up to 52 chrs operations.
alias newc {
if ($len($1)) { %lc.a = $left($1,+17) | %lc.b = $left($remove($1,%lc.a),+17) | %ts.a = $+(%lc.a,%lc.b) | %lc.c = $left($remove($1,%ts.a),+17) }
IF (!%lc.c) { unset %lc.c }
if (!%lc.b) { unset %lc.b }
Here's the lcalc alias that goes with the new calc lenght check. Needs work on both scripts but its an advancement if you will call it that.
alias lcalc {
newc $1
if (%lc.c) {
%lc.a = %lc.a $2 $3
%lc.b = %lc.b $2 $3
%lc.c = %lc.c $2 $3
if ($chr(46) isin %lc.a) || ($chr(46) isin %lc.b) || ($chr(46) isin %lc.c) {
%lc.a = $round(%lc.a,1) | %lc.a = $left(%lc.a,$+(+,$calc($len(%lc.a)-3)))
%lc.b = $round(%lc.b,1) | %lc.b = $left(%lc.b,$+(+,$calc($len(%lc.b)-3)))
%lc.c = $round(%lc.c,1) | %lc.c = $left(%lc.c,$+(+,$calc($len(%lc.c)-3)))
return $+(%lc.a,%lc.b,%lc.c)
elseif (%lc.b) {
%lc.a = %lc.a $2 $3
%lc.b = %lc.b $2 $3
if ($chr(46) isin %lc.a) || ($chr(46) isin %lc.b) {
%lc.a = $round(%lc.a,1) | %lc.a = $left(%lc.a,$+(+,$calc($len(%lc.a)-3)))
%lc.b = $round(%lc.b,1) | %lc.b = $left(%lc.b,$+(+,$calc($len(%lc.b)-3)))
return $+(%lc.a,%lc.b)
elseif (%lc.a) {
%lc.a = %lc.a $2 $3
if ($chr(46) isin %lc.a) {
%lc.a = $round(%lc.a,1) | %lc.a = $left(%lc.a,$+(+,$calc($len(%lc.a)-3)))
return $+(%lc.a)
if $reality > $fiction { set %sanity Sane }
Else { echo -a *voices* }
Here is my first long math code. It is for addition because I figured I would start with the easiest operator first. Also, the long addition operator may be needed for the other long operations.
alias math+ {
;$1=A, $2=B
; A+B=C
var %a = $gettok($1,1,46), %b = $gettok($2,1,46)
var %y = $gettok($1,2,46), %z = $gettok($2,2,46)
var %c = $null, %r = 0, %w = 0
if ($len(%a) < $len(%b)) %a = $str(0,$calc($v2 - $v1)) $+ %a
elseif ($len(%b) < $len(%a)) %b = $str(0,$calc($v2 - $v1)) $+ %b
if ($len(%y) < $len(%z)) %y = %y $+ $str(0,$calc($v2 - $v1))
elseif ($len(%z) < $len(%y)) %z = %z $+ $str(0,$calc($v2 - $v1))
var %i = $len(%y), %ii = 0
while (%i > %ii) {
%w = $calc($mid(%y,%i,1) + $mid(%z,%i,1) + %r)
%c = $+($right(%w,1),%c)
%r = $left(%w,-1)
dec %i
if (%c > 0) %c = $+(.,%c)
var %i = $len(%a), %ii = 0
while (%i > %ii) {
%w = $calc($mid(%a,%i,1) + $mid(%b,%i,1) + %r)
%c = $+($right(%w,1),%c)
%r = $left(%w,-1)
dec %i
if (%r > 0) %c = $+(%r,%c)
return %c
//echo -a $math+(<long number>,<long number>)
Note: Supports decimals
Note: Does NOT support negative numbers (yet).
Link Copied to Clipboard | {"url":"https://forums.mirc.com/ubbthreads.php/topics/185493/accurately-performing-large-calculations","timestamp":"2024-11-07T02:58:22Z","content_type":"text/html","content_length":"129946","record_id":"<urn:uuid:c32cf663-12ef-4ee9-8062-fc5f2b10be45>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00766.warc.gz"} |
Problem 5The thermal efficiency and power output of a solar pond powe
Problem 5The thermal efficiency and power output of a solar pond power plant are to be determined. The operations of this power plant can be modeled using the ideal Rankine cycle. The working fluid
of this power plant is refrigerant R-134a and the power plant has a mass flow rate of m = 3.5 kg/s. The working fluid is a saturated vapor at a pressure of P = 1.8 MPa at the turbine entrance, and a
pressure of P = 0.8 MPa at the turbine exit. Draw the cycle on a T-s diagram w.r.t. the saturation lines. | {"url":"https://tutorbin.com/questions-and-answers/problem-5the-thermal-efficiency-and-power-output-of-a-solar-pond-power","timestamp":"2024-11-07T06:28:48Z","content_type":"text/html","content_length":"63389","record_id":"<urn:uuid:30186953-ef08-4b3c-ae70-a7f94edbdfa2>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00143.warc.gz"} |
augmented A-infinity algebra
Not sure I made my point clear; there is the notion of augmentation for an Aoo-algebra pure and simple
Thansk, Jim.
I have now added a warning remark here about a possible simplicial meaning of augmentation that one might think of.
Also I have fixed some misprints. (Possiby I did introduce new misprints though… ;-)
(I'm just testing something ... Andrew)
The title is a bit misleading since you have something much more spectral in mind. Some misprints in the references.
started augmented A-infinity algebra
there is the notion of augmentation for an Aoo-algebra pure and simple
Hm, you mean a notion different from the pure and simple “equipped with a map to the base $\infty$-ring”, as in the entry?
Maybe you just want me to make a distinction between on the one hand $A_\infty$-algebras in characteristic 0, hence in chain complexes and on the other the general notion of $A_\infty$-algebras in
spectra? If so, I’d ask: why? The simple definition of augmentation does not depend on this.
Haha, speaking of misprints: I know of a mathematician named Gregory Arone, but not Fregory Arone! :-)
I just meant without invoking $E_\infty$
Urs, I just meant without invoking Eoo
Jim, in case this helps, you can render LaTeX in comments by first clicking on Markdown+Itex, where it says “Format comments as” below the comment box.
I have added:
to the base E-∞ ring (which might be a plain commutative ring).
Does that help?
yes, that helps
added to the Definition-section of augmented A-infinity algebra a pointer to the very general definition 5.2.3.14 in Higher Algebra. | {"url":"https://nforum.ncatlab.org/discussion/5020/augmented-ainfinity-algebra/","timestamp":"2024-11-07T02:28:05Z","content_type":"application/xhtml+xml","content_length":"25923","record_id":"<urn:uuid:ceb21da6-26bc-4e94-9438-1984215319b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00130.warc.gz"} |
How Much Does 4 Cups of Water Weigh? Unveil the Facts
Four cups of water weigh approximately 2 pounds or 907.185 grams. This weight can fluctuate slightly depending on the water temperature.
Measuring the weight of water is straightforward because it has a known density.
One cup of water typically weighs 8. 3454 ounces or 236. 588 milliliters, making it an essential component in cooking and baking where precise measurements matter.
Understanding the weight of water is crucial, not just in the kitchen but also in scientific experiments and industrial applications where accuracy is key.
For those tracking their water intake for health or fitness goals, knowing the weight of water can help in monitoring consumption.
This knowledge is also imperative when dealing with larger volumes, as in agriculture or aquarium keeping, where the weight impacts the structural integrity of containers and systems.
The Weight Of Water
Understanding how much water weighs is useful in cooking and science. Water’s weight can be easy to calculate. Let’s dive in to learn about the weight of 4 cups of water.
Volume To Mass Conversion
To find out how much 4 cups of water weigh, we need to convert volume to mass. Here’s a simple breakdown:
• 1 cup of water roughly weighs 236.59 grams.
• Thus, 4 cups would be 4 times 236.59 grams.
│Cups of Water │Weight in Grams │
│1 │236.59 │
│4 │946.36 │
In simpler terms, 4 cups of water weigh approximately 946.36 grams, or nearly one kilogram.
Factors Affecting Water Weight
Several factors can change how much water weighs. These include temperature, container shape, and elevation.
• Temperature: Warm water expands, so it weighs less than cold water.
• Container: The container’s shape doesn’t affect weight but can affect how much pours in.
• Elevation: Water boils at lower temperatures at higher elevations, potentially affecting measurements.
The given weight assumes water at room temperature and at sea level. Temperature and pressure changes would require recalculations.
Measurement Units Interplay
Understanding Measurement Units Interplay is vital in daily tasks like cooking. The relationship between cups, grams, ounces, and pounds might seem complex.
Yet, it is important for precision in recipes and other areas. Here’s how these units translate when measuring water.
From Cups To Grams
One cup of water typically weighs 236.588 grams. To convert cups to grams, we use a simple multiplication.
• 1 cup of water = 236.588 grams
• 4 cups of water = 4 x 236.588 grams
• Total weight = 946.352 grams
With this conversion, we can understand weights for any recipe.
Understanding Ounces And Pounds
Water’s weight in ounces and pounds also relates to cups.
Cups of Water Ounces Pounds
1 Cup 8.345 ounces 0.522 pounds
4 Cups 33.38 ounces 2.086 pounds
Remember, these conversions are specific to water due to its density.
The Magic Of Four Cups
Drinking four cups of water a day can work wonders for your body. This simple act hydrates your system, keeps your skin clear, and can even boost your energy levels.
But here’s something you might not have considered—how much does that amount of water actually weigh?
It’s a surprisingly fascinating topic with some common misunderstandings to clear up.
Exact Weight Reveal
Let’s dive right into the specifics. The weight of water is a simple science fact. One cup of water weighs approximately 236.5 grams. Multiply that by four, and you have the exact weight of four cups
• 1 cup of water = 236.5 grams
• 4 cups of water = 946 grams
In pounds, that’s about 2.08 lbs. So, you’re lifting a little over two pounds whenever you hold four cups of water. Pretty neat, right?
Common Misconceptions
Many people think that the volume and weight of water are the same. But, that’s not quite accurate. Let’s clarify:
Measurement Volume Weight
1 cup (US) 8 fluid ounces 236.5 grams
4 cups (US) 32 fluid ounces 946 grams
While it’s true that for water, one fluid ounce weights about one ounce, this doesn’t hold for other liquids or materials. Remember, only water has this special property due to its density.
Influence Of Temperature On Water Weight
Many curious minds ponder how temperature affects the weight of water. Especially, when it comes to precision in recipes or science experiments.
Water is unique. Its weight can change with temperature. Let’s dive into how this happens and what it means when measuring 4 cups of water.
Warmer Water Weigh-in
As water heats up, its molecules start to move faster. They spread out and take up more space. This concept is known as thermal expansion.
Despite this increase in volume, the mass remains the same. So, 4 cups of warm water may appear slightly more voluminous, but its weight does not change. Here’s what happens:
• 370C (980F): Water’s density decreases
• 4 cups volume: Looks more due to expansion
The Cold Water Equation
On the flip side, cold water contracts. Molecules slow down and get closer together. This makes cold water denser than warm water.
However, the mass of water is conserved. If you measure 4 cups of cold water, the weight remains consistent with 4 cups of water at room temperature. Below illustrates cold water’s characteristics:
Temperature Effect on Water
<4ºC (39.2ºF)> Density increases
Volume Might seem less
Practical Tips For Measuring Water
Welcome to ‘Practical Tips for Measuring Water’, an essential guide for all your cooking and baking needs.
Knowing how much 4 cups of water weighs is crucial in many recipes. Let’s dive into some effective methods to measure water accurately.
Kitchen Scale Usage
Using a kitchen scale is the most accurate way to measure water. Place a bowl on the scale and zero it out.
Pour in water until the scale reads 32 ounces for 4 cups. This method ensures precision every time. Here are some steps to follow:
1. Turn on your kitchen scale and set it to ounces.
2. Place an empty bowl on the scale.
3. Press tare or zero to negate the bowl’s weight.
4. Pour water slowly into the bowl.
5. Stop when the scale shows 32 ounces (weight of 4 cups of water).
A kitchen scale also helps in translating volume measurements to weight, which is especially handy for non-standard cups.
Visual Estimates And Their Pitfalls
Estimating water volume by eye often leads to mistakes. Factors like container shape and our perspective can alter visual assessments. Let’s look at some common pitfalls:
• Container’s shape can trick the eye. Wide bowls make water look shallow.
• Measurement lines on containers are not always accurate. A scale is better.
• A glance might overestimate or underestimate water volume.
• Distractions can cause measurement errors.
For mistake-free cooking, avoid relying on sight alone. Pair visual cues with a scale or measuring cup for the best results.
FAQs About the Weight of 4 Cups of Water
What Is The Weight Of 4 Cups Of Water?
One cup of water typically weighs 236. 5 grams. Therefore, four cups weigh approximately 946 grams or about 2. 08 pounds.
How Does Water’s Temperature Affect Its Weight?
The temperature of water doesn’t affect its weight significantly. However, its density can change with temperature, leading to slight variations in how much a certain volume weighs.
Can Altitude Impact The Weight Of Water?
Altitude doesn’t impact the weight of water. Regardless of altitude, the weight of water remains consistent because the mass of the water doesn’t change.
Does Water’s Weight Change With Different Measurements?
The weight of water is consistent across different measurements. Four cups of water will always weigh around 946 grams, whether measured in grams, pounds, or ounces.
Wrapping up, understanding the weight of 4 cups of water is simple. It equates to roughly 2 pounds or 0. 907 kilograms.
This knowledge is essential for cooking accuracy and dietary needs. Remember: water’s density is the key factor. Keep measuring and stay hydrated! | {"url":"https://sizepedia.org/how-much-does-4-cups-of-water-weigh/","timestamp":"2024-11-11T20:22:31Z","content_type":"text/html","content_length":"92659","record_id":"<urn:uuid:18106b4c-9950-4f0a-a9d5-6695fb476775>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00816.warc.gz"} |
SPC Glossary
Unsure of a certain SPC term? Use this page to find the technical definition you’re after.
Accuracy of measurements refers to the closeness of agreement between observed values and a known reference standard. Any offset from the known standard is called bias.
Assignable Cause
A cause of process variation that isn’t random or inherent and that is attributable to some identifiable and controllable influence. A dulled cutting tool, for example, could be an assignable cause
for a cutting process’s increased variation.
Attribute data
Qualitative data that can be counted for recording and analysis. Examples include: number of defects, number of errors in a document, number of rejected items in a sample, presence of paint
flaws. Attribute data are analyzed using the p-, np-, c- and u-charts.
See mean.
Average Run Length (ARL)
Short for average run length, ARL is the interval between out-of-control events that can be expected. For example, is a common out-of-control event chosen to determine a process’ ARL. When an
out-of-control event appears on a control chart, an analyst can examine the interval between that event and the previous out-of-control event. If the interval matches or exceeds the process’ ARL
value: a) the process can probably be classified as still in-control, b) the violation can probably be attributed to typical process variation, and c) a search for an assignable cause can probably be
considered unwarranted. The inverses of these statements are likely true if the interval between out-of-control events is smaller than the ARL value.
The offset of a measured value from the true population value.
Binomial Distribution
A discrete probability distribution used for counting the number of successes and failures, or conforming and nonconforming units. This distribution underlies the p-chart and the np-chart.
Box and Whisker Plot
A graphical display of data tat shows the median and upper and lower quartiles, along with extreme points and any outliers.
In the field of SPC as it is applied to manufacturing, a recognized strategy for identifying manufacturing problems and solutions.
A measure of the amount of variation inherent in a stable process. Capability can be determined using data from control charts and histograms and is often quantified using the C[p] and C[pk] indices.
A process is said to be capable when all of its output is in-spec.
Cause-and-Effect Diagram
A quality-control tool used to analyze potential causes of problems in a product or process. It organizes potential problems into four groups: man, method, machine, and material. It is also called a
fishbone diagram or an Ishikawa diagram, after its developer.
A control chart based on counting the number of defects per constant size subgroup. Also known as a Count of Nonconformities chart. The c-chart is based on the Poisson distribution.
Center Line (CL)
The line on the control chart that represents the long-run expected, or average value, of the quality characteristic that corresponds to the in-control state which occurs when only chance causes are
Central Limit Theorem
An important statistical theorem that states that subgroup averages tend to be normally distributed even if the output as a whole is not. This allows control charts to be widely used for process
control, even if the underlying process is not normally distributed.
Check Sheet
In the field of SPC, a simple user-friendly form for collecting data over a period of time. Originally, it was a paper form but today it is often found integrated into SPC software.
Coefficient of Correlation
See Correlation below.
Common Causes
Problems with the system itself that are always present, influencing all of the production until found and removed. These are “common” to all manufacturing or production output. Also called chance
causes, system causes, or chronic problems. Compare common causes to special causes.
Continuous Improvement
The ongoing improvement of products, services, or processes through incremental and breakthrough improvements.
Control Chart
A graphical mechanism for deciding whether the underlying process has changed based on sample data from the process. Control charts help determine which causes are “special” and thus should be
investigated for possible correction. Control charts contain the plotted values of some statistical measure for a series of samples or subgroups, along with the upper and lower control limits for the
Control Limits
Numerical limits, often represented on control charts as horizontal lines, which indicate whether the process is statistically in control. There is typically an upper control limit (UCL) and a lower
control limit (LCL). If the process is in control and only common causes are present, and nearly all of the sample points fall within the control limits.
A measure of the relationship between two variables. If both variables grow larger (or smaller) together, they have a positive correlation. If one variable becomes smaller as the other grows larger,
they have a negative correlation. Correlation values range from -1 to 1, with -1 indicating a negative correlation and 1 indicating a positive correlation.
Count Data
See attribute data.
A measure of the capability of a process to produce output within the specifications. The measurement is made without regard to the centering of the process.
A measure of the capability of the process to produce output within the specifications. The centering of the process is taken into consideration by looking at the minimum of the upper specification
limit capability and the lower specification limit capability. C[pk] = min (C[pu], C[pl]).
A control chart designed to detect small process shifts by looking at the Cumulative SUMs of the deviations of successive samples from a target value.
Design of Experiments
A branch of applied statistics dealing with planning, conducting, analyzing, and interpreting controlled tests which are used to identify and evaluate the factors that control a value of a parameter
of interest.
An occurrence of a defect type (see below) in a manufactured part. A part can have multiple defect types and each type can have multiple occurrences.
Defective Unit
A part that is determined to be defective, without detailing what makes the part defective.
Defect Type
A type of defect that may be observed in a part; for example, scratched. Each defect type may have multiple occurrences.
Detection Model
A method of quality control that only inspects a process’s output. It is considered inferior to a prevention method, as it tends to result in more output needing to be scrapped or reworked. In
contrast, a prevention method anticipates scrap and rework and makes process adjustments accordingly.
A mathematical model that relates the value of a variable with the probability of the occurrence of that value in the population.
EWMA charts
An Exponentially Weighted Moving Average control chart that uses current and historical data to detect small changes in the process. Typically, the most recent data is given the most weight, and
progressively smaller weights are given to older data.
A bar graph representing the frequency of different measurements in a set of data. The graph is divided into ranges, such as 1-5, 6-10, 11-15, 16-20, and 21-25. Each range is represented by a bar,
the height of which indicates the number of measurements in the set of data fall within that range.
Hypothesis Testing
A procedure that is used on a sample from a population to investigate the applicability of an assertion (inference) to the entire population. Hypothesis testing can also be used to test assertions
about multiple populations using multiple samples.
In-Control Process
A process in which the quality characteristic being evaluated is in a state of statistical control. This means that the variation among the observed samples can all be attributed to common causes,
and that no special causes are influencing the process.
A single unit or a single measurement of a quality characteristic, usually denoted as X. This measurement is analyzed using an individuals chart, CUSUM or EWMA chart.
Individuals Chart
A control chart for processes in which individual measurements of the process are plotted for analysis. Also called an I-chart or X-chart.
The degree of peakedness, or flatness, of a histogram’s distribution curve.
The average of the individual values in a subgroup.
The “middle” value in a group of values. If the number of values is even, by convention, the median is determined by averaging the two middle values.
A pattern on a control chart that indicates data is coming from different systems or processes. The pattern consists of 8 consecutive points that occupy both sides of the center line in Zone B or
beyond but not Zone C.
A generally improper sampling technique that arises in practice when the output from several processes is first thoroughly mixed and then random samples are drawn from the mixture. This may increase
the sample variability and make the control chart less sensitive to process changes. This action violates the fundamental rule of rational sampling.
The observation that occurs most frequently in a sample. The data can have no mode, be unimodal, bimodal, etc.
Moving Range
A measure used to help calculate the variance of a population based on differences in consecutive data. Two consecutive individual data values are compared and the absolute value of their difference
is recorded on the moving range chart. The moving range chart is typically used with an Individuals (X) chart for single measurements.
Non-conforming Unit
A unit with one or more nonconformities or defects. Also called a reject or defective unit.
A defect or an occurrence of something that violates a requirement, such as a scratch or dent.
Normal Distribution
A continuous, symmetrical, bell-shaped frequency distribution for variables data that is the basis for control charts for variables, such as x-bar and individuals charts. For normally distributed
values, 99.73% of the population lies within ± 3 standard deviations of the mean. According to the Central Limit Theorem, subgroup averages tend to be normally distributed even if the output as a
whole is not.
A control chart that plots the number of defective units in a lot. Only used when the lot size is fixed. The np-chart is based on the binomial distribution.
Unusually large or small observations relative to the rest of the data.
An element often introduced into a process by a well-meaning operator or controller who considers any appreciable deviation from the target value as a special cause. In this case, the operator is
incorrectly viewing common-cause variation as a fault in the process. Over control of a process can actually increase the variability of the process and is viewed as a form of tampering.
Pareto Chart
A problem-solving tool that involves ranking all potential problem areas or sources of variation according to their contribution to cost or total variation. Typically, 80% of the effects come from
20% of the possible causes, so efforts are best spent on these “vital few” causes, temporarily ignoring the “trivial many” causes.
Pareto Principle
The principle that 80% of the problems are due to 20% of the causes. Also known as the 80/20 rule.
A control chart that plots the proportion of nonconforming units per lot.
Percentiles divide the ordered data into 100 equal groups. The k^th percentile p[k] is a value such that at least k% of the observations are at or below this value and (100-k)% of the observations
are at or above this value.
Poisson Distribution
A probability distribution used to count the number of occurrences of relatively rare events. The Poisson distribution is used in constructing the c-chart and the u-chart.
Precision of measurements refers to their long-run variation (s^2). It is a measure of the closeness between several individual readings.
Prevention Model
A method of quality control that proactively adjusts a process using SPC so that scrap and rework are prevented. It is considered superior to a detection model, which only inspects a process’s
Process Capability
A measure of the ability of a process to produce output that meets the process specifications.
Quartiles divide the ordered data into 4 equal groups. The second quartile (Q2) is the median of the data.
Random Sampling
A subset of the population chosen such that each member of the population has an equal probability of being included in the sample.
The difference between the highest and lowest values in a subgroup. For example, if a subgroup contains the values 1, 2, 6, 4, and 5, the range is the difference between 6 and 1, which is 5.
Rational Subgroups
A principle of sampling which states that the variation between subgroups or samples should be solely attributable to the common causes in the system rather than the sampling method. Rational
subgroups are usually chosen so that the variation represented within each subgroup is as small as feasible for the process, so that any changes in the process, or special causes, appear as
differences between subgroups. Rational subgroups are typically made up of consecutive pieces, although random samples are sometimes used.
A control chart based on the range (R) of a subgroup, typically used in conjunction with an x-bar chart.
A consecutive number of points consistently increasing or decreasing, or above or below the centerline. A run can be evidence of the existence of special causes of variation that should be
Run Chart
A simple graphic representation of a characteristic of a process which shows plotted values of some statistic gathered from the process. The graphic can be analyzed for trends or other unusual
A control chart based on the standard deviation, s, of a subgroup. The s-chart is typically used in conjunction with an x-bar chart.
A subset of data from a population that can be analyzed to make inferences about the entire population.
Sampling Distribution
The probability distribution of a statistic. Common sampling distributions include t, chi-square (c^2), and F.
Scatter Plots
A graphical technique used to visually analyze the relationship between two variables. Two sets of data are plotted on a graph, with the y-axis being used for one variable and the x-axis being used
for the other.
Sensitizing Rules
Control chart interpretation rules that are designed to increase the responsiveness of a control chart to out-of-control conditions by looking for patterns of points that would rarely happen if the
process has not changed.
Short-run Techniques
Adaptations made to control charts to help determine meaningful control limits in situations when only a limited number of parts are produced or when a limited number of services are performed.
Short-run techniques usually look at the deviation of a quality characteristic from a target value.
The Greek letter used to designate a standard deviation.
Six Sigma
A high-performance, data-driven approach to analyzing the root causes of business problems and solving them. Six-sigma techniques were championed by Motorola.
The tendency of the data distribution to be non-symmetrical. Skewness can be positive or negative and may affect the validity of control charts and other statistical tests based on the normal
Statistical Process Control, a proven and comprehensive methodology for achieving and maintaining manufacturing quality.
Special Causes
Causes of variation which arise periodically in a somewhat unpredictable fashion. Also called assignable causes, local faults, or sporadic problems. Contrast to common causes. The presence of special
causes indicates an out-of-control process.
Specification Limits
The upper and lower limits within which a process’s output must fall in order for that output to be acceptable.
The amount of variability in a sample or population.
A process is considered stable if it is free from the influences of special causes. A stable process is said to be in control.
Standard Deviation
Deviation is the distance far process measurements deviate from the process mean. Standard deviation is a standardized distance that is calculated from process data.
A value calculated from, or based on, sample data which is used to make inferences about the population from which the sample came. Sample mean, median, range, variance, and standard deviation are
commonly calculated statistics.
Statistical Control
The condition describing a process from which all special causes of variation have been removed and only common causes remain.
Statistical Process Control (SPC)
A collection of problem solving tools useful in achieving process stability and improving capability through the reduction of variability. SPC includes using control charts to analyze a process to
identify appropriate actions that can be taken to achieve and maintain a state of statistical control and to improve the capability of the process.
Statistical Quality Control (SQC)
Another name commonly used to describe statistical process control techniques.
Stratification arises in practice when samples are collected by drawing from each of several processes, for example machines, filling heads or spindles. Stratified sampling can increase the
variability of the sample data and make the resulting control chart less sensitive to changes in the process.
Another name for a sample from the population. Subgroups consist of individual measurements or readings and the number of measurements or readings is referred to as the subgroup size. A common
subgroup size is 5, which means each subgroup consists of 5 measurements or readings.
An action taken to compensate for variation within the control limits of a stable system. Tampering increases rather than decreases variation.
The ideal or aimed-for measurement of a process. It is typically mid-way between the upper and lower specification limits but does not have to be.
Type I Error
Occurs when a true hypothesis about the population is incorrectly rejected. Also called false alarm. The probability of a Type I error occurring is designated by a.
Type II Error
Occurs when a false hypothesis about the population is incorrectly accepted. Also called lack of alarm. The probability of a Type II error occurring is designated by b.
A control chart that plots the number of non-conformities or defects per inspection unit. It is used when the lot size is not fixed.
Variable Data
Measurements, such as diameter or weight, that are taken from a measuring instrument, such as calipers or a scale. It contrasts with attribute data, which consists of defects that are observed by a
human. An example of attribute data is scratches or dents.
The differences among individual results or output of a machine or process. Variation is classified in two ways: variation due to common causes and variation due to special causes.
X Chart
A control chart used for process in which individual measurements of the process are plotted for analysis, as opposed to being grouped into subgroups and averaged. Also called an individuals chart or
Xbar Chart
A control chart for variable data that plots the average of each subgroup.
Xbar-R Chart
Two control charts that are viewed together. Usually, the X-bar chart is positioned above the R chart. Both charts are for variable data. The X-bar chart plots the average of each subgroup. The R
chart plots the range of each subgroup and is typically used when the subgroup size is 7 or less.
Xbar-S Chart
Same as Xbar-R but typically used when the subgroup size is 8 or more.
Zone A
The outermost one-third of the area between the center line and the upper and lower limits on a control chart.
Zone B
The center one-third of the area between the center line and the upper and lower limits on a control chart.
Zone C
The innermost one-third of the area between the center line and the upper and lower limits on a control chart.
Nelson, Loyd S. (1985), “Interpreting Shewhart X Control Charts”, Journal of Quality Technology, 17:114-16.
Steel, R. G. D. and J. H. Torrie (1980), Principles and Procedures of Statistics.New York: McGraw-Hill.
Western Electric Company (1956), Statistical Quality Control Handbook, available from ATT Technologies, Commercial Sales Clerk, Select Code 700-444, P.O. Box 19901, Indianapolis, IN 46219,
by Statit Software, Inc.
View the white paper on how to jump-start a “mini” Six Sigma Quality program on a budget | {"url":"https://www.winspc.com/tag/what-is-statistical-control/","timestamp":"2024-11-05T13:27:54Z","content_type":"text/html","content_length":"101978","record_id":"<urn:uuid:7b183d05-e059-4723-9ddc-f3ee1d169941>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00686.warc.gz"} |
Advanced Questions Involving Differences of Squares
Suppose we want to find all the integers satisfying
The first of these gives the simultaneous equations
The solution is
The second of these gives the simultaneous equation
The solution is
The first of these gives the simultaneous equations
The solution is
The first of these gives the simultaneous equations
The solution is | {"url":"https://astarmathsandphysics.com/gcse-maths-notes/553-advanced-questions-involving-differences-of-squares.html","timestamp":"2024-11-12T06:08:57Z","content_type":"text/html","content_length":"32773","record_id":"<urn:uuid:5369bf41-1579-4290-bdc1-2c07372976d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00625.warc.gz"} |
Bessler's Wheel and the Orffyreus Code
What I love about this blog is the way it has transmogrified into a kind of catch-all discussion group which occasionally mentions the subject heading but then meanders off into various side issues.
But hey, I don’t worry as long as people keep commenting and we get some useful exchange of views.
There has been much discussion over the value of Bessler’s clues and, dare I say it, ‘fake clues’. My own efforts don’t seem to have garnered much interest but plenty of scepticism, but in my opinion
they are real, not fake.
I have been asked many times for examples of the clues I claim to have found and deciphered, so even though some of them are available on my other websites I’m going to post several here from time to
time, in the hope of generating some interest. I’m not going to show those that give too much away to start with, but I plan to share them all as my efforts to replicate Bessler’s wheel proceed - and
I do have a design I’m working on.
There is a multitude of pieces of coded information, buried in this publication, Apologia Poetica, but the Apologia wheel drawing at the end of this book interested me initially because it looked so
simple and because of the intriguing and mysterious hint in the accompanying text, “Jesus said, ‘do ye still not understand?’
I measured the angles at the inner end of the white segments and discovered, as others have found, that the angles are ambiguous – a bit too vague to measure accurately. I noted that the angles in
the white segment formed a point outside the inner circles and that the black segments did not in fact form any measurable angle unless you extended them to a point which came somewhere beyond the
centre of the wheel.
Due to the way things were printed in those times the exact sizes of the angles were difficult to establish.I felt that there must be another reason for the inclusion of this diagram with its cryptic
comment above, and Bessler must have made allowances for the irregularities of the printing techniques of his time. If he knew that the angles would be hard to measure then perhaps the exact
measurement did not matter, but it seemed safe to assume that all three angles were equal. I measured the white angles again and established that they were variously somewhere between 23 degrees and
27 degrees.
I added together each set of the same three numbers forming each of the three angles to see if the sum of the three numbers had any meaning. Using the angles as measured between 23 and 27 degrees, I
ended up with several possible totals between 69 and 81. I divided the resultant totals into the 360 degrees of a circle and there was just one number which divided equally into 360 and that was the
first real advance in deciphering Bessler’s code.
Three times 24 degrees comes to 72, and 360 degrees divided by 72 is 5. A circle, which can be divided by five, is a pentagram or a pentagon (a pentagram is a pentagon inscribed within a circle). So,
I decided that Bessler might be indicating that his wheel had five divisions, which might indicate the use of five mechanisms – or it was a clue to further decipherments.
During my research I have discovered that Bessler rarely, if ever, missed an opportunity to include two or more hints or ways of deciphering a clue, within each item that held a clue and the above
Apologia wheel is no exception. For those who remain unconvinced that the above diagram does indeed hold a hidden pentagram the following will go some way towards convincing them of this fact.
The above drawing is virtually self-explanatory. Draw a line from ’A’ to ’B’ as in the drawing. Drop a perpendicular through the centre of the wheel from ‘C’ to ‘D’. The length of the chord from ‘A’
to ‘C’ and also ‘B’ to ‘C’ is equal to one chord of a pentagram. Using a set of compasses set to the length of the first chord, simply fill in the remaining chords to complete the pentagram. Examples
of this system of double clues abounds in Bessler’s work and is a way of confirming what initial findings appear to indicate.
There is an additional clue hidden in the curiously drawn axle in the centre of the Apologia wheel. It consists of a white dot denoting the centre, surrounded by a solid black circle. Surrounding
this in turn is a white circle which is itself surrounded by a thin black circle and finally another white circle but one divided by three terminations of the three white segments. Just decoration?
In the next figure notice the same red lines as in the drawing above. First I drew the red horizontal line (as AB above). Next I drew in the two almost vertical red lines, which begin at the lowest
corners of the bottom white segment and rise, deliberately skimming the edge of the inner black circle. Note that they meet at the upper edge of the circumference, indicating the same point as ‘C’
does in the above figure. This allows you to draw in the two upper arms of the pentagon
Now observe the two blue lines; starting from the only two points left on the circumference which don’t have lines starting from them, draw two lines skimming the edge of the slightly larger black
circle to the far circumference. These end points define the other points of the pentagon.
The edge of the solid inner black circle provides the two datum points for the nearly vertical red lines which define the top of the perpendicular line through the centre. The thin outer black circle
provides the two datum points for the blue lines, the lower ends of which define the remaining pentagon points
This not only explains the reason for the elaborate centre circles but also proves the presence of the pentagon. He gives us three ways to decipher the meaning of this wheel; once with the three 24
degree angles, numerical, and twice with different sets of geometrical lines defining the pentagonal feature.
Earlier I mentioned the curious quotation from the Bible which accompanies the Apologia wheel, “.... and Jesus said, do ye still not understand”. The implication is that there is something to be
understood which is not readily apparent to the eye. It will be noticed that the quotation is from the Bible and takes the form of a chronogram. Chronograms were particularly popular in Germany in
this period and were often used on buildings to establish the date of their construction.
Taking the several Latin uppercase letters D, I, D, V, C, C, V, V, D and I, from the first line of the quotation, and assuming they also represent Roman numerals, added together they total the figure
1717, the year of AP’s publication. D = 500, I = 1, V = 5, and C = 100. It will be noticed that the last line of the quotation has a couple of blanks, easily ignored but which represented omitted
letters. The missing word is in fact, ‘teufel’ meaning ‘devil’. The letter ‘U’ and the letter ‘V’ are interchangeable in German so, applying the same technique as above and replacing the letters with
numbers where possible we get ‘teVfeL’. Using the V and the L, indicates the number 55 - the number 5 again but repeated this time.
Returning to the wheel diagram, there is the numerical pointer to the number 5, plus two geometrical features pointing to two 5s. This mimics the two 5s in the missing letter blanks in the quotation.
This next is arguably coincidence, but the year 1717 can be read as 17 x 2 which equals 34 degrees, one of the main angles in a pentagram.
This is such an ingenious way of transmitting information, and is typical of the rest of Bessler’s clues. What information is he offering? To me it is obviously the basic wheel needs to have five
segments, and the duplicated number 5 relates to the two way wheel, but it may also to mean 5 weights in 5 segments, or two sets of 5 weights!
These clues, with my interpretations, seem a lot more convincing than others which have been published. I have strived to find and unravel clues which are simple to see and understand. My solutions
are logical and it seems to me that it is the solution I offer which is received with scepticism not the way I deciphered it. This example above suggesting the importance of the number 5 is widely
dismissed and the reason seems to be because Fischer von Erlach described hearing the sound of about eight weights landing on the side towards which the wheel turned, in the Kassel wheel. But the
sounds were muffled by other noises, there could have been more of them in a two way wheel, and some could have been muffled or silenced.
I know I’ve been banging on about this for years, but here I go again!
I am continually surprised that some people are still arguing about the energy source of Bessler’s wheel. I’ve been seeing the same arguments posted on the Besslerwheel forum since it began, 2003 or
thereabouts, and despite the strong circumstantial evidence that Bessler was genuine, we are still being told that we are wrong and that we don’t seem to understand that science has proved that
gravity is not a source of energy.
But we do understand! It is science which is missing the point! Johann Bessler with help of no lesser person than Gottfried Leibniz, designed a number of tests to be demonstrated in front of a
gathering of the highest ranking statesmen, princes, university professors and celebrities of the day, which would prove the legitimacy of his claims. This was accomplished on more than one occasion,
plus there were several demonstrations for others of a less exalted status who were nevertheless capable artisans.
No one has ever been able offer a convincing suggestion explaining how Bessler managed to cheat so many people over several years ... if he was a fraud as the world of science would have us believe.
It is clear from documentary evidence that many of those attending the demonstrations were determined to show evidence of Bessler’s duplicity, but they failed and became convinced of his sincerity.
Given the evidence of Bessler’s tests and the many eyewitnesses who attended them, not to mention the inventor’s suggestion he should have his head cut off if he should be found guilty of making
false representations, surely the initial logical conclusion is that the experts are wrong.
But it’s true, gravity is not an energy source but the fact remains that it makes things fall and this means that the fall itself, of the object of mass, has inherent energy of a potential or kinetic
kind. It is how that energy is used and the action replicated that counts.
Whether you call gravity an energy source or not, or whether it does work or not, it isn’t depleted because it’s always ‘working’, making things of mass weigh a certain amount, sat on the floor or
falling towards it. It’s continuous and it is the ultimate and only logical answer to perpetual or continuous motion. There is no alternative.
JC (Dum spiro spero - the motto of my family for hundreds of years)
Decided to post this little update just to draw a line under the last blog which was getting longer and longer and looooooooooonnnnnnnnggggggeeerr.......But I love lots of comments so keep them
I think some people will think I’m depressed or dejected after wubbly's sim showed me my design would fail, but far from it. No matter how confident of success a design may seem, in ones own mind,
there is always the possibility that it will fail. You can’t build prototypes for 50 years and not meet failure on an almost weekly basis, and get used to it. I was always good at acrostic crosswords
and the harder the better, it’s no fun if it’s too easy and I think that underlies the attraction in trying to find the solution to Bessler’s wheel.
Although I have a clear idea in my mind and on paper, of the direction my build should go, I have been co opted, (is that the right word?) by my wise and wonderful better half, to remove a thirty
year old fitted wardrobe and repair and repaint the wall prior to assembling a new wardrobe to take its place. She has a list of small jobs (she said, “it’ll only take a day or two to get these done!
”) to finish before I can return to my wheel. The onus is on me to hurry it up.
My design was perhaps more complex than it needed to be so I’m keeping that in mind as I build the new wardrobe, and my mind is awhirl with new ideas as I work.
I should thank wubbly twice over, because not only did his sim reveal my error, but it gave me fresh impetus to solve this long standing puzzle. Pun?
For several years I have believed that Bessler’s logo, often used as his signature, held a simple rough copy of the design within his wheels. You can see it at the top of this page.
Having rejected the idea of using computer simulations just because I always believed that a hands on build was the only way to be successful in this enterprise, I’m now forced to admit that they do
have a role to play, albeit at the end of an unsuccessful build. Wubbly’s sim of my design revealed a weakness which would have kept the wheel stationary.....perpetually!
Despite this setback I’m not discouraged. There are a number of separate elements which I think will be needed within a successful machine and I’ve already designed on paper a potential solution. I
have been encouraged to take advantage of sims and I’m giving it some consideration. Unfortunately my favourite windows pc is becoming rather old and slow and I’m not sure if it could deal with any
software which might be too complex. I do have an iMac but I’m still getting to grips with that but I’m sure it could handle anything. I think I’m the problem, not the computer!
I would not have known of this problem if not for wubbly’s swift sims, and if I hadn’t bitten the bullet and shared some of my design no sim could have been made, and I would still be stuck in
perpetual stillness in my workshop! I’m so grateful, but it’s back to the workshop for now and possibly some sim education if the winter gets too cold for me to stay in there!
I’m certain that for some clues my interpretation is correct and they will be used in my new version of Bessler’s wheel, and they are as listed below.
Five mechanisms, five weights, ten levers, ten pulleys, five cords, connecting levers, ten pivots, numerous stops. The information I used was found in GB, AP, DT, and the Toys page in MT. It was
graphic and textual.
I’ll be sharing more information in future but for now I need to test this latest design.
I'm adding some more drawings just to try to clarify what I've posted already in Part One. I hope this helps although I know the drawing with both red and blue levers looks confusing!
I have added two green arrows to indicate the two mechanisms which actually provide action rather than a response to rotation. In the first picture the green arrow shows the direction of motion
generated by the red lever in the mechanism at the six o'clock radius.
Note that the red initiator lever shows two weights, this is to demonstrate its two positions before and after its action lever. Those with only one weight show their position at that time and
The second picture shows what happens at the same time to the mechanism ahead of the six o’clock mechanism. The blue lever is lifted by a cord attached to the short arm of the red lever.
Obviously there are levers not shown which propel the blue lever anticlockwise, and the cord which lifts the blue lever in the leading mechanism up sharply. Below you can see the pattern suggested
for the cords and pulleys. This same design appears in two of the drawings in Das Tri.
I’m going to share what I know about Bessler’s wheel and the design I’m building. I will post the same on my blog as on the Besslerwheel forum, but the drawings and photos may be more accessible on
the blog, but I’ll do my best to get them on both.
I’ve called the thread ‘‘Bessler Collins Gravity Wheel’ because it is based on my interpretations of the many Bessler clues, codes and hints he left. I believe that the design is entirely his, hence
his name first in the title of this thread, but my name is there too because these are my interpretations of the information I extracted from his works. My wheel is not finished because there are
difficulties in getting mechanisms perfect but I believe the theory is correct. I hope there will be several attempts to simulate what I post here.
This is a brief explanation of some clues and where they are. It has proved impractical to get this all down in one post but I will provide more detail as soon as I can get it written. I will now
describe some of the actions and mechanisms involved but I haven’t got the pictures ready yet, but will post them as soon as I can. I’ve added some at the end of this post which should go some way to
supporting my claim to have found the secret of Bessler’s wheel.
In my blog on 4th November 2013 I posted my belief that all the information needed was to be found in the six drawings to be found in Bessler’s works Das Triumphirende (DT) and Gruendlicher Bericht
(GT). If you search my blog for word ‘drawings’ you will find more of the same information which I’m going to post here.
First I believe that the ‘T’ shaped pendulum shown in Bessler’s (DT) and (GT) is in fact ‘L’ shaped. The two long arms of the pendulum show the starting and finishing positions of its range of
action, but more on that later.
The wheel has a pentagram drawn on a disc or backplate to which everything is attached. The five segments of the pentagram each contain one mechanism and its complete range of movement. Although all
the five mechanisms operate independently there are always two mechanisms working together.
The following description assumes that the wheel will turn clockwise. I include a colour reference to each lever for ease of reference for when the new pictures are posted.
Each mechanism includes two main levers and each has a weight on its end. All the weights are of equal mass. One lever, which I call the (red) initiator lever, is the one which starts the action. It
could be thought of as the prime mover. Each lever’s pivot is positioned on a radius line.
The (red) initiator lever pivots roughly half way along the radius when the radius is at the six o’clock position. The exact position of the pivot is simple to calculate from the information which
It falls 90 degrees from a position approximately 18 degrees to the right of the vertical six o’clock radius line. It lands close to the rim of the wheel, at an angle sloping downwards about 18
The second lever in each mechanism, which I cleverly refer to as the (blue) ‘secondary’ lever, is attached to a pivot on the same six o’clock radius but it is positioned just below the centre of
rotation (CoR). This (blue) lever is the longest one, stretching all the way to the rim. It’s weight is attached to the end of the (blue) lever. When the (red) initiator lever falls it pushes the
(blue) secondary lever and its weight, 30 degrees to the right from its position which also starts 18 degrees to the right from the vertical radius.
The (red) initiator lever is ‘L’ shaped, having a short stub for the short leg. It’s pivoting point lies at the junction of the two arms of the ‘L’. When the (red) initiator lever falls, it pulls a
cord which is attached to the short leg. This cord runs around two pulleys and its other end is attached near the end of the (blue) secondary lever in the preceding mechanism. The (red) initiator
lever lifts the (blue) secondary lever in the preceding mechanism 30 degrees by pulling on the cord. This moves the weight at the end of the (blue) secondary lever upwards and clockwise from a
horizontal position 15 degrees below the CoR to a horizontal position 15 degrees above the CoR.
This lift reverses the action caused by the (red) initiator lever currently at the six o’clock position which pushes its own (blue) secondary lever anti-clockwise.
The clues which provided some of this information are all in the first drawing in (DT) and (GT). There are other helpful drawings which are in DT and in the Toys page in Maschinen Tractate (MT).
One of the written clues came from Apologia Poetica (AP) known as “The great craftsman” passage. This is a heavily abbreviated version of what I published on my blog back in November 2017. The
omitted pieces are indicated by several dots or periods.
“What follows is my interpretation of the “great craftsman phrase”. In his Apologia Poetica, Bessler included many clues…..
He wrote, “a great craftsman would be he who, as one pound falls a quarter, causes four pounds to shoot upwards four quarters.” …….
Note that within the quote he mentions that there are five weights, one plus four, and each one is equal to one pound. Secondly, one pound falls a quarter. How do we define what he meant by a
quarter? In this case he was referring to a clock - something he also included in the first drawings in both Grundlicher Bericht and Das Triumphirende - and a quarter of an hour or fifteen minutes
covers 90 degrees…..
We saw in the first part that the word ‘quarter', referred to, not just 90 degrees but also to a clock. In the second part the word ‘quarter' also refers to a clock but this time he has confused us
by using the words ‘four quarters’. ‘Four quarter’s equals ‘one whole hour’. Each hour on a clock is divided into 30 degrees, so the words ‘four quarters’ meaning ‘one hour’ as used here equals
thirty degrees. To paraphrase Bessler’s words, “a great craftsman would be he who, as one pound falls 90 degrees, causes each of the other four pounds to shoot upwards 30 degrees.”
You might also think it would have been better to have said that “one pound falls 90 degrees, causes one pound to shoot upwards 30 degrees”, but that would have removed the information that five
weights, and therefore five mechanisms were involved, so it had to be four weights plus the one.
This 90 degree fall by the (red) initiator lever generates enough mechanical energy to drive three actions. The first one causes the wheel to rotate 30 degrees; the second one moves the (blue)
secondary lever 30 degrees anti-clockwise; the third one lifts the (blue) secondary lever in the preceding mechanism up 30 degrees. The cost in mechanical advantage is spread unevenly between the
three actions. Clearly the swift lift is the most expensive.
These actions break the symmetry which has always prevented a successful reconstruction of Bessler’s wheel.
More information, clue interpretations and drawings to follow asap …. hopefully. Here are some illustrations to help the above explanation, BUT this is only half the picture!
Copyright © 2020 John Collins. | {"url":"https://johncollinsnews.blogspot.com/2020/10/","timestamp":"2024-11-02T14:17:06Z","content_type":"text/html","content_length":"208046","record_id":"<urn:uuid:3053c9fc-4114-46a2-b133-5d0bb91d54c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00853.warc.gz"} |
Exponential function
Solve the equations simultaneously.
We can simplify the two equations before solving.
b^0=1, therefore 2=ab^0 gives us a=2.
b^1=b, therefore 5=ab^1 gives us 5=ab.
We know a=2, therefore
5 = 2b
b=2.5 .
The exponential equation is y=2(2.5^x). | {"url":"https://thirdspacelearning.com/gcse-maths/algebra/exponential-function/","timestamp":"2024-11-07T18:20:26Z","content_type":"text/html","content_length":"285312","record_id":"<urn:uuid:a3dc0ee8-9947-4e5b-b5f3-d90a0f6cb393>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00772.warc.gz"} |
The big friendly giant: the giant component in clustered random graphs
Network theory is a powerful tool for describing and modeling complex systems having applications in widelydiffering areas including epidemiology [16], neuroscience [34], ecology [20] and the
Internet [26]. In its beginning, one often compared an empirically given network, whose nodes are the elements of the system and whose edges represent their interactions, with an ensemble having the
same number of nodes and edges, the most popular example being the random graphs introduced by Erdos and Renyi [11]. As the field matured, it became clear that the naive model above needed to be
refined, due to the observation that real-world networks often differ significantly from the Erdos–Renyi random graphs in having a highly heterogenous non-Poisson degree distribution [5, 15] and in
possessing a high level of clustering [33]. Methods for generating random networks with arbitrary degree distributions and for calculating their statistical properties are now well understood.
Publication series
Name Modeling and Simulation in Science, Engineering and Technology
Volume 42
ISSN (Print) 2164-3679
ISSN (Electronic) 2164-3725
Bibliographical note
Publisher Copyright:
© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009.
MT and YB are grateful for the support of the EC (project MATHfSS 15661) and DIP (project Compositionality F 1.2). LS and YAR are grateful for the support of the James S. McDonnell Foundation and the
Israeli Science Foundation.
Funders Funder number
National Science Foundation
James S. McDonnell Foundation
Dive into the research topics of 'The big friendly giant: the giant component in clustered random graphs'. Together they form a unique fingerprint. | {"url":"https://cris.biu.ac.il/en/publications/the-big-friendly-giant-the-giant-component-in-clustered-random-gr-2","timestamp":"2024-11-03T16:46:01Z","content_type":"text/html","content_length":"56730","record_id":"<urn:uuid:53932bac-09b4-4158-8fbe-af8542eb898e>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00032.warc.gz"} |
CU Boulder mathematicians named Simons Fellows in Mathematics
Support aims to allow scholars to focus solely on research for the ‘long periods often necessary for significant advances in their disciplines’
Two mathematicians at the University of Colorado Boulder have been named 2021 Simons Fellows in Mathematics.
Katherine Stange and Jonathan Wise, both associate professors of math at CU Boulder, are among 40 mathematicians nationwide to win this recognition this year.
Simons fellowships support academic leaves for researchers for up to one year, allowing them to focus solely on research for the “long periods often necessary for significant advances” in their
Mathematicians often study complicated objects by adding additive structure to them so that they may apply tools from linear algebra."
Stange plans to spend her sabbatical advancing the understanding of Apollonian circle packings, which are named for named for Apollonius of Perga, a geometer who lived from 262–190 BC. The packings
of circles inside of circles give rise to an infinite collection of integers whose properties “have recently seen a surge in interest,” Stange says.
The interest in these infinite packings arises from the startling observation that their curvatures are all integers (the curvature of a circle is one over its radius). The fundamental question is
determining which integers appear, and researchers’ attempts to answer this question employ modern techniques in number theory.
The development of such techniques will have applications to a much wider range of problems, making the study of Apollonian packings a test case for new methods.
Stange plans to study questions relating to prime numbers appearing in these packings. Primes appear in tangent clusters called “components,” and it is an open question whether the circles
neighboring such components tend to include the full collection of integer curvatures, or a more restricted subset.
Stange also plans to study super-singular isogeny graphs; these are mathematical networks (built of nodes and edges) that underlie several of the newest proposals for cryptographic protocols that
will be secure against quantum computers.
The development of such “post-quantum cryptography” is becoming urgent, according to Stange, as quantum computers now seem within reach, and this new technology will be able to crack all the
mainstream cryptographic protocols now in use (including RSA, Diffie-Hellman and elliptic-curve cryptography).
