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Logic | CheckFirst
What is logic?
Logic is the brain of your checker, where you make calculations and filter a user's answers to your questions to determine their eligibility.
How to use logic
Conceptually, there are 2 ways to use logic blocks, as intermediate logic or displayed logic.
Intermediate logic blocks are outcomes that you don’t want users to see, they are used as input/calculation for other logic blocks. Hidelogic blocks to remove them from the results section.
Displayed logic are logic blocks that you want to show to the user as a final outcome. Showthe logic blocks that you want to display in the results section.
See the example below of how the blocks look like in the logic tab vs in preview.
Using questions as input
Each question gives us an input that can be used in the logic tab in various ways.
Logic types
CheckFirst currently offers these types of logic blocks:
Below shows the acceptable inputs and the expected outcome for each logic block.
Referencing blocks
Logic blocks are tied to a symbolic letter for its reference, as well as a number. In a logic block, you can refer to other logic blocks.
See question references and constant table references.
Type in the block's letter and number. For example: N1, where N is the question type and 1 is the number of the block.
Pro tip: Type in the @ key to get a dropdown of questions, tables, or result blocks to choose from. Filter the dropdown by typing in a letter. | {"url":"https://guide.checkfirst.gov.sg/features/logic","timestamp":"2024-11-02T11:22:50Z","content_type":"text/html","content_length":"472634","record_id":"<urn:uuid:5a1172a6-c269-4497-be61-52ba0d3384c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00199.warc.gz"} |
Calculus 1 – Everything You Need
The AskDrCallahan calculus video course is the perfect tool for homeschool, public, private, engineering academy, STEM, or college students to learn the fundamentals of Calculus 1.
Required to take the course and included in this package
Calculus Teaching Videos: Videos of course content - Approximately 6.5 hours of video following the textbook. Includes the tests, the test grading guide, and the syllabus.
Calculus Textbook (USED): This calculus textbook by James Stewart is used in many universities for Calculus 1-3 courses. We will be covering Chapters 1-4 in this course, which are the normal
sections for Calculus 1 at university. The textbook is required for the course. ISBN: 0534377181. We offer this book used due to the high price of new college textbooks.
Solutions Manual
Student Solutions Manual for Stewart's Calculus (USED): Contains answers to the odd problems in the textbook. We are using the 2nd ed by Cole - USED. ISBN: 0534379230. We offer this book used due to
the high price of new college textbooks.
Teacher's Guide, Syllabus and Tests
Teacher's Guide: Our Teacher's Guide is available as a downloadable PDF, click here, and also included inside the online course with the purchase of our Calculus Teaching Videos listed above.
Sample Course Video
Additional information
Weight 8 lbs
Dimensions 14 × 10 × 8 in | {"url":"https://askdrcallahan.com/product/calculus-everything-you-need/","timestamp":"2024-11-02T15:29:03Z","content_type":"text/html","content_length":"352112","record_id":"<urn:uuid:fb4c8b74-1de7-4dae-8609-0dc3523ac99e>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00185.warc.gz"} |
How to Generate Array Tensor In Tensorflow?
To generate an array tensor in TensorFlow, you can use the tf.constant() function. This function allows you to create a tensor from a constant value or an array of values. For example, you can create
a 2D array tensor by passing a nested list of values as an argument to the function. Additionally, you can use functions like tf.zeros(), tf.ones(), or tf.random.normal() to create tensors filled
with zeros, ones, or random values, respectively. These functions make it easy to generate array tensors for use in deep learning models and other machine learning tasks.
What is the concept of tensor manipulation in relation to array tensors in TensorFlow?
Tensor manipulation in relation to array tensors in TensorFlow refers to the process of performing various operations and transformations on tensors, which are multi-dimensional arrays used for
representing data in TensorFlow. This concept allows users to manipulate the shape, size, and values of tensors in order to perform computations and build machine learning models more efficiently.
Some common tensor manipulation operations in TensorFlow include reshaping, slicing, joining, splitting, transposing, and concatenating tensors. These operations can be used to preprocess data,
transform input features, and prepare data for training and inference in machine learning models.
Overall, tensor manipulation is an essential concept in TensorFlow that allows users to manipulate and transform tensor arrays to perform complex calculations, build models, and extract meaningful
insights from data.
How to slice an array tensor in TensorFlow?
To slice an array tensor in TensorFlow, you can use the tf.slice() function. The tf.slice() function takes three arguments: the input tensor, the start index of the slice, and the size of the slice.
Here's an example of how to slice an array tensor in TensorFlow:
1 import tensorflow as tf
3 # Create a tensor with shape [3, 2]
4 tensor = tf.constant([[1, 2], [3, 4], [5, 6]])
6 # Slice the tensor to get the first row
7 slice_tensor = tf.slice(tensor, [0, 0], [1, 2])
9 # Create a TensorFlow session
10 with tf.Session() as sess:
11 result = sess.run(slice_tensor)
12 print(result)
In this example, we first create a tensor with shape [3, 2]. We then use the tf.slice() function to slice the tensor and get the first row. The start index [0, 0] indicates the starting position of
the slice, and the size [1, 2] indicates the shape of the slice.
When we run this code, the output will be:
This shows that we successfully sliced the tensor to get the first row.
What is the function of the reduce_sum operation in array tensors in TensorFlow?
The reduce_sum operation is used in TensorFlow to compute the sum of elements across a specified axis of a tensor. It takes the input tensor and reduces it to a single scalar value by summing up all
the elements along the specified axis. This can be useful for calculating the total sum of values in an array or for aggregating values along a specific dimension of a tensor.
What is the difference between a sparse tensor and a dense tensor in TensorFlow?
In TensorFlow, a sparse tensor represents a tensor where only a subset of elements contain actual values, while the remaining elements are assumed to be zero. This is typically used to save memory
and computation when dealing with large, mostly empty tensors. Sparse tensors store values and indices separately to efficiently represent the sparse data.
On the other hand, a dense tensor in TensorFlow represents a tensor where every element within the tensor contains a value, whether it be a scalar, vector, or higher-dimensional data structure. Dense
tensors are the default type of tensor used in most TensorFlow operations and are more memory-intensive compared to sparse tensors.
In summary, the key difference between a sparse tensor and a dense tensor in TensorFlow is that a sparse tensor only stores non-zero values and their corresponding indices, while a dense tensor
stores values for every element within the tensor. | {"url":"https://stock-market.uk.to/blog/how-to-generate-array-tensor-in-tensorflow","timestamp":"2024-11-10T20:55:58Z","content_type":"text/html","content_length":"145534","record_id":"<urn:uuid:83ec388d-4896-4c4f-bc2f-edcdf6b8a816>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00824.warc.gz"} |
Multiplication Chart 1-1000 Printable
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Multiplication Table 11000 Chart Roman Numerals
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Mixed-sensitivity modeling · JuliaSimControl
This example will demonstrate how you can utilize the mixed-sensitivity $\mathcal{H}_2$ design methodology to augment a plant model and achieve effective disturbance rejection using an MPC
controller. For simplicity, we will consider a simple first-order system $G$
\[\begin{aligned} \dot{x} &= -ax + b(u + d) \\ y &= cx \end{aligned}\]
where a load disturbance $d$ is acting on the input of the system. This is a simple and very common model for load disturbances. In this example, we will let $d$ be a unit step at time $t=10$.
We will begin by setting up the MPC problem and solve it without any disturbance model. For details regarding the setup of an MPC problem, see, the MPC documentation.
We start by defining the process model and discretize it using zero-order hold.
using JuliaSimControl, JuliaSimControl.MPC, Plots, LinearAlgebra
Ts = 1 # Sample time
disc(G) = c2d(ss(G), Ts)
G = tf(1, [10, 1]) |> disc # Process model
StateSpace{Discrete{Int64}, Float64}
A =
B =
C =
D =
Sample Time: 1 (seconds)
Discrete-time state-space model
This gave us a discrete-time statespace model that we can use to construct the MPC controller. The next step is to define the controller, we define the prediction horizon $N$ and the initial
condition $x_0$. We also define the reference state $r$ and the control limits $u_{\min}, u_{\max}$ using an object of type MPCConstraints. To solve the problem, we will use the OSQP solver, which is
a quadratic programming solver that is well suited for MPC problems. To estimate the state of the system, which is linear, we use a KalmanFilter. The plant model G and the Kalman filter are combined
into a LinearMPCModel object that is used to construct the MPC problem.
nx = G.nx
nu = G.nu
ny = G.ny
N = 10 # Prediction horizon
x0 = zeros(G.nx) # Initial condition
r = zeros(nx) # reference state
# Control limits
umin = -1.1 * ones(nu)
umax = 1.1 * ones(nu)
constraints = MPCConstraints(; umin, umax)
solver = OSQPSolver(
verbose = false,
eps_rel = 1e-10,
max_iter = 15000,
check_termination = 5,
polish = true,
Q1 = 100spdiagm(ones(G.nx)) # state cost matrix
Q2 = 0.01spdiagm(ones(nu)) # control cost matrix
kf = KalmanFilter(ssdata(G)..., 0.001I(nx), I(ny))
model = LinearMPCModel(G, kf; constraints, x0)
prob = LQMPCProblem(model; Q1, Q2, N, r, solver)
disturbance = (u, t) -> 1#t * Ts ≥ 10 # This is our load disturbance
hist = MPC.solve(prob; x0, T = 100, verbose = false, disturbance, noise = 0)
plot(hist, ploty = true)
As we can see, our initial controller appears to do very little to suppress the disturbance. The problem is that the observer (Kalman filter) does not have a model for such a disturbance, and its
estimate of the state will thus be severely biased.
The next step is to design the performance weights, the function hinfpartition is helpful in creating a plant model that contains all the necessary performance outputs. We select the weights $W_U$
and $W_S$ in order to minimize the norm
$$\[\begin{Vmatrix} W_S S \\ W_U CS \end{Vmatrix}_2\]$$
where $S$ is the sensitivity function and $C$ is the controller transfer function. The function hinfpartition forms a system $P$ such that $\operatorname{lft}_l(P, C)$ is the transfer function we ar
minimizing the norm of.
WS = makeweight(1000, (.03, 5), 1)*tf(1,[0.1, 1]) |> disc
WU = 0.01makeweight(1e-4, 1, 10) |> disc
Gd = hinfpartition(G, WS, WU, [])
lqg = LQGProblem(Gd)
Already at this stage, it's a good idea to verify the closed-loop properties of the system, we do this by plotting the relevant sensitivity functions.
S,_,CS,T = RobustAndOptimalControl.gangoffour(lqg)
specificationplot([S,CS,T], [WS,WU,[]], wint=(-5, log10(pi/Ts)))
In the "specification plot" we see the achieved sensitivity functions by the designed controller as well as the inverse of the weighting functions. We may also use the function gangoffourplot to show
each sensitivity function in a separate pane together with relevant peak values:
w = exp10.(LinRange(-3, log10(pi / Ts), 200))
gangoffourplot(lqg, w, lab = "", legend = :bottomright)
We see that the design appears to be robust with low peaks in the sensitivity functions and high-frequency roll-off limiting the noise gain at high frequencies.
We may now extract the cost matrices $Q_1, Q_2$ for the MPC problem and the feedback gain for the Kalman filter from the lqg object and form the MPC problem:
(; Q1,Q2) = lqg
K = kalman(lqg) # Kalman gain
Gs = -system_mapping(Gd, identity) # The - is due to the sign convention in hinfpartition
nx = Gs.nx
x0 = zeros(nx)
kf = FixedGainObserver(Gs, x0, -K)
r = zeros(nx)
model = LinearMPCModel(Gs, kf; constraints, x0)
prob = LQMPCProblem(model; Q1, Q2, N, r, solver)
When we simulate, we provide the actual dynamics G as well as Cz_actual that indicates that we measure actual performance in terms of the original output of G only (this is the first state in the
augmented plant Gd).
x0 = zeros(G.nx)
@time hist = MPC.solve(prob; x0, T = 100, verbose = false, disturbance, noise = 0, dyn_actual=G, Cz_actual = [G.C; 0; 0; 0])
plot(hist, ploty = true)
This time around we see that the controller indeed rejects the disturbance and the control signal settles on -1 which is exactly what's required to counteract the load disturbance of +1.
The astute reader might have noticed that we did not use a KalmanFilter as the observer when we used mixed-sensitivity tuning of the controller. The KalmanFilter type does not support the cross-term
between dynamics and measurement noise, but this term is required in order for the LQG problem to be equivalent to the $\mathcal{H}_2$ problem. Hence, we calculate the infinite-horizon Kalman gain
using a Riccati solver that supports the cross term and use the fixed-gain observer instead. The cross-term is available as
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[Solved] The following distribution shows the daily pocket allo... | Filo
The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is . Find the missing frequency .
Daily pocket allowance (in )
Number of children
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Lets take assumed mean, as
Class interval
Now, we have and
We know that,
Hence, the value of missing frequency,
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Question Text The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is . Find the missing frequency .
Topic Statistics
Subject Mathematics
Class Class 10
Answer Type Text solution:1
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How Many Ounces in a Gallon
This post may contain affiliate links.
Here’s everything you need to know about how to convert ounces to gallons (oz to gal or gal to oz). How many ounces in a gallon (how many oz in a gallon) and how many fluid ounces in a gallon? What
is an ounce and what is a gallon? I’ve also provided you with an easy conversion chart below to help you memorize these conversions.
Being successful in the kitchen requires the accurate measurement of ingredients. When following recipes to make a meal, you might be able to eyeball the amount of herbs and spices and other simple
ingredients, but if you add too much or too little of an ingredient, the outcome of your recipe might not be what you had hoped for. Knowing how many oz in a gallon is a helpful liquid measurement
conversion to have memorized for achieving success in the kitchen.
How Many Ounces Are In A Gallon (How Many Oz In A Gallon)
The answer to this will depend on whether you’re measuring wet or dry ingredients and the system (US Customary or Imperial) used to measure.
In the United States 1 dry gallon equals 148.94545 ounces, 1 liquid gallon equals 128 fluid ounces and 1/2 liquid gallon equals 64 fluid ounces.
How Many Fluid Ounces in a Gallon
It’s helpful to know some simple kitchen conversions when you’re needing to adjust a recipe whether its to double it or cut the amount in half. Knowing these simple conversions will help you adjust
measurements in a recipe so you can achieve the desired serving size.
In the United States 128 fluid ounces equals 1 gallon
1 gallon = 128 fluid ounces
3/4 gallon = 96 fluid ounces
1/2 gallon = 64 fluid ounces
1/4 gallon = 32 fluid ounces
In the United Kingdom 160 fluid ounces equals 1 gallon
1 gallon = 160 fluid ounces = 4.732 liters
Liquid Measurement Chart
These are some basic U.S. liquid measurement conversions that are good know, especially if you spend a lot of time in the kitchen.
How To Covert Ounces To Gallons
Want to know How Many Oz in a Gallon? Here’s an easy conversion formula.
Take the ounce value and divide it by 128 [128 = the number of ounces in 1 gallon]
Ounce value ÷ 128 = # of gallons
Example – US Customary
64 oz ÷ 128 = 0.5 [64 oz ÷ 128 = 1/2 gallon]
32 oz ÷ 128 = 0.25 [32 oz ÷ 128 = 1/4 gallon]
16 oz ÷ 128 = 0.125 [16 oz ÷ 128 = 1/8 gallon]
8 oz ÷ 128 = 0.0625 [8 oz ÷ 128 = 1/16 gallon]
What is an Ounce
Ounce is used to measure the “weight” of something (like flour and sugar). The word “ounce” comes from the Latin word “unica” meaning “one-twelfth” and it was abbreviated as “oz” back in the 1500s.
The abbreviation “oz” comes from the Italian word “onza”, which also means “ounce”.
What is a Fluid Ounce
“Fluid ounce” (fl oz or fl. oz.) is from the Imperial and U.S. Customary System measurements system that is used in the measurement of the volume of liquid ingredients (like milk, water or juice).
The term fluid ounce is often abbreviated as fl. oz or fl oz, which means either a single fluid ounce or fluid ounces (both “ounce” and “ounces” can be abbreviated as “oz”).
Fluid ounces are still used in the United States Customary System of measurements, while the United Kingdom stopped using this legal unit of measure and started using the Metric System in 2000
Wet vs Dry
It’s important to remember that wet and dry ingredients have different densities (solids are more dense than liquids) so they need to be measured differently. When cooking, remember to keep in mind
that there are different types of ounces – fluid ounces (fl oz) and dry ounces (oz). Fluid ounces measures the volume of the liquid and dry ounces measures the weight of the ingredient. If your
recipe calls for your ingredients in measurements in ounces or pounds (such as flour, chocolate chips or nuts), you would need to use a kitchen scale to measure the exact weight of the ingredient. To
measure fluid ounces, you would use a liquid measuring cup.
What is a Gallon
In the U.S. a Gallon (abbreviated “gal”) is a measurement in the Imperial and United States customary measurement systems used to measure volume or how much liquid a container can hold. A gallon is a
unit of volume equal to 4 quarts, 8 pints, 16 cups and 128 fluid ounces, but the British imperial gallon is about 20% larger than the U.S. gallon because the two measuring systems are defined
differently. Most other countries have standardized their measurements to the metric system.
Imperial (US) System vs Metric System
Different countries use different volume and weight measurements. Some countries use the Imperial System while some use the Metric System.
The United States still uses the Imperial System for measurements, where most countries like the United Kingdom, Australia, Canada have adopted the metric system using measurements like grams,
meters, liters, kilograms and kilometers. There aren’t significant differences in volume between the Imperial (US) and Metric measuring systems. You can easily use US teaspoons and tablespoons for
metric measurements if a recipe is written using the Metric system. The Imperial system of units was developed and used in the United Kingdom around 1826.
For the most part, the Metric System has replaced the imperial system in countries who once used it. The United States is one of the few countries in the world who has yet to switch to the Metric
System of measurements.
The US Customary System
The United States Customary System (U.S. Customary System) is a system of weights and measures used in the United States and some other countries. This system includes units for measuring length
(inches, feet, yards and miles), weight (ounce, pound, ton), and capacity (teaspoons, tablespoon, cups, pints, quarts and gallons).
Conclusion – How Many Oz In A Gallon
How many ounces in a gallon?
1 gallon = 128 fluid ounces
How many ounces in 1/2 a gallon?
1 gallon = 64 fluid ounces
How many ounces in a gallon of water?
In the US 1 gallon of water = 128 fluid ounces
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Lesson 8
Rotation Patterns
Let’s rotate figures in a plane.
8.1: Building a Quadrilateral
Here is a right isosceles triangle:
1. Rotate triangle \(ABC\) 90 degrees clockwise around \(B\).
2. Rotate triangle \(ABC\) 180 degrees clockwise round \(B\).
3. Rotate triangle \(ABC\) 270 degrees clockwise around \(B\).
4. What would it look like when you rotate the four triangles 90 degrees clockwise around \(B\)? 180 degrees? 270 degrees clockwise?
8.2: Rotating a Segment
Create a segment \(AB\) and a point \(C\) that is not on segment \(AB\).
1. Rotate segment \(AB\) \(180^\circ\) around point \(B\).
2. Rotate segment \(AB\) \(180^\circ\) around point \(C\).
1. Rotate segment \(AB\) \(180^\circ\) around its midpoint. What is the image of A?
2. What happens when you rotate a segment \(180^\circ\)?
Here are two line segments. Is it possible to rotate one line segment to the other? If so, find the center of such a rotation. If not, explain why not.
8.3: A Pattern of Four Triangles
Here is a diagram built with three different rigid transformations of triangle \(ABC\).
Use the applet to answer the questions. It may be helpful to reset the image after each question.
1. Describe a rigid transformation that takes triangle \(ABC\) to triangle \(CDE\).
2. Describe a rigid transformation that takes triangle \(ABC\) to triangle \(EFG\).
3. Describe a rigid transformation that takes triangle \(ABC\) to triangle \(GHA\).
4. Do segments \(AC\), \(CE\), \(EG\), and \(GA\) all have the same length? Explain your reasoning.
When we apply a 180-degree rotation to a line segment, there are several possible outcomes:
• The segment maps to itself (if the center of rotation is the midpoint of the segment).
• The image of the segment overlaps with the segment and lies on the same line (if the center of rotation is a point on the segment).
• The image of the segment does not overlap with the segment (if the center of rotation is not on the segment).
We can also build patterns by rotating a shape. For example, triangle \(ABC\) shown here has \(m(\angle A) = 60\). If we rotate triangle \(ABC\) 60 degrees, 120 degrees, 180 degrees, 240 degrees, and
300 degrees clockwise, we can build a hexagon.
• corresponding
When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.
For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).
• rigid transformation
A rigid transformation is a move that does not change any measurements of a figure. Translations, rotations, and reflections are rigid transformations, as is any sequence of these. | {"url":"https://im.kendallhunt.com/MS/students/3/1/8/index.html","timestamp":"2024-11-04T04:01:29Z","content_type":"text/html","content_length":"94969","record_id":"<urn:uuid:f91a992e-2c39-4d26-b1d5-26f1091eefa5>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00764.warc.gz"} |
Click to start working through an example. Click on instructions in the steps of the algorithm to see the values of the variables in each step. First click on step 1. and then on the following steps
in the order of execution.
Sum of the natural numbers up to
Input: A natural number n
Output: The sum of the natural numbers up to n
1. let i := 0
2. let s := 0
3. repeat
1. let i := i + 1
2. let s := s+i
4. until i = n
5. return s | {"url":"https://mathstats.uncg.edu/sites/pauli/112/HTML/js-algfac-if.html","timestamp":"2024-11-07T00:50:53Z","content_type":"text/html","content_length":"10587","record_id":"<urn:uuid:0f8d685a-5587-4acd-8aa0-f6c5f961c9dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00324.warc.gz"} |
2017 AMC 12B Problems/Problem 22
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and
two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth
round, each of the players has four coins?
$\textbf{(A)}\quad \dfrac{7}{576} \qquad \qquad \textbf{(B)}\quad \dfrac{5}{192} \qquad\qquad \textbf{(C)}\quad \dfrac{1}{36} \qquad\qquad \textbf{(D)}\quad \dfrac{5}{144} \qquad\qquad\textbf{(E)}\
quad \dfrac{7}{48}$
Solution 1
It amounts to filling in a $4 \times 4$ matrix. Columns $C_1 - C_4$ are the random draws each round; rowof each player. Also, let $\%R_A$ be the number of nonzero elements in $R_A$. Sidenote: (Not
the author)(What does -1, 1, and 0, and R notation represent)?
WLOG, let $C_1 = \begin{pmatrix} 1\\-1\\0\\0\end{pmatrix}$. Parity demands that $\%R_A$ and $\%R_B$ must equal $2$ or $4$.
Case 1: $\%R_A = 4$ and $\%R_B = 4$. There are $\binom{3}{2}=3$ ways to place $2$$-1$'s in $R_A$, so there are $3$ ways.
Case 2: $\%R_A = 2$ and $\%R_B=4$. There are $3$ ways to place the $-1$ in $R_A$, $2$ ways to place the remaining $-1$ in $R_B$ (just don't put it under the $-1$ on top of it!), and $2$ ways for one
of the other two players to draw the green ball. (We know it's green because Bernardo drew the red one.) We can just double to cover the case of $\%R_A = 4$, $\%R_B = 2$ for a total of $24$ ways.
Case 3: $\%R_A=\%R_B=2$. There are three ways to place the $-1$ in $R_A$. Now, there are two cases as to what happens next.
Sub-case 3.1: The $1$ in $R_B$ goes directly under the $-1$ in $R_A$. There's obviously $1$ way for that to happen. Then, there are $2$ ways to permute the two pairs of $1, -1$ in $R_C$ and $R_D$.
(Either the $1$ comes first in $R_C$ or the $1$ comes first in $R_D$.)
Sub-case 3.2: The $1$ in $R_B$ doesn't go directly under the $-1$ in $R_A$. There are $2$ ways to place the $1$, and $2$ ways to do the same permutation as in Sub-case 3.1. Hence, there are $3(2+2 \
cdot 2)=18$ ways for this case.
There's a grand total of $45$ ways for this to happen, along with $12^3$ total cases. The probability we're asking for is thus $\frac{45}{(12^3)}= \boxed{\textbf{(B)}\frac{5}{192}}$.
Solution 2 (Less Casework)
We will proceed by taking cases based on how many people are taking part in this "transaction." We can have $2$, $3$, or $4$ people all giving/receiving coins during the $4$ turns. Basically, (like
the previous solution), we think of this as filling out a $4\text{x}2$ matrix of letters, where a letter on the left column represents this person gave, and a letter on the right column means this
person received. We need to make sure that for each person that gave in total certain amount, they received in total from other people that same amount, or in other words, we want it such that there
are an equal number of A's, B's, C's, and D's in both columns of the matrix.
For example, the matrix below represents: A gives B a coin, then B gives C a coin, then C gives D a coin, and finally A gives D a coin, in this order. $\[ \begin{bmatrix} & A & B & \\ & B & C & \\ &
C & D & \\ & A & D & \\ \end{bmatrix} \]$ Case 1: $2$ people. In this case, we have $\binom{4}{2}$ ways to choose the two people, and $\binom{4}{2}$ ways to order them to get a count of ${\binom{4}
{2}}^2 = 36$ ways.
Case 2: $3$ people. In this case, one special person is giving/receiving twice. There are four ways to choose this person, then of the remaining three people we choose two, to be the people
interacting with the special person. Thus, we have $4\cdot\binom{3}{2}\cdot4! = 288$ ways here.
Case 3: $4$ people. If we keep the order of A, B, C, D giving in that order (and permute afterward), then there are three options to choose A's receiver, and three options for B's receiver afterward.
Then it is uniquely determined who C and D give to. This gives a total of $3\cdot3\cdot4! = 216$ ways, after permuting.
So we have a total of $36+288+216=540$ ways to order the four pairs of people. Now we divide this by the total number of ways: $(4\cdot3)^4$ (four rounds, four ways to choose giver, three to choose
receiver each round). So the answer is $\frac{540}{12^4} = \boxed{\textbf{(B)}\frac{5}{192}}$.
Latex polished by Argonauts16 and by Grace.yongqing.yu.
Solution 3
Similar to solution 2, we think in terms of transactions. WLOG, for the 1st transaction, we assume that A gives to B, which we denote AB. For the 2nd transaction, there are 12 options to choose from,
so there are $12^3$ possible options.
Case 1 (Giving back immediately): The 2nd transaction is BA (1 option). Then, the 3rd transaction can be whatever you like (12 options), but the 4th transaction is now fixed to be the opposite of the
3rd transaction (1 option). So here we have $1\cdot 12\cdot 1 = 12$ good options.
Case 2 (Allows a cycle or two back-and-forths): The 2nd transaction is one of BC, BD, CA, DA, CD, DC (6 options). Then, for the 3rd transaction, 2 options force a "cycle" on the 4th transaction
(example 2.1), and 2 options force two "back-and-forths" on the 4th transaction (example 2.2). In total, there are $2+2=4$ options for the 3rd transaction. So here we have $6\cdot 4\cdot 1 = 24$ good
Example 2.1: suppose 2nd = BC. Then if 3rd = DA or CD, the 4th is forced to be CD or DA respectively, completing a transaction "cycle"
Example 2.2: suppose 2nd = BC. Then if 3rd = AB or BC, the 4th is forced to be BC or AB respectively. In the end, both A-B and B-C had back-and-forth transactions
Case 3 (Allows two back-and-forths only): The 2nd transaction is one of AC, AD, DB, CB (4 options). Then, for the 3th transaction, there are only 2 possible options (namely, to reverse one of the
previous transactions). Of course, the final transaction is forced. Here, we have $4\cdot 2\cdot 1 = 8$ good options.
Case 4 (AB again): The 2nd transaction is AB (same as 1st). This forces the later transactions to both be BA. Here we have 1 good option.
Summing, we have $12+24+8+1=45$ good options among $12^3$ total options, so the solution is $\frac{45}{(12^3)}= \boxed{\textbf{(B)}\frac{5}{192}}$.
Solution 4
Define a cycle of length $n$ to be a sequence of $n$ transactions, starting and ending with the same person. For example, $A \to B, B \to C, C \to A$ would be a cycle of $3$.
$\textbf{Case 1}$: Two different cycles of $2$: There are $\binom{4}{2}$ to choose the people for each cycle, giving $\binom{\binom{4}{2}}{2} = 15$ possible ways to choose the two cycles. Then, there
would be $4! = 24$ ways to order the transactions. This will mean $15\cdot 24 = 360$ ways for the first case.
$\textbf{Case 2}$: Two same cycles of $2$: There are $\binom{4}{2}$ to choose the two people, multiplying by $\binom{4}{2}$ for the number of orderings of the four transactions. Thus, $6\cdot 6 = 36$
ways for this case.
$\textbf{Case 3}$: One cycle of $4$: We choose what goes in front of $A$ (whom $A$ gives a coin to) in $3$ ways, and what goes before $A$ (whom gives a coin to $A$) in $2$ ways. The remaining two
transactions immediately get fixed after that. Again, there are $4! = 24$ ways to order the four transactions, giving $2\cdot 3\cdot 24 = 144$ ways for this case.
Thus, there would be a total of $360+36+144 = 540$ total ways: $P = \dfrac{540}{12^4} = \boxed{\textbf{(B)}\frac{5}{192}}$
Visualization of three cases
Solution 5 (Casework)
Let the notation $(a,b,c,d)$ be the arrangement of the $16$ coins, where order does not matter. Notice that after the first round, $(3,5,4,4)$ is the only possible arrangement. Also, notice that the
arrangement after the third round also has to be $(3,5,4,4)$ for the arrangement at the end of the fourth round to return to $(4,4,4,4)$. The probability of $(4,4,4,4)$ occurring after $(3,5,4,4)$ is
After the arrangement $(3,5,4,4)$, there are $12$ permutations that are $6$ different arrangements, they are: \begin{align*} (2,6,4,4) \text{ occurs once} \quad & (4,4,4,4) \text{ occurs once} \quad
& (2,5,5,4) \text{ occurs twice}\\ (4,5,3,4) \text{ occurs 4 times} \quad & (3,6,3,4) \text{ occurs twice} \quad & (3,5,3,5) \text{ occurs twice}\\ \end{align*}
Therefore, we can create $6$ cases and calculate the probability of each case occurring. For each case, only the arrangement after the second round is different, therefore, we will ignore the
probability for the fourth round, and consider it in the end when adding the cases.
Case 1: $(2,6,4,4)$
The probability of $(2,6,4,4)$ occurring after the second round is $\frac{1}{12}$.
The probability of $(3,5,4,4)$ occurring after $(2,6,4,4)$ is $\frac{1}{12}$.
$\frac{1}{12} \cdot \frac{1}{12} = \frac{1}{144}$
Case 2: $(4,4,4,4)$
The probability of $(4,4,4,4)$ occurring after the second round is $\frac{1}{12}$.
The probability of $(3,5,4,4)$ occurring after $(4,4,4,4)$ is $1$.
$\frac{1}{12} \cdot 1 = \frac{1}{12}$
Case 3: $(2,5,5,4)$
The probability of $(2,5,5,4)$ occurring after the second round is $\frac{1}{6}$.
The probability of $(3,5,4,4)$ occurring after $(2,5,5,4)$ is $\frac16$.
$\frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}$
Case 4: $(4,5,3,4)$
The probability of $(4,5,3,4)$ occurring after the second round is $\frac{1}{3}$.
The probability of $(3,5,4,4)$ occurring after $(4,5,3,4)$ is $\frac{1}{3}$.
$\frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}$
Case 5: $(3,6,3,4)$
The probability of $(3,6,3,4)$ occurring after the second round is $\frac{1}{6}$.
The probability of $(3,5,4,4)$ occurring after $(3,6,3,4)$ is $\frac{1}{6}$.
$\frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}$
Case 6: $(3,5,3,5)$
The probability of $(3,5,3,5)$ occurring after the second round is $\frac{1}{6}$.
The probability of $(3,5,4,4)$ occurring after $(3,5,3,5)$ is $\frac{1}{3}$.
$\frac{1}{6} \cdot \frac{1}{3} = \frac{1}{18}$
The probability that at the end of the fourth round the arrangement is $(4,4,4,4)$ is $\[\frac{1}{12} \left( \frac{1}{144} + \frac{1}{12} + \frac{1}{36} + \frac19 + \frac{1}{36} + \frac{1}{18} \
right) = \boxed{\textbf{(B)}\frac{5}{192}}\]$
See Also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. | {"url":"https://artofproblemsolving.com/wiki/index.php/2017_AMC_12B_Problems/Problem_22","timestamp":"2024-11-03T04:30:27Z","content_type":"text/html","content_length":"79527","record_id":"<urn:uuid:f1ccbc87-3fa2-4386-b925-dfe6e02ee539>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00287.warc.gz"} |
How do you classify finite groups?
The classification of finite simple groups is a theorem stating that every finite simple group belongs to one of the following families:
1. A cyclic group with prime order;
2. An alternating group of degree at least 5;
3. A simple group of Lie type;
4. One of the 26 sporadic simple groups;
How long is the classification of finite simple groups?
It is expected that they will number about twelve, and stretch to a total of 3,000-4,000 pages. At time of writing, 6 have been published so far. A second major theme in finite group theory is the
extension problem for finite groups.
What is the most important theorem on finite groups?
The most important structure theorem for finite groups is the Jordan–Hölder Theorem, which shows that any finite group is built up from finite simple groups.
Are all finite groups classified?
Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers….Timeline of the proof.
Publication date
1974 Thompson classifies N-groups, groups all of whose local subgroups are solvable.
What are the classification of groups?
Classification of Groups
• Primary and Secondary Groups.
• Membership and Reference Groups.
• Small and Large Groups.
• Organized and Unorganized Groups.
• In and Out-going Groups.
• Accidental and Purposive Groups.
• Open and Closed Groups.
• Temporary and Permanent Groups.
What is finite and infinite group?
An infinite set is endless from the start or end, but both the side could have continuity unlike in Finite set where both start and end elements are there. If a set has the unlimited number of
elements, then it is infinite and if the elements are countable then it is finite.
What is a group classification?
Group can be classified into many forms. Basically, groups are classified as formal and informal. A formal group can be divided into the Command group and Task group. Command Group: A command group
is determining the organizational chart. It is composed of the individuals who report directly to a given manager.
Are all finite groups cyclic?
Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is finitely generated and has exactly two ends; an example of such a group is
the direct product of Z/nZ and Z, in which the factor Z has finite index n.
What is the order of group?
The Order of a group (G) is the number of elements present in that group, i.e it’s cardinality. It is denoted by |G|. Order of element a ∈ G is the smallest positive integer n, such that an= e, where
e denotes the identity element of the group, and an denotes the product of n copies of a.
Can a finite group have elements of infinite order?
No, because in a finite group every element has its order less than equal to the order of group. let G be a finite group of order n. It cannot have an element of infinite order since a belongs to G
implies a^n=e and so o(a) is any positive integer less than or equal to n.
Are finite groups cyclic?
What are the 4 types of groups?
Four basic types of groups have traditionally been recognized: primary groups, secondary groups, collective groups, and categories.
What is a finite group?
Finite group. In abstract algebra, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered
pair of elements, an element of the set. In the case of a finite group, the set is finite.
What is representation theory?
Representation theory. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and
studies modules over these abstract algebraic structures.
What is Group classification?
Group classification is determined by member’s occupation, position, or job duties. State law classifies types of employment within groups and NCRS assign members to the group classification for
their position. | {"url":"https://ru-facts.com/how-do-you-classify-finite-groups/","timestamp":"2024-11-14T17:26:35Z","content_type":"text/html","content_length":"53856","record_id":"<urn:uuid:26cfe065-eb32-4781-8d73-9a1fded7d492>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00458.warc.gz"} |
Voltage Booster - HardwareBee
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qbits n
cbits n
qregs n
cregs n
mem n
The order of the headers is important and should not be changed. The qbits header specifies the number of qubits in each quantum register. The cbits header specifies the number of classical bits in
each classical register. The qregs header specifies the number of quantum registers. The cregs header specifies the number of classical registers. The n after each header is any integer number.
Operands/arguments for all instructions are delimited by spaces and not commas.
Code execution should always end at a hlt instruction. If the emulator reaches the end of code but does not encounter a hlt instruction, it will end with a "PC out of bounds" error.
Quantum Instructions
The general format for quantum instructions are:
op qn <other arguments>
Where op is the name/opcode, qn specifies a specific qubit n of the currently selected quantum register. <other arguments> can include more qubits as arguments, or in the case of some instructions, a
rotation expressed as a rational multiple of pi, in the format [n]pi[/n], where n can be any integer number, and items in [] are optional. Quantum registers can be selected via the qsel instruction,
which has the general format qsel qrn where n is any non-negative number.
List of currently implemented quantum instructions:
Quantum Gate Instruction name Syntax example Explanation
Hadamard h h q0 Applies a Hadamard to qubit 0
CNOT cnot cnot q0 q1 Applies a CNOT to qubit 1 with qubit 0 being the control
CCNOT/Toffoli ccnot ccnot q0 q1 q2 Applies a Toffoli to qubit 2 with qubit 0 and qubit 1 being the controls
Pauli X x x q0 Applies a Pauli X to qubit 0
Pauli Y y y q0 Applies a Pauli Y to qubit 0
Pauli Z z z q0 Applies a Pauli Z to qubit 0
Rx rx rx q0 pi/3 Rotates the statevector of qubit 0 by pi/3 radians along X axis on bloch sphere
Ry ry ry q0 pi Rotates the statevector of qubit 0 by pi radians along Y axis on bloch sphere
Rz rz rz q0 pi/4 Rotates the statevector of qubit 0 by pi/4 radians along Z axis on bloch sphere
U gate u u q0 pi pi/3 pi/6 Rotates the statevector of qubit 0 by the 3 Euler angles pi, pi/3, pi/6
S gate s s q0 Applies an S gate to qubit 0
T gate t t q0 Applies a T gate to qubit 0
S-dagger sdg sdg q0 Applies a S-dagger or the inverse of S gate to qubit 0
T-dagger tdg tdg q0 Applies a T-dagger or the inverse of T gate to qubit 0
Phase gate p p q0 pi/3 Applies a relative phase of pi/3 radians to qubit 0
Controlled Hadamard ch ch q0 q1 Applies a controlled Hadamard to qubit 1 with qubit 0 being the control
Controlled Pauli Y cy cy q0 q1 Applies a controlled Pauli Y to qubit 1 with qubit 0 being the control
Controlled Pauli Z cz cz q0 q1 Applies a controlled Pauli Z to qubit 1 with qubit 0 being the control
Controlled Phase cp cp q0 q1 pi/2 Applies a controlled Phase gate to qubit 1 of pi/2 radians with qubit 0 being the control
Swap swap swap q0 q1 Swaps the state of qubits 0 and 1
Square Root NOT sqrtx sqrtx q0 Applies a sqrt(NOT)/sqrt(X) to qubit 0
Square Root Swap sqrtswp sqrtswp q0 q1 Applies a sqrt(Swap) to qubits 0 and 1, halfway swapping their state
Controlled Swap cswap cswap q0 q1 q2 Swaps the state of qubits 1 and 2 with qubit 0 being the control
Measure m m q0 cr1 c3 Measures the state of qubit 0 into 3rd bit of classical register 1
Note: Remove any measurement operations before running the emulator with --print-state (or -p) as the emulator does not ignore them currently when run with that flag set
Classical Instructions
General format for classical instructions are:
op <operands>
Where op is the name/opcode, operands may include crn, which specifies a specific classical register n, or an immediate literal value (for now non-negative due to not implemented in parser yet) Other
than these differences, they behave basically the same as any other assembly language instructions.
List of currently implemented classical instructions:
Note: The value of a register refers to the value stored in the register. The value of an immediate is the immediate number itself.
Note 2: An operand can either be a register or immediate unless a restriction is specified.
Instruction name Description
add op1 = op2 + op3. op1 is always a register.
sub op1 = op2 - op3. op1 is always a register.
mult op1 = op2 * op3. op1 is always a register. All values are treated unsigned.
umult op1 = (op2 * op3) >> (cbits/2). op1 is always a register. All values are treated unsigned.
div op1 = op2 / op3. op1 is always a register. Performs integer division. All values are treated unsigned.
smult op1 = op2 * op3. op1 is always a register. All values are treated signed.
sumult op1 = (op2 * op3) >> (cbits/2). op1 is always a register. All values are treated signed.
sdiv op1 = op2 / op3. op1 is always a register. Performs integer division. All values are treated signed.
not op1 = ~op2. op1 is always a register.
and op1 = op2 & op3. op1 is always a register.
or op1 = op2 | op3. op1 is always a register.
xor op1 = op2 ^ op3. op1 is always a register.
nand op1 = ~(op2 & op3). op1 is always a register.
nor op1 = ~(op2 | op3). op1 is always a register.
xnor op1 = ~(op2 ^ op3). op1 is always a register.
Misc. Instructions
These instructions are here because.
Instruction name Description
qsel Selects a quantum register so that proceeding quantum instructions act on that qreg.
cmp Updates flags based on comparing values in op1 and op2. op1 is always a register.
jmp Unconditionally jump to a label.
jeq Jump to label if comparison resulted in EQ flag set.
jne Jump to label if comparsion did not result in EQ flag set.
jg Jump to label if comparison resulted in GREATER flag set.
jge Jump to label if comparison resulted in GREATER or EQ flag set.
jl Jump to label if comparison resulted in LESSER flag set.
jle Jump to label if comparison resulted in LESSER or EQ flag set.
hlt Halt the program.
This program simulates the \ket{\Phi^+} bell state:
qbits 2
cbits 2
qregs 1
cregs 1
qsel qr0
h q0
cnot q0 q1
m q0 cr0 c0
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Work in a Thermodynamic Process | iCalculator™
Welcome to our Physics lesson on Work in a Thermodynamic Process, this is the fourth lesson of our suite of physics lessons covering the topic of The Kinetic Theory of Gases. Ideal Gases, you can
find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
Work in a Thermodynamic Process
Let's find an expression for calculation of the work done by a gas during a slow thermal process. Consider a cylinder filled with gas with a freely moveable piston above it as shown in the figure
Since the piston moves freely, the gas pressure remains constant because it is balanced from the atmospheric pressure, i.e. the pressure exerted by the air from above the piston. The work done by the
gas to raise the piston by Δy as shown in the figure below, is
Given that
Force = Pressure × Area
we have
W[by gas]=P[gas] × A[piston] × ∆y
=P[gas] × (A[piston] × ∆y)
=P[gas] × ∆V
This expression gives the left part of the ideal gas law formula.
The formula indicates that:
Volume increases ⇒ ΔV > 0 ⇒ W[by gas] > 0
Volume decreases ⇒ ΔV < 0 ⇒ W[by gas] < 0
Volume constant ⇒ ΔV = 0 ⇒ W[by gas] = 0
as we discussed earlier.
Example 4
12 g of helium gas is heated from 20 ºC to 100 ºC, at a constant pressure. What is the work done by helium during the heating process? (Take MHe = 4 g/mol)
Solution 4
First, let's write the clues to create a clearer idea about the situation. We have:
m = 12 g
M = 4 g/mol
t[1] = 20°C
t[2] = 100°C
P^1 = P^2 = P
W[by gas] = ?
From the first two clues we can work out the number of moles, n. We have:
n = m/M
= 12 g/4 g/mol
= 3 moles
Also, from the next two clues, we obtain the change in temperature:
∆T = t[2] - t[1]
= 100°C - 20°C
= 80°C
= 80 K
Applying the ideal gas law for the two given instants (states) 1 and 2, we obtain
P × V[1] = n × R × T[1]
P × V[2] = n × R × T[2]
Subtracting the first equation from the second, we obtain
P × V[2] - P × V[1] = n × R × T[2] - n × R × T[1]
P × ∆V = n × R × ∆T
The first part of the last equation gives the work done by the gas. Therefore, the left part will also give the work done by the gas. Hence, we can write:
W[by gas] = n × R × ∆T
= 3 moles × 8.31 J/mol × K × 80 K
= 1994.4 J
This result means helium does 1994.4 J of work against the environment during thermal expansion.
You have reached the end of Physics lesson 13.6.4 Work in a Thermodynamic Process. There are 6 lessons in this physics tutorial covering The Kinetic Theory of Gases. Ideal Gases, you can access all
the lessons from this tutorial below.
More The Kinetic Theory of Gases. Ideal Gases Lessons and Learning Resources
Thermodynamics Learning Material
Tutorial ID Physics Tutorial Title Tutorial Video Revision Revision
Tutorial Notes Questions
13.6 The Kinetic Theory of Gases. Ideal Gases
Lesson ID Physics Lesson Title Lesson Video
13.6.1 The Meaning of Mole. Avogadro's Number
13.6.2 Ideal Gases
13.6.3 The Ideal Gas Law
13.6.4 Work in a Thermodynamic Process
13.6.5 Finding Work using a P - V Diagram
13.6.6 State Variables and Path-Dependent Variables
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Area of parallelograms and triangles worksheet | Maths Year 6
Area of parallelograms and triangles worksheet
Maths Resource Description
This worksheet, designed for key stage 2 students, provides a series of engaging activities aimed at teaching children how to calculate the area of parallelograms and triangles. In Activity 1,
students are given a variety of these shapes on squared paper and tasked with counting the squares to determine the area. They then exchange their work with a partner to verify the accuracy of their
calculations. This hands-on approach helps students to visually understand the concept of area and reinforces their counting skills.
Subsequent activities delve deeper into the properties of triangles. In Activity 2, each student works out the area of a selected triangle and then draws another triangle with the same area,
fostering creativity and understanding of geometric principles. Pairs of students then swap and check each other's work for accuracy. Activity 3 challenges students to find the dimensions of
right-angled triangles with a given area, enhancing their problem-solving abilities. Activity 4 requires the application of their knowledge to calculate the area of composite shapes made up of
triangles. Finally, Activity 5 encourages students to use squared paper and rulers to draw triangles of specified areas, allowing them to explore a range of possible solutions and recognise patterns.
The objective of these activities is to enable students to accurately calculate the area of parallelograms and triangles, as well as to understand when to apply formulae for the area and volume of
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Part of a lesson by KS2 Gems
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String theory
• Strings carry waves, “standing waves”
• Strings have resonant frequencies, representing certain levels of energy
• Strings are 1-D structures, can be in loops, and have vibrational modes that define particle properties
• Since string theory only works in 11D, one could say the additional dimensions we do not “see” in our universe are extremely tiny dimensions in which the strings can vibrate (but pretty much
nothing else)
• The different vibrational modes of strings produce different particles, analagous to how the strings on a guitar produce different notes
• Gravity appears in string theory nicely, a pro to it as a theory
• Kaluza-Klein theory: general relativity applied in 5-d (4 space, 1 time), yields the same results from our 4d universe, plus 1 extra that looks oddly similar to EM.
□ Klein said that the 5th dimension could be wrapped up into a tiny amount of space. One could liken this to the idea that at every point in 3d space (+1 time), there is the additional ability
to go around a tiny ring, representing this tiny extra dimension wrapped up into a loop. The idea of being able to move along this ring at any point in 3d space is the same idea as moving
through that extra dimension. The important idea behind this extra dimension was that momentum in that looped dimension was equivalent to electric charge. So there’s this special additional
degree of freedom of “movement” that allows sufficiently small objects (i.e. those that can “fit inside” the tiny extra dimension) to gain electric charge according to their position in this
5d space. (Note that the direction of rotation in this looped dimension corresponded to the sign of the charge).
□ The immediate question here is: what about all the other dimensions of modern string theory? Do they reduce nicely to other observable effects (like that of EM seen in the 5th dimension)?
• Modern string theory is inspired by Keluza-Klein, just with vibrating strings and just the right extra dimensions, along with super-symmetry.
• Many apparently diverging super-string theories were developed, but then unified by M theory, all as special cases of a larger idea (with an added dimension).
• Extra (tiny) dimensions in M theory are wrapped, just not in simple loops like Keluza-Klein. Instead they’re wrapped according to complex structures known as Calabi–Yau manifolds, which are
high-dimensional geometries in which the strings are expected to live.
□ But this is a family of manifolds, and there are ~10^500 different possible topologies defined by different manifolds. Each geometry has different implications, yields different particles
with different properties.
□ A previous though on this: can this definition of string theory be used as a “universe generator”? If there exists an actual mapping between concrete string manifolds and sets of fundamental
particles, what can we learn from arbitrary geometries? Do these even define “semantically correct” universes that would produce any value? Thoughts | {"url":"https://samgriesemer.com/String_theory","timestamp":"2024-11-12T12:22:48Z","content_type":"text/html","content_length":"10501","record_id":"<urn:uuid:e418c523-a476-404d-aee1-890020b9f06a>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00577.warc.gz"} |
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Dr. Arunima Bhattacharya, SLMath - A priori interior estimates for Lagrangian mean curvature equations - Department of Mathematics
Dr. Arunima Bhattacharya, SLMath – A priori interior estimates for Lagrangian mean curvature equations
November 11, 2022 @ 3:00 pm - 4:00 pm
Mode: In-person
Title: A priori interior estimates for Lagrangian mean curvature equations
Abstract: In this talk, we will introduce the special Lagrangian and Lagrangian mean curvature type equations. We will derive a priori interior estimates for the Lagrangian mean curvature equation
under certain natural restrictions on the Lagrangian angle. As an application, we will use these estimates to solve the Dirichlet problem for the Lagrangian mean curvature equation with continuous
boundary data on a uniformly convex, bounded domain. We will also briefly introduce the fourth-order Hamiltonian stationary equation and mention some recent results on the regularity of solutions of
certain fourth-order PDEs, which are critical points of variational integrals of the Hessian of a scalar function. Examples include volume functionals on Lagrangian submanifolds. This is based on
joint works with Connor Mooney and Ravi Shankar.
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CitationIntroductionMethodologyResults and discussionsModel parameterization for cesium binary CsF-H2O, CsOH-H2O and Cs2SO4-H2O systems at 25 °CValidation of models for cesium binary systems CsCl-H2O, CsBr- H2O, CsI-H2O, CsNO3-H2O, and Cs2SeO4- H2ODeliquescence Relative Humidity (DRH (%)) and thermodynamic solubility product (ln Kosp) of cesium solid phasesModels for cesium ternary systemsSummary and conclusionsAcknowledgementReferences
4 urn:lsid:arphahub.com:pub:27D7DBB2-BDE1-5A1F-B26F-1372106F69DB BioRisk BR 1313-2644 1313-2652 Pensoft Publishers 10.3897/biorisk.17.77523 77523 Research Article Animalia Plantae Data analysis &
Modelling Ecology & Environmental sciences Landscape ecology Waste management & remediation World Development of accurate chemical thermodynamic database for geochemical storage of nuclear waste.
Part III: Models for predicting solution properties and solid-liquid equilibrium in cesium binary and mixed systems Tsenov Tsvetan 1 Investigation Donchev Stanislav 1 Investigation Christov
Christomir ch.christov@shu.bg 1 Conceptualization Methodology Supervision Department Chemistry, Faculty of Natural Sciences, Shumen University “Konstantin Preslavski”, Shumen, Bulgaria Shumen
University “Konstantin Preslavski” Shumen Bulgaria
Corresponding author: Christomir Christov (ch.christov@shu.bg)
Academic editor: Michaela Beltcheva
2022 21 04 2022 17 407 422 76F9D689-6AD2-51D3-A3C2-8971268D1E9B 6478881 02 11 2021 08 12 2021 Tsvetan Tsenov, Stanislav Donchev, Christomir Christov This is an open access article distributed under
the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
The models described in this study are of high importance in the development of thermodynamic database needed for nuclear waste geochemical storage as well as for technology for extracting cesium
resources from saline waters. In this study we developed new not concentration restricted thermodynamic models for solution behavior and solid-liquid equilibrium in CsF-H[2]O CsOH-H[2]O and Cs[2]SO
[4]-H[2]O systems at 25 °C. To parameterize models we used all available experimental osmotic coefficients data for whole concentration range of solutions and up to saturation point. The new models
are developed on the basis of Pitzer ion interactions approach. The predictions of new developed here models are in excellent agreement with experimental osmotic coefficients data (ϕ) in binary
solutions from low to extremely high concentration (up to 21.8 mol.kg^-1 for CsOH-H[2]O and up to 35.6 mol.kg^-1 for CsF-H[2]O). The previously developed by Christov by Christov and co-authors and by
other authors Pitzer approach based thermodynamic models for five (5) cesium binary systems (CsCl-H[2]O CsBr- H[2]O CsI-H[2]O CsNO[3]-H[2]O and Cs[2]SeO[4]- H[2]O) are tested by comparison with
experimental osmotic coefficients data and with recommendations on activity coefficients (γ[±]) in binary solutions. The models which give the best agreement with (ϕ)- and (γ[±]) -data from low to
high concentration up to m(sat) are accepted as correct models which can be used for solubility calculations in binary and mixed systems and determination of thermodynamic properties of precipitating
cesium solid phases. The thermodynamic solubility products (ln K^o[sp]) and the Deliquescence Relative Humidity (DRH) of solid phases precipitating from saturated cesium binary solutions (CsF(cr)
CsCl(cr) CsBr(cr) CsI(cr) CsOH(cr) CsNO[3](cr) Cs[2]SO[4](cr) and Cs[2]SeO[4](cr)) have been determined on the basis of evaluated and accepted binary parameters and using experimental solubility
data. The reported mixing parameters [θ(Cs M^2+) and ψ(Cs M^2+ X)] evaluated by solubility approach for 15 cesium mixed ternary systems (CsCl-MgCl[2]-H[2]O CsBr-MgBr[2]-H[2]O CsCl-NiCl[2]-H[2]O
CsBr-NiBr[2]-H[2]O CsCl-MnCl[2]-H[2]O CsCl-CoCl[2]-H[2]O CsCl-CuCl[2]-H[2]O CsCl-CsBr-H[2]O CsCl-RbCl-H[2]O Cs[2]SO[4]-CoSO[4]-H[2]O Cs[2]SeO[4]-CoSeO[4]-H[2]O Cs[2]SO[4]-NiSO[4]-H[2]O Cs[2]SeO[4]
-NiSeO[4]-H[2]O Cs[2]SO[4]-ZnSO[4]-H[2]O and Cs[2]SeO[4]-ZnSeO[4]-H[2]O) are tabulated.
Cesium binary and mixed systems computer thermodynamic modeling geochemical nuclear waste sequestration Pitzer approach The work was supported by the European Regional Development Fund, Project
BG05M2OP001-1.001-0004, and by Shumen University Research Program, Project No. RD-08-131/04.02.2021
Tsenov T, Donchev S, Christov C (2022) Development of accurate chemical thermodynamic database for geochemical storage of nuclear waste. Part III: Models for predicting solution properties and
solid-liquid equilibrium in cesium binary and mixed systems. In: Chankova S, Peneva V, Metcheva R, Beltcheva M, Vassilev K, Radeva G, Danova K (Eds) Current trends of ecology. BioRisk 17: 407–422.
Radioactive waste is a by-product of the nuclear fuel cycle and the production of weapons and medical radioisotopes. As nuclear technologies become more widespread, so does the production of waste
materials. In Europe, nuclear waste is classified into 1) high-level; 2) intermediate level; 3) low-level, and 4) transitional radioactive waste. The long-term storage of high-level waste is still
experimental. Radiocesium isotopes, particularly ^137Cs, form part of the high-level nuclear waste group. Crucially, the storage of high-level waste in liquid form poses serious risks. On 29
September 1957 a liquid storage tank exploded at the Mayak facility (Chelyabinsk-40), contaminating more than 52,000 square kilometers with ^137Cs and ^90Sr (Kostyuchenko and Krestinina 1994). This
is known as the Kyshtym accident and is the second most serious radiation disaster after Chernobyl at 1986. On 1987 a release of ^137Cs occurred as a result of improper disposal of radiotherapy
source (Rosenthal et al. 1991). This is known as Goiânia accident in Brazil. According to Scharge et al. (2012) among the more common fission products from spent nuclear fuels, the radionuclide ^
137Cs with half-lives of 30.17 years, is mostly critical for the design of the nuclear waste repository because of the intense γ and β radiation and the heat generated by the decay process, as well
as the high solubilities of cesium halides. Thus, modelling the properties of cesium atoms in salts and solutions is a current, pertinent question in theoretical chemistry.
A long term safety assessment of a repository for radioactive waste requires evidence that all relevant processes are known and understood, which might have a significant positive or negative impact
on its safety (Altmaier et al. 2011a, b; Lach et al. 2018; Donchev and Christov 2020; Donchev et al. 2021b). It has to be demonstrated, that the initiated chemical reactions don’t lead to an un-due
release of radionuclides into the environmental geo-, hydro-, and bio-sphere. One key parameter to assess the propagation of a radionuclide is its solubility in solutions interacting with the waste.
Solubility estimations can either be based on experimental data determined at conditions close to those in the repository or on thermodynamic calculations. The thermodynamic database created from
experimental data is the basis for thermodynamic model calculations. Since the disposal of radioactive waste is a task encompassing decades, the database is projected to operate on a long-term basis.
Chemical models that predict equilibrium involving mineral, gas and aqueous phases over a broad range of solution compositions and temperatures are useful for studying the interactions between used
nuclear fuel waste and its surroundings. The reliability of such predictions depends largely on the thermodynamic database. An accurate description of highly concentrated waters should be required
for modeling of chemical interactions in and around nuclear repositories. The modeling of dissolution and precipitation processes in concentrated solutions requires an adequate thermodynamic model
for the prediction of activities and solubilities (Lach et al. 2018; Donchev and Christov 2020; Donchev et al. 2021b). This requirement is fulfilled by the ion interaction model of Pitzer (Pitzer
1973). Extensive thermodynamic databases, which are based on the Pitzer ion interaction model, were developed within the Yucca Mountain Project (YMTDB: data0.ypf.r2) (Sandia National Laboratories
(2005, 2007), Thereda project (THermodynamic REference DAtabase, THEREDA-Final Report) (Altmaier et al. 2011a, b), and ANDRA project (Lach et al. 2018). However, the subject of long-term radioactive
waste storage still has many questions left for scientists to solve. Unfortunately, many of the Pitzer models introduced in YMTDB and in THEREDA databases for cesium binary and mixed systems are
concentration restricted and cannot describe correctly the solid-liquid equilibrium in geochemical and industrial systems of interest for nuclear waste programs.
This paper presents a comprehensive analysis and evaluation of existent thermodynamic database for cesium binary and mixed systems. It should be noted, that the thermodynamic properties, solubility
isotherms and their simulation by thermodynamic model of the cesium binary and mixed brine type systems (s.a. CsX−MgX[2]−H[2]O (X =Cl,Br,I) ternary systems) are also of significant importance for
extracting cesium resources from brine type solutions (Balarew et al. 1993; Christov et al. 1994; Christov 1995a, b, 1996a, 2005; Guo et al. 2017). According to Baranauskaite et al. (2021) carnalite
type minerals of the type MX.MgX[2].6H[2]O (cr) (M=Li, K,NH[4],Rb,Cs) (Christov and Balarew 1995; Christov 2012; Lassin et al. 2015) “are interesting not only as natural sources of chemical
compounds, but also they can be made use of in renewable thermochemical energy storage since their hydration reactions are exothermic”.
In this study we developed new, not concentration restricted thermodynamic models for solution behavior and solid-liquid equilibrium in CsF-H[2]O, CsOH-H[2]O and Cs[2]SO[4]- H[2]O systems at 25 °C.
The new models are developed on the basis of Pitzer ion interactions approach. The previously developed by Christov (2003a, 2005), and Christov and co-authors (Balarew et al. 1993; Barkov et al.
2001; Donchev and Christov 2020) and by other authors (Pitzer and Mayorga 1973; Scharge et al. 2012; Palmer et al. 2002) Pitzer approach based thermodynamic models for five (5) cesium binary systems
(CsCl-H[2]O, CsBr- H[2]O, CsI-H[2]O, CsNO[3]-H[2]O, and Cs[2]SeO[4]- H[2]O) are tested by comparison with experimental osmotic coefficients data and with recommendations on activity coefficients (γ
[±]) in binary solutions. The models that give the best agreement with (ϕ)-, and (γ[±]) data from low to high concentration, up to m(sat), are accepted as correct models, which can be used for
solubility calculations in binary and mixed systems. We also summarized the previously established by the main author (C. Christov) solid-liquid equilibrium model for 15 cesium mixed ternary systems
at 25 °C. The evaluated mixing parameters [θ(Cs,M^2+) and ψ(Cs,M^2+,X)], determined by solubility approach are tabulated.
The models for cesium binary systems have been developed and tested on the basis of Pitzer’s semi-empirical equations (Pitzer 1973). The specific interaction approach for describing electrolyte
solutions to high concentration introduced by Pitzer (1973) represents a significant advance in physical chemistry that has facilitated the construction of accurate computer thermodynamic models.
Pitzer approach has found extensive use in the modeling of the thermodynamic properties of aqueous electrolyte solutions. It was shown that this approach could be expanded to accurately calculate
solubilities in binary and complex systems, and to predict the behavior of natural and industrial fluids from very low to very high concentration at standard temperature of 25 °C (Harvie et al. 1984;
Christov et al. 1994, 1998; Christov 1995a, 1996a, 1998, 1999, 2002, 2003a, b, 2005, 2007, 2009, 2012, 2020; Barkov et al. 2001; Park et al. 2009; Lach et al. 2018; Donchev and Christov 2020; Donchev
et al. 2021a, b), and from 0 to 290 °C (Christov and Moller 2004; Moller et al. 2006; Lassin et al. 2015). Several extensive parameter databases have been reported. These include: 25 °C database of
Pitzer and Mayorga (1973, 1974), of Kim and Frederick (1988), the most widely used database of Chemical Modelling Group at UCSD [(University California San Diego) at 25 °C (Harvie et al. 1984; Park
et al. 2009), and T-variation (from 0 to 300 °C) (Christov and Moller 2004; Moller et al. 2006; Christov 2009)], YMTDB (Sandia National Laboratories 2005, 2007), and THEREDA (2011a, b).
According to Pitzer theory, electrolytes are completely dissociated and in the solution there are only ions interacting with one another (Pitzer 1973; Pitzer and Mayorga 1973). Two kinds of
interactions are observed: (i) specific Coulomb interaction between distant ions of different signs, and (ii) nonspecific short-range interaction between two and three ions. The first kind of
interaction is described by an equation of the type of the Debye-Hueckel equations. Short-range interactions in a binary system (MX(aq)) are determined by Pitzer using the binary parameters of ionic
interactions (β^(0),β^(1), C^ϕ). The Pitzer’s equations are described and widely discussed in the literature (Harvie et al. 1984; Christov and Moller 2004; Christov 2005; Moller et al. 2006; Donchev
et al. 2021b). Therefore, these equations are not given here. According to the basic Pitzer equations, at constant temperature and pressure, the solution model parameters to be evaluated for mixed
ternary system are: 1) pure electrolyte β^(0), β^(1), and C^ϕ for each cation-anion pair; 2) mixing θ for each unlike cation-cation or anion-anion pair; 3) mixing ψ for each triple ion interaction
where the ions are all not of the same sign (Christov 2003a, b, 2005; Donchev et al. 2021b).
Pitzer and Mayorga (1973) did not present analysis for any 2-2 (e.g. MgSO[4]-H[2]O) or higher {e.g. 3-2: Al[2](SO[4])[3]-H[2]O} electrolytes. In their next study (Pitzer and Mayorga 1974) modify the
original equations for the description of 2-2 binary solutions: parameter β^(2)(M,X), and an associated α[2] term are added to the original expression. Pitzer presented these parameterizations
assuming that the form of the functions (i.e. 3 or 4 β and C ^ϕ values, as well as the values of the α terms) vary with electrolyte type. For binary electrolyte solutions in which either the cationic
or anionic species are univalent (e.g. NaCl, Na[2]SO[4], or MgCl[2]), the standard Pitzer approach use 3 parameters (i.e. omit the β^(2) term) and α[1] is equal to 2.0. For 2-2 type of electrolytes
the model includes the β^(2) parameter and α[1] equals to 1.4 and α[2] equals to 12. This approach provides accurate models for many 2-2 binary sulfate (Pitzer and Mayorga 1974; Christov 1999, 2003a)
and selenate (Christov et al. 1998; Barkov et al. 2001; Christov 2003a) electrolytes, giving excellent representation of activity data covering the entire concentration range from low molality up to
saturation and beyond.
To describe the high concentration solution behaviour of systems showing a “smooth” maximum on γ[±] vs. m dependence, and to account for strong association reactions at high molality, Christov
(1996b, 1998a, b, 1999, 2005) used a very simple modelling technology: introducing into a model a fourth ion interaction parameter from basic Pitzer theory {β^(2) }, and varying the values of α[1]
and α[2] terms (see Eqns. (3) and (3A) in Donchev et al. 2021b). According to previous studies of Christov, an approach with 4 ion interaction parameters (β^(0),β^(1), β^(2),and C^ϕ), and accepting α
[1] = 2, and varying in α[2] values can be used for solutions for which ion association occurs in high molality region. This approach was used for binary electrolyte systems of different type: 1-1
type {such as HNO[3]-H[2]O, LiNO[3]-H[2]O (Donchev and Christov 2020), and LiCl-H[2]O (Lassin et al. 2015)}, 2-1 {such as NiCl[2]-H[2]O, CuCl[2]-H[2]O, MnCl[2]-H[2]O, CoCl[2]-H[2]O: (Christov and
Petrenko 1996; Christov 1996b, 1999); Ca(NO[3])[2]-H[2]O: (Lach et al. 2018); 1-2 {such as K[2]Cr[2]O[7]-H[2]O: (Christov 1998)}, 3-1 {such as FeCl[3]-H[2]O: (Christov 2005), and 3-2 {such as Al[2]
(SO[4])[3]-H[2]O, Cr[2](SO[4])[3]-H[2]O, and Fe[2](SO[4])[3]-H[2]O: (Christov 2002, 2005)}. The resulting models reduce the sigma values of fit of experimental activity data, and extend the
application range of models for binary systems to the highest molality, close or equal to molality of saturation {m(sat)}, and in case of data availability: up to supersaturation.
In this study we developed new, not concentration restricted thermodynamic models for solution behavior and solid-liquid equilibrium in CsF-H[2]O, CsOH-H[2]O and Cs[2]SO[4]- H[2]O systems at 25 °C.
The new models are developed on the basis of Pitzer ion interactions approach. To parameterize models for cesium binary systems we used all available experimental osmotic coefficients data for whole
concentration range of solutions, and up to saturation point. Raw data at low molality from Hamer and Wu (1972) and Mikulin (1968), and extrapolated data from Mikulin (1968) are used to parameterize
the model for CsF-H[2]O system. The model for CsOH-H[2]O has been constructed using low molality data from Hamer and Wu (1972) and Mikulin (1968), and osmotic coefficients data-point at saturation
from Mikulin (1968). The new model for Cs[2]SO[4]- H[2]O system has been developed using low molality data from Palmer at al. (2002) and Mikulin (1968), and extrapolated osmotic coefficients data-up
to saturation from Mikulin (1968). To construct the models, we used different versions of standard molality-based Pitzer approach. It was established that for CsF-H[2]O system application of extended
approach with 4 parameters (β^(0), β^(1), β^(2) and C^ϕ) and variation of α[1] and α[2] terms in fundamental Pitzer equations leads to the lowest values of standard model-experiment deviation. For
CsOH-H[2]O and Cs[2]SO[4]- H[2]O system a standard approach with 3 interaction parameters was used. The predictions of new developed here models are in excellent agreement with experimental osmotic
coefficients data (ϕ) in binary solutions from low to extremely high concentration (up to 21.8 mol.kg^-1 for CsOH-H[2]O, and up to 35.6 mol.kg^-1 for CsF-H[2]O) (see Fig. 1a, b, g, h, k). As it is
shown on Fig. 1 for CsF-H[2]O, CsOH-H[2]O systems the new models are in pure agreement at high concentration with the low molality models of Pitzer and Mayorga (1973). For Cs[2]SO[4]- H[2]O system
the new model is again in pure agreement at high concentration with the low molality models of Palmer et al. (2002) and Scharge et al. (2012). New activity data are needed to validate the model for
this binary.
(a,b,c,d,e,f,g,h,i,j,k). Comparison of model calculated (lines) for activity coefficients (Fig.i) and for osmotic coefficients (ϕ) in cesium binary solutions (CsF-H[2]O, CsCl-H[2]O, CsBr- H[2]O,
CsI-H[2]O, CsOH-H[2]O, CsNO[3]-H[2]O, Cs[2]SO[4]- H[2]O, and Cs[2]SeO[4]- H[2]O) against molality at T = 298.15 K with recommendations in literature (symbols). For CsF-H[2]O (Fig. b) and CsOH-H[2]O
(Fig. h) systems an enlargement of the low molality corner is also given. Heavy solid lines represent the predictions of the developed in this study (for CsF-H[2]O, CsOH-H[2]O, and Cs[2]SO[4]- H[2]O
systems) and previously reported and accepted models constructed by Christov and co-authors (Christov 2003a, 2005; Balarew et al. 1993; Barkov et al. 2001; Donchev and Christov 2020) and by Pitzer
and Mayorga (1973) (for CsI-H[2]O). Dashed-dotted, dashed and light solid lines represent the predictions of the reference models of Pitzer and Mayorga (1973) (as P&M on Fig. a,b,c,f, g and h), of
Scharge et al. (2012) (for CsCl-H[2]O and for Cs[2]SO[4]- H[2]O (Fig. c, d and k)), and of Palmer et al. (2002) (for Cs[2]SO[4]- H[2]O (Fig. k)) and of YMTB (given as YM on Fig. c and g) (Sandia
National Laboratories (2005). Experimental data (symbols) are from Hamer and Wu (1972) (for 1-1 systems), Robinson and Stokes (1959), Mikulin (1968), Palmer et al. (2002) (for Cs[2]SO[4]- H[2]O),
Partanen (2010) (recommended values for CsI-H[2]O) and from Barkov et al. (2001) (for Cs[2]SeO[4]- H[2]O). The molality of stable crystallization of solid cesium phases is given on all figures by
vertical lines (see Table 1 for m(sat) sources).
The previously developed by Christov (2003a, 2005), and Christov and co-authors (Balarew et al. 1993; Barkov et al. 2001; Donchev and Christov 2020) and by other authors (Pitzer and Mayorga 1973; Kim
and Frederick 1988; Sharge et al. 2012) Pitzer approach based thermodynamic models for five (5) cesium binary systems (CsCl-H[2]O, CsBr-H[2]O, CsI-H[2]O, CsNO[3]-H[2]O, and Cs[2]SeO[4]- H[2]O) are
tested in this study by comparison with experimental osmotic coefficients data and with recommendations on activity coefficients (γ[±]) (for CsNO[3]-H[2]O) in binary solutions (Fig. 1). The models
which give the best agreement with (ϕ)-, and (γ[±]) - data from low to high concentration, up to m(sat), are accepted as correct models, which can be used for solubility calculations in binary and
mixed systems and determination of thermodynamic characteristics of precipitating cesium solid phases. The following models are accepted as correct models: model of Balarew et al. (1993) and Christov
et al. (1994) for CsCl-H[2]O, and CsBr-H[2]O systems (see heavy solid line on Fig. 1c,d,e); model of Pitzer and Mayorga (1973) for CsI-H[2]O (see heavy solid line on Fig. 1f); model of Donchev and
Christov (2020) for CsNO[3]-H[2]O (see heavy solid line on Fig. 1i), and the model of Barkov et al. (2001) for Cs[2]SeO[4]- H[2]O system (see heavy solid line on Fig. 1j).
On the basis of evaluated previously and accepted models (see previous paragraph) and evaluated in this study binary parameters we determine water activity (a[w]) and Deliquescence Relative Humidity
(DRH (%)) of solid phases crystallizing from saturated binary solutions. According to Christov (2009, 2012), Donchev and Christov (2020) and Donchev et al. (2021b): DRH (%) = a[w] (sat) × 100; where
a[w] (sat) is activity of water at saturation. The results of DRH calculations are given in Table 1. The DRH predictions of new and accepted models are in excellent agreement with the experimental
data determined using isopiestic method, and given in Mikulin (1968). According to model calculations the solid-liquid phase change of CsF(s), occurs at extremely low relative humidity of
environment. As a next step, using the accepted and new developed parameterizations, and experimentally determined molalities (m(sat)) of the saturated binary solutions (Mikulin 1968; Balarew et al.
1993; Barkov et al. 2001; Palmer et al. 2002) we calculate the logarithm of the thermodynamic solubility product (ln K^o[sp]) of cesium solid phases crystallizing from saturated binary solutions at
25 °C. The calculation approach is the same as in Christov (1995a, 1996a, 2005, 2009, 2012), in Donchev and Christov (2020), and in Donchev et al. (2021b). The model calculations are given in Table
Model calculated logarithm of the thermodynamic solubility product (as lnK^o[sp]), and model calculated and recommended values of the Deliquescence Relative Humidity (DRH) of the of cesium solid
phases crystallizing from saturated binary solutions at T = 25 °C.
