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Differentiability With Respect To The Initial Condition For Hamilton-jacobi Equations
Esteve-Yague C, Zuazua Iriondo E (2022)
Publication Type: Journal article
Publication year: 2022
Book Volume: 54
Pages Range: 5388-5423
Journal Issue: 5
DOI: 10.1137/22M1469353
We prove that the viscosity solution to a Hamilton-Jacobi equation with a smooth convex Hamiltonian of the form H(x, p) is differentiable with respect to the initial condition. More-over, the
directional Gateaux derivatives can be explicitly computed almost everywhere in R-N by means of the optimality system of the associated optimal control problem. We also prove that, in the
one-dimensional case in space and in the quadratic case in any space dimension, these directional Gateaux derivatives actually correspond to the unique duality solution to the linear transport
equation with discontinuous coefficient, resulting from the linearization of the Hamilton-Jacobi equation. The motivation behind these differentiability results arises from the following optimal
inverse-design problem: given a time horizon T > 0 and a target function uT, construct an initial condition such that the corresponding viscosity solution at time T minimizes the L-2-distance to u
(T). Our differentiability results allow us to derive a necessary first-order optimality condition for this optimization problem and the implementation of gradient-based methods to numerically
approximate the optimal inverse design.
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How to cite
Esteve-Yague, C., & Zuazua Iriondo, E. (2022). Differentiability With Respect To The Initial Condition For Hamilton-jacobi Equations. SIAM Journal on Mathematical Analysis, 54(5), 5388-5423. https://
Esteve-Yague, Carlos, and Enrique Zuazua Iriondo. "Differentiability With Respect To The Initial Condition For Hamilton-jacobi Equations." SIAM Journal on Mathematical Analysis 54.5 (2022):
BibTeX: Download | {"url":"https://cris.fau.de/publications/283175916/","timestamp":"2024-11-12T07:19:18Z","content_type":"text/html","content_length":"11192","record_id":"<urn:uuid:f7f4be85-37b1-43c6-87b0-cddebb1e791f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00454.warc.gz"} |
3.9: Implicit Differentiation
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Learning Objectives
• Find the derivative of a complicated function by using implicit differentiation.
• Use implicit differentiation to determine the equation of a tangent line.
We have already studied how to find equations of tangent lines to functions and the rate of change of a function at a specific point. In all these cases we had the explicit equation for the function
and differentiated these functions explicitly. Suppose instead that we want to determine the equation of a tangent line to an arbitrary curve or the rate of change of an arbitrary curve at a point.
In this section, we solve these problems by finding the derivatives of functions that define \(y\) implicitly in terms of \(x\).
Implicit Differentiation
In most discussions of math, if the dependent variable \(y\) is a function of the independent variable \(x\), we express y in terms of \(x\). If this is the case, we say that \(y\) is an explicit
function of \(x\). For example, when we write the equation \(y=x^2+1\), we are defining y explicitly in terms of \(x\). On the other hand, if the relationship between the function \(y\) and the
variable \(x\) is expressed by an equation where \(y\) is not expressed entirely in terms of \(x\), we say that the equation defines \(y\) implicitly in terms of \(x\). For example, the equation \
(y−x^2=1\) defines the function \(y=x^2+1\) implicitly.
Implicit differentiation allows us to find slopes of tangents to curves that are clearly not functions (they fail the vertical line test). We are using the idea that portions of \(y\) are functions
that satisfy the given equation, but that y is not actually a function of \(x\).
In general, an equation defines a function implicitly if the function satisfies that equation. An equation may define many different functions implicitly. For example, the functions
\[y=\begin{cases}\sqrt{25−x^2}, & \text{if }−25≤x<0\\ −\sqrt{25−x^2}, & \text{if }0≤x≤25\end{cases}\nonumber\]
which are illustrated in Figure \(\PageIndex{1}\), are just three of the many functions defined implicitly by the equation \(x^2+y^2=25\).
Figure \(\PageIndex{1}\):The equation \(x^2+y^2=25\) defines many functions implicitly.
If we want to find the slope of the line tangent to the graph of \(x^2+y^2=25\) at the point \((3,4)\), we could evaluate the derivative of the function \(y=−\sqrt{25−x^2}\) at \(x=3\). On the other
hand, if we want the slope of the tangent line at the point \((3,−4)\), we could use the derivative of \(y=−\sqrt{25−x^2}\). However, it is not always easy to solve for a function defined implicitly
by an equation. Fortunately, the technique of implicit differentiation allows us to find the derivative of an implicitly defined function without ever solving for the function explicitly. The process
of finding \(\dfrac{dy}{dx}\) using implicit differentiation is described in the following problem-solving strategy.
Problem-Solving Strategy: Implicit Differentiation
To perform implicit differentiation on an equation that defines a function \(y\) implicitly in terms of a variable \(x\), use the following steps:
1. Take the derivative of both sides of the equation. Keep in mind that \(y\) is a function of \(x\). Consequently, whereas \[\dfrac{d}{dx}(\sin x)=\cos x\nonumber\] and \[\dfrac{d}{dx}(\sin y)=\cos
y\cdot\dfrac{dy}{dx}\nonumber\] because we must use the chain rule to differentiate \(\sin y\) with respect to \(x\).
2. Rewrite the equation so that all terms containing \(dy/dx\) are on the left and all terms that do not contain \(dy/dx\) are on the right.
3. Factor out \(dy/dx\) on the left.
4. Solve for \(dy/dx\) by dividing both sides of the equation by an appropriate algebraic expression.
Example \(\PageIndex{1}\): Using Implicit Differentiation
Assuming that \(y\) is defined implicitly by the equation \(x^2+y^2=25\), find \(\dfrac{dy}{dx}\).
Follow the steps in the problem-solving strategy.
\(\dfrac{d}{dx}(x^2+y^2)=\dfrac{d}{dx}(25)\) Step 1. Differentiate both sides of the equation.
\(\dfrac{d}{dx}(x^2)+\dfrac{d}{dx}(y^2)=0\) Step 1.1. Use the sum rule on the left.On the right \(\dfrac{d}{dx}(25)=0\).
\(2x+2y\dfrac{dy}{dx}=0\) Step 1.2. Take the derivatives, so \(\dfrac{d}{dx}(x^2)=2x\) and \(\dfrac{d}{dx}(y^2)=2y\dfrac{dy}{dx}\).
\(2y\dfrac{dy}{dx}=−2x\) Step 2. Keep the terms with \(\dfrac{dy}{dx}\) on the left.Move the remaining terms to the right.
\(\dfrac{dy}{dx}=−\dfrac{x}{y}\) Step 4. Divide both sides of the equation by \(2y\).(Step 3 does not apply in this case.)
Note that the resulting expression for \(\dfrac{dy}{dx}\) is in terms of both the independent variable \(x\) and the dependent variable \(y\). Although in some cases it may be possible to express \(\
dfrac{dy}{dx}\) in terms of \(x\) only, it is generally not possible to do so.
Example \(\PageIndex{2}\): Using Implicit Differentiation and the Product Rule
Assuming that \(y\) is defined implicitly by the equation \(x^3\sin y+y=4x+3\), find \(\dfrac{dy}{dx}\).
\(\dfrac{d}{dx}(x^3\sin y+y)=\dfrac{d}{dx}(4x+3)\) Step 1: Differentiate both sides of the equation.
\(\dfrac{d}{dx}(x^3\sin y)+\dfrac{d}{dx}(y)=4\) Step 1.1: Apply the sum rule on the left.On the right, \(\dfrac{d}{dx}(4x+3)=4\).
\((\dfrac{d}{dx}(x^3)⋅\sin y+\dfrac{d}{dx}(\sin y)⋅x^3)+\dfrac{dy}{dx}=4\) Step 1.2: Use the product rule to find \(\dfrac{d}{dx}(x^3\sin y)\).Observe that \(\dfrac{d}{dx}(y)=\dfrac{dy}{dx}\).
\(3x^2\sin y+(\cos y\dfrac{dy}{dx})⋅x^3+\dfrac{dy}{dx}=4\) Step 1.3: We know \(\dfrac{d}{dx}(x^3)=3x^2\).Use the chain rule to obtain \(\dfrac{d}{dx}(\sin y)=\cos y\dfrac{dy}{dx}\).
\(x^3\cos y\dfrac{dy}{dx}+\dfrac{dy}{dx}=4−3x^2\sin y\) Step 2: Keep all terms containing \(\dfrac{dy}{dx}\) on the left. Move all other terms to the right.
\(\dfrac{dy}{dx}(x^3\cos y+1)=4−3x^2\sin y\) Step 3: Factor out \(\dfrac{dy}{dx}\) on the left.
\(\dfrac{dy}{dx}=\dfrac{4−3x^2\sin y}{x^3\cos y+1}\) Step 4: Solve for \(\dfrac{dy}{dx}\) by dividing both sides of the equation by \(x^3\cos y+1\).
Example \(\PageIndex{3}\): Using Implicit Differentiation to Find a Second Derivative
Find \(\dfrac{d^2y}{dx^2}\) if \(x^2+y^2=25\).
In Example \(\PageIndex{1}\), we showed that \(\dfrac{dy}{dx}=−\dfrac{x}{y}\). We can take the derivative of both sides of this equation to find \(\dfrac{d^2y}{dx^2}\).
\(\begin{align*} \dfrac{d^2y}{dx^2}&=\dfrac{d}{dy}\left(−\dfrac{x}{y}\right) & & \text{Differentiate both sides of }\dfrac{dy}{dx}=−\dfrac{x}{y}.\\[4pt]
&=−\dfrac{\left(1⋅y−x\dfrac{dy}{dx}\right)}{y^2} & & \text{Use the quotient rule to find }\dfrac{d}{dy}\left(−\dfrac{x}{y}\right).\\[4pt]
&=\dfrac{−y+x\dfrac{dy}{dx}}{y^2} & & \text{Simplify.}\\[4pt]
&=\dfrac{−y+x\left(−\dfrac{x}{y}\right)}{y^2} & & \text{Substitute }\dfrac{dy}{dx}=−\dfrac{x}{y}.\\[4pt]
&=\dfrac{−y^2−x^2}{y^3} & & \text{Simplify.} \end{align*}\)
At this point we have found an expression for \(\dfrac{d^2y}{dx^2}\). If we choose, we can simplify the expression further by recalling that \(x^2+y^2=25\) and making this substitution in the
numerator to obtain \(\dfrac{d^2y}{dx^2}=−\dfrac{25}{y^3}\).
Exercise \(\PageIndex{1}\)
Find \(\dfrac{dy}{dx}\) for \(y\) defined implicitly by the equation \(4x^5+\tan y=y^2+5x\).
Follow the problem solving strategy, remembering to apply the chain rule to differentiate \(\tan y\) and \(y^2\).
Finding Tangent Lines Implicitly
Now that we have seen the technique of implicit differentiation, we can apply it to the problem of finding equations of tangent lines to curves described by equations.
Example \(\PageIndex{4}\): Finding a Tangent Line to a Circle
Find the equation of the line tangent to the curve \(x^2+y^2=25\) at the point \((3,−4)\).
Although we could find this equation without using implicit differentiation, using that method makes it much easier. In Example \(\PageIndex{1}\), we found \(\dfrac{dy}{dx}=−\dfrac{x}{y}\).
The slope of the tangent line is found by substituting \((3,−4)\) into this expression. Consequently, the slope of the tangent line is \(\dfrac{dy}{dx}\Big|_{(3,−4)}=−\dfrac{3}{−4}=\dfrac{3}{4}\).
Using the point \((3,−4)\) and the slope \(\dfrac{3}{4}\) in the point-slope equation of the line, we obtain the equation \(y=\dfrac{3}{4}x−\dfrac{25}{4}\) (Figure).
Figure \(\PageIndex{2}\): The line \(y=\dfrac{3}{4}x−\dfrac{25}{4}\) is tangent to \(x^2+y^2=25\) at the point (3, −4).
Example \(\PageIndex{5}\): Finding the Equation of the Tangent Line to a Curve
Find the equation of the line tangent to the graph of \(y^3+x^3−3xy=0\) at the point \(\left(\frac{3}{2},\frac{3}{2}\right)\) (Figure). This curve is known as the folium (or leaf) of Descartes.
Figure \(\PageIndex{3}\): Finding the tangent line to the folium of Descartes at \(\left(\frac{3}{2},\frac{3}{2}\right)\).
Begin by finding \(\dfrac{dy}{dx}\).
Next, substitute \(\left(\frac{3}{2},\frac{3}{2}\right)\) into \(\dfrac{dy}{dx}=\dfrac{3y−3x^2}{3y^2−3x}\) to find the slope of the tangent line:
Finally, substitute into the point-slope equation of the line to obtain
Example \(\PageIndex{6}\): Applying Implicit Differentiation
In a simple video game, a rocket travels in an elliptical orbit whose path is described by the equation \(4x^2+25y^2=100\). The rocket can fire missiles along lines tangent to its path. The object of
the game is to destroy an incoming asteroid traveling along the positive \(x\)-axis toward \((0,0)\). If the rocket fires a missile when it is located at \(\left(3,\frac{8}{3}\right)\), where will it
intersect the \(x\)-axis?
To solve this problem, we must determine where the line tangent to the graph of
\(4x^2+25y^2=100\) at \(\left(3,\frac{8}{3}\right)\) intersects the \(x\)-axis. Begin by finding \(\dfrac{dy}{dx}\) implicitly.
Differentiating, we have
Solving for \(\dfrac{dy}{dx}\),
we have
The slope of the tangent line is \(\dfrac{dy}{dx}\Bigg|_{\left(3,\frac{8}{3}\right)}=−\dfrac{9}{50}\). The equation of the tangent line is \(y=−\dfrac{9}{50}x+\dfrac{183}{200}\). To determine where
the line intersects the \(x\)-axis, solve \(0=−\dfrac{9}{50}x+\dfrac{183}{200}\). The solution is \(x=\dfrac{6}{13}\). The missile intersects the \(x\)-axis at the point \(\left(\frac{61}{3},0\right)
Exercise \(\PageIndex{2}\)
Find the equation of the line tangent to the hyperbola \(x^2−y^2=16\) at the point \((5,3)\).
Key Concepts
• We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations).
• By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.
implicit differentiation
is a technique for computing \(\dfrac{dy}{dx}\) for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable \(y\) as a
function) and solving for \(\dfrac{dy}{dx}\)
Contributors and Attributions
• Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org. | {"url":"https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q1/03%3A_Derivatives/3.09%3A_Implicit_Differentiation","timestamp":"2024-11-07T10:54:44Z","content_type":"text/html","content_length":"145936","record_id":"<urn:uuid:e2a55742-207d-4ea2-a639-7547a20ca441>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00520.warc.gz"} |
Lesson 13
Solving Systems of Equations
13.1: True or False: Two Lines (5 minutes)
The purpose of this warm-up is to get students to reason about solutions to equations by looking at their structure and reading their graphs. While some students may solve each equation to find if it
is true or false without relating it to the graphs, encourage all students to show why their answer is correct based on the graphs of the equations during the whole-class discussion.
Arrange students in groups of 2. Display the image for all to see. Give students 2 minutes of quiet work time to begin the task individually and then 1 minute to discuss their responses with a
partner followed by a whole-class discussion.
Student Facing
Use the lines to decide whether each statement is true or false. Be prepared to explain your reasoning using the lines.
1. A solution to \(8=\text-x+10\) is 2.
2. A solution to \(2=2x+4\) is 8.
3. A solution to \(\text-x+10=2x+4\) is 8.
4. A solution to \(\text-x+10=2x+4\) is 2.
5. There are no values of \(x\) and \(y\) that make \(y=\text-x+10\) and \(y=2x+4\) true at the same time.
Activity Synthesis
Display the task image for all to see. Ask students to share their solutions and to reference the lines in their explanations. Emphasize the transitive property when students explain that since \(y=8
\) at the point of intersection of \(y = 2x + 4\) and \(y=\text- x+10\), then both \(2x + 4 = 8\) and \(\text- x + 10 = 8\) are true, which leads to \(\text- x + 10 = 2x + 4\). If students do not
mention this idea, bring it to their attention.
Ask students to solve \(\text- x + 10 = 2x + 4\) for \(x\) if they’ve not already done so and confirm that \(x=2\) is the \(x\)-coordinate of the solution to the system of equations.
13.2: Matching Graphs to Systems (15 minutes)
This activity represents the first time students solve a system of equations using algebraic methods. They first match systems of equations to their graphs and then calculate the solutions to each
system. The purpose of matching is so students have a way to check that their algebraic solutions are correct, but not to shortcut the algebraic process since the graphs themselves do not include
enough detail to accurately guess the coordinates of the solution.
Keep students in groups of 2. Give 2–3 minutes of quiet work time for the first problem and then ask students to pause their work. Select 1–2 students per figure to explain how they matched it to one
of the systems of equations. For example, a student may identify the system matching Figure A as the only system with an equation that has negative slope. Give students 5–7 minutes of work time with
their partner to complete the activity followed by a whole-class discussion. If students finish early and have not already done so on their own, ask them how they could check their solutions and
encourage them to do so.
If using the digital activity, implement the lesson as indicated above. The only difference between the print and digital version is the digital lesson has an applet that will simulate the graphs so
the students have another way of checking their solutions.
Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. Invite students to
illustrate connections between slopes and \(y\)-intercepts of each line to the corresponding parts of each equation using the same color.
Supports accessibility for: Visual-spatial processing
Conversing: MLR8 Discussion Supports. Use this routine to support small-group discussion as students describe the reasons for their matches. Arrange students in groups of 2. Invite Partner A to begin
with this sentence frame: “Figure ____ matches with the system of equations ____, because ____.” Invite the listener, Partner B, to press for additional details referring to specific features of the
graphs (e.g. positive slope, negative y-intercept, coordinates of the intersection point, etc). Students should switch roles for each figure. This will help students justify how features of the graph
can be used to identify matching equations.
Design Principle(s): Support sense-making; Cultivate conversation
Student Facing
Here are three systems of equations graphed on a coordinate plane:
1. Match each figure to one of the systems of equations shown here.
1. \(\begin{cases} y=3x+5\\ y=\text- 2x+20 \end{cases}\)
2. \(\begin{cases} y=2x-10\\ y=4x-1 \end{cases}\)
3. \(\begin{cases} y=0.5x+12\\ y=2x+27 \end{cases}\)
2. Find the solution to each system and then check that your solution is reasonable on the graph.
□ Notice that the sliders set the values of the coefficient and the constant term in each equation.
□ Change the sliders to the values of the coefficient and the constant term in the next pair of equations.
□ Click on the spot where the lines intersect and a labeled point should appear.
Keep students in groups of 2. Give 2–3 minutes of quiet work time for the first problem and then ask students to pause their work. Select 1–2 students per figure to explain how they matched it to one
of the systems of equations. For example, a student may identify the system matching Figure A as the only system with an equation that has negative slope. Give students 5–7 minutes of work time with
their partner to complete the activity followed by a whole-class discussion. If students finish early and have not already done so on their own, ask them how they could check their solutions and
encourage them to do so.
If using the digital activity, implement the lesson as indicated above. The only difference between the print and digital version is the digital lesson has an applet that will simulate the graphs so
the students have another way of checking their solutions.
Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. Invite students to
illustrate connections between slopes and \(y\)-intercepts of each line to the corresponding parts of each equation using the same color.
Supports accessibility for: Visual-spatial processing
Conversing: MLR8 Discussion Supports. Use this routine to support small-group discussion as students describe the reasons for their matches. Arrange students in groups of 2. Invite Partner A to begin
with this sentence frame: “Figure ____ matches with the system of equations ____, because ____.” Invite the listener, Partner B, to press for additional details referring to specific features of the
graphs (e.g. positive slope, negative y-intercept, coordinates of the intersection point, etc). Students should switch roles for each figure. This will help students justify how features of the graph
can be used to identify matching equations.
Design Principle(s): Support sense-making; Cultivate conversation
Student Facing
Here are three systems of equations graphed on a coordinate plane:
1. Match each figure to one of the systems of equations shown here.
1. \(\begin{cases} y=3x+5\\ y=\text-2x+20 \end{cases}\)
2. \(\begin{cases} y=2x-10\\ y=4x-1 \end{cases}\)
3. \(\begin{cases} y=0.5x+12\\ y=2x+27 \end{cases}\)
2. Find the solution to each system and check that your solution is reasonable based on the graph.
Activity Synthesis
The goal of this discussion is to deliberately connect the current topic of systems of equations to the previous topic of solving equations with variables on both sides. For each of the following
questions, give students 30 seconds of quiet think time and then invite students per question to explain their answer. The final question looks ahead to the following activity.
• “Do you need to see the graphs of the equations in a system in order to solve the system?" (No, but the graphs made me feel more confident that my answer was correct.)
• “How do you know your solution doesn’t contain any errors?" (I know my solution does not have errors because I substituted my values for \(x\) and \(y\) into the equations and they made both
equations true.)
• “How does solving systems of equations compare to solving equations with variables on both sides like we did in earlier lessons?" (They are very similar, only with a system of equations you are
finding an \(x\) and a \(y\) to make both equations true and not just an \(x\) to make one equation true.)
• “When you solved equations with variables on both sides, some had one solution, some had no solutions, and some had infinite solutions. Do you think systems of equations can have no solutions or
infinite solutions?" (Yes. We have seen some graphs of parallel lines where there were no solutions and some graphs of lines that are on top of one another where there are infinite solutions.)
13.3: Different Types of Systems (15 minutes)
While students have encountered equations with different numbers of solutions in earlier activities, this is the first activity where students connect systems of equations with their previous
thinking about equations that have no solution, one solution, or infinitely many solutions. The purpose of this activity is for students to connect the features of the graph of the equations of a
system to the number of solutions of a system (MP7). While students are not asked to solve the systems of equations, they may choose to rewrite the equations in equivalent forms as they work to graph
the lines.
Depending on instructional time available, you may wish to alter the activity and ask students to solve one or more of the systems of equations algebraically.
Remind students of the activity they did sorting equations with a single variable where each equation had either one solution, no solution, or infinitely many solutions. Tell them that, just like
one variable equations, systems of equations can also have either one solution, no solution, or infinitely many solutions. Point out that in the previous activity, each of the three systems of
equations had one solution, which they found algebraically by solving the system, and so the graphs of the equations of the system showed one point where the lines intersected. Ask students what they
think the graphs of equations from systems with no or infinitely many solutions might look like. Allow 30 seconds of quiet think time before inviting a few students to suggest possibilities for each
type of system while recording and displaying their ideas for all to see. Remind students of the activities in previous lessons where they have seen these situations and their graphs (a bike race
between Elena and Jada had infinite solutions and stacking different sized cups had none).
Arrange students in groups of 2–3. Provide each group with access to straightedges and scissors as well as one copy of the blackline master. Encourage partners to split the work by cutting apart the
problems, each taking one to three graphs, and then trading pages within their group to check the work. Give 4–6 minutes for groups to complete the graphs and remind students to use straightedges for
precision while graphing.
Before beginning the final problem, have each group trade their work with another group and place a question mark next to the graphs they are not sure are correct. Give groups 3–4 minutes to
revise as needed and write their descriptions for the second problem followed by a whole-class discussion.
If using the digital activity, use the discussion structure above. The digital applet will make the graphing and solving of systems go quickly so students can spend more time analyzing the solutions.
Using technology to graph allows students to focus on the main purpose of the lesson and also recognize the value in technology when solving systems in addition to appreciating when the graphing
method is efficient. In this activity, one of the main purposes is to notice what is common among systems with the same number of solutions. Therefore, it may be useful to ask students to justify why
the lines graphed with no obvious intersections are actually parallel.
Student Facing
Your teacher will give you a page with 6 systems of equations.
1. Graph each system of equations by typing each pair of the equations in the applet, one at a time.
2. Describe what the graph of a system of equations looks like when it has . . .
1. 1 solution
2. 0 solutions
3. infinitely many solutions
Use the applet to confirm your answer to question 2.
Remind students of the activity they did sorting equations with a single variable where each equation had either one solution, no solution, or infinitely many solutions. Tell them that, just like
one variable equations, systems of equations can also have either one solution, no solution, or infinitely many solutions. Point out that in the previous activity, each of the three systems of
equations had one solution, which they found algebraically by solving the system, and so the graphs of the equations of the system showed one point where the lines intersected. Ask students what they
think the graphs of equations from systems with no or infinitely many solutions might look like. Allow 30 seconds of quiet think time before inviting a few students to suggest possibilities for each
type of system while recording and displaying their ideas for all to see. Remind students of the activities in previous lessons where they have seen these situations and their graphs (a bike race
between Elena and Jada had infinite solutions and stacking different sized cups had none).
Arrange students in groups of 2–3. Provide each group with access to straightedges and scissors as well as one copy of the blackline master. Encourage partners to split the work by cutting apart the
problems, each taking one to three graphs, and then trading pages within their group to check the work. Give 4–6 minutes for groups to complete the graphs and remind students to use straightedges for
precision while graphing.
Before beginning the final problem, have each group trade their work with another group and place a question mark next to the graphs they are not sure are correct. Give groups 3–4 minutes to
revise as needed and write their descriptions for the second problem followed by a whole-class discussion.
If using the digital activity, use the discussion structure above. The digital applet will make the graphing and solving of systems go quickly so students can spend more time analyzing the solutions.
Using technology to graph allows students to focus on the main purpose of the lesson and also recognize the value in technology when solving systems in addition to appreciating when the graphing
method is efficient. In this activity, one of the main purposes is to notice what is common among systems with the same number of solutions. Therefore, it may be useful to ask students to justify why
the lines graphed with no obvious intersections are actually parallel.
Student Facing
Your teacher will give you a page with some systems of equations.
1. Graph each system of equations carefully on the provided coordinate plane.
2. Describe what the graph of a system of equations looks like when it has . . .
1. 1 solution
2. 0 solutions
3. infinitely many solutions
Student Facing
Are you ready for more?
The graphs of the equations \(Ax + By = 15\) and \(Ax - By = 9\) intersect at \((2,1)\). Find \(A\) and \(B\). Show or explain your reasoning.
Activity Synthesis
The goal of this discussion is for students to draw conclusions about the relationship between the number of solutions a system of equations has and the appearance of the graphs of the equations in
the system. Select 2–3 students to share and explain their answers to the second problem.
If no students mention it, bring in slope language and how inspecting the slopes of the equations before graphing or solving can give clues to the possible number of solutions the system has. In
particular, students should notice that systems with lines that have different slopes have a single solution, lines that have the same slope and different \(y\)-intercept have no solution, and lines
that have the same slope and \(y\)-intercept will have infinitely many solutions.
Assign a number of solutions (one, none, or infinite) to each group and ask them to write a system of equations that would have that number of solutions. Have a few groups share their systems and
describe how the graphs of the systems would look. In particular, ask each group to describe how the slope and \(y\)-intercept of their written lines would be seen in the graph and how the number of
solutions would appear on the graph. Following the description, display the graph of the system using a digital resource, if possible, or a general sketch on a set of displayed axes.
Writing: MLR1 Stronger and Clearer Each Time. Use this routine to give students an opportunity to revise and improve their response to the final question, “Describe what the graph of a system of
equations looks like when it has one solution, zero solutions, and infinitely many solutions.” Give students time to meet with 2–3 partners, to share and get feedback on their response. Encourage the
listener to press for supporting details and evidence by asking, “How do the slopes compare?”, “How do the \(y\)-intercepts compare?” or “What do you notice about the slopes and the \(y\)-intercepts?
” Students can borrow ideas and language from each partner to strengthen the final product. This will help students produce a written generalization for how to identify the number of solutions for a
system of equations by using the features of a graph.
Design Principle(s): Optimize output (for generalization)
Lesson Synthesis
To highlight the connection between the number of solutions to a system of equations and features of its graph and equations, ask:
• “How can you know the number of solutions for a system of equations from its graph?” (If the two lines intersect at a point, there is one solution. If the two lines are parallel and do not
intersect, there are no solutions. If the two lines are drawn through the same points, there are infinitely many solutions.)
• “How can you know the number of solutions for a system of equations from their equations?” (If the two equations have different slopes, there is one solution. If the two equations have the same
slope and different \(y\)-intercepts, there are no solutions. If the two equations have the same slope and the same \(y\)-intercept, there are infinitely many solutions.)
If students do not make the connection themselves, remind them of their earlier conclusions about the number of solutions and equation in one variable has.
13.4: Cool-down - Two Lines (5 minutes)
Student Facing
Sometimes it is easier to solve a system of equations without having to graph the equations and look for an intersection point. In general, whenever we are solving a system of equations written as
\(\displaystyle \begin{cases} y = \text{[some stuff]}\\ y = \text{[some other stuff]} \end{cases}\)
we know that we are looking for a pair of values \((x,y)\) that makes both equations true. In particular, we know that the value for \(y\) will be the same in both equations. That means that
\(\displaystyle \text{[some stuff]} = \text{[some other stuff]}\)
For example, look at this system of equations:
\(\displaystyle \begin{cases} y = 2x + 6 \\ y = \text-3x - 4 \end{cases}\)
Since the \(y\) value of the solution is the same in both equations, then we know \(\displaystyle 2x + 6 = \text-3x -4\)
We can solve this equation for \(x\):
\( 2x+6 &= \text-3x-4&& \\ 5x+6 &=\text-4\ &&\text{add \(3x\) to each side}\\ 5x &=\text-10\ &&\text{subtract 6 from each side}\\ x &=\text-2\ &&\text{divide each side by 5}\ \)
But this is only half of what we are looking for: we know the value for \(x\), but we need the corresponding value for \(y\). Since both equations have the same \(y\) value, we can use either
equation to find the \(y\)-value:
\(\displaystyle y = 2(\text-2) + 6\)
\(\displaystyle y = \text-3(\text-2) -4\)
In both cases, we find that \(y = 2\). So the solution to the system is \((\text-2,2)\). We can verify this by graphing both equations in the coordinate plane.
In general, a system of linear equations can have:
• No solutions. In this case, the lines that correspond to each equation never intersect.
• Exactly one solution. The lines that correspond to each equation intersect in exactly one point.
• An infinite number of solutions. The graphs of the two equations are the same line! | {"url":"https://curriculum.illustrativemathematics.org/MS/teachers/3/4/13/index.html","timestamp":"2024-11-02T21:16:56Z","content_type":"text/html","content_length":"124629","record_id":"<urn:uuid:4e0fc202-ee74-42e9-92c2-9dd091c3145f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00630.warc.gz"} |
class hyperspy.api.signals.BaseSignal(data, **kwds)#
Bases: FancySlicing, MVA, MVATools
General signal created from a numpy or cupy array.
>>> data = np.ones((10, 10))
>>> s = hs.signals.BaseSignal(data)
Whether the signal is ragged or not.
Signal indexer/slicer.
Navigation indexer/slicer.
The metadata of the signal.
The original metadata of the signal.
Create a signal instance.
The signal data. It can be an array of any dimensions.
axes[dict/axes], optional
List of either dictionaries or axes objects to define the axes (see the documentation of the AxesManager class for more details).
attributesdict, optional
A dictionary whose items are stored as attributes.
metadatadict, optional
A dictionary containing a set of parameters that will to stores in the metadata attribute. Some parameters might be mandatory in some cases.
original_metadatadict, optional
A dictionary containing a set of parameters that will to stores in the original_metadata attribute. It typically contains all the parameters that has been imported from the original data
raggedbool or None, optional
Define whether the signal is ragged or not. Overwrite the ragged value in the attributes dictionary. If None, it does nothing. Default is None.
property T#
The transpose of the signal, with signal and navigation spaces swapped. Enables calling transpose() with the default parameters as a property of a Signal.
add_gaussian_noise(std, random_state=None)#
Add Gaussian noise to the data.
The operation is performed in-place (i.e. the data of the signal is modified). This method requires the signal to have a float data type, otherwise it will raise a TypeError.
The standard deviation of the Gaussian noise.
random_stateNone, int or numpy.random.Generator, default None
Seed for the random generator.
This method uses numpy.random.normal() (or dask.array.random.normal() for lazy signals) to generate the noise.
add_marker(marker, plot_on_signal=True, plot_marker=True, permanent=False, plot_signal=True, render_figure=True)#
Add one or several markers to the signal or navigator plot and plot the signal, if not yet plotted (by default)
The marker or iterable (list, tuple, …) of markers to add. See the Markers section in the User Guide if you want to add a large number of markers as an iterable, since this will be
much faster. For signals with navigation dimensions, the markers can be made to change for different navigation indices. See the examples for info.
If True, add the marker to the signal. If False, add the marker to the navigator
If True, plot the marker.
If False, the marker will only appear in the current plot. If True, the marker will be added to the metadata.Markers list, and be plotted with plot(plot_markers=True). If the signal
is saved as a HyperSpy HDF5 file, the markers will be stored in the HDF5 signal and be restored when the file is loaded.
>>> im = hs.data.wave_image()
>>> m = hs.plot.markers.Rectangles(
... offsets=[(1.0, 1.5)], widths=(0.5,), heights=(0.7,)
... )
>>> im.add_marker(m)
Add permanent marker:
>>> rng = np.random.default_rng(1)
>>> s = hs.signals.Signal2D(rng.random((100, 100)))
>>> marker = hs.plot.markers.Points(offsets=[(50, 60)])
>>> s.add_marker(marker, permanent=True, plot_marker=True)
Removing a permanent marker:
>>> rng = np.random.default_rng(1)
>>> s = hs.signals.Signal2D(rng.integers(10, size=(100, 100)))
>>> marker = hs.plot.markers.Points(offsets=[(10, 60)])
>>> marker.name = "point_marker"
>>> s.add_marker(marker, permanent=True)
>>> del s.metadata.Markers.point_marker
Adding many markers as a list:
>>> rng = np.random.default_rng(1)
>>> s = hs.signals.Signal2D(rng.integers(10, size=(100, 100)))
>>> marker_list = []
>>> for i in range(10):
... marker = hs.plot.markers.Points(rng.random(2))
... marker_list.append(marker)
>>> s.add_marker(marker_list, permanent=True)
add_poissonian_noise(keep_dtype=True, random_state=None)#
Add Poissonian noise to the data.