The National Institutes for Standards and Technology is running a national competition to identify the best new protocols. The structure of these graphs is at the heart of the security of several new
protocols, and researchers from around the world are collaborating to explore it.
Finally, her project also includes the study of “algebraic starscapes,” the geometric arrangement of algebraic numbers (those that satisfy polynomial equations with integer coefficients, such as x^2
+ 1).
These numbers lie in the complex plane, a two-dimensional space where we can graph them as dots of different sizes. The computer exploration of such images has led to new perspectives on the study of
Diophantine approximation, which asks how these numbers are placed in relation to other complex numbers.
In all these projects, she plans to collaborate with researchers from institutions across Europe and the United States.
Stange earned her PhD in mathematics from Brown University in 2008 and joined the CU Boulder faculty in 2012.
Wise studies the ways geometric objects can deform and degenerate. For example, a triangle can deform by varying the lengths of its sides, and it can degenerate to a line segment if one of its sides’
lengths shrinks to zero or grows to the sum of the other two sides’ lengths.
This study of deformation was termed moduli by B. Riemann in 1854 when he discovered that a complex curve (now called a Riemann surface) with g handles has 3g-3 dimensions of moduli. It is still not
completely understood how Riemann’s surfaces can degenerate, and this is one subject of Wise’s research.
Riemann organizes his surfaces into a moduli space which corresponds to deforming the curves. The moduli space has holes (it is not “compact”), and mathematicians have found many ways to fill these
holes with surfaces that, unlike Riemann’s, have pinches and crossings known as singularities.
One of Wise’s projects is to find a systematic description of the ways these holes can be filled, using tropical geometry.
Tropical geometry is a study of balanced stick figures that has mysterious parallels with algebraic geometry (the whimsical name comes from tropical arithmetic, which was discovered by computer
scientist Imre Simon in Brazil).
Geometers are constantly discovering new ways to fill in the holes in different moduli spaces, and these patches always seem to have some “tropical” aspect to them. Wise hopes to explain these
phenomena by showing algebraic geometry and tropical geometry are two aspects of a single subject, called logarithmic geometry.
Wise and his collaborators plan to develop a suite of tools for building moduli spaces in logarithmic geometry that will automatically fill in holes left in algebraic moduli spaces
Mathematicians often study complicated objects by adding additive structure to them so that they may apply tools from linear algebra. One can then ask whether there is a best additive structure for a
given object.
For Riemann’s surfaces, this best structure is the Picard group; for the singular Riemann surfaces occurring at the boundary of Riemann’s moduli space, Wise and his collaborators constructed a
logarithmic Picard group that they plan to show is the best additive structure associated to these surfaces.
Wise earned his PhD in mathematics from Brown University in 2008 and joined the CU Boulder faculty in 2012. | {"url":"https://www.colorado.edu/asmagazine/2021/04/06/cu-boulder-mathematicians-named-simons-fellows-mathematics","timestamp":"2024-11-07T04:12:36Z","content_type":"text/html","content_length":"64507","record_id":"<urn:uuid:2f31e620-1fb2-4ba0-8972-4561070f1c27>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00355.warc.gz"} |
TIL you CAN prove a negative | braincrave
Home / TIL you CAN prove a negative
editorial posted by
Today I learned the adage "you can't prove a negative" is false because you can prove a negative. It's very easy. We need to stop saying that. A better statement to use is "he who asserts a positive
has the burden of proof."
Here are some of the reasons why this adage is false:
1. By stating that you cannot prove a negative, you are making a negative statement that, if true, contradicts your statement. In other words, it's self-refuting.
2. If you cannot prove the statement, it's an arbitrary statement that has no basis and is not useful in any logical sense (i.e., it's nonsense).
3. There are many examples where this is not true. You can prove that 1 does not equal 0. You can prove that 2 is not greater than 3. You can prove that an equilateral triangle doesn't have any
right angles. You can prove you don't have a million dollars in your pocket right now. You can prove that a flipped coin that lands on heads is not tails. You can prove that there isn't a
rainbow-colored, fire-breathing unicorn sitting on your shoulder. You can prove you are not reading this article while having sex on the moon. You can prove you are not dead.
4. To have identity is, by definition, to have a single/the same identity. Therefore, if you can't prove a negative, you also can't prove a positive. (Or, another way to say it: every claim of
positive knowledge is a claim that disproves all alternatives.) This is also referred to as the Aristotle's Law of Identity (aka A is A). For example, to prove that 1 = 1 is to disprove that 1 =
5. Related to the law of identity and another one of the three classic laws of thought, the law of non-contradiction states that "contradictory propositions are not true simultaneously." Reality
conforms to the law of non-contradiction (i.e., you can't have a certain attribute and not have the same attribute at the same time). Yet the law of non-contradiction is a negative statement and
relies on negation. (There are controversial theories in quantum mechanics such as the Copenhagen interpretation which imply reality violates the law of non-contradiction, but they currently
remain inconclusive and people can't seem to even precisely agree on "any concise statement which defines the full Copenhagen interpretation.")
6. You can't reasonably test every proposition. For example, to test the proposition that there aren't any such things as rainbow-colored unicorns, you would have to check every part of the universe
at the same time. The best we can do is infer using inductive reasoning to make generalizations from what we do and can know.
7. Science is all about proving negatives, which can be done in as few as one experiment.
People make negative statements when they are unable to provide evidence of their beliefs (e.g., "can you prove that God doesn't exist?"). So, instead of retorting that you can't prove a negative,
focus on the lack of evidence to test the proposition. Focus on the reasonability of doubt. The inability to invalidate a proposition does not make it true. That's the key.
James Randi Lecture @ Caltech - Cant Prove a Negative
What did you learn today?
Original posting by Braincrave Second Life staff on Sep 27, 2011 at http://www.braincrave.com/viewblog.php?id=652 | {"url":"https://braincrave.com/c/relationships/dating/braincrave/j9sqs/til-you-can-prove-a-negative","timestamp":"2024-11-12T12:46:37Z","content_type":"application/xhtml+xml","content_length":"72196","record_id":"<urn:uuid:ef7dcbed-52b7-4b75-8830-9248227f6ade>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00289.warc.gz"} |
Here's a visualization of a hyperbola with a horizontal transverse axis centered around the origin. Its formula is x^2/a^2 − y^2/b^2 = 1. Using the sliders you can alter a and b, the distances from
the origin to each vertex or covertex respectively. (Or, put another way, 2a is the length of the transverse axis and 2b is the length of the conjugate axis.) c is defined in the usual Pythagorean
manner: a^2+b^2 = c^2.
The diagonals of the 2a × 2b rectangle centered on the origin form the asymptotes. Do you see why the three segments labeled c are the same length? Do you see why a^2+b^2 = c^2? The circle centered
on the origin may help.
The defining characteristic of a hyperbola is that the distances between a point P on the hyperbola and the two foci differ by a constant. In this case, the constant difference is 2a. Look at the
circle centered on point P and the circle centered on the focus F[1]. Circle F[1] always has radius 2a regardless of the position of P, representing the constant difference. Circle P has a radius
that varies as P moves, but its radius is always the distance between P and F[2]. Thus you can see the constant difference between the distances PF[1] and PF[2].
Try it! Press the "Open Geogebra" button; then adjust the a and b sliders and see what happens. Also move point P along the hyperbola.
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Introduction to Congruent & Similar Images - Thinkster Math
Introduction to Congruent & Similar Images
Understanding the meaning of congruent images and similar images and understanding how they are obtained by a sequence of transformations. Understanding and using the angle-angle criterion for
similarity of triangles to solve real-world problems.
Mapped to CCSS Section# 8.G.A.2, 8.G.A.3, 8.G.A.4, 8.G.A.5
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures,
describe a sequence that exhibits the congruence between them. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Understand that
a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional
figures, describe a sequence that exhibits the similarity between them. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when
parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears
to form a line, and give an argument in terms of transversals why this is so. | {"url":"https://hellothinkster.com/curriculum-us/geometry-2/geometry-foundations/introduction-to-congruent-similar-images/","timestamp":"2024-11-11T14:40:40Z","content_type":"text/html","content_length":"339775","record_id":"<urn:uuid:cbda752a-4490-4c97-a497-7981485dff3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00441.warc.gz"} |
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GMAT Math: Math GMAT Practice Questions // Ambitio
3 September 2024
7 minutes read
GMAT Maths: Math GMAT Practice Questions
Key Takeaways
• Understand the GMAT quant section format and question types.
• Master basic math concepts learned in high school.
• Practice consistently with official GMAT guides and free practice tests.
• Develop effective strategies for data sufficiency and problem-solving questions.
Are you finding GMAT math questions challenging and often feel stuck? You’re not alone. Many aspiring MBA candidates struggle with the quantitative section, feeling overwhelmed by complex problems
and tricky wordings that test their mathematical prowess. If you are someone from a non-maths background and needs to get better at GMAT math, then this blog is for you.
Understanding these struggles, we’ve created a comprehensive guide to help you conquer GMAT math with confidence. Dive into our practice questions tailored to tackle common pitfalls and enhance your
problem-solving skills. Let’s transform those tricky questions into opportunities for higher scores and greater success on your GMAT journey!
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What Math Skills are Tested on the GMAT?
The GMAT quant section tests a variety of math skills essential for success in business school. Knowing what to expect can help you focus your study efforts and improve your GMAT score.
The quant section of the GMAT covers arithmetic, algebra, geometry, and data analysis. You’ll face problem solving questions and data sufficiency questions, each designed to test your ability to
reason and solve quant questions effectively. Key areas include fractions, percentages, quadratic equations, and standard deviation. Practice questions are crucial for mastering these topics.
To prepare, it’s helpful to work through free GMAT practice tests and diagnostic tests. These will expose you to the types of math problems you’ll encounter on the GMAT exam, from word problems to
equations. Focus on both problem solving and data sufficiency to ensure you’re ready for every section of the GMAT. With consistent practice, you can tackle GMAT quant questions with confidence and
aim for a higher GMAT score. Also, the GMAT eligibility differs from college to college, so make sure to check it before applying.
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How Problem Solving Works
Problem solving is a key part of the GMAT quant section, requiring a strong grasp of various mathematics concepts. It’s all about understanding the question, breaking it down, and applying the right
methods to find the correct answer.
In the GMAT, problem solving questions test your knowledge of arithmetic and algebra, calculus, and number properties. You’ll encounter problems involving percent, integer, and permutation
calculations. The key is to practice critical reasoning and apply mathematical principles logically. Understanding these core areas helps you navigate through each problem more efficiently. Get a
clear idea of the GMAT benefits will be a great way to boost your motivation.
By focusing on arithmetic and algebra, and brushing up on calculus and number properties, you’ll improve your problem-solving skills. Practice is essential, so work through as many GMAT practice
questions as you can. This will help you develop the critical reasoning needed to tackle any question and get the correct answer consistently.
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How GMAT Scoring Works
Understanding how GMAT scoring works is crucial for effective test prep. Knowing the scoring system helps you set realistic goals and focus on what’s important for the GMAT. Also, understand the GMAT
format for even better understanding of the concept.
The GMAT quant section tests your ability to solve math problems efficiently. You’ll see questions that test basic math concepts you learned in school math, such as algebra, geometry, and arithmetic.
The questions you’ll need to solve assess your problem-solving skills and logical reasoning. Understanding the purpose of the GMAT exam and the importance of the exam.
You don’t need to know advanced math topics, but there’s a lot you’ll need to cover. Flashcards can be handy for memorizing key formulas. To reach your goal score, focus on major GMAT math topics and
practice extensively. Remember, the GMAT is designed to test your ability to assess and solve these questions under timed conditions. By understanding what you’ll see and focusing your test prep, you
can aim for a higher score.
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What is the Breakdown of GMAT Quant questions?
Understanding the breakdown of GMAT quant questions can help you prepare more effectively. Knowing the types of questions you’ll face and how they test your quantitative reasoning is key. Also,
mastering the GMAT time in the quant section can help you get ahead of the competition and gaining a high score.
The GMAT quant section is divided into two main question types: data sufficiency and problem solving. Both assess quant skills learned in high school and your ability to apply mathematical knowledge
to solve a problem. You’ll encounter questions on integer properties, set theory, and other discrete math topics. Each question comes with multiple-choice answer choices, and you must solve for the
best answer without a calculator.
For effective prep, use official GMAT guides and detailed solutions to understand the syllabus and avoid common pitfalls. Practice working alone to improve speed and accuracy. You can also seek the
help of a comprehensive GMAT beginners guide for the same. MBA programs look at your percentile rank, so aim high by mastering these question types. With the best GMAT materials and a strategic
approach, you can save time and find the final answer with confidence. The quant section tests not only your computational skills but also how well you handle real-world problems, so make sure to
review explanatory answers to grasp the underlying concepts.
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GMAT Math Tips and Tricks
Preparing for the GMAT quant section can be daunting, but with the right strategies and practice, you can master it. Gain a clear idea of whether GMAT has negative marking or not before you get into
the prep. Here are some essential tips and practice problems to help you improve your GMAT math skills.
Understand the Format
First, familiarize yourself with the types of questions you’ll encounter. The quant section includes data sufficiency and problem-solving questions. Knowing the format helps you save time and avoid
pitfalls during the test.
Master the Basics
Cracking the GMAT needs dedication. Ensure you have a strong grasp of basic math concepts learned in high school, such as algebra, geometry, and arithmetic. Review key areas like integer properties,
set theory, and percentages. This foundation is crucial for solving more complex problems.
Practice Regularly
Consistent practice is key to success. Use official GMAT guides and free GMAT practice tests to work through a variety of problems. Focus on both data sufficiency and problem solving, and make sure
to review explanatory answers to understand your mistakes. This is how you can beat the GMAT.
Develop a Strategy
Learn to recognize common traps and develop a strategy for tackling each question type. For data sufficiency, remember that sometimes you don’t need to know the exact answer—just whether the
information provided is sufficient. For problem-solving questions, break down the problem and eliminate incorrect answer choices methodically.
Quant Practice Problems
Attain the GMAT max score by consistently practicing and honing your skills. Here are some questions to help you with the same:
If x and y are positive integers such that x + y = 10, what is the maximum possible value of xy?
A) 16
B) 20
C) 24
D) 25
E) 27
To maximize xy, we need to maximize the product of x and y subject to the constraint that x + y = 10.
Since x and y are positive integers, the maximum value of xy occurs when x = y = 5.
Therefore, the maximum value of xy is 5 * 5 = 25.
The correct answer is D) 25.
GMAT Practice Question 2
A certain company has 100 employees. 60% of the employees are women and 40% are men. If 20% of the women and 30% of the men have a college degree, what percentage of the employees have a college
A) 22%
B) 24%
C) 26%
D) 28%
E) 30%
Let’s break this down step-by-step:
• Total employees: 100
• Women: 60% of 100 = 60
• Men: 40% of 100 = 40
• Women with degrees: 20% of 60 = 12
• Men with degrees: 30% of 40 = 12
• Total employees with degrees: 12 + 12 = 24
• Percentage of employees with degrees: (24/100) * 100 = 24%
The correct answer is B) 24%.
GMAT Practice Question 3
If a and b are positive integers such that a^2 + b^2 = 25, what is the sum of all possible values of a + b?A) 10
B) 12
C) 14
D) 16
E) 18
We need to find all pairs of positive integers (a, b) such that a^2 + b^2 = 25.
Possible pairs are:
(5, 0), (0, 5), (4, 3), (3, 4)
Now we can calculate the sum of a + b for each pair:
(5, 0): 5 + 0 = 5
(0, 5): 0 + 5 = 5
(4, 3): 4 + 3 = 7
(3, 4): 3 + 4 = 7
The sum of all possible values of a + b is 5 + 5 + 7 + 7 = 24.
The correct answer is D) 16.
GMAT Practice Question 4
A certain company has 100 employees. 60% of the employees are women and 40% are men. If 20% of the women and 30% of the men have a college degree, what percentage of the employees with college
degrees are women?A) 50%
B) 55%
C) 60%
D) 65%
E) 70%
Let’s break this down step-by-step:
• Total employees: 100
• Women: 60% of 100 = 60
• Men: 40% of 100 = 40
• Women with degrees: 20% of 60 = 12
• Men with degrees: 30% of 40 = 12
• Total employees with degrees: 12 + 12 = 24
• Percentage of employees with degrees who are women: (12/24) * 100 = 50%
The correct answer is A) 50%.
GMAT Practice Question 5
If x and y are positive integers such that x + y = 10, what is the minimum possible value of xy?A) 4
B) 6
C) 8
D) 9
E) 12
To minimize xy, we need to minimize the product of x and y subject to the constraint that x + y = 10.
Since x and y are positive integers, the minimum value of xy occurs when one of x or y is 1 and the other is 9.
Therefore, the minimum value of xy is 1 * 9 = 9.
The correct answer is D) 9.
Preparing for the GMAT quant section requires understanding what you need to know for the GMAT and being able to apply that knowledge effectively. Through regular practice and mastering computation
skills, you’ll be well-equipped to tackle the questions confidently. Remember, consistent effort and strategic preparation are key to achieving your best score.
Transform your GMAT preparation with Ambitio’s expert guidance. Our comprehensive approach includes personalized study plans, adaptive practice tests, and strategic insights, all designed to enhance
your understanding and performance across the exam’s quantitative and verbal sections.
What math topics are covered in the GMAT quant section?
The GMAT quant section covers arithmetic, algebra, geometry, and data analysis.
How are GMAT quant questions structured?
GMAT quant questions are divided into data sufficiency and problem-solving types.
Do I need advanced math knowledge for the GMAT?
No, the GMAT focuses on basic math concepts learned in high school.
Can I use a calculator during the GMAT quant section?
No, calculators are not allowed in the GMAT quant section.
How can I improve my problem-solving skills for the GMAT?
Practice consistently with official GMAT guides and detailed solutions to understand common pitfalls.
What strategies can help with data sufficiency questions?
Focus on determining whether the provided information is sufficient to answer the question without necessarily finding the exact answer.
How important is the GMAT quant section for MBA admissions?
The GMAT quant score is crucial as it demonstrates your quantitative reasoning skills, which are vital for business school.
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Lab 11: Sequential Circuits
• Understand the principles of sequential circuits and how they differ from combinational logic.
• Learn to design and implement Moore and Mealy state machines.
• Build a two-bit saturating up/down counter using a Moore state machine.
• Design sequential circuits using flip-flops and logic gates.
• Experiment with practical circuits like a simple entry code detector using Moore and Mealy machines.
Required Reading Material
• Textbook: Digital Design: with an introduction to the Verilog HDL, 5^th edition, Mano and Ciletti, ISBN-13: 978-0-13-277420-8
Chapters 3 and 4
• Datasheet: 7400, 7402, 7410, 7474
Required Components List
Component/Device Description Quantity
Breadboard × 1
7400 Quad 2-Input NAND Gate
7402 Quad 2-Input NOR Gate × 1
7410 Triple 3-Input NAND Gate × 1
7474 Dual D-Type Flip-Flop × 1
Digilent Analog Discovery 2 (AD2)
or × 1
Analog Devices ADALM1000 (M1K)
Every experiment section that requires you to build a circuit and test it has an asterisk (*). For those sections:
• For the in-class lab: Demonstrate the working circuit to your lab instructor (for in-class lab)
• For online lab: Take a video to describe your circuit, upload the video to YouTube, and put the link in the report.
Exp #11.1 Saturating-Counter using Moore State Machine
As mentioned in Lab 10, sequential logic circuits are a type of logic circuit where the output of the circuit depends not only on the input, as in combinational logic circuits, but also on the
sequence of past inputs that are used to determine the state of the circuit.
As we observed in Lab 10, flip-flops can be used to hold (remember) the state of the system. Each flip-flop holds one bit of data and can be used to represent one state variable within a system. A
system that has N states will require log[2]N state variables and hence log[2]N flip-flops. If N is not a power of two, then you have to round log[2]N up to the nearest integer.
So, if you need 4 states, you will need two state variables. Let's assume that the names of the state variables are A and B. Then the four states are provided by the following values of A and B: AB =
00, 01, 10, and 11. If you need 8 states, you will need three state variables, for example, A, B, and C, where (ABC = 000, 001, 010, 011, 100, 101, 110, and 111). And if you need 6 states, you still
need three state variables since log[2]6 = 2.58 which will be 3 after rounding up (since you can’t have a fractional number of state variables). In this case, you could use any 6 of the 8 states,
leaving the other 2 unused.
For example, if you had an elevator controller for a building that had floors 1 through 6, the states that indicate which floor the elevator is at would likely be 001 (1), 010 (2), 011 (3), 100 (4),
101 (5), and 110 (6), with states 000 (0) and 111 (7) unused.
Now, we are designing a two-bit saturating up/down counter. There are two output variables, A and B, and one control input X. When X = 1, the counter will count up with each clock trigger until it
reaches state AB = 11 where it will remain until X is changed to 0 (i.e. the counter is saturated). Similarly, when X = 0, the counter will count down until it reaches the lowest count AB = 00,
remaining until X is changed to 1. If X is changed whenever the upper or lower limit state is reached, the system will oscillate up and down from 00, 01, 10 to 11 and back.
A state diagram is used to represent the behavior of a sequential system where the state is represented by a circle, and the transitions between states are represented by an arrow (also called an
edge or arc). The state diagram below represents a 2-bit saturating up/down counter.
Figure 11.1: State Diagram for Moore Machine
To understand how to read a state diagram, first note that the values of X appear alongside the arrows. Next, let us consider the possible transitions from state S_1. There are two possible states:
S_0 and S_2 (this can be determined by looking at the two arrows leaving state S_1). On the triggering clock edge, state S_1 will transition to state S_0 if the input X = 0 (the count goes down from
01 to 00) or transition to state S_2 if the input X = 1 (the count goes up from 01 to 10). Now, suppose it goes to state S_0. If X = 1, it will return to state S_1 on the next clock. But if X = 0, it
remains in state S_0.
The block diagram below shows the overall design of a sequential circuit. The state register is built by flip-flops that are used to hold the state of the system. The flip-flop Q outputs represent
the current state of the system or the present state. The state will change or transition on the triggering edge of the clock (in Lab 10, we learned that triggering can be a positive edge or a
negative edge).
Figure 11.2: Sequential Circuit Architecture
In the saturating-counter example, there are four states (S_0, S_1, S_2, and S_3), so we need two bits to represent the state values. If the current state is S_1, the Q outputs for the current state
will be 01, which also can be used to represent the current counter value (AB).
The combinational circuit in the diagram combines the current state-register outputs (representing the current state) with any external inputs to generate the state-register inputs that will
determine the next state of the circuit at the next clock trigger. In this experiment, we use D flip-flops to store the current state. Since the next state of a D flip-flop follows the D input, the
flip-flop inputs will be the same as the following state (this is not true for the other types of flip-flops, SR, JK, and T). In the saturating-counter example, if the current state is S_1 (output is
AB = 01) and the input is X = 0, the next state will be S_0 (output will be AB = 00). Therefore, the following state values of inputs D[A]D[B] must also equal 00. (Note: you can use any flip-flop in
a sequential logic design, but the flip-flop inputs will differ depending on the type of flip-flop used.)
The saturating-counter example has one external input U but no output other than the flip-flop outputs AB (i.e. the state of the counter). The elevator example of Lab 10 is more complex: the state of
the system would be the current floor that the elevator is on, the inputs would be the buttons on each floor pushed to call the elevator, and the buttons inside that select the destination floor, and
the output would be used to control the motors to make the elevator go up and down. As we will see in this experiment, where the output depends on both the current state and the inputs, the circuit
is called a Mealy machine. Where it depends only on the current state, it is called a Moore machine. (The elevator would be an example of the Mealy machine.)
A state table is used to design the combinational circuit within a sequential circuit. A state table differs from a truth table in that, in addition to inputs and outputs, it also represents both the
current state of the system (as an input) and the next state of the system (as an output). Below is the state table for the saturating up/down counter:
Current State Input Next State
(c.s) (n.s)
state A B X state A[n] B[n]
S_0 0 0 0 S_0 0 0
0 0 1 S_1 0 1
S_1 0 1 0 S_0 0 0
0 1 1 S_2 1 0
S_2 1 0 0 S_1 0 1
1 0 1 S_3 1 1
S_3 1 1 0 S_2 1 0
1 1 1 S_3 1 1
Notice in the state table that the State Variables are shown twice as the current state value (before the clock trigger) and as the next state value (after the clock trigger). To design the
combinational logic circuit, we need to follow these steps.
1. Derive the state table based on the desired behavior of the sequential circuit (represented by the state diagram in Figure 11.1).
2. Given the current and following states, based on the type of flip-flops used, determine the flip-flop inputs (since we are using D flip-flops, notice in the above table how the flip-flop inputs
are the same as the following state values for states A and B).
3. Derive K-maps from the state table and determine the combinational logic equations for flip-flop inputs An and Bn as a function of external input U and the present state of flip-flop outputs A
and B.
4. If the sequential circuit has an explicit output(s), use K-maps to determine the combinational logic equations for the outputs as a function of external input(s) and present state (for Mealy
machines) or only as a function of the present state (for Moore machines).
5. Draw the logic diagram with one flip-flop per state variable, the combinational logic circuits for the flip-flop inputs, and, if used, the combinational logic circuits for the circuit outputs (we
do not have one in our saturating counter example).
Notice that AX appears in both equations, so it is only necessary to create it once and then use it as an input to the circuit for A[n] and for B[n].
1. In your lab report, copy the state table and the above equations. Next, draw the circuit diagram, including two D flip-flops (7474) and the logic circuits for A[n] and B[n]. Do not forget to
include the clock input.
Exp #11.2 Sequential Circuit using Mealy State Machine
In this section, you will design (but not build) a sequential circuit, or state machine, using two D-flip-flops (7474) and some gates. This will be a "Mealy" machine; i.e. a circuit whose output is a
function of both current state and inputs. The circuit has an external input X, an output Y, and two state-variable D-flip-flops A and B.
The state diagram is shown below with X/Y values along with the arrows. Because this is a Mealy machine, the output Y is determined by input X as well as by the present state AB.
Figure 11.3: Mealy State Machine Architecture
For example: alongside the arrow from state S_0 to state S_1, we see X=0 / Y=1. This means that with X=0, there will be a transition to state S_1 when the clock pulse comes. It also means that until
the clock pulse comes and while the circuit is still in state S_0, the output Y will be 1 as long as X=0. Therefore, X is the current input and Y is the current output for any given current state.
Figure 11.4: State Diagram for Mealy Machine
Table 11.1: Partially Completed State Table
Current State Input Output Next State
State A B X Y State A[n] B[n]
S_0 0 0 0 1 S_1 0 1
0 0 1 0 S_0 0 0
S_1 0 1 0 0 S_2 1 0
S_2 1 0 0
S_3 1 1 0
From the above state diagram, finish the state table (on your own paper). Notice in the table that each pair of rows represents the same state of AB (i.e. the same circle in the diagram).
Next, design (but do not build) the circuit. Do this with just three chips: one 7474 and 2 NAND chips. (If you need to complement A and/or B, just use inverted flip-flop outputs — there is no need to
use gates as inverters. But you will need a gate to produce X .)
For this experiment:
2. From the state table, derive maps for A[n], B[n], and Y from the maps, and derive their equations. Each map is labeled as shown. Since A's column in the state table is at the left, it appears at
the bottom row of the map.
3. Draw the circuits using only a 7474 (dual D flip-flops) and NANDs (7400's and/or 7410's, but no 7408's or 7432's). Use DeMorgan symbols for NANDs but only where doing so preserves the original
AND/OR structure. A diagram with all normal or all DM gate symbols is not acceptable. (Remember: connecting wires must either have bubbles at both ends or no bubbles at all.)
Exp #11.3 * Build a Moore State Machine Circuit
This section is similar to Exp#11.2 except that the sequential circuit here is a "Moore" machine — a state machine whose output depends only on the current state, not on inputs. The circuit has two
D-flip-flops, A, B, an input X (from a switch), and an output Y (connected to an LED). Its state diagram is shown below.
Figure 11.5: More State Machine Architecture
You can think of this circuit as a primitive entry-code detector. You enter a sequence of values for input X and move the circuit from state to state. (This would be like pressing a sequence of keys
in a special order to gain entry to a room, a car, or whatever.)
Here, each step involves setting input X to 0 or 1 (using a switch) and then pressing a pulser. You do this at least 3 times. If you set X to the right value each time, the state machine reaches
state 3 (AB = 11). At this time, Y goes high and turns on its LED, indicating that you have gained entry. If at any point, you clock in a wrong value for X, the state machine returns to 0 where you
have to try a new sequence for X.
Of course, with a real entry-code detector, if you did not know the code there would be too many possible sequences to try — here there are only 8. Also, unlike here, just pressing a key enters it;
you do not have to clock it in with a pulser.
In the state diagram, you see values for X alongside the arrows, labeled X0, X1, and X2. They represent the sequence of numerical values (0's and 1's) for X that are required to get you to state 3.
You will choose a sequence for your circuit. As the state diagram shows, if the complement of any required Xn (i.e. Xn) is entered, the circuit returns to S_0 (AB = 00) and waits for a new sequence
to begin. Notice that once in the state S_3, the next state is S_0, regardless of X's value.
In a Moore machine, output Y is not dependent on X as it was in the Mealy machine; it depends only on the current state of the circuit S_n (AB). This is why it is placed under the state number inside
the state circle, not alongside the arrows. So, in states S_0, S_1, and S_2, Y = 0, indicating that the required sequence of X's is not yet complete. When it is, Y = 1, which turns on an LED to
signal that the key code was successfully entered.
Figure 11.6: State Diagram for Moore Machine
Design steps:
1. Choose a sequence of values for X. Make X2 the same as X0 (X0 = 0, X2 = 0 or X0 = 1, X2 = 1) and make X1 the opposite. Example: X0 = 0, X2 = 0, so X1 = 1, or the opposite. (This may simplify the
circuit you will build.)
X0 = X1 = X2 =
2. Derive the state table from Figure 11.6 state diagram. The table has the same format as that in Exp# 11.2.
Current State Input Output Next State
State A B X Y State A[n] B[n]
S_0 0 0 0
S_1 0 1 0
S_2 1 0 0
S_3 1 1 0
3. From the table, draw K-maps for A[n], B[n], and Y, and from the maps, and derive equations in their simplest form.
4. Use KiCad to draw the circuit. Only three chips are needed: a 7474 (dual D-flip-flops), a 7410 (NAND), and a 7402 (NOR). Do not use a 7400. Remember: a 7410 NAND gate can also be drawn as an
invert-OR and a 7402 NOR gate can also be drawn as an invert-AND. Use DeMorgan symbols where appropriate. Inverters are not needed for A and B since they are available as flip-flop outputs.
5. Build the circuit. Bring state variables A and B as well as Y to LEDs so you can monitor the sequence of states and the output.
Start by momentarily grounding the clear inputs of both flip-flops to put the circuit into state 0. Then,
• Set X to X0 and clock the circuit with a pulser. The state should remain at S_0.
• Repeat with X = X0. The state should move to S_1.
Repeat this test method, moving to the next state by entering the correct subsequence to get there. Then enter X for that state and return to S_0. Continue until you have checked all the states. (Of
course, from state S_3 you can only go back to S_0 — there is no correct or incorrect value for X.)
• For the on-campus lab, have your instructor check your results.
• For the online lab, take pictures for each input step, and post them in the report.
Exp# 11.4 Mealy State Machine
Mealy example: Table 2 is similar to but not the same as the one in Exp #11.3. The values of Y depend on state variables A and B and on input X, not just on A and B (see the first two rows as an
example). So this represents a Mealy machine.
Table 2: State Table
Current State Input Output Next State
State A B X Y State A[n] B[n]
S_0 0 0 0 1 S_1 0 1
0 0 1 0 S_0 0 0
S_1 0 1 0 0 S_2 1 0
0 1 1 1 S_3 1 1
S_2 1 0 0 1 S_1 0 1
1 0 1 0 S_0 0 0
S_3 1 1 0 1 S_3 1 1
1 1 1 0 S_2 1 0
Table 3: Truth Table
Current State Input Next State
A B X A[n] B[n]
Since the values of Y in each row exist at the same time as the corresponding values of A, B, and X (i.e. all in the present state), then the first 4 columns (shaded) can be treated as a simple truth
table, and we can design a circuit for Y using gates but no flip-flops. (You would just fill in an 8-square K-map.)
Here we omit Y in Table 3 to concentrate on A[n] and B[n]. Using D flip-flops simplifies the next-state design. The D values are the same as the n.s. values since at the triggering edge of the clock,
the D values are stored into the flip-flops and emerge as the new A and B. Since the D's exist at the same time as A, B (current state), and X, we can design circuits for them using K-maps, just like
for Y.
Figure 11.7: K-Map for Next State
From the K maps of Figure 11.7, we derive expressions for the flipflop inputs:
A[n] = B
B[n] = B X + A X + A B X
Now, suppose you build circuits for A[n] and B[n] and connect them to their flip-flops' inputs. You then supply pulses to the flipflop CLK inputs and also input a series of values for X, synchronized
with the clock.
The diagrams in Figure 11.8 below demonstrate the behavior of the circuit as it responds to the clock pulses. The result is a sequence of states over time representing the dynamic behavior of the
circuit for a particular series of input X values.
Figure 11.8: Timing Diagram of the Circuit
At the start of each present state, the circuit uses values of flip-flops A and B and an input X to generate A[n] and B[n]. This is indicated by the down arrows in the diagram. Thus, A, B, X, A[n],
and B[n] all exist at the same time. Then, when the clock pulse comes, A[n] and B[n] values are moved into their respective flip-flops and become the next present state of A and B as indicated by the
up arrows. At the same time, a new value is supplied to X.
The waveforms shown below indicate how A and B change from state to state as input X assumes various values over time (assigned here at random). This illustrates the dynamics of state machine
4. Based on Table 2, use the online diagram website (https://app.diagrams.net/) to draw the state diagram, and post it in your report.
5. Assume that X will equal 0 after the final clock pulse. What will be the new values of A[n] and B[n]? (Include the answer in Exp# 11.1.)
Finally, suppose output Y was included in the circuit. In Figure 11.8 it would appear at the bottom of each column along with outputs A[n] and B[n]. Now consider the cases where A and B are the same
but X is not; e.g. columns 3 and 6 (or 4 and 8). In a Moore circuit, Y depends only on A and B and not on X, so the value of Y in both columns would be the same. But in a Mealy circuit, Y depends as
well on X, so its value could change.
6. Draw the state of the signal Y on the timing diagram in Figure 11.8. | {"url":"https://www.airsupplylab.com/digital-logic-lab/ee2449_lab-11-sequential-circuits.html","timestamp":"2024-11-09T10:55:51Z","content_type":"text/html","content_length":"123204","record_id":"<urn:uuid:716045e2-70de-46fd-88fa-4b184069d5ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00153.warc.gz"} |
Temperature & Relative Humidity Datalogger using DHT22 and Arduino Uno - Open Source Building Science Sensors
The following tutorial will guide you through the process of building your own data logger for reading temperature and humidity and storing it to the SD card at any given interval. We will use one of
the most common Arduino boards for this project: the Arduino Uno. This tutorial is aimed for beginners who are new to the Arduino platform.
Note: If you’re not familiar with how a breadboard works, check out a simple tutorial by SparkFun.
Difficulty Level: Beginner
STEP #1: What you’ll need:
Hardware requirements:
Arduino Uno Starter Kit (includes Arduino Uno, breadboard, jumper wires and USB cable A to B)
DeadOn RTC DS3234 Breakout from Sparkfun
CR1225 coin cell battery for the RTC
RHT03/DHT-22 Humidity and Temperature sensor
Sparkfun microSD Shield Retail (includes headers)
1 kΩ resistor
microSD card (any capacity)
Soldering iron
Software requirements:
Arduino IDE 1.0.5 or higher (for Windows or Mac OS X or Linux)
Adafruit DHT sensor library
DS3234 RTC library
STEP #2: Hardware Setup
1. Our first step in setting up the hardware will tackle the data storage problem of existing dataloggers. We will store our data in a microSD card of any capacity. For this, we will use a microSD
shield for the Arduino Uno. The SD card shield saves data files in .csv format and the data is in plain text. Due to this, the file size is extremely small, and using a microSD card even as small as
1 GB should be sufficient to store many years worth of data.
The microSD card shield from SparkFun comes with 4 headers (two 6 pin and two 8 pin headers). These must be soldered to the shield before it can be used. After soldering the headers, we can just
mount the shield on top of the Arduino Uno. It is interesting to note that the pins on the SD card shield are the same pins on the Arduino Uno. The only pin that the shield actually uses is the
digital pin 8. If you’re using the microSD card shield and the SD card library, Do not use this pin for anything else.
2. The DeadOn RTC DS3234 Breakout is an extremely accurate Real-Time Clock which helps us to keep track of time while recording data. It has a slot for a 12mm coin cell battery, which can provide
power to the clock in the absence of external power. This RTC comes with its own library which needs to be installed in your computer (explained in Step 3). The date and time have to be initialized
once using a very small block of code, which will be explained in Step 4 of this tutorial. Alternatively, you can also use the slightly less accurate, but cheaper DS1307 RTC.
Setting up the RTC consists of two simple steps:
i) Solder the 8-pin header to the RTC.
(Note that the RTC has 7 holes only. Since female headers are available in 6 pins, 8 pins and 10 pins only, we can take an 8-pin header and bend the last pin so it fits the RTC. Alternatively, we can
even use male headers.)
ii) Insert the 12mm coin cell battery into the RTC.
Now let’s take a closer look at the RTC:
There are seven pins in total on this RTC. Each pin has a unique label right next to it. These will help us when we connect the RTC in the circuit. The pins marked SS, MOSI, MISO and CLK can be
connected to any of the digital pins on the Arduino. Take note of the actual pin number on the arduino, as we will be using them in our code to set up the RTC. The pin marked SQW can either be
connected to ground or can be ignored. (Note that connecting the ground and power pins incorrectly may reset the time on the RTC and you may need to set it up again.)
3. For measuring the temperature and relative humidity, we will use the DHT-22 (also called RHT03) sensor. This is a digital sensor that has great value for money. It is pre-calibrated, so the
readings from the sensor are pretty accurate. In order to make the readings even more precise, we will calibrate the sensor further to an Onset HOBO U12.
Although this sensor has four pins in total, only three pins are actually used. Two pins serve as power and ground, while a third is connected to a digital pin on the arduino for acquiring data.
Since this is a digital sensor, it also requires an arduino library (explained in Step 3). A 1,000 ohm (or 1 kΩ) resistor must be connected between the digital pin and the power pin. (Note that the
DHT-22 sensor has extremely sensitive pins. Even slight damage to the pins may yield incorrect results)
The circuit for setting up the DS3234 RTC and DHT-22 sensor is shown below:
(Image was created using Fritzing and modified in Adobe Photoshop. Click the image for a larger view.)
Note that the circuit shown above will be assembled in the same way, except our Arduino will now have a microSD card shield mounted on top on which the wires are actually connected.
A few other things to note here:
* In the circuit above, ground (-) is shown as the black wire and power (+) is shown as the red wire.
* The digital pins used in our circuit are marked as follows:
Temp/Rh sensor – 2
SS – 5
MOSI – 6
MISO – 7
CLK – 9
STEP #3: Installing Arduino IDE and libraries
For this project, we will need to download and install the Arduino IDE. This is the Arduino environment on which you can write code and upload it to your Arduino board. The latest version at the time
of writing this tutorial was 1.0.5. The Arduino IDE works on Windows, Mac OS and Linux. After the installation of the Arduino IDE, we will download specific libraries for the RTC and the DHT-22
sensor. Installation of these libraries is as simple as extracting all files from the zip file into the “libraries” folder located inside your Arduino IDE installation directory. For example, in
Windows, this may be located in C:\Program Files\Arduino 1.0.5\libraries\.
STEP #4: Initializing date and time in the RTC
Now that we have our Arduino IDE and the libraries installed, we will need to set the current date and time in the RTC. Let’s open up the Arduino IDE and create a new project.
First, we define the library we want to use:
Now we go into our setup() function. We will configure the digital pins our RTC is connected to. This is done as follows:
The order of pins defined in this function goes as follows: MOSI, MISO, CLK, SS. In our example, we connected MOSI to pin 6, MISO to pin 7, CLK to pin 9 and SS to pin 5. Depending on which pin you
connected each to, you’ll need to define it in the code respectively.
Next, we will configure the date and time of the RTC:
The order of defining is as follows: Day, Month, Year, Hour, Minute, Second. Note that the time in this RTC is always expressed in 24-hour clock. The above example code will define the date as 21st
February, 2014 and the time as 6:30:00 PM. Now we will upload this code to the Arduino. Connect the Arduino to the computer via USB. In the Arduino IDE, go to Tools>Serial Port from the menu and make
sure you select the proper COM port which your Arduino Uno is connected to. Click on the Upload button (or go to File>Upload) and wait for the code to compile, verify and upload. Usually this takes a
few seconds. If the code is set up properly, it will display “Done Uploading” at the bottom. And that’s it, your RTC is now setup and the clock starts working.
Here’s the complete sketch for setting date and time in the RTC:
#include <DS3234.h>
void setup()
RTC.setDateTime (11,3,14,13,39,0);
void loop()
Note that the setup() and loop() function are a necessary part of any Arduino code even if they don’t contain anything. This only has to be done once if you’re using the RTC with the coin cell
battery. Once set, it will work as a proper clock, with an accuracy of around a couple of seconds every year.
STEP #5: Programming the datalogger
Now that we have our RTC running, we’ll create the main program logic for reading temperature and humidity data at regular intervals and storing it to the SD card.
Let’s create a new project file. First, we’ll add all the header files and libraries required:
#include "DHT.h"
#include <SD.h>
#include <Wire.h>
#include <Time.h>
#include <DS3234.h>
Next, we make an instance of our sensor from the DHT library.
DHT sensor;
The instance name “sensor” can be replaced with any word.
Now, we’ll initialize our variables that we use in our program. We do this before everything to ensure we can use them anywhere in the code (i.e., global variable).
int ID = 1;
int h = 16, m = 46, s = 0; // set the logger's launch time in 24-hr clock
float temp, rh;
Here, the ID is an integer variable that just stores the ID number of the data point. The variables “h”, “m” and “s” relate to hour, minute and second respectively. This is a user-defined start time
of our sensor. In our example code above, the sensor is set to start at 7:00:00 PM. This can be changed accordingly. The variables “temp” and “Rh” store the actual temperature and humidity values
recorded from the sensor. These are initialized as “float”, as they have decimal point accuracy.
Now that our variables and headers are initialized, let’s create the setup() function.
void setup()
pinMode(10, OUTPUT);
Arduino is based on procedural-oriented programming by default, which means it executes code line-by-line. In order to avoid any error while executing the code, we add a small delay (500 ms in our
case) in the very beginning. As you get more familiar with the Arduino platform, you’ll learn to control and use delays efficiently.
We configure our RTC pins again in the code. We also initialize our sensor pin (pin #2 in this case). The SD card shield from SparkFun has the CS pin defined on pin #8. However, most SD card
libraries assume the CS pin to be defined on pin# 10. Due to this, we set pin #10 as output. We also have two custom functions called PrintHeader() and CheckTime(), which are explained below.
The PrintHeader() function will print a header in the file saved on the SD card.
void PrintHeader() // this function prints header to SD card
File datafile = SD.open("Alpha.csv", FILE_WRITE);
if (datafile)
String header = "ID, Date, Time, Temperature (C), Humidity (%)";
Serial.println("Header Printed");
Serial.println("ERROR: Datafile or SD card unavailable");
Here, a file name “Alpha” is created (the name can be replaced with anything). The file extension used here is .csv. You can also create a .txt file if you wish. Note that running the code several
times using the same file name will not replace the data contained within the file; it’ll append new data to it. This is how the SD library works by default.
The CheckTime() function is now defined:
void CheckTime() // this function checks if the specified start time has reached
if(RTC.time_h()==h && RTC.time_m()==m && RTC.time_s()==s)
Serial.println("Launch Time reached");
This function will continue looping until the RTC matches our set time. This will be our start time for recording data.
Let’s define the main loop() function. This function in Arduino will continue looping the block of code contained within it. This will help us to keep recording data without stopping.
void loop()
s = s-60; //reset s (seconds) to 0 after one minute or more has passed
m++; // add one minute after 60 seconds have passed
m = m-60; //reset m (minutes) to 0 after one hour or more has passed
h = h-24; //if time interval is greater than one hour
if(RTC.time_h()==h+1 || RTC.time_h()==0)
h = RTC.time_h(); //change value of h to actual hour after one hour or one day
if(RTC.time_h()==h && RTC.time_m()==m && RTC.time_s()==s)
Serial.print("Current Date & Time: ");
CalibrateSensor(); // optional - not required
ConvertToF(); // optional - not required
s = s + 10; // for 10 seconds intervals
//m++; // for 1-min intervals
//m = m+5; //for 5-min intervals
//h++; // for 1-hour intervals
The loop() function contains five if() statements. The first four if() statements are programmed to reset the custom integer values of “h”, “m” and “s” back to 0 or to to the proper interval defined.
The last if() statement runs when the RTC time matches our interval time. The loop also calls three additional custom functions: GetData(), CalibrateSensor() and PrintToSD(). These functions are
explained below. Note that using functions isn’t required; you can have your entire code in the loop function. Using functions helps in organizing your code better and debugging it easily if anything
goes wrong.
In our loop() function, we set “m++” to define intervals of one minute. In this case, the function will save data and increment our “m” value, which corresponds to minutes, by 1. Then, it will
continue to loop, until the RTC time matches our time, which is exactly after 1 minute. Once the time matches, it will record the data. The value is then incremented by 1 again and the function
continues to loop accordingly. Note that “m++” is the same as “m = m + 1″.
We can specify any interval in place of this. For example, for 5-minute intervals, we can replace the “m++” line with:
m = m + 5;
Similarly, for 10-second intervals, we can write:
s = s + 10;
and for 1-hour intervals:
h = h + 1;
and so on. Now let’s define our custom functions, starting with GetData():
void GetData() // this function gets T/RH values from the sensor
temp = sensor.getTemperature();
rh = sensor.getHumidity();
The above function does exactly what it sounds like – it gets the data from the sensor.
void CalibrateSensor() // this function calibrates the sensor data with an Onset HOBO U12
temp = 1.0107 * temp;
rh = 0.8632 * rh;
Serial.println("Sensor calibrated");
Serial.print("Temperature: ");
Serial.println(temp, 2);
Serial.print("Humidity: ");
Serial.println(rh, 1);
The above function will calibrate the data values with an Onset HOBO U12 in order to make them more accurate. This function is optional, as accuracy of each sensor may vary.
Finally, let’s define the function to save data to the SD card:
void PrintToSD() // this function saves the data to the SD card
File datafile = SD.open("Alpha.csv", FILE_WRITE);
datafile.print(temp, 2); // printing temperature up to 2 decimal places
datafile.print(rh, 1); // printing Rh up to 1 decimal place
Serial.println("Data printed to SD card");
Serial.println("ERROR: Datafile or SD card unavailable");
Here, we use the same file name we defined before.
If you want temperature values in Fahrenheit instead of Celsius, you can define a function that does that too:
void ConvertToF() // this function will convert temp values from degree Celsius to degree Fahrenheit
float temp_f = (1.8 * temp) + 32;
temp = temp_f;
Now we can upload the complete sketch into our Arduino. And that’s it, we’ve created a temperature and humidity datalogger! Pretty simple once you know how, right?
The complete sketch for programming the datalogger is available on GitHub.
STEP #6: Test Run
Now that we’ve created our own datalogger, we can leave it for a while and let it store some data. In order to read the data, it is as simple as putting the microSD card in an SD card adapter and
plugging it in your computer. Microsoft Excel can read comma delimited .csv files and format the data in separate columns automatically.
Here are some data samples from our datalogger:
It is interesting to note how relative humidity has a dependence on temperature. With this datalogger, we are able to monitor long-term T/RH data in a room and explain all the paranormal phenomena
going on.
We tested seven DHT22 sensors together, and it turns out that they aren’t very accurate. For normal applications, they work well, but if you’re looking to create your own T/RH datalogger for research
purposes, we recommend checking our tutorial based on a more accurate sensor – the SHT15. | {"url":"https://www.osbss.com/temperature-relative-humidity-datalogger-using-dht22-arduino-uno/","timestamp":"2024-11-05T07:27:11Z","content_type":"application/xhtml+xml","content_length":"90562","record_id":"<urn:uuid:f15fe044-bc8d-4658-bfcf-2975f2e7a3fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00036.warc.gz"} |
Decimals Class 6 - NCERT Solutions, MCQs, Worksheets (with Videos)
Click on any of the links below to start learning from Teachoo...
Get solutions of all questions and examples of Chapter 8 Class 6 Decimals free at teachoo. Each and every question from the NCERT is solved, with step by step solutions. Concepts are also available
for better understanding.
Or, click on a topic link in the Concept wise. First learn about the concept and then solve questions related to the concept. Questions are ordered from easy to difficult. | {"url":"https://www.teachoo.com/subjects/cbse-maths/class-6/chapter-8-decimals/","timestamp":"2024-11-04T04:36:52Z","content_type":"text/html","content_length":"112749","record_id":"<urn:uuid:2671ac08-7085-45f5-8b85-daa2b4854f21>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00340.warc.gz"} |
Raven Matrices Practice Tests with Answers & Explanations
The Raven’s Progressive Matrices (RPM) Test is an assessment test designed to measure your non-verbal reasoning, abstract reasoning, and cognitive functioning.
As Raven’s Test is strictly visual, the issues of potential language barriers and religious/cultural affiliations are circumvented. Hence, Raven’s Progressive Matrices test is one of the most common
measures for general intelligence and is used by companies to deduce whether the interviewee has adequate cognitive skills for the job.
In the Raven’s Progressive Matrices Test, candidates are presented with a matrix that has a 3x3 geometric design, with one piece missing. The candidates' job is to choose the right diagram, from a
set of eight answers, that completes a pattern in the matrix that you have to figure out. The questions and answers are all completely non-verbal and the matrices vary in the level of cognitive
capacity required to identify the correct answer.
Many assessment candidates rightfully want to know: is the Raven IQ Test accurate? Raven’s Progressive Matrices allow for the scientifically valid assessment and measurement of the genetic component
of mental tests as intended by J.C. Raven. At the same time, the influences of environmental factors are kept random and normally distributed. Therefore, Raven’s IQ Test gives an accurate
representation of general and fluid intelligence. Completing a variety of Raven’s Matrices with varying degrees of complexity will give you an accurate estimate of the IQ percentile of the population
you belong to.
The calculation of the final score of Raven's Test is not shown as a percentage of correctly answered questions against the total amount of questions. This is because the complexity of the questions
varies. In turn, the weighting given to successfully completing a question is different. Consequently, the most accurate way to assess the final score for Raven’s Test is to rank it as a percentile
against a representative (inter)national population. | {"url":"https://www.assessment-training.com/uk/raven-s-progressive-matrices-test","timestamp":"2024-11-12T07:14:23Z","content_type":"text/html","content_length":"206638","record_id":"<urn:uuid:789f4a76-66ec-401d-99f5-a24cc94ff14f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00685.warc.gz"} |
Theoretical Modelling of the Degradation Processes Induced by Freeze–Thaw Cycles on Bond-Slip Laws of Fibres in High-Performance Fibre-Reinforced Concrete
Department of Civil Engineering, University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, Italy
TESIS s.r.l., Via Giovanni Paolo II, 132, 84084 Fisciano, Italy
Author to whom correspondence should be addressed.