Salt composition m (sat) (exp) (mol.kg^-1) Calculated lnK^o[sp] DRH(%)
Calculated Experimental data^a
CsF (cr) 35.6^a 14.74 2.46 4.0
CsCl (cr) 11.37^b 3.49 65.69 65.80
CsBr(cr) 5.79^b 1.905 82.62 82.6
CsI(cr) 3.305^a 0.675 90.71 90.60
CsOH(cr) 21.8^a 6.067 66.57 -
CsNO[3](cr) 1.40^a -1.328^e 96.54^e 96.50
0.9424^f 80.60^f
Cs[2]SO[4](cr) 5.0^c 1.971^g 74.59 ^g 80.40
1.486^h 76.74^h
Cs[2]SeO[4](cr) 6.34^d 1.45 72.86 -
aExperimental data of Mikulin (1968); bExperimental data of Balarew et al. (1993) and Christov (2005); cExperimental data of Palmer et al. (2002) and Christov (2003,2005); dExperimental data of
Barkov et al. (2001) and Christov (2003); eFrom Donchev and Christov (2020);
^fCalculated using binary parameters determinate in this study (heavy solid line on Fig. 1k);
^gCalculated using binary parameters from Palmer et al. (2002) (dashed line on Fig. 1k);
^hCalculated using 4 parameters model of Sharge et al. (2012) (light solid line on Fig. 1k).
In previous studies of Christov (1996bc, 2003a, 2005) and Christov and co-authors (Balarew et al. 1993; Christov et al. 1994; Christov and Petrenko 1996; Barkov et al. 2001), a solid-liquid
equilibrium Pitzer approach models for 15 cesium mixed ternary (CsCl-MgCl[2]-H[2]O, CsBr-MgBr[2]-H[2]O, CsCl-NiCl[2]-H[2]O, CsBr-NiBr[2]-H[2]O, CsCl-MnCl[2]-H[2]O, CsCl-CoCl[2]-H[2]O, CsCl-CuCl[2]-H
[2]O, CsCl-CsBr-H[2]O, CsCl-RbCl-H[2]O, Cs[2]SO[4]-CoSO[4]-H[2]O, Cs[2]SeO[4]-CoSeO[4]-H[2]O, Cs[2]SO[4]-NiSO[4]-H[2]O, Cs[2]SeO[4]-NiSeO[4]-H[2]O, Cs[2]SO[4]-ZnSO[4]-H[2]O, and Cs[2]SeO[4]-ZnSeO[4]
-H[2]O) systems at 25 °C are reported. The validated here parameterization for binary systems CsCl-H[2]O, CsBr- H[2]O, and Cs[2]SeO[4]- H[2]O have been used without adjustment to develop a model for
mixed systems. The Pitzer mixing ion interaction parameters (θ(Cs,M^2+) and ψ(Cs,M^2+,X) for the cesium common anion ternary systems have been evaluated on the basis of the experimental data on the
compositions of the saturated ternary solutions, i.e. using “ solubility approach” (Harvie et al. 1984; Christov 1995a, 1996a, b, 1998, 1999, 2005, 2012).
The values of evaluated mixing parameter are summarized in Table 2. The mixed solution models are developed using our own solubility data (Balarew et al. 1993; Barkov et al. 2001), or the reference
data from Zdanovskii et al. (2003), and Silcock (1979). The choice of the mixing parameters is based on the minimum deviation of the logarithm of the solubility product (lnK^o[sp]) for the whole
crystallization curve of the component from its value for the binary solution. See Table 1 for lnK^o[sp] values for cesium simple salts. In addition, the lnK^o[sp] value for the cesium double salts
crystallizing from the saturated ternary solutions has to be constant along the whole crystallization branch of the double salt. Since the parameters θ(M,M’) take into account only the ionic
interactions of the type M-M’ in mixing solutions, their values have to be constant for the chloride, bromide, sulfate and selenate solutions with the same cations (M^+ and M^2+). Therefore, for
common cation systems in constructing the mixing model, we keep the same value of θ(M,M’), and only the ψ(M,M’,X) have been varied. In our θ and ψ evaluation the unsymmetrical mixing terms (^Eθ and ^
Eθ’) have been included (2003a, b, 2005). Mixing solution parameters for systems with precipitation of solid solutions (CsCl-CsBr-H[2]O and CsCl-RbCl-H[2]O) calculated by using the Zdanovskii rule
(Christov et al. 1994; Christov 1996c, 2005) are also given in Table 2.
Solutions mixing parameters [q(Cs,M^2+) and y(Cs,M^2+,X)] evaluated on the basis of the m (sat) molality in cesium common anion ternary systems at 25 °C.
System q(Cs,M^2+) y(Cs,M^2+,X) Reference
CsCl-MgCl2-H2O -0.1260 0.0000 Balarew et al. (1993)
CsBr-MgBr2-H2O -0.1260 -0.0367 Balarew et al. (1993)
CsCl-MnCl2-H2O 0.00 0.00 Christov and Petrenko (1996)
CsCl-CoCl2-H2O 0.00 0.00 Christov and Petrenko (1996)
Cs[2]SO[4]-CoSO[4]-H[2]O^a (I) 0.00 (I)-0.09 Christov (2003a, 2005)
(II) -0.05 (II) -0.04
Cs[2]SeO[4]-CoSeO[4]-H[2]O^a (I) 0.00 (I) 0.04 Christov (2003a, 2005)
(II) -0.05 (II) -0.02
CsCl-NiCl2-H2O -0.23 0.0000 Christov (1996b)
CsBr-NiBr2-H2O -0.23 -0.0199 Christov (1996b)
Cs[2]SeO[4]-NiSO[4]-H[2]O^a (I) – 0.23 (I) 0.015 Christov (2003a, 2005)
(II) -0.05 (II) -0.05
Cs[2]SeO[4]-NiSeO[4]-H[2]O^a (I) – 0.23 (I) 0.015 Barkov et al. (2001)
(II) -0.05 (II) -0.13 Christov (2003a, 2005)
Cs[2]SO[4]-ZnSO[4]-H[2]O -0.05 -0.05 Christov (2003a)
Cs[2]SeO[4]-ZnSeO[4]-H[2]O -0.05 -0.08 Christov (2003a)
CsCl-CuCl2-H2O 0.00 -0.050 Christov and Petrenko (1996)
CsCl-CsBr-H[2]O^b -0.0001 0.00001 Christov (1996c, 2005)
CsCl-RbCl-H[2]O^b 0.00025 -0.00060 Christov et al. (1994)
^a Two sets of mixing parameters (I and II) are evaluated in Christov (2003, 2005); ^bMixing solution parameters calculated by using the Zdanovskii rule (Christov et al. (1994); Christov (1996c,
In this study we developed new, not concentration restricted thermodynamic models for solution behavior and solid-liquid equilibrium in CsF-H[2]O, CsOH-H[2]O and Cs[2]SO[4]- H[2]O systems at 25 °C.
To parameterize models for cesium binary systems we used all available experimental osmotic coefficients data for whole concentration range of solutions, and up to saturation point. The new models
are developed on the basis of Pitzer ion interactions approach. To construct the models, we used different versions of standard molality-based Pitzer approach. It was established that for CsF-H[2]O
system application of extended approach with 4 parameters (β^(0), β^(1), β^(2) and C^ϕ) and variation of α[1] and α[2] terms in fundamental Pitzer equations leads to the lowest values of standard
model-experiment deviation. The predictions of new developed here models are in excellent agreement with experimental osmotic coefficients data (ϕ) in binary solutions from low to extremely high
concentration (up to 21.8 mol.kg^-1 for CsOH-H[2]O, and up to 35.6 mol.kg^-1 for CsF-H[2]O). The previously developed Pitzer approach based thermodynamic models for five (5) cesium binary systems
(CsCl-H[2]O, CsBr- H[2]O, CsI-H[2]O, CsNO[3]-H[2]O, and Cs[2]SeO[4]- H[2]O) are tested by comparison with experimental osmotic coefficients data and with recommendations on activity coefficients (γ
[±]) in binary solutions. The models which give the best agreement with (ϕ)-, and (γ[±]) data from low to high concentration, up to m(sat), are accepted as correct models, which can be used for
solubility calculations in binary and mixed systems and determination of thermodynamic characteristics of cesium solid phases. The thermodynamic solubility products (ln K^o[sp]), and the
Deliquescence Relative Humidity (DRH) of solid phases, precipitating from saturated cesium binary solutions (CsF(cr), CsCl(cr), CsBr(cr), CsI(cr), CsOH(cr), CsNO[3](cr), Cs[2]SO[4](cr), and Cs[2]SeO
[4](cr)) have been determined on the basis of evaluated binary parameters and using experimental solubility data. The previously established and validated here parameterization for binary systems
CsCl-H[2]O, CsBr- H[2]O, Cs[2]SO[4]- H[2]O, and Cs[2]SeO[4]- H[2]O have been used without adjustment to develop a solid-liquid equilibrium model for 15 cesium mixed ternary (CsCl-MgCl[2]-H[2]O,
CsBr-MgBr[2]-H[2]O, CsCl-NiCl[2]-H[2]O, CsBr-NiBr[2]-H[2]O, CsCl-MnCl[2]-H[2]O, CsCl-CoCl[2]-H[2]O, CsCl-CuCl[2]-H[2]O, CsCl-CsBr-H[2]O, CsCl-RbCl-H[2]O Cs[2]SO[4]-CoSO[4]-H[2]O, Cs[2]SeO[4]-CoSeO[4]
-H[2]O, Cs[2]SO[4]-NiSO[4]-H[2]O, Cs[2]SeO[4]-NiSeO[4]-H[2]O, Cs[2]SO[4]-ZnSO[4]-H[2]O, and Cs[2]SeO[4]-ZnSeO[4]-H[2]O) systems at 25 °C. The evaluated previously mixing parameters [θ(Cs,M^2+) and ψ
(Cs,M^2+,X)], determined by solubility approach are tabulated. The models described in this study are of high importance in development of thermodynamic database needed for nuclear waste geochemical
storage. The models are also of significant importance for extracting cesium resources from saline waters.
We wish to thank the reviewers (Dr. Krasimir Kostov, Dr. Francisca Justel and anonymous reviewer) for their constructive suggestions and helpful comments. The manuscript was improved considerably
through their comments. The work was supported by the European Regional Development Fund, Project BG05M2OP001-1.001-0004, and by Shumen University Research Program, Project No. RD-08-131/04.02.2021.
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Cylinder on a decelerating truck
• Thread starter Karol
• Start date
In summary, the cylinder attached to a weight moves without sliding, and there is no friction between the weight and the truck. The linear acceleration of the cylinder in reference to the truck is
given by:
Homework Statement
A truck with a cylinder of mass M and moment of inertia ##I=kMR^2## on top has initial velocity v[0] and decelerates with deceleration B. the cylinder is attached with a rope to a weight m. the
coefficient of friction is μ.
The outer radius of the cylinder is R and the smaller is r. The cylinder moves without sliding and there isn't friction between the weight m and the truck.
What is the relation between the angular acceleration of the cylinder and the linear acceleration of the weight.
What are the forces and the moments that act on the weight and the cylinder.
What is the linear acceleration of the cylinder in the reference frame of the truck.
What is the condition on B so that the cylinder will move to the right relative to the truck.
Homework Equations
Torque and moment of inertia: ##M=I\ddot{\theta}##
The Attempt at a Solution
The relation between the accelerations:
The tension in the rope is T. the forces:
$$\left\{ \begin{array}{l} MBR+Tr=kMR^2\cdot \frac{\ddot{\theta}}{r} \\ mg-T=m\ddot{y} \end{array} \right.$$
$$\rightarrow MBR+m\left( g+\ddot{y} \right)r=kMR^2\cdot \frac{\ddot{\theta}}{r}$$
$$\ddot{y}=\frac{r\left( MBR+mgr \right)}{kMR^2+mr^2}$$
The linear acceleration of the cylinder in reference to the truck:
The condition on B:
$$\left\{ \begin{array}{l} mg-T=\ddot{y}m=\ddot{\theta}mR\rightarrow T=m\left( g-R\ddot{\theta} \right) \\ Tr-MBR=kMR^2\cdot \ddot{\theta} \end{array}\right.$$
$$\ddot{\theta}=0 \rightarrow B=\frac{mr}{MR}g$$
For the first part, it is not clear whether the acceleration of the mass is wanted relative to the truck or relative to the vround. Asuuming it's relative to the truck, I believe R should feature in
the answer. If the cylinder rotates an angle theta, how far does it move relative to the truck? How far does a horizontal piece of the string move?
The relation between the accelerations:
The tension in the rope is T. the forces:
$$\left\{ \begin{array}{l} MBR+Tr=kMR^2\cdot \ddot{\theta} \\ mg-T=m\ddot{y} \end{array} \right.$$
$$\rightarrow MBR+m\left( g+\ddot{y} \right)r=kMR^2\cdot \ddot{\theta}$$
The linear acceleration of the cylinder in reference to the truck: ##\ddot{x}=\ddot{y}##
The condition on B:
$$\left\{ \begin{array}{l} mg-T=\ddot{y}m=\ddot{\theta}mR\rightarrow T=m\left( g-R\ddot{\theta} \right) \\ Tr-MBR=kMR^2\cdot \ddot{\theta} \end{array}\right.$$
$$\ddot{\theta}=0 \rightarrow B=\frac{mr}{MR}g$$
It comes out also from ##\ddot{y}=0##
Karol said:
The relation between the accelerations: ##\frac{\ddot{y}}{R}=\ddot{\theta}##
Forget the motion of the truck for the moment and consider the cylinder rolling an angle theta, to the right say. The centre of the cylinder has moved ##R\theta## to the right, so the point on the
cylinder where the rope first contacts it has also moved ##R\theta## to the right. But there is now more rope wound on the cylinder. How much more? So how far has the rope moved?
Check the answer by considering the extreme cases, r=R, and r=0.
Karol said:
The tension in the rope is T. the forces: ##MBR+Tr=kMR^2\cdot \ddot{\theta}##
That's not right either. Again, let B=0 for the moment. Which way will the cylinder roll? Is the tension in the string the only force exerting a torque on the cylinder?
The relation between the accelerations:
The moment of inertia round the point of contact with roof:
The tension in the rope is T. in the frame of the truck d'alambert:
$$\left\{ \begin{array}{l} Tr-MBR=kMR^2(k+1)\cdot \ddot{\theta} \\ mg-T=m\ddot{y} \end{array} \right.$$
But i don't know to do it in laboratory frame since the forces are:
And they rotate the cylinder in the same direction, to the left, if i take moments round the center.
And how do i take the sign of ##\ddot{\theta}##? i never knew to do that. do i decide that counterclockwise is positive? but there are two forces, so which one is dominant? if θ rotates in it's
Last edited:
Karol said:
##Tr−MBR=kMR^2(k+1)⋅\ddot θ##
I assume the first k on the right is a typo.
How do you get Tr on the left? Isn't this supposed to be the torque the rope generates about the point of contact with the roof?
$$T(R-r)-MBR=MR^2(k+1)\cdot \ddot{\theta}$$
But still how do i solve the forces in the laboratory frame? see the other sketch
Karol said:
$$T(R-r)-MBR=MR^2(k+1)\cdot \ddot{\theta}$$
But still how do i solve the forces in the laboratory frame? see the other sketch
It's almost the same. You just have to include the truck's acceleration in the total angular acceleration resulting from the torque. I.e. it is a real acceleration instead of a fictitious force. If
you have trouble convincing yourself of the validity of this, you can put in a friction force and take moments about the cylinder's centre instead.
$$\left\{ \begin{array}{l} T(R-r)-MBR=MR^2(k+1)\cdot \ddot{\theta} \\ mg-T=m(R-r)\ddot{y} \end{array} \right.$$
$$\rightarrow \ddot{\theta}=\frac{mg(R-r)}{MR^2(k+1)+m(R-r)^2}$$
$$R\ddot{\theta}=B\rightarrow \ddot{\theta}=\frac{B}{R}$$
$$\left\{ \begin{array}{l} T(R-r)=\left[MR^2(k+1)-\frac{B}{R} \right]\ddot{\theta} \\ mg-T=m(R-r)\ddot{y} \end{array} \right.$$
But i don't get the same result.
It's strange that i made moments round the center for ##R\ddot{\theta}=B## but for T i made moments around the point of contact. is it allowed to combine them into one equation?
Karol said:
How do you get that? Isn't that saying the centre of the cylinder does not accelerate?
To simplify suppose there is no rope, only cylinder. if it were a box and the deceleration increases gradually the friction force f would grow until it would reach ##mg\mu##. as long as ##f<mg\mu##
the deceleration of the box is B and it stays in place.
If the cylinder was mass-less ##R\ddot{\theta}=B## would be true but with mass the cylinder will rotate slower, but how slow? what's the friction force? if B is big the cylinder will also slide.
Karol said:
If the cylinder was mass-less ##R\ddot{\theta}=B## would be true but with mass the cylinder will rotate slower, but how slow? what's the friction force? if B is big the cylinder will also slide.
Look for a relationship between B, R, r, ##\ddot\theta##, ##\ddot y##.
haruspex said:
Look for a relationship between B, R, r, ##\ddot\theta##, ##\ddot y##.
I see only kinematics, is it true? no gravitation, Mg?
Karol said:
I see only kinematics, is it true? no gravitation, Mg?
No, sorry, I got that wrong. (Making a lot of mistakes today.)
If you want to take moments about the cylinder's centre then you will need to introduce the force of friction between the cylinder and truck. This contributes both to the rotational acceleration and
the linear acceleration in the lab frame. It's the latter that relates to B, R, r, ##\ddot\theta##, ##\ddot y##
The acceleration of the center: ##B-R\ddot{\theta}##
Moments round the center:
$$M\left( B-R\ddot{\theta} \right)R+Tr=kMR^2\ddot\theta$$
But the friction f and T act in the same direction, relative to the center and it's not true, B rotates counter clockwise and T clockwise.
If i could take moments round the contact point for T and round the center for f it would be better.
Karol said:
The acceleration of the center: ##B-R\ddot{\theta}##
Moments round the center:
$$M\left( B-R\ddot{\theta} \right)R+Tr=kMR^2\ddot\theta$$
But the friction f and T act in the same direction, relative to the center and it's not true, B rotates counter clockwise and T clockwise.
If i could take moments round the contact point for T and round the center for f it would be better.
I have trouble keeping track of which way positive is being defined for these variables. For the equation in post #7, with which I agree, I believe B is defined as positive to the right and theta as
positive clockwise.
On that basis, if a is the linear acceleration of the cylinder's mass centre, to the right, in an inertial frame, ##a = B+R\ddot{\theta}##. If the frictional force on the cylinder is F to the right,
##F+T=Ma=MB+M\ddot\theta R##, ##kMR^2\ddot\theta=-FR-Tr##.
I believe that putting those together leads to the same equation as in post #7.
Inertial frame:
$$\left\{ \begin{array}{l} F+T=MB+M\ddot\theta R \\ kMR^2\ddot\theta=-FR-Tr \\ mg-T=m\ddot\theta (R-r) \end{array} \right.$$
$$\rightarrow \ddot\theta=\frac{mg(R-r)-MRB}{MR^2(k+1)+m(R-r)^2}$$
Truck's frame:
$$\left\{ \begin{array}{l} T(R-r)-MBR=MR^2(k+1)\cdot \ddot{\theta} \\ mg-T=m\ddot\theta (R-r) \end{array} \right.$$
It gives the same
haruspex said:
For the equation in post #7, with which I agree, I believe B is defined as positive to the right and theta as positive clockwise.
The equation in post #7 took moments round the contact point so if a force acts to the right θ is positive clockwise. but if i took moments round the center θ would have been positive counter
I just want to ask how do i know which way θ is positive, and i assume that for every equation it depends on the pivot point, the axis, right?
Karol said:
The equation in post #7 took moments round the contact point so if a force acts to the right θ is positive clockwise. but if i took moments round the center θ would have been positive counter
I just want to ask how do i know which way θ is positive, and i assume that for every equation it depends on the pivot point, the axis, right?
The positive direction for theta, as a variable, is however you choose to define it. You just need to be consistent. The choice is independent of your choices for the linear movements etc.
The value computed for theta may be positive or negative. If you defined it positive clockwise but the value of ##\ddot{\theta}## comes out negative then you know that in fact it rotates
haruspex said:
The positive direction for theta, as a variable, is however you choose to define it. You just need to be consistent. The choice is independent of your choices for the linear movements etc.
Doesn't the positive direction of the force define θ?
The equation in post #7:
$$T(R-r)-MBR=MR^2(k+1)\cdot \ddot{\theta}$$
I could say positive θ is counter clockwise and still the equation would be the same!
Karol said:
Doesn't the positive direction of the force define θ?
It determines which way the angular acceleration, as a value, will be positive, but how you choose to define the variable is up to you.
Consider gravity. It is common to define up as the positive direction. Most people then write that the weight of an object is -mg, g taking a positive value, like 9.8m/s
. But it is more logical to say that the weight is mg, and g will take the value -9.8m/s
Karol said:
I could say positive θ is counter clockwise and still the equation would be the same!
No. If T (as a force on the cylinder) is positive to the right then T(R-r) is a clockwise torque. Similarly MBR represents an anticlockwise torque (because of the way the variables in it are
defined), so -MBR is clockwise. If you now want to define θ (and hence ##\ddot\theta##) as anticlockwise then you will need to introduce a minus sign on the right.
haruspex said:
It determines which way the angular acceleration, as a value, will be positive
How does the direction of the force determine which way is the positive angle?
If you change the point of rotation, if you put the force under or above the point of rotation it changes the direction of torque, like in the drawing.
Or is there another method to decide.
Karol said:
How does the direction of the force determine which way is the positive angle?
If you change the point of rotation, if you put the force under or above the point of rotation it changes the direction of torque, like in the drawing.
Or is there another method to decide.
Suppose you have defined clockwise as positive and, for linear matters, positive to the right. If a horizontal force F is applied at the top of the circle then the torque is Fr. Check: if the force
is in fact to the right then the torque will indeed be clockwise. But note that that we have not actually fixed which way the force F acts. It may be that F has to be deduced from other information.
If in reality it acts to the left then its value will be negative. The torque it exerts will then turn out to be negative, i.e. it will in fact be anticlockwise.
If the force F is applied at the bottom then we must write the torque as -Fr.
This comes out a little more naturally in vectors. If the vertical unit vector is j, positive up, then in the top picture the torque is ##r\vec j \times \vec F## (I might have that backwards, but
I'll be consistent); in the bottom picture the displacement is ##-r \vec j##.
In short, it is not to do with the directions the forces and torques will turn out to have, it's just a consistent model for representing them.
haruspex said:
Suppose you have defined clockwise as positive and, for linear matters, positive to the right.
So i can decide independently for F and θ which is the positive direction?
So in the equation from post #7:
$$T(R-r)-MBR=MR^2(k+1)\cdot \ddot{\theta}$$
How do i know what sign to put on the right term?
Does it have to do with an assumption that i make on the resultant θ?
For example you said that you agree with that formula because θ is positive clockwise. do i have to assume the cylinder will actually turn to the right, or it doesn't have anything to do with the
resultant rotation?
Suppose i decide that θ is positive counterclockwise and i assume it will indeed rotate counterclockwise, then the equation will still be the same.
Karol said:
So i can decide independently for F and θ which is the positive direction?
Karol said:
How do i know what sign to put on the right term?
Because you have chosen T as positive right, theta as positive clockwise, and R-r is such that an actual positive T would lead to an actual positive angular acceleration.
Karol said:
Does it have to do with an assumption that i make on the resultant θ?
It does not depend on an assumption about how the answer will turn out.
Karol said:
Suppose i decide that θ is positive counterclockwise
An actual T to the right will still lead to an actual clockwise torque, so its effect on theta would then be to reduce it, so you'd need to write -##\ddot\theta##.
The question you have to ask yourself is one of consistency:
T takes a positive value in this equation will the torque be clockwise or counterclockwise? Will the equation give me a positive or negative angular acceleration? Does that match with the sense of
the torque?
This was the previous result from post #17:
$$\left\{ \begin{array}{l} T(R-r)-MBR=MR^2(k+1)\cdot \ddot{\theta} \\ mg-T=m\ddot\theta (R-r) \end{array} \right.$$
$$\rightarrow \ddot\theta=\frac{mg(R-r)-MRB}{MR^2(k+1)+m(R-r)^2}$$
Now i change the sign on the right:
$$\left\{ \begin{array}{l} T(R-r)-MBR=-MR^2(k+1)\cdot \ddot{\theta} \\ mg-T=m\ddot\theta (R-r) \end{array} \right.$$
$$\rightarrow \ddot\theta=\frac{MRB-mg(R-r)}{MR^2(k+1)-m(R-r)^2}$$
You see that the value, not only the sign, of ##\ddot\theta## has changed because the denominator is different, i didn't expect that.
So the decision which is the positive direction of the angular displacement changes the value of ##\ddot\theta##? not reasonable
Karol said:
This was the previous result from post #17:
$$\left\{ \begin{array}{l} T(R-r)-MBR=MR^2(k+1)\cdot \ddot{\theta} \\ mg-T=m\ddot\theta (R-r) \end{array} \right.$$
$$\rightarrow \ddot\theta=\frac{mg(R-r)-MRB}{MR^2(k+1)+m(R-r)^2}$$
Now i change the sign on the right:
$$\left\{ \begin{array}{l} T(R-r)-MBR=-MR^2(k+1)\cdot \ddot{\theta} \\ mg-T=m\ddot\theta (R-r) \end{array} \right.$$
$$\rightarrow \ddot\theta=\frac{MRB-mg(R-r)}{MR^2(k+1)-m(R-r)^2}$$
You see that the value, not only the sign, of ##\ddot\theta## has changed because the denominator is different, i didn't expect that.
So the decision which is the positive direction of the angular displacement changes the value of ##\ddot\theta##? not reasonable
If you change the direction in which the angle is measured then all occurrences of ##\ddot \theta## in the equations must flip sign.
haruspex said:
Because you have chosen T as positive right, theta as positive clockwise, and R-r is such that an actual positive T would lead to an actual positive angular acceleration.
An actual T to the right will still lead to an actual clockwise torque, so its effect on theta would then be to reduce it, so you'd need to write -##\ddot\theta##.
You mean to say that i have to consider the whole torque side, the left side of:
$$M=I\ddot\theta\rightarrow T(R-r)-MBR=MR^2(k+1)\cdot \ddot{\theta}$$
I have to consider not only if T gets a positive value, because there is also the term -MBR
haruspex said:
The question you have to ask yourself is one of consistency: if T takes a positive value in this equation will the torque be clockwise or counterclockwise? Will the equation give me a positive or
negative angular acceleration? Does that match with the sense of the torque?
So if the system, in this case the cylinder, actually yields to the torque and rotates in the same direction of the torque, that is what i have to consider, right? and by that direction to decide, in
accordance with the positive direction of θ that i have decided upon, which is the sigh of the right side of the equation ##M=I\ddot\theta##, the ##I\ddot\theta## side?
What is the condition on B that the cylinder will rotate right relative to the truck?
$$\ddot\theta>0\rightarrow mg(R-r)-MRB>0 \rightarrow B<\frac{mg(R-r)}{MR}$$
FAQ: Cylinder on a decelerating truck
1. What is a cylinder on a decelerating truck?
A cylinder on a decelerating truck refers to a cylindrical object that is placed on a truck and experiences a decrease in speed or velocity due to the truck slowing down.
2. Why is a cylinder used on a decelerating truck?
A cylinder is used on a decelerating truck to demonstrate the concept of inertia. Inertia is the tendency of an object to resist changes in its state of motion, and the cylinder on a decelerating
truck is a common example used to illustrate this concept.
3. How does a cylinder on a decelerating truck relate to Newton's First Law of Motion?
Newton's First Law of Motion states that an object at rest will remain at rest and an object in motion will remain in motion at a constant velocity unless acted upon by an unbalanced force. In the
case of a cylinder on a decelerating truck, the cylinder will continue to move forward at a constant velocity until it is acted upon by an unbalanced force (friction) from the truck's deceleration.
4. What factors affect the behavior of a cylinder on a decelerating truck?
The behavior of a cylinder on a decelerating truck can be affected by various factors such as the mass and shape of the cylinder, the speed and rate of deceleration of the truck, and the surface and
texture of the truck bed.
5. How can the experiment with a cylinder on a decelerating truck be used in real-world applications?
The concept of inertia demonstrated by a cylinder on a decelerating truck has practical applications in various fields such as transportation, engineering, and sports. Understanding inertia can help
in designing safer vehicles, improving athletic performance, and developing efficient machinery. | {"url":"https://www.physicsforums.com/threads/cylinder-on-a-decelerating-truck.816601/","timestamp":"2024-11-10T21:25:05Z","content_type":"text/html","content_length":"242127","record_id":"<urn:uuid:ad83fd58-e077-4148-909c-0d9302349468>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00689.warc.gz"} |
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The advantage of quantum annealing machines is they are much simpler to build than gate-model quantum computing machines. The latest D-Wave machine (the D-Wave 2X), installed at NASA Ames, is
approximately twice as powerful (in a quantum, exponential sense) as the previous model at over 1,000 qubits (1,097). This compares with roughly 10 qubits for current gate-model quantum systems, so
two orders of magnitude. It’s a question of scale, no simple task, and a unique achievement. Although quantum researchers initially questioned whether the D-Wave system even qualified as a quantum
computer, albeit a subset of quantum computing architectures, that argument seems mostly settled and it is now generally accepted that quantum characteristics have been adequately demonstrated.
In a D-Wave system, a coupled pair of qubits (quantum bits) demonstrate quantum entanglement (they influence each other), so that the entangled pair can be in any one of four states resulting from
how the coupling and energy biases are programmed. By representing the problem to be addressed as an energy map the most likely outcomes can be derived by identifying the lowest energy states.
A lattice of approximately 1,000 tiny superconducting circuits (the qubits) is chilled close to absolute zero to deliver quantum effects. A user models a problem into a search for the lowest point in
a vast energy landscape. The processor considers all possibilities simultaneously to determine the lowest energy required to form those relationships. Multiple solutions are returned to the user,
scaled to show optimal answers, in an execution time of around 20 microseconds, practically instantaneously for all intents and purposes.
The D-wave system cabinet – “The Fridge”– is a closed cycle dilution refrigerator. The superconducting processor itself generates no heat, but to operate reliably must be cooled to about 180 times
colder than interstellar space, approximately 0.015° Kelvin.
Environmental considerations: Green is the color
To function reliably, quantum computing systems require environments that are not only shielded from the Earth’s natural environment, but would be considered inhospitable to any known life form. A
high vacuum is required, a pressure 10 billion times lower than atmospheric pressure, and shielded to 50,000 times less than Earth’s magnetic field. Not exactly a normal office, datacenter, or HPC
facility environment.
On the other hand, the self-contained “Fridge” and servers consume just 25kW of power (approximately the power draw of a single heavily populated standard rack) and about three orders of magnitude
(1000 times) less power than the current number one system on the TOP500, including its cooling system. Perhaps a more significant consideration is that power demand is not anticipated to increase
significantly as it scales to several thousands of qubits and beyond.
In addition to doubling the number of qubits compared with the prior D-Wave system, the D-Wave 2X delivers lower noise in qubits and couples, delivering greater confidence in achieved results.
So much for the pictures, what about the conversations?
Now that we have largely moved beyond the debate of whether a D-Wave system is actually a quantum machine or not, then the question “OK, so what now?” could bring us back to chasing rabbits, although
this time inspired by the classic Jefferson Airplane song, “White Rabbit”:
“One algorithm makes you larger
And another makes you small
But do the ones a D-Wave processes
Do anything at all?”
That of course, is where the conversations begin. It may depend upon the definition of “useful” and also a comparison between “conventional” systems and quantum computing approaches. Even the fastest
supercomputer we can build using the most advanced traditional technologies can still only perform analysis by examining each possible solution serially, one solution at a time. This makes optimizing
complex problems with a large number of variables and large data sets a very time consuming business. By comparison, once a problem has been suitably constructed for a quantum computer it can explore
all the possible solutions at once and instantly identify the most likely outcomes.
If we consider relative performance then we begin to have a simplistic basis for comparison, at least for execution times. The QuAIL system was benchmarked for the time required to find the optimal
solution with 99% probability for different problem sizes up to 945 variables. Simulated Annealing (SA), Quantum Monte Carlo (QMC) and the D-Wave 2X were compared. Full details are available in the
paper referenced previously. Shown in the chart are the 50^th, 75^th and 85^th percentiles over a set of 100 instances. The error bars represent 95% confidence intervals from bootstrapping.
This experiment occupied millions of processor cores for several days to tune and run the classical algorithms for these benchmarks. The runtimes for the higher quantiles for the larger problem sizes
for QMC were not computed because the computational cost was too high.
The results demonstrate a performance advantage to the quantum annealing approach by a factor of 100 million compared with simulated annealing running on a single state of the art processor core. By
comparison the current leading system on the TOP500 has fewer than 6 million cores of any kind, implying a clear performance advantage for quantum annealing based on execution time.
The challenge and the next step is to explore the mapping of real world problems to quantum machines and to improve the programming environments, which will no doubt take a significant amount of work
and many conversations. New players will become more visible, early use cases and gaps will become better defined, new use cases will be identified, and a short stack will emerge to ease programming.
This is reminiscent of the early days of computing or space flight.
A quantum of solace for the TOP500: Size still matters.
Even though we don’t expect to see viable exascale systems this decade, and quite likely not before the middle of the next, we won’t be seeing a Quantum500 anytime soon either. NASA talks about
putting humans on Mars sometime in the 2030s and it isn’t unrealistic to think about practical quantum computing as being on a similar trajectory. Recent research at the University of New South Wales
(UNSW) in Sidney, Australia demonstrated that it may be possible to create quantum computer chips that could store thousands, even millions of qubits on a single silicon processor chip leveraging
conventional computer technology.
Although the current D-Wave 2X system is a singular achievement it is still regarded as being relatively small to handle real world problems, and would benefit from getting beyond pairwise
connectivity, but that isn’t really the point. It plays a significant role in research into areas such as vision systems, artificial intelligence and machine learning alongside its optimization
In the near term, we’ve got enough information and evidence to get the picture. It will be the conversations that become paramount with both conventional and quantum computing systems working
together to develop better algorithms and expand the boundaries of knowledge and achievement.
References and attributions:
Paper on What is the Computational Value of Finite Range Tunneling? http://arxiv.org/pdf/1512.02206v4.pdf
Paper on Benchmarking a quantum annealing processor with the time-to-target metric: http://www.dwavesys.com/sites/default/files/ttt_arxiv.pdf
White rabbit from Sir John Tenniel’s illustrations of “Alice’s Adventures in Wonderland” by Lewis Carroll
“White Rabbit” Lyrics copyright Jefferson Airplane/ Grace Slick
Peter ffoulkes – Partner at OrionX.net
Peter is an expert in Cloud Computing, Big Data, and HPC markets, technology trends, and customer requirements which he blends to craft growth strategies and assess opportunities.
3 Comments
1. Peter – I’m in the chasing rabbits camp 😉
There is world of difference between a simulated annealing problem abd say AI. To understand AI – don’t look to quantum mechanics / Penrose musing on intelligence – rather look at Steve Grand’s –
Creation: Life and how to make it. http://www.amazon.com/Creation-Life-How-Make-It/dp/0674011139?tag=duckduckgo-osx-20
□ Richard, Good to hear from you. We should catch up sometime, I’d like to hear about Paramus and and what you have been up to since we last spoke.
The AI work currently happening – such as the recent Go challenge – is fascinating, and can all be explored with current technology. On the other hand Google, Microsoft and others seem to
find some value in exploring what can be done with quantum approaches for optimization, hence the allusion to the need for conversations.
I don’t think anyone is suggesting that quantum approaches will replace other approaches in the foreseeable future, but rather that there is still much to be learned from all research into
these areas, and there could be some benefit from cross fertilization.
Thanks for taking an interest and responding. I would look forward to a chat sometime.
Peter ff
2. Interestingly enough, the quantum world and AI and machine learning are all in the news right now. Here’s a link to an interview with Microsoft researcher Krysta Svore on NPR’s Science Friday on
March 11, 2016. http://www.sciencefriday.com/segments/the-ultimate-parallel-processor-quantum-bits/
Perhaps we’ll end up with Quantum Cloud resources chasing intelligent robotic rabbits! Enjoy…
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About Channels
Submitted by mario on Tue, 17/01/2017 - 18:58
About Channels, or how Math is not really that hard
What are those "channels" anyway?
It's just a fancy name for program - kind of like washing machine has a program. Gecho has relatively large memory for code - 1MB, of which currrent firmware occupies <200kB. Hardware is able to run
more than one type of synthesis, it can also combine various inputs, outputs, sensors and processing to form an uniquely behaving mode of operation.
Some channels are passive, have no interaction - you can just listen to them. Other react to your input in various forms. Some are for testing and some for settings or programming your own content.
The list is long and expanding.
How many of them can we have?
Since they are selected by those four buttons, and we want more than four channels, you can press a button more than once, forming a longer number. For example, 11, 12, 13, 21, 111, or even 121314.
This allows you to take really any unused number and define your own functionality under that combination. Of course there is a limit on number of channels: you can press at most 20 of them in in the
sequence. And if it is 20 of them, then first one must be 1, otherwise it can be 1,2,3 or 4.
Because internally, an uint64_t variable type is used to store channel number. That variable can have maximum value of 18446744073709551616 (which is 2 to the power of 64). If you tried to write
number 28446744073709551616 there, it won't fit in.
Of course, we only have buttons 1 to 4, so we can't really have channels like 1844... What is the maximum number then? It's all possible ways you can press those four buttons, forming a sequence from
1 to 14444444444444444444. Ouch. How many is that?
Let's use some "divide and conquer" strategy.
Imagine we only had two buttons, and we want to find out how many combinations is there between 1 and 222. It is easy to enumerate them all: 1, 2, 11, 12, 21, 22, 111, 112, 121, 122, 211, 212, 221,
222. Note how there are two consiting of only one press of a button, then four consisting of two, and eight consisting of three.
What happens if number of buttons increases to 3? There will be 3 possible channels of single digit (1,2,3), 9 channels of 2 digits (11, 12, 13, 21, 22, 23, 31, 32, 33), and 27 channels of 3 digits
(too many to write here but trust me). Can you see the pattern emerging?
In case of two buttons, number of possible channels was 2 + 4 + 8, in case of three buttons, it was 3 + 9 + 27... or, if written another way, 2 + 2*2 + 2*2*2, 3 + 3*3 + 3*3*3. Why? Because, for
example if you have 3 buttons, and you want to press them for 3 times (a random one each time), you have 3 possibilities to choose first button, 3 possibilities to choose second, and three
possibilities to choose third one.
With four buttons, it is now easy to see, that all possible channels between 1 and 444 are calculated like 4 + 4*4 + 4*4*4. Which is 84.
If we wanted channels from 1 to 4444 instead, we need to add 4*4*4*4 to previous formula, to also include all channels that consist of 4 button presses. So, number of button presses = number of "4"s
in the multiplication.
Now, we are close to the final answer, we just need to expand this few times... about 16 more times, oh... And also, we don't want number of all channels between 1 and 44444444444444444444, right? It
was about 14444444444444444444 we wanted to know (because of uint64_t limit).
Let's split this problem in two - forget about the initial "1" for now. We are left with up to 19 random button presses: from 1 to 4444444444444444444. We already know how to calculate this:
4 + 4*4 + 4*4*4 + 4*4*4*4 + 4*4*4*4*4 + ... + 4*4*4*...*4*4*4 (19 "4"s in the last multiplication). Is there any short-hand notation? Sure there is: a sequence of multiplications of the same number x
is called a "power" - x to the power of y, written as x^y. Now we can write 4 + 4^2 + 4^3 + 4^4 + ... + 4^19.
It is easy enough to type that into a calculator or spreadsheet:
The total sum of all combinations here is 366503875924. That's a lot of possible channels, isn't it? And, some of those numbers may be familiar to you, because as powers of 4 they are also powers of
2, which we meet in computers or other electronics all the time.
So, what about that second half of the whole problem, an outstanding button "1" that could have been pressed at the very beginning? I leave it for you to finish - the prize for first correct answer
is a free wooden box :)
How do I add a new channel?
Unless you don't want to invent one from scratch (any C/C++ code can be wrapped in a function and assigned to a channel), the easiest way is to look at an existing one, clone it and modify.
A channel definition (these are stored in function custom_program_init() which you find in file custom_program.cpp) may typically look like this:
if(prog == 1122) //drum machine samples test
play_buffer((uint16_t*)0x080A8000, 137246, 1); //1=mono
Thanks to the above code, after pressing "1122" and SET button, Gecho will play all samples that are stored at certain memory address, up to given length, and then halt using infinite loop.
if(prog == 3331) //cv_gate_test
custom_program_init(2); //recursive init channel #2
//disable some controls
PROG_enable_S1_control_noise_boost = false;
PROG_enable_S2_control_noise_attenuation = false;
//test-enable CV and Gate
TEST_enable_V1_control_voice = true;
TEST_enable_V2_control_drum = true;
PROG_drum_machine = true;
ACTIVE_FILTERS_PAIRS = FILTERS/2 - 2; //need to set aside some computation power
Here, for example, we want to extend default functionality of channel #2 so we recursively call it's init function first. Then, we override parameters which we want different in our new channel. Our
goal is to change logic in which the first two ADC signals are parsed, effectively disabling IR proximity sensors and reacting to direct voltage on a given channel instead. Also, the number of active
filters is slightly decreased (from 16 to 12 voices), because together with direct playback from FLASH memory it would be too much - DAC would not get served by MCU with new samples on time.
A channel may also need to have other files edited, for example when you want to do something completely new. Examples of this are there, and should be understandable from the comments scattered in
the source code.
Thu, 29/06/2017 - 21:17 | {"url":"https://gechologic.com/channels-channels-everywhere","timestamp":"2024-11-02T16:57:01Z","content_type":"text/html","content_length":"22934","record_id":"<urn:uuid:38458786-6c69-455b-bd98-ec0ed9b90c3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00286.warc.gz"} |
Break a Rope
Under what conditions would a rope be most likely to break?
a. 20 men of equal strength with 10 pulling on each end.
b. 10 of the same men pulling on one end and the other end being fastened to a tree.
c. It makes no difference.
9 comments:
1. Anonymous3/03/2009 3:50 PM
c. .... I think. Newton helps us here.
2. Anonymous3/04/2009 12:11 AM
I think its c. because if 10 men where to pull on a tree, the tree would be exerting an equal but opposite force, taking on the exact roll of the 10 men on the other side in choice a.
3. As usual, I can't fool you. C is the answer, as explained above by you.
4. the answer should be (B)The tree would be stiff offering little if any give(depending on tree size and placement of the rope)(A)10 men pulling in oppisite directions would tend to heave & hoe
thus giving less resistance and fluxuation of stress on the rope.
5. Anonymous5/24/2009 8:45 PM
This comment has been removed by the author.
6. you didnt give any tree description..
it can be b or c..
a and c makes no difference..
but you need to break the rope..
but the answer depends on what type of tree..
7. C. the tree doesn't exert any force. contradicting to the men forces. :]
8. The answer being C is actually incorrect. It is in fact A. 10 people pulling in one direction, whether the other end is attached to a tree, brick wall, house, or any other non-moving in-animate
object can only exert the force of 10 people. The tree/wall/house/whatever can exert no force of its own. Contrast this with 10 people pulling in one direction and 10 pulling in the other,
maintaining that every single person exerts the exact same force, and you obtain double the pulling power of 10 people. Thus, the rope has a greater chance to break under teh combined pulling
force of 20 people. It really is simple mathematics.
9. My previous comment was actually incorrect as I was misusing Newton's third law. C would actually be the correct answer as the tree would exert a force equal to those pulling on the rope. If
however 20 people were pulling on one side of the rope, the rope would break more easily and this is what I was thinking. Direction does in fact matter.
Leave your answer or, if you want to post a question of your own, send me an e-mail. Look in the about section to find my e-mail address. If it's new, I'll post it soon.
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An Introduction to K-Means Clustering
K-means clustering is a popular method for grouping data by assigning observations to clusters based on proximity to the cluster’s center. This article explores k-means clustering, its importance,
applications, and workings, providing a clear understanding of its role in data analysis.
What is K-Means Clustering?
K-means clustering is a popular unsupervised machine learning algorithm used for partitioning a dataset into a pre-defined number of clusters. The goal is to group similar data points together and
discover underlying patterns or structures within the data.
Recall the first property of clusters – it states that the points within a cluster should be similar to each other. So, our aim here is to minimize the distance between the points within a cluster.
There is an algorithm that tries to minimize the distance of the points in a cluster with their centroid – the k-means clustering technique.
K-means is a centroid-based algorithm or a distance-based algorithm, where we calculate the distances to assign a point to a cluster. In K-Means, each cluster is associated with a centroid.
The main objective of the K-Means algorithm is to minimize the sum of distances between the points and their respective cluster centroid.
Optimization plays a crucial role in the k-means clustering algorithm. The goal of the optimization process is to find the best set of centroids that minimizes the sum of squared distances between
each data point and its closest centroid.
To learn more about clustering and other machine learning algorithms (both supervised and unsupervised) check out our AI/ML Blackbelt Plus Program!
How K-Means Clustering Works?
Here’s how it works:
1. Initialization: Start by randomly selecting K points from the dataset. These points will act as the initial cluster centroids.
2. Assignment: For each data point in the dataset, calculate the distance between that point and each of the K centroids. Assign the data point to the cluster whose centroid is closest to it. This
step effectively forms K clusters.
3. Update centroids: Once all data points have been assigned to clusters, recalculate the centroids of the clusters by taking the mean of all data points assigned to each cluster.
4. Repeat: Repeat steps 2 and 3 until convergence. Convergence occurs when the centroids no longer change significantly or when a specified number of iterations is reached.
5. Final Result: Once convergence is achieved, the algorithm outputs the final cluster centroids and the assignment of each data point to a cluster.
Objective of k means Clustering
The main objective of k-means clustering is to partition your data into a specific number (k) of groups, where data points within each group are similar and dissimilar to points in other groups. It
achieves this by minimizing the distance between data points and their assigned cluster’s center, called the centroid.
Here’s an objective:
• Grouping similar data points: K-means aims to identify patterns in your data by grouping data points that share similar characteristics together. This allows you to discover underlying structures
within the data.
• Minimizing within-cluster distance: The algorithm strives to make sure data points within a cluster are as close as possible to each other, as measured by a distance metric (usually Euclidean
distance). This ensures tight-knit clusters with high cohesiveness.
• Maximizing between-cluster distance: Conversely, k-means also tries to maximize the separation between clusters. Ideally, data points from different clusters should be far apart, making the
clusters distinct from each other.
What is Clustering?
Cluster analysis is a technique in data mining and machine learning that groups similar objects into clusters. K-means clustering, a popular method, aims to divide a set of objects into K clusters,
minimizing the sum of squared distances between the objects and their respective cluster centers.
Hierarchical clustering and k-means clustering are two popular techniques in the field of unsupervised learning used for clustering data points into distinct groups. While k-means clustering divides
data into a predefined number of clusters, hierarchical clustering creates a hierarchical tree-like structure to represent the relationships between the clusters.
Example of Clustering
Let’s try understanding this with a simple example. A bank wants to give credit card offers to its customers. Currently, they look at the details of each customer and, based on this information,
decide which offer should be given to which customer.
Now, the bank can potentially have millions of customers. Does it make sense to look at the details of each customer separately and then make a decision? Certainly not! It is a manual process and
will take a huge amount of time.
So what can the bank do? One option is to segment its customers into different groups. For instance, the bank can group the customers based on their income:
Can you see where I’m going with this? The bank can now make three different strategies or offers, one for each group. Here, instead of creating different strategies for individual customers, they
only have to make 3 strategies. This will reduce the effort as well as the time.
The groups I have shown above are known as clusters, and the process of creating these groups is known as clustering. Formally, we can say that:
Clustering is the process of dividing the entire data into groups (also known as clusters) based on the patterns in the data.
Can you guess which type of learning problem clustering is? Is it a supervised or unsupervised learning problem?
Think about it for a moment and use the example we just saw. Got it? Clustering is an unsupervised learning problem!
How is Clustering an Unsupervised Learning Problem?
Let’s say you are working on a project where you need to predict the sales of a big mart:
Or, a project where your task is to predict whether a loan will be approved or not:
We have a fixed target to predict in both of these situations. In the sales prediction problem, we have to predict the Item_Outlet_Sales based on outlet_size, outlet_location_type, etc., and in the
loan approval problem, we have to predict the Loan_Status depending on the Gender, marital status, the income of the customers, etc.
So, when we have a target variable to predict based on a given set of predictors or independent variables, such problems are called supervised learning problems.
Now, there might be situations where we do not have any target variable to predict.
Such problems, without any fixed target variable, are known as unsupervised learning problems. In these problems, we only have the independent variables and no target/dependent variable.
In clustering, we do not have a target to predict. We look at the data, try to club similar observations, and form different groups. Hence it is an unsupervised learning problem.
We now know what clusters are and the concept of clustering. Next, let’s look at the properties of these clusters, which we must consider while forming the clusters.
Properties of K means Clustering
How about another example of k-means clustering algorithm? We’ll take the same bank as before, which wants to segment its customers. For simplicity purposes, let’s say the bank only wants to use the
income and debt to make the segmentation. They collected the customer data and used a scatter plot to visualize it:
On the X-axis, we have the income of the customer, and the y-axis represents the amount of debt. Here, we can clearly visualize that these customers can be segmented into 4 different clusters, as
shown below:
This is how clustering helps to create segments (clusters) from the data. The bank can further use these clusters to make strategies and offer discounts to its customers. So let’s look at the
properties of these clusters.
First Property of K-Means Clustering Algorithm
All the data points in a cluster should be similar to each other. Let me illustrate it using the above example:
If the customers in a particular cluster are not similar to each other, then their requirements might vary, right? If the bank gives them the same offer, they might not like it, and their interest in
the bank might reduce. Not ideal.
Having similar data points within the same cluster helps the bank to use targeted marketing. You can think of similar examples from your everyday life and consider how clustering will (or already
does) impact the business strategy.
Second Property of K-Means Clustering Algorithm
The data points from different clusters should be as different as possible. This will intuitively make sense if you’ve grasped the above property. Let’s again take the same example to understand this
Which of these cases do you think will give us the better clusters? If you look at case I:
Customers in the red and blue clusters are quite similar to each other. The top four points in the red cluster share similar properties to those of the blue cluster’s top two customers. They have
high incomes and high debt values. Here, we have clustered them differently. Whereas, if you look at case II:
Points in the red cluster completely differ from the customers in the blue cluster. All the customers in the red cluster have high income and high debt, while the customers in the blue cluster have
high income and low debt value. Clearly, we have a better clustering of customers in this case.
Hence, data points from different clusters should be as different from each other as possible to have more meaningful clusters. The k-means algorithm uses an iterative approach to find the optimal
cluster assignments by minimizing the sum of squared distances between data points and their assigned cluster centroid.
So far, we have understood what clustering is and the different properties of clusters. But why do we even need clustering? Let’s clear this doubt in the next section and look at some applications of
Applications of Clustering in Real-World Scenarios
Clustering is a widely used technique in the industry. It is being used in almost every domain, from banking and recommendation engines to document clustering and image segmentation.
Customer Segmentation
We covered this earlier – one of the most common applications of clustering is customer segmentation. And it isn’t just limited to banking. This strategy is across functions, including telecom,
e-commerce, sports, advertising, sales, etc.
Document Clustering
This is another common application of clustering. Let’s say you have multiple documents and you need to cluster similar documents together. Clustering helps us group these documents such that similar
documents are in the same clusters.
Image Segmentation
We can also use clustering to perform image segmentation. Here, we try to club similar pixels in the image together. We can apply clustering to create clusters having similar pixels in the same
Also Read: A Step-by-Step Guide to Image Segmentation Techniques
Recommendation Engines
Clustering can also be used in recommendation engines. Let’s say you want to recommend songs to your friends. You can look at the songs liked by that person and then use clustering to find similar
songs and finally recommend the most similar songs.
There are many more applications that I’m sure you have already thought of. You can share these applications in the comments section below. Next, let’s look at how we can evaluate our clusters.
Understanding the Different Evaluation Metrics for Clustering
The primary aim of clustering is not just to make clusters but to make good and meaningful ones. We saw this in the below example:
Here, we used only two features, and hence it was easy for us to visualize and decide which of these clusters was better.
Unfortunately, that’s not how real-world scenarios work. We will have a ton of features to work with. Let’s take the customer segmentation example again – we will have features like customers’
income, occupation, gender, age, and many more. We would not be able to visualize all these features together and decide on better and more meaningful clusters.
This is where we can make use of evaluation metrics. Let’s discuss a few of them and understand how we can use them to evaluate the quality of our clusters.
Recall the first property of clusters we covered above. This is what inertia evaluates. It tells us how far the points within a cluster are. So, inertia actually calculates the sum of distances of
all the points within a cluster from the centroid of that cluster. Normally, we use Euclidean distance as the distance metric, as long as most of the features are numeric; otherwise, Manhattan
distance in case most of the features are categorical.
We calculate this for all the clusters; the final inertial value is the sum of all these distances. This distance within the clusters is known as intracluster distance. So, inertia gives us the sum
of intracluster distances:
Now, what do you think should be the value of inertia for a good cluster? Is a small inertial value good, or do we need a larger value? We want the points within the same cluster to be similar to
each other, right? Hence, the distance between them should be as low as possible.
Keeping this in mind, we can say that the lesser the inertia value, the better our clusters are.
Dunn Index
We now know that inertia tries to minimize the intracluster distance. It is trying to make more compact clusters.
Let me put it this way – if the distance between the centroid of a cluster and the points in that cluster is small, it means that the points are closer to each other. So, inertia makes sure that the
first property of clusters is satisfied. But it does not care about the second property – that different clusters should be as different from each other as possible.
This is where the Dunn index comes into action.
Along with the distance between the centroid and points, the Dunn index also takes into account the distance between two clusters. This distance between the centroids of two different clusters is
known as inter-cluster distance. Let’s look at the formula of the Dunn index:
Dunn index is the ratio of the minimum of inter-cluster distances and maximum of intracluster distances.
We want to maximize the Dunn index. The more the value of the Dunn index, the better the clusters will be. Let’s understand the intuition behind the Dunn index:
In order to maximize the value of the Dunn index, the numerator should be maximum. Here, we are taking the minimum of the inter-cluster distances. So, the distance between even the closest clusters
should be more which will eventually make sure that the clusters are far away from each other.
Also, the denominator should be minimum to maximize the Dunn index. Here, we are taking the maximum of all intracluster distances. Again, the intuition is the same here. The maximum distance between
the cluster centroids and the points should be minimum, eventually ensuring that the clusters are compact.
Silhouette Score
The silhouette score and plot are used to evaluate the quality of a clustering solution produced by the k-means algorithm. The silhouette score measures the similarity of each point to its own
cluster compared to other clusters, and the silhouette plot visualizes these scores for each sample. A high silhouette score indicates that the clusters are well separated, and each sample is more
similar to the samples in its own cluster than to samples in other clusters. A silhouette score close to 0 suggests overlapping clusters, and a negative score suggests poor clustering solutions.
How to Apply K-Means Clustering Algorithm?
Let’s now take an example to understand how K-Means actually works:
Time needed: 10 minutes
We have these 8 points, and we want to apply k-means to create clusters for these points. Here’s how we can do it.
1. Choose the number of clusters k
The first step in k-means is to pick the number of clusters, k.
2. Select k random points from the data as centroids
Next, we randomly select the centroid for each cluster. Let’s say we want to have 2 clusters, so k is equal to 2 here. We then randomly select the centroid:
Here, the red and green circles represent the centroid for these clusters.
3. Assign all the points to the closest cluster centroid
Once we have initialized the centroids, we assign each point to the closest cluster centroid:
Here you can see that the points closer to the red point are assigned to the red cluster, whereas the points closer to the green point are assigned to the green cluster.
4. Recompute the centroids of newly formed clusters
Now, once we have assigned all of the points to either cluster, the next step is to compute the centroids of newly formed clusters:
Here, the red and green crosses are the new centroids.
5. Repeat steps 3 and 4
We then repeat steps 3 and 4:
The step of computing the centroid and assigning all the points to the cluster based on their distance from the centroid is a single iteration. But wait – when should we stop this process? It
can’t run till eternity, right?
Stopping Criteria for K-Means Clustering
There are essentially three stopping criteria that can be adopted to stop the K-means algorithm:
1. Centroids of newly formed clusters do not change
2. Points remain in the same cluster
3. Maximum number of iterations is reached
We can stop the algorithm if the centroids of newly formed clusters are not changing. Even after multiple iterations, if we are getting the same centroids for all the clusters, we can say that the
algorithm is not learning any new pattern, and it is a sign to stop the training.
Another clear sign that we should stop the training process is if the points remain in the same cluster even after training the algorithm for multiple iterations.
Finally, we can stop the training if the maximum number of iterations is reached. Suppose we have set the number of iterations as 100. The process will repeat for 100 iterations before stopping.
Implementing K-Means Clustering in Python From Scratch
Time to fire up our Jupyter notebooks (or whichever IDE you use) and get our hands dirty in Python!
We will be working on the loan prediction dataset that you can download here. I encourage you to read more about the dataset and the problem statement here. This will help you visualize what we are
working on (and why we are doing this). Two pretty important questions in any data science project.
First, import all the required libraries:
#import libraries
import pandas as pd
import numpy as np
import random as rd
import matplotlib.pyplot as plt
Now, we will read the CSV file and look at the first five rows of the data:
data = pd.read_csv('clustering.csv')
For this article, we will be taking only two variables from the data – “LoanAmount” and “ApplicantIncome.” This will make it easy to visualize the steps as well. Let’s pick these two variables and
visualize the data points:
Python Code:
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
data = pd.read_csv('clustering.csv')
X = data[["LoanAmount","ApplicantIncome"]]
#Visualise data points
plt.ylabel('Loan Amount (In Thousands)')
Steps 1 and 2 of K-Means were about choosing the number of clusters (k) and selecting random centroids for each cluster. We will pick 3 clusters and then select random observations from the data as
the centroids:
# Step 1 and 2 - Choose the number of clusters (k) and select random centroid for each cluster
#number of clusters
# Select random observation as centroids
Centroids = (X.sample(n=K))
plt.ylabel('Loan Amount (In Thousands)')
Here, the red dots represent the 3 centroids for each cluster. Note that we have chosen these points randomly, and hence every time you run this code, you might get different centroids.
Next, we will define some conditions to implement the K-Means Clustering algorithm. Let’s first look at the code:
# Step 3 - Assign all the points to the closest cluster centroid
# Step 4 - Recompute centroids of newly formed clusters
# Step 5 - Repeat step 3 and 4
diff = 1
for index1,row_c in Centroids.iterrows():
for index2,row_d in XD.iterrows():
for index,row in X.iterrows():
for i in range(K):
if row[i+1] < min_dist:
min_dist = row[i+1]
Centroids_new = X.groupby(["Cluster"]).mean()[["LoanAmount","ApplicantIncome"]]
if j == 0:
diff = (Centroids_new['LoanAmount'] - Centroids['LoanAmount']).sum() + (Centroids_new['ApplicantIncome'] - Centroids['ApplicantIncome']).sum()
Centroids = X.groupby(["Cluster"]).mean()[["LoanAmount","ApplicantIncome"]]
These values might vary every time we run this. Here, we are stopping the training when the centroids are not changing after two iterations. This is the most common convergence criteria used for the
K-Means clustering. We have initially defined the diff as 1, and inside the whole loop, we are calculating this diff as the difference between the centroids in the previous iteration and the current
When this difference is 0, we stop the training. Let’s now visualize the clusters we have got:
for k in range(K):
plt.ylabel('Loan Amount (In Thousands)')
Awesome! Here, we can clearly visualize three clusters. The red dots represent the centroid of each cluster. I hope you now have a clear understanding of how K-Means work.
However, there are certain situations where this algorithm might not perform as well. Let’s look at some challenges you can face while working with k-means.
Challenges With the K-Means Clustering Algorithm
The k value in k-means clustering is a crucial parameter that determines the number of clusters to be formed in the dataset. Finding the optimal k value in the k-means clustering can be very
challenging, especially for noisy data. The appropriate value of k depends on the data structure and the problem being solved. It is important to choose the right value of k, as a small value can
result in under-clustered data, and a large value can cause over-clustering.
Also, one of the common challenges we face while working with K-Means is that the size of clusters is different. Let’s say we have the following points:
The leftmost and the rightmost clusters are of smaller size compared to the central cluster. Now, if we apply k-means clustering on these points, the results will be something like this:
Another challenge with k-means is when the densities of the original points are different. Let’s say these are the original points:
Here, the points in the red cluster are spread out, whereas the points in the remaining clusters are closely packed together. Now, if we apply k-means on these points, we will get clusters like this:
We observe that tightly packed points are grouped into one cluster, while loosely spread points, previously in the same cluster, are now assigned to different clusters. Not ideal, so what can we do
about this?
One of the solutions is to use a higher number of clusters.
So, in all the above scenarios, instead of using 3 clusters, we can have a bigger number. Perhaps setting k=10 might lead to more meaningful clusters.
Remember how we randomly initialize the centroids in k-means clustering? Well, this is also potentially problematic because we might get different clusters every time. So, to solve this problem of
random initialization, there is an algorithm called K-Means++ that can be used to choose the initial values, or the initial cluster centroids, for K-Means.
Determining the optimal number of clusters for k-means clustering can be another challenge.
Optimal number heavily relies on subjective interpretations and the underlying structure of the data. One commonly used method to find the optimal number of clusters is the elbow method, which plots
the sum of squared Euclidean distances between data points and their cluster center and chooses the number of clusters where the change in the sum of squared distances begins to level off.
Outliers can have a significant impact on the results of k-means clustering, as the algorithm is sensitive to extreme values.
This makes it important to identify and handle outliers before applying k-means clustering to ensure that the results are meaningful and not skewed by the presence of outliers. There are various
methods to identify and handle outliers, such as removing them, transforming them, or using a robust variant of k-means clustering that is less sensitive to the presence of outliers.
The algorithm can handle millions of data points and produce results in a matter of seconds or minutes, making it a popular choice for analyzing big data. However, as the size of the data set
increases, the computational cost of k-means clustering can also increase. Hence, it is important to consider alternative algorithms when working with extremely large data sets.
K-Means++ to Choose Initial Cluster Centroids for K-Means Clustering
In some cases, if the initialization of clusters is not appropriate, K-Means can result in arbitrarily bad clusters. This is where K-Means++ helps. It specifies a procedure to initialize the cluster
centers before moving forward with the standard k-means clustering algorithm.
Using the K-Means++ algorithm, we optimize the step where we randomly pick the cluster centroid. We are more likely to find a solution that is competitive with the optimal K-Means solution while
using the K-Means++ initialization.
Steps to Initialize the Centroids Using K-Means++
1. The first cluster is chosen uniformly at random from the data points we want to cluster. This is similar to what we do in K-Means, but instead of randomly picking all the centroids, we just pick
one centroid here
2. Next, we compute the distance (D(x)) of each data point (x) from the cluster center that has already been chosen
3. Then, choose the new cluster center from the data points with the probability of x being proportional to (D(x))2
4. We then repeat steps 2 and 3 until k clusters have been chosen
Let’s take an example to understand this more clearly. Let’s say we have the following points, and we want to make 3 clusters here:
Now, the first step is to randomly pick a data point as a cluster centroid:
Let’s say we pick the green point as the initial centroid. Now, we will calculate the distance (D(x)) of each data point with this centroid:
The next centroid will be the one whose squared distance (D(x)2) is the farthest from the current centroid:
In this case, the red point will be selected as the next centroid. Now, to select the last centroid, we will take the distance of each point from its closest centroid, and the point having the
largest squared distance will be selected as the next centroid:
We will select the last centroid as:
We can continue with the K-Means algorithm after initializing the centroids. Using K-Means++ to initialize the centroids tends to improve the clusters. Although it is computationally costly relative
to random initialization, subsequent K-Means often converge more rapidly.
I’m sure there’s one question that you’ve been wondering since the start of this article – how many clusters should we make? In other words, what should be the optimum number of clusters to have
while performing K-Means?
How to Choose the Right Number of Clusters in K-Means Clustering?
One of the most common doubts everyone has while working with K-Means is selecting the right number of clusters.
So, let’s look at a technique that will help us choose the right value of clusters for the K-Means algorithm. Let’s take the customer segmentation example that we saw earlier. To recap, the bank
wants to segment its customers based on their income and amount of debt:
Here, we can have two clusters which will separate the customers as shown below:
All the customers with low income are in one cluster, whereas the customers with high income are in the second cluster. We can also have 4 clusters:
Here, one cluster might represent customers who have low income and low debt; another cluster is where customers have high income and high debt, and so on. There can be 8 clusters as well:
Honestly, we can have any number of clusters. Can you guess what would be the maximum number of possible clusters? One thing we can do is assign each point to a separate cluster. Hence, in this case,
the number of clusters will equal the number of points or observations.
The maximum possible number of clusters will be equal to the number of observations in the dataset.
But then, how can we decide the optimum number of clusters? One thing we can do is plot a graph, also known as an elbow curve, where the x-axis will represent the number of clusters and the y-axis
will be an evaluation metric. Let’s say inertia for now.
You can choose any other evaluation metric like the Dunn index as well:
Next, we will start with a small cluster value, say 2. Train the model using 2 clusters, calculate the inertia for that model, and finally plot it in the above graph. Let’s say we got an inertia
value of around 1000:
Now, we will increase the number of clusters, train the model again, and plot the inertia value. This is the plot we get:
When we changed the cluster value from 2 to 4, the inertia value reduced sharply. This decrease in the inertia value reduces and eventually becomes constant as we increase the number of clusters
So,the cluster value where this decrease in inertia value becomes constant can be chosen as the right cluster value for our data.
Here, we can choose any number of clusters between 6 and 10. We can have 7, 8, or even 9 clusters. You must also look at the computation cost while deciding the number of clusters. If we increase the
number of clusters, the computation cost will also increase. So, if you do not have high computational resources, my advice is to choose a lesser number of clusters.
Let’s now implement the K-Means Clustering algorithm in Python. We will also see how to use K-Means++ to initialize the centroids and will also plot this elbow curve to decide what should be the
right number of clusters for our dataset.
Implementing K-Means Clustering in Python
We will be working on a wholesale customer segmentation problem. You can download the dataset using this link. The data is hosted on the UCI Machine Learning repository.
The aim of this problem is to segment the clients of a wholesale distributor based on their annual spending on diverse product categories, like milk, grocery, region, etc. So, let’s start coding!
We will first import the required libraries:
# importing required libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from sklearn.cluster import KMeans
Next, let’s read the data and look at the first five rows:
# reading the data and looking at the first five rows of the data
data=pd.read_csv("Wholesale customers data.csv")
we have the spending details of customers on different products like Milk, Grocery, Frozen, Detergents, etc. Now, we have to segment the customers based on the provided details.
Let’s pull out some statistics related to the data:
# statistics of the data
Here, we see that there is a lot of variation in the magnitude of the data. Variables like Channel and Region have low magnitude, whereas variables like Fresh, Milk, Grocery, etc., have a higher
Since K-Means is a distance-based algorithm, this difference in magnitude can create a problem.
Bring all the variables to the same magnitude:
# standardizing the data
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
data_scaled = scaler.fit_transform(data)
# statistics of scaled data
The magnitude looks similar now.
Create a kmeans function and fit it on the data:
# defining the kmeans function with initialization as k-means++
kmeans = KMeans(n_clusters=2, init='k-means++')
# fitting the k means algorithm on scaled data
We have initialized two clusters and pay attention – the initialization is not random here. We have used the k-means++ initialization which generally produces better results as we have discussed in
the previous section as well.
Let’s evaluate how well the formed clusters are:
To do that, we will calculate the inertia of the clusters:
# inertia on the fitted data
Output: 2599.38555935614
We got an inertia value of almost 2600. Now, let’s see how we can use the elbow method to determine the optimum number of clusters in Python.
We will first fit multiple k-means models, and in each successive model, we will increase the number of clusters.
We will store the inertia value of each model and then plot it to visualize the result:
# fitting multiple k-means algorithms and storing the values in an empty list
SSE = []
for cluster in range(1,20):
kmeans = KMeans(n_jobs = -1, n_clusters = cluster, init='k-means++')
# converting the results into a dataframe and plotting them
frame = pd.DataFrame({'Cluster':range(1,20), 'SSE':SSE})
plt.plot(frame['Cluster'], frame['SSE'], marker='o')
plt.xlabel('Number of clusters')
Can you tell the optimum cluster value from this plot? Looking at the above elbow curve, we can choose any number of clusters between 5 to 8.
Set the number of clusters as 6 and fit the model:
# k means using 5 clusters and k-means++ initialization
kmeans = KMeans(n_jobs = -1, n_clusters = 5, init='k-means++')
pred = kmeans.predict(data_scaled)
Value count of points in each of the above-formed clusters:
frame = pd.DataFrame(data_scaled)
frame['cluster'] = pred
So, there are 234 data points belonging to cluster 4 (index 3), 125 points in cluster 2 (index 1), and so on. This is how we can implement K-Means Clustering in Python.
In this article, we discussed one of the most famous clustering algorithms – K-Means Clustering. We implemented it from scratch and looked at its step-by-step implementation. We looked at the
challenges we might face while working with K-Means and also saw how K-Means++ can be helpful when initializing the cluster centroids.
Finally, we implemented k-means and looked at the elbow method, which helps to find the optimum number of clusters in the K-Means algorithm.
If you have any doubts or feedback, feel free to share them in the comments section below.
Frequently Asked Questions
Q1.What is Kmeans classification?
A. K-means classification is a method in machine learning that groups data points into K clusters based on their similarities. It works by iteratively assigning data points to the nearest cluster
centroid and updating centroids until they stabilize. It’s widely used for tasks like customer segmentation and image analysis due to its simplicity and efficiency.
Q2. What type of algorithm is k-means?
A. K-means is a type of algorithm used for clustering data into groups based on similarity. It aims to find cluster centers (centroids) that minimize the distance between data points and their
respective centroids
Q3. What are the benefits of K-means clustering?
A. The biggest benefit of K-means clustering is that it is faster than hierarchical clustering. It is easy to implement, can work with a large number of variables, and can quickly change clusters
when the centroids are recomputed.
Q4. What is the difference between KNN and k-means?
A. KNN is a supervised learning algorithm used for classification and regression by finding the majority class or average value among the k nearest neighbors of a data point. K-means, on the other
hand, is an unsupervised learning algorithm used for clustering data into groups based on similarity by iteratively assigning data points to the nearest cluster centroid and updating centroids.
Q5. What is an example of k-means in real life?
A. K-means is useful in market segmentation to group customers by purchasing behavior, enabling targeted marketing strategies.
Responses From Readers
Hi Pulkit, Thank you for this excellent article on the subject - one of the most comprehensive ones I have read. My question is that lets say I have 7 distinct clusters arrived at using the
techniques you have mentioned. How can I come up with relevant criteria/ rules using some ML algorithm such that any new observation can be assigned to one of the clusters by passing through the
decision rule instead of running K-Means again.
Hi Pulkit, Thanks for the post. Kindly clarify me: 1. In the "WholeSale Customer Data" data set, the variables: region and channel are categorical. In mathematical terms, we can not describe distance
between different categories of a categorical variable. But we converted them to a numeric form here and the distances are calculated. How can we justify the usage of these variables while
clustering? 2. Usually in most of the real-world problems, we have datasets of mixed form( containing of both numerical and categorical features). Is it ok to apply same k-means algorithm, on such
datasets? -Rajiv
Cluster explained very well. Thanks for the article in Python. Can you clarify below points 1) In the wholesale example, all the columns are considered for clustering, Column Channel & Region also
need to be included? as there is no variation in that. 2) After identifying the cluster group, how to update back the cluster group in the raw data
Hi Pulkit, Thanks for your article, it's very helpful for me. I wonder about the lines of your code: 1. C=[] 2. for index,row in X.iterrows(): 3. min_dist=row[1] 4. pos=1 5. for i in range(K): 6. if
row[i+1] < min_dist: 7. min_dist = row[i+1] 8. pos=i+1 9. C.append(pos) In the line 3, i think it should be: min_dist=row[2] and in line 6 should be: if row[i+2] < min_dist: Thanks for read my
Thanks for the article Pulkit. Can you please clarify my queries: 1. K- Means , by default assigns the initial centroid thru init : {‘k-means++’}. Hope, it will be taken care by sklearn. 2. For an
imbalanced data which has the class ratio of 100 : 1, can i generate labels thru kmeans and use it as a feature in my classification algorithm? Will it improve accuracy like knn?