This method works in-place. The resulting data type is int64. If this is different from the original data type then a warning is added to the log.
keep_dtypebool, default True
If True, keep the original data type of the signal data. For example, if the data type was initially 'float64', the result of the operation (usually 'int64') will be converted to
random_stateNone, int or numpy.random.Generator, default None
Seed for the random generator.
This method uses numpy.random.poisson() (or dask.array.random.poisson() for lazy signals) to generate the Poissonian noise.
apply_apodization(window='hann', hann_order=None, tukey_alpha=0.5, inplace=False)#
Apply an apodization window to a Signal.
Select between {'hann' (default), 'hamming', or 'tukey'}
Only used if window='hann' If integer n is provided, a Hann window of n-th order will be used. If None, a first order Hann window is used. Higher orders result in more homogeneous
intensity distribution.
Only used if window='tukey' (default is 0.5). From the documentation of scipy.signal.windows.tukey():
★ Shape parameter of the Tukey window, representing the fraction of the window inside the cosine tapered region. If zero, the Tukey window is equivalent to a rectangular window.
If one, the Tukey window is equivalent to a Hann window.
If True, the apodization is applied in place, i.e. the signal data will be substituted by the apodized one (default is False).
outBaseSignal (or subclass), optional
If inplace=False, returns the apodized signal of the same type as the provided Signal.
>>> import hyperspy.api as hs
>>> wave = hs.data.wave_image()
>>> wave.apply_apodization('tukey', tukey_alpha=0.1).plot()
as_lazy(copy_variance=True, copy_navigator=True, copy_learning_results=True)#
Create a copy of the given Signal as a LazySignal.
Whether or not to copy the variance from the original Signal to the new lazy version. Default is True.
Whether or not to copy the navigator from the original Signal to the new lazy version. Default is True.
Whether to copy the learning_results from the original signal to the new lazy version. Default is True.
The same signal, converted to be lazy
as_signal1D(spectral_axis, out=None, optimize=True)#
Return the Signal as a spectrum.
The chosen spectral axis is moved to the last index in the array and the data is made contiguous for efficient iteration over spectra. By default, the method ensures the data is stored
optimally, hence often making a copy of the data. See transpose() for a more general method with more options.
spectral_axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
If True, the location of the data in memory is optimised for the fastest iteration over the navigation axes. This operation can cause a peak of memory usage and requires considerable
processing times for large datasets and/or low specification hardware. See the Transposing (changing signal spaces) section of the HyperSpy user guide for more information. When
operating on lazy signals, if True, the chunks are optimised for the new axes configuration.
>>> img = hs.signals.Signal2D(np.ones((3, 4, 5, 6)))
>>> img
<Signal2D, title: , dimensions: (4, 3|6, 5)>
>>> img.as_signal1D(-1+1j)
<Signal1D, title: , dimensions: (6, 5, 4|3)>
>>> img.as_signal1D(0)
<Signal1D, title: , dimensions: (6, 5, 3|4)>
as_signal2D(image_axes, out=None, optimize=True)#
Convert a signal to a Signal2D.
The chosen image axes are moved to the last indices in the array and the data is made contiguous for efficient iteration over images.
image_axestuple (of int, str or DataAxis)
Select the image axes. Note that the order of the axes matters and it is given in the “natural” i.e. X, Y, Z… order.
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
If True, the location of the data in memory is optimised for the fastest iteration over the navigation axes. This operation can cause a peak of memory usage and requires considerable
processing times for large datasets and/or low specification hardware. See the Transposing (changing signal spaces) section of the HyperSpy user guide for more information. When
operating on lazy signals, if True, the chunks are optimised for the new axes configuration.
When data.ndim < 2
>>> s = hs.signals.Signal1D(np.ones((2, 3, 4, 5)))
>>> s
<Signal1D, title: , dimensions: (4, 3, 2|5)>
>>> s.as_signal2D((0, 1))
<Signal2D, title: , dimensions: (5, 2|4, 3)>
>>> s.to_signal2D((1, 2))
<Signal2D, title: , dimensions: (2, 5|4, 3)>
blind_source_separation(number_of_components=None, algorithm='sklearn_fastica', diff_order=1, diff_axes=None, factors=None, comp_list=None, mask=None, on_loadings=False,
reverse_component_criterion='factors', whiten_method='PCA', return_info=False, print_info=True, **kwargs)#
Apply blind source separation (BSS) to the result of a decomposition.
The results are stored in self.learning_results.
Read more in the User Guide.
number_of_componentsint or None
Number of principal components to pass to the BSS algorithm. If None, you must specify the comp_list argument.
algorithm{"sklearn_fastica" | "orthomax" | "FastICA" | "JADE" |
``”CuBICA”`` | ``”TDSEP”``} or object, default “sklearn_fastica”
The BSS algorithm to use. If algorithm is an object, it must implement a fit_transform() method or fit() and transform() methods, in the same manner as a scikit-learn estimator.
diff_orderint, default 1
Sometimes it is convenient to perform the BSS on the derivative of the signal. If diff_order is 0, the signal is not differentiated.
diff_axesNone, list of int, list of str
■ If None and on_loadings is False, when diff_order is greater than 1 and signal_dimension is greater than 1, the differences are calculated across all signal axes
■ If None and on_loadings is True, when diff_order is greater than 1 and navigation_dimension is greater than 1, the differences are calculated across all navigation axes
■ Otherwise the axes can be specified in a list.
factorsBaseSignal or numpy.ndarray
Factors to decompose. If None, the BSS is performed on the factors of a previous decomposition. If a Signal instance, the navigation dimension must be 1 and the size greater than 1.
comp_listNone or list or numpy.ndarray
Choose the components to apply BSS to. Unlike number_of_components, this argument permits non-contiguous components.
maskBaseSignal or subclass
If not None, the signal locations marked as True are masked. The mask shape must be equal to the signal shape (navigation shape) when on_loadings is False (True).
on_loadingsbool, default False
If True, perform the BSS on the loadings of a previous decomposition, otherwise, perform the BSS on the factors.
reverse_component_criterion{“factors”, “loadings”}, default “factors”
Use either the factors or the loadings to determine if the component needs to be reversed.
whiten_method{"PCA" | "ZCA"} or None, default “PCA”
How to whiten the data prior to blind source separation. If None, no whitening is applied. See whiten_data() for more details.
return_info: bool, default False
The result of the decomposition is stored internally. However, some algorithms generate some extra information that is not stored. If True, return any extra information if available.
In the case of sklearn.decomposition objects, this includes the sklearn Estimator object.
print_infobool, default True
If True, print information about the decomposition being performed. In the case of sklearn.decomposition objects, this includes the values of all arguments of the chosen sklearn
Any keyword arguments are passed to the BSS algorithm.
None or subclass of sklearn.base.BaseEstimator
If True and ‘algorithm’ is an sklearn Estimator, returns the
Estimator object.
change_dtype(dtype, rechunk=False)#
Change the data type of a Signal.
dtypestr or numpy.dtype
Typecode string or data-type to which the Signal’s data array is cast. In addition to all the standard numpy Data type objects (dtype), HyperSpy supports four extra dtypes for RGB
images: 'rgb8', 'rgba8', 'rgb16', and 'rgba16'. Changing from and to any rgb(a) dtype is more constrained than most other dtype conversions. To change to an rgb(a) dtype, the
signal_dimension must be 1, and its size should be 3 (for rgb) or 4 (for rgba) dtypes. The original dtype should be uint8 or uint16 if converting to rgb(a)8 or rgb(a))16, and the
navigation_dimension should be at least 2. After conversion, the signal_dimension becomes 2. The dtype of images with original dtype rgb(a)8 or rgb(a)16 can only be changed to uint8
or uint16, and the signal_dimension becomes 1.
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
>>> s = hs.signals.Signal1D([1, 2, 3, 4, 5])
>>> s.data
array([1, 2, 3, 4, 5])
>>> s.change_dtype('float')
>>> s.data
array([1., 2., 3., 4., 5.])
cluster_analysis(cluster_source, source_for_centers=None, preprocessing=None, preprocessing_kwargs=None, number_of_components=None, navigation_mask=None, signal_mask=None, algorithm=None,
return_info=False, **kwargs)#
Cluster analysis of a signal or decomposition results of a signal Results are stored in learning_results.
cluster_sourcestr {"bss" | "decomposition" | "signal"} or BaseSignal
If “bss” the blind source separation results are used If “decomposition” the decomposition results are used if “signal” the signal data is used Note that using the signal or
BaseSignal can be memory intensive and is only recommended if the Signal dimension is small BaseSignal must have the same navigation dimensions as the signal.
source_for_centersNone, str {"decomposition" | "bss" | "signal"} or BaseSignal
default : None If None the cluster_source is used If “bss” the blind source separation results are used If “decomposition” the decomposition results are used if “signal” the signal
data is used BaseSignal must have the same navigation dimensions as the signal.
preprocessingstr {"standard" | "norm" | "minmax"}, None or object
default: ‘norm’ Preprocessing the data before cluster analysis requires preprocessing the data to be clustered to similar scales. Standard preprocessing adjusts each feature to have
uniform variation. Norm preprocessing adjusts treats the set of features like a vector and each measurement is scaled to length 1. You can also pass one of the scikit-learn
preprocessing scale_method = import sklearn.processing.StandadScaler() preprocessing = scale_method See preprocessing methods in scikit-learn preprocessing for further details. If
object, must be sklearn.preprocessing-like.
preprocessing_kwargsdict or None, default None
Additional parameters passed to the supported sklearn preprocessing methods. See sklearn.preprocessing scaling methods for further details
number_of_componentsint, default None
If you are getting the cluster centers using the decomposition results (cluster_source_for_centers=”decomposition”) you can define how many components to use. If set to None the
method uses the estimate of significant components found in the decomposition step using the elbow method and stored in the learning_results.number_significant_components attribute.
This applies to both bss and decomposition results.
navigation_masknumpy.ndarray of bool
The navigation locations marked as True are not used.
signal_masknumpy.ndarray of bool
The signal locations marked as True are not used in the clustering for “signal” or Signals supplied as cluster source. This is not applied to decomposition results or
source_for_centers (as it may be a different shape to the cluster source)
algorithm{"kmeans" | "agglomerative" | "minibatchkmeans" | "spectralclustering"}
See scikit-learn documentation. Default “kmeans”
return_infobool, default False
The result of the cluster analysis is stored internally. However, the cluster class used contain a number of attributes. If True (the default is False) return the cluster object so
the attributes can be accessed.
Additional parameters passed to the clustering class for initialization. For example, in case of the “kmeans” algorithm, n_init can be used to define the number of times the algorithm
is restarted to optimize results.
None or object
If 'return_info' is True returns the Scikit-learn cluster object used for clustering. Useful if you wish to examine inertia or other outputs.
Other Parameters:
Number of clusters to find using the one of the pre-defined methods “kmeans”, “agglomerative”, “minibatchkmeans”, “spectralclustering” See sklearn.cluster for details
Return a “shallow copy” of this Signal using the standard library’s copy() function. Note: this will return a copy of the signal, but it will not duplicate the underlying data in memory, and
both Signals will reference the same data.
crop(axis, start=None, end=None, convert_units=False)#
Crops the data in a given axis. The range is given in pixels.
Specify the data axis in which to perform the cropping operation. The axis can be specified using the index of the axis in axes_manager or the axis name.
The beginning of the cropping interval. If type is int, the value is taken as the axis index. If type is float the index is calculated using the axis calibration. If start/end is None
the method crops from/to the low/high end of the axis.
The end of the cropping interval. If type is int, the value is taken as the axis index. If type is float the index is calculated using the axis calibration. If start/end is None the
method crops from/to the low/high end of the axis.
Default is False. If True, convert the units using the convert_units() method of the AxesManager. If False, does nothing.
property data#
The underlying data structure as a numpy.ndarray (or dask.array.Array, if the Signal is lazy).
decomposition(normalize_poissonian_noise=False, algorithm='SVD', output_dimension=None, centre=None, auto_transpose=True, navigation_mask=None, signal_mask=None, var_array=None, var_func=None,
reproject=None, return_info=False, print_info=True, svd_solver='auto', copy=True, **kwargs)#
Apply a decomposition to a dataset with a choice of algorithms.
The results are stored in self.learning_results.
Read more in the User Guide.
normalize_poissonian_noisebool, default False
If True, scale the signal to normalize Poissonian noise using the approach described in [*].
algorithmstr {"SVD", "MLPCA", "sklearn_pca", "NMF", "sparse_pca",
``”mini_batch_sparse_pca”``, ``”RPCA”``, ``”ORPCA”``, ``”ORNMF”``} or object, default ``”SVD”``
The decomposition algorithm to use. If algorithm is an object, it must implement a fit_transform() method or fit() and transform() methods, in the same manner as a scikit-learn
estimator. For cupy arrays, only “SVD” is supported.
output_dimensionNone or int
Number of components to keep/calculate. Default is None, i.e. min(data.shape).
centreNone or str {"navigation", "signal"}, default None
■ If None, the data is not centered prior to decomposition.
■ If “navigation”, the data is centered along the navigation axis. Only used by the “SVD” algorithm.
■ If “signal”, the data is centered along the signal axis. Only used by the “SVD” algorithm.
auto_transposebool, default True
If True, automatically transposes the data to boost performance. Only used by the “SVD” algorithm.
navigation_masknumpy.ndarray or BaseSignal
The navigation locations marked as True are not used in the decomposition.
signal_masknumpy.ndarray or BaseSignal
The signal locations marked as True are not used in the decomposition.
Array of variance for the maximum likelihood PCA algorithm. Only used by the “MLPCA” algorithm.
var_funcNone, callable() or numpy.ndarray, default None
If None, ignored If callable, applies the function to the data to obtain var_array. Only used by the “MLPCA” algorithm. If numpy array, creates var_array by applying a polynomial
function defined by the array of coefficients to the data. Only used by the “MLPCA” algorithm.
reprojectNone or str {“signal”, “navigation”, “both”}, default None
If not None, the results of the decomposition will be projected in the selected masked area.
return_info: bool, default False
The result of the decomposition is stored internally. However, some algorithms generate some extra information that is not stored. If True, return any extra information if available.
In the case of sklearn.decomposition objects, this includes the sklearn Estimator object.
print_infobool, default True
If True, print information about the decomposition being performed. In the case of sklearn.decomposition objects, this includes the values of all arguments of the chosen sklearn
svd_solver{“auto”, “full”, “arpack”, “randomized”}, default “auto”
For cupy arrays, only “full” is supported.
copybool, default True
■ If True, stores a copy of the data before any pre-treatments such as normalization in s._data_before_treatments. The original data can then be restored by calling
■ If False, no copy is made. This can be beneficial for memory usage, but care must be taken since data will be overwritten.
Any keyword arguments are passed to the decomposition algorithm.
tuple of numpy.ndarray or sklearn.base.BaseEstimator or None
■ If True and ‘algorithm’ in [‘RPCA’, ‘ORPCA’, ‘ORNMF’], returns the low-rank (X) and sparse (E) matrices from robust PCA/NMF.
■ If True and ‘algorithm’ is an sklearn Estimator, returns the Estimator object.
■ Otherwise, returns None
Return a “deep copy” of this Signal using the standard library’s deepcopy() function. Note: this means the underlying data structure will be duplicated in memory.
derivative(axis, order=1, out=None, **kwargs)#
Calculate the numerical derivative along the given axis, with respect to the calibrated units of that axis.
For a function \(y = f(x)\) and two consecutive values \(x_1\) and \(x_2\):
\[\frac{df(x)}{dx} = \frac{y(x_2)-y(x_1)}{x_2-x_1}\]
axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
order: int
The order of the derivative.
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
All extra keyword arguments are passed to numpy.gradient()
Note that the size of the data on the given axis decreases by the given order. i.e. if axis is "x" and order is 2, if the x dimension is N, then der’s x dimension is N - 2.
This function uses numpy.gradient to perform the derivative. See its documentation for implementation details.
diff(axis, order=1, out=None, rechunk=False)#
Returns a signal with the n-th order discrete difference along given axis. i.e. it calculates the difference between consecutive values in the given axis: out[n] = a[n+1] - a[n]. See
numpy.diff() for more details.
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
The order of the discrete difference.
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
BaseSignal or None
Note that the size of the data on the given axis decreases by the given order. i.e. if axis is "x" and order is 2, the x dimension is N, der’s x dimension is N - 2.
If you intend to calculate the numerical derivative, please use the proper derivative() function instead. To avoid erroneous misuse of the diff function as derivative, it raises an error when
when working with a non-uniform axis.
>>> import numpy as np
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.diff(0)
<BaseSignal, title: , dimensions: (|1023, 64, 64)>
estimate_elbow_position(explained_variance_ratio=None, log=True, max_points=20)#
Estimate the elbow position of a scree plot curve.
Used to estimate the number of significant components in a PCA variance ratio plot or other “elbow” type curves.
Find a line between first and last point on the scree plot. With a classic elbow scree plot, this line more or less defines a triangle. The elbow should be the point which is the furthest
distance from this line. For more details, see [1].
explained_variance_ratio{None, numpy array}
Explained variance ratio values that form the scree plot. If None, uses the explained_variance_ratio array stored in s.learning_results, so a decomposition must have been performed
Maximum number of points to consider in the calculation.
The index of the elbow position in the input array. Due to zero-based indexing, the number of significant components is elbow_position + 1.
V. Satopää, J. Albrecht, D. Irwin, and B. Raghavan. “Finding a “Kneedle” in a Haystack: Detecting Knee Points in System Behavior,. 31st International Conference on Distributed Computing
Systems Workshops, pp. 166-171, June 2011.
estimate_number_of_clusters(cluster_source, max_clusters=10, preprocessing=None, preprocessing_kwargs=None, number_of_components=None, navigation_mask=None, signal_mask=None, algorithm=None,
metric='gap', n_ref=4, show_progressbar=None, **kwargs)#
Performs cluster analysis of a signal for cluster sizes ranging from n_clusters =2 to max_clusters ( default 12) Note that this can be a slow process for large datasets so please consider
reducing max_clusters in this case. For each cluster it evaluates the silhouette score which is a metric of how well separated the clusters are. Maximima or peaks in the scores indicate good
choices for cluster sizes.
cluster_sourcestr {“bss”, “decomposition”, “signal”} or BaseSignal
If “bss” the blind source separation results are used If “decomposition” the decomposition results are used if “signal” the signal data is used Note that using the signal can be
memory intensive and is only recommended if the Signal dimension is small. Input Signal must have the same navigation dimensions as the signal instance.
max_clustersint, default 10
Max number of clusters to use. The method will scan from 2 to max_clusters.
preprocessingstr {“standard”, “norm”, “minmax”} or object
default: ‘norm’ Preprocessing the data before cluster analysis requires preprocessing the data to be clustered to similar scales. Standard preprocessing adjusts each feature to have
uniform variation. Norm preprocessing adjusts treats the set of features like a vector and each measurement is scaled to length 1. You can also pass an instance of a sklearn
preprocessing module. See preprocessing methods in scikit-learn preprocessing for further details. If object, must be sklearn.preprocessing-like.
preprocessing_kwargsdict or None, default None
Additional parameters passed to the cluster preprocessing algorithm. See sklearn.preprocessing preprocessing methods for further details
number_of_componentsint, default None
If you are getting the cluster centers using the decomposition results (cluster_source_for_centers=”decomposition”) you can define how many PCA components to use. If set to None the
method uses the estimate of significant components found in the decomposition step using the elbow method and stored in the learning_results.number_significant_components attribute.
navigation_maskbool numpy array, defaultNone
The navigation locations marked as True are not used in the clustering.
signal_masknumpy.ndarray of bool, default None
The signal locations marked as True are not used in the clustering. Applies to “signal” or Signal cluster sources only.
metric{'elbow' | 'silhouette' | 'gap'}, default 'gap'
Use distance,silhouette analysis or gap statistics to estimate the optimal number of clusters. Gap is believed to be, overall, the best metric but it’s also the slowest. Elbow
measures the distances between points in each cluster as an estimate of how well grouped they are and is the fastest metric. For elbow the optimal k is the knee or elbow point. For
gap the optimal k is the first k gap(k)>= gap(k+1)-std_error For silhouette the optimal k will be one of the “maxima” found with this method
n_refint, default 4
Number of references to use in gap statistics method Gap statistics compares the results from clustering the data to clustering uniformly distributed data. As clustering has a random
variation it is typically averaged n_ref times to get an statistical average.
show_progressbarNone or bool
If True, display a progress bar. If None, the default from the preferences settings is used.
Parameters passed to the clustering algorithm.
Estimate of the best cluster size.
Other Parameters:
Number of clusters to find using the one of the pre-defined methods “kmeans”,”agglomerative”,”minibatchkmeans”,”spectralclustering” See sklearn.cluster for details
estimate_poissonian_noise_variance(expected_value=None, gain_factor=None, gain_offset=None, correlation_factor=None)#
Estimate the Poissonian noise variance of the signal.
The variance is stored in the metadata.Signal.Noise_properties.variance attribute.
The Poissonian noise variance is equal to the expected value. With the default arguments, this method simply sets the variance attribute to the given expected_value. However, more generally
(although then the noise is not strictly Poissonian), the variance may be proportional to the expected value. Moreover, when the noise is a mixture of white (Gaussian) and Poissonian noise,
the variance is described by the following linear model:
\[\mathrm{Var}[X] = (a * \mathrm{E}[X] + b) * c\]
Where a is the gain_factor, b is the gain_offset (the Gaussian noise variance) and c the correlation_factor. The correlation factor accounts for correlation of adjacent signal elements that
can be modeled as a convolution with a Gaussian point spread function.
If None, the signal data is taken as the expected value. Note that this may be inaccurate where the value of data is small.
a in the above equation. Must be positive. If None, take the value from metadata.Signal.Noise_properties.Variance_linear_model if defined. Otherwise, suppose pure Poissonian noise (
i.e. gain_factor=1). If not None, the value is stored in metadata.Signal.Noise_properties.Variance_linear_model.
b in the above equation. Must be positive. If None, take the value from metadata.Signal.Noise_properties.Variance_linear_model if defined. Otherwise, suppose pure Poissonian noise (
i.e. gain_offset=0). If not None, the value is stored in metadata.Signal.Noise_properties.Variance_linear_model.
c in the above equation. Must be positive. If None, take the value from metadata.Signal.Noise_properties.Variance_linear_model if defined. Otherwise, suppose pure Poissonian noise (
i.e. correlation_factor=1). If not None, the value is stored in metadata.Signal.Noise_properties.Variance_linear_model.
export_bss_results(comp_ids=None, folder=None, calibrate=True, multiple_files=True, save_figures=False, factor_prefix='bss_factor', factor_format='hspy', loading_prefix='bss_loading',
loading_format='hspy', comp_label=None, cmap=<matplotlib.colors.LinearSegmentedColormap object>, same_window=False, no_nans=True, per_row=3, save_figures_format='png')#
Export results from ICA to any of the supported formats.
If None, returns all components/loadings. If an int, returns components/loadings with ids from 0 to the given value. If a list of ints, returns components/loadings with ids provided
in the given list.
The path to the folder where the file will be saved. If None the current folder is used by default.
The prefix that any exported filenames for factors/components begin with
The extension of the format that you wish to save the factors to. Default is 'hspy'. See loading_format for more details.
The prefix that any exported filenames for factors/components begin with
The extension of the format that you wish to save to. default is 'hspy'. The format determines the kind of output:
■ For image formats ('tif', 'png', 'jpg', etc.), plots are created using the plotting flags as below, and saved at 600 dpi. One plot is saved per loading.
■ For multidimensional formats ('rpl', 'hspy'), arrays are saved in single files. All loadings are contained in the one file.
■ For spectral formats ('msa'), each loading is saved to a separate file.
If True, one file will be created for each factor and loading. Otherwise, only two files will be created, one for the factors and another for the loadings. The default value can be
chosen in the preferences.
If True, the same figures that are obtained when using the plot methods will be saved with 600 dpi resolution
Other Parameters:
If True, calibrates plots where calibration is available from the axes_manager. If False, plots are in pixels/channels.
If True, plots each factor to the same window.
the label that is either the plot title (if plotting in separate windows) or the label in the legend (if plotting in the same window)
The colormap used for images, such as factors, loadings, or for peak characteristics. Default is the matplotlib gray colormap (plt.cm.gray).
The number of plots in each row, when the same_window parameter is True.
The image format extension.
The following parameters are only used when save_figures = True
export_cluster_results(cluster_ids=None, folder=None, calibrate=True, center_prefix='cluster_center', center_format='hspy', membership_prefix='cluster_label', membership_format='hspy', comp_label
=None, cmap=<matplotlib.colors.LinearSegmentedColormap object>, same_window=False, multiple_files=True, no_nans=True, per_row=3, save_figures=False, save_figures_format='png')#
Export results from a cluster analysis to any of the supported formats.
if None, returns all clusters/centers. if int, returns clusters/centers with ids from 0 to given int. if list of ints, returnsclusters/centers with ids in given list.
The path to the folder where the file will be saved. If None the current folder is used by default.
The prefix that any exported filenames for cluster centers begin with
The extension of the format that you wish to save to. Default is “hspy”. See loading format for more details.
The prefix that any exported filenames for cluster labels begin with
The extension of the format that you wish to save to. default is “hspy”. The format determines the kind of output.
For image formats ('tif', 'png', 'jpg', etc.),
plots are created using the plotting flags as below, and saved at 600 dpi. One plot is saved per loading.
For multidimensional formats ('rpl', 'hspy'), arrays
are saved in single files. All loadings are contained in the one file.
For spectral formats ('msa'), each loading is saved to a
separate file.
If True, on exporting a file per center will be created. Otherwise only two files will be created, one for the centers and another for the membership. The default value can be chosen
in the preferences.
If True the same figures that are obtained when using the plot methods will be saved with 600 dpi resolution
Other Parameters:
These parameters are plotting options and only used when
if True, calibrates plots where calibration is available from the axes_manager. If False, plots are in pixels/channels.
if True, plots each factor to the same window.
The label that is either the plot title (if plotting in separate windows) or the label in the legend (if plotting in the same window)
The colormap used for the factor image, or for peak characteristics, the colormap used for the scatter plot of some peak characteristic.
the number of plots in each row, when the same_window=True.
The image format extension.
export_decomposition_results(comp_ids=None, folder=None, calibrate=True, factor_prefix='factor', factor_format='hspy', loading_prefix='loading', loading_format='hspy', comp_label=None, cmap=
<matplotlib.colors.LinearSegmentedColormap object>, same_window=False, multiple_files=True, no_nans=True, per_row=3, save_figures=False, save_figures_format='png')#
Export results from a decomposition to any of the supported formats.
If None, returns all components/loadings. If an int, returns components/loadings with ids from 0 to the given value. If a list of ints, returns components/loadings with ids provided
in the given list.
The path to the folder where the file will be saved. If None, the current folder is used by default.
The prefix that any exported filenames for factors/components begin with
The extension of the format that you wish to save the factors to. Default is 'hspy'. See loading_format for more details.
The prefix that any exported filenames for factors/components begin with
The extension of the format that you wish to save to. default is 'hspy'. The format determines the kind of output:
■ For image formats ('tif', 'png', 'jpg', etc.), plots are created using the plotting flags as below, and saved at 600 dpi. One plot is saved per loading.
■ For multidimensional formats ('rpl', 'hspy'), arrays are saved in single files. All loadings are contained in the one file.
■ For spectral formats ('msa'), each loading is saved to a separate file.
If True, one file will be created for each factor and loading. Otherwise, only two files will be created, one for the factors and another for the loadings. The default value can be
chosen in the preferences.
If True the same figures that are obtained when using the plot methods will be saved with 600 dpi resolution
Other Parameters:
If True, calibrates plots where calibration is available from the axes_manager. If False, plots are in pixels/channels.
If True, plots each factor to the same window.
the label that is either the plot title (if plotting in separate windows) or the label in the legend (if plotting in the same window)
The colormap used for images, such as factors, loadings, or for peak characteristics. Default is the matplotlib gray colormap (plt.cm.gray).
The number of plots in each row, when the same_window parameter is True.
The image format extension.
The following parameters are only used when save_figures = True
fft(shift=False, apodization=False, real_fft_only=False, **kwargs)#
Compute the discrete Fourier Transform.
This function computes the discrete Fourier Transform over the signal axes by means of the Fast Fourier Transform (FFT) as implemented in numpy.
A Signal containing the result of the FFT algorithm
If performing FFT along a non-uniform axis.
Requires a uniform axis. For further information see the documentation of numpy.fft.fftn()
>>> import skimage
>>> im = hs.signals.Signal2D(skimage.data.camera())
>>> im.fft()
<ComplexSignal2D, title: FFT of , dimensions: (|512, 512)>
>>> # Use following to plot power spectrum of `im`:
>>> im.fft(shift=True, apodization=True).plot(power_spectrum=True)
If the signal was previously unfolded, fold it back
Return the blind source separation factors.
BaseSignal (or subclass)
Return the blind source separation loadings.
BaseSignal (or subclass)
get_bss_model(components=None, chunks='auto')#
Generate model with the selected number of independent components.
componentsNone, int or list of int, default None
If None, rebuilds signal instance from all components If int, rebuilds signal instance from components in range 0-given int If list of ints, rebuilds signal instance from only
components in given list
BaseSignal or subclass
A model built from the given components.
Euclidian distances to the centroid of each cluster
Hyperspy signal of cluster distances
Return cluster labels as a Signal.
mergedbool, default False
If False the cluster label signal has a navigation axes of length number_of_clusters and the signal along the the navigation direction is binary - 0 the point is not in the cluster, 1
it is included. If True, the cluster labels are merged (no navigation axes). The value of the signal at any point will be between -1 and the number of clusters. -1 represents the
points that were masked for cluster analysis if any.
The cluster labels
Return the cluster centers as a Signal.
signal{“mean”, “sum”, “centroid”}, optional
If “mean” or “sum” return the mean signal or sum respectively over each cluster. If “centroid”, returns the signals closest to the centroid.
get_current_signal(auto_title=True, auto_filename=True, as_numpy=False)#
Returns the data at the current coordinates as a BaseSignal subclass.
The signal subclass is the same as that of the current object. All the axes navigation attributes are set to False.
If True, the current indices (in parentheses) are appended to the title, separated by a space, otherwise the title of the signal is used unchanged.
If True and tmp_parameters.filename is defined (which is always the case when the Signal has been read from a file), the filename stored in the metadata is modified by appending an
underscore and the current indices in parentheses.
as_numpybool or None
Only with cupy array. If True, return the current signal as numpy array, otherwise return as cupy array.
csBaseSignal (or subclass)
The data at the current coordinates as a Signal
>>> im = hs.signals.Signal2D(np.zeros((2, 3, 32, 32)))
>>> im
<Signal2D, title: , dimensions: (3, 2|32, 32)>
>>> im.axes_manager.indices = (2, 1)
>>> im.get_current_signal()
<Signal2D, title: (2, 1), dimensions: (|32, 32)>
Return the decomposition factors.
signalBaseSignal (or subclass)
Return the decomposition loadings.
signalBaseSignal (or subclass)
Generate model with the selected number of principal components.
componentsNone, int or list of int, default None
■ If None, rebuilds signal instance from all components
■ If int, rebuilds signal instance from components in range 0-given int
■ If list of ints, rebuilds signal instance from only components in given list
BaseSignal or subclass
A model built from the given components.
Get the dimension parameters from the Signal’s underlying data. Useful when the data structure was externally modified, or when the spectrum image was not loaded from a file
Return explained variance ratio of the PCA components as a Signal1D.
Read more in the User Guide.
Explained variance ratio.
get_histogram(bins='fd', range_bins=None, max_num_bins=250, out=None, **kwargs)#
Return a histogram of the signal data.
More sophisticated algorithms for determining the bins can be used by passing a string as the bins argument. Other than the 'blocks' and 'knuth' methods, the available algorithms are the same
as numpy.histogram().
Note: The lazy version of the algorithm only supports "scott" and "fd" as a string argument for bins.
binsint or sequence of float or str, default “fd”
If bins is an int, it defines the number of equal-width bins in the given range. If bins is a sequence, it defines the bin edges, including the rightmost edge, allowing for
non-uniform bin widths.
If bins is a string from the list below, will use the method chosen to calculate the optimal bin width and consequently the number of bins (see Notes for more detail on the
estimators) from the data that falls within the requested range. While the bin width will be optimal for the actual data in the range, the number of bins will be computed to fill the
entire range, including the empty portions. For visualisation, using the 'auto' option is suggested. Weighted data is not supported for automated bin size selection.
Maximum of the ‘sturges’ and ‘fd’ estimators. Provides good all around performance.
‘fd’ (Freedman Diaconis Estimator)
Robust (resilient to outliers) estimator that takes into account data variability and data size.
An improved version of Sturges’ estimator that works better with non-normal datasets.
Less robust estimator that that takes into account data variability and data size.
Estimator based on leave-one-out cross-validation estimate of the integrated squared error. Can be regarded as a generalization of Scott’s rule.
Estimator does not take variability into account, only data size. Commonly overestimates number of bins required.
R’s default method, only accounts for data size. Only optimal for gaussian data and underestimates number of bins for large non-gaussian datasets.