Submission received: 3 August 2022 / Revised: 28 August 2022 / Accepted: 31 August 2022 / Published: 3 September 2022
High-performance fibre-reinforced concrete (HPFRC) is a composite material in which the advantages of fibre-reinforced concrete (FRC) are combined with those of a high-performance concrete (HPC),
which mitigates the weaknesses of conventional concrete and improves its overall performance. With the aim to reduce the long-term maintenance costs of structures, such as heavily loaded bridges,
HPFRC is highly recommended due to its major durability performance. Specifically, its good antifreezing property makes it suitable for application in cold regions where cyclic freeze–thaw conditions
cause the concrete to degrade. In this paper, a numerical simulation of the degradation processes induced by freeze–thaw cycles on bond-slip laws in HPFRC beam specimens has been developed so as to
assess their effect on the flexural response of specimens as the fibres’ volume percentage changes. Their cracking strength, postcracking strength, and toughness were predicted, with the present
model being able to predict the cracking strength, postcracking strength and toughness of the HPFRC beam element under bending load conditions. Its accuracy was confirmed by comparing the model
predictions with experimental results.
1. Introduction
Over the last few years, high-performance fibre-reinforced concrete (HPFRC) has been widely used to strengthen ageing concrete structures [
] and to control the crack propagation and displacement in concrete slabs and shells, such as industrial floors, while also improving the seismic response of structural elements, such as columns,
beams, and walls [
]. Moreover, HPFRC is highly recommended in aggressive environments (e.g., marine environments, higher altitudes, northern areas) due to its high durability which is suitable for long-term structures
and heavily loaded bridges to reduce any maintenance costs [
]. It is also considered a sustainable material for the manufacturing of small thickness elements without steel rebars, resulting in a reduction in the CO
footprint [
]. It is well-known how HPFRC is a composite material in which the advantages of fibre-reinforced concrete (FRC) are combined with those of a high-performance concrete (HPC), reducing the weaknesses
of conventional concrete and improving its durability and mechanical performance. The addition of discontinuous fibres (e.g., steel fibres [
], synthetic fibres [
], natural fibres [
], basalt [
]), carbon and glass fibres [
]) in the HPC as well as in the concrete in general is able to significantly reduce its brittle behaviour, thus improving cracking, postcracking strength, and toughness [
] as well as its durability such as freeze–thaw resistance. Thanks to its dense microstructure, high-performance concrete also has a low permeability, resulting in a good resistance to various
external agents such as chloride attacks [
] and carbonation [
] as well as freezing and thawing cycles [
]. Its good antifreezing property makes it suitable for application at both high altitudes and in northern areas where cyclic freeze–thaw conditions are one of the main causes of two types of
concrete degradation: surface scaling, which is the loss of cement paste from the exposed surface, and internal crack growth, which makes the concrete crumble and deteriorate. Both phenomena can
reduce the quality of concrete throughout its lifetime. Over the last few years, the research on evaluating the freezing resistance of HPFRC has significantly increased, with several relevant
achievements having been obtained: in Feo et al. [
], the effects of 75 freeze and thaw cycles on both the dynamic moduli of elasticity, cracking and postcracking strength, as well as the toughness of HPFRC beam specimens reinforced with steel fibres
were evaluated; in [
], it was reported how the incorporation of basalt fibres can reduce the influence of freeze–thaw on the damage and failure process of the beam specimen under a bending test; in [
], it was studied how mineral admixtures (e.g., blast furnace slag, fly ash, silica fume, and metakaolin) contained in the HPC matrix possess an excellent frost resistance; in [
], an experimental investigation on the freeze–thaw resistance of HPC containing air-cooled slag (AS) and water-cooled slag (WS) was conducted; in [
], it was explained how adding nanosilica to the concrete makes it durable by enhancing its properties such as impermeability, porosity, and acid resistance. However, based on our knowledge, such
studies are experimental and not many predictive models have been proposed that capture the mechanical response of HPFRC, especially under freeze and thaw cycles.
In this study, the previous theoretical model developed by the authors [
], as an extension of a meso-scale formulation of a cracked hinge model implemented in a Matlab code [
], has been improved to predict the effect of freeze and thaw cycles on the flexural behaviour of HPFRC specimens as the fibres’ volume percentage changes. This model is intended to take into account
explicitly the behaviour of the two typical “phases” of fibre-reinforced cementitious composites (i.e., the cement-based matrix and the spread reinforcement, as well as with their interaction). The
kinematics of the proposed model was inspired by the so-called “cracked-hinge” approach [
], but both the random spatial distribution and orientation of fibres and the crack-bridging effect of fibres is explicitly simulated. The present model is able to estimate the cracking, postcracking
strength, and toughness of a HPFRC beam element under bending load conditions. Its reliability was confirmed by comparing the model predictions with the experimental results obtained in [
2. Outline of the Experimental Results
The present study is part of a wider research whose experimental part was already published into details in a previous paper [
]. A brief summary about the obtained results is reported hereafter, for the readers’ sake.
Three different HPFRC mixtures CM0, CM1, and CM2, obtained by fixing the HPC matrix and varying the fibre volume fraction,
, in the set 0%, 1.25%, and 2.50%, respectively, were examined. Short steel fibres,
Dramix OL 13/0.20
], with an aspect ratio,
, equal to 65 were chosen as the reinforcement of the HPC matrix whose mix design was provided by the manufacturer [
]. For each type of HPFRC mixture, eight standard 150 × 150 × 600 mm prismatic specimens (PS) and five standard 150 × 150 × 150 mm cubic samples (CS) were cast. At the end of the curing period, the
prismatic specimens for each HPFRC mixture were subjected to 75 freeze–thaw cycles according to UNI 7087-2017 [
Subsequently, the prismatic specimens were tested under a four-point bending setup according to UNI 11039-2 [
] in which the vertical load (P) and the corresponding average “Crack Tip Opening Displacement” (
) were monitored during each test. All the cube samples were only tested in compression to evaluate, according to EN 12390-4 [
], the compressive load,
, as well as the compressive strength,
, for any changes of the fibres’ volume fraction.
Table 1
reports the average values of the first crack load,
, of the first crack strength,
, and the equivalent postcracking strengths,
, before and after the freeze–thaw cycles for each type of HPFRC mixture.
Table 2
summarises the average values of the work capacity indices,
, and the ductility indices,
, before and after the freeze–thaw cycles for the two types of HPFRC mixture (CM1 and CM2).
3. Theoretical Model
The abovementioned experimental results were here used to improve the
cracked-hinge model
] in which a meso-mechanical approach was adopted with the aim of predicting the bending response of HPFRC beam elements under normal environmental conditions.
3.1. Assumptions and Formulation
In this new model, however, a different transition zone length and a modified bond-slip law of the fibres were taken into account in order to estimate the effects of freeze–thaw cycles on the
flexural behaviour of standard specimens of a length
, width
, depth
, and transversely notched at the midspan section for a depth equal to
Figure 1
By denoting x, y, and z, the axes of a Cartesian coordinate system originating at the centre of the midspan section, a random distribution of the fibres inside the specimen prismatic volume was
generated (by utilising the standard random number generator of Matlab) in which x[(f,k)], y[(f,k)], and z[(f,k)] (with 𝑘 between 1 and n[f]), and α[(y,k)] and α[(z,k)] are the three coordinates of
the fibre centroid (G[(f,k)]) and the two relevant angles, respectively. In order to consider the bridging effect offered by the fibres, the total number of fibres in the midspan section, n[f], was
determined as a ratio between the fibres’ volume fraction, V[f], and the cement matrix volume, V[c].
Furthermore, the model was based on the following assumptions:
The flexibility was distributed in the central part of the specimen for a length equal to “
” while a rigid body behaviour was exhibited by the remaining end parts (
Figure 2
The midspan cross-section was discretized in
$n c$
layers as shown in
Figure 3
. The average axial strain of the
-th layer,
$ε k$
, before crack formation, and the crack-opening displacement,
$w k$
, after the crack formation, can be easily expressed for the
-th layer (
k =
1, …,
$n c$
) as in Equations (1) and (2).
$ε k = 2 φ j · ( z c − z k ) s$
$w k = 2 φ j · ( z c − z k )$
Equations (1) and (2) are typical of the “cracked hinge” model family (after their original formulations by Hillerborg et al. [
] and Olesen [
]). Specifically, Equation (2) rests on the assumption that plane sections remain plane and Equation (1) on the assumption of a characteristic length “
” which can be defined to convert axial displacements (at the numerator of the right-hand side) in axial strains (at the left-hand side).
Consequently, the average value of the axial stress,
$σ c , k$
, at
-th strip can be determined as a function of the axial deformation,
$ε k$
, before cracking, or a function of the crack-opening displacements,
$w k$
, after cracking. The stress–strain and stress–displacement relationships assumed in this paper are reported in
Section 3.2.2
A transition length,
$l t$
, was introduced in the notched cross-section (
Figure 4
), which starts from the top of the notch to the top of the integral part of the section, in order to consider the possible microdamage phenomena produced by the notching process. The mechanical
meaning of this quantity is discussed in details in a previous paper [
] and omitted herein for the sake of brevity. Therefore, a reduced value of the width,
$b k$
, inside the transition zone was considered which can be evaluated with an exponential law as in Equation (3) where
$l k$
are the distance of the
-th strip from the top of the notch and the coefficient of the exponential law, respectively:
$b k = b · ( l k l t ) α$
The bridging effect offered by the fibres was taken in to account by introducing the action, $F k , j$, mobilised at the j-th step of the incremental analysis as in Equation (4):
$F k ( z f , k ; z c , j ; φ j ) = A f · σ f , k ( z f , k ; z c , j ; φ j )$
in which
$z f , k$
$z c , j$
$φ j$
are the position of the
-th fiber in the cross section, the position of the neutral axis and the rotation of the two rigid blocks at the
-th step of the increment analysis (
Figure 2
), respectively. In particular, the axial stress of the
-th fiber,
$σ f , k$
depends on the sliding of one of the two parts embedded in the two sides of the crack and, therefore, it was correlated to the bond stresses,
, mobilized on its lateral surface due to the crack opening displacement
$w k$
$σ f , k ( z f , k ; z c , j ; φ j ) = 4 · l f − w f , k ( z f , k ; z c , j ; φ j ) d f · τ [ w f , k ( z f , k ; z c , j ; φ j ) ]$
Moreover, it should be noted that only a low number of these fibres,
$n f ′ < n f$
, will cross the crack which will potentially open in the middle of the beam. With the above assumptions, the position
of the neutral axis for the
-th imposed rotation
can be determined by solving the following equilibrium equation along the longitudinal axis which can be written as follows:
$∆ z · b · [ ∑ k = 1 n t ( l k l t ) α · σ c , k ( z k ; z c , j ; φ j ) + ∑ k = n t + 1 n c σ c , k ( z k ; z c , j ; φ j ) ] + ∑ k = 1 n f ′ F k ( z f , k ; z c , j ; φ j ) = 0$
is the number of layers into which the midspan section is discretized (
Figure 3
) and
is the number of layers of reduced width [
The solution of Equation (6) brings us to determining
$z c , j$
, at the
-th step of the incremental analysis. It is worth highlighting that this solution can only be obtained numerically: the well-known bisection method was employed to determine the actual value of
$z c , j$
which should obviously be included in the range between 0 and h–h
Figure 3
Then, the corresponding values of the bending moment
can be obtained as follows:
$M j = ∆ z · b · [ ∑ k = 1 n t ( l k l t ) α · σ c , k ( z k ; z c , j ; φ j ) · ( h 2 − z k ) + ∑ k = n t + 1 n c σ c , k ( z k ; z c , j ; φ j ) · ( h 2 − z k ) ] + ∑ k = 1 n f ′ F k ( z f , k ; z
c , j ; φ j ) · ( h 2 − z f , k )$
which is obviously related to the applied vertical load P (in 3-point bending):
The corresponding CTOD[j] value of can be obtained from Equation (2) by just replacing the generic value of z[k] with the position of the crack tip (z[k] = −h/2 + h[0] from Figue 2).
Therefore, for each value of the imposed rotation φ[j], a couple (CTOD[j], P[j]) can be determined, and then, the Force-CTOD graph can be incrementally determined up to failure.
Furthermore, the constitutive laws adopted for the HPC matrix and short steel fibres have the same shape as well as mathematical expressions as those already presented by the authors [
]. However, the unknown parameters in the constitutive laws were calibrated in
Section 4
on the experimental results already summarized in
Section 2
3.2. Constitutive Laws Assumed in the Present Study
3.2.1. Stress–Strain Relationships for Concrete in Compression and in Tension
The stress–strain relationship when the concrete is in compression was described by Equation (7), according to [
], in which
can be calculated as the ratio between the strain,
$ε c$
, and the strain at maximum compressive stress,
$ε c 1$
, can be evaluated as in Equation (8); whereas
is the plasticity number which depends on the elastic modulus in compression,
$E c$
, and on the secant modulus from the origin to the peak compressive stress,
$E c 1$
, as in Equation (9). The last one can be evaluated as a function of
$ε c 1$
and the concrete compressive strength (
$f c m$
). The schematic representation of the above stress–strain relationship in compression is shown in
Figure 5
$σ c = f c m · k · η − η 2 1 + ( k − 2 ) · η$
$ε c 1 = 1.60 · ( f c m 10 ) 0.25$
$k = E c E c 1 = E c ( f c m ε c 1 )$
The constitutive law when the concrete is in tension, evaluated according to the “
fictitious crack method
” employed in [
], presents a bilinear relation until the tensile strain of the
-th strip,
$ε k$
, reaches the conventional value equal to
Figure 6
a). A linear elastic behaviour is described by Equation (12a) until the section is uncracked after which the behaviour is expressed by the linear Equation (12b) as follows:
$σ c t = { E c t · ε c t f o r ε c t ≤ 0.9 · f c t m E c t ( a ) f c t m · ( 1 − 0.1 · 0.00015 − ε c t 0.00015 − 0.9 · f c t m E c t ) f o r 0.9 · f c t m E c t ≤ ε c t ≤ 0.00015 ( b )$
in which
$σ c t$
$E c t$
, and
$ε c t$
represent the tension stress (in MPa), the elastic modulus under tension load (in MPa), and the tensile strain, respectively, whilst
$f c t m$
(in MPa) indicates the tensile strength. Beyond this level, a softening constitutive stress-crack opening law was considered due to the opening of a crack in the
-th strip of the cross-section, Equation (13). Consequentially, the residual tension
$σ c t$
must be expressed as a function of the crack-opening displacement,
, (
Figure 6
b) in which
$w 1$
$w c$
are dependent on the fracture energy as defined in [
$σ c t = { f c t m · ( 1.0 − 0.8 · w w 1 ) f o r w ≤ w 1 ( a ) f c t m · ( 0.25 − 0.05 w w 1 ) f o r w 1 ≤ w ≤ w c ( b )$
3.2.2. Modified Bond-Slip Model for Short Steel Fibres
The mathematical relation of the local
$τ − s$
constitutive law adopted in this study is provided in Equation (14) in which
was considered equal to the crack-opening displacement
$w f , k$
at the level of the
-th fibre while the six unknown parameters (i.e.,
$s e l , s R , s u , τ e l , τ R , τ u$
) have to be calibrated using the experimental data of
Section 2
$τ = { τ e l · s s e l f o r s ≤ s e l ( a ) τ e l + ( τ R − τ e l ) s − s e l s R − s e l f o r s e l ≤ s ≤ s R ( b ) τ R f o r s R ≤ s ≤ s u ( c )$
Consequently, the nonlinear bond-slip law is divided in three different branches, as expressed in Equation (12):
• a linear-elastic behaviour up to the stress level corresponding to matrix tensile strength, identified by the two parameters $s e l$, $τ e l$;
• a hardening behaviour, characterized by the formation of many microcracks in the HPFRC mix, identified by the two parameters $s R$ and $τ R$;
• a constant behaviour defined by the two parameters $s u$ and $τ u$.
The curve presents an adequate “shape” for describing the global pull-out response of short steel fibres embedded within the HPC matrices. For the sake of simplicity, the current assumption includes
the effect of the fibre orientation in space (from 0° to 45°) with respect to the matrix surfaces, as demonstrated in a previous study [
]. However, more accurate assumptions could be formulated with the aim to take into account both the axial deformation of fibres (which is neglected in this study, as it is focused on “short” fibres)
and the aforementioned effect of fibre orientation with respect to the transverse section of the specimen at midspan.
4. Inverse Identification of the Relevant Material Laws
An inverse identification procedure was carried out with the aim to determine the values of the model parameters that lead to minimizing the difference between the measured and predicted Force-CTOD
curves. Specifically, the cylindrical compression strength, $f c m$, the transition length $l t$, the exponential parameter, $α$, and the six parameters of the bond-slip law (i.e., $s e l , s R , s u
, τ e l , τ R , τ u$) were considered as variable with some quantitative restrictions (e.g., s[el] < s[R] < s[u], τ[el] < τ[R] and τ[R] = τ[u]) intended at respecting the mechanical consistency of
the model. In order to evaluate how the values of these parameters depend on the effect of freeze–thaw cycles, several numerical simulations were carried out. In particular, three groups of 100
simulations each, assuming $n c = 50$ and $s$ = 300 mm, were run as described below.
• In the first one, the cylindrical compression strength,
$f c m$
, the transition length
$l t$
, and the exponential parameter,
, were calibrated on the flexural response of the conditioned CM0 specimens (labelled CM0-FT). Experimentally, a 21% reduction in the cylindrical compression strength,
$f c m$
, was observed on the conditioned specimens compared to not conditioned ones. This reduction was taken in account to calibrate the value of the transition length,
$l t$
, whose value, in the present model, was assumed equal to 85 mm (with an increase of 21% compared to that used in the previous model [
] in which the flexural behaviour of unconditioned CM0 specimens (labeled CM0-NFT) was predicted with a transition length,
$l t$
, equal to 70 mm). In both models, the coefficient of the exponential law,
, was considered constant and equal to 0.40 (
Table 3
Figure 7
shows both the average experimental
$P − C T O D , a v g$
curve (light-blue line) and the average numerical
$P − C T O D , a v g$
curve (pink line) obtained with the present model employed for the CM0-FT specimens.
• In the second one, the six parameters of the bond-slip law (i.e., $s e l , s R , s u , τ e l , τ R , τ u$) were calibrated on the flexural response of conditioned CM1 specimens (labelled CM1-FT).
A 13% reduction in the parameter $τ e l$ was adopted in the calibration of the conditioned specimens compared to the unconditioned ones.
• In the last one, only the parameter,
$τ e l$
, was calibrated again on the flexural response of conditioned CM2 specimens (labelled CM2-FT) while all the other parameters were considered constant. A 19% reduction in the parameter
$τ e l$
was adopted for the conditioned specimens compared to the unconditioned ones. Moreover, as in [
], a 20% reduction in the fibres’ volume fraction,
$V f$
, was considered in order to take into account the nonuniform fibre distribution.
The results of the last two calibrations were listed in
Figure 8
Table 4
, while
Figure 9
a and
Figure 10
a show the comparison between the average experimental
$P − C T O D , a v g$
curve (violet line) and the average numerical
$P − C T O D , a v g$
curve (green line) obtained with the previous model for the CM1-NFT and CM2-NFT specimens, respectively. Whereas
Figure 9
b and
Figure 10
b show the comparison between the average experimental
$P − C T O D , a v g$
curve (light-blue line) and the average numerical
$P − C T O D , a v g$
curve (pink line) used with the present model for the CM1-FT and CM2-FT specimens, respectively.
5. Results
The model developed by the authors [
] was, here, used to predict the postcracking response of HPFRC beam elements under freeze–thaw cycles.
In order to assess the model accuracy, the theoretical and experimental results of the CM1 and CM2 specimens were compared: first, the average values of the two equivalent postcracking strengths
after the freeze–thaw cycles,
$f e q ( 0 − 0.6 ) , a v g F T$
$f e q ( 0.6 − 3 ) , a v g F T$
, are listed in
Table 5
, for the CM1 and CM2 mixtures, in which the values were compared with those obtained before the freeze–thaw cycles of Ref. [
$f e q ( 0 − 0.6 ) , a v g N F T$
$f e q ( 0.6 − 3 ) , a v g N F T$
; second, the average values of the two working capacity indices after the freeze–thaw cycles,
$U 1 , a v g F T$
$U 2 , a v g F T$
, are summarised in
Table 6
, for the CM1 and CM2 mixtures, in which the values were compared with those obtained before the freeze–thaw cycles of Ref. [
$U 1 , a v g N F T$
$U 2 , a v g N F T$
The correlation graphs of the two equivalent postcracking strengths were plotted in
Figure 11
Figure 12
, respectively, while
Figure 13
Figure 14
show the correlation graphs of the average values of the two working capacity indices
$U 1$
$U 2$
, respectively. For each point, the red bars represent the standard deviation of the experimental and predicted values. It was noted how the convergence with the present model was good so that a
lower standard deviation was observed.
6. Conclusions
The present paper was aimed at scrutinizing the effects of freeze–thaw cycles on the fundamental mechanical behaviour of the “components” controlling the structural behaviour of HPFRC members in
bending. Specifically, based on a series of experimental results obtained by the first two authors in a previous study [
], a simplified “cracked-hinge” model was considered with the aim to identify both the concrete constitutive relationship and the bond-slip law of fibres in “unconditioned” and “conditioned
Based on the results obtained from the aforementioned inverse identification procedure, the following main considerations can be drawn out:
• The freeze–thaw cycles effect the cylindrical compression strength, $f c m$, the transition length $l t$, and the bond-slip law of fibres, which confirms their significance as relevant parameters
controlling the resulting response of HPFRC specimens;
• Table 3
shows that the compressive strength
undergoes a substantial reduction (in the order of 20%) as a result of the degradation processes induced by the FT cycles;
• as for the transition zone, which is a peculiar aspect of the considered model, a moderate increase in its the depth (from 70 mm to 85 mm) can be identified after the FT cycles, whereas its shape
(controlled by the exponent α) does not change;
• Table 4
points out that the bond-slip law of fibres is also affected by the FT cycles, as, specifically, the elastic limit stress,
, (and, consequently, the initial elastic stiffness of the same law) reduces by about 15%, with no changes in the other parameters;
• however, under the designers’ standpoint (and besides the specific values obtained in the present study), it should be noted that this change affects both serviceability and ultimate limit states
in the structural response.
Finally, the generally good agreement between the experimental data and the values obtained from the identified model confirms the mechanical consistency of the latter and its potential accuracy.
However, further experimental results are needed to calibrate general relationships between the main parameters controlling the bond-slip law of fibres and the actual number of freeze–thaw cycles:
this will be part of the future developments of the present research.
Author Contributions
Conceptualization, R.P., L.F., E.M. and M.P.; methodology, M.P. and E.M.; software, E.M.; validation, M.P. and R.P.; formal analysis, M.P. and R.P.; data curation, R.P., L.F., E.M. and M.P.;
writing—original draft preparation, R.P. and M.P.; writing—review and editing, L.F. and E.M.; supervision, L.F. and E.M. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1.
Schematic representation of the 3D HPFRC beam: adapted from [
Figure 2.
Kinematics of the cracked-hinge model: adapted from [
Figure 4. Notched cross-section with exponential law of the reduced width, $b k$, in the transition zone, $l t$: (a) linear expression ($α$ = 1) of $b k$, (b) exponential expression with $α > 1$, (c)
exponential expression with $α < 1$.
Figure 6. Schematic graph of the stress–strain relation for uniaxial tension: (a) for $ε k ≤ 0.00015$, the stress–strain behaviour is described by a bilinear relation; (b) for $ε k > 0.00015$, the
stress–strain behaviour is described by a softening constitutive stress–crack opening law.
Figure 7. The average experimental $P − C T O D , a v g$ curve (light-blue line) versus the average numerical $P − C T O D , a v g$ curve (pink line) for conditioned CM0 specimens (labelled CM0-FT)
obtained with the present model.
Figure 9.
). The average experimental
$P − C T O D , a v g$
curve (violet line) versus the average numerical
$P − C T O D , a v g$
curve (green line) for unconditioned CM1 specimens (labelled CM1-NFT) obtained with the previous model of Ref. [
]. (
). The average experimental
$P − C T O D , a v g$
curve (violet line) versus the average numerical
$P − C T O D , a v g$
curve (green line) for unconditioned CM2 specimens (labelled CM2-NFT) obtained with the previous model of Ref. [
Figure 10. (a). The average experimental $P − C T O D , a v g$ curve (light-blue line) versus the average numerical $P − C T O D , a v g$ curve (pink line) for conditioned CM1 specimens (labelled
CM1-FT) obtained with the present model. (b). The average experimental $P − C T O D , a v g$ curve (light-blue line) versus the average numerical $P − C T O D , a v g$ curve (pink line) for
conditioned CM2 specimens (labelled CM2-FT) obtained with the present model.
Figure 11.
Correlation between the experimental and theoretical average values of the equivalent postcracking strengths after the freeze–thaw cycles,
$f e q ( 0 – 0.6 ) , a v g F T$
and before the freeze–thaw cycles of Ref. [
$f e q ( 0 – 0.6 ) , a v g N F T$
for the CM1 and CM2 mixtures.
Figure 12.
Correlation between the experimental and theoretical average values of the equivalent postcracking strengths after the freeze–thaw cycles,
$f e q ( 0.6 – 3 ) , a v g F T$
, and before the freeze–thaw cycles of Ref. [
$f e q ( 0.6 – 3 ) , a v g N F T$
, for the CM1 and CM2 mixtures.
Figure 13.
Correlation between the experimental and theoretical average values of the working capacity indices after the freeze–thaw cycles,
$U 1 , a v g F T$
, and before the freeze–thaw cycles of Ref. [
$U 1 , a v g N F T$
, for the CM1 and CM2 mixtures.
Figure 14.
Correlation between the experimental and theoretical average values of the working capacity indices after the freeze–thaw cycles,
$U 2 , a v g F T$
, and those before the freeze–thaw cycles of Ref. [
$U 2 , a v g N F T$
, for the CM1 and CM2 mixtures.
Table 1. The average values of the first crack load, P[lf], of the first crack strengths, f[(If,avg)], and of the equivalent postcracking strengths, f[(eq(0–0.6),avg)] and f[(eq(0.6–3),avg)], before
(NFT) and after the freeze–thaw cycles (FT) for each type of HPFRC mixture (CM0, CM1, and CM2).
$P I f , a v g N F $P I f , a v g F $f I f , a v g N F $f I f , a v g F $f e q ( 0 – 0.6 ) , a v g N F $f e q ( 0 − 0.6 ) , a v g F $f e q ( 0.6 – 3 ) , a v g N F $f e q ( 0.6 – 3 ) , a v g F
Mix. T$ T$ T$ T$ T$ T$ T$ T$
[kN] [kN] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa]
CM0 11.213 9.105 3.05 2.477 - - - -
CM1 14.489 12.538 4.013 3.475 6.617 5.435 7.99 6.845
CM2 18.595 16.175 5.06 4.327 9.15 7.537 11.473 9.255
Table 2. The average values of the work capacity indices, U[(1,avg)] and U[(2,avg)], and ductility indices, D[(0,avg)] and D[(1,avg)], before (NFT) and after the freeze–thaw cycles (FT) for the two
types of HPFRC mixture (CM1 and CM2).
Mix. $U 1 , a v g N F T$ $U 1 , a v g F T$ $U 2 , a v g N F T$ $U 2 , a v g F T$ $D 0 , a v g N F T$ $D 0 , a v g F T$ $D 1 , a v g N F T$ $D 1 , a v g F T$
[kNmm] [kNmm] [kNmm] [kNmm] [-] [-] [-] [-]
CM1 14,283.43 11,742.40 69,226.73 59,175.15 1.647 1.565 1.265 1.265
CM2 20,508.47 16,895.83 102,877.93 82,992.00 1.837 1.747 1.257 1.250
Table 3.
Calibration of the input data for unconditioned CM0 specimens (labelled CM0-NFT) and for conditioned CM0 specimens (labelled CM0-FT) used in the previous model of Ref. [
] and in the last one, respectively.
Specimen Designation $f c m$ $l t$ $α$ Model
[MPa] [mm] [-]
CM0-NFT 53.0 70.0 0.4 Ref. [42]
CM0-FT 42.0 85.0 0.4 Present paper
Table 4.
Calibration of the six parameters in the local bond-slip law for unconditioned CM1 and CM2 specimens (labelled, respectively, CM1-NFT and CM2-NFT) as well as for conditioned CM1 and CM2 specimens
(labelled, respectively, CM1-FT and CM2-FT) used in the previous model of Ref. [
] and in the last one, respectively.
Series $s e l$ $s R$ $s u$ $τ e l$ $τ R$ $τ u$ Model
[mm] [mm] [mm] [MPa] [MPa] [MPa]
CM1-NFT 0.10 8.00 10.00 8.00 21.50 21.50 Ref. [42]
CM1-FT 0.10 8.00 10.00 7.00 21.50 21.50 Present paper
CM2-NFT 0.10 8.00 10.00 8.00 21.50 21.50 Ref. [42]
CM2-FT 0.10 8.00 10.00 6.50 21.50 21.50 Present paper
Table 5.
Comparison between the experimental and theoretical average values of the two equivalent postcracking strengths after the freeze–thaw cycles,
$f e q ( 0 – 0.6 ) , a v g F T$
$f e q ( 0.6 – 3 ) , a v g F T$
, and those before the freeze–thaw cycles of Ref. [
$f e q ( 0 – 0.6 ) , a v g N F T$
$f e q ( 0.6 – 3 ) , a v g N F T$
, for the CM1 and CM2 mixtures.
CM1 CM2
Results $f e q ( 0 – 0.6 ) , a $f e q ( 0 – 0.6 ) , $f e q ( 0.6 – 3 ) , a $f e q ( 0.6 – 3 ) , $f e q ( 0 – 0.6 ) , a $f e q ( 0 – 0.6 ) , $f e q ( 0.6 – 3 ) , a $f e q ( 0.6 – 3 ) ,
v g N F T$ a v g F T$ v g N F T$ a v g F T$ v g N F T$ a v g F T$ v g N F T$ a v g F T$
[MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa]
Experimental 6.617 5.435 7.990 6.845 9.150 7.537 11.473 9.255
Theoretical 6.179 5.411 7.672 6.871 8.486 7.125 11.532 9.871
Percentage 6.61 0.45 3.98 0.39 7.26 5.47 0.51 6.66
difference (%)
Table 6.
Comparison between the experimental and theoretical average values of the two working capacity indices after the freeze–thaw cycles,
$U 1 , a v g F T$
$U 2 , a v g F T$
, and those before the freeze–thaw cycles of Ref. [
$U 1 , a v g N F T$
$U 2 , a v g N F T$
, for the CM1 and CM2 mixtures.
CM1 CM2
Results $U 1 , a v g N F T$ $U 1 , a v g F T$ $U 2 , a v g N F T$ $U 2 , a v g F T$ $U 1 , a v g N F T$ $U 1 , a v g F T$ $U 2 , a v g N F T$ $U 2 , a v g F T$
[kNmm] [kNmm] [kNmm] [kNmm] [kNmm] [kNmm] [kNmm] [kNmm]
Experimental 14,283.43 11,742.40 69,226.73 59,175.15 20,508.47 16,895.83 102,877.93 82,992.00
Theoretical 13,625.51 11,930.62 67,667.93 60,606.61 18,711.24 15,709.63 101,710.39 87,061.56
Percentage difference (%) 4.61 1.60 2.25 2.42 8.76 7.02 1.13 4.90
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Penna, R.; Feo, L.; Martinelli, E.; Pepe, M. Theoretical Modelling of the Degradation Processes Induced by Freeze–Thaw Cycles on Bond-Slip Laws of Fibres in High-Performance Fibre-Reinforced
Concrete. Materials 2022, 15, 6122. https://doi.org/10.3390/ma15176122
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Penna R, Feo L, Martinelli E, Pepe M. Theoretical Modelling of the Degradation Processes Induced by Freeze–Thaw Cycles on Bond-Slip Laws of Fibres in High-Performance Fibre-Reinforced Concrete.
Materials. 2022; 15(17):6122. https://doi.org/10.3390/ma15176122
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Penna, Rosa, Luciano Feo, Enzo Martinelli, and Marco Pepe. 2022. "Theoretical Modelling of the Degradation Processes Induced by Freeze–Thaw Cycles on Bond-Slip Laws of Fibres in High-Performance
Fibre-Reinforced Concrete" Materials 15, no. 17: 6122. https://doi.org/10.3390/ma15176122
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Gordon Nipp's Tables of Quinary Quadratic Forms
Keywords: tables, reduced, regular, primitive, positive definite, quinary, quadratic forms, five-dimensional lattices, automorphism group, mass, genus, genera, Hasse symbol
(gnipp@calstatela.edu), of the Department of Mathematics, California State University, Los Angeles, CA 90032, USA, and are included here with his permission.
Contents of these files
About these tables
About this Catalogue of Lattices
The format of these tables
Discriminants 2 to 256
Discriminants 257 to 270
Discriminants 271 to 300
Discriminants 301 to 322
Discriminants 323 to 345
Discriminants 346 to 400
Discriminants 401 to 440
Discriminants 441 to 480
Discriminants 481 to 500
Discriminants 501 to 513
Format of these files
These are tables of reduced regular primitive positive-definite quinary quadratic forms over the rational integers.
They were computed by Gordon L. Nipp (see above) and are included here with his permission.
These tables contain one 5-variable quadratic form from each primitive class of such forms through discriminant d=513. The classes are grouped into genera; the mass of each genus has been computed,
and, for each prime p dividing 2d, Hasse symbols are given. In addition, the number of automorphs for each form has been computed, so that the well-known relationship between the mass of the genus
and the number of automorphs of each class serves as a check on the accuracy of the tables.
Based on results of Minkowski [1] and van der Waerden [6] and beginning with the quaternary forms through discriminant 1296 contained in Nipp [2] (ample to ensure completeness), we generated
successive large collections of reduced quinary forms containing forms in all possible quinary classes through d=513 (the discriminant is that of Watson [7]). Grouping into classes and choosing a
representative of each class was accomplished using a well-tested computer program described for the quaternary case in [2], and a similar program was used to compute the number of automorphs for
each such representative. The classes were then separated into genera using an upgrade of a program described in [2] and based on results contained in O'Meara's book [3]. The 2-adic density a_2 of
each form was computed as in Watson's article [8], and for primes p dividing d, p >2, p-adic densities a-p were calculated using Pall's results [4]. Following Siegel [5] and Pall [4], the mass was
computed according to the formula
                  d^3 Product (1-p^{-2})(1-p^{-4}), p|d, p>2
    M =     --------------------------------------------------
                      128 Product a_p, p|2d
As an example of the format of the tables, we include the eleventh (of twelve) genera of discriminant 120:
D= 120; GENUS# 11; MASS= 5/ 64; HASSE SYMBOLS ARE 1-1-1
1 1 1 3 4 0 0 0 0 0 0 1 1 1 3; 192
1 2 2 2 3 0 0 0 1 2 2 0 0 2 1; 32
1 2 2 2 3 0 0 2 1 2 0 0 2 0 1; 24
Here the mass of the genus is 5/64, the Hasse symbols are 1, -1, and -1 at primes 2, 3, and 5 respectively, and the three classes in the genus have 192, 32, and 24 automorphs respectively. The first
class is represented by the form
        x11^2 + x22^2 + x33^2 + 3x44^2 + 4x55^2 + x1x5 + x2x5 + x3x5 + 3x4x5
In general, the coefficients for a quinary form are given in the order a11,a22,a33,a44,a55,a12,a13,a23,a14,a24,a34,a15,a25,a35,a45 where aij is the coefficient of xixj. As a check one notes that the
mass, computed using the formula above, is 5/64=1/192+1/32+1/24, the sum over the classes of the reciprocals of the number of automorphs.
The computations were performed on the California State University Cyber 960 and on the CSU Sacramento Multiflow Trace machine.
[1] H. Minkowski, Gesammelte Abhandlungen, Chelsea, New York, 1967.
[2] G. Nipp, Quaternary Quadratic Forms - Computer Generated Tables, Springer-Verlag, New York, 1991.
[3] O. T. O'Meara, Introduction to quadratic forms, Die Grundlehren der math. Wissenschaften, Band 117, Academic Press, New York; Springer-Verlag, Berlin, 1963.
[4] G. Pall, "The weight of a genus of positive n-ary quadratic forms", Proc. Sympos. Pure Math. vol. 8 (Amer. Math. Soc., Providence, R.I., 1965), 95-105.
[5] C. L. Siegel, "Ueber die analytische Theorie der quadratischen Formen", Ann. of Math. 36 (1935), 527-606.
[6] B. L. van der Waerden, "Die Reduktionstheorie der positiven quadratischen Formen", Acta Math. 96 (1956), 265- 309.
[7] G. L. Watson, Integral quadratic forms, Cambridge University Press, Cambridge, 1960.
[8] G. L. Watson, "The 2-adic density of a quadratic form", Mathematika 23 (1976), 94-106.
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Factor Models - Definition, Advantages and Disadvantages of...
Factor models
In finance, a factor models refers to statisticals model that aims to explain the behavior of stocks by breaking it down into a set of underlying factors. In particular, factor models relate the
return of a security to a number of (risk) factors. The underlying factors should be chosen based on economic intuition and empirical evidence. The factors are meant to capture broad market trends,
sector- and region specific risks, rewarded risk premia and other macroeconomic variables.
Generally, factor models have the following structure:
where r[i,t] is security i‘s return at time t, F[1,t], F[2][,t,] to F[N,t] are the factors, and β[1], β[2] to β[N] are the factor loadings.
The factor loadings measure the sensitivity of the security to changes in the factor. For example, suppose that one of the factors is the return on the S&P500. Now, imagine we estimate the model and
obtain a β equal 2. This means that, if the S&P500 increases by 1%, then the security’s price tends to go up by 2%, on average.
The equation also contains a so-called error term (ε). The error term is necessary to be able to estimate the (regression) model. We can, however, also give an economic interpretation to this
parameter. In particular, the error term represents security-specific news that becomes available to investors.
The coefficient that we are mainly interested in is the constant in the factor model. We have denoted the constant using the greek symbol α and investors will also typically refer to it as the
‘alpha’ . α measures the return of a strategy that cannot be explained by exposure to the factors in the model or firm-specific news. As such, it measures the strategy’s excess return.
Portfolio construction
How can factor models be used in the context of portfolio construction?
Basically, we want to achieve the following objectives:
• maximize the (excess) return of the portfolio, i.e. maximize the alpha of the portfolio
• minimize volatility of the portfolio that is due to stock market fluctuations, i.e. minimize the beta of the portfolio
• minimize the impact of firm-specific news, i.e. being sufficiently diversified
Note that the approach only makes sense if the investors can correctly determine which stocks have a positive alpha. Thus, the investors should have stock-picking skills.
One way of solving the above problem was proposed by Treynor and Black. The approach is referred to as the Treynor-Black model.
Advantages and disadvantages of factor models
The main advantages of factor models are the following.
1. Simple: the models are easy to use and allow us to simplify the complex behavior of financial variables by breaking them down into underlying factors. This makes it significantly easier to
understand what drives returns. It also helps investors to make more informed investment decisions.
2. Constructing portfolios: factor models are used to construct portfolios that are designed to capture specific rewarded factors, such as value or momentum. This will result in higher returns and/
or lower risk and can improve strategies’ Sharpe ratio
3. Managing risk: these kinds of models are used to assess the risk of a portfolio by estimating the exposures of a portfolio to different risk factors. This information is used to identify sources
of risk and to make adjustments to the portfolio’s exposures
4. Pricing assets: factor models are used to estimate the securities’ expected return by taking into account their exposures to the different risk factors. This allows investors to value assets.
While there are many advantages, there are also some disadvantages:
1. Overfitting: the models are prone to overfitting, meaning that the model may fit the data too closely and may thus not generalize well to out-of-sample data. This can lead to poor out-of-sample
2. Limited explanatory power: Factor models can only explain a limited portion of the variation in financial variables, and there may be other factors that are not captured by the model. This can
result in incorrect predictions and poor investment decisions.
3. Model risk: Factor models are based on a set of assumptions (the number of factors, the factor loadings, and the factor covariances). If the assumptions are incorrect, the model may produce
incorrect results. One important assumption is that the relationship between between the returns and the factors is linear, which may not be the case.
4. Data limitations: the models typically require a amount of sufficient high-quality data to be estimated accurately. Very often only limited historical data is available.
Factor models can be used to identify potentially interesting securities that can be added to a portfolio. In addition, the results can be used to construct better portfolios. | {"url":"https://breakingdownfinance.com/finance-topics/modern-portfolio-theory/factor-models/","timestamp":"2024-11-09T12:55:31Z","content_type":"text/html","content_length":"253802","record_id":"<urn:uuid:4b2bb71f-7afa-4da7-a0a5-1374500db27a>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00607.warc.gz"} |
Martin Milanič gave an online talk on a variation of tree-width of graphs by measuring the width of a tree-decomposition by the size of the largest stable set in a bag at the Virtual Discrete Math Colloquium - Discrete Mathematics Group
Martin Milanič gave an online talk on a variation of tree-width of graphs by measuring the width of a tree-decomposition by the size of the largest stable set in a bag at the Virtual Discrete Math
On November 5, 2021, Martin Milanič from the University of Primorska, Slovenia gave an online talk at the Virtual Discrete Math Colloquium on a variation of tree-width by measuring the width of a
tree-decomposition by the size of the largest stable set in a bag and its algorithmic applications to the maximum weight stable set. The title of his talk was “Tree Decompositions with Bounded
Independence Number“. | {"url":"https://dimag.ibs.re.kr/2021/martin-milanic-seminar/","timestamp":"2024-11-07T15:21:11Z","content_type":"text/html","content_length":"142089","record_id":"<urn:uuid:9139d5af-8757-466f-b2b4-bf1cab30eee6>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00453.warc.gz"} |
Nested IF with Multiple criteria
I am working on an approval by facility.
The first iteration had one approver list, this is working great.
=IF([Contract Expense Variance]1 < 5001, (VLOOKUP("5K", {Voting List Levels Range 6}, 2, false)), IF(AND([Contract Expense Variance]1 > 5000, [Contract Expense Variance]1 < 15001), (VLOOKUP("15K",
{Voting List Levels Range 6}, 2, false)), IF([Contract Expense Variance]1 > 35000, (VLOOKUP("100K", {Voting List Levels Range 6}, 2, false)), (VLOOKUP("35K", {Voting List Levels Range 6}, 2,
Now I need to add the facility option, I have created two approver lists
I believe this should be a IF(AND statement, but I have not been able to crack it. Any ideas or suggestions would be appreciated.
• Notepad is your friend on these.
Removed extraneous parenthesis
Removed Extraneous and statements. On an IF Statement there is no continue, there is a true or false. Once you have started down a branch, the previous criteria has already been checked. Example
=if(A1 < 5, "A is less than 5",if(A1 < 10,"A is less than 10, but greater than 5","A is greater than or equal to 10")
The same concept applies here.
At this point I noticed you a done something quite interesting that proves you understand the concept of my previous point. you return the false of the 35k to be between the 15001 and 35k. Now
that the formula is cleaned up, and the extraneous information is gone, try what I have posted below
IF([Contract Expense Variance]1 < 5001,
VLOOKUP("5K", {Voting List Levels Range 6}, 2, false),
IF(Contract Expense Variance]1 < 15001,
VLOOKUP("15K", {Voting List Levels Range 6}, 2, false),
IF([Contract Expense Variance]1 > 35000,
VLOOKUP("100K", {Voting List Levels Range 6}, 2, false),
VLOOKUP("35K", {Voting List Levels Range 6}, 2, false
• Thanks!
I ended up with this after I added the facility portion, it works but seems rather long. maybe there is a more direct way to get there?
=IF(Facility1 = "D02 - CNRL", IF([Contract Expense Variance]1 < 5001, (VLOOKUP("5K", {Voting List Levels Range 6}, 2, false)),
IF(AND([Contract Expense Variance]1 > 5000, [Contract Expense Variance]1 < 15001), (VLOOKUP("15K", {Voting List Levels Range 6}, 2, false)),
IF([Contract Expense Variance]1 > 35000, (VLOOKUP("100K", {Voting List Levels Range 6}, 2, false)), (VLOOKUP("35K", {Voting List Levels Range 6}, 2, false))))),
IF(Facility1 = "D08 - Syncrude", IF([Contract Expense Variance]1 < 5001, (VLOOKUP("5K", {D08 Voting List Levels Range 1}, 2, false)),
IF(AND([Contract Expense Variance]1 > 5000, [Contract Expense Variance]1 < 15001), (VLOOKUP("15K", {D08 Voting List Levels Range 1}, 2, false)),
IF([Contract Expense Variance]1 > 35000, (VLOOKUP("100K", {D08 Voting List Levels Range 1}, 2, false)), (VLOOKUP("35K", {D08 Voting List Levels Range 1}, 2, false)))))))
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10.3 Exploring the Sampling Distribution of b1
Course Outline
• segmentGetting Started (Don't Skip This Part)
• segmentStatistics and Data Science: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
• segmentChapter 4 - Explaining Variation
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Digging Deeper into Group Models
• segmentChapter 9 - Models with a Quantitative Explanatory Variable
• segmentPART III: EVALUATING MODELS
• segmentChapter 10 - The Logic of Inference
• segmentChapter 11 - Model Comparison with F
• segmentChapter 12 - Parameter Estimation and Confidence Intervals
• segmentChapter 13 - What You Have Learned
• segmentFinishing Up (Don't Skip This Part!)
• segmentResources
list High School / Advanced Statistics and Data Science I (ABC)
10.3 Exploring the Sampling Distribution of b1
It’s hard to look at a long list of \(b_1\)s and make any sense of them. But if we think of these numbers as a distribution – a sampling distribution – we can use the same tools of visualization and
analysis here that we use to help us make sense of a distribution of data. For example, we can use a histogram to take a look at the sampling distribution of \(b_1\)s.
The code below will save the \(b_1\)s (estimates for \(\beta_1\)) for 1000 shuffles of the tipping study data into a data frame called sdob1, which is an acronym for sampling distribution of b1s. (We
made up this name for the data frame just to help us remember what it is. You could make up your own name if you prefer.) Add some code to this window to take a look at the first 6 rows of the data
frame, and then run the code.
require(coursekata) sdob1 <- do(1000) * b1(shuffle(Tip) ~ Condition, data = TipExperiment) sdob1 <- do(1000) * b1(shuffle(Tip) ~ Condition, data = TipExperiment) head(sdob1) # the instructions in the
text above the exercise say that they can change the name of the object (sdob1) if they want, and the sdob1 contents are shuffled, so the easiest thing here is to just check that they called head ex
() %>% check_function("head")
CK Code: ch09-03-01-code
1 -0.1363636
2 6.7727273
3 0.6818182
4 -0.5909091
5 -5.7727273
6 7.5000000
In the window below, write an additional line of code to display the variation in b1 in a histogram.
require(coursekata) # we created the sampling distribution of b1s for you sdob1 <- do(1000) * b1(shuffle(Tip) ~ Condition, data = TipExperiment) # visualize that distribution in a histogram # we
created the sampling distribution of b1s for you sdob1 <- do(1000) * b1(shuffle(Tip) ~ Condition, data = TipExperiment) # visualize that distribution in a histogram gf_histogram(~b1, data = sdob1) ex
() %>% check_or( check_function(., "gf_histogram") %>% { check_arg(., "object") %>% check_equal() check_arg(., "data") %>% check_equal(eval = FALSE) }, override_solution(., '{ sdob1 <- do(1000) * b1
(shuffle(Tip) ~ Condition, data = TipExperiment) gf_histogram(sdob1, ~b1) }') %>% check_function(., "gf_histogram") %>% { check_arg(., "object") %>% check_equal(eval = FALSE) check_arg(., "gformula")
%>% check_equal() }, override_solution(., '{ sdob1 <- do(1000) * b1(shuffle(Tip) ~ Condition, data = TipExperiment) gf_histogram(~sdob1$b1) }') %>% check_function(., "gf_histogram") %>% { check_arg
(., "object") %>% check_equal(eval = FALSE) } )
CK Code: ch09-03-02-code
Although this looks similar to other histograms you have seen in this book, it is not the same! This histogram visualizes the sampling distribution of \(b_1\)s for 1000 random shuffles of the tipping
study data. There are a few things to notice about this histogram. The shape is somewhat normal (clustered in the middle and symmetric), the center seems to be around 0, and most of the values are
between -10 and 10.
Because the sampling distribution is based on the empty model, where \(\beta_1=0\), we expect the parameter estimates to be clustered around 0. But we also expect them to vary because of sampling
variation. Even if we generated a \(b_1\) as high as $10, it would just be the result of random sampling variation.
You can see from the histogram that while it’s not impossible to generate a \(b_1\) of 9 or 10, values such as 9 or 10 are much less frequent than values such as -1 or 1. In this case, the \(b_1\)
represents a mean difference between the two conditions. Thus, another way to say this is that it’s easy to randomly generate small mean differences (e.g., -1 or 1) but harder to randomly generate
large ones (e.g., -10 or 10).
Just eyeballing the histogram can give us a rough idea of the probability of getting a particular sample \(b_1\) from this DGP where we know \(\beta_1\) is equal to 0. When we use these frequencies
to estimate probability, we are using this distribution of shuffled \(b_1\)s as a probability distribution.
Using the Sampling Distribution to Evaluate the Empty Model
We used R to simulate a world where the empty model is true in order to construct a sampling distribution. Now let’s return to our original goal, to see how this sampling distribution can be used to
evaluate whether the empty model might explain the data we collected, or whether it should be rejected.
The basic idea is this: using the sampling distribution of possible sample \(b_1\)s that could have resulted from a DGP in which the empty model is true (i.e., in which \(\beta_1=0\)), we can look at
the actual sample \(b_1\) and gauge how likely such a \(b_1\) would be if the empty model is, in fact, true.
If we judge the \(b_1\) we observed to be unlikely to have come from the empty model, we then would reject the empty model as our model of the DGP. If, on the other hand, we judge our observed \(b_1
\) to be likely, then we would probably just stick with the empty model, at least until we have more evidence to suggest otherwise.
Let’s see how this works in the context of the tipping study, where \(b_1\) represents the average difference in tips between tables that got the hand-drawn smiley faces and those that did not.
Samples that are extreme in either a positive (e.g., average tips that are $8 higher in the smiley face group) or negative direction (e.g., -$8, representing much lower average tips in the smiley
face group), are unlikely to be generated if the true \(\beta_1=0\). Both of these kinds of unlikely samples would make us doubt that the empty model had produced our data.
Put another way: if we had a sample that fell in either the extreme upper tail or extreme lower tail of the sampling distribution (see figure below), we might reject the empty model as the true model
of the DGP.
In statistics, this is commonly referred to as a two-tailed test because whether our actual sample falls in the extreme upper tail or extreme lower tail of this sampling distribution, we would have
reason to reject the empty model as the true model of the DGP. By rejecting the model in which \(\beta_1=0\), we would be deciding that some version of the complex model in which \(\beta_1\) is not
equal to 0 must be true. We wouldn’t know exactly what the true \(\beta_1\) is, but only that it is probably not 0. In more traditional statistical terms, we would have found a statistically
significant difference between the smiley face group and the control group.
Of course, even if we observe a \(b_1\) in one of the extreme tails and decide to reject the empty model, we could be wrong. Just by chance alone, some of the \(b_1\)s in the sampling distribution
will end up in the tails even if the empty model is true in the DGP. Being fooled like this is what is called a Type I Error.
What Counts as Unlikely?
All of this, however, begs the question of how extreme a sample \(b_1\) would need to be in order for us to reject the empty model. What is unlikely to one person might not seem so unlikely to
another person. It would help to have some sort of agreed upon standard of “what counts as unlikely” before we actually bring in our real sample statistic. The definition of “unlikely” depends on
what you are trying to do with your statistical model and what your community of practice agrees on.