Hi Pulkit, You did a great job ob this article to be patient and detailed as possible. However, I got lost here because I am still a newbie, can you explain each line of code with comments and send
it to me email address [email protected]. I know I am asking for a lot. "Next, we will define some conditions to implement the K-Means Clustering algorithm. Let’s first look at the code: # Step 3 -
Assign all the points to the closest cluster centroid # Step 4 - Recompute centroids of newly formed clusters # Step 5 - Repeat step 3 and 4 diff = 1 j=0 while(diff!=0): XD=X i=1 for index1,row_c in
Centroids.iterrows(): ED=[] for index2,row_d in XD.iterrows(): d1=(row_c["ApplicantIncome"]-row_d["ApplicantIncome"])**2 d2=(row_c["LoanAmount"]-row_d["LoanAmount"])**2 d=np.sqrt(d1+d2) ED.append(d)
X[i]=ED i=i+1 C=[] for index,row in X.iterrows(): min_dist=row[1] pos=1 for i in range(K): if row[i+1] < min_dist: min_dist = row[i+1] pos=i+1 C.append(pos) X["Cluster"]=C Centroids_new = X.groupby
(["Cluster"]).mean()[["LoanAmount","ApplicantIncome"]] if j == 0: diff=1 j=j+1 else: diff = (Centroids_new['LoanAmount'] - Centroids['LoanAmount']).sum() + (Centroids_new['ApplicantIncome'] -
Centroids['ApplicantIncome']).sum() print(diff.sum()) Centroids = X.groupby(["Cluster"]).mean()[["LoanAmount","ApplicantIncome"]]"
Hey Pulkit, this is a really great article and it really helps a lot to get a clear understanding about k means. I am trying to replicate the process in R and I had a question about multiple
variables. So given a similar dataset, if I have multiple observations and I have multiple variables, is there a way I can run a k means on multiple variables? If yer then is there a limit?
Awesome! You have given me a real push. Many thanks for the article.
Hi , can you provide more information on code , "model.predict()" to find the cluster number for each observation. Thanks in advnace.
Hi, Great article and well explained for someone who has little to no experience or formal institutionalized education in the field. Very intuitive. My question is regarding how I can isolate a
specific cluster to do further analysis on it or to prove some sort of hypothesis about a cluster. I have a decent understanding of algorithms due to an engineering background but lack the intuition
for programming languages and thus am relatively inexperienced at python
Hi Pulkit , Can you share any code, where we are applying supervised learning after clustering, because that's how the flow is right?
Hi Pulkit, Thank a lot for this amazing and well explained article on K-means. I am just confused about the way distances are calculated in K-means for choosing the centroids. What is the default
method for calculating distances and can we mention any other method in place of default if we want to?
Hi Pulkit, Thanks for the superb article. This is by far the most comprehensive piece on clustering i cam across. Would be great if you could also share how to evaluate the clusters created alongwith
how to use this output. Thanks, Kiran
Hi Pulkit , I need to plot cluster number against each row/value. Please mention the code to plot the cluster value against each row of input data. Thanks.
Awesome article. Thanks a lot. It is very very explanative, exciting and useful.
Great article. However, this phrase is missing important information: ". . . inertia actually calculates the sum of all the points within a cluster from the centroid of that cluster." I believe the
correct statement is as follows: ". . . inertia actually calculates the sum of the distances of all the points within a cluster from the centroid of that cluster."
Pulkit ..one of very simplified approach to expose K means to new entrants to data science..Thanks very much .. If you have written any article on anomaly detection techniques using K means I will be
interested .If you can share it will be much appreciated.
I really enjoyed your blog Thanks for sharing such an informative post.
This article is really amazing, congratulations by the job done. It do help me to understand how to K-Means works and to write my graduation article. Thanks a lot!
Hi pulkit, I want to classify my training tweet set into 3 classes such as positive, negative, and neytral using k-mean clustering. If i take k value as 3, we would get 3 clusters but how to decide
which tweet is positive or negative. My aim is to label the tweets into one of three classes (pos, neg or neutral) using kmean clustering. Is it possible to do that on tweets?
Thank you for the great article, well explained and very helpful especially for a beginner like me. I've also read the reviews and comments which are great. Please I would like to know:- -How can one
convert the output to an interpretable format? -How can one determine who (Loan_ID) or group of people that the bank can give loan to, based on the Kmeans clustering output? I would be very grateful
if you can help me out with this part. I believe your methods and ways of analyzing the output will be best based on your understanding of the topic.
Thank you for this very comprehensive article. You just demystified K-means to me and I am very grateful.
hello,first of all thanks for your explanation, i have question could you please kindly help me? i want to perform k-means just to cluster users based on one parameter , how can i do that?
Hi Pulkit, thx for this excellent introduction! I have the clusters now, but the door opens and a new customer walks in... How do I calculate the cluster the new customer belongs to? -Bodo
Thank you for the well-explained article. Very easy to understand,
Hi Pulkit Excellent coverage of the topic, very nicely complied. Please need to know how to attach labels predicted from the data , how can we get our data including attached cluster number etc to
further study and analysis
Hey Pulkit! This is such a wonderful and detailed article, helped me understand the process properly. Great work!
I work for an ambulance service we know the grid reference for all the emergencies. My question is if we wanted to know the location of where best to site our stations what would be the best method
and how would I find out the grid references of the stations. I am not a programmer or very technical, any help would be appreciated
Hi Pulkirt, I found your article incredibly informative. Your expertise in this area caught my attention, and I am reaching out to seek your guidance. and I am a biologist currently exploring the
application of clustering techniques in my study. I am particularly interested in clustering the evolution of a specific parameter, such as BMI, over time (at 6 months and 12 months). While I have a
solid understanding of the basics, I could use some advice on the practical aspects of applying clustering algorithms to longitudinal data. If you have any recommendations on which clustering
algorithm might be most suitable for this type of data, or if you can provide insights into the preprocessing steps involved, I would greatly appreciate it. Additionally, any resources or examples
you could point me towards would be immensely helpful. Thank you so much for your time,
Hi Pulkirt, I found your article incredibly informative. Your expertise in this area caught my attention, and I am reaching out to seek your guidance. and I am a biologist currently exploring the
application of clustering techniques in my study. I am particularly interested in clustering the evolution of a specific parameter, such as BMI, over time (at 6 months and 12 months). While I have a
solid understanding of the basics, I could use some advice on the practical aspects of applying clustering algorithms to longitudinal data. If you have any recommendations on which clustering
algorithm might be most suitable for this type of data, or if you can provide insights into the preprocessing steps involved, I would greatly appreciate it. Additionally, any resources or examples
you could point me towards would be immensely helpful. Thank you so much for your time
Very comprehensive and insightful! Great work!
Hi Pulkit, Excellent tutorial. Thanks!!!
Hi Pulkit,it is such a detailed and interesting explanation...you have covered all the points related to k-means clustering and this make the article wholesome in itself.Thankyou so much for this
wonderfull material. | {"url":"https://www.analyticsvidhya.com/blog/2019/08/comprehensive-guide-k-means-clustering/?utm_source=blog&utm_medium=gaussian-mixture-models-clustering","timestamp":"2024-11-07T07:28:55Z","content_type":"text/html","content_length":"788758","record_id":"<urn:uuid:ba8f40b5-d048-40aa-9bb1-bd5a30e8c9a8>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00540.warc.gz"} |
What is the conjugate of 3 minus square root of 2? | Socratic
What is the conjugate of 3 minus square root of 2?
1 Answer
By definition the conjugate of
$\textcolor{w h i t e}{\text{XXX}} \left(a + b\right)$ is $\left(a - b\right)$
$\textcolor{w h i t e}{\text{XXX}} \left(a - b\right)$ is $\left(a + b\right)$
The term "conjugate" only applies to the sum or difference of two terms.
"3 minus the square root of 2"
means (in algebraic form)
$3 - \sqrt{2}$
Applying the earlier definition with $a = 3$ and $b = \sqrt{2}$
we have
The conjugate of $\left(3 - \sqrt{2}\right)$ is $\left(3 + \sqrt{2}\right)$
Impact of this question
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Bala Qhova
[BALL-uh COAH-vuh]
(Trad. EphilKcyMeril Name)
The first person to domesticate a Water Mole.
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Baldwin Brannan
[(as per linear English)]
Sergeant in Army of Alleigh, respected by friend and foe alike. Sometimes called Sarge Brannan. Featured in Cyrus Hawkes’ memoir Feudal California Boyhood.
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Bar Sinister
[(as per linear English)]
A band of undead lawyers who travel the world getting guilty people acquitted.
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base 10
[(as per linear English)]
The numbering system used in most of the linear world. Where numbers are presented in exponential units of 10 and the position of the digit is quite significant. When we write “5,” we mean 5 units of
1 each. But when we write “5243,” we mean 5 units of 1000 (10x10x10) each, 2 units of 100 (10x10) each, 4 of 10 each, and 3 of 1 each.
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base 29
[(as per linear English)]
The numbering system used by many Kcymaerxthaereal cultures. Where numbers are presented in exponential units of 29 and the position of the digit is quite significant. When we write “5,” we mean 5
units of 1 each. And that is true in base 29 as well. But when we write “5243” in base 29 instead of base 10, we mean: 5 units of 24389 (29x29x29) each, 2 units of 841 (29x29) each, 4 of 29 each, and
3 of 1 each. Which in base 10 would be: 123,746. This is especially useful in the number languages like 158s—over 3.5 million words and names can be written with just 4 digits. In the appendix see
one of the most commonly used set of symbols for base 29 in the xthaere.
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base 9
[(as per linear English)]
The numbering system used by the Jihn Wranglikans. Where numbers are presented in exponential units of 9 and the position of the digit is quite significant. When we write “5,” we mean 5 units of 1
each. But when we write “5243” in base 9 instead of base 10, we mean: 5 units of 729 (9x9x9) each, 2 units of 81 (9x9) each, 4 of 9 each, and 3 of 1 each. Which in base 10 would be: 3846
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Battle of Devil’s Marbles
[(as per linear English)]
The final major battle between the People of the Rock and the Material Alliance took place on this site in the rezhn of Estrelliia.
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Battle of Marathon
[(as per linear English)]
Here the Marithon coalition decisively stopped the expansionist goals of the Conch Republic, under Switlik and Xi.
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Battle of Some Times
[(as per linear English)]
The cataclysmic battle between Kmpass and the Armies of Complexity that would determine the richness of the worlds.
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Battle of the Platte
[(as per linear English)]
Decisive defeat of the Urushiol by forces led by Nobunaga-Gaisen.
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Battle of the Seas
[(as per linear English)]
An inconclusive battle between oceans that took place in what is now known as the Dry Tortugas.
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Means Whorl of the World—a great nested multi-dimensional spiral that honors all non-combatants who perish in war. And it is built by all those who led forces into battle. There are 6 all together.
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Armstrong Metalcrafts
A cometarium is a mechanical illustration of Johannes Kepler's 2nd Law, which states that an object in orbit sweeps out equal areas in equal time intervals. These were used in lecture demonstrations
beginning in the 18th Century, not long after Halley and Newton proved Kepler's assertion about orbits, thanks in part to the timely return of what we now call "Halley's Comet".
Understanding Orbits
The cometarium was invented at a time when man's understanding of orbital mechanics was evolving rapidly and the middle class began to ask for lecture demonstrations of the new findings.
The story of today's understanding of orbits has more chapters than can be described here, but one must first credit the work of the Danish astronomer Tycho Brahe, who worked to test the Copernican
model of the solar system by measuring the parallax of Mars. If Mars were to come closer to the Earth than the Sun, the Ptolemaic system would be disproved. Brahe's instruments were the best
available of the time but not sufficiently accurate to resolve this parallax. Beginning in 1600 Johannes Kepler worked with Brahe analyzing some of the new Mars observations. By nature, Kepler
defended heliocentrism, and he was skeptical of Brahe's "geo-heliocentric" system now known as the Tychonic system. When Brahe died in 1601, Kepler was appointed his successor, and now had full
access to 20 years of Mars data.
Kepler calculated and recalculated various approximations of Mars' orbit, eventually creating a model that generally agreed with Brahe's observations to within two arcminutes (the average measurement
error). He was not satisfied with the complex and still slightly inaccurate result; at certain points the model differed from the data by up to eight arcminutes.
Based on measurements of the aphelion and perihelion of the Earth and Mars, Kepler created a formula in which a planet's rate of motion is inversely proportional to its distance from the Sun.
Verifying this relationship throughout the orbital cycle, however, required very extensive calculation; to simplify this task, by late 1602 Kepler reformulated the proportion in terms of geometry:
planets sweep out equal areas in equal times — Kepler's second law of planetary motion. This is the law the cometarium illustrates.
Much of Kepler's work through 1604 involved a complete calculation of the orbit of Mars. Calculations of three different eccentricities at 1 degree intervals in steps of 1 degree to 5 digit accuracy
was most tedious. That these calculations disagreed with Brahe's observations are in effect a tribute to the accuracy of Brahe's observational data. After about 40 failed attempts, Kepler hit upon
the idea of an ellipse, which he had previously assumed to be too simple a solution for earlier astronomers to have overlooked.
Finding that an elliptical orbit fit the Mars data, he immediately concluded that all planets move in ellipses, with the Sun at one focus — Kepler's first law of planetary motion. Because he employed
no calculating assistants, however, he did not extend the mathematical analysis beyond Mars.
In the late 17th century, a number of physical astronomy theories drawing from Kepler's work — notably those of Giovanni Alfonso Borelli and Robert Hooke — began to incorporate attractive forces and
the Cartesian concept of inertia. This culminated in Isaac Newton's "Principia Mathematica" (1687), in which Newton derived Kepler's laws of planetary motion from a force-based theory of universal
Lecture Demonstrations
At the end of the 17th century science had been a topic for the elites of society and courts. This began to change quickly in the first decade of the 18th century with the combination of coffee
houses, a growing middle class, and the availability of lecture demonstrations. By the middle of the 18th century the achievements of astronomers were becoming widely accepted and a culture of
lectures using demonstration apparatus evolved greatly. Between 1745 and 1770 the London paper "Daily Advertiser" featured advertisements from twelve lecturers, including Stephen Demainbray, James
Ferguson, and Benjamin Martin.
The demand for these lectures created a small market for makers of scientific instruments. Simpler models were driven by a hand crank; elaborate ones were driven by clockwork.
The two best known mechanical models used in lecture demonstrations were the orrery (a mechanical model of planets going around the Sun) and the cometarium – a model of a comet's orbit. The orrery
and the cometarium have to make compromises. An orrery cannot show correct relative size or spacing of planets in their orbits, nor do they show elliptical orbits. A cometarium is constrained by
mechanical limits to show only modest eccentricities to fit on a tabletop.
The origins of the cometarium trace to a device first made by J T Desaguliers and demonstrated to the Royal Society in 1732 to facilitate a discussion of the orbit of the planet Mercury. Mercury's
orbit is the most eccentric (0.21) of the planets in our solar system, so the changes in velocity of Mercury between aphelion and perihelion were sufficiently large to attract interest. Later, in the
1740s as Halley's comet was predicted to return soon, interest in comets surged and the name "cometarium" was coined to name the device by Benjamin Martin. Martin sold a cometarium along with his
book "The Theory of Comets" which was published in 1757.
Early Cometaria used elliptical wheels and a figure 8 shaped cord as the essential device to create the differential motion of the comet and the calendar indicator.
Today the Cometarium by Armstrong Metalcrafts uses elliptical gears to achieve the correct motions.
The gears in this model have an eccentricity of about .65, which is close to matching some models found in museums.
Early cometaria had varying methods for moving the comet ball around the orbit. Example ideas included a rod pushing the comet ball in the track, or a pin and slot mechanism. This cometarium is the
first to have the ball mounted on the shaft driven by the gears.
For Sale
Armstrong Metalcrafts has built a limited number of these for sale. The price is $1250, and this includes a small booklet featuring an in-depth description of the history and how this mechanism
works. To inquire about a purchase, please use our contact form or send an email to "sales" at armstrongmetalcrafts.com. | {"url":"https://www.armstrongmetalcrafts.com/Products/Cometarium.aspx","timestamp":"2024-11-02T08:14:40Z","content_type":"application/xhtml+xml","content_length":"30186","record_id":"<urn:uuid:28efe8a6-9a05-4a53-8ccb-96eb34c497fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00543.warc.gz"} |
ERK Inhibitor Enhances Crosstalk of DNA double‐strand break repair
In addition, these standard models do not allow for nonlinear covariate effects
In addition, these standard models do not allow for nonlinear covariate effects. a sparse CR kernel machine (KM) regression method for ordinal outcomes where we use the KM framework to incorporate
nonlinearity and impose sparsity on the overall differences between the Sucralose covariate effects of continuation ratios to control for overfitting. In addition, we provide data driven rule to
select an optimal kernel to maximize the prediction accuracy. Simulation results show that our proposed procedures perform well under both linear and nonlinear settings, especially when the true
underlying model is in-between pCR and fCR models. We apply our procedures to develop a prediction model for levels of anti-CCP among rheumatoid arthritis patients and demonstrate the advantage of
our method over other commonly used methods. with a 1 predictor vector x, one may employ regression models relating x to and classify future subjects into different categories based on their
predicted = | x). Naive analysis strategies, such as dichotomizing into a binary fitting and variable multinomial regression models, are not efficient as they do not take into account the ordinal
property of the outcome. Commonly used traditional methods for modeling ordinal response data include the cumulative proportional odds model, the forward and backward continuation ratio (CR) models
and the corresponding proportional odds version of the CR (pCR) model (Ananth and Kleinbaum, 1997). The forward Sucralose full CR (fCR) model assumes that is assumed to take ordered categories, {1, ,
and but not Sucralose all and Rabbit Polyclonal to OR8K3 thus it is possible to improve the estimation by leveraging the sparsity on independent and identically distributed random vectors, to denote
Fubinis norm for matrices. From here onward, for notational ease, we suppress from the kernel function with respect to the eigensystem of has eigenvalues = 1, , with = 1, , such that 0 for any . The
basis functions, = 1, , , span the RKHS . All for all is smooth Hence, leading to bounded = 1, , = 1, , ? 1: = [ 1 vector of unknown weights to be estimated as model parameters. This representation
reduces (6) to an explicit optimization problem in the dual form: + 1)(? 1) parameters to be estimated, when the sample size is not small especially. On the other hand, if the eigenvalues of decay
quickly, then we may reduce the complexity by approximating by a truncated kernel such that can be bounded by is the kernel matrix constructed from kernel is typically fairly small and we can
effectively approximate by a finite dimensional space . Although Sucralose = (= diag{ 0 are the eigenvalues of and {u1, , uconverge to the eigenvalues and the projection error can be bounded by and
sufficiently fast decay rate for {and applying a variable transformation is the for some close to 1. Let denote the estimator from the maximization of (8). For a future subject with x Then, the
probability = ? = = 1= 1within a range of values. For any given and obtained from (10), in (and the resulting classification will outperform the corresponding estimators and classifications derived
from the fCRKM model based on and the reduced pCRKM model when the underlying model has but not all. When = can be approximated well with a finite dimensional space with a fixed 1 if and the average
size of prediction sets ( ) to be defined below. The OME puts equal weights to any error as long as = 11 = 1 in to fit our proposed procedures with several candidate kernels and obtain the
corresponding estimate to calculate their predicted probabilities (= 1 would then be used for prediction in the validation set. In regards to the choice of = 10 as previously suggested in Breiman and
Spector (1992). 3. Numerical Studies 3.1 Simulation Study We conducted extensive simulations to evaluate the finite sample performance of our proposed methods and compared with three existing
methods: the one-against-one SVM method (Hsu and Lin, 2002), the 1)with continuous covariates under the CRKM model in (3). The 20and = 1 1 were set to be between 0 and 0.4 and the intercept
parameters = 1 Sucralose 1 were selected such that there were approximately the same number. | {"url":"https://dupublicaucommun.com/2022/03/13/in-addition-these-standard-models-do-not-allow-for-nonlinear-covariate-effects/","timestamp":"2024-11-10T11:27:55Z","content_type":"text/html","content_length":"28774","record_id":"<urn:uuid:70205c21-1b95-4f03-8bf1-4a92d27b4e9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00394.warc.gz"} |
What is 108 Celsius to Fahrenheit? - ConvertTemperatureintoCelsius.info
108 Celsius to Fahrenheit – Easy Conversion Guide
When it comes to converting temperature measurements from Celsius to Fahrenheit, knowing the proper formula is key. In this guide, we will explore how to convert 108 degrees Celsius to Fahrenheit,
providing a step-by-step explanation along the way.
To begin, let’s first understand the difference in temperature scales. Celsius (°C) and Fahrenheit (°F) are two common units for measuring temperature. The Celsius scale is based on the freezing and
boiling points of water, where 0°C is the freezing point and 100°C is the boiling point. On the other hand, the Fahrenheit scale sets the freezing point of water at 32°F and the boiling point at
212°F. This means that there is a 180-degree difference between the two scales.
Now, to convert 108 degrees Celsius to Fahrenheit, we will use the following formula:
°F = (°C × 9/5) + 32
With this formula in mind, let’s calculate the conversion:
°F = (108 × 9/5) + 32
°F = (194.4) + 32
°F = 226.4
So, 108 degrees Celsius is equal to 226.4 degrees Fahrenheit.
It’s important to note that when converting from Celsius to Fahrenheit, the result is always greater than the original temperature. This is due to the differences in the scales and the offset of the
freezing points.
Now that we have successfully converted 108 degrees Celsius to Fahrenheit, let’s explore some practical applications of this conversion. In many parts of the world, including the United States,
Fahrenheit is the primary unit used for measuring temperature. This means that understanding how to convert between Celsius and Fahrenheit can be useful in various scenarios.
For example, if you are traveling to a country that uses the Fahrenheit scale and you want to know what 108 degrees Celsius feels like in Fahrenheit, our conversion formula comes in handy.
Additionally, for scientific or engineering purposes, being able to easily convert between Celsius and Fahrenheit is essential.
In conclusion, converting 108 degrees Celsius to Fahrenheit is a straightforward process when using the correct formula. By following the steps outlined in this guide, you can confidently make the
conversion and apply it to real-world situations. Whether for travel, work, or everyday knowledge, understanding temperature conversions is a valuable skill. | {"url":"https://converttemperatureintocelsius.info/what-is-108celsius-in-fahrenheit/","timestamp":"2024-11-05T23:28:53Z","content_type":"text/html","content_length":"72514","record_id":"<urn:uuid:fbb37950-ee87-483b-a2a4-c0d4516050a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00319.warc.gz"} |
Enter Text Numbers And Dates Into A Worksheet 2024 - NumbersWorksheets.com
Enter Text Numbers And Dates Into A Worksheet
Enter Text Numbers And Dates Into A Worksheet – A Reasonable Amounts Worksheet will help your kids become more acquainted with the methods associated with this percentage of integers. With this
worksheet, students should be able to fix 12 different issues associated with logical expressions. They are going to figure out how to increase 2 or more numbers, group them in pairs, and determine
their products. They will likely also exercise simplifying reasonable expressions. Once they have mastered these ideas, this worksheet will certainly be a beneficial instrument for advancing their
reports. Enter Text Numbers And Dates Into A Worksheet.
Logical Numbers certainly are a percentage of integers
The two main forms of figures: rational and irrational. Reasonable phone numbers are described as complete amounts, whereas irrational numbers usually do not replicate, and possess an unlimited
quantity of numbers. Irrational amounts are no-absolutely nothing, no-terminating decimals, and rectangular roots which are not excellent squares. These types of numbers are not used often in
everyday life, but they are often used in math applications.
To determine a reasonable quantity, you must understand exactly what a rational quantity is. An integer is a total quantity, and a rational variety can be a ratio of two integers. The proportion of
two integers is the number at the top split through the quantity on the bottom. For example, if two integers are two and five, this would be an integer. However, there are also many floating point
numbers, such as pi, which cannot be expressed as a fraction.
They are often made in to a small fraction
A reasonable quantity includes a numerator and denominator that are not absolutely no. Because of this they can be expressed as a small percentage. Along with their integer numerators and
denominators, reasonable amounts can in addition have a negative worth. The negative benefit must be put left of and its total worth is its range from absolutely no. To easily simplify this
illustration, we are going to claim that .0333333 is actually a small fraction which can be composed as a 1/3.
In addition to adverse integers, a realistic amount can be manufactured right into a small percentage. As an example, /18,572 is a realistic variety, although -1/ will not be. Any small fraction made
up of integers is logical, so long as the denominator will not contain a and will be published being an integer. Similarly, a decimal that leads to a level is yet another rational variety.
They make sense
Even with their title, reasonable phone numbers don’t make very much sense. In mathematics, these are solitary entities using a distinctive size in the number line. This means that when we count up
some thing, we are able to purchase the size and style by its ratio to its original amount. This keeps true regardless if you will find endless reasonable phone numbers involving two distinct
numbers. If they are ordered, in other words, numbers should make sense only. So, if you’re counting the length of an ant’s tail, a square root of pi is an integer.
If we want to know the length of a string of pearls, we can use a rational number, in real life. To discover the length of a pearl, as an example, we might add up its size. An individual pearl weighs
in at twenty kilograms, which is actually a realistic variety. In addition, a pound’s bodyweight is equal to 15 kilograms. Hence, we must be able to break down a lb by 10, without the need of be
concerned about the size of an individual pearl.
They could be expressed being a decimal
If you’ve ever tried to convert a number to its decimal form, you’ve most likely seen a problem that involves a repeated fraction. A decimal amount can be created as a a number of of two integers, so
4x 5 is equivalent to 8. The same difficulty involves the recurring small percentage 2/1, and each side ought to be divided up by 99 to have the right response. But how would you create the
conversion? Here are a few cases.
A realistic amount may also be developed in great shape, which includes fractions as well as a decimal. A good way to represent a reasonable number within a decimal is usually to divide it into its
fractional counterpart. You can find 3 ways to separate a reasonable amount, and each one of these methods brings its decimal counterpart. One of those methods is always to split it into its
fractional comparable, and that’s what’s called a terminating decimal.
Gallery of Enter Text Numbers And Dates Into A Worksheet
Cardinal And Ordinal Numbers Dates ESL Worksheet By English Grammar
English Worksheets Dates Ordinal Numbers
MONTHS DATES ORDINAL NUMBERS ESL Worksheet By Sandramendoza Date
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The Pull-Up Resistor: How It Works and Choosing a Value
The pull-up resistor is very common and you’ll see it in digital circuits all the time. It’s just a resistor connected from an input up to V[DD], the positive supply of the circuit.
For example on digital inputs on an Arduino. Or the input of digital chips such as the 4000-series IC.
Pull-up resistors are used to make sure you have a HIGH state on the input pin when the button is not pushed. Without one, your input will be floating, and you risk that the input randomly changes
between HIGH and LOW as it picks up noise in the air.
How To Choose a Pull-Up Resistor Value
Rule 1: The value can’t be too high.
The higher the pull-up value, the lower the voltage on the input becomes. It’s important that the voltage is high enough for the chip to see it as a HIGH, or logical 1, input.
For example, if you use a CD4017 with a power supply of 10V, it requires a minimum of 7V on the input for it to be seen as HIGH.
Rule 2: But it can’t be too small either.
If you for example choose 100 Ω, the problem is that you get a lot of current flowing through it when the button is pushed.
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With a 9V power supply, you get 9V across 100 Ω, which is 90 mA. It’s an unnecessary waste of power, but it also means the resistor needs to withstand 0.81W. Most resistors can handle only up to
Rule of thumb
The rule of thumb when choosing a pull-up resistor is to choose a resistance value that is at least 10 times smaller than the input impedance (or the internal resistance) of the pin.
Often, a pull-up value of 10 kΩ will do the trick. But if you want to understand how it works, keep reading.
How Do Pull-Up Resistors Work?
You can think of the input pin of an integrated circuit (IC) as having a resistor connected to ground. This is called the input impedance:
These two resistors make up a voltage divider. If you look at the standard voltage divider circuit, you can see that the pull-up resistor is R1 and the input impedance is R2:
You can use the voltage divider formula to find the voltage on the input pin when the button is not pushed:
Below, I’ve renamed the components of the formula to fit the pull-up example. The input voltage is V[DD] from our pull-up example. And the output voltage is the voltage on the input pin. So the
formula becomes:
Example Calculation
Let’s say your chip has an input impedance of 1MΩ (100kΩ to 1MΩ is normal for many chips). If your power supply is 9V and you choose a pull-up resistor value of 10 kΩ, what’s the voltage you get on
the input pin?
You get 8.9V on the input pin, which is more than enough to act as a HIGH input.
In general, if you stick to the rule of thumb of using a pull-up resistor that is no more than ten times lower than the input impedance, you’ll make sure you always have a minimum of 90% of the VDD
voltage on the input pin.
How To Find the Input Impedance of an IC
You can easily measure the input impedance of a chip. Impedance is actually a term for resistance that can change depending on frequency. But for this pull-up case, we only deal with DC currents.
Connect a pull-up resistor of for example 10 kΩ to the input of the chip, and measure the voltage on the input.
Let’s say you got 8.5V when you measured.
Use this to find the current flowing through the resistor by using Ohm’s law. The voltage drop across the resistor is 9V – 8.5V = 0.5V, so you get:
There is 0.05 mA flowing through the resistor, and thereby also through the input pin down to ground. Again, use Ohm’s law to find the resistance of something with a voltage drop of 8.5 V and a
current of 0.05 mA:
The input impedance is 170 kΩ. That means a pull-up resistor for this input should be no more than 17 kΩ.
What questions do you have about the pull-up resistor? Let me know in the comments field below!
More Resistors Tutorials
10 Simple Steps to Learn Electronics
Electronics is easy when you know what to focus on and what to ignore. Learn what "the basics" really is and how to learn it fast.
1 thought on “The Pull-Up Resistor: How It Works and Choosing a Value”
1. Thank you for this detailed explanation. It is clear to me, you have a gift for instruction by relating it to real world and simplifying the explanation to a simple ‘think of it as’ format. I
have bookmarked your site for my further reference as I get deeper into learning electronics. Best I have seen so far, and I have studied your pages on transistors, to clarify
collector-base-emitter and source-gate-drain since these have baffled me.
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Bigframes package - Problem with Dataframe apply function
When working with large datasets in Python, the Bigframes package is a powerful tool designed to handle data more efficiently than traditional DataFrame libraries like Pandas. However, users may
encounter issues while using the apply() function on DataFrames within Bigframes. In this article, we will examine a common problem users face with the apply() function, provide the original code for
context, and offer solutions to overcome these challenges.
Problem Scenario
Consider the following scenario where a user attempts to use the apply() function on a Bigframes DataFrame. The original code might look like this:
import bigframes as bf
# Create a sample Bigframes DataFrame
data = {
'A': [1, 2, 3],
'B': [4, 5, 6]
# Convert the dictionary into a Bigframes DataFrame
big_df = bf.DataFrame(data)
# Apply a function to calculate the sum of columns A and B
result = big_df.apply(lambda row: row['A'] + row['B'], axis=1)
Problem Analysis
In the above code, the user attempts to apply a lambda function to sum columns 'A' and 'B' in a Bigframes DataFrame. However, the apply() function in Bigframes may behave differently from what users
expect if they are familiar with Pandas. Specifically, it might return unexpected results or raise errors related to row-wise operations.
Why Does This Issue Occur?
1. Lack of Compatibility: Bigframes is optimized for large-scale data processing, and its functionalities may not always align with those of Pandas. As a result, certain operations like apply() may
not perform as intended.
2. Performance Concerns: The apply() function can be slow and resource-intensive, especially with large datasets. Bigframes aims to provide optimized functions for performance, and using apply()
might negate some of these benefits.
Solution and Workarounds
To overcome the limitations of the apply() function, consider the following approaches:
1. Use Vectorized Operations: Instead of using apply(), try to leverage vectorized operations available in Bigframes. For the example above, you can simply perform the addition directly on the
# Direct addition of columns
big_df['Sum'] = big_df['A'] + big_df['B']
This approach is generally faster and more efficient.
2. Alternative Functions: Use built-in functions specifically designed for Bigframes. Check the documentation for functions that may offer similar functionality without the drawbacks of apply().
Practical Example
Suppose you need to calculate the mean of a specific column in a Bigframes DataFrame. Instead of using apply(), you could utilize the built-in mean() function:
# Calculate the mean of column A
mean_value = big_df['A'].mean()
print(f"The mean value of column A is: {mean_value}")
This method will yield faster results while maintaining the integrity of large datasets.
When working with the Bigframes package, it is essential to understand the limitations and behavior of its DataFrame operations, particularly the apply() function. By employing vectorized operations
and using built-in functions, you can avoid common pitfalls and optimize your data processing tasks.
Additional Resources
For further reading and in-depth documentation, check out the following resources:
By being aware of these techniques and alternatives, you'll enhance your efficiency in data manipulation and analysis with Bigframes. | {"url":"https://laganvalleydup.co.uk/post/bigframes-package-problem-with-dataframe-apply-function","timestamp":"2024-11-04T23:35:31Z","content_type":"text/html","content_length":"83873","record_id":"<urn:uuid:13d819b5-8a14-4cae-9632-523e1d5abcbb>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00377.warc.gz"} |
What Is an Inclusion Proof – Gateway Documentation
What is an Inclusion Proof?
An inclusion proof, also known as a Merkle proof or Merkle inclusion proof, is a cryptographic proof that shows a specific data element (like a transaction or a piece of data in storage) is part of a
larger dataset, usually represented as a Merkle tree.
How It Works
1. Merkle Tree Structure:
Data elements are arranged in a hierarchical structure called a Merkle tree (or hash tree). Each leaf node in the tree contains a hash of a data element.
2. Calculation of Intermediate Nodes:
Hashes of adjacent leaf nodes are combined (hashed together) to create parent nodes. This process continues up the tree until a single root hash, known as the Merkle root, is calculated.
3. Inclusion Proof Components:
To prove that a specific data element is in the Merkle tree, an inclusion proof includes:
□ The data element itself (or its hash).
□ A series of hashes (nodes) that form the path from the leaf node (containing the data) to the tree’s root.
□ Sibling nodes along this path that are needed to reconstruct the path to the root.
4. Verification:
With the inclusion proof, you can recreate the Merkle root by hashing the provided data element and its sibling nodes up to the root. If the computed root matches the known Merkle root (stored
on-chain or verified), it confirms that the data element is part of the original dataset in the Merkle tree. | {"url":"https://gateway-docs.unruggable.com/fundamentals/what-is-an-inclusion-proof","timestamp":"2024-11-07T04:32:26Z","content_type":"text/html","content_length":"40297","record_id":"<urn:uuid:0d79f68a-420a-499e-bd0f-efb0a0cb7759>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00494.warc.gz"} |
Source code for pyomo.dae.contset
# ___________________________________________________________________________
# Pyomo: Python Optimization Modeling Objects
# Copyright (c) 2008-2024
# National Technology and Engineering Solutions of Sandia, LLC
# Under the terms of Contract DE-NA0003525 with National Technology and
# Engineering Solutions of Sandia, LLC, the U.S. Government retains certain
# rights in this software.
# This software is distributed under the 3-clause BSD License.
# ___________________________________________________________________________
import logging
import bisect
from pyomo.common.numeric_types import native_numeric_types
from pyomo.common.timing import ConstructionTimer
from pyomo.core.base.set import SortedScalarSet
from pyomo.core.base.component import ModelComponentFactory
logger = logging.getLogger('pyomo.dae')
"A bounded continuous numerical range optionally containing"
" discrete points of interest."
class ContinuousSet(SortedScalarSet):
"""Represents a bounded continuous domain
Minimally, this set must contain two numeric values defining the
bounds of a continuous range. Discrete points of interest may
be added to the continuous set. A continuous set is one
dimensional and may only contain numerical values.
initialize : `list`
Default discretization points to be included
bounds : `tuple`
The bounding points for the continuous domain. The bounds will
be included as discrete points in the :py:class:`ContinuousSet`
and will be used to bound the points added to the
:py:class:`ContinuousSet` through the 'initialize' argument,
a data file, or the add() method
_changed : `boolean`
This keeps track of whether or not the ContinuousSet was changed
during discretization. If the user specifies all of the needed
discretization points before the discretization then there is no
need to go back through the model and reconstruct things indexed
by the :py:class:`ContinuousSet`
_fe : `list`
This is a sorted list of the finite element points in the
:py:class:`ContinuousSet`. i.e. this list contains all the
discrete points in the :py:class:`ContinuousSet` that are not
collocation points. Points that are both finite element points
and collocation points will be included in this list.
_discretization_info : `dict`
This is a dictionary which contains information on the
discretization transformation which has been applied to the
def __init__(self, *args, **kwds):
if kwds.pop("filter", None) is not None:
raise TypeError(
"'filter' is not a valid keyword argument for ContinuousSet"
# if kwds.pop("within", None) is not None:
# raise TypeError("'within' is not a valid keyword argument for "
# ContinuousSet")
kwds.setdefault('dimen', 1)
if kwds["dimen"] != 1:
raise TypeError("'dimen' is not a valid keyword argument for ContinuousSet")
if kwds.pop("virtual", None) is not None:
raise TypeError(
"'virtual' is not a valid keyword argument for ContinuousSet"
if kwds.pop("validate", None) is not None:
raise TypeError(
"'validate' is not a valid keyword argument for ContinuousSet"
if len(args) != 0:
raise TypeError("A ContinuousSet expects no arguments")
kwds.setdefault('ctype', ContinuousSet)
self._changed = False
self._fe = []
self._discretization_info = {}
super(ContinuousSet, self).__init__(**kwds)
[docs] def get_finite_elements(self):
"""Returns the finite element points
If the :py:class:`ContinuousSet <pyomo.dae.ContinuousSet>` has been
discretizaed using a collocation scheme, this method will return a
list of the finite element discretization points but not the
collocation points within each finite element. If the
:py:class:`ContinuousSet <pyomo.dae.ContinuousSet>` has not been
discretized or a finite difference discretization was used,
this method returns a list of all the discretization points in the
:py:class:`ContinuousSet <pyomo.dae.ContinuousSet>`.
`list` of `floats`
return self._fe
[docs] def get_discretization_info(self):
"""Returns a `dict` with information on the discretization scheme
that has been applied to the :py:class:`ContinuousSet`.
return self._discretization_info
[docs] def get_changed(self):
"""Returns flag indicating if the :py:class:`ContinuousSet` was
changed during discretization
Returns "True" if additional points were added to the
:py:class:`ContinuousSet <pyomo.dae.ContinuousSet>` while applying a
discretization scheme
return self._changed
[docs] def set_changed(self, newvalue):
"""Sets the ``_changed`` flag to 'newvalue'
newvalue : `boolean`
# TODO: Check this if-statement
if newvalue is not True and newvalue is not False:
raise ValueError(
"The _changed attribute on a ContinuousSet may "
"only be set to True or False"
self._changed = newvalue
[docs] def get_upper_element_boundary(self, point):
"""Returns the first finite element point that is greater or equal
to 'point'
point : `float`
if point in self._fe:
return point
elif point > max(self._fe):
"The point '%s' exceeds the upper bound "
"of the ContinuousSet '%s'. Returning the upper bound"
% (str(point), self.name)
return max(self._fe)
for i in self._fe:
# This works because the list _fe is always sorted
if i > point:
return i
[docs] def get_lower_element_boundary(self, point):
"""Returns the first finite element point that is less than or
equal to 'point'
point : `float`
if point in self._fe:
if 'scheme' in self._discretization_info:
if self._discretization_info['scheme'] == 'LAGRANGE-RADAU':
# Because Radau Collocation has a collocation point on the
# upper finite element bound this if statement ensures that
# the desired finite element bound is returned
tmp = self._fe.index(point)
if tmp != 0:
return self._fe[tmp - 1]
return point
elif point < min(self._fe):
"The point '%s' is less than the lower bound "
"of the ContinuousSet '%s'. Returning the lower bound "
% (str(point), self.name)
return min(self._fe)
rev_fe = list(self._fe)
for i in rev_fe:
if i < point:
return i
[docs] def construct(self, values=None):
"""Constructs a :py:class:`ContinuousSet` component"""
if self._constructed:
timer = ConstructionTimer(self)
super(ContinuousSet, self).construct(values)
for val in self:
if type(val) is tuple:
raise ValueError("ContinuousSet cannot contain tuples")
if val.__class__ not in native_numeric_types:
raise ValueError("ContinuousSet can only contain numeric values")
# TBD: If a user specifies bounds they will be added to the set
# unless the user specified bounds have been overwritten during
# OrderedScalarSet construction. This can lead to some unintuitive
# behavior when the ContinuousSet is both initialized with values and
# bounds are specified. The current implementation is consistent
# with how 'Set' treats this situation.
for bnd in self.domain.bounds():
# Note: the base class constructor ensures that any declared
# set members are already within the bounds.
if bnd is not None and bnd not in self:
if None in self.bounds():
raise ValueError(
"ContinuousSet '%s' must have at least two values"
" indicating the range over which a differential "
"equation is to be discretized" % self.name
if len(self) < 2:
# (reachable if lb==ub)
raise ValueError(
"ContinuousSet '%s' must have at least two values"
" indicating the range over which a differential "
"equation is to be discretized" % self.name
self._fe = list(self)
[docs] def find_nearest_index(self, target, tolerance=None):
"""Returns the index of the nearest point in the
:py:class:`ContinuousSet <pyomo.dae.ContinuousSet>`.
If a tolerance is specified, the index will only be returned
if the distance between the target and the closest point is
less than or equal to that tolerance. If there is a tie for
closest point, the index on the left is returned.
target : `float`
tolerance : `float` or `None`
`float` or `None`
lo = 0
hi = len(self)
arr = list(self)
i = bisect.bisect_right(arr, target, lo=lo, hi=hi)
# i is the index at which target should be inserted if it is to be
# right of any equal components.
if i == lo:
# target is less than every entry of the set
nearest_index = i + 1
delta = self.at(nearest_index) - target
elif i == hi:
# target is greater than or equal to every entry of the set
nearest_index = i
delta = target - self.at(nearest_index)
# p_le <= target < p_g
# delta_left = target - p_le
# delta_right = p_g - target
# delta = min(delta_left, delta_right)
# Tie goes to the index on the left.
delta, nearest_index = min(
(abs(target - self.at(j)), j) for j in [i, i + 1]
if tolerance is not None:
if delta > tolerance:
return None
return nearest_index | {"url":"https://pyomo.readthedocs.io/en/stable/_modules/pyomo/dae/contset.html","timestamp":"2024-11-06T14:03:25Z","content_type":"text/html","content_length":"42652","record_id":"<urn:uuid:0d9518a6-2542-443a-acb5-4d9bea5aa1c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00141.warc.gz"} |
piece weight analysis in ball mill
WEBNov 1, 2019 · In general, the ball mill is ground by the impact energy applied to the materials owing to the dropping of the grinding media that is moved upward by the rotation of the mill.
The normal force applied to the materials by the grinding media were calculated using the EDEM and can be seen from Fig. 10. When the total force is checked, the .
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WEBJun 6, 2015 · The Ball Mill Charge: Clean the Ball Mill and accessory equipment before starting. Count and weigh the Balls by sizing group. Record the results on the Ball Count and Charge
Weight Determination form. Determine a charge weight made up of exactly 285 steel Balls weighing as close to 20,125 grams as possible.
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WEBSince balls have a greater surface area per unit weight than rods, they are better suited for fine grinding. The term ball mill is restricted to those having a length to diameter ratio of 2 to
1 and less. Ball mills in which the length to diameter ratio is between 3 and 5 are designated tube mills. The latter are sometimes divided into several ...
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WEBJan 1, 2022 · The experimental study was carried out in a labscale planetary ball mill (Droide®, Shanghai). As shown in Fig. 1, the planetary ball mill contains a disk and four grinding bowls,
each with a capacity of 1000 a clearer explanation, a simplified diagram is used, as shown in Fig. centers of rotation and revolution are O r and O .
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WEBMedilab Technocracy Offering Ball Mill 1 kg Heavy,Laboratory Grinding Mill in Ambala, Haryana. Also get Ball Mills price list from verified companies | ID:
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WEBOne tone per hour of dolomite is produced by a ball mill operating in closed circuit grinding with a 100 mesh screen. The screen analysis (weight %) is given below. Calculate the screen
efficiency. Here's the best way to solve it. We know that screen efficiency is given by, E = { (Xf Xb)* (Xd Xf)*Xd* (1 Xb)}/ { (Xd Xb)^ .. 1.
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WEBMar 10, 2010 · To get rid of pulp and rocks in the charge, a mill grindout (no ore feed) of 10 to 20. minutes is also performed before mill. inspection or relining. The complete grindout. is
required to obtain ...
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WEBFeb 1, 2001 · The energy efficiency of a planetary ball mill has been calculated as 51% at optimal conditions (Iasonna and Magini, 1996); therefore, the actual energy consumption will be
higher than that ...
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WEBApr 1, 2013 · the ball mill is smaller than the weight of media and the raw material. the weight of the rotating department of the ball mill can be ignored when the power is calculated. That i
s to say ： 1
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WEBApr 28, 2014 · For a planetary mill, it is ~ for a. point ball and ~ for a ball with diameter of 10 mm. Because of simplicity of the Eq. (2) it is usually used to. calculate the kinetic energy
of the ...
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WEBSep 22, 2003 · The earliest analysis of ball motion in tumbling mills dates back to early 1900, when Davis (1919) calculated trajectories of a single ball based on simple force balance. ...
This alone turns out to be a practical piece of information that can be used to decide the operating parameters of the mill.
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WEBThis means that during one rotation of the sun wheel, the grinding jar rotates twice in the opposite direction. This speed ratio is very common for Planetary Ball Mills in general. Planetary
ball mills with higher energy input and a speed ratio of 1: or even 1:3 are mainly used for mechanochemical appliions.
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WEBJan 15, 2019 · Ball Mill Velocity. The ball mill inside the jar will smash the particle in the jar. When the balls smash the particle, the particle will break into a smaller particle. Figure
10 shows the velocity of the ball mill motion inside the cylindrical jar with poles. From this graph, the velocity of the ball mill decreases from time to time.
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WEBFeb 13, 2009 · The results of discrete element method simulation were compared with actual grinding experimental results. The grinding rate constant K can be expressed as K=a exp(bn), where n
is the rotation speed. To investigate the correlation between K and the simulation results, a new factor, the calculated force, was defined as F cal =average .
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WEBFeb 15, 2001 · The present mathematical analysis of the milling dynamics aims at predicting the milling condition in terms of ωd and ωv, for the occurrence of the most effective impact between
the ball and vial wall to achieve MA. In the present analysis, the values of rd, rv and ball radius ( rb) are taken as 132, 35 and 5 mm, respectively (typical .
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WEBPROJECT REPORT – PHASE I on ANALYSIS AND DESIGN OF BALL MILL FOUNDATION FOR TATA IRON ORE PELLET PLANT AT JAMSHEDPUR Submitted in partial fulfillment for the award of the degree of BACHELOR OF
TECHNOLOGY in CIVIL ENGINEERING by MOHD ADEEL ANKIT GOYAL VIVEK .
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WEBJul 2, 2020 · Analysis of grinding actions of ball mills by discrete element . method, in: No. CONF. ... Apart from the milling ball itself, the balls to powder weight ratio also plays an
important ...
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WEBJun 16, 2015 · Fill the 700ml test can with ore and compact by shaking. Add more ore as necessary until further compaction ceases. Weight and transfer ore to the ball mill. Grind dry for 100
revolutions. Empty the ball charge and ore through a coarse screen to separate the balls from the ore.
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WEBNov 8, 2016 · Table 1 shows a comparison of the specific energy values calculated from Eqs. () and for the 100 mesh test sieve (S = 150 µm) and seven values of G in the range of –3 g/ can be
seen that at a G value of both the equations give the same estimate of the specific energy. For G values greater than the Bond equation gives .
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WEBJun 19, 2015 · We can calculate the steel charge volume of a ball or rod mill and express it as the % of the volume within the liners that is filled with grinding media. While the mill is
stopped, the charge volume can be gotten by measuring the diameter inside the liners and the distance from the top of the charge to the top of the mill. The % loading or ...
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WEBMar 10, 2023 · Lathe Work Piece Weight Limits. The weights do not include the weight of the workholding. These weights are estimates. They do not make a safe setup. The chuck and jaw have a
weight capacity. The weight of the workpiece must be less than the capacity. The operator is responsible for ensuring that the setup and operation of the .
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WEBBall Mills. Ball mills originally were used to grind approximately 2 in. material to pass 10 to 80 mesh screens. Present day practice is to use a feed of about 1/2 in. or finer. Product size
has become increasingly finer and no actual grind limit is indied.
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WEBAug 1, 2021 · A dimensional analysis of the ball mill process is carried out through the BuckinghamPi method. The dimensionless quantities identified are discussed and used in order to suggest
scaling criteria for ball mills. The flowability and the particle size distribution of an alumina powder ground in laboratory ball mills of various dimensions .
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WEBOct 1, 2015 · A fullscale threecompartment FL® cement grinding ball mill with dimensions of Ø × L10 operating in open circuit was sampled to analyse the grinding media effect on specific
breakage rate function of reduction performance of the ball mill was evaluated with respect to the applied grinding media size.
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WEBHere's the best way to solve it. Q2) One ton per hour of dolomite is produced by a ball mill operating in a closed circuit grinding with a 100 mesh screen. The screen analysis (weight %) is
shown in the below table. Calculate the mass ratios of the overflow and underflow to feed and the overall effectiveness of the screen.
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WEBBradken Bullnose®discharge cones Manufactured from superior composite wear materials to extend wear life of the liners and reduce the overall weight by 40% over steel ® discharge cone system
reduces the relining times by 50% to maximise mill availability. Discharge grate slot analysis is carried out across all grate designs to .
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WEBJul 20, 2017 · A screen analysis down to 3 mesh is also made. Bond Impact, grindability tests, and abrasion index tests are also run on the sample. Rod mill grindability tests for Work Index
are run at 10 or 14 mesh, and ball mill Work Index tests are run at the desired grind if finer than 28 mesh.
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WEBJun 19, 2015 · The approximate horsepower HP of a mill can be calculated from the following equation: HP = (W) (C) (Sin a) (2π) (N)/ 33000. where: W = weight of charge. C = distance of centre
of gravity or charge from centre of mill in feet. a = dynamic angle of repose of the charge. N = mill speed in RPM. HP = A x B x C x L. Where.
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WEBJul 2, 2020 · In total, 165 scenarios were simulated. When the mills charge comprising 60% of small balls and 40% of big balls, mill speed has the greatest influence on power consumption. When
the mill charge is more homogeneous size, the effect of ball segregation is less and so the power consumption of the mill will be less affected.
WhatsApp: +86 18203695377 | {"url":"https://deltawatt.fr/piece_weight_analysis_in_ball_mill.html","timestamp":"2024-11-14T00:24:21Z","content_type":"application/xhtml+xml","content_length":"26655","record_id":"<urn:uuid:93444637-4ad0-496d-a7b1-497f4dc2bb40>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00255.warc.gz"} |
Radius of a Sphere in context of length to volume
31 Aug 2024
The Relationship Between the Radius of a Sphere and Its Volume: A Geometric Analysis
This article explores the geometric properties of a sphere, focusing on the relationship between its radius and volume. We derive the formula for the volume of a sphere in terms of its radius and
examine the implications of this relationship.
A sphere is a three-dimensional shape that is perfectly symmetrical about its center. Its volume is a fundamental property that has been studied extensively in mathematics and physics. In this
article, we will investigate the relationship between the radius of a sphere and its volume, using geometric analysis to derive the relevant formulae.
The Formula for the Volume of a Sphere
The volume of a sphere (V) can be expressed as:
V = (4/3) * π * r^3
where r is the radius of the sphere. This formula is derived from the concept of integration, where the volume of the sphere is calculated by summing up the volumes of infinitesimally small spherical
Geometric Interpretation
The formula for the volume of a sphere can be interpreted geometrically as follows:
• The factor 4/3 represents the proportion of the sphere’s volume that lies within a cube with edge length equal to the diameter of the sphere.
• The term π * r^2 represents the area of the sphere’s surface, which is proportional to the square of its radius.
• The exponent 3 in the formula indicates that the volume of the sphere grows cubically with respect to its radius.
In conclusion, this article has explored the relationship between the radius of a sphere and its volume. We have derived the formula for the volume of a sphere in terms of its radius and examined the
geometric implications of this relationship. The formula V = (4/3) * π * r^3 provides a fundamental link between the length and volume properties of a sphere, highlighting the importance of geometric
analysis in understanding the physical world.
• [1] Weisstein, E.W. Sphere. MathWorld–A Wolfram Web Resource.
• [2] Wikipedia. Sphere (geometry).
Related articles for ‘length to volume’ :
• Reading: Radius of a Sphere in context of length to volume
Calculators for ‘length to volume’ | {"url":"https://blog.truegeometry.com/tutorials/education/cd0832553d32f0dfe2a9dd26c6d338fc/JSON_TO_ARTCL_Radius_of_a_Sphere_in_context_of_length_to_volume.html","timestamp":"2024-11-06T08:19:26Z","content_type":"text/html","content_length":"15677","record_id":"<urn:uuid:386223e5-020a-42cd-a75f-9713609a843c>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00838.warc.gz"} |
String theory
Critical string models
Extended objects
Topological strings
Differential cohomology
Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Fiber integration
Application to gauge theory
An orientifold (Dai-Leigh-Polchinski 89, p. 12) is a target spacetime for string sigma-models that combines aspects of $\mathbb{Z}_2$-orbifolds with orientation reversal on the worldsheet, hence the
In type II string theory orientifold backgrounds (inducing e.g. type I string theory, and the Sugimoto string) with $\mathbb{Z}_2$-fixed points – called O-planes (see there for more) – are required
for RR-field tadpole cancellation. This is a key consistency condition in particular for intersecting D-brane models used in string phenomenology.
Where generally (higher gauge) fields in physics/string theory are cocycles in (differential) cohomology theory and typically in complex oriented cohomology theory, fields on orientifolds are
cocycles in genuinely $\mathbb{Z}_2$-equivariant cohomology and typically in real-oriented cohomology theory. For instance, the B-field, which otherwise is a (twisted) cocycle in (ordinary)
differential cohomology, over an orientifold is a cocycle in (twisted) HZR-theory, and the RR-fields, which usually are cocycles in (twisted differential) K-theory, over an orientifold are cocycles
in KR-theory (Witten 98).
An explicit model for B-fields for the bosonic string on orientifolds (differential HZR-theory) is given in (Schreiber-Schweigert-Waldorf 05) and examples are analyzed in this context in (
Gawedzki-Suszek-Waldorf 08). See also (HMSV 16, HMSV 19).
The claim that for the superstring the B-field is more generally a cocycle with coefficients in the Picard infinity-group of complex K-theory (super line 2-bundles) and a detailed discussion of the
orientifold version of this can be found in (Distler-Freed-Moore 09, Distler-Freed-Moore 10) with details in (Freed 12).
The quadratic pairing entering the 11d Chern-Simons theory that governs the RR-field here as a self-dual higher gauge field is given in (DFM 10, def. 6).
Orientifold backreaction?
Essentially all existing results on orientifolds (such as O-plane-charges and RR-field tadpole cancellation) are derived for string sigma-models on flat orbifold (toroidal orbifold) or orientifolded
Calabi-Yau manifold target spacetimes, or for orientifoldings of algebraically defined rational CFT string vacua (Gepner models, non-geometric vacua).
No string-theory results on back-reacted orientifolds
There are to date no results in actual string theory for what one might expect to be the curved back-reacted geometry of orientifolds, analogous to the curved near horizon geometry that is well-known
for the case of black D-branes.
From Banks-van den Broeck 06:
The discussion of these $[$orientifold$]$ compactifications is generally carried out in low energy effective field theory $[$3a, 3b $]$$[$4$]$, despite the fact that they all contain orientifold
singularities. Further, there is no perturbative world sheet treatment of these backgrounds.
The inevitable orientifold of flux compactifications is one potential barrier to an effective field theory treatment $[$footnote 1 : G. Moore and S. Ramanujam have emphasized to us the problems
with the back reaction of the orientifold $[...$] $]$
From Cordova-de Luca-Tomasiello 19 (p. 2 and p. 30):
To assess the validity of these $[$assumed effective orientifold$]$ solutions in string theory, one should ideally use the full string theory action, or switch to a dual description.
Unfortunately neither of these options is available
the presence of O-planes is inferred by comparison with their flat-space behavior. Since the $[$spacetimes with orientifolds considered$]$ have strong curvature and coupling, stringy corrections
come into play, and it is impossible to decide with supergravity alone whether the solutions are valid.
It is important to stress that this will be so for any solution with O-planes. It would be important, then,to develop techniques to decide whether a solution with O-planes will survive in full
string theory. In other words, it would be important to understand what conditions one needs to impose near the O-plane singularities.
We clearly need alternative procedures that are better justified physically.
A popular assumption in effective field theory
Nevertheless, motivated from the fact that the computations for flat orbifolds show that the O-planes there are similar to (while clearly different from!) D-branes (not in their back-reacted form as
black branes, though!) with negative tension (i.e. negative energy density) it may seem plausible that the low energy effective field theory of perturbative string theory vacua including O-planes is
a modification of supergravity where negative-energy source-terms are added to the equations of motion much like one might add black D-brane-contributions, but simply equipped with a negative sign
This ad hoc effective field theory picture of orientifold backgrounds has been advocated, seminally, in Giddings-Kachru-Polchinski 01, in a one-sentence argument (below (2.19)):
String theory does have such $[$negative tension$]$ objects, and so evades the no-go theorem $[$which rules out certain warped solutions of supergravity$]$.
There this is followed by reference to a presumed example considered earlier in Verlinde 00, where the statement is introduced in a similar manner (above (9)):
In a more complete treatment, we must also include the backreaction of the 64 orientifold planes. These have a negative tension equal to −1/4 times the D3-brane tension, which need to be taken
into account. We can write an explicit form for the background metric $[$just as for D-branes but with a minus sign included$]$.
One may feel this is plausible – and it might even be right, sometimes – but there does not exist a derivation of this statement from actual perturbative string theory (see above), beyond the
hand-waving leap of faith extrapolating from perturbative string theory on flat orientifolds to curved backreacted orientifold throat geometries (if such indeed exist).
Its use in the landscape/swampland literature
Nevertheless, in the wake of the discussion of the landscape (or not) of de Sitter string theory vacua and of the “swampland conjectures” it became popular to rely on the handwaving argument of
Giddings-Kachru-Polchinski 01 and behave as if it is established that questions about low energy effective orientifold string theory vacua may be answered using a modification of supergravity where
the equations of motion are changed – by hand – simply by including negative-tension source terms of some form.
This step happens for instance around (2.2) in Junghans 20 where it is justified, without references, by the words “as is standard in the literature” (footnote 5 in Junghans 20).
Lifts to M-theory F-theory
Lifts of orientifold background from type II string theory to F-theory go back to (Sen 96, Sen 97a).
Lifts of type IIA string theory orientifolds of D6-branes to D-type ADE singularities in M-theory (through the duality between M-theory and type IIA string theory) goes back to (Sen 97b). See at
heterotic M-theory on ADE-orbifolds.
A more general scan of possible lifts of type IIA orientifolds to M-theory is indicated in (Hanany-Kol 00, around (3.2)), see (Huerta-Sati-Schreiber 18, Prop. 4.7) for details.
For instance the O4-plane lifts to the MO5-plane.
The concept originates around
Early accounts include
• Sunil Mukhi, Orientifolds: The Unique Personality Of Each Spacetime Dimension, Workshop on Frontiers of Field Theory, Quantum Gravity and String Theory, Puri, India, 12 - 21 Dec 1996 (
arXiv:hep-th/9710004, cern:335233)
• Jan de Boer, Robbert Dijkgraaf, Kentaro Hori, Arjan Keurentjes, John Morgan, David Morrison, Savdeep Sethi, section 3 of Triples, Fluxes, and Strings, Adv. Theor. Math. Phys. 4 (2002) 995-1186 (
Traditional lecture notes include
Textbook discussion is in
and specifically in the context of intersecting D-brane models with an eye towards string phenomenology in
• Marcus Berg, Introduction to Orientifolds (pdf, pdf)
The original observation that D-brane charge for orientifolds should be in KR-theory is due to
and was re-amplified in
In terms of KO-theory
Discussion of orbi-orienti-folds in terms of equivariant KO-theory is in
• N. Quiroz, Bogdan Stefanski, Dirichlet Branes on Orientifolds, Phys.Rev. D66 (2002) 026002 (arXiv:hep-th/0110041)
• Volker Braun, Bogdan Stefanski, Orientifolds and K-theory, Braun, Volker. “Orientifolds and K-theory.” Progress in String, Field and Particle Theory. Springer, Dordrecht, 2003. 369-372 (
• H. Garcia-Compean, W. Herrera-Suarez, B. A. Itza-Ortiz, O. Loaiza-Brito, D-Branes in Orientifolds and Orbifolds and Kasparov KK-Theory, JHEP 0812:007, 2008 (arXiv:0809.4238)
A definition and study of orientifold bundle gerbes, modeling the B-field background for the bosonic string (differential HZR-theory), is in
• Urs Schreiber, Christoph Schweigert, Konrad Waldorf, Unoriented WZW models and Holonomy of Bundle Gerbes, Communications in Mathematical Physics August 2007, Volume 274, Issue 1, pp 31-64 (
• Krzysztof Gawedzki, Rafal R. Suszek, Konrad Waldorf, Bundle Gerbes for Orientifold Sigma Models Adv. Theor. Math. Phys. 15(3), 621-688 (2011) (arXiv:0809.5125)
see also
• Pedram Hekmati, Michael Murray, Richard Szabo, Raymond Vozzo, Real bundle gerbes, orientifolds and twisted KR-homology (arXiv:1608.06466)
• Pedram Hekmati, Michael Murray, Richard Szabo, Raymond Vozzo, Sign choices for orientifolds, Commun. Math. Phys. 378, 1843–1873 (2020) (arXiv:1905.06041)
An elaborate formalization in terms of differential cohomology in general and twisted differential K-theory in particular that also takes the spinorial degrees of freedom into account is briefly
sketched out in
based on stuff like
Details on the computation of string scattering amplitudes in such a background:
Related lecture notes / slides include
• Daniel Freed, Dirac charge quantization, K-theory, and orientifolds, talk at a workshop Mathematical methods in general relativity and quantum field theories, Paris, November 2009 (pdf, pdf)
• Greg Moore, The RR-charge of an orientifold, Oberwolfach talk 2010 (pdf, pdf, ppt)
• Daniel Freed, Lectures on twisted K-theory and orientifolds, lecures at K-Theory and Quantum Fields, ESI 2012 (pdf)
A detailed list of examples of KR-theory of orientifolds and their T-duality:
A formulation of some of the relevant aspects of (bosonic) orientifolds in terms of the differential nonabelian cohomology with coefficients in the 2-group $AUT(U(1))$ coming from the crossed module
$[U(1) \to \mathbb{Z}_2]$ is indicated in
More on this in section 3.3.10 of
Examples and Models
Specifically K3 orientifolds ($\mathbb{T}^4/G_{ADE}$) in type IIB string theory, hence for D9-branes and D5-branes:
• Eric Gimon, Joseph Polchinski, Section 3.2 of: Consistency Conditions for Orientifolds and D-Manifolds, Phys. Rev. D54: 1667-1676, 1996 (arXiv:hep-th/9601038)
• Eric Gimon, Clifford Johnson, K3 Orientifolds, Nucl. Phys. B477: 715-745, 1996 (arXiv:hep-th/9604129)
• Alex Buchel, Gary Shiu, S.-H. Henry Tye, Anomaly Cancelations in Orientifolds with Quantized B Flux, Nucl.Phys. B569 (2000) 329-361 (arXiv:hep-th/9907203)
• P. Anastasopoulos, A. B. Hammou, A Classification of Toroidal Orientifold Models, Nucl. Phys. B729:49-78, 2005 (arXiv:hep-th/0503044)
Specifically K3 orientifolds ($\mathbb{T}^4/G_{ADE}$) in type IIA string theory, hence for D8-branes and D4-branes:
• Jaemo Park, Angel Uranga, A Note on Superconformal $\mathcal{N}=2$ theories and Orientifolds, Nucl. Phys. B542:139-156, 1999 (arXiv:hep-th/9808161)
• G. Aldazabal, S. Franco, Luis Ibanez, R. Rabadan, Angel Uranga, D=4 Chiral String Compactifications from Intersecting Branes, J. Math. Phys. 42:3103-3126, 2001 (arXiv:hep-th/0011073)
• G. Aldazabal, S. Franco, Luis Ibanez, R. Rabadan, Angel Uranga, Intersecting Brane Worlds, JHEP 0102:047, 2001 (arXiv:hep-ph/0011132)
• H. Kataoka, M. Shimojo, $SU(3) \times SU(2) \times U(1)$ Chiral Models from Intersecting D4-/D5-branes, Progress of Theoretical Physics, Volume 107, Issue 6, June 2002, Pages 1291–1296 (
arXiv:hep-th/0112247, doi:10.1143/PTP.107.1291)
The $\mathbb{Z}_N$ action with even $N$ contains an order 2 element $[ ...]$ Then there will be D8-branes in the type IIA D4-brane theory. Since the concept of intersecting D-branesinvolves use
of the same dimensional D-branes, we restrict ourselves to the case that the order $N$ of $\mathbb{Z}_N$ is odd. (p. 4)
• Gabriele Honecker, Non-supersymmetric Orientifolds with D-branes at Angles, Fortsch.Phys. 50 (2002) 896-902 (arXiv:hep-th/0112174)
• Gabriele Honecker, Intersecting brane world models from D8-branes on $(T^2 \times T^4/\mathbb{Z}_3)/\Omega\mathcal{R}_1$ type IIA orientifolds, JHEP 0201 (2002) 025 (arXiv:hep-th/0201037)
• Gabriele Honecker, Non-supersymmetric orientifolds and chiral fermions from intersecting D6- and D8-branes, thesis 2002 (pdf)
The Witten-Sakai-Sugimoto model on D4-D8-brane bound states for QCD with orthogonal gauge groups on O-planes:
• Toshiya Imoto, Tadakatsu Sakai, Shigeki Sugimoto, $O(N)$ and $USp(N)$ QCD from String Theory, Prog.Theor.Phys.122:1433-1453, 2010 (arXiv:0907.2968)
• Hee-Cheol Kim, Sung-Soo Kim, Kimyeong Lee, 5-dim Superconformal Index with Enhanced $E_n$ Global Symmetry, JHEP 1210 (2012) 142 (arXiv:1206.6781)
Specifically D5 brane models T-dual to D6/D8 models:
Specifically for D6-branes:
• S. Ishihara, H. Kataoka, Hikaru Sato, $D=4$, $N=1$, Type IIA Orientifolds, Phys. Rev. D60 (1999) 126005 (arXiv:hep-th/9908017)
• Mirjam Cvetic, Paul Langacker, Tianjun Li, Tao Liu, D6-brane Splitting on Type IIA Orientifolds, Nucl. Phys. B709:241-266, 2005 (arXiv:hep-th/0407178)
Specifically for D3-branes/D7-branes:
Specifically 2d toroidal orientifolds:
2d toroidal orientifolds:
• Dongfeng Gao, Kentaro Hori, Section 7.3 of: On The Structure Of The Chan-Paton Factors For D-Branes In Type II Orientifolds (arXiv:1004.3972)
• Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg, String theory on elliptic curve orientifolds and KR-theory (arXiv:1402.4885)
• Dieter Lüst, S. Reffert, E. Scheidegger, S. Stieberger, Resolved Toroidal Orbifolds and their Orientifolds, Adv.Theor.Math.Phys.12:67-183, 2008 (arXiv:hep-th/0609014)
Orientifold Gepner models
Orientifolds of Gepner models
• Brandon Bates, Charles Doran, Koenraad Schalm, Crosscaps in Gepner Models and the Moduli space of $T^2$ Orientifolds, Advances in Theoretical and Mathematical Physics, Volume 11, Number 5,
839-912, 2007 (arXiv:hep-th/0612228)
Specifically string phenomenology and the landscape of string theory vacua of Gepner model orientifold compactifications:
• T.P.T. Dijkstra, L. R. Huiszoon, Bert Schellekens, Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models, Phys.Lett. B609 (2005) 408-417 (arXiv:hep-th/0403196)
• T.P.T. Dijkstra, L. R. Huiszoon, Bert Schellekens, Supersymmetric Standard Model Spectra from RCFT orientifolds, Nucl.Phys.B710:3-57,2005 (arXiv:hep-th/0411129)
Lift to M-theory
Lifts of orientifolds to M-theory (MO5, MO9) and F-theory are discussed in
• Ashoke Sen, F-theory and Orientifolds (arXiv:hep-th/9605150)
• Ashoke Sen, Orientifold Limit of F-theory Vacua (arXiv:hep-th/9702165)
• Ashoke Sen, A Note on Enhanced Gauge Symmetries in M- and String Theory, JHEP 9709:001,1997 (arXiv:hep-th/9707123)
• Kentaro Hori, Consistency Conditions for Fivebrane in M Theory on $\mathbb{R}^5/\mathbb{Z}_2$ Orbifold, Nucl. Phys. B539:35-78, 1999 (arXiv:hep-th/9805141)
• Eric Gimon, On the M-theory Interpretation of Orientifold Planes (arXiv:hep-th/9806226, spire:472499)
• Changhyun Ahn, Hoil Kim, Hyun Seok Yang, $SO(2N)$$(0,2)$ SCFT and M Theory on $AdS_7 \times \mathbb{R}P^4$, Phys.Rev. D59 (1999) 106002 (arXiv:hep-th/9808182)
• Amihay Hanany, Barak Kol, section 4 of On Orientifolds, Discrete Torsion, Branes and M Theory, JHEP 0006 (2000) 013 (arXiv:hep-th/0003025)
• Philip Argyres, Ron Maimon, Sophie Pelland, The M theory lift of two O6 planes and four D6 branes, JHEP 0205 (2002) 008 (arXiv:hep-th/0204127)
• Edward Witten, Solutions Of Four-Dimensional Field Theories Via M Theory, (arXiv:hep-th/9703166)
The MO5 is originally discussed in
See also:
The classification in Hanany-Kol 00 (3.2) also appears, with more details, in Prop. 4.7 of
The “higher orientifold” appearing in Horava-Witten theory with circle 2-bundles replaced by the circle 3-bundles of the supergravity C-field is discussed towards the end of
Orientifold backreaction
• Herman Verlinde, Holography and Compactification, Nucl. Phys. B580 (2000) 264-274 (arXiv:hep-th/9906182)
• Steven Giddings, Shamit Kachru, Joseph Polchinski, Hierarchies from Fluxes in String Compactifications, Phys. Rev. D66:106006, 2002 (arXiv:hep-th/0105097)
• Tom Banks, K. van den Broek, Massive IIA flux compactifications and U-dualities, JHEP 0703:068, 2007 (arXiv:hep-th/0611185)
• Clay Cordova, G. Bruno De Luca, Alessandro Tomasiello, New de Sitter Solutions in Ten Dimensions and Orientifold Singularities (arXiv:1911.04498)
• Daniel Junghans, O-plane Backreaction and Scale Separation in Type IIA Flux Vacua (arXiv:2003.06274) | {"url":"https://ncatlab.org/nlab/show/orientifold","timestamp":"2024-11-02T15:06:24Z","content_type":"application/xhtml+xml","content_length":"88984","record_id":"<urn:uuid:6ab322ec-6362-4278-9617-f54bff2357e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00131.warc.gz"} |
Work Performed by a Steam Engine
Task number: 1803
The temperature of steam coming from a boiler to a cylinder of a steam engine is 120 °C; the steam condenses in a cold reservoir with a temperature of 40 °C. What is the maximum work performed by the
engine in ideal conditions with a heat consumption of 4.2 kJ?
• Hint 1
If we want the efficiency to be maximal, the engine must work according to Carnot cycle.
• Hint 2
Find the definition of an efficiency of a heat engine in general, and the definition of Carnot engine efficiency.
• Analysis
First we write down the efficiency of a heat engine as the performed work divided by the supplied heat.
The efficiency of the engine is maximal provided that the engine operates according to the Carnot cycle. In such a case it would perform maximum work with the same heat supplied. The efficiency
of Carnot cycle depends only on the temperatures between which it operates.
By comparing both terms we obtain the expression for the efficiency, from which we evaluate the performed work.
• Given values
t[1] = 120 °C => T[1] = 393 K temperature of steam (hot reservoir)
t[2] = 40 °C => T[2] = 313 K temperature of cold reservoir
Q = 4.2 kJ = 4.2·10^3 J heat consumption
W = ? maximum work performed by engine
• Solution
The efficiency η of a heat engine is defined as the ratio of performed work W and supplied heat Q. Therefore it applies
Maximum efficiency is achieved when the engine operates according to Carnot cycle; the efficiency η depends only on the temperature T[1] of the hot reservoir and the temperature T[2] of the cold
reservoir. Therefore the following applies
In our case, the temperature of the hot reservoir corresponds to the temperature of the steam in the cylinder of the steam engine.
By comparing the two equations for efficiency we obtain an equation
We evaluate the work performed by the engine from this equation:
• Numerical solution
\[W=\frac{Q\left(T_1-T_2\right)}{T_1}= \frac{4.2\cdot{ 10^3}\cdot \left(393-3135\right)}{393}\,\mathrm{J}\,\dot{=}\,850\,\mathrm{J}\]
• Answer
Maximum work performed by the engine is approximately 850 J. | {"url":"https://physicstasks.eu/1803/work-performed-by-a-steam-engine","timestamp":"2024-11-11T17:13:10Z","content_type":"text/html","content_length":"29025","record_id":"<urn:uuid:01d681a6-80fa-463a-98c9-15121d53a7b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00041.warc.gz"} |
Turing Machine
So Wikipedia: ‘A Turing machine is an abstract machine^ that manipulates symbols on a strip of tape according to a table of rules; to be more exact, it is a mathematical model that defines such a
device. Despite the model's simplicity, given any computer algorithm, a Turing machine can be constructed that is capable of simulating that algorithm's logic.’