Square root (of data size) estimator, used by Excel and other programs for its speed and simplicity.
Knuth’s rule is a fixed-width, Bayesian approach to determining the optimal bin width of a histogram.
Determination of optimal adaptive-width histogram bins using the Bayesian Blocks algorithm.
range_binstuple or None, optional
the minimum and maximum range for the histogram. If range_bins is None, (x.min(), x.max()) will be used.
max_num_binsint, default 250
When estimating the bins using one of the str methods, the number of bins is capped by this number to avoid a MemoryError being raised by numpy.histogram().
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
other keyword arguments (weight and density) are described in numpy.histogram().
A 1D spectrum instance containing the histogram.
>>> s = hs.signals.Signal1D(np.random.normal(size=(10, 100)))
>>> # Plot the data histogram
>>> s.get_histogram().plot()
>>> # Plot the histogram of the signal at the current coordinates
>>> s.get_current_signal().get_histogram().plot()
Get the noise variance of the signal, if set.
Equivalent to s.metadata.Signal.Noise_properties.variance.
varianceNone or float or BaseSignal (or subclass)
Noise variance of the signal, if set. Otherwise returns None.
ifft(shift=None, return_real=True, **kwargs)#
Compute the inverse discrete Fourier Transform.
This function computes the real part of the inverse of the discrete Fourier Transform over the signal axes by means of the Fast Fourier Transform (FFT) as implemented in numpy.
shiftbool or None, optional
If None, the shift option will be set to the original status of the FFT using the value in metadata. If no FFT entry is present in metadata, the parameter will be set to False. If
True, the origin of the FFT will be shifted to the centre. If False, the origin will be kept at (0, 0) (default is None).
return_realbool, default True
If True, returns only the real part of the inverse FFT. If False, returns all parts.
other keyword arguments are described in numpy.fft.ifftn()
sBaseSignal (or subclass)
A Signal containing the result of the inverse FFT algorithm
If performing IFFT along a non-uniform axis.
Requires a uniform axis. For further information see the documentation of numpy.fft.ifftn()
>>> import skimage
>>> im = hs.signals.Signal2D(skimage.data.camera())
>>> imfft = im.fft()
>>> imfft.ifft()
<Signal2D, title: real(iFFT of FFT of ), dimensions: (|512, 512)>
indexmax(axis, out=None, rechunk=False)#
Returns a signal with the index of the maximum along an axis.
axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
sBaseSignal (or subclass)
A new Signal containing the indices of the maximum along the specified axis. Note: the data dtype is always int.
>>> s = BaseSignal(np.random.random((64,64,1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.indexmax(0)
<Signal2D, title: , dimensions: (|64, 64)>
indexmin(axis, out=None, rechunk=False)#
Returns a signal with the index of the minimum along an axis.
axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
sBaseSignal (or subclass)
A new Signal containing the indices of the minimum along the specified axis. Note: the data dtype is always int.
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.indexmin(0)
<Signal2D, title: , dimensions: (|64, 64)>
integrate1D(axis, out=None, rechunk=False)#
Integrate the signal over the given axis.
The integration is performed using Simpson’s rule if axis.is_binned is False and simple summation over the given axis if True (along binned axes, the detector already provides integrated
counts per bin).
axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
sBaseSignal (or subclass)
A new Signal containing the integral of the provided Signal along the specified axis.
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.integrate1D(0)
<Signal2D, title: , dimensions: (|64, 64)>
integrate_simpson(axis, out=None, rechunk=False)#
Calculate the integral of a Signal along an axis using Simpson’s rule.
axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
sBaseSignal (or subclass)
A new Signal containing the integral of the provided Signal along the specified axis.
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.integrate_simpson(0)
<Signal2D, title: , dimensions: (|64, 64)>
interpolate_on_axis(new_axis, axis=0, inplace=False, degree=1)#
Replaces the given axis with the provided new_axis and interpolates data accordingly using scipy.interpolate.make_interp_spline().
:class:`hyperspy.axes.DataAxis` or :class:`hyperspy.axes.FunctionalDataAxis`
Axis which replaces the one specified by the axis argument. If this new axis exceeds the range of the old axis, a warning is raised that the data will be extrapolated.
axisint or str, default 0
Specifies the axis which will be replaced using the index of the axis in the axes_manager. The axis can be specified using the index of the axis in axes_manager or the axis name.
inplacebool, default False
If True the data of self is replaced by the result and the axis is changed inplace. Otherwise self is not changed and a new signal with the changes incorporated is returned.
degree: int, default 1
Specifies the B-Spline degree of the used interpolator.
BaseSignal (or subclass)
A copy of the object with the axis exchanged and the data interpolated. This only occurs when inplace is set to False, otherwise nothing is returned.
property is_rgb#
Whether or not this signal is an RGB dtype.
property is_rgba#
Whether or not this signal is an RGB + alpha channel dtype.
property is_rgbx#
Whether or not this signal is either an RGB or RGB + alpha channel dtype.
map(function, show_progressbar=None, num_workers=None, inplace=True, ragged=None, navigation_chunks=None, output_signal_size=None, output_dtype=None, lazy_output=None, **kwargs)#
Apply a function to the signal data at all the navigation coordinates.
The function must operate on numpy arrays. It is applied to the data at each navigation coordinate pixel-py-pixel. Any extra keyword arguments are passed to the function. The keywords can
take different values at different coordinates. If the function takes an axis or axes argument, the function is assumed to be vectorized and the signal axes are assigned to axis or axes.
Otherwise, the signal is iterated over the navigation axes and a progress bar is displayed to monitor the progress.
In general, only navigation axes (order, calibration, and number) are guaranteed to be preserved.
Any function that can be applied to the signal. This function should not alter any mutable input arguments or input data. So do not do operations which alter the input, without
copying it first. For example, instead of doing image *= mask, rather do image = image * mask. Likewise, do not do image[5, 5] = 10 directly on the input data or arguments, but make a
copy of it first. For example via image = copy.deepcopy(image).
If True, display a progress bar. If None, the default from the preferences settings is used.
If True, the output will be returned as a lazy signal. This means the calculation itself will be delayed until either compute() is used, or the signal is stored as a file. If False,
the output will be returned as a non-lazy signal, this means the outputs will be calculated directly, and loaded into memory. If None the output will be lazy if the input signal is
lazy, and non-lazy if the input signal is non-lazy.
If True, the data is replaced by the result. Otherwise a new Signal with the results is returned.
Indicates if the results for each navigation pixel are of identical shape (and/or numpy arrays to begin with). If None, the output signal will be ragged only if the original signal is
Set the navigation_chunks argument to a tuple of integers to split the navigation axes into chunks. This can be useful to enable using multiple cores with signals which are less that
100 MB. This argument is passed to rechunk().
Since the size and dtype of the signal dimension of the output signal can be different from the input signal, this output signal size must be calculated somehow. If both
output_signal_size and output_dtype is None, this is automatically determined. However, if for some reason this is not working correctly, this can be specified via output_signal_size
and output_dtype. The most common reason for this failing is due to the signal size being different for different navigation positions. If this is the case, use ragged=True. None is
See docstring for output_signal_size for more information. Default None.
Number of worker used by dask. If None, default to dask default value.
All extra keyword arguments are passed to the provided function
If the function results do not have identical shapes, the result is an array of navigation shape, where each element corresponds to the result of the function (of arbitrary object type),
called a “ragged array”. As such, most functions are not able to operate on the result and the data should be used directly.
This method is similar to Python’s map() that can also be utilized with a BaseSignal instance for similar purposes. However, this method has the advantage of being faster because it iterates
the underlying numpy data array instead of the BaseSignal.
Currently requires a uniform axis.
Apply a Gaussian filter to all the images in the dataset. The sigma parameter is constant:
>>> import scipy.ndimage
>>> im = hs.signals.Signal2D(np.random.random((10, 64, 64)))
>>> im.map(scipy.ndimage.gaussian_filter, sigma=2.5)
Apply a Gaussian filter to all the images in the dataset. The signal parameter is variable:
>>> im = hs.signals.Signal2D(np.random.random((10, 64, 64)))
>>> sigmas = hs.signals.BaseSignal(np.linspace(2, 5, 10)).T
>>> im.map(scipy.ndimage.gaussian_filter, sigma=sigmas)
Rotate the two signal dimensions, with different amount as a function of navigation index. Delay the calculation by getting the output lazily. The calculation is then done using the compute
>>> from scipy.ndimage import rotate
>>> s = hs.signals.Signal2D(np.random.random((5, 4, 40, 40)))
>>> s_angle = hs.signals.BaseSignal(np.linspace(0, 90, 20).reshape(5, 4)).T
>>> s.map(rotate, angle=s_angle, reshape=False, lazy_output=True)
>>> s.compute()
Rotate the two signal dimensions, with different amount as a function of navigation index. In addition, the output is returned as a new signal, instead of replacing the old signal.
>>> s = hs.signals.Signal2D(np.random.random((5, 4, 40, 40)))
>>> s_angle = hs.signals.BaseSignal(np.linspace(0, 90, 20).reshape(5, 4)).T
>>> s_rot = s.map(rotate, angle=s_angle, reshape=False, inplace=False)
If you want some more control over computing a signal that isn’t lazy you can always set lazy_output to True and then compute the signal setting the scheduler to ‘threading’, ‘processes’,
‘single-threaded’ or ‘distributed’.
Additionally, you can set the navigation_chunks argument to a tuple of integers to split the navigation axes into chunks. This can be useful if your signal is less that 100 mb but you still
want to use multiple cores.
>>> s = hs.signals.Signal2D(np.random.random((5, 4, 40, 40)))
>>> s_angle = hs.signals.BaseSignal(np.linspace(0, 90, 20).reshape(5, 4)).T
>>> s.map(
... rotate, angle=s_angle, reshape=False, lazy_output=True,
... inplace=True, navigation_chunks=(2,2)
... )
>>> s.compute()
max(axis=None, out=None, rechunk=False)#
Returns a signal with the maximum of the signal along at least one axis.
axisint, str, DataAxis or tuple
Either one on its own, or many axes in a tuple can be passed. In both cases the axes can be passed directly, or specified using the index in axes_manager or the name of the axis. Any
duplicates are removed. If None, the operation is performed over all navigation axes (default).
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
sBaseSignal (or subclass)
A new Signal containing the maximum of the provided Signal over the specified axes
>>> import numpy as np
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.max(0)
<Signal2D, title: , dimensions: (|64, 64)>
mean(axis=None, out=None, rechunk=False)#
Returns a signal with the average of the signal along at least one axis.
axisint, str, DataAxis or tuple
Either one on its own, or many axes in a tuple can be passed. In both cases the axes can be passed directly, or specified using the index in axes_manager or the name of the axis. Any
duplicates are removed. If None, the operation is performed over all navigation axes (default).
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
sBaseSignal (or subclass)
A new Signal containing the mean of the provided Signal over the specified axes
>>> import numpy as np
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.mean(0)
<Signal2D, title: , dimensions: (|64, 64)>
property metadata#
The metadata of the signal.
min(axis=None, out=None, rechunk=False)#
Returns a signal with the minimum of the signal along at least one axis.
axisint, str, DataAxis or tuple
Either one on its own, or many axes in a tuple can be passed. In both cases the axes can be passed directly, or specified using the index in axes_manager or the name of the axis. Any
duplicates are removed. If None, the operation is performed over all navigation axes (default).
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
sBaseSignal (or subclass)
A new Signal containing the minimum of the provided Signal over the specified axes
>>> import numpy as np
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.min(0)
<Signal2D, title: , dimensions: (|64, 64)>
nanmax(axis=None, out=None, rechunk=False)#
Identical to max(), except ignores missing (NaN) values. See that method’s documentation for details.
nanmean(axis=None, out=None, rechunk=False)#
Identical to mean(), except ignores missing (NaN) values. See that method’s documentation for details.
nanmin(axis=None, out=None, rechunk=False)#
Identical to min(), except ignores missing (NaN) values. See that method’s documentation for details.
nanstd(axis=None, out=None, rechunk=False)#
Identical to std(), except ignores missing (NaN) values. See that method’s documentation for details.
nansum(axis=None, out=None, rechunk=False)#
Identical to sum(), except ignores missing (NaN) values. See that method’s documentation for details.
nanvar(axis=None, out=None, rechunk=False)#
Identical to var(), except ignores missing (NaN) values. See that method’s documentation for details.
normalize_bss_components(target='factors', function=<function sum>)#
Normalize BSS components.
target{“factors”, “loadings”}
Normalize components based on the scale of either the factors or loadings.
functionnumpy callable(), default numpy.sum
Each target component is divided by the output of function(target). The function must return a scalar when operating on numpy arrays and must have an axis argument.
normalize_decomposition_components(target='factors', function=<function sum>)#
Normalize decomposition components.
target{“factors”, “loadings”}
Normalize components based on the scale of either the factors or loadings.
functionnumpy callable(), default numpy.sum
Each target component is divided by the output of function(target). The function must return a scalar when operating on numpy arrays and must have an axis argument.
normalize_poissonian_noise(navigation_mask=None, signal_mask=None)#
Normalize the signal under the assumption of Poisson noise.
Scales the signal using to “normalize” the Poisson data for subsequent decomposition analysis [*].
navigation_mask{None, bool numpy array}, default None
Optional mask applied in the navigation axis.
signal_mask{None, bool numpy array}, default None
Optional mask applied in the signal axis.
property original_metadata#
The original metadata of the signal.
plot(navigator='auto', axes_manager=None, plot_markers=True, **kwargs)#
Plot the signal at the current coordinates.
For multidimensional datasets an optional figure, the “navigator”, with a cursor to navigate that data is raised. In any case it is possible to navigate the data using the sliders. Currently
only signals with signal_dimension equal to 0, 1 and 2 can be plotted.
navigatorstr, None, or BaseSignal (or subclass).
Allowed string values are ``’auto’``, ``’slider’``, and ``’spectrum’``.
■ If 'auto':
★ If navigation_dimension > 0, a navigator is provided to explore the data.
★ If navigation_dimension is 1 and the signal is an image the navigator is a sum spectrum obtained by integrating over the signal axes (the image).
★ If navigation_dimension is 1 and the signal is a spectrum the navigator is an image obtained by stacking all the spectra in the dataset horizontally.
★ If navigation_dimension is > 1, the navigator is a sum image obtained by integrating the data over the signal axes.
★ Additionally, if navigation_dimension > 2, a window with one slider per axis is raised to navigate the data.
★ For example, if the dataset consists of 3 navigation axes “X”, “Y”, “Z” and one signal axis, “E”, the default navigator will be an image obtained by integrating the data over
“E” at the current “Z” index and a window with sliders for the “X”, “Y”, and “Z” axes will be raised. Notice that changing the “Z”-axis index changes the navigator in this
★ For lazy signals, the navigator will be calculated using the compute_navigator() method.
■ If 'slider':
★ If navigation dimension > 0 a window with one slider per axis is raised to navigate the data.
■ If 'spectrum':
★ If navigation_dimension > 0 the navigator is always a spectrum obtained by integrating the data over all other axes.
★ Not supported for lazy signals, the 'auto' option will be used instead.
■ If None, no navigator will be provided.
Alternatively a BaseSignal (or subclass) instance can be provided. The navigation or signal shape must match the navigation shape of the signal to plot or the navigation_shape +
signal_shape must be equal to the navigator_shape of the current object (for a dynamic navigator). If the signal dtype is RGB or RGBA this parameter has no effect and the value is
always set to 'slider'.
axes_managerNone or AxesManager
If None, the signal’s axes_manager attribute is used.
plot_markersbool, default True
Plot markers added using s.add_marker(marker, permanent=True). Note, a large number of markers might lead to very slow plotting.
Only for image navigator, additional keyword arguments for matplotlib.pyplot.imshow().
normstr, default 'auto'
The function used to normalize the data prior to plotting. Allowable strings are: 'auto', 'linear', 'log'. If 'auto', intensity is plotted on a linear scale except when power_spectrum
=True (only for complex signals).
The string must contain any combination of the 'x' and 'v' characters. If 'x' or 'v' (for values) are in the string, the corresponding horizontal or vertical axis limits are set to
their maxima and the axis limits will reset when the data or the navigation indices are changed. Default is 'v'.
Only when plotting an image: additional (optional) keyword arguments for matplotlib.pyplot.imshow().
plot_bss_factors(comp_ids=None, calibrate=True, same_window=True, title=None, cmap=<matplotlib.colors.LinearSegmentedColormap object>, per_row=3, **kwargs)#
Plot factors from blind source separation results. In case of 1D signal axis, each factors line can be toggled on and off by clicking on their corresponding line in the legend.
If comp_ids is None, maps of all components will be returned. If it is an int, maps of components with ids from 0 to the given value will be returned. If comp_ids is a list of ints,
maps of components with ids contained in the list will be returned.
If True, calibrates plots where calibration is available from the axes_manager. If False, plots are in pixels/channels.
if True, plots each factor to the same window. They are not scaled. Default is True.
Title of the matplotlib plot or label of the line in the legend when the dimension of factors is 1 and same_window is True.
The colormap used for the factor images, or for peak characteristics. Default is the matplotlib gray colormap (plt.cm.gray).
The number of plots in each row, when the same_window parameter is True.
plot_bss_loadings(comp_ids=None, calibrate=True, same_window=True, title=None, with_factors=False, cmap=<matplotlib.colors.LinearSegmentedColormap object>, no_nans=False, per_row=3, axes_decor=
'all', **kwargs)#
Plot loadings from blind source separation results. In case of 1D navigation axis, each loading line can be toggled on and off by clicking on their corresponding line in the legend.
If comp_ids=None, maps of all components will be returned. If it is an int, maps of components with ids from 0 to the given value will be returned. If comp_ids is a list of ints, maps
of components with ids contained in the list will be returned.
if True, calibrates plots where calibration is available from the axes_manager. If False, plots are in pixels/channels.
If True, plots each factor to the same window. They are not scaled. Default is True.
Title of the matplotlib plot or label of the line in the legend when the dimension of loadings is 1 and same_window is True.
If True, also returns figure(s) with the factors for the given comp_ids.
The colormap used for the loading image, or for peak characteristics,. Default is the matplotlib gray colormap (plt.cm.gray).
If True, removes NaN’s from the loading plots.
The number of plots in each row, when the same_window parameter is True.
One of: 'all', 'ticks', 'off', or None Controls how the axes are displayed on each image; default is 'all' If 'all', both ticks and axis labels will be shown If 'ticks', no axis
labels will be shown, but ticks/labels will If 'off', all decorations and frame will be disabled If None, no axis decorations will be shown, but ticks/frame will
plot_bss_results(factors_navigator='smart_auto', loadings_navigator='smart_auto', factors_dim=2, loadings_dim=2)#
Plot the blind source separation factors and loadings.
Unlike plot_bss_factors() and plot_bss_loadings(), this method displays one component at a time. Therefore it provides a more compact visualization than then other two methods. The loadings
and factors are displayed in different windows and each has its own navigator/sliders to navigate them if they are multidimensional. The component index axis is synchronized between the two.
One of: 'smart_auto', 'auto', None, 'spectrum' or a BaseSignal object. 'smart_auto' (default) displays sliders if the navigation dimension is less than 3. For a description of the
other options see the plot() documentation for details.
See the factors_navigator parameter
Currently HyperSpy cannot plot a signal when the signal dimension is higher than two. Therefore, to visualize the BSS results when the factors or the loadings have signal dimension
greater than 2, the data can be viewed as spectra (or images) by setting this parameter to 1 (or 2). (The default is 2)
See the factors_dim parameter
plot_cluster_distances(cluster_ids=None, calibrate=True, same_window=True, with_centers=False, cmap=<matplotlib.colors.LinearSegmentedColormap object>, no_nans=False, per_row=3, axes_decor='all',
title=None, **kwargs)#
Plot the euclidian distances to the centroid of each cluster.
In case of 1D navigation axis, each line can be toggled on and off by clicking on the corresponding line in the legend.
if None (default), returns maps of all components using the number_of_cluster was defined when executing cluster. Otherwise it raises a ValueError. if int, returns maps of cluster
labels with ids from 0 to given int. if list of ints, returns maps of cluster labels with ids in given list.
if True, calibrates plots where calibration is available from the axes_manager. If False, plots are in pixels/channels.
if True, plots each factor to the same window. They are not scaled. Default is True.
Title of the matplotlib plot or label of the line in the legend when the dimension of distance is 1 and same_window is True.
If True, also returns figure(s) with the cluster centers for the given cluster_ids.
The colormap used for the factor image, or for peak characteristics, the colormap used for the scatter plot of some peak characteristic.
If True, removes NaN’s from the loading plots.
the number of plots in each row, when the same_window parameter is True.
Controls how the axes are displayed on each image; default is ‘all’ If ‘all’, both ticks and axis labels will be shown If ‘ticks’, no axis labels will be shown, but ticks/labels will
If ‘off’, all decorations and frame will be disabled If None, no axis decorations will be shown, but ticks/frame will
plot_cluster_labels(cluster_ids=None, calibrate=True, same_window=True, with_centers=False, cmap=<matplotlib.colors.LinearSegmentedColormap object>, no_nans=False, per_row=3, axes_decor='all',
title=None, **kwargs)#
Plot cluster labels from a cluster analysis. In case of 1D navigation axis, each loading line can be toggled on and off by clicking on the legended line.
if None (default), returns maps of all components using the number_of_cluster was defined when executing cluster. Otherwise it raises a ValueError. if int, returns maps of cluster
labels with ids from 0 to given int. if list of ints, returns maps of cluster labels with ids in given list.
if True, calibrates plots where calibration is available from the axes_manager. If False, plots are in pixels/channels.
if True, plots each factor to the same window. They are not scaled. Default is True.
Title of the matplotlib plot or label of the line in the legend when the dimension of labels is 1 and same_window is True.
If True, also returns figure(s) with the cluster centers for the given cluster_ids.
The colormap used for the factor image, or for peak characteristics, the colormap used for the scatter plot of some peak characteristic.
If True, removes NaN’s from the loading plots.
the number of plots in each row, when the same_window parameter is True.
Controls how the axes are displayed on each image; default is ‘all’ If ‘all’, both ticks and axis labels will be shown If ‘ticks’, no axis labels will be shown, but ticks/labels will
If ‘off’, all decorations and frame will be disabled If None, no axis decorations will be shown, but ticks/frame will
Plot the cluster metrics calculated using the estimate_number_of_clusters() method
plot_cluster_results(centers_navigator='smart_auto', labels_navigator='smart_auto', centers_dim=2, labels_dim=2)#
Plot the cluster labels and centers.
Unlike plot_cluster_labels() and plot_cluster_signals(), this method displays one component at a time. Therefore it provides a more compact visualization than then other two methods. The
labels and centers are displayed in different windows and each has its own navigator/sliders to navigate them if they are multidimensional. The component index axis is synchronized between
the two.
centers_navigator, labels_navigatorNone, {"smart_auto" | "auto" | "spectrum"} or BaseSignal, default "smart_auto"
"smart_auto" displays sliders if the navigation dimension is less than 3. For a description of the other options see plot documentation for details.
labels_dim, centers_dimsint, default 2
Currently HyperSpy cannot plot signals of dimension higher than two. Therefore, to visualize the clustering results when the centers or the labels have signal dimension greater than 2
we can view the data as spectra(images) by setting this parameter to 1(2)
plot_cluster_signals(signal='mean', cluster_ids=None, calibrate=True, same_window=True, title=None, per_row=3)#
Plot centers from a cluster analysis.
If “mean” or “sum” return the mean signal or sum respectively over each cluster. If “centroid”, returns the signals closest to the centroid.
If None, returns maps of all clusters. If int, returns maps of clusters with ids from 0 to given int. If list of ints, returns maps of clusters with ids in given list.
If True, calibrates plots where calibration is available from the axes_manager. If False, plots are in pixels/channels.
If True, plots each center to the same window. They are not scaled.
Title of the matplotlib plot or label of the line in the legend when the dimension of loadings is 1 and same_window is True.
The number of plots in each row, when the same_window parameter is True.
Plot cumulative explained variance up to n principal components.
Number of principal components to show.
Axes object containing the cumulative explained variance plot.
plot_decomposition_factors(comp_ids=None, calibrate=True, same_window=True, title=None, cmap=<matplotlib.colors.LinearSegmentedColormap object>, per_row=3, **kwargs)#
Plot factors from a decomposition. In case of 1D signal axis, each factors line can be toggled on and off by clicking on their corresponding line in the legend.
If comp_ids is None, maps of all components will be returned if the output_dimension was defined when executing decomposition(). Otherwise it raises a ValueError. If comp_ids is an
int, maps of components with ids from 0 to the given value will be returned. If comp_ids is a list of ints, maps of components with ids contained in the list will be returned.
If True, calibrates plots where calibration is available from the axes_manager. If False, plots are in pixels/channels.
If True, plots each factor to the same window. They are not scaled. Default is True.
Title of the matplotlib plot or label of the line in the legend when the dimension of factors is 1 and same_window is True.
The colormap used for the factor images, or for peak characteristics. Default is the matplotlib gray colormap (plt.cm.gray).
The number of plots in each row, when the same_window parameter is True.
plot_decomposition_loadings(comp_ids=None, calibrate=True, same_window=True, title=None, with_factors=False, cmap=<matplotlib.colors.LinearSegmentedColormap object>, no_nans=False, per_row=3,
axes_decor='all', **kwargs)#
Plot loadings from a decomposition. In case of 1D navigation axis, each loading line can be toggled on and off by clicking on the legended line.
If comp_ids is None, maps of all components will be returned if the output_dimension was defined when executing decomposition(). Otherwise it raises a ValueError. If comp_ids is an
int, maps of components with ids from 0 to the given value will be returned. If comp_ids is a list of ints, maps of components with ids contained in the list will be returned.
if True, calibrates plots where calibration is available from the axes_manager. If False, plots are in pixels/channels.
if True, plots each factor to the same window. They are not scaled. Default is True.
Title of the matplotlib plot or label of the line in the legend when the dimension of loadings is 1 and same_window is True.
If True, also returns figure(s) with the factors for the given comp_ids.
The colormap used for the loadings images, or for peak characteristics. Default is the matplotlib gray colormap (plt.cm.gray).
If True, removes NaN’s from the loading plots.
The number of plots in each row, when the same_window parameter is True.
One of: 'all', 'ticks', 'off', or None Controls how the axes are displayed on each image; default is 'all' If 'all', both ticks and axis labels will be shown. If 'ticks', no axis
labels will be shown, but ticks/labels will. If 'off', all decorations and frame will be disabled. If None, no axis decorations will be shown, but ticks/frame will.
plot_decomposition_results(factors_navigator='smart_auto', loadings_navigator='smart_auto', factors_dim=2, loadings_dim=2)#
Plot the decomposition factors and loadings.
Unlike plot_decomposition_factors() and plot_decomposition_loadings(), this method displays one component at a time. Therefore it provides a more compact visualization than then other two
methods. The loadings and factors are displayed in different windows and each has its own navigator/sliders to navigate them if they are multidimensional. The component index axis is
synchronized between the two.
factors_navigatorstr, None, or BaseSignal (or subclass)
One of: 'smart_auto', 'auto', None, 'spectrum' or a BaseSignal object. 'smart_auto' (default) displays sliders if the navigation dimension is less than 3. For a description of the
other options see the plot() documentation for details.
loadings_navigatorstr, None, or BaseSignal (or subclass)
See the factors_navigator parameter
factors_dim, loadings_dimint
Currently HyperSpy cannot plot a signal when the signal dimension is higher than two. Therefore, to visualize the BSS results when the factors or the loadings have signal dimension
greater than 2, the data can be viewed as spectra (or images) by setting this parameter to 1 (or 2). (The default is 2)
plot_explained_variance_ratio(n=30, log=True, threshold=0, hline='auto', vline=False, xaxis_type='index', xaxis_labeling=None, signal_fmt=None, noise_fmt=None, fig=None, ax=None, **kwargs)#
Plot the decomposition explained variance ratio vs index number.
This is commonly known as a scree plot.
Read more in the User Guide.
nint or None
Number of components to plot. If None, all components will be plot
logbool, default True
If True, the y axis uses a log scale.
thresholdfloat or int
Threshold used to determine how many components should be highlighted as signal (as opposed to noise). If a float (between 0 and 1), threshold will be interpreted as a cutoff value,
defining the variance at which to draw a line showing the cutoff between signal and noise; the number of signal components will be automatically determined by the cutoff value. If an
int, threshold is interpreted as the number of components to highlight as signal (and no cutoff line will be drawn)
hline: {‘auto’, True, False}
Whether or not to draw a horizontal line illustrating the variance cutoff for signal/noise determination. Default is to draw the line at the value given in threshold (if it is a
float) and not draw in the case threshold is an int, or not given. If True, (and threshold is an int), the line will be drawn through the last component defined as signal. If False,
the line will not be drawn in any circumstance.
vline: bool, default False
Whether or not to draw a vertical line illustrating an estimate of the number of significant components. If True, the line will be drawn at the the knee or elbow position of the curve
indicating the number of significant components. If False, the line will not be drawn in any circumstance.
xaxis_type{‘index’, ‘number’}
Determines the type of labeling applied to the x-axis. If 'index', axis will be labeled starting at 0 (i.e. “pythonic index” labeling); if 'number', it will start at 1 (number
xaxis_labeling{‘ordinal’, ‘cardinal’, None}
Determines the format of the x-axis tick labels. If 'ordinal', “1st, 2nd, …” will be used; if 'cardinal', “1, 2, …” will be used. If None, an appropriate default will be selected.
Dictionary of matplotlib formatting values for the signal components
Dictionary of matplotlib formatting values for the noise components
figmatplotlib.figure.Figure or None
If None, a default figure will be created, otherwise will plot into fig
axmatplotlib.axes.Axes or None
If None, a default ax will be created, otherwise will plot into ax
remaining keyword arguments are passed to matplotlib.figure.Figure
Axes object containing the scree plot
To generate a scree plot with customized symbols for signal vs. noise components and a modified cutoff threshold value:
>>> s = hs.load("some_spectrum_image")
>>> s.decomposition()
>>> s.plot_explained_variance_ratio(
... n=40,
... threshold=0.005,
... signal_fmt={'marker': 'v', 's': 150, 'c': 'pink'},
... noise_fmt={'marker': '*', 's': 200, 'c': 'green'}
... )
print_summary_statistics(formatter='%.3g', rechunk=False)#
Prints the five-number summary statistics of the data, the mean, and the standard deviation.
Prints the mean, standard deviation (std), maximum (max), minimum (min), first quartile (Q1), median, and third quartile. nans are removed from the calculations.
The number formatter to use for the output
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
property ragged#
Whether the signal is ragged or not.
rebin(new_shape=None, scale=None, crop=True, dtype=None, out=None)#
Rebin the signal into a smaller or larger shape, based on linear interpolation. Specify either new_shape or scale. Scale of 1 means no binning and scale less than one results in up-sampling.
For each dimension specify the new_shape. This will internally be converted into a scale parameter.
For each dimension, specify the new:old pixel ratio, e.g. a ratio of 1 is no binning and a ratio of 2 means that each pixel in the new spectrum is twice the size of the pixels in the
old spectrum. The length of the list should match the dimension of the Signal’s underlying data array. Note : Only one of ``scale`` or ``new_shape`` should be specified, otherwise the
function will not run
Whether or not to crop the resulting rebinned data (default is True). When binning by a non-integer number of pixels it is likely that the final row in each dimension will contain
fewer than the full quota to fill one pixel. For example, a 5*5 array binned by 2.1 will produce two rows containing 2.1 pixels and one row containing only 0.8 pixels. Selection of
crop=True or crop=False determines whether or not this “black” line is cropped from the final binned array or not. Please note that if ``crop=False`` is used, the final row in each
dimension may appear black if a fractional number of pixels are left over. It can be removed but has been left to preserve total counts before and after binning.
Specify the dtype of the output. If None, the dtype will be determined by the behaviour of numpy.sum(), if "same", the dtype will be kept the same. Default is None.
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
The resulting cropped signal.
If trying to rebin over a non-uniform axis.
>>> spectrum = hs.signals.Signal1D(np.ones([4, 4, 10]))
>>> spectrum.data[1, 2, 9] = 5
>>> print(spectrum)
<Signal1D, title: , dimensions: (4, 4|10)>
>>> print ('Sum =', sum(sum(sum(spectrum.data))))
Sum = 164.0
>>> scale = [2, 2, 5]
>>> test = spectrum.rebin(scale)
>>> print(test)
<Signal1D, title: , dimensions: (2, 2|5)>
>>> print('Sum =', sum(sum(sum(test.data))))
Sum = 164.0
>>> s = hs.signals.Signal1D(np.ones((2, 5, 10), dtype=np.uint8))
>>> print(s)
<Signal1D, title: , dimensions: (5, 2|10)>
>>> print(s.data.dtype)
Use dtype=np.unit16 to specify a dtype
>>> s2 = s.rebin(scale=(5, 2, 1), dtype=np.uint16)
>>> print(s2.data.dtype)
Use dtype=”same” to keep the same dtype
>>> s3 = s.rebin(scale=(5, 2, 1), dtype="same")
>>> print(s3.data.dtype)
By default dtype=None, the dtype is determined by the behaviour of numpy.sum, in this case, unsigned integer of the same precision as the platform integer
>>> s4 = s.rebin(scale=(5, 2, 1))
>>> print(s4.data.dtype)
Reverse the independent component.
component_numberlist or int
component index/es
>>> s = hs.load('some_file')
>>> s.decomposition(True)
>>> s.blind_source_separation(3)
Reverse component 1
>>> s.reverse_bss_component(1)
Reverse components 0 and 2
>>> s.reverse_bss_component((0, 2))
Reverse the decomposition component.
component_numberlist or int
component index/es
>>> s = hs.load('some_file')
>>> s.decomposition(True)
Reverse component 1
>>> s.reverse_decomposition_component(1)
Reverse components 0 and 2
>>> s.reverse_decomposition_component((0, 2))
rollaxis(axis, to_axis, optimize=False)#
Roll the specified axis backwards, until it lies in a given position.
axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name. The axis to roll backwards. The positions of the other axes do
not change relative to one another.
to_axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name. The axis is rolled until it lies before this other axis.