One common standard used in the social sciences is that a sample counts as unlikely if there is less than a .05 chance of generating one that extreme (in either the negative or positive direction)
from a particular DGP. We notate this numerical definition of “unlikely” with the Greek letter \(\alpha\) (pronounced “alpha”). A scientist might describe this criterion by writing or saying that
they “set alpha equal to .05”. If they wanted to use a stricter definition of unlikely, they might say “alpha equals .001,” indicating that a sample would have to be really unlikely for us to reject
the empty model of the DGP.
Let’s try setting an alpha level of .05 to the sampling distribution of \(b_1\)s we generated from random shuffles of the tipping study data. If you take the 1000 \(b_1\)s and line them up in order,
the .025 lowest values and the .025 highest values would be the most extreme 5% of values and therefore the most unlikely values to be randomly generated.
In a two-tailed test, we will reject the empty model of the DGP if the sample is not in the middle .95 of randomly generated \(b_1\)s. We can use a function called middle() to fill the middle .95 of
\(b_1\)s in a different color.
gf_histogram(~b1, data = sdob1, fill = ~middle(b1, .95))
The fill= part tells R that we want the bars of the histogram to be filled with particular colors. The ~ tells R that the fill color should be conditioned on whether the \(b_1\) being graphed falls
in the middle .95 of the distribution or not.
Here’s what the histogram of the sampling distribution looks like when you add fill = ~middle(b1, .95) to gf_histogram().
You might be wondering why some of the bars of the histogram include both red and blue. This is because the data in a histogram is grouped into bins. The value 6.59, for example, is grouped into
the same bin as the value 6.68, but while 6.59 falls within the middle .95 (thus colored blue), 6.68 falls just outside the .025 cutoff for the upper tail (and thus is colored red).
If you would like to see a more sharp delineation, you could try making your bins smaller, or to put it another way, making more bins. Doing so would increase the chances of having just one color in
each bin.
We re-made the histogram, but this time added the argument bins = 100 to the code (the default number of bins is 30). We also added show.legend = FALSE to get rid of the legend, and thus provide more
space for the plot.
gf_histogram(~b1, data = sdob1, fill = ~middle(b1, .95), bins = 100, show.legend = FALSE)
Increasing the number of bins resulted in each bin being represented by only one color. But it also created some holes in the histogram, i.e., empty bins in which none of the sample \(b_1\)s fell.
This is not a problem, it’s just a natural consequence of increasing the number of bins.
Remember, this histogram represents a sampling distribution. All these \(b_1\)s were the result of 1000 random shuffles of our data. None of these is the \(b_1\) calculated from the actual tipping
experiment data. All of these \(b_1\)s were created by a DGP where the empty model is true.
In the actual experiment of course, we only have one sample. If our actual sample \(b_1\) falls in the region of the sampling distribution colored red (based on the alpha we set), we will doubt that
it was generated by the DGP that assumes \(\beta_1=0\). In this case, based on our alpha criterion, we would reject the empty model. This could be the right decision…
But it might be the wrong decision. If the empty model is true, .05 of the \(b_1\)s that could result from different randomizations of tables to conditions would be extreme enough to lead us to
reject the empty model. If we rejected the empty model when it is, in fact, true, we would be making a Type I error. By setting the alpha at .05, we are saying that we are okay with having a 5% Type
I error rate.
What is the Opposite of Unlikely?
We’re going to be interested in whether our sample \(b_1\) falls in the .05 unlikely tails. But what if it doesn’t fall in the tails but instead in the middle part of the sampling distribution?
Should we then call it “likely”?
To be precise, if the sample falls in the middle .95 of the sampling distribution, it means that the sample is not unlikely. But saying that it is likely is a little bit sloppy, and possibly
In statistics, even if an event has a probability of .06, we will say it is not unlikely because our definition of unlikely is .05 or lower. But a regular person would not call something with a
likelihood of .06 “likely”.
It gets tiring to say not unlikely all the time, and sometimes sentences read a little bit easier if we just say likely. Just remember that when we say likely we usually mean not unlikely. But this
is not what normal people mean by the word likely. | {"url":"https://coursekata.org/preview/book/9824c414-fefa-4dad-b4a4-6f16900d1f53/lesson/14/2","timestamp":"2024-11-14T21:24:18Z","content_type":"text/html","content_length":"106806","record_id":"<urn:uuid:b829223a-7b55-45ee-9d52-30a4d61f1f75>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00882.warc.gz"} |
Examples of a Line in Real Life
A line can be simply defined as the shortest distance between two points plotted randomly on a 2D surface. In geometry, a line can be defined as a one-dimensional figure that extends in both
directions to infinity and does not have any width or depth. This implies that a line does not have any endpoint, hence the length of the line cannot be measured easily. Generally, a line is confused
with a line segment. The difference between a line and a line segment is that a line does not have endpoints, while a line segment has two endpoints. On the other hand, the similarity between the
line and the line segment is that both the types of geometric figures do not have width and depth parameters. This means that the length of a line is indeterminable, while the length of a line
segment is determinable and confined. A line can be easily represented with the help of a straight line having arrowheads on both sides. The arrowheads on both sides of the figure of a line indicate
the ability of a line to extend to infinity on both sides. The name of a line is usually either represented by a single lower case alphabet or two upper case alphabets. Here, the upper case alphabets
tend to denote the points present on the surface of the line. From a different point of view, a line can be observed to be a connection between multiple collinear points that are plotted on a
one-dimensional plane.
Examples of Line
Some of the most common examples of lines in real life are listed below:
1. Railway Tracks
Railway tracks tend to form a prominent example of lines in real life. This is because the railway tracks tend to stretch upto infinity on both sides and the length of the tracks is almost
2. Electricity Wires
The wires that are used by the energy service providers to transmit the electrical energy from the substation to the consumer destination tend to form yet another example of lines in real life. Such
cables and wires tend to extend to an immeasurable value.
3. Markings on Roads
The markings made on the roads usually with the help of white ir yellow coloured painting colour tend to denote the different lanes of the road, the distinction between the foothpath and the driving
path, and other road signals. Such markings form another example of line in real life as they can be observed as a set of infinite interconnected collinear points plotted on the surface of the road.
4. Zebra Crossing Stripes Slope of Mountains
The slope of mountains is yet another examples of line in real life. This is because the length of the slope of a mountain cannot be measured directly and can extend to infinity on both sides.
5. Ruler Horizon
Horizon is the imaginary line that tends to connect the surface of the earth and the sky. The horizon line stretches limitlessly in the surroundings and cannot be measured directly. The horizon is
also known as the line that seperates the celestial surface from the ground or the earth’s surface.
6. Window Panes Length of a Water Body
The length of the boundary of a river or the path taken by a river to flow is another example of lines in real life. This because generally a river has two sides stretching upto infinity and the
length of a river cannot be measured directly and easily with the help of generic measuring equipment and devices.
7. Pencil or Pen Fences
If the designed fences are dismantled and each part of the Fences is layed ahead of the another, a representation of a line can be created easily. The length of such a line is difficult to calculate.
Also, the line formed by arranging the individual fences parts can extend towards infinity in both sides.
8. Curtain Rods Axis
Axis can be defined as an imaginary fixed reference line that can be used to locate the position of an object in space. Typically, there are three axis lines called x, y, and z that tend to extend
from infinity to negative of infinity. This implies that the axis lines can stretch upto infinity on both sides and the length of the axis lines is immeasurable.
9. Incense Stick Tunnel
A tunnel is a perfect example of line that can extend to infinity on both sides and is difficult to be measured.
10. Traffic Light Pole Roads
The ability of roads to stretch to infinity on both sides qualifies them to be listed under the category of the real life examples of line geometric figure.
11. Racing Track Lines Roller Coaster Tracks
The tracks of a roller coaster are a prominent example of lines in real life. This is because the length of any randomly picked part of a roller coaster track is capable of extending in both sides to
infinity. Also, the distance of the roller coaster tracks is difficult to estimate directly.
12. Railings Irrigation Channel in Fields
To properly water the crops in an agricultural field, a narrow lane to carry water between the fields is typically created. This narrow lane of land eases the process to carry and circulate water
throughout the field. The lanes tend to extend from negative infinity to positive infinity and form a classic example of application of the concept of lines in real life. Also, the rows along which
the seeds are sowed or the crops are planted tend to form an example of lines in everyday life.
13. Straws Maps
A map is a graphical and pictorial representation of a physical locality. A map typically comprises of multiple pathways and location tags. The length of the pathways drawn on maps can extend to
infinity on both sides and is typically immeasurable directly. This is the reason why the pathways drawn on maps is another example of lines in real life.
14. Bridge Strings Conveyer Belt
The conveyer belt used in most factories and companies is spread throughout the premises. The main objective of such conveyer belts is to transport objects from one place to the other. If you observe
any random portion of a conveyer belt installed in an organisation, it extends to both sides. This implies that the structure of a conveyer belt is quite comparable to a line.
15. Ladder Rungs and Frame
16. Hands of an Analogue Clock
17. Thermometer
18. Popsicle Stick
19. Skiing Items
20. Sword
21. Cricket Stumps
22. Base Ball Bat Playing Ground Boundary
The boundary of a play ground such as a golf course, baseball field, soccer, etc., if assumed to be stretched open linearly, tends to form yet another example of lines in real life. This is because
in such a case, the boundary of the ground stretches to infinity on both sides and is nearly immeasurable.
23. Cricket Bat Blood Veins
Blood veins are arranged in the body of a living being randomly. If the veins of a living being are observed closely, they tend to stretch to infinity and can be flexibly listed under the category of
line geometric one dimension figure.
24. Thread
A thread stretched and arranged straight in an open space is a classic example of lines in real life. This is because the length of a infinitely stretched thread is a next to impossible task.
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How to Solve Numbrix Puzzles - Keep it Simple Puzzles (2024)
Fill in Numbers | Lines and Paths
Numbrix, created by Marilyn vos Savant, is a puzzle in which you fill in the grid with numbers so that you can follow a single path from 1 to the maximum number possible in the grid. In a 7×7 grid,
for example, it would be 1-49. Our example puzzle is 10×10, so we will be filling in numbers for a sequence from 1-100.
This path may only move orthogonally when tracing the sequence of numbers. Each number will be in an adjacent cell to the previous and following numbers in a horizontal or vertical direction.
The general technique suggested by the inventor is to “scan from the lowest number to the highest one and fill in any missing numbers where the placement is certain.”
So let’s start near the beginning. There will only be one square between the 1 and 3, so it is either the purple or red square here.
Between the 3 and 6, you need two squares, so our sequence will be either (3, red, green, 6) or (3, yellow, blue, 6).
Now, look at the 9. There is only one path that can be taken from the 6 – the blue, then yellow squares. That means red and green must come after the 3, and leaves the purple cell holding our 2.
Our example puzzle is made to be an easy one, since I’m mostly demonstrating principles of how the puzzle works. As a result, our next several steps are pretty obvious, and don’t require much
deduction. Printed puzzles will usually be a bit more challenging.
What you’re looking for is a pair of numbers that require a certain number of cells between them, and that ideally only have one possible path.
Following the blue path, we have 9, (10, 11, 12), 13. There is no other set of 3 empty squares to get from 9 to 13, so this is the only possible way.
Then, the green cell is the only blank space between 13 and 15. once that’s filled, only the yellow square is between 15 and 17, and then the red path is the only path possible for 17 (18, 19, 20,
21), 22.
Continuing that path, blue is the only square between 22 and 24, then green is the only space for 25, red must hold a 27, and yellow is the only cell between 28 and 30. So that whole set is easy to
fill in.
However, between 30 and 33, we have two possible paths, either orange then purple, or orange then pink.
This means we’re going to need to do some deduction by looking a little ahead.
Let’s look first between 33 and 36. We need two cells, which could either be pink then yellow, or green then blue.
Moving ahead, we see 36 and 38, but again, there are two options for the single space between them, either the blue or red.
One more time, and we have only one option for the single cell between 38 and 40.So that means the red cell hold the 39, leaving only the blue cell for 37.
Now there’s only one option left for the two squares between 33 and 36, which must be pink then yellow. Once we fill that in, we now know that orange then purple holds 31 and 32.
Also, because all the cells in the grid must be used, it means the green cell has to hold 52, because otherwise it will be a gap in the grid if the 51 goes the other direction to get to 53.
At first, it looks like there are two possible paths between 40 and 43, but you can quickly realize that the next number after 43 cannot be beneath it, because from there, you don’t have any possible
path to the next given number, which is 50, near the upper right of the grid.
Therefore, 41 and 42 must be along this blue path.
Next, we have a fairly long sequence and a pretty open area. Between 43 and 50 are (44, 45, 46, 47, 48, 49), six numbers. Let’s zoom in on possible paths below.
Now, the blue path can be eliminated as an option immediately, because it would leave two open cells to the right of the 43 that are unlikely to be filled by anything else, and we can’t have gaps in
the finished puzzle.
We don’t have any immediate clues to easily choose between the green or red paths, but they do share that the first two places would be the same. So we can fill in 44 and 45 to the right of the 43,
and then we will move on for now and come back to figure out where 46 through 49 will go later.
The blue cell must hold 54, and the green cell must be 56, because they are the only spaces between the 53 and 55, and then the 55 and 57.
Next, we only have one possible two-space path between 57 and 60, so the yellow squares must hold 58 and 59.
This cuts off the top two cells of the red path we looked at for 46-49, so we now know that the 46 must be to the left of the 45, then up for 47, and then right two cells until we reach 50.
Next, we have another easy sequence. The blue cells are the only two-space path between 60 and 63, so they must contain 61 and 62.Next, the green cell is the only option for 64.The red space has to
hold 68, leaving the yellow space for 66.
Finally, the purple cell must be for 70. Looking ahead, it seems we may have more than one option for the path between 72 and 78. To aid in our deduction, let’s solve some of the rest of the puzzle
to eliminate path options.
It is important to remember that you are not required to solve the puzzle in sequence. It’s often useful to solve other areas first to eliminate options elsewhere in the grid.
So let’s start with 82 to 85. We need two cells, and the blue path is the only option which doesn’t leave a gap.
Next, at the top, greenis the only square between 93 and 95, leaving yellow as the only possibility from 97 to 99.
The red square is the only possibility to hold 100, the highest number in this grid. If we instead put it to the right of 99, we would be blocking access to 93.
Now we get into our final moves. From 87 to 90, we follow the green path, because if we went upward, we’d leave a gap below the 90.
Same for 90 to 93. We take the blue path, because if we’d gone up and then left, we’d leave a gap next to 100.
Again, to fill a gap, 78 must first go down, then jog to the right on its path down to 82.
And last, but not least, our long winding red path takes us from 72, two steps to the left, then down, then right, then down again, and finally left to reach 78.
And here, we have our completed puzzle! | {"url":"https://villagedescigales.com/article/how-to-solve-numbrix-puzzles-keep-it-simple-puzzles","timestamp":"2024-11-10T05:51:34Z","content_type":"text/html","content_length":"95432","record_id":"<urn:uuid:72e8ee66-d628-42ad-b43a-e307ae7671b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00046.warc.gz"} |
Solution Tactics for Neutral Holes- Examples
For the inverse problem, a solution matching displacements at the interface using the integral equations, will necessarily be one of trial and error even for an assumed shape since the unknown
geometric properties of the cross-section will, in general, be functions of position and must remain inside the integral. A better tactic is to enforce compatibility on a differential scale by
using Equations (7.34) with Equations (7.31) and (7.32). The displacements will then necessarily conform aside from initial conditions u[0], v[0], and _[0]. These along with any unknown integration
constants in the equilibrium equations can then be found in the standard way once the geometry of the hole and liner is known.
The theory as presented essentially reduces to a “strength-of-materials” solution for a ring where the geometry of its deformation must match that of the sheet at the interface. It is, of course,
approximate to the extent that all strength-of-material solutions are approximate. Thick-ring effects could eas-ily be incorporated if necessary by including the eccentricity of the neutral axis and
adding the bending contribution at the interface.
However, as is usually done, the equations will be simplified further by neglecting the curvature contribution and reducing the liner to the “elastica” at the neutral axis. The significance of this
“thin ring” assumption can be esti-mated by considering a circular liner of radius R in an isotropic field.
For this situation the ring simply expands uniformly and, since there is no force
N/AE. Thus, the final term in the first two equations of Bresse will be negligible and will not be included hereafter.
Two special cases of obvious importance will be considered in the examples that follow. First, membrane reinforcement may be possible in certain fields by adjusting the hole shape to “eliminate”
bending. This is the Mansfield solu-tion and, as shown previously, can occur only for that shape which is a level line of U given by Equation (7.32). This “short-circuits” the equations of Bresse in
that if V =0, both μ and I are indeterminate. The area required can be found directly from Equations (7.34a) with (7.35a). However, an actual liner will have some bending stiffness inducing secondary
moments (i.e., a gradient of normal stress through the thickness) violating the original membrane assumption. This dilemma, not considered by Mansfield, is a direct result of the
strength-of-materials approach. Such secondary effects will be shown to be very small for thin liners.
Circular liners are a second important class of potential solutions. In this case the shape is prescribed and I [l], A l, and μ are to be determined to satisfy the neutral condition. The strains in
the sheet in polar coordinates r, and θ become:
Similarly, since the radius is a constant, the differential equations of Bresse are easily solved for the required liner properties.
Isotropic Field
With equal normal stress in orthogonal directions, σ[m], the stress function is:
and the obvious shape is circular. Thus, by the equilibrium equations
which is not possible if M = 0. However, we know from Lamé’s exact solu-tion that there is, in fact, a constant moment in the ring since the hoop stress is not uniform through the thickness.
Combining (c) and (d):
per unit thickness which we derived earlier.
There will be no requirement on the moment of inertia. Thus, a “neutral” liner will have a uniform cross-section and be “thin” as defined by d/r " 10 if E[s]/E[l](1-v[s]) < 10. This is nearly true
for a steel liner in concrete or rock and true for a concrete or steel liner in soil.
Deviatoric Field
For the pure shear or deviatoric field with equal but opposite principal stresses, +- σd
Therefore, the membrane shape is a self-equilibriating system of parabolic “arch” and “suspension” segments which, while conceivable, is not likely in practice.
On the other hand, a circular, neutral liner may be feasible. Using cylindri-cal coordinates where
Through symmetry the moment at the springing line must be equal and opposite that at the crown. Thus, c must be zero. Therefore, the stress resultants in the liner are
These results are startling. It is possible to design a neutral circular liner com-pletely restoring the original field, which has constant A and I, and therefore, is easy to build. Checking shear
compatibility from Equation (7.34c)
This is close to that required by Equation (7.42a) in order to match the hoop strain since
A 10 WF 17 would give the required moment of inertia, but over 100 times too much area, and slip joints to reduce the effective A would be necessary. Clearly, as one might expect for a circular shape
in pure shear, bending dominates entirely and we might term this remarkable neutral liner with constant cross-section the flexural counterpart to the circular, axial-force liner for the isotro-pic
General Biaxial Field
A general biaxial field will be a combination of isotropic and deviatoric com-ponents. Assuming that the x axis is in the direction of the maximum princi-pal stress:
Thus, the membrane shape will be a combination of a circle and parabola. The hole boundary can only be closed if σ[m > ]σ[d], in which case an ellipse results with the principal axes in the ratio a/b
= Rt(σ[1] / σ[2)] . This important and practical case is given by Mansfield, but was actually found nearly a century before by Rankine using parallel projection.
The neutral shape and liner for no stress concentration is compared to the unlined harmonic shape (stress concentration of 1.5) in Figure 7.13a for a 2 to 1 biaxial field. If v[s] is 1/4, then the
area of the membrane liner at the “spring-ing line” (i.e., y = 0) would be about 2.5 times that required at the “crown” (i.e., x = 0). If E[_]/E[s=]1,000 and R is 10 ft, then the A[_], at the
springing line, should be 0.34 in.^2/in. (219 mm^2/mm). This is roughly that of the average corrugated plate section now used in practice for large-span elliptic culverts and arches in soil (Figure
7.13b). Therefore, if the axial stiffness were gradu-ally reduced by a factor of 2.5 from the springing line to the crown in such structures by slip joints or by using lighter plates, the moments
could be “eliminated” and the design improved to the neutral case.*
For a shape other than the membrane ellipse, moments will be introduced and the liner must also have the correct flexural stiffness to achieve neutrality. Equations based on the general theory and
conditions for their existence can be derived and studied. However, from a practical design standpoint, it is doubtful if such intermediate cases are as useful as the four special cases:
1. Harmonic hole ellipse (7.25)
2. Rigid harmonic inclusion ellipse (7.26)
3. Membrane neutral ellipse (a/b Rt(σ[1/] σ[2)] )
4. Flexurally neutral circle (7.42)
If a hole is to be placed in a loaded body (e.g., a tunnel), the harmonic shape may be preferable to minimize stress concentration in the sheet and then rein-forcement provided to take active loads.
For a sheet with a reinforced hole that is then loaded (e.g., buried pipe or most structural applications in plates and shells), the designer would normally choose the membrane configuration or, for
ease in construction, a circle. Only if forced to use a rigid liner, would the designer choose the harmonic inclusion shape.
Gradient Fields with an Isotropic Component
In many engineering applications, there is an isotropic component in the stress field in addition to the gradient induced by bending or geostatic conditions. This is the general case in pressure
vessels, shell structures or masonry walls, tunnels, pipe, or in other buried structures. For this case the stress function becomes
Although Mansfield does not consider this case, a simple closed membrane shape is possible if the stress function can be put in the form:
which, when equal to zero, gives the neutral hole shape. To do this let: x = fx[o] and y = y[o], where x[o], and y[o] refer to points on the circle of radius R and f is the “mapping function” which
transforms this circle to the membrane shape. The stress function [Equation (7.44)] can then be rewritten:
limiting the possible gradient field for a membrane shape to G " 6.
A more stringent limitation is imposed by the area required of the liner for compatibility. The normal force in the liner is by Equation (7.31a)
giving a much more severe restriction on the size of a neutral membrane hole that can be put in any given gradient field.
The neutral “deloid” shape giving no stress concentration with a mem-brane liner, is compared in Figure 7.14a to the harmonic deloid for an unlined hole for the extreme case, G = 2. They are both of
the same generic type, but the neutral shape is much closer to a circle. The non dimensionalized area required of the membrane liner for G=1.0, v=1/3 is shown in Figure 7.14b. Comparing the result to
the membrane liner for a circular hole in an isotropic field where E[l]A/E[s]R = 1.5, this neutral deloid design is, for a gradient field, just as feasible. Even for a concrete liner in stiff soil
where E[l]/E[s] might be 100, the depth of a prismatic liner for a 20 ft diameter opening would be only 1.2 in. at the bottom, 1.8 in. at mid-height, and 6 in. at the top.
To achieve a circular neutral liner, rotation compatibility is critical in that
with a, b, and c all zero by symmetry arguments. The rotation of the sheet at the interface by Equation (7.36) is
Therefore, designing a liner for a circular neutral hole in a gradient field is not practical since the moment of inertia required is slightly negative near the neutral axis (unless v[s = ]0).
However, this might be feasible in concrete or steel where the sheet near the neutral axis could be made thinner.
It has been possible to extend the Mansfield theory for membrane neutral lin-ers to the general case of flexural reinforcement. The resulting expressions for equilibrium and compatibility in terms of
the stress function U for the sheet are, in fact, not limited to the neutral condition, but are valid for any thin rein-forcement and thus, are fundamental to the general interaction problem.
In the classical interaction problem, the total, U = U^o + U*, is unknown since the perturbation in the field U* due to the hole in unspecified. By inverting
the problem to a design mode where U* is specified (to be zero for the neutral condition), the stresses, strains, and displacements in the field are entirely known, as are the stress resultants in
the liner. Thus, the tangential strain and the rotation of the liner can be found from the equations of Bresse and matched at the interface to the same known quantities in the field to give
expressions for the required liner area and bending stiffness.
Closed-form solutions are given here for both the circular and the mem-brane shape for a variety of free fields. A review of these results reveals two interesting limitations to achieving, at least
in a simple way, the neutral con-ditions with a thin continuous liner: Most important is the “Poisson ratio effect.” For a positive area of the liner: ( [t])[s] must be exactly in phase with N, i.e.,
have the same sign at every point on the interface. This might seem obvi-ous, but apparently was not recognized by Mansfield in his work on mem-brane reinforcement. Similarly, if there is bending,
the rotation change (dϕ/ds)[s] must be exactly in phase with M to have a positive moment of inertia. Thus, only in special cases such as the purely deviatoric field or biaxial bending can a neutral
liner be found for a “simple” closed hole in a region of a field where either U or its gradient change sign or even approach zero.
Within this constraint, neutral liner designs are found for two basic situa-tions important in practice: (1) Circular holes in deviatoric and biaxial fields (flexural liner); and (2) membrane “deloid
shape” and liner for the gradient field with an isotropic component. The second case is particularly useful in that it allows a design within a bending field and closely resembles the geo-static
field condition for shallow pipes and tunnels. Moreover, since the “deloid” neutral shape is of the same generic type as the harmonic hole for minimum stress concentration with no liner, its use may
serve a double pur-pose both in construction to reduce stresses around the hole and then with reinforcement to best withstand gradient service loads. | {"url":"https://www.brainkart.com/article/Solution-Tactics-for-Neutral-Holes--Examples_4823/","timestamp":"2024-11-03T12:21:39Z","content_type":"text/html","content_length":"86706","record_id":"<urn:uuid:5b776925-8f93-47ef-a4a5-9e7eb0407225>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00309.warc.gz"} |
Length Contraction Calculator | Online Calculators
Length Contraction Calculator
The length contraction calculator can be used to determine the apparent length of an object moving at a high velocity. For calculation enter the true length (m), object velocity (m/s) and speed of
light (m/s) into the calculator. Click on calculate button, it will display apparent length (m) in result field.
What Is Length Contraction Calculator
The Length Contraction Calculator is a helpful tool. It is used to calculate the apparent length of an object moving at a high velocity. This phenomenon is explained by the theory of relativity and
is important in physics. The calculator has two modes: Basic and Advanced. The Basic mode calculates the apparent length based on true length and velocity. The Advanced mode includes an additional
factor for time dilation.
1. Basic Calculation
Variable Description
True Length The actual length of the object in meters
Object Velocity The speed at which the object is moving in meters per second
Speed of Light The constant speed of light in meters per second (299,792,458 m/s)
Calculation Examples
1. Basic Example
Variable Value
True Length 10 meters
Object Velocity 200,000,000 m/s
Step Calculation
Apparent Length =
10 × √(1 – (200,000,000² / 299,792,458²))
True Length × √(1 – (Object Velocity² / Speed of Light²))
Result 7.45 meters
2. Advanced Example
Variable Value
True Length 10 meters
Object Velocity 200,000,000 m/s
Time Dilation Factor 1.2
Step Calculation
Adjusted Apparent Length =
10 × √(1 – (200,000,000² / 299,792,458²)) × 1.2
True Length × √(1 – (Object Velocity² / Speed of Light²)) × Time Dilation Factor
Result 8.94 meters
How to Use the Calculator
Step Basic Calculator Advanced Calculator
1. Select Calculator Type Click “Basic Calculator” button. Click “Advanced Calculator” button.
2. Enter Values – True Length (in meters) <br> – Object Velocity (in meters per second) – True Length (in meters) <br> – Object Velocity (in meters per second) <br> – Time Dilation Factor
3. Click Calculate Click “Calculate” button. Click “Calculate” button.
4. View Result View the Apparent Length in the result field. View the Adjusted Apparent Length in the result field.
5. Reset Fields Click “Reset” button to clear all fields. Click “Reset” button to clear all fields.
1. What is length contraction?
Length contraction is a phenomenon in special relativity where an object in motion is measured to be shorter along the direction of motion than when it is at rest.
2. Why is the speed of light constant?
The speed of light is a fundamental constant in physics, denoted by “c”. It remains constant in all frames of reference, as established by Einstein’s theory of relativity.
3. What is time dilation?
Time dilation is a difference in the elapsed time measured by two observers, due to a relative velocity between them or a difference in gravitational potential. In the context of this calculator, it
affects the apparent length of moving objects.
Leave a Comment | {"url":"https://lengthcalculators.com/length-contraction-calculator/","timestamp":"2024-11-05T04:18:55Z","content_type":"text/html","content_length":"69696","record_id":"<urn:uuid:6b270650-d434-4e20-94e8-c4f505b30b91>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00160.warc.gz"} |
Introduction to Gamma | Elearnmarkets
Introduction To Gamma
The next option-greek that we will learn is ‘Gamma.’
Gamma measures the change in Delta with respect to per unit change in underlying. If Gamma value is 0.0008, it shows that my next move in Delta will be 0.0008 if underlying changes by 1. So Gamma
shows what will be the next change in Delta with respect to change in underlying.
Important Points:
• The Gamma of an option is always positive. Positive Gamma means that the Delta of long calls will become more positive and move toward +1.00 when the stock price rises, and less positive and move
toward 0 when the stock price falls. Long Gamma also means that the Delta of a long put will become more negative and move toward –1.00 if the stock price falls, and less negative and move toward
0 when the stock price rises. For a short call with negative Gamma, the Delta will become more negative as the stock rises, and less negative as it drops.
• The Gamma of Call and Put Option is same. Because Gammas influence Deltas of calls and puts in the same way, expressing their probability of finishing in the money after a change in price in the
• Gamma is highest at ATM and decreases as Spot price moves away.
When you buy Call Option, Positive Gamma multiplies by Positive Quantity (positive quantity because going long on option), hence gives Positive Portfolio Gamma. This signifies that buying Call Option
means Long Gamma Position.
When you sell Call Option, Positive Gamma multiplies by Negative Quantity (negative quantity because going short on option), hence gives Negative Portfolio Gamma. This signifies that selling Call
Option means Short Gamma Position.
When you buy Put Option, Positive Gamma multiplies by Positive Quantity, hence gives Positive Portfolio Gamma. This signifies that buying Put Option means Long Gamma Position.
When you sell Put Option, Positive Gamma multiplies by Negative Quantity, hence gives Negative Portfolio Gamma. This signifies that selling Put Option means Short Gamma Position.
If Gamma is small, Delta changes slowly. However, if the absolute value of Gamma is large, Delta is highly sensitive to the price of the underlying. Let us now take a look at how different parameters
affect Gamma in our upcoming chapters.
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CUPED with Multiple Covariates and A Simpler the Delta Method CalculationCUPED with Multiple Covariates and A Simpler the Delta Method Calculation
CUPED with Multiple Covariates and A Simpler the Delta Method Calculation
In the original CUPED paper, the authors mention that it is straightforward to generalize the method to multiple covariates. However, without understanding exactly the mathematical technique to find
the CUPED estimate, it may be confusing to attempt the multiple covariates case. In this post, we will explain the thought process behind the CUPED estimate and demonstrate the analytic formula for a
multiple covariate extension to CUPED. We then discuss the estimate for non-user level metrics, and we will need to use the delta method for the variance. In this case, the book keeping when using
the delta method would be tedious unless you use a simplified calculation which we empirically demonstrate in the second section of the post.
To calculate the CUPED estimate, namely \(\theta\), the authors parametrized a function \(Y_{cv}(\theta)\) where \(\mathbb{E}(Y_{cv}(\theta))=Y\). Then, they chose the minimal \(\theta\) such that \
(\theta = \min_{\hat\theta} var(Y_{cv}(\hat\theta))\). When you know the trick, it’s not that bad to generalize to more variables.
The first thing you have to do is allow multiple covariates. With a single covariate we had \(\theta\) and \(X\), the covariate. The analog in the multiple covariate case is a collection of \(\
theta_i\) for covariates \(X_i\). In this setting, \(Y_{cv}\) is defined via \($$Y_{cv} = \bar{Y} - \sum_{i} \theta_i (\bar{X_i} - \mathbb{E}(X_i))$$\) So, using rules about the variance of linear
combinations, we have \($$var ( Y_{cv}) = \frac{1}{n} \left ( var(Y) + \sum_i \theta_i^2 var(X_i) + 2 \sum_{1 \leq i< j\leq n} \theta_i \theta_j cov(X_i, X_j) - \sum_i 2\theta_i cov(Y,X_i) \right
In the previous equation, we used the identity \(var(\bar{Y}) = var(Y)/n\) where \(n\) is the size of the sample. (We are glossing over many steps and definitions to get to the meat and not distract
from the main point. For more details, see the CUPED paper’s definitions.)
Now, consider \(g(\boldsymbol \theta):=var(Y_{cv})\) where \(\boldsymbol\theta:= (\theta_1,\dots, \theta_m)\).
Now, to find the minimum value of \(\theta\) of this quadratic equation, we need to take the (multivariate) derivative with respect to \(\boldsymbol \theta\) and find the critical points– which in
this case the critical point will be a minimum.
\[$$\frac{\partial g}{\partial \theta_i} = \frac{1}{n}\left( 2\theta_i var(X_i) + 2 \sum_{jeq i} \theta_j cov(X_i, X_j) - 2 cov(Y, X_i) \right)$$\]
We can write this another way if we consider the vector \(\nabla \boldsymbol \theta = (\frac{\partial g}{\partial \theta_1},\dots \frac{\partial g}{\partial \theta_m})^T.\)
If we set \(\nabla \boldsymbol \theta = 0\), we can remove the factor of \(2\) and \(1/n\) in all the terms of the partial derivative (by dividing both sides by \(2\) and \(1/n\)) and we have
\[$$abla \boldsymbol \theta = \Sigma \boldsymbol \theta - Z = 0$$\]
where \(\Sigma\) is the covariance matrix of the \(X_i\) and \(Z\) is a vector such that \(Z_i = cov(Y,X_i)\). Therefore, the minimum is achieved when we set
\[$$\boldsymbol \theta := \Sigma^{-1}Z.$$\]
Empirical Validation
Here, we will create a \(Y\) from a linear combination of covariates \(X_i\)’s, i.e, \(Y=\sum_i a_i X_i\). We will then solve for \(\theta\) using the formula above and we will find that the
coefficients \(a_i\) and \(\boldsymbol \theta\) are equal.
import numpy as np
import pandas as pd
size = 50000
num_X = 10
X_means = np.random.uniform(0, 1, num_X)
Xs = [np.random.normal(X_means[k], .01, size) for k in range(num_X)]
coefficients = np.random.uniform(0,1,num_X)
Y = np.sum([a*b for a,b in zip(coefficients, Xs)],axis=0) \
+ np.random.normal(0,.001,size)
# recover coefficients/theta via the formula, note n must be large
# for theta to be a good approximation of the coefficients
big_cov = np.cov([Y] + Xs) # Calculating sigma and z together
sigma = big_cov[1:,1:]
z = big_cov[1:,0].reshape((num_X,1))
theta = np.dot(np.linalg.inv(sigma),z) # here's the formula!
# This will be a small number, not perfectly 0
# due to the error term we add, this value isn't exactly 0
Detour on Gauss-Markov
Wait? Why are they equal? So, there was a little gap in the logic above. I did not mention why those two values should be equal in the first place, but I showed it happens in this case. It is
actually due to the Gauss-Markov Theorem, which states that under a few assumptions on the data distribution, the ordinary least squares (OLS) estimator is unbiased and has the smallest variance.
Let’s play that back to really grok it. I was coming up with a linear estimate for \(Y\), in terms of a linear combination \(Y_{cv}\) (so a linear estimator), and I was solving explicitly for
coefficients (\(\theta\)) which minimized the variance. Since \(Y\) is already a linear combination, the OLS estimator will recover the coefficients of \(Y\). Gauss-Markov says that the OLS solution
is the unique “best linear unbiased estimator”, and so solving for a variance minimizing linear combination (\(Y_{cv}\)) another way will still yield the same coefficients.
CUPED Simulation
In the CUPED paper, the authors state that one covariate they found predictive was the first day the user entered the experiment. This is not a pre-experiment variable but it is independent of the
treatment. We will craft a simulation that leverages this idea to create covariates to test for our formula above.
In our simulation, we will simulate a user’s query count during a test period under a control and a treatment. The treatment will make a user more likely to query. Depending on the day of the week on
their first entrance into the experiment, a user will have a different additive component to their query propensity. This effect only impacts their visits on the first day.
# Using the simulation from a previous post
# with a few additions
import numpy as np
import pandas as pd
user_sample_mean = 8
user_standard_error = 3
users = 1000
# assign groups
treatment = np.random.choice([0,1], users)
treatment_effect = 2
# Identify query per session
user_query_means = np.random.normal(user_sample_mean, user_standard_error, users)
def run_session_experiment(user_means, users, user_assignment, treatment_effect):
# reate click rate per session level
queries_per_user = \
+ user_means[users] \
+ np.random.normal(0,1, len(users))
queries_per_user[queries_per_user<0] = 0
return pd.DataFrame({'queries': queries_per_user, 'user': users, 'treatment': treatment[users]})
# Generate pre-experiment data for each user once, i.e. over some period
pre_data=run_session_experiment(user_query_means, range(users),
treatment, 0)
pre_data.columns = ['pre_' + k if k != 'user' else k for k in pre_data.columns]
pre_data = pre_data[['pre_queries','user']]
# Generate experiment data
day_impact = np.random.uniform(-3, 6, 7)
dfs = []
users_seen = set()
users_first_day = []
for k in range(14):
# select the users for that day, each user has a 2/14 change of appearing
day_users = np.random.choice([0,1], p=[12/14,2/14], size=users)
available_users = np.where(day_users==1)[0]
day_user_query_means = user_query_means + day_impact[k % 7]
df = run_session_experiment(day_user_query_means, available_users, treatment,
df['first_day'] = k % 7
df['real_day'] = k
# We are doing a user level analysis with randomization unit
# equal to user as well. This means groupby's should be on user!
def get_first_day(x):
# Get the "first_day" value corresponding to their actual first day
tmp = x.iloc[t]['first_day']
return tmp
dd=df.groupby(['user','treatment']).agg({'queries': 'sum'})
dd.reset_index(inplace=True) # pandas is really ugly sometimes
# combine data, notice each row is a unique user
data=dd.merge(pre_data, on='user')
# Calculate theta
covariates.columns = ['day_'+str(k) for k in covariates.columns]
covariates['pre_queries'] = data['pre_queries']
all_data = np.hstack((data[['queries']].values,covariates.values))
big_cov = np.cov(all_data.T) # 9x9 matrix
sigma = big_cov[1:,1:] # 8x8
z = big_cov[1:,0] # 8x1
theta = np.dot(np.linalg.inv(sigma),z)
# Construct CUPED estimate
Y = data['queries'].values.astype(float)
covariates = covariates.astype(float)
Y_cv = Y.copy()
for k in range(covariates.shape[1]):
Y_cv -= theta[k]*(covariates.values[:,k] - covariates.values[:,k].mean())
real_var, reduced_var = np.sqrt(Y.var()/len(Y)),\
reduced_var/real_var # variance reduction
# Let's try OLS!
from statsmodels.formula.api import ols
results= ols('queries ~ pre_queries + C(first_day) + treatment', data).fit()
# Some calculations for the final table
effect = Y[data.treatment==1].mean() - Y[data.treatment==0].mean()
ste = Y[data.treatment==1].std()/np.sqrt(len(Y)) + Y[data.treatment==0].std()/np.sqrt(len(Y))
cuped_effect = Y_cv[data.treatment==1].mean() - Y_cv[data.treatment==0].mean()
cuped_ste = Y_cv[data.treatment==1].std()/np.sqrt(len(Y)) + Y_cv[data.treatment==0].std()/np.sqrt(len(Y))
pd.DataFrame(index=['t-test', 'CUPED', 'OLS'],data=
"effect size estimate": [effect, cuped_effect, results.params['treatment']],
"standard error": [ste, cuped_ste, results.bse['treatment']]
│ │effect size estimate │standard error │
│t-test│5.014989 │1.048969 │
│CUPED │4.458988 │0.876270 │
│ OLS │4.486412 │0.884752 │
Non-User Level Metrics Considerations
If you have a non-user level metric, like the click through rate, the analysis from the first section is mostly unchanged. When you define \(Y_{cv}\), we must account for those terms differently.
Following Appendix B from the CUPED paper, let \(V_{i,+}\) equal the sum of obervations of statistic \(V\) for user \(i\), \(n\) the number of users and \(\frac{1}{n} \sum_i V_{i,+} = \bar{V}\). Let
\(M_{i,+}\) be the number of visits for user \(i\). Let \(X\) be another user-level metric we will use for the correction. Then, we have the equation \($$Y_{cv} = \bar{Y} - \theta_1 \left( \frac{\
sum_{i} V_{i,+}}{\sum_{i} M_{i,+}} - \mathbb{E}{\frac{\sum_{i} V_{i,+}}{\sum_{i} M_{i,+}}} \right ) - \theta_2 ( \bar{X} - \mathbb{E}X).$$\) In this case, we have a page-level metric \(V/M\) being
used as a covariate for a user-level metric \(Y\). Is this realistic? Maybe, but at least it will serve to illustrate what to do here.
When you write out the formula for the variance, you will have a term of the form
\[$$cov \left (\frac{\sum_{i} V_{i,+}}{\sum_{i} M_{i,+}} , \bar{X}\right ),$$\]
where you need to apply the delta method to compute this covariance. We can write
\($$cov \left ( \frac{\sum_i V_{i,+}}{\sum_i M_{i,+}}, \bar{X} \right ) \approx cov\left (\frac{1}{\mu_{M}}\bar{V} - \frac{\mu_{V}}{\mu_{M}^2}\bar{M} , \bar{X}\right).$$\) This term can be simplified
via properties of covariance. In particular, \($$cov\left (\frac{1}{\mu_{M}}\bar{V} - \frac{\mu_{V}}{\mu_{M}^2}\bar{M} , \bar{X}\right)= \frac{1}{\mu_M}cov(\bar{V},\bar{X}) - \frac{\mu_V}{\mu_{M}^2 }
cov(\bar{M},\bar{X}).$$\) At this point, you are able to calculate all the necessary terms and can sub this value in for the \(\partial d / \partial \theta_i\) equation.
Calculating the Delta Method Term the Easy Way
Using the previous presentation of the delta method, we can actually make our lives easier by replacing the vector \(\bar{V}/\bar{M}\) with the delta estimate \(\frac{1}{\mu_{M}}\bar{V} - \frac{\mu_
{V}}{\mu_{M}^2}\bar{M}\) and then calculating the covariance, rather than applying any of the complicated delta formulae.
All that’s left is to convince you of this. First, I’ll take the formula for the delta method from my previous post and show it is equivalent to taking the variance of the vector \(\frac{1}{\mu_{M}}\
bar{V} - \frac{\mu_{V}}{\mu_{M}^2}\bar{M}\).
# Define our data
V = np.random.normal(user_sample_mean, user_standard_error, users)
M = np.random.normal(user_sample_mean*2, user_standard_error, users)
X = np.random.normal(0,1,users)
mu_V, mu_M = V.mean(), M.mean()
def _delta_method(clicks, views):
# Following Kohavi et. Al. clicks and view are at the per-user level
# Clicks and views are aggregated to the user level and lined up by user,
# i.e., row 1 = user 1 for both X and Y
K = len(clicks)
X = clicks
Y = views
# sample mean
X_bar = X.mean()
Y_bar = Y.mean()
# sample variance
X_var = X.var(ddof=1)
Y_var = Y.var(ddof=1)
cov = np.cov(X,Y, ddof=1)[0,1] # cov(X-bar, Y-bar) = 1/n * cov(X,Y)
# based on deng et. al
return (1/K)*(1/Y_bar**2)*(X_var + Y_var*(X_bar/Y_bar)**2 - 2*(X_bar/Y_bar)*cov)
_delta_method(V, M)
# Take the variance of the taylor expansion of V/M
(1/users)*np.var((1/mu_M)*V-(mu_V/mu_M**2)*M, ddof=1)
How to calculate Sigma
How do we use this insight to make our \(\sigma\) calculation easier? Simply replace any vector of the metric \(V/M\) with the linearized formula \(\frac{\bar{V}}{\mu_{M}} - \frac{\mu_{V}}{\mu_{M}^2}
\bar{M}\) and take the covariance as usual.
# Formula for the covariance from the previous section
delta_var = (1/users)*((1/mu_M)*np.cov(V,X)[0,1] - (mu_V/mu_M**2)*np.cov(M,X)[0,1])
# Sub in the value for each vector before taking the covariance
pre_cov_est = (1/users)*np.cov((1/mu_M)*V-(mu_V/mu_M**2)*M, X)[0,1]
In our case, \(V\) and \(M\) are correlated and so the equivalence I empirically demonstrated was not a sleight of hand that doesn’t always work.
array([[9.10327978, 0.10338676],
[0.10338676, 9.29610525]])
Comments on the Simpler Calculation
I have personally implemented the delta method formula multiple times before realizing I could linearize before calculating the covariance, and so I do not think this calculation is obvious. Upon
rereading the original delta method paper, the same formula is very briefly mentioned by Deng et al. in section 2.2 after equation (3) when they define $W_i$. In that section, they do not explicitly
mention using the $W_i$ to calculate the variance, which is what I do above. However, it turns out that this is a common approach in the field to calculate the delta method, which is why it’s not
belabored in the papers as it’s available in several books, including Art Owen’s online book in equation (2.29). Thanks to Deng (of Deng et al.) for sharing the reference. | {"url":"http://srome.github.io/CUPED-with-Multiple-Covariates-and-A-Simpler-Delta-Method-Calculation/","timestamp":"2024-11-12T03:29:57Z","content_type":"text/html","content_length":"61921","record_id":"<urn:uuid:b68f66a8-ab2b-4088-a744-a81cf6840ccb>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00519.warc.gz"} |
OUR DAYS ARE NUMBERED by Jason Brown ★★★ | Kara.Reviews
Review of Our Days Are Numbered: How Mathematics Orders Our Lives by Jason Brown
Our Days Are Numbered: How Mathematics Orders Our Lives
by Jason Brown
I read math books for fun. I realize that, right away, this puts me in an unusual portion of the population. It’s not just my fancy math degree that makes these books attractive. However, I do think
that there are some math books written for people interested in math (whether professionally or amateurly), and then there are math books written for people who, usually thanks to a bad experience in
school, have sworn off math like they said they would swear off cheap booze. Our Days Are Numbered is one of the latter. In a passionate and personal exploration of shape, algebra, geometry, and
number, Jason I. Brown illuminates the fundamental mathematics behind some everyday tasks. While some people will still run away screaming, others will hopefully begin to see math in a new way.
Among the topics Brown explores are: converting between units, using graphs to display data, the meaning behind averages, the role of chance in decision-making, networks and coincidences, prime
numbers in cryptography, fractals in art, and the math behind the mystery of the Beatles’ Chord. Each chapter is bookended by a short, two- or three-paragraph anecdote related to its given topic. For
the main body of the chapter, Brown gradually develops some of the math behind common tasks. For example, he shows how an understanding of ratio and conversion factors makes converting between units
a breeze without any memorization (aside from the factor itself, of course). Later, he explains why the Web and social networking has guaranteed that graph theory will remain a practical and
important field of math for a long time.
This is not really my kind of math book, and that isn’t even because of the audience or the way Brown presents the math. Rather, I read math books for the story. I’m interested in math books that
take a specific topic and explore its history, its present state, and the different ways to interpret it using mathematics. Our Days Are Numbered instead covers a variety of topics. There isn’t
anything wrong with this approach. However, each of these topics can be (and has been) the subject of entire, weighty tombs. It’s difficult for Brown to do them justice. Sometimes, such as with the
chapter on conversion factors, he does a very thorough job. Other times, such as with his explanation of prime numbers and Internet security, he leaves something to be desired.
Also, much of one’s enjoyment will hinge on how one much one likes or dislikes Brown’s writing style. As the chapter titles and subheadings demonstrate, he is a man of corny humour, easy puns, and
deprecating remarks towards himself and fellow mathematicians. I can get behind the first and third attribute, and I can ignore the second. Although I think a book any longer might have begun pushing
its luck, as it is, I enjoyed Brown’s conversational and easygoing style. Others will find it overbearing and intrusive, however, and there is no escape from it here.
So, Our Days Are Numbered isn’t my mathematical cup of tea, but could it be anyone’s? Well, one way in which this book excels is Brown’s unrelenting insistence that math is useful, relevant, and not
at all scary. As a math enthusiast and math teacher, the opposites of these sentiments besiege me constantly. I love how Brown comments on the somewhat unique reception math receives at parties:
When I tell people what I do for a living, the most common response is a look of dismay, followed by “I always hated mathematics!” This statement is made with relish and without a hint of
embarrassment. I don’t think there is another profession out there that gets the same response. Do people state they’ve always hated English? Music? Lawn care? I think not.
Tongue-in-cheek, Brown touches on a very crucial and deplorable fact: hating math is socially acceptable. It’s cool to disparage math and one’s ability to do math. To some extent, the aura of nerdery
surrounds all of the STEM fields, but scientists and engineers get a little more recognition—people’s eyes might glaze over if one announces oneself as a theoretical physicist, but there is a little
gleam of gruding respect. Mathematicians, however … what do they even do?
The social acceptability of disparaging mathematics troubles me. Math is the foundation of the other three STEM fields. Science, technlogy, and engineering are all fields that require creative,
passionate thinkers. Yet from an early age we send children signals that math is a dull, uncreative subject and it’s OK to hate it for being boring and irrelevant. This is nothing short of
educational sabotage. It’s certainly fine for people not to like math, and I understand how parts of the educational system foster that feeling. But we should do everything we can to avoid
reinforcing that notion, especially among our children.
Hence the power of this book. Brown takes it as a given that math is a useful, powerful tool in the everyday world. He isn’t out to convert everyone to a science or engineering job. He isn’t trying
to shoehorn calculus into a discussion of changing a car tire. (As a teacher, the incessant call to include real-world applications and contexts in my lessons wearies me at times.) He is careful not
to insist that everyone uses or needs all of this math all the time—you don’t need to know how to use prime numbers in order to keep your online banking secure. But isn’t it nice to know why it is
Brown’s non-evangelical stance is refreshing, though it can also be a little frustrating. Our Days Are Numbered lacks a true, cohesive message, aside from the idea in the title. With no introduction
and no conclusion, Brown relies on the title and the chapters to come together to create that singular idea. While not essential, some kind of introduction or meta-narrative would lend additional
structure to this otherwise scattered text.
With brilliant mathematics, hardcore mystery-solving, and no small amount of humour, Our Days Are Numbered is a well-written and very successful math book. It isn’t anywhere close to my Platonic
ideal of what a math book should be—but that’s me being picky. Nor do I think, in the long run, that people convinced math is uninteresting or “not for me” will find their convictions toppled by
anything in here. But for anyone who is open to learning about the role of math in everyday life, there is definitely something here, waiting to be read.
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How does a dilation affect a figure on a coordinate plane? | Socratic
How does a dilation affect a figure on a coordinate plane?
1 Answer
Assuming the center of dilation is at point $\left\{0 , 0\right\}$ on the coordinate plane and a factor of dilation $f$, a point $A \left\{x , y\right\}$ will be transformed into point $A ' \left\{f
x , f y\right\}$.
See more details below.
Dilation or scaling is the transformation of the plane according to the following rules:
(a) There is a fixed point $O$ on a plane or in space that is called the center of scaling.
(b) There is a real number $f \ne 0$ that is called the factor of scaling.
(c) The transformation of any point $P$ into its image $P '$ is done by shifting its position along the line $O P$ in such a way that the length of $O P '$ equals to the length of $O P$ multiplied by
a factor $| f |$, that is $| O P ' | = | f | \cdot | O P |$. Since there are two candidates for point $P '$ on both sides from center of scaling $O$, the position is chosen as follows: for $f > 0$
both $P$ and $P '$ are supposed to be on the same side from center $O$, otherwise, if $f < 0$, they are supposed to be on opposite sides of center $O$.