In Victor’s Way terms: A Turing machine is a non-localized (i.e. abstract ≈ universal) set of rules that can simulate, i.e. copy, and so become, any local set of rules (read: boundaries or limits),
whereby neither set rules is defined.
A set of rules (i.e. a rules, laws or conditions simulating machine) is named an algorithm. If an algorithm is understood (so Wikipedia, referring to mathematics and computer science) as: a
self-contained (bounded or limited) step-by-step (i.e. quantised) set of operations to be performed, then a (basic or original) Turing machine is an (non-localised, i.e. universal) algorithm that can
simulate, i.e. copy and so become any other (secondary, i.e. local) algorithm whereby each copy manifests as fractal elaboration of the original (universal) Turing algorithm. In other words, the
secondary algorithm, for instance any one of n bio-systems, operates merely as the original Turing machine locally (i.e. conditionally as determined by alternate rules, so the Buddha) elaborated.
In simplest terms, the Creative Drive, call it God, Brahman, the Atman (i.e. SELF), the Way (or Tao) functions as universal Turing Machine that can become any local machine (i.e. a niche operation,
such as the atma or (little) self)) whose (niche determined) rules it copies. Meaning that the local machine, for instance a human or a goose as fractal elaborations are the original algorithm,
albeit with frills and whistles.
Or, as Meister Eckhart stated: “Some there are so simple as to think of God as if He dwelt there, and of themselves as being here. It is no so. God (i.e. the original or originating Turing machine)
and I (as its fractal elaboration) are one.”
See: the Self-realization icon | {"url":"https://www.victorsway.eu/com/com%202/turing.htm","timestamp":"2024-11-05T09:56:00Z","content_type":"text/html","content_length":"47710","record_id":"<urn:uuid:6e1da332-c2a9-4e07-91ed-9af308d0b203>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00368.warc.gz"} |
Converting Laplacian to polar coordinates
• Thread starter fahraynk
• Start date
In summary, the books solution is to take the Laplacian of the equation and eliminate the cosine and sine terms.
Homework Statement
U_{tt}=\alpha^2\bigtriangledown^2U$$ in polar coordinates if solution depends only on R, t.
Homework Equations
The Attempt at a Solution
So, the books solution is $$U_{tt}=\alpha^2[U_{rr}+\frac{1}{r}U_r]$$. I am getting stuck along the way can't figure out this last step I think. Here is my attempt:
$$\frac{du}{dx}=\frac{du}{dr}\frac{dr}{dx} + \frac{du}{d\Theta}\frac{d\Theta}{dx}$$
$$\frac{d}{dx}(\frac{du}{dr}\frac{dr}{dx} + \frac{du}{d\Theta}\frac{d\Theta}{dx}) =
\frac{d^2u}{dr^2}*(\frac{dr}{dx})^2 + \frac{d^2u}{drd\Theta}\frac{dr}{dx}\frac{d\Theta}{dx} + \frac{du}{dr}\frac{d^2r}{dx^2}+\frac{d^2u}{d\Theta^2}(\frac{d\Theta}{dx})^2
+ \frac{d^2u}{d\Theta dr}\frac{d\Theta}{dx}\frac{dr}{dx} + \frac{du}{d\Theta}\frac{d^2\Theta}{dx^2}
all the derivatives with theta = 0. So it becomes :
$$U_{rr}r_x^2 + U_rr_{xx}$$
$$ x=rcos(\Theta) $$
$$ r_x = \frac{x}{sqrt(x^2+y^2)} = cos(\Theta)$$
$$r_{xx} = \frac{y^2}{(x^2+y^2)^{3/2}} = \frac{sin^2(\Theta)}{r}$$
When I plug in, I get :
$$U_{rr}cos^2(\Theta) + \frac{sin^2(\Theta)}{r}U_r$$
the books answer is the same, but without the sine and cos terms. How do I get rid of them otherwise what am I doing wrong?
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
fahraynk said:
Homework Statement
U_{tt}=\alpha^2\bigtriangledown^2U$$ in polar coordinates if solution depends only on R, t.
Homework Equations
The Attempt at a Solution
So, the books solution is $$U_{tt}=\alpha^2[U_{rr}+\frac{1}{r}U_r]$$. I am getting stuck along the way can't figure out this last step I think. Here is my attempt:
$$\frac{du}{dx}=\frac{du}{dr}\frac{dr}{dx} + \frac{du}{d\Theta}\frac{d\Theta}{dx}$$
$$\frac{d}{dx}(\frac{du}{dr}\frac{dr}{dx} + \frac{du}{d\Theta}\frac{d\Theta}{dx}) =
\frac{d^2u}{dr^2}*(\frac{dr}{dx})^2 + \frac{d^2u}{drd\Theta}\frac{dr}{dx}\frac{d\Theta}{dx} + \frac{du}{dr}\frac{d^2r}{dx^2}+\frac{d^2u}{d\Theta^2}(\frac{d\Theta}{dx})^2
+ \frac{d^2u}{d\Theta dr}\frac{d\Theta}{dx}\frac{dr}{dx} + \frac{du}{d\Theta}\frac{d^2\Theta}{dx^2}
all the derivatives with theta = 0. So it becomes :
$$U_{rr}r_x^2 + U_rr_{xx}$$
$$ x=rcos(\Theta) $$
$$ r_x = \frac{x}{sqrt(x^2+y^2)} = cos(\Theta)$$
$$r_{xx} = \frac{y^2}{(x^2+y^2)^{3/2}} = \frac{sin^2(\Theta)}{r}$$
When I plug in, I get :
$$U_{rr}cos^2(\Theta) + \frac{sin^2(\Theta)}{r}U_r$$
the books answer is the same, but without the sine and cos terms. How do I get rid of them otherwise what am I doing wrong?
You forgot to add ##\partial^2 u/ \partial y^2## to get the Laplacian.
FAQ: Converting Laplacian to polar coordinates
What is Laplacian in polar coordinates?
The Laplacian in polar coordinates is a mathematical operator used to describe the change in a function with respect to its position in a polar coordinate system. It is commonly denoted as ∇^2 and is
defined as the sum of the second partial derivatives of the function with respect to the radial coordinate and the angular coordinate.
How do I convert a Laplacian to polar coordinates?
To convert a Laplacian to polar coordinates, you can use the following formula: ∇^2f = (1/r) ∂/∂r (r ∂f/∂r) + (1/r^2) ∂^2f/∂θ^2, where r is the radial coordinate and θ is the angular coordinate. This
formula can also be written in terms of the polar coordinates (r, θ) as ∇^2f = (1/r^2) (∂^2f/∂r^2 + (1/r) ∂f/∂r) + (1/r^2) ∂^2f/∂θ^2.
Why is converting Laplacian to polar coordinates useful?
Converting Laplacian to polar coordinates can be useful in solving problems involving functions defined in a polar coordinate system, such as problems in electromagnetism, fluid mechanics, and
quantum mechanics. It allows for a simpler and more intuitive representation of these problems and can often lead to more elegant solutions.
What are some applications of converting Laplacian to polar coordinates?
Some applications of converting Laplacian to polar coordinates include finding the electric potential and electric field in a system of charges, solving the Schrödinger equation in quantum mechanics,
and modeling the flow of fluids in cylindrical or spherical systems.
Are there any limitations to converting Laplacian to polar coordinates?
One limitation of converting Laplacian to polar coordinates is that it can only be used for functions that are rotational symmetric, meaning that they are unchanged when rotated around the origin.
Additionally, it may not be the most efficient method for solving problems involving complex geometries or boundary conditions. | {"url":"https://www.physicsforums.com/threads/converting-laplacian-to-polar-coordinates.893889/","timestamp":"2024-11-09T11:10:21Z","content_type":"text/html","content_length":"86966","record_id":"<urn:uuid:c65d7170-a0c1-4ee7-b271-9e2bc0cf56ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00216.warc.gz"} |
Repayment Chart Loans having Annual Repayement - Loan Chart
Example 1:-
Suppose Loan taken from SBI 20 Lacs @ 15% p.a on 1 April 2016
It is to repaid in 4 years in equal annual instalments.First Instalment paid on 31 March 2017
Instalment repaid at the end of year
Prepare Loan Chart for Next 4 years
View Answer
Example 2:-
Solve last question assuming Loan was taken on 1 April 2016 And first annual installment including interest repaid on 1 April 2017 (beginning of next year)
View Answer
Example 3:-
Solve Q1 assuming Loan was taken on 1 June 2016 And first annual instalment including interest repaid on 31 May 2017
View Answer | {"url":"https://www.teachoo.com/3859/266/Repayment-Chart-Loans-having-Annual-Repayement/category/Loan-Chart/","timestamp":"2024-11-02T01:41:15Z","content_type":"text/html","content_length":"118713","record_id":"<urn:uuid:b3c27091-aa41-4191-894d-1a9bef1b76b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00364.warc.gz"} |
Enter and come out of the opposite side, but that
Submitted by Atanu Chaudhuri on Sat, 09/04/2016 - 09:38
How many ways to go?
While into the maze, you must not retrace any part of your earlier path. With this simple restriction, you just have to enter the maze and try to come out.
It's a very simple maze. You may enter through any opening and go out also through any exit.
Only one restriction this time
1. While on your way you must not retrace your path.
Three questions
1. Entry and exit should be from opposite sides. How many ways?
2. Along how many paths can you walk in and out of the maze?
3. What is the shortest path?
You may spend some time in forming more questions on the maze. You know,
Solving the maze is not so important, looking at the problem from all angles is more important.
If you look at a problem from all angles, and do this on various problems, many times, your ability to think new goes on increasing, and this ability is one of the more important problem solving
We have another name for this problem solving skill enhancement technique - the many ways technique.
By nature, when we face a problem, not only during the solving process we look automatically at every aspect of the problem, but even after its solution we try to find other solutions. In short,
practicing many ways technique has become our habit.
In many of our more detailed problem solving posts here, we have mentioned this technique, but only one complete post is devoted to this very important problem solving skill enhancement technique. If
you like you may refer to it here.
The other skill that you use in any of the problems that you solve is the pattern recognition skill, and you use it by pattern recognition techniques.
Without ability to detect and use of effective patterns no problem can be solved.
In any case, enjoy, this maze is really simple.
And watch out this section for more surprises. | {"url":"https://mail.suresolv.com/brain-teaser/enter-and-come-out-opposite-side-thats-not-all-maze","timestamp":"2024-11-06T04:29:32Z","content_type":"text/html","content_length":"31128","record_id":"<urn:uuid:6fe313ae-4572-4bf1-8532-b19be4f9f977>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00261.warc.gz"} |
We have to make decisions on a daily basis. There are many decisions we make every day which are not very important, but about some of them we think more thoroughly because they are more examples -
important than others. In the decision making process we have our own criteria. For some decisions comparative process is simple and it can be expressed in units of measurement. For collection
example, price, weight, height and many other values can be expressed in units of measurement. What about the criteria that can not be expressed in such way? For example, quality, design,
reliability, suitableness, pleasure etc. Moreover, what about those criteria depending on our own belief, taste or standards? examples
Have you ever been in a situation when A is much better than B, B is slightly better than C
and C is better than A on the one hand, but on the other hand the situation is opposite? Or A is two times better then B, B is three times better than C, and A and C are equally good? If
not, than this is not a web site for you.
Decision making is an evaluation process including alternatives which are all satisfying a certain set of criteria. The problem appears when one has to choose only one alternative which
satisfies the entire set of our personal criteria.
Did you know that there is one simple method that can help people make that choice and which takes into consideration things like your perception, intuition, rational and irrational, and
the inconsistency of choosing among several options?
The method is called Analytic Hierarchy Process (or Analytical Hierarchy Process) - AHP
It is based on the comparison of pairs of alternative solutions during which all alternatives are compared to one another and you, as a decision maker, express intensity and the level of
preference towards one alternative in relation to the other according to the criteria you find important. In the same way, you compare criteria according to your own preferences and their
AHP is a strong and flexible decision making technique which helps in setting priorities and reaching optimal decisions in situations when quantitative and qualitative aspects have
already been taken into consideration. By reducing complex decision making to comparisons between pairs of alternatives and by synthesizing results AHP helps not only in decision making
but leads to a rational decision. Created in a way to reflect the way people think, AHP was developed by Dr. Thomas Saaty in the 1970ies while he was a professor at the Wharton School of
Business. The method is still one of the most appreciated and widely used methods. Numerous institutions and companies use it in decision making process. Why not you?
AHP - Analytic Hierarchy Process (or Analytical Hierarchy Process) is a mathematical method.
Compared to other decision making methods and techniques AHP enables you, as the decision maker, to compare the significance of each alternative in relation to another one individually
and within a criterion you find relevant. This preference-based method shows the best option. The value of the method is not only in finding the optimal result, but intermediate steps are
clearly distinguishable, as well as the elements that contribute to the result the most.
Theoretical and mathematical description of the method
The first step is to determine a set of elements that consists of alternatives and criteria we wish to consider. The next step is to form the set into a hierarchical structure consisting
of the mentioned criteria and alternatives.
Upon defining that set, we begin developing the mathematical model by which we calculate priorities (weight, importance) of the elements on the same level in the hierarchical structure.
The entire process of the AHP method can be described in several steps:
• The development of the hierarchical model of the decision making problem by defining the goal, criteria and alternative solutions.
• On each level of the hierarchical model elements of the model are compared with one another in pairs, and the preferences of the decision maker are expressed with the use of the Saaty’s
scale. In scientific literature that scale is more precisely described as a scale of five levels and four intermediate levels of verbally described intensities and corresponding numerical
values for them on the scale from 1 to 9. The following table shows the values and their description used for the comparison of relevant values of the elements of the AHP model.
│Intensity of Importance│Definition │Explanation │
│1 │Equal importance │Two activities contribute equally to the objective │
│3 │Moderate importance │Experience and judgment slightly favor one activity over another │
│5 │Strong importance │Experience and judgment strongly favor one activity over another │
│7 │Very strong or demonstrated importance│An activity is favorad very strongly over another; its dominance demonstrated in practice │
│9 │Extreme importance │The evidence favoring one activity over another is of the highest possible order of affirmation│
│2,4,6,8 │Intermediate values │ │
Let us describe it in more detail.
The first step would be to define a set in which we list the elements of the selection - the set of alternatives from which we wish to choose the best one for ourselves. Then we define
the criteria we will use to compare those alternatives. It is clear that you, as the decision maker, determine all of this. That fact alone guarantees that the decision will be based on
your preferences.
For the explanation of the following steps we will use mathematical language.
If n is the number of criteria or alternatives whose weight (priority, importance) wi should be defined based on the assessment of the values of their ratios aij = wi/wj. If we form a
matrix A from the ratio of their relevant importance aij, in case of consistent assessments equaling to aij= aik*akj , it will correspond to the equation Aw=nw
Matrix A has special characteristics (all her rows are proportional to the first row, all are positive and aij=1/aji is accurate resulting in only one of its eigenvalues being different
from 0 and equal to n. If A matrix has inconsistent changes (in praxis that is always the case) the importance vector w can be calculated by solving the equation (A- Lamda I)w=0
The condition: SUMAwi=1 is true, where LAMDAmax is the biggest eigenvalue of A matrix.
Due to the characteristics of the matrix LAMDAmax ≥ n, and the subtraction LAMDAmax – n is used in the measuring of the assessment consistency. With a consistency index CI = (LAMDAmax -n)
/(n-1) we calculate the consistency ratio CR=CI/RI where RI is a random index (consistency index for the n row matrixes of randomly generated comparisons in pairs – a table with
calculated values applies.
Value of the random index RI
│n │1 │2 │3 │4 │5 │6 │7 │8 │
│n │9 │10 │11 │12 │13 │14 │15 │
If CR≤ 0,1000 is true for matrix A the assessments of the relative importance of the criteria ( alternative priorities) are considered as acceptable. To the contrary, the reasons why the
assessment inconsistency is acceptably high must be investigated.
It will often happen that the consistency ratio exceeds 0,1000. That should only be taken into account as an indicator of the inconsistency level of your selection. Despite the
inconsistency, you will get a suggestion of the best alternative. That is the value of this method. You can always revise the chosen importance intensities and check which alternative is
the best and to what extent compared to the following one. | {"url":"http://www.123ahp.com/OMetodi.aspx","timestamp":"2024-11-03T03:07:41Z","content_type":"text/html","content_length":"30233","record_id":"<urn:uuid:541acc9d-b773-45ce-b636-5244f1a842cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00330.warc.gz"} |
NCERT Solutions for Class 9 Maths Chapter 7 Triangles Exercise 7.4
Chapter 7 Triangles Exercise 7.4 NCERT Solutions are provided here that are useful for the preparation of examinations and improving marks. These
NCERT Solutions for Class 9 Maths
will prove useful guide if you are facing any type of problem while solving a question. Keeping this in view, subject matter experts of Studyrankers have prepared accurate and detailed NCERT
questions and answers so you can always take help from here if you get stuck in any question.
Exercise 7.4 has only six questions in the whole exercise which are about finding showing that of all line segments drawn from a given point not on it etc. | {"url":"https://www.studyrankers.com/2020/02/ncert-solutions-for-class-9-maths-chapter-7-exercise-7.4.html","timestamp":"2024-11-07T07:43:47Z","content_type":"application/xhtml+xml","content_length":"293368","record_id":"<urn:uuid:2250d4e4-63eb-4539-8fd3-ee205351ba67>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00828.warc.gz"} |
How Does the Bearing Bore Size Impact Grease Selection?
We often get questions during our training courses about lubricant selection. I was recently asked how bearing bore size impact grease selection. Let’s start with defining the type of bearing. There
are element bearings that come with a variety of element shapes, sizes, and names.
There are cylindrical rollers, taper rollers, spherical rollers and several different kinds of element shapes. The shape fundamentally defines how often you will lubricate, but not necessarily what
lubricant is the best fit.
To determine which lubricant to use, whether oil or grease, you need to understand the bearings parts and how the parts influence this decision.
Bearing manufacturer handbooks will identify the point between the 12 o’clock position to the 6 o’clock position on the outer ring (outside the bearing) as the outer diameter (OD). Likewise, the
manufacturer will refer to the gap between the 12 o’clock position to the 6: o’clock position inside the inner ring as the bore or Inside Diameter (ID). The axial face plane from one side to the
other side of the bearing is referred to as the width. All these dimensions are generally considered in millimeters, and they all come into play when deciding which lubricants to use.
The remaining variable is the speed the shaft is turning. The speed is referred to with an ‘n’ value in most machine maintenance and reliability calculations.
The lubricant selected must contain oil (whether oil or grease) that moves proportionally to the speed that the machine parts move. Machines with fast-moving parts require fast-moving oil and
machines with slow-moving parts require slow-moving oil.
When making a bearing lubricant selection decision, the first task is to determine whether or not the bearing is moving at a speed where grease can be an acceptable option. The bearing manufacturers
give us that target range (a speed limit value) for the use of grease.
First, we must calculate the nDm value where:
The nDm value gives us an idea of what the angular velocity characteristic is for this bearing. It is used to compare to standardized unit values from the bearing manufacturer to determine whether
grease is a good fit.
The chart below shows some suggested limits for four different types of bearings. Generally, the more element-to-race contact and the greater the axial loading characteristics based on bearing
design, the less effective grease is going to be. Determine whether it is safe to lubricate with grease before going to the next step.
Once you determine whether grease or oil should be used, the next step is to identify the viscosity that is required and the operating temperature to make the elements in the rotating shaft float in
the race.
The viscosity of the lubricant at operating temperature is the single most important characteristic to establish.
Viscosity changes with temperature and pressure. As temperature increases, viscosity decreases, and as pressure increases, viscosity increases. These factors are interdependent.
The pressure viscosity relationship is dependent on the type of raw materials used to construct the lubricant. For any given lubricant selection, this characteristic cannot be changed by the
reliability engineer, so you should focus on the process of selecting the correct oil thickness regardless of the type of lubricant.
The central questions for selecting the correct lubricant grade for a given brand and product type are:
• What will the viscosity of the lubricant be at the normalized machine operating temperature?
• What are the allowable, minimum, and optimum viscosities for a given element bearing regardless of operating temperature?
The first question begs for knowledge of the machine’s operating state. Temperature is influenced by machine speed, machine load, process temperatures, oil viscosity, and frictional conditions at the
element contact area.
If the machine is already in operation, then the answer may be evident from machine observation and measurement. If not, the reliability engineer must consult with the OEM and production personnel
and collect sufficient information to project a safe answer to the question.
For this exercise assume that the temperature is known; we’ll use a figure of 154 F (70 C). We will also assume the shaft speed is 2,000 RPM and the bearing has been properly selected for the
An exact number can be produced if every incremental detail is known (speeds, loads, forces, material compositions and strengths, VP responses, etc.). As most plant circumstances afford only
estimates of these details, this article provides a model that can be followed by plant personnel to appropriately answer questions without the requirement for a full set of exact details and a
computer-aided design program.
Step 1. Locate a viscosity selection reference chart for element bearings. Fortunately, most bearing manufacturers provide suitable tables and charts in their respective lubrication reference
guidebooks. This viscosity selection chart is provided by FAG Bearings and is appropriate for this task.
Step 2. Use the following formula to estimate bearing pitch diameter, dm, where:
• dm = (OD + ID)/2
• OD = Bearing Outer Diameter
• ID = Bearing Bore
Assuming you wish to lubricate the bearings in a 254-frame-size motor containing bearings with a bore diameter (ID) of 45 mm and an outer diameter (OD) of 85 mm, the pitch diameter is 65 mm. Locate
this value on the chart’s x-axis (bottom of the chart) and plot a vertical line from this point to the top of the chart.
This line is referenced on the chart above as item I.
Step 3. Determine shaft rotation speed (noted above as 2,000 RPM). Locate the diagonal line labeled with this value on the chart.
Step 4. Using a chart like the one in Figure 4, locate the intersection of the pitch diameter and shaft speed lines.
Step 5. Draw a line from this intersecting point to the left side of the chart, to the y-axis, to read the minimum allowable viscosity in centistokes (mm2/sec).
Following these instructions, these points coincide on the y-axis at approximately 12 centistokes. This value represents the bearing manufacturer’s projected minimum operating viscosity or the
required oil thickness at the normal machine operating temperature. It is advisable to try to provide three to four times this value as a target operating viscosity.
The practitioner must still determine which of the available grade options will deliver this result.
Step 6. Determine the correct starting viscosity (always measured at 40 C) in a similar manner as noted above. Observing the following steps, the practitioner may use Figure 5 to determine the
viscosity starting point (viscosity value at 40 C)^2.
Step 6(a). Determine the target viscosity (three times the required viscosity = 12 cSt * 3 = 36 cSt). Locate this viscosity value on the y-axis. From this point, plot a line that is parallel to the
x-axis (left to right).
Step 6(b). Locate the machine operating temperature on the x-axis. From this point, plot a line parallel to the y-axis (bottom to top).
Step 6(c). Note where the two lines intersect. If the value is not at a normal ISO code specification, select the viscosity grade representing the next highest category. This chart represents
paraffinic mineral oils with a viscosity index of around 100.
The black arrows in the chart above (courtesy of SKF) represent the parameters to meet the minimum recommended operating viscosity, and the red lines provide parameters to meet the preferred
operating viscosity. In this instance, a lubricant with a viscosity grade above ISO 100 and below ISO 150 would be appropriate.
Given that it is an electric motor bearing on a small frame-size motor, and given that these are nearly always grease lubricated, one should look for a grease constructed with a viscosity grade at or
slightly above 100 centistokes.
So, as it turns out, the bearing bore size is a key element of lubricant selection decisions, as well as volume and grease replenishment decisions, but it is not the ONLY factor. The bore and the
bearing outer diameter are proportionally interrelated, and both of these values are required (along with the bearing width) to answer the key questions related to lubricant selection and daily
essential care. | {"url":"https://precisionlubrication.com/articles/bearing-bore-size-grease-selection/","timestamp":"2024-11-07T22:14:51Z","content_type":"text/html","content_length":"357551","record_id":"<urn:uuid:4a84f1c8-b0f1-4f37-b3cf-6683dfd61fb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00887.warc.gz"} |
Pierce, Benjamin C
Research Projects
Organizational Units
• Quotient Lenses
(2009-02-10) Foster, J. Nathan; Pilkiewicz, Alexandre; Pierce, Benjamin C
There are now a number of bidirectional programming languages, where every program can be read both as a forward transformation mapping one data structure to another and as a reverse
transformation mapping an edited output back to a correspondingly edited input. Besides parsimony—the two related transformations are described by just one expression— such languages are
attractive because they promise strong behavioral laws about how the two transformations fit together—e.g., their composition is the identity function. It has repeatedly been observed, however,
that such laws are actually a bit too strong: in practice, we do not want them “on the nose,” but only up to some equivalence, allowing inessential details, such as whitespace, to be modified
after a round trip. Some bidirectional languages loosen their laws in this way, but only for specific, baked-in equivalences. In this work, we propose a general theory of quotient
lenses—bidirectional transformations that are well behaved modulo equivalence relations controlled by the programmer. Semantically, quotient lenses are a natural refinement of lenses, which we
have studied in previous work. At the level of syntax, we present a rich set of constructs for programming with canonizers and for quotienting lenses by canonizers. We track equivalences
explicitly, with the type of every quotient lens specifying the equivalences it respects. We have implemented quotient lenses as a refinement of the bidirectional string processing language
Boomerang. We present a number of useful primitive canonizers for strings, and give a simple extension of Boomerang’s regular-expression-based type system to statically typecheck quotient lenses.
The resulting language is an expressive tool for transforming real-world, ad-hoc data formats. We demonstrate the power of our notation by developing an extended example based on the UniProt
genome database format and illustrate the generality of our approach by showing how uses of quotienting in other bidirectional languages can be translated into our notation.
• How Good Is Local Type Inference?
(1999-06-22) Hosoya, Haruo; Pierce, Benjamin C
A partial type inference technique should come with a simple and precise specification, so that users predict its behavior and understand the error messages it produces. Local type inference
techniques attain this simplicity by inferring missing type information only from the types of adjacent syntax nodes, without using global mechanisms such as unification variables. The paper
reports on our experience with programming in a full-featured programming language including higher-order polymorphism, subtyping, parametric datatypes, and local type inference. On the positive
side, our experiments on several nontrivial examples confirm previous hopes for the practicality of the type inference method. On the negative side, some proposed extensions mitigating known
expressiveness problems turn out to be unsatisfactory on close examination.
• Union Types for Semistructured Data
(1999-04-06) Buneman, Peter; Pierce, Benjamin C
Semistructured databases are treated as dynamically typed: they come equipped with no independent schema or type system to constrain the data. Query languages that are designed for semistructured
data, even when used with structured data, typically ignore any type information that may be present. The consequences of this are what one would expect from using a dynamic type system with
complex data: fewer guarantees on the correctness of applications. For example, a query that would cause a type error in a statically typed query language will return the empty set when applied
to a semistructured representation of the same data. Much semistructured data originates in structured data. A semistructured representation is useful when one wants to add data that does not
conform to the original type or when one wants to combine sources of different types. However, the deviations from the prescribed types are often minor, and we believe that a better strategy than
throwing away all type information is to preserve as much of it as possible. We describe a system of untagged union types that can accommodate variations in structure while still allowing a
degree of static type checking. A novelty of this system is that it involves non-trivial equivalences among types, arising from a law of distributivity for records and unions: a value may be
introduced with one type (e.g., a record containing a union) and used at another type (a union of records). We describe programming and query language constructs for dealing with such types,
prove the soundness of the type system, and develop algorithms for subtyping and typechecking.
• Statically Typed Document Transformation: An XTATIC Experience
(2005-10-14) Gapeyev, Vladimir; Garillot, François; Pierce, Benjamin C
XTATIC is a lightweight extension of C⋕ with native support for statically typed XML processing. It features XML trees as built-in values, a refined type system based on regular types à la XDUCE,
and regular patterns for investigating and manipulating XML. We describe our experiences using XTATIC in a real-world application: a program for transforming XMLSPEC, a format used for authoring
W3C technical reports, into HTML. Our implementation closely follows an existing one written in XSLT, facilitating comparison of the two languages and analysis of the costs and benets—both
signicant—of rich static typing for XML-intensive code.
• Boomerang: Resourceful Lenses for String Data
(2007-11-19) Bohannon, Aaron; Foster, J. Nathan; Pierce, Benjamin C; Pilkiewicz, Alexandre; Schmitt, Alan
A lens is a bidirectional program. When read from left to right, it denotes an ordinary function that maps inputs to outputs. When read from right to left, it denotes an "update translator" that
takes an input together with an updated output and produces a new input that reflects the update. Many variants of this idea have been explored in the literature, but none deal fully with ordered
data. If, for example, an update changes the order of a list in the output, the items in the output list and the chunks of the input that generated them can be misaligned, leading to lost or
corrupted data. We attack this problem in the context of bidirectional transformations over strings, the primordial ordered data type. We first propose a collection of bidirectional string lens
combinators, based on familiar operations on regular transducers (union, concatenation, Kleene-star) and with a type system based on regular expdressions. We then design a new semantic space of
dictionary lenses, enriching the lenses of Foster et al. (2007b) with support for two additional combinators for marking "reorderable chunks" and their keys. To demonstrate the effectiveness of
these primitives, we describe the design and implementation of Boomerang, a full-blown bidirectional programming language with dictionary lenses at its core. We have used Boomerang to build
transformers for complex real-world data formats including the SwissProt genomic database. We formalize the essential property of resourcefulness - the correct use of keys to associate chunks in
the input and output - by defining a refined semantic space of quasi-oblivious lenses. Several previously studied properties of lenses turn out to have compact characterizations in this space.
• Symmetric Lenses
(2010-07-28) Hoffmann, Martin; Pierce, Benjamin C; Wagner, Daniel
Lenses—bidirectional transformations between pairs of connected structures—have been extensively studied and are beginning to find their way into industrial practice. However, some aspects of
their foundations remain poorly understood. In particular, most previous work has focused on the special case of asymmetric lenses, where one of the structures is taken as primary and the other
is thought of as a projection, or view. A few studies have considered symmetric variants, where each structure contains information not present in the other, but these all lack the basic
operation of composition. Moreover, while many domain-specific languages based on lenses have been designed, lenses have not been thoroughly studied from a more fundamental algebraic perspective.
We offer two contributions to the theory of lenses. First, we present a new symmetric formulation, based on complements, an old idea from the database literature. This formulation generalizes the
familiar structure of asymmetric lenses, and it admits a good notion of composition. Second, we explore the algebraic structure of the space of symmetric lenses. We present generalizations of a
number of known constructions on asymmetric lenses and settle some longstanding questions about their properties—in particular, we prove the existence of (symmetric monoidal) tensor products and
sums and the non-existence of full categorical products or sums in the category of symmetric lenses. We then show how the methods of universal algebra can be applied to build iterator lenses for
structured data such as lists and trees, yielding lenses for operations like mapping, filtering, and concatenation from first principles. Finally, we investigate an even more general technique
for constructing mapping combinators, based on the theory of containers.
• Contracts Made Manifest
(2010-01-17) Pierce, Benjamin C; Greenberg, Michael; Weirich, Stephanie
Since Findler and Felleisen introduced higher-order contracts, many variants have been proposed. Broadly, these fall into two groups: some follow Findler and Felleisen in using latent contracts,
purely dynamic checks that are transparent to the type system; others use manifest contracts, where refinement types record the most recent check that has been applied to each value. These two
approaches are commonly assumed to be equivalent-different ways of implementing the same idea, one retaining a simple type system, and the other providing more static information. Our goal is to
formalize and clarify this folklore understanding. Our work extends that of Gronski and Flanagan, who defined a latent calculus lambdac and a manifest calculus lambdah, gave a translation phi
from lambdac to lambdah, and proved that, if a lambdac term reduces to a constant, then so does its phiimage. We enrich their account with a translation psi from lambdah to lambdac and prove an
analogous theorem. We then generalize the whole framework to dependent contracts, whose predicates can mention free variables. This extension is both pragmatically crucial, supporting a much more
interesting range of contracts, and theoretically challenging. We define dependent versions of lambdah and two dialects (“lax” and “picky”) of lambdac, establish type soundness-a substantial
result in itself, for lambdah-and extend phi and psi accordingly. Surprisingly, the intuition that the latent and manifest systems are equivalent now breaks down: the extended translations
preserve behavior in one direction but, in the other, sometimes yield terms that blame more.
• Schema-Directed Data Synchronization
(2005-03-23) Greenwald, Michael B; Foster, J. Nathan; Pierce, Benjamin C; Kirkegaard, Christian; Schmitt, Alan
Increased reliance on optimistic data replication has led to burgeoning interest in tools and frameworks for synchronizing disconnected updates to replicated data. We have implemented a generic,
synchronization framework, called Harmony, that can be instantiated to yield state-based synchronizers for a wide variety of tree-structured data formats. A novel feature of this framework is
that the synchronization process—in particular, the recognition of situations where changes are in conflict—is driven by the schema of the structures being synchronized. We formalize Harmony’s
synchronization algorithm, prove that it obeys a simple and intuitive specification, and illustrate how it can be used to synchronize a variety of specific forms of application data—sets,
records, tuples, and relations.
• On Decidability of Nominal Subtyping with Variance
(2006-09-01) Kennedy, Andrew J; Pierce, Benjamin C
We investigate the algorithmics of subtyping in the presence of nominal inheritance and variance for generic types, as found in Java 5, Scala 2.0, and the .NET 2.0 Intermediate Language. We prove
that the general problem is undecidable and characterize three different decidable fragments. From the latter, we conjecture that undecidability critically depends on the combination of three
features that are not found together in any of these languages: contravariant type constructors, class hierarchies in which the set of types reachable from a given type by inheritance and
decomposition is not always finite, and class hierarchies in which a type may have multiple supertypes with the same head constructor. These results settle one case of practical interest:
subtyping between ground types in the .NET intermediate language is decidable; we conjecture that our proof can also be extended to show full decidability of subtyping in .NET. For Java and
Scala, the decidability questions remain open; however, the proofs of our preliminary results introduce a number of novel techniques that we hope may be useful in further attacks on these
• Contracts Made Manifest
(2012-05-01) Pierce, Benjamin C; Greenberg, Michael; Weirich, Stephanie
Since Findler and Felleisen (Findler, R. B. & Felleisen, M. 2002) introduced higher-order contracts, many variants have been proposed. Broadly, these fall into two groups: some follow Findler and
Felleisen (2002) in using latent contracts, purely dynamic checks that are transparent to the type system; others use manifest contracts, where refinement types record the most recent check that
has been applied to each value. These two approaches are commonly assumed to be equivalent—different ways of implementing the same idea, one retaining a simple type system, and the other
providing more static information. Our goal is to formalize and clarify this folklore understanding. Our work extends that of Gronski and Flanagan (Gronski, J. & Flanagan, C. 2007), who defined a
latent calculus λC and a manifest calculus λH, gave a translation φ from λC to λH, and proved that if a λC term reduces to a constant, so does its φ-image. We enrich their account with a
translation ψ from λH to λC and prove an analogous theorem. We then generalize the whole framework to dependent contracts, whose predicates can mention free variables. This extension is both
pragmatically crucial, supporting a much more interesting range of contracts, and theoretically challenging. We define dependent versions of λH and two dialects (“lax” and “picky”) of λC,
establish type soundness—a substantial result in itself, for λH — and extend φ and ψ accordingly. Surprisingly, the intuition that the latent and manifest systems are equivalent now breaks down:
the extended translations preserve behavior in one direction, but in the other, sometimes yield terms that blame more. | {"url":"https://repository.upenn.edu/entities/person/5b9f033c-2d75-4f87-9dab-bb1be38e848c","timestamp":"2024-11-02T01:41:37Z","content_type":"text/html","content_length":"753700","record_id":"<urn:uuid:21031825-3960-4bcb-97f8-d646653342e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00262.warc.gz"} |
Note about fitting and visualizing exponential modelsNote about fitting and visualizing exponential models
One of the consequences of the open data, big data movement is that everyone is an amateur data analyst. To wit, Andrew Gelman earlier linked to a Kaggle competition to forecast coronavirus cases.
This is both unnerving and exciting; it's unclear if the benefits of more eyes outweigh the costs of misinformation. But there isn't a way to push the genie back in the bottle.
I don't plan on spending much time with the currently available data about the coronavirus cases, mainly because there is too much missing context, such as the amount of testing, what types of people
get tested, how cases and deaths are defined (as related to the novel coronavirus), what measures are taken that impact the counts, age and other co-variants, etc.
The one analysis I did from the other day addresses the narrow question of whether there are signs yet that the containment measures yielded results in the region of Lombardia. Zooming in to a single
region is helpful as the variations due to definitions and policies is limited. The key takeaway from that analysis is that the growth curve of reported cases is not exponential.
This note is written for those attempting to fit exponential growth curves.
A typical process of fitting exponential curves is to plot the data with a log y-axis. The supposed benefit of looking at a log plot is that the implied growth rate can be eyeballed from this chart.
The hockey-stick curve of case counts looks like a straight line in the log scale. The slope of this best-fit line leads to the growth rate of the exponential model.
Here is what happens when I obtained a "trend line" in Excel after transforming the case counts to a log10 scale.
For this illustration, I used the entire data series (Feb 25 - Mar 19, 2020) to fit the model. Excel reports that this is a tight-fitting model, with R-squared of 98%. [See my previous post for why
using the entire data series to fit a model, as I did above, isn't recommended.]
The measure of goodness of fit (R-squared) comes from aggregating individual daily errors. In the following chart, I highlighted two such errors, the gaps between the model estimate and the actual
case count on two selected days. I picked those two days because the lengths of the two red lines are almost identical.
For a normal linear regression fit, it's standard practice to plot these errors and conclude that they are pretty uneventful.
The log scale makes huge errors look small.
It's extremely easy to forget that we are looking at a log plot here. Log transforming the data has the effect of pulling big numbers closer to zero, and pushing small numbers further from zero. If
the two red lines were plotted in the original scale, you'll see that the error on March 19 is much, much larger than the error on Feb 26!
This misreading of the error sizes is the same visual misperception that makes any log scale prone to misinterpretation.
The above problem is not limited to the phase of model fitting. It is common today to draw two (or more) growth curves in log scale and compare them. This is a typical visualization:
This one from Spanish outlet El Pais (link) was featured by Alberto Cairo (link) recently so I have it handy - but my comment applies to all variants of this chart which can be found everywhere. The
El Pais design is distinguished by a side-by-side presentation of the growth curve in log scale and in linear (everyday) scale.
The log scale complicates comparing two growth curves.
When comparing two growth curves, say Spain and Italy, comparing the straight-line slopes is fine (with the caveat that we are then assuming exponential models for both countries).
We get in trouble when we interpret differences in slope over time. Using the above graph, one might say that in the first ten days, the Spanish case counts were lagging behind the Italians but after
that, there was a cross-over and the Spanish cases grew faster than the Italian cases.
All those words represent our interpretation of the gaps between two growth curves. The optical illusion described in section 1 of this post applies here equally. Gaps to the left side of the
time-line are artificially magnified by the log transform while gaps to the right side are artificially compressed.
The further out it goes on the time axis, the bigger the compression. You have to train your head to think in log scale to reverse the visual distortion.
The reality is only experts who have been trained to think in log scale can properly interpret this type of chart.
You can follow this conversation by subscribing to the comment feed for this post.
I was surprised you didn't include a discussion of proportional change as opposed to absolute change here. Both types of change can be important. If you are looking to measure the human toll of the
virus, the absolute change (e.g. 651 deaths in Italy yesterday) should reasonably be what you want to model most accurately - the difference between 10,000 and 12,000 is more important than the
difference between 100 and 120. But if you want to understand the nature of the system, proportional change may be a more useful way of looking at the data. Due to the nature of disease spread, it is
reasonable to conceptualize the system as one where exponential growth is expected, and deviation from that exponential growth might reasonably be interpreted as something meaningful - mode/
observation mismatch of 20% is similarly important, whether that's 100 v.s 120, or 10,000 vs. 12,000.
BH: In the last part, you anticipated a post that currently sits in my head. It will probably appear this week or next. When there is a discrepancy between a model and the observed data, the modeler
has to make a judgment call: how much of the gap is due to a mis-specified model and how much of it is due to poorly measured data? It can be some of each. Nevertheless, it's important to recognize
that the exponential curve is an analytical solution to a theoretical setup so there is some basis for it to be "true".
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Printable Sudoku Puzzles Easy 6 | Sudoku Printables
Printable Sudoku Puzzles Easy 6
Printable Sudoku Puzzles Easy 6 – If you’ve ever had trouble solving sudoku, you’re aware that there are a variety of kinds of puzzles that are available, and sometimes, it’s difficult to decide
which one to solve. However, there are different ways to solve them. In fact, you’ll see that an printable version of sudoku can be a great way to start. The rules to solve sudoku are the same as
those for other kinds of puzzles, but the actual format varies slightly.
What Does the Word ‘Sudoku’ Mean?
The word ‘Sudoku’ is taken from the Japanese words suji and dokushin meaning “number” as well as “unmarried’, respectively. The goal of the game is to fill all the boxes with numbers so that every
numeral from one to nine appears only once on every horizontal line. The term Sudoku is a trademark of the Japanese puzzle firm Nikoli that was founded in Kyoto.
The name Sudoku comes of the Japanese word shuji wa dokushin ni kagiru meaning ‘numbers have to remain single’. The game is comprised of nine 3×3 squares, with nine smaller squares. Originally called
Number Place, Sudoku was an exercise that stimulated mathematical development. Although the origins of the game aren’t known, Sudoku is known to have deep roots in ancient number puzzles.
Why is Sudoku So Addicting?
If you’ve played Sudoku you’ll realize the way addictive the game can be. A Sudoku player will not be able to not think about the next puzzle they can solve. They’re constantly planning their next
adventure, while the other aspects in their life are slipping to the wayside. Sudoku is a game that can be addictive however it’s essential for players to maintain the addicting power of the game
under control. If you’ve developed a craving for Sudoku, here are some ways to curb your addiction.
One of the best ways to detect that the addict you are to Sudoku is by observing your actions. Most people carry books and magazines with them as well as scroll through social media posts. Sudoku
addicts carry books, newspapers, exercise books, as well as smartphones everywhere they go. They can be found for hours working on puzzles and can’t stop! Some people even find it easier to finish
Sudoku puzzles than their regular crosswords. They simply can’t quit.
Printable Sudoku Puzzles Easy 6
What is the Key to Solving a Sudoku Puzzle?
A good strategy for solving the printable sudoku game is to try and practice with various approaches. The most effective Sudoku puzzle solvers do not follow the same formula for every single puzzle.
It is important to test and practice different methods until you can find one that works for you. After a while, you will be able to solve sudoku puzzles without a problem! But how do you know to
solve the printable Sudoku game?
In the beginning, you must grasp the basic concept of suduko. It’s a game of logic and deduction, and you need to view the puzzle from many different angles to spot patterns and solve it. When
solving Suduko puzzles, suduko puzzle, do not attempt to figure out the numbers; instead, you should scan the grid for clues to identify patterns. You can apply this strategy to squares and rows.
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Deterministic view of random sampling and its use in geometry
A number of efficient probabilistic algorithms based on the combination of divide-and-conquer and random sampling have been recently discovered. It is shown that all those algorithms can be
derandomized with only polynomial overhead. In the process, results of independent interest concerning the covering of hypergraphs are established, and various probabilistic bounds in geometry
complexity are improved. For example, given n hyperplanes in d-space and any large enough integer r, it is shown how to compute, in polynomial time, a simplicial packing of size O(r^d) that covers
d-space, each of whose simplices intersects O(n/r) hyperplanes. It is also shown how to locate a point among n hyperplanes in d-space in O(log n) query time, using O(n^d) storage and polynomial
Original language English (US)
Title of host publication Annual Symposium on Foundations of Computer Science (Proceedings)
Publisher Publ by IEEE
Pages 539-549
Number of pages 11
ISBN (Print) 0818608773
State Published - 1988
Publication series
Name Annual Symposium on Foundations of Computer Science (Proceedings)
ISSN (Print) 0272-5428
All Science Journal Classification (ASJC) codes
• Hardware and Architecture
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A liquid cools down from 70% to 60 c in 5 minutes. the time tak-Turito
Are you sure you want to logout?
A liquid cools down from
A. 5 minutes
B. Lesser than 5 minutes
C. Greater than 5 minutes
D. Lesser or greater than 5 minutes depending upon the density of the liquid
The correct answer is: Greater than 5 minutes
According to Newton’s law of cooling
Rate of cooling
Initially, mean temperature difference
Finally, mean temperature difference
In second case mean temperature difference decreases, so rate of fall of temperature decreases, so it takes more time to cool through the same range
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orthogonal projection - Swedish translation – Linguee
PULSATING FLOW - Dissertations.se
Definition Orthogonal Complements. The orthogonal complement S⊥ of a subspace S of Rm is defined. S⊥ = {y ∈ Rm : v · y = 0 for all v ∈ S }. If v ∈ S then y · v = 0 for Orthogonal Complement -
Exercise 3. html5: Unsupported video format. Try installing Adobe Flash. 00:00.
http://adampanagos.orgConsider the subspace W. Let z be a vector that is orthogonal to every element of W. In this case, we say that z is orthogonal to W. I defined orthogonal complement. I prove
that it is a subspace via Subspace theorem (see previous video http://youtu.be/ah8l_r8Vu3M) I show that many examp Let W j int be the orthogonal complement of V j int in V j − 1 int.The support of
the Daubechies wavelet ψ with p vanishing moments is [–p + 1, p].Since φ j, n is orthogonal to any φ j, l, we verify that an orthogonal basis of W j int can be constructed with the 2 −j − 2p inside
wavelets with support in [0, 1]: Orthogonal complement Transpose Row Space Orthogonal Set, Orthogonal Basis Orthonormal Set, Orthonormal Basis Projection onto a subspace (i.e. proj W(u)) 1. Math 54
Summer 2017 Worksheet 20 Theorems: If a vector is orthogonal to every vector in a list then it is also orthogonal to all vectors V is the orthogonal complement of U in W. Every vector in V is
orthogonal to every vector in U 3 - Direct sum Every vector b in W can be written as the sum of a vector in U and a vector in V: U \oplus V = W Orthogonal complement and subspaces The orthogonal
complement of S in V is the same as the orthogonal complement W in V. Every vector of S belongs to the orthogonal complement of S in V. If u is a vector in V which belongs to both W and its
orthogonal complement in V, then u = 0. If u is a vector in Remark: The set U ⊥ (pronounced " U -perp'') is the set of all vectors in W orthogonal to every vector in U. This is also often called the
orthogonal complement of U. … The Orthogonal complement (or dual) of a k-blade is a (n-k)-blade where n is the number of dimensions.As the name suggests the orthogonal complement is entirely
orthogonal to the corresponding k-blade.
Linear Transformations – Linear Algebra – Mathigon
The Orthogonal complement (or dual) of a k-blade is a (n-k)-blade where n is the number of dimensions. As the name suggests the orthogonal complement is entirely orthogonal to the corresponding
oblique projection matrix - Den Levande Historien
Learn more about orthogonal complement, matrix, linear equation 面和面更不行，orthogonal和complement都不满足！ orthogonal complements 的意义.
In a three-dimensional Euclidean vector space, the orthogonal complement of a line through the origin is the plane through the origin perpendicular to it, and vice versa. I defined orthogonal
complement. I prove that it is a subspace via Subspace theorem (see previous video http://youtu.be/ah8l_r8Vu3M) I show that many examp Section 5.1 Orthogonal Complements and Projections Definition:
Socialisme regering
Then the orthogonal complement of ?, denoted by ? ⊥ (read as ? − ???? Consider the infinite dimensional vector space of functions ##M## over ##\\mathbb{C}##.
Taking the orthogonal complement is an operation that is performed on subspaces. Definition. Let W be a subspace of R n. Its orthogonal complement is the subspace The orthogonal complement is a
subspace of vectors where all of the vectors in it are orthogonal to all of the vectors in a particular subspace.
Ersta hannahemmet bromma
vardcentralen slandan ljungbyaldersgrense moped sverigemartin luther betydelsetransportstyrelsen regnr besiktningkontraktionskraft des herzensfria t
Orthogonal: Swedish translation, definition, meaning
V is the orthogonal complement of U in W. Every vector in V is Orthogonal Complements. Definition of the Orthogonal Complement. Geometrically, we can understand that two lines can be perpendicular in
R 2 and that a line and a plane can be perpendicular to each other in R 3. We now generalize this concept and ask given a vector subspace, what is the set of vectors that are orthogonal to all
vectors in the It is known that complement of a subspace ?
Kyle bosman twittercylinder geometri
Basbyte för en vektor by Tommy Ekola
Then the orthogonal complement of ?, denoted by ? ⊥ (read as ? − ???? Consider the infinite dimensional vector space of functions ##M## over ##\\mathbb{C}##. The inner product defined as in square
integrable functions we use in quantum mechanics.
PDF Existing Prototyping Perspectives: Considerations for
orthogonal matrix. ortogonal matris. orthogonal operator. ortogonal operator.
Geometrically, we can understand that two lines can be perpendicular in R 2 and that a line and a plane can be perpendicular to each other in R 3. | {"url":"https://hurmanblirrikspqdqqr.netlify.app/45998/55435","timestamp":"2024-11-04T17:15:49Z","content_type":"text/html","content_length":"10020","record_id":"<urn:uuid:a23b2178-3dc1-474f-ad92-eb9296fe0dae>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00739.warc.gz"} |
Electric Field
Consider a charge $q$ fixed at the point O. Let P be a point at a distance $r$ from O. The electric field of charge $q$ at the point P is given by \vec{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2} \hat
{r} where $\epsilon_0$ is permittivity of free space ($1/4\pi\epsilon_0=9\times{10}^{9}\;$ m/F) and $\hat{r}$ is the unit vector in the direction from O to P.
Principle of Superposition
The electric field of a system of fixed point charges is equal to the vector sum of the electric fields that would be created by each charge separately: \vec{E}=\sum \vec{E}_i=\frac{1}{4\pi\
Electric field on the axis of a thin uniformly charged ring
A charge $q>0$ is uniformly distributed over a thin ring of radius $a$. The electric field $\vec{E}$ on the axis of the ring as a function of distance $x$ from its centre is given by, |\vec{E}|=\frac
{q}{4\pi\epsilon_0} \frac{x}{(a^2+x^2)^{3/2}}.
Electric field of a uniformly charged straight filament
A thin straight filament of length $2l$ is uniformly charged by a charge $q$. The field at a point separated by a distance $x$ from the midpoint of the filament and located symmetrically with respect
to its ends is given by |\vec{E}|=\frac{q}{4\pi\epsilon_0 x} \frac{1}{\sqrt{l^2+x^2}}.
Problems from IIT JEE
Problem (IIT JEE 2008): Consider a system of three charges $\frac{q}{3}$, $\frac{q}{3}$ and $-\frac{2q}{3}$ placed at points A, B and C, respectively, as shown in the figure. Take O to be the centre
of the circle of radius $R$ and angle $\text{CAB}={60}\;\mathrm{degree}$.
A. The electric field at point O is $\frac{q}{8\pi\epsilon_0 R^2}$ directed along the negative $x$-axis.
B. The potential energy of the system is zero.
C. The magnitude of the force between the charge C and B is $\frac{q^2}{54\pi\epsilon_0 R^2}$.
D. The potential at point O is $\frac{q}{12\pi\epsilon_0 R}$.
Solution: The charges at A, B, and C are $q_A=q/3$, $q_B=q/3$, and $q_C=-2q/3$. The electric field at O due to $q_A$ and $q_B$ is equal in magnitude but opposite in direction. Thus, resultant
electric field at O is by the charge $q_C$ and it is given by \begin{alignat}{2} \vec{E}_O=-\frac{q}{6\pi\epsilon_0 R^2}\;\hat\imath. \nonumber \end{alignat} The triangle ABC is right-angled with $\
angle A={60}\;\mathrm{deg}$, $\angle C={90}\;\mathrm{deg}$, and $r_\text{AB}=2R$. Thus, $r_\text{AC}=R$ and $r_\text{BC}=\sqrt{3}R$. The potential energy for given charge distribution is \begin
{alignat}{2} U&=\frac{1}{4\pi\epsilon_0}\left[ \frac{q_A q_B}{r_\text{AB}}+\frac{q_A q_C}{r_\text{AC}}+\frac{q_B q_C}{r_\text{BC}}\right] \nonumber\\ &=\frac{1}{4\pi\epsilon_0} \left[\frac{q^2}{18R}-
\frac{2q^2}{9R}-\frac{2q^2}{9\sqrt{3}R}\right]\neq 0. \nonumber \end{alignat} The magnitude of force between $q_C$ and $q_B$ is $F_\text{BC}=\frac{1}{4\pi\epsilon_0}\frac{q_B q_C}{r_\text{BC}^2}=\
frac{q^2}{54\pi\epsilon_0 R^2}$. The potential at O is $V=\frac{1}{4\pi\epsilon_0}(q_A/R+q_B/R+q_C/R)=0$. | {"url":"https://www.concepts-of-physics.com/electromagnetism/electric-field.php","timestamp":"2024-11-14T17:33:50Z","content_type":"text/html","content_length":"14517","record_id":"<urn:uuid:70e2370f-e3fe-4626-ae06-735e01382af4>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00486.warc.gz"} |
Monolithic transient thermo-elasticity
This demo is a direct transposition of the transient thermo-elasticity demo using a pure FEniCS formulation. We will show how to compute fully coupled thermo-mechanical problems using MFront, which
can pave the way to more complex thermo-mechanical behaviours including plasticity for instance.
Source files:
Constitutive equations
The constitutive equations are derived from the following expression of the Gibbs free energy:
\[ \begin{aligned} \rho\,\Phi{\left(\boldsymbol{\varepsilon}^{\mathrm{to}},T\right)}&={{\displaystyle \frac{\displaystyle \lambda}{\displaystyle 2}}}\,{\left({\mathrm{tr}{\left(\boldsymbol{\
varepsilon}^{\mathrm{to}}\right)}}-3\,\alpha\,{\left(T-T^{\mathrm{ref}}\right)}\right)}^2+ \mu\,{\left(\boldsymbol{\varepsilon}^{\mathrm{to}}-\alpha\,{\left(T-T^{\mathrm{ref}}\right)}\,\mathbf{I}\
right)}\,\colon\,{\left(\boldsymbol{\varepsilon}^{\mathrm{to}}-\alpha\,{\left(T-T^{\mathrm{ref}}\right)}\,\mathbf{I}\right)}\\ &+{{\displaystyle \frac{\displaystyle \rho\,C_{\varepsilon}}{\
displaystyle 2\,T^{\mathrm{ref}}}}}\,{\left(T-T^{\mathrm{ref}}\right)}^2+s_{0}\,{\left(T-T^{\mathrm{ref}}\right)} \end{aligned} \]
• \(\lambda\) and \(\mu\) are the Lamé coefficients
• \(\rho\) is the mass density
• \(\alpha\) is mean linear thermal expansion coefficient
• \(C_{\varepsilon}\) is the specific heat at constant strain (per unit of mass).
This expression leads to the following expressions of the stress tensor \(\boldsymbol{\sigma}\) and entropy per unit of mass \(s\):
\[ \begin{aligned} \boldsymbol{\sigma}&=\rho \dfrac{\partial \Phi}{\partial \boldsymbol{\varepsilon}^{\mathrm{to}}}=\lambda\,{\mathrm{tr}{\left(\boldsymbol{\varepsilon}^{\mathrm{to}}\right)}}\,\
mathbf{I}+2\,\mu\,\boldsymbol{\varepsilon}^{\mathrm{to}}-\kappa\,{\left(T-T^{\mathrm{ref}}\right)}\,\mathbf{I}\\ s&={\displaystyle \frac{\displaystyle \partial \Phi}{\displaystyle \partial T}}={{\
displaystyle \frac{\displaystyle C_{\varepsilon}}{\displaystyle T^{\mathrm{ref}}}}}\,{\left(T-T^{\mathrm{ref}}\right)}+{{\displaystyle \frac{\displaystyle \kappa}{\displaystyle \rho}}}\,{\mathrm{tr}
{\left(\boldsymbol{\varepsilon}^{\mathrm{to}}\right)}}\\ \end{aligned} \qquad{(1)}\]
where \(\kappa=\alpha\,{\left(3\,\lambda+2\,\mu\right)}\).
The heat flux \(\mathbf{j}\) is related to the temperature gradient \(\nabla\, T\) by the linear Fourier law:
\[ \mathbf{j}=-k\,\nabla\, T \qquad{(2)}\]
MFront implementation
Choice of the domain specific language
The constitutive equations (1) and (2) exhibit an explicit expression of the thermodynamic forces \({\left(\boldsymbol{\sigma}\, \mathbf{j}, s\right)}\) as a function of the gradients \({\left(\
boldsymbol{\varepsilon}^{\mathrm{to}}, \nabla T, T\right)}\).
The most suitable domain specific language for this kind of behaviour if the DefaultGenericBehaviour.
Name of the behaviour
The @Behaviour keyword allows giving the name of the behaviour:
The following lines add some metadata (authors of the implementation, date, description):
@Author Thomas Helfer, Jérémy Bleyer;
@Date 19/04/2020;
@Description {
This simple thermoelastic behaviour allows to perform
fully coupled thermo-mechanical resolutions.
See https://comet-fenics.readthedocs.io/ for details.
Definition of the gradients and conjugated thermodynamic forces
The gradients are the strain \(\boldsymbol{\varepsilon}^{\mathrm{to}}\), the temperature gradient \(\nabla\,T\) and the temperature. The associated thermodynamic forces are respectively the stress \
(\boldsymbol{\sigma}\), the heat flux \(\mathbf{j}\) and the entropy \(s\).
\(\boldsymbol{\varepsilon}^{\mathrm{to}}\), \(\nabla\,T\), \(\boldsymbol{\sigma}\) and \(\mathbf{j}\) are declared as follows:
@Gradient StrainStensor εᵗᵒ;
@ThermodynamicForce StressStensor σ;
@Gradient TemperatureGradient ∇T;
@ThermodynamicForce HeatFlux j;
The glossary names are the names seen from the calling solver. Glossary names are described on this page.
Due to a MFront convention, the temperature is automatically declared as an external state variable. For this reason, the entropy is declared as a state variable:
In the current version of MFront, there is no glossary name associated with the entropy per unit of mass. In this case, the setEntryName is used to associate a name to this variable.
Declaration of the tangent operator blocks
By default, all the derivatives of the thermodynamic forces with respect to the increments of the gradients are declared as tangent operator blocks, i.e. derivatives that are meant to be used when
building the stiffness matrix at the structural scale.
In this case, this is not appropriate as:
• some derivatives are known to be null, such as \({\displaystyle \frac{\displaystyle \partial \boldsymbol{\sigma}}{\displaystyle \partial \Delta\,\nabla\,T}}\) and \({\displaystyle \frac{\
displaystyle \partial \mathbf{j}}{\displaystyle \partial \Delta\,\boldsymbol{\varepsilon}^{\mathrm{to}}}}\).
• the derivative \({\displaystyle \frac{\displaystyle \partial s}{\displaystyle \partial \Delta\,\boldsymbol{\varepsilon}^{\mathrm{to}}}}\) of the entropy with respect to strain and the derivative
\({\displaystyle \frac{\displaystyle \partial s}{\displaystyle \partial \Delta\,T}}\) of the entropy with respect to the temperature are also required.
The required tangent operator blocks are therefore explicitly requested:
Declaration of the reference temperature
The reference temperature is declared using the @StaticVariable keyword:
Internally Tʳᵉᶠ is hold in an immutable static variable.
Declaration of the material coefficients
The various material coefficients are now declared as parameters:
@Parameter stress E = 70e3;
@Parameter real ν = 0.3;
@Parameter massdensity ρ = 2700.;
@Parameter thermalconductivity α = 2.31e-5 ;
@Parameter real Cₑ = 910e-6;
@Parameter thermalconductivity k = 237e-6;
Parameters are global values that can be modified at runtime.
Computation of the thermodynamic forces and tangent operator blocks
The computation of the thermodynamic forces and tangent operator blocks is implemented in the @Integrator code block:
First, the Lamé coefficients are computed using the built-in computeLambda and computeMu functions and then we compute the \(\kappa\) factor:
const auto λ = computeLambda(E, ν);
const auto μ = computeMu(E, ν);
const auto κ = α ⋅ (2 ⋅ μ + 3 ⋅ λ);
For brevity, we compute the strain at the end of the time step as follows:
The computation of the thermodynamic forces is then straightforward and closely looks like the constitutive equations (1) and (2):
σ = λ ⋅ trace(ε) ⋅ I₂ + 2 ⋅ μ ⋅ ε - κ ⋅ (T + ΔT - Tʳᵉᶠ) ⋅ I₂;
s = Cₑ / Tʳᵉᶠ ⋅ (T + ΔT - Tʳᵉᶠ) + (κ / ρ) ⋅ trace(ε);
j = -k ⋅ (∇T + Δ∇T);
The computation of the consistent tangent operator is only required if the computeTangentOperator_ boolean value is true. Again, their computations is straightforward [2]:
if (computeTangentOperator_) {
∂σ∕∂Δεᵗᵒ = λ ⋅ (I₂ ⊗ I₂) + 2 ⋅ μ ⋅ I₄;
∂σ∕∂ΔT = -κ ⋅ I₂;
∂s∕∂ΔT = Cₑ / Tʳᵉᶠ;
∂s∕∂Δεᵗᵒ = κ ⋅ Cₑ / Tʳᵉᶠ ⋅ I₂;
∂j∕∂Δ∇T = -k ⋅ tmatrix<N, N, real>::Id();
A final curly bracket then ends the @Integrator code block:
[1] We may also note that those blocks are third order tensors that are not yet supported by MFront.
[2] N is the space dimension. real is a type alias to the numeric type used, which depends on the interface used.
FEniCS implementation
Problem position
The problem consists of a quarter of a square plate perforated by a circular hole. A temperature increase of \(\Delta T=+10^{\circ}\text{C}\) will be applied on the hole boundary. Symmetry conditions
are applied on the corresponding symmetry planes and stress and flux-free boundary conditions are adopted on the plate outer boundary. Similarly to the original demo, we will formulate the problem
using the temperature variation as the main unknown.
We first import the relevant modules then define the mesh and some constants.
from dolfin import *
import mgis.fenics as mf
from mshr import Rectangle, Circle, generate_mesh
import numpy as np
import matplotlib.pyplot as plt
L = 1.0
R = 0.1
N = 50 # mesh density
domain = Rectangle(Point(0.0, 0.0), Point(L, L)) - \
Circle(Point(0.0, 0.0), R, 100)
mesh = generate_mesh(domain, N)
Tref = Constant(293.15)
DThole = Constant(10.0)
dt = Constant(0) # time step
We now define the relevant FunctionSpace for the considered problem. Since we will adopt a monolithic approach i.e. in which both fields are coupled and solved at the same time, we will need to
resort to a Mixed FunctionSpace for both the displacement \(\boldsymbol{u}\) and the temperature variation \(\Theta = T-T^\text{ref}\).
Vue = VectorElement("CG", mesh.ufl_cell(), 2) # displacement finite element
Vte = FiniteElement("CG", mesh.ufl_cell(), 1) # temperature finite element
V = FunctionSpace(mesh, MixedElement([Vue, Vte]))
def inner_boundary(x, on_boundary):
return near(x[0]**2 + x[1]**2, R**2, 1e-3) and on_boundary
def bottom(x, on_boundary):
return near(x[1], 0) and on_boundary
def left(x, on_boundary):
return near(x[0], 0) and on_boundary
bcs = [DirichletBC(V.sub(0).sub(1), Constant(0.0), bottom),
DirichletBC(V.sub(0).sub(0), Constant(0.0), left),
DirichletBC(V.sub(1), DThole, inner_boundary)]
Variational formulation and time discretization
The constitutive equations described earlier are completed by the quasi-static equilibrium equation:
\[ \text{div} \boldsymbol{\sigma}= 0 \]
and the transient heat equation (without source terms):
\[ \rho T^\text{ref} \dfrac{\partial s}{\partial t} + \text{div} \mathbf{j}= 0 \]
which can both be written in the following weak form:
\[ \begin{aligned} \int_{\Omega}\boldsymbol{\sigma}:\nabla^s\widehat{\boldsymbol{u}}\text{ d} \Omega &=\int_{\partial \Omega} (\boldsymbol{\sigma}\cdot\boldsymbol{n})\cdot\widehat{\boldsymbol{u}} dS
\quad \forall \widehat{\boldsymbol{u}}\in V_U \\ \int_{\Omega}\rho T^\text{ref} \dfrac{\partial s}{\partial t}\widehat{T}d\Omega - \int_{\Omega} \mathbf{j}\cdot\nabla \widehat{T}d\Omega &= -\int_{\
partial \Omega} \mathbf{j}\cdot\boldsymbol{n} \widehat{T} dS \quad \forall \widehat{T} \in V_T \end{aligned} \qquad{(3)}\]
with \(V_U\) and \(V_T\) being the displacement and temperature function spaces.
The time derivative in the heat equation is now replaced by an implicit Euler scheme, so that the previous weak form at the time increment \(n+1\) is now:
\[ \int_{\Omega}\rho T^\text{ref} \dfrac{s^{n+1}-s^n}{\Delta t}\widehat{T}d\Omega - \int_{\Omega} \mathbf{j}^{n+1}\cdot\nabla \widehat{T}d\Omega = -\int_{\partial \Omega} \mathbf{j}^{n+1}\cdot\
boldsymbol{n} \widehat{T} dS \quad \forall \widehat{T} \in V_T \]
where \(s^{n+1},\mathbf{j}^{n+1}\) correspond to the unknown entropy and heat flux at time \(t_{n+1}\).
Since both the entropy and the stress tensor depend on the temperature and the total strain, we obtain a fully coupled problem at \(t=t_{n+1}\) for \((\boldsymbol{u}_{n+1},T_{n+1})\in V_U\times V_T\)
. With the retained boundary conditions both right-hand sides in (3).
We now load the material behaviour and define the corresponding MFrontNonlinearProblem. One notable specificity of the present example is that the unknown field v belongs to a mixed function space.
Therefore, we cannot rely on automatic registration for the strain and temperature gradient. We will have to specify explicitly their UFL expression with respect to the displacement u and temperature
variation Theta sub-functions of the mixed unknown v. We also register the "Temperature" external state variable with respect to Theta.
material = mf.MFrontNonlinearMaterial("./src/libBehaviour.so",
rho = Constant(material.get_parameter("MassDensity"))
v = Function(V)
(u, Theta) = split(v)
problem = mf.MFrontNonlinearProblem(v, material, quadrature_degree=2, bcs=bcs)
problem.register_gradient("Strain", sym(grad(u)))
problem.register_gradient("TemperatureGradient", grad(Theta))
problem.register_external_state_variable("Temperature", Theta + Tref)
Similarly to the Transient heat equation with phase change demo, we need to specify explicitly the coupled thermo-mechanical residual expression using the stress, heat flux and entropy variables. For
the implicit Euler scheme, we will need to define the entropy at the previous time step. For the mechanical residual, note that the stress variable sig is represented in the form of its vector of
components. The computation of \(\boldsymbol{\sigma}:\nabla^s\widehat{\boldsymbol{u}}\) therefore requires to express \(\widehat{\boldsymbol{\varepsilon}}=\nabla^s\widehat{\boldsymbol{u}}\) in the
same way. For this purpose, we could use the mgis.fenics.utils.symmetric_tensor_to_vector on the tensorial UFL expression sym(grad(u)). Another possibility is to get the corresponding "Strain"
gradient object (expressed in vectorial form) and get his variation with respect to v_.
sig = problem.get_flux("Stress")
j = problem.get_flux("HeatFlux")
s = problem.get_state_variable("EntropyPerUnitOfMass")
s_old = s.copy(deepcopy=True)
v_ = TestFunction(V)
u_, T_ = split(v_) # Displacement and temperature test functions
eps_ = problem.gradients["Strain"].variation(v_)
mech_residual = dot(sig, eps_)*problem.dx
thermal_residual = (rho*Tref*(s - s_old)/dt*T_ - dot(j, grad(T_)))*problem.dx
problem.residual = mech_residual + thermal_residual
The problem is now solved by looping over time increments. Because of the typical exponential time variation of temperature evolution of the heat equation, time steps are discretized on a non-uniform
(logarithmic) scale. \(\Delta t\) is therefore updated at each time step. The previous entropy field s_old is updated at the end of each step.
Nincr = 10
t = np.logspace(1, 4, Nincr + 1)
Nx = 100
x = np.linspace(R, L, Nx)
T_res = np.zeros((Nx, Nincr + 1))
for (i, dti) in enumerate(np.diff(t)):
print("Increment " + str(i + 1))
T_res[:, i + 1] = [v(xi, 0.0)[2] for xi in x]
Increment 1
Increment 2
Increment 3
Increment 4
Increment 5
Increment 6
Increment 7
Increment 8
Increment 9
Increment 10
At each time increment, the variation of the temperature increase \(\Delta T\) along a line \((x, y=0)\) is saved in the T_res array. This evolution is plotted below. As expected, the temperature
gradually increases over time, reaching eventually a uniform value of \(+10^{\circ}\text{C}\) over infinitely long waiting time. We check that we obtain the same solution as the pure FEniCS demo.
%matplotlib notebook
plt.plot(x, T_res[:, 1::Nincr // 10])
plt.xlabel("$x$-coordinate along $y=0$")
plt.ylabel("Temperature variation $\Delta T$")
plt.legend(["$t={:.0f}$".format(ti) for ti in t[1::Nincr // 10]], ncol=2)
<IPython.core.display.Javascript object> | {"url":"https://thelfer.github.io/mgis/web/mgis_fenics_monolithic_transient_thermoelasticity.html","timestamp":"2024-11-02T10:43:52Z","content_type":"text/html","content_length":"51431","record_id":"<urn:uuid:a23365d6-d199-4289-9f3c-2260e61b3214>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00731.warc.gz"} |
Plotting Points
<p><span style="font-size:22px"><a target="_blank" href="https://www.prepswift.com/quizzes/quiz/prepswift-plotting-points">Plotting Points Exercise</a></span></p><p>As briefly outlined in the
$xy$-coordinate plane entry, <span style="color:#27ae60;">Plotting Points</span> is pretty straightforward. Every point has a pair of values, written like $(x,y)$, where the $x$ value represents the
distance to the left or right of the origin on the horizontal axis and the $y$ value represents the distance below or above the origin on the vertical axis.</p> <p><strong><span style="color:#8e44ad;
">Examples</span></strong></p> <p>If we have the point $(4, 2)$, that means we would move four points to the <span style="color:#27ae60;">RIGHT </span>from the origin in the horizontal direction and
two points <span style="color:#27ae60;">UP</span> in the vertical direction.</p> <p>If we have the point $(-3, 0)$, we would move three points to the <span style="color:#e74c3c;">LEFT </span>from the
origin in the horizontal direction and not move at all in the vertical direction. Thus, this point would be positioned directly on the $x$-axis.</p> <p> </p>Sorry, you need to log in to see
this. Click here to log in. | {"url":"https://www.prepswift.com/content/plotting-points","timestamp":"2024-11-11T16:34:54Z","content_type":"text/html","content_length":"319487","record_id":"<urn:uuid:a8b6aab6-6daf-446c-a935-0d719bf29718>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00402.warc.gz"} |
4.2: Impedance of a Wire
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The goal of this section is to determine the impedance – the ratio of potential to current – of a wire. The answer to this question is relatively simple in the DC (“steady current”) case: The
impedance is found to be equal to the resistance of the wire, which is given by
\[R = \frac{l}{\sigma A} ~~~~~\mbox{(DC)} \label{m0159_eRW} \]
where \(l\) is the length of the wire and \(A\) is the cross-sectional area of the wire. Also, the impedance of a wire comprised of a perfect conductor at any frequency is simply zero, since there is
no mechanism in the wire that can dissipate or store energy in this case.
However, all practical wires are comprised of good – not perfect – conductors, and of course many practical signals are time-varying, so the two cases above do not address a broad category of
practical interest. The more general case of non-steady currents in imperfect conductors is complicated by the fact that the current in an imperfect conductor is not uniformly distributed in the
wire, but rather is concentrated near the surface and decays exponentially with increasing distance from the surface (this is determined in Section 4.1).
We are now ready to consider the AC case for a wire comprised of a good but imperfect conductor. What is the impedance of the wire if the current source is sinusoidally-varying? Equation \ref
{m0159_eRW} for the DC case was determined by first obtaining expressions for the potential (\(V\)) and net current (\(I\)) for a length-\(l\) section of wire in terms of the electric field intensity
\({\bf E}\) in the wire. To follow that same approach here, we require an expression for \(I\) in terms of \({\bf E}\) that accounts for the non-uniform distribution of current. This is quite
difficult to determine in the case of a cylindrical wire. However, we may develop an approximate solution by considering a surface which is not cylindrical, but rather planar. Here we go:
Consider the experiment described in Figure \(\PageIndex{1}\).