If True, the location of the data in memory is optimised for the fastest iteration over the navigation axes. This operation can cause a peak of memory usage and requires considerable
processing times for large datasets and/or low specification hardware. See the Transposing (changing signal spaces) section of the HyperSpy user guide for more information. When
operating on lazy signals, if True, the chunks are optimised for the new axes configuration.
sBaseSignal (or subclass)
Output signal.
>>> s = hs.signals.Signal1D(np.ones((5, 4, 3, 6)))
>>> s
<Signal1D, title: , dimensions: (3, 4, 5|6)>
>>> s.rollaxis(3, 1)
<Signal1D, title: , dimensions: (3, 4, 5|6)>
>>> s.rollaxis(2, 0)
<Signal1D, title: , dimensions: (5, 3, 4|6)>
save(filename=None, overwrite=None, extension=None, file_format=None, **kwds)#
Saves the signal in the specified format.
The function gets the format from the specified extension (see Supported formats in the User Guide for more information):
☆ 'hspy' for HyperSpy’s HDF5 specification
☆ 'rpl' for Ripple (useful to export to Digital Micrograph)
☆ 'msa' for EMSA/MSA single spectrum saving.
☆ 'unf' for SEMPER unf binary format.
☆ 'blo' for Blockfile diffraction stack saving.
☆ Many image formats such as 'png', 'tiff', 'jpeg'…
If no extension is provided the default file format as defined in the preferences is used. Please note that not all the formats supports saving datasets of arbitrary dimensions, e.g. 'msa'
only supports 1D data, and blockfiles only supports image stacks with a navigation_dimension < 2.
Each format accepts a different set of parameters. For details see the specific format documentation.
filenamestr or None
If None (default) and tmp_parameters.filename and tmp_parameters.folder are defined, the filename and path will be taken from there. A valid extension can be provided e.g.
'my_file.rpl' (see extension parameter).
overwriteNone or bool
If None, if the file exists it will query the user. If True(False) it does(not) overwrite the file if it exists.
extensionNone or str
The extension of the file that defines the file format. Allowable string values are: {'hspy', 'hdf5', 'rpl', 'msa', 'unf', 'blo', 'emd', and common image extensions e.g. 'tiff', 'png'
, etc.} 'hspy' and 'hdf5' are equivalent. Use 'hdf5' if compatibility with HyperSpy versions older than 1.2 is required. If None, the extension is determined from the following list
in this order:
1. the filename
2. Signal.tmp_parameters.extension
3. 'hspy' (the default extension)
chunkstuple or True or None (default)
HyperSpy, Nexus and EMD NCEM format only. Define chunks used when saving. The chunk shape should follow the order of the array (s.data.shape), not the shape of the axes_manager. If
None and lazy signal, the dask array chunking is used. If None and non-lazy signal, the chunks are estimated automatically to have at least one chunk per signal space. If True, the
chunking is determined by the the h5py guess_chunk function.
save_original_metadatabool , defaultFalse
Nexus file only. Option to save hyperspy.original_metadata with the signal. A loaded Nexus file may have a large amount of data when loaded which you may wish to omit on saving
use_defaultbool , defaultFalse
Nexus file only. Define the default dataset in the file. If set to True the signal or first signal in the list of signals will be defined as the default (following Nexus v3 data
write_datasetbool, optional
Only for hspy files. If True, write the dataset, otherwise, don’t write it. Useful to save attributes without having to write the whole dataset. Default is True.
close_filebool, optional
Only for hdf5-based files and some zarr store. Close the file after writing. Default is True.
file_format: string
The file format of choice to save the file. If not given, it is inferred from the file extension.
Set the noise variance of the signal.
Equivalent to s.metadata.set_item("Signal.Noise_properties.variance", variance).
varianceNone or float or BaseSignal (or subclass)
Value or values of the noise variance. A value of None is equivalent to clearing the variance.
Set the signal_origin metadata value.
The signal_origin attribute specifies if the data was obtained through experiment or simulation.
Typically 'experiment' or 'simulation'
Set the signal type and convert the current signal accordingly.
The signal_type attribute specifies the type of data that the signal contains e.g. electron energy-loss spectroscopy data, photoemission spectroscopy data, etc.
When setting signal_type to a “known” type, HyperSpy converts the current signal to the most appropriate BaseSignal subclass. Known signal types are signal types that have a specialized
BaseSignal subclass associated, usually providing specific features for the analysis of that type of signal.
HyperSpy ships with a minimal set of known signal types. External packages can register extra signal types. To print a list of registered signal types in the current installation, call
print_known_signal_types(), and see the developer guide for details on how to add new signal_types. A non-exhaustive list of HyperSpy extensions is also maintained here: hyperspy/
signal_typestr, optional
If no arguments are passed, the signal_type is set to undefined and the current signal converted to a generic signal subclass. Otherwise, set the signal_type to the given signal type
or to the signal type corresponding to the given signal type alias. Setting the signal_type to a known signal type (if exists) is highly advisable. If none exists, it is good practice
to set signal_type to a value that best describes the data signal type.
Let’s first print all known signal types:
>>> s = hs.signals.Signal1D([0, 1, 2, 3])
>>> s
<Signal1D, title: , dimensions: (|4)>
>>> hs.print_known_signal_types()
| signal_type | aliases | class name | package |
| DielectricFunction | dielectric function | DielectricFunction | exspy |
| EDS_SEM | | EDSSEMSpectrum | exspy |
| EDS_TEM | | EDSTEMSpectrum | exspy |
| EELS | TEM EELS | EELSSpectrum | exspy |
| hologram | | HologramImage | holospy |
We can set the signal_type using the signal_type:
>>> s.set_signal_type("EELS")
>>> s
<EELSSpectrum, title: , dimensions: (|4)>
>>> s.set_signal_type("EDS_SEM")
>>> s
<EDSSEMSpectrum, title: , dimensions: (|4)>
or any of its aliases:
>>> s.set_signal_type("TEM EELS")
>>> s
<EELSSpectrum, title: , dimensions: (|4)>
To set the signal_type to “undefined”, simply call the method without arguments:
>>> s.set_signal_type()
>>> s
<Signal1D, title: , dimensions: (|4)>
split(axis='auto', number_of_parts='auto', step_sizes='auto')#
Splits the data into several signals.
The split can be defined by giving the number_of_parts, a homogeneous step size, or a list of customized step sizes. By default ('auto'), the function is the reverse of stack().
axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name. If 'auto' and if the object has been created with stack() (and
stack_metadata=True), this method will return the former list of signals (information stored in metadata._HyperSpy.Stacking_history). If it was not created with stack(), the last
navigation axis will be used.
number_of_partsstr or int
Number of parts in which the spectrum image will be split. The splitting is homogeneous. When the axis size is not divisible by the number_of_parts the remainder data is lost without
warning. If number_of_parts and step_sizes is 'auto', number_of_parts equals the length of the axis, step_sizes equals one, and the axis is suppressed from each sub-spectrum.
step_sizesstr, list (of int), or int
Size of the split parts. If 'auto', the step_sizes equals one. If an int is given, the splitting is homogeneous.
list of BaseSignal
A list of the split signals
If trying to split along a non-uniform axis.
>>> s = hs.signals.Signal1D(np.random.random([4, 3, 2]))
>>> s
<Signal1D, title: , dimensions: (3, 4|2)>
>>> s.split()
[<Signal1D, title: , dimensions: (3|2)>, <Signal1D, title: , dimensions: (3|2)>, <Signal1D, title: , dimensions: (3|2)>, <Signal1D, title: , dimensions: (3|2)>]
>>> s.split(step_sizes=2)
[<Signal1D, title: , dimensions: (3, 2|2)>, <Signal1D, title: , dimensions: (3, 2|2)>]
>>> s.split(step_sizes=[1, 2])
[<Signal1D, title: , dimensions: (3, 1|2)>, <Signal1D, title: , dimensions: (3, 2|2)>]
Remove single-dimensional entries from the shape of an array and the axes. See numpy.squeeze() for more details.
A new signal object with single-entry dimensions removed
>>> s = hs.signals.Signal2D(np.random.random((2, 1, 1, 6, 8, 8)))
>>> s
<Signal2D, title: , dimensions: (6, 1, 1, 2|8, 8)>
>>> s = s.squeeze()
>>> s
<Signal2D, title: , dimensions: (6, 2|8, 8)>
std(axis=None, out=None, rechunk=False)#
Returns a signal with the standard deviation of the signal along at least one axis.
axisint, str, DataAxis or tuple
Either one on its own, or many axes in a tuple can be passed. In both cases the axes can be passed directly, or specified using the index in axes_manager or the name of the axis. Any
duplicates are removed. If None, the operation is performed over all navigation axes (default).
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
sBaseSignal (or subclass)
A new Signal containing the standard deviation of the provided Signal over the specified axes
>>> import numpy as np
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.std(0)
<Signal2D, title: , dimensions: (|64, 64)>
sum(axis=None, out=None, rechunk=False)#
Sum the data over the given axes.
axisint, str, DataAxis or tuple
Either one on its own, or many axes in a tuple can be passed. In both cases the axes can be passed directly, or specified using the index in axes_manager or the name of the axis. Any
duplicates are removed. If None, the operation is performed over all navigation axes (default).
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
A new Signal containing the sum of the provided Signal along the specified axes.
If you intend to calculate the numerical integral of an unbinned signal, please use the integrate1D() function instead. To avoid erroneous misuse of the sum function as integral, it raises a
warning when working with an unbinned, non-uniform axis.
>>> import numpy as np
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.sum(0)
<Signal2D, title: , dimensions: (|64, 64)>
swap_axes(axis1, axis2, optimize=False)#
Swap two axes in the signal.
axis1: :class:`int`, :class:`str`, or :class:`~hyperspy.axes.DataAxis`
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
axis2: :class:`int`, :class:`str`, or :class:`~hyperspy.axes.DataAxis`
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
If True, the location of the data in memory is optimised for the fastest iteration over the navigation axes. This operation can cause a peak of memory usage and requires considerable
processing times for large datasets and/or low specification hardware. See the Transposing (changing signal spaces) section of the HyperSpy user guide for more information. When
operating on lazy signals, if True, the chunks are optimised for the new axes configuration.
sBaseSignal (or subclass)
A copy of the object with the axes swapped.
Transfer data array from host to GPU device memory using cupy.asarray. Lazy signals are not supported by this method, see user guide for information on how to process data lazily using the
Raise expection if cupy is not installed.
Raise expection if signal is lazy.
Transfer data array from GPU device to host memory.
Raise expection if signal is lazy.
transpose(signal_axes=None, navigation_axes=None, optimize=False)#
Transposes the signal to have the required signal and navigation axes.
signal_axesNone, int, or iterable type
The number (or indices) of axes to convert to signal axes
navigation_axesNone, int, or iterable type
The number (or indices) of axes to convert to navigation axes
If True, the location of the data in memory is optimised for the fastest iteration over the navigation axes. This operation can cause a peak of memory usage and requires considerable
processing times for large datasets and/or low specification hardware. See the Transposing (changing signal spaces) section of the HyperSpy user guide for more information. When
operating on lazy signals, if True, the chunks are optimised for the new axes configuration.
With the exception of both axes parameters (signal_axes and navigation_axes getting iterables, generally one has to be None (i.e. “floating”). The other one specifies either the required
number or explicitly the indices of axes to move to the corresponding space. If both are iterables, full control is given as long as all axes are assigned to one space only.
>>> # just create a signal with many distinct dimensions
>>> s = hs.signals.BaseSignal(np.random.rand(1,2,3,4,5,6,7,8,9))
>>> s
<BaseSignal, title: , dimensions: (|9, 8, 7, 6, 5, 4, 3, 2, 1)>
>>> s.transpose() # swap signal and navigation spaces
<BaseSignal, title: , dimensions: (9, 8, 7, 6, 5, 4, 3, 2, 1|)>
>>> s.T # a shortcut for no arguments
<BaseSignal, title: , dimensions: (9, 8, 7, 6, 5, 4, 3, 2, 1|)>
>>> # roll to leave 5 axes in navigation space
>>> s.transpose(signal_axes=5)
<BaseSignal, title: , dimensions: (4, 3, 2, 1|9, 8, 7, 6, 5)>
>>> # roll leave 3 axes in navigation space
>>> s.transpose(navigation_axes=3)
<BaseSignal, title: , dimensions: (3, 2, 1|9, 8, 7, 6, 5, 4)>
>>> # 3 explicitly defined axes in signal space
>>> s.transpose(signal_axes=[0, 2, 6])
<BaseSignal, title: , dimensions: (8, 6, 5, 4, 2, 1|9, 7, 3)>
>>> # A mix of two lists, but specifying all axes explicitly
>>> # The order of axes is preserved in both lists
>>> s.transpose(navigation_axes=[1, 2, 3, 4, 5, 8], signal_axes=[0, 6, 7])
<BaseSignal, title: , dimensions: (8, 7, 6, 5, 4, 1|9, 3, 2)>
Undo Poisson noise normalization and other pre-treatments.
Only valid if calling s.decomposition(..., copy=True).
unfold(unfold_navigation=True, unfold_signal=True)#
Modifies the shape of the data by unfolding the signal and navigation dimensions separately
Whether or not to unfold the navigation dimension(s) (default: True)
Whether or not to unfold the signal dimension(s) (default: True)
Whether or not one of the axes needed unfolding (and that unfolding was performed)
It doesn’t make sense to perform an unfolding when the total number of dimensions is < 2.
Modify the shape of the data to obtain a navigation space of dimension 1
Whether or not the navigation space needed unfolding (and whether it was performed)
Modify the shape of the data to obtain a signal space of dimension 1
Whether or not the signal space needed unfolding (and whether it was performed)
unfolded(unfold_navigation=True, unfold_signal=True)#
Use this function together with a with statement to have the signal be unfolded for the scope of the with block, before automatically refolding when passing out of scope.
>>> import numpy as np
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> with s.unfolded():
... # Do whatever needs doing while unfolded here
... pass
If this Signal has been plotted, update the signal and navigator plots, as appropriate.
valuemax(axis, out=None, rechunk=False)#
Returns a signal with the value of coordinates of the maximum along an axis.
axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
sBaseSignal (or subclass)
A new Signal containing the calibrated coordinate values of the maximum along the specified axis.
>>> import numpy as np
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.valuemax(0)
<Signal2D, title: , dimensions: (|64, 64)>
valuemin(axis, out=None, rechunk=False)#
Returns a signal with the value of coordinates of the minimum along an axis.
axisint, str, or DataAxis
The axis can be passed directly, or specified using the index of the axis in the Signal’s axes_manager or the axis name.
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
BaseSignal or subclass
A new Signal containing the calibrated coordinate values of the minimum along the specified axis.
var(axis=None, out=None, rechunk=False)#
Returns a signal with the variances of the signal along at least one axis.
axisint, str, DataAxis or tuple
Either one on its own, or many axes in a tuple can be passed. In both cases the axes can be passed directly, or specified using the index in axes_manager or the name of the axis. Any
duplicates are removed. If None, the operation is performed over all navigation axes (default).
outBaseSignal (or subclass) or None
If None, a new Signal is created with the result of the operation and returned (default). If a Signal is passed, it is used to receive the output of the operation, and nothing is
Only has effect when operating on lazy signal. Default False, which means the chunking structure will be retained. If True, the data may be automatically rechunked before performing
this operation.
sBaseSignal (or subclass)
A new Signal containing the variance of the provided Signal over the specified axes
>>> import numpy as np
>>> s = BaseSignal(np.random.random((64, 64, 1024)))
>>> s
<BaseSignal, title: , dimensions: (|1024, 64, 64)>
>>> s.var(0)
<Signal2D, title: , dimensions: (|64, 64)> | {"url":"https://hyperspy.org/hyperspy-doc/v2.0/reference/api.signals/BaseSignal.html","timestamp":"2024-11-09T10:24:33Z","content_type":"text/html","content_length":"613671","record_id":"<urn:uuid:07e672f6-2a0f-4ae6-aa92-30303972ba44>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00054.warc.gz"} |
Expressions/Equations/Equivalence – Video in the Middle
The teacher, Maryann, launches the Cubes in a Line lesson by showing her students two cubes and asking the question, “If I put two cubes together, how many faces are there?” We drop in as several
students share their responses and the class discussion ensues. | {"url":"https://videointhemiddle.org/math-topic/expressions-equations-equivalence/","timestamp":"2024-11-11T20:06:22Z","content_type":"text/html","content_length":"49285","record_id":"<urn:uuid:e264ed52-73e7-4fe0-83dc-e970f5e12f5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00797.warc.gz"} |
Re: [Numpy-discussion] numpy ufuncs and COREPY - any info?
25 May 2009 25 May '09
10:59 a.m.
For some reason the list seems to occasionally drop my messages... Francesc Alted wrote:
A Friday 22 May 2009 13:52:46 Andrew Friedley escrigué:
I'm the student doing the project. I have a blog here, which contains some initial performance numbers for a couple test ufuncs I did:
Another alternative we've talked about, and I (more and more likely) may look into is composing multiple operations together into a single ufunc. Again the main idea being that memory
accesses can be reduced/eliminated.
IMHO, composing multiple operations together is the most promising venue for leveraging current multicore systems.
Agreed -- our concern when considering for the project was to keep the scope reasonable so I can complete it in the GSoC timeframe. If I have time I'll definitely be looking into this over the
summer; if not later.
Another interesting approach is to implement costly operations (from the point of view of CPU resources), namely, transcendental functions like sin, cos or tan, but also others like sqrt or pow)
in a parallel way. If besides, you can combine this with vectorized versions of them (by using the well spread SSE2 instruction set, see [1] for an example), then you would be able to achieve
really good results for sure (at least Intel did with its VML library ;)
[1] http://gruntthepeon.free.fr/ssemath/
I've seen that page before. Using another source [1] I came up with a quick/dirty cos ufunc. Performance is crazy good compared to NumPy (100x); see the latest post on my blog for a little more info.
I'll look at the source myself when I get time again, but is NumPy using a Python-based cos function, a C implementation, or something else? As I wrote in my blog, the performance gain is almost too
good to believe. [1] http://www.devmaster.net/forums/showthread.php?t=5784 Andrew | {"url":"https://mail.python.org/archives/list/numpy-discussion@python.org/message/XIQ77H35AI7SLCQWLSNWH7GUW6QFS3TM/","timestamp":"2024-11-06T14:26:34Z","content_type":"text/html","content_length":"14744","record_id":"<urn:uuid:a36cd3e0-8947-47f3-861f-e1f91cc056b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00438.warc.gz"} |
Center of Mass in Physics
These are my notes on the center of mass in physics.
My book on Cloud Computing with Amazon. It is only $0.99 and buying it helps support my family and I. Thank you in advance.
A discussion of the application of the law of conservation of momentum starts
with a consideration of the center of mass of a collection of particles. For
discrete mass points, the center of mass is defined as:
\[r_{cm} = \frac{1}{m} \sum_{i} m_{i} r_{i}\]
Example 1
Calculate the center of mass for a distribution of mass points.
20kg at (3,3), 12kg at (-1,-1), and 18kg at (-2,-2)
The \(r_{cm}\) form indicates that the calculation is to be done in vector
notation, so the units are often left out of calculations involving unit vectors
and added at the end.
\[r_{cm} = \frac{1}{50}[20(3i+3j) + 18(-2i-2j) + 12(i-j)]\]
\[r_{cm} = \frac{1}{50}[36i+12j] = \frac{36}{50}i + \frac{12}{50}j = 0.72i +
These mass points act as if all their mass (50kg) were at the point (0.72,
0.24). If these three masses were placed on a plate of negligible mass, the
balance point would be at (0.72, 0.24).
The law of conservation of momentum can be viewed as a consequence of the
statement: the total mass of a collection of particles times the acceleration of
the center of mass equals the applied or external force, or the sum of the
forces on all the individual component masses. This is a vector equation:
\[Ma_{cm} = F_{ext}\]
Example 2
For the same collection of masses at the same points, add forces to each mass,
and find the resulting acceleration of the center of mass.
20 kg at 45 degrees with a force of 30 N at (3,3)
18 kg at 180 degrees with a force of 24 N at (-2,-2)
12 kg at 270 degrees with a force of 40 N at (-1,-1)
Remember, the acceleration of the center of mass is the total mass times the
vector sum of all these forces.
\[(50kg)a_{cm} = [30(\cos 45)i + 30(\sin 45)j - 24i - 40j]N = (-3i - 19j)N\]
\[a_{cm} = (\frac{-3}{50}i - \frac{19}{50}j) m/s^2\]
If the preceding statement \(F_{ext}=ma_{cm}\) is viewed as
\(F_{ext}=\frac{d}{dt}(mv_{cm}\), then if \(F_{ext}=0\), then \(mv_{cm}\) must
be a constant. The derivative of a constant is zero, or viewed graphically, if
the curve of \(mv\) versus time is a constant, then the slope is zero.
Stated another way, for a system with no external forces, the sum of the
momentum vectors \(m1v1+m2v2+...\), which add to \(mv_{cm}\) must all add to
Example 3
A 5.0 g pellet is compressed against a spring in a gun of mass 300 g. The spring
is released and the gun allowed to recoil with no friction as the pellet leaves
the gun. If the speed of the recoiling gun is 8.0 m/s, what is the speed of the
The problem is solved by application of the law of conservation of momentum.
This law can be applied because there is no external force. Since there is no
external force, all the mv's must add to zero.
\[m_{g}v_{g} = m_{p}v_{p}\]
\[m_{g}v_{g} - m_{p}v_{p} = 0\]
The conservation of momentum statement on the left is based on the simple
observation "bullet goes one way, gun goes the other" while the formal statement
that the mv's add to zero is on the right. With a well labeled diagram of the
situation, the statement on the left is probably easier to visualize. The
momenta are equal and opposite. Putting in the numbers we have :
\[300g(8.0m/s) = 5.0g*v_{p} \text{ so } v_{p} = 480m/s\]
As a check, note that the momentum of the gun \(P_{g} = m_{g}v_{g} = 2.4kg*m/s\)
and the momentum of the pellet \(P_{p} = m_{p}v_{p} = 2.4kg*m/s\) are
numerically equal because they are in the opposite direction, add to zero.
The energy of each is \(mv^2/2\) or \(p^2/2m\). so for the gun,
\[KE_{gun} = \frac{(2.4kg*m/s)^2}{(2*0.30kg)} = 9.6j\]
Performing the same calculation for the pellet:
\[KE_{pellet} = \frac{(2.4kg*m/s)^2}{(2*0.01kg)} = 288l\]
The total energy stored in the spring is the sum of these energies.
Example 4
Make the pellet gun of previous example fully automatic and capable of firing 10
pellets per second. Calculate the force these pellets make on a target where the
pellets do not bounce.
This problem is solved by calculating the average momentum transferred to the
target per unit of time. The momentum of each pellet is \(2.4kg*m/s\). The force
on the target is calculated from the simple expression:
\[F = \frac{\Delta p}{\Delta t} = \frac{10(2.4kg*m/s)}{1.0s} = 24N\]
The total momentum transferred each second is 10 individual momenta of each
Example 5
A 75 kg hockey player traveling at 12 m/s collides with a 90 kg player
traveling, at right angles to the first, at 15 m/s. The players stick together.
Find their resulting velocity and direction. Assume the ice surface to be
This problem can be analyzed by conservation of momentum. Calculate the momenta,
and draw a vector diagram.
\[p_{1} = 75kg(12m/s) = 900kg*m/s\]
\[p_{2} = 90kg(15m/s) = 1350kg*m/s\]
The angle of the two hockey players is from:
\[\tan \theta = 1350/900 = 1.5 \text{ or } \theta = 56 \]
and the resulting momentum is:
\[p = \sqrt{1350^2 + 900^2} kg*m/s = 1620kg*m/s\]
The players move off with velocity:
\[v = \frac{P}{m1+m2} = 9.83m/s\]
at an angle of 56 degrees to the original direction of the 75 kg player.
Example 6
James Bond is skiing along being pursued by Goldfinger, also on skis. Assume no
friction. mr. bond, at 100 kg, fires backward a 40 g bullet at 800 m/s.
Goldfinger, at 120 kg, fires forward at Bond with a similar weapon. What is the
relative velocity change after the exchange of six shots each. No bullets hit
bond or Goldfinger.
The problem is analyzed with conservation of momentum. The \(m_{b}v_{b}\) of the
bullet fired by Bond increases his momentum by \(m_{B} \Delta v_{B}\). Remember
that each bullet Bond fires increases his velocity. Set \(m_{b}v_{b} = m_{B}
\Delta v_{B}\) and solve for \(\Delta v_{B}\).
\[40*10^{-3}kg(800m/s) = (100kg) \Delta v_{B}\]
\[\Delta v_{B} = 0.32 m/s\]
The \(40*10^{-3}kg\) bullet is small compared to the 100 kg of Bond, and it
would not affect the calculation. The \(\Delta v_{B}\) notation is used to
indicate that each bullet fired by Bond causes a change in his velocity.
Goldfinger has his momentum decreased. In his case, \(m_{b}v_{b} = m_{G} \Delta
v_{G}\). Putting in the numbers:
\[32kg*m/s = (120kg) \Delta v_{G}\]
\[\Delta v_{G} = 0.26m/s\]
Bond goes faster and Goldfinger goes slower, with the total change in velocity
0.58 m/s for each pair of shots fired. For six shots, this amounts to a
difference of 3.48 m/s. If Bond and Goldfinger have been traveling at the same
speed, then after this exchange Bond would have a relative speed advantage of
3.48 m/s.
Example 7
A 3000 kg closed boxcar traveling at 3.0 m/s is overtaken by a 1000 kg open
boxcar traveling at 5.0 m/s. The cars couple together. Find the resulting speed
of the combination.
The momentum before coupling is the same as the momentum after coupling.
\[3000kg(3.0m/s) + 1000kg(5.0m/s) = (4000kg)v\]
\[v = 3.5 m/s\]
Example 8
Continuing with the previous example, rain falls into the open boxcar so that
the mass increases at 1.0 kg/s. What is the velocity of the boxcars at 500 s?
The total momentum of the boxcars is 4000kg(3.5m/s)=14000kg*m/s. Assume that
there is no horizontal component of the rain to change the momentum in the
direction of motion of the boxcars. The mass increases by (1.0kg/s)3.5m/s=500kg.
The momentum is a constant, so the new velocity is:
\[14000kg*m/s = (4500kg)v_{R}\]
\[v_{R} = 3.11 m/s\]
Example 9
For the situation described in the previous problem, what is the rate of change
in velocity for the boxcars?
This is a very interesting calculus problem that involves taking the total
derivative. Since there are no external forces, the total change in mv must
equal zero.
\[d(mv) = mdv + vdm = 0\]
\[mdv = -vdm\]
Now write m as a function of time.
\[m = m_{o} + rt = 4000kg + (1.0kg/s)t\]
The derivative of m is dm = rdt.
Using the two previous equation and rearranging:
\[\frac{dv}{v} = -\frac{dm}{m} = -\frac{r}{m_{o}+rt}dt\]
Introduce a change of variables:
\[u = m_{o} + rt\]
with du = rdt
\[\frac{dv}{v} = -\frac{du}{u}\]
integrating, ln v = -ln u + ln K because it is a convenient form for the
constant. Now rearrange:
\[ln v + ln u = K\]
\[ln uv = ln K\]
\[uv = K\]
Change the variable back to t, so that \((m_{o} + rt)v = k\)
Evaluate the constant from the condition that at t=0,
\[m_{o} = 14000kg*m/s\]
\[K = 14000kg*m/s\]
The relation between v and t is:
\[v = \frac{K}{m_{o} + rt} = \frac{14000kg*m/s}{(4000kg + 1.0kg/s*t)}\]
The velocity at t=500s is:
\[v_{t=500} = \frac{14000kg*m/s}{4500kg} = 3.11 m/s\]
conservation of momentum and a little calculus produce the v versus t relation.
Example 10
A 3.0 kg cat is in a 24 kg boat. The cat is 10 m from the shore. The cat walks
3.0 m toward the shore. How far is the cat from the shore? Assume no friction
between boat and water.
There are no external forces, so the center of mass of the cat-boat system is
constant. Knowing that the center of mass doesn't move is all that is necessary
to do this problem.
Write the center of mass of the cat-boat system before the cat walks. M is the
mass of the boat and m is the cat. Then write the center of mass of the cat-boat
system after the cat walks.
\[x_{cm} = \frac{Mx_{b}+mx_{c}}{M+m}\]
Because there are no external forces, the centers of mass are the same, so:
\[Mx_{b} + mx_{c} = Mx'_{b} + mx'_{c}\]
Watch the algebra, and solve for \(x'_{c}\).
\[x'_{c} = 10m-3.0m+(x'_{b}-x_{b})\]
Now substitute from \(M(x'_{b}-x_{b}) = m(x_{c}-x'_{c}\)
\(x'_{c} = 7+(1/8)(10-x'_{c})\) and \(8x'_{c} = 56+(10-x'_{c})\)
\[x'_{c} = 7.33m\] | {"url":"https://sciencebyjason.com/center-of-mass-in-physics.html","timestamp":"2024-11-10T18:19:33Z","content_type":"text/html","content_length":"23022","record_id":"<urn:uuid:9a634233-7e7d-42bf-930d-8a309f026ce7>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00387.warc.gz"} |
Re: Col Cumulative Sum without calling the current row's valueRe: Col Cumulative Sum without calling the current row's value
Hello, I am quite new to writing JMP scripts and primarily use "Edit Formula" for my work. I have been browsing solutions related to Col Cumulative Sum and could not find one for my specific problem.
The simplified version of my data table is as follows.
│Row│Week│Col Cumulative Sum │My Wish │
│1 │1 │1 │0 │
│2 │2 │3 │1 │
│3 │3 │6 │3 │
│4 │4 │10 │6 │
│5 │5 │15 │10 │
│6 │6 │21 │15 │
The formula below works if the current row value of "Week" is fixed or if I have two separate columns "Week" and "My Wish."
If( Row() != 1,
Lag( Col Cumulative Sum( :Week ), 1 ),
The issue is that the value of the previous row's "Week" is fixed but the current row's is not. And I only have one column "Week" to work with. I am developing a formula that simulates progression
based on the previous simulation value. I am simulating the current row's "Week" value based on the previous row's.
Suppose I am working on Row 4 in the column "Week." I want to cumulatively sum all the previous "Week" values without calling the current "Week" value in the function. Otherwise, it creates an
illegal reference or cycling problem in my case.
I tried using subscript instead of lag since the lag function calls the current row's "Week" value. It did not work.
If( Row() != 1,
Col Cumulative Sum( :Week )[Row() - 1],
I hope my explanation was clear enough. Thank you for your time and expertise in advance. I use JMP Pro 17. | {"url":"https://community.jmp.com/t5/Discussions/Col-Cumulative-Sum-without-calling-the-current-row-s-value/m-p/739100","timestamp":"2024-11-09T12:37:20Z","content_type":"text/html","content_length":"769475","record_id":"<urn:uuid:02adf63b-5804-4d0c-b8b8-62639cee4262>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00005.warc.gz"} |
SOLVED: Can someone help me with this and let me know if I filled this out...
| SkillsMatt
Assignment Instructions/ Description
Can someone help me with this and let me know if I filled this out correctly? This is due July 24th, 2023.�Image transcription textName:
Maria Lebedev
July 24, 2023
Housing Affordability and Mortgage Qualification
Purpose: To estimate the amount of affordable mortgage payment, mortgage amount, and home purchase price.
Financial Planning Activities: Enter the amounts requested to estimate the amount of affordable mortgage payment, mortgage amount, and home purchase price.
Suggested Websites:
Suggested App:
Mortgage Calculator
Debt-to-Income Ratios
ratio (28%)**
ratio (36%)**
Step 1
Determine your monthly gross income (annual income divided by 12).
Enter annual income
Step 2
With a down payment of at least 10 percent, lenders use 28 percent (front-end ratio) of monthly gross income as a
guideline for TIPI (taxes, insurance, principal, and interest) and 36 percent (back-end ratio) of monthly gross income
as a guideline for TIPI plus other debt payments.
$1, 120.00
Step 3
Subtract other monthly debt payments (e.g., payments on an auto loan)
an estimate of the monthly costs of property taxes and homeowner's insurance.
Affordable monthly mortgage payment
> > > > > > > >
> > > > > > >
Step 4
Your Personal Financial Plan
Divide this amount by the monthly mortgage payment per $1,000 based on current mortgage rates-an 8 percent, 30-
year loan, for example (see Exhibit 7-7)-and multiply by $1,000.
Monthly mortgage payment factor (exact) per
Enter rate:
Exact factor is
each $1,000 in principal:
Enter term (years):
Affordable mortgage amount
> > > > > > > >
> > > > > > > >
Step 5
Divide your affordable mortgage amount by 1 minus the fractional portion of your down payment (e.g., 1 - 0.1 with a 10
percent down payment).
Enter down payment (%'age):
able home purchase price
** Note: The two ratios lending institutions use (step 2) and other loan requirements are likely to vary based on a
variety of factors, including the type of mortgage, the amount of the down payment, your income level, credit score,
and current interest rates. If you have other debts, lenders will calculate both ratios and then use the one that allows
you greater flexibility in borrowin
What's Next for Your Personal Financial Plan?