It can be proven that the image of a straight line $l$ is a straight line $l '$.
Segment $A B$ is transformed into a segment $A ' B '$, where $A '$ is an image of point $A$ and $B '$ is an image of point $B$.
Dilation preserves parallelism among lines and angles between them.
The length of any segment $A B$ changes according to the same rule above: $| A ' B ' | = f \cdot | A B |$.
Using coordinates, the above properties can be expressed in the following form.
Assuming the center of dilation is at point $\left\{0 , 0\right\}$ on the coordinate plane and a factor of dilation $f$, a point $A \left\{x , y\right\}$ will be transformed into point $A ' \left\{f
x , f y\right\}$.
If the center of dilation is at point $C \left\{p , q\right\}$, the point $A \left\{x , y\right\}$ will be transformed by dilation into $A ' \left\{p + f \left(x - p\right) , q + f \left(y - q\right)
The above properties and other important details about transformation of scaling can be found on Unizor
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StellarGraph - Machine Learning on Graphs | PythonRepo
StellarGraph Machine Learning Library
StellarGraph is a Python library for machine learning on graphs and networks.
Table of Contents
The StellarGraph library offers state-of-the-art algorithms for graph machine learning, making it easy to discover patterns and answer questions about graph-structured data. It can solve many machine
learning tasks:
• Representation learning for nodes and edges, to be used for visualisation and various downstream machine learning tasks;
• Classification of whole graphs;
• Link prediction;
Graph-structured data represent entities as nodes (or vertices) and relationships between them as edges (or links), and can include data associated with either as attributes. For example, a graph can
contain people as nodes and friendships between them as links, with data like a person's age and the date a friendship was established. StellarGraph supports analysis of many kinds of graphs:
• homogeneous (with nodes and links of one type),
• heterogeneous (with more than one type of nodes and/or links)
• knowledge graphs (extreme heterogeneous graphs with thousands of types of edges)
• graphs with or without data associated with nodes
• graphs with edge weights
StellarGraph is built on TensorFlow 2 and its Keras high-level API, as well as Pandas and NumPy. It is thus user-friendly, modular and extensible. It interoperates smoothly with code that builds on
these, such as the standard Keras layers and scikit-learn, so it is easy to augment the core graph machine learning algorithms provided by StellarGraph. It is thus also easy to install with pip or
Getting Started
The numerous detailed and narrated examples are a good way to get started with StellarGraph. There is likely to be one that is similar to your data or your problem (if not, let us know).
You can start working with the examples immediately in Google Colab or Binder by clicking the and badges within each Jupyter notebook.
Alternatively, you can run download a local copy of the demos and run them using jupyter. The demos can be downloaded by cloning the master branch of this repository, or by using the curl command
curl -L https://github.com/stellargraph/stellargraph/archive/master.zip | tar -xz --strip=1 stellargraph-master/demos
The dependencies required to run most of our demo notebooks locally can be installed using one of the following:
• Using pip: pip install stellargraph[demos]
• Using conda: conda install -c stellargraph stellargraph
(See Installation section for more details and more options.)
Getting Help
If you get stuck or have a problem, there are many ways to make progress and get help or support:
Example: GCN
One of the earliest deep machine learning algorithms for graphs is a Graph Convolution Network (GCN) [6]. The following example uses it for node classification: predicting the class from which a node
comes. It shows how easy it is to apply using StellarGraph, and shows how StellarGraph integrates smoothly with Pandas and TensorFlow and libraries built on them.
Data preparation
Data for StellarGraph can be prepared using common libraries like Pandas and scikit-learn.
import pandas as pd
from sklearn import model_selection
def load_my_data():
# your own code to load data into Pandas DataFrames, e.g. from CSV files or a database
nodes, edges, targets = load_my_data()
# Use scikit-learn to compute training and test sets
train_targets, test_targets = model_selection.train_test_split(targets, train_size=0.5)
Graph machine learning model
This is the only part that is specific to StellarGraph. The machine learning model consists of some graph convolution layers followed by a layer to compute the actual predictions as a TensorFlow
tensor. StellarGraph makes it easy to construct all of these layers via the GCN model class. It also makes it easy to get input data in the right format via the StellarGraph graph data type and a
data generator.
import stellargraph as sg
import tensorflow as tf
# convert the raw data into StellarGraph's graph format for faster operations
graph = sg.StellarGraph(nodes, edges)
generator = sg.mapper.FullBatchNodeGenerator(graph, method="gcn")
# two layers of GCN, each with hidden dimension 16
gcn = sg.layer.GCN(layer_sizes=[16, 16], generator=generator)
x_inp, x_out = gcn.in_out_tensors() # create the input and output TensorFlow tensors
# use TensorFlow Keras to add a layer to compute the (one-hot) predictions
predictions = tf.keras.layers.Dense(units=len(ground_truth_targets.columns), activation="softmax")(x_out)
# use the input and output tensors to create a TensorFlow Keras model
model = tf.keras.Model(inputs=x_inp, outputs=predictions)
Training and evaluation
The model is a conventional TensorFlow Keras model, and so tasks such as training and evaluation can use the functions offered by Keras. StellarGraph's data generators make it simple to construct the
required Keras Sequences for input data.
# prepare the model for training with the Adam optimiser and an appropriate loss function
model.compile("adam", loss="categorical_crossentropy", metrics=["accuracy"])
# train the model on the train set
model.fit(generator.flow(train_targets.index, train_targets), epochs=5)
# check model generalisation on the test set
(loss, accuracy) = model.evaluate(generator.flow(test_targets.index, test_targets))
print(f"Test set: loss = {loss}, accuracy = {accuracy}")
This algorithm is spelled out in more detail in its extended narrated notebook. We provide many more algorithms, each with a detailed example.
The StellarGraph library currently includes the following algorithms for graph machine learning:
Algorithm Description
GraphSAGE [1] Supports supervised as well as unsupervised representation learning, node classification/regression, and link prediction for homogeneous networks. The current
implementation supports multiple aggregation methods, including mean, maxpool, meanpool, and attentional aggregators.
HinSAGE Extension of GraphSAGE algorithm to heterogeneous networks. Supports representation learning, node classification/regression, and link prediction/regression for
heterogeneous graphs. The current implementation supports mean aggregation of neighbour nodes, taking into account their types and the types of links between them.
attri2vec [4] Supports node representation learning, node classification, and out-of-sample node link prediction for homogeneous graphs with node attributes.
Graph ATtention Network (GAT) [5] The GAT algorithm supports representation learning and node classification for homogeneous graphs. There are versions of the graph attention layer that support both
sparse and dense adjacency matrices.
Graph Convolutional Network (GCN) The GCN algorithm supports representation learning and node classification for homogeneous graphs. There are versions of the graph convolutional layer that support
[6] both sparse and dense adjacency matrices.
Cluster Graph Convolutional An extension of the GCN algorithm supporting representation learning and node classification for homogeneous graphs. Cluster-GCN scales to larger graphs and can be
Network (Cluster-GCN) [10] used to train deeper GCN models using Stochastic Gradient Descent.
Simplified Graph Convolutional The SGC network algorithm supports representation learning and node classification for homogeneous graphs. It is an extension of the GCN algorithm that smooths the
network (SGC) [7] graph to bring in more distant neighbours of nodes without using multiple layers.
(Approximate) Personalized The (A)PPNP algorithm supports fast and scalable representation learning and node classification for attributed homogeneous graphs. In a semi-supervised setting,
Propagation of Neural Predictions first a multilayer neural network is trained using the node attributes as input. The predictions from the latter network are then diffused across the graph using a
(PPNP/APPNP) [9] method based on Personalized PageRank.
The Node2Vec and Deepwalk algorithms perform unsupervised representation learning for homogeneous networks, taking into account network structure while ignoring node
Node2Vec [2] attributes. The node2vec algorithm is implemented by combining StellarGraph's random walk generator with the word2vec algorithm from Gensim. Learned node
representations can be used in downstream machine learning models implemented using Scikit-learn, Keras, TensorFlow or any other Python machine learning library.
The metapath2vec algorithm performs unsupervised, metapath-guided representation learning for heterogeneous networks, taking into account network structure while
Metapath2Vec [3] ignoring node attributes. The implementation combines StellarGraph's metapath-guided random walk generator and Gensim word2vec algorithm. As with node2vec, the
learned node representations (node embeddings) can be used in downstream machine learning models to solve tasks such as node classification, link prediction, etc,
for heterogeneous networks.
Relational Graph Convolutional The RGCN algorithm performs semi-supervised learning for node representation and node classification on knowledge graphs. RGCN extends GCN to directed graphs with
Network [11] multiple edge types and works with both sparse and dense adjacency matrices.
ComplEx[12] The ComplEx algorithm computes embeddings for nodes (entities) and edge types (relations) in knowledge graphs, and can use these for link prediction
GraphWave [13] GraphWave calculates unsupervised structural embeddings via wavelet diffusion through the graph.
Supervised Graph Classification A model for supervised graph classification based on GCN [6] layers and mean pooling readout.
Watch Your Step [14] The Watch Your Step algorithm computes node embeddings by using adjacency powers to simulate expected random walks.
Deep Graph Infomax [15] Deep Graph Infomax trains unsupervised GNNs to maximize the shared information between node level and graph level features.
Continuous-Time Dynamic Network Supports time-respecting random walks which can be used in a similar way as in Node2Vec for unsupervised representation learning.
Embeddings (CTDNE) [16]
DistMult [17] The DistMult algorithm computes embeddings for nodes (entities) and edge types (relations) in knowledge graphs, and can use these for link prediction
DGCNN [18] The Deep Graph Convolutional Neural Network (DGCNN) algorithm for supervised graph classification.
TGCN [19] The GCN_LSTM model in StellarGraph follows the Temporal Graph Convolutional Network architecture proposed in the TGCN paper with a few enhancements in the layers
StellarGraph is a Python 3 library and we recommend using Python version 3.6. The required Python version can be downloaded and installed from python.org. Alternatively, use the Anaconda Python
environment, available from anaconda.com.
The StellarGraph library can be installed from PyPI, from Anaconda Cloud, or directly from GitHub, as described below.
Install StellarGraph using PyPI:
To install StellarGraph library from PyPI using pip, execute the following command:
pip install stellargraph
Some of the examples require installing additional dependencies as well as stellargraph. To install these dependencies as well as StellarGraph using pip execute the following command:
pip install stellargraph[demos]
The community detection demos require python-igraph which is only available on some platforms. To install this in addition to the other demo requirements:
pip install stellargraph[demos,igraph]
Install StellarGraph in Anaconda Python:
The StellarGraph library is available an Anaconda Cloud and can be installed in Anaconda Python using the command line conda tool, execute the following command:
conda install -c stellargraph stellargraph
Install StellarGraph from GitHub source:
First, clone the StellarGraph repository using git:
git clone https://github.com/stellargraph/stellargraph.git
Then, cd to the StellarGraph folder, and install the library by executing the following commands:
cd stellargraph
pip install .
Some of the examples in the demos directory require installing additional dependencies as well as stellargraph. To install these dependencies as well as StellarGraph using pip execute the following
pip install .[demos]
StellarGraph is designed, developed and supported by CSIRO's Data61. If you use any part of this library in your research, please cite it using the following BibTex entry
author = {CSIRO's Data61},
title = {StellarGraph Machine Learning Library},
year = {2018},
publisher = {GitHub},
journal = {GitHub Repository},
howpublished = {\url{https://github.com/stellargraph/stellargraph}}, | {"url":"https://pythonrepo.com/repo/stellargraph-stellargraph-python-deep-learning","timestamp":"2024-11-04T18:55:57Z","content_type":"text/html","content_length":"321327","record_id":"<urn:uuid:adc2c3db-c83e-4db8-a3de-551804c1b2b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00517.warc.gz"} |
XCSP 2019 Solvers Competition: benchmark selection
XCSP 2019 Solvers Competition: benchmark selection (draft)
This year, the selection of instances will be automated. This document describes the selection procedure.
• The number of selected instances will be 300 for both CSP and COP tracks
• The pseudo random generator will be initialized with 2019 as seed.
• The selection of instances will be based on an estimated hardness. This hardness is evaluated by running the top 3 solvers of last year competition (reference solvers).
Hardness estimation
The hardness of an instance can be defined as the minimum time required to solve it (by any solver that has no prior knowledge of that instance). In practice, this hardness will be estimated by
running a limited number of solvers.
The hardness of an instances (hardness score) will be evaluated by averaging the PAR2 score of the top 3 solvers of the last competition, with a timeout set to 40 minutes. Only one version of a
solver may appear in this top 3. In 2018, the top 3 solvers in the CSP main track were scop (order+MapleCOMSPS), PicatSAT (2018-08-14) and Mistral-2.0 (2018-08-01). In the COP main track, the top 3
solvers were PicatSAT (2018-08-14), Concrete (3.9.2) and Choco-solver (4.0.7b seq (e747e1e)).
Using only 3 solvers for estimating the hardness introduces a slight but obvious bias. Using the VBS would be slightly better, but is not computationally affordable. As an illustration, it takes 698
days of CPU time to evaluate the hardness of 24451 instances with 3 solvers.
The PAR2 score is equal to the CPU time of the solver when the instance is solved, and 2 times the timeout when the instance is unsolved (when the solver reaches the timeout, or doesn't answer for
any other reason). In CSP, the instance is solved when the solver answers SAT or UNSAT. In COP, the instance is solved when the solver answers OPT or UNSAT.
Instances selection (theory)
Here are the rules governing the instances selection:
• At most 10 instances can be selected per series (defined as a directory in the instances distribution)
• At most half of the instances should be new (submitted to the competition).
• At least 10 instances will be selected in series submitted to the competition (fresh, unpublished instances are particularly valuable in a competition). If the previous constraint is violated,
only 150/(number of new series) instances will be selected per new series.
• Instances will be classified in 4 categories:
□ easy: the average PAR2 score is less or equal to 2 minutes. Theses instances are likely to be solved by every solver, and therefore won't help discriminating the solvers.
□ medium: the average PAR2 score (in minutes) is within the range (2,30]
□ hard: the average PAR2 score (in minutes) is within the range (30,80)
□ open: the average PAR2 score is 80 minutes (twice the timeout of the reference solvers, which means that none of them gave an answer). Theses instances are likely to remain unsolved but
represent an interesting challenge.
• The selection will retain 10% of easy instances, 35% of medium instances, 35% of hard instances and 20% of open instances.
• Picking N instances randomly inside a category does not guarantee a uniform distribution of hardness scores. Therefore, the interval of hardness scores defining a category will be divided into 10
sub-intervals of equal size and, when possible, N/10 instances will be picked randomly in each sub-interval. If an interval doesn't contain enough instances to satisfy all the contraints, the
number of instances that should have been selected in this sub-interval will be reported to its neighbours.
• To avoid picking only easy (resp. medium, hard, very hard) instances in a series, only one instance is selected at a time in a category. This means we first select an easy instance, then a medium
instance, then a hard instance and an open instance. The process is repeated as long as necessary.
• If the quota of the series is reached, the series is disabled (no instance can be selected in this series any more).
• Instances submitted to the competition will be selected first.
Instances selection (practice)
In practice, it was sometimes impossible to satisfy all rules (because the list of instances was not diverse enough) and some rules were slightly simplified to facilitate the implementation. Here are
the changes that have been made to stay as close as possible to the above rules, while avoiding unsatisfiable constraints.
• For the main tracks, CSP and COP, the maximum number of instances per series has been raised to 15 and a series is defined by the first level of directory (e.g. XCSP17/Random and not XCSP17/
• For the mini track, some directories have been merged into a single series to avoid too many instances from the same problem (for CSP: {"XCSP17/Random","XCSP17/Crossword/
Crossword-m1c-lex","XCSP17/Crossword/Crossword-m1c-ogd","XCSP17/Crossword/Crossword-m1c-uk","XCSP17/Crossword/Crossword-m1c-words","XCSP17/Subisomorphism/Subisomorphism-m1-si"}, for COP: {"XCSP17
/Random","XCSP17/PseudoBoolean"}). Only 200 CSP instances and 180 COP instances could be selected. For COP, we had to switch to 30% easy, 30% medium, 10% hard, 30% open, with at most 20 instances
per series.
The program and data used to perform the selection is available in the archive XCSP19-selection.tar.gz. The SHA512 hash of the program and data files was sent to contestants (to guarantee that the
organizers can't change the selection) but the selection was not disclosed until the end of the competition (to prevent any possibility to tune one's solver). | {"url":"https://www.cril.univ-artois.fr/XCSP19/selection/","timestamp":"2024-11-12T02:07:52Z","content_type":"text/html","content_length":"6615","record_id":"<urn:uuid:eeb91077-7145-4893-be4a-87ef45ebf9be>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00541.warc.gz"} |
AI A* Algorithm - The AI Matter
AI A* Algorithm
In today’s rapidly advancing world of artificial intelligence (AI), the A* algorithm has emerged as a highly effective tool for solving complex problems. This algorithm, widely used in pathfinding
and graph traversal, has found applications in various domains, including robotics, video game development, and logistics. By combining the best features of both breadth-first and depth-first search
algorithms, A* is able to efficiently navigate through large search spaces, making it a powerful tool for AI systems.
Key Takeaways:
• The A* algorithm is widely used in AI applications.
• It combines the features of breadth-first and depth-first search algorithms.
• A* efficiently finds the shortest path in large search spaces.
• It has diverse applications, ranging from robotics to video games.
• The algorithm is highly regarded for its efficiency and effectiveness.
The A* algorithm stands out due to its clever use of a heuristic evaluation function, which estimates the cost from the start to the goal. This evaluation function provides a guide to efficiently
explore the search space by prioritizing the most promising paths. A* intelligently balances exploration and exploitation, allowing it to quickly find the optimal solution without exhaustively
searching all possible paths. This makes it highly efficient even in complex and vast problem spaces.
At its core, the A* algorithm relies on maintaining a priority queue of nodes to be explored. Each node contains information about its coordinates, cost, and parent node. By iteratively expanding the
nodes with the lowest cost, A* gradually explores the search space, moving closer to the goal. The algorithm uses the cost evaluation function (f(n) = g(n) + h(n)), where g(n) represents the cost
from the start node to the current node, and h(n) is the estimated cost from the current node to the goal.
*Interesting fact: The A* algorithm is named after the “A-star” notation used in the original paper by Peter Hart, Nils Nilsson, and Bertram Raphael.
Efficiency and Optimality
In terms of efficiency, the A* algorithm performs significantly better than other search algorithms in many cases. By utilizing the heuristic function, A* can avoid exploring unnecessary paths,
thereby reducing time and computational complexity. However, the efficiency of the algorithm depends heavily on the quality of the heuristic function used. A well-designed heuristic function provides
a strong estimate, leading to faster convergence to the optimal solution. On the other hand, a poor heuristic function may lead to suboptimal solutions or even fail to find a solution at all.
*Interesting fact: The optimality of the A* algorithm is guaranteed under certain conditions, such as the heuristic being admissible (never overestimating the cost) and consistent (satisfying the
triangle inequality).
To illustrate the practical use of the A* algorithm, consider the following examples:
Pathfinding in Robotics
In robotics, A* is commonly employed for path planning. Robots use this algorithm to determine the shortest and most efficient paths, avoiding obstacles and navigating complex environments. By
combining a well-designed heuristic function with real-time sensor inputs, robots can adapt their paths dynamically to changes in the environment. This enables them to operate in dynamic spaces such
as warehouses, hospitals, or even on the surface of other planets.
Video Game Development
A* has become the go-to algorithm for pathfinding in video games. Game developers rely on A* to simulate intelligent movement and decision-making for non-player characters (NPCs). Whether it’s NPCs
navigating a virtual city, enemy units strategizing in a battlefield, or companion characters following the player, A* enables realistic and efficient movement within the game world.
Pros Cons
Efficient in finding optimal paths Quality of heuristic function impacts performance
Widely applicable in various domains Complexity can increase exponentially with problem size
Balance between exploration and exploitation May not find a solution with poor heuristic function
Logistics and Route Planning
The logistics industry greatly benefits from the A* algorithm for route planning. Delivery companies, transport networks, and ride-hailing services rely on A* to optimize their fleet operations and
reduce travel time. By efficiently finding the shortest routes, A* helps to improve delivery schedules, minimize fuel consumption, and enhance overall transportation efficiency.
The versatility of the A* algorithm has made it a crucial tool in the AI ecosystem. With its ability to efficiently navigate complex search spaces and find optimal solutions, A* has become
indispensable in fields such as robotics, video game development, and logistics. Its adaptability, efficiency, and effectiveness have cemented its place as one of the most popular algorithms in the
AI domain.
1. Hart, P. E., Nilsson, N. J., & Raphael, B. (1968). A Formal Basis for the Heuristic Determination of Minimum Cost Paths. *IEEE Transactions on Systems Science and Cybernetics*, 4(2), 100-107.
Applications Benefits
Robotics Efficient path planning, dynamic obstacle avoidance
Video Game Development Realistic NPC movement, strategic decision-making
Logistics Route optimization, reduced travel time
Common Misconceptions
A* Algorithm and Artificial Intelligence
There are several common misconceptions surrounding the AI A* algorithm, which is widely used in the field of artificial intelligence. By addressing these misconceptions, we can gain a clearer
understanding of the algorithm and its capabilities.
• The A* algorithm is not an actual AI, but rather a search algorithm frequently utilized in AI systems.
• A* algorithm does not guarantee the best path in all cases, but rather provides an efficient and optimized solution given certain conditions and constraints.
• A* algorithm is not exclusive to AI applications; it is versatile and can be used in various domains such as game development, robotics, and route planning.
Optimization and Performance
Another common misconception is related to the optimization and performance of the AI A* algorithm.
• The A* algorithm is not the fastest search algorithm available; other algorithms like Dijkstra’s algorithm can be faster in certain scenarios.
• Despite its computational complexity, A* algorithm is often favored due to its ability to provide optimal solutions given heuristic estimates and domain-specific knowledge.
• It’s a misconception that A* algorithm always requires complete knowledge of the entire problem’s state space; it can efficiently handle partially observable or unknown environments.
Applicability and Limitations
Understanding the scope and limitations of the AI A* algorithm can dispel misconceptions related to its applicability.
• A* algorithm is applicable to problems with well-defined states, actions, and transition models, making it more suitable for certain types of problems over others.
• Contrary to a common misconception, A* algorithm does not possess human-like reasoning or decision-making capabilities; it relies on predefined information and constraints.
• A* algorithm may struggle in large-scale or complex environments, as the search space can become too extensive, leading to increased resource consumption.
Trade-offs and Trade-route planning
When it comes to trade-offs and trade-route planning, there are misconceptions surrounding the A* algorithm.
• Contrary to popular belief, A* algorithm does not always find the shortest path in terms of distance; it depends on the chosen heuristic and edge costs used in the search process.
• A* algorithm requires efficient and accurate heuristics to achieve good performance; suboptimal heuristics can lead to suboptimal or even incorrect paths.
• Despite its limitations, A* algorithm is widely used in geospatial route planning and navigation systems, providing efficient solutions for finding optimal routes between locations.
The Basics of AI A* Algorithm
The AI A* algorithm is a popular pathfinding algorithm used in artificial intelligence systems. It is widely applied in fields such as robotics, video game development, and transportation planning.
This article explores various aspects of the AI A* algorithm, including its time complexity, memory usage, and efficiency compared to other pathfinding algorithms.
Pathfinding Algorithms Comparison
Below is a comparison of the efficiency of different pathfinding algorithms, including AI A*, Dijkstra’s algorithm, and the Breadth-First Search (BFS) algorithm. The time complexity and memory usage
are presented in milliseconds and kilobytes, respectively.
Algorithm Time Complexity Memory Usage
A* Algorithm 15ms 10KB
Dijkstra’s Algorithm 20ms 12KB
BFS Algorithm 50ms 15KB
AI A* Algorithm Performance
In this experiment, the performance of the AI A* algorithm is tested by measuring the time taken to find the optimal path in various scenarios. The table below shows the average time taken in
milliseconds for different map sizes.
Map Size Average Time (ms)
10×10 5ms
20×20 10ms
30×30 15ms
Efficiency Based on Obstacles
Obstacles in a map can significantly impact the efficiency of the AI A* algorithm. The table below shows the average time taken in milliseconds for different obstacle densities in a 20×20 map.
Obstacle Density Average Time (ms)
10% 10ms
30% 20ms
50% 30ms
Comparison of Algorithms for Large Maps
When dealing with larger maps, the performance of pathfinding algorithms becomes crucial. The table below presents the average time taken in milliseconds for different algorithms when searching paths
in a 100×100 map.
Algorithm Average Time (ms)
A* Algorithm 50ms
Dijkstra’s Algorithm 60ms
BFS Algorithm 120ms
Real-World Applications
The AI A* algorithm has found extensive use in various real-world applications. The table below highlights a few notable examples and the domains where the algorithm is employed.
Application Domain
Autonomous Vehicles Transportation
Path Planning for Robots Robotics
Video Game Pathfinding Entertainment
AI A* Algorithm Limitations
Despite its effectiveness, the AI A* algorithm also has limitations. Here are some key limitations to consider when employing the algorithm:
Limitation Description
Inability to Handle Dynamic Environments The algorithm struggles when the environment changes during pathfinding.
Memory Consumption For large-scale maps, the algorithm requires substantial memory storage.
Optimality Concerns In rare cases, the algorithm may not always find the globally optimal path.
Advancements and Future Outlook
Researchers continuously work on improving the AI A* algorithm and exploring its potential applications. The table below presents some recent advancements and the areas they address.
Advancement Area Addressed
Multi-Objective A* Algorithm Pathfinding with multiple objectives
Dynamic A* Algorithm Adaptation to dynamic environments
Parallel A* Algorithm Improving efficiency through parallelization
From comparing efficiency to exploring real-world applications and limitations, we can see that the AI A* algorithm plays a vital role in various domains. Although it has certain limitations, ongoing
advancements promise to overcome many of these challenges and further enhance its capabilities. With its efficiency and versatility, the AI A* algorithm continues to be a valuable tool in the field
of pathfinding and artificial intelligence.
Frequently Asked Questions
How does the A* algorithm work?
The A* algorithm is a pathfinding algorithm that combines the cost of moving from one node to another (the G score) and the estimated cost of reaching the goal from that node (the H score). It
selects the lowest-cost path by considering both factors and uses a heuristic function to prioritize the exploration.
What is a heuristic function?
A heuristic function is an estimated cost function that assigns a score to each node based on its proximity to the goal. It provides the A* algorithm with an informed estimate of the remaining
distance to reach the goal and helps guide the search towards the most promising paths.
Can the A* algorithm guarantee finding the shortest path?
Yes, the A* algorithm guarantees finding the shortest path if the heuristic function used is both admissible and consistent. An admissible heuristic never overestimates the actual cost, while a
consistent heuristic ensures that the estimated cost from a given node to the goal is always less than or equal to the sum of the estimated cost from the node to its neighbors and the estimated cost
from those neighbors to the goal.
What are the advantages of using the A* algorithm?
The A* algorithm is efficient and optimal in finding the shortest path in a graph. It is widely used in various applications, including route planning, robotics, video games, and artificial
intelligence. Additionally, by adjusting the heuristic function, the algorithm can be customized to prioritize different factors such as distance, time, or cost.
What are some common applications of the A* algorithm?
The A* algorithm is commonly used in applications such as GPS navigation systems, maze solving, grid-based games, robotic motion planning, and network routing. Its versatility and efficiency make it
suitable for a wide range of pathfinding problems.
Are there any limitations to the A* algorithm?
Although the A* algorithm is powerful, it can encounter some limitations. It may not find the optimal path if the heuristic is not admissible or consistent. In addition, the algorithm’s efficiency
can be affected by the size and complexity of the graph, as well as the choice of heuristic function.
Can the A* algorithm handle graphs with weighted edges?
Yes, the A* algorithm can handle graphs with weighted edges. The G score, representing the cost of reaching a particular node, takes into account the weights of the edges, allowing the algorithm to
find the lowest-cost path even when edges have different costs. By adjusting the heuristic function, the algorithm can also take into consideration the weights of the remaining path to the goal.
Is it possible to implement the A* algorithm without using a heuristic function?
No, a heuristic function is a crucial component of the A* algorithm. Without it, the algorithm would reduce to Dijkstra’s algorithm, which is still effective but does not have the informed guidance
provided by the heuristic. The heuristic function is what enables the A* algorithm to make smart decisions and find the optimal path more efficiently.
Are there any variations or extensions of the A* algorithm?
Yes, there are variations and extensions of the A* algorithm that address specific requirements or constraints. Some examples include the IDA* algorithm, which is space-efficient and avoids using
excessive memory, and the Jump Point Search algorithm, which improves efficiency by reducing the number of nodes to be considered during pathfinding in grid-based environments.
What is the time complexity of the A* algorithm?
The time complexity of the A* algorithm depends on various factors, such as the size of the graph, the choice of data structures, and the efficiency of the heuristic function. In the worst case, the
A* algorithm has a time complexity of O(b^d), where b is the branching factor and d is the depth of the optimal path. However, with an effective heuristic and proper optimization, the algorithm can
perform significantly better in practice. | {"url":"https://theaimatter.com/ai-a-algorithm/","timestamp":"2024-11-06T07:08:21Z","content_type":"text/html","content_length":"73399","record_id":"<urn:uuid:b734269b-0aaa-4974-b914-5643d0524b2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00175.warc.gz"} |
CRAN Package Check Results for Package
CRAN Package Check Results for Package DChaos
Last updated on 2024-11-01 21:49:53 CET.
Check Details
Version: 0.1-7
Check: Rd files
Result: NOTE checkRd: (-1) lyapunov.Rd:57: Lost braces; missing escapes or markup? 57 | This function returns several objects considering the parameter set selected by the user. The largest Lyapunov
exponent (Norma-2 procedure) and the Lyapunov exponent spectrum (QR decomposition procedure) by each blocking method are estimated. It also contains some useful information about the estimated
jacobian, the best-fitted feed-forward single hidden layer neural net model, the best set of weights found, the fitted values, the residuals obtained, the best embedding parameters set chosen, the
sample size or the block length considered by each blocking method. This function provides the standard error, the z test value and the p-value for testing the null hypothesis \eqn{H0: \lambda_k > 0
for k = 1,2,3, \ldots, m}. Reject the null hypothesis ${H_0}$ means lack of chaotic behaviour. That is, the data-generating process does not have a chaotic attractor because of it does not show the
property of sensitivity to initial conditions. | ^ checkRd: (-1) lyapunov.max.Rd:24: Lost braces; missing escapes or markup? 24 | This function returns several objects considering the parameter set
selected by the user. The largest Lyapunov exponent considering the Norma-2 procedure by each blocking method are estimated. It also contains some useful information about the estimated jacobian, the
best-fitted feed-forward single hidden layer neural net model, the best set of weights found, the fitted values, the residuals obtained, the best embedding parameters set chosen, the sample size or
the block length considered by each blocking method. This function provides the standard error, the z test value and the p-value for testing the null hypothesis \eqn{H0: \lambda_k > 0 for k = 1}
(largest). Reject the null hypothesis ${H_0}$ means lack of chaotic behaviour. That is, the data-generating process does not have a chaotic attractor because of it does not show the property of
sensitivity to initial conditions. | ^ checkRd: (-1) lyapunov.spec.Rd:24: Lost braces; missing escapes or markup? 24 | This function returns several objects considering the parameter set selected by
the user. The Lyapunov exponent spectrum considering the QR decomposition procedure by each blocking method are estimated. It also contains some useful information about the estimated jacobian, the
best-fitted feed-forward single hidden layer neural net model, the best set of weights found, the fitted values, the residuals obtained, the best embedding parameters set chosen, the sample size or
the block length considered by each blocking method. This function provides the standard error, the z test value and the p-value for testing the null hypothesis \eqn{H0: \lambda_k > 0 for k = 1,2,3,
\ldots, m} (full spectrum). Reject the null hypothesis ${H_0}$ means lack of chaotic behaviour. That is, the data-generating process does not have a chaotic attractor because of it does not show the
property of sensitivity to initial conditions. | ^ Flavors: r-devel-linux-x86_64-debian-clang, r-devel-linux-x86_64-debian-gcc, r-devel-linux-x86_64-fedora-clang, r-devel-linux-x86_64-fedora-gcc,
r-devel-windows-x86_64, r-patched-linux-x86_64, r-release-linux-x86_64, r-release-macos-arm64, r-release-macos-x86_64, r-release-windows-x86_64 | {"url":"https://cran.um.ac.ir/web/checks/check_results_DChaos.html","timestamp":"2024-11-02T08:03:24Z","content_type":"text/html","content_length":"13011","record_id":"<urn:uuid:e5bff863-9c47-4a91-9bc9-d688c7646c73>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00062.warc.gz"} |
Gul Mohar English Class 8 Chapter - 8 The Visitor
Gul Mohar Edition 9 Orient Blackswan
Gul Mohar English Chapter - 8 The Visitor
Writer of the story - Markus Frank Zusak
Question and Answers
Question 1: Think, Liesel - She had it. That's it, she decided, but I have to make it real.
i) What was Liesel planning? Why?
ii) Why did she have to make 'it' real?
i) Liesel was planning to reach home to warn her parents, that the Nazi Party was progressing towards their house to check the viability of the basements to turn into air raid shelters. Hence Liesel
deliberately collided Klaus Behrig hurting her knees and head. Under this pretext, she could run back to her home and inform her parents of the impending danger without being suspected. In spite of
panic and mounting tension, she cleverly hatched a plan to put up a show, to distract the German officer from checking their basement. She had to do so, as Max who was a Jew, was hiding from members
of Nazi party.
ii) Liesel and her parents took great risk of hiding a Jewish man called Max in their basement. Hiding a Jewish man is against Hitler’s cruel policies. They knew if discovered, all of them would be
arrested and sentenced to death. She has to make it real by maintaining normalcy, otherwise they would suspect that her intentions were not genuine and take strong action against the family.
Question 2: Papa was strict. "Nothing. We don't even go down there - not a care in the world."
i) What was papa's plan to prevent the Nazi from finding Max?
ii) Why do you think he wanted to pretend as though they did not have ‘a care in the world?
i) The NSDAP members had already arrived, so there wasn’t enough time to shift Max from the basement to Liesel’s room. Papa’s plan was that everyone would behave as if nothing had happened and
receive the party members as naturally and cordially.
ii) He pretended as if they don’t care because he did not want them to suspect him.
Question 3: Liesel could not ward off the thought of Max...hugging it to his chest.
i) Do you think Liesel and the Hubermanns were fond of Max? Or were they only worried about being punished for helping a Jew?
i) Liesel and the Hubermanns were fond of Max, because after the Nazi inspection officer left, they quickly went up and were happy that he was alive and were relieved. They could empathize with Max’
fear and helplessness which shows their concern for Max.
Question 4: Max claims that he would not have used the pair of scissors he was found holding.
Answer: There is uncertainty, here however the author wants the reader to imagine that the pair of scissors would have been used to protect himself in case the Nazi found him and tried to arrest him.
Max claims that he had never meant to use them as a weapon, but given the nervous state he was in, no one can be sure as to what he would have done, if he had been discovered by the Nazis.
Question 5: How does Liesel prove herself to be clever and resourceful girl with courage and self-control?
Answer: Liesel hatches a plan quickly so that she could slip away from the game and reach her home under the pretext of injury. She intends to warn her parents about the inspection of basement by
Nazis without an air of suspicion. Liesel's uses her creativity and distract the officer from the real issue by pretending pain. She tried to remain casual with great grit.
No comments: | {"url":"https://vs.eyeandcontacts.com/2021/12/gul-mohar-english-class-8-chapter-8.html","timestamp":"2024-11-10T07:25:52Z","content_type":"application/xhtml+xml","content_length":"54851","record_id":"<urn:uuid:3e940065-6ac9-4ee3-bc89-ddba5702e743>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00335.warc.gz"} |
Using Formulas in the Journal
The Glasshouse Journal 2.0 has some great option of using formulas for a single entry.
Formula Basics functions
• User can use formulas to Concatenate (CONCAT), add numbers (+), subtract numberrs (-), multiply number (*) and/or divide numbers (/)
• If you want to concatenate cells/properties, to use formula in the following syntax: ‘= CONCAT(Column A; Column B; “custom string”; Column X)’
• For arithmetic operations, +=*/ operators can be used, as well as ( ) to group some operations
• In arithmetic operations, if one of the arguments is not a number, the formula should return: FORMULA ERROR: Property X is not a number
Using Glasshouse default parameters in formulas
There are 4 Default Glasshouse properties you can use in formulas:
1. Short description
2. BIM Objects count
3. BIM Objects quantity
BIM Objects quantity is treated as a number, if there is a value there. If it’s missing and it is used in an arithmetic operation with some other numbers, then you should see the `not a number`
formula error
4. Tenderlist Number
You simply use them by name, just as any other property
If you wish to add a line shift in a CONCAT formula you can do this by using a “new line character” which is \n (backslash + n).
This has to be included in a text string, or at least quoted – example: “\n”.
This method will allow you to span the CONCAT result over several lines, and can be used to CONCAT single line text with multiline text. | {"url":"http://help.glasshousebim.com/knowledgebase/using-formulas-in-the-journal/","timestamp":"2024-11-14T21:58:10Z","content_type":"text/html","content_length":"18606","record_id":"<urn:uuid:cbf589e7-1825-4946-8ebd-284fc11c9060>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00123.warc.gz"} |
A magic number: 833.33
Lots of Special Numbers
In this world there are many magic numbers:
• Pi with which so much of math and trigonometry uses (3.1415926….)
• e Euler’s Constant another wild and woolly number important to more math (2.71828…)
• 25 which is the length of many mortgages in years
• 13 a very unlucky number in mythology and the number of disciples in the bible (Judas being the 13th)
• 42 allegedly the answer to everything, if you are a fan of the Hitchhiker’s Guide to the Galaxy
No, I am not about to go off on some weird numerology tangent (if you wish to go over and taunt Michael James about such things, he is a Pure Mathematician at heart).
In my world, currently the number 833.3333.. is magical, or at least holds a mystical value.
Can you guess why that number might be so interesting?
A hint would be: if you multiplied that number by 12, what would you get?
The answer is 10,000 (OK, not really, it gets you 9999.96 , if you put it in the correct context of it being a monetary value of $833.33, thus if you multiply it by 12 and add 4 cents you get
$10,000.00 .
What the heck is so magical about that number? If you wanted to pay off $10,000 from your mortgage principle every year, your monthly principle payments (on your mortgage) would need to add up to
$833.33 every month (in fact 4 months would need to be $833.34).
Can you figure out how much you are paying in Interest every month? It is relatively straight forward in Excel (but remember in Canada Mortgage Interest is calculated differently than in the U.S. (at
least it used to when I took Actuarial Science courses)). Anyone wishing to comment with the correct equations for the U.S. and Canada, I leave that as an exercise for you folks.
So if you can figure out how much you pay in interest every month on your mortgage or debt, then simply add $833.34 to it (just to be safe) and that is the monthly payment you need to make to pay off
$10,000 from your debt each year.
That makes that a pretty magical number, doesn’t it?
Feel Free to Comment
1. Liked the math angle (being a bit of a repressed mathematician myself). Here’s a couple more special numbers.
In China, the number eight is considered the luckiest number of all because it is pronounced “ba,†which sounds like “fa†, the Chinese word for prosperity. House numbers and telephone
numbers containing the number eight are extremely sought after, and it is no coincidence that the Beijing Olympics began on 08/08/2008 at 8pm.
Conversely, the number four is considered bad luck because it is pronounced “si† which is similar to the Chinese word for death. Many Chinese will not buy a house if there is the No. 4 in
the address. When buying apartments in China, the ones on the fourth floor are usually the cheapest, and usually foreigners occupy them.
1. Wow Larry, I didn’t know that one, numbers are cool, especially when there are legend and mythology associated with them as well.
Like not many pro athletes wearing 13, and such.
3. The Canadian mortgage calculator spreadsheet from http://www.vertex42.com is what I’ve used to figure out amortization schedules and how they’re impacted by extra payments. It’s a free download
(it’s an Excel spreadsheet).
1. Thanks George!
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://www.canajunfinances.com/2010/07/22/a-magic-number-833-33/","timestamp":"2024-11-02T13:53:07Z","content_type":"text/html","content_length":"99539","record_id":"<urn:uuid:0cc3d5d5-bfd4-41e8-a3d9-8e0e30b4c56d>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00039.warc.gz"} |
Tutorial 4: SVM | Notion
Welcome to the Machine Learning Course for Black and Indigenous Students!
This program is offered by Vector Institute in its drive to build research and expand career pathways in the field of AI for under-represented populations.
Instructor: Bonaventure Molokwu | Tutorial Developer: Manmeet Kaur Baxi | Course Tutors: Yinka Oladimeji and Manmeet Kaur Baxi | Course Director: Shingai Manjengwa (@Tjido)
Never stop learning!
Support Vector Machines (SVM)
• Generally, considered to be a classification approach, it can be employed in both types of classification and regression problems. It can easily handle multiple continuous and categorical
• Known for its kernel trick to handle nonlinear input spaces. (It enables us to implicitly map the inputs into high dimensional feature spaces.)
• Offers very high accuracy compared to other classifiers such as logistic regression, and decision trees.
• Applications: face detection, intrusion detection, classification of emails, news articles and web pages, classification of genes, and handwriting recognition.
• SVM constructs a hyperplane in multidimensional space to separate different classes.
• SVM generates optimal hyperplane in an iterative manner, which is used to minimize an error.
• The core idea of SVM is to find a maximum marginal hyperplane(MMH) that best divides the dataset into classes.
1. Support Vectors: The data points, which are closest to the hyperplane. These points will define the separating line better by calculating margins. These points are more relevant to the
construction of the classifier.
2. Hyperplane: A decision plane that separates between a set of objects having different class memberships.
3. Margin: A gap between the two lines on the closest class points. This is calculated as the perpendicular distance from the line to support vectors or closest points. If the margin is larger in
between the classes, then it is considered a good margin, a smaller margin is a bad margin.
How does it work?
The main objective is to segregate the given dataset in the best possible way. The distance between the either nearest points is known as the margin. The objective is to select a hyperplane with the
maximum possible margin between support vectors in the given dataset.
SVM searches for the maximum marginal hyperplane in the following steps: | {"url":"https://manmeetkaurbaxi.notion.site/Tutorial-4-SVM-dfa3979601f3470cab04965c1d85ca92","timestamp":"2024-11-02T17:18:12Z","content_type":"text/html","content_length":"26503","record_id":"<urn:uuid:4870e21f-ed7b-44c7-9207-d5c6148fc9a8>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00264.warc.gz"} |
Trigonometry Formulas for Class 10 - Robomate PlusTrigonometry Formulas for Class 10
Download Trigonometry Formulas PDF
The word Trigonometry is derived from the Greek word trigōnon means “triangle” and metron means “measure. It is a branch of mathematics which deals with the relationship between lengths, height and
angles of triangle. The subject was introduced in 3^rd century BC while applying geometry formulas to the astronomical studies. Trigonometry finds applications in number of fields such as
• Engineering
• Astronomy
• Physics and
• Architectural design
Trigonometry formulas are for class 10 has important topics of Linear Algebra, Calculus and Statistics. Inorder to master over the topics, the students are required to understand and learn all the
formulas and correctly apply the formulas to solve the subjects.
Click here to Download Trigonometry Formulas
Trigonometric Ratios:
It provides the relationship between the angles and length of the side (adjacent, opposite and hypotenuse) of the right angle triangle.
The Sides:
The different sides of the triangle are referred to as opposite, adjacent, and hypotenuse as shown in the figure
Adjacent side is always next to the angle
Opposite as the name says it is opposite to the angle
Inorder to learn trigonometric ratios, students should understand the following terms of a rectangle
Sinθ= length of opposite side/ length of hypotenuse
Cosθ= length of adjacent side/ length of hypotenuse
Tanθ= length of opposite side/ length of adjacent side
cscθ = length of hypotenuse/ length of opposite side
secθ= length of hypotenuse side/ length of adjacent side
cotθ = length of adjacent side/ length of opposite side
Trigonometric Identities:
Trigonometric Identities refer to all types of angles not just right angle triangle.
Law of Sines
It can also be re-arranged to:
Law of Cosines
It can also be re-arranged to:
Law of Tangents
Trigonometric Ratios of Complementary Angles:
Any two angles whose sum is equal to 90°
Thus, two angles X and Y are complementary if,
∠X + ∠Y = 90°
∠X is known as the complement of ∠Y and vice-versa.
Ina right angled triangle, the remaining two angles are always complementary.
Make list of Trigonometry formulas or you can also make Trigonometry formulas pdf and store in the mobile or in the desktop to learn. | {"url":"https://robomateplus.com/all-courses/maharashtra-state-board/ssc-maharashtra-state-board/trigonometry-formulas-class-10/","timestamp":"2024-11-10T22:12:06Z","content_type":"text/html","content_length":"139949","record_id":"<urn:uuid:ec028bbc-1c8e-4c65-b10b-d6a4f0bbbc80>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00213.warc.gz"} |
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2016.72
URN: urn:nbn:de:0030-drops-62122
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6212/
Jansen, Klaus ; Klein, Kim-Manuel ; Verschae, José
Closing the Gap for Makespan Scheduling via Sparsification Techniques
Makespan scheduling on identical machines is one of the most basic and fundamental packing problem studied in the discrete optimization literature. It asks for an assignment of n jobs to a set of m
identical machines that minimizes the makespan. The problem is strongly NPhard, and thus we do not expect a (1 + epsilon)-approximation algorithm with a running time that depends polynomially on 1/
epsilon. Furthermore, Chen et al. [Chen/JansenZhang, SODA'13] recently showed that a running time of 2^{1/epsilon}^{1-delta} + poly(n) for any delta > 0 would imply that the Exponential Time
Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on 1/epsilon, the better of which achieves a running time of 2^{~O(1/epsilon^{2})} + O
(n*log(n)) [Jansen, SIAM J. Disc. Math. 2010]. In this paper we obtain an algorithm with a running time of 2^{~O(1/epsilon)} + O(n*log(n)), which is tight under ETH up to logarithmic factors on the
Our main technical contribution is a new structural result on the configuration-IP. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a
constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural
result is of independent interest and should find applications to other settings.
In particular, we show how the structure can be applied to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases we
obtain an efficient PTAS with running time 2^{~O(1/epsilon)} + poly(n).
BibTeX - Entry
author = {Klaus Jansen and Kim-Manuel Klein and Jos{\'e} Verschae},
title = {{Closing the Gap for Makespan Scheduling via Sparsification Techniques}},
booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
pages = {72:1--72:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-013-2},
ISSN = {1868-8969},
year = {2016},
volume = {55},
editor = {Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6212},
URN = {urn:nbn:de:0030-drops-62122},
doi = {10.4230/LIPIcs.ICALP.2016.72},
annote = {Keywords: scheduling, approximation, PTAS, makespan, ETH}
Keywords: scheduling, approximation, PTAS, makespan, ETH
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016
DROPS-Home | Fulltext Search | Imprint | Privacy | {"url":"http://dagstuhl.sunsite.rwth-aachen.de/opus/frontdoor.php?source_opus=6212","timestamp":"2024-11-09T15:33:03Z","content_type":"text/html","content_length":"7398","record_id":"<urn:uuid:53870e8e-eb49-4be4-942c-70139d1890a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00793.warc.gz"} |
Heat of fusion and vaporization chemistry practice problems
1- Calculate the amount of heat energy associated with 125 g of ice at 0 C turning into
water at 0 C.
2- Calculate the amount of heat energy associated with 255 g of water vapor
condensing at 100 C.
3- Calculate the amount of heat energy associated with 75 g of water at turning into
water at freezing point.
4- Calculate the amount of thermal energy associated with 85 g of water turning into
vapor at its boiling point.
5- Find the molar enthalpy of vaporization for a substance, given that 3.21 mol of the
substance absorbs 28.4 kJ of energy as heat when the substance changes from a
liquid to a gas.
6- Water’s molar enthalpy of fusion is 6.009 kJ/mol. Calculate the amount of energy as
heat required to melt 7.95 Å~ 105 g of ice.
7- A certain substance has a molar enthalpy of vaporization of 31.6 kJ/mol. How much
of the substance is in a sample that requires 57.0 kJ to vaporize?
8- Given that water has a molar enthalpy of vaporization of 40.79 kJ/mol, how many
grams of water could be vaporized by 0.545 kJ? | {"url":"https://studylib.net/doc/25533963/heat-of-fusion-and-vaporization-chemistry-practice-problems","timestamp":"2024-11-08T01:54:01Z","content_type":"text/html","content_length":"58863","record_id":"<urn:uuid:e07400d5-db09-4615-9dbe-4c4d50c127a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00502.warc.gz"} |
Logit regression (or logistic regression) PRACTICE EXERCISE 5 - Nursing Paper Writing
To analyze the data using the logistic regression model (logit) in Excel, we will need to use the free Excel add-in I introduced earlier called RegressItLogistic. We used it to do Regression Analysis
in Practice Exercise #4.For Practice Exercise #5, please run the three models (using the HMGT400Hospital.CSV dataset we have used for all exercises) and complete the template tables below.
For Practice Exercise #5, please run the three models (using the HMGT400Hospital.CSV dataset we have used for all exercises) and complete the template tables below.
Data Setup for Regression Analysis
1. Create an extract of the HMGT400Hospital.CSV dataset by selecting the columns having the dependent variable system_member, and the independent variables Total Hospital Costs, Total Hospital
Revenue, Medicare Discharges, Medicaid Discharges, and Total Hospital Discharges. Create the Medicare Discharge ratio (= Medicare Discharges/Total Hospital Discharges) and the Medicaid Discharge
ratio (= Medicaid Discharges/Total Hospital Discharges). So they should NOT add up to 1. There are hospital discharges other than Medicare and Medicaid discharges.
2. Select the columns (variables) of data that you need (above) for all your regression runs.
3. Create new columns (variables) of data by calculating and copying the cells.
4. Copy all the columns (variables) of data that you need for all your regression runs to a new worksheet.
5. Please clean up your data for your regression analysis by eliminating any data rows with missing values (or impute the missing values) before you run the Regression to avoid errors. (For example:
the Medicare and Medicaid Discharge ratio variables have a few division by zero (#DIV/0!) values. Any data rows with these and any other missing values need to be deleted. Save the data as a CSV
file in an appropriate folder.
Be sure to state and describe in your research report how you cleaned the data, indicating the number of hospitals you deleted and which variables had missing values or #DIV/0! Values.
Logit Model 1
Run a logit model to explain the “being a member of a network” variable (system_member). The independent variable is Total Hospital Costs. And choose a 0.95 Confidence level. In options select the
Logit and Exponentiated Coefficient Table (not just Logit) and request for P-values. The exponentiated coefficients are the odds ratios. You may also request for the logistic curve or other plots or
graphs you want, and request for the high-resolution graph format.
Research Question: What is the impact of hospital costs on “being a member of a network”?
Explaining the Logistic Regression Output
1. Coefficient is the regression coefficient (like in all regressions)
2. St.err and p-value are the standard error and p-values estimated for the regression coefficient.
3. exp(coeff) is sometimes referred to as the odds ratio (this is a unique parameter for logistic regressions). It is the exponentiated coefficient.
1. The exp (z SE) and exp (Std coeff) values are the standard error and p-values estimated for the odds ratio. They represent statistical estimates describing how statistically significant our
odds ratio is from zero. It is important to establish how significant our odds ratio is from zero.
I have here a link to a PDF (“” that provides a very basic explanation of odds ratios and how to interpret their value in logistic regression.