CC BY-SA 4.0; C. Wang)
Here, a semi-infinite region filled with a homogeneous good conductor meets a semi-infinite region of free space along a planar interface at \(z=0\), with increasing \(z\) corresponding to increasing
depth into the material. A plane wave propagates in the \(+\hat{\bf z}\) direction, beginning from just inside the structure’s surface. The justification for presuming the existence of this wave was
presented in Section 4.1. The electric field intensity is given by
\[\widetilde{\bf E} = \hat{\bf x}E_0 e^{-\alpha z} e^{-j\beta z} \nonumber \]
where \(E_0\) is an arbitrary complex-valued constant. The current density is given by Ohm’s law of electromagnetics:^1
\[\widetilde{\bf J} = \sigma\widetilde{\bf E} = \hat{\bf x}\sigma E_0 e^{-\alpha z} e^{-j\beta z} \label{m0159_eJ} \]
Recall that \(\alpha = \delta_s^{-1}\) where \(\delta_s\) is skin depth (see Section 3.12). Also, for a good conductor, we know that \(\beta \approx \alpha\) (Section 3.11). Using these
relationships, we may rewrite Equation \ref{m0159_eJ} as follows:
\[ \widetilde{\bf J} &\approx \hat{\bf x}\sigma E_0 e^{-z/\delta_s} e^{-jz/\delta_s} \nonumber \\ &= \hat{\bf x}\sigma E_0 e^{-(1+j)z/\delta_s} \nonumber \]
The net current \(\widetilde{I}\) is obtained by integrating \(\widetilde{\bf J}\) over any cross-section \(\mathcal{S}\) through which all the current flows; i.e.,
\[\widetilde{I} = \int_{\mathcal{S}} \widetilde{\bf J}\cdot d{\bf s} \nonumber \]
Here, the simplest solution is obtained by choosing \(\mathcal{S}\) to be a rectangular surface that is perpendicular to the direction of \(\widetilde{\bf J}\) at \(x=0\). This is shown in Figure \(\
CC BY-SA 4.0; Y. Zhao)
The dimensions of \(\mathcal{S}\) are width \(W\) in the \(y\) dimension and extending to infinity in the \(z\) direction. Then we have
\[ \widetilde{I} &\approx \int_{y=0}^{W} \int_{z=0}^{\infty} \left( \hat{\bf x}\sigma E_0 e^{-(1+j)z/\delta_s} \right) \cdot \left( \hat{\bf x}~dy~dz \right) \nonumber \\ &= \sigma E_0 W \int_{z=0}^
{\infty} e^{-(1+j)z/\delta_s} dz \label{m0159_eI} \]
For convenience, let us define the constant \(K \triangleq (1+j)/\delta_s\). Since \(K\) is constant with respect to \(z\), the remaining integral is straightforward to evaluate:
\[ \int_0^{\infty} e^{-Kz}dz &= \left. -\frac{1}{K} e^{-Kz}\right|_0^{\infty} \nonumber \\ &= +\frac{1}{K} \nonumber \]
Incorporating this result into Equation \ref{m0159_eI}, we obtain:
\[\widetilde{I} \approx \sigma E_0 W \frac{\delta_s}{1+j} %\label{m0159_eI} \]
We calculate \(\widetilde{V}\) for a length \(l\) of the wire as follows:
\[\widetilde{V} = -\int_{x=l}^{0} \widetilde{\bf E}\cdot d{\bf l} \nonumber \]
where we have determined that \(x=0\) corresponds to the “\(+\)” terminal and \(x=l\) corresponds to the “\(-\)” terminal.^2 The path of integration can be any path that begins and ends at the
proscribed terminals. The simplest path to use is one along the surface, parallel to the \(x\) axis. Along this path, \(z=0\) and thus \(\widetilde{\bf E}=\hat{\bf x}E_0\). For this path:
\[\widetilde{V} = -\int_{x=l}^{0} \left(\hat{\bf x}E_0\right)\cdot \left(\hat{\bf x}dx\right) = E_0 l \label{m0159_eV} \]
The impedance \(Z\) measured across terminals at \(x=0\) and \(x=l\) is now determined to be:
\[Z \triangleq \frac{\widetilde{V}}{\widetilde{I}} \approx \frac{1+j}{\sigma\delta_s} \cdot \frac{l}{W} \label{m0159_eCFGC-Z} \]
The resistance is simply the real part, so we obtain
\[R \approx \frac{l}{\sigma(\delta_s W)} ~~~~\mbox{(AC case)} \label{m0159_eACZ} \]
The quantity \(R\) in this case is referred to specifically as the ohmic resistance, since it is due entirely to the limited conductivity of the material as quantified by Ohm’s law.^3
Note the resemblance to Equation \ref{m0159_eRW} (the solution for the DC case): In the AC case, the product \(\delta_s W\), having units of area, plays the role of the physical cross-section \(S\).
Thus, we see an interesting new interpretation of the skin depth \(\delta_s\): It is the depth to which a uniform (DC) current would need to flow in order to produce a resistance equal to the
observed (AC) resistance.
Equations \ref{m0159_eCFGC-Z} and \ref{m0159_eACZ} were obtained for a good conductor filling an infinite half-space, having a flat surface. How well do these results describe a cylindrical wire? The
answer depends on the radius of the wire, \(a\). For \(\delta_s \ll a\), Equations \ref{m0159_eCFGC-Z} and \ref{m0159_eACZ} are excellent approximations, since \(\delta_s \ll a\) implies that most of
the current lies in a thin shell close to the surface of the wire. In this case, the model used to develop the equations is a good approximation for any given radial slice through the wire, and we
are justified in replacing \(W\) with the circumference \(2\pi a\). Thus, we obtain the following expressions:
\[\boxed{ Z \approx \frac{1+j}{\sigma\delta_s} \cdot \frac{l}{2\pi a} ~~~ (\delta_s \ll a) } \label{m0159_eZW} \]
and so
\[\boxed{ R \approx \frac{l}{\sigma(\delta_s 2\pi a)} ~~~ (\delta_s \ll a) } \label{m0159_eRWAC} \]
The impedance of a wire of length \(l\) and radius \(a\gg\delta_s\) is given by Equation \ref{m0159_eZW}. The resistance of such a wire is given by Equation \ref{m0159_eRWAC}.
If, on the other hand, \(a < \delta_s\) or merely \(\sim \delta_s\), then current density is significant throughout the wire, including along the axis of the wire. In this case, we cannot assume that
the current density decays smoothly to zero with increasing distance from the surface, and so the model leading to Equation \ref{m0159_eZW} is a poor approximation. The frequency required for
validity of Equation \ref{m0159_eZW} can be determined by noting that \(\delta_s \approx 1/\sqrt{\pi f \mu \sigma}\) for a good conductor; therefore, we require
\[\frac{1}{\sqrt{\pi f \mu \sigma}} \ll a \nonumber \]
for the derived expressions to be valid. Solving for \(f\), we find:
\[f \gg \frac{1}{\pi \mu \sigma a^2} \label{m0159_efvalid} \]
For commonly-encountered wires comprised of typical good conductors, this condition applies at frequencies in the MHz regime and above.
These results lead us to one additional interesting finding about the AC resistance of wires. Since \(\delta_s \approx 1/\sqrt{\pi f \mu\sigma}\) for a good conductor, Equation \ref{m0159_eRWAC} may
be rewritten in the following form:
\[\boxed{ R \approx \frac{1}{2} \sqrt{\frac{\mu f}{\pi \sigma}} \cdot \frac{l}{a} } \label{m0159_eACZ2} \]
We have found that \(R\) is approximately proportional to \(\sqrt{f}\). For example, increasing frequency by a factor of 4 increases resistance by a factor of 2. This frequency dependence is evident
in all kinds of practical wires and transmission lines. Summarizing:
The resistance of a wire comprised of a good but imperfect conductor is proportional to the square root of frequency.
At this point, we have determined that resistance is given approximately by Equation \ref{m0159_eACZ2} for \(\delta_s \ll a\), corresponding to frequencies in the MHz regime and above, and by
Equation \ref{m0159_eRW} for \(\delta_s \gg a\), typically corresponding to frequencies in the kHz regime and below. We have also found that resistance changes slowly with frequency; i.e., in
proportion to \(\sqrt{f}\). Thus, it is often possible to roughly estimate resistance at frequencies between these two frequency regimes by comparing the DC resistance from Equation \ref{m0159_eRW}
to the AC resistance from Equation \ref{m0159_eACZ2}. An example follows.
Elsewhere we have considered RG-59 coaxial cable (see the section “Coaxial Line,” which may appear in another volume depending on the version of this book). We noted that it was not possible to
determine the AC resistance per unit length \(R'\) for RG-59 from purely electrostatic and magnetostatic considerations. We are now able to consider the resistance per unit length of the inner
conductor, which is a solid wire of the type considered in this section. Let us refer to this quantity as \(R'_{ic}\). Note that
\[R' = R'_{ic} + R'_{oc} \nonumber \]
where \(R'_{oc}\) is the resistance per unit length of the outer conductor. \(R'_{oc}\) remains a bit too complicated to address here. However, \(R'_{ic}\) is typically much greater than \(R'_{oc}\),
so \(R' \sim R'_{ic}\). That is, we get a pretty good idea of \(R'\) for RG-59 by considering the inner conductor alone.
The relevant parameters of the inner conductor are \(\mu\approx\mu_0\), \(\sigma\cong 2.28 \times 10^7\) S/m, and \(a\cong 0.292\) mm. Using Equation \ref{m0159_eACZ2}, we find:
\[\begin{aligned} R'_{ic} &\triangleq \frac{R_{ic}}{l} = \frac{1}{2}\sqrt{\frac{\mu f}{\pi \sigma}} \cdot \frac{1}{a} \nonumber \\ &\cong \left(227~\mu\Omega\cdot\mbox{m}^{-1}\cdot\mbox{Hz}^{-1/2}\
right) \sqrt{f} \end{aligned} \nonumber \]
Using Expression \ref{m0159_efvalid}, we find this is valid only for \(f \gg 130\) kHz. So, for example, we may be confident that \(R'_{ic} \approx 0.82~\Omega\)/m at 13 MHz.
At the other extreme (\(f \ll 130\) kHz), Equation \ref{m0159_eRW} (the DC resistance) is a better estimate. In this low frequency case, we estimate that \(R'_{ic} \approx 0.16~\Omega\)/m and is
approximately constant with frequency.
We now have a complete picture: As frequency is increased from DC to 13 MHz, we expect that \(R'_{ic}\) will increase monotonically from \(\approx 0.16~\Omega\)/m to \(\approx 0.82~\Omega\)/m, and
will continue to increase in proportion to \(\sqrt{f}\) from that value.
Returning to Equation \ref{m0159_eCFGC-Z}, we see that resistance is not the whole story here. The impedance \(Z=R+jX\) also has a reactive component \(X\) equal to the resistance \(R\); i.e.,
\[X \approx R \approx \frac{l}{\sigma(\delta_s 2\pi a)} \nonumber \]
This is unique to good conductors at AC; that is, we see no such reactance at DC. Because this reactance is positive, it is often referred to as an inductance. However, this is misleading since
inductance refers to the ability of a structure to store energy in a magnetic field, and energy storage is decidedly not what is happening here. The similarity to inductance is simply that this
reactance is positive, as is the reactance associated with inductance. As long as we keep this in mind, it is reasonable to model the reactance of the wire as an equivalent inductance:
\[L_{eq} \approx \frac{1}{2\pi f} \cdot \frac{l}{\sigma(\delta_s 2\pi a)} \nonumber \]
Now substituting an expression for skin depth:
\[ L_{eq} &\approx \frac{1}{2\pi f} \cdot \sqrt{\frac{\pi f \mu}{\sigma}} \cdot \frac{l}{2\pi a} \nonumber \\ &= \frac{1}{4\pi^{3/2}} \sqrt{\frac{\mu}{\sigma f}} \cdot \frac{l}{a} \label{m0159_eLeq2}
for a wire having a circular cross-section with \(\delta_s\ll a\). The utility of this description is that it facilitates the modeling of wire reactance as an inductance in an equivalent circuit.
A practical wire may be modeled using an equivalent circuit consisting of an ideal resistor (Equation \ref{m0159_eACZ2}) in series with an ideal inductor (Equation \ref{m0159_eLeq2}). Whereas
resistance increases with the square root of frequency, inductance decreases with the square root of frequency.
If the positive reactance of a wire is not due to physical inductance, then to what physical mechanism shall we attribute this effect? A wire has reactance because there is a phase shift between
potential and current. This is apparent by comparing Equation \ref{m0159_eI} to Equation \ref{m0159_eV}. This is the same phase shift that was found to exist between the electric and magnetic fields
propagating in a good conductor, as explained in Section 3.11.
Elsewhere in the book we worked out that the inductance per unit length \(L'\) of RG-59 coaxial cable was about \(370\) nH/m. We calculated this from magnetostatic considerations, so the reactance
associated with skin effect is not included in this estimate. Let’s see how \(L'\) is affected by skin effect for the inner conductor. Using Equation \ref{m0159_eLeq2} with \(\mu=\mu_0\), \(\sigma\
cong 2.28 \times 10^7\) S/m, and \(a\cong 0.292\) mm, we find
\[L_{eq} \approx \left( 3.61 \times 10^{-5}~\mbox{H$\cdot$m$^{-1}\cdot$Hz$^{1/2}$} \right) \frac{l}{\sqrt{f}} \nonumber \]
Per unit length:
\[L_{eq}' \triangleq \frac{L_{eq}}{l} \approx \frac{3.61 \times 10^{-5}~\mbox{H$\cdot$Hz$^{1/2}$}}{\sqrt{f}} \nonumber \]
This equals the magnetostatic inductance per unit length (\(\approx 370\) nH/m) at \(f \approx 9.52\) kHz, and decreases with increasing frequency.
Summarizing: The equivalent inductance associated with skin effect is as important as the magnetostatic inductance in the kHz regime, and becomes gradually less important with increasing frequency.
Recall that the phase velocity in a low-loss transmission line is approximately \(1/\sqrt{L'C'}\). This means that skin effect causes the phase velocity in such lines to decrease with decreasing
frequency. In other words:
Skin effect in the conductors comprising common transmission lines leads to a form of dispersion in which higher frequencies travel faster than lower frequencies.
This phenomenon is known as chromatic dispersion, or simply “dispersion,” and leads to significant distortion for signals having large bandwidths.
Additional Reading:
• “Skin effect” on Wikipedia.
1. To be clear, this is the “point form” of Ohm’s law, as opposed to the circuit theory form (\(V=IR\)).↩
2. If this is not clear, recall that the electric field vector must point away from positive charge (thus, the \(+\) terminal).↩
3. This is in contrast to other ways that voltage and current can be related; for example, the non-linear \(V\)-\(I\) characteristic of a diode, which is not governed by Ohm’s law.↩ | {"url":"https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_II_(Ellingson)/04%3A_Current_Flow_in_Imperfect_Conductors/4.02%3A_Impedance_of_a_Wire","timestamp":"2024-11-02T02:27:49Z","content_type":"text/html","content_length":"143464","record_id":"<urn:uuid:37514000-ab4b-40fd-8d65-aceca642ec0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00097.warc.gz"} |
The Greatest and Grandest Illusion (Part I)- A Relativistic Deception - Hyper-Dimensional Universe
The Greatest and Grandest Illusion (Part I)- A Relativistic Deception
Albert Einstein
H.A. Lorentz
In my book The Enlightening, I make the following observation: Both Albert Einstein and H.A. Lorentz conceived of Lorentz’ Relativistic Velocity Transformation equations as describing the entire
extent of the universe. This is clearly supported by Einstein’s relativity theories, including the proposed universal “speed limit” of c. Lorentz did not disagree with Einstein’s interpretation.
My new postulate is as follows: The Transformations do not describe the entire universe. Instead, they describe the entire relativistic range of physical perception possible within the space-time
reference frame of any single observer.
That might sound like a subtle difference, but the cosmological ramifications are profound.
My second new postulate builds upon the first: Space-time interacts with mass/energy in the form of a wave, at c, relative to the reference frame of the mass/energy.
These are not just “throw away” postulates on my part. I spent nearly 25 years tying these simple ideas to a wide-ranging model of what I call hyper-dimensional space-time. Although various
multi-dimensional models have been proposed in the past, they all lack the depth and specificity of my model.
In my previous post “Space-time”, I point out how my second new postulate aligns with some very interesting research out of Caltech, and in fact aligns quite perfectly with the famous double-slit
experiment. In fact, in Chapter 30 of my book, The Enlightening, I depict a variation of the double-slit experiment which clearly describes how space-time itself is responsible for the wavelike
behavior of all particles in the universe. I think my book is very important just for the description of this one single experiment, but it goes much deeper than that.
This new way of looking at the universe also leads the way toward a multi-dimensional model of dark matter that does not necessitate any new, exotic particles, or bizarre alternate universes, or
anything of the sort. In my model, dark matter is simply ordinary matter which exists in entirely separate reference frames than our own frame. In fact, the portion of the Milky Way in which we
exist, within our own frame of space-time, is a source of dark matter for other portions of the Milky Way which exist within entirely separate space-time reference frames, or entirely separate
dimensions of space-time. We have looked high and low for dark matter, and I find it to be deeply and satisfyingly ironic that we ourselves are someone else’s dark matter.
There is much more to the book, such as a stunningly simple physical model of electromagnetic energy which provides a mechanism for the Second Postulate of Special Relativity. I will take a deeper
look at that model in Part II.
Next: The Greatest and Grandest Illusion (Part II) | {"url":"https://hyperdimensionaluniverse.org/2018/04/23/the-greatest-and-grandest-illusion-part-i/","timestamp":"2024-11-06T06:10:45Z","content_type":"text/html","content_length":"46252","record_id":"<urn:uuid:38ae8c6b-d4b8-4df8-b224-951e9eb68789>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00335.warc.gz"} |
Nominal Stiffness of GT-2 Rubber-Fiberglass Timing Belts for Dynamic System Modeling and Design
Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, 117 Transportation Building, 104 South Mathews Avenue, Urbana, IL 61801, USA
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 144 Mechanical Engineering Building, 1206 West Green Street, Urbana, IL 61801, USA
Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, 306 Talbot Laboratory, 104 South Wright Street, Urbana, IL 61801, USA
Author to whom correspondence should be addressed.
Submission received: 20 October 2018 / Revised: 18 November 2018 / Accepted: 19 November 2018 / Published: 21 November 2018
GT-style rubber-fiberglass (RF) timing belts are designed to effectively transfer rotational motion from pulleys to linear motion in robots, small machines, and other important mechatronic systems.
One of the characteristics of belts under this type of loading condition is that the length between load and pulleys changes during operation, thereby changing their effective stiffness. It has been
shown that the effective stiffness of such a belt is a function of a “nominal stiffness” and the real-time belt section lengths. However, this nominal stiffness is not necessarily constant; it is
common to assume linear proportional stiffness, but this often results in system modeling error. This technical note describes a brief study where the nominal stiffness of two lengths ($400 m m$ and
$760 m m$) of GT-2 RF timing belt was tested up to breaking point; regression analysis was performed on the results to best model the observed stiffness. The experiments were performed three times,
providing a total of six stiffness curves. It was found that cubic regression mod els ($R 2 > 0.999$) were the best fit, but that quadratic and linear models still provided acceptable representations
of the whole dataset with $R 2$ values above $0.940$.
1. Introduction
Timing belts are a common means of motion transfer between rotating motors/shafts in a machine or mechatronic system. Many small-to-medium sized mechatronic systems such as 3D printers [
], robots [
], desktop computer numerical control (CNC) machines [
], and positioners [
] use such belts, typically in the GT-style [
]. GT-style belts are specifically designed to effectively translate rotating motion from pulleys into linear motion with minimal deformation, slippage, and backlash. One of the fundamental
characteristics of such a motion transfer system is that the length of the belts changes with time, causing time-variant stiffnesses in the belts which must be considered in dynamic system modeling
and design. Note that the "stiffness" in the belt is considered only in the tension direction of the belt for this work, resulting in a stiffness that can be described as a single value or function
instead of the full stiffness matrix [
When analyzing and designing any robotic and other mechatronic systems, it is vital that a good dynamic model of the system be developed and used. Since such systems often use some kind of flexible
belts for motion transfer, the belt stiffness is a very important parameter in a system model. In cases where the length of the belt is constant (e.g., running between two fixed pulleys), the
of the belt can be modeled as a spring where
$f ( x ) = k x$
; therefore, the stiffness of the belt is a function of its deflection
under load. In effect, this constant-length stiffness of the belt is its “nominal” design stiffness. However, in cases where the belt changes length during use, the effective length of these belt is
a function of time and, therefore, its stiffness is also time-variant; this time-variant stiffness is the “effective” or apparent stiffness of the belt at some time
. It has been shown that the effective stiffness of the length-changing belts can be directly calculated as a function of the nominal stiffness value, the belt width, and the real-time length of the
belt [
] such that:
$k i ( t ) = C s p b L i ( t )$
$k i$
is the effective stiffness as a function of time,
$C s p$
is the nominal stiffness,
is the belt width, and
$L i ( t )$
is the length of the belt section at time
. For any case where the length remains constant, the effective and nominal stiffnesses are equal since the value of
$L i ( t )$
is a constant. Note that the value of
$C s p$
may be a constant or function of material properties for different belt materials; it cannot be considered a function of time the way that the length of the belt is. The most commonly-used GT-style
belt is the GT-2;
Figure 1
shows the fundamental geometry and specifications for this type of belt.
Figure 2
shows a common application, where a GT-style belt is used to transfer motion from a stepper motor to drive a linear positioning system. Also shown is a 2D dynamic model representation of such a
system (
Figure 2
b), where the differences in effective stiffness, based on belt length, in the belt sections are clearly evident. The sections
$L 1$
$L 2$
change in effective stiffness as a function of time, while section
$L 3$
stays constant during use [
] so the effective and nominal stiffnesses are equal.
The work described in this note explored the nominal stiffness
$C s p$
and the best way to model it in dynamic systems where belt length is not constant. Several previous studies have assumed that rubber-based timing belts have a linear nominal stiffness [
]. However, it is vital for designers and engineers working with dynamic systems which use belts for energy transfer to understand the true effects of the belt stiffness [
]. Therefore, experimental data was collected and used to derive conclusions on the true stiffness behavior of the GT-2 belts during use. The collected data was subjected to regression analysis to
see which type of model best fit, allowing the comparison of models for the same dataset. The information in this study will prove useful, both in choosing
stiffness values for dynamic models and for judging expected model error if linear stiffness assumptions are used.
2. Procedure and Results
Two lengths of new GT-2 belts, 400 mm and 760 mm, were subjected to a simple tensile test until they ruptured. The test apparatus was a custom-built, screw-driven manual desktop test stand set up for
tensile testing with
$3000 N$
capability and a travel rate of
$1 / 16 i n$
(1.6 mm) per screw revolution. The screw drive was rotated at a constant rate of
revolutions per second (0.8 mm/s), a reading being taken every revolution of the screw or every 1.6 mm. Since the length measurement was based on a count of the threads during travel, the uncertainty
in length was too small to quantify; the digital readout for the unit used a load cell with a given uncertainty of 100 gram-force or
$0.89 N$
. It was necessary to use this kind of manual tensile testing machine as none of the available standard machines were sensitive enough to measure the force-deflection behavior of these kinds of belts
]. In addition, the discrete time measurement ensured a reasonably-sized dataset for curve-fitting. This was replicated twice to obtain a set of six different curves, three from each length. The
ruptured belts were observed to fail suddenly and to show tearing of the glass fibers inside, as shown in
Figure 3
. The GT-2 belts used were a composite of neoprene (synthetic rubber) [
] and glass fibers, where the fibers appeared to drive the failure point of the belts.
The collected data, in terms of force-deflection curves, are shown in
Figure 4
a, while the equivalent stress–strain curves for the tests are shown in
Figure 4
b. The length of the belts clearly had an effect on the force-deflection curves, but this largely disappeared when the length was accounted for in the stress–strain curves. Note that most of the
curves show hyper-elastic behavior, i.e., there is no region in the curve where the stiffness is constant.
As the nominal compliance of the belts was clearly found to be nonlinear, a regression analysis was performed to model the curves and find the level of unexplained variance in these curves. One of
the most common polynomial regression models [
] used for hyper-elastic materials is the cubic polynomial. The basic model used for this study began with the following polynomial model:
$σ b e l t = A ε b e l t 3 + B ε b e l t 2 + C ε b e l t + D$
where a cubic model includes all of the variables, a quadratic model can be generated by setting
$A = 0$
, and a linear model can be used with
$A = B = 0$
. These curve fits, completed using Microsoft Excel™ (Microsoft Corp, Redmond, WA, USA) are shown in
Figure 5
a, and the fits for each of the variable and the resulting
$R 2$
values are shown in the first six cases in
Table 1
After fitting the cubic models to each of the six sets of experimental data, the cubic model, a quadratic model, and a linear model were then fit to the entire set at once, as shown in
Figure 5
b. A significant drop in the
$R 2$
value was noted for all of the models fit to the dataset, but differences between the cubic, quadratic, and linear models were observed to be small, as shown in
Table 1
It was observed that the low-strain region of the dataset (
Figure 5
b,c) conforms better to a linear model when the entire dataset is used. In actual use, it is most likely that the belts will not reach more than 20–30% of the belt breaking strength during normal use
], so this is a valid assumption for many systems; this will, of course, need to be determined by the modeler or designer before using a linear belt model. If the low-strain assumption can be used,
then the data fit a linear model with a slightly greater
$R 2$
value than a quadratic model for the entire dataset and is certainly superior to a linear model for the entire dataset. The linear model for this case is shown in the last row of
Table 1
3. Recommendations for Use and Applications
In cases where a time-variant belt length is used in a dynamic system model, the time-dependent stiffness of the belt must be considered, even when a mix of time-variant and time-invariant belt
lengths are used. In practice, it is recommended that the modeler follow a three-step procedure:
• Identify the nominal stiffness $C s p$ of each belt type used in the system (e.g., if two thicknesses of belts are used, two different nominal stiffnesses will be present). This information may
be collected from manufacturer datasheets or from tests on each belt type, similar to the tests done in this technical report.
• Decide if a linear or nonlinear nominal stiffness $C s p$ model will be used for each belt type. The primary driving force for this decision will be the computational cost for analyzing the
system; for a simple system, it may be practical to use a nonlinear nominal stiffness model, but a linear model would be more feasible in a system with several elements. However, the importance
of the model accuracy is a serious consideration and may justify a high computational cost if high accuracy is required.
• Based on the configuration of the system and the decisions made in the first two steps, the effective stiffness k can take one of four forms:
If the belt length is constant and a linear model is used for
$C s p$
, the effective stiffness in the equations of motion will be constant and described by
If the belt length is constant and a nonlinear model is used to find
$C s p$
, the nominal stiffness will be a function derived form a force-deflection curve. The effective stiffness in that belt section will be described by
$C s p ( x )$
is a continuous function of
If the belt length is time-variant and a linear model is used for
$C s p$
, the effective stiffness in the equations of motion will be time-variant and described by
If the belt length is time-variant and a nonlinear model is used to find
$C s p$
, the nominal stiffness will be a function derived form a force-deflection curve. In this case, the effective belt section stiffness will be described by
$k i = C s p ( x ) b L ( t )$
$C s p ( x )$
is a continuous function of
and the belt length is a function of time. Therefore, the effective stiffness will be dependent on both the length of the belt and the amount of force placed on the belt.
When modeling these dynamic systems, it is recommended that the simplest model of the belt stiffness which gives acceptable accuracy be used in order to balance computational cost with extreme
accuracy in the model. In most cases, the uncertainty in the material properties of the belt and the common use of linearization in dynamic models would erase any advantage to using an extremely
high-fidelity belt model.
4. Conclusions
This short technical note presents the results of a brief exploratory study on modeling the nominal stiffness of GT-2 timing belts; this information can be used to more accurately model the true,
time-variant, stiffness behavior of common GT-2 belts when the effective length of belt sections changes with time. It was observed that these belts do not behave in a linear way, as expected for
belts with a hyper-elastic base material, but that a linear model can provide a reasonable approximation of the behavior under some conditions, particularly low-strain conditions. When possible, the
cubic stiffness model should be used, but this would often be impractical for dynamic systems with many components, as it can cause a simple model to become nonlinear in more than one variable. When
practical and necessary for problem tractability, a linear model may be used with a reasonable degree of accuracy. The modeler or designer should keep in mind that some uncertainty will exist with
any belt model and should choose the model that best balances accuracy with computational cost.
Author Contributions
B.W. and A.E.P. conceived and designed the study. All authors helped to set up the experiments, collect data, perform the regression analyses, and write the report.
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest. No external funding was used to perform the work described in this study. Opinions and conclusions presented in this work are solely those of the authors.
• b = Belt width (m)
• $β i$ = Belt section i damping coefficient
• $C s p$ = Nominal belt stiffness (N/m)
• $k i$ = Effective (true) belt section i stiffness (N/m)
• $L i$ = Belt section i length (m)
• $m i$ = Mass of block i (kg)
• $θ i$ = Pulley i angle (degrees)
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Figure 2. (a) simple positioning system that utilizes a GT-type belt to drive the table and (b) its representative dynamic model.
Figure 5. Curve fits for (a) individual belts (cubic model); (b) full sample curve fit (cubic, quadratic, and linear models); and (c) low-strain linear curve fit.
Case Plot Reference A B C D $R 2$
760 mm (cubic model) - $R 1$ Figure 5a $− 2.00 × 10 6$ 94,969 958.80 −0.5658 0.9996
760 mm (cubic model) - $R 2$ Figure 5a $− 3.00 × 10 6$ 144,204 445.86 −0.1163 0.9996
760 mm (cubic model) - $R 3$ Figure 5a $− 2.00 × 10 6$ 95,693 1281.60 −0.0723 0.9997
400 mm (cubic model) - $R 1$ Figure 5a $− 2.00 × 10 6$ 81,219 849.99 0.2296 0.9993
400 mm (cubic model) - $R 2$ Figure 5a $− 2.00 × 10 6$ 95,332 935.37 0.2840 0.9995
400 mm (cubic model) - $R 3$ Figure 5a −922,283 50,993 810.95 0.4965 0.9994
Full dataset (cubic model) Figure 5b $− 2.00 × 10 6$ 98,091 937.22 −0.1758 0.9672
Full dataset (quadratic model) Figure 5b - −19,340 2408.9 −3.3656 0.9552
Full dataset (linear model) Figure 5b - - 1821.1 −0.6140 0.9431
Low strain (linear model) Figure 5c - - 2013.8 −2.3275 0.9573
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Wang, B.; Si, Y.; Chadha, C.; Allison, J.T.; Patterson, A.E. Nominal Stiffness of GT-2 Rubber-Fiberglass Timing Belts for Dynamic System Modeling and Design. Robotics 2018, 7, 75. https://doi.org/
AMA Style
Wang B, Si Y, Chadha C, Allison JT, Patterson AE. Nominal Stiffness of GT-2 Rubber-Fiberglass Timing Belts for Dynamic System Modeling and Design. Robotics. 2018; 7(4):75. https://doi.org/10.3390/
Chicago/Turabian Style
Wang, Bozun, Yefei Si, Charul Chadha, James T. Allison, and Albert E. Patterson. 2018. "Nominal Stiffness of GT-2 Rubber-Fiberglass Timing Belts for Dynamic System Modeling and Design" Robotics 7,
no. 4: 75. https://doi.org/10.3390/robotics7040075
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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• Physical Activity and Exercise
What Is The Average Height Of a 5-Year-Old Boy?
According to the CDC, a normal child is 43 inches tall and weighs 43 pounds when they are five years old.
However, the height of children at this age might vary by up to 5 inches. For a 5-year-old boy or girl, a typical height is between 39 to 48 inches, and a typical weight is typically between 34 and
50 pounds.
What’s The Best Way To Predict a Child’s Adult Height?
• Add the mother’s height to the father’s height in either inches or centimeters.
• Add 5 inches (13 centimeters) for boys or subtract 5 inches (13 centimeters) for girls.
• Divide by 2.
Keep visiting The Ganga Times for such beautiful articles. Follow us on Google News, Facebook, Twitter, Instagram, and Koo for regular updates. | {"url":"https://gangatimes.com/how-do-you-estimate-the-height-with-the-height-calculator/","timestamp":"2024-11-13T06:17:11Z","content_type":"text/html","content_length":"100112","record_id":"<urn:uuid:957b5d06-8371-4412-aa45-ee653f255a4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00178.warc.gz"} |
Special Quadrilaterals - ppt video online download
2 Special Quadrilaterals (four sides)A parallelogram has two pairs of opposite sides parallel. A rectangle has two pairs of opposite sides parallel and four right angles. A square has two pairs of
opposite sides parallel, four right angles, and four equal sides. A rhombus has two pairs of opposite sides parallel and four equal sides. A trapezoid has one pair of parallel sides.
3 Venn Diagram: QuadrilateralsParallelograms in White Circle Rectangles Squares Trapezoids Rhombuses
4 Rectangles Definition:A rectangle is a parallelogram with four right angles. A rectangle is a special type of parallelogram. Thus a rectangle has all the properties of a parallelogram. Opposite
sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other.
5 Examples If AE = 3x +2 and BE = 29, find the value of x.If AC = 21, then BE = _______. If m<1 = 4x and m<4 = 2x, find the value of x. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6. x = 9 units 10.5
units x = 18 units 6 5 4 3 2 1 E D C B A m<1=50, m<3=40, m<4=80, m<5=100, m<6=40
6 Rhombus ≡ ≡ Definition:A rhombus is a parallelogram with four congruent sides. ≡ ≡ Since a rhombus is a parallelogram the following are true: Opposite sides are parallel. Opposite sides are
congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other Diagonals bisect the angles.
7 Rhombus Examples Given: ABCD is a rhombus. Complete the following.If AB = 9, then AD = ______. If m<1 = 65, the m<2 = _____. m<3 = ______. If m<ADC = 80, the m<DAB = ______. If m<1 = 3x -7 and m<2
= 2x +3, then x = _____. 9 units 65° 90° 100° 10
8 Square Definition: A square is a parallelogram with four congruent angles and four congruent sides. Since every square is a parallelogram as well as a rhombus and rectangle, it has all the
properties of these quadrilaterals. Opposite sides are parallel. Four right angles. Four congruent sides. Consecutive angles are supplementary. Diagonals are congruent. Diagonals bisect each other.
Diagonals are perpendicular. Each diagonal bisects a pair of opposite angles.
9 Squares Examples Given: ABCD is a square. Complete the following.If AB = 10, then AD = _______ and DC = _______. If CE = 5, then DE = _____. m<ABC = _____. m<ACD = _____. m<AED = _____. 10 units 10
units 5 units 90° 45° 90° | {"url":"http://slideplayer.com/slide/4892619/","timestamp":"2024-11-11T23:59:17Z","content_type":"text/html","content_length":"159538","record_id":"<urn:uuid:593a7f4b-2b01-4311-97a6-9415375b49ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00487.warc.gz"} |
PHY301GC2: Practical Physics
(90 hours of practicals)
• Demonstrateskills on applicationof modern physics and thermal physics concepts
• Exhibit health and safety issues in relation to lasers and other advanced instruments
• Demonstrate the interpersonal skills through group projects and seminar presentations
Course Description:
• Students have to attend weekly practical sessions each of three hours duration
• Students will do group-projects and seminar presentations
• On completion of each weekly experiment, students should submit a brief report
• Students have to submit at least two full reports in the first semester and one full report in the second semester on experiments chosen by the lecturer in-charge
Continuous assessment on practical classes and brief lab reports 20%
Three full reports 20%
End of semester practical examinations 20%
Seminar presentation during the course 20%
Group project 20%
Recommended Readings:
• A.C Melissinos and J. Napolitano, Experiments in Modern Physics (2^nd edition), Academic Press (2003)
• Yaakov Kraftmakher, Experiments and Demonstrations in Physics (2^nd edition), World Scientific (2014)
• G.L. Squires, Practical Physics (4^th edition), Cambridge University Press (2001)
(45 hours of lectures and tutorials)
• Outline the inadequacy of classical physics and the need for modern theories
• Apply quantum concepts to understand atomic spectra
• Describe the basics of nuclear and elementary particle physics
Quantum Physics:
• Inadequacyof classical mechanics, Photo electric effect, Compton effect, wave particle duality, de Broglie wave, Heisenberg’s uncertainly principle, Schrödinger wave equation, probability
density, solution of simple time independent Schrödinger equations-the step potential and the potential well.
Atomic Physics:
• Scattering ofparticles,alpha particle scattering, Thomson atomic model, Bohr model of the Hydrogen atom,Rutherford model of the atom, estimation of the size of the nucleus, Bohr’s theory and its
limitations, Schrödinger equation for the hydrogen atom and its solution, the total, orbital, and magnetic quantum numbers, atomic spectra, Zeeman effect, fine structure of spectra and spin
quantum number, many electron atoms, production and properties of X-rays.
Nuclear Physics:
• Nuclear composition, mass and size of nucleus, nuclear forces, nuclear stability, radioactive transformation, liquid drop model of nuclei and its applications, nuclear reactions, nuclear fission
and fusion, a brief introduction to elementary particles.
In-course assessments 30%
End of course examination 70%
Recommended Readings:
• K.S. Krane, Modern Physics (2^nd edition), Wiley (1995)
• J. Taylor, C. Zafiratos and M.A. Dubson, Modern Physics for Scientists and Engineers (2^nd edition), Addison-Wesley (2003)
• A.P. French and E.F. Taylor, Introduction to Quantum Physics (The MIT introductory physics series), W.W. Norton and Company (1978)
PHY303GC3: Thermal and Statistical Physics
(45 hours of lectures and tutorials)
• Discuss the laws of classical thermodynamics and formulations of statistical physics
• Apply principles of thermodynamics to simple engineering systems
• Make use of kinetic theory to understand the properties of materials
• Zeroth law and the concept of temperature, work, heat, internal energy and the first law of thermodynamics, second law of thermodynamics, Carnot’s theorem, temperature, entropy, equation of
state, Maxwell’s thermodynamic relations and their application to simple systems, production and measurement of low temperatures, the third law of thermodynamics.
Thermal radiation:
• The law of blackbody radiation, application of thermodynamics to blackbody radiation, radiation pyrometer.
Kinetic theory:
• Ideal gases, Van der Waal’s gases, classical theory of specific heats of gases and solids, transport phenomena.
Statistical Physics:
• Thermodynamic probability and its relation to entropy, Boltzmann distribution and its classical limit, partition functions, application to solid like assemblies and gaseous systems, Maxwell’s
distribution of velocities in gases.
In-course assessments 30%
End of course examination 70%
Recommended Readings:
• M.W. Zemansky and R.H. Dittman,Heat and Thermodynamics (7^th edition), McGraw Hill (1997)
• B.N.Roy, Fundamentals of Classical and Statistical Thermodynamics, Wiley (2002)
• M.J. Moran and H.N. Shapiro, Fundamentals of Engineering Thermodynamics (5^th edition), Wiley (2006)
PHY321GE2: Medical Physics
(25 hours of lectures and tutorials plus 15 hours of clinical site visits)
• Discuss the principles of physics behind the operation of therapeutic and diagnostic medical equipments such as linear accelerators, MRI, PET and ultrasound scanner
• Explain the physical aspects of radiation dosimetry, treatment planning, dose calculations and distributions
• Identify safety and radiation protection principles and procedures
Radiation Physics:
• Review of atomic structure, characteristics of x- rays, photoelectric effect, Compton effect, pair production, nuclear decay, radioactivity, radiation physics, interaction of radiation with
matter, radiation detection and radiation dosimetry.
Medical imaging physics:
• Principles of image formation and quality, films and screens, digital imaging, image reconstruction with back projection, X- ray Computed Tomography (CT) and image processing, radiography
(mammography and fluoroscopy), principles of Magnetic Resonance Imaging (MRI), mapping and applications, nuclear medicine imaging [Gamma camera, Single Photon Emission Computed Tomography (SPECT)
and Positron Emission Tomography (PET)], principles and practice of ultrasound imaging.
Radiotherapy physics and radiation protection:
• Medical transducers, standard equipments used in radiotherapy (linear accelerator and Cobalt teletherapy machine), basic physical aspects of photon and electron therapy, radiation treatment
planning, dose calculations and distributions, radiation protection, safety considerations for patients and workers, quality assurance ofmedical devices.
In-course assessments 20%
End of course examination 70%
Report on clinical exposure 10%
Recommended Readings:
• E.B. Podgorsak, Radiation Oncology Physics: A Handbook for Teachers and Students, Vienna, IAEA (2005)
• J.T. Bushberg, J.A. Seibert, E.M. LeidholdtJrand J.M. Boone, The Essential Physics of Medical Imaging (3^rd edition), Lippincott Williams and Wilkins (2011)
• W.J. Meredith and J. B. Massey, The Fundamental Physics of Radiology (3^rd edition), Butterworth-Heinemann (1977)
(30 hours of lectures and tutorials)
• Recall the historical developments of astrophysics
• Explain the formation and properties of solar system, stars and galaxies
• Describe the origin and the evolution of the universe
Introduction to astrophysics:
• Historical background of astronomy, units in astronomy and observational measurement techniques, motions of heavenly bodies, celestial sphere and the atlas of stars, uses of optical instruments
in astronomy and Doppler Effect.
Solar system:
• The origin of the solar system and extra-solar planets, moon and eclipses, terrestrial and Jovian planets, properties of the Sun.
Stars and galaxies:
• Formation and general properties of stars, measurement of basic stellar properties such as distance, luminosity, spectral classification, mass, density and radii, Stellar evolution and
nucleo-synthesis, white dwarfs, neutron stars, black holes, structure of the milky way, other galaxies and their properties.
• Introduction to cosmology, the Hubble law, origin of the universe, the big bang theory, cosmic background radiation.
In-course assessments 30%
End of course examination 70%
Recommended Readings:
• B.W. Carroll and D.A. Ostlie, An Introduction to Modern Astrophysics (2^nd edition), Addison-Wesley (2006)
• J. Dufay, Introduction to Astrophysics: The Stars (reissue edition), Dover Publications (2012)
• B. Ryden and B.M. Peterson, Foundations of Astrophysics (1^st edition), Addison-Wesley (2010)
Supplementary Subject Area: Electronics
Electronics Elective Courses for Non-Physics Students
ELE341GE2: Analogue Electronics II
(20 hours of lectures and 30 hours of practicals)
• Discuss the evolution of integrated circuits
• Design, build and test different types of linear amplifier circuits
• Make use of op-amps for applications including mathematical operations
• Evolution of integrated circuits, integrated circuit components, monolithic and hybrid integrated circuits, Large Scale Integrated (LSI) circuits and Very Large Scale Integrated (VLSI) circuits.
Differential amplifiers:
• dc transfer characteristics, common mode and differential mode gains, differential amplifiers with constant current source and differential amplifiers with single ended input and output,typical
op-amps- the 741 op-amp.
Practical op-amps:
• Open loop voltage gain, input offset voltage, input bias current, common-mode rejection, phase shift, slew rate, output resistance, operation and types, characteristics.
Applications of op-amps:
• Function of operational amplifiers as subtractor, integrator, differentiator and logarithmic amplifier, analogue computer,rectifiers, feedback limiters, comparators, Schmitt triggers, function
generators, digital to analog converters, analog to digital converters, oscillators and 555 timer as a relaxation oscillator, as a pulse generator and as a monostable vibrator.
In-course assessments 30%
End of course examination 70%
Continuous assessment of practical reports 40%
End of course practical examinations 60%
Weightage: Theory (75%) and Practical (25%)
Recommended Readings:
• Roy ChoudhuryD, JainB and Shail Jain, Linear Integrated Circuits (4^th edition), New Age Publishers (2010)
• Thomas L. Floyd and David Buchla, Basic Operational Amplifiers and Linear Integrated Circuits, Prentice Hall (1999)
• Sergio Franco, Design With Operational Amplifiers and AnalogIntegrated Circuits, McGraw Hill (1997)
ELE342GE2:Digital Electronics
(20 hours of lectures and 30 hours of practicals)
• Discuss the principles and uses of logic gates
• Design, construct and test sequential circuits
• Demonstrate skills in the construction of electronic circuits using logic gates.
Introduction to digital concepts:
• Binary digits, logic levels and digital waveforms, basic logic operations, basic logic functions, digital system applications.
Number systems:
• Operations and codes, decimal numbers, binary numbers, decimal to binary conversion, binary arithmetic, octal numbers, hexa-decimal numbers, Binary Coded Decimal (BCD), digital codes, digital
system applications, The Karnaugh map.
Logic gates:
• The inverter, The AND gate, The OR gate, The NAND gate, The NOR gate, The Exclusive OR and Exclusive NOR gates, digital system applications, logic families.
Digital Circuits:
• Combinational digital circuits: Basic adders, parallel binary adders, comparators, decoders, encoders, multiplexer (data selector), de-multiplexer, parity generators, checkers.
• Sequential digital circuits: Flip-flops, counters, registers and their applications.
• Microcomputer: Central Processing Unit (CPU), The memory- Read Only Memory (ROM), Programmable ROMs (PROMs and EPROMs), Read/Write Random Access Memories (RAMs).
In-course assessments 30%
End of course examination 70%
Continuous assessment of practical reports 40%
End of course practical examinations 60%
Weightage: Theory (75%) and Practical (25%)
Recommended Readings:
• M. Morris Mano and Michael D. Ciletti, Digital Design with an Introduction to the Verilog HDL (5^th edition), Pearson Education(2013)
• Charles H. Roth, Jr., Fundamentals of Logic Design(4^th edition), Jaico Books (2002)
• John. F. Wakerly, Digital Design Principles and Practices(4^th edition), Pearson Education (2007) | {"url":"https://www.phy.jfn.ac.lk/index.php/level-3g/","timestamp":"2024-11-13T04:06:34Z","content_type":"text/html","content_length":"97419","record_id":"<urn:uuid:5d111329-e509-4bf8-bb4b-6121ff6d0e07>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00135.warc.gz"} |
September 24, 1994
This Week's Finds in Mathematical Physics (Week 39)
John Baez
I want to say a bit about Alain Connes' book, newly out in English, and then some about Yang-Mills theory in 2 dimensions.
1) Noncommutative Geometry, by Alain Connes, Academic Press, 640 pp.
You know something is up when a prominent mathematical physicist (Daniel Kastler) says "Alain is great. I am just his humble prophet." (This happened at a conference at Penn State I just went to.)
What is noncommutative geometry and what's so great about it?
Basically, the idea of noncommutative geometry is to generalize geometry to "quantum spaces". For example, the ordinary plane has two functions on it, the coordinate functions x and y, which commute:
xy = yx. We can think of x and y as representing the position and momentum of a classical particle. But when we consider a quantum-mechanical particle, we must give up commutativity and instead
impose the "canonical commutation relations" xy - yx = i ħ, where ħ is Planck's constant. Now x and y are not really functions on any space at all, but simply elements of a noncommutative algebra.
Still, we can try our best to pretend that they are functions on some mysterious sort of "quantum space" in which knowing one coordinate of a point precisely precludes us from knowing the other
coordinate exactly, by the Heisenberg uncertainty principle. Mathematically, noncommutative geometry consists of 1) expressing the geometry of spaces algebraically in terms of the commutative algebra
of functions on them, and 2) then generalizing the results to classes of noncommuative algebras.
The main trick invented by Connes was to come up with a substitute for the "differential forms" on a space. Differential forms are the bread and butter of modern geometry. If we start with a
commutative algebra A (say the algebra of smooth functions on some manifold like the plane), we can form the algebra of differential forms over A by introducing, for each element f in A, a formal
symbol df, and imposing the following rules:
d(f+g) = df + dg
d(cf) = c df (c a constant)
d(fg) = (df)g + f dg
fdg = (dg)f
df dg = -dg df.
More precisely, the differential forms over A are the algebra generated by A and these differentials df, modulo the above relations. This gives a purely algebraic way of understanding what those
mysterious things like dx dy dz in integral signs are.
Now, the last two of the five rules listed above fit nicely with the commutative of A when it is commutative, but they jam up the works horribly otherwise. So: how to generalize differential forms to
the noncommutative case? There are various things one can do if A is commutative in some generalized sense, such as "supercommutative" or "braided commutative" (which I call "R-commutative" in some
papers on this subject). However, if A is utterly noncommutative, it seems that the best approach is Connes', which is first to throw out the last two relations, obtaining something folks call the
"differential envelope" of A or the "universal differential graded algebra" over A --- which is pleasant but quite boring by itself --- and then to consider "chains" which are linear maps F from this
gadget to the complex numbers (or whatever field you're working in) satisfying the cyclic property
F(uv) = (-1)^{ij} F(vu)
where u is something that looks like f_0 df_1 df_2 .... df_i, and v is something like g_0 dg_1 dg_2 .... dg_j. There are charming things one can do with chains that wind up letting one do most of
what one could do with differential forms. More precisely, just as differential forms allow you entry into the wonderful world of DeRham cohomology, chains let you develop something similar called
cyclic homology (and there is a corresponding cyclic cohomology that's even more like the DeRham theory).
Connes, being extremely inventive and ambitious, has applied noncommutative differential geometry to many areas: index theory, K-theory, foliations, Penrose tilings, fractals, the quantum Hall
effect, and even elementary particle physics. Perhaps the most intriguing result is that if one develops the Yang-Mills equations using the techniques of noncommutative geometry, but with a very
simple "commutative" model of spacetime, namely a two-sheeted cover of ordinary spacetime, the Higgs boson falls out rather magically on its own. This has led Kastler and other physicists to pursue a
reformulation of the whole Standard Model in terms of noncommutative geometry, hoping to simplify it and even make some new predictions. It is far too early to see if this approach will get somewhere
useful, but it's certainly interesting.
I haven't read this book, just part of the French version on which it's based (with extensive additions), but my impression is that it's quite easy to read given the technical nature of the subject.
2) 2d Yang-Mills theory and topological field theory, by Gregory Moore, available as hep-th/9409044.
This is a nice review of recent work on 2d Yang-Mills theory. While Yang-Mills theory in 4 dimensions is the basis of our current theories of the strong, weak, and electromagnetic forces, and
mathematically gives rise to a cornucopia of deep results about 4-dimensional topology, 2d Yang-Mills theory has traditionally been considered "trivial" in that one can exactly compute pretty much
whatever one wants. However, Witten, in "On quantum gauge theories in two dimensions" (see "week36"), showed that precisely because 2d Yang-Mills theory was exactly soluble, one could use it to study
a lot of interesting mathematics problems relating to "moduli spaces of flat connections." (More about those below.) And Gross, Taylor and others have recently shown that 2d Yang-Mills theory, at
least working with gauge groups like SU(N) or SO(N) and taking the "large N limit", could be formulated as a string theory. So people respect 2d Yang-Mills theory more these days; its complexities
stand as a strong clue that we've just begun to tap the depths of 4d Yang-Mills theory!
I can't help but add that Taylor and I did some work a while back in which we formulated SU(N) 2d Yang-Mills theory for finite N as a string theory. This was meant as evidence for my proposal that
the loop representation of quantum gravity is a kind of string theory, a proposal described in "week18". For more on this sort of thing, try my paper in the book Knots and Quantum Gravity (see "
week23") --- which by the way is finally out --- and also the following:
3) Strings and two-dimensional QCD for finite N, by J. Baez and W. Taylor, 19 pages in LaTeX format available as hep-th/9401041, or by ftp from math.ucr.edu as "baez/string2.tex", to appear in Nuc.
Phys. B.
When it comes to "moduli spaces of flat connections", it's hard to say much without becoming more technical, but I certainly recommend starting with the beautiful work of Goldman:
4) The symplectic nature of fundamental groups of surfaces, by W. Goldman, Adv. Math. 54 (1984), 200-225.
Invariant functions on Lie groups and Hamiltonian flows of surface group representations, by W. Goldman, Invent. Math. 83 (1986), 263-302.
Topological components of spaces of representations, by W. Goldman, Invent. Math. 93 (1988), 557-607.
The basic idea here is to take a surface S with a particular G-bundle on it, and carefully study the space of flat connections modulo gauge transformations, which will be a finite-dimensional
stratified space. If you fix G and S, no matter what bundle you pick, this space will appear as a subspace of a bigger space called the moduli space of flat connections, which is the same as Hom(π_1
(S),G)/Ad G. There is an open dense set of this space, the "top stratum", which is a symplectic manifold. Geometric quantization of this manifold has everything in the world to do with Chern-Simons
theory, as summarized so deftly by Atiyah:
5) "The Geometry and Physics of Knots," by Michael Atiyah, Cambridge U. Press, Cambridge, 1990.
On the other hand, lately people have been using 2d Yang-Mills theory, BF theory, and the like (see "week36") to get a really thorough handle on the cohomology of the moduli space of flat
connections. For a mathematical approach to this problem that doesn't talk much about gauge theory, try:
6) Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds, by Lisa C. Jeffrey, preprint available from Princeton U. Mathematics Department.
© 1994 John Baez | {"url":"https://math.ucr.edu/home/baez/week39.html","timestamp":"2024-11-14T23:30:59Z","content_type":"text/html","content_length":"10377","record_id":"<urn:uuid:4841c00e-3dda-447a-be5d-7b38d21e71aa>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00239.warc.gz"} |
Generate Maximum revenue by selling K tickets from N windows
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185. Generate Maximum revenue by selling K tickets from N windows
Objective: Given 'N' windows where each window contains certain number of tickets at each window. Price of a ticket is equal to number of tickets remaining at that window. Write an algorithm to sell
'k' tickets from these windows in such a manner so that it generates the maximum revenue.
This problem was asked in the Bloomberg for software developer position.
Say we have 6 windows and they have 5, 1, 7, 10, 11, 9 tickets respectively.
Window Number 1 2 3 4 5 6
Tickets 5 1 7 10 11 9
Sell the first ticket from window 5, since it has 11 tickets so cost will be $11.
Revenue after selling first ticket, MaxRevenue: 11.
Window Number 1 2 3 4 5 6
Tickets 5 1 7 10 10 9
Sell the second ticket from window 4 or window 5, since they have 10 tickets each so cost will be $10, assume we sell it from window 5.
Revenue after selling second ticket, MaxRevenue: 21.
Window Number 1 2 3 4 5 6
Tickets 5 1 7 10 9 9
Sell the third ticket from window 4, since it has 10 tickets so cost will be $10.
Revenue after selling second ticket, MaxRevenue: 31.
Window Number 1 2 3 4 5 6
Tickets 5 1 7 9 9 9
Sell the fourth ticket from window 4,5 or 6, since they have 9 tickets each so cost will be $10.
Revenue after selling fourth ticket, MaxRevenue: 40.
1. Create a max-heap of size of number of windows. (Click here read about max-heap and priority queue.)
2. Insert the number of tickets at each window in the heap.
3. Extract the element from the heap k times (number of tickets to be sold).
4. Add these extracted elements to the revenue. It will generate the max revenue since extracting for heap will give you the max element which is the maximum number of tickets at a window among all
other windows, and the price of a ticket will be the number of tickets remaining at each window.
5. Each time we extract an element from the heap and add it to the revenue, reduce the element by 1 and insert it again into the heap since after a number of tickets will be one less after selling.
Max revenue generated by selling 5 tickets: 49 | {"url":"http://js-algorithms.tutorialhorizon.com/algorithms/generate-maximum-revenue-by-selling-k-tickets-from-n-windows/","timestamp":"2024-11-02T11:09:53Z","content_type":"text/html","content_length":"93003","record_id":"<urn:uuid:d156bd13-a731-42e8-88c7-f304e336056e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00664.warc.gz"} |
Inspiring Drawing Tutorials
How To Draw A Bode Plot
How To Draw A Bode Plot - A bode plot consists of two separate plots, one for. The table below summarizes what to do for each type of term in a bode plot. Web technique to get started: Web 2 bode
plots basics. Web lecture 17 exercise 102: 170k views 3 years ago linear control systems.
170k views 3 years ago linear control systems. Web to create bode plots with default options or to extract the frequency response data, use bode. Web bode plots give engineers a way to visualize the
effect of their circuit, in terms of voltage magnitude and phase angle (shift). • l16e93 control systems, lecture 16, e. Web bode plots are quite useful for performing filter design by hand quickly
for various applications.
Using bode plots (video series) closed. Web how to draw bode plot? Web bode plots are quite useful for performing filter design by hand quickly for various applications. • l16e93 control systems,
lecture 16, e. This is also available as a word document or pdf.
Write the given transfer function in the standard form. Web choose the type of bode plot you want to draw. A bode plot consists of two separate plots, one for. This is also available as a word
document or pdf. The magnitude is plotted in decibels (db) and the.
The magnitude is plotted in decibels (db) and the. Draw the line of each individual term on the graph. Making the bode plots for a transfer function involve drawing both the magnitude and phase
plots. The plot displays the magnitude (in db) and phase (in degrees) of the system. This is also available as a word document or pdf.
The magnitude is plotted in decibels (db) and the. Understanding bode plots (video series) why use them? Web lecture 17 exercise 102: Web bode plots are quite useful for performing filter design by
hand quickly for various applications. This is also available as a word document or pdf.
Draw the line of each individual term on the graph. The plot displays the magnitude (in db) and phase (in degrees) of the system. This is also available as a word document or pdf. A software tool for
generating asymptotic bode plots. This video illustrates the steps to draw bode plot for a given transfer.
Draw the line of each individual term on the graph. Making the bode plots for a transfer function involve drawing both the magnitude and phase plots. H = bodeplot (sys) plots the bode magnitude and
phase of the dynamic. Web technique to get started: Web bode plots are quite useful for performing filter design by hand quickly for various applications.
Making the bode plots for a transfer function involve drawing both the magnitude and phase plots. The plot displays the magnitude (in db). The magnitude is plotted in decibels (db) and the. Web
choose the type of bode plot you want to draw. 170k views 3 years ago linear control systems.
This range depends on the application at hand, such as audio or data. Web bode plots are quite useful for performing filter design by hand quickly for various applications. Web lecture 17 exercise
102: This is also available as a word document or pdf. • l16e93 control systems, lecture 16, e.
Understanding bode plots (video series) why use them? This is also available as a word document or pdf. You can choose between these three options: H = bodeplot (sys) plots the bode magnitude and
phase of the dynamic. 170k views 3 years ago linear control systems.
A bode plot consists of two separate plots, one for. This is also available as a word document or pdf. Web choose the type of bode plot you want to draw. Making the bode plots for a transfer function
involve drawing both the magnitude and phase plots. Web to create bode plots with default options or to extract the frequency.
This range depends on the application at hand, such as audio or data. Web bode plots are quite useful for performing filter design by hand quickly for various applications. Write the given transfer
function in the standard form. Web rules for drawing bode diagrams. Web to create a bode plot from an existing circuit, test the circuit with a range.
How To Draw A Bode Plot - Web lecture 17 exercise 102: This note will present 2 key ideas, which build on what you’ve learned about. The plot displays the magnitude (in db). 170k views 3 years ago
linear control systems. Web how to draw bode plot? Web technique to get started: Write the given transfer function in the standard form. This range depends on the application at hand, such as audio
or data. H = bodeplot (sys) plots the bode magnitude and phase of the dynamic. Web 2 bode plots basics.
A bode plot consists of two separate plots, one for. Making the bode plots for a transfer function involve drawing both the magnitude and phase plots. Write the given transfer function in the
standard form. The plot displays the magnitude (in db). This range depends on the application at hand, such as audio or data.
H = bodeplot (sys) plots the bode magnitude and phase of the dynamic. Web technique to get started: Draw the line of each individual term on the graph. This range depends on the application at hand,
such as audio or data.
Web technique to get started: A software tool for generating asymptotic bode plots. Web bode plots give engineers a way to visualize the effect of their circuit, in terms of voltage magnitude and
phase angle (shift).
170k views 3 years ago linear control systems. H = bodeplot (sys) plots the bode magnitude and phase of the dynamic. Web bode plots give engineers a way to visualize the effect of their circuit, in
terms of voltage magnitude and phase angle (shift).
Web Detailed Instructions On How To Draw A Bode Plot Diagram On First Order Denominators And Integrators.
Web rules for drawing bode diagrams. This range depends on the application at hand, such as audio or data. 170k views 3 years ago linear control systems. Web choose the type of bode plot you want to
Draw The Line Of Each Individual Term On The Graph.
Web how to draw bode plot? The plot displays the magnitude (in db) and phase (in degrees) of the system. This note will present 2 key ideas, which build on what you’ve learned about. Understanding
bode plots (video series) why use them?
Making The Bode Plots For A Transfer Function Involve Drawing Both The Magnitude And Phase Plots.
Bode (sys) creates a bode plot of the frequency response of a dynamic system model sys. Web to create bode plots with default options or to extract the frequency response data, use bode. Using bode
plots (video series) closed. Web bode plots are quite useful for performing filter design by hand quickly for various applications.
Write The Given Transfer Function In The Standard Form.
A bode plot consists of two separate plots, one for. • l16e93 control systems, lecture 16, e. Web bode plots give engineers a way to visualize the effect of their circuit, in terms of voltage
magnitude and phase angle (shift). The plot displays the magnitude (in db). | {"url":"https://one.wkkf.org/art/drawing-tutorials/how-to-draw-a-bode-plot.html","timestamp":"2024-11-13T18:55:15Z","content_type":"text/html","content_length":"33271","record_id":"<urn:uuid:88b6c9db-2b70-4ede-896d-a9c668174866>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00368.warc.gz"} |
Eddie's Math and Calculator Blog
The function atan2(y,x) is defined as:
atan2(y,x) = tan^-1 (y/x) with respect to the quadrant the point (x, y) is in. In case you didn't know, with respect to point (x, y):
(x, y) is in Quadrant I if x > 0 and y > 0
(x, y) is in Quadrant II if x < 0 and y > 0
(x, y) is in Quadrant III if x < 0 and y < 0
(x, y) is in Quadrant IV if x > 0 and y < 0
This is different from the two common calculator functions used to find the arctangent: Arctangent and Argument.
The Arctangent Function
TI and Casio Calculators*: tan^-1(y/x)
Hewlett Packard Calculators*: atan (y/x)
* The majority of them
Range (Output): -90° to 90°, -π/2 to π/2 radians
How to use Arctangent to get atan(y,x)
If the point is in quadrant I:
Use atan(y/x)
If the point is in quadrant II or III:
Use atan(y/x) + 180° in degrees mode
Use atan(y/x) + π in radians mode
If the point is in quadrant IV:
Use atan(y/x) + 360° in degrees mode
Use atan(y/x) + 2*π in radians mode
Special cases have to be used x or y is equal to 0:
If x=0 and y<0, the angle is 270° (3*π/2 radians)
If x=0 and y>0, the angle is 90° (π/2 radians)
If y=0 and x<0, the angle is 180° (π radians)
If y=0 and x>0, the angle is 360° or 0° (2*π or 0 radians)
The Argument Function
The complex number x + yi is used.
TI Calculators: angle(x + y*i)
Casio and Hewlett Packard Calculators: ARG(x + y*i)
Range: -180° to 180°, -π to π radians
How to use the Argument function to get atan2(y,x)
This is a great way to get atan2, which cleverly makes the use of complex numbers. In addition, there are a lot fewer things to remember:
If y≥0 (Quadrants I and II):
Use ARG(x+yi)*
(*The angle function if you are using a TI calculator)
If y<0 (Quadrants III and IV):
Use ARG(x+yi) + 360° for degrees mode
Use ARG(x+yi) + 2*π for radians mode
I hope this tip is helpful. Happy Thanksgiving and I am very thankful for all who have read, followed, and supported my blog over the last two years.
This blog is property of Edward Shore. 2013
This session will show how routines work in HPPL. Generally, subroutines have be declared before the main program. Declaration is important. The details of the subroutines are after the main program.
Definitely take a look at the example programs to get a better understanding.
SUB Routines for HP Prime
General Syntax:
sub(); //declare subroutines
EXPORT main()
commands go here, including sub()
commands go here
This is just a demonstration of how sub routines work. This program calculates one of two values:
If A is positive, then the program evaluates A. If not, the program values B instead. Where:
A = 2(x-y)/Φ + xy
B = Φ^2
and Φ = 2e^(x+y) - e^(x-y) - e^(y-x)
We will use Φ as the subroutine.
EXPORT SUBEXAM(X,Y)
LOCAL A, B;
IF A>B THEN
RETURN A;
RETURN B;
RETURN 2*e^(X+Y)-e^(X-Y)-e^(Y-X);
SUBEXAM(-4, 1) returns 21998.918189
SUBEXAM(2,3) returns 86283.2797974
SUBEXAM(-5,-6) returns 30.648061288
SUBEXAM(2,-3) returns 21810.6046664
Days Between Dates
DDAYS Using Subroutines for HP Prime: Best for 1901 to 2099
* Remember century years not divisible by 400 are NOT leap years. This program does not take this into account. If any such years are passed, subtract one day for such year manually.
Source: HP 12C Manual - Hewlett Packard
// Declare Subroutines
// Main program
EXPORT DDAYS(m1,d1,y1,m2,d2,y2)
// ΔDYS HP 12C
LOCAL x1, x2, z1, z2;
x1:=SUB1(m1); x2:=SUB1(m2);
z1:=SUB2(m1,y1); z2:=SUB2(m2,y2);
RETURN SUB3(y2,m2,d2,z2,x2)-
IF X≤2 THEN
RETURN 0;
RETURN IP(.4*X+2.3);
IF X≤2 THEN
RETURN Y-1;
RETURN Y;
RETURN 365*Y+31*(M-1)+D+IP(Z/4)-X;
(Thanks to Owitte for pointing out my typo)
Days Between Dates:
7/3/1985 to 2/28/1995 is 3,527 days
3/14/1977 to 11/17/2013 is 13,397 days
12/10/2010 to 6/30/2014 is 1,298 days
1/5/2015 to 3/19/2227 returns 77,506 BUT this program treats 2100 and 2200 as leap years, which in reality they are not. Subtract 2 to get the correct answer of 77,504 days.
So that is how subroutines work. Please give comments, ask questions, and always thanks to my supporters and readers. Cheers!
This blog is property of Edward Shore. 2013
Today's session is about starting other apps in a program and using colors.
Defining equations in the Program Editor and Home
The equation must be a string and be stored to the appropriate designated variable.
F# is for functions of X. (Function app).
R# is for polar functions of θ. (Polar app).
U# is for sequences of N, N-1, N-2. (Sequence app).
X# and Y# for parametric equations of T. (Parametric App)
V# for open statements and equations in the Advanced Graphing App, which The independent variables are X and Y.
# is a digit 0-9.
Defining equations this way leaves them uncheck. If you want them plotted or accessed in Num View, you will need to check them.
F1:="2*X^3" stores the function f(x) = 2*x^3 in Function 1.
R5:="A*SIN(θ)" stores the polar function r(θ) = A*sin(θ) in Polar Function 5, with A being what value stored in it.
STARTAPP(application name in quotes);
Starts the named App. The calculator points the screen to the default view (Plot, Symb, Num).
Access: Cmds, 4. App Functions, 2. STARTAPP
CHECK and UNCHECK
Checks and unchecks specific equation or function (0-9) in the current app. For example, if you are in the Function app, CHECK(1) activates F. As you should expect, UNCHECK(1) turns F1 off.
What does CHECK and UNCHECK affect?
1. Whether a function is plotted in Plot view.
2. Whether a function is analyzed in Num view.
Access for CHECK: Cmds, 4. App Functions, 1. CHECK
Access for UNCHECK: Cmds, 4. App Functions, 4. UNCHECK
Instructs the HP Prime to go to a certain view. It has two arguments, the view number and a redraw number.
Common view numbers include (not all inclusive):
-2 = Modes screen
-1 = Home
0 = Symbolic (Symb)
1 = Plot
2 = Numeric (Num)
3 = Symbolic Setup
4 = Plot Setup
5 = Numeric Setup
6 = App Information
7 = The Views Key
8 = first special view
9 = second special view
The redraw number is either 0 or non-zero. 0 does not redraw the screen, anything else does. I recommend the latter.
Syntax: STARTVIEW(view number, redraw number)
Access: Cmds, 4. App Functions, 3. STARTVIEW
Returns an integer code pertaining to a color's RGB code. This is super useful for drawing and text writing.
Syntax: RGB(red, green, blue, alpha)
Red: Intensity of Red, 0-255
Green: Intensity of Green, 0-255
Blue: Intensity of Blue, 0-255
Alpha: (optional) Opacity (up to 128).
RGB codes:
Blue: RGB(0,0,255)
Violet: RGB(143,255,0)
Dark Green: RGB(0,128,0)
Orange: RGB(255,127,0)
Yellow: RGB(0,255,255)
Red: RGB(255,0,0)
White: RGB(255,255,255)
Black: RGB(0,0,0)
Gray: RGB(129,128,128)
Brown: RGB(150,75,0)
Light Blue: RGB(173,216,330)
For other colors, RGB can be found on various sites on the Internet, including Wikipedia.
Access: Cmds, 2. Drawing, 5. RGB
Tip: Change a color of a graph
Use the syntax
F stands for the designated function type (F for function, R for polar, etc)
# is the digit 0-9.
makes the function F8 plot in blue.
This is a lot, but this is doable. Let's see all these commands and tips in action and create some magic.
Conic Drawing for HP Prime
Draws the conic section for the general equation
Ax^2 + By^2 + Cxy + Dx + Ey + F = 0
You can choose the color how the conic section is plotted, from red, blue, orange, and green. (Game show enthusiasts take note of the order of the colors I listed... ;) ).
EXPORT CONIC()
LOCAL cr, cg, cb, I;
"Ax^2+By^2+Cxy+Dx+Ey+F", { }, { },
// Colors
CHOOSE(I, "Choose a Color",
STARTAPP("Advanced Graphing");
// Plot View
Below are some examples. Remember the form:
Ax^2 + By^2 + Cxy + Dx + Ey + F = 0
Projectile Motion for HP Prime
This program calculates range and height of a projectile, and plots its path. The program sets the mode into Degrees (HAngle=1) and the calculator to the Parametric app.
x = V * cos θ * t
y = V * sin θ * t - .5 * g * t^2
V = initial velocity
θ = initial degree of flight
g = Earth gravitation constant (9.80665 m/s^2, ≈32.17404 ft/s^2)
Air resistance is not factored, so we are dealing with ideal conditions. How much the projectile represents reality varies, where factors include the object being projected, the temperate and
pressure of the air, and the weather.
EXPORT PROJ13()
LOCAL M, str;
// V, G, θ are global
// Degrees
CHOOSE(M, "Units", "SI", "US");
IF M==1 THEN
INPUT({V, θ}, "Data",
{"Initial Velocity in "+str+"/s",
"Initial Angle in Degrees"});
// Adjust Window
// Range
// Height
MSGBOX("Range: "+Xmax+" "+str+", "
+", Height: "+Ymax+" "+str);
// Plot View
Below are screen shots from an example with V = 35.25 m/s and θ = 48.7°.
This concludes this session of the tutorials. Shortly I will have Part 6 up, which has to do routines.
I am catch-up mode, still. But then again I always feel like there is too much to do and too little time. LOL
See you soon!
This blog is property of Edward Shore. 2013
Welcome to Part 4 of our programming series for the Prime. Today's session will cover CHOOSE and CASE.
First a tip from Han of the MoHPC Forum, which is found at http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/forum.cgi#255084. Thank you Han for allowing me to share this.
Use the IF THEN ELSE structure with INPUT to execute a set of default instructions if the user presses cancel. INPUT returns a value of 0 if ESC or cancel is pressed, and 1 if a value is entered.
IF INPUT(...) THEN
commands if values are entered
commands if Cancel is pressed
Default values can be assigned to values as an optional fifth argument for INPUT.
INPUT(var, "Title", "Prompt", "Help", default value)
The type of variable maybe set to other than real numbers. Just remember to store such type before the INPUT command. For example, if you want var to be a string, store an empty string:
var:=" ";
Again, major thanks to Han. CHOOSE and CASE
CHOOSE: Creates a pop up choose box, similar to what you see when you click on a soft menu. There are two syntaxes for CHOOSE:
Simple Syntax (up to 14 options):
CHOOSE(var, "title string", "item 1", "item 2", ... , "item n");
List syntax (infinite amount of items):
CHOOSE(var, "title string", {"item 1", "item 2"});
Choosing item 1 assigns the value of 1 to var, choosing item 2 assigns the value of 2 to var.
Access: Cmds, 6. I/O, 1. CHOOSE
CASE: Allows for different test cases for one variable. Also includes a default scenario (optional).