Identify actions you might need to take to qualify for a mortgage.
Discuss your mortgage qualifications with a mortgage broker or other lender.... Show more� | {"url":"https://www.skillsmatt.com/tutors-problem/36239/can-someone-help-me-with-this-and-let-me-know-if-i-filled-this-out","timestamp":"2024-11-07T00:06:38Z","content_type":"text/html","content_length":"66279","record_id":"<urn:uuid:c5bdf2e4-d761-4110-87fe-60a4b1500bde>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00555.warc.gz"} |
Financial Ratio Analysis: A Comprehensive Guide
2.2.1 Ratio Analysis
In the realm of finance and investment, understanding a company’s financial health and performance is crucial for making informed decisions. Financial ratio analysis serves as a pivotal tool in this
process, offering insights into various aspects of a company’s operations and financial standing. This section delves into the purpose, categories, calculations, interpretations, and limitations of
financial ratio analysis, equipping you with the knowledge to assess a company’s performance effectively.
The Purpose of Financial Ratio Analysis
Financial ratio analysis is a method of evaluating the relationships between different pieces of financial data extracted from a company’s financial statements. By analyzing these relationships,
stakeholders can gain a clearer picture of a company’s financial health, operational efficiency, and overall performance. Ratios help in simplifying complex financial data, making it easier to
compare and contrast different companies or track a company’s performance over time.
Categories of Financial Ratios
Financial ratios are broadly categorized into five main types, each serving a distinct purpose in financial analysis:
1. Liquidity Ratios: These ratios measure a company’s ability to meet its short-term obligations. They are crucial for assessing whether a company has enough resources to cover its immediate
2. Solvency Ratios: Also known as leverage ratios, these assess a company’s long-term financial stability and its ability to meet long-term obligations. They provide insights into the company’s debt
levels relative to its equity.
3. Profitability Ratios: These ratios evaluate a company’s capacity to generate earnings relative to its revenue, assets, equity, and other financial metrics. They are key indicators of financial
success and operational efficiency.
4. Efficiency Ratios: Also referred to as activity ratios, these indicate how effectively a company utilizes its assets and manages its operations. They highlight areas where a company can improve
its operational efficiency.
5. Market Valuation Ratios: These ratios provide insights into investor perceptions and the market value of a company. They are often used to assess whether a company’s stock is overvalued or
How Ratios Are Used to Assess a Company’s Performance
Ratios facilitate two primary types of analysis:
• Trend Analysis: By comparing ratios over different periods, stakeholders can identify trends in a company’s performance. This helps in understanding whether the company’s financial health is
improving or deteriorating over time.
• Benchmarking: Ratios allow for comparisons between companies, providing a benchmark against industry standards or competitors. This is particularly useful for investors and creditors in making
informed decisions.
Calculating and Interpreting Key Financial Ratios
Let’s explore the calculation and interpretation of some key financial ratios:
Liquidity Ratios
Current Ratio
The current ratio measures a company’s ability to cover its short-term liabilities with its short-term assets.
$$ \text{Current Ratio} = \frac{\text{Current Assets}}{\text{Current Liabilities}} $$
• Interpretation: A current ratio above 1 indicates that the company has more current assets than current liabilities, suggesting good short-term financial health. However, an excessively high
ratio may indicate inefficient use of assets.
Solvency Ratios
Debt-to-Equity Ratio
This ratio assesses a company’s financial leverage by comparing its total debt to its total equity.
$$ \text{Debt-to-Equity Ratio} = \frac{\text{Total Debt}}{\text{Total Equity}} $$
• Interpretation: A lower debt-to-equity ratio is generally preferred as it indicates less reliance on borrowing. However, the acceptable level varies by industry.
Profitability Ratios
Return on Equity (ROE)
ROE measures a company’s ability to generate profits from its shareholders’ equity.
$$ \text{ROE} = \frac{\text{Net Income}}{\text{Shareholders' Equity}} $$
• Interpretation: A higher ROE indicates efficient use of equity to generate profits. It is a key measure of financial performance and shareholder value.
Efficiency Ratios
Asset Turnover Ratio
This ratio evaluates how efficiently a company uses its assets to generate sales.
$$ \text{Asset Turnover Ratio} = \frac{\text{Net Sales}}{\text{Average Total Assets}} $$
• Interpretation: A higher asset turnover ratio indicates better utilization of assets in generating revenue.
Market Valuation Ratios
Price-to-Earnings (P/E) Ratio
The P/E ratio compares a company’s current share price to its earnings per share (EPS).
$$ \text{P/E Ratio} = \frac{\text{Market Price per Share}}{\text{Earnings per Share}} $$
• Interpretation: A high P/E ratio may suggest that the market expects future growth, while a low P/E could indicate undervaluation or potential issues.
Limitations of Ratio Analysis
While financial ratio analysis is a powerful tool, it has its limitations:
• Differences in Accounting Policies: Companies may use different accounting methods, affecting the comparability of ratios.
• Window Dressing: Companies might manipulate financial statements to present a more favorable financial position.
• Lack of Context: Ratios provide quantitative data but lack qualitative insights, which are crucial for comprehensive analysis.
• Industry Variations: Ratios that are healthy for one industry may not be suitable for another.
Financial ratio analysis is an essential component of financial analysis, offering valuable insights into a company’s performance and financial health. However, it should be used as a starting point,
complemented by qualitative assessments and industry-specific considerations for a holistic evaluation.
Quiz Time!
📚✨ Quiz Time! ✨📚
### What is the primary purpose of financial ratio analysis? - [x] To evaluate relationships between financial data and assess a company's performance - [ ] To predict future stock prices - [ ] To
calculate taxes owed by a company - [ ] To determine the market value of a company's assets > **Explanation:** Financial ratio analysis is used to evaluate relationships between different pieces of
financial data, allowing stakeholders to assess a company's performance and financial health. ### Which category of financial ratios measures a company's ability to meet short-term obligations? - [x]
Liquidity Ratios - [ ] Solvency Ratios - [ ] Profitability Ratios - [ ] Efficiency Ratios > **Explanation:** Liquidity ratios measure a company's ability to meet its short-term obligations,
indicating its short-term financial health. ### What does a high current ratio indicate? - [x] Good short-term financial health - [ ] Excessive debt - [ ] Poor asset utilization - [ ] High
profitability > **Explanation:** A high current ratio indicates that a company has more current assets than current liabilities, suggesting good short-term financial health. ### How is the
debt-to-equity ratio calculated? - [x] Total Debt / Total Equity - [ ] Net Income / Shareholders' Equity - [ ] Current Assets / Current Liabilities - [ ] Net Sales / Average Total Assets >
**Explanation:** The debt-to-equity ratio is calculated by dividing total debt by total equity, assessing a company's financial leverage. ### What does a high ROE indicate? - [x] Efficient use of
equity to generate profits - [ ] High levels of debt - [ ] Poor asset utilization - [ ] Low profitability > **Explanation:** A high ROE indicates that a company is efficiently using its shareholders'
equity to generate profits, reflecting strong financial performance. ### What is a limitation of financial ratio analysis? - [x] Differences in accounting policies - [ ] Accurate prediction of future
stock prices - [ ] Comprehensive qualitative insights - [ ] Consistent results across all industries > **Explanation:** One limitation of financial ratio analysis is that differences in accounting
policies can affect the comparability of ratios across companies. ### Which ratio is used to assess how efficiently a company uses its assets to generate sales? - [x] Asset Turnover Ratio - [ ]
Current Ratio - [ ] Debt-to-Equity Ratio - [ ] Return on Equity (ROE) > **Explanation:** The asset turnover ratio evaluates how efficiently a company uses its assets to generate sales, indicating
operational efficiency. ### What does a high P/E ratio suggest? - [x] Market expects future growth - [ ] Company is undervalued - [ ] Poor financial health - [ ] High levels of debt >
**Explanation:** A high P/E ratio suggests that the market expects future growth, indicating investor confidence in the company's potential. ### Why is it important to complement ratio analysis with
qualitative assessments? - [x] Ratios lack qualitative insights - [ ] Ratios provide complete information - [ ] Ratios are always accurate - [ ] Ratios are industry-specific > **Explanation:** Ratios
provide quantitative data but lack qualitative insights, making it important to complement them with qualitative assessments for a comprehensive evaluation. ### True or False: Financial ratio
analysis can predict future stock prices. - [ ] True - [x] False > **Explanation:** Financial ratio analysis cannot predict future stock prices. It is used to assess a company's current financial
health and performance. | {"url":"https://csccourse.ca/2/2/1/","timestamp":"2024-11-08T20:47:55Z","content_type":"text/html","content_length":"92323","record_id":"<urn:uuid:d068b05b-070c-4bca-8fc7-03c613c7c41f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00121.warc.gz"} |
Multiplication Table Games 2️⃣✖️3️⃣ - Times Tables Kids
1-2-5-10 Multiplication Tables Games
2-3-4-5 Multiplication Tables Games
6-7-8-9 Multiplication Tables Games
The multiplication table is a very basic and necessary table in learning arithmetic and science.
Knowing the table by heart also helps with calculations required on a daily basis such as shopping.
To learn the multiplication table, it is important to understand the operation of multiplication. Then, practice and solve as many exercises as possible until you calculate each exercise with
confidence and success. At this point, you will already begin to remember the results of the exercises by heart. It is important to keep practicing until you remember the entire table by heart and
without mistakes.
There are methods for learning the multiplication table easily and quickly. The easiest multiples to learn are multiples 10,5,2,1. Therefore, it is recommended to start learning the multiplication
table with these multiples, and only after mastering them, move on to the other multiples.
Click here for multiplication table study methods. In each multiplication, you will find method(s) that will help you learn it easily and quickly.
Tips for learning the multiplication table:
* It is recommended to use the word “times” instead of the word “double”. For example, instead of saying “five times three” say “five times three”. This wording reminds us of the essence of the
operation and helps to move from multiplication to addition in solving the exercise.
* The order of the terms in a multiplication exercise is not important and does not change the result, but the order determines the appropriate addition exercise, and sometimes one order is easier to
calculate than the opposite order. Therefore, reverse the order of the members in the exercise if it helps to calculate it more easily. For example, “7 times 5” is easier to calculate than “5 times
* Use your fingers. Raise as many fingers as one number that appears in the exercise and count all the fingers you raised several times according to the second number in the exercise. For example, in
the 6×4 exercise, lift 4 fingers and count them 6 times or lift 6 fingers and count them 4 times.
Multiplication Table Quiz - 10 Questions
Answer and click on the send answer button
Multiplication Table Quiz Game - 60 Seconds Speed Test
Click the “Start Game” button and answer as many questions as possible in 60 seconds.
Continue studying the multiplication tables | {"url":"https://timestablekids.com/multiplication-table-games/","timestamp":"2024-11-11T14:09:26Z","content_type":"text/html","content_length":"355987","record_id":"<urn:uuid:4b2b7f39-af85-4c8e-8f4a-5f169506f492>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00861.warc.gz"} |
Efficiency quiz question that Sam asked about.
The ONE correct answer is O(n). Because the inner loop executes a constant number of times, its execution is not dependent on the value of n. Therefore, the answer is O(n) because of the outer loop.
Now in Moodle, the only correct answer accepted is O(n).
As far as your score, if you answered O(n^2), I'm not reducing your score, but you should understand why O(n^2) is not correct. | {"url":"http://lovelace.augustana.edu/q2a/index.php/3923/efficiency-quiz-question-that-sam-asked-about?show=3924","timestamp":"2024-11-04T17:06:19Z","content_type":"text/html","content_length":"22670","record_id":"<urn:uuid:b669f0b1-07eb-4f7b-b152-946ebecec6bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00448.warc.gz"} |
Reparametrize The Curve In Terms Of Arc Length (Resolved)
Learning how to reparametrize a curve in terms of arc length is an important concept in curve creation. This document will guide you through the steps in understanding and performing the
What Is Reparametrization?
Reparametrization is the process of changing the parameters of a given curve in one coordinate space, either by adding a new coordinate system or by changing the parameters of the existing coordinate
system. This means that you can change the size, shape, and orientation of a curve by changing its parameters.
The Concept Behind Reparametrization in Terms of Arc Length
In mathematical terms, reparametrization in terms of arc length is the process of taking a given curve and dividing it into a sequence of arcs so that each arc has the same length. The new parameters
of the curve will be associated with the arc lengths, allowing for a smooth and continuous change in the shape of the curve.
Step-by-Step Guide to Reparametrize a Curve
First, draw out the curve on a graph paper. Make sure to label each point and the coordinates of each point. This will help in later steps.
Calculate the total length of the curve. This is done by measuring the distance between two consecutive points and adding them together for the total length.
Divide this total length into a number of equal segments or arcs. This is done by taking the total length and dividing it by the number of arcs.
Using the new set of points associated with the arcs, create a new parameterization of the curve. This is done by mapping the coordinates of the points onto the new parameterization.
Finally, evaluate the new parameterization of the curve. This can be done by calculating the length of each arc using the equation of the curve.
FAQ Section
Q1: What Is Reparametrization?
Reparametrization is the process of changing the parameters of a given curve in one coordinate space, either by adding a new coordinate system or by changing the parameters of the existing coordinate
Q2: What Is the Concept Behind Reparametrization in Terms of Arc Length?
The concept behind reparametrization in terms of arc length is the process of taking a given curve and dividing it into a sequence of arcs so that each arc has the same length. The new parameters of
the curve will be associated with the arc lengths, allowing for a smooth and continuous change in the shape of the curve.
Q3: What Are the Steps Involved When Reparametrizing a Curve?
The steps involved in reparametrizing a curve include drawing the curve on a graph paper, calculating the total length of the curve, dividing it into a number of equal segments or arcs, mapping the
coordinates of the points onto the new parameterization, and evaluating the new parameterization of the curve.
Q4: How Do I Calculate the Length of Each Arc?
The length of each arc can be calculated using the equation of the curve. This will depend on the type of curve and its equation.
Q5: Where Can I Find More Information About Reparametrization?
More information about reparametrization can be found in mathematical textbooks, online tutorials and reference guides, or on the websites of universities and colleges that offer mathematics courses.
Related Links
For more information about curves and reparametrization, check out the following related links: | {"url":"https://lxadm.com/reparametrize-the-curve-in-terms-of-arc-length/","timestamp":"2024-11-03T00:24:58Z","content_type":"text/html","content_length":"56167","record_id":"<urn:uuid:5f5401df-c129-4397-98f3-0a20a827a165>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00120.warc.gz"} |
ICMS 2020 - Call for Session Proposals
The 7th International Congress on Mathematical Software will consist of several topical sessions. Each session will provide an overview of the challenges, achievements and progress in a subfield of
mathematical software research, development and use. The program committee will consist of the session organizers. We solicit session proposals.
For inspiration, have a look at the sessions of past ICMS: ICMS 2018, ICMS 2016,
You are invited to propose a session if you
• are active in mathematical software research, development and use,
• want to serve the research community by nurturing and facilitating mathematical software work in your area, and
• would like to focus only on the scientific matters in the organization (not on other matters such as administrative, logistic, etc).
How to propose a session
1. Prepare a session proposal with the following contents.
□ title of the session
□ name(s) of the organizer(s), with contact addresses and emails
□ aim and scope of the session (at most 150 words)
2. Submit it
3. The decision on the proposal will be made
□ by the program chair, the general chair and the advisory committee
□ until 16 December 2019
How to organize a session
• Maintain a session web page (see template in html-format or markdown-format).
• Send a call for short abstracts (about 200 words) to the potential speakers in the topic area of the session.
• Review the submitted abstracts and make decision on their acceptance, as soon as each one arrives.
• Post accepted short abstracts on the session web page.
• Complete the process by 24 February 2020
• During the meeting: manage the session, and arrange for chair(s) for each time slot.
Extended abstracts for the proceedings may be submitted via EasyChair (until 16 March 2020; details TBA) by those who were accepted as speakers in any session. Session organisers can have papers
themselves (acceptance agreed by an appropriate programme chair).
Format of a session
• A session will consist of one or more time slots.
• A time slot will consist of 2 talks (of 25+5 minutes)
• We encourage that each session begins with one general overview talk (may be given by a session organizer).
“Talks” may also include software presentations. Demos aiming at a wide audience should submitted to the Software Fair.
Possible topics for sessions
• These are not exclusive. You can propose any mathematical topic.
• These are not required titles of sessions. You can propose any title.
• These are provided as initial hint for topics and titles.
• logic
□ theorem proving
□ formalization of mathematics
□ logic minimization
□ quantifier elimination
□ ….
• number theory
□ diophantine equations
□ algebraic numbers theory
□ analytic number theory
□ elliptic curves
□ ….
• combinatorics
□ partition
□ graph
□ matroid
□ finite summation, difference equations
□ arithmetic combinatorics
□ algebraic combinatorics
□ analytic combinatorics
□ topological combinatorics
□ …
• algebra
□ group theory
□ linear algebra
□ polynomial algebra
□ differential algebra
□ homological algebra
□ non-commutative algebra
□ tensor algebra
□ ….
• analysis
□ numerical analysis
□ functional analysis
□ differential/integral equations
□ special functions
□ ….
• geometry
□ computational geometry
□ polyhedral geometry
□ algebraic geometry
□ differential geometry
□ algebraic topology
□ differential topology
□ …
• inter-disciplinary
□ statistics
□ optimization
□ cryptography
□ coding
□ scientific computation
□ engineering computation
□ mathematical document processing
□ education
□ …
• mathematical problem solving platform
□ AI, machine learning and big-data methods in mathematics
□ computer understanding and natural language processing of mathematics
□ mathematical theory exploration
□ mathematical knowledge management
□ user interface
□ programming language
□ kernel design
□ …
© 2024. All rights reserved. | {"url":"http://icms-conference.org/2020/call-for-session-proposals/","timestamp":"2024-11-03T16:36:40Z","content_type":"text/html","content_length":"10238","record_id":"<urn:uuid:30bf5161-87ca-4e4e-b694-771fbf2d1b94>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00102.warc.gz"} |
Part 2 - Metrics | Less
Hint #1
In the first section of solving this part, we going to split up the output from Part 1 (use an Output tool at the end of Part one above) into two streams - one for the Gross Profit Margin (GPM, top)
and one for the Operating Profit Margin (OPM, bottom).
We start with a Filter to remove all the irrelevant rows - we only need Sales and Cost of Sales to caluclate GPM and Sales and >EBIT< to calculate OPM. We remove some unnecessary columns with Change
Columns tools.
The complexity comes from the Transpose. Later on in the model, we need to make a formula to calculate of margins - we need the data (e.g. Sales and Cost of Sales) to be on the same row to do so. The
Transpose is the first step to achieving that. We begin transforming our value columns (Y2018, Y2019, Y2020) to rows keeping the Account constant. The Transpose is made to do that. See the config
Hint #2
The second section of this part is about calculating our metrics. We do that with a Compare tool and a Calculate. The Compare tool is what really helps us achieve what we want. The entire second
section looks like this:
The Compare tool helps us move data between rows, i.e. vertically in our dataset. Take a look below at the configuration of the first compare (Tool ID 144):
Let's begin with the output column prev_Value. This one is fetching the previous cell of the column Value. So the number from Value column on row 1 will show up in the number of prev_Value on row 2.
We also use the Option Setting "Group By" to reset our calculation everytime we see a new year in the column Columns. In doing all of this, we move the Sales and the Cost of Sales to the same row -
and now we can calculate with it! The calculations - Tool ID 145 and 159 - are faily simply. You can see the formula in the first screenshot here. Afterwards we remove the rows without any
calculation with the Filter tools.
Hint #3
The final step we simply use a Combine tool to merge the two streams of data back together. We Combine using an Inner Join on the Year column. It looks like this:
In totality, our full model looks like this: | {"url":"https://resources.less.tech/less-tech/exercises/automated-financial-analysis/hints/part-2-metrics","timestamp":"2024-11-14T09:01:48Z","content_type":"text/html","content_length":"364545","record_id":"<urn:uuid:80611fb2-fe1c-435d-bdf2-90cb6a5f018d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00685.warc.gz"} |
Atmospheric Profiles
ClearSky contains functions for common atmospheric profiles and quantities.
Many radiative transfer calculations require an atmospheric temperature profile. ClearSky is designed to work with any arbitrary temperature profile if it can be defined as a function of pressure, T
For convenience, dry and moist adiabatic profiles are available through the DryAdiabat and MoistAdiabat types, which are function-like types. There is also a tropopause function.
Function-like type for initializing and evaluating a dry adiabatic temperature profile. Optionally, a uniform upper atmospheric temperature can be set below a specified temperature or pressure.
DryAdiabat(Tₛ, Pₛ, cₚ, μ; Tstrat=0.0, Ptropo=0.0, Pₜ=1.0e-9)
• Tₛ: surface temperature [K]
• Pₛ: surface pressure [K]
• cₚ: specific heat of the atmosphere [J/kg/K]
• μ: molar mass of the atmosphere [kg/mole]
If Tstrat is greater than zero, the temperature profile will never drop below that temperature. If Ptropo is greater than zero, the temperature profile at pressures lower than Ptropo will be equal to
the temperature at exactly Ptropo. Tstrat and Ptropo cannot both greater than zero.
Pₜ defines the highest pressure [Pa] in the temperature profile. If should generally be small but cannot be zero.
Once constructed, use a DryAdiabat like a function to compute temperature at a given pressure.
Tₛ = 288; #surface temperature [K]
Pₛ = 1e5; #surface pressure [Pa]
cₚ = 1040; #specific heat of air [J/kg/K]
μ = 0.029; #molar mass of air [kg/mole]
#construct the dry adiabat with an upper atmosphere temperature of 190 K
D = DryAdiabat(Tₛ, Pₛ, cₚ, μ, Tstrat=190);
#temperatures at 40-10 kPa
D.([4e4, 3e4, 2e4, 1e4])
Function-like type for initializing and evaluating a moist adiabatic temperature profile. Optional uniform upper atmospheric temperature below a specified temperature or pressure.
MoistAdiabat(Tₛ, Pₛ, cₚₙ, cₚᵥ, μₙ, μᵥ, L, psat; Tstrat=0, Ptropo=0, N=1000, Pₜ=1.0e-9)
• Tₛ: surface temperature [K]
• Pₛ: surface pressure [K]
• cₚₙ: specific heat of the non-condensible atmospheric component (air) [J/kg/K]
• cₚᵥ: specific heat of the condensible atmospheric component [J/kg/K]
• μₙ: molar mass of the non-condensible atmospheric component (air) [kg/mole]
• μᵥ: molar mass of the condensible atmospheric component [kg/mole]
• L: condsible component's latent heat of vaporization [J/kg]
• psat: function defining the saturation vapor pressure for a given temperature, psat(T)
If Tstrat is greater than zero, the temperature profile will never drop below that temperature. If Ptropo is greater than zero, the temperature profile at pressures lower than Ptropo will be equal to
the temperature at exactly Ptropo. Tstrat and Ptropo cannot both greater than zero.
Pₜ defines the highest pressure [Pa] in the temperature profile. If should generally be small but cannot be zero.
The profile is evaluated along a number of pressure values in the atmosphere set by N. Those points are then used to construct a cubic spline interpolator for efficient and accurate temperature
calculation. Experience indicates that 1000 points is very accurate and also fast.
Once constructed, use a MoistAdiabat like a function to compute temperature at a given pressure.
Tₛ = 288; #surface temperature [K]
Pₛ = 1e5; #surface pressure [Pa]
cₚₙ = 1040; #specific heat of air [J/kg/K]
cₚᵥ = 1996; #specific heat of H2O [J/kg/K]
μₙ = 0.029; #molar mass of air [kg/mole]
μᵥ = 0.018; #molar mass of H2O [kg/mole]
L = 2.3e6; #H2O latent heat of vaporization [J/kg]
#a saturation vapor pressure function for H2O is built in
psat = psatH2O;
#construct the moist adiabat with a tropopause pressure of 1e4 Pa
M = MoistAdiabat(Tₛ, Pₛ, cₚₙ, cₚᵥ, μₙ, μᵥ, L, psat, Ptropo=1e4);
#temperatures at 30-5 kPa
M.([3e4, 2e4, 1e4, 5e3])
Compute the temperature [K] and pressure [Pa] at which the tropopause occurs in an adiabatic temperature profile. This function can be called on a DryAdiabat or a MoistAdiabat if it was constructed
with nonzero Tstrat or Ptropo. Returns the tuple (T,P).
In case a pressure profile with constant scale height isn't sufficient, hydrostatic profiles with arbitrary temperature and mean molar mass functions are available through the Hydrostatic type and
related functions.
Function-like type for initializing and evaluating a hydrostatic pressure profile with arbitrary temperature and mean molar mass profiles. A Hydrostatic object maps altitude to pressure. Internally,
a pressure vs altitude profile is generated and used for interpolation.
Hydrostatic(Pₛ, Pₜ, g, fT, fμ, N=250)
• Pₛ: surface pressure [Pa]
• Pₜ: top of profile pressure [Pa]
• g: gravitational acceleration [m/s$^2$]
• fT: temperature [K] as a function of presssure, fT(P)
• fμ: mean molar mass [kg/mole] as a function of temperature and pressure, fμ(T,P)
• N: optional, number of interpolation nodes
For a constant molar mass or temperature, you can use anonymous functions directly. For example, to construct a hydrostatic pressure profile for a crude Earth-like atmosphere:
#moist adiabatic temperature profile
M = MoistAdiabat(288, 1e5, 1040, 1996, 0.029, 0.018, 2.3e6, psatH2O, Ptropo=1e4);
#hydrostatic pressure profile with constant mean molar mass
H = Hydrostatic(1e5, 1, 9.8, M, (T,P)->0.029);
#evaluate pressures at a few different altitudes
H.([0, 1e3, 1e4])
hydrostatic(z, Pₛ, g, fT, fμ)
Compute the hydrostatic pressure [Pa] at a specific altitude using arbitrary atmospheric profiles of temperature and mean molar mass. This function integrates the hydrostatic relation,
``\frac{dP}{dz} = \frac{\mu g}{R T}
from the surface to a height of $z$, where $R$ is the universial gas constant.
• z: altitude [m] to compute pressure at
• Pₛ: surface pressure [Pa]
• g: gravitational acceleration [m/s$^2$]
• fT: temperature [K] as a function of pressure, fT(P)
• fμ: mean molar mass [kg/mole] as a function of pressure and temperature fμ(T,P)
altitude(P, Pₛ, g, fT, fμ)
Compute the altitude [m] at which a specific hydrostatic pressure occurs using arbitrary atmospheric profiles of temperature and mean molar mass. This function applies a root finder to the
hydrostatic function.
• P: pressure [Pa] to compute altitude at
• Pₛ: surface pressure [Pa]
• g: gravitational acceleration [m/s$^2$]
• fT: temperature [K] as a function of pressure, fT(P)
• fμ: mean molar mass [kg/mole] as a function of pressure and temperature fμ(T,P)
altitude(H::Hydrostatic, P)
Compute the altitude at which a specific pressure occurs in a Hydrostatic pressure profile. A root finder is applied to the object.
Compute the saturation partial pressure of water vapor at a given temperature using expressions from
This function uses equation 10 in the paper above when $T >= 273.15$ K and equation 7 otherwise.
Compute the saturation pressure of carbon dioxide at a certain pressure using equation 19 from
The equation is inverted to express temperature as a function of pressure.
ozonelayer(P, Cmax=8e-6)
Approximate the molar concentration of ozone in Earth's ozone layer using an 8 ppm peak at 1600 Pa which falls to zero at 100 Pa and 25500 Pa. Peak concentration is defined by Cmax. This
approximation is discussed in
• Jacob, D. Introduction to Atmospheric Chemistry. (Princeton University Press, 1999).
condensibleprofile(Γ::AbstractAdiabat, fPₛ)
Create a function defining concentration vs pressure for a condensible with uniform upper-atmosphere (stratosphere) concentration. The new concentration profile is created with reference to an
existing adiabatic profile (DryAdiabat or MoistAdiabat), which must have Ptropo != 0 or Tstrato != 0. Lower atmospheric concentration is determined by the temperature dependent partial pressure
function fPₛ(T). The concentration is P/(fPₛ + P), where P is the dry/non-condensible pressure.
haircut!(T, P, fTₛ)
Put a temperature floor on a temperature profile using the saturation temperature function fTₛ(P).
Missing docstring for rayleighCO2. Check Documenter's build log for details. | {"url":"https://markbaum.xyz/ClearSky.jl/atmospheric_profiles/","timestamp":"2024-11-14T08:51:05Z","content_type":"text/html","content_length":"24808","record_id":"<urn:uuid:c60d80e0-7984-4ae4-8b61-26d5caa7350f>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00624.warc.gz"} |
Using a Triangle of Forces to Solve Equilibrium Problems
Question Video: Using a Triangle of Forces to Solve Equilibrium Problems Mathematics • Second Year of Secondary School
Use the following diagram to find the tension in π Άπ ΅. Round your answer to two decimal places.
Video Transcript
Use the following diagram to find the tension in π Άπ ΅. Round your answer to two decimal places.
Remember, if a pair of forces are acting on a rigid body and that body is in equilibrium, then a third force π must also be acting on the body, which is equal in magnitude and opposite in
direction to the resultant of those two forces. In this case, the pair of forces are the tensions with magnitudes π sub one and π sub two. The third force is the downward force of 10 newtons,
meaning that the magnitude of the resultant π must also be 10 newtons.
So if we know the angle between the two forces, letβ s call that π , we can use this equation to find the magnitude of the resultant. Substituting what we know about this system and we get 10
equals the square root of π sub one squared plus π sub two squared plus two times π sub one times π sub two cos π .
Since in this case the triangle is equilateral, π sub one and π sub two are equal. So we can replace these with π , square both sides, and begin to evaluate the right. We can now factor two π
squared, so 100 equals two π squared times one plus cos π . Making π squared the subject by dividing through by two times one plus cos π gives π squared equals 50 over one plus cos
π .
Next, we might spot that we can find the value of π using the length of the sides in the triangle. We label our triangle as shown. And we get 50 squared equals 30 squared plus 30 squared minus two
times 30 times 30 cos π . Thatβ s 2500 equals 1800 minus 1800 cos π , and we can rearrange to find that cos of π equals negative seven over 18. Letβ s substitute that into our earlier
expression. Thatβ s π squared equals 50 over one plus negative seven over 18, which is 900 over 11.
Finally, we know that the tension in π Άπ ΅ is π sub one, which is equal to π , so we just need to square root. The square root of 900 over 11 is 9.0453. Correct to two decimal places, thatβ
s 9.05, so the tension in π Άπ ΅ is 9.05 newtons. | {"url":"https://www.nagwa.com/en/videos/757147057427/","timestamp":"2024-11-04T07:25:51Z","content_type":"text/html","content_length":"251073","record_id":"<urn:uuid:368c031e-d62d-480a-a3b1-420bcb53a82b>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00779.warc.gz"} |
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mainform is shown below
If the user clicks on the "Add New Employee" button a new form is displayed as shown in the fig | {"url":"https://codifytutor.com/marketplace?page=14","timestamp":"2024-11-02T12:27:05Z","content_type":"text/html","content_length":"37800","record_id":"<urn:uuid:dad1454f-817c-4a6f-8c3c-24844e4cd24e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00603.warc.gz"} |
What is the average starting salary for a teacher - CAREER KEG
What is the average starting salary for a teacher
May 28, 2022 Uncategorized No Comments
What is the average starting salary for a teacher? This varies quite a bit based on what state you’re in and what type of teacher you plan on becoming. Different states have different requirements
you will have to meet as well.
The average starting salary for a teacher in the United States is $40,555 according to The Bureau of Labor Statistics. It was reported that the average high school biology teacher made $48,025 or
$23.62 per hour in 2011, with a median income of $48,220. These salaries vary according to the location and educational setting as well as the experience level of new teachers and difficulty of
degree program.
What is the average starting salary for a teacher
On average, teachers with a bachelor’s degree earn a starting salary of $37,692. While the average salary for all teachers is $58,950, experienced teachers earn an average salary of $78,957. The
highest-paid teachers are in New York and California while those in Oklahoma and Mississippi make the least. However, factors like individual school districts, cost of living and union negotiations
can all have an impact on teacher salaries. For example, the highest teacher salaries in Connecticut are found in affluent Greenwich ($103K) while the lowest are found near New Haven ($50K).
The average starting salary for a teacher with a bachelor’s degree is $37,692.
The average starting salary for a teacher with a bachelor’s degree is $37,692.
The average starting salary for a teacher with a master’s degree is $43,118.
The average starting salary for a teacher with a master’s degree is $43,118.
One of the most important factors in determining your salary as a teacher is whether or not you have a master’s degree. According to the Bureau of Labor Statistics, around 75% of all teachers have at
least an undergraduate degree and 21% have earned a master’s degree.
You can expect to earn about $4,000 more per year than those with just a bachelor’s degree. A master’s in education typically takes around two years to complete after earning your bachelor’s degree,
but some schools offer accelerated programs that allow students to complete both degrees within five years.
Teachers’ salaries vary depending on the state where they live and teach.
The average teacher salary varies by state, so it’s wise to research your state’s average salary when deciding where to teach. While some states pay teachers significantly more than others, there are
also many factors that play into a teacher’s paycheck beyond the state they work in.
For example, some states offer higher salaries for teachers who have advanced degrees such as master’s degrees or doctorate degrees. Teachers who have specialized training or experience can expect an
even higher salary than those with no special education credentials. Additionally, certain school districts may offer additional benefits such as bonuses or incentives for working at a certain school
or teaching certain subjects that aren’t available elsewhere
(Average teacher salaries in all 50 states).