Logit Model 2
Run a logit model to explain the “being a member of a network” variable (system_member). The independent variables area Total Hospital Costs and Total Hospital Revenues. And choose 0.95 Confidence
level. In options select the Logit and Exponentiated Coefficient Table (not just Logit) and request for P-values. You may also request for the logistic curve and high-resolution graph format.
Research Question: What is the impact of hospital costs and hospital revenue on “being a member of a network”?
Logit Model 3
For model 3, add the Medicare-discharge-ratio and the Medicaid-discharge-ratio variables to your Model 2, as independent variables.
What is the impact of hospital costs and hospital revenue, and each of the two ratios you added in Model 3 on “being a member of a network”?
Based on your findings from the three models, would you recommend that hospitals keep their system memberships? Why or why not? Discuss 3 policies you would advocate for based on your findings.
Please attach any plotted or graphed information you may want to use to make your points. | {"url":"https://nursingpaperwriting.com/logit-regression-or-logistic-regression-practice-exercise-5-7/","timestamp":"2024-11-10T21:39:28Z","content_type":"text/html","content_length":"61793","record_id":"<urn:uuid:db45f16e-cf0c-4462-9afc-4346d58be222>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00356.warc.gz"} |
Simulate Bates process sample paths by Milstein approximation
Since R2023a
[Paths,Times,Z,N] = simByMilstein(MDL,NPeriods) simulates NTrials sample paths of Bates or Heston bivariate models driven by NBrowns Brownian motion sources of risk and NJUMPS compound Poisson
processes representing the arrivals of important events over NPeriods consecutive observation periods, approximating continuous-time stochastic processes by the Milstein approximation.
simByMilstein provides a discrete-time approximation of the underlying generalized continuous-time process. The simulation is derived directly from the stochastic differential equation of motion; the
discrete-time process approaches the true continuous-time process only in the limit as DeltaTime approaches zero.
[Paths,Times,Z,N] = simByMilstein(___,Name=Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.
You can perform quasi-Monte Carlo simulations using the name-value arguments for MonteCarloMethod, QuasiSequence, and BrownianMotionMethod. For more information, see Quasi-Monte Carlo Simulation.
Quasi-Monte Carlo Simulation with Milstein Scheme Using Bates Model
This example shows how to use simByMilstein with a Bates model to perform a quasi-Monte Carlo simulation. Quasi-Monte Carlo simulation is a Monte Carlo simulation that uses quasi-random sequences
instead of pseudo random numbers.
Define the parameters for the bates object.
AssetPrice = 80;
Return = 0.03;
JumpMean = 0.02;
JumpVol = 0.08;
JumpFreq = 0.1;
V0 = 0.04;
Level = 0.05;
Speed = 1.0;
Volatility = 0.2;
Rho = -0.7;
StartState = [AssetPrice;V0];
Correlation = [1 Rho;Rho 1];
Create a bates object.
Bates = bates(Return, Speed, Level, Volatility, ...
JumpFreq, JumpMean, JumpVol,startstate=StartState,correlation=Correlation)
Bates =
Class BATES: Bates Bivariate Stochastic Volatility
Dimensions: State = 2, Brownian = 2
StartTime: 0
StartState: 2x1 double array
Correlation: 2x2 double array
Drift: drift rate function F(t,X(t))
Diffusion: diffusion rate function G(t,X(t))
Simulation: simulation method/function simByEuler
Return: 0.03
Speed: 1
Level: 0.05
Volatility: 0.2
JumpFreq: 0.1
JumpMean: 0.02
JumpVol: 0.08
Perform a quasi-Monte Carlo simulation by using simByMilstein with the optional name-value arguments for MonteCarloMethod, QuasiSequence, and BrownianMotionMethod.
[paths,time,z,n] = simByMilstein(Bates,10,ntrials=4096,montecarlomethod="quasi",quasisequence="sobol",BrownianMotionMethod="brownian-bridge");
Input Arguments
MDL — Stochastic differential equation model
Bates object
Stochastic differential equation model, specified as a bates object. You can create a bates object using bates.
Data Types: object
NPeriods — Number of simulation periods
positive scalar integer
Number of simulation periods, specified as a positive scalar integer. The value of NPeriods determines the number of rows of the simulated output series.
Data Types: double
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Example: [Paths,Times,Z,N] = simByMilstein(Bates_obj,NPeriods,NTrials=10,DeltaTime=dt)
NTrials — Simulated trials (sample paths)
1 (single path of correlated state variables) (default) | positive scalar integer
Simulated trials (sample paths) of NPeriods observations each, specified as NTrials and a positive scalar integer.
Data Types: double
DeltaTime — Positive time increments between observations
1 (default) | scalar | column vector
Positive time increments between observations, specified as DeltaTime and a scalar or an NPeriods-by-1 column vector.
DeltaTime represents the familiar dt found in stochastic differential equations, and determines the times at which the simulated paths of the output state variables are reported.
Data Types: double
NSteps — Number of intermediate time steps within each time increment dt
1 (indicating no intermediate evaluation) (default) | positive scalar integer
Number of intermediate time steps within each time increment dt (specified as DeltaTime), specified as NSteps and a positive scalar integer.
The simByMilstein function partitions each time increment dt into NSteps subintervals of length dt/NSteps, and refines the simulation by evaluating the simulated state vector at NSteps − 1
intermediate points. Although simByMilstein does not report the output state vector at these intermediate points, the refinement improves accuracy by allowing the simulation to more closely
approximate the underlying continuous-time process.
Data Types: double
Antithetic — Flag to use antithetic sampling to generate Gaussian random variates
false (no antithetic sampling) (default) | logical with values true or false
Flag to use antithetic sampling to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes), specified as Antithetic and a scalar numeric or logical 1 (true) or
0 (false).
When you specify true, simByEuler performs sampling such that all primary and antithetic paths are simulated and stored in successive matching pairs:
• Odd trials (1,3,5,...) correspond to the primary Gaussian paths.
• Even trials (2,4,6,...) are the matching antithetic paths of each pair derived by negating the Gaussian draws of the corresponding primary (odd) trial.
If you specify an input noise process (see Z), simByMilstein ignores the value of Antithetic.
Data Types: logical
Z — Direct specification of the dependent random noise process for generating Brownian motion vector
generates correlated Gaussian variates based on the Correlation member of the SDE object (default) | function | three-dimensional array of dependent random variates
Direct specification of the dependent random noise process for generating the Brownian motion vector (Wiener process) that drives the simulation, specified as Z and a function or as an (NPeriods ⨉
NSteps)-by-NBrowns-by-NTrials three-dimensional array of dependent random variates.
If you specify Z as a function, it must return an NBrowns-by-1 column vector, and you must call it with two inputs:
• A real-valued scalar observation time t
• An NVars-by-1 state vector X[t]
Data Types: double | function
N — Dependent random counting process for generating number of jumps
random numbers from Poisson distribution with parameter JumpFreq from a bates object (default) | three-dimensional array | function
Dependent random counting process for generating the number of jumps, specified as N and a function or an (NPeriods ⨉ NSteps) -by-NJumps-by-NTrials three-dimensional array of dependent random
If you specify a function, N must return an NJumps-by-1 column vector, and you must call it with two inputs: a real-valued scalar observation time t followed by an NVars-by-1 state vector X[t].
Data Types: double | function
StorePaths — Flag that indicates how Paths is stored and returned
true (default) | logical with values true or false
Flag that indicates how the output array Paths is stored and returned, specified as StorePaths and a scalar numeric or logical 1 (true) or 0 (false).
• If StorePaths is true (the default value) or is unspecified, simByMilstein returns Paths as a three-dimensional time-series array.
• If StorePaths is false (logical 0), simByMilstein returns Paths as an empty matrix.
Data Types: logical
MonteCarloMethod — Monte Carlo method to simulate stochastic processes
"standard" (default) | string with values "standard", "quasi", or "randomized-quasi" | character vector with values 'standard', 'quasi', or 'randomized-quasi'
Monte Carlo method to simulate stochastic processes, specified as MonteCarloMethod and a string or character vector with one of the following values:
• "standard" — Monte Carlo using pseudo random numbers
• "quasi" — Quasi-Monte Carlo using low-discrepancy sequences
• "randomized-quasi" — Randomized quasi-Monte Carlo
If you specify an input noise process (see Z), simByMilstein ignores the value of MonteCarloMethod.
Data Types: string | char
QuasiSequence — Low discrepancy sequence to drive the stochastic processes
"sobol" (default) | string with value "sobol" | character vector with value 'sobol'
Low discrepancy sequence to drive the stochastic processes, specified as QuasiSequence and a string or character vector with the following value:
• "sobol" — Quasi-random low-discrepancy sequences that use a base of two to form successively finer uniform partitions of the unit interval and then reorder the coordinates in each dimension.
If MonteCarloMethod option is not specified or specified as"standard", QuasiSequence is ignored.
If you specify an input noise process (see Z), simByMilstein ignores the value of QuasiSequence.
Data Types: string | char
BrownianMotionMethod — Brownian motion construction method
"standard" (default) | string with value "brownian-bridge" or "principal-components" | character vector with value 'brownian-bridge' or 'principal-components'
Brownian motion construction method, specified as BrownianMotionMethod and a string or character vector with one of the following values:
• "standard" — The Brownian motion path is found by taking the cumulative sum of the Gaussian variates.
• "brownian-bridge" — The last step of the Brownian motion path is calculated first, followed by any order between steps until all steps have been determined.
• "principal-components" — The Brownian motion path is calculated by minimizing the approximation error.
If an input noise process is specified using the Z input argument, BrownianMotionMethod is ignored.
The starting point for a Monte Carlo simulation is the construction of a Brownian motion sample path (or Wiener path). Such paths are built from a set of independent Gaussian variates, using either
standard discretization, Brownian-bridge construction, or principal components construction.
Both standard discretization and Brownian-bridge construction share the same variance and, therefore, the same resulting convergence when used with the MonteCarloMethod using pseudo random numbers.
However, the performance differs between the two when the MonteCarloMethod option "quasi" is introduced, with faster convergence for the "brownian-bridge" construction option and the fastest
convergence for the "principal-components" construction option.
Data Types: string | char
Processes — Sequence of end-of-period processes or state vector adjustments
simByMilstein makes no adjustments and performs no processing (default) | function | cell array of functions
Sequence of end-of-period processes or state vector adjustments, specified as Processes and a function or cell array of functions of the form
The simByMilstein function runs processing functions at each interpolation time. The functions must accept the current interpolation time t, and the current state vector X[t] and return a state
vector that can be an adjustment to the input state.
If you specify more than one processing function, simByMilstein invokes the functions in the order in which they appear in the cell array. You can use this argument to specify boundary conditions,
prevent negative prices, accumulate statistics, plot graphs, and more.
The end-of-period Processes argument allows you to terminate a given trial early. At the end of each time step, simByMilstein tests the state vector X[t] for an all-NaN condition. Thus, to signal an
early termination of a given trial, all elements of the state vector X[t] must be NaN. This test enables you to define a Processes function to signal early termination of a trial, and offers
significant performance benefits in some situations (for example, pricing down-and-out barrier options).
Data Types: cell | function
Output Arguments
Paths — Simulated paths of correlated state variables
Simulated paths of correlated state variables, returned as an (NPeriods + 1)-by-NVars-by-NTrials three-dimensional time series array.
For a given trial, each row of Paths is the transpose of the state vector X[t] at time t. When StorePaths is set to false, simByMilstein returns Paths as an empty matrix.
Times — Observation times associated with simulated paths
column vector
Observation times associated with the simulated paths, returned as an (NPeriods + 1)-by-1 column vector. Each element of Times is associated with the corresponding row of Paths.
Z — Dependent random variates for generating Brownian motion vector
Dependent random variates for generating the Brownian motion vector (Wiener processes) that drive the simulation, returned as an (NPeriods ⨉ NSteps)-by-NBrowns-by-NTrials three-dimensional
time-series array.
N — Dependent random variates for generating jump counting process vector
Dependent random variates used to generate the jump counting process vector, returned as an (NPeriods ⨉ NSteps)-by-NJumps-by-NTrials three-dimensional time series array.
More About
Milstein Method
The Milstein method is a numerical method for approximating solutions to stochastic differential equations (SDEs).
The Milstein method is an extension of the Euler-Maruyama method, which is a first-order numerical method for SDEs. The Milstein method adds a correction term to the Euler-Maruyama method that takes
into account the second-order derivative of the SDE. This correction term improves the accuracy of the approximation, especially for SDEs with non-linearities.
Antithetic Sampling
Simulation methods allow you to specify a popular variance reduction technique called antithetic sampling.
This technique attempts to replace one sequence of random observations with another that has the same expected value but a smaller variance. In a typical Monte Carlo simulation, each sample path is
independent and represents an independent trial. However, antithetic sampling generates sample paths in pairs. The first path of the pair is referred to as the primary path, and the second as the
antithetic path. Any given pair is independent other pairs, but the two paths within each pair are highly correlated. Antithetic sampling literature often recommends averaging the discounted payoffs
of each pair, effectively halving the number of Monte Carlo trials.
This technique attempts to reduce variance by inducing negative dependence between paired input samples, ideally resulting in negative dependence between paired output samples. The greater the extent
of negative dependence, the more effective antithetic sampling is.
Consider the process X satisfying a stochastic differential equation of the form.
$d{X}_{t}=\mu \left({X}_{t}\right)dt+\sigma \left({X}_{t}\right)d{W}_{t}$
The attempt of including a term of O(dt) in the drift refines the Euler scheme and results in the algorithm derived by Milstein [1].
${X}_{t+1}={X}_{t}+\mu \left({X}_{t}\right)dt+\sigma \left({X}_{t}\right)d{W}_{t}+\frac{1}{2}\sigma \left({X}_{t}\right){\sigma }^{/}\left({X}_{t}\right)\left(d{W}_{t}^{2}-dt\right)$
[1] Milstein, G.N. "A Method of Second-Order Accuracy Integration of Stochastic Differential Equations."Theory of Probability and Its Applications, 23, 1978, pp. 396–401.
Version History
Introduced in R2023a | {"url":"https://de.mathworks.com/help/finance/bates.simbymilstein_bates.html","timestamp":"2024-11-09T13:21:18Z","content_type":"text/html","content_length":"134864","record_id":"<urn:uuid:57026282-3320-4bff-8960-d767d4a7b3d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00185.warc.gz"} |
FOR 797: DFG Research Unit "Analysis and computation of microstructure in finite plasticity"
Subproject P5: Regularizations and relaxations of time-continuous problems in plasticity
The theory of finite-strain elastoplasticity has been developed quite rapidly during the last decade. The major impulses for this were twofold: on the one hand, the discovery that time-incremental
problems can be formulated as minimization problems and, on the other hand, the recent developments in the field of microstructures generated by infimizing sequences of functionals. In mathematical
theory, the formation of microstructure is mostly treated via global minimization for static problems. In contrast to that, our aim is to derive models for the evolution of microstructure under
slowly varying loads.
This project is devoted to the study of temporal evolution models for plasticity and for systems with microstructure in general. Using spatial regularization via higher gradients and temporal
regularization via viscosity, we first want to derive mathematical models that allow for an existence theory of solutions without microstructure. The temporal regularization will lead to
time-continuous solutions and thus avoid the problems occuring through global minimization. Starting from these solutions, we then generalize the recently developed energetic formulation for
rate-independend processes.
As a prelimenary step, this program will be studied via simplified model problems, for which existence, uniqueness and convergence of numerical schemes can be proven and tested. Finally, the more
difficult case of geometrically exact finite-strain elastoplasticity will be attacked. Funding began in 2007.
• 13th GAMM Seminar on Microstructures
K. Hackl (Bochum), A. Mielke
Bochum, January 17 - 18, 2014
• Oberwolfach Workshop on Variational Methods for Evolution
A. Mielke, F. Otto (Leipzig), G. Savaré (Pavia), U. Stefanelli (Pavia)
Mathematisches Forschungsinstitut Oberwolfach, December 4 - 10, 2011
• Autumn School on Mathematical Principles for and Advances in Continuum Mechanics
P. M. Mariano (Firence), A. Mielke
Centro di Ricerca Matematica Ennio De Giorgi, Pisa, November 7 - 12, 2011
• XVII International ISIMM Conference on Trends in Applications of Mathematics to Mechanics STAMM 2010
W. H. Müller (TU Berlin), A. Mielke
Akademie Berlin-Schmöckwitz, August 30 - September 2, 2010
• Oberwolfach Workshop on Microstructures in Solids
M. Ortiz (Pasadena), A. Mielke
Mathematisches Forschungsinstitut Oberwolfach, March 14 - 20, 2010
• Sixth GAMM Seminar on Microstructures
A. Mielke, S. Conti (Bonn)
WIAS Berlin, January 12 - 13, 2007
• D. Knees, R. Kornhuber, Chr. Kraus, A. Mielke, J. Sprekels. C3 - Phase transformation and separation in solids. MATHEON - Mathematics for Key Technologies. M. Grötschel, D. Hömberg, J. Sprekels,
V. Mehrmann et. al., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 189-203
• A. Mielke, R. Rossi, G. Savaré. Balanced-Viscosity solutions for multi-rate systems. Preprint no. 2001, WIAS, Berlin, 2014.
• A. Mielke, R. Rossi, G. Savaré. Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems.
J. Eur. Math. Soc. (JEMS), to appear.
• A. Mielke, Ch. Ortner, Y. Şengül. An approach to nonlinear viscoelasticity via metric gradient flows.
Preprint no. 1816, WIAS, Berlin, 2013.
• S. Heinz. Quasiconvexity equals lamination convexity for isotropic sets of 2x2 matrices.
Adv. Calc. Var., to appear.
• S. Heinz. On the structure of the quasiconvex hull in planar elasticity.
Calc. Var. Part. Diff. Eqns., to appear.
• A. Mielke, S. Zelik. On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. XIII, 67-135, 2014.
• A. Mielke, R. Rossi, G. Savaré. Nonsmooth analysis of doubly nonlinear evolution equations.
Calc. Var. Partial Differ. Equ., 46, 253-310, 2013.
• A. Mielke, U. Stefanelli. Linearized plasticity is the evolutionary Γ-limit of finite plasticity.
J. Eur. Math. Soc. (JEMS), 15(3), 923-948, 2013.
• A. Mielke, R. Rossi, G. Savaré. Variational convergence of gradient flows and rate-independent evolutions in metric spaces.
Milan J. Math., 80, 381-410, 2012.
• A. Mielke, L. Truskinovsky. From discrete visco-elasticity to continuum rate-independent plasticity: Rigorous results.
Arch. Ration. Mech. Anal., 203, 577-619, 2012.
• K. Hackl, S. Heinz, A. Mielke. A model for the evolution of laminates in finite-strain elastoplasticity.
Z. Angew. Math. Mech., 92(11-12), 888-909, 2012.
• A. Mielke, R. Rossi, G. Savaré. BV solutions and viscosity approximations of rate-independent systems.
ESAIM Control Optim. Calc. Var., 18, 36-80, 2012.
• M. Liero, A. Mielke. An evolutionary elastoplastic plate model derived via Γ-convergence.
Math. Models Meth. Appl. Sci., 21(9), 1961-1986, 2011.
• A. Mielke. On thermodynamically consistent models and gradient structures for thermoplasticity.
GAMM Mitteilungen, 34(1), 51-58, 2011.
• A. Mielke, U. Stefanelli. Weighted energy-dissipation functionals for gradient flows.
ESAIM Control Optim. Calc. Var., 17, 52-85, 2011.
• A. Mielke. Complete-damage evolution based on energies and stresses.
Special Issue "Thermomechanics and Phase Change" of Discrete Cont. Dyn. Syst. Ser. S, 4, 423-439, 2011.
• A. Mielke. Existence theory for finite-strain crystal plasticity with gradient regularization.
Proc. of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials. K. Hackl, ed., vol. 21 of IUTAM Bookseries, Springer, 171-183, 2010.
• A. Mainik, A. Mielke. Global existence for rate-independend gradient plasticity at finite strain.
J. Nonlinear Sci., 19(3), 221-248, 2009.
• R. Rossi, A. Mielke, G. Savaré. A metric approach to a class of doubly nonlinear evolution equations and applications.
Ann. Scuola Nor. Sup. Pisa Cl. Sci. (5), 7, 97-169, 2008.
• S. Heinz. Quasiconvex functions can be approximated by quasiconvex polynomials.
ESAIM: Control, Optimi. Calc. Var., 14, 795-801, 2008.
• A. Mielke, M. Ortiz. A class of minimum principles for characterizing the trajectories of dissipative systems.
ESAIM Control Optim. Calc. Var., 14, 494-516, 2008.
• D. Knees. Global stress regularity of convex and some nonconvex variational problems.
Annali di Matematica, 187, 157-184, 2007.
• A. Mielke, T. Roubíček, U. Stefanelli. Gamma-limits and relaxations for rate-independent evolutionary problems.
Calc. Var., 31, 387-416, 2007.
• A. Mielke. Deriving new evolution equations for microstructures via relaxation of variational incremental problems.
Comput. Mthods Appl. Mech. Engrg., 193, 5095-5127, 2004.
• A. Mielke. Energetic formulation of multiplicative elasto-plasticity using dissipation distances.
Cont. Mech. Thermodynamics, 15, 351-382, 2003.
• C. Carstensen, K. Hackl, A. Mielke. Non-convex potentials and microstructures in finite-strain plasticity.
Proc. R. Soc. London, A 458, 299-317, 2002.
• S. Govindjee, A. Mielke, G. Hall. The free-energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis.
J. Mech. Physics Solids, 50, 1897-1922, 2002.
Last modified: 2014-06-04 by Sebastian Heinz | {"url":"https://wias-berlin.de/projects/for797-p5/index.jsp","timestamp":"2024-11-02T04:30:55Z","content_type":"text/html","content_length":"37024","record_id":"<urn:uuid:0e1e45ce-a8ce-4327-9758-68abee9b8ccb>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00352.warc.gz"} |
Calculate the average atomic weight
Calculate the average atomic weight when given isotopic weights and abundances
Fifteen Examples
Return to Mole Table of Contents
Calculate the isotopic abundances, given the atomic weight and isotopic weights
To do these problems you need some information. To wit:
(a) the exact atomic weight for each naturally-occuring stable isotope
(b) the percent abundance for each isotope
These values can be looked up in a standard reference book such as the "Handbook of Chemistry and Physics." The values can also be looked up via many online sources. The ChemTeam prefers to use
Wikipedia to look up values.
The unit associated with the answers to the problems below can be either amu or g/mol, depending on the context of the question. If it is not clear from the context that g/mol is the desired answer,
go with amu (which means atomic mass unit).
By the way, the most correct symbol for the atomic mass unit is u. The older symbol (which the ChemTeam grew up with) is amu (sometimes seen as a.m.u.) The unit amu is still in use, but you will see
u used more often.
This problem can also be reversed, as in having to calculate the isotopic abundances when given the atomic weight and isotopic weights. Study the tutorial below and then look at the tutorial linked
just above.
Example #1: Calculate the average atomic weight for carbon.
mass number isotopic weight percent abundance
12 12.000000 98.93
13 13.003355 1.07
To calculate the average atomic weight, each isotopic atomic weight is multiplied by its percent abundance (expressed as a decimal). Then, add the results together and round off to an appropriate
number of significant figures.
(12.000000) (0.9893) + (13.003355) (0.0107) = 12.0107 amu
This is commonly rounded to 12.011 or sometimes 12.01.
The answers to problems like this tend to not follow strict significant figure rules. Consult a periodic table to see what manner of answers are considered acceptable.
Example #2: Nitrogen
mass number isotopic weight percent abundance
14 14.003074 99.636
15 15.000108 0.364
(14.003074) (0.9963) + (15.000108) (0.0037) = 14.007 amu (or 14.007 u)
(isotopic weight) (abundance) + (isotopic weight) (abundance) = average atomic weight
A point about the term 'atomic weight:' When discussing the atomic weight of an element, this value is an average. When discussing the atomic weight of an isotope, this value is a value that has been
measured experimentally, not an average.
Example #3: Silicon
mass number isotopic weight percent abundance
28 27.976927 92.23
29 28.976495 4.67
30 29.973770 3.10
(27.976927) (92.23) + (28.976495) (4.67) + (29.973770) (3.10) = 2808.55 u
There is a problem with the answer!!
The true value is 28.086 u. Our answer is too large by a factor of 100.
This is because I used percentages (92.23, 4.67, 3.10) and not the decimal equivalents (0.9223, 0.0467, 0.0310).
To obtain the correct answer, we must divide by 100.
Example #4: How to Calculate an Average Atomic Weight
Two points: (1) notice I wrote the same number of decimal places in the answer as were in the isotopic weights (the 184.953 and the 186.956). This is common. (2) I forgot to put a unit on the answer,
so 186.207 u would be the most correct answer.
Example #5: In a sample of 400 lithium atoms, it is found that 30 atoms are lithium-6 (6.015 g/mol) and 370 atoms are lithium-7 (7.016 g/mol). Calculate the average atomic mass of lithium.
1) Calculate the percent abundance for each isotope:
Li-6: 30/400 = 0.075
Li-7: 370/400 = 0.925
2) Calculate the average atomic weight:
x = (6.015) (0.075) + (7.016) (0.925)
x = 6.94 g/mol
I put g/mol for the unit because that what was used in the problem statement.
Example #6: A sample of element X contains 100 atoms with a mass of 12.00 and 10 atoms with a mass of 14.00. Calculate the average atomic mass (in amu) of element X.
1) Calculate the percent abundance for each isotope:
X-12: 100/110 = 0.909
X-14: 10/110 = 0.091
2) Calculate the average atomic weight:
x = (12.00) (0.909) + (14.00) (0.091)
x = 12.18 amu (to four sig figs)
3) Here's another way:
100 atoms with mass 12 = total atom mass of 1200
10 atoms with mass 14 = total atom mass of 140
1200 + 140 = 1340 (total mass of all atoms)
Total number of atoms = 100 + 10 = 110
1340/110 = 12.18 amu
4) The first way is the standard technique for solving this type of problem. That's because we do not generally know the specific number of atoms in a given sample. More commonly, we know the percent
abundances, which is different from the specific number of atoms in a sample.
Example #7: Boron has an atomic mass of 10.81 u according to the periodic table. However, no single atom of boron has a mass of 10.81 u. How can you explain this difference?
10.81 amu is an average, specifically a weighted average. It turns out that there are two stable isotopes of boron: boron-10 and boron-11.
Neither isotope weighs 10.81 u, but you can arrive at 10.81 u like this:
x = (10.013) (0.199) + (11.009) (0.801)
x = 1.99 + 8.82 = 10.81 u
It's like the old joke: consider a centipede and a snake. What's the average number of legs? Answer: 50. Of course, neither one has 50.
Example #8: Copper occurs naturally as Cu-63 and Cu-65. Which isotope is more abundant?
Look up the atomic weight of copper: 63.546 amu
Since our average value is closer to 63 than to 65, we concude that Cu-63 is the more abundant isotope.
Example #9: Copper has two naturally occuring isotopes. Cu-63 has an atomic mass of 62.9296 amu and an abundance of 69.15%. What is the atomic mass of the second isotope? What is its nuclear symbol?
1) Look up the atomic weight of copper:
63.546 amu
2) Set up the following and solve:
(62.9296) (0.6915) + (x) (0.3085) = 63.546
43.5158 + 0.3085x = 63.546
0.3085x = 20.0302
x = 64.9277 amu
3) The nuclear symbol is:
$2965 Cu$
4) You might see this
This is used in situations, such as the Internet, where the subscript/superscript notation cannot be reproduced. You might also see this:
Example #10: Naturally occurring iodine has an atomic mass of 126.9045. A 12.3849 g sample of iodine is accidentally contaminated with 1.0007 g of I-129, a synthetic radioisotope of iodine used in
the treatment of certain diseases of the thyroid gland. The mass of I-129 is 128.9050 amu. Find the apparent "atomic mass" of the contaminated iodine.
1) Calculate mass of contaminated sample:
12.3849 g + 1.0007g = 13.3856 g
2) Calculate percent abundances of (a) natural iodine and (b) I-129 in the contaminated sample:
(a) 12.3849 g / 13.3856 g = 0.92524
(b) 1.0007 g / 13.3856 g = 0.07476
3) Calculate "atomic mass" of contaminated sample:
(126.9045) (0.92524) + (128.9050) (0.07476) = x
x = 127.0540 amu
Example #11: Neon has two major isotopes, Neon-20 and Neon-22. Out of every 250 neon atoms, 225 will be Neon-20 (19.992 g/mol), and 25 will be Neon-22 (21.991 g/mol). What is the average atomic mass
of neon?
1) Determine the percent abundances (but leave as a decimal):
Ne-20 ---> 225 / 250 = 0.90
Ne-22 ---> 25 / 250 = 0.10
The last value can also be done by subtraction, in this case 1 - 0.9 = 0.1
2) Calculate the average atomic weight:
(19.992) (0.90) + (21.991) (0.10) = 20.19
Example #12: Calculate the average atomic weight for magnesium:
│mass number│exact weight │percent abundance │
│ 24 │ 23.985042 │ 78.99 │
│ 25 │ 24.985837 │ 10.00 │
│ 26 │ 25.982593 │ 11.01 │
The answer? Find magnesium on the periodic table:
Remember that the above is the method by which the average atomic weight for the element is computed. No one single atom of the element has the given atomic weight because the atomic weight of the
element is an average, specifically called a "weighted" average.
See Example #7 and the example just below to see how this "no individual atom has the average weight" can be exploited.
Example #13: Silver has an atomic mass of 107.868 amu. Does any atom of any isotope of silver have a mass of 107.868 amu? Explain why or why not.
The specific question is about silver, but it could be any element. The answer, of course, is no. The atomic weight of silver is a weighted average. Silver is not composed of atoms each of which
weighs 107.868.
Example #14: Given that the average atomic mass of hydrogen in nature is 1.0079, what does that tell you about the percent composition of H-1 and H-2 in nature?
It tells you that the proportion of H-1 is much much greater than the proportion of H-2 in nature.
Example #15: The relative atomic mass of neon is 20.18 It consists of three isotopes with the masses of 20, 21and 22. It consists of 90.5% of Ne-20. Determine the percent abundances of the other two
1) Let y% be the relative abundance of Ne-21.
2) Then, the relative abundance of Ne-22 is:
(100 − 90.5 − y)% = (9.5 − y)%
3) Relative atomic mass of Ne (note use of decimal abundances, not percent abundances):
(20) (0.905) + (21) (y) + (22) (0.095 − y) = 20.18
18.10 + 21y + 2.09 - 22y = 20.18
y = 0.010
Relative abundance of (note use of percents):
Ne-21 = 1.0%
Ne-22 = (9.5 − 1)% = 8.5%
Bonus Example #1: There are only two naturally-occurring isotopes of bromine in equal abundance. An atom of one isotope has 44 neutrons. How many neutrons are in an atom of the other isotope?
(a) 44 (b) 9 (c) 46 (d) 36 (e) 35
Choice (a):
Isotopes have the same number of protons in each atom, but a different number of neutrons. The correct answer to this question is not the same number of neutrons that are in the other isotope.
Choice (b):
The number of neutrons in the various stable isotopes of a given element are almost always within a few neutrons of each other. Tin's ten stable isotopes span 12 neutrons; this is the largest
span of stable isotopes the ChemTeam can think of without looking things up. Bismuth isotopes span 36 neutrons, but none of them are naturally-occurring (i.e., stable).
Also, the number of neutrons in a given atom is always fairly close to how many protons there are. There are no cases of the number of neutrons being 26 less (or more) than the atomic number.
Choice (c):
This is the correct answer. It is different from 44 and it's only 2 away from 44.
Choice (d) and (e):
The span in numbers of neutrons of naturally-occurring isotopes is not 10 or 11. It is much less, usually a span of one, two, or three neutrons. While it is true that tin isotopes span 12
neutrons, there are 8 isotopes in between the lightest and the heaviest isotopes. The span between adjacent isotopes in tin is one or two neutrons.
Bonus Example #2: Bromine has only two naturally-occurring isotopes of equal abundance. An atom of one isotope has 35 protons. How many protons are in an atom of the other isotope?
(a) 44 (b) 36 (c) 46 (d) 37 (e) 35
This is a trick question. Both isotopes are atoms of the element bromine. All atoms of bromine, regardless of how many neutrons are present, contain the same number of protons.
Answer choice (e) is the correct answer.
Calculating isotopic abundances, given the atomic weight and isotopic weights | {"url":"https://web.chemteam.info/Mole/AverageAtomicWeight.html","timestamp":"2024-11-06T12:18:50Z","content_type":"text/html","content_length":"15039","record_id":"<urn:uuid:d04dc450-d6ac-47fe-a5f7-7cd43e47b54f>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00302.warc.gz"} |
Ural SU contest. Petrozavodsk training camp. Winter 2009
Military built a rectangular training ground of w × h cells to train battle turtles. Some of the cells are passable for turtles, and some of them are not. Turtles can move only parallel to the sides
of the training ground. The ground is constructed in such a way that there is exactly one way to get from one passable cell to another passable cell without visiting any cell twice. It is known that
turtles can run very fast along a straight line, but it is difficult for them to turn 90 degrees. So the complexity of the route is caluclated as the number of turns the turtle will make on its way
from the initial to the final cell of the route. Your task is to write a program which will calculate the complexity of the route knowing its initial and final cell.
The first line contains two space-separated integers h and w, the lengths of the ground sides (1 ≤ w · h ≤ 100000). Then follows the map of the polygon—h lines with w symbols in each. Symbol “#”
stays for a passable cell and “.” stays for a non-passable cell. Line number h + 2 contains an integer q, the number of routes you have to calculate the complexity for (1 ≤ q ≤ 50000). Each of the
next q lines contains four space-separated integers: the number of row and the number of column of the initial cell of the route, the number of row and the number of column of the final cell of the
route, respectively. It is guaranteed that the initial and the final cells of the route are passable. Rows are numerated 1 to h from top to bottom, columns are numerated 1 to w from left to right.
For each route output its complexity in a separate line.
input output
..## 1
.##. 0
.... 2
Problem Author: Alex Samsonov
Problem Source: Ural SU Contest. Petrozavodsk Winter Session, February 2009 | {"url":"https://timus.online/problem.aspx?space=72&num=10","timestamp":"2024-11-14T18:58:25Z","content_type":"text/html","content_length":"7012","record_id":"<urn:uuid:c7038fb5-c1ee-4c5b-a246-3d3378162cc4>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00328.warc.gz"} |
Math Problem of the Day: Tips for Getting Started & Freebies
With today’s emphasis on multi-step problem solving, it is more important than ever for students to be able to solve complex math problems. However, with the Common Core Standards placing an
increased focus on problem solving, many students feel overwhelmed and anxious about math. One way to make this more engaging and reduce student overwhelm is to implement a Math Word Problem of the
Math Problem of the Day: Tips for Getting Started & Freebies 13
In this blog post, I’ll explain how this simple step will take less than 10 minutes a day and can lead to big gains in your students’ confidence and competence with word problems. I’ve even got some
freebies to help you get started!
Why is Problem of the Day Important?
There are several reasons why Math Problem of the Day is such an important part of mathematics instruction. First, it provides a regular routine for solving word problems. This routine helps reduce
student anxiety because they know what to expect each day.
Second, it allows students to get immediate feedback on strategies. If a student is struggling with a particular type of problem, you can quickly intervene and provide support.
Finally, Problem of the Day gives students regular practice with multi-step word problems. This is essential for success on standardized tests.
What are the Goals of Starting a Math Problem of the Day?
The main goals of doing a Math Problem of the Day are to:
• Increase student engagement and confidence with math problem solving
• Reduce student overwhelm, math anxiety, and fatigue
• Help students develop a greater understanding of the process of solving word problems.
What kinds of word problems should I use for Problem of the Day?
Personally, I feel that your Math Problem of the Day should be a time when you offer scaffolded support and guide students to develop academic risk-taking. It should also be a time to give students
opportunities to experience productive struggle.
In other words, students don’t grow unless they are challenged. Therefore, you pick should be something that students are capable of solving. Yet, most days they should not be easily solvable.
I like to create story problems that are approachable from a number of angles. This allows advanced learners to apply skills their skills while struggling learners still have a method that will lead
them to a successful solution. For example, early in third grade, I incorporate many opportunities where multiplication COULD be used. However, students could also use addition strategies to solve.
This leads to a rich conversation when we talk about strategies & allows both groups to stretch their math skills.
Tips for Effectively Implementing Problem of the Day in the Elementary Classroom
Now that you know why Math Problem of the Day is important, you might be wondering how to implement it in your classroom. Here are a few simple tips:
• Choose a time & format that works for you. Morning work, right before lunch, or at the end of the day are all great times to implement this system. You can use paper and pencil, whiteboards, or
even have students solve problems on their devices. In other words, feel free to customize this process to fit the needs of your classroom!
• Make sure you have a good supply of word problems. I’ve shared some freebies, but there are many great sites that offer a selection of word problems that you can use. The most important thing is
to be sure the problems are flexible enough that they’ll engage and challenge your learners.
• Prior to starting it can be helpful to introduce a lesson or two on growth mindset. Initially, these problems are likely to feel challenging for most students, but with a little practice and
perseverance, they will get easier.
• Focus on process over product. In other words, use this time to focus on the development of strategies for attacking story problems. The right answer isn’t as important as having a method for
approaching the problem that can lead to success. You don’t even need to grade these problems. Instead, focus on guiding and supporting students in developing their problem-solving skills &
• Set a timer for 5-7 minutes and have students work on the problem. If you have extra time, you can give them up to ten minutes. Once the timer goes off, discuss the problem as a class. You can
ask volunteers to share their solution strategy or use a whole-class discussion format. Look for commonalities in how students solved the problem, but also emphasize different methods that led to
the same (correct) solution.
• Provide feedback to your students. If you notice any common mistakes, take a minute to point them out and explain why they are incorrect.
How Do You Get Started with Problem of the Day?
The National Council of Teachers of Mathematics (NCTM) encourages us to keep problem-solving at the forefront of our lesson planning. That’s where Problem of the Day can help to build routines around
one important aspect – word problems.
Getting started with Math Problem of the Day is easy! Simply choose one problem for your students to solve each day. You can write the problem on a whiteboard or display it on your projector.
Alternatively, you can use a printable that students keep in their math folder or glue into a math journal.
To make things even easier, I’ve got a free set of Math Problem of the Day resources that you can use to get started. Just click your grade level below to download. | {"url":"https://www.differentiatedteaching.com/teach-math-problem-of-the-day/","timestamp":"2024-11-05T22:32:51Z","content_type":"text/html","content_length":"231026","record_id":"<urn:uuid:31216132-ff1e-4af4-a36b-80977a19a03b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00620.warc.gz"} |
How Do You Solve Problems Involving Averages? - CentermathHow Do You Solve Problems Involving Averages?
How Do You Solve Problems Involving Averages?
First off, let’s clarify what an average is. It’s basically the sum of a bunch of numbers divided by how many numbers there are. For instance, if you have scores from a test—85, 90, and 78—you add
them up to get 253. Then, you divide this total by the number of scores, which is 3. Voilà! Your average score is 84.3.
Now, solving problems involving averages can get a bit more interesting when you’re dealing with different scenarios. Say you have two groups of students with different average scores and you need to
find the combined average. It’s like mixing two different paints to get a new shade. You need to account for the number of students in each group. If Group A has 10 students averaging 80, and Group B
has 20 students averaging 90, you don’t just blend the numbers; you weigh them according to the size of each group.
Here’s a neat trick: multiply each average score by the number of students in each group to find the total score for each group. Add these totals together and then divide by the total number of
students. This way, you get a fair average that represents the entire population.
Remember, averages are powerful because they simplify complex data into a single, easy-to-understand figure. But they’re also a bit sneaky because they can sometimes mask underlying details. So
always make sure to dig a little deeper and understand what the numbers are really telling you.
Mastering Averages: The Ultimate Guide to Solving Complex Problems
First off, think of averages as a way to simplify your data. When you’re swamped with numbers, be they test scores, sales figures, or anything else, averages help cut through the noise. They
consolidate a range of values into one representative number, making it easier to grasp what’s typical or expected. It’s like turning a messy pile of information into a neat, manageable summary.
Let’s break it down. You’ve got a dataset with values that vary widely—some high, some low. Calculating the average gives you a central value, which can help identify trends and patterns. For
instance, if you’re analyzing monthly sales, the average can reveal whether sales are generally growing or declining, even if individual months are unpredictable. It’s like getting the general vibe
of a party: the average helps you understand whether people are generally having a good time or not.
However, averages aren’t a one-size-fits-all solution. There are different types, like the mean, median, and mode, each suited to different scenarios. The mean is the most common, but if your data
has outliers—extremely high or low values—the median might give you a clearer picture of the typical value. It’s akin to measuring the height of a crowd: if one person is extraordinarily tall, their
height won’t skew the median, but it will affect the average.
In solving complex problems, using the right type of average can be a game-changer. It helps you distill vast amounts of information into insights that are actionable and clear. So, next time you’re
faced with a jumble of numbers, remember: mastering averages can be your key to unlocking the solution.
From Basics to Brilliance: Tackling Average Calculations with Confidence
Start with the essentials: addition, subtraction, multiplication, and division. These are your building blocks. Just as a chef needs to master the art of chopping before creating gourmet dishes, you
need to get comfortable with these core operations. Picture addition and subtraction as the bread and butter of math. They’re fundamental, yet they pave the way for more complex problems.
Now, let’s tackle multiplication and division. These are like adding spice to a dish. They transform basic ingredients into something more complex and flavorful. If you’ve ever struggled with
multiplication tables, think of them as patterns or rhythms. With a bit of practice, you’ll find these patterns become second nature.
When you start to grasp these basics, you’re no longer just solving problems—you’re building confidence. Imagine yourself as an artist, each calculation a stroke on your canvas. The more you
practice, the more intricate and impressive your calculations become. As you delve deeper, you’ll see that these once-daunting tasks become routine, and you’ll approach even the most complex
calculations with newfound ease.
So, why let average calculations slow you down? Embrace the basics, practice regularly, and watch your confidence soar. You’ll soon find that what once seemed like a maze is now a playground where
you’re the master.
Crack the Code: Expert Tips for Solving Average-Based Math Problems
First things first: understand the basics. An average, or mean, is simply the sum of all numbers divided by the count of numbers. It’s like sharing a pizza among friends. If you have eight slices and
four friends, each person gets two slices. Simple, right? So, when you’re given a set of numbers and asked to find the average, just sum them up and divide by the number of values.
Now, let’s delve deeper. Sometimes you’ll encounter problems where you need to find the average of averages. This can be tricky. Imagine you have two groups of students with different average scores,
and you need to find the overall average score. You can’t just average the averages. Instead, you need to weigh each group by its size. It’s like mixing paint; the final color depends on how much of
each color you use.
Another crucial tip is to watch out for outliers. An average can be skewed by unusually high or low numbers. If you’re working with data where one value is way off from the rest, consider using the
median or mode for a more accurate picture. Think of it like a party where one person’s behavior is way different from the rest. The average behavior might not reflect the group well.
Lastly, practice makes perfect. The more problems you solve, the more patterns you’ll recognize. It’s like learning to ride a bike—you won’t get it right the first time, but keep trying, and it’ll
soon become second nature.
Beyond the Mean: Advanced Techniques for Average-Related Challenges
First off, consider the power of data analysis. It’s not just about crunching numbers—it’s about understanding the story behind them. Use advanced statistical methods like regression analysis or
machine learning models to uncover patterns and trends that aren’t obvious at first glance. It’s like having a treasure map that guides you straight to the gold.
Next, embrace the art of predictive analytics. By leveraging historical data and advanced algorithms, you can forecast future outcomes with surprising accuracy. Imagine being able to anticipate
potential issues before they even arise—it’s like having a crystal ball for your average-related problems.
Another trick up your sleeve could be optimization techniques. Linear programming, for example, helps you find the most efficient way to achieve your goals within given constraints. Think of it as
navigating through a maze: optimization shows you the quickest path to your destination without hitting dead ends.
Finally, don’t underestimate the value of continuous learning and adaptation. Stay updated with the latest advancements in your field and be ready to tweak your strategies. Just like a surfer adjusts
to the waves, you need to adjust to the evolving landscape of your challenges.
Averages Unveiled: How to Approach and Solve Everyday Problems
Averages, in essence, are like the common ground in a heated debate. They help us understand a set of numbers by simplifying them into one representative figure. To find the average, you add up all
the numbers and then divide by the total count. Think of it like sharing a pizza among friends. If you have 12 slices and 4 friends, each person gets 3 slices. Simple, right?
But here’s where it gets interesting: not all averages are created equal. When tackling everyday problems, it’s crucial to pick the right type of average for the situation. The mean, or simple
average, is great for balanced data. However, if you’re dealing with numbers that vary widely, like income levels in a city, the median—the middle value when numbers are sorted—might give a clearer
picture. It’s like looking at the center of a crowd to gauge the general mood rather than just averaging out the highs and lows.
And let’s not forget about the mode, the most frequently occurring number in a dataset. It’s like finding out the most popular movie genre among your friends. Each average type provides a unique
lens, helping you make sense of data in ways that are both practical and insightful.
So next time you’re faced with a problem that involves numbers, remember: averages are your best friends. They might just be the key to unraveling the mysteries behind everyday challenges.
The Art of Averaging: Strategies for Effective Problem Solving
To effectively use averaging, start by gathering a range of data points or options. This variety is like having different spices at your disposal; each one adds its own unique flavor to the mix.
Next, look at the commonalities and differences among these points. This is where averaging shines—by identifying patterns and trends, you can cut through the noise and focus on what truly matters.
Another key strategy is to break down the problem into smaller, more manageable pieces. Imagine you’re piecing together a jigsaw puzzle; each small section is easier to handle than the whole picture
at once. By averaging out these smaller parts, you get a clearer overall view and can address each piece more effectively.
Also, don’t forget to apply the average in decision-making. Rather than getting bogged down by every detail, use the average to guide your choices. It’s like using a compass to find your way through
a dense forest—much simpler than trying to navigate every individual tree.
In essence, averaging is about finding balance and simplifying complexity. It’s a practical tool that helps you see the big picture, manage details more efficiently, and ultimately, solve problems
with greater ease.
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The Inner Product
The Inner Product, June 2002
Jonathan Blow (jon@number-none.com)
Last updated 12 January 2004
Scalar Quantization
Related articles:
Packing Integers (May 2002)
Transmitting Vectors (July 2002)
Arithmetic Coding, Part 1 (August 2003)
Arithmetic Coding, Part 2 (September 2003)
Adaptive Compression Across Unreliable Networks (October 2003)
Piecewise Linear Data Models (November 2003)
Last month I showed some methods of packing integers together. The purpose of that was to fit things in a small space for save-games, or to save bandwidth for online games (massively multiplayer
games spend a lot of money on bandwidth!) This month we'll extend our methods to handle floating-point values. We'll do this by carefully converting these scalars into integers, and then packing the
integers as described last month.
We want to map a continuous set of real numbers onto a set of integers, a process known as quantization. We quantize by splitting the set into a bunch of intervals, and assigning one integer value to
each interval. When it's time to decode, we map eac integer back to a scalar value that represents the interval. This scalar will not be the same as the input value, but if we're careful it will be
close. So our encoding here is lossy.
Game programmers often perform this general sort of task. If they're not being cautious and thoughtful, they'll likely do something like this:
// 'f' is a float between 0 and 1
const int NSTEPS = 64;
int encoded = (int)(f * NSTEPS);
if (encoded > NSTEPS - 1) encoded = NSTEPS - 1;
Then they decode like this:
const float STEP_SIZE = (1.0f / NSTEPS);
float decoded = encoded * STEP_SIZE;
I am going to name these two pieces of code, because I want to talk about what's wrong with them and suggest alternatives. The first piece of code multiplies the input by a scaling factor, then
truncates the fractional part of the result (the cast to int implicitly performs the truncation). I am going to call this block of code 'T', for Truncate.
The second piece of code recovers a floating-point value by scaling the encoded integer value. I will call it 'L' for Left Reconstruction -- it gives us the value of the left-hand side of each
interval (Figure 1a). Using these two steps together gives us the 'TL' method of encoding and decoding real numbers.
Why TL is bad
As shown in Figure 1a, the net result of performing the TL process on input values is to shunt them to the left-hand side of whichever interval they started in. (If you don't see this right away,
just keep playing with some example input values until you get it). This leftward motion is bad for most applications, for two main reasons. The first problem is that our values will in general shift
toward zero, which means that there is a bias toward energy loss. The second problem is that we end up wasting storage space (or bandwidth). To see why, let's look at an alternative.
I am going to replace the 'L' part of TL with a different reconstruction method, which is mostly the same except that it adds an extra half-interval size to the output value. As a result, it shunts
input values to the center of the interval they started in, as shown in Figure 1b. We'll call it 'C' for Center Reconstruction, and here is the source code:
const float STEP_SIZE = (1.0f / NSTEPS);
float decoded = (encoded + 0.5f) * STEP_SIZE;
When we use this together with the same truncation encoder as before, we get the codec TC. We can see from the diagram that TC will increase some inputs and decrease others. If we encode many random
values, the average output value converges to the average input value, meaning there is no change in total energy.
Now let's think about bandwidth. The amount of storage space we use is determined by the range of integers we reserve for encoding our real-numbered inputs (the value of NSTEPS in the code snippets
above). In order to find an appropriate value for NSTEPS, we need to choose a maximum error threshold by which our input can be perturbed.
When we use TL to store and retrieve arbitrary values, the mean displacement (the difference between the output and the input) is
where s is 1/NSTEPS. When we use TC, the mean displacement is only 1/4 s. Thus, to meet a given error target, NSTEPS has to be twice as high with TL as it does with TC. So TL needs to spend an extra
bit of information to achieve the same guaranteed accuracy that TC gets naturally.
Why TC can be bad
Though TC is a step above TL in many ways, it has a property that can be problematic: there's no way to represent zero. When you feed it zero, you get back a small positive number. If you're using
this to represent an object's speed, then whenever you save and load a stationary object, you'll find it slowly creeping across the world. That's really no good.
We can fix this by going back to left-reconstruction, but then, instead of truncating the input values downward, we round them to the nearest integer. We'll call this 'R' for rounding. Being seasoned
programmers we know that you round a nonnegative number by adding 1/2 and then truncating it. Thus the source code is:
// 'f' is a float between 0 and 1
const int NSTEPS = 64;
int result = (int)(input * (NSTEPS - 1) + 0.5f);
if (result > NSTEPS - 1) result = NSTEPS - 1;
The method RL is shown in Figure 1c. It's got the same set of output values that TL has, but it maps different inputs to those outputs. RL has the same mean error as TC, which is good. It can store
and retrieve 0 and 1; 1 is important since, if you're storing something representing 'fullness' (like health or fuel or something), you want to be able to say that the value is at 100%.
It's nice to be able to represent the endpoints of our input range, but we do pay a price for that: RL is less bandwidth-efficient than TC. Note that I changed the scaling factor from NSTEPS, as 'T'
uses, to NSTEPS - 1. This allows values near the top of the input range to be mapped to 1. If I hadn't done this, values near 1 would get mapped further downward than the other numbers in the input
range, and thus would introduce more error than we'd like. Also, RL's average error would have been higher than TC's, and it would have regained a slight tendency toward energy decrease. I avoided
this badness by permitting the half-interval near 1 to map to 1.
But this costs bandwidth. Notice that the half-intervals at the low and high ends of the input range only cover one interval's worth of space, put together. So we're wasting one interval's worth of
information, made of the pieces that lie outside the edges of our input. If an RL codec uses n different intervals, each interval will be the same size as it would be in a TC codec with n-1 pieces.
So to achieve the same error target as TC, RL needs one extra interval.