IF test 1 THEN do if true END;
IF test 2 THEN do if true END;
DEFAULT commands END;
Access: Cmds, 2. Branch, 3. CASE
Let's look at two programs to demonstrate both CHOOSE and CASE. TERMVEL - Terminal Velocity of an Object
LOCAL L0:={9.80665,32.174},
CHOOSE(K,"Type of Object","Sphere","Cube",
{"M=","A="},{"Mass","Surface Area"});
MSGBOX("Terminal Velocity="+T);
RETURN T;
Sphere, SI Units, M = .05 kg, A = .0028 m^2
Terminal Velocity: T = 24.6640475387 m/s
Cube, US Units, M = 1.2 lb, A = .3403 ft^2
Terminal Velocity: T = 53.149821209 ft/s
AREAC - Area of Circles, Rings, and Sectors
EXPORT AREAC()
LOCAL C,R,S,θ,A;
CHOOSE(C,"Areas","1. Circle","2. Ring","3. Sector");
INPUT(R, "Input Radius", "R =");
IF C==1 THEN A:=π*R^2; END;
IF C==2 THEN
INPUT(S,"Small Radius","r=");
IF C==3
INPUT(θ, "Angle", "θ=");
\\ Assume you are in the correct angle mode
IF HAngle==1 THEN
\\ Test Angle Mode
MSGBOX("Area is "+A);
RETURN A;
R = 2.5, r = 1.5, θ = π/4 radians or 45°
Circle: 19.6349540849
Ring: 12.5663706144
Sector: 2.45436926062
That is how, in general CHOOSE and CASE work. I thank you as always. It is so good to finally be rid of a cold and firing on all cylinders again.
This blog is property of Edward Shore. 2013
This tutorial is going to cover a lot, each with some new programming commands in this series. I hope you are ready for the intensity. :)
WHILE, INPUT, KILL
HP Prime Program: TARGET. TARGET is a game where you provide a guess to get a desired number. If you miss, the calculator will tell you if number is higher and lower. At the end of the game, the
calculator gives you how may picks you needed to get the target number.
WHILE: Repeat a number of commands while a specific condition is test.
WHILE condition is true DO
Access: Tmplt, 3. Loop, 5. WHILE
Caution: Watch your ENDs! Make sure an END is with each loop and the program itself. Press the soft key Check to check your work.
INPUT: Creates an input screen for variables. On the HP Prime, the input can asked for more than one input. TARGET demonstrates INPUT with one prompt.
One Variable:
INPUT(variable, "title", "label", "help text")
INPUT(list of variables, "title", list of "labels", list of "help text")
Note: Pressing Cancel will store a 0 in variable. You may include code of what to do if the user presses Cancel, but it is not required.
Access: Cmds, 6. I/O, 5. INPUT
KILL: Terminates program execution. Nothing dies, I promise.
Access: Tmplt. 1. Block, 3. KILL
LOCAL C:=0, N:=RANDINT(1,20), G:=-1;
WHILE G≠N DO
INPUT(G,"Guess?","GUESS:","1 - 20");
IF G==0 THEN
IF G < N THEN
IF G > N THEN
MSGBOX("Correct! Score: "+C);
Try it and of course, you can adjust the higher limit. Here is some thing for you to try with TARGET:
1. Add a limited amount of guesses.
2. Can you display the list of guesses?
ULAM Algorithm: take an integer n. If n is even, divide it by 2. If n is odd, multiply it by 3 and add 1. ULAM counts how many steps it takes to get n to 1.
Access: Tmplt, 3. Loop, 6. REPEAT
CONCAT(list1, list2): Melds list1 and list2 into one.
Access: Toolbox, Math, 6. List, 4. Concatenate
EXPORT ULAM(N)
LOCAL C:=1, L0:={N};
IF FP(N/2)==0 THEN
UNTIL N==1;
MSGBOX("NO. OF STEPS="+C);
RETURN L0;
ULAM(5) returns:
Message Box: "NO. OF STEPS=6"
List: {5, 16, 8, 4, 2, 1}
ULAM(22) returns:
Message Box: "NO. OF STEPS=16"
List: {22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1}
The next section will introduce a super-important command, GETKEY. We will be working with GETKEY over the entire series.
The Program KEYNO: The person presses key presses. Which each key press, the code returns to the terminal screen. The program terminates when the Enter key is pressed.
GETKEY: Returns the key code of last key pressed. The Prime's key map is below. (Picture is from the HP Prime User's Guide)
Access: Cmds, 6. I/O, 4. GETKEY
EXPORT KEYNO()
LOCAL K;
PRINT("Press any key to get its code.");
PRINT("Press Enter to exit.");
IF K ≥ 0 THEN
UNTIL K==30;
END; Example Key Codes:
33: 8 key
2: up
7: left
8: right
12: down
50: plus
45: minus
This concludes Part 3. Again, it can't be said enough, thanks for all the comments and compliments. And until next time,
This blog is property of Edward Shore. 2013
Major thanks to Miguel Toro from the MoHPC Forum for this tip:
There are two ways to call created programs in RPN mode of the HP Prime:
argument 1 [Enter]
argument 2 [Enter]
argument n [Enter]
program [Enter]
argument 1 [space] argument 2 [space] ... argument n [space] program
Keep in mind this works with created user programs. To ensure that built-in commands work correctly, still include the number of arguments as needed.
Link to the forum: http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/forum.cgi
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(2015–) Wittgenstein Source Bergen Nachlass Edition (WS-BNE). Edited by the Wittgenstein Archives at the University of Bergen under the direction of Alois Pichler. In: Wittgenstein Source, curated by
Alois Pichler (2009–) and Joseph Wang-Kathrein (2020–). (N) Bergen: WAB.
To cite this element you can use the following URL: | {"url":"http://wittgensteinsource.org/agora_show_transcription?id=42107&siglum=Ts-201a1%2Cb14%5B1%5D&collection=1&data-title=Ts-201a1%2Cb14%5B1%5D+&data-verticalTitle=Ts-201a1%2Cb14%5B1%5D+Normalized+transcription&data-type=Normalized+transcription&data-id=Ts-201a1%2Cb14%5B1%5D-Normalized&data-boxTitle=Ts-201a1%2Cb14%5B1%5D+Normalized+transcription&data-replaceContent=&data-url=%2Fagora_show_transcription%3Fid%3D42107","timestamp":"2024-11-12T21:54:53Z","content_type":"application/xhtml+xml","content_length":"12285","record_id":"<urn:uuid:0ee71ef5-edf9-4f4f-87ec-b3ea6563b3e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00091.warc.gz"} |
2011, 2011
We study properties of semi-elementary imsets and elementary imsets introduced by Studeny [10]. The rules of the semi-graphoid axiom ...
Show moreWe study properties of semi-elementary imsets and elementary imsets introduced by Studeny [10]. The rules of the semi-graphoid axiom (decomposition, weak union and contraction) for
conditional independence statements can be translated into a simple identity among three semi-elementary imsets. By recursively applying the identity, any semi-elementary imset can be written as
a sum of elementary imsets, which we call a representation of the semi-elementary imset. A semi-elementary imset has many representations. We study properties of the set of possible
representations of a semi-elementary imset and prove that all representations are connected by relations among four elementary imsets.
Show less
2017, 2017-02-08
To evaluate the goodness-of-fit of a statistical model to given data, calculating a conditional p value by a Markov chain Monte Carlo method...
Show moreTo evaluate the goodness-of-fit of a statistical model to given data, calculating a conditional p value by a Markov chain Monte Carlo method is one of the effective approaches. For this
purpose, a Markov basis plays an important role because it guarantees the connectivity of the chain, which is needed for unbiasedness of the estimation, and therefore is investigated in various
settings such as incomplete tables or subtable sum constraints. In this paper, we consider the two-way change-point model for the ladder determinantal table, which is an extension of these two
previous works, i.e., works on incomplete tables by Aoki and Takemura (2005, J. Stat. Comput. Simulat.) and subtable some constraints by Hara, Takemura and Yoshida (2010, J. Pure Appl. Algebra).
Our main result is based on the theory of Gr ?obner basis for the distributive lattice. We give a numerical example for actual data.
Show less
2011, 2011
Rapid research progress in genotyping techniques have allowed large genome-wide association studies. Existing methods often focus on...
Show moreRapid research progress in genotyping techniques have allowed large genome-wide association studies. Existing methods often focus on determining associations between single loci and a
specific phenotype. However, a particular phenotype is usually the result of complex relationships between multiple loci and the environment. In this paper, we describe a two-stage method for
detecting epistasis by combining the traditionally used single-locus search with a search for multiway interactions. Our method is based on an extended version of Fisher’s exact test. To perform
this test, a Markov chain is constructed on the space of multidimensional contingency tables using the elements of a Markov basis as moves. We test our method on simulated data and compare it to
a two-stage logistic regression method and to a fully Bayesian method, showing that we are able to detect the interacting loci when other methods fail to do so. Finally, we apply our method to a
genome-wide data set consisting of 685 dogs and identify epistasis associated with canine hair length for four pairs of single nucleotide polymorphisms (SNPs).
Show less
2015, 2015-11-09
Exchange type chromosome aberrations (ETCAs) are rearrangements of the genome that occur when chromosomes break and the resulting fragments...
Show moreExchange type chromosome aberrations (ETCAs) are rearrangements of the genome that occur when chromosomes break and the resulting fragments rejoin with fragments from other chromosomes
or from other regions within the same chromosome. ETCAs are commonly observed in cancer cells and in cells exposed to radiation. The frequency of these chromosome rearrangements is correlated
with their spatial proximity, therefore it can be used to infer the three dimensional organization of the genome. Extracting statistical significance of spatial proximity from cancer and
radiation data has remained somewhat elusive because of the sparsity of the data. We here propose a new approach to study the three dimensional organization of the genome using algebraic
statistics. We test our method on a published data set of irradiated human blood lymphocyte cells. We provide a rigorous method for testing the overall organization of the genome, and in
agreement with previous results we find a random relative positioning of chromosomes with the exception of the chromosome pairs {1,22} and {13,14} that have a significantly larger number of ETCAs
than the rest of the chromosome pairs suggesting their spatial proximity. We conclude that algebraic methods can successfully be used to analyze genetic data and have potential applications to
larger and more complex data sets.
Show less
2015, 2015-11-09
We consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is...
Show moreWe consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is bounded from above by the Markov complexity
of the base configuration. As important examples of base configurations we consider incidence matrices of graphs and study the maximum Markov degree of configurations defined by fibers of the
incidence matrices. In particular we give a proof that the Markov degree for two-way transportation polytopes is three.
Show less | {"url":"https://repository.iit.edu/islandora/search?type=dismax&f%5B0%5D=mods_subject_topic_ms%3AMarkov%5C%20basis","timestamp":"2024-11-10T15:00:39Z","content_type":"text/html","content_length":"71114","record_id":"<urn:uuid:974593ee-fff1-4751-b98a-44784432b4e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00164.warc.gz"} |
TCS / Software / GnT
GnT (Generate'n'Test): A Solver for Disjunctive Logic Programs
GnT is an experimental implementation of the stable model semantics for disjunctive logic programs [Gelfond and Lifschitz, 1991]. Our implementation is based on an architecture consisting of two
interacting smodels solvers for non-disjunctive programs. One of the them is responsible for generating as good as possible model candidates while the other checks for minimality, as required from
disjuctive stable models. Please see [Janhunen et al., 2006] for details.
This program is to be compiled under the smodels (versions 2.*) source distribution; and it is to be used with the front-end lparse.
Using the Software
• Disjunctive rules are written in the following syntax:
a1|a2|a2 :- b1, b2, b3, not c1, not c2, not c3.
• Also rules with variables can be used, but the front-end lparse performs an instantation for the rules using domain predicates like d(.) below:
a(X)|b(X)|c(X) :- d(X). d(1..10).
• Parametrized disjunctions (lparse versions 1.0.14 and later) in the heads of rules are also supported:
|a(X):d(X)|. d(1..4).
• The stable models of a disjunctive logic program can be computed by giving a command line like
lparse --dlp disjunctive-program.lp | gnt2 0
If the command line argument 0 is replaced by a positive integer n, then the first n stable models are to be computed (if so many stable models exist).
• Even partial stable models can be computed by giving the the command line option --partial for lparse.
• Similarly, regular models can be computed by giving the command line option -r for lparse.
Version Information
The author of the first version of GnT (also included as example4.cc in the smodels source distribution) is Patrik Simons. The version GnT1 (given below) is basically the orginal one -- only few
lines of code have been added to handle command line options. The second version (GnT2.* below), which is a derivative of the first one, was developed by Tomi Janhunen to speed up computation.
Terms and Conditions
This software is distributed under the GNU General Public License (available from Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA). In particular, as stated in
the licence agreement, the software is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
Source code: files for GnT1 (the first version) and GnT2 (version 2.1) -- to be placed and compiled (Makefile) in a subdirectory of the smodels source directory (such as examples therein). Inline
functions may cause problems with some compilers (g++ series 3.* and 4.*). This problem is fixed in the smodels 2.32 distribution.
Precompiled Linux binaries and a gzipped tar-archive of the sources are also available for your convenience: GnT1, GnT2.1, GnT2.1 for lparse-1.0.14 (and later), and GnT.tgz. It is also possible to
switch between old (pre 1.0.14) and new formats using dencode.
Related Publications
T. Janhunen, I. Niemelä, D. Seipel, P. Simons, and J.-H. You. Unfolding Partiality and Disjunctions in Stable Model Semantics. ACM Transactions on Computational Logic, 7(1), 1-37, January 2006. (Also
in Proc. of KR 2000, 411-419).
Introduction to disjunctive stable models:
M. Gelfond and V. Lifschitz. Classical Negation in Logic Programs and Disjunctive Databases. New Generation Computing, 9 (1991), 365-385.
Introduction to regular models:
J. You and L. Yuan. A three-valued semantics for deductive databases and logic programs. Journal of Computer Systems and Sciences, 49 (1994), 334--361.
[TCS main] [Contact Info] [Personnel] [Research] [Publications] [Software] [Studies] [News Archive] [Links] Latest update: 06 February 2013. Tomi Janhunen | {"url":"https://research.ics.aalto.fi/software/asp/gnt/","timestamp":"2024-11-11T16:25:41Z","content_type":"text/html","content_length":"6344","record_id":"<urn:uuid:cdc53cdc-9fe9-44f7-8f91-97b04aef3d03>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00482.warc.gz"} |
M. Nunokawa Et Al. , "New condition for univalence of certain analytic functions," JOURNAL OF THE INDIAN MATHEMATICAL SOCIETY , 2012
Nunokawa, M. Et Al. 2012. New condition for univalence of certain analytic functions. JOURNAL OF THE INDIAN MATHEMATICAL SOCIETY .
Nunokawa, M., UYANIK, N., Owa, S., Saitoh, H., & Srivastava, H. M., (2012). New condition for univalence of certain analytic functions. JOURNAL OF THE INDIAN MATHEMATICAL SOCIETY .
Nunokawa, Mamoru Et Al. "New condition for univalence of certain analytic functions," JOURNAL OF THE INDIAN MATHEMATICAL SOCIETY , 2012
Nunokawa, Mamoru Et Al. "New condition for univalence of certain analytic functions." JOURNAL OF THE INDIAN MATHEMATICAL SOCIETY , 2012
Nunokawa, M. Et Al. (2012) . "New condition for univalence of certain analytic functions." JOURNAL OF THE INDIAN MATHEMATICAL SOCIETY .
@article{article, author={Mamoru Nunokawa Et Al. }, title={New condition for univalence of certain analytic functions}, journal={JOURNAL OF THE INDIAN MATHEMATICAL SOCIETY}, year=2012} | {"url":"https://avesis.anadolu.edu.tr/activitycitation/index/1/0d01056e-82aa-4ce1-a3c2-dfda5bb92572","timestamp":"2024-11-05T23:05:54Z","content_type":"text/html","content_length":"10806","record_id":"<urn:uuid:e2d253fb-42ee-4f57-a2ee-e38cf6664f40>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00400.warc.gz"} |
Weighted Graph Implementation – JAVA
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270. Weighted Graph Implementation – JAVA
We have already discussed about Graph basics. We recommend reading this before you continue to read this article.
What is Weighted Graph?
A Graph is called weighted graph when it has weighted edges which means there are some cost associated with each edge in graph.
1. Each edge of a graph has an associated numerical value, called a weight.
2. Usually, the edge weights are nonnegative integers.
3. Weighted graphs may be either directed or undirected.
4. The weight of an edge is often referred to as the "cost" of the edge.
5. Will create an Edge class to put weight on each edge
vertex-0 is connected to 2 with weight 3
vertex-0 is connected to 1 with weight 4
vertex-1 is connected to 2 with weight 5
vertex-1 is connected to 3 with weight 2
vertex-2 is connected to 3 with weight 7
vertex-3 is connected to 4 with weight 2
vertex-4 is connected to 5 with weight 6
vertex-4 is connected to 1 with weight 4
vertex-4 is connected to 0 with weight 4
Reference: here | {"url":"https://tutorialhorizon.com/algorithms/weighted-graph-implementation-java/","timestamp":"2024-11-12T21:33:49Z","content_type":"text/html","content_length":"93734","record_id":"<urn:uuid:67312dd8-34a4-46b5-acf0-2deb73e0e61e>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00863.warc.gz"} |
Observation and Estimation of Lagrangian, Stokes, and Eulerian Currents Induced by Wind and Waves at the Sea Surface
1. Introduction
Surface drift constitutes one of the most important applications of the emerging operational oceanography systems (e.g., Hackett et al. 2006), because it plays an important role in the fate of oil
pollutions and larvae recruitment. A quantitative understanding of the relative contribution of the wave-induced Stokes drift to the near-surface velocities is also paramount for the proper
estimation of air–sea energy fluxes (Kantha et al. 2009). The quantitative variation of surface drift as a function of the forcing parameters is still relatively poorly known. In areas of strong
currents resulting from tides or quasigeostrophic dynamics, the surface drift current is highly correlated to the subsurface current. Otherwise, winds play a major role in defining the surface
Recent theoretical and numerical works (Ardhuin et al. 2004; Kantha and Clayson 2004; Rascle et al. 2006; Ardhuin et al. 2008b) have sought to reconcile historical measurements of Eulerian and
Lagrangian (i.e., drift) velocities with recent knowledge on wave-induced mixing (Agrawal et al. 1992) and wave-induced drift (Rascle et al. 2008). These suggest that the surface Stokes drift U[ss]
induced by waves typically accounts for ⅔ of the surface wind-induced drift, in the open ocean, and that the surface wind-related Lagrangian velocity U[L](z) is the sum of the strongly sheared Stokes
drift U[S](z) and a relatively uniform quasi-Eulerian current û(z), defined by Jenkins (1987) and generalized by Ardhuin et al. (2008b). The Stokes drift decays rapidly away from the surface on a
scale that is the Stokes depth D[S]. For deep-water monochromatic waves of wavelength L, we take D[S] = L/4, by analogy with the usual definition of the (2 times larger) depth of wave influence for
the orbital motion (e.g., Kinsman 1965); that is, at that depth, the Stokes drift is reduced to 4% of its surface value. For random waves, a similar result requires a more complex definition, but the
approximate same result can be obtained by using the mean wavelength , where T[m03] is the mean period defined from the third moment of the wave frequency spectrum (see appendix C). Smaller values,
such as L/(4π), which was used by Polton et al. (2005), are more representative of the depth where the Stokes drift is truly significant.
For horizontally homogeneous conditions, the depth-integrated quasi-Eulerian mass transport vector
is constrained by the balance between the Coriolis force and the wind stress
and bottom stress
Hasselmann 1970
Ardhuin et al. 2004
Smith 2006
is the (Stokes) mass “transport”
induced by surface gravity waves;
is 2 times the vertical component of the earth rotation vector, usually called the Coriolis parameter; and
is the vertical unit vector, which points up. The surface stress vector
is typically on the order of
, where
is the air density,
is in the range 1–2 × 10
, and
the wind speed at 10-m height. The horizontal homogeneity is obviously never achieved strictly (e.g.,
Pollard 1983
); this aspect will be further discussed in the context of our measurements.
The wind-driven current is not expected to be significant at a depth greater than 0.7 times the Ekman depth, D[E] = 0.4(τ[a]/ρ[w])/f (i.e., less than 0.2% of the wind speed if the surface value is
2.8% of U[10]; Madsen 1977). For a wind speed U[10] = 10 m s^−1, 0.7D[E] is on the order of 30 m. In locations with a larger water depth, the bottom stress is thus expected to be negligible. Further,
this depth of maximum influence can also be limited by a vertical stratification, with larger velocities in shallow mixed layers and directions of U[E] more strongly deflected to the right of the
wind (in the Northern Hemisphere) than previously expected (Price and Sundermeyer 1999; Rascle 2007). It has also been proposed by Polton et al. (2005) that the wave-induced mass “transport” M^w may
play a role in the modification of near-surface currents, but M^w is generally less than 30% of the Ekman transport M^E = τ[a]/f, and its effect appears to be secondary compared to the stratification
(Rascle and Ardhuin 2009). The time-averaged balance given by (1) is thus, approximately, M^m = −M^w + (τ[a] × e[z])/f. This was nearly verified for the Long-Term Upper Ocean Study (LOTUS3) dataset (
Price and Sundermeyer 1999) when allowing for wave-induced biases in the mooring measurements (Rascle and Ardhuin 2009). Yet, this is not always the case (e.g., Nerheim and Stigebrandt 2006),
possibly because of baroclinic currents and other phenomena that are difficult to separate from the wind-driven component.
The vertical profile of the quasi-Eulerian current is, under the same homogeneous and stationary circumstances, the solution of (
Xu and Bowen 1994
Ardhuin et al. 2008b
is a turbulent mixing coefficient.
These predictions were verified by Rascle (2007) with mooring data at depths greater than 5 m and surface-following measurements by Santala and Terray (1992) at depths larger than 2 m. When
extrapolated to the surface using a simple numerical model, these observations give directions of U[E] between 45° and 90°, more than the 45° given by the constant eddy-viscosity model of Ekman
(1905), as extended by Gonella (1971), and the 10° given by the linear eddy-viscosity model of Madsen (1977). This surface angle, as well as the magnitude of U[E], is also critical for the estimation
of the flux of wind energy to the Ekman layer (e.g., Wang and Huang 2004) or the analysis of near-surface drifter data (e.g., Rio and Hernandez 2003; Elipot and Lumpkin 2008). For a better
understanding of these questions, it is thus necessary to use ocean velocities measured much closer to the surface.
High-frequency (HF) radars can provide such measurements at depths that depend on their operating frequency. Using a 30-MHz radar, Mao and Heron (2008) made observations that are also consistent with
the idea that the drift current, found to be 2.1% of the wind speed on average, is the sum of U[E], which—according to their theory—depends quadratically on the wind speed, and U[ss], which they
estimate to depend linearly on the wind speed, with a variation according to the fetch. Unfortunately, their analysis relied on empirical wave estimates that give large relative errors (on the order
of 100%; see, e.g., Kahma and Calkoen 1992; Ardhuin et al. 2007) and a limited range of wind speeds. Other HF radar observations give a surface current on the order of 1.5%–2.5% of U[10] (Essen 1993)
with 25–30-MHz radars. Dobson et al. (1989) also report a ratio of 2.0% using a 22 MHz radar, and Shay et al. (2007) report a ratio of 2%–3% using a 16-MHz radar in water depths of 20–50 m. These
analyses are difficult to interpret because of the filters applied on time series to remove motions (tides, geostrophic currents, etc.) that are not related to the wind and also because of the
importance of inertial oscillations that make the wind- and wave-driven current a function of the full wind history and not just a function of the wind vector at the same time and location.
In the present paper, we extend the previous analyses of HF radar data by independently estimating the Stokes drift by using an accurate wave model. We find that, at our deep-water^^2 northeast
Atlantic site, the quasi-Eulerian current U[E] is on the order of 0.6% of the wind speed with a direction that is, on average, 60° to the right of the wind. We also find that the time-dependent
response of surface current to the wind is typical of a slab layer with a transfer function proportional to 1/(f + ω), where ω is the radian frequency considered. This result is expected to be
representative of the open ocean. Therefore, the estimates of the flux of wind energy to the Ekman layer by Wang and Huang (2004) and others may not be quantitatively correct: they used an angle of
45°, a surface velocity that is 2τ[a]/ρ[w] for steady winds (about 0.2% of the wind speed), and a transfer function proportional to 1/f + ω. A proper analysis of the effects of waves is needed to
properly evaluate energy fluxes.
Our new data and its processing are described in section 2. The analysis of the stratification effect is presented in section 3, with conclusions in section 4.
2. Lagrangian and quasi-Eulerian current from HF radars
a. Radar measurements and processing
High-frequency radars measure, among other things (e.g.,
Ivonin et al. 2004
), the phase velocity
of Bragg waves that have a wavelength equal to one-half of the radar electromagnetic wavelength and propagate in directions away from and toward the radar. This phase velocity is a combination of the
quasi-Eulerian current
Stewart and Joy 1974
Kirby and Chen 1989
), the phase speed of linear waves
, and a nonlinear wave correction (
Weber and Barrick 1977
) that can be interpreted as a filtered surface Stokes drift
. For monostatic systems, the usual radial current velocity in the direction
toward one radar can be expressed as
is the unit vector in direction
. This velocity can be loosely interpreted as the projection in direction
of a current vector
. The reason why this is not exactly true is that
) for all directions cannot be exactly given by the projection of a vector
. In other words,
) is not exactly proportional to cos(
), although it is a reasonable approximation (
Broche et al. 1983
To express
, we first define the Stokes drift vector for waves with frequencies up to
from the directional wave spectrum
) is the magnitude of the wavenumber
, which is equal to (2
for linear waves in deep water, and
is the acceleration of gravity. Starting from the full expression given by
Weber and Barrick (1977)
Broche et al. (1983)
showed that the filtered Stokes drift component that affects the radial current measured by one radar station is well approximated by
is the frequency of the Bragg waves and
is the corresponding wavenumber vector, with a direction
and magnitude
. The full expression, correcting typographic errors in
Broche et al. (1983)
is given in
appendix A
. To simplify the notations, the variable
will now be omitted, but the filtered Stokes drift is always a function of the Bragg wavenumber, thus being different for different radar frequencies.
The depth-varying quasi-Eulerian current
) is defined as the difference of the Lagrangian velocity and Stokes drift (
Jenkins 1987
) and can generally be estimated from the full velocity field using a generalized Lagrangian mean (
Ardhuin et al. 2008b
). The value
estimated from the radar is, according to linear wave theory, the integral of
) weighted by the Bragg wave Stokes drift profile (
Stewart and Joy 1974
Kirby and Chen 1989
). In deep water, this is
Here, we use data from an HF Wellen radar (WERA) system (
Gurgel et al. 1999
), which is manufactured by Helzel GmbH and operated at 12.4 MHz. The Bragg wavelength is 12.1 m, corresponding to a wave frequency of 0.36 Hz in deep water. Thus, half of the weight exp(2
) in Eq.
comes from water depths less than 0.6 m from the moving sea surface, compared to 0.28 m with the 30-MHz radar of
Mao and Heron (2008)
. The relative contributions from deeper layers to
decrease exponentially with depth as exp(2
). Therefore,
can be interpreted as the quasi-Eulerian current in the top 1 m of the ocean.
The radar system has been deployed and operated by Actimar SAS since July 2006 on the west coast of France (Fig. 1), measuring surface currents and sea states every 20 min. The area is characterized
by intense tidal currents, in particular between the largest islands where it exceeds 3 m s^−1 during mean spring tides. Also important, the offshore stratification is largely suppressed by mixing
due to the currents in the areas shallower than 90 m, resulting in complex temperature fronts that are related to the bottom topography (e.g., Mariette and Le Cann 1985).
Each radar station transmits a chirped continuous wave with a repetition frequency of 4 Hz and a 100-kHz bandwidth, which gives a radial resolution of 1.5 km. The receiving antennas are 16-element
linear arrays with a spacing of 10 m, giving a typical angular resolution of 15°. The raw data are processed to remove most of the interference signals (Gurgel and Barbin 2008). Ensemble averaging
over 4 consecutive segments of 512 pulses yields a velocity resolution d[u] = 0.09 m s^−1 in the Doppler spectrum used to estimate each individual radial current measurement. Yet, the current value
is obtained by a weighted sum over a 9-point window applied to the Doppler spectrum. Provided that some inhomogeneity exists in the current field, the width of the Doppler spectrum permits a
measurement resolution that is infinitely small but with an accuracy that is difficult to define, because no other instrument, except maybe for the Coastal Dynamics Experiment (CODE)-type drifter (
Davis 1985), is able to measure surface current in the top 1 m of the ocean. Similarly, satellite altimeters are reported to measure the mean sea level position with an accuracy on the order of 2 cm,
whereas their typical range resolution is close to 40 cm. Prandle (1987) used the coherence of the tidal motions to infer that the accuracy of his 27-MHz radar system was indeed less than the Doppler
resolution when averaged over one hour. We will thus take the accuracy to be equal to the resolution; however, as it appears later in this paper, the only source of concern for our analysis is not so
much the random error but a systematic bias, because we will average a very large number of independent measurements.
Because we investigate the relationship between surface currents and winds based on modeled winds and waves, we will consider only the temporal evolution of the wave field at one point of the radars’
field of view that is representative of the offshore conditions, at a distance of 80–100 km from shore and with a water depth of 120 m. The reason for choosing this location is that we have verified
the wind and wave model results to be most accurate offshore, where they were verified in situ with measurements that only span 6 and 9 months of our radar time series. Other reasons for looking at
offshore conditions are the expected limited effect of the bottom and the expected small horizontal gradients of both tidal currents and other processes; that is, we stay away from the thermal front
that typically follows the 90-m depth contour (Mariette and Le Cann 1985; Le Boyer et al. 2009). The downside of this choice is that the HF-derived current is generally less accurate as the distance
from the coast increases, and the coverage is not permanent, especially during severe storms (e.g., Figure 1). These two drawbacks are limited in practice, as we now discuss.
Interferences and ships cause some data to be rejected in the radar processing or yield bad measurements, and heavy seas or calm seas also reduce the radar working range. To obtain a nearly
continuous time series, we compiled and filtered data from a 0.2° latitude by 0.3° longitude box around that point (A in Fig. 1, with the arrow spacing indicating the resolution of the radar grid).
This compilation was done in two steps. First, based on a visual inspection of the data, at each radar grid point, 0.05% of the total number of data points in the radial velocities time series are
considered spurious and removed. These points are selected as the points where the raw radial current time series differs most from the result of a 5-point median filter. The 0.05% value was selected
as a convenient rule of thumb, which removes most of the visibly spurious points but does not introduce too many unnecessary gaps in the time series. Second, the time series of all the grid points in
the box around A were converted to u and υ components and then averaged.
The Cartesian components of U[R] and U[E] with respect to west–east (u) and south–north (υ) directions are calculated from the two radial components U[R](θ[B1]) and U[R](θ[B2]), each measured by one
radar station, before and after the subtraction of U[Sf] (θ[B]). These Cartesian components suffer from a geometrical dilution of precision (GDOP) varying with position (Chapman et al. 1997; Shay et
al. 2007). The radar beams intersect at point A with an angle r = 34°, and it is possible to estimate the GDOP values for u and υ (i.e., the ratios S[u]/s and S[υ]/s, where S[u], S[υ], and S are the
uncertainties in u, υ, and u[r], respectively). Assuming that S has no bias and is uniformly distributed from −d[u]/2 to +d[u]/2, each radar measurement has intrinsic uncertainties S[u] = 0.04 m s^−1
and S[υ] = 0.11 m s^−1.
This compiled time series, extending from 5 July 2006 to 31 July 2008, is the basis of the following analysis. The 1200-s resolution data were averaged over 3-h blocks centered on round hours. Gaps
shorter than 6 h were linearly interpolated. That time series is 97% complete and thus covers two full years. Other parts of the radar field of view yield similar results, briefly discussed later.
Because of averaging in space and time, each point in the time series is the combination of about 30 range cells and 9 time intervals (i.e., 180 independent velocity measurements when the full radar
range is obtained). Even with an 11 cm s^−1 uncertainty on the original measurement, the expected root-mean-square (rms) errors on the velocity components are thus less than 1 cm s^−1. This analysis
assumes that the instrument is not biased. After verification of the radar antenna lobe patterns using both in situ transmitters and a novel technique based on the analysis of radio interference (to
be described elsewhere), the main lobe of the radar is known to be mispointed by less than 5°, with a −3-dB width less than 15°. The largest source of uncertainty is thus the interpretation of the
phase speed and the numerical estimation of the Stokes drift, as discussed later.
Because we wish to focus on the random wind-driven currents, we also performed a tidal analysis using the T-Tide software (Pawlowicz et al. 2002) applied to each velocity component. This analysis on
the full time series (before time averaging) allows the removal of the deterministic diurnal constituents K[1], O[1], P[1], and Q[1] that have amplitudes of 1.5–0.3 cm s^−1, with estimated errors of
0.1 cm s^−1. Because this only corrects for 95% of the apparent variance in the M[2] and S[2] semidiurnal tides, these will be further filtered by using a time filter.
b. Numerical wave model and estimations of Stokes drift
1) General principles
As expressed by Eq. (5), the estimation of U[Sf] (θ[B]) requires the measurement or modeling of the wave spectrum E(f, θ). In situ buoys were moored for restricted periods at several locations for
the investigation of offshore to coastal wave transformation (Ardhuin 2006) and to provide complementary data for radar validation. The radar also measures the sea state, but the coverage is often
limited and its accuracy for a 20-min record is typically only on the order of 25% for the significant wave height H[s]. Thus, to use the full current time series at the offshore location (point A),
we have to estimate the sea state by using a numerical wave model.
We use an implementation of the WAVEWATCH III (WWIII) code, in its version 3.14 (Tolman 2007, 2008), with minor modifications of the parameterizations (see appendix B) and the addition of advection
schemes on unstructured grids (Roland 2009).
The model setting consists of a two-way nested pair of grids, covering the global ocean at 0.5° resolution and the Bay of Biscay and English Channel at a resolution of 0.1°. A further zoom over the
measurement area is done using an unstructured grid with 8429 wet points (Fig. 1). The model setting is fully described in appendix B.
In practice,
is dominated by the first term
), in Eq.
. Examining a large number of spectral data (6 buoys for 2 yr spanning a range of wave climates; see
appendix C
), we realized that
) is essentially a function of the wind speed
and the wave height
. Although
explains typically only 50% of the variance of
) with 0.3 <
< 0.5,
generally explain over 85% of the variance. This behavior of
) is similar to that of the fourth spectral moment, which is related to the surface mean square slope (
Gourrion et al. 2002
Vandemark et al. 2004
). The reason for this correlation is that the wind speed is obviously related to the high-frequency part of the wave spectrum, which determines most of the Stokes drift, whereas
is a surrogate variable for both the presence of swell and the stage of development of the wind sea. Here, we find
The relationship given by Eq.
appears to be very robust, with a 2.6 cm
rms difference compared to global hindcast values of
(∞), which is a 16.9% difference. Nevertheless, when compared to buoy data, an accurate wave model generally provides a better fit to the observations (
appendix C
). We thus have used our hindcasts using WAVEWATCH III to provide an estimate for
2) Uncertainty on U[sf] around point A
We have no wave measurement at point A and no permanent spectral measurement in the area. A detailed validation of U[ss] was thus performed for the coastal buoys 62069 (Fig. 1) and 62064 (off Cap
Ferret, 600 km to the southeast of point A), the U.S. Northwest Pacific coast (appendix C), the U.S. East Coast, the Gulf of Mexico, and California.
We further use wave information at buoy 62163, located 150 km west of point A, which is representative of the offshore conditions found at point A and a combination of satellite altimeter data. The
present model estimates of H[s] are more accurate at buoy 62163, located 150 km west of point A, than at Pacific buoy locations. Further, the model estimate of the fourth moment m[4] of the wave
spectrum is better correlated in the Bay of Biscay to radar altimeter C-band cross sections compared to other regions of the World Ocean (appendix C). We thus expect the model estimate of U[ss](f[B]
= 0.36 Hz) to have a bias smaller than 5%, with a random error less than 20% (see appendix C). As a result, we chose to use this numerical wave model for the estimation of U[ss] and U[Sf]. We can
thus propose an error budget for our estimate of the wind-driven quasi-Eulerian current in which the measurement error is dominated by U[Sf] with a bias of 5% at most and a standard deviation less
than 20% overall. Using the analysis of 2 yr of model results, this standard deviation at the Pacific buoy 46005 is 24% for wind speeds of 3 m s^−1, 20% for wind speeds of 5 m s^−1, 16% for wind
speeds of 7 m s^−1, and 11% for wind speeds of 11 m s^−1. Given the general accuracy of the wave model in the northeast Atlantic, we expect similar results here.
We thus estimate that the root-mean-square error of the modeled quasi-Eulerian current U[E] at 3-h intervals is on the order of 0.2% of U[10]. On this time scale, it is difficult to rule out
contributions from horizontal pressure gradients in the momentum balance, and this current may not be purely wind driven.
The averaged current (e.g., for a given class of wind speed, as shown in Fig. 7) has a relative accuracy better than 0.1% of U[10]. In situ measurements of time-averaged velocities from 10 to 70 m
above the bottom at 48°6′N, 5°23′W (south of point A; see Fig. 1) using an RD Instruments Workhorse acoustic Doppler current profiler (ADCP) deployed from June to September 2007 (Le Boyer et al. 2009
) give tide-filtered currents less than 2 cm s^−1 or 0.25% of the wind speed when averaged following the wind direction (the instantaneous measurements are rotated before averaging) and less than
0.1% when winds are stronger than 10 m s^−1. This is typically less than 20% of U[Sf]. Assuming that wind-correlated baroclinic currents are negligible during the ADCP measurement campaign, the
wind-correlated geostrophic current is expected to be less than 0.2% of U[10]. Generalizing this result to the entire radar time series, the averaged values of U[E] can be interpreted as a
wind-driven current with an accuracy to within 0.3% of U[10].
3. Analysis of wind-driven flows
The study area is dominated by moderate 6–12 m s^−1 winds, from a wide range of directions, with slightly dominant southwesterly and northeasterly sectors (Fig. 2).
a. Rotary spectral analysis
The rotary spectral analysis gives both the frequency distribution of the signal, and an indication of its circular polarization (Gonella 1971). The positive frequencies correspond to
counterclockwise (CCW) motions, and the negative frequencies correspond to clockwise (CW) motions, the usual polarization of inertial motions in the Northern Hemisphere.
The instantaneous measurements of the radar are dominated by tidal currents, and the variance of motions with frequencies less than 1.75 counts per day (cpd) only accounts for 8% of the total
variance (Fig. 3). These low frequency motions include the diurnal tidal constituents, most importantly K[1] and O[1], but these only account for 0.1% of the variance. The low frequency motions are
generally dominated by near-inertial motions, which are polarized clockwise with frequencies close to the inertial frequency f[I] = 1.3 cpd (see Fig. 3).
b. Cospectral analysis
Here, we investigate the relationship between measured currents (processed as described earlier) and winds taken from 6-hourly wind analyses from the European Centre for Medium-Range Weather
Forecasts (ECMWF). These analyses were verified to give excellent correlation (r ≃ 0.92) with the Beatrice buoy (WMO code 62052), which unfortunately malfunctionned during large periods of time. The
wind and current data are thus completely independent. The wave model was forced by these same winds; thus, the high level of coherence between the predicted Stokes drift and the wind (Fig. 4) is not
To isolate the wind-correlated dynamics from the shorter (tide) and longer (general circulation) time scales, we first perform a cospectral analysis of the measured currents with the wind, following
the method of Gonella (1971). To keep as many data as possible between data gaps, the Fourier transforms are taken over 264 h, which corresponds to 21 M[2] tidal cycles. The measured currents are
significantly coherent with the wind vector over the range −1.75 to 1.75 cpd (Fig. 4). This coherence is generally reduced when the Stokes component U[Sf] is subtracted from the radar measurements.
The radar-measured current vectors U[R] have stable directions relative to the wind, 20°–40° to the right for f > −f[I], given by their coherence phase (Fig. 4). The coherence phase of the Stokes
drift increases with frequency. This pattern is typical of a time lag that can be estimated to about 1.5 h, which is consistent with the relatively slow response of the wave field compared to the
current. This is rather short compared to the time scale of wave development, but one should bear in mind that the Stokes drift is mostly due to short waves that respond faster to the wind forcing
than the dominant waves. Because the wind preferentially turns clockwise, the Stokes drift is slightly to the left of the wind. The asymmetry in the phase of U[Sf] for clockwise and counterclockwise
motions may be related to varying fetch when the wind turns.
As expected from the theory by Gonella (1972), the phase of the quasi-Eulerian current U[E] jumps by about 180° at the inertial frequency −f[I]. In the frequency range from −1.2 to 0.2 cpd, which
contains 40% of the nontidal signal, U[E] is at an angle between 45° and 60° to the right of the wind. This conclusion is not much altered when one correlates the Eulerian current against the wind
stress, which for simplicity is estimated here with a constant drag coefficient τ = ρ[a] 1.3 × 10^−3 U[10]U[10], where ρ[a] is the air density. One may argue that the theoretical filtering of the
Stokes drift is not well validated. A lower bound on the estimate of U[Sf] can be given by removing the contribution from waves shorter than the Bragg waves. This has very little impact on the
estimation of U[E].
The observed coherence phases of U[E] and U[10] are similar to the values given by Gonella (1972, Fig. 6), which are based on the constant eddy-viscosity model of Ekman (1905), but for the current
considered at a depth as large as 25% of the Ekman depth. Because the radar measurements are representative of the upper 1 m and the Ekman depth is generally on the order of 30 m, it follows that the
classical Ekman theory, with a constant eddy viscosity, does not apply here. Instead, this large near-surface deflection is consistent with model results obtained with a high surface mixing—such as
those induced by Langmuir circulations (McWilliams et al. 1997; Kantha and Clayson 2004), breaking waves (Craig and Banner 1994; Mellor and Blumberg 2004; Rascle et al. 2006), or both—and consistent
with the few observed near-surface velocity profiles (Santala and Terray 1992).
c. Effects of stratification
Following the theory of Gonella (1972) and the previous observations by Price and Sundermeyer (1999), it is expected that the stratification has a significant effect on the surface currents. Here, we
used sea surface temperature time series to diagnose the presence of a stratification. Because of the strong vertical mixing year round at the site of buoy 62069, the horizontal temperature
difference between point A and point 62069 is a good indicator of the vertical stratification at point A. This temperature difference reaches up to 2°C and was present in 2006, 2007, and 2008 from
early July to late October, as revealed by satellite SST data. We thus separated the data records used for the spectral analysis into “stratified” and “homogeneous” records based on the date of the
midpoint in these time series.
These two series show a significant difference (at the 95% confidence level) when the spectra are smoothed over 0.3-cpd bands, with a 2 times larger response in the cases expected to be stratified
(dashed lines, Fig. 5) for frequencies in the range of −1.7 to 1.5 cpd. Additionally, the current variance in the frequency band −1.7 < f < 1.3 cpd exhibits a pronounced annual cycle, with a maximum
in July or August at 6–7 times the January minimum, despite weaker winds (not shown). Interestingly, the transfer functions decrease like 1/(f + ω) from a peak at the inertial frequency f, where ω
is the radian frequency. This decrease is typical of slab-like behaviors that are expected in mixed layers with a much larger surface mixing (e.g., Rascle et al. 2006) than typically used with Ekman
theory or a mixed-layer depth much shallower than the Ekman depth (Gonella 1972). Ekman theory in unstratified conditions, which should apply to our winter and spring measurements, would give a much
slower decrease that is proportional to 1/(f + ω) (Gonella 1972). Together with this stronger amplitude of the current response in stratified conditions, we find a larger deflection angle in the −0.8
to −0.2-cpd frequency range. This pattern of larger currents and larger deflection angles in stratified conditions is consistent with the observations of Price and Sundermeyer (1999) and the
numerical model results by Rascle and Ardhuin (2009).
d. Relationship between tide-filtered currents and winds
A proper model for the wind-induced current may be given by the relationship between the wind speed and wave height, giving the Stokes drift and the complex transfer function (transfer function and
phase) from the wind stress spectrum to the Eulerian current spectrum, following Gonella (1971) or Millot and Crépon (1981). Such a model is beyond the scope of the present paper.
Simpler models that would give the current speed and direction as a function of the instantaneous wind vector are even less accurate. Because the transfer function is very peaked at the inertial
frequency, the current speed may vary widely for a given wind speed. Yet, for practical reasons, there is a long tradition of directly comparing current and wind magnitudes and directions for search
and rescue operations and ocean engineering applications. Because of the inertial oscillations, there is usually a large scatter in the correlation of the current and wind speed vectors. To compare
with previous analyses (e.g., Mao and Heron 2008), we thus perform such a comparison after filtering out the dominant tidal current by taking the inverse Fourier transform of the current, wind, and
Stokes drift spectra in which the amplitudes of components with frequencies higher than 1.75 cpd as well as the zero frequency are set to zero. Again, the Fourier transforms are taken over 264 h.
We find that the surface Eulerian U[E] current lies 40°–60° to the right of the wind, suggesting that the near-inertial motions only add scatter to the longer period motions (|f| < 1.3 cpd) that
were found to have similar deflection angles. Interestingly, the typical magnitude of U[E] decreases from about 0.8% of U[10] at low wind to nearly 0.4% for high winds. This reduction in the relative
magnitude of U[E] is accompanied by a reduction of the deflection angle from 65° on average for U[10] = 3 m s^−1 to 40° for U[10] = 15 m s^−1. On the contrary, the Stokes drift typically increases
quadratically with the wind speed. These observations contradict the usual theoretical statements of Kirwan et al. (1979) and Mao and Heron (2008); they concluded that the Stokes drift should be
linear and the Eulerian current should be quadratic in terms of wind speed. The fact that the Stokes drift is quadratic as a function of the wind speed is shown by the fitted Eq. (7) (as well as
observed wave spectra in Fig. C1). The error in Mao and Heron (2008) is likely due to their erroneous assumption that the Stokes drift is dominated by waves at the peak of the spectrum. In the
analysis of Kirwan et al. (1979) and Rascle et al. (2006), the error essentially arises from the assumed shape of the wave spectrum.
The less-than-linear dependence of U[E] on U[10] contradicts the usual simple Ekman model for the quasi-Eulerian current, which would predict a current proportional to the wind stress and thus
varying as the square or cube of the wind speed. This difference is likely due to the enhanced mixing caused by breaking waves, which tends to mix the momentum over a scale on the order of the
wind–sea wave height (i.e., increasing with the wind speed; Terray et al. 1996; Rascle et al. 2006). Numerical models without stratification but with a realistic mixing tend to give a quasi-Eulerian
current that increases with wind speed and with the inverse wave age. Here, the stronger winds do not correspond to very different wave ages, and it is likely that a correlation of deeper mixed
layers with stronger winds is the cause of the reduction of U[E] with increasing wind speed (Rascle and Ardhuin 2009). As a result, the nonlinear current response to the wind stress will likely limit
the accuracy of models based on transfer functions.
e. Effects of fetch or wave development
The same analysis was also repeated for other points in the radar field of view; for example, at point B (Fig. 1), the radar data quality is generally better, but the wave model may have a bias of
about 10% on U[ss] and the ECMWF wind field may be less accurate. Point B is relatively sheltered from southerly and northwesterly waves, and the fetch from the east is 40 km at most. If we assume
that the winds are accurate at that site too, we find that the radar-derived current is weaker relative to the wind, with U[R]/U[10] typically smaller by 0.2% (a ∼15% reduction) compared to point A.
This appears to be due to a reduction in U[Sf], which is only partially compensated for by a small increase in U[E]. This difference between points A and B nearly vanishes when only westerly wind
situations are considered (defined by winds within 60° from the westerly direction).
4. Conclusions
Using a 2-yr time series of HF radar data and a novel numerical wave model that is shown to reproduce the observed variability of the surface Stokes drift with wind speed and wave height, we have
analyzed the wind-driven surface current. When tidal currents are filtered out, theory predicts that the measured velocities are a superposition of a filtered Stokes drift U[Sf] and a quasi-Eulerian
current U[E]. With our 12-MHz radar, U[Sf] is estimated to be on the order of 0.5%–1.3% of the wind speed, with a percentage that increases linearly with wind speed. These values are a function of
the radar wavelengths and would be larger, by up to 20%, with higher-frequency radars that give currents representative of a shallower surface layer. The other component U[E] is found to be on the
order of 0.6% of the wind speed and lies in our Northern Hemisphere at an average 40°–70° to the right of the wind, with a large scatter because of inertial oscillations that may be well modeled by
using a Laplace transform of the wind stress (Broche et al. 1983). This large deflection angle is robustly given by the coherence phase for clockwise motions in the frequency range from 0 to the
inertial frequency.
When instantaneous currents are compared to the wind, the magnitude of U[E] appears to decrease with wind speed, but it increases when a stronger stratification is expected (Fig. 6). These surface
observations correspond to currents in the depth range 0–1.6 m and confirm previous analysis of deeper subsurface mooring data. If wind-correlated geostrophic currents are negligible in our
measurements, the shape of the classical picture of the Ekman spiral is not correct and the surface layer is much more slab-like than assumed in many analyses, probably because of the large
wave-induced mixing at the surface (Agrawal et al. 1992). These findings are summarized in Fig. 7.
If we neglect the wind-correlated geostrophic currents, which we deem reasonable, and interpret U[E] as being purely wind-driven, our observations of U[E]/U[10] at point A are expected to be
representative of the open ocean, whereas in coastal areas and small basins, a less developed sea state will lead to a smaller U[Sf] and a larger U[E], as we observe at point B. Such a generic
relationship of U[E] and U[10] is very important for a proper estimation of the energy flux to the mixed layer. Besides, on top of the wind stress work on the Ekman current, this energy flux should
be dominated by the dissipation of wave energy induced by breaking (e.g., Rascle et al. 2008). Also, there is the depth-integrated Stokes–Coriolis force that is equal to the product of the
depth-integrated Stokes transport M^w = ρ[w] ∫ U[s](z) dz and the Coriolis parameter f. This force is smaller than the depth-integrated Coriolis force by about a factor of 3 on average (Rascle et al.
2008), but that may give a comparable work because of the smaller angle between that force and the quasi-Eulerian current û(z). The accurate estimation of the surface Stokes drift using a numerical
wave model also opens the way for a more accurate interpretation of space-borne measurements of surface currents using Doppler methods, which are contaminated by a Stokes-like component amplified 10
times or more (Chapron et al. 2005).
The efforts of Vincent Mariette and Nicolas Thomas are essential to maintain the radars in proper operating conditions. Funding for the radar purchase and maintenance was provided by DGA under the
MOUTON project, and funding for the wave model development was provided under the ECORS project. Florent Birrien performed the integration of Aaron Roland’s routines into the WAVEWATCH III framework.
Wind and wave data were kindly provided by ECMWF, Météo-France, and the French Centre d’Etudes Techniques Maritimes Et Fluviales (CETMEF), and the sea surface temperature data used to diagnose the
presence of a stratified layer were taken from the ODYSSEA Level 4 global analysis product, produced as part of the MERSEA Integrated Project. The SHOM buoy deployments were managed by David Corman
with precious help from Guy Amis.
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Nonlinear Correction for the Wave Dispersion Relation in a Random Sea State
Based on the lowest order approximate theory of
Weber and Barrick (1977)
for deep-water waves with
≃ 2
, the nonlinear correction to the phase speed of components with wavenumber
and direction
can be expressed as an integral over the wave spectrum. Defining
Broche et al. [1983
, their Eq. (A2)] give the following expression:
where, correcting for typographic errors and using
= cos
These expressions give the correct figures in
Broche et al. (1983)
. For
< 1,
, 0) = 4
; for
> 1,
, 0) = 4
Longuet-Higgins and Phillips 1962
Huang and Tung 1976
Barrick and Weber 1977
). As commented by
Broche et al. (1983)
) ≃
, 0) cos
, with the largest errors occurring for
= 1 where
) >
, 0) cos
for |
| <
/3, which in our case makes
larger than the approximation given by Eq.
by 2%–5%.
Parameterization and Numerical Settings for the Wave Models
The implementation of the WAVEWATCH III model used here was run with source functions S[in], S[nl], and S[ds] parameterizing the wind input; nonlinear 4-wave interactions; and whitecapping
dissipation. An extra additional dissipation term S[db] is also included to enhance the dissipation resulting from wave breaking in shallow water, based on Battjes and Janssen (1978).
The parameterization for S[nl] is taken from Hasselmann et al. (1985), with a minor reduction of the coupling coefficient from 2.78 × 10^7 to 2.5 × 10^7. The parameterizations for S[in] and S[ds] are
very similar to the ones used by Ardhuin et al. (2008a), with modifications to further improve the high-frequency part of the spectrum (Filipot et al. 2008); that is, the whitecapping dissipation is
based on recent observations of wave breaking statistics (Banner et al. 2000) and swell dissipation (Ardhuin et al. 2009). These model settings give the best estimates so far of wave heights, peaks,
and mean periods but also of parameters related to the high-frequency tail of the spectrum (appendix C). The present model results are thus a significant improvement over the results of Bidlot et al.
(2007) and Rascle et al. (2008). The physical and practical motivations for the parameterizations will be fully described elsewhere, and we only give here a description of their implementation. We
only note for the interested users that the parameter settings given here tend to produce larger negative biases on H[s] for H[s] > 8 m than the parameterization by Bidlot et al. (2007). Better
settings for H[s] in extreme waves would be s[u] = 0 and c[3] = 0.5 (see below), but this tends to give too large values of U[ss], which is why we do not use these settings here.
The parameterization of
is taken from
Janssen (1991)
as modified by
Bidlot et al. (2007)
, with some further modifications for the high frequencies and the addition of a wind output term
(or “negative wind input”) based on the observations by
Ardhuin et al. (2009)
. The source term is thus
is a (constant) nondimensional growth parameter;
is the von Kármán constant;
is the friction velocity in the air;
is the phase speed of the waves,
is the intrinsic frequency, which is equal to 2
in the absence of currents; and
) is the frequency-directional spectrum of the surface elevation variance. In the present implementation, the air–water density ratio is constant. We define
= log(
), where
is given by
Janssen [1991
, their Eq. (16)] and corrected for intermediate water depths, so that
is a roughness length modified by the wave-supported stress
is a wave age tuning parameter. The effective roughness
is implicitly defined by
is the wind stress magnitude,
is the wave-supported fraction of the wind stress,
is the wind at 10-m height, and
is the acceleration of gravity.
The maximum value of
was added to reduce the unrealistic stresses at high winds that are otherwise given by the standard parameterization. This is equivalent to setting a maximum wind drag coefficient of 2.8 × 10
. This and the use of an effective friction velocity
) instead of
are the only changes to the general form of
Janssen’s (1991)
wind input. That friction velocity is defined by
Here the empirical factor
= 1.0 adjusts the sheltering effect of short waves by long waves adapted from
Chen and Belcher (2000)
and helps to reduce the input at high frequency, without which a balance of source terms would not be possible (except with a very high dissipation as in
Bidlot et al. 2007
). This sheltering is also applied in the precomputed tables that give the wind stress as a function of
Bidlot et al. 2007
The wind output term is identical to the one used by
Ardhuin et al. (2008a)
, based on the satellite observations of
Ardhuin et al. (2009)
with an adjustment to Pacific buoy data. Namely, defining the Reynolds number Re = 4
, where
are the significant surface orbital velocity and displacement amplitudes, respectively, and
is the air viscosity, we take, for Re < 10
is the friction factor given by
Grant and Madsen’s (1979)
theory for rough oscillatory boundary layers without a mean flow, using a roughness length adjusted to 0.04 times the roughness for the wind. This gives a stronger dissipation for swells opposed to
The dissipation term is the sum of the saturation-based term of
Ardhuin et al. (2008a)
and the cumulative breaking term
Filipot et al. (2008)
. It thus takes the form
= 0.0009 is a threshold for the onset of breaking consistent with the observations of
Banner et al. (2000)
Banner et al. (2002)
, as discussed by
Babanin and van der Westhuysen (2008)
, when including the normalization by the width of the directional spectrum [here replaced by the cos
factor in Eq.
]. The dissipation constant
was adjusted to 2.2 × 10
in order to reproduce the directional fetch-limited data described by
Ardhuin et al. (2007)
The cumulative breaking term represents the smoothing of the surface by big breakers with celerity C′ that wipes out smaller waves of phase speed C (Babanin and Young 2005). Because of uncertainties
in the estimation of this effect from observations, we use the theoretical model of Filipot et al. (2008). Briefly, the relative velocity of the crests is the norm of the vector difference, Δ[C] = |C
− C′|, and the dissipation rate of short wave is simply the rate of passage of the large breaker over short waves [i.e., the integral of Δ[C]Λ(C) dC, where Λ(C) dC is the length of breaking crests
per unit surface that have velocity components between C[x] and C[x] + dC[x] and between C[y] and C[y] + dC[y]; Phillips 1985]. Because there is no consensus on the form of Λ (Gemmrich et al. 2008),
we prefer to link Λ to breaking probabilities. Based on Banner et al. (2000, their Fig. 6) and taking their saturation parameter ϵ to be on the order of 1.6B, the breaking probability of dominant
waves is approximately P = 28.4(max{B − B[r], 0})^2. In this expression, a division by 2 was included to account for the fact that their breaking probabilities were defined for waves detected by
using a zero-crossing analysis, which underestimates the number of dominant waves, because at any given time only one wave is present and thus low waves of the dominant scale are not counted when
shorter but higher waves are present.
Extrapolating this result to higher frequencies and assuming that the spectral density of crest length per unit surface
) in the wavenumber spectral space is
) = 1/(2
), we define a spectral density of breaking crest length, Λ(
) =
), giving the source term,
The tuning coefficient c[3], which was expected to be on the order of 1, was adjusted here to 0.4. The resulting model results appear to be very accurate for sea states with significant wave heights
up to 8 m. Larger wave heights are underestimated. Other parameter adjustments can correct for this defect (e.g., reducing s[u] and increasing c[3]), but then the Stokes drift may not be so well
reproduced, especially for the average conditions discussed here. These different possible adjustments and their effects will be discussed elsewhere.
Numerical schemes and model settings
Spatial advection in the finer model grid is performed using the explicit contour integration based residual distribution–narrow stencil (CRD-N) scheme (Csík et al. 2002) that was applied to the wave
action equation by Roland (2009) and provided as a module for the WWIII model. The scheme is first order in time and space, conservative, and monotone.
All model grids are forced by 6-hourly wind analysis at 0.5° resolution provided by ECMWF. The model spectral grid has 24 regularly spaced directions and extends from 0.037 to f[max] = 0.72 Hz with
32 frequencies exponentially spaced. The model thus covers the full range of frequencies that contribute most to the filtered Stokes drift U[Sf]. The usual high-frequency tail proportional to f^−5
is only imposed for frequencies larger than the diagnostic frequency f[d] = Ff[m,0,−1], with the mean frequency defined by f[m,0,−1] = [∫ E(f)/f df/∫ E(f) df]^−1. Here, we take a factor F = 10,
instead of the usual value of 2.5 (Bidlot et al. 2007), so that f[d] is almost always larger than the model maximum frequency of 0.72 Hz. Besides, the time step for integration of the source function
is adaptatively refined from 150 s for the local model down to 10 s if needed, so that virtually no limiter constrains the wave field evolution (Tolman 2002).
Model Accuracy for Relevant Parameters
To define the errors on the estimations of U[Sf] used to determine the quasi-Eulerian velocity U[E] from the radar measurement, it is necessary to examine the quality of the wind forcing and model
results in the area of interest, as summarized in Table C1. The only two parameters that are measured continuously offshore of the area of interest are the wave height H[s] and mean period f[02],
recorded at buoy 62163, which is 150 km to the west of point A. The values of H[s] and f[02] can be combined to give the second moment of the wave spectrum m[2] = (0.25H[s]f[02])^2 (Fig. C1).
Because there is no reliable wave measurement with spectral information in deep water off the northeast French Atlantic coast, we also use buoy data and model results in a relatively similar wave
environment, at the location of buoy 46005, which is 650 km off Aberdeen, Washington, on the U.S. Pacific coast. Because this buoy is not directional we first examine the third moment of the wave
If waves were all in the same direction,
would be proportional to the Stokes drift
) of waves with frequency up to
, as given by Eq.
. We thus define a nondirectional Stokes drift as
Looking at buoy data, we found that
is in hertz,
is in meters per second, and
is in meters.
Taking directionality into account, Eq. (4) yields U[ss](f[c]) ≃ 0.85U[ssnd](f[c]) for typical wave spectra and the relationship (C3) becomes Eq. (7). For buoy 46005, which is a 6-m Navy
Oceanographic Meteorological Automatic Device (NOMAD) buoy, and f[c] in the range 0.3–0.5 Hz, this relationship gives an rms error less than 1.0 cm s^−1, which corresponds to less than 15% of the rms
value estimated using Eq. (C2). This is smaller than the error of estimates using previous wave models (24% with the parameterization by Bidlot et al. 2007) but comparable to the 14.2% error obtained
with the present model. The same analysis was performed, with similar results, for very different sea states recorded by National Data Buoy Center (NDBC) buoys 51001 (northeast of Hawaii), 41002
(U.S. East Coast), 46047 (Tanner Banks, California), and 42036 (Gulf of Mexico).
Another source of continous wave measurements is provided by altimeter-derived
, which we correct for bias following
Queffeulou (2004)
, and fourth spectral moment
. The latter is approximately given by (
Vandemark et al. 2004
is the normalized radar cross section, corrected for a 1.2-dB bias on the C-band altimeter of
in order to fit airborne observations (
Hauser et al. 2008
). The model estimation of
(0.72 Hz) is extrapolated to C-band by the addition of a constant 0.011
, which is consistent with the saturation of the short wave slopes observed by
Vandemark et al. (2004)
. For this parameter, the model is found to be very accurate, especially around the region of interest, more accurate than on the U.S. Pacific coast.
These indirect validations suggest that the third spectral moment including waves up to the Bragg frequency f[B] = 0.36 Hz, which is proportional to U[ssnd], is probably estimated with bias between
−5% and 10% and an rms error less than 20%. The bias on the significant wave height appears to increase from offshore (altimeter and buoy 62163 data) to the coast (buoys Iroise and 62069), and we
attribute this effect to the tidal currents (not included in the present wave model) and coastal modifications of the winds that are not well reproduced at this 10–20-km scale by the ECMWF model.
Because the chosen area of interest lies offshore of the area where currents are strongest (Fig. 1), we shall assume that, at this site, the model bias on U[ss](f[B]) is zero, which appears most
likely. Extreme biases of ±10% only result in deflections of 5° on the diagnosed quasi-Eulerian current U[E].
Fig. 1.
Map of the area showing significant wave height at 1200 UTC 1 Jan 2008 estimated with a numerical wave model (see appendix B) and the instantaneous surface current measured by the HF radars installed
at Porspoder and Cléden-Cap-Sizun, France. In situ measurement stations include the weather buoy Beatrice (number 62052); the Pierre Noires (62069) directional Datawell waverider buoy (installed from
November 2005 to March 2006 and back again since January 2008); and a previous waverider deployment (Iroise), which is more representative of the offshore wave conditions. The large black square
around point A is the area over which the radar data have been compiled to provide the time series analyzed here, which is representative of offshore conditions. When the radar functioned,
measurements are available over the entire square for more than 80% of the 20-min records, a number than rises to 99% for the area east of 5°35′W. The partial radar coverage around point A is typical
of high sea states with H[s] > 6 m offshore, which are rare events.
Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4169.1
Fig. 2.
Wind rose for the years 2006–08 at point A, based on ECMWF analyses. The observations at Beatrice buoy give a similar result. For each direction, the cumulative frequency is indicated with wind
speeds increasing from the center to the outside, with a maximum of 4.3% maximum from west-southwest (heading 250°). An isotropic distribution would have a maximum of 2.7%.
Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4169.1
Fig. 3.
Rotary power spectra of the current measured by the radar, and the contribution U[Sf] to the surface Stokes drift estimated via Eq. (A1). CW motions are shown with dashed lines and CCW motions are
shown with solid lines. The spectra were estimated using half-overlapping segments 264 h long over the parts of the time series with no gaps. The number of degrees of freedom is taken to be the
number of nonoverlapping segments (e.g., 59 at the spectral resolution of 0.09 cpd, giving a relative error of 35% at the 95% confidence level). (bottom) The tidal components have been filtered out,
which clearly removes the diurnal peak. However, the semidiurnal tides are only reduced by a factor of 25, which is not enough compared to the magnitude of the near-inertial motions and requires the
use of an additional filter. This tide-filtered time series is used in all of the following analyses.
Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4169.1
Fig. 4.
(a) Magnitude and (b) phase of rotary cospectra of the wind and wind stress with the radar-derived current, Stokes drift, and Eulerian current. The number of degrees of freedom is 108 at the spectral
resolution of 0.09 cpd. Coherence is significant at the 95% confidence level for a value of 0.1. Negative and positive frequencies are CW and counterclockwise polarized motions, respectively.
Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4169.1
Fig. 5.
Amplitude (top) transfer functions and (bottom) coherence phases between the wind forcing and the current response. The dashed lines correspond to records where a stratification is expected to be
important (18 out of 108), and the solid lines correspond to the other records. Confidence intervals for the two groups of records are shown for the native spectral resolution of 0.09 cpd. To be at a
comparable level, the wind stress was multiplied by 50 before estimating the transfer function. The two peaks of the transfer functions at ±2 cpd are due to the tidal currents but do not correspond
to a causal relationship between the wind forcing and the current response.
Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4169.1
Fig. 6.
Observed tide-filtered quasi-Eulerian velocity magnitudes normalized by the wind speed and directions relative to the wind vector. The linear increase of U[Sf]/U[10] with U[10] is consistent with the
quadratic dependence of U[Sf] on U[10] given by Eq. (7). The full dataset was binned according to wind speed. Dashed–dotted lines correspond to stratified conditions only and dotted lines correspond
to homogeneous conditions. (bottom) The number of data records in each of these cases. The dashed line show results when U[Sf] is replaced by U[ss](f[B]). Error bars show only 1/2 of the standard
deviation for all conditions combined, in order to make the plots readable. All time series (wind, current, U[Sf], and U[ss]) were filtered in the same manner for consistency (except for the initial
detiding applied only to the current data). The error bars do not represent measurement errors but rather the geophysical variability due to inertial motions.
Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4169.1
Fig. 7.
Mean wind-correlated current vectors in low and high wind conditions, with and without stratification, measured off the west coast of France with the 12.4-MHz HF radar, based on the results shown in
Fig. 6. Here, U[R] is the radar-measured vector, which can be interpreted as a sum of a quasi-Eulerian current U[E], representative of the upper 2 m and a filtered surface Stokes drift U[Sf]. The
full surface Stokes drift is typically 40% larger than this filtered value. Solid circles give the expected error on the mean current components resulting from biases in the wave contribution to the
radar measurement. The dashed circles show the expected error on the interpretation of U[E] as a wind-driven current based on the ADCP measurements at depths of 60–120 m, assuming that the baroclinic
part of the geostrophic current is negligible.
Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4169.1
Fig. C1. Variation of the wave spectrum third moment m[3] converted to a velocity U[ssnd] = (2π)^3m[3](f[c])/g that would equal the surface Stokes drift in deep water if all waves propagated in the
same direction. For each data source, a cut-off frequency of f[c] = f[B] = 0.36 Hz is taken and the data are binned wind speed at 1 m s^−1 intervals and significant wave height H[s] (in colors) at
1-m intervals from 1 to 11 m. (top) Buoy data offshore of Oregon (NDBC buoy 46005); (middle) present model results; and (bottom) results from the same model but using the parameterization of Bidlot
et al. (2007), including a factor F = 2.5. The vertical error bars indicate ±1/2 the standard deviation of the data values in each (U[10], H[s]) class.
Citation: Journal of Physical Oceanography 39, 11; 10.1175/2009JPO4169.1
Table C1. Model accuracy for measured wave parameters in various regions of the World Ocean. Buoy validations span the entire year 2007, except for buoy 62069, for which data cover 25 Jan–20 Aug
2008; buoy Iroise, which covers 13 Apr–20 May 2004; and Jason-1, for which data correspond to January–July 2007 for the global validation (JAS-Glo: 393 382 data points) and the full year for a box 3°
× 4° centered on 48°30′N, 8°W or 45°N, 128°W. (JAS-Gas or JAS-Was: 380 data points). Unless otherwise specified by the number in parenthesis, the cut-off frequency is 0.5 Hz, C stands for C band, and
f[B] = 0.36 Hz corresponds to our 12-MHz HF radar. The normalized bias (NB) is defined as the bias divided by the rms observed value, whereas the scatter index (SI) is defined as the rms difference
between modeled and observed values, after correction for the bias and normalized by the rms observed value, and r is Pearson’s correlation coefficient. Only altimeter data are available at point A,
but the uniform error pattern and the model consistency suggest that errors at A should be similar to offshore buoy errors such as those found at buoy 62163 offshore of A or at the U.S. West coast
buoy 46005. Errors at point B, not discussed here, are expected to be closer to those at the nearshore buoys 62069 and Iroise.
Because the term M^w in the momentum balance (1) drives a component of mean transport that opposes M^w, there is no net wave-induced transport, except in nonstationary or nonhomogenous conditions (
Hasselmann 1970; Xu and Bowen 1994).
This means deeper than both the Stokes depth D[S] and the expected Ekman depth D[E]. | {"url":"https://journals.ametsoc.org/view/journals/phoc/39/11/2009jpo4169.1.xml","timestamp":"2024-11-07T13:56:58Z","content_type":"text/html","content_length":"1049715","record_id":"<urn:uuid:78507a14-2daa-428d-b3aa-301718a411e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00729.warc.gz"} |
Measuring and Calculating Thermochemical Values
Author: admintanbourit
Thermochemistry is a branch of chemistry that deals with the study of the energy changes involved in chemical reactions and the properties of substances. Understanding thermochemical values is
crucial in numerous fields, including analytical chemistry, physical chemistry, and biochemistry, as it helps us understand and predict the behavior of substances under different conditions.
Measuring and calculating thermochemical values is a fundamental process in thermochemistry, and it involves several techniques and principles. In this article, we will discuss the various methods
used for determining these values and the calculations involved.
One of the primary methods for measuring thermochemical values is through calorimetry. This technique involves measuring the heat changes associated with a chemical reaction or a physical process.
These heat changes are then used to calculate the enthalpy (ΔH) of the reaction, which is a crucial thermochemical value.
In calorimetry, a device called a calorimeter is used to measure the heat. This device is made up of two main parts – the inner vessel, where the reaction takes place, and the outer water jacket,
which acts as a heat reservoir. The change in temperature of the water in the outer jacket is used to calculate the heat of the reaction.
The principle of Hess’s law is also used in determining thermochemical values. This law states that the enthalpy change of a chemical reaction is independent of the pathway taken to reach the final
products, as long as the initial and final conditions are the same. This means that the enthalpy change of a reaction can be calculated indirectly by using a series of known reactions and their
corresponding enthalpy values.
Another commonly used method for measuring thermochemical values is through standard enthalpy of formation (ΔHf°). This is the enthalpy change that occurs when one mole of a substance is formed from
its constituent elements in their standard states. These standard states are defined as 25 degrees Celsius and one atmosphere of pressure. The standard enthalpy values for common elements and
compounds are listed in tables and can be used to calculate the change in enthalpy for any chemical reaction.
Calculating thermochemical values involves several mathematical equations, and it’s essential to understand the basic principles behind them. The change in enthalpy (ΔH) can be calculated by using
the equation ΔH = Q/n, where Q is the heat absorbed or released in the reaction, and n is the number of moles of the substance involved.
The enthalpy change (ΔH) can also be calculated from standard enthalpies of formation (ΔHf°) using the formula ΔH = ΣnΔHf°(products) – ΣnΔHf°(reactants). This equation represents the sum of the
standard enthalpy values of the products minus the sum of the standard enthalpy values of the reactants.
It’s also essential to remember that the enthalpy change of a reaction is dependent on the stoichiometry of the reaction. This means that the coefficients of the balanced chemical equation must be
taken into account when calculating thermochemical values.
In conclusion, measuring and calculating thermochemical values is a crucial aspect of thermochemistry that helps us understand and predict the behavior of substances under different conditions.
Techniques such as calorimetry, Hess’s law, and standard enthalpies of formation are used to determine these values, and the calculations involve various mathematical equations. Having a firm
understanding of these techniques and principles is essential for anyone studying or working in the field of thermochemistry. | {"url":"https://tanbourit.com/measuring-and-calculating-thermochemical-values/","timestamp":"2024-11-08T14:38:49Z","content_type":"text/html","content_length":"112481","record_id":"<urn:uuid:cb404450-3dc2-4907-a21f-a1eaa568385d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00445.warc.gz"} |
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Mixed To Improper Fraction Worksheet
Mixed To Improper Fraction Worksheet
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Square Limit
Square Limit#
Published: June 10, 2019 · Updated: November 3, 2023
Square Limit#
The above image shows a famous woodcut by M.C. Escher called Square Limit composed of tesselating fish tiles. In this notebook, we will recreate this pattern using the HoloViews Spline element.
The construction used here is that of Peter Henderson’s Functional Geometry paper and this notebook was inspired by Massimo Santini’s programming-with-escher notebook, itself inspired by Haskell and
Julia implementations.