As you might expect, the average teacher salary varies depending on where you work. For example, teachers in New York earn an average of $77,902 per year, while those in Mississippi earn an average
of $42,548 per year. However, even among states with similar wage structures and cost of living factors (such as California and Florida), there was still a significant discrepancy between salaries:
California had an average starting salary for a teacher of $59,000 and Florida had an average starting salary for a teacher of $48,000.
The reason for this variance is that each state has its own salary schedule based on experience level and education level; therefore certain jobs will pay more than others in certain states but not
necessarily across all states.
New York teachers earned the highest average salary among all states, at $76,959 per year.
New York teachers earn the highest average salary among all states, at $76,959 per year. Average salaries for teachers in South Dakota and West Virginia are significantly lower than those in New
York: $57,958 per year and $53,765 per year respectively. In contrast with these two states, California’s teacher salaries are much higher than those in other states. The average salary for a teacher
there is $68,409 per year — which is almost as high as the national average and comparable to some East Coast state averages like Maryland ($61,839) or Connecticut ($61,459).
South Dakota ranked first among all states in the ratio of salaries of experienced teachers to starting salaries, with an experienced-to-starting salary ratio of 3.23. Experienced teachers earned
$71,865 and new teachers averaged $22,304 annually.
As expected, the average salary for a teacher in South Dakota was higher than the national average. The state ranked first among all states in the ratio of salaries of experienced teachers to
starting salaries, with an experienced-to-starting salary ratio of 3.23. Experienced teachers earned $71,865 and new teachers averaged $22,304 annually.
This is not surprising given that South Dakota is one of only two states without any mandated minimum salary for teachers (alongside Alabama). However, according to data from the Bureau of Labor
Statistics’ Occupational Employment Statistics survey from May 2016 through April 2017—the most recent period available—the average annual salary for elementary school teachers across America was
$48,859 compared with $38,828 for high school teachers (both figures include special education positions as well), which makes sense given that elementary school teachers tend to have less experience
and skills than high school instructors do when they start out teaching.
The average salary depends on your location and experience
The average salary for teachers depends on several factors:
• Where you live. The cost of living varies greatly between states and cities, so the average starting salary will be higher in some areas than others. For example, New York City has a high cost of
living but also offers larger salaries due to the number of teaching jobs available in the area.
• Your level of experience. Teachers with more experience tend to earn higher salaries than those who are just starting out. A teacher with a bachelor’s degree earns an average salary of $37,692
per year, while one with a master’s degree earns an average salary of $43,118 per year (not including bonuses).
You can see by these numbers that there is a wide range of salaries for teachers, but the starting salary at $37,692 is a good amount. We would hope that after four years of college and student
teaching that you get to be paid more than $10,000! For more information on this topic you can check out our article on how much do teachers make. | {"url":"https://infolearners.com/what-is-the-average-starting-salary-for-a-teacher/","timestamp":"2024-11-07T23:02:53Z","content_type":"text/html","content_length":"58916","record_id":"<urn:uuid:b7de31d7-70fd-443c-8e62-d63aaaebffd4>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00841.warc.gz"} |
Assignments | Analytical Subsonic Aerodynamics | Aeronautics and Astronautics | MIT OpenCourseWare
There are only two problems for this course. They are listed below.
Problem 1: Two-Dimensional Subsonic Flow Over Slender Bodies
Using regular perturbation methods, derive the partial differential equations and boundary conditions for the perturbation velocity potentials φ[n], n=0, 1, and 2.
Hint: For n=0, the PDE is: (1 - M[∞]^2)φ[0xx] + φ[0xy] = 0
Problem 2: Slender Body in Subsonic Flow
Consider a subsonic flow over a slender axially body of profile section
where t is hte maximum thickness, L is the total length of the body, and (t/L) <<1.
(a) Sketch the profile.
(b) Find the perturbation potential φ.
(c) Find the perturbation velocity component u=∂φ/∂x
(d) Find C[p.] | {"url":"https://ocw.mit.edu/courses/16-121-analytical-subsonic-aerodynamics-fall-2017/pages/assignments/","timestamp":"2024-11-01T20:54:22Z","content_type":"text/html","content_length":"41667","record_id":"<urn:uuid:2f3aef54-0da6-427e-a380-4f222b994ee9>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00681.warc.gz"} |
- T
Lesson 1
Moving in Circles
1.1: Which One Doesn't Belong: Reading Clocks (5 minutes)
This warm-up prompts students to compare four clock faces. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and
talk about characteristics of the items in comparison to one another, such as how students describe circular motion or the directionality of the clock hands. The context of a clockface is used
throughout this unit, so this warm-up is an opportunity for students to start building familiarity with the context (MP1).
Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups,
ask each student to share their reasoning why a particular item does not belong, and together, find at least one reason each item doesn’t belong.
Student Facing
Which one doesn’t belong?
Anticipated Misconceptions
Some students may be unfamiliar with telling time using an analog clock. For these students, display the time shown on each clock image for all to see. Encourage students to discuss with each other
the process for telling time with this type of clock so they may be more familiar with it in future lessons.
Activity Synthesis
Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is
no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology they use, such as angle, arc, or arc length. Also, press students on unsubstantiated claims.
1.2: Around and Around (15 minutes)
The goal of this activity is to get students thinking about an input-output relationship whose outputs repeat at regular intervals. This leads to defining period and naming the functions that
represent these types of relationships as periodic functions. Clocks provide a familiar repeating context for students to reason about. Several relationships between time, angles created by clock
hands, and the height of the end of a clock hand will be explored in this unit.
Monitor for students who use clear language to describe the repetition in the height of the ladybug. Students will refine their language regarding periodic functions throughout the unit, so here
their language can be less formal and more context-based. For example, in reflecting on the differences and similarities between the motion of the second hand and the motion of the minute hand,
students can use the context to reason about period and amplitude informally without using those terms.
Display a blank clock face, such as the one shown here, for all to see throughout the activity.
Ask students to read the situation and first problem. Give students quiet work time and then time to share their work with a partner. Invite students to share their answers and reasoning for each of
the 4 times before asking them to continue with the rest of the task.
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation
such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
A ladybug lands on the end of a clock’s second hand when the hand is pointing straight up. The second hand is 1 foot long and when it rotates and points directly to the right, the ladybug is 10 feet
above the ground.
1. How far above the ground is the ladybug after 0, 30, 45, and 60 seconds have passed?
Pause here for a class discussion.
2. Estimate how far above the ground the ladybug is after 10, 20, and 40 seconds. Be prepared to explain your reasoning.
3. If the ladybug stays on the second hand, describe how its distance from the ground will change over the next minute. What about the minute after that?
4. At exactly 3:15, the ladybug flies from the second hand to the minute hand, which is 9 inches long.
1. How far off the ground is the ladybug now?
2. At what time will the ladybug be at that height again if it stays on the minute hand? Be prepared to explain your reasoning.
Activity Synthesis
The purpose of this discussion is for students to share their observations about the height of a ladybug on the clock hands over time and introduce students to the idea of period and periodic
Select previously identified students to share their responses. If no students notice how the heights of the ladybug on the second hand are the same for times such as 20 and 40 seconds after 0
seconds, point this out and invite students to identify other times where the ladybug would be the same height over the ground. An important takeaway here is that the heights repeat every minute.
Some students may also notice the symmetric nature of the heights from maximum to minimum and back to maximum, but it is okay to not focus on this aspect yet since it will be a focus of future
Tell students that the type of function represented by the height of the ladybug on the end of the clock hand over time can be described by a periodic function. Periodic functions are ones in which
the values of the function repeat at regular intervals. An important feature of a periodic function is the period, which is the length of the interval the function repeats. In this situation, we say
that the second hand has a period of 60 seconds. Movement around and around a circle is one type of periodic function and we’ll explore more types throughout the unit.
Conclude the discussion by inviting students to describe how the motion of the ladybug on the minute hand is the same or different from the motion of the ladybug on the second hand. Possible
responses to highlight include:
• The period of the height of the ladybug on the minute hand is 60 minutes.
• The ladybug travels vertically between a maximum of 10.75 feet to a minimum of 9.25 feet.
Conversing: MLR2 Collect and Display. During the synthesis, listen for and collect language students use to share their observations about the height of a ladybug on the clock hands over time.
Write students’ words and phrases on a visual display and refer back to it throughout the lesson. Amplify words and phrases such as “the height repeats,” “regular intervals,” and “up and down every
_____ (length of time).” When defining the terms periodic function and period, use student language from the display. This will provide students with a resource to draw language from during
small-group and whole-group discussions throughout the lesson.
Design Principle(s): Maximize meta-awareness; Support sense-making
1.3: Where is the Point? (15 minutes)
Building on their work in the previous activity, the goal of this activity is for students to make connections between points on a circle, the coordinates of those points if the circle is centered at
the origin, and how right triangles and the Pythagorean Theorem can help determine unknown information. While the focus in this activity is on using the Pythagorean Theorem, in future lessons,
students will incorporate the trigonometric work they learned in a previous course and use cosine, sine, tangent, and the Pythagorean Theorem to determine the location of specific points on a circle.
The idea that we can choose to overlay the coordinate plane to reason about the location of a point on a circle is part of reasoning abstractly and quantitatively (MP2). By adding the familiar
structure, we can use a greater number of mathematical tools to determine information about a situation.
Monitor for students with clear explanations about why point \(S\) could be in 2 different places on the circle, particularly students who have created a sketch to help them think about the
Arrange students in groups of 2. Display the clock with point \(P\) at the 2 for all to see. Ask, “What do you need to find the location of the point \(P\) marked on the clock?”
After a brief quiet think time, invite students to share their ideas. Students may suggest things like the height of the clock off the ground, the radius of the clock, a ruler, and so on. A key idea
here is that in order to say where the point is, we need something to measure from and some type of scale to measure with.
Display a new image like the one given here where the clock is centered at the origin with a radius of 5 units. Invite students to work with a partner to determine how they can calculate the \(y\)
-coordinate of the point \(P\).
After a brief work think time, select students to share their solutions, recording their reasoning for all to see on or near the image. While students may reason about the \(y\)-coordinate in many
ways, focus on those who recognize that we can use a right triangle and the Pythagorean Theorem to identify the value of the \(y\)-coordinate. If no students suggest doing so, draw in the right
triangle with hypotenuse 5, known side length 3, and right angle on the horizontal axis and then invite students to consider again how they could determine the \(y\)-coordinate.
Representation: Internalize Comprehension. Use annotations to highlight connections between representations in a problem. As students share their reasoning about what they need to find the location
of point \(P\), scribe their thinking on a visible display. Display and scribe the student’s thinking for the new image of the clock centered at the origin of a coordinate plane as well.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
1. What is the radius of the circle?
2. If \(Q\) has a \(y\)-coordinate of -4, what is the \(x\)-coordinate?
3. If \(B\) has a \(y\)-coordinate of 4, what is the \(x\)-coordinate?
4. A circle centered at \((0,0)\) has a radius of 10. Point \(S\) on the circle has an \(x\)-coordinate of 6. What is the \(y\)-coordinate of point \(S\)? Explain or show your reasoning.
Student Facing
Are you ready for more?
1. How many times a day do the minute hand and the hour hand on a clock point in the same direction?
2. At what times do they point in the same direction?
Anticipated Misconceptions
Some students may be unsure where to begin working with the given information. Encourage these students to draw in the right triangle, recalling the problem discussed during the launch of the
Activity Synthesis
The purpose of this discussion is for students to share how they calculated the unknown values. Highlight students who drew in right triangles as a strategy. If time allows, pair partners up into
groups of 4 to first share strategies with each other before selecting students to share their responses, including any visuals made, with the class.
For the last question, an important takeaway for students is that without more information, point \(S\) could be in 1 of two places on the circle since there are two quadrants where the \(x\)-value
is positive. This repeating feature of coordinates on a circle is one students will work with more in the future and connects to the periodic nature of trigonometric functions.
Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. Encourage students who have visuals to display to connect their strategies and explanations
multi-modally by gesturing. After students share how they calculated the unknown values, ask another student to restate what they heard using precise mathematical language. Ask the original speaker
if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an
opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making; Cultivate conversation
Lesson Synthesis
The purpose of this discussion is for students to reflect on circular motion and the \((x,y)\) coordinates of a point on a circle that is centered at the origin.
Invite students to consider what is true about the \(x\)- and \(y\)-coordinates of the ladybug if the clock is centered at \((0,0)\). In particular, invite students to consider how the values of the
coordinates change as time passes. After a brief quiet think time, invite students to share their responses, recording them for all to see. Here are some things students may notice:
• The values of the coordinates of the ladybug repeat each time they make a full circle: every minute for the second hand and every hour for the minute hand.
• We could use right triangles and the Pythagorean Theorem to figure out the \(x\)-coordinate of the ladybug if we knew the \(y\)-coordinate (the height) at that time.
• The \(y\)-coordinate of the ladybug was the same at times like 10 seconds and 50 seconds after 0. Except for straight up and down, every point on the clock has a “matching” point on the opposite
• The \(x\)-coordinate of the ladybug was the same at times like 10 seconds and 20 seconds after 0. Except for straight left and right, every point on the clock has a “matching” point on the
opposite side.
If students do not mention all the points on the list, there is no need to bring them up at this time. They will have more opportunities to consider these ideas in future lessons.
If time allows, invite students to propose other situations we could model with a function whose outputs repeat. For example, the height of a person on a Ferris wheel, phases of the Moon, frequency
of a sound wave, or Earth’s distance from the Sun.
1.4: Cool-down - Two Particular Points (5 minutes)
Student Facing
Consider the height of the end of a second hand on a clock over a full minute. It starts pointing up, then rotates to point down, then rotates until it is pointing straight up again. This motion
repeats once every minute.
If we imagine the clock centered at \((0,0)\) on the coordinate plane, then we can study the movement of the end of the second hand by thinking about its \((x,y)\) coordinates on the plane. Over one
minute, the \(y\)-coordinate starts at its highest value (when the hand is pointing up), decreases to its lowest value (when the hand is pointing down), and then returns to its highest value. This
happens once every minute that passes.
While we have worked with many types of functions, such as rational or exponential, none of them are characterized by output values that repeat over and over again, so we can’t use them to model the
height of the end of the second hand. This means we need to use a new type of function. A function whose values repeat at regular intervals is called a periodic function, and the length of the
interval at which a periodic function repeats is called the period. We will study several types of periodic functions in this unit. | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/3/6/1/index.html","timestamp":"2024-11-11T06:58:10Z","content_type":"text/html","content_length":"131381","record_id":"<urn:uuid:eaa4f427-e950-4fdf-b7fd-9067a48e19f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00430.warc.gz"} |
Area Of Trapezium: Definition, Properties, Formula And Examples
A trapezium is a four-sided shape with one set of parallel sides. The parallel sides are called the bases of the trapezium, and the non-parallel sides are called the legs. The altitude of a trapezium
is the perpendicular distance between the bases. The area of a trapezium is equal to the average of the bases multiplied by the altitude.
Trapeziums are a common shape in nature and in everyday life. For example, the wings of a butterfly, the roof of a house, and the teeth of a saw are all trapeziums. Trapeziums are also used in many
different structures and machines, such as bridges, airplanes, and cars.
Area of Trapezium
A trapezium is a 2D figure, it is a quadrilateral and has 4 sides out of which 2 sides are parallel to each other. The area of a trapezium is equal to the sum of the areas of the two triangles and
the area of the rectangle. A trapezium where the two parallel sides are equal and form equal angles at one of the bases is called the isosceles trapezium.
Properties of Area of Trapezium
Some of the properties of a trapezium are listed below:
1. The sum of the angles of a trapezium is 360º
2. A trapezium is not a parallelogram (as only one pair of opposite sides is parallel in a trapezium and we require both pairs to be parallel in a parallelogram).
3. The 4 sides of a trapezium are unequal unless it is an isosceles trapezium in which the 2 parallel sides are equal.
4. The diagonals of a trapezium bisect each other.
5. Two pairs of adjacent angles of a trapezium sum up to 180 degrees.
The formula of the Area of Trapezium
In order to calculate the area of a trapezium, you need to draw a perpendicular between the two parallel lines. The perpendicular will be donated as the height ‘h’ which is the distance between the
parallel sides.
Hence, the area of a trapezium is given by the formula:
Area of Trapezium = 1/2 x distance between the parallel sides x Sum of parallel sides
Area = 1/2 x h x (AB + DC)
Area of Trapezium Examples
Q1: The length of the two parallel sides of a trapezium are given in the ratio 3: 2 and the distance between them is 8 cm. If the area of trapezium is 400 cm², find the length of the parallel sides.
Let the 2 parallel sides as 3x and 2x.
Then, as the area of trapezium is 1/2 x distance between the parallel sides x Sum of parallel sides.
400= 1/2 x (3x + 2x) x 8
400 = 1/2 x 5x x 8
400 = 20x => x = 20 cm
The length of the parallel sides are 60 cm and 40 cm.
Q2. Two parallel sides of a trapezium are of lengths 27 cm and 19 cm respectively, and the distance between them is 14 cm. Find the area of the trapezium.
Area of the trapezium = ¹/₂ × (sum of parallel sides) × (distance between them)
Area of the trapezium = {¹/₂ × (27 + 19) × 14} cm² = 322 cm²
Q3. The area of a trapezium is 352 cm² and the distance between its parallel sides is 16 cm. If one of the parallel sides is of length 25 cm, find the length of the other.
Let the length of the required side be x cm.
Then, area of the trapezium = {¹/₂ × (25 + x) × 16} cm²
Area of the trapezium = (200 + 8x) cm².
But, the area of the trapezium = 352 cm² (given)
Therefore, 200 + 8x = 352
⇒ 8x = (352 – 200)
⇒ 8x = 152
⇒ x = (152/8)
⇒ x = 19.
The length of the other side is 19 cm. | {"url":"https://www.sscadda.com/area-of-trapezium/","timestamp":"2024-11-06T08:46:13Z","content_type":"text/html","content_length":"628834","record_id":"<urn:uuid:3208c7ad-d3d9-4171-b6e0-34cbc44f940d>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00190.warc.gz"} |
Week 1: Learning the Theory - BASIS Independent Schools
Week 1: Learning the Theory
March 1, 2024
Hello my fellow human beings. Welcome back to my blog, where I will be tracking my progress on my EM field simulation project. This week, I worked on reviewing some theoretical knowledge on the
behavior of electromagnetic fields. This knowledge will be necessary in the future when I develop, test, and debug the simulation algorithm.
For the first two days, I reviewed the basics of electrostatics. This subfield of electrodynamics aims to describe the behavior of electric fields when all the charges are at rest (hence electro
statics). The main building blocks of electrostatics are Gauss’ Law and Coulomb’s Law. Both laws relate the behavior of the electric field to the distribution of the electric charge producing the
field. Together, the two laws predict that the electrostatic field is both spherically symmetrical and, like the Newtonian gravitational field, obeys the inverse square law.
Gauss’ Law is Lorentz invariant, which means that the law remains the same when shifting relativistically from one inertial frame to another. Coulomb’s Law, however, is not Lorentz invariant. This
means that electrostatic theory fails for moving charges. As we take into account systems of charges moving relativistically, the situation becomes more complicated. According to Maxwell’s equations,
charges moving relativistically produce both electric and magnetic fields. Moreover, the force experienced by a charge moving through an EM field is given by the Lorentz force law, which essentially
expresses that the total force can be divided into a magnetic and an electric force.
For the next two days of the project, I reviewed the basics of magnetostatics, which investigates the behavior of magnetic fields produced by steadily moving charges. The main building blocks of
magnetostatics are Ampere’s Law and the Biot-Savart Law, which are essentially analogous to Gauss’ Law and Coulomb’s Law for electrostatics. Together, these laws state that magnetic fields arise from
moving charges and that there are no magnetic monopoles.
Of course, both magnetostatics and electrostatics are both subsets of the full theory of classical electrodynamics in which certain assumptions are made to simplify the physical analysis. However,
situations in both areas are easy to solve by hand and offer a more intuitive understanding of electrodynamics. This makes electrostatic and magnetostatic systems optimal for constructing simple
tests for my algorithm.
In my next week, I will review the other two Maxwell’s equations (in addition to the ones we already have reviewed, which are Gauss’ and Ampere’s Laws). Moreover, I will begin finding and comparing
algorithms for solving Maxwell’s equations. Goodbye, my fellow human beings.
View more of Alan X.'s posts.
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Coplanarity Two Lines - Ways to Identify Coplanar Lines and Solved Examples
Coplanarity of Two Lines In 3D Geometry
Coplanar lines in 3-dimensional geometry are a common mathematical theory. To recall, a plane is 2-D in nature stretching into infinity in the 3-D space, while we have employed vector equations to
depict straight lines.
In this chapter, we will further look into what condition is mandatory to be fulfilled for two lines to be coplanar. We will learn to prove how two lines are coplanar using the condition in Cartesian
form and vector form using important concepts and solved examples for your better understanding.
How do we Identify Coplanar Lines?
Why do we want for example lines m →, n and MN→MN → to be coplanar? Let’s take into account the following two cases.
(1) If m ∥n m→∥n→, then the lines are parallel and thus coplanar. Remember that, in such a case, the 3 vectors are also coplanar irrespective of the 3rd vector.
(2) Otherwise, we would require differentiating between bisecting lines (coplanar) and skew lines (not coplanar). If the lines are bisecting, then all their points will lie in the same plane as m m→
and n n→, thus MN→MN→ should lie in that same plane.
What is the Condition of Vectors Coplanarity?
• For 3-vectors: The 3 vectors are said to be coplanar if their scalar triple product equals 0. Also, if three vectors are linearly dependent, then they are coplanar.
• For n-vectors: Vectors are said to be coplanar if no more than two amongst those vectors are linearly independent.
Coplanarity of Lines Using Condition in Vector Form
Let’s take into account the equations of two straight lines as below:
• r1 = p1 + λq1
• r2 = p2 + λq2
Wondering what the above equations suggest? It implies that the 1st line crosses through a point, L, whose position vector is provided by l1 and is parallel to m1. In the same manner, the 2nd line
passes through another point whose position vector is provided by l2 and is parallel to m2.
The condition for coplanarity under the vector form is that the line connecting the 2 points should be perpendicular to the product of the two vectors namely, p1 and p2. To represent this, we know
that the line connecting the two said points can be expressed in the vector form as (l2 – l1). So, we have:
(l2 – l1). (P1 x p2) = 0
Coplanarity of Lines Using Condition in Vector Form
Coplanarity in Cartesian is a derivative of the vector form. Let’s take into account the two points L (a1, b1, c1) & P (a2, b2, c2) in the Cartesian plane. Let there be 2 vectors p1 and p2. Their
direction ratios are provided as x1, y1, z1 and x2, y2, z2 respectively.
The vector equation of the line connecting L and P can be provided by:
LP = (a2 – a1)i + (b2 – b1)j + (c2– c1)k
p1 = x1i + y1j + z1k
p2 = x2i + y2j + z2k
We must now apply the above condition under the vector form in order to derive our condition in Cartesian form. By the condition stated above, the two lines are coplanar if LM. (p1 a p2) = 0. Hence,
in the Cartesian form, the matrix representing this equation is provided as 0.
Solved Examples
Question 1: Prove that the Lines [a + 3]/3 = [b – 1]/1 = [c – 5]/5 and [a + 1]/ -1 = [b – 2]/2 = [c – 5]/5 are Coplanar?
Answer: On comparing the equations, we get:
[a1, b1, c1] = {-3, 1, 5} and [a2, b2, c2] = {-1, 2, 5}.
Now, using the condition of Cartesian form, we shall solve the matrix:
= 2 [5 – 10] – 1 [-15 + 5] + 0 [-6 + 1]
= -10 + 10 = 0
Because the solution of the matrix provides a zero, we can say that the lines given are coplanar
FAQs on Coplanarity Two Lines
Q1. What Do We Understand By Coplanar Lines?
Answer: Coplanar lines are simply the lines that lie on the same plane. Imagine a sheet of paper or cardboard. Whatever lines are constructed on that sheet will be coplanar since they are lying on
the same plane, or on the same flat surface.
Q2. What Do We Understand By Non Coplanar Lines?
Answer: On contrary to the coplanar lines, these are the lines that do not lie on the same plane or a flat surface. Such a plane is said to be non-coplanar. Consider the image given below. The points
E and D are non-coplanar because they lie on different planes or different surfaces while points A, B and C are coplanar given that they lie on the same surface.
[Image will be Uploaded Soon]
Q3. What is the Significance of 3-dimensional Space When Dealing with Coplanar Lines?
Answer: When we deal with coplanar lines or seek to check if two lines are coplanar, then we need to consider and work in the 3-dimensional space. Else there is nothing to check. Only in the 3-D can
we have more than one plane. Planes can be parallel to each other or they can bisect each other. Also, remember that anything that is 2-dimensional in characteristic will be coplanar since there is
only one plane in the 2-D space.
Q4. Give an Example of Coplanar Lines in 2-dimensional Space?
Answer: Imagine a piece of paper. Whatever you draw on it will be 2-D, and everything on it will be coplanar since everything is joined by the flat sheet of paper. | {"url":"https://www.vedantu.com/maths/coplanarity-two-lines","timestamp":"2024-11-14T01:55:10Z","content_type":"text/html","content_length":"285461","record_id":"<urn:uuid:3ce11063-325e-4001-8cf2-9ea79bc4f4ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00553.warc.gz"} |
Tim Button (University College London): Publications - PhilPeople
• 3225
forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL
and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), symbolizing English in TFL and FOL, and Fitch-style natural deduction proof
systems for both TFL and FOL. It also deals with some advanced topics such as modal logic, soundness, and fu…
Read more
• 2534
Livro-texto de introdução à lógica, com (mais do que) pitadas de filosofia da lógica, produzido como uma versão revista e ampliada do livro Forallx: Calgary. Trata-se da versão de 13 de outubro
de 2022. Comentários, críticas, correções e sugestões são muito bem-vindos.
• 2495
Tim Button explores the relationship between words and world; between semantics and scepticism. A certain kind of philosopher – the external realist – worries that appearances might be radically
deceptive. For example, she allows that we might all be brains in vats, stimulated by an infernal machine. But anyone who entertains the possibility of radical deception must also entertain a
further worry: that all of our thoughts are totally contentless. That worry is just incoherent. We cannot, then, …
Read more
• 923
Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the
result of tapping any earlier-found object with any magic wand (from a given stock of magic wands). By varying the number and behaviour of the wands, we can flesh out this idea in many different
ways. This paper's main Theorem is that any loosely constructive way of fleshing out this idea is synonymous with…
Read more
• 859
No-futurists ('growing block theorists') hold that that the past and the present are real, but that the future is not. The present moment is therefore privileged: it is the last moment of time.
Craig Bourne (2002) and David Braddon-Mitchell (2004) have argued that this position is unmotivated, since the privilege of presentness comes apart from the indexicality of 'this moment'. I
respond that no-futurists should treat 'x is real-as-of y' as a nonsymmetric relation. Then different moments are re…
Read more
• 757
Universals are putative objects like wisdom, morality, redness, etc. Although we believe in properties (which, we argue, are not a kind of object), we do not believe in universals. However, a
number of ordinary, natural language constructions seem to commit us to their existence. In this paper, we provide a fictionalist theory of universals, which allows us to speak as if universals
existed, whilst denying that any really do.
• 744
This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity
arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments
for arithmetic and set theory.
• 727
Amie Thomasson and Eli Hirsch have both attempted to deflate metaphysics, by combining Carnapian ideas with an appeal to ordinary language. My main aim in this paper is to critique such
deflationary appeals to ordinary language. Focussing on Thomasson, I draw two very general conclusions. First: ordinary language is a wildly complicated phenomenon. Its implicit ontological
commitments can only be tackled by invoking a context principle; but this will mean that ordinary language ontology is not a…
Read more
• 715
Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable
models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of
Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001…
Read more
• 700
In Truth by Analysis (2012), Colin McGinn aims to breathe new life into conceptual analysis. Sadly, he fails to defend conceptual analysis, either in principle or by example.
• 699
Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions:
according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical
light on set theory. We then argue that, while CTT ’s type-restrictions are unjustifiable, the type-restrictions i…
Read more
• 676
Hilary Putnam once suggested that “the actual existence of sets as ‘intangible objects’ suffers… from a generalization of a problem first pointed out by Paul Benacerraf… are sets a kind of
function or are functions a sort of set?” Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets.
But these do not raise any obvious philosophical or foundational puzzles. For that, we first need to provide a full-…
Read more
• 658
In “Models and Reality” (1980), Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. Here, I will develop that sketch. By combining Putnam’s
model-theoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem-Gödel Antinomy. In brief: our mathematical concepts are perfectly precise;
however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is und…
Read more
• 631
We offer two arguments against the halving repose to Sleeping Beauty. First, we show that halving violates the Epistemological Sure-Thing Principle, which we argue is a necessary constraint on
any reasonable probability assignment. The constraint is that if hypothetically on C you assign to A the same probability you assign to A hypothetical on not-C, you must assign that probability
to A simpliciter. Epistemically, it's a sure thing for you that A has this probability. Second, we show that halv…
Read more
• 622
Tennenbaum's Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have recently claimed that this result has important philosophical consequences: in
particular, it offers us a way of responding to model-theoretic worries about how we manage to grasp the standard model. We disagree. If there ever was such a problem about how we come to grasp
the standard model, then Tennenbaum's Theorem does not help. We show this by examining a parallel argument, from …
Read more
• 614
Recent work on hypercomputation has raised new objections against the Church–Turing Thesis. In this paper, I focus on the challenge posed by a particular kind of hypercomputer, namely, SAD
computers. I first consider deterministic and probabilistic barriers to the physical possibility of SAD computation. These suggest several ways to defend a Physical version of the Church–Turing
Thesis. I then argue against Hogarth's analogy between non-Turing computability and non-Euclidean geometry, showing t…
Read more
• 597
The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: 'Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found
before S, we find a set whose members are exactly those sets. We find nothing else at S.' Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and
suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories due to Scott, M…
Read more
• 586
It is a metaphysical orthodoxy that interesting non-symmetric relations cannot be reduced to symmetric ones. This orthodoxy is wrong. I show this by exploring the expressive power of symmetric
theories, i.e. theories which use only symmetric predicates. Such theories are powerful enough to raise the possibility of Pythagrapheanism, i.e. the possibility that the world is just a vast,
unlabelled, undirected graph.
• 565
Hilary Putnam’s BIV argument first occurred to him when ‘thinking about a theorem in modern logic, the “Skolem–Löwenheim Theorem”’ (Putnam 1981: 7). One of my aims in this paper is to explore the
connection between the argument and the Theorem. But I also want to draw some further connections. In particular, I think that Putnam’s BIV argument provides us with an impressively versatile
template for dealing with sceptical challenges. Indeed, this template allows us to unify some of Putnam’s most e…
Read more
• 558
Tallant (2007) has challenged my recent defence of no-futurism (Button 2006), but he does not discuss the key to that defence: that no-futurism's primitive relation 'x is real-as-of y' is not
symmetric. I therefore answer Tallant's challenge in the same way as I originally defended no-futurism. I also clarify no-futurism by rejecting a common mis-characterisation of the growing-block
theorist. By supplying a semantics for no-futurists, I demonstrate that no-futurism faces no sceptical challenges…
Read more
• 545
In the early-to-mid 1930s, Wittgenstein investigated solipsism via the philosophy of language. In this paper, I want to reopen Wittgenstein's ‘grammatical’ examination of solipsism.Wittgenstein
begins by considering the thesis that only I can feel my pains. Whilst this thesis may tempt us towards solipsism, Wittgenstein points out that this temptation rests on a grammatical confusion
concerning the phrase ‘my pains’. In Section 1, I unpack and vindicate his thinking. After discussing ‘my pains’,…
Read more
• 545
Keränen (2001) raises an argument against realistic (ante rem) structuralism: where a mathematical structure has a non-trivial automorphism, distinct indiscernible positions within the structure
cannot be shown to be non-identical using only the properties and relations of that structure. Ladyman (2005) responds by allowing our identity criterion to include 'irreflexive two-place
relations'. I note that this does not solve the problem for structures with indistinguishable positions, i.e. positio…
Read more
• 520
Can we quantify over everything: absolutely, positively, definitely, totally, every thing? Some philosophers have claimed that we must be able to do so, since the doctrine that we cannot is
self-stultifying. But this treats restrictivism as a positive doctrine. Restrictivism is much better viewed as a kind of militant quietism, which I call dadaism. Dadaists advance a hostile
challenge, with the aim of silencing everyone who holds a positive position about ‘absolute generality’
• 489
Prior’s Tonk is a famously horrible connective. It is defined by its inference rules. My aim in this article is to compare Tonk with some hitherto unnoticed nasty connectives, which are defined
in semantic terms. I first use many-valued truth-tables for classical sentential logic to define a nasty connective, Knot. I then argue that we should refuse to add Knot to our language. And I
show that this reverses the standard dialectic surrounding Tonk, and yields a novel solution to the problem of ma…
Read more
• 471
Minimalists, such as Paul Horwich, claim that the notions of truth, reference and satisfaction are exhausted by some very simple schemes. Unfortunately, there are subtle difficulties with
treating these as schemes, in the ordinary sense. So instead, minimalists regard them as illustrating one-place functions, into which we can input propositions (when considering truth) or
propositional constituents (when considering reference and satisfaction). However, Bertrand Russell's Gray's Elegy argument …
Read more
• 426
Whatever the attractions of Tolkein's world, irrealists about fictions do not believe literally that Bilbo Baggins is a hobbit. Instead, irrealists believe that, according to The Lord of the
Rings {Bilbo is a hobbit}. But when irrealists want to say something like “I am taller than Bilbo”, there is nowhere good for them to insert the operator “according to The Lord of the Rings”.