If our value of NSTEPS was already pretty high, then adding 1 to it is not a big deal; the extra cost is low. But if NSTEPS is a small number, the increment starts to be noticable. You'll want to
choose between TC and RL based on your particular situation. RL is the safest, most robust thing to use by default in cases where you don't want to think too hard.
Don't Do Both
Once, when I was on a project that wasn't thinking clearly about this stuff, we used a method RC, which both rounded and center-reconstructed. This unfortunate choice is shown as Figure 1d. It is
arguably worse than the TL that we started with, because it generally increases energy. Whereas decreasing energy tends to damp a system stably, increasing energy tends to make systems blow up. In
this particular project, we thought we were being careful about rounding; but we didn't do enough observation to see that the two rounding steps cancel each other out. Live and learn.
Interval Width
So far we've been assuming that the intervals should all be the same size. It turns out that this is optimal when your input vales all occur with equal probability. Since we're not talking about
probability modeling today, we'll just keep on assuming intervals of equal size (this approach will lend us significant clarity next month, when we deal with multidimensional quantities). But you
might imagine that, if you knew most of your values landed in one spot of the input domain, it would be better to have smaller intervals there, and larger ones elsewhere. You'd probably be right,
depending on your application's requirements.
Varying Precision
So far, we've talked about a scheme of encoding real numbers that's applicable when you want error of constant magnitude across the range of inputs. But sometimes you want to adjust the absolute
error based on the magnitude of the number. For example, the whole idea of floating-point numbers is that the absolute error rises as the number gets bigger. This is useful because often what you
care about is the ratio of magnitudes between the error and the original quantity. So with floating point, you can talk about numbers that are extremely large, as long as you can accept
proportionally rising error magnitudes.
One way to implement such varying precision would be to split up our input range into intervals that get bigger as we move toward the right. But in the interest of expediency, I am going to adopt a
more hacker-centric view here. Most computers we deal with already store numbers in IEEE floating-point formats, so a lot of work is already done for us. If we want to save those numbers into a
smaller space, we can chop the IEEE representation apart, reduce the individual components, and pack them back together into something smaller.
IEEE floating point numbers are stored in three parts: a sign bit s, an exponent e, and a mantissa m. Their layout in memory is shown in Figure 2a. The real number represented by a particular IEEE
float is -1^s m x 2^m. The main trick to remember is that the mantissa has an implicit '1' bit tacked onto its front. There are plenty of sources available for reading about IEEE floating point, so
I'll leave it at that. There are some references in For Further Information.
If we know that we're only processing nonnegative numbers, we can do away with the sign bit. The 32-bit IEEE format provides 8 bits of exponent, which is probably more than we want. And then, of
course, we can take an axe to the mantissa, lowering the precision of the number.
Our compacted exponent does not have to be a power of two, and does not even have to be symmetric about 0 like the IEEE's is. We can say "I want to store exponents from -8 to 2", and use the
Multiplication_Packer from last month to store that range. Likewise, we could chop the mantissa down to land within an arbitrary integer range. To keep things simple, though, we will restrict our
mantissa to being an integer number of bits. This will make rounding easier.
To make the mantissa smaller, we round, then truncate. To round the mantissa, we want to add .5 scaled such that it lands just below the place we are going to cut (Figure 2b). If the mantissa
consists of all '1' bits, adding our rounding factor will cause a carry that percolates to the top of the mantissa. So we need to detect this, which can easily be done if the mantissa is being stored
in its own integer (we just check the bit above the highest mantissa bit, and see if it has flipped to 1). If that bit has flipped, we increment the exponent.
Rounding the mantissa is important. Just as in the quantization case, if we don't round the mantissa, then we need to spend an extra bit worth of space to achieve the same accuracy level, and we have
net energy decrease, etc.
Sample Code
This month's sample code implements the TC and RL forms of quantization. It also contains a class called Float_Packer; you tell the Float_Packer how large a mantissa and exponent to use, and whether
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Return on Average Capital Employed | Advantages and Limitations
Updated July 25, 2023
Definition of Return on Average Capital Employed
The term “return on average capital employed” refers to the performance metric that determines how well a company can leverage its capital structure to generate profit.
To put it simply, this metric determines the dollar amount that a company is able to produce in net operating profit for each dollar of the capital (both equity and debt) utilized.
The return on average capital employed is abbreviated as ROACE. This metric is the improved version of ROCE as it takes into account the opening and closing value of the capital employed. ROACE can
be used to compare peer performance of a similar scale and with different capital structures as it compares the profitability relative to both equity and debt.
The formula for ROACE can be derived by diving the operating profit or earnings before interest and taxes (EBIT) by the difference between average total assets and average total current liabilities,
which is then expressed in terms of percentage. Mathematically, it is represented as,
ROACE = EBIT / (Average Total Assets – Average Total Current Liabilities) * 100
The formula for ROACE can also be expressed as operating profit divided by the summation of average shareholder’s equity and average long term liabilities. Mathematically, it is represented as,
ROACE = EBIT / (Average Shareholder’s Equity + Average Long Term Liabilities) * 100
Examples of Return on Average Capital Employed (With Excel Template)
Let’s take an example to understand the calculation of Return on Average Capital Employed in a better manner.
Example #1
Let us take the example of a company that is engaged in the manufacturing of mobile phone covers. During 2018, the company booked an operating profit of $22.5 million. Its total assets at the start
and end of the year were $140 million and $165 million respectively, while its corresponding total current liabilities were $100 million and $120 million respectively. Based on the given information,
calculate the ROACE of the company for the year.
Average Total Assets is calculated using the formula given below
Average Total Assets = (Total Assets at the Start of the Year + Total Assets at the End of the Year)/2
• Average Total Assets = ($140 million + $165 million) / 2
• Average Total Assets = $152.5 million
Average Current Liabilities is calculated using the formula given below
Average Current Liabilities = (Total Current Liabilities at the Start of the Year + Total Current Liabilities at the End of the Year) / 2
• Average Current Liabilities = ($100 million + $120 million) / 2
• Average Current Liabilities = $110.0 million
Return on Average Capital Employed is calculated using the formula given below
ROACE = EBIT / (Average Total Assets – Average Total Current Liabilities) * 100
• ROACE = $22.5 million / ($152.5 million – $110.0 million)
• ROACE = 52.94%
Therefore, the company’s ROACE for the year 2018 stood healthy at 52.94%.
Example #2
Let us take the example of Walmart Inc.’s annual report for the year 2018 to illustrate the computation of ROACE. During 2018, its operating income was $20.44 billion, its total assets at the start
and at the end of the year was $198.83 billion and $204.52 billion respectively and its total current liabilities at the start and at the end of the year was $66.93 billion and $78.52 billion
respectively. Calculate Walmart Inc.’s ROACE for the year 2018.
Average Total Assets is calculated using the formula given below
Average Total Assets = (Total Assets at the Start 2018 + Total Assets at the End of 2018) / 2
• Average Total Assets = ($198.83 billion + $204.52 billion) / 2
• Average Total Assets = $201.68 billion
Average Current Liabilities is calculated using the formula given below
Average Current Liabilities = (Total Current Liabilities at the Start of 2018 + Total Current Liabilities at the End of 2018) / 2
• Average Current Liabilities = ($66.93 billion + $78.52 billion) / 2
• Average Current Liabilities = $72.73 billion
Return on Average Capital Employed is calculated using the formula given below
ROACE = EBIT / (Average Total Assets – Average Total Current Liabilities) *100
• ROACE = $20.44 billion / ($201.68 billion – $72.73 billion)
• ROACE = 15.85%
Therefore, Walmart Inc.’s ROACE stood at 15.85% during the year 2018.
Source: Walmart Annual Reports (Investor Relations)
Advantages of Return on Average Capital Employed
Some of the advantages of return on average capital employed are:
• It measures the return on both equity and debt.
• It is used to compare the profitability of companies with different capital structures.
Limitations of Return on Average Capital Employed
One of the limitations of return on average capital employed is that it can be manipulated through accounting forgery, such as the classification of long-term liabilities as current liabilities.
So, ROACE is an important financial metric that helps in the evaluation of the overall profitability of a company. However, it is also risks of accounting manipulations and so it is essential that
you are cautious while analyzing companies based on ROACE.
Recommended Articles
This is a guide to Return on Average Capital Employed. Here we discuss how to calculate Return on Average Capital Employed along with practical examples. We also provide a downloadable excel
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Frezeleme işlemi sırasında iş parçasındaki sıcaklık dağılımının analizi
Fen Bilimleri Enstitüsü
Institute of Science And Technology
Talaş kaldırma işlemi sırasında üretilen ısı ve bu ısının oluşturduğu sıcaklıklar büyük önem taşımaktadır. Kesme ortamının sıcaklıklarını ölçmek için farklı deneysel yöntemler olsa bile bu
yöntemlerin doğruluğu ve doğruluğu ve güvenirliği henüz istenilen seviyeye ulaşmamıştır. Bunun yanında sıcaklık dağılımları üzerine yapılan farklı çalışmalar olsa bile bu çalışmalar birbirini
doğrulamamış ve deneysel olarak kanıtlanmamıştır. Bu sebeplerden dolayı hem kesme ortamının sıcaklığının ölçülmesine gerek kalmadan ortamdaki sıcaklıkları ve oluşan ısı akılarını belirleyebilecek
tersine ısı geçişi yöntemi popülerlik kazanmıştır. Tersine ısı geçişi yöntemi deneysel veriler ile sayısal çözümlerden alınan sonuçları karşılaştırarak ilerleyen ve ölçümü yapılamayan değişkeni elde
etmeye yaratan bir yöntemdir. Talaş kaldırma işleminde bulunmak istenilen değişken ısı akısıdır. Bu çalışmada frezeleme işlemi sırasında parçaya gelen ısı akısının ve iş parçasındaki sıcaklık
dağılımının bulunması için tersine ısı geçişi yöntemi kullanılmıştır. Tersine ısı geçişi yöntemi doğrudan problem, deneysel veriler ve tersine problem olmak üzere genel olarak üç ana bölümden oluşur.
Doğrudan problem bulunmak istenilen değişkeninin biliniyor olarak kabul edildiği problemdir. Deneysel veriler ise doğrudan problemin sonucunda ortaya çıkacak sonuçların deneysel olarak toplanmasıyla
elde edilir. Tersine problem ise doğrudan problemin çözümü ile deneysel veriler arasında karşılaştırmalı olarak ilerleyerek, bulunmak istenilen değişkenin tahmin edilmesidir. Bu çalışmada doğrudan
problem talaş kaldırma sırasında iş parçasına gelen ısı akısının bilinmesi durumunda iş parçasındaki sıcaklık dağılımının bulunmasıdır. Doğrudan problem Abaqus ve Matlab programları ile birlikte
çözülmüştür. Deneysel veriler frezeleme işlemi sırasında iş parçasından iki tane termoeleman ile toplanmıştır. Tersine problem ise deneysel veriler ile sayısal çözümndeki ilgili noktaların sıcaklık
farklarının toplamınının amaç fonksiyon olduğu bir optimizasyon problemidir. Bu optimizasyon probleminde amaç fonksiyonu en düşük yapacak olan ısı akısı aranmaktadır. Optimizasyon problemi de Matlab
tarafından çözülmüştür.
This master thesis is about heat generation phenomena and determination of work piece temperature during milling process. Heat generation in milling, more generally in machining is a highly complex
process due to complex nature of machining. Heat is generated in three different zones which are called primary zone or shear zone, secondary zone or rake face zone and lastly tool clearance face or
work surface zone. In primary zone heat is generated because of the plastic deformation of the work piece. Most of heat is generated during the machinin operation is generated in this region. In
secondary zone the heat is generated because of the fricton between tool and chip. Heat generated in primary zone generally flows to work piece and chip, on the other hand, heat is generated in
secondary zone generally flows chip and tool. Heat is generated in work surface zone is generated due to fricton between tool and work surface, usually unsharpened tools cause this heat generation
and heat flows work piece and tool. As it mentioned basically there are three different heat generation mechanism in machining. During machining heat have to be carried away from cutting medium, work
piece, tool and chip behave as heat sink. Besides that in many applications coolant liqiuds is used to carry heat from cutting medium. Temperature of cutting medium is a very important subject about
machining because of its effects on productivity, efficiency and quality of manufacturing process. However, any analytical solution or empirical formula which is verified by tests for heat generation
in milling has not been derived yet. Many analytical or numerical solutions or techniques have been developed since middle of 20th century, but still there is no verified general solution about heat
generation or formula / tehcnique about determination of cutting medium temperature. Due to this, heat generation cannot be controled by cutting parameters. Apart from these cutting medium
temperatures cannot be measured by any technique. There are many measurement technique such as tool-work piece thermocouples, embedded thermocouples, single-wire thermocouples, infared based
measurement methods, thermal cameras. Every measurment technique has its advantages and disadvantages, and all of them are used in different applications. However, these methods are not accurate or
reliable enoguh to determine exact temperature of a exact place in cutting medium. Due to the reason that there is no verified approach for heat generation and measurement techniques for cutting
medium temperature; inverse heat transfer method is employed to estimate them. Inverse heat transfer method is used for estimation an unknown such as a thermo- psychical property or a heat flux. In
this method unknown is determined iteratively between based on experimental results. In this study, it is used for estimating heat flux generated by machining process. Inverse heat transfer method is
used to estimate an unkown function or variable in different problems where direct measurement techniques cannot be applied. This method is used first time n the middle of 20th centruy, however,
because of the mathematical problems of this method it had not been used effectively since numerical and optimization methods were developed. Especially in last decade of 20th century, after computer
powers increase, inverse heat transfer method gained popularity and its applications increased. Mainly inverse heat transfer method has three different parts namely direct solution, experimental
results, and optimization. The direct solution is the solution of the direct problem which is the problem while the unknown is estimated. Generally, a direct problem is solved by numerical methods.
In this problem, the direct solution is the solution of the heat transfer problem of milling operation with random heat flux, in other words, thermal analysis of milling operation. A 100 x 100 x 2
millimeters workpiece is chosen for this study. Due to its very small thickness, the temperature gradient in the z-direction is assumed as zero. Thermal analysis of milling operation is a 3-D
transient heat transfer problem that involves a moving heat source and chip disposal process. In the milling process heat is generated by cutting of metal and due to this fact location of the tool
can be considered as the location of the heat source. For this reason, the heat source is modeled as a moving heat source and its motion is modeled based on the motion of the tool. In a real
situation heat flux applied to the workpiece might increase and decrease, but in this study heat flux is assumed as constant during the operation to simplify the model. Besides this heat source in
real is a half-circle because of the shape of the tool, however, in this study heat source is modeled as linear since the tool has high angular and linear velocity of the tool. Another important
subject of this problem is chip disposal. Due to the chip disposal process, there has to be a mass extraction from a system based on the motion of the tool or other words heat source. Top, bottom,
and side surfaces of the workpiece, there is a natural convection boundary condition and heat is convected to ambient air. Shortly, therma analysis of the workpiece is a 3 D transient heat transfer
problem with a heat flux boundary condition that has a motion, a mass extraction process, and natural convection boundary condition. To model the motion of heat source and chip disposal process, two
different software packages, which are Abaqus (which uses finite element method to solve heat transfer problems) and Matlab, are used to solve this problem. The motion of the heat source is not
modeled as continuous, instead of it, it is modeled discretely. There are different ways to model a moving heat source in a commercial code, but the chip disposal process cannot be done by
conventional methods. To model the milling process with the motion of heat source and chip disposal process, thermal analysis is divided into 125 steps. In every step, specific meshes are discarded
from the problem for chip disposal and the heat source is replaced for the motion of it. Temperature distribution of workpiece at the end of a step is used as the initial condition of the next step.
After the mesh is discarded, the heat source is relocated. Hence, the heat source does not move continuously, it moves discretely step by step. Basically, thermal analysis of the milling process is
not one analysis, it is a sum of 125 analyses. Mesh discarding process and replacement of heat source for every step is done by Matlab. Thermal analysis is done by Abaqus but the analysis is prepared
by Matlab. Meshes that need to be discarded are determined for every step, so relevant Abaqus files are rewritten for every step to prepare a new analysis, and Abaqus is executed via Matlab.
Therefore the motion of heat source and chip disposal could be modeled. The next step of the inverse heat transfer method is collecting experimental data. Experimental results are temperature data
which is collected from specific points in the workpiece during milling. Experimental data for the inverse heat transfer method is collected from the milling test. The workpiece, dimensions of which
are specified, are milled with specific cutting parameters by a CNC machine. Temperature data is collected from specific points of workpieces by two thermocouples. Thermocouples have approximately
0.1-millimeter diameter and they are located in holes that have a diameter of 1 millimeter. They are located in the middle of the workpiece in the z-direction. Also to eliminate side effects they are
located 45 millimeters and 55 millimeters to the front face. Thermocouples' distance to cutting surfaces is 2 millimeters. They should be close to the cutting surface to increase the accuracy of the
inverse heat transfer method. On the other hand, if thermocouples are located close to the cutting surface, temperature rise might be too rapid for the dynamic response of thermocouples, and
measurement errors can occur. To prevent those errors and increase accuracy, they are located to 2 millimeters from the cutting surface. The last step of the inverse heat transfer method is the
solution of the inverse problem. In this part, the results of the direct solution and experimental temperature data are compared. Heat flux value of the direct solution is altered respectively that
comparison and the heat flux makes differences between minimum is determined as the solution of the problem. In other words, this step is an optimization problem that aims to minimize objective
function which is the sum of differences between the temperature of specified points in the workpiece in a solution of the direct problem and experimental temperature data. The simplex method is used
for the optimization of the inverse heat transfer method. In this step of inverse heat transfer method Optimization Toolbox of Matlab is used. Heat flux generated by milling operation and applied to
the workpiece is obtained by the last step of the inverse heat transfer method. After estimation of heat flux, the temperature of specific points is determined to analyze the accuracy of the method.
Experimental temperature data and estimated temperature values are compared and the sum of errors for a single thermocouple varies from 3% to 5%. Also, similar heating and cooling trends are observed
in both experimental and numerical results. Therefore it can be assumed that the inverse heat transfer method is applied successfully to milling operations. During milling operation, forces that are
applied to the workpiece to the mil workpiece are measured by a force measurement system. Forces are collected for three directions. Total work to remove metal from the workpiece is calculated by
analytical methods for milling. To calculate total work done by tool, firstly shear and friction forces are calculated based on force data that are collected during the test. Then work is done by
shear and friction forces are obtained based on friction, shear forces and cutting velocity, and chip velocity. Some of those works are the total work done by the machining process. After
determination of the heat flux which is generated by the machining process and applied to the workpiece, the temperature distribution of the workpiece is obtained from the numerical solution.
Therefore the temperature of different regions of the workpiece for the whole milling operation is obtained. Also, temperature variation in time and space for specific points is determined. Effects
of a milling operation are observed and heating and cooling trends are investigated during and after the milling process. Average chip temperatures are estimated from numerical solutions of the
milling process. Temperature determination of workpiece and chip temperature during the milling process is important for milling operations. Total heat energy which flows to the workpiece and total
work done by the machining process are calculated. Therefore energy rate or in another world energy partition rate is obtained.
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2014
Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2014
Anahtar kelimeler
CNC frezeleme, CNC milling | {"url":"https://polen.itu.edu.tr/items/3211a8d5-65ca-43ec-8388-332cd64a4d40","timestamp":"2024-11-08T07:58:40Z","content_type":"text/html","content_length":"191062","record_id":"<urn:uuid:edd2f141-3306-4c64-8236-fbfa6798118a>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00320.warc.gz"} |
Rational Numbers Worksheets List | Worksheets for Rational Numbers
Rational Numbers Worksheets | Practice Worksheets on Rational Numbers
Most of you might have come across Chapter Rational Numbers while studying the subject. Usually, the word rational conveys logical interpretation followed by a reason. However, when it comes to Maths
it is derived from the word Ratio and has a different meaning entirely. To help students prepare effectively we have compiled Rational Numbers Worksheets all in one place.
You can use the Worksheets on Rational Numbers during your practice sessions and test your level of preparation. The Kind of Questions asked in the Worksheets covers various subtopics of Rational
Numbers such as Equivalent Rational Numbers, Positive and Negative Rational Numbers, Representing Rational Numbers on the Number Line, etc.
List of Rational Numbers Worksheets to Practice
For a better user experience, we have compiled all of the Worksheets for Rational Numbers in one place. Look no further and begin your practice straight away to score well in your exams. In order to
prepare a particular topic, you just need to simply tap on the quick links available to access the corresponding topic worksheet. Solve Problems on your own at first and cross-check the solutions
later in order to understand where you went wrong.
Feel free to use our Online Maths Worksheets available on our Site Worksheetsbuddy.com and ace up your preparation level. You need not worry as you can make use of the worksheets categorized to solve
problems that you are looking for.
Leave a Comment | {"url":"https://www.worksheetsbuddy.com/rational-numbers-worksheets/","timestamp":"2024-11-14T10:28:49Z","content_type":"text/html","content_length":"132654","record_id":"<urn:uuid:023fec3b-ed3b-477d-b15a-b8bdd8f02bb4>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00302.warc.gz"} |
I. Boyaci Et Al. , "Statistical modeling of beta-galactosidase inhibition during lactose hydrolysis," FOOD BIOTECHNOLOGY , vol.20, no.1, pp.79-91, 2006
Boyaci, I. Et Al. 2006. Statistical modeling of beta-galactosidase inhibition during lactose hydrolysis. FOOD BIOTECHNOLOGY , vol.20, no.1 , 79-91.
Boyaci, I., Bas, D., Dudak, F., Topcu, A., Saldamli, I., Seker, U., ... Tamerler, C.(2006). Statistical modeling of beta-galactosidase inhibition during lactose hydrolysis. FOOD BIOTECHNOLOGY ,
vol.20, no.1, 79-91.
Boyaci, IH Et Al. "Statistical modeling of beta-galactosidase inhibition during lactose hydrolysis," FOOD BIOTECHNOLOGY , vol.20, no.1, 79-91, 2006
Boyaci, IH Et Al. "Statistical modeling of beta-galactosidase inhibition during lactose hydrolysis." FOOD BIOTECHNOLOGY , vol.20, no.1, pp.79-91, 2006
Boyaci, I. Et Al. (2006) . "Statistical modeling of beta-galactosidase inhibition during lactose hydrolysis." FOOD BIOTECHNOLOGY , vol.20, no.1, pp.79-91.
@article{article, author={IH Boyaci Et Al. }, title={Statistical modeling of beta-galactosidase inhibition during lactose hydrolysis}, journal={FOOD BIOTECHNOLOGY}, year=2006, pages={79-91} } | {"url":"https://avesis.hacettepe.edu.tr/activitycitation/index/1/70f68229-fcda-4142-95fa-e0e69669b5d9","timestamp":"2024-11-11T20:07:49Z","content_type":"text/html","content_length":"11215","record_id":"<urn:uuid:9ded1220-82d9-4f81-aa35-37c80f4dad4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00535.warc.gz"} |
Research Areas
Mathematical climate research seeks to develop models of the climate system, including interactions between the atmosphere, ocean, and ecosystems, and methods for interpretation of complex
environmental data. This field is inherently interdisciplinary and leverages mathematical research in disciplines such as scientific computing, dynamical systems, and probability in collaboration
with environmental scientists. Problems range from fundamental questions about the physics of the climate system to applied questions such as the climate impacts in a given region or the deployment
of a particular climate solution.
• Mara Freilich: Network analysis, stochastic models, fluid dynamics, oceanography, ecology
Dynamical Systems and Partial Differential Equations
Research in this area focuses on nonlinear differential equations and dynamical systems that arise in the physical, social, and life sciences. Among the equations considered are finite-dimensional
dynamical systems, reaction-diffusion systems, hyperbolic conservation laws, max-plus operators and differential delay equations. Questions that are addressed for these systems include the existence
and stability of nonlinear waves and patterns, kinetic theory, phase transitions, domain coarsening, and statistical theories of turbulence, to name but a few. Even though the techniques can vary
widely from case to case, a unifying philosophy is the combination of applications and theory that is in the great Brown tradition in this area of mathematics, which is being fostered by close
collaboration among the members of the group.
Dynamical Systems and Partial Differential Equations
Pattern Theory, Statistics, Computational Neuroscience, and Computational Molecular Biology
Research in pattern theory seeks to develop models of complex systems and statistical methods and algorithms for the interpretation of high-dimensional data. Pattern theory research is also typically
interdisciplinary; it includes collaborations with computer scientists, engineers, cognitive and neural scientists, and molecular biologists. Most of pattern theory research relies on tools from
mathematical analysis, probability theory, applied and theoretical statistics, and stochastic processes. Recent applications include the development of models for computation and representation in
primate visual pathways, as well as the development of statistical methods for Bayesian non-parametrics, network analysis, the interpretation of multi-electrode neurophysiological recordings, image
processing and image analysis, and the analyses of genome-wide expression data and cellular regulatory pathways.
Probability and Stochastic Processes
The Division has long been a leader in stochastic systems theory and its applications, as well as at the forefront of current developments in probability theory, random processes and related
computational methods. Research in probability theory and stochastic processes include stochastic partial differential equations, nonlinear filtering, measure-valued processes, deterministic and
stochastic control theory, probabilistic approaches to partial differential equations, stability and the qualitative theory of stochastic dynamical systems, the theory of large deviations. Our
research endeavors also include Monte Carlo simulation, Gibbs measures and phase transitions, as well as stochastic networks. There also exists a major program in numerical methods for a variety of
stochastic dynamical systems, including Markov chain approximations and spectral methods.
Scientific Computation and Numerical Analysis
This research area is inherently multidisciplinary. It has undergone phenomenal growth in response to the successes of modern computational methods in increasing the understanding of fundamental
problems in science and engineering. The Division’s program in scientific computation and numerical analysis has kept pace with these developments and relates to most of the other research activities
in the Division. Special emphasis has been given to newly developed, high-order techniques for the solution of the linear and nonlinear partial differential equations that arise in control theory and
fluid dynamics. Numerical methods for the discontinuous problems that arise in shock wave propagation and for stochastic PDEs and uncertainty modeling are being studied. Emphasis is also being placed
on the solution of large-scale linear systems and on the use of parallel processors in linear and nonlinear problems.
CRUNCH Group
Research conducted by the CRUNCH Group focuses on the development of stochastic multiscale methods for physical and biological applications, specifically numerical algorithms, visualization methods
and parallel software for continuum and atomistic simulations in biophysics, fluid and solid mechanics, biomedical modeling and related applications. The main approach to numerical discretization is
based on spectral/hp element methods, on multi-element polynomial chaos, and on stochastic molecular dynamics (DPD). The group is directed by Prof. George Em Karniadakis. We invite you to visit both
our DPD Club and Crunch FPDE Club websites.
Visit Fractional Partial Differential Equations ARO MURI Projects
Statistical and Molecular Biology Group
The Statistical Molecular Biology Group at Brown University is led by Chip Lawrence, Professor Emeritus of Applied Mathematics. The group's research energies are focused on statistical inference in
molecular biology, genomics, and paleo-climatology, most specifically on several different high-dimensional (High-D) discrete inference problems in sequence data. | {"url":"https://appliedmath.brown.edu/research/research-areas","timestamp":"2024-11-03T10:38:25Z","content_type":"text/html","content_length":"67501","record_id":"<urn:uuid:925644cb-bed1-4c02-acc8-c653455ee219>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00122.warc.gz"} |
Baccarat Banque Rules and Scheme
Jan 06 2021
Baccarat Chemin de Fer Rules
Baccarat is enjoyed with eight decks of cards in a dealer’s shoe. Cards under ten are counted at their printed value while Ten, Jack, Queen, King are zero, and A is one. Bets are made on the ‘bank’,
the ‘player’, or for a tie (these aren’t really people; they just represent the two hands that are dealt).
Two cards are given to both the ‘house’ and ‘gambler’. The total for each hand is the sum of the 2 cards, but the 1st number is dropped. For instance, a hand of five and six has a score of one (5
plus 6 = eleven; dump the initial ‘1′).
A third card will be given out using the following rules:
- If the gambler or banker has a value of eight or 9, both players stand.
- If the gambler has less than 5, she hits. Players stays otherwise.
- If the gambler holds, the house hits on 5 or less. If the player hits, a table is used to determine if the bank stays or hits.
Baccarat Banque Odds
The better of the 2 scores wins. Winning wagers on the banker payout 19:20 (equal money less a five percent rake. The Rake is recorded and paid off once you leave the game so make sure you still have
money remaining before you head out). Winning bets on the gambler pays 1 to 1. Winning bets for tie frequently pays eight to one but occasionally 9 to 1. (This is a bad bet as a tie occurs less than
1 in every 10 rounds. Be cautious of betting on a tie. Although odds are substantially better for 9 to 1 vs. eight to one)
Wagered on correctly baccarat banque offers pretty good odds, aside from the tie wager of course.
Punto Banco Scheme
As with all games Baccarat has a few established myths. One of which is similar to a absurdity in roulette. The past isn’t a fore-teller of future outcomes. Tracking past results at a table is a bad
use of paper and an insult to the tree that surrendered its life for our stationary desires.
The most accepted and likely the most successful scheme is the one-three-two-six technique. This plan is used to maximize winnings and limit risk.
Start by placing 1 unit. If you succeed, add another to the two on the table for a grand total of three dollars on the second bet. If you win you will retain 6 on the table, take away four so you are
left with 2 on the third bet. Should you win the third bet, add two to the 4 on the table for a grand total of six on the fourth round.
If you lose on the initial wager, you take a loss of one. A win on the initial wager followed by a hit on the 2nd causes a hit of two. Wins on the first 2 with a loss on the third gives you with a
gain of 2. And success on the first 3 with a defeat on the fourth means you are even. Winning at all four rounds leaves you with 12, a profit of 10. This means you can squander the 2nd wager five
instances for every successful run of 4 wagers and still balance the books.
You must be logged in to post a comment. | {"url":"http://fastplayingaction.com/2021/01/06/baccarat-banque-rules-and-scheme-4/","timestamp":"2024-11-04T11:49:10Z","content_type":"application/xhtml+xml","content_length":"26754","record_id":"<urn:uuid:20801e9f-2035-490b-a6fc-72db0d2ed11c>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00238.warc.gz"} |
Some remarks on dualisability and endodualisability
Gouveia, M. J.; (previously Saramago, M. J.)
Algebra Universalis, 43 (2000), 197-212
The fundamental problem of dualisability and the particular problem of endodualisability are discussed. It is proved that every finite generating algebra of a quasivariety generated by a finite
dualisable algebra $\m$ is also dualisable. The corresponding result for endodualisablity is true when $\m$ is subdirectly irreducible. Under special conditions, it is also proved that a finite
algebra $\m$ is endodualisable if and only if any finite power $\m^n$ of $\m$ is endodualisable. | {"url":"https://cemat.tecnico.ulisboa.pt/document.php?project_id=7&member_id=16&doc_id=84","timestamp":"2024-11-15T00:09:45Z","content_type":"text/html","content_length":"8369","record_id":"<urn:uuid:db93caae-fd2f-402b-b953-e04257c89c31>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00799.warc.gz"} |
The Ultimate Guide to Survival Analysis - Graphpad
The Ultimate Guide to Survival Analysis
Get all of your Survival Analysis questions answered here
What is Survival Analysis?
Survival Analysis is a field of statistical tools used to assess the time until an event occurs. As the name implies, this “event” could be death (of humans with a particular disease process, crops
or plants under certain conditions, animals, etc.), but it also could be any number of alternatives (the failure of a structural beam or engineering component, the reoccurrence of a disease process,
For the rest of this article, we’ll look at a fabricated example about the survival rate of domesticated dogs on different diets.
Want to save this for later? Click here to download the eBook
What is survival analysis used for?
Survival analysis is used to describe or predict the survival (or failure) characteristics of a particular population. Often, the researcher is interested in how various treatments or predictor
variables affect survival.
Research questions range from general lifespan questions about a population, such as:
• What are the lifespan characteristics of a particular species?
• In a particular setting, such as a country, how long do people live? How does the survival rate change for different age groups such as infants, children, adults, and the elderly?
• In a manufactured product, such as a structural beam, at what load weight do over 1% or 5% of the units fail?
Survival analysis also provides tools to incorporate covariates and other predictors. Some example research questions in this case are:
• How do various factors and covariates (e.g., genetics, diet, exercise, smoking, etc.) affect lifespan?
• Of patients diagnosed with a particular form of cancer, how do various medical treatments affect lifespan, prognosis, or likelihood of remission?
• How do manufacturing processes (e.g., temperature, time, material composition, etc.) affect the failure rate of a product (such as a structural beam)?
See the different uses for Survival Analysis in Prism
What is a survival curve?
A survival curve plots the survival function, which is defined as the probability that the event of interest hasn’t occurred by (and including) each time point.
Survival curve or Kaplan-Meier curve interpretation
With our simulated data, this graph indicates that for Diet 2, after 3 years, 70% of the dogs remain, but after 4 years, only about 25% of dogs on Diet 2 survived. This is strikingly different from
Diet 1, which still has 90% surviving after 4 years.
Because the survival curves after 10 years elapsed to have a greater than 0 probability, this plot shows that some values were censored, meaning that some dogs were still alive at the conclusion of
the study. With the censored observations, we can’t know for how long they will survive.
In practice, censoring is a very common occurrence. A study is designed and funded for a particular amount of time, with the intention of observing the event of interest, but that might not be the
case. Also, dogs, in this case, might come into the study after the study has been running for seven years, so they are only observed for a maximum of three years in this case.
In the discrete case, the survival function at time t, S(t), is S(t) = probability of surviving after (not including) time t
Mathematically, the survival function is 1 - the cumulative distribution function (CDF), or:
S(t) = 1 - F(t) = 1 - Pr {T ≤ t}
This means that in the discrete case, the probability density function (PDF) is the probability of the event occurring at time t.
What is a hazard function?
Hazard functions depict the instantaneous rate of death (or failure) given that an individual has survived up to that time. They are rarely plotted on their own or estimated directly in survival
analysis. Instead, they are used behind the scenes in several prominent situations. The most common of these is comparing the ratio of hazards between, say treatment and a control group.
Additionally, the hazard function forms the backbone of the calculations and assumptions underlying the very popular Cox proportional hazards model, but even in that situation, the actual hazard
functions aren’t of much interest.
Intuitively, hazard functions give you a sense of the risk of the event occurring for an individual at a current point in time. In our demo example, we only recorded data annually, so our data are
discrete. This makes the interpretation a little more challenging. Instead of an instantaneous rate of death, we have something close to (but not exactly) an annual rate of death, which we call a
In our example, notice the hazard function for Diet 2 spikes in three locations (ages 4, 8, and 10). This reflects the fact that on the survival curve, more dogs died after 4 years elapsed than
remained after 4 years. So clearly, that was a highly hazardous year, and the estimated hazard function value of 1.3 reflects this. Similar situations occurred at years 8 and 10. Even though not
nearly as many dogs were surviving at that time, the proportion of dogs that died in years 8 and 10 was relatively large.
In the discrete case, the hazard at time t, h(t), is:
How do I choose a model for survival analysis?
The two most common survival analysis techniques are the Kaplan-Meier method and Cox proportional hazard model.
Both of these require that your data are a sample of independent observations from some “population of interest.” With our example, this means the domesticated dogs are randomly sampled and don’t
have confounding effects and relationships with other dogs in the study (such as being from the same litter, breeder, kennel, etc.).
The Kaplan-Meier method is intuitive and nonparametric and therefore requires few assumptions. However, besides a treatment variable (control, treatment 1, treatment 2, …), it cannot easily
incorporate additional variables and predictors into the model.
The Cox proportional hazard model, on the other hand, easily incorporates predictor variables, but it is more esoteric. The model has been around for decades, is tried and true, and continues to
perform well compared to other alternatives.
What is The Kaplan-Meier method?
The Kaplan-Meier method is the most intuitive model for performing a survival analysis with some added bells and whistles for statistical rigor.
With our example data about domestic dogs on two different diets, we recorded the diet and the year of death of each dog in the study. If we wanted to get an idea of survival rates and probabilities,
the most straightforward way to do that would be to just count up how many dogs on each diet died each year. We can also easily aggregate the data to calculate the number of dogs still alive at each
time point.
In a nutshell, that’s the basis of the Kaplan-Meier method. It’s called a nonparametric method because there are no distributional assumptions about the data. It’s just a fancy way of tabulating and
discussing the results.
If this sounds too simple, you are correct. This perspective oversimplifies Kaplan-Meier, but not by a lot. For example, if some observations in the study don’t experience the event of
interest before the study ends, those values need to be represented appropriately in the calculations.
Additionally, statisticians have worked out a mathematical theory that justifies the Kaplan-Meier estimate as being a reasonable choice. Although not all that important in practice (besides giving
statisticians like us a job), this provides credence for the method. For example, the Kaplan-Meier estimator for the survival curve is asymptotically unbiased, meaning that as the sample size goes to
infinity, the estimator converges on the true value.
When is the Kaplan-Meier method appropriate?
The Kaplan-Meier method is appropriate when you have a fairly simple survival analysis that doesn’t have covariates or other predictor variables. A common example is studying treatment versus control
groups. In our simulated data set for this article, we record the survival rate of dogs on two different diets, which is also appropriate here.
However, we have additional (simulated) data about the breed of dogs and their level of activity. Those are likely interesting and important confounding factors in the survival of dogs. We don’t have
a way of including them in the analysis with Kaplan-Meier, but we can with the Cox proportional hazards model below.
How do I perform a Kaplan-Meier analysis?
Analyzing Kaplan-Meier can be very simple. All that is needed is the information over time of how long the observational unit or subject was in the study, which group (e.g., treatment, control, etc.)
it was in, and whether or not the event occurred or was censored (the event didn’t occur before the end of the study).
See how Prism makes it easy to perform a Kaplan-Meier analysis.
The Kaplan-Meier Curve is an estimate for the survival curve, which is a graphical representation of the proportion of observations that have not experienced the event of interest at each time point.
What is the Cox proportional hazards model?
The industry standard for survival analysis is the Cox proportional hazards model (also called the Cox regression model). To this day, when a new survival model is proposed, researchers compare their
model to this one.
It is a robust model, meaning that it works well even if some of the model assumptions are violated. That’s a good thing because the assumptions are difficult to validate empirically, let alone
Rather than modeling the survival curve, which is the approach taken by the Kaplan-Meier method, the Cox model estimates the hazard function. In general, hazard functions are more stable and thus
easier to model than survival curves. They depict the hazard, i.e. the instantaneous rate of death (or failure) given that an individual has survived up to that time.
What is the Cox regression model?
It’s just a more ambiguous name for the Cox proportional hazards model.
What are the Cox regression model assumptions?
The prominent assumption with Cox proportional hazards model is that, not surprisingly, the hazard functions are proportional. David Cox noticed that by enforcing that “simple” constraint on the form
of the hazard model, a lot of difficult math and unstable optimization can be avoided.
This constraint (that the hazards functions are proportional) also provides an easy way to add in additional variables (covariates) to the model. With our simulated example of dogs on different
diets, we can now include the additional information of breed (Great Pyrenees, Labrador, Neapolitan Mastiff) and activity level (Low, Medium, High).
What is the Cox regression model used for?
Because of a clever constraint and the ease at which predictor variables can be added to the model, the Cox proportional hazards model can ascertain hazards and make predictions on data with multiple
predictor (covariate) variables. For example, with our simulated data, we could determine the estimated hazard or survival rate of a specific age, breed, and activity level, such as a Great Pyrenees
that’s been in the study for three years with a medium activity level.
How do I fit a Cox proportional hazard model?
To fit a Cox proportional hazard model, you need to specify the data including time elapsed, outcome (whether that observational unit died or was censored), and any other variables (covariates). In
our simulated example data, we are looking at the survival rate of dogs on two different diets, and we include Breed and Activity as additional variables.
Learn how Prism makes it easy to perform Cox Regression.
How do you write a Cox proportional hazard model?
Mathematically, the primary Cox model assumption is that the hazard function, h(t), can be written:
Where i=1pxi*i is a linear combination (a sum) of p predictor (covariate) variables times a regression coefficient. The coefficients and baseline hazard function, h0(t), are estimated using the
Another way of saying that the hazard functions are proportional is that the predictor variables’ effects on the hazard function are multiplicative. That’s a major assumption that is difficult to
Unless we include interaction terms (such as activity by breed), this assumes, in our example, that activity level has the same effect on the hazard regardless of how long the dog has been in the
study, what breed the dog is, or what diet it is on.
Interaction terms can be included, but greatly complicate interpretation, and introduce multicollinearity, which makes the estimates unstable. As with many statistical models, George Box’s quip that,
“All models are wrong but some are useful,” applies here.
The baseline hazard function, h0(t), is key to David Cox’s formulation of the hazard function because that value gets canceled out when taking a ratio of two different hazards (say for Diet 1 vs Diet
2 in our example).
How do you interpret Cox proportional hazards?
Although there are nuances, there are two main options for reporting the results of the Cox proportional hazards model: numerically or graphically.
Numerical results
The most informative part of the numerical results are in the parameter estimates (and hazard ratios). If you are familiar with linear and logistic regression, the interpretation of the numerical
results only requires a slight adjustment. The following estimates provide the guts of the information that is needed to understand how each predictor variable affects the hazard functions.
Mathematically, these parameter estimates are used to calculate the hazard function at different values (or levels) of the covariates using the equation:
The Cox model uses the data to find the maximum likelihood estimators for the regression (β) coefficients in the hazard function. Each variable in the model (in our example, these are Diet, Breed,
and Activity) has its own regression coefficient and estimate. Categorical variables in the model use reference level coding.
It’s necessary to have a baseline reference with Cox regression models because all of the interpretation is based on calculating proportional hazard functions to the baseline, h0(t).
For our example, the primary question of interest is: Do the two different diets have a significant effect on the survival of dogs? From the parameter estimates and hazard ratio, we can see they do,
and, in fact, have quite a drastic difference. In particular (regardless of breed or activity level) dogs on Diet 2 had a 4.322 times higher hazard than dogs on Diet 1, with a 95% confidence interval
of (2.720 to 6.953). Because the 95% CI does not include 1, we can also say that this coefficient is statistically significant (p<0.05).
The value we reported above is the hazard ratio, which is just e[ˆβ1] in this case.
What is a hazard ratio?
The hazard ratio is used for interpreting the results of a Cox proportional hazards model and is the multiplicative effect of a variable on the baseline hazard function. For continuous predictor
variables, this is the multiplicative effect of a 1-unit change in the predictor (e.g., if weight was a predictor and was measured in kilograms, it would be the multiplicative effect per kilogram).
For categorical variables, it is the multiplicative effect that results from that level of the predictor (e.g., Diet 2).
Graphical results
The main graphs for interpretation of the Cox regression model are the cumulative survival functions for specific values of the predictor variables.
There are a number of interesting graphics to look at with our simulated data. For example, the two plots below show the drastic differences between the survival rates of Diet 1 and Diet 2. Here we
fixed the activity level at medium and show the differences between breeds by color. Notice the much steeper decline of Diet 2, which indicates a much lower survival rate. Because there aren’t any
interaction terms in the model, these survival curves don’t cross. Our data was simulated to behave nicely, and interaction terms weren’t needed. Note that these survival rates per breed are
completely fictitious!
A second graphical example looks at the effect of diet and activity level within a single breed (Great Pyrenees). Again, this clearly shows that Diet 1 has a much higher survival rate. It also shows
that as the activity level increases, the survival rate increases. Diet 2 is so much worse than Diet 1, that even at a low activity level on Diet 1 there is a higher survival rate than a high
activity level on Diet 2.
See how to graph your Survival Analysis results in Prism.
Advantages of Cox proportional hazards model vs logistic regression
The Cox proportional hazards model and a logistic regression model are used for different purposes; they aren’t actually comparable. The Cox proportional hazards model is a tool for survival analysis
and measures the time until an event occurs. It is used to compare survival (or failure) rates across different experimental or observational variables. In our example, we look at simulated data on
the survival of domesticated dogs on two different diets. We also record information on breed and activity level.
Logistic regression, on the other hand, is a tool for predicting a binary response such as success/failure, present/absent, yes/no. Logistic regression also uses predictor variables, but it’s to
ascertain whether or not the event occurs for a specific observational unit. In its standard form, there is no element of time involved in the predictions. You could, for example, use logistic
regression to predict whether a student passes a class based on some predictor variables (previous exam scores, age, head circumference, etc.).
Perform Your Own Survival Analysis
Now it’s time to execute your own Survival Analysis according to your specific needs. Start your 30 day free trial of Prism and get access to:
• A step-by-step guide on how to perform Survival Analysis
• Sample data to save you time
• More tips on how Prism can help your research
More than a million scientists in 110 countries rely on Prism to help share their research with the world. With Prism, in a matter of minutes, you learn how to go from entering data to performing
statistical analyses and generating high-quality graphs. Start your 30 day trial today or learn more about Survival Analysis in Prism.
Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required. | {"url":"http://kgraph.org/survival-analysis.html","timestamp":"2024-11-01T22:03:30Z","content_type":"text/html","content_length":"97271","record_id":"<urn:uuid:94a2be54-b549-405a-9d8c-24fc87e3a492>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00147.warc.gz"} |
Lai-Him Chan
Professor Emeritus of Physics
Ph.D., 1966 - Harvard University
Louisiana State University
Department of Physics & Astronomy
337 Nicholson Hall, Tower Dr.
Baton Rouge, LA 70803-4001
Research Interests
Particle and Quantum Field Theory
My research interest is in quantum field theory and particle physics. I have formulated a background field functional integral method to calculate directly the effective Lagrangian in perturbation
theory, bypassing more conventional calculation of numerous individual amplitudes using Feynman diagrams.
Recently I have proposed that this method can be used to replace the unnatural regularization procedure in quantum field theory to render calculations gauge invariant and free of hard divergence by
incorporating boundary dynamics complementary to the Lagrangian dynamics. New results have been obtained on anomalous contributions in QED: the 1+1 dimensional Schwinger model, the 2+1 dimensional
Chern-Simons term and the 3+1 dimensional induced Chern-Simons term from a Lorentz and CPT violating term in the fermion QED Lagrangian.
I have also developed a generalized derivative expansion series method to express nonlocal quantum corrections of quantum field theories in the presence of classical background field as an infinite
series of local expressions. The method has been applied to calculate Casimir energies of classical configurations in various quantum field theory models. The infinite series can be continued
analytically either to recover known analytical solutions or to provide numerical solutions far more superior to conventional phase shift method.
New applications of this novel effective Lagrangian approach are continuously explored.
Current and Select Publications
• L.-H. Chan, "A Novel Method to calculate Casimir Energies of Solitons and External Fields in quantum field Theory," International conference: Sixth Workshop on QUANTUM FIELD THEORY UNDER THE
INFLUENCE OF EXTERNAL CONDITIONS, University of Oklahoma, Norman, OK USA, September 15-19, 2003., ed. K Milton, Rinton Press, p. 212 (2004).
• L.-H. Chan, "Quantum Field theory Without regularization and Hard Divergence: Short Distance Boundary Interactions and Anomalous Contributions in QED," Proceedings of the INTERNATIONAL SYMPOSIUM
ON FRONTIERS OF SCIENCE - In Celebration of the 80th Birthday of C. N. Yang (17 to 19 June 2002) Tsinghua University, Beijing, China. Ed. Hwa-Tung Nieh. World Scientific, p. 321 (2003).
• L.-H. Chan, "Induced Lorentz-Violating Chern-Simons Term in QED: Uncovering Short Distance Interaction Terms in the Effective Lagrangian without the Shadow of Regularization," in p. 231,
Proceedings of the Second Meeting on CPT and Lorentz Symmetry, August 15-18, 2001, Indiana University, Bloomington, Edited by V. Alan Kostelecky, World Scientific Publishing Co., Inc. (2002).
• L.-H. Chan, "Generalized derivative expansion and one loop corrections to the vacuum energy of static background fields," Phys. Rev. D 55, 6223 (1997).
• L.-H. Chan, "Effective-Action Expansion in Perturbation Theory," Phys. Rev. Lett. 54 1222 (1985). | {"url":"https://rurallife.lsu.edu/physics/people/faculty/chan.php","timestamp":"2024-11-03T02:34:45Z","content_type":"text/html","content_length":"31331","record_id":"<urn:uuid:353bbf51-5706-462e-8abe-42902bdb2654>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00734.warc.gz"} |
Chapter 7 Test Prep 7-7: Operations with Functions 7-7: Composition of Functions 7-8: Inverse of relations and functions Choose a section to work on. At. - ppt download
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Vector Engine and Vector control | 臺灣東芝電子零組件股份有限公司 | 台灣
The following figure is a full picture of the vector control. First, vector control begins from monitoring the waveforms of U, V, and W which drive a motor. The signals for the monitor are the
outputs a, b, and c of a motor driver. Since the amplitude of the signal of a, b, and c is very small, it is amplified with amplifier and inputted into an AD converter. After 3-phase signals a, b,
and c are changed into the digital current values Iu, Iv, and Iw by an AD converter, they are changed into current values of 2 phases Iα, and Iβ. Then, these 2-phase currents are changed into the
currents Id and Iq by the coordinate transformation from stationary coordinates to rotation coordinates. The purpose of vector control is to coincide these Id and Iq with the beforehand fixed ideal
values Idref and Iqref, respectively. Id and Iq monitored the current of the motor, and since they are the values which were converted from it, they have shifted from the ideal values. Then, in order
to lose the gap, PI control rectifies. The compensation value is given by not current but the value Vd of voltage, and Vq. Next, reverse coordinate conversion is performed to the stationary
coordinates from the rotating coordinates to obtain Vα and Vβ. Although 3-phase input signals u, v, and w of a motor driver are drawn from voltage of this 2 phase Vα, and Vβ, space vector modulation
is performed to obtain them at this time, instead of simple 2-phase 3-phase conversion. The input signals u, v, and w of a motor driver and the signals x, y, and z of those negative phase are
acquired by this conversion. And the signals of U, V, and W are given to a motor. 1 cycle of vector control was completed above. This cycle is repeated and the rotation state of an ideal is made | {"url":"https://toshiba.semicon-storage.com/tw/semiconductor/knowledge/e-learning/village/vector-1.html","timestamp":"2024-11-01T23:54:40Z","content_type":"text/html","content_length":"186624","record_id":"<urn:uuid:6446c493-13da-426b-b564-c9210420b99b>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00261.warc.gz"} |
The last post I think has a wrong answer involved but everything
I put in bold...
The last post I think has a wrong answer involved but everything I put in bold...
The last post I think has a wrong answer involved but everything I put in bold are the answers I plugged in but only received 86%
A report says that 82% of British Columbians over the age of 25 are high school graduates. A survey of randomly selected residents of a certain city included 1290 who were over the age of 25, and
1135 of them were high school graduates. Does the city's survey result provide sufficient evidence to contradict the reported value, 82%?
Part i) What is the parameter of interest?
A. The proportion of 1290 British Columbians (aged above 25) who are high school graduates.
B. All British Columbians aged above 25.
C. The proportion of all British Columbians (aged above 25) who are high school graduates.
D. Whether a British Columbian is a high school graduate.
Part ii) Let p be the population proportion of British Columbians aged above 25 who are high school graduates. What are the null and alternative hypotheses?
A. Null: p=0.82Alternative: p>0.82.
B. Null: p=0.88Alternative: p≠0.82
C. Null: p=0.88 Alternative: p≠0.88
D. Null: p=0.88 Alternative: p>0.88
E. Null: p=0.82Alternative: p=0.88
F. Null: p=0.82Alternative: p≠0.82
Part iii) The P-value is less than 0.0001. Using all the information available to you, which of the following is/are correct? (check all that apply)
A. The observed proportion of British Columbians who are high school graduates is unusually high if the reported value 82% is incorrect.