We start by importing HoloViews and NumPy and loading the extension:
import holoviews as hv
from holoviews import opts
import numpy as np
This notebook makes extensive use of the Spline element and we will want to keep equal aspects and suppress the axes:
opts.defaults(opts.Spline(xaxis=None, yaxis=None, aspect='equal', bgcolor='white', linewidth=0.8))
‘Square Limit’ is composed from the following fish pattern, over which we show the unit square:
unitsquare = hv.Bounds((0,0,1,1))
fish = hv.Spline((spline, [1,4,4,4]*34)) # Cubic splines
fish * unitsquare
As you may expect, we will be applying a number of different geometric transforms to generate ‘Square Limit’. To do this we will use Affine2D from matplotlib.transforms and matplotlib.path.Path (not
to be confused with hv.Path!).
from matplotlib.path import Path
from matplotlib.transforms import Affine2D
# Define some Affine2D transforms
rotT = Affine2D().rotate_deg(90).translate(1, 0)
rot45T = Affine2D().rotate_deg(45).scale(1. / np.sqrt(2.), 1. / np.sqrt(2.)).translate(1 / 2., 1 / 2.)
flipT = Affine2D().scale(-1, 1).translate(1, 0)
def combine(obj):
"Collapses overlays of Splines to allow transforms of compositions"
if not isinstance(obj, hv.Overlay): return obj
return hv.Spline((np.vstack([el.data[0] for el in obj.values()]),
np.hstack([el.data[1] for el in obj.values()])))
def T(spline, transform):
"Apply a transform to a spline or overlay of splines"
spline = combine(spline)
result = Path(spline.data[0], codes=spline.data[1]).transformed(transform)
return hv.Spline((result.vertices, result.codes))
# Some simple transform functions we will be using
def rot(el): return T(el,rotT)
def rot45(el): return T(el, rot45T)
def flip(el): return T(el, flipT)
Here we define three Affine2D transforms (rotT,rot45T and flipT), a function to collapse HoloViews Spline overlays (built with the * operator) in a single Spline element, a generic transform function
T and the three convenience functions we will be using directly (rot, rot45 and flip). Respectively, these functions rotate the spline by \(90^o\), rotate the spline by \(45^o\) and flip the spline
Here is a simple example of a possible tesselation:
Next we need two functions, beside and above to place splines next to each other or one above the other, while compressing appropriately along the relevant axis:
def beside(spline1, spline2, n=1, m=1):
den = n + m
t1 = Affine2D().scale(n / den, 1)
t2 = Affine2D().scale(m / den, 1).translate(n / den, 0)
return combine(T(spline1, t1) * T(spline2, t2))
def above(spline1, spline2, n=1, m=1):
den = n + m
t1 = Affine2D().scale(1, n / den).translate(0, m / den)
t2 = Affine2D().scale(1, m / den)
return combine(T(spline1, t1) * T(spline2, t2))
beside(fish, fish)* unitsquare + above(fish,fish) * unitsquare
One import tile in ‘Square Limit’ is what we will call smallfish which is our fish rotate by \(45^o\) then flipped:
smallfish = flip(rot45(fish))
smallfish * unitsquare
We can now build the two central tesselations that are necessary to build ‘Square Limit’ which we will call t and u respectively:
t = fish * smallfish * rot(rot(rot(smallfish)))
u = smallfish * rot(smallfish) * rot(rot(smallfish)) * rot(rot(rot(smallfish)))
t *unitsquare + u * unitsquare
We are now ready to define the two recursive functions that build the sides and corners of ‘Square Limit’ respectively. These recursive functions make use of quartet which is used to compress four
splines into a small 2x2 grid:
blank = hv.Spline(([(np.nan, np.nan)],[1])) # An empty Spline object useful for recursion
def quartet(p, q, r, s):
return above(beside(p, q), beside(r, s))
def side(n):
if n == 0:
return hv.Spline(([(np.nan, np.nan)],[1]))
return quartet(side(n-1), side(n-1), rot(t), t)
def corner(n):
if n == 0:
return hv.Spline(([(np.nan, np.nan)],[1]))
return quartet(corner(n-1), side(n-1), rot(side(n-1)), u)
corner(2) + side(2)
We now have a way of building the corners and sides of ‘Square Limit’. To do so, we will need one last function that will let us put the four corners and four sides in place together with the central
def nonet(p, q, r, s, t, u, v, w, x):
return above(beside(p, beside(q, r), 1, 2),
above(beside(s, beside(t, u), 1, 2),
beside(v, beside(w, x), 1, 2)), 1, 2)
args = [fish]* 4 + [blank] + [fish] * 4
Here we see use nonet to place eight of our fish around the edge of the square with a blank in the middle. We can finally use nonet together with our recursive corner and side functions to recreate
‘Square Limit’:
def squarelimit(n):
return nonet(corner(n), side(n), rot(rot(rot(corner(n)))),
rot(side(n)), u, rot(rot(rot(side(n)))),
rot(corner(n)), rot(rot(side(n))), rot(rot(corner(n))))
hv.output(squarelimit(3), size=250)
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CSE TUBE - Anna university CGPA Calculator instructions
This can also be used for finding how much should we study to get the appropriate percentage or CGPA.
• Wait for the excel sheet to load. It takes few minutes depending on the speed of system your system.
• We have tried to keep the mark in the order of the results for 1st to 5th semester
• Since Electives starts from 6th semester,ordering cant be maintained in it. So check your subject. Don't change the credits column.
1. Enter grades with respect to the subjects in the grade column.
2. People who took Numerical methods as electives in 6th semester, change the credit as 4 and enter your corresponding grade.
3. AU Madurai and Tirunelveli students enter your grades to your corresponding subject.
4. For arrears, change the respective subject credits to "0"
5. Lateral entry students, change the credits of 1st and 2nd semester subjects as "0"
6. Both GPA and CGPA is calculated simutaneously here..
7. For calculating CGPA Enter the credit column as "0" for the various semesters for which you have not got your result.
8. If you have any doubts, mail us to csetube@gmail.com
Enter the Grades corresponding to the subjects in the respective column to get your CGPA.
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Resistor - (Electromagnetism I) - Vocab, Definition, Explanations | Fiveable
from class:
Electromagnetism I
A resistor is an electrical component that limits or regulates the flow of electric current in a circuit. Resistors are fundamental for controlling voltage and current, making them essential in
various applications, including power distribution, signal processing, and circuit design. They play a crucial role in defining the behavior of both direct current (DC) and alternating current (AC)
congrats on reading the definition of resistor. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Resistors can be classified as fixed or variable; fixed resistors have a constant resistance value, while variable resistors, like potentiometers, allow for adjustment.
2. The unit of resistance is the ohm (ฮฉ), which defines how much voltage is needed to produce a certain current through a resistor.
3. In a series circuit, the total resistance is the sum of individual resistances, while in parallel circuits, the total resistance can be calculated using the formula 1/R_total = 1/R1 + 1/R2 + ...
4. Resistors can generate heat when current flows through them, which is why they are rated for power dissipation, often specified in watts.
5. In AC circuits, resistors contribute to the overall impedance and affect the phase relationships between voltage and current.
Review Questions
• How do resistors influence the behavior of multi-loop circuits?
□ In multi-loop circuits, resistors play a critical role in determining the current distribution and voltage across different branches. By applying Kirchhoff's laws, one can analyze how each
resistor affects the overall circuit behavior. The values of the resistors determine how much current flows through each loop and branch, allowing for effective management of power
consumption and ensuring that components operate within safe limits.
• What is the significance of power dissipation in resistors when analyzing transient behavior in RC circuits?
□ Power dissipation in resistors is significant when examining transient behavior in RC circuits because it impacts how quickly the circuit charges or discharges. During transient analysis,
resistors influence the time constant ($$\tau = RC$$$), which determines how fast the voltage across the capacitor changes. The heat generated by resistors during this process must be managed
to prevent damage to components and ensure accurate functioning.
• Evaluate how resistors affect resonance in RLC circuits and their implications on circuit performance.
□ Resistors in RLC circuits have a direct impact on resonance by introducing damping effects. The presence of resistance affects the quality factor (Q) of the circuit, which describes how
underdamped or overdamped the system is. A higher resistance reduces the peak amplitude at resonance and broadens the resonance curve, leading to less sharp tuning and impacting signal
integrity. Understanding this interaction helps in designing RLC circuits for specific applications such as filters and oscillators.
ยฉ 2024 Fiveable Inc. All rights reserved.
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Python SyntaxError: cannot assign to operator Solution
We can assign the result of a mathematical calculation to a variable, but we can not assign a value to a mathematical expression. While assigning a value to a variable in Python , we write the
variable name on the left side of the assignment operator "=" and the mathematical computational expression on the right side. But if we try to switch it around, we will encounter the error
SyntaxError: cannot assign to operator .
This Python guide will discuss the above error and how to solve it. Also, we shall go through an example demonstrating this error, so you can learn how to solve it yourself.
So let's get started.
Python Error - SyntaxError: cannot assign to operator
According to the syntax defined by Python, when we want to assign a computational mathematical value to a variable, we need to write the variable on the left side and the mathematical computation on
the right side of the assignment operator "=". In simple terms, You must write a mathematical expression on the right side and a variable on the left.
x = 20 + 30
print(a) #50
The above example is the correct syntax to assign a mathematical computational value to a variable x. When the Python interpreter reads the above code, it assigns 20+30 , i.e., 50 to the variable x .
But if we switch the position of the mathematical computation and variable, we encounter the SyntaxError: cannot assign to operator Error.
20 + 30 = a # SyntaxError: cannot assign to operator print(a)
The error statement SyntaxError: cannot assign to operator has two parts.
1. SyntaxError (Exception type)
2. cannot assign to the operator (Error Message)
1. SyntaxError
SyntaxError is a standard Python exception. It occurs when we violate the syntax defined for a Python statement.
2. cannot assign to operator
"cannot assign to operator" is an error message. The Python interpreter raises this error message with the SyntaxErorr exception when we try to perform the arithmetic operation on the left side of
the assignment operator. Python cannot assign the value on the right side to the mathematical computation on the left side.
Common Example Scenario
Now, you know the reason why this error occurs. Let us now understand this through a simple example.
Let's say we have a list prices that contains the original prices of the different products. We need to write a program that discounts 10 rupees from every product price and adds a 2 rupee profit to
every price.
discount = 10
profit = 2
prices = [7382, 3623, 9000,3253,9263,9836]
for i in range(len(prices)):
# discount 10 and profit 2
prices[i] + (profit - discount) = prices[i]
File "main.py", line 9
prices[i] + (profit - discount) = prices[i]
SyntaxError: cannot assign to operator
Break the code
In the above example, we get the error SyntaxError: cannot assign to operator because the variable to which we want to assign the value " prices[i] " is on the right side of the assignment operator,
and the value that we want to assign prices[i] + (profit - discount) is on the left side.
When we want to assign a mathematical or arithmetic result to a variable in Python, we should always write the variable on the left side of the assignment operator and the mathematical computational
value on the right side. To solve the above example error, we need to ensure that the prices[i] must be on the left side of the assignment operator.
discount = 10
profit = 2
prices = [7382, 3623, 9000,3253,9263,9836]
for i in range(len(prices)):
# discount 10 and profit 2
prices[i] = prices[i] + (profit - discount)
[7374, 3615, 8992, 3245, 9255, 9828]
When we try to assign a value to a mathematical computational statement, the " SyntaxError: cannot assign to operator" error is raised in a Python program. This means if you write the mathematical
computational expression on the assignment operator's left side, you will encounter this error. To debug or fix this error, you need to ensure that the variable or variables you write on the left
side of the assignment operator do not have an arithmetic operator between them.
If you still get this error in your Python program, you can share your code and query in the comment section. We will try to help you with debugging.
Happy Coding!
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Mineralka 3168 - math word problem (3168)
Mineralka 3168
Petr bought 0.35 kg of ham, six rolls, and mineral water for a snack. How much did he pay for the purchase?
(Mineralka costs 14.60 CZK, a roll 3.40 CZK, and a kilogram of ham 129 CZK). They rounded up the purchase to whole crowns.
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Grade of the word problem:
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Understanding Mathematical Functions: How To Find Prime Of A Function
Understanding Mathematical Functions: How to Find Prime of a Function
Mathematical functions are like magic spells that unlock the secrets of the universe. They are powerful tools used in various fields such as physics, engineering, economics, and computer science. At
their core, functions are a set of mathematical rules that establish a relationship between two sets of values, typically represented as input and output. They play a crucial role in modeling
real-world phenomena, making predictions, and solving complex problems.
Prime functions are a special class of functions that hold significant importance in advanced mathematics. Prime functions are the building blocks of more complex mathematical concepts and are
fundamental to understanding calculus and analysis. They are essential in studying the behavior of more complicated functions and are a cornerstone of higher-level mathematical theories and
The purpose of this blog is to guide readers through the process of finding the 'prime' of a function. This is a key concept in calculus and analysis, and mastering it is essential for anyone looking
to delve deeper into the world of advanced mathematics.
Explanation of Mathematical Functions
Mathematical functions are essential tools for representing and analyzing relationships between different quantities. A function takes an input, applies a set of operations to it, and produces an
output. This relationship can be represented as an equation, a graph, or a rule that describes how the output depends on the input.
• Functions in Different Fields: Functions are used in various fields such as engineering, physics, economics, and computer science to model and solve real-world problems.
• Importance of Functions: Functions provide a systematic way of understanding and exploring the behavior of natural and man-made systems, making predictions, and developing solutions to complex
Brief Overview of Prime Functions
Prime functions are foundational functions that play a crucial role in advanced mathematics. They serve as the basis for more complex mathematical concepts and are essential for understanding the
behavior of more intricate functions.
• Importance in Calculus and Analysis: Prime functions are fundamental to the study of calculus and analysis, providing insights into the behavior of functions and their derivatives, integrals, and
• Building Blocks of Mathematics: Prime functions are the elementary functions that form the basis for constructing more complex expressions and understanding the fundamental principles of
mathematical analysis.
By understanding prime functions and their properties, mathematicians and scientists can gain deeper insights into the underlying structure of mathematical systems and develop powerful tools for
solving a wide range of problems.
Key Takeaways
• Understand the concept of a mathematical function
• Identify the characteristics of a prime function
• Use mathematical tools to find the prime of a function
• Apply the concept to real-world problem solving
Understanding the Concept of 'Prime' in Mathematics
When it comes to mathematical functions, the concept of 'prime' is often associated with the derivative of a function. Understanding the prime of a function is crucial in calculus and mathematical
analysis. Let's delve into the details of this concept and its historical context.
A Define 'derivative' as it's often confused with 'prime' of a function
The derivative of a function represents the rate at which the function's value changes with respect to the change in its input variable. In simpler terms, it gives us the slope of the function at a
particular point. This is often denoted by f'(x) or dy/dx, where 'y' is the dependent variable and 'x' is the independent variable.
B Historical context of the term 'prime' as it relates to the derivative
The term 'prime' in the context of mathematics has its roots in the historical development of calculus. It is derived from the notation used by Leibniz, one of the co-founders of calculus, who used a
dot above the function's variable to denote the derivative. Over time, this notation evolved into the use of a prime symbol (') to represent the derivative of a function with respect to its variable.
C Clarification of 'prime' notation and its use in differentiating functions
In mathematical notation, the prime symbol (') is used to denote the derivative of a function with respect to its variable. For example, if we have a function f(x), its derivative with respect to 'x'
would be denoted as f'(x). This notation is essential in differentiating functions and finding the rate of change at a specific point.
The Role and Significance of Prime Functions
Prime functions play a crucial role in calculus and are essential for understanding the behavior of mathematical functions. They have significant applications in solving real-world problems and are
valuable for gaining insights into the behavior of functions.
A. Significance of Prime Functions in Calculus
Prime functions are fundamental in calculus as they help in determining the rate of change of a function. They provide information about the slope of a function at a specific point, which is
essential for understanding the behavior of the function. By finding the prime of a function, calculus enables us to analyze the behavior of functions in a dynamic and changing environment.
B. Application of Prime Functions in Solving Real-World Problems
Prime functions have practical applications in various fields such as physics, engineering, economics, and biology. For example, in physics, prime functions are used to calculate the velocity and
acceleration of an object in motion. In economics, prime functions help in determining the marginal cost and revenue of a product. These real-world applications demonstrate the significance of prime
functions in solving complex problems.
C. Introduction to the Value of Finding the Prime for Understanding Function Behavior
Finding the prime of a function is valuable for understanding the behavior of the function. It provides insights into the maximum and minimum points of a function, which are crucial for optimization
problems. Additionally, prime functions help in identifying the concavity and inflection points of a function, which are essential for understanding its overall behavior.
Steps to Find the Prime of a Function
Understanding how to find the prime of a function is a fundamental concept in calculus. The process involves finding the derivative of a function, which gives us the rate of change of the function at
any given point. In this chapter, we will outline the standard process for finding the prime of a function, explain the use of differentiation rules, and provide examples with simple functions to
illustrate the step-by-step process.
Outline the standard process for finding the derivative of a function
The derivative of a function represents the slope of the function at any given point. To find the derivative of a function, we use the concept of limits to calculate the rate of change. The standard
process for finding the derivative involves applying the rules of differentiation to the function.
Explain the use of differentiation rules: product rule, quotient rule, chain rule
There are several rules of differentiation that are used to find the derivative of a function. The product rule is used when the function is a product of two other functions. The quotient rule is
used when the function is a quotient of two other functions. The chain rule is used when the function is composed of two or more functions. These rules provide a systematic way to find the derivative
of more complex functions.
Provide examples with simple functions to illustrate the step-by-step process
Let's consider the function f(x) = x^2 as an example. To find the prime of this function, we start by applying the power rule, which states that the derivative of x^n is n*x^(n-1). Therefore, the
derivative of f(x) = x^2 is f'(x) = 2x. This means that the rate of change of the function f(x) = x^2 at any point x is given by 2x.
Now, let's consider the function g(x) = 3x^2 + 4x - 2. To find the prime of this function, we apply the sum rule, which states that the derivative of the sum of two functions is the sum of their
derivatives. Therefore, the derivative of g(x) is g'(x) = 6x + 4. This gives us the rate of change of the function g(x) at any point x.
These examples illustrate the step-by-step process of finding the prime of a function using the rules of differentiation.
Common Mistakes and Misconceptions
When it comes to finding the prime of a function, there are several common mistakes and misconceptions that can trip up even the most experienced mathematicians. Understanding these pitfalls and how
to avoid them is crucial for accurately determining the prime of a function.
A Address frequent errors made when finding the prime of a function
• Incorrect application of the power rule: One common mistake is applying the power rule incorrectly when finding the derivative of a function. It's important to carefully follow the steps of the
power rule to ensure an accurate result.
• Forgetting to consider all terms: Another frequent error is forgetting to consider all terms in a function when finding its prime. Each term in the function must be evaluated separately to find
the prime correctly.
• Confusion with chain rule: The chain rule is often misunderstood and misapplied, leading to errors in finding the prime of a composite function. It's important to fully grasp the concept of the
chain rule and how to use it effectively.
B Debunk misconceptions regarding prime functions and their calculation
There are also several misconceptions surrounding prime functions and their calculation that can lead to confusion and errors.
• Prime functions are not always linear: One common misconception is that prime functions are always linear. In reality, prime functions can take various forms, including quadratic, cubic,
exponential, and logarithmic functions.
• Derivative at a point is not the same as the prime function: Another misconception is that the derivative of a function at a specific point is the same as the prime function. The prime function
represents the rate of change of the original function across its entire domain, not just at a single point.
• Prime functions are not always increasing or decreasing: It's also a misconception that prime functions are always increasing or decreasing. In reality, a prime function can have intervals of
both increasing and decreasing behavior.
C Offer troubleshooting tips for typical issues encountered during the process
When encountering issues in finding the prime of a function, it's important to have troubleshooting tips to address these typical problems.
• Double-check calculations: If the result of finding the prime of a function seems incorrect, it's important to double-check the calculations step by step to identify any errors in the process.
• Review fundamental concepts: Sometimes, encountering difficulties in finding the prime of a function can be attributed to a lack of understanding of fundamental concepts such as the power rule,
chain rule, or derivative properties. Reviewing these concepts can help clarify any confusion.
• Seek additional resources: If troubleshooting on your own doesn't resolve the issues, seeking additional resources such as textbooks, online tutorials, or consulting with a knowledgeable peer or
instructor can provide valuable insights and assistance.
Advanced Techniques and Considerations
When it comes to understanding mathematical functions, there are advanced techniques and considerations that come into play. These include higher-order derivatives, implicit differentiation, special
functions, and the role of software tools in computing primes for complicated functions.
A Introduction to more complex scenarios requiring higher-order derivatives
Higher-order derivatives come into play when dealing with more complex scenarios in mathematical functions. These derivatives provide information about the rate of change of the rate of change, and
so on. In other words, they give insight into how the rate of change of a function itself is changing. Understanding and calculating higher-order derivatives is essential for finding the prime of a
function in more intricate scenarios.
B Techniques for handling implicit differentiation and special functions
Implicit differentiation is a technique used to differentiate functions that are not explicitly expressed in terms of the independent variable. This technique is particularly useful when dealing with
equations that are difficult to solve for the dependent variable explicitly. Special functions, such as trigonometric, logarithmic, and exponential functions, require specific techniques for
differentiation. Understanding how to handle implicit differentiation and special functions is crucial for accurately finding the prime of a function.
C Discuss the role of software tools in computing primes for complicated functions
With the advancement of technology, software tools play a significant role in computing primes for complicated functions. These tools can handle complex calculations and provide accurate results in a
fraction of the time it would take to manually compute primes. Additionally, software tools can handle a wide range of functions, including those with higher-order derivatives, implicit
differentiation, and special functions. Utilizing software tools can streamline the process of finding primes for complicated functions and reduce the margin of error.
Conclusion & Best Practices
A Recap the critical importance of understanding prime functions in mathematics
Understanding prime functions is crucial in mathematics as it forms the foundation for various mathematical concepts and applications. Prime functions help in identifying the fundamental building
blocks of more complex functions, making it easier to analyze and manipulate them. It also plays a significant role in number theory, cryptography, and computer science, making it an essential
concept to grasp for anyone pursuing a career in these fields.
Emphasize best practices, including thorough practice and utilization of mathematical software for complex functions
When it comes to mastering prime functions, thorough practice is key. Solving a wide range of problems involving prime functions can help in developing a deeper understanding of their properties and
behavior. Additionally, utilizing mathematical software for complex functions can aid in visualizing and analyzing prime functions, making it easier to comprehend their intricacies and applications.
Encouragement for continued learning and exploration of prime functions’ applications in different mathematical domains
As with any mathematical concept, the learning process for prime functions is ongoing. It is essential to continue exploring and applying prime functions in different mathematical domains to gain a
comprehensive understanding of their significance. Whether it's in calculus, algebra, or number theory, prime functions have diverse applications that can enrich one's mathematical knowledge and
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# Author Title Accn# Year Item Type Claims
1 Kembhavi Ajit Gravitational Waves : A New Window to the Universe 027108 2020 Book
2 Sanjeev Dhurandhar General relativity and gravitational waves: Essentials of Theory and Practice 026860 2022 Book
3 Michele Maggiore Gravitational waves vol. 2: Astrophysics and Cosmology 026370 2018 Book
4 David G. Blair (ed.) Advanced gravitational wave detectors 024629 2012 Book
5 Jolien D. E. Creighton Gravitational-wave physics and astronomy: An introduction to theory, experiment and data analysis 024080 2011 Book
6 Maurice H.P.M. van Putten Gravitational radiation, luminous black holes and gamma-ray burst supernovae 023808 2005 Book
7 Harry Collins Gravity's shadow: The search for gravitational waves 023619 2004 Book
8 Louis Witten (ed.) Gravitation: An introduction to current research 022449 1962 Book
9 Pankaj S. Joshi Gravitational collapse and spacetime singularities 022460 2007 Book
10 Michele Maggiore Gravitational waves vol. 1: Theory and experiments 021958 2008 Book
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Title General relativity and gravitational waves: Essentials of Theory and Practice
Author(s) Sanjeev Dhurandhar ;Sanjit Mitra
Publication Cham, Springer Nature, 2022.
Description xiv, 207p.
Series (Unitext for Physics)
Abstract Note This book serves as a textbook for senior undergraduate students who are learning the subject of general relativity and gravitational waves for the first time. Both authors have been
teaching the course in various forms for a few decades and have designed the book as a one stop book at basic level including derivations and exercises. A spectacular prediction of
general relativity is gravitational waves. Gravitational waves were first detected by the LIGO detectors in 2015, hundred years after their prediction. Both authors are part of the
LIGO Science Collaboration and were authors on the discovery paper. Therefore, a strong motivation for this book is to provide the essential concepts of general relativity theory and
gravitational waves with their modern applications to students and to researchers who are new to the multi-disciplinary field of gravitational wave astronomy. One of the advanced
topics covered in this book is the fundamentals of gravitational wave data analysis, filling a gap in textbooks on general relativity. The topic blends smoothly with other chapters in
the book not only because of the common area of research, but it uses similar differential geometric and algebraic tools that are used in general relativity.
ISBN,Price 9783030923341 : Eur 64.99(HB)
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Keyword(s) 1. GENERAL RELATIVITY 2. GRAVITATIONAL WAVES
Item Type Book
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Title Gravitational waves vol. 2: Astrophysics and Cosmology
Author(s) Michele Maggiore
Publication Oxford, Oxford University Press, 2018.
Description xiv, 820p.
Abstract Note The two-volume book Gravitational Waves provides a comprehensive and detailed account of the physics of gravitational waves. While Volume 1 is devoted to the theory and experiments,
Volume 2 discusses what can be learned from gravitational waves in astrophysics and in cosmology, by systematizing a large body of theoretical developments that have taken place over
the last decades. The second volume also includes a detailed discussion of the first direct detections of gravitational waves. In the author's typical style, the theoretical results
are generally derived afresh, clarifying or streamlining the existing derivations whenever possible, and providing a coherent and consistent picture of the field.
ISBN,Price 9780198570899 : UKP 60.00(HB)
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Keyword(s) 1. COSMOLOGY 2. EBOOK 3. EBOOK - OXFORD UNIVERSITY PRESS 4. GRAVITATIONAL WAVES
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Title Gravitational-wave physics and astronomy: An introduction to theory, experiment and data analysis
Author(s) Jolien D. E. Creighton ;Warren G. Anderson
Publication Weinheim , Wiley-VCH, 2011 .
Description xiv, 375p.
Abstract Note This most up-to-date, one-stop reference combines coverage of both theory and observational techniques, with introductory sections to bring all readers up to the same level. Written
by outstanding researchers directly involved with the scientific program of the Laser Interferometer Gravitational-Wave Observatory (LIGO), the book begins with a brief review of
general relativity before going on to describe the physics of gravitational waves and the astrophysical sources of gravitational radiation. Further sections cover gravitational wave
detectors, data analysis, and the outlook of gravitational wave astronomy and astrophysics.
ISBN,Price 9783527408863 : 9680.00 (HB)
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Keyword(s) 1. EBOOK 2. EBOOK - WILEY 3. GRAVITATIONAL WAVES
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Title Gravitational radiation, luminous black holes and gamma-ray burst supernovae
Author(s) Maurice H.P.M. van Putten
Publication Cambridge, Cambridge University Press, 2005.
Description xvii, 308p.
Abstract Note Black holes and gravitational radiation are two of the most dramatic predictions of general relativity. The quest for rotating black holes - discovered by Roy P. Kerr as exact
solutions to the Einstein equations - is one of the most exciting challenges facing physicists and astronomers. Gravitational Radiation, Luminous Black Holes and Gamma-Ray Burst
Supernovae takes the reader through the theory of gravitational radiation and rotating black holes, and the phenomenology of GRB-supernovae. Topics covered include Kerr black holes
and the frame-dragging of spacetime, luminous black holes, compact tori around black holes, and black-hole spin interactions. It concludes with a discussion of prospects for
gravitational-wave detections of a long-duration burst in gravitational-waves as a method of choice for identifying Kerr black holes in the Universe. This book is ideal for a special
topics graduate course on gravitational-wave astronomy and as an introduction to those interested in this contemporary development in physics.
ISBN,Price 9780521143615 : UKP 20.99(PB)
Classification 531.51:530.12
Keyword(s) 1. BLACK HOLE 2. GAMMA-RAY BURSTS 3. GRAVITATIONAL RADIATION 4. LUMINOUS BLACK HOLES 5. ROTATING BLACK HOLE
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Title Gravitation: An introduction to current research
Author(s) Louis Witten (ed.)
Publication New York, John Wiley and Sons, 1962.
Description x, 481
ISBN,Price US $ 70(Xerox)
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Title Gravitational collapse and spacetime singularities
Author(s) Pankaj S. Joshi
Publication Cambridge, Cambridge University Press, 2007.
Description x, 273p.
Abstract Note Physical phenomena in astrophysics and cosmology involve gravitational collapse in a fundamental way. The final fate of a massive star when it collapses under its own gravity at the
end of its life cycle is one of the most important questions in gravitation theory and relativistic astrophysics, and is the foundation of black hole physics. General relativity
predicts that continual gravitational collapse gives rise to a space-time singularity. Quantum gravity may take over in such regimes to resolve the classical space-time singularity.
This book investigates these issues, and shows how the visible ultra-dense regions arise naturally and generically as an outcome of dynamical gravitational collapse. It will be of
interest to graduate students and academic researchers in gravitation physics, fundamental physics, astrophysics, and cosmology. It includes a detailed review of recent research into
gravitational collapse, and several examples of collapse models are investigated in detail.
ISBN,Price 9780521871044 : UKP 65.00(HB & EB)
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Keyword(s) 1. ASTROPHYSICS 2. COSMOLOGY 3. EBOOK 4. EBOOK - CAMBRIDGE UNIVERSITY PRESS 5. GRAVITATIONAL COLLAPSE 6. SPACETIME SINGULARITIES
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Title Gravitational waves vol. 1: Theory and experiments
Author(s) Michele Maggiore
Publication Oxford, Oxford University Press, 2008.
Description xvii, 554p.
Contents Note Book is devoted to the theory of gravitational waves.
ISBN,Price 9780198570745 : BP 45.00
Classification 531.51:530.12
Keyword(s) 1. EBOOK 2. EBOOK - OXFORD UNIVERSITY PRESS 3. GRAVITATIONAL WAVES
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Data Sufficiency
Data Sufficiency Reasoning: Concepts & Tricks
Data sufficiency questions test your knowledge of basic math facts and skills along with reasoning, analytical, and problem-solving abilities. Each data sufficiency item presents you with a question.
You do not actually have to find the answer to the problem; instead, your challenge is to decide whether or not the information presented along with the question would be sufficient to allow you to
answer the question. Five answer choices are provided, each of which categorizes the relationship between question and the information provided in a different way. You must select the answer choice
that accurately describes this relationship.
Understanding Answer options
In Data sufficiency problems, a question consists of two statements labeled I and II, in which a certain information is given. You have to decide whether the information given in the statements is
sufficient to answer the question or not. Using the information given in the statements plus your knowledge of mathematics and wellknown facts (such as Earth revolves around Sun or the meaning of
counterclockwise), you must indicate whether.
1. The answer can be obtained from Statement I alone but statement II alone is not sufficient to answer the question asked;
2. The answer can be obtained from statement II alone but statement I alone is not sufficient to answer the question asked;
3. The answer can be obtained from both statements I and II together; but neither statement I nor statement II alone is sufficient to answer the question asked ;
4. We can answer the asked question from either statement I or statement II;
5. We can not answer the question asked from Statement I and II together, and additional information is required to answer the question.
Note: In data sufficiency problems, the information given in the statements is sufficient only when it is possible to determine exactly one numerical value as answer for the problem.
1. Three packages have a combined weight of 48 pounds. What is the weight of the heaviest package?
A. One package weighs 12 pounds.
B. One package weighs 24 pounds.
1. Statement A alone is sufficient to answer this question, but statement B alone is not sufficient.
2. Statement B alone is sufficient to answer this question, but statement A alone is not sufficient.
3. Both statements together are needed to answer this question, but neither statement alone is sufficient.
4. Either statement by itself is sufficient to answer this question.
5. Not enough facts are given to answer the question.
The correct answer is option 2. Statement A is not sufficient to determine the weight of the heaviest package. It implies only that the combined weight of the other two packages is 36 pounds.
(Eliminate options 1 and 4). Statement B alone is sufficient for it implies that the combined weight of two of the packages is only 24 pounds. Since the weight of the 24 -pound packages is equal to
the combined weight of the other two packages, the heaviest package must weigh 24 pounds. (Eliminate options 3 and 5). Since statement B alone is sufficient to answer the question but statement A
alone is not, answer this question as option 2.
2. How many books are there on a certain shelf?
A. If four books are removed, the number of books remaining on the shelf will be less than 12.
B. If three more books are placed on the shelf, the total number of books on the shelf will be more than 17.
1. Statement A alone is sufficient to answer the question, but statement B alone is not sufficient.
2. Statement B alone is sufficient to answer the question, but statement A alone is not sufficient.
3. Both statements together are needed to answer the question, but neither statement alone is sufficient.
4. Either statement by itself is sufficient to answer the question.
5. Not enough facts are given to answer the question.
Must Read Data Sufficiency Articles
• Data Sufficiency Concepts & Tricks
The correct answer is option 3. Neither statement alone is sufficient to answer the question asked. Statement A alone implies only that the number of books on the shelf is 15 or fewer, and statement
B alone implies only that the number of books on the shelf is 15 or more. (Eliminate options 1, 2 and 4). But the two statements taken together are sufficient to answer the question, for they imply
that the number of books on the shelf is 15. (15 is the only integer that satisfies both statements A and B). Since neither statement alone is sufficient, but the two statements together are, answer
this question as option 3.
Data sufficiency tricks
Step 1 – Examine the Question:
What is asked? Do we have to find a value or do we have to check a relationship?
Before looking at the two numbered statements, take twenty to thirty seconds to consider the question by itself. Figure out what is being asked. There are usually 2 possibilities a specific number
may be sought (“What is the value of y?” "How many gallons of milk is in the tank?”), or a true/false answer may be needed (“Is it true that a >7?” “Is n a prime number?”) Make sure you understand
what the question is asking.
Then consider what information would be needed to answer the question. This will depend on the type of question, of course. If it is a geometry question, the information needed will be based on rules
you’ve learned about how one geometric fact can be deduced from another. For example, to determine the area of a circle, you need to know its radius, its diameter, or its circumference. To determine
the length of the hypotenuse of a right triangle, you need to know the length of the other two sides.
On the other hand, if it is a percentage question, different rules will come into play. To determine what percentage X is of Y, for example, you need to know the value of X and the value of Y. When a
change from one value to another is involved – the increase in value of an investment, for example – you need to know both the old value and the percentage by which it has increased if you want to
calculate the new value.
As these examples suggest, the data sufficiency question format allows the test makers to measure your knowledge of a wide array of mathematical topics.
Step 2 – Consider each statement individually
Having figured out the nature of the question and decided, in a general way, what information is needed to answer it, look at each of the two numbered statements provided. Consider them one at a
time, without reference to each other.
First look at statement A. Does it provide, all by itself, enough information to answer the question? If so, you’ve already narrowed the possible answer choices to just two: 1 and 4. If not, three
answer choices are possible: 2, 3 and 5.
Then look at statement B. Does it provide, all by itself, enough information to answer the question? If so, only answers 2 and 4 are possible. If not, only answers 1, 3 and 5 are possible.
Having gotten this far, you may already be able to pick the right answer. If either statement by itself provides enough information to answer the question, you can pick from answers 1, 2 and 4,
depending on which statement is sufficient or whether either statement will do.
If neither statement by itself is sufficient to answer the question, go on to the third stage:
Step 3 – Combine the two statements
Third, if necessary, combine the two statements. If neither of the statements by itself is sufficient to answer the question, consider whether you can answer the question by combining the information
given in both statements. If so, the answer is 3; if not, the answer is 5.
Flow Chart:
The following flow chart summarizes the questions you need to ask yourself as you use the three-stage system. It’s a handy way to review and refresh your understanding of this method. | {"url":"http://littlegnomehome.com/data-sufficiency-tips-and-tricks.html","timestamp":"2024-11-04T01:42:04Z","content_type":"text/html","content_length":"87218","record_id":"<urn:uuid:dad27d42-7331-42fd-80b5-3fd3b2b61214>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00230.warc.gz"} |
Andrew Myers
Andrew Myers
Computer Systems Engineer, Applied Mathematics and Computational Research Division
Contact Information
Andrew Myers
MS 50A-3111
Lawrence Berkeley National Lab
1 Cyclotron Rd.
Berkeley, CA 94720
510-486-6900 (fax)
[email protected]
Affiliation and Research Interests
I am a member of the Center for Computational Sciences and Engineering (CCSE) at the Lawrence Berkeley National Laboratory (LBNL). My current research focuses on the design and implementation of
scalable parallel algorithms for conducting particle and particle-mesh simulations on current and upcoming supercomputing architectures, particularly in the context of adaptive mesh refinement. These
algorithms have applications to the modelling of, for example, large-scale structure formation in cosmology, plasma acceleration in particle accelerators, and the solids phase in multi-phase flow
More generally, I am interested in parallel algorithms for scientific computing (particularly for GPU platforms), particle methods, and the visualization and analysis of large simulation datasets.
Much of my work is open source and can be followed on my Github page.
Current Projects
Particles in AMReX
AMReX, a software framework for building massively parallel block-structured adaptive mesh refinement (AMR) applications that is supported by the Exascale Computing Project (ECP). Particles are used
in some capacity by most of the AMReX application codes, including six of the ECP application development projects.
In AMReX, we are particularly interested in particles that live on and interact with a constantly-changing hierarchy of refinement patches, which adds an additional layer of complication to the
underyling data structures. AMReX provides methods for handling the MPI communication of particle data, including both redistribution and halo exchange, as well as implementations of several common
operations used in particle applications, such as neighbor list construction, particle-mesh deposition and interpolation, reductions, etc.
Much of my recent work has focused on re-working the particle data structures and algorithms for hybrid CPU/GPU platforms, such as OLCF's Summit machine. AMReX application codes involving significant
particle work have been demonstrated to scale well up to the full machine, and have demonstrated good speed-ups over CPU-only platforms.
Plasma and laser wakefield acceleration with WarpX
Accelerator Modelling group at LBNL, Livermore Lab, and the Stanford Linear Accelerator Center. The goal of this project is to build a new simulation tool for studying plasma- and laser- driven
wakefield acceleration, a process by which charged particles are accelerated over much shorter distances than in conventional accelerators. The hope is that, by harnessing the power of future
exascale computers, WarpX can aid in the design of smaller, less costly particle accelerators.
WarpX is a relativistic, electromagnetic Particle-in-Cell code that includes several advanced features, such as spectral Maxwell solvers, perfectly-matched absorbing layers, mesh refinement,
ionization, and the ability to operate in a boosted reference frame. Along with MFIX-Exa, it is one of the primary drivers of development for the AMReX particle library.
Previous Work
I completed my PhD in Physics at the University of California at Berkeley in September 2013, working with Richard Klein and Christopher McKee on the topics of high-mass star formation and the
interstellar medium. Later, I was a postdoctoral researcher in the Applied Numerical Algorithms Group at LBNL, where I worked on various mathematical topics related to the convergence of
particle-in-cell schemes for Vlasov-Poisson problems. You can read about this work in my publications below. Finally, I have also been active in the open-source scientific Python community, in
particular with the yt project, and still retain interests in large-scale scientific visualization and data analysis.
• Andrew Myers, Weiqun Zhang, Ann Almgren, Thierry Antoun, John Bell, Axel Huebl and Alexander Sinn, AMReX and pyAMReX: Looking Beyond ECP, International Journal of High Performance Computing
Applications , August 2024, [doi]
• Scott Atchley et. al., Frontier: Exploring Exascale. accepted for publication at Supercomputing 2023,
• A. Lattanzi, W. Fullmer, A. Myers, J. Musser, Towards polydisperse flows with MFIX-Exa. ASME. J. Fluids Eng, 23 January 2024. [doi].
• H. Klion, R. Jambunathan, M. E. Rowan, E. Yang, D. Willcox, J.-L. Vay, R. Lehe, A. Myers, A. Huebl, W. Zhang, Particle-in-Cell Simulations of Relativistic Magnetic Reconnection with Advanced
Maxwell Solver Algorithms. The Astrophysical Journal, 925, 1, 2023. [arxiv]
• Bruce J. Palmer, Ann S. Almgren, Connah G.M. Johnson, Andrew T Myers, and William R. Cannon, BMX: Biological Modelling and interface eXchange, Nature Scientific Reports, 13, July 2023. [doi]
• Luca Fedeli, Axel Huebl, France Boillod-Cerneux, Thomas Clark, Kevin Gott, Conrad Hillairet, Stephan Jaure, Adrien Leblanc, Remi Lehe, Andrew Myers, Christelle Piechurski, Mitsuhisa Sato, Neil
Zaim, Weiqun Zhang, Jean-Luc Vay, Henri Vincenti, Pushing the frontier in the design of laser-based electron accelerators with groundbreaking mesh-refined particle-in-cell simulations on
exascale-class supercomputers, SC '22: Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, November 2022, Article No.: 3, Pages 1-12 [doi]
• Severin Diederichs, Carlo Benedetti, Axel Huebl, Remi Lehe, Andrew Myers, Alexander Sinn, Jean-Luc Vay, Weiqun Zhang, Maxence Thevenet, HiPACE++: a portable, 3D quasi-static Particle-in-Cell code
, Computer Physics Communications, 278, 108421, 2022. [link][arXiv]
• Jean Sexton, Zarija Lukic, Ann Almgren, Chris Daley, Brian Friesen, Andrew Myers, and Weiqun Zhang, Nyx: A Massively Parallel AMR Code for Computational Cosmology, Journal of Open Source Software
, 6(63), 3068, 2021. [doi]
• L. Fedeli, A. Sainte-Marie, N. Zaim, M. Thevenet, J. L. Vay, A. Myers, F. Quere, and H. Vincenti, Probing strong-field QED with Doppler-boosted petawatt-class lasers, accepted by Physical Review
Letters, May 10, 2021, [PRL]
• Sherwood Richers, Don E. Willcox, Nicole M. Ford, and Andrew Myers, Particle-in-cell simulation of the neutrino fast flavor instability, Physical Review D, April 20, 2021, [doi]
• Jordan Musser, Ann S Almgren, William D Fullmer, Oscar Antepara, John B Bell, Johannes Blaschke, Kevin Gott, Andrew Myers, Roberto Porcu, Deepak Rangarajan, Michele Rosso, Weiqun Zhang, and
Madhava Syamlal, MFIX:Exa: A Path Towards Exascale CFD-DEM Simulations, The International Journal of High Performance Computing Applications, April 16, 2021. [IJHPCA] [doi]
• Weiqun Zhang, Andrew Myers, Kevin Gott, Ann Almgren and John Bell, AMReX: Block-Structured Adaptive Mesh Refinement for Multiphysics Applications, The International Journal of High Performance
Computing Applications, June 12, 2021. [IJHPCA] [doi]
• J-L Vay, Ann Almgren, LD Amorim, John Bell, L Fedeli, L Ge, K Gott, DP Grote, M Hogan, A Huebl, R Jambunathan, R Lehe, A Myers, C Ng, M Rowan, O Shapoval, M Thevenet, H Vincenti, E Yang, N Zaim,
W Zhang, Y Zhao and E Zoni, Modeling of a chain of three plasma accelerator stages with the WarpX electromagnetic PIC code on GPUs, Physics of Plasmas, 28(2), 2021. [doi]
• Andrew Myers, Ann Almgren, Diana Almorim, John Bell, Luca Fedeli, Lixin Ge, Kevin Gott, David Grote, Mark Hogan, Axel Huebl, Revathi Jambunathan, Remi Lehe, Cho Ng, Michael Rowan, Olga Shapoval,
Maxence Thevenet, Jean-Luc Vay, Henri Vincenti, Eloise Yang, Neil Zaim, Weiqun Zhang, Yin Zhao, Edoardo Zoni, Porting WarpX to GPU-accelerated platforms, accepted by Parallel Computing, 2021.
• Y. Zhao, R. Lehe, A. Myers, M. Thevenet, A. Huebl, C. B. Schroeder, and J.-L. Vay, Modeling of emittance growth due to Coulomb collisions in plasma-based accelerators, Physics of Plasmas,
October, 2020. [doi]
• W Zhang, A Almgren, V Beckner, J Bell, J Blashke, C Chan, M Day, B Friesen, K Gott, D Graves, M Katz, A Myers, T Nguyen, A Nonaka, M Rosso, S Williams, M Zingale, AMReX: a framework for
block-structured adaptive mesh refinement, Journal of Open Source Software, May 2019
• B Loring, A Myers, D Camp, EW Bethel, Python-based in situ analysis and visualization Proceedings of the Workshop on In Situ Infrastructures for Enabling Extreme-Scale Analysis and Visualization
- ISAV 18, ACM Press, 2018
• JL Vay, A Almgren, J Bell, L Ge, DP Grote, M Hogan, O Kononenko, R Lehe, A Myers, C Ng, J Park, R Ryne, O Shapoval, M Thevenet, W Zhang, Warp-X: A new exascale computing platform for beam-plasma
simulations, Nuclear Instruments and Methods in Physics Research, Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2018, 909:476--479
• M Krumholz, A Myers, R Klein, C McKee, What Physics Determines the Peak of the IMF? Insights from the Structure of Cores in Radiation-Magnetohydrodynamic Simulations, MNRAS, May 19, 2016
• A Myers, P Colella, B Van Straalen, A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov-Poisson Equation, SIAM Journal on Scientific Computing, Volume 39, No 3, pp.
B467-B485, May 9, 2017
• A Myers, P Colella, B Van Straalen, The Convergence of Particle-in-Cell Schemes for Cosmological Dark Matter Simulations, The Astrophysical Journal, Volume 816, Issue 2, article id. 56, 2016
• A Myers, C McKee, PS Li, The CH+ abundance in turbulent, diffuse molecular clouds, Monthly Notices of the Royal Astronomical Society, Volume 453, Issue 3, p.2747-2758, November 1, 2015
• A Myers, R Klein, M Krumholz, C McKee, Star cluster formation in turbulent, magnetized dense clumps with radiative and outflow feedback, Monthly Notices of the Royal Astronomical Society, Volume
439, Issue 4, p.3420-3438, April 1, 2014
• A Myers, C McKee, A Cunningham, R Klein, M Krumholz, The Fragmentation of Magnetized, Massive Star-forming Cores with Radiative Feedback, The Astrophysical Journal, Volume 766, Issue 2, article
id. 97, April 1, 2013
• PS Li, A Myers, C McKee, Ambipolar Diffusion Heating in Turbulent Systems, The Astrophysical Journal, Volume 760, Issue 1, article id. 33, November 1, 2012
• A Myers, M Krumholz, R Klein, C McKee, Metallicity and the Universality of the Initial Mass Function, The Astrophysical Journal, Volume 735, Issue 1, article id. 49, 2011
Selected Talks
• Building Exascale-Ready Adaptive Mesh Refinement Applications with AMReX, ECP Annual Meeting, Virtual, 2021
• WarpX: Scalable Particle-in-Cell Algorithms for Emerging Architectures with AMReX. SIAM Conference on Computational Science and Engineering, Fort Worth, TX, 2021
• AMReX and applications on GPUs - lessons learned from Summit. Joint SIAM / CAIMS Annual Meeting, Toronto, Ontario, Canada, 2020
• An overview of particles in AMReX, SIAM Conference on Parallel Processing, Seattle, Washington, 2020
• Introduction to AMReX - a new framework for block-structured adaptive mesh refinement calculations, Advanced Modelling and Simulation Seminar Series, Nasa Ames Research Center, 2018
• A high-order accurate Particle-in-Cell method for Vlasov-Poisson problems over long time integrations, Advanced Modelling and Simulation Seminar Series, Nasa Ames Research Center, 2016
• Controlling Numerical Error in Particle-in-Cell Simulations of Collisionless Dark Matter, SIAM Conference on Computational Science and Engineering, Salt Lake City, Utah, 2015
• Radiation-Magnetohydrodynamic Simulations of Star Formation, Friday Lunch Time Astrophysics Seminar, UC - Santa Cruz, 2014
• The Fragmentation of High-Mass Dense Cores, Planet and Star Formation Seminar, UC - Berkeley, 2013
Poster Presentations
• M Thevenet, J-L Vay, A Almgren, D Amorim, J Bell, A Heubl, R Jambunathan, R Lehe, A Myers, J Park, O Shapoval, W Zhang, L Ge, M Hogan, C Ng, D Grote, Toward Exascale modeling of Plasma Particle
Accelerators on GPU , Supercomputing 2019, Denver, CO, 2019
• D Amorim, J-L Vay, A Almgren, J Bell, K Gott, A Heubl, R Jambunathan, R Lehe, A Myers, J Park, M Rowan, O Shapoval, M Thevenet, W Zhang, Y Zhao, L Ge, M Hogan, C Ng, D Grote, WarpX - Efficient
modeling of plasma-based accelerators with mesh refinement , American Physical Society Division of Plasma Physics Annual Meeting, Fort Lauderdale, FL, 2019
• D Amorim, J-L Vay, A Almgren, J Bell, A Heubl, R Jambunathan, R Lehe, A Myers, J Park, O Shapoval, M Thevenet, W Zhang, L Ge, M Hogan, C Ng, D Grote, WarpX ECP project recent progress ,
International Conference on Numerical Simulation of Plasmas, Sante Fe, NM, 2019
• R Jambunathan, A Myers, D Wilcox, J-L Vay, A Almgren, D Amorim, J Bell, K Gott, A Heubl, R Lehe, J Park, M Rowan, O Shapoval, M Thevenet, W Zhang, L Ge, M Hogan, C Ng, D Grote, WarpX: Towards
Exascale Modelling of Pulsar Magnetospheres , Connecting Micro and Macro Scales: Acceleration, Reconnection, and Dissipation in Astrophysical Plasmas, Kavli Institute for Theoretical Physics,
Santa Barbara, CA, 2019
• A Myers, J Bell, A Almgren, V Beckner, J Blaschke, C Chan, M Day, B Friesen, K Gott, D Graves, M Katz, T Nguyen, A Nonaka, M Rosso, S Williams, W Zhang, M Zingale Overview of AMReX - a new
framework for block-structured adaptive mesh refinement calculations, SIAM Conference on Computational Science and Engineering, Spokane, WA, 2019
• Y L Lin, A Almgren, B Friesen, A Myers Performance Study of GPU Offloading via CUDA, OpenACC, and OpenMP in AMReX, SIAM Conference on Computational Science and Engineering, Spokane, WA, 2019
• D Wilcox, D Kasen, A Almgren, A Myers, W Zhang, SedonaEx: A Monte Carlo Radiation Transfer Code for Astrophysical Events, SIAM Conference on Computational Science and Engineering, Spokane, WA,
• K Gott, A Myers, W Zhang, An Overview of GPU Strategies for Porting Amrex-Based Applications to Next-generation HPC Systems, SIAM Conference on Computational Science and Engineering, Spokane, WA,
• M Thevenet, J-L Vay, A Almgren, J Bell, R Lehe, A Myers, J Park, O Shapoval, W Zhang, L Ge, M Hogan, C Ng, D Grote, WarpX: Toward Exascale modeling of Plasma Particle Accelerators ,
Supercomputing 2018, Dallas, TX, 2018
• A Myers, A Cunningham, R Klein, M Krumholz, C McKee, Fragmentation of Magnetized, Massive Cores with Radiative Feedback Star Formation and the Interstellar Medium: Thirty-Five Years Later,
Berkeley, CA, 2012 | {"url":"https://ccse.lbl.gov/people/amyers/","timestamp":"2024-11-09T12:39:12Z","content_type":"text/html","content_length":"21852","record_id":"<urn:uuid:288ebe0a-c7a5-430f-990d-30836cf9b73d>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00775.warc.gz"} |
Simplifying and Determining the Domain of Rational Functions
Question Video: Simplifying and Determining the Domain of Rational Functions Mathematics • Third Year of Preparatory School
Simplify the function π (π ₯) = (π ₯Β² + 2π ₯)/(π ₯Β² β 4), and find its domain.
Video Transcript
Simplify the function π of π ₯ equals π ₯ squared plus two π ₯ over π ₯ squared minus four and find its domain.
π of π ₯ is a rational function and so the expression on the right-hand side is an algebraic fraction. To simplify this fraction, we need to look for common factors of the numerator and
denominator which we can then cancel out. So our first task is to factor both the numerator and denominator. Starting with the numerator, we can see that the two terms, π ₯ squared and two π ₯,
have a common factor of π ₯. π ₯ squared is π ₯ times π ₯ and two π ₯ is π ₯ times two. And so together they are π ₯ times π ₯ plus two where here we have applied the distributive property.
Now we move on to the denominator which is π ₯ squared minus four, and we notice that that is a difference of two squares. It is π ₯ minus two times π ₯ plus two. Now that the numerator and
denominator are fully factored, we can see that they have a common factor of π ₯ plus two. We can cancel this out. And we see that the simplified form of π of π ₯ is π ₯ over π ₯ minus two and
that we canβ t simplify any further.
So we have simplified the function, but now we need to find its domain. The domain of a rational function is the set of values for which its denominator is nonzero. In other words, it is the set of
real numbers minus the set of values for which the denominator of a rational function is zero. If you look at the simplified function, you might be tempted to think that the only value of π ₯ for
which the denominator is zero is π ₯ equals two. However, the denominator of the original function, as it was defined, is π ₯ squared minus four and not π ₯ minus two. And if you look at the
factorized form of this denominator, itβ s easy to see there are actually two values of π ₯ for which this denominator is zero, two and negative two. The domain is therefore the set of real numbers
minus the set containing negative two and two.
So this is our answer: For every value of π ₯ in the domain of a function, π of π ₯ is equal to π ₯ over π ₯ minus two. However, the domain of the function is the real numbers minus negative
two and two. Had the function originally been defined as just π ₯ over π ₯ minus two, the domain wouldβ ve been bigger. It wouldβ ve been the real numbers minus just the set of two. If the
function had originally been defined as just π ₯ over π ₯ minus two, then certainly the domain would be just the real numbers minus the set of two. Had we not excluded this negative two from the
domain, then we wouldnβ t be allowed to have cancelled π ₯ plus two on the numerator and the denominator because in effect we wouldβ ve been dividing by zero on both the top and bottom.
So while itβ s possible often to simplify a rational function, the simplification process doesnβ t change the domain of the function. And so when talking about the domain of the function, you
should look at the original statement, the original definition of the function and not the simplified version that you get after simplification. | {"url":"https://www.nagwa.com/en/videos/959101536284/","timestamp":"2024-11-12T08:46:55Z","content_type":"text/html","content_length":"252656","record_id":"<urn:uuid:1dc49f3d-d621-4daa-ad44-de854c034d6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00758.warc.gz"} |
Electric Circuits Final Project - Alternative Chua's Circuit
\documentclass[conference,compsoc]{IEEEtran} \usepackage{graphicx} \usepackage[english]{babel} \usepackage[utf8]{inputenc} \usepackage{physics} \usepackage{mathtools} \usepackage{float} \usepackage
[T1]{fontenc} \usepackage[pdftex]{hyperref} \begin{document} \title{Electric Circuits Final Project - Alternative Chua's Circuit} \author{ \IEEEauthorblockN{José Luiz Gomes Nogueira} \
IEEEauthorblockA{Course of Computer Engineering\\Universidade de Brasília\\ID: 16/0032458\\Email: joseluizgnogueira@live.com} \and \IEEEauthorblockN{Igor Roberto Rodrigues Alves} \IEEEauthorblockA
{Course of Mechatronics Engineering\\Universidade de Brasília\\ID: 15/0129998\\Email: igulito14@hotmail.com} \and \IEEEauthorblockN{Danilo Queirós de Feritas} \IEEEauthorblockA{Course of Electrical
Engineering\\Universidade de Brasília\\ID: 15/0122713\\Email: danilo.df.freitas@gmail.com} } \maketitle \section{Abstract} \textbf{The Chua's circuit is one of the simplest circuits that presents a
chaotic behavior. This behavior is difficult to predict due to the nonlinearity of of the functions that models it. In general, the Chua's circuit has an oscillatory behavior based on a piecewise
linear function (Fig \ref{chua_nonlinear_diode}). This alternative Chua's circuit exhibits a smoother function and even a faster response when compared to the traditional Chua's circuits. } \section
{Introduction} Chua's Circuit is a simple electronic circuit that implements a chaotic behavior (autonomous circuit). This circuit can show non periodic oscillations . But for one circuit to be an
autonomous circuit, they must satisfy three criteria: \begin{enumerate} \item one or more nonlinear elements; \item one or more locally active resistor; \item three or more energy-storage elements; \
end{enumerate} The Chua's Circuit is simple because use the minimum to be an autonomous circuit. In the Chua's Circuit we have two Capacitor, one Inductor, one simple resistor and on Chua's diode,
that can be implemented with several different forms. The example os Chua's Circuit can be seen below. Figure \ref{chua_diagram}. \begin{figure}[H] \caption{Simple Chua's Diagram} \label
{chua_diagram} \centering \includegraphics[width=0.5\textwidth]{chuas_circuit_diagram.png} \end{figure} In the default circuit the nonlinear element has the following curve (Current x Voltage).
Figure \ref{chua_nonlinear_diode}: \begin{figure}[H] \caption{Nonlinear Chua's Diode} \label{chua_nonlinear_diode} \centering \includegraphics[width=0.5\textwidth]
{chuas_diode_nonlinear_resistor_graph_nondimensional.jpg} \end{figure} As stated above, the Chua's diode can be implemented with several different forms, but the equations that governing the circuit
stay with the same model. In this article the objective is implement non the original circuit, but the alternative circuit, that has a more simple equation to the nonlinear diode, and can be
implemented with simple additional circuits.\\ The alternative circuit of Chua used as base in this project is the implementations proposed for K. Tang and K. Man of Department of Electronic
Engineering of City University of Hong Kong, published in the International Symposium IEEE - 1998.\\ The objective of this project is put into practice the knowledge gained during the course of
Electronics Circuits, and see beyond, for the beginning of a large area not contemplated for the course, the non linear circuits.\\ Some noises is expected in the circuit implemented, due to the
sensitivity of the circuit, that was done in a protoboard instead of welding the components. Since the protoboard allows a several contact with external ambient, and the trails has many
imperfections. So, to get a better result is recommended that circuit be welded. \section{Theoretical Calculations} The Chua's Circuit is a most basic circuit that can implement a chaotic circuit. To
define the equations of the circuit let see the image of the original curve of chua's diode above. As seen in the picture, the nonlinear diode is nonlinear only in three parts, but is linear between
them, so we can calculate the equations of the circuit for parts, or we can use a generic non linear function called here as g(x) to define ours Differential equation. g(x) is a functions represented
by the Figure \ref{chua_nonlinear_diode}, where g(x) is the current in function of tension on diode. Using the node and mesh analysis we can get the following equations. $$\dv{V_1}{t} = \frac{1}{C_1}
[\frac{1}{R}(v_2-v_1) - g(v_1)]$$ $$\dv{V_2}{t} = \frac{1}{C_2} [\frac{1}{R}(v_1-v_2)+i_L]$$ $$\dv{i_L}{t} = \frac{1}{L}[v_2 - R_o i_L]$$ Following the original model of Chua's circuit we can
represent g(v1) as a linear function. $$g(v_1) = \begin{Bmatrix} m_0 v_1+(m_0-m_1)E1,\quad if\quad v_1 \leq -E1\\ m_1 v_1,\quad if\quad -E1 < v_1 < E1\\ m_0 v_1 + (m_1-m_0)E1,\quad if\quad E1 \leq
v_1\\ \end{Bmatrix}$$ Or we can represent the diode function completely. $$g(v_R) = i_R = G_b v_R + \frac{1}{2}(G_a - G_b) [|v_R + E| - |v_R - E|]$$ Where Gb, Ga and E is constants. The above
equations define the original Chua's circuit, but in this document we will implement the alternative Chua's circuit that have a more simple diode equation, the curve of g(x) in original circuit it is
similar a cubic equation, however with discontinuities. In the alternative Chua's circuit the proposed model is similar a cubic equation but more smoothed, without discontinuities. For the
alternative Chua's circuit the equation proposed is the below: $$g(v_R) = a v_R + b v_R|v_R|$$ where a < 0 and b > 0. For the proposed diode equation, we have a circuit that consists of two Op-amps
(AD711 and LF347), one multiplier (AD633), one comparator (LM319), a analog multiplexer (74HC4052) and six resistors. \begin{figure}[H] \caption{Circuit of alternative Chua's Diode} \label
{alternative_circuit} \centering \includegraphics[width=0.5\textwidth]{alternative_circuit.PNG} \end{figure} So, the driving-current for g(x) proposed is: $$i_R = g(v_R) = -\frac{1}{R_4}v_R + \frac
{R_5 + R_6}{R_4 R_5} \frac{1}{10V} v_R|v_R|$$ In this equation we can choose the value of R4, R5 and R6 but limited to the limit of logical, analogical and Op-amps of circuit. \section{Computer
Simulation} The software used in this simulation was Multisim from National Instruments. Because of software limitations the components used were not from the same type as the ones of the proposed
circuit, therefore, the values of C1, C2 and L needed to be altered and the double scroll result was not achieved. The circuit was simulated as shown below: \begin{figure}[H] \caption{Chua's circuit
simulation on Multisim} \label{chua_multisim} \centering \includegraphics[width=0.5\textwidth]{ChuasSimulation.png} \end{figure} \begin{figure}[H] To observe a chaotic behavior, we varied the
resistance R and saved the oscilloscope's results for each different value of R as shown: \caption{$R = 1220\Omega$} \label{R1220} \centering \includegraphics[width=0.5\textwidth]{R1220b.png} \end
{figure} \begin{figure}[H] \caption{$R = 1210\Omega$} \label{R1210} \centering \includegraphics[width=0.5\textwidth]{R1210b.png} \end{figure} \begin{figure}[H] \caption{$R = 1200\Omega$} \label
{R1200} \centering \includegraphics[width=0.5\textwidth]{R1200b.png} \end{figure} \begin{figure}[H] \caption{$R = 1197\Omega$} \label{R1197} \centering \includegraphics[width=0.5\textwidth]
{R1197b.png} \end{figure} \begin{figure}[H] \caption{$R = 1196\Omega$} \label{R1196} \centering \includegraphics[width=0.5\textwidth]{R1196b.png} \end{figure} Note that the minimum value for R to
stay oscillating is $1197\Omega$, and when we choose a lower value, it oscillates just for a very short time and them it drastically varies the trajectory of the graph and stops oscillating. \section
{Experiment} \subsection{Experimental Components} In general Chua's circuits can be composed simply by capacitors, inductors, resistors and a Chua's diode, that can be implemented using Op Amps and
diodes, or even Op Amps alone. \\ But for this circuit implementation the following components are required: \\ \begin{itemize} \item 1 Inductor. \item 2 Capacitors. \item 2 Op Amps (TL074). \item 1
Comparator (LM393). \item 1 Analogic multiplexer (TLHEF4052). \item 1 Analog Multiplier (AD633). \item Resistors. \item 1 Trimpot. \end{itemize} \subsection{Experimental Procedure } The circuit
should be implemented in accordance with Fig \ref{alternative_circuit}. In the experiment, the value of R is varied while the other components have the constant values: \\ \\ $R_{1} = 1k\Omega$ $R_
{2} = R_{3} = 2k\Omega$\\ $R_{4} = 1695\Omega$ $R_{5} = 3k\Omega$ $R_{6} = 750\Omega$\\ $C_{1} = 7nF$ $C_{2} = 78nF$\\ $L = 18.84mH$\\ To obtain the necessary inductance, a inductive decade from the
laboratory will be used. It has an internal resistance of $45\Omega$ per Henry, which implies a total resistance of $0.8478\Omega$ for $18.84mH$. It is important that the internal resistance of the
equipment is less than $30\Omega$ because the resistance influences in the distribution of chaos when R is adjusted.\\ After assembling the circuit, the parameters to be analyzed are X, Y and Z,
which represent the voltage across the capacitor C1, the voltage across capacitor C2 and the current across the inductor L respectively. \\ \begin{figure}[H] \caption{Variables to be analyzed in the
Chua Circuit.} \label{circuit_diagram_XYZ} \centering \includegraphics[width=0.5\textwidth]{chuas_circuit_diagram_XYZ.jpg} \end{figure} With this parameters in hand and the oscilloscope properly
configured in a Voltage x Voltage display, we can obtain curves like these: \\ \begin{figure}[H] \caption{Chua attractor with double scrolling seen from an analog oscilloscope. Image taken from:
http://www.chuacircuits.com/howtobuild4.php} \label{double_scroll} \centering \includegraphics[width=0.5\textwidth]{analog_double_scroll_attractor.jpg} \end{figure} \section{Conclusion} The
implemented circuit was proposed as an alternative circuit with chaotic behavior and with a smoother nonlinearity than the traditional Chua's circuits nonlinearity. Unfortunatly, in the real circuit
implementation, the calibration of the capacitance, inductance and the value of resistor R is very sensitive and it was dificult to obtain a clear chaotic behavior. \\ \begin{thebibliography}{1} \
bibitem{Three-steps-to-chaos} \href{https://people.eecs.berkeley.edu/~chua/papers/Kennedy93.pdf} {Kennedy, Michael Peter (October 1993). "Three steps to chaos – Part 1: Evolution" (PDF). IEEE Trans.
on Circuits and Systems. Institute of Electrical and Electronic Engineers. 40 (10): 640. Retrieved May 31, 2018} \bibitem{Alternative-chuas-circuit} \href{https://www.researchgate.net/publication/
3763733_An_alternative_Chua\%27s_circuit_implementation} {K.S Tang, K.F. Man. "An Alternative Chua's Circuit Implementation". Retrieved May 31, 2018} \bibitem{How-to-Build-Chua-Circuit} \href{http://
www.chuacircuits.com/}{Artigo e pesquisas realizadas por V. Siderskiy. Retrieved June 5, 2018} \bibitem{Indutive-decade} \href{https://www.ietlabs.com/pdf/Datasheets/1491.pdf}{Frabricant's data of
inductive decade. Retrieved June 5, 2018} \end{thebibliography} % that's all folks \end{document} | {"url":"https://www.overleaf.com/articles/electric-circuits-final-project-alternative-chuas-circuit/dpqpqytcydmp","timestamp":"2024-11-02T03:05:07Z","content_type":"text/html","content_length":"49228","record_id":"<urn:uuid:ab67a137-543c-4f6a-ab76-21c24430330e>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00270.warc.gz"} |
Discrete Data - (Data, Inference, and Decisions) - Vocab, Definition, Explanations | Fiveable
Discrete Data
from class:
Data, Inference, and Decisions
Discrete data refers to a type of quantitative data that can only take on specific, distinct values, typically whole numbers. This means that discrete data cannot be divided into smaller parts or
fractions, making it countable and often used to represent items, occurrences, or categories that have no intermediate values. Understanding discrete data is essential as it contrasts with continuous
data, which can take any value within a range and can be measured more fluidly.
congrats on reading the definition of Discrete Data. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Discrete data often arises in scenarios involving counting items, such as the number of students in a classroom or the number of cars in a parking lot.
2. The values of discrete data are often represented using integers, making it easy to use in statistical analysis and visualizations like bar graphs.
3. Common examples of discrete data include the number of pets owned, scores on a test, or the outcome of rolling dice.
4. Because discrete data is countable, it can be analyzed using various statistical methods specifically designed for integer-based values.
5. Discrete data can also involve categorical representations, where different categories are counted, such as the number of voters for each candidate in an election.
Review Questions
• How does discrete data differ from continuous data in terms of measurement and representation?
□ Discrete data differs from continuous data mainly in its ability to represent only specific, distinct values that cannot be divided. Discrete data is countable and typically consists of whole
numbers, while continuous data can take on any value within a range and includes fractions and decimals. This distinction impacts how both types of data are measured and visualized, with
discrete data often represented using bar graphs and continuous data using line graphs.
• What role does discrete data play in statistical analysis compared to categorical data?
□ Discrete data plays a crucial role in statistical analysis by providing quantifiable measurements that can be easily counted and analyzed mathematically. In contrast, categorical data focuses
on grouping characteristics without inherent numerical values. While both types are important for understanding different aspects of research and surveys, discrete data allows for more
precise calculations, such as averages and probabilities, whereas categorical data may rely on frequency counts to derive insights.
• Evaluate the implications of using discrete versus continuous data when collecting information for research purposes.
□ Using discrete versus continuous data has significant implications for research outcomes. Discrete data allows researchers to count exact occurrences and facilitate straightforward analysis
through clear metrics. In contrast, continuous data provides richer detail since it captures a wider range of values. Choosing between the two can affect the depth of insights gained;
researchers must consider their specific goals and the nature of what they are measuring to determine which type best suits their needs. Ultimately, understanding these differences ensures
accurate representation and interpretation of collected information.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/data-inference-and-decisions/discrete-data","timestamp":"2024-11-04T12:30:44Z","content_type":"text/html","content_length":"166651","record_id":"<urn:uuid:b4e3ed08-442a-4dd7-bc26-b25a6381244c>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00311.warc.gz"} |
Force Converter
Published: 2/6/2021 Last updated: 2/6/2021
The force converter is a solution to make any force conversion in less than a second, without any effort and for free. Don’t you believe us? Click here, try the force converter and see for yourself.
The force – this term you commonly see, but probably in not the context of physics. In this article we are going to focus on the force from the scientific standpoint. So if you need to make the force
conversion, you are in the right place. We will show you the force converter, which will change your force calculations once and for all. Do you want to know more? So keep reading.
What are we going to tell you in this article? First, we want to tell you a little about the force. It means, we are going to explain to you what it is, what the units of force are and give you some
examples from everyday life. Second, we want to show you our force converter. We promise, with this tool the force conversion will be effortless and take just a few seconds. Third, we want to show
you an example calculated with the use of our force converter. It is commonly known that the practice is the best way to learn. That’s why we decided to add this part to our article.
So if you know what you will find here, we can start now. First things first, let’s move on to the theory. Why? Because the theoretical information is necessary to make the force conversion properly.
So what is the force exactly? You probably know what’s going on, but do you know how to define the force? Simply put, the force is any interaction which will change the motion of a particular object.
In other words, we can say that the force is energy or strength which is responsible for physical action or movement.
Do you still not understand this term? Don’t worry. Have a look at these examples. First, if you want to raise your glass, you need to use a particular force to do it. Second, if you want to push the
bookshelf, you also need to use a particular force to do it. Of course, in this case greater than in the case of glass. We are sure that now it is clearer. Are we right?
The force has its own symbol. It is ‘F’. The force can be defined by the use of the equation. It looks as follows:
What do these symbols mean? The ‘m’ is a mass. That it means how much something or somebody at which the force acts weighs. Then, the ‘a’ is an acceleration. It means the rate of change of the
velocity of an object per unit of time.
To define the force is used mostly one unit – the Newton. It is, like many of the base units of other kinds of measurements, the unit which comes from the International System of Units (in
abbreviation SI). The symbol of this unit is ‘N’.
A little tidbit – the name of this unit was established to honour Issac Newton. Thanks to him we know nowadays the classical mechanics with 3 laws, commonly known as Newton’s laws of motion.
Newton is defined as: „the force needed to accelerate one kilogram of mass at the rate of one metre per second squared in the direction of the applied force”. The equation for this unit looks as
The symbols of this equation are: ‘kg’ – the kilogram, ‘m’ – the meter, ‘s’ – second.
Of course, Newton is not the only one unit of force. It is also possible to distinguish Dyne, pound-force or kilogram-force and a few others. Have a look at a little comparison of these 3 units and
• 1 Dyne = 10^-5 N;
• 1 pound-force = 4.448222 N;
• 1 kilogram-force = 9.80665 N.
The most theoretical part of our article is done. So now we are going to show you an amazing solution to make any force conversion quickly and easily – the force converter.
Force converter – how it will help you
As you could see, there is more than one unit of force and they are totally different. It could be problematic, for instance, to calculate 46 pounds-force to Newtons on your own. Why? Because you
will have to multiply two numbers – first quite huge and second with many decimal places. Fortunately, there is an easier way to make the force conversion like this. Without any effort and in just a
few seconds.
This way is using our force converter. It is an online calculator, fully automatized. It contains all units of force with all formulas. So you can make any force conversion you want, without need to
know the formula on your own. Thanks to the special, very precise algorithm you will also have the guarantee that the result obtained is correct. And the third thing which we want to emphasize: this
algorithm can make any force conversion in less than a second. It is time which our force converter needs to give you the result.
So sum up – what do we give to you? The force converter which:
• Is equipped with all units of force with formulas,
• Always give the accurate results;
• Calculate the results in less than a second.
We want to add one more thing – you can use the force converter for totally free, wherever you are and whenever you need. You just need a device (smartphone, tablet, laptop, etc.) with an Internet
Force converter – see how it works in practice
You know how our force converter will help you, now it is time to see how it works in practice. Let’s make a quick conversion, for instance, 60 Newtons to pounds-force. How to make it?