This is an instance of the operator problem. In this paper, I outline and criticise Sainsbury's (2006) spotty scope a…
Read more
• 421
On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical
logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas
due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; a…
Read more
• 413
Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bar-bones story introduces the idea of a potential hierarchy of sets:
'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence.
Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-…
Read more
• 398
Wittgenstein’s atomist picture, as embodied in his Tractatus, is initially very appealing. However, it faces the famous colour-exclusion problem. In this paper, I shall explain when the atomist
picture can be defended in the face of that problem; and, in the light of this, why the atomist picture should be rejected. I outline the atomist picture in Section 1. In Section 2, I present a
very simple necessary and sufficient condition for the tenability of the atomist picture. The condition is: logi…
Read more
• 384
There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much
philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas.
Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discr…
Read more | {"url":"https://philpeople.org/profiles/tim-button/publications?order=viewings","timestamp":"2024-11-10T09:52:39Z","content_type":"text/html","content_length":"155307","record_id":"<urn:uuid:838f7c22-fb9f-4409-bb06-3ba2104020d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00243.warc.gz"} |
NCERT 6TH CLASS MATHS DATA HANDLING PART - ll
Important Questions
Multiple Choice Questions:
Q.1 The maximum marks obtained by any student is
(a) 95
(b) 78
(c) 75
(d) 25.
Q.2 The minimum marks obtained by any student is
(a) 95
(b) 78
(c) 75
(d) 25
Q.3 How many students got the same marks?
(a ) 2
(b) 3
(c) 4
(d) 5
Q.4 The difference between the maximum and mini-mum marks obtained is
(a) 60
(b) 50
(c) 70
(d) 80.
Q.5 How many students got 75 or more marks?
(a) 1
(b) 2
(c) 3
(d) 4
Q.6 How many students got marks below 60?
(a) 1
(b) 2
(c) 3
(d) 4
Q.7 How many students got marks between 60 and 75?
(a) 1
(b) 2
(c) 3
(d) 4.
Q.8 Following frequency distribution table shows marks (out of 50) obtained in English
by 45 students of class VI. Which two classes have the same frequency?
(a) 10 - 20 and 40 - 50
(b) 10 - 20 and 20 - 30
(c) 20 - 30 and 40 – 50
(d) None of these
Q.9 The pictograph shows the numbers of goals scored by four soccer teams in a season.
How many goals did Kickers score?
(a) 20
(b) 10
(c) 15
(d) None of these
Q.10 A _______ is a collection of numbers gathered to give some information.
(a) Tally mark
(b) Data
(c) None of these
(d) Frequency
Match The Following:
Fill in the blanks:
1. Representation of data with the help of tally marks is called _________.
2. In a bar graph width of rectangle is always _________.
3. The tally mark represents ________.
4. In a bar graph, _______ can be drawn horizontally and vertically.
True /False:
1. A bar graph represents data in the form of pictures, object or parts of objects.
2. Data is a collection of numerical figures giving required information.
3. In a bar graph width of rectangle is always equal.
4. The tally mark
Very Short Questions:
1. A collection of numbers gathered to give some information is called?
2. For a math assignment a group of students had to draw their favorite shapes.
The following pictures represent their choices. Each picture stands for 25 shapes.
3. A die was thrown 35 times and the following numbers were obtained:
5, 1, 4, 2, 3, 2, 6, 6, 1, 4, 2, 5, 4, 5, 3, 6, 1, 5
2, 6, 2, 5, 4, 1, 3, 2, 1, 4, 1, 6, 2, 6, 3, 3, 3
Prepare a frequency table for the data.
4. The result of a Mathematics test is as follows:
80, 90, 70, 80, 80, 60, 80, 70, 90, 65, 100, 60, 70, 60, 70, 85, 65, 70, 70, 85, 90, 60, 65, 80, 60
Make a frequency table for the above data and answer the following questions:
(a) What is the maximum marks obtained?
(b) How many students score less than 75 marks?
(c) How many students scored 80 marks or above?
(d) How many students appeared in the test?
5. The colors of fridges preferred by people living in a locality are shown by the following
pictograph Which colour most liked by the people?
6. In a village six fruit merchants sold the following number of fruit baskets in a particular season:
Observe this pictograph and answer the following questions:
(a) Which merchant sold the maximum number of baskets?
(b) How many fruit baskets were sold by Answer?
(c) The merchants who have sold 600 or more number of baskets are planning to buy a godown
for the next season. Can you name them?
7. The bar graph shows the number of toys produced by a factory during a certain week:
Answer the following questions:
(a) On which day the maximum number of toys were produced?
(b) On which day equal number of toys were produced?
(c) What is the total number of toys produced during the week?
(d) In which day minimum number of toys were produced?
Short Questions:
1. Mr. Rajan made a pictograph given below to show the number of cars washed at a car
washing station during three days of a week.
From the pictograph, find that:
(a) How many cars were washed on
(i) Friday
(ii) Saturday
(iii) Sunday?
(b) On which day the maximum number of cars were washed at the station?
(c) On which day the minimum number of cars were washed at the station?
(d) How many more cars were washed on Saturday than on Friday?
2. Read the pictograph given below and answer the following questions:
Persons employed in one year
(a) What is the number of persons employed in government service?
(b) How many more person were employed in government service than in private service?
(c) In which service, were the maximum number of persons employed?
3. In March 2012, children for six colonies of Meerut were given pulse polio Drops.
The colony wise number of children were as follows:
Represent the data by pictograph.
4. The given bar graph represents the frequency of a, e, i, o, and u in a piece of English writing.
(a) Which letter occurred the maximum number of times?
(b) Which letter occurred 40 times?
(c) Which letter occurred less than 30 times?
(d) Write down the five letters in the decreasing order of frequencies.
Long Questions:
1. The marks obtained by six students in Mathematics are given below. Represent the. data by a bar graph.
Use a scale of 0.5 cm for each name on the horizontal axis and 0.5 cm for 10 marks on the vertical axis.
2. A survey of 120 school students was done to find which activity they prefer to do in their free time:
Draw a bar graph to illustrate the above data taking scale of 1 unit length = 5 students.
Which activity is preferred by most of the students other than playing?
Assertion and Reason Questions:
(1.) Assertion (A) – The maximum marks obtained by any student is 95 out of 100
Reason (R) – Data is a collection of numbers gathered to give some information.
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
(2.) Assertion (A) – The minimum marks obtained by any student is 100 out of 100
Reason (R) – Data is a collection of numbers gathered to give some information.
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true but R is not the correct explanation of A
(c) A is true but R is false
(d) A is false but R is true
ANSWER KEY -
Multiple Choice questions:
1. (a) 95
2. (d) 25
3. (a) 2
62, 62
4. (c) 70
95 – 25 = 70
5. (c) 3
95, 78, 75
6. (d) 4.
55, 36, 42, 25
7. (c) 3
8. (a) 10 – 20 and 40 – 50
9. (a) 20
10. (b) Data
Match The Following:
Fill in the blanks:
1. Representation of data with the help of tally marks is called frequency distribution table.
2. In a bar graph width of rectangle is always equal.
3. The tally mark
4. In a bar graph, bar can be drawn horizontally and vertically.
True /False:
1. False. A pictograph represents data in the form of pictures, object or parts of objects.
2. True
3. True
4. True
Very Short Answer:
1. Data is collection of numbers gathered to give some information.
2. Total pictures = 25
Each picture stands for 25 shapes.
So, Total shapes students drew altogether = 25 × 25 = 625 shapes
3. From the given data, we have the following table.
4. From the above information, we have the following table.
(a) Maximum marks obtained by a student = 100
(b) 5 + 3 + 6 = 14 students obtained marks less than 75.
(c) 5 + 2 + 3 + 1 = 11 students scored marks 80 or above 80.
(d) Total 25 students were appeared in the test.
5. Number of people liked Red colour = 5 × 10 + 5 = 55
Number of people liked White colour = 2 × 10 = 20
Number of people liked Green colour = 3 × 10 = 30
Number of people liked Blue colour = 5 × 10 = 20
Hence, Red colour most liked by the people.
6. (a) martin sold the maximum number of baskets.
(b) 7 × 100 = 700 fruit baskets were sold by Answer.
(c) Answer, Martin and Ranjit Singh are planning to buy a godown for the next season.
7. (a) The maximum number of toys were produced on Tuesday.
(b) Wednesday and Thursday, Friday and Saturday have equal number of toys were produced.
(c) Total number of toys produced in the week = 175 + 225 + 150 + 150 + 125 + 125 = 900
(d) Minimum number of toys were produced on Friday and Saturday.
Short Answer:
1. (a) (i) On Friday – 4 × 5 = 20 cars
(ii) On Saturday – 9 × 5 = 45 cars
(iii) On Sunday – 7 × 5 = 35 cars.
(b) On Saturday, the maximum number of cars,
i.e., 9 × 5 = 45 were washed at the stations.
(c) On Friday, the minimum number of cars,
i.e., 4 × 5 = 20 were washed on the station.
(d) 45 – 20 = 25 more cars were washed on Saturday than on Friday.
2. (a) Number of persons employed in government service = 10 x 3000 = 30,000
(b) 10 x 3000 – 6 x 3,000 = 30,000 – 18,000 = 12,000 persons were employed more
in government service than in private service.
(c) In government service, the maximum number of persons were employed.
3. Pictograph:
4. (a) a letter occurred the maximum number of times.
(b) i letter occurred 40 times.
(c) u letter occurred less than 30 times.
(d) a, e, o, i, u is the decreasing order of their frequencies.
Long Answer:
1. The required bar graph is given as below:
2. (1) Draw two perpendicular lines – one vertical and one horizontal.
(2) Along horizontal line mark the “Preferred activity” and along vertical line mark the “No. of students”.
(3) Take bars of same width keeping uniform gap between them.
(4) Take scale of 1 unit length = 5 students along the vertical line and then mark the corresponding values.
(5) Calculate the heights of the bars for various activities preferred as shown below:
(6) Now draw various bars.
The activity “Reading story books” is preferred by most of the students other than playing.
Assertion and Reason Answers:
(1) (a) A is false but R is true
(2) (b) A is false but R is true | {"url":"https://studymaterialkota.com/blog/detail/ncert-6th-class-maths-data-handling-part-ll","timestamp":"2024-11-13T23:11:41Z","content_type":"text/html","content_length":"53535","record_id":"<urn:uuid:e05da1ef-8fde-4cef-a927-c650aafedb50>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00204.warc.gz"} |
RE: G40 update: Testers we need your help!
What I feel is the real problem here is the problem with bases. Bases are for one, too expensive, and two, if the Allies can liberate each-others territories, whats restricting them from building
bases on them? America should be able to build bases all over French territory, and even be able to Build on UK as well, on the historical basis that the US kept the UK war effort running with
endless military supplies.
I’m all for a Sierra Leon change, but what all of us really want is more flexibility of bases.
posted in Axis & Allies Global 1940
RE: Soviet Purchase Strategy
Id like to challenge YGs solid defensive line and propose a new idea of Defend the North, Attack the South. The focus of this strategy is to overwhelm the axis with too many fronts, forcing
Germany to distract too many of its forces away from Leningrad. By focusing power into Romania, Bulgaria, Albania and even Greece, many of Germany’s early game infantry and artillery will be
wasted or extended too far south in an attempt to regain valuable Russian NOs. This strategy needs an unpopular UK Greece landing to help with pincer attacks and a lot of offensive buys but
could prove to be a valuable fresh blood strategy if done right.
Germany has the advantage of massive mobility and ability to add extra firepower with the air force. I the UK goes to Greece on turn 1, I don’t think that it would be that much of a headache.
They already have 5 troops stationed in Bulgaria. A couple of planes added to the fight will make it easy to destroy the UK guys in Greece. It seems a big waste of 13ish IPCs that need to be
used in to initially smack Italy in Africa.
I also don’t see how a Russian army could get into Romania, Bulgaria, or Albania against a competent German player. Anything standing next to Germany can be obliterated when Germany finally
starts an attack into Russia. It isn’t until R4 or R5 that Russia can even contemplate a significant counter attack.
You must be playing against German opponents who have a balanced play with half the money spent on fleet. I spend 90% of the money against Russia, leaving only a bit left over to purchase a few
infantry to counter early land invasions. I can always get to Moscow around G6, but often decide to divert around it for an economic victory.
More like I can barely find an opponent to play…
Well, sounds like a bust so ill just disappear awkwardly back into the shadows for a while…
posted in Axis & Allies Global 1940
RE: Misprint? Why is Sierra Leone neutral in 1940?
One thing that I’ve wanted to see is the incorporation of Pro-axis neutrals in South America. I first thought that it really might not be very historically accurate and that it would tip the balance
in favor of allies too much, but seeing as the allies are under powered in this game it may help a little.
posted in Axis & Allies Global 1940
Sea Zones make no sense
After reading Black Elks post about Sierra Leon, it made me think about the board in general and open a thread specifically to discuss Sea Zones. Do Sea Zones represent any actual naval ranges? Any
sea zones that make no sense?
Hawaii to Japan, US to Spain and not UK?
posted in Axis & Allies Global 1940
RE: Soviet Purchase Strategy
The calculator is just a tool, and pretty limited one at that. I think it’s overrated and gets over-relied upon.
Anyways, this thread is getting WAY off topic.
I couldn’t agree more
And could you guys pound my alternate strategy some more?
posted in Axis & Allies Global 1940
RE: Soviet Purchase Strategy
Id like to challenge YGs solid defensive line and propose a new idea of Defend the North, Attack the South. The focus of this strategy is to overwhelm the axis with too many fronts, forcing Germany
to distract too many of its forces away from Leningrad. By focusing power into Romania, Bulgaria, Albania and even Greece, many of Germany’s early game infantry and artillery will be wasted or
extended too far south in an attempt to regain valuable Russian NOs. This strategy needs an unpopular UK Greece landing to help with pincer attacks and a lot of offensive buys but could prove to be a
valuable fresh blood strategy if done right.
posted in Axis & Allies Global 1940
RE: Angels Landing: SeaLion gone Archangel
What you can also reckon too is the possibility of sneaking tanks and mechs away to the far east, where even though all Russian territories are worth one, the sheer number of them are a huge income.
posted in Axis & Allies Global 1940
Angels Landing: SeaLion gone Archangel
I just wanted to discuss the tricky play of feinting SeaLion to actually land full force in Archangel. I’ve never really seen the downside to this play other than it is extremely predictable and
dangerous as you leave all of your transports wide open/trapped up north. As seeing that Germany’s main goal is to conquer Moscow asap, landing a Certain-Death invasion force two spaces away from the
capital seems like a much better option than letting Russia “RedTurtle” while you slowly march through the winter. Have any of you actually attempted this? I think the term Operation “Angels Landing”
would be hilariously fitting.
Operation “Up and Over”
Operation “Artic Storm”
Operation “Red Wedding”
Operation “Kremlin’s Back Door”
(Sorry if this tactic has already been termed or if it has already been played to death)
posted in Axis & Allies Global 1940
RE: Transports are too expensive
Actually, going back to the old Transports costing 8 would probably favour the allies massivly.
The old transports had a 1 in combat and was a hitpoint to take as casuality.
Lets take US as an example. If they where to make a fleet with 8 transports to threathen to land in europe. With transports costing 7, they have to use about 56 IPC for the transports. They will
probably need about 4 other transports as well, so in total, they pay about 82 IPC for the pleasure. If transports cost 8, they pay 96 IPC. So where is the gain? Well, the gain is that they can
now reduce the number of DDs needed to stand against luftwaffe. If we assume that 8 TTs is with the main fleet, then they would need about 5-6 fewer DDs in the main fleet. If we say they need 4
DDs less (then, they have the same number of combat dice in the main fleet, but with 4 extra HP), then they save 32 IPC in DDs. So buying TTs at 8 with 1 hp and 1 Combatdie will make the us
invastionfleet at least 16 IPCs cheaper, probably more in the range of 24 to 30 IPCs cheaper.
The point to making transport cheaper would also be to help out Germany with Sea Lion, as it nearly always proves to be too costly of an operation.
posted in House Rules
RE: French Liberation
So have any of you found it more or less beneficial to have France producing troops and being even of a small effort? I’m really interested in whether France as a nation is actually an important
factor in this game. I’d always just assume that making France was more of an interesting idea and was never actually a nation that was meant to be played, even during late stages of the game.
posted in Axis & Allies Global 1940 | {"url":"https://www.axisandallies.org/forums/user/jeroldthegreat","timestamp":"2024-11-13T02:13:40Z","content_type":"text/html","content_length":"163890","record_id":"<urn:uuid:c7c6743f-4ce3-479e-8c44-f4c4c4825c17>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00311.warc.gz"} |
Better Benchmarks Through Graphs
This is a blog post version of a talk I gave at the Northwest Database Society meeting last week. The slides are here, but I don’t believe the talk was recorded.
I believe that one of the things that’s holding back databases as an engineering discipline (and why so much remains stubbornly opinion-based) is a lack of good benchmarks, especially ones available
at the design stage. The gold standard is designing for and benchmarking against real application workloads, but there are some significant challenges achieving this ideal. One challenge^1 is that,
as in any system with concurrency, traces capture the behavior of the application running on another system, and they might have issued different operations in a different order running on this one
(for example, think about how in most traces it’s hard to tell the difference between application thinking and application waiting for data, which could heavily influence results if we’re trying to
understand the effect of speeding up the waiting for data portion). Running real applications is better, but is costly and raises questions of access (not all customers, rightfully, are comfortable
handing their applications over to their DB vendor).
Industry-standard benchmarks like TPC-C, TPC-E, and YCSB exist. They’re widely used, because they’re easy to run, repeatable, and form a common vocabulary for comparing the performance of systems. On
the other hand, these benchmarks are known to be poorly representative of real-world workloads. For the purposes of this post, mostly that’s because they’re too easy. We’ll get to what that means
later. First, here’s why it matters.
Designing, optimizing, or improving a database system requires a lot of choices and trade-offs. Some of these are big (optimistic vs pessimistic, distributed vs single machine, multi-writer vs
single-writer, optimizing for reads or writes, etc), but there are also thousands of small ones (“how much time should I spend optimizing this critical section?”). We want benchmarks that will shine
light on these decisions, allowing us to make them in a quantitative way.
Let’s focus on just a few of the decisions the database system engineer makes: how to implement atomicity, isolation, and durability in a distributed database. Three of the factors that matter there
are transaction size (how many rows?), locality (is the same data accessed together all the time?), and coordination (how many machines need to make a decision together?). Just across these three
factors, the design that’s best can vary widely.
If we think of these three factors as ones that define a space^2. At each point in this space, keeping other concerns constant, some design is best. Our next challenge is generating synthetic
workloads—fake applications—for each point of the space. Standard approaches to benchmarking sample this space sparsely, and the industry-standard ones do it extremely poorly. In the search for a
better way, we can turn, as computer scientist so often do, to graphs.
In this graph, each row (or other object) in our database is a node, and the edges mean transacted with. So two nodes are connected by a (potentially weighted) edge if they appear together in a
transaction. We can then generate example transactions by taking a random walk through this graph of whatever length we need to get transactions of the right size.
The graph model seems abstract, but is immediately useful in allowing us to think about why some of the standard benchmarks are so easy. Here’s the graph of write-write edges for TPC-C neworder (with
one warehouse), for example.
Notice how it has 10 disjoint islands. One thing that allows us to see is that we could immediately partition this workload into 10 shards, without ever having to execute a distributed protocol for
atomicity or isolation. Immediately, that’s going to look flattering to a distributed database architecture. This graph-based way of thinking is generally a great way of thinking about the
partitionability of workloads. Partitioning is trying to draw a line through that graph which cuts as few edges as possible^3.
If we’re comfortable that graphs are a good way of modelling this problem, and random walks over those graphs^4 are a good way to generate workloads with a particular shape, we can ask the next
question: how do we generate graphs with the properties we want? Generating graphs with particular shapes is a classic problem, but one approach I’ve found particularly useful is based on the
small-world networks model from Watts and Strogatz^6. This model gives us a parameter $p$ which, which allows us to vary between ring lattices (the simplest graph with a particular constant degree),
and completely random graphs. Over the range of $p$, long-range connections form across broad areas of the graph, which seem to correlate very well with the contention patterns we’re interested in
That gives us two of the parameters we’re interested in: transaction size is set by the length of random walks we do, and coordination which is set by adjusting $p$. We haven’t yet solved locality.
In our experiments, locality is closely related to degree distribution, which the Watts-Strogatz model doesn’t control very well. We can easily control the central tendency of that distribution (by
setting the initial degree of the ring lattice we started from), but can’t really simulate the outliers in the distribution that model things like hot keys.
In the procedure for creating these Watts-Strogatz graph, the targets of the rewirings from the ring lattice are chosen uniformly. We can make the degree distribution more extreme by choosing
non-uniformly, such as with a Zipf distribution (even though Zipf seems to be a poor match for real-world distributions in many cases). This lets us create a Watt-Strogatz-Zipf model.
Notice how we have introduced a hot key (near the bottom right). Even if we start our random walk uniformly, we’re quite likely to end up there. This kind of internal hot key is fairly common in
relational transactional workloads (for example, secondary indexes with low cardinality, or dense auto-increment keys).
This approach to generating benchmark loads has turned out to be very useful. I like how flexible it is, how we can generate workloads with nearly any characteristics, and how well it maps to other
graph-based ways of thinking about databases. I don’t love how the relationship between the parameters and the output characteristics is non-linear in a potentially surprising way. Overall, this post
and talk were just scratching the surface of a deep topic, and there’s a lot more we could talk about.
Play With the Watts-Strogatz-Zipf Model
$p$ parameter:
Zipf exponent:
1. There’s an excellent discussion of more problems with traces in Traeger et al’s A Nine Year Study of File System and Storage Benchmarking.
2. I’ve drawn them here as orthogonal, which they aren’t in reality. Let’s hand-wave our way past that.
3. This general way of thinking dates back to at least 1992’s On the performance of object clustering techniques by Tsangaris et al (this paper’s Expansion Factor, from section 2.1, is a nice way of
thinking about distributed databases scalability in general). Thanks to Joe Hellerstein for pointing this paper out to me. More recently, papers like Schism and Chiller have made use of it.
4. There’s a lot to be said about the relationship between the shape of graphs and the properties of random walks over those graphs. Most of it would need to be said by somebody more competent in
this area of mathematics than I am.
5. The degree distribution of these small-world networks is a whole deep topic of its own. Roughly, there’s a big spike at the degree of the original ring lattice, and the distribution decays
exponentially away from that (with the exponent related to $p$). | {"url":"https://brooker.co.za/blog/2024/02/12/parameters.html","timestamp":"2024-11-09T07:39:14Z","content_type":"application/xhtml+xml","content_length":"17671","record_id":"<urn:uuid:d7932dde-2e56-4de7-ba8a-4e0dc1feddf8>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00840.warc.gz"} |
Adding And Subtracting Integers - Steps, Examples & Questions
To add, start at the first number and move to the second number; to subtract, start from the second number and move to the first number.
From positive 9 move in the negative direction until you get to 7. You move 2 places to the left, which is -2.
Can fractions and decimals be negative?
Yes, there are negative fractions and decimals. Numbers to the left of 0 on the number line are negative.
Can you always use the number line to add and subtract integers?
Yes, using the number line when adding and subtracting integers will always work. However, it might not always be the fastest way to get the answer.
What is a zero pair?
A zero pair is a number and its opposite. For example, 5 and -5 are a zero pair. The opposite of positive integers is negative integers.
What is the additive inverse?
The sum of zero pairs is an additive inverse because the sum is 0.
How will adding and subtracting integers help in algebra?
Addition of integers and subtraction of integers help when simplifying algebraic expressions and also when factoring algebraic expressions.
Do you have to write the positive sign in front of positive numbers?
The positive sign does not necessarily need to be written in front of a number. For example, +5 is the same as 5. The positive sign is understood.
-1 degree
1 degree
7 degree
-7 degree | {"url":"http://karenguzak.net/index-1244.html","timestamp":"2024-11-03T02:31:49Z","content_type":"text/html","content_length":"249824","record_id":"<urn:uuid:13d1cf0e-6373-470b-a1dc-d09d552d381d>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00011.warc.gz"} |
"One thing you can't datOmatic
recycle is wasted time" Perpetual tables
Calculate it!
Anonymous Chinese new year finder
Tags: Perpetual calendar (Engl.), Calendario perpetuo (It., Sp., Port.),
Calendrier perpétuel (Fr.), Ewiger Kalender (Ger.), Eeuwigdurende kalender (Du.),
Evighetskalender (Sw.), Kalendarz wieczny (Pol.), věčný kalendář (Cz.),
Вечный календарь (Ru.), 万年历 (Chin.), 万年暦 (Jap.), सतत कैलेंडर (Hin.),
לוח עד (Hebr.), التقويم الدائم (Arab.)
Date Finder 1 (javascript)
Which day of the week were you born on?
Find the day of the week for any date... Fill in the date you want to check then click "Get Date"
In quale giorno della settimana sei nato?
Per sapere in quale giorno della settimana cade una data basta compilare il modulo e premere "Get Date"
falls on
cade il...
Perpetual Calendar 2 (tables)
By means of the tables below, you can determine the day of the week for specific dates that might be of your interest. Just follow the easy instructions. First, locate the number at the intersection
of the month column and day row in the table 1 (example: the column for June and the row for day 16th cross at the number 6). Then, locate the number at the intersection of the year column and
century row in the table 2 (example: the column for year 77th and the row for century 18th cross at the number 3). Finally, locate in table 3 the letter at the intersection of both numbers you've
found (here, numbers 6 and 3 cross at D). The letter D corresponds to Saturday (if the day of the week you're looking for is in January or February of a leap year, you have to bring forward of 1 day
the result).
Per trovare un giorno della settimana, cercate nella tabella 1 il numero segnato dall'incontro della colonna dei mesi e della linea dei giorni; poi, nella tabella 2, il numero all'incrocio della
colonna degli anni e della linea dei secoli... Infine, nella tabella 3, la lettera che si trova all'incontro delle due linee contrassegnate dai numeri che avete trovato. Ogni lettera corrisponde ad
un giorno.
Perpetual Calendar 3 (calculate it!)
The following formula - named Zeller's Rule - allows you to calculate a day of the week for any date:
F = k + [(13 x m-1)/5] + D + [D/4] + [C/4] - 2 x C
k is the day of the month. Let's use January 27, 2024 as an example. For this date, k = 27.
m is the month number. Months have to be counted specially: March is 1, April is 2, and so on to February, which is 12 (this makes the formula simpler, because on leap years February 29 is counted as
the last day of the year). Because of this rule, January and February are always counted as the 11th and 12th months of the previous year. In our example, m = 11.
D is the last two digits of the year. Because of the month numbering, D = 23 in our example, even though we are using a date from 2024.
C stands for century: it's the first two digits of the year. In our case, C = 20.
Now let's substitute our example numbers into the formula:
F = k + [(13 x m-1)/5] + D + [D/4] + [C/4] - 2 x C
= 27 + [(13 x 11-1)/5] + 23 + [23/4] + [20/4] - 2 x 20
= 27 + [28.4] + 23 + [5.75] + [5] - 40
[dropp every number after the decimal point]
= 27 + 28 + 23 + 5 + 5 - 40 = 48.
Once we have found F, we divide it by 7 and take the remainder (if the remainder is negative, add 7). A remainder of 0 corresponds to Sunday, 1 means Monday, etc. For our example, 48 / 7 = 6,
remainder 6, so January 27, 2024 will be a Saturday. Then, have a nice week-end!
Chinese New Year's Day Finder
Any Chinese year invariably begins with the second new-moon day after the winter solstice (December 21st). For instance, in the year 2011, the next new moon after winter solstice was January 4th, and
the second one was on February 3rd. Consequently, this date corresponds to the Chinese New Year 2011.
However, the precise rules for determining the Chinese New Year’s day are far more complex. One problem with any lunar calendar system is that in some years there are 13 new moons. The Chinese deal
with this by slotting in an extra intercalary month.
So, the Chinese New Year's day is movable — just as Easter Day, which is also tributary of the moon — and takes place somewhere between January 21 and February 20 according to astronomic
The Chinese zodiac is a cycle of 12 years, each placed under the sign of one of the twelve symbolic animals: Rat, Buffalo (or Ox), Tiger, Cat (or Rabbit or Hare), Dragon, Snake, Horse, Goat (or Sheep
or Ram), Monkey, Rooster, Dog, Pig (or Boar). Chinese years also evolve in cycles of ten years each. Every set of two consecutive years is governed by a Chinese cosmic element. There are five
elements in all: Wood, Fire, Earth, Metal, Water. | {"url":"https://archimedes-lab.org/datOmatic.html","timestamp":"2024-11-04T08:09:34Z","content_type":"text/html","content_length":"56297","record_id":"<urn:uuid:2f4a550c-6a9c-4126-b4d0-0e8d01faf3a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00717.warc.gz"} |
DRDO(CEPTAM) Placement Paper : Whole-Testpaper
Saturday, March 2, 2013
Saturday, March 2, 2013
DRDO (CEPTAM)electronics and electrical Questions:
1.The current I in the given network.
a) 1A
b) 3A
c) 5A
d) 7A
2.For the Delta- Wye transformation in given figure, the value of the resistance R is.
a) 1/3 ohms
b) 2/3 ohms
c) 3/2 ohms
d) 3 ohms
3.In the given network, the Thevenin’s equivalent as seen by the load resistance Rl is
a) V=10 V, R= 2ohms
b) V=10V, R=3 ohms
c) V=15V, R= 2ohms
d) V=15V, R=3 ohms
4.The current I in a series R-L circuit with R=10 ohms and L=20mH is given by i=2sin500t A. If v is the voltage across the R-L combination then i
a) lags v by 45 degree
b) is in-phase with v
c) leads v by 45
d) lags v by 90
5.In thr given network, the mesh current I and the input impedance seen by the 50 V source, respectively, are
a) 125/13 A and 11/8 ohms
b) 150/13 A and 13/8 ohms
c) 150/13 A and 11/8 ohms
d) 125/13 A and 13/8 ohms
6.A voltage source having a source impedance Z = R + jX can deliver maximum Average power to a load impedance Z, when
a) Z = R + jX
b) Z = R
c) Z = jX
d) Z = R –jX
7.In the given circuit, the switch S is closed at t=0. Assuming that there is no initial Charge in the capacitor, the current i(t) for t>0 is
a) V/R e^ (-2t/RC)
b) V/R e^ (-t/RC)
c) V/2R e^ (-2t/RC)
d) V/2R e^ (-t/RC)
8.For the circuit in given figure, if e(t) is a ramp signal, the steady state value of the Output voltage v(t) is
a) 0 b) LC c) R/L d) RC
9.For the series RLC circuit in given figure, if w=1000 rad/sec, then the current I (in Amperes) is
a) 2 ∟-15
b) 2 ∟15
c) √2∟-15
d) √2∟15
10.The Y-parameter matrix (mA/V) of the two-port given network is
a) [2 -1 -1 2]
b) [2 1 -1 2]
c) [1 -2 -1 2]
d) [2 1 1 2]
11.The maximum number of trees of the given graph is
a) 16 b) 25 c) 100 d) 125
12.Given figure shows a graph and one of its trees. Corresponding to the tree, the group of branches that CAN NOT constitute a fundamental cut set is
a) 1,2,3
b) 1,4,6,8,3
c) 5,6,8,3
d) 4,6,7,
13.The Y-parameter matrix of a network is given by Y=[1 1 -1 1] A/V. The Z11 parameter of the same network is
a) ½ ohms
b) 1/√2 ohms
c) 1 ohms
d) 2 ohms
14.For the given circuit, the switch was kept closed for a long time before opening it at t=0. The voltage v(0+) is
a) -10 V b) -1 V c) 0V d) 10 V
15.The input impedance of a series RLC circuit operating at frequency W=√2w, w being the resonant frequency, is
a) R-j(wL/√2) ohms
b) R+j(wL/√2) ohms
c) R-j√2wL ohms
d) R-j√2wL ohms
16.The threshold voltage V is negative for
a) an n-channel enhancement MOSFET
b) an n-channel depletion MOSFET
c) an p-channel depletion MOSFET
d) an p-channel JFET
17.At a given temperature, a semiconductor with intrinsic carrier concentration ni= 10 ^ 16 / m^3 is doped with a donor dopant of concentration Nd = 10 ^ 26 /m^3. Temperature remaining the same,
the hole concentration in the doped semiconductor is
a) 10 ^ 26 /m^3
b) 10 ^ 16 /m^3
c) 10 ^ 14 /m^3
d) 10 ^ 6 /m^3}
18.At room temperature, the diffusion and drift constants for holes in a P-type semiconductor were measured to be Dp = 10 cm^2/s and µp = 1200 cm^2/V-s, respectively. If the diffusion constant of
electrons in an N-type semiconductor at the same temperature is Dn = 20 cm^2/s, the drift constant for electrons in it is
a) µn = 2400 cm^2/V-s
b) µn = 1200 cm^2/V-s
c) µn = 1000 cm^2/V-s
d) µn = 600 cm^2/V-s
19.A common LED is made up of
a) intrinsic semiconductor
b) direct semiconductor
c) degenerate semiconductor
d) indirect semiconductor
20.When operating as a voltage regulator, the breakdown in a Zener diode occurs due to the
a) tunneling effect
b) avalanche breakdown
c) impact ionization
d) excess heating of the junction.