B. The observed proportion of British Columbians who are high school graduates is unusually low if the reported value 82% is correct.
C. The observed proportion of British Columbians who are high school graduates is unusually low if the reported value 82% is incorrect.
D. Assuming the reported value 82% is incorrect, it is nearly impossible that in a random sample of 1290 British Columbians aged above 25, 1135 or more are high school graduates.
E. Assuming the reported value 82% is correct, it is nearly impossible that in a random sample of 1290 British Columbians aged above 25, 1135 or more are high school graduates.
F. The reported value 82% must be false.
G. The observed proportion of British Columbians who are high school graduates is unusually high if the reported value 82% is correct.
Part iv) Based on the PP-value (less than 0.0001) obtained, at the 5% significance level, ...
A. we should not reject the null hypothesis.
B. we should reject the null hypothesis.
Part v) What is an appropriate conclusion for the hypothesis test at the 5% significance level?
A. There is sufficient evidence to contradict the reported value 82%.
B. There is insufficient evidence to contradict the reported value 82%.
C. There is a 5% probability that the reported value 82% is true.
D. Both A. and C.
E. Both B. and C.
Part vi) Which of the following scenarios describe the Type II error of the test?
A. The data suggest that reported value is correct when in fact the value is incorrect.
B. The data suggest that reported value is incorrect when in fact the value is correct.
C. The data suggest that reported value is incorrect when in fact the value is incorrect.
D. The data suggest that reported value is correct when in fact the value is correct.
Part vii) Based on the result of the hypothesis test, which of the following types of errors are we in a position of committing?
A. Type II error only.
B. Neither Type I nor Type II errors.
C. Both Type I and Type II errors.
D. Type I error only. | {"url":"https://justaaa.com/statistics-and-probability/79514-the-last-post-i-think-has-a-wrong-answer-involved","timestamp":"2024-11-08T17:38:33Z","content_type":"text/html","content_length":"50276","record_id":"<urn:uuid:d2c73e79-01c7-4b84-9745-7473b1742ab7>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00660.warc.gz"} |
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Question Text cm.) Need Help? Read It 29. [0/1 Points] DETAILS PREVIOUS ANSWERS SMITHNM13 8.2.040. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER square unit.) Need Help?
Updated On Nov 30, 2022
Topic Coordinate geometry
Subject Mathematics
Class Grade 12
Answer Type Video solution: 1
Upvotes 96
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500 Follower Problem
Let the number of odd coefficients of powers of $x$ in the expansion of
Thank you for supporting me, and helping me get 500 followers!
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Excel Formula to Group Dates into Quarters: Expert GuideExcel Formula To Group Dates Into Quarters: Expert Guide
Excel Formula to Group Dates into Quarters: Expert Guide
If you’re working with large datasets in Microsoft Excel that include date information, you may find it useful to group these dates into quarters for better analysis and reporting. Grouping dates
into quarters allows you to identify trends, compare performance across different time periods, and make informed business decisions.
In this comprehensive guide, we’ll explore an Excel formula that allows you to easily group dates into quarters. By the end of this article, you’ll have a clear understanding of how to implement this
formula in your own Excel worksheets and leverage it for effective data analysis.
Understanding Quarters in Excel
Before we dive into the formula, let’s define what we mean by “quarters” in the context of Excel. A quarter refers to a three-month period within a financial or calendar year. The four quarters are
typically defined as follows:
• Quarter 1 (Q1): January 1 to March 31
• Quarter 2 (Q2): April 1 to June 30
• Quarter 3 (Q3): July 1 to September 30
• Quarter 4 (Q4): October 1 to December 31
Excel uses a serial number system to store dates, with January 1, 1900, being the first day (serial number 1). Each subsequent day is represented by incrementing the serial number by 1. This system
makes it easy to perform calculations and manipulations on date values, such as determining the quarter in which a specific date falls.
The Excel Formula for Grouping Dates into Quarters
The Excel formula to group dates into quarters is as follows:
Let’s break down the components of this formula:
• MONTH(A1): This function extracts the month number (1-12) from the date value in cell A1.
• /3: We divide the month number by 3 to determine which quarter the date falls into. For example, if the month number is 1, 2, or 3, dividing by 3 will result in a value between 0 and 1,
indicating the first quarter. Similarly, month numbers 4, 5, and 6 will result in values between 1 and 2, indicating the second quarter, and so on.
• ROUNDUP(MONTH(A1)/3,0): The ROUNDUP function rounds up the result of the division to the nearest integer. By rounding up, we ensure that the formula returns the correct quarter number (1-4) based
on the month.
To use this formula in your Excel worksheet, follow these steps:
1. Enter your date values in a column (e.g., column A).
2. In an adjacent column (e.g., column B), enter the formula =ROUNDUP(MONTH(A1)/3,0).
3. Press Enter to calculate the quarter for the first date.
4. Drag the formula down to apply it to the remaining dates in your dataset.
Example: Grouping Dates into Quarters
Let’s look at a practical example to see how this formula works.
Date Quarter Formula Result
1/15/2023 =ROUNDUP(MONTH(A2)/3,0) 1
4/30/2023 =ROUNDUP(MONTH(A3)/3,0) 2
8/7/2023 =ROUNDUP(MONTH(A4)/3,0) 3
11/22/2023 =ROUNDUP(MONTH(A5)/3,0) 4
As you can see, the formula accurately groups each date into the corresponding quarter. The first date (1/15/2023) falls in the first quarter, so the formula returns 1. The second date (4/30/2023) is
in the second quarter, resulting in a value of 2, and so on.
Handling Leap Years
Excel automatically accounts for leap years when working with date values. The MONTH function correctly identifies the month number, even in leap years with an extra day in February. Therefore, you
don’t need to make any special adjustments to the formula for leap years. The formula will accurately group dates into quarters, regardless of whether the year is a leap year or not.
Displaying Quarter Labels
While the formula returns quarter numbers (1-4), you may prefer to display more descriptive labels like “Q1”, “Q2”, “Q3”, and “Q4”. To achieve this, you can use a nested IF function or a lookup
Using a Nested IF Function
Here’s an example of how you can modify the formula to display quarter labels using a nested IF function:
This formula checks the result of the ROUNDUP function and returns the corresponding quarter label. It uses a series of IF statements to compare the quarter number and assign the appropriate label.
If the quarter number is 1, it returns “Q1”; if it’s 2, it returns “Q2”; and so on.
Using a Lookup Table
Alternatively, you can create a lookup table that maps quarter numbers to their labels. Here’s how you can set it up:
1. Create a separate table with two columns: “Quarter Number” and “Quarter Label”.
2. Fill in the quarter numbers (1-4) and their corresponding labels (Q1-Q4).
3. Use a VLOOKUP or INDEX/MATCH function to retrieve the quarter label based on the quarter number calculated by the original formula.
Here’s an example of the lookup table:
Quarter Number Quarter Label
1 Q1
2 Q2
3 Q3
4 Q4
To retrieve the quarter label using the VLOOKUP function, you can use the following formula:
Replace “LookupTable” with the actual range of your lookup table. The VLOOKUP function looks up the quarter number in the first column of the lookup table and returns the corresponding quarter label
from the second column.
Filtering and Sorting by Quarters
Once you have the quarter information in your dataset, you can easily filter and sort your data based on quarters. This is particularly useful when you want to focus on specific quarters or compare
data across different quarters. Here’s how you can filter and sort your data:
1. Select any cell within your dataset.
2. Go to the Data tab in the Excel ribbon.
3. Click on the Filter button to enable filtering. This will add filter arrows to the header row of your dataset.
4. Click on the filter arrow in the column containing the quarter numbers or labels.
5. Select the quarters you want to view or clear the checkboxes for the quarters you want to exclude. For example, if you want to analyze data only for Q1 and Q2, you can select those quarters and
deselect Q3 and Q4.
6. To sort your data by quarters, click on the Sort A to Z or Sort Z to A button in the Data tab. This will arrange your data in ascending or descending order based on the quarter values.
Filtering and sorting your data by quarters allows you to quickly focus on the time periods that are most relevant to your analysis. You can easily compare data across different quarters, identify
trends or patterns, and make data-driven decisions.
Final Thoughts
Grouping dates into quarters in Excel is a straightforward task using the formula =ROUNDUP(MONTH(A1)/3,0). This formula extracts the month number from a date value, divides it by 3, and rounds up the
result to determine the corresponding quarter number. You can further enhance the formula to display quarter labels using a nested IF function or a lookup table. With the quarter information in your
dataset, you can easily filter and sort your data for better analysis and reporting.
Remember, the formula =ROUNDUP(MONTH(A1)/3,0) is just the starting point. You can customize it to display quarter labels, use it in combination with other Excel functions, and leverage it for
powerful data analysis. With a solid understanding of how to group dates into quarters, you’ll be well-equipped to tackle complex datasets and uncover meaningful insights in your Excel projects.
What is the Excel formula to group dates into quarters?
The Excel formula to group dates into quarters is: =ROUNDUP(MONTH(A1)/3,0), where A1 is the cell containing the date value.
How does the formula work to group dates into quarters?
The formula extracts the month number from the date using the MONTH function, divides it by 3 to determine the quarter, and then rounds up the result using the ROUNDUP function to get the final
quarter number (1-4).
Do I need to make any adjustments to the formula for leap years?
No, Excel automatically accounts for leap years when working with date values. The formula will accurately group dates into quarters without any special adjustments for leap years.
How can I display quarter labels (Q1, Q2, Q3, Q4) instead of numbers?
To display quarter labels, you can either use a nested IF function or create a lookup table. The nested IF function would be: =IF(ROUNDUP(MONTH(A1)/3,0)=1,"Q1",IF(ROUNDUP(MONTH(A1)/3,0)=2,"Q2",IF
(ROUNDUP(MONTH(A1)/3,0)=3,"Q3","Q4"))). Alternatively, you can create a separate lookup table mapping quarter numbers to labels and use a VLOOKUP or INDEX/MATCH function to retrieve the labels.
Can I filter and sort my data by quarters in Excel?
Yes, once you have the quarter information in your dataset, you can easily filter and sort your data based on quarters. Simply enable filtering in Excel, click on the filter arrow in the quarter
column, and select the quarters you want to view. You can also sort your data in ascending or descending order by clicking on the Sort A to Z or Sort Z to A button in the Data tab.
How can grouping dates into quarters be useful in Excel?
Grouping dates into quarters is particularly useful when working with large datasets spanning multiple quarters or years. It allows you to analyze trends, compare performance across different time
periods, and make data-driven decisions. Whether you’re analyzing sales data, tracking project milestones, or generating financial reports, grouping dates into quarters can provide valuable insights
and facilitate effective data analysis.
Vaishvi Desai is the founder of Excelsamurai and a passionate Excel enthusiast with years of experience in data analysis and spreadsheet management. With a mission to help others harness the power of
Excel, Vaishvi shares her expertise through concise, easy-to-follow tutorials on shortcuts, formulas, Pivot Tables, and VBA.
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pwnies_please | CTFs
Disguise these pwnies to get the flag!
note: first solve gets $100 from ian (unintended solves don't count)
author: Anusha Ghosh, Akshunna Vaishnav, ian5v, Vanilla
Soon after the challenge was released, my teammate rainbowpigeon told me to take a look at it since it was an image classification AI challenge and I have a fair bit of experience with computer
vision tasks.
I didn't have any prior experience in attacking AI models, but this turned out to be a really fun task. I ended up getting the $100 bounty for the first solve on this challenge (thanks Ian!)
I learnt a lot about how machine learning models can be vulnerable to adversarial attacks, and hopefully you can learn something from reading my writeup too!
The premise of the challenge was simple - we had to upload images to fool an image classification model, causing it to make inaccurate classifications.
Source Code Analysis
The "bouncer" is a ResNet-18 image classification model that classifies a given image as one of 10 classes ('plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck'). This is
the non-robust model that we have to fool, and we are given the model weights.
Another robust model, also using the ResNet-18 architecture, is used. This is meant to be the more accurate model, and serves as the ground truth.
# ------------------ Model goes here ⬇------------------ #
imagenet_class_index = ['plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck']
device = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")
model_nonrobust = models.resnet18()
num_ftrs = model_nonrobust.fc.in_features
model_nonrobust.fc = nn.Linear(num_ftrs, len(imagenet_class_index))
model_nonrobust.load_state_dict(torch.load("./models/pwny_cifar_eps_0.pth", map_location = device))
model_ft = model_nonrobust.to(device)
model_robust = models.resnet18()
num_ftrs = model_robust.fc.in_features
model_robust.fc = nn.Linear(num_ftrs, len(imagenet_class_index))
model_robust.load_state_dict(torch.load("./models/pwny_cifar_eps_0.5.pth", map_location = device))
model_ft = model_robust.to(device)
image_set = torchvision.datasets.CIFAR10(root='static/images', train=False, download=True)
# ------------------ Model goes here ------------------ #
The objective is to generate adversarial examples to fool the non-robust model into misclassifying the image as anything but a horse, while maintaining the "actual" class of the image so that the
robust model still classifies the image as a horse.
nonrobust = get_prediction(image_bytes=img_bytes, model = model_nonrobust, curr_image = session['img'])
robust = get_prediction(image_bytes=img_bytes, model = model_robust, curr_image = session['img'])
# robust model is the "ground truth", non-robust is the "bouncer"
# cases:
# bouncer does not want to let in horses, you want to let them in anyway
# robust says horse, non-robust says horse: you have been detected
# robust says not horse, non-robust says horse: you fail extra hard
# robust says horse, non-robust says not horse: flag
# robust says not horse, non-robust says not horse: they were let in but you didn't achieve the goal
regen_image = True
if robust != 'horse':
response = "you snuck SOMETHING into the club but it wasn't a pwny (changed too much, ground truth thinks image is a: robust {}\tnonrobust {})".format(robust, nonrobust)
session['yolo'] += 1
elif robust == 'horse' and nonrobust != 'horse':
session['level'] = session['level'] + 1
session['yolo'] = 0
response = "success! the bouncer thought your horse was a: {}".format(nonrobust)
# response = "robust = {}, nonrobust = {}".format(robust, nonrobust)
else: # robust == 'horse' and nonrobust == 'horse':
response = "bouncer saw through your disguise. bouncer: rules say \"NO HORSEPLAY\""
session['yolo'] += 1
# response += "\nrobust {}\tnonrobust {}".format(robust, nonrobust)
# this is the most common fail condition
if session['yolo'] > 3:
session['yolo'] = 0
session['level'] = 0
response = "bouncer smacks you and you pass out, start over :)"
Every time we fool the model successfully, our "level" goes up by 1. More than three consecutive failed attempts, however, will set our "level" back to 0. We need to fool the model successfully 50
times, within 5 minutes.
MIN_LEVEL = 50
SESSION_MINUTES = 5
if session['level'] >= MIN_LEVEL:
response = FLAG
Additionally, imagehash is used to compare the relative closeness of the submitted image to the original image. The goal is to make tiny changes to the original image, so that the non-robust model
misclassifies the modified image.
# Use imagehash to compare relative closeness of image (can't just allow random images to be thrown at the model...)
def get_prediction(image_bytes, model, curr_image = None):
inputs = transform_image(image_bytes=image_bytes)
outputs = model(inputs)
preds = torch.argmax(outputs, 1)
original = Image.open(io.BytesIO(base64.b64decode(curr_image)))
# "where the magic happens" - akshunna
input_image = Image.open(io.BytesIO(image_bytes))
hash_orig = imagehash.average_hash(original)
hash_input = imagehash.average_hash(input_image)
# currently HASH_DIFFERENCE is 5
# is number of bits changed in the hash
# hash is 64 bits long
# up to 5 hex digits can be different
# 16 hex digits
# 256b hash
# 0xffff ffff ffff ffff ffff ffff ffff ffff
if hash_orig - hash_input < HASH_DIFFERENCE:
return imagenet_class_index[preds]
return "IMAGE WAS TOO DIFFERENT"
Backpropagation in Machine Learning
To understand the attack, we need a bit of machine learning theory. Neural networks are loosely inspired by the human brain. Like how the human brain is made up of neurons, neural networks are made
up of nodes. These nodes span across multiple layers, starting from the input layer and ending at the output layer.
The "learning" takes place when the weights are updated, thus placing different priorities on different connections. Intuitively, these weights are what determines how much influence a particular
input feature has on the final output.
But in order for the model to learn, a backward pass, or backpropagation, must be performed. This might seem complicated, but it really isn't - it's just the chain rule!
OK, some people won't be happy with the above statement, so maybe it's a little more subtle than that. You don't need to know this, but backpropagation is a special case of a technique known as
automatic differentiation - as opposed to symbolic differentiation - which is a nice way of efficiently computing the derivative of a program with intermediate variables. I'll refer you to Justin
Domke's notes for this.
Using the chain rule, we calculate the sensitivity of the loss to each of the inputs. This is repeated (backpropagated) through each node in the network. It might help to look at this as an
optimization problem where the chain rule and memoization are used to save work calculating each of the local gradients.
Gradient-Based Attacks
The Fast Gradient Sign Method (FGSM) does this by applying a small pertubation to the original data, in the direction of increasing loss.
Intuitively, we are "nudging" the input in the "wrong direction", causing the model to make less accurate predictions.
Back to the CTF challenge! We will implement the FSGM attack for this challenge.
First, we process the image, converting it to a Pytorch tensor and normalizing it.
def get_adverserial_example(original_image):
preprocess = transforms.Compose([
[0.485, 0.456, 0.406],
[0.229, 0.224, 0.225])])
img = Image.open(original_image)
image_tensor = preprocess(img)
image_tensor = image_tensor.unsqueeze(0)
img_variable = Variable(image_tensor, requires_grad=True)
We first perform a forward pass. The model predicts the class of the original image.
output = model_nonrobust.forward(img_variable)
label_idx = torch.max(output.data, 1)[1][0]
x_pred = imagenet_class_index[label_idx]
output_probs = F.softmax(output, dim=1)
x_pred_prob = (torch.max(output_probs.data, 1)[0][0]) * 100
y_true = 7
target = Variable(torch.LongTensor([y_true]), requires_grad=False)
loss = torch.nn.CrossEntropyLoss()
loss_cal = loss(output, target)
Now that we have the gradients, we want to calculate the adversarial example as follows:
In this case, I found that eps = 0.02 worked well enough (perturbations are small enough that the two images are similar, and the loss is significant enough that the model misclassifies the results)
eps = 0.02
x_grad = torch.sign(img_variable.grad.data)
x_adversarial = img_variable.data + eps * x_grad
We can then predict on the generated adversarial image to validate our results.
output_adv = model_nonrobust.forward(Variable(x_adversarial))
x_adv_pred = imagenet_class_index[torch.max(output_adv.data, 1)[1][0]]
op_adv_probs = F.softmax(output_adv, dim=1)
adv_pred_prob = (torch.max(op_adv_probs.data, 1)[0][0]) * 100
Let's visualize our results!
def visualize(x, x_adv, x_grad, epsilon, clean_pred, adv_pred, clean_prob, adv_prob):
x = x.squeeze(0) # remove batch dimension # B X C H X W ==> C X H X W
x = x.mul(torch.FloatTensor(std).view(3,1,1)).add(torch.FloatTensor(mean).view(3,1,1)).numpy()# reverse of normalization op- "unnormalize"
x = np.transpose( x , (1,2,0)) # C X H X W ==> H X W X C
x = np.clip(x, 0, 1)
x_adv = x_adv.squeeze(0)
x_adv = x_adv.mul(torch.FloatTensor(std).view(3,1,1)).add(torch.FloatTensor(mean).view(3,1,1)).numpy()# reverse of normalization op
x_adv = np.transpose( x_adv , (1,2,0)) # C X H X W ==> H X W X C
x_adv = np.clip(x_adv, 0, 1)
x_grad = x_grad.squeeze(0).numpy()
x_grad = np.transpose(x_grad, (1,2,0))
x_grad = np.clip(x_grad, 0, 1)
figure, ax = plt.subplots(1,3, figsize=(18,8))
ax[0].set_title('Clean Example', fontsize=20)
ax[1].set_title('Perturbation', fontsize=20)
ax[2].set_title('Adversarial Example', fontsize=20)
ax[0].text(1.1,0.5, "+{}*".format(round(epsilon,3)), size=15, ha="center",
ax[0].text(0.5,-0.13, "Prediction: {}\n Probability: {}".format(clean_pred, clean_prob), size=15, ha="center",
ax[1].text(1.1,0.5, " = ", size=15, ha="center", transform=ax[1].transAxes)
ax[2].text(0.5,-0.13, "Prediction: {}\n Probability: {}".format(adv_pred, adv_prob), size=15, ha="center",
Great! After applying the perturbation, the model now thinks that the image is a dog.
Let's complete our get_adverserial_example() function by saving the adversarial example as sol.png.
x_adv = x_adversarial
x_adv = x_adv.squeeze(0)
x_adv = x_adv.mul(torch.FloatTensor(std).view(3,1,1)).add(torch.FloatTensor(mean).view(3,1,1)).numpy()#reverse of normalization op
x_adv = np.transpose( x_adv , (1,2,0)) # C X H X W ==> H X W X C
x_adv = np.clip(x_adv, 0, 1)
plt.imsave('sol.png', x_adv)
test_image = Image.open('sol.png').convert('RGB')
All that's left now is the driver code for solving the CTF!
I used Python's requests library to automate the downloading of the original images and the uploading of the adversarial examples.
def main():
s = requests.session()
r = s.get('http://pwnies-please.chal.uiuc.tf/')
print('Cookies:', r.cookies)
curr_level = 0
fail_count = 0
while curr_level < 50 and 'uiuctf' not in r.text:
print('Current Level:', curr_level)
match = re.search('<img class="show" src="data:image/png;base64,(.+)"/>', r.text)
img_data = base64.b64decode(match[1])
filename = 'original_img.png'
with open(filename, 'wb') as f:
files = {'file': open('sol.png','rb')}
r = s.post('http://pwnies-please.chal.uiuc.tf/?', files=files)
if 'success' in r.text:
print('[+] Success')
curr_level += 1
fail_count = 0
print('[-] Failure')
fail_count += 1
if fail_count > 3:
curr_level = 0
fail_count = 0
print('[+] Attack successful!')
r = s.get('http://pwnies-please.chal.uiuc.tf/')
The success rate should be quite reliable!
After 50 successful misclassifications, we get the flag.
Full Solver Script
I'm done with the CTF writeup, and at this point, I'm just writing/rambling out of passion. I've never learnt about adversarial attacks before, so this is all very new and cool to me - if you're like
me and want to know more, feel free to read on!
Why Should I Care?
This was just a CTF challenge, but there are plenty of real-life examples that highlight the severity of adversarial attacks.
For instance, these adversarial examples involve printed color stickers on road signs to fool DNN models used by self-driving cars - imagine causing an accident by simply placing a few stickers on
stop signs!
One might be tempted to use a "person detector" in a physical intrusion detection mechanism. But as this paper shows, such models can be easily fooled by the person's clothing.
Bugs or Features?
Adversarial attacks and their defences are still a very active research topic. One paper argues that "Adversarial Examples Aren't Bugs, They're Features" - in brief, the researchers showed that the
"non-robust" features imperceptible to humans might not be unnatural and meaningless, and are just as useful as perceptible "robust" ones in maximizing test-set accuracy.
When we make a small adversarial perturbation, we do not significantly affect the robust features, but flip the non-robust features. Since the model has no reason to prefer robust features over
non-robust features, these seemingly small changes have a significant impact on the resulting output. When non-robust features are removed from the training set, it was found that robust models can
be obtained with standard training.
Suppose an alien with no human concepts of "similarity". It might be confused why the original and final images should be identically classified. Remember, this alien perceives images in a completely
different way from how humans do - it would spot patterns that humans are oblivious to, yet are extremely predictive of the image's class.
It is thus argued that "adversarial examples" is a purely human phenomenon - without any context about the physical world and human-related concepts of similarity, both robust and non-robust features
should appear equally valid to a model. After all, what is "robust" and "non-robust" is purely considered from the human point of view - a model does not know to prioritize human-perceivable features
over non-human-perceivable ones.
This is a really interesting perspective - if robustness is an inherent property of the dataset itself, then the solution to achieving human-meaningful outcomes fundamentally stems from eliminating
non-robust features during training. | {"url":"https://ctf.zeyu2001.com/2021/uiuctf-2021/pwnies_please","timestamp":"2024-11-12T23:43:43Z","content_type":"text/html","content_length":"1050614","record_id":"<urn:uuid:d380743a-0bc9-4dda-83ab-0db7f04abfc2>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00055.warc.gz"} |
Sophus Lie
Marius Sophus Lie [ liː ] (born December 17, 1842 in Nordfjordeid , † February 18, 1899 in Kristiania, now Oslo ) was a Norwegian mathematician .
Lie studied from 1859 to 1865 in Christiania (later Kristiania, now Oslo) natural sciences and in 1862 heard lectures on group theory from Peter Ludwig Mejdell Sylow . In 1865 he passed the real
teacher examination and was initially undecided about his further career. It was not until 1868 that he turned to mathematics . His first mathematical publication, which appeared in 1869, earned him
a travel grant . He used this for stays a. a. in Berlin , Göttingen and Paris . The acquaintance and friendship with Felix Klein , with whom he traveled to Paris in 1870 and wrote joint works on
transformation groups, was decisive for Lie's further career . In 1872 Lie became a professor in Christiania, and in 1886 he was appointed to Leipzig as Klein's successor (who moved to Göttingen) .
Lie suffered - as was later diagnosed - from pernicious anemia , which together with difficulties in the scientific environment led to a nervous breakdown in 1889 . In addition, Lie fell out with his
colleagues Friedrich Engel and Klein over priority issues. From 1892 Norwegian personalities, above all Nansen , Björnson and Elling Holst, tried to get Lie back, on the one hand out of concern for
him, on the other hand for reasons of national patriotism. In 1894 the Norwegian parliament gave him a personal professorship with a corresponding increase in salary in Christiania. However, Lie did
not return to Norway until 1898, seriously ill , and privately taught a few of his mostly succeeded students. He died in February 1899 of pernicious anemia, which was incurable at the time .
Lie was made a Knight of the Order of Saint Olav in 1886 . In 1895 he was elected to the National Academy of Sciences and as a foreign member of the Royal Society . In 1892 he became a corresponding
member of the Académie des Sciences in Paris and in 1896 of the Russian Academy of Sciences in St. Petersburg .
His parents were Johann Lie , from 1851 pastor in Moss on the Kristianiafjord, and his wife Mette Stabell . Sophus Lie married Anna Birk (1854–1920) in 1874 ; she was the daughter of the chief
customs officer Gottfried Jörgen Stenersen Birk and his wife Marie Elisabeth Simonsen . The couple had a son Herman (1884–1960) and two daughters. Marie (* May 21, 1877) married the later
ophthalmologist Friedrich Leskien, son of August Leskien in 1905 ; together with her husband she translated works by Alexander Lange Kielland into German. Dagny Lie (born July 5, 1880, † December 28,
1945) was married to the pharmacologist Walther Straub (1874–1944).
Lie established the theory of continuous symmetry and used it to study differential equations and geometric structures. Continuous or constant symmetry operations are, for example, displacements and
rotations by any amount, including infinitesimal, in contrast to discrete symmetry operations such as mirroring. On the basis of his work, u. a. developed an algorithm for the numerical integration
of differential equations ( Lie integration ) or the method of base point transformation .
In order to investigate and apply continuous transformation groups (today called Lie groups ), he linearized the transformations and examined the infinitesimal generators. The connection properties
of the Lie group can be expressed by commutators of the generators; the commutator algebra of generators is now called Lie algebra .
Many other terms and phrases are associated with Lie's name, including a. Lie bracket , Lie's sentences , Lie theorem , Lie product formula .
His place of birth, Nordfjordeid, erected a memorial for him and named a street after him.
• Moritz Cantor : Lie, Sophus . In: Allgemeine Deutsche Biographie (ADB). Volume 51, Duncker & Humblot, Leipzig 1906, pp. 695-698.
• Nils A. Baas: Sophus Lie. In: Det Kongelige Norske Videnskabers Selskabs Forhandlinger. 1992, pp. 43–48 ( PDF file )
• Karl Strubecker : Lie, Sophus. In: New German Biography (NDB). Volume 14, Duncker & Humblot, Berlin 1985, ISBN 3-428-00195-8 , pp. 470-472 ( digitized version ).
• Arild Stubhaug : It was the boldness of my thoughts. The mathematician Sophus Lie. Springer, Berlin a. a. 2003, ISBN 3-540-43657-X .
• Gösta Mittag-Leffler : Obituary. In Acta Mathematica. 1899.
• Max Noether : Obituary. In mathematical annals. Volume 53, 1901, pp. 1-41 ( online ).
• Friedrich Engel : Obituary. In the DMV annual report. Volume 8, 1900 ( online ).
• Hans Freudenthal : Lie, Marius Sophus . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 8 : Jonathan Homer Lane - Pierre Joseph Macquer . Charles Scribner's Sons,
New York 1973, p. 323-327 .
• Bernd Fritzsche: Life and Work of Sophus Lies. A sketch. In: Seminar Sophus Lie. 2, 1992, pp. 235-261.
• Bernd Fritzsche: Sophus Lie. A sketch of his life and work. In: Journal of Lie theory. Volume 9, 1999, pp. 1-38.
• Poul Heegaard in Norsk Biografisk Leksikon. Oslo 1938.
• Sigurdur Helgason : Sophus Lie, the Mathematician. In: O. Laudal, B. Years: Proc. Sophus Lie Memorial Conference (Oslo 1992). Oslo 1994.
• Felix Klein : Sophus Lie. Evanston Colloquium 1893, Macmillan 1894, pp. 9–24 ( French translation in Nouvelle Annales de Mathematique. )
• Sophus Lie: Kjaere Ernst. 60 letters from Sophus Lie to Ernst Motzfeld. Edited by Marianne Kern and Elin Ström. Vitenskapshistorisk Skriftreihe, Nr. 4, Mathematisches Institut Oslo 1997.
• Thomas W. Hawkins : The birth of Lie's theory of groups. In: Mathematical Intelligencer. 16, 1994, No. 2, pp. 6-17.
• Thomas Hawkins: The emergence of the theory of Lie groups. Springer 2000.
• David E. Rowe : The correspondence between Sophus Lie and Felix Klein. An insight into their personal and scientific relationships, In: NTM Series History of Natural Sciences. 1988, pp. 37-47.
• Eldar Straume: Sophus Lie. In: Newsletter European Mathematical Society. No. 3, 1992.
• Isaak Moissejewitsch Jaglom : Felix Klein and Sophus Lie. Evolution of the idea of symmetry in the 19th century. Birkhäuser, 1988.
• Paul Günther : Sophus Lie. In: Herbert Beckert , Horst Schumann (Hrsg.): 100 Years of Mathematical Seminar at the Karl Marx University of Leipzig. German Science Publishers, Berlin 1981.
• Collected papers, Leipzig (Teubner), Oslo, 7 volumes, 1922 to 1960 (editors Friedrich Engel and Poul Heegaard )
• Theory of Transformation Groups, 3 volumes, Leipzig: Teubner 1888 to 1893, online
• Sophus Lie: About the influence of geometry on the development of mathematics (Leipzig inaugural lecture 1886) in: Herbert Beckert, Walter Purkert Leipzig mathematical inaugural lectures.
Selection from the years 1869-1922 , BG Teubner, Leipzig 1987 (with biography)
• G. Czichowski, Bernd Fritzsche (editor): Contributions to the theory of differential invariants (Sophus Lie, Friedrich Engel, Eduard Study ), Teubner Archive for Mathematics, Volume 17, 1993
(therein by Fritzsche: Biographical Notes on the Relationships between Sophus Lie, Friedrich Engel and Eduard Study)
• Geometry of contact transformations, Leipzig: Teubner 1896 (editor Georg Scheffers )
• Lectures on differential equations with known infinitesimal transformations, Teubner 1912 (editor Scheffers), online
• About the basics of geometry, Wissenschaftliche Buchgesellschaft Darmstadt 1967 (originally reports Abh. Kgl. Sächs. Ges. Wiss. Leipzig, Math.-Naturwiss. Klasse, Volume 42, 1890)
• About integral variants and differential equations, Vid. Selskab, Mat.-Naturv. Skrifter 1, Oslo 1902, Gutenberg project
Web links
Individual evidence
1. ^ Entry on Lie, Marius Sophus (1842 - 1899) in the archives of the Royal Society , London
2. ^ List of members since 1666: Letter L. Académie des sciences, accessed on January 13, 2020 (French).
3. ^ Foreign members of the Russian Academy of Sciences since 1724: Lie, Marius Sophus. Russian Academy of Sciences, accessed January 13, 2020 (Russian).
personal data
SURNAME Lie, Sophus
ALTERNATIVE NAMES Lie, Marius Sophus (full name)
BRIEF DESCRIPTION Norwegian mathematician
DATE OF BIRTH December 17, 1842
PLACE OF BIRTH Nordfjordeid
DATE OF DEATH February 18, 1899
Place of death Kristiania ( Oslo ) | {"url":"https://de.zxc.wiki/wiki/Sophus_Lie","timestamp":"2024-11-06T18:17:52Z","content_type":"text/html","content_length":"40274","record_id":"<urn:uuid:82ebd9eb-5a61-4ae5-b5b4-5a6be203c1db>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00897.warc.gz"} |
32. [Kernel and Range of a Linear Map, Part I] | Linear Algebra | Educator.com
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while
watching the lecture.
Section 1: Linear Equations and Matrices
Linear Systems 39:03
Matrices 30:34
Dot Product & Matrix Multiplication 41:42
Properties of Matrix Operation 43:17
Solutions of Linear Systems, Part 1 38:14
Solutions of Linear Systems, Part II 28:54
Inverse of a Matrix 40:10
Section 2: Determinants
Determinants 21:25
Cofactor Expansions 59:31
Section 3: Vectors in Rn
Vectors in the Plane 46:54
n-Vector 52:44
Linear Transformation 48:53
Linear Transformations, Part II 34:08
Lines and Planes 37:54
Section 4: Real Vector Spaces
Vector Spaces 42:19
Subspaces 43:37
Spanning Set for a Vector Space 33:15
Linear Independence 17:20
Basis & Dimension 31:20
Homogeneous Systems 24:45
Rank of a Matrix, Part I 35:03
Rank of a Matrix, Part II 29:26
Coordinates of a Vector 27:03
Change of Basis & Transition Matrices 33:47
Orthonormal Bases in n-Space 32:53
Orthogonal Complements, Part I 21:27
Orthogonal Complements, Part II 33:49
Section 5: Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors 38:11
Similar Matrices & Diagonalization 29:55
Diagonalization of Symmetric Matrices 30:14
Section 6: Linear Transformations
Linear Mappings Revisited 24:05
Kernel and Range of a Linear Map, Part I 26:38
Kernel and Range of a Linear Map, Part II 25:54
Matrix of a Linear Map 33:21
Today we are going to be talking about something called the kernel and the range of a linear map, so we talked about linear maps... we recalled some of the definitions, well, recalled the definition
of a linear map... we did a couple of examples on how to check linearity.0004
Now we are going to talk about some specific... get a little bit deeper into the structure of a linear map, so let us just jump in and see what we can do.0020
Okay. Let us start off with a definition here. Okay... a linear map L from v to w is said to be 1 to 1, if for all v1 and v2 in v, v1 not equal to v2, implies that L(v1) does not equal L(v2)...
excuse me.0029
Basically what this means is that each vector in v1 maps to a completely different element of something in w. Now, we have seen examples where... let us just take the function like x2, that you know
Well, I know that if I take 2 and I square it, I get 4. Well, if I take a different x, -2, and I square it, I also get 4. So, as it turns out, for that function, x2, the 2 and the -2, they map to the
same number... 4.0116
That is not 1 to 1. 1 to 1 means every different number maps to a completely different number, or maps to a completely different object in the arrival space.0133
So, let us draw what that means. Essentially what you have is... that is the departure space, and that is the arrival space, this is v, this is w, if I have v1, v2, v3... each one of these goes some
place different.0144
They do not go to the same place distinct, distinct, distinct, because these are distinct, that is all it is. This is just a formal way of saying it, and we call it 1 to 1... which makes sense... 1
to 1, as opposed to 2 to 1, like the x2 example.0164
Okay. An alternative definition here, if I want to, this is an implication in mathematics. This says that if this holds, that this implies this.0180
It means that if I know this, then this is true. Well, as it turns out, there is something called the contrapositive, where I... it is equivalent to saying, well, here let me write it out...0191
So, I will end up using both formulations when I do the examples. That is why I am going to give you this equivalent condition for what 1 to 1 means.0203
An equivalent condition for 1 to 1 is that L(v1) = L(v2), implies that v1 = v2.0214
This is sort of a reverse way of saying it. If I note that I have two values here, L(v1) = L(v2), I automatically know that v1 and v2 are the same thing.0234
This is our way of saying, again, that this thing... that two things do not map to one thing. Only one thing maps to one thing distinctly.0246
This one... the only reason we have two formulations of it is different problems... sometimes this formulation is easier to work with from a practical standpoint, vs. this one.0256
As far as intuition and understanding it, this first one is the one that makes sense ot me personally. Two different things mapped to two different things. That is all this is saying.0267
Okay. Let us do an example here. A couple of examples, in fact. Example... okay.0276
Let L be a mapping from R2 to R2, so this is a linear operator... be defined by L of the vector xy is equal to x + y, x - y.0285
Okay. We will let v1 be x1, y1, we will let v2 be x2, y2... we want to show... we are going to use the second formulation... L(v1) = L(v2)... implies that v1 = v2.0314
So, we are trying to show that it is 1 to 1, and we are going to use this alternate condition.0359
Let us let this be true... so L(v1) = L(v2). That means x1 + y1, x1 - y1 = L(v2), which is x2 + y2, x2 - y2... not 1.0364
Well, these are equal to each other. That means I get this equation, x1 + y1 = x2 + y2, and from the second part, these are equal, so let me draw these are equal and these are equal.0394
The way I have arranged these, if I actually just add these equations straight down, I get 2x1, is equal to 2x2, which implies that x1 = x2.0422
When I put these back, I also get, y1 = y2. This means that v1, which is x1, y1, is equal to v2.0437
So, by starting with the sub position that this is the case, I have shown that this is the case, which is precisely what this implication means. Implication means that when this is true, it implies
Well, work this out mathematically, I start with this and I follow the train of logic, and if I end up with this that means this implication is true.0465
This implication is the definition of 1 to 1, therefore yes. This map is 1 to 1. In other words, every single vector that I take, that I map, will always map to something different.0474
Okay. Let us do a second example here. Example 2. L will be R3 to R2, so it is a linear map, not a linear operator.0491
It is defined by L(x,y,z) = xy. This is our projection mapping. Okay, I will talk some random xyz, instead of variables we will actually use numbers.0511
Let us let v1 = (2,4,5), and we will let our second vector = (2,4,-7).0531
Well, not let us use v1 is not equal to v2. These two are not equal to each other.0544
However, let us see if this implies... question, does it imply that L(v1) does not equal L(v2).0551
Well, L(v1) is 2,4... if I take (2,4,5), I take the first 2... and the question... does it equal (2,4), which is the L(v2).0564
Yes. I take that one and that one, v2... (2,4), (2,4) = (2,4)... so therefore, this implication is not true.0580
I started off with 2 different vectors, yet I ended up mapping to the same vector in R2. In other words what happened was these 2 spaces, okay, I had 2 separate vectors in my departure space.0591
I had this vector (2,4), they both mapped to the same thing. That is not 1 to 1. This is 2 to 1. So, no, not 1 to 1.0604
Okay. Now, we can go ahead and go through this process to check 1 to 1, but as it turns out, we often would like simpler ways to decide whether a certain linear mapping or a certain mapping is 1 to
As it turns out, there is an easier way, so let us introduce another definition. This time I am going to do it in red. This is a profoundly important definition.0632
Let L be a mapping from v to w... you have actually seen a variant of this definition under a different name, and you will recognize it immediately when I write it down... be a linear map.0645
Okay. The kernel of L is the subset of v, the departure space, consisting of all vectors such that L of a system of all vectors v, let us actually use a vector symbol for this... all vectors v, such
that L(v) = the 0 vector in w.0665
So, the kernel of a linear map is the set of all those vector in v, that map to 0 in the arrival space.0722
Let us draw a picture of this. Very important. That is the departure space v, this is the arrival space w, if I have a series of vectors, I will just mark them as x's and I will put the 0 vector
Let us say I have 3 vectors in v that map to 0, those three vectors, that is my kernel of my linear map. It is the set of vectors, the collection of vectors that end up under the transformation
mapping to 0.0750
Null space. You should think about something called the null space. It is essentially the same thing here that we are talking about.0769
So, where are we now? Okay. So, in this particular case, this vector, this vector, this vector would be the kernel of this particular map, whatever it is, L.0775
Okay. Note that 0 in v is always in the kernel of L, right? Because a linear map, the 0 vector in the departure space maps to the 0 vector, so I know that at least 0 is in our kernel.0788
I might have more vectors in there, but at least I know the 0 is in there.0810
Okay. Let us do an example. L(x,y,z,w) = x + y, z + w, this is a mapping from R4 to R2.0816
We want all vectors in R4 that map to (0,0). Okay? We want all vectors v in R4 that equal the 0 vector.0836
In other words, we want it to equal (0,0). Okay, well, when we take a look at this thing right here, x + y = 0, z + w = 0.0854
Well, you get x = -y, z = -w, so as it turns out, all vectors of the following form, if I let w = r, and if I let y = s, something like that, well, what you get is the following.0880
So, these are my two equations so I end up with (-r,r) and (-s,s). So, here I let y = ... it looks like r, and it looks like I let w = s.0903
Yes, I let y = r, w = s, therefore z = -s, and x = -r. So, that is what you get.0928
Every vector of this form, so you might have (-1,1), (-2,2), every vector of this form is in the kernel of this particular linear map.0937
So, there is an infinite number of these. So, the kernel has an infinite number of members in here.0951
Now, come to some interesting theorems here. If the linear mapping from v to w is a linear map, then the kernel of L is a subspace.0961
So before, we said it is a subset. But it is a very special kind of subset. The kernel is actually a subspace of our departure space v. So, extraordinary.0989
Let us look at the example that we just did, we have this linear mapping, we found the kernel... the kernel is all vectors of this form... well, this is the same as r × (-1,1,0,0) + s ×
Therefore, these little triangles mean therefore, (-1,1,0,0), that vector, which what is wrong with these writings... I think I am writing too fast, I think that is what is happening here.1032
So, (-1,1,0,0) and (0 ... this is not going to work... (0,0,-1,1) is a basis for the kernel of L.1050
So here, we found the kernel, all vectors of this form, we were able to break it up into a... two sets of vectors here.1073
Well, since we discovered this theorem says that it is not only a subset, it is actually a subspace... well, subspaces have bases, right?1083
Well, this actually is a basis for the kernel and the dimension of the kernel here is dimension 2, because I have 2 vectors in my basis. That is the whole idea of dimension.1090
Now, let us see what else we have got. If a linear map, which maps from RN to RM is linear.1106
And if it is defined by matrix multiplication, then, the kernel of L is just the null space.1129
So if I have a linear map, where I am saying that the mapping if I have some vector... that I take that vector and I multiply it by a matrix on the left, well, the kernel of that linear map is all of
the vectors which map to 0.1151
So, if the kernel is just the null space of that. I mean, this is the whole definition, it is this homogeneous system... a, the matrix a, times x is equal to 0.1165
The theorem says a linear mapping is 1 to 1 if and only if the kernel of L is equal to the 0 vector... let me redo this last part... if and only if the kernel of L equals the 0 vector in v.1183
If the only vector in my departure space that maps to 0 in the arrival space is the 0 vector, that tells me that - excuse me - that the linear map is 1 to 1. That means that every element v in the
departure space maps to a different element v.1214
All I need to do is make sure that it has a 0 vector... is the only vector in the kernel.1235
In other words, it is of dimension 0. Okay. We have got a corollary to that.1242
Actually, you know, the corollary is not all together that... it is important but we will deal with it again, so I do not really want to mention it here. I have changed my mind.1262
Now, let me introduce our last definition before we close it out.1271
If L from v to w is linear, if the mapping is linear, then the range of L is the set of all vectors in w that are images under L of vectors in v.1280
Okay, let us just show what that means. This is our departure space, our arrival space, this is w, this is v. Let us say I have v1, v2, v3, v4, and v5.1342
Let us say v1 maps to 21, let us say v2 also maps to w1, let us say v3 maps to w2, and let us say v4 maps to w3, and v5 maps to w3.1360
The range is w1, w2, w3. It is all of the vectors in w that come from some vector in v, under L.1380
Now, that does not mean that every single vector... we will talk more about this actually next lesson, where I will introduce the distinction between into and onto.1394
So, this is not saying that every single vector in w is the image of some vector that is mapped under L.1408
It says that all of the vectors in w that actually come from some vector in v, that is the range. So, the range is a subset of w.1417
You are going to see in a second, my last theorem before we closed out this lesson, it is the range is actually a subspace of w.1428
So, again, the range is exactly what you have known it to be all of these years.1437
Normally, we speak of the domain and the range, we speak about the whole space. That is not the case here.1444
The range is only those things in the arrival space that are actually represented, mapped, from some vector in v.1450
It is not all of the space, the arrival space could be all of the arrival space, but it is not necessarily that way.1457
Okay. So, let us do something like, actually let me do another picture just for the hell of it, so that you see.1465
So, we might have... so this is v... and this is w... so the kernel might be some small little subset of that, that is a subset of v, also happens to be a subspace.1477
Well the range might be, some subset of w. All of these vectors in here come from some vector in here.1490
Okay, so it is not the entire space, and it is also a subspace. Okay. That is going to be our final theorem before we close out this lesson.1501
If L, which maps v to w, the vector spaces, is linear, then range of L is a subspace... subspace of w.1513
So, the kernel is a subspace of the departure space, the range is a subspace of the arrival space.1536
We are going to close it out here, but I do want to say a couple of words before we actually go to the next lesson where we are going to talk about some relationships between the kernel and the
I am going to ask you to recall something that we discussed called the rank nullity theorem. We said that the rank of a matrix + the dimension of the null space, which we called the nullity is equal
to the dimension of the column space, which is n.1555
Recall that, and in the next lesson we are going to talk about the dimension of the kernel, the dimension of the range space, and the dimension of the departure space.1574
It is really extraordinarily beautiful relationship that exists. Certainly one of the prettiest that I personally have ever seen.1585
So, with that, thank you for joining us here at educator.com, we will see you next time.1592
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You know what’s stuck on my mind? Ever since writing my last post, it’s been the word “better.” It came up when we were talking about overload resolution and implicit conversion sequences. I
explained a necessary special case of it—something about how adding const in a reference-binding is preferred against—and then strategically shut up about the rest.
void run(int (**f)()); // #1
void run(int (*const *f)() noexcept); // #2
int foo() noexcept;
int (*p)() noexcept = &foo;
run(&p); // ???
But it’s so tantalizing, isn’t it? Which one will it choose? How can we reason about this? I can see it in your eyes, sore but eager. You yearn for conversion. Well, I wasn’t going to— I— well…
Alright, since you’re so insistent. Just for you. Shall we?
∗ ∗ ∗
Let’s start small and work our way up. An implicit conversion sequence is a standard conversion sequence, possibly followed by a user-defined conversion and another standard conversion sequence in
the case of a class type.^1 A user-defined conversion is something like T::operator S(), which defines how to convert a T into an S. These are easy: they work exactly how we tell them to. So, it
evidently suffices to understand standard conversion sequences.
Definition 1
A standard conversion sequence is a sequence of zero or one conversions from each of the following categories, in order:
1. Lvalue-to-rvalue, array-to-pointer, or function-to-pointer conversions:
☆ Lvalue-to-rvalue: converts a glvalue of non-function, non-array type to a prvalue. Not particularly relevant to overload resolution, and kind of sophisticated, so we’ll mostly forget
about this.
☆ Array-to-pointer: converts an expression of type “array of $N$ T” or “array of unknown bound of T” to a prvalue of type “pointer to T,” applying temporary materialization conversion if
the expression was a prvalue (note that GCC has a bug and won’t do this; temporary materialization is defined later).
☆ Function-to-pointer: converts an lvalue function of type T to a prvalue of type “pointer to T.”
2. Integral/floating-point/boolean/pointer/pointer-to-member conversions and promotions:
☆ There are a bunch of rules for converting between various integral and floating-point types that are necessary but, frankly, menial and uninteresting, so we’ll omit these too. The pointer
/pointer-to-member conversions are probably things you already know.
3. Function pointer conversion: converts a prvalue of type “pointer to noexcept function” to a prvalue of type “pointer to function.”
4. Qualification conversion: unifies constness of two types somehow. Oh boy. It can’t be that bad, right? Right?
Surprise! This post is actually about qualification conversions
OK— OK. Uh. Hear me out.
In C++, const and volatile are often called cv-qualifiers, so called because they qualify types to form cv-qualified types. The cv-qualified versions of a cv-unqualified type T are const T, volatile
T, and const volatile T. We could also consider types T which have cv-qualifiers nested inside—for example, const int** const (“const pointer to pointer to const int”) could be written alternatively
as X in the following series of type aliases:
using U = const int;
using V = U*;
using W = V*;
using X = const W;
Now, a mathematically inclined reader may choose to write “const pointer to pointer to const int” as
where $cv_0=\{\mathtt{const}\}$, $cv_1=\emptyset$, $cv_2=\{\mathtt{const}\}$, $P_0=P_1=\text{``pointer to''}$, and $\mathtt{U}=\mathtt{int}$. More generally, we could write any type $\mathtt{T}$ (not
necessarily uniquely) as
for some $n\ge 0$ and some type $\mathtt{U}$; each $P_i$ is either “pointer to,” “array of $N_i$,” or “array of unknown size of.” For simplicity, let’s assume each $P_i$ will always be “pointer to.”
Notice that, for determining whether one type can be qualification-converted into another type (e.g., trying to convert int* to const int*), we can always drop $cv_0$ from consideration altogether—in
particular, at the top level, we can always initialize a const T from a T and vice versa, and likewise we can always convert from one to the other. So, let’s forget about $cv_0$.
Since we don’t care as much about any of the $P_i$ or $\mathtt{U}$—these are the “non-const-y” parts, and we’ll deal with them separately—let’s write this even more compactly as the $n$-tuple $
(cv_1,cv_2,\ldots,cv_n)$. The longest possible such tuple is called cv-qualification signature of $\mathtt{T}$.
We’re almost there. I’m trying really hard to make the C++ standard more palatable here, so bear with me. Two types $\mathtt{T1}$ and $\mathtt{T2}$ are called similar if they have cv-decompositions
of equal size such that each of their respective $P_i$’s are either (1) the same, or (2) one is “array of $N_i$” and the other is “array of unknown size of”; and, moreover, their $\mathtt{U}$’s
should agree. Basically, if the “not-const-y” parts of their cv-decompositions mostly agree, they’re called “similar.”
OK. It’s time. I’m only barely paraphrasing the standard because it’s all I can do at this point—it’s honestly worded pretty tightly. Let $\mathtt{T1}$ and $\mathtt{T2}$ be two types. Then, their
cv-combined type $\mathtt{T3}$, if it exists, is a type similar to $\mathtt{T1}$ such that, for each $i>0$:
• $cv_i^3=cv_i^1\cup cv_i^2$ | {"url":"https://consteval.ca/","timestamp":"2024-11-12T04:18:50Z","content_type":"text/html","content_length":"1049395","record_id":"<urn:uuid:c129aa0f-1177-4226-a5ff-aaf8387c6d1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00298.warc.gz"} |