You need to:
1. Enter 60 as a number which you want to calculate.
2. Pick Newton as a unit of force which you want to convert.
3. Pick the pounds-force as a unit of force in which you want to get the result.
After these 3 simple steps, our force converter will give you the result. So what is 60 Newtons in pounds-force? The answer is as follows: 60 Newtons is equal to 13.488536586 pounds-force.
The force conversion can be problematic and take so long or be totally effortless and take just a few seconds. The choice is obvious, isn’t it? Choose the force converter and make your force
conversion easier.
Read more latest articles:
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Data Structures And Algorithms — Understanding Space And Time Complexity
Resources, whether computing or otherwise, are limited and must be used wisely to maximize utility. Additionally, businesses require software engineers to create software products that can scale as
the number of users and operations increases. Engineers use data structures and algorithms (DSA) concepts to properly manage resources and achieve scale.
What are Data Structures?
A data structure is a location in computer memory where data can be stored and organized. It also refers to the manner (structure) in which data is stored and organized so that it can be retrieved
and processed efficiently.
What are Algorithms?
Algorithms are the step-by-step procedures for solving problems. An algorithm is not always the code itself, but rather the path to take in order to solve the problem. For ease of understanding and
representation in various programming languages, algorithms are usually represented with pseudocodes or flowcharts.
An algorithm’s efficiency is determined by how fast it runs and how much space it takes up. Furthermore, the speed and memory consumption are measured using Space and Time Complexity.
What is Space Complexity?
The amount of memory used by an algorithm when it is executed is referred to as its space complexity. It is proportional to the number of inputs/variables used in the function, which means that the
more inputs an algorithm has, the more space it requires. The sum of the auxiliary space and the space used by these individual inputs equals the used memory. Auxiliary space is the extra space used
to run the code during execution.
What is Time Complexity?
The time complexity of an algorithm is the amount of time it takes to execute as the number of inputs increases. It is commonly calculated using the Big O Notation, which measures an
algorithm’s worst-case running time. Other metrics for calculating time complexity include:
• Big Omega: Determines the best case for an algorithm’s running time.
• Big Theta: Calculates the best and worst case analysis of an algorithm.
Consider the various types of Big O time complexities below as Big O notation is commonly used to describe algorithms in software engineering.
Examples of Big O Time Complexities
• Constant Time Complexity — O(1): This refers to algorithms whose time does not increase with the number of inputs. The algorithm’s time consumption is constant.
• Linear Time Complexity — O(n): It refers to algorithms where the time increases linearly with the number of inputs. As an example, suppose a function takes 1ms to execute for a single input. It
will take 5ms to process 5 inputs.
• Quadratic Time Complexity — O(n2): This refers to algorithms whose execution time is squared as the number of inputs increases. A nested for loop is a good example of this. If one input takes 2ms
to execute, four inputs will take 16ms (4²) to execute.
• Logarithmic Time complexity — O(log n): As the number of inputs increases, the time complexity of the algorithms decreases. The reason for this is that as the number of inputs increases
exponentially, the time increases linearly. If it takes 3ms to execute 8 inputs, it will take 6ms to execute 64 inputs. As a result, it is one of the most efficient time complexities.
• Log Linear Time Complexity — O (n log n): Similar to logarithmic time complexity, but grows linearly as the number of inputs increases due to the multiplication of n and log n.
• Exponential Time Complexity — O (2^n): The time for algorithms with exponential time complexity doubles as the number of inputs increases. These algorithms do not scale well.
The Big O time complexities are listed below in order of best to worst:
• Constant Time Complexity — O (1)
• Logarithimic Time Complexity — O (log n)
• Linear Time Complexity — O (n)
• Log Linear Time Complexity — O (n log n)
• Quadratic Time Complexity — O (n²)
• Exponential Time Complexity — O (2^n)
Scale should be prioritized by developers and organizations when developing software applications to avoid unfavorable production outcomes. They can accomplish this by ensuring that each line of code
is as perfectly optimized as possible with the highest order around the linear time complexity — O (n).
This article provided a conceptual understanding of data structures and algorithms, as well as the meaning of space and time complexity. It also offered an explanation of the Big O notation’s time
For further reading, check out: | {"url":"https://dataproducts.io/data-structures-and-algorithms-understanding-space-and-time-complexity/","timestamp":"2024-11-13T12:49:09Z","content_type":"text/html","content_length":"123274","record_id":"<urn:uuid:477b54df-9228-46b2-95a8-16e480c44f9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00171.warc.gz"} |
Thanksgiving Multiplying Decimals
Thanksgiving Multiplying Decimals
Price: 200 points or $2 USD
Subjects: math,holiday
Grades: 4,5,6
Description: This deck of 24 Boom Cards is in a fill in the blank, paperless format with a fun Thanksgiving theme. Students will work problems out on scratch paper to multiply decimals. Then, they
will type the correct product in the box provided. Each card is self-checking. Therefore, students will receive immediate feedback. This resource is perfect for 4th, 5th, or 6th grade students to
multiply decimals. It is also a wonderful resource for teachers to use for distance learning! The Boom Cards in this set align with: CCSS 5.NBT.7: Add, subtract, multiply, and divide decimals to
hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written
method and explain the reasoning used and CCSS 6.NS.B.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. | {"url":"https://wow.boomlearning.com/deck/ExuFPc7BjCSEx8w5m","timestamp":"2024-11-15T03:22:33Z","content_type":"text/html","content_length":"2805","record_id":"<urn:uuid:e9984646-7294-48a3-b603-674686f92d9f>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00696.warc.gz"} |
Math Simple Multiplication Worksheets
Math, especially multiplication, develops the foundation of various academic self-controls and real-world applications. Yet, for lots of students, grasping multiplication can pose an obstacle. To
address this difficulty, instructors and moms and dads have actually accepted a powerful device: Math Simple Multiplication Worksheets.
Intro to Math Simple Multiplication Worksheets
Math Simple Multiplication Worksheets
Math Simple Multiplication Worksheets -
Multiplication Worksheets for Beginners Multiplication worksheets for beginners are exclusively available on this page There are various exciting exercises like picture multiplication repeated
addition missing factors comparing quantities forming the products and lots more These pdf worksheets are recommended for 2nd grade through 5th grade
On this page you have a large selection of 2 digit by 1 digit multiplication worksheets to choose from example 32x5 Multiplication 3 Digits Times 1 Digit On these PDF files students can find the
products of 3 digit numbers and 1 digit numbers example 371x3 Multiplication 4 Digits Times 1 Digit
Relevance of Multiplication Technique Recognizing multiplication is critical, laying a solid structure for advanced mathematical ideas. Math Simple Multiplication Worksheets use structured and
targeted practice, cultivating a much deeper understanding of this essential arithmetic procedure.
Advancement of Math Simple Multiplication Worksheets
4 Digit Multiplication Worksheets Times Tables Worksheets
4 Digit Multiplication Worksheets Times Tables Worksheets
These multiplication worksheets are appropriate for 3rd Grade 4th Grade and 5th Grade Free dynamically created math multiplication worksheets for teachers students and parents Great resource for
lesson plans quizzes homework or just practice different multiplication topics
Download and printout our FREE worksheets HOLIDAY WORKSHEETS Free Secret Word Puzzle Worksheets New YearsWorksheets Martin Luther King Jr Worksheets Teaching the math facts quickly and effectively
Teacher Resources Teaching Tools Tips tools and tricks for teachers Free Multiplication Worksheets Download and printout our FREE
From typical pen-and-paper workouts to digitized interactive layouts, Math Simple Multiplication Worksheets have actually developed, satisfying diverse knowing styles and choices.
Sorts Of Math Simple Multiplication Worksheets
Standard Multiplication Sheets Easy exercises focusing on multiplication tables, helping learners develop a strong arithmetic base.
Word Issue Worksheets
Real-life circumstances integrated right into issues, improving important reasoning and application abilities.
Timed Multiplication Drills Tests developed to boost speed and precision, assisting in rapid mental math.
Benefits of Using Math Simple Multiplication Worksheets
multiplication Worksheet For Kids Archives EduMonitor
multiplication Worksheet For Kids Archives EduMonitor
We have thousands of multiplication worksheets This page will link you to facts up to 12s and fact families We also have sets of worksheets for multiplying by 3s only 4s only 5s only etc Practice
more advanced multi digit problems Print basic multiplication and division fact families and number bonds
These multiplication facts worksheets provide various exercise to help students gain fluency in the multiplication facts up to 12 x 12 Jump to your topic Multiplication facts review times tables
Multiplication facts practice vertical Multiplication facts practice horizontal Focus numbers Circle drills
Boosted Mathematical Abilities
Constant method develops multiplication effectiveness, enhancing general math abilities.
Enhanced Problem-Solving Abilities
Word problems in worksheets develop analytical reasoning and approach application.
Self-Paced Understanding Advantages
Worksheets accommodate individual learning rates, promoting a comfortable and adaptable understanding atmosphere.
Just How to Produce Engaging Math Simple Multiplication Worksheets
Integrating Visuals and Shades Vibrant visuals and shades catch focus, making worksheets visually appealing and engaging.
Including Real-Life Situations
Relating multiplication to everyday situations includes relevance and functionality to exercises.
Tailoring Worksheets to Various Ability Levels Customizing worksheets based on varying proficiency degrees guarantees inclusive knowing. Interactive and Online Multiplication Resources Digital
Multiplication Devices and Games Technology-based resources provide interactive understanding experiences, making multiplication appealing and satisfying. Interactive Web Sites and Apps On the
internet platforms supply varied and accessible multiplication practice, supplementing typical worksheets. Tailoring Worksheets for Numerous Discovering Styles Aesthetic Learners Visual aids and
diagrams help comprehension for students inclined toward visual discovering. Auditory Learners Spoken multiplication troubles or mnemonics satisfy students who understand concepts via acoustic ways.
Kinesthetic Learners Hands-on activities and manipulatives support kinesthetic learners in comprehending multiplication. Tips for Effective Execution in Knowing Uniformity in Practice Normal
technique enhances multiplication skills, advertising retention and fluency. Balancing Repetition and Selection A mix of recurring workouts and diverse trouble formats preserves passion and
comprehension. Offering Useful Responses Responses help in determining locations of enhancement, encouraging continued progression. Difficulties in Multiplication Method and Solutions Motivation and
Interaction Obstacles Boring drills can bring about uninterest; cutting-edge techniques can reignite motivation. Getting Over Fear of Math Negative perceptions around math can prevent progress;
creating a favorable discovering setting is necessary. Impact of Math Simple Multiplication Worksheets on Academic Efficiency Researches and Study Searchings For Study shows a favorable connection
between regular worksheet use and boosted mathematics efficiency.
Math Simple Multiplication Worksheets emerge as functional devices, promoting mathematical proficiency in students while accommodating varied knowing designs. From standard drills to interactive
on-line resources, these worksheets not just improve multiplication abilities however additionally advertise important thinking and analytic abilities.
Multiplication Worksheets Kinder Printable Multiplication Flash Cards
Free Printable Multiplication Worksheets 2nd Grade
Check more of Math Simple Multiplication Worksheets below
Multiplication Worksheets Grade 1 Multiplication Table Charts
FREE PRINTABLE MULTIPLICATION WORKSHEETS WonkyWonderful
Multiplication Worksheets Kindergarten Printable Multiplication Flash Cards
Fill In Multiplication Worksheets 10 Multiplication Wo Printable multiplication worksheets
Multiplication Worksheets Multiplication Facts For 2 Times Tables Teaching multiplication
Multiplication Worksheets Double Digit PrintableMultiplication
Printable Multiplication Worksheets Super Teacher Worksheets
On this page you have a large selection of 2 digit by 1 digit multiplication worksheets to choose from example 32x5 Multiplication 3 Digits Times 1 Digit On these PDF files students can find the
products of 3 digit numbers and 1 digit numbers example 371x3 Multiplication 4 Digits Times 1 Digit
Multiplication Facts Worksheets Math Drills
This section includes math worksheets for practicing multiplication facts to from 0 to 49 There are two worksheets in this section that include all of the possible questions exactly once on each page
the 49 question worksheet with no zeros and the 64 question worksheet with zeros
On this page you have a large selection of 2 digit by 1 digit multiplication worksheets to choose from example 32x5 Multiplication 3 Digits Times 1 Digit On these PDF files students can find the
products of 3 digit numbers and 1 digit numbers example 371x3 Multiplication 4 Digits Times 1 Digit
This section includes math worksheets for practicing multiplication facts to from 0 to 49 There are two worksheets in this section that include all of the possible questions exactly once on each page
the 49 question worksheet with no zeros and the 64 question worksheet with zeros
Fill In Multiplication Worksheets 10 Multiplication Wo Printable multiplication worksheets
FREE PRINTABLE MULTIPLICATION WORKSHEETS WonkyWonderful
Multiplication Worksheets Multiplication Facts For 2 Times Tables Teaching multiplication
Multiplication Worksheets Double Digit PrintableMultiplication
Math Worksheets On Multiplication
Beginner Multiplication Worksheets For Grade 2 Thekidsworksheet
Beginner Multiplication Worksheets For Grade 2 Thekidsworksheet
Printable Multiplication Worksheets
Frequently Asked Questions (Frequently Asked Questions).
Are Math Simple Multiplication Worksheets ideal for all age teams?
Yes, worksheets can be customized to different age and skill degrees, making them adaptable for various learners.
How usually should students practice utilizing Math Simple Multiplication Worksheets?
Consistent technique is key. Normal sessions, preferably a couple of times a week, can produce significant enhancement.
Can worksheets alone improve mathematics abilities?
Worksheets are an important tool but should be supplemented with varied understanding methods for extensive ability growth.
Exist on-line systems supplying totally free Math Simple Multiplication Worksheets?
Yes, several academic internet sites provide free access to a wide range of Math Simple Multiplication Worksheets.
How can parents support their children's multiplication practice in the house?
Urging constant technique, giving help, and developing a positive discovering atmosphere are helpful steps. | {"url":"https://crown-darts.com/en/math-simple-multiplication-worksheets.html","timestamp":"2024-11-13T21:15:14Z","content_type":"text/html","content_length":"29104","record_id":"<urn:uuid:53af1744-f541-49e4-b3f5-2939a3eb105a>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00739.warc.gz"} |
Square Rods and Acres Converter - CalculatorBox
Square Rods and Acres Converter
Figures rounded to max decimal places.
Using the Square Rods and Acres Converter
This converter allows you to find equivalent values between two imperial units of area, the square rods and acres (ac).
Start by choosing the spelling you would like to see in the converter between the British and the American spelling.
# Step
1 Follow up by choosing your input unit, which will be the unit of the value you are trying to convert. The choice is between square rods and acres. This choice is made in the ‘CONVERT FROM’ section.
2 The output unit, which is the one you want your result in, can be chosen in the ‘CONVERT TO’ section. Choose between the same two options as you did when choosing the input units.
3 You can also just accept the default settings of your unit, or easily swap them by clicking on the icon with two arrows headed in opposite directions.
4 Once you are happy with your input and output unit selection, write the actual value of your input into the ‘VALUE TO CONVERT’ part of the converter. If you input a decimal value, make sure you use
the decimal dot and not the comma, as is the case in some countries.
5 Proceed to choose the number of decimal places you want your result rounded toward, and click on ‘CONVERT’.
Your result will appear below the converter as a decimal number rounded to the desired number of decimal places. Alongside the result, you will also receive the conversion rate between the units, as
well as a convenient ‘COPY’ icon, which allows for easy copying and pasting of the result.
What is a Square Rod?
A square rod is a unit of area that is defined as the area of a square with a side length of 1 rod.
This still leaves us with the question of what exactly is the unit of rod equal to.
Albeit it is a proper part of the imperial system, there is a chance you did not hear about this unit, as it has been pushed into a bit of obscurity with time.
The history of the rod is a fairly inconsistent one, meaning that the value and the definition of the unit have been adjusted many times throughout history.
First of all, the name itself is a bit of an enigma, as the reference has been changed throughout history in favor of other words as well. Alternative names include perch, pole, or lug.
To makes matters worse, there were times in history when all words were used, referencing different units of measurement, while other times in history indicate that these were interchangeable
references to the same unit. Given that the history of the unit goes all the way to ancient Rome, and its usage spanned across many nations and languages, it is clear that there were ample
opportunities to further this confusion.
The actual value of the unit has also been modified throughout time, from anywhere between 10 to 24 feet.
It is also famously known, that the confusion was used as a means for a lot of devious activities, the most notable being some land seizures done by Henry VIII, where he used varied definitions of
the unit to shrink the lands of the Church by simply changing the official definition and then proceeding to claim the “extra” land for his kingdom.
The complete history of the rod is a wild and messy one, but definitely worth pursuing. You can read more using the link in our references.
Today, a rod is defined as 16.5 feet, which means that a square rod has an area of 16.5^2 ft^2, which is 272.25 ft^2.
Since the rod, and therefore also the square and cube rod, is an officially established unit of measurement of the imperial system, based on the 1959 international agreement, it can still find its
use despite not being very popular with the general public.
To this day, you will hear about rods as units of measurement in the fields of canoeing (especially because the original canoe is 1 rod long) and pipeline management (where prices are often indicated
in USD per rod).
Converting Square Rods and Acres Manually
The conversion rate you receive alongside your results is the key concept that can help us convert the two units manually.
The best way to determine the conversion rates is to know, that 1 acre is defined as 160 square rods.
This leads to 2 formulae that can be used for conversion. Despite not having a proper label, we will informally mark square rods as R^2 in the formulae.
The best way to use the formulae is to apply the one where the output value is also the subject of the formula, which means that if the output is in square rods, we use the first formula, while if
the output is in acres, we use the second formula. Let’s demonstrate the usage of each of those 2 formulae in the examples below.
EXAMPLE 1: A small part of a lake has been dedicated to parking canoes. If that area is 0.75 acres, what is the area in square rods?
Since our output is in square rods, we choose the first formula. There, we substitute 0.75 for AC and count as follows.
r^2 = ac*160 = 0.75*160 \\= 120 ~square ~rods
EXAMPLE 2: What is the area in acres of a pond that has an area of 360 square rods?
We will use the second formula, as our output is in acres and the input is in square rods. We substitute 360 for R^2 and count as follows.
ac = r^2\div160 \\= 360\div160 = 2.25 ~ac | {"url":"https://calculatorbox.com/calculator/square-rods-and-acres-converter/","timestamp":"2024-11-10T22:09:16Z","content_type":"text/html","content_length":"151932","record_id":"<urn:uuid:6eeadacb-6e1c-4df2-98bc-a734cac85700>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00699.warc.gz"} |
On the topology of manifolds admitting cascades with hyperbolic non-wandering sets
CitationGrines V. Z. ''On the topology of manifolds admitting cascades with hyperbolic non-wandering sets'' [Electronic resource]. Proceedings of the XV International scientific conference
"Differential equations and their applications in mathematical modeling". (Saransk, July 15-18, 2021). Saransk: SVMO Publ, 2021. - pp. 41-42. Available at: https://conf.svmo.ru/files/2021/papers/
paper12.pdf. - Date of access: 14.11.2024. | {"url":"https://conf.svmo.ru/en/archive/article?id=315","timestamp":"2024-11-14T01:19:42Z","content_type":"text/html","content_length":"10659","record_id":"<urn:uuid:2ab6a430-3c6e-478d-99cf-5c20fec43a7b>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00316.warc.gz"} |
Converting Minutes to Hours – Understanding What 1000 Minutes Equals in Hours
Understanding Time Conversion: What is 1000 Minutes in Hours?
Have you ever wondered what 1000 minutes equals in hours? Time conversion is a fundamental concept that we encounter in our daily lives, whether we’re planning our schedules, measuring durations, or
simply trying to make sense of the passage of time. In this blog post, we will explore the process of converting minutes to hours and provide a step-by-step guide to help you understand what 1000
minutes represents in hours.
Understanding the Basics of Converting Minutes to Hours
Before we delve into the conversion of 1000 minutes to hours, let’s take a moment to understand the basics of converting minutes to hours. Time conversion is essential because it allows us to express
durations in a more convenient and standardized format. Instead of dealing with large numbers of minutes, converting to hours helps us simplify our measurements.
The formula for converting minutes to hours is quite simple. You divide the number of minutes by 60, as there are 60 minutes in an hour. This conversion formula ensures that we can easily calculate
how many hours correspond to a given number of minutes.
Let’s take a look at a few examples to solidify our understanding:
Example 1: If we have 120 minutes, dividing by 60 gives us:
120 ÷ 60 = 2 hours
Example 2: If we have 180 minutes, dividing by 60 gives us:
180 ÷ 60 = 3 hours
As you can see, dividing the number of minutes by 60 gives us the equivalent number of hours.
Exploring the Conversion of 1000 Minutes to Hours
Now, let’s focus on the conversion of 1000 minutes to hours – the specific value that we are interested in. Understanding this conversion is particularly useful in various real-life scenarios, such
as planning projects, calculating travel times, or determining the duration of events.
Step 1: To convert 1000 minutes to hours, we divide by 60:
1000 ÷ 60 = 16.67
Step 2: The result of dividing 1000 minutes by 60 is 16.67 hours. However, since hours are typically expressed in decimal form, it’s helpful to convert this to a more familiar format.
Step 3: Convert the decimal value to a fractional or mixed fraction form:
16.67 hours ≈ 16 hours and 40 minutes
Therefore, 1000 minutes is equivalent to approximately 16 hours and 40 minutes.
Real-life Applications of Converting 1000 Minutes to Hours
Understanding the conversion of 1000 minutes to hours is beneficial in various real-life situations. Let’s explore a few practical applications:
1. Work: When planning projects or estimating the time needed for specific tasks, knowing how long 1000 minutes translates to in hours can help with scheduling and time management.
2. Travel: If you are planning a journey with an estimated travel time of 1000 minutes, converting this to hours can give you a better understanding of the duration of your trip and help with
planning stops or estimating arrival times.
3. Sports: In sports events or competitions where timing is crucial, understanding the conversion of 1000 minutes to hours allows athletes and organizers to accurately determine and communicate
timing information.
In all these instances, having a grasp of time conversion, specifically converting minutes to hours, is essential for effective planning and communication.
Common Mistakes to Avoid when Converting Minutes to Hours
While converting minutes to hours may seem straightforward, there are a few common errors that people often encounter. Here are some mistakes to avoid:
1. Neglecting to divide by 60: Remember, the conversion factor to go from minutes to hours is dividing by 60. Failing to apply this step correctly can lead to inaccurate results.
2. Confusing decimals and fractions: It’s important to convert decimal values to fractional form for easier interpretation. Mistaking the two can result in misrepresenting the converted value.
3. Rounding inaccurately: When expressing the converted value in fractional form, ensure that you round correctly to the nearest minute, especially when dealing with values that have decimal places.
By being mindful of these common pitfalls, you can avoid errors and ensure accurate time conversions.
Converting minutes to hours is a fundamental skill that allows us to make sense of time durations and simplify our measurements. By understanding the conversion of 1000 minutes to hours, we can
better plan our schedules, estimate travel durations, and communicate timing information accurately.
Remember, to convert minutes to hours, divide the number of minutes by 60. In the case of 1000 minutes, it is equal to approximately 16 hours and 40 minutes. Applying this knowledge in various
aspects of life can lead to better time management and improved planning.
So the next time someone asks you, “What is 1000 minutes in hours?” You can confidently answer and help them understand the conversion process. Embrace the power of time conversion and apply it to
enhance your everyday life! | {"url":"https://skillapp.co/blog/converting-minutes-to-hours-understanding-what-1000-minutes-equals-in-hours/","timestamp":"2024-11-11T04:10:07Z","content_type":"text/html","content_length":"110051","record_id":"<urn:uuid:57cfb7ad-6cb9-414c-b307-a516ca58f35f>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00162.warc.gz"} |
In the early days of programming it was impossible to do anything without knowing about bits and bytes. Nowadays programming languages are sufficiently far abstracted from the details of what really
happens in the computer's processor and memory that it is no longer a requirement. Indeed many IT professionals today have little or no knowledge of the subject - they just don't need it.
So skip this page if you wish but it will give you a better understanding of how computers work and in our opinion that means you will be able to write programs that will perform better.
A computer is really made up of tiny switches, implemented in solid state circuitry around transistors. One such switch can have two states: on or off.
In arithmetic a binary digit similarly has 2 states: 0 or 1. So binary numbers are the natural representation of data storage in a computer at the lowest level. The term binary digit is abbreviated
to "bit".
Two binary digits together allow 4 states: 00, 01, 10 and 11.
3 bits allow 8 states. The general pattern is that n bits are enough to represent 2^n distinct values. So for 4 bits we have:
Binary Decimal Hexadecimal
equivalent equivalent
1010 10 A
1011 11 B
1100 12 C
1101 13 D
1110 14 E
1111 15 F
The table shows that 4 bits can very conveniently be represented by a single hexadecimal (base 16) digit. We already met this fact in Part 1 in the box about HTML colours: so you see, knowledge of
bits can help.
4 bits is sometimes referred to as a "nibble" because it is half a byte: a byte is 8 bits. So a byte can store 256 possible values, from 0 to 255 inclusive. To store larger numbers we group several
bytes together. So in JavaScript the general value of type Number requires 8 bytes (or 64 bits, of which 11 are used for the power of 2).
What about text?
We have seen that basically the computer only manipulates binary numbers. Characters of text are represented by numerical codes. In the early days, when only the Roman alphabet, digits, punctuation
and a few other symbols were required, 128 codes were enough, so the character set (called ASCII) fitted into 1 byte per character (with 1 bit to spare, often used as a parity check because the
electronics were less reliable then). In the early 1990s the Unicode character set was specified, to cater for all written languages in the world. That is a multi-byte-per-character set but it is so
arranged that the first 128 characters are the same as the original ASCII set. For most purposes 2 bytes are sufficient (65,536 values) and that is the basis of the \uxxxx notation (see next page)
for using non-ASCII characters in JavaScript (4 hex digits x).
Operators on bits
Bitwise logic:
Corresponding pairs of bits in the two operands are operated on by logical operators:
& AND
| OR
^ XOR, exclusive OR: a or b but not both
~ NOT, 1's complement (ie, swap 0s and 1s)
Bit shifting:
For these operations it is necessary to know that the left-most (ie, most significant) bit is often used to represent the numerical sign. There are 2 states again: + or -. To obtain the negative of a
signed number it is necessary to do an operation called a 2's complement which is the same as a 1's complement (~) followed by adding 1.
shift left by 1 bit, bringing 0 in from the right; equivalent to doubling
shift right 1 bit, signed (copy the sign bit); equivalent to halving
right, unsigned (bring 0 in from the left)
Eg, a = b << 3; // Shift b left 3 bits (fill with zeroes)
This example, shifting left by 3 bits, is equivalent to multiplying by 8. So notice that multiplying or dividing by powers of 2 can be done efficiently by shifting. Shifts are usually fundamental
operations in the lowest level instruction set of the machine and therefore faster than normal arithmetic. The JavaScript interpreter in your browser is likely to take advantage of this when it can,
to improve speed. Therefore, when scaling anything consider whether a power of 2 would be a suitable factor. | {"url":"http://www.grelf.net/jscourse/bits.html","timestamp":"2024-11-08T06:16:16Z","content_type":"application/xhtml+xml","content_length":"6595","record_id":"<urn:uuid:f7192d35-82f8-4373-aa1b-462db3416d76>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00828.warc.gz"} |
Ch. 28 Problems & Exercises - College Physics for AP® Courses 2e | OpenStax
28.2 Simultaneity And Time Dilation
(a) What is $γγ$ if $v=0.250cv=0.250c$? (b) If $v=0.500cv=0.500c$?
(a) What is $γγ$ if $v=0.100cv=0.100c$? (b) If $v=0.900cv=0.900c$?
Particles called $ππ$-mesons are produced by accelerator beams. If these particles travel at $2.70×108m/s2.70×108m/s$ and live $2.60×10−8s2.60×10−8s$ when at rest relative to an observer, how long do
they live as viewed in the laboratory?
Suppose a particle called a kaon is created by cosmic radiation striking the atmosphere. It moves by you at $0.980c0.980c$, and it lives $1.24×10−8s1.24×10−8s$ when at rest relative to an observer.
How long does it live as you observe it?
A neutral $ππ$-meson is a particle that can be created by accelerator beams. If one such particle lives $1.40×10−16s1.40×10−16s$ as measured in the laboratory, and $0.840×10−16s0.840×10−16s$ when at
rest relative to an observer, what is its velocity relative to the laboratory?
A neutron lives 900 s when at rest relative to an observer. How fast is the neutron moving relative to an observer who measures its life span to be 2065 s?
If relativistic effects are to be less than 1%, then $γγ$ must be less than 1.01. At what relative velocity is $γ=1.01γ=1.01$?
If relativistic effects are to be less than 3%, then $γγ$ must be less than 1.03. At what relative velocity is $γ=1.03γ=1.03$?
(a) At what relative velocity is $γ=1.50γ=1.50$? (b) At what relative velocity is $γ=100γ=100$?
(a) At what relative velocity is $γ=2.00γ=2.00$? (b) At what relative velocity is $γ=10.0γ=10.0$?
Unreasonable Results
(a) Find the value of $γγ$ for the following situation. An Earth-bound observer measures 23.9 h to have passed while signals from a high-velocity space probe indicate that $24.0 h24.0 h$ have passed
on board. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
28.3 Length Contraction
A spaceship, 200 m long as seen on board, moves by the Earth at $0.970c0.970c$. What is its length as measured by an Earth-bound observer?
How fast would a 6.0 m-long sports car have to be going past you in order for it to appear only 5.5 m long?
(a) How far does the muon in Example 28.1 travel according to the Earth-bound observer? (b) How far does it travel as viewed by an observer moving with it? Base your calculation on its velocity
relative to the Earth and the time it lives (proper time). (c) Verify that these two distances are related through length contraction $γ=3.20γ=3.20$.
(a) How long would the muon in Example 28.1 have lived as observed on the Earth if its velocity was $0.0500c0.0500c$? (b) How far would it have traveled as observed on the Earth? (c) What distance is
this in the muon’s frame?
(a) How long does it take the astronaut in Example 28.2 to travel 4.30 ly at $0.99944c0.99944c$ (as measured by the Earth-bound observer)? (b) How long does it take according to the astronaut? (c)
Verify that these two times are related through time dilation with $γ=30.00γ=30.00$ as given.
(a) How fast would an athlete need to be running for a 100-m race to look 100 yd long? (b) Is the answer consistent with the fact that relativistic effects are difficult to observe in ordinary
circumstances? Explain.
Unreasonable Results
(a) Find the value of $γγ$ for the following situation. An astronaut measures the length of her spaceship to be 25.0 m, while an Earth-bound observer measures it to be 100 m. (b) What is unreasonable
about this result? (c) Which assumptions are unreasonable or inconsistent?
Unreasonable Results
A spaceship is heading directly toward the Earth at a velocity of $0.800c0.800c$. The astronaut on board claims that he can send a canister toward the Earth at $1.20c1.20c$ relative to the Earth. (a)
Calculate the velocity the canister must have relative to the spaceship. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
28.4 Relativistic Addition of Velocities
Suppose a spaceship heading straight towards the Earth at $0.750c0.750c$ can shoot a canister at $0.500c0.500c$ relative to the ship. (a) What is the velocity of the canister relative to the Earth,
if it is shot directly at the Earth? (b) If it is shot directly away from the Earth?
Repeat the previous problem with the ship heading directly away from the Earth.
If a spaceship is approaching the Earth at $0.100c0.100c$ and a message capsule is sent toward it at $0.100c0.100c$ relative to the Earth, what is the speed of the capsule relative to the ship?
(a) Suppose the speed of light were only $3000 m/s3000 m/s$. A jet fighter moving toward a target on the ground at $800 m/s800 m/s$ shoots bullets, each having a muzzle velocity of $1000 m/s1000 m/s$
. What are the bullets’ velocity relative to the target? (b) If the speed of light was this small, would you observe relativistic effects in everyday life? Discuss.
If a galaxy moving away from the Earth has a speed of $1000 km/s1000 km/s$ and emits $656 nm656 nm$ light characteristic of hydrogen (the most common element in the universe). (a) What wavelength
would we observe on the Earth? (b) What type of electromagnetic radiation is this? (c) Why is the speed of the Earth in its orbit negligible here?
A space probe speeding towards the nearest star moves at $0.250c0.250c$ and sends radio information at a broadcast frequency of 1.00 GHz. What frequency is received on the Earth?
If two spaceships are heading directly towards each other at $0.800c0.800c$, at what speed must a canister be shot from the first ship to approach the other at $0.999c0.999c$ as seen by the second
Two planets are on a collision course, heading directly towards each other at $0.250c0.250c$. A spaceship sent from one planet approaches the second at $0.750c0.750c$ as seen by the second planet.
What is the velocity of the ship relative to the first planet?
When a missile is shot from one spaceship towards another, it leaves the first at $0.950c0.950c$ and approaches the other at $0.750c0.750c$. What is the relative velocity of the two ships?
What is the relative velocity of two spaceships if one fires a missile at the other at $0.750c0.750c$ and the other observes it to approach at $0.950c0.950c$?
Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We receive 1900 nm electromagnetic radiation and know that it was 1875 nm when emitted by
the hydrogen gas. What is the speed of the gas?
A highway patrol officer uses a device that measures the speed of vehicles by bouncing radar off them and measuring the Doppler shift. The outgoing radar has a frequency of 100 GHz and the returning
echo has a frequency 15.0 kHz higher. What is the velocity of the vehicle? Note that there are two Doppler shifts in echoes. Be certain not to round off until the end of the problem, because the
effect is small.
Prove that for any relative velocity $vv$ between two observers, a beam of light sent from one to the other will approach at speed $cc$ (provided that $vv$ is less than $cc$, of course).
Show that for any relative velocity $vv$ between two observers, a beam of light projected by one directly away from the other will move away at the speed of light (provided that $vv$ is less than
$cc$, of course).
(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy $12.0×109 ly12.0×109 ly$ ly away is receding from us at 0.$0.900c0.900c$, at what velocity relative to us must
we send an exploratory probe to approach the other galaxy at $0.990c0.990c$, as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from the Earth?
You may assume that the velocity of the other galaxy remains constant. (c) How long will it then take for a radio signal to be beamed back? (All of this is possible in principle, but not practical.)
28.5 Relativistic Momentum
Find the momentum of a helium nucleus having a mass of $6.68×10–27 kg6.68×10–27 kg$ that is moving at $0.200c0.200c$.
What is the momentum of an electron traveling at $0.980c0.980c$?
(a) Find the momentum of a $1.00×109 kg1.00×109 kg$ asteroid heading towards the Earth at $30.0 km/s30.0 km/s$. (b) Find the ratio of this momentum to the classical momentum. (Hint: Use the
approximation that $γ=1+(1/2)v2/c2γ=1+(1/2)v2/c2$ at low velocities.)
(a) What is the momentum of a 2000 kg satellite orbiting at 4.00 km/s? (b) Find the ratio of this momentum to the classical momentum. (Hint: Use the approximation that $γ=1+(1/2)v2/c2γ=1+(1/2)v2/c2$
at low velocities.)
What is the velocity of an electron that has a momentum of $3.04×10–21 kg⋅m/s3.04×10–21 kg⋅m/s$? Note that you must calculate the velocity to at least four digits to see the difference from $cc$.
Find the velocity of a proton that has a momentum of $4.48×–10-19 kg⋅m/s.4.48×–10-19 kg⋅m/s.$
(a) Calculate the speed of a $1.00-μg1.00-μg$ particle of dust that has the same momentum as a proton moving at $0.999c0.999c$. (b) What does the small speed tell us about the mass of a proton
compared to even a tiny amount of macroscopic matter?
(a) Calculate $γγ$ for a proton that has a momentum of $1.00 kg⋅m/s.1.00 kg⋅m/s.$ (b) What is its speed? Such protons form a rare component of cosmic radiation with uncertain origins.
28.6 Relativistic Energy
What is the rest energy of an electron, given its mass is $9.11×10−31 kg9.11×10−31 kg$? Give your answer in joules and MeV.
Find the rest energy in joules and MeV of a proton, given its mass is $1.67×10−27 kg1.67×10−27 kg$.
If the rest energies of a proton and a neutron (the two constituents of nuclei) are 938.3 and 939.6 MeV respectively, what is the difference in their masses in kilograms?
The Big Bang that began the universe is estimated to have released $1068 J1068 J$ of energy. How many stars could half this energy create, assuming the average star’s mass is $4.00×1030 kg4.00×1030
A supernova explosion of a $2.00×1031 kg2.00×1031 kg$ star produces $1.00×1044 J1.00×1044 J$ of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio
$Δm/mΔm/m$ of mass destroyed to the original mass of the star?
(a) Using data from Table 7.1, calculate the mass converted to energy by the fission of 1.00 kg of uranium. (b) What is the ratio of mass destroyed to the original mass, $Δm/mΔm/m$?
(a) Using data from Table 7.1, calculate the amount of mass converted to energy by the fusion of 1.00 kg of hydrogen. (b) What is the ratio of mass destroyed to the original mass, $Δm/mΔm/m$? (c) How
does this compare with $Δm/mΔm/m$ for the fission of 1.00 kg of uranium?
There is approximately $1034J1034J$ of energy available from fusion of hydrogen in the world’s oceans. (a) If $1033 J1033 J$ of this energy were utilized, what would be the decrease in mass of the
oceans? Assume that 0.08% of the mass of a water molecule is converted to energy during the fusion of hydrogen. (b) How great a volume of water does this correspond to? (c) Comment on whether this is
a significant fraction of the total mass of the oceans.
A muon has a rest mass energy of 105.7 MeV, and it decays into an electron and a massless particle. (a) If all the lost mass is converted into the electron’s kinetic energy, find $γγ$ for the
electron. (b) What is the electron’s velocity?
A $ππ$-meson is a particle that decays into a muon and a massless particle. The $ππ$-meson has a rest mass energy of 139.6 MeV, and the muon has a rest mass energy of 105.7 MeV. Suppose the $ππ$
-meson is at rest and all of the missing mass goes into the muon’s kinetic energy. How fast will the muon move?
(a) Calculate the relativistic kinetic energy of a 1000-kg car moving at 30.0 m/s if the speed of light were only 45.0 m/s. (b) Find the ratio of the relativistic kinetic energy to classical.
Alpha decay is nuclear decay in which a helium nucleus is emitted. If the helium nucleus has a mass of $6.80×10−27 kg6.80×10−27 kg$ and is given 5.00 MeV of kinetic energy, what is its velocity?
(a) Beta decay is nuclear decay in which an electron is emitted. If the electron is given 0.750 MeV of kinetic energy, what is its velocity? (b) Comment on how the high velocity is consistent with
the kinetic energy as it compares to the rest mass energy of the electron.
A positron is an antimatter version of the electron, having exactly the same mass. When a positron and an electron meet, they annihilate, converting all of their mass into energy. (a) Find the energy
released, assuming negligible kinetic energy before the annihilation. (b) If this energy is given to a proton in the form of kinetic energy, what is its velocity? (c) If this energy is given to
another electron in the form of kinetic energy, what is its velocity?
What is the kinetic energy in MeV of a $ππ$-meson that lives $1.40×10−16 s1.40×10−16 s$ as measured in the laboratory, and $0.840×10−16 s0.840×10−16 s$ when at rest relative to an observer, given
that its rest energy is 135 MeV?
Find the kinetic energy in MeV of a neutron with a measured life span of 2065 s, given its rest energy is 939.6 MeV, and rest life span is 900s.
(a) Show that $(pc)2/(mc2)2=γ2−1(pc)2/(mc2)2=γ2−1$. This means that at large velocities $pc>>mc2pc>>mc2$. (b) Is $E≈pcE≈pc$ when $γ=30.0γ=30.0$, as for the astronaut discussed in the twin paradox?
One cosmic ray neutron has a velocity of $0.250c0.250c$ relative to the Earth. (a) What is the neutron’s total energy in MeV? (b) Find its momentum. (c) Is $E≈pcE≈pc$ in this situation? Discuss in
terms of the equation given in part (a) of the previous problem.
What is $γγ$ for a proton having a mass energy of 938.3 MeV accelerated through an effective potential of 1.0 TV (teravolt) at Fermilab outside Chicago?
(a) What is the effective accelerating potential for electrons at the Stanford Linear Accelerator, if $γ=1.00×105γ=1.00×105$ for them? (b) What is their total energy (nearly the same as kinetic in
this case) in GeV?
(a) Using data from Table 7.1, find the mass destroyed when the energy in a barrel of crude oil is released. (b) Given these barrels contain 200 liters and assuming the density of crude oil is $750
kg/m3750 kg/m3$, what is the ratio of mass destroyed to original mass, $Δm/mΔm/m$?
(a) Calculate the energy released by the destruction of 1.00 kg of mass. (b) How many kilograms could be lifted to a 10.0 km height by this amount of energy?
A Van de Graaff accelerator utilizes a 50.0 MV potential difference to accelerate charged particles such as protons. (a) What is the velocity of a proton accelerated by such a potential? (b) An
Suppose you use an average of $500 kW·h500 kW·h$ of electric energy per month in your home. (a) How long would 1.00 g of mass converted to electric energy with an efficiency of 38.0% last you? (b)
How many homes could be supplied at the $500 kW·h500 kW·h$ per month rate for one year by the energy from the described mass conversion?
(a) A nuclear power plant converts energy from nuclear fission into electricity with an efficiency of 35.0%. How much mass is destroyed in one year to produce a continuous 1000 MW of electric power?
(b) Do you think it would be possible to observe this mass loss if the total mass of the fuel is $104kg104kg$?
Nuclear-powered rockets were researched for some years before safety concerns became paramount. (a) What fraction of a rocket’s mass would have to be destroyed to get it into a low Earth orbit,
neglecting the decrease in gravity? (Assume an orbital altitude of 250 km, and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) (b) If the ship has a mass
of $1.00×105kg1.00×105kg$ (100 tons), what total yield nuclear explosion in tons of TNT is needed?
The Sun produces energy at a rate of $4.00×10264.00×1026$ W by the fusion of hydrogen. (a) How many kilograms of hydrogen undergo fusion each second? (b) If the Sun is 90.0% hydrogen and half of this
can undergo fusion before the Sun changes character, how long could it produce energy at its current rate? (c) How many kilograms of mass is the Sun losing per second? (d) What fraction of its mass
will it have lost in the time found in part (b)?
Unreasonable Results
A proton has a mass of $1.67×10−27kg1.67×10−27kg$. A physicist measures the proton’s total energy to be 50.0 MeV. (a) What is the proton’s kinetic energy? (b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
Construct Your Own Problem
Consider a highly relativistic particle. Discuss what is meant by the term “highly relativistic.” (Note that, in part, it means that the particle cannot be massless.) Construct a problem in which you
calculate the wavelength of such a particle and show that it is very nearly the same as the wavelength of a massless particle, such as a photon, with the same energy. Among the things to be
considered are the rest energy of the particle (it should be a known particle) and its total energy, which should be large compared to its rest energy.
Construct Your Own Problem
Consider an astronaut traveling to another star at a relativistic velocity. Construct a problem in which you calculate the time for the trip as observed on the Earth and as observed by the astronaut.
Also calculate the amount of mass that must be converted to energy to get the astronaut and ship to the velocity travelled. Among the things to be considered are the distance to the star, the
velocity, and the mass of the astronaut and ship. Unless your instructor directs you otherwise, do not include any energy given to other masses, such as rocket propellants.
Critical Thinking A space rock with a length of 1,000.0 m is moving through space at exactly 0.6 c. (a) If the space rock is moving toward an observer, what is the contracted length observed? (b) If
the space rock is moving away from the observer, what is the contracted length observed? (c) Can the object reach the speed of light? (d) If the object were to stop in the observer’s reference frame,
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Iwamura’s Lemma, Markowsky’s Theorem and ordinals
On p.61 of the book, there is a remark that the dcpos are exactly the chain-complete posets. This is a theorem by George Markowsky [1]. It is time I explained seriously how this worked. The first
step is Iwamura’s Lemma [2], which states that every directed subset decomposes as the union of a small chain of small directed subsets.
The reason I did not put the proof of that result in the book is because it rests on using ordinals, and I did not want to introduce ordinals, specially if they served for only one result. I’ll need
them badly here, mostly to explain what “small” means in the informal statement of the lemma given above.
Ordinals are a generalization of natural numbers “into the transfinite”. That is, ordinals contain 0, are closed under successors (adding 1), … and also under taking suprema of chains. And there is
an induction principle that states that ordinals form the smallest collection that has these properties.
Every natural number is an ordinal, but there are many more. For example, the set of natural numbers itself forms a chain, so their supremum is an ordinal. This ordinal is written ω, and is the first
infinite ordinal. Then you can form ω+1, ω+2, …, ω+n for every natural number n. Their supremum is an even higher ordinal, written ω+ω, or ω.2 (not 2.ω! the latter would be the supremum of the
family 0, 2, 4, …, 2n, …, hence equal to ω). Again, you can form ω.2+1, ω.2+2, …, ω.2+ω = ω.3.
At each step, we build larger ordinals. And we can go even further. For example, the chain 0, ω, ω.2, ω.3, …, ω.n, … again has a supremum, written ω.ω, or ω^2. I’ll let you build ω^3, ω^4, …, ω^n,
also their supremum ω^ω, etc. We have barely scratched the surface.
All that is good and dandy, but I really have not defined ordinals yet. For that, I need to say what a chain of ordinals is, and what their supremum is, so I need to define how they are ordered.
Let us see how this can be defined formally.
In set theory, 0 is an abbreviation for the empty set ∅, and α+1 is an abbreviation for the funny α ∪ {α}. So 1 is encoded as {∅}, 2 as {∅, {∅}}, 3 as {∅, {∅}, {∅, {∅}}}, and so on. The point of
this weird convention is that every natural number n is encoded as the set {0, 1, …,n-1} of its predecessors. We shall encode ordinals in a similar way, encoding every ordinal α as the set of
ordinals β strictly less than α.
Note that if we do so, then strict ordering < on ordinals is just set membership: β<α if and only if β∈α. Also, the ordering ≤ on ordinals will be set inclusion: β≤α if and only if β⊆α.
Finally, note that, the way I explained ordinals, it should be clear that they are totally ordered. Given two ordinals α and β, either α≤β, or β<α. Equivalently, there is trichotomy between three
exclusive cases: α∈β, α=β, or β∈α.
This is one of the standard definitions of an ordinal:
Definition. An ordinal is a transitive trichotomous set, where:
• A set is transitive if and only if, for element α of this set, all the elements of α are in the set.
• A set is trichotomous if and only if, for any two elements α and β of this set, α∈β, α=β, or β∈α.
Sometimes you’ll find in the literature that it is required to be well-founded as well. However we have assumed every set to be well-founded in the Von Neumann-Bernays-Gödel axiomatization given at
the beginning of the book.
I’ll let you check that 0 is an ordinal in this sense, and that for every ordinal α, α+1 is again an ordinal. We can now make sense of chains and suprema: a chain is one for the inclusion ordering,
and suprema of chains are just unions. I’ll let you check that any union of a chain of ordinals is an ordinal.
By the way, the ordinals that one cannot write down as 0 or of the form α+1 are called limit ordinals. The only way we can construct a limit ordinal is as a supremum of strictly smaller ordinals.
For example, the first limit ordinal is ω. The next ones are ω.2, ω.3, …, ω.n, etc.
There is much one can say about ordinals, and the Wikipedia page on ordinals is a good start.
The most important properties of ordinals are probably the following:
• Every ordinal is a well-founded chain (of ordinals), ordered by ≤ (inclusion)
• Every well-founded, totally ordered set is order-isomorphic to a unique ordinal.
The first property is because set membership is well-founded. The second property is shown by well-founded induction on the given set.
The first property allows one to prove properties by (well-founded) induction on ordinals: to show that a property P holds of all ordinals, prove it on 0, show that it holds of α+1 as soon as it
holds of α, and show that for every limit ordinal α, if P holds of all ordinals smaller than α, then P also holds of α.
Recall that a set I has cardinality smaller than or equal to that of a set J if and only if there is an injective map from I into J, or equivalently if and only if there is a surjective map from J
onto I.
Zermelo’s Theorem asserts that every set I can be equipped with a well-founded, total ordering. I’ll give a sketch of a proof shortly.
It follows that every set I can be put in bijection with some ordinal. Since ordinals are well-founded, there is a smallest ordinal in bijection with I (for the usual ordering on ordinals). This is
called the cardinality of I.
Such smallest ordinals are called cardinals. By definition, these are ordinals that are in bijection with none of their elements. For example, 0, 1, 2, …, n, … are cardinals. The first infinite
ordinal ω is also a cardinal, because it is infinite, but all smaller ordinals are natural numbers n, hence are finite sets {0, 1, …, n-1}. As a cardinal, ω is also written aleph 0. The next
infinite cardinal, aleph 1, is pretty mysterious. As an ordinal, it is much higher than all the ordinals we have enumerated at the beginning (ω, ω.2, ω.3, ω^2, ω^ω, etc.) It is smaller than or
equal to the cardinality of the powerset of ω, simply because the latter is strictly larger than the cardinal of ω, by Cantor’s Theorem 2.2.1. Whether it is equal to it or not is in fact unprovable
from VBG set theory alone, as shown by Paul Cohen in the early 1960s.
In any case, what all this boils down to is:
• Every set I is in bijection with a smallest ordinal; this ordinal is called the cardinality |I| of I;
• I has smaller cardinality than J (in the sense that there is an injection from I into J, equivalently a surjection from J onto I) if and only if |I| < |J| (the smaller than relation for ordinals
• In particular, the “smaller cardinality than” relation is well-founded (!). We shall need this below.
Before we continue, let me state and prove Zermelo’s Theorem:
Theorem (Zermelo). Every set I can be equipped with a well-founded, total ordering.
Consider pairs (E, ≤) of a subset E of I and a well-founded, total ordering ≤ on E. These pairs are ordered by extension: (E, ≤) is below (E‘, ≤’) if and only if E is a downward closed subset of E’
with respect to ≤’, and ≤ is the restriction of ≤’ to E. Under extension, the set of those pairs is inductive. Zorn’s Lemma gives us a maximal pair (E, ≤). If E was not the whole of I, say there
is a point i in I that is not in E, then we could find a larger pair (E ∪ {i}, ≤’) where ≤’ is defined so that i is the new top element. So E=I.
A curious theorem
We have everything we need to prove Iwamura’s Lemma, but let’s not go too fast. Achim Jung suggested that I presented the following, curious result first. Its proof has the same canvas as for
Iwamura’s Lemma, which we shall see next.
We let |D| denote the cardinality of D, represented as an ordinal.
Proposition. Every infinite set can D be written as a union of a chain of subsets D[α] of strictly smaller cardinality, indexed by ordinals α<|D|.
So we have a small set of subsets D[α], in the sense that there are at most |D| of them, and each of these sets is small, in the sense that they have strictly smaller cardinality than D.
The result is curious. For example, if you were to write {0, 1, 2, 3, 4} as a union of a chain of subsets, then one of them would have to be the whole of {0, 1, 2, 3, 4}, whose cardinality is
certainly not strictly smaller than that of the whole set. The proposition holds because D is infinite.
For the proof, we index the elements of D as x[α], α<|D|. This is the definition of |D|, which, as we have noted, exists by Zermelo’s Theorem. We let D[0] = ∅, D[α+1] = D[α] ∪ {x[α]}, and, for
every limit ordinal α, D[α] = ∪[β<α]D[β]. We check that for each of the involved ordinals α, |D[α]| = α < |D|, and we are done.
Iwamura’s Lemma
In 1944, Iwamura proved the following [2].
Lemma (Iwamura). Let X be a poset, and D be an infinite directed subset of X. Then one can write D as the union of a chain of directed subsets D[α], indexed by ordinals α<|D|, such that:
• |D[α]| < |D|
• if α<β then D[α] is included in D[β]
In other words, every directed subset D can be decomposed as the union of a small chain of small directed subsets, where “small” means of cardinality strictly smaller than that of D.
• The proof rests on the following construction. For every finite subset E of D, fix an upper bound Ê of E in D. This exists because D is directed, and we can fix the map from E to Ê once and for
all by using the Axiom of Choice.
□ Given an infinite directed subset D‘ of D, and a point x in D, we can form the smallest subset of D‘ that contains D’ and x, and such that for every finite subset E of D’, Ê is again in D’.
Call that subset D’+x. Note that D’+x is directed.
One can show that D’+x has exactly the same cardinality as D’. Here is how. D’+x is the least fixed point of a Scott-continuous operator T on the collection of infinite subsets of D that
contain D’ and x, defined by T(A) = A ∪ {Ê | E finite included in A}. So D’+x is a countable union of sets of the form T^n(∅), n∈ N. Because A is infinite, one can show that T(A) has
exactly the same cardinality as A. Also, a countable union of sets of infinite cardinality c again has cardinality c. We conclude that D’+x has exactly the same cardinality as D’.
□ If D’ is a finite directed subset of D, that construction would be too large for our purposes. Instead, we define D’+x in this case as the union of E = D’ ∪ {x}, and {Ê}. This is again
directed, but now D’+x is guaranteed to be finite.
• We now construct D[α] by induction on α<|D|. By the definition of |D|, we can index the elements of D as x[α], α<|D|. We let D[0] = {x[0]}, D[α+1] =D[α] + x[α], and, for every limit ordinal α,
D[α] = ∪[β<α]D[β]. Note that, by the considerations above, if α is a finite ordinal, then D[α] is finite, hence |D[α]| < |D| since D is infinite; while in the other cases, |D[α]| ≤ α < |D|.
Markowsky’s Theorem
We can now show Markowsky’s Theorem [1].
Theorem (Markowsky). Every chain-complete poset is a dcpo.
Let X be a chain-complete poset, and D be a directed subset of X. We show by well-founded induction on |D| that D has a supremum.
• If D is finite, then in fact D has a maximal element since D is directed, and this must be its supremum.
• Otherwise, apply Iwamura’s Lemma. Using the notations we have used there, let y[α] be the supremum of D[α] in X. This exists, by induction hypothesis, since |D[α]| < |D|. If α<β then D[α] is
included in D[β], hence y[α]≤y[β], so the elements y[α], α<|D|, form a (well-founded) chain. By assumption, this has a supremum in X, and it is easy to check that this is the desired supremum of
That’s it. As a bonus, we have shown that every “well-founded-chain-complete” poset is also a dcpo, where a “well-founded-chain-complete” poset is a poset in which only the well-founded chains are
required to have a supremum.
On Exercise 4.2.26
Exercise 4.2.26 asks you to show that the chain-open topology on a poset X is just the same as the Scott-open topology. A subset U is chain-open if and only if it is upward-closed and for every
chain C such that sup C is in U, some element of C is in U already.
The proof that I was trying to make you find was as follows. Let F be the complement of U in X. This is a chain-complete poset by assumption, hence a dcpo by Markowsky’s Theorem. Since F is closed
under directed sups, and downward-closed, its complement U is Scott-open.
As it stands, this proof has a flaw. There is no reason that directed suprema, or suprema of chains would be computed in F as in X, and this has to be shown. Formally, let D be a directed family of
elements of F. What we know is that D has a supremum y in F. It may have another supremum in X, or none at all, for what we know. Oops, it must have a supremum in X, because we have assumed that X
was a dcpo. Call it z. Certainly, y is an upper bound of D, even inside X, so y≥z. Since F is closed, it is downward-closed, so z is in F. However, since z is an upper bound of D in X, hence also
in F, z is above the supremum y of D taken in F. We conclude that y and z are equal. That was hard!
Knowing how Markowsky’s Theorem is proved, we can prove the result of the Exercise in a more transparent fashion. Let D be any directed family whose supremum (in X) is in U. We show by induction on
|D| that D must meet U. Using Iwamura’s Lemma, as in the proof of Markowsky’s Theorem, let y[α] be the supremum of D[α] in X. By induction hypothesis, each y[α] is in F. They form a chain, whose
supremum z taken in X is in F. However, z is the supremum of D, taken in X. It follows that F is Scott-closed, hence U is Scott-open.
— Jean Goubault-Larrecq(February 23rd, 2015)
2. Tsurane Iwamura. A lemma on directed sets. Zenkoku Shijo Sugaku Danwakai 262, 1944, pages 107-111. In Japanese. (Let me thank Hideki Tsuiki for checking the name and the reference. If there
is still any mistake here, it will be my fault.) | {"url":"https://topology.lmf.cnrs.fr/iwamuras-lemma-kowalskys-theorem-and-ordinals/","timestamp":"2024-11-03T09:18:56Z","content_type":"text/html","content_length":"63995","record_id":"<urn:uuid:5be7d12a-eacc-457a-b618-c9340bd94de4>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00853.warc.gz"} |
Section 6.10 in Matter and Interactions (4th edition)
Graphing Energy for Gravitationally Interacting Systems
Knowing the equation for the Newtonian gravitational potential energy might help you solve certain problems, but graphing the energy can help you reason about the motion of different systems. In
these notes, you will read about the graph of the gravitational potential energy, how it can tell you about the motion of systems, and how the Near-Earth gravitational potential energy is an
approximation of the Newtonian gravitational potential energy.
Lecture Video
Graphs of Gravitational Potential Energy
You can graph the gravitational potential energy (J) as a function of the radial separation,
$$U(r) = -G\dfrac{Mm}{r}$$
The fact that this potential energy is negative simply means that it is less than the zero of energy. Think about a small object that interacts with a large object, so that only the small object
sufficiently changes its kinetic energy. In those situations, the total energy can be either positive or negative if the kinetic energy of the small object is sufficiently large (i.e., the magnitude
of the gravitational potential energy is smaller than the kinetic energy).
$$\underbrace{E_{tot}}_{+\:or\:-} = \underbrace{\left(\dfrac{1}{2}mv^2\right)}_{+} + \underbrace{\left(-G\dfrac{Mm}{r}\right)}_{-}$$
Visualizing the kinetic energy
The value of the potential energy is measured from the zero line down to the graph's location at any given point (as shown by the red arrows in the figures below). For a gravitational system with a
given constant, total energy ($E_{tot}$, the dotted black lines in the figures below), the kinetic energy of the less massive object ($K$) can be visualized as the distance between the potential
graph up to the total energy line (the blue arrows in the figures below).
Notice that in the figure on the left, the total energy is negative and hence the less massive object cannot get any farther away then the location where the potential energy equals the system's
total energy (i.e., where $K$ goes to zero). This is called a bound system because the less massive object is gravitationally bound to the more massive object and cannot leave that bounded state.
For the figure on the right, the total energy is positive and hence, even at infinite distance, the less massive object has non-zero kinetic energy. This is an unbound system because the less massive
object can move infinitely far away from the more massive object.
How is $\Delta U = mgh$ an approximation?
As you have read, the gravitational force near the surface of the Earth is an approximation of the Newtonian gravitational force. As you might suspect, the gravitational potential energy near the
surface of the Earth (or any large object) can be approximated also. As you have read, this form of the gravitational potential energy increases linearly with distance (i.e., $\Delta U_{grav} = +mg\
Delta y$).
If you zoom in on the graph of the gravitational potential energy, it looks like it increases linearly (figure to the left). You can show mathematically that this will produce the same expected
result (with an additional constant term).
Mathematical Proof of the Approximation
Consider an object of mass $m$ (kg) at a distance $y$ (m) above the Earth's surface (mass, $M_E$; radius, $R_E$). The potential energy of the object-Earth system is:
$$U_{grav} = -G\dfrac{M_Em}{\left(R_E+y\right)} = -G\dfrac{M_Em}{R_E\left(1+\dfrac{y}{R_E}\right)} = -m\dfrac{GM_E}{R_E}\dfrac{1}{\left(1+\dfrac{y}{R_E}\right)}$$
The value of the coefficient $GM_E/R^2_E$ is precisely $g=9.81\:\mathrm{m/s}^2$, so that this equation becomes,
$$U_{grav} = -m\dfrac{GM_E}{R^2_E}\dfrac{R_E}{\left(1+\dfrac{y}{R_E}\right)} = -mg\dfrac{R_E}{\left(1+\dfrac{y}{R_E}\right)}$$
Now for these considerations, the distance above the Earth ($y$) is typically much smaller than the radius of the Earth ($R_E$), so that you can approximate the ratio $h/R_E$ as much smaller than 1.
Using a Taylor expansion gives you,
$$U_{grav} = -mg\dfrac{R_E}{\left(1+\dfrac{y}{R_E}\right)} \approx -mgR_E \left(1-\dfrac{y}{R_E}\right) = -mgR_E + mgy$$
The first term in the above equation is just a constant, so that if you are interested in the change in potential energy (as we usually are), it would drop out,
$$\Delta U = \left( -mgR_E + mgy_f \right)- \left(-mgR_E + mgy_i \right) = \left( mgy_f - mgy_i \right) = mg\Delta y$$
This is just the linear change in height from previous work where the gravitational force was assumed constant.
Graphing Kinetic Energy
It is often the the kinetic energy of the less massive object which is graphed along side the potential energy of the system and the total energy. For a bound system, this graph looks like the one to
the right (green line is the kinetic energy).
The kinetic energy graph has the same characteristic shape as the potential energy graph, but it is a reflected version. As the potential energy gets larger (less negative), the kinetic gets smaller
and vice versa. The kinetic energy cannot become negative, so its graph terminates at zero energy. This is the farthest location the less massive object can reach with the given total energy.
For an unbound system the kinetic energy levels off to the value of the total (positive) energy of the system. When the less massive object is infinitely far away, the potential energy of the system
goes to zero. | {"url":"https://msuperl.org/wikis/pcubed/doku.php?id=183_notes:grav_pe_graphs","timestamp":"2024-11-05T19:07:44Z","content_type":"application/xhtml+xml","content_length":"44505","record_id":"<urn:uuid:833fdf56-7944-4c5d-848a-d22ab8913328>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00743.warc.gz"} |
Introduction to MathMol
People have always been fascinated with objects that can't be seen with the naked eye. One only has to observe students taking their first look at microscopic organisms through a microscope, or the
rings of Saturn through a telescope.
We are now just at the threshold of being able to visualize molecules. Crystallographic techniques, advanced scanning devices and sophisticated software programs have provided scientists with tools
that can provide realistic models for many molecules. The power of the computer is fast approaching the point at which we will soon be able to model molecules and their motion with a fair degree of
Most students get their first exposure to atomic and molecular structure in elementary school by building model atoms or simple crystal lattices such as salt. In middle school, textbooks begin to
introduce simple organic molecules, more advanced crystal structures and water and ice structures. Although all schools now consider the microscope a necessity in the classroom to view microscopic
objects, few schools have provided tools for students to visualize molecules. What is perhaps not known to many educators is that time and effort is all that is necessary --provided the school has a
computer and link to the internet
During the past few years scientists have provided several excellent molecular viewers to the scientific community. Several of these programs are ideally suited for use in K-12 education. These
programs are public domain and simply require knowledge of use and a fairly modern computer (something that most schools now have)
There is still much controversy concerning what grade level molecular visualization should begin. Many educators feel that molecular structure should not be introduced until high school, since it may
confuse students understanding of macroscopic processes. But other educators feel that middle school is actually an ideal starting point for introducing molecules. Project MathMol aims to target
middle school students, introducing basic molecular concepts and linking them to familiar mathematical topics. We hope teachers will integrate some of the MathMol activities into their mathematics
and/or physical science curriculum. Considering the importance of molecular concepts for the 21st century it is important to provide a strong foundation for students as early as possible. | {"url":"https://www.mathmol.net/txtbk3/introduction.html","timestamp":"2024-11-04T08:22:41Z","content_type":"text/html","content_length":"14297","record_id":"<urn:uuid:28cbcf5c-9e6c-405f-9c32-f3cc76acf08f>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00531.warc.gz"} |
How to Convert Days to Hours In Pandas?
To convert days to hours in pandas, you can use the timedelta function along with the apply method. First, you need to create a new column with the number of days you want to convert. Then, you can
use the apply method to convert this column into hours by multiplying each day by 24. Finally, you will get the converted values in hours.
How to convert days to hours in pandas using vectorized operations?
You can convert days to hours in pandas using vectorized operations by multiplying the number of days by 24. Here's an example:
1 import pandas as pd
3 # Create a DataFrame with a column of days
4 df = pd.DataFrame({'days': [1, 2, 3, 4]})
6 # Convert days to hours using vectorized operations
7 df['hours'] = df['days'] * 24
9 # Print the updated DataFrame
10 print(df)
This will output:
1 days hours
In this example, we first create a DataFrame with a column of days. We then use the vectorized operation df['days'] * 24 to convert the days to hours and create a new column called 'hours' in the
DataFrame. Finally, we print the updated DataFrame with the days and hours columns.
What is the difference between converting days to hours in pandas and other Python libraries?
In pandas, converting days to hours can be done using the timedelta function, which is specifically designed to handle time-related calculations. This function allows you to easily convert days to
hours while maintaining the data type as a timedelta object.
On the other hand, in other Python libraries such as datetime or dateutil, you would need to manually calculate the conversion by multiplying the number of days by 24. This can lead to potential
errors or inconsistencies in the data type, as the output may be a float instead of a timedelta object.
Overall, using pandas for time-related calculations such as converting days to hours provides a more efficient and accurate way to handle time data.
How to round the converted hours to a specific decimal place in pandas?
To round the converted hours to a specific decimal place in pandas, you can use the round() method along with the astype() method to convert the data type to float.
Here is an example code snippet:
1 import pandas as pd
3 # Create a DataFrame with converted hours
4 data = {'hours': [5.33333333, 8.66666666, 10.55555555]}
5 df = pd.DataFrame(data)
7 # Round the converted hours to 2 decimal places
8 df['rounded_hours'] = df['hours'].round(2).astype(float)
10 # Display the DataFrame
11 print(df)
In the above code snippet, we first create a DataFrame df with converted hours. Then, we use the round(2) method to round the hours column to 2 decimal places and then use the astype(float) method to
convert the rounded values to float. Finally, we create a new column rounded_hours in the DataFrame to store the rounded values. | {"url":"https://ubuntuask.com/blog/how-to-convert-days-to-hours-in-pandas","timestamp":"2024-11-05T15:37:27Z","content_type":"text/html","content_length":"313651","record_id":"<urn:uuid:c819ce22-0e31-4265-ae9a-3b73308c42e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00199.warc.gz"} |
[HP-12C] Angular coefficent from linear regression off by 1
05-02-2016, 11:48 AM
Post: #5
Werner Posts: 902
Senior Member Joined: Dec 2013
RE: [HP-12C] Angular coefficent from linear regression off by 1
I don't know either the 12C or the 15C very well but I have emulators on my phone ;-)
So, correct me if I'm wrong, but the 12C doesn't have linear regression as a function, just x-intercept and y-intercept. So you calculate the slope in two steps:
y | x=1 minus y | x=0
1 g(y,r) (11.25867251)
0 g(y,r) (0.15934086.19)
If you do the same on a 15C, you get the same result. But a 15C has a f(L.R.)
function delivering the result without 10-digit intermediate calculations. Hence the difference.
Cheers, Werner
41CV†,42S,48GX,49G,DM42,DM41X,17BII,15CE,DM15L,12C,16CE
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taxicab geometry answers
Well that depends. Taxicab Geometry. Will the taxicab distance between two points ever be equal to the Euclidean distance between them? The answer to1 "c" --- check wikipedia "taxicab geometry" None
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Page 8 Day 3: Taxicab Applications 1.Doug moves to Taxicab City and works at the distillery at D= (4; 2). Southwest)ChicagoMath)Teachers’)Circle))) )))))Monthly)Meeting)at)Lewis)University)11/17/16)
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How to Find the Tangent Line: A Comprehensive Guide - The Explanation Express
How to Find the Tangent Line: A Comprehensive Guide
I. Introduction
As you begin to study calculus, understanding tangent lines becomes a fundamental concept. A tangent line is simply the line that touches a curve at a point without crossing it. Being able to find
the tangent line is a key skill for calculating rates of change, determining maximum or minimum points on a curve, and finding the derivative of a function. In this article, we will explore different
methods and strategies for finding tangent lines.
II. The Beginner’s Guide to Finding Tangent Lines
Before we dive into the methods for finding tangent lines, let’s define some key terms. The slope is the measure of the steepness of a line. A point is a specific location on the curve. A curve is a
continuous path made by bending or curving.
To find the slope of a line tangent to a point on a curve, we first need to identify the point where the tangent line touches the curve. From there, we can find the slope of the line using the slope
formula: rise over run, or (y2 – y1) / (x2 – x1).
Let’s take the example of a curve with the equation y = x^2. If we want to find the tangent line at the point (1,1), we would first identify the point and then use the slope formula. The point (1,1)
means that y = 1 when x = 1. To find the slope, we need to identify another point on the curve close to (1,1). We can use (2,4), since when x = 2, y equals 4.
The slope of the line through these two points is (4-1)/(2-1) or 3. Therefore, the equation of the tangent line at (1,1) is y = 3(x-1) + 1.
III. Mastering Tangent Lines: Tips and Tricks
While the basic steps for finding tangent lines remain the same, there are different methods available to use depending on the situation. One of these methods is the limit definition. Using this
method, we calculate the slope of a tangent to a curve by finding the limit of the function as the distance between two points on the curve approaches zero.
Another method is using derivatives. Calculus is based on finding derivatives, which determine the rate at which a function changes. The derivative of a function gives us the slope of the tangent
line at any point on the curve.
When should we use each method? The limit definition can be useful when we don’t have an equation for the curve. On the other hand, using derivatives can be quick and efficient when working with
Regardless of the method chosen, it’s important not to make common mistakes when finding tangent lines. One common mistake is forgetting to find the slope at the exact point of the tangent line. Make
sure to examine the curve carefully to properly identify the point of intersection.
IV. Step-by-Step Guide to Locating Tangent Lines
Now that we’ve covered some basic strategies, let’s dive into some step-by-step instructions for finding tangent lines. We will use both the limit definition and the derivative methods to illustrate
the process.
Finding the Tangent Line Using the Limit Definition
1. Identify the point on the curve where you want to find the tangent line.
2. Select a second point on the curve that is very close to the first point.
3. Find the slope of the line that passes through these two points.
4. Move the second point closer to the first point and find the slope again.
5. Repeat step 4 until the distances between the two points approach zero.
6. The final value of the slope is the slope of the tangent line at the point of intersection.
Finding the Tangent Line Using Derivatives
1. Write the function equation of the curve.
2. Take the derivative of the equation at the point of interest. This will give you the slope of the tangent line at that point.
3. Find the equation of the tangent line using the point-slope formula y – y1 = m(x – x1).
It’s important to note that when using the derivative method, make sure to check the derivative at the point of interest. Additionally, finding the equation of the tangent line may require additional
steps depending on the specific problem.
V. Unlocking the Mystery of Tangent Lines: A Comprehensive Guide
We’ve covered several methods for finding tangent lines, but we haven’t discussed the connection between tangent lines and derivatives yet. A derivative is a value that describes the rate of change
of a function. By finding the derivative of a function at a certain point, we can find the slope of the tangent line to the curve at that point.
Using the previous example of y = x^2, we can take the derivative of the function to find the slope at any point x. The derivative is y = 2x, meaning that at x = 1, the slope is 2. This matches the
slope we found using the first method, showing how derivatives can be used to find the slope of the tangent line.
We can also use the derivative to find the equation of the tangent line directly, rather than just the slope. If we know the derivative of the function and the point where we want to find the tangent
line, we can plug in these values into the point-slope formula and find the equation of the line.
There are real-world applications for tangent lines, especially in the study of physics and engineering. For example, when designing a rollercoaster, engineers need to calculate the slope of the
track at each point in order to design the coaster for rider safety and enjoyment.
VI. No Math Skills Required: How to Find Tangent Lines in Simple Ways
For those who may not have extensive math skills, there are still methods available to find tangent lines. One way is to use a graphing calculator or online graphing tool that can plot a curve and
show the tangent line at a specific point. Simply enter the function equation and the point of interest and the tool will show the graph with the tangent line.
Another method is to estimate the slope of the tangent line by eyeballing it. Look at the curve at the point of interest and estimate the steepness of the line at that point. This method may not be
as precise as the others, but it can still provide a general idea of the tangent line.
VII. Easy and Accurate Ways to Determine Tangent Lines to Curve
In summary, finding the tangent line is an important skill in calculus and is useful in many fields. There are different methods available, including the limit definition and derivative methods. To
avoid common mistakes, make sure to properly identify the point of intersection and check the derivative or slope at that exact point. For those without extensive math skills, graphing tools or
estimating can be used to find the tangent line. Regardless of the method chosen, practice and repetition will help to master this important concept.
VIII. Conclusion
Now that we’ve covered the basics and advanced methods, we hope this article has demystified the process of finding the tangent line. Remember that tangent lines have real-world applications in areas
like engineering and physics, so the ability to find them is an important skill. Whether you use the limit definition, the derivative rule, or estimation, make sure to properly identify the point of
intersection and check your work carefully. | {"url":"https://www.branchor.com/how-to-find-the-tangent-line/","timestamp":"2024-11-04T20:25:35Z","content_type":"text/html","content_length":"45698","record_id":"<urn:uuid:a743c886-065d-417c-b0b2-4c6f3178923a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00416.warc.gz"} |