21.If the common base DC current gain of a BJT is 0.98, its common emitter DC current gain is
a) 51 b) 49 c) 1 d) 0.02
22.Negative resistance characteristics is exhibited by a
a) Zener diode
b) Schottky diode
c) photo diode
d) Tunnel diode
23.Let En and Ep, respectively, represent the effective Fermi levels for electrons and holes during current conduction in a semiconductor. For lasing to occur in a P-N junction of band-gap energy
1.2 eV, (En - Ep) should be
a) greater than 1.2eV
b) less than 1.2eV
c) equal to 1.1eV
d) equal to 0.7eV
24.In a P-well fabrication process, the substrate is
a) N-type semiconductor and is used to build P-channel MOSFET
b) P-type semiconductor and is used to build P-channel MOSFET
c) N-type semiconductor and is used to build N-channel MOSFET
d) P-type semiconductor and is used to build N-channel MOSFET
25.In a MOS capacitor with n-type silicon substrate, the Fermi potential ¢ = -0.41 V and the flat-band voltage Vfb = 0V. The value of the threshold voltage Vt is
a) -0.82 V b) -0.41 V c) 0.41 V d) 0.82
Refer given figure for question 26 and 27. Assume D1 and D2 to be ideal diodes.
26.Which one of the following statements is true?
a) Both D1 and D2 are ON.
b) Both D1 and D2 are OFF.
c) D1 is ON and D2 is OFF.
d) D2 is ON and D1 is OFF.
27.Values of Vo and I, respectively, are
a) 2V and 1.1 mA
b) 0V and 0 mA
c) -2V and 0.7 mA
d) 4V and 1.3 mA
28.In a BJT CASCODE pair, a
a) common emitter follows a common base
b) common base follows a common collector
c) common collector follows a common base
d) common base follows a common emitter
29.Inside a 741 op-amp, the last functional block is a
a) differential amplifier
b) level shifter
c) class-A power amplifier
d) class-AB power amplifier
30.For the MOSFET in the given circuit, the threshold voltage Vt = 0.5V, the process parameter KP = 150 µA/V^2 and W/L = 10. The values of Vd and Id, respectively, are
a) Vd = 4.5 V and Id = 1 mA
b) Vd = 4.5 V and Id = 0.5 mA
c) Vd = 4.8 V and Id = 0.4 mA
d) Vd = 6 V and Id = 0 mA
31.A negative feedback is applied to an amplifier with the feedback voltage proportional to the output current. This feedback increases the
a) input impedance of the amplifier
b) output impedance of the amplifier
c) distortion in the amplifier
d) gain of the amplifier
32.The early effect in a BJT is modeled by the small signal parameter
a) r0 b) r∏ c) gm d) β
33.For a given filter order, which one of the following type of filters has the least amount of ripple both in pass-band and stop-band?
a) Chebyshev type I
b) Bessel
c) Chebyshev type II
d) Elliptic
34.For a practical feedback circuit to have sustained oscillation, the most appropriate value of the loop gain T is
a) 1 b) -1 c) -1.02 d) 1.02
35.Assume the op-amps in given figure to be ideal. If the input signal vi is a sinusoid of 2V peak-to-peak and with zero DC component, the output signal vo is
a) sine wave
b) square wave
c) pulse train
d) triangular wave
36.In a common source amplifier, the mid-band voltage gain is 40 dB and the upper cutoff frequency is 150kHz. Assuming single pole approximation for the amplifier the unity gain frequency fT is
a) 6 MHz b) 15 MHz c) 150 MHz d) 1.5 GHz
37.An op-amp is ideal except for finite gain and CMRR. Given the open loop differential gain Ad=2000, CMRR = 1000, the input to the noninverting terminal is 5.002 V and the input to the inverting
terminal is 4.999 V, the output voltage of the op-amp is
a) 14 V b) 24 V c) -6 V c) -8 V
38.The op-amp in the circuit in given figure has a non-zero DC offset. The steady state value of the output voltage Vo is
a) –RC dvs(t)/ dt b) – (1/RC)|vs(t)dt c) –V d) +V
39.For the circuit in given figure, if the value of the capacitor C is doubled, the duty-cycle of the output waveform Vo
a) increases by a factor of 2
b) increases by a factor of 1.44
c) remains constant
d) decreases by a factor of 1.44
40.Assume the op-amp in the given circuit to be ideal. The value of the output voltage Vo is
a) 3.2 Vi b) 4 Vi c) 9 Vi d) 10 Vi
41.The complement of the Boolean expression F = (X + Y¯ + Z)(X¯ + Z¯)(X + Y) is
a) XYZ+XZ¯+Y¯Z
b) X¯YZ¯+XZ+X¯Y¯
c) X¯YZ¯+XZ+YZ
d) XYZ+X¯Y¯
42.The Boolean function F(A,B,C,D) = ∑(0,6,8,13,14) with don’t care conditions d(A,B,C,D) = ∑(2,4,10) can be simplified to
a) F = B¯D¯+CD¯+ABC¯
b) F = B¯D¯+CD¯+ABC¯D
c) F = AB¯D¯+CD¯+ABC¯
d) F = B¯D¯+CD¯+ABCD
43.The Boolean function F = A¯D¯+B¯D can be realized by one of the following figures
44. For the multiplexer in given figure, the Boolean expression for the output Y is
a) A¯B¯+B¯C¯+AC b) AB¯+B¯C¯+AC¯ c) AB¯+B¯C+AC d) A¯B¯+B¯C+A¯C
45. Which one of the following is TRUE?
a) Both latch and flip-flop are edge triggered.
b) A latch is level triggered and a flip-flop is edge triggered.
c) A latch is edge triggered and a flip-flop is level triggered.
d) Both latch and flip-flop are level triggered.
46. In a schottky TTL gate, the Schottky diode
e) increases the propagation delay
f) increases the power consumption
g) prevents saturation of the output transistor
h) keeps the transistor in cutoff region
47. For which one of the following ultraviolet light is used to erase the stored contents
a) PROM b) EPROM c) EEPROM d) PLA
48. Which one of the following is NOT a synchronous counter
a) Johnson counter
b) Ring counter
c) Ripple counter
d) Up-down counter
49. In 8085 microprocessor, the accumulator is a
a) 4 bit register
b) 8 bit register
c) 16 bit register
d) 32 bit register
50. In the register indirect addressing mode of 8085 microprocessor, data is stored
a) at the address contained in the register pair
b) in the register pair
c) in the accumulator
d) in a fixed location of the memory
51. The output w[n] of the system shown in given figure is
a) x[n] b) x[n-1] c) x[n] – x[n-1] d) 0.5(x[n-1] + x[n])
52. Which one of the following is a periodic signal
a) x(t) = 2 e^j(t+(π/4))
b) x[n] = u[n] + u[-n]
c) x[n] = ∑{∂[n-4k]-∂[n-1-4k]} where k = -∞to ∞
d) x(t) = e^ (-1+j)t
53. If the input-output relation of a system is y(t) = ∫x(t) dt where t = -∞ to 2t
a) linear, time-invariant and unstable
b) linear, non-causal and unstable
c) linear, causal and time invariant
d) non-causal, time invariant and unstable
54. Which one of the can be the magnitude of the transfer function | H(jw) | of a causal system
55. Consider the function H(jw) = H1(w) + jH2(w), where H1(w) is an odd function and H2(w) is an even function. The inverse Fourier transform of H(jw) is
a) a real and odd function
b) a complex function
c) a purely imaginary function
d) a purely imaginary and odd function
56. The laplace transform of given signal is
a) –A((1-e^cs)/s)
b) A((1-e^cs)/s)
c) A((1-e^-cs)/s)
d) –A((1-e^-cs)/s)
57. If X(z) is the z-transform of x[n] = (1/2)^ |n|, the ROC of X(z) is
a) |z| > 2 b) |z| < 2 c) 0.5<|z|<2 d) the entire z-plane
58. In a linear phase system, τg the group delay and τp the phase delay are
a) constant and equal to each other b) τg is a constant and τp is proportional to w c) a constant and τg is proportional to w d) τg is proportional to w and τp is proportional to w
59. A signal m(t), band-limited to a maximum frequency of 20 kHz is sampled at a frequency fs kHz to generate s(t). An ideal low pass filter having cut-off frequency 37 kHz is used to reconstruct
m(t) from s(t). The maximum value of fs required to reconstruct m(t) without distortion is
a) 20 kHz
b) 40kHz
c) 57 kHz
d) 77 kHz
60. If the signal x(t) shown in given figure is fed to an LTI system having impulse response h(t) as shown in given figure, the value of the DC component present in the output y(t) is
a) 1 b) 2 c) 3 d) 4
61. The characteristic equation of an LTI system is given as s^3 + Ks^2 + 5s + 10. When the system is marginally stable, the value of K and the sustained oscillation frequency w, respectively,
a) 2 and 5
b) 0.5 and √5
c) 0.5 and 5
d) 2 and √5
62. The time required for the response of a linear time-variant system to reach half the final value for the first time is
a) delay time
b) peak time
c) rise time
d) decay time
63. The signal flow graph of the given network is
64. Let c(t) be the unit step response of a system with transfer function K(s+a)/(s+K). If c(0+)=2 and c(∞)=10, then the values of a and K, respectively, are
a) 2 and 10
b) -2 and 10
c) 10 and 2
d) 2 and -10
65. The loop transfer function of an LTI system is G(s)H(s)= K(s+1)(s+5) / s(s+2)(s+3). For K>0, the point on the real axis that DOES NOT belong to the root locus of the system is
a) -0.5 b) -2.5 c) -3.5 d) -5.5
66. The state space equation of the circuit shown in given figure for x1=v0, x2=I is
67. The open loop gain of a unity feedback system is G(s)=wn^2 / s(s+2wn). The unit step response c(t) of the system is
69. The angles of the asymptotes of the root loci of the equation s^3 + 5s^2 + (K+2)s + K = 0, for 0<=K<∞, are
a) 0 and 270
b) 0 and 180
c) 90 and 270
d) 90 and 180
70. The bode plot corresponding to a proportional derivative controller is the one shown in given figure
71. In frequency modulation, the instantaneous
a) amplitude of the carrier signal is varied with the instantaneous amplitude of the message signal
b) amplitude of the carrier signal is varied with the instantaneous frequency of the message signal
c) frequency of the carrier signal is varied with the instantaneous amplitude of the message signal
d) frequency of the carrier signal is varied with the instantaneous frequency of the message signal
72. If X is a zero mean Gaussian random variable, then P{X<=0} is
a) 0 b) 0.25 c) 0.5 d) 1
73. If a single-tone amplitude modulated signal at a modulation depth of 100% transmits a total power of 15W, the power in the carrier component is
a) 5W b) 10W c) 12W d) 15W
74. In a super heterodyne receiver, rejection of the image signal can be achieved by using a
a) higher local oscillatorn frequency b) crystal oscillator c) narrow band IF filter d) narrow band filter at RF stage
75. The number of bbits per sample of a PCM system depends upon the
a) sampler type b) quantizer type c) number of levels of the quantizer d) sampling rate
76. Which one of the following is used for the detection of AM-DSB-SC signal
a) Ratio detector b) Foster-Seeley discriminator c) Product demodulator d) Balanced-slpoe detector
77. Which one of the following signal pairs can represent a BPSK signal
a) A cos2πfct, A sinπfct
b) A cos2πfct, - A sinπfct
c) - A cos2πfct, A sinπfct
d) A sin2πfct, A cosπfct
78. Which one of the following can be used for the detection of the noncoherent BPSK signal
a) matched filter b) phase-locked loop c) envelope detector
d) product demodulator
79. Bits of duration Tb are to be transmitted using a BPSK modulation with a carrier of frequency Fc Hz. The power spectral density of the transmitted signal has the first null at the normalized
a) |F – Fc|Tb = 0 b) |F – Fc|Tb = 1 c) |F – Fc|Tb = 2 d) |F – Fc|Tb = 4
80. The probability of bit error of a BPSK modulation scheme, with transmitted signal energy per bit Eb, in an additive white Gaussian noise channel having one-sided power spectral density N0, is
a) (1/2) erfc(Eb/2N0) b) (1/2) erfc√(Eb/2N0) c) (1/2) erfc(Eb/N0) d) (1/2) erfc√ (Eb/N0)
81. For a given transmitted pulse p(t), 0<=t<=T, the impulse response of a filter matched to the received signal is
a) –p(t-T), 0<=t<=T b) –p(T-t), 0<=t<=T c) p(t-T), 0<=t<=T d) p(T-t), 0<=t<=T
82. The multiple access communication scheme in which each user is allocated the full available channel spectrum for a specified duration of time is known as
a) CDMA b) FDMA c) TDMA d) MC-CDMA
83. GSM system uses TDMA with
a) 32 users per channel b) 16 users per channel c) 8 users per channel d) 4 users per channel
84. If Rx(τ) is the auto-correlation function of a zero-mean wide-sense stationary random process X, then which one of the following is NOT true?
a) Rx(τ) = Rx(-τ) b) Rx(τ) = -Rx(-τ) c) σx^2 = Rx(0) d) |Rx(τ)| <=Rx(0)
85. If E denotes the expectation operator, then E[X-EX]^3 of a random variable X is
a) EX^3 – E^3X b) EX^3 + 2E^3X – 3EX Ex^2 c) 3EX^3 – E^3X d) 2EX^3 + E^3X – 3EX EX^2
86. A discrete memory less source produces symbols m1,m2,m3 and m4 with probabilities 1/2, 1/4 , 1/8 and 1/8, respectively. The entropy of the source is
a) ¼ b) 1 c) 7/4 d) 2
87. A channel has a signal-to-noise ratio of 63 and bandwidth of 1200 Hz. The maximum data rate that can be sent through the channel with arbitrary low probability of error is
a) 600 bps b) 1200 bps c) 4800 bps d) 7200 bps
88. For the vectors A = X ax + Y ay and B = Z az, del . (A X B) is
a) 0 b) 1 c) XZ d) YZ
89. Which one of the following relations represents Strokes’ theorem (symbols have their usual meaning)?
a) ∫s del X A.ds = 0 b) ∫L A.dl = ∫s del X A.ds c) ∫s A X dS = -∫v (del X A)dv d) ∫v del.Adv = ∫s A.ds
90. Which one of the following relations is not correct (symbols have their usual meaning)?
a) del X E = - ∂B/∂t b) del X H = J + ∂E/∂t c) del.D = ρv d) del.B = 0
91. The electric field component of a uniform plane wave propagating in a loss less magnetic dielectric medium is given by E(t,z)=ax 5cos(10^9 t – 20/3 z)V/m. If η0 represents the intrinsic
impedance of the free space, the corresponding magnetic field component is given by
a) H(t,z)= ay 5/2 η0 cos(10^9t – 20/3 z)A/m
b) H(t,z)= ay 10/ η0 cos(10^9t – 20/3 z)A/m
c) H(t,z)= az 5/2 η0 cos(10^9t – 20/3 z)A/m
d) H(t,z)= az 10/ η0 cos(10^9t – 20/3 z)A/m
92. The skin depth of a non-magnetic conducting material at 100 MHz is 0.15 mm. The distance which a plane wave of frequency 10 GHz travels in this material before its amplitude reduces by a
factor of e^-1 is
a) 0.0015 mm b) 0.015 mm c) 0.15 mm d) 1.5 mm
93. A loss less transmission line has a characteristic impedance of 100 ohms and an inductance per unit length of 1 μH/m. If the line is operated at 1 GHz, the propagation constant β is
a) 2π rad/m b) 20π/3 rad/m c) 20π rad/m d) 2π *10^5 rad/m
94. When a load resistance Rl is connected to a loss less transmission line of characteristic impedance 75 ohms, it results in a VSWR of 2. The load resistance is
a) 100 ohms b) 75√2 ohms c) 120 ohms d) 150 ohms
95. A two-port network characterized by the S-parameter matrix, [S] = [0.3 L0 0.9 L90
0.9 L90 0.2 L0] Is
a) both reciprocal and loss less
b) reciprocal, but not loss less
c) loss less, but not reciprocal
d) neither reciprocal nor loss less
96. A loss less air filled rectangular waveguide has internal dimensions of a cm * b cm. If a=2b and the cutoff frequency of the TE02 mode is 12 GHz, the cutoff frequency of the dominant mode is
a) 1 GHz b) 3 GHz c) 6 GHz d) 9 GHz
97. A Hertzian dipole antenna is placed at the origin of a coordinate system and it is oriented along z-axis. In which one of the following planes the radiation pattern of the antenna has a
circular shape?
a) x=0 b) y=0 c) z=0 d) ø=45
98. Which one of the following statements is not true?
a) Antenna losses are taken into account in calculating its power gain
b) For an antenna which does not dissipate any power, the directive gain and the power gain are equal
c) Directivity of an antenna is the maximum value of its directive gain
d) The directive gain of a Hertzian dipole is same in all direction
99. The directivity of a half dipole antenna is
a) 1.0 b) 1.5 c) 1.64 d) 2
100. Which one of the following is not true for a step index optical fibre?
a) It can support multiple modes
b) HE11 mode is its lowest order mode
c) The refractive index of the cladding is higher than that of the core
d) At a given wavelength, single mode operation is possible by proper choice of core diameter, core and cladding refractive indices.
GENERAL ABILITY TEST:
101. Sarnath is situated in the state of
a) MP b) Bihar c) Punjab d) UP
102. Green house effect is due to the increase of atmospheric
a) CO2 level b) SO2 level c) CO level d) N2 level
103. In the month of July, it is winter in
a) New York b) Beijing c) Sydney d) London
104. The chairman of the Planning commission of India is
a) The prime minister b) The vice-president c) The union finance minister d) The union commerce minister
105. The satellite launch vehicle that placed a number of satellites ito orbit in May 2008 is
a) PSLV-C7 b) PSLV-C8 c) PSLV-C9 d) PSLV-C10
106.DRDO was formed in
a) 1947 b) 1950 c) 1954 d) 1958
107. SAMYUKTA is developed for the use of
a) Navy b) Army c) Air force d) RAC
108. DARL 202 is a variety of
a) pea b) garlic c) capsicum d) tomato
109. TRISHUL is
a) a surface to surface battlefield missile
b) a quick reaction surface to air missile
c) an intermediate range ballistic missile
d) a supersonic cruise missile
110. HUMSA is a
a) sonar b) tank c) mine d) night vision device
111. The value of 1+2i / 3-4i + 2-I / 5i , where i^2 is -1, is
a) -5/2 b) 5/2 c) 2/5 d) -2/5
112. The particular solution of the differential equation d^2y/dx^2 + 2 dy/dx + 5y = 0 satisfying the conditions y(0)=0 and y’(0)=1 is
a) y=1/2 e^-x cos2x b) y=1/2 e^-x sin4x c) y=1/2 e^-x sin2x d) y=1/2 e^-x cos4x
113. For the vectors A=3i-2j+k and B=2i-k, the value of (A*B).A is
a) 0 b) 1 c) 2 d) 3
114. The orthogonal trajectory of the family of curves x^2-y^2 = a (where a is a constant) and passing through the point (1,1) is
a) y=-1/x b) y=1/x c) y=-x d) y=x
115. The value of the line integral ∫ y^2 dx + 2xydy over the curve x=accost, y=asint is
a) 0 b) 1 c) 2 d) 4
116. The n-th partial sum of the infinite series 1/1*2 + 1/2*3 + 1/3*4+……1/n*(n+1)……..
a) 1/n+1 b) n+2/n+1 c) n/n+1 d) n-1/n+1
117. The complex-valued function f(z)=e^z is analytic for
a) no z b) all z c) real z only d) imaginary z only
118. The inverse of the matrix [ cos A sin A
-sin A cos A] is
a) [ -cos A Sin A b) [cos A sin A} c) [cos A -sin A d) [cos A -sin A
sin A cos A] sin A -cos A] -sin A cos A] sin A cos A]
119. Consider the function f(x) defined as
F(x) = 3x-1, x<0
0, x=0
2x+5, x>0
In the following table, List I shows 4 expressions for limits of f(x) and List II indicates the values of the limits
List I List II
P.Lim x->2 f(x) 1. -1
Q.Lim x->0+ f(x) 2. 9
R.Lim x->0- f(x) 3. -10
S.Lim x->-3 f(x) 4. 5
The correct matches are
a) P-2,Q-4,R-1,S-3 B) P-2,Q-4,R-3,S-1 C) P-4,Q-2,R-1,S-3 D) P-4,Q-2,R-3,S-1
120. Two events A and B with probability 0.5 and 0.7, respectively, have joint probability of 0.4. The probability that neither A nor B happens is
a) 0.2 b) 0.4 c) 0.6 d) 0.8
121. Consider the differential equation
X^2 d^2/dx^2 + x dy/dx + (x^2 - 4)y = 0. The statement which is not true for it is
a) It is a linear second order ordinary differential equation
b) It can not be reduced to a differential equation with constant coefficients
c) X=0 is a regular singular point
d) It is a non-homogeneous second order ordinary differential equation
122. The sum of two numbers is 16 and the sum of their squares is a minimum. The two numbers are
a) 10,6 b) 9,7 c) 8,8 d) 5,11
123. The value of the definite integral 0∫(π/2)^(1/3) x^2 sin(x^3)dx is
a) -1/3 b) 0 c) 1 d) 1/3
124. A circle C2 is concentric with the circle C1 : x^2 + y^2 -4x +6y -12 =0 and has a radius twice that of C1. The equation of the circle C2 is
a) x^2 + y^2 -4x +6y -13 =0 b) x^2 + y^2 -4x +6y -87 =0 c) x^2 + y^2 -4x +6y -100 =0 d) x^2 + y^2 -4x +6y -88 =0
125. Consider the quadratic equation x^2 + px + q =0. If p and q are roots of the equation, the values of p and q are
a) p=0, q=0 only b) p=1, q=-2 only c) p=0, q=0 and p=1, q=-2 d) p=0, q=0 and p=-2, q=1
126. Consider the list of words: etiquette, accommodate, forty, exaggerate, continous, independent, receipt. The number of misspelt words are
a) 1 b) 2 c) 3 d) 4
127. Consider the following sentences
1. A few friends he has are all very rich.
2. Do not insult the weak.
3. The later of the two persons was more interesting.
4. All the informations were correct.
Out of these sentences, the grammatically correct sentence is
a) 1 b) 2 c) 3 d) 4
128. The appropriate auxiliary verb to fill in the blank of the sentence “Gandhi knew that he __ soon be jailed.”is
a) would b) will c) shall d) may
129. The number of missing punctuation marks in the sentence “Rajesh along with Amit went to the market.”is
a) 0 b) 1 c) 2 d) 3
130. The meaning of the word PLAGIARISM is
a) theft of public money b) theft of ideas c) belief in one god d) belief in many gods
132. ACROPHOBIA is the abnormal fear of
a) open spaces b) height c) fire d) water
133. The appropriate pair of prepositions to fill in the blank in the sentence “He was angry __ me, because my remarks were aimed __ him.”is
a) at,to b) with, at c) with, to d) at, for
134. The appropriate word(s) to fill up the blank in the sentence “ I remember __ voices in the middle of the night.”is (are)
a) hear b) to hear c) hearing d) heard
135. The passive voice form of the sentence “I have known him for a long time.”is
a) He is known to me for a long time.
b) He is known by me for a long time.
c) He has been known to me for a long time.
d) He has been known by me for a long time.
136. If kennel is to a dog, then __ is to a hen.
a) nest b) coop c) hole d) stable
137. If NATION is to 5236765, then NOTION is to
a) 573675 b) 563765 c) 576375 d) 557365
138. The next two numbers of the series 3,5,11,21 are
a) 34 and 52 b) 34 and 53 c) 35 and 52 d) 35 and 53
139. A, B and C are three places in India with longitudes 80E, 85 E and 90 E respectively. Which one of the following statements about the local times of the places is true?
a) Local time of C is ahead of that of B.
b) Local time of B is ahead of that of C.
c) Local time of A is ahead of that of C.
d) A, B and C all have the same local time.
140. In this question, notations +, / and * are used as follows
A + B means A is the husband of B.
A / B means A is the sister of B.
A * B means A is the son of B.
With these relations, the relationship
denoted by P / Q * R is
a) P is son of R
b) P is daughter of R
c) P is uncle of R
d) P is father of R
141. If DELHI is written as EDHIL, then PARIS is written as
a) APRIS b) SARIP c) SAPIR d) APISR
142. The number of prime numbers between 10 and 50 is
a) 10 b) 11 c) 12 d) 13
143. The odd one in the list : LAN, TCP/IP, HACKER and KILLER is
a) LAN b) TCP/IP c) KILLER d) HACKER
144. SAW is to carpenter as SCALPEL is to
a) surgeon b) mason c) plumber d) tailor
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Richard E. Bellman Control Heritage Award
Award/Recognition Menu
The Richard E. Bellman Control Heritage Award is given for distinguished career contributions to the theory or application of automatic control. It is the highest recognition of professional
achievement for US control systems engineers and scientists.
The awardee is expected to make a short acceptance speech at the AACC Awards Ceremonies during the ACC.
The recipient must have spent a significant part of his/her career in the USA.
Nomination Details
Form Instructions:
The following information will be necessary in order to complete the nomination application form:
• All fields marked by a red asterisk (*) is required.
• Three letters of support are required with option to submit no more than 5. You will need to provide the Supporter's Name, Affiliation, Email address and upload the letter of support.
• You may save the application at anytime during the process. You will receive an email on how to access your draft to update the nomination application.
• Upon submission, a confirmation email will be sent for to you for your reference.
Nomination Deadline
December 1
It is a great honor to receive the Richard E. Bellman Control Heritage Award. I am delighted to share this recognition with my students, postdocs, collaborators, and colleagues. I thank them and my
teachers, mentors, and family for inspiring me, for encouraging me, for their generosity, and for their contagious joy in the exchange of ideas. They have made and continue to make my intellectual
and professional journey wonderful and significant.
I also feel fortunate to have made my intellectual home in the controls field with you as my community. The controls field provides us a robust foundation for the mathematically inclined and
intellectually curious, a vibrant community for intellectual exchange and life-long learning, a powerful set of tools for asking new questions and innovating new ways of thinking and doing, and a
point of departure for making connections and contributions to fascinating, important and challenging questions in engineering and technology as well as in the natural and social sciences and even in
the arts.
As a graduate student I learned control theory and I was exposed to creative research, which gave me the opportunity to develop my own taste for research problems. I learned how to ask questions that
are important and challenging but also tractable. I learned how to address concrete problems and how to abstract out a general story that could have further impact and connect in new creative ways
across disciplines. I learned about the thrill of discovering and making good on the ubiquity and utility of feedback in nature and engineering.
Almost 25 years ago, as part of a project led by Steve Morse, I started to work on questions on how and why fish move in schools, in collaboration with biologists with expertise in animal behavior. I
found it incredibly fascinating, and I remember worrying to Krishnaprasad, my PhD advisor, that I may have made a mistake in not studying ecology and evolutionary biology in graduate school. He very
quickly assured me that I could still study these fields, not as a graduate student but as a control theorist and that I was well prepared given how central feedback dynamics are to understanding
collective animal behavior.
And I continue to explore and exploit(!) new opportunities that the control field affords through my own work and collaborations and through exposure to the creative work of our community.
There is much that distinguishes our field, but I would point to three important features that make the controls field uniquely relevant and important to the challenges of the day:
1. The ubiquity and utility of feedback. Feedback is everywhere in technology and designed systems. Feedback is also everywhere in our bodies, and across the natural world. Feedback determines
stability, adaptability, speed, accuracy, robustness, resilience and more.
2. Control-theoretic abstractions. Our abstractions provide the means to new fundamental understanding and approaches to principled design, the means to generalize from one concrete problem and
specialize to another, and the means to translate insights, models, and approaches across seemingly different domains.
3. Tractability and rigor. In control we insist on tractability and rigor because with them we can provide guarantees, we can identify limitations and how much room there is for improvement, and we
can make systematic our insights, algorithms and design methodologies.
These three features – ubiquity and utility of feedback, control-theoretic abstractions, and our insistence on tractability and rigor – together make our field relevant to so many others, to fields
investigating both technological and natural systems.
They give us the freedom to choose topics and questions that we find fascinating and rich. And they give us opportunity to make contributions, through both theory and application, towards a wide
range of challenges. We as a community are already having an impact on addressing a vast array of highly topical problems, including security and safety of technology, robotics and autonomous
vehicles, transportation, human-machine interaction, machine learning, energy and the environment, biological systems from cells to populations, biomedical innovations, neuroscience and neuromorphic
engineering, social networks and more.
But there is more to do as a community to improve on the relevance and impact of our work.
1. We can and should make our work better accessible to broader audiences. There is significant interest in what we do and a real need for collaborators in other fields to appreciate the value of
what we do. But people outside our field often find our papers impenetrable. We need to change that.
2. We can and should embrace the questions our collaborators in other fields are asking and invest in understanding their fields. Collaborations are most successful when everyone learns something
new. More than just finding an application for a tool we’ve developed, we can and should dig in and explore how we can make a real difference. It’s amazing how easy it is to get engaged in the
questions being asked in other fields. And it means that we aren’t merely bringing our perspective to questions in another field but also bringing something fundamental back to our community.
3. We can and should remain humble. Control theory has much to offer but we need to remember that we have much to learn. Humility is key in making connections across disciplinary divides.
4. We can and should pro-actively promote and welcome a more diverse community. A more diverse community means a richer diversity of expertise, a richer diversity of questions and challenges that
drive us, a richer diversity of ways of thinking and approaching research, and a richer diversity of life experience and perspectives.
I am proud to be the second woman to receive this award, and I acknowledge all the women in control who came before me and paved the way for the opportunities that I have had. Thank you to Bozenna
Pasik-Duncan, Linda Bushnell, Anu Annaswamy, Dawn Tilbury, and others who have been tireless advocates for women in control.
We have much to do and it is an exciting time for the controls field. I look forward to joining you in next adventures. Thank you very much for this great honor.
For fundamental contributions to geometric control theory, networked multiagent systems, and for bridging control theory with ecological systems, neuroscience, and the arts.
First of all: I'm very sorry to not be there in person. Over the last few years I've really missed my control colleagues from around the world.
I'm very honored and grateful to receive the Richard E. Bellman award. Bellman is one of the most frequently occurring names in my papers and books. He's right behind Gauss and Lyapunov, so he's in
very good company. Bellman is also one of my heros, for formulating sequential decision making, also known as control, in such a clear way, as well as developing an elegant solution method that is
often practical.
On the other hand, it's a bit scary to receive an award with the word 'heritage' in it. Is there a hidden message there? Like, maybe it's time for me to retire?
I've outlined my trajectory from pure math to EECS and control before, so I'll skip right to the many thanks I need to give.
My first thanks are to Charlie Desoer, Leon Chua, Shankar Sastry, Pravin Varaiya, and Alberto Sangiovanni. They are why I transferred from Math to EECS at Berkeley more than 40 years ago. I've never
regretted that move. At the same time, I'm very happy that my initial training was in pure math, with Andy Gleason.
I thank all of my former and current PhD students, post-docs, research visitors, and research collaborators and co-authors. 'Thanks' really isn't the right word; these are the people who did the
work. I'm proud of what we've done and created. *I* had a lot of fun, and I suspect my research collaborators mostly did too. (Maybe not when we were going over the 10th revision of a paper.) Special
thanks to my collaborator Lieven Vandenberghe, with whom I've written two books, with another one in the oven.
I want to thank the students in my classes. My classes are big, and I've been teaching for almost 40 years now, so there are a lot of them. The students come from a wide range of fields and
backgrounds. They were the training and test sets for explaining ideas. And they suggested plenty of new and interesting ideas. I have learned a lot from them.
I am very grateful to my colleagues in EE at Stanford. I really like my colleagues. And that's after serving as department chair. (You have to have been chair to fully understand that.) They are
awesome, and I'm proud to be their colleague.
I are grateful for my colleagues in control and optimization
across the world. Control is a great field, which I fell into at Berkeley during my PhD, under the influence of Charlie Desoer and Shankar Sastry. Control is my intellectual home, it’s where I grew
up, and where I’ve made a huge number of very good life-long friends. I’m really proud to be a part of this community.
Thanks also to my optimization colleagues, especially Arkadi Nemirovsky and Yuri Nesterov. And super special thanks to Boris Polyak, who was, like Bellman, a hero of mine.
I'm also very grateful to have spent time in industry, in multiple areas. I've learned an enormous amount from my industry friends, especially how someone with industry knowledge and experience,
along with some good street-fighting skills, can do very well in a practical setting.
And finally, family. It has been a fantastic adventure, and I could not be more grateful to have shared it with Anna, my wife of more than 35 years, and our two great 'kids' Nick and Nora, now in
their early 30s.
Thank you all again.
For pioneering contributions and sustained leadership in the development and application of advanced optimization algorithms.
Richard Bellman was a paragon of deep foundational thinking and interdisciplinary work, so I am deeply grateful to receive an award that honors him. It is especially meaningful that the prize is
awarded by the Automatic Control Council, which brings together such disparate areas of applications in engineering, mathematics, and the sciences.
Exactly 50 years ago, as an undergraduate in Buenos Aires looking for a senior project, I discovered the work of Rudolf Kalman, which sparked a stable and robust attraction to control theory that
continues to this day. One of my professors met Kalman at a conference, which led to Kalman inviting me to be his student. Kalman’s rigorous mathematical approach inspired research excellence, deep
thinking, and clear exposition. The 1970s witnessed an explosion of new and exciting ideas in systems theory, and many of the leaders in the field visited Kalman's Center. I was extremely lucky to
have the opportunity to learn from all of them.
After my PhD, I went to Rutgers, where I was fortunate to collaborate with Hector Sussmann, and to learn so much from him.
Five years ago, I was recruited by Northeastern University, where I have fantastic colleagues, especially Mario Sznaier and Bahram Shafai.
Of course, I am grateful to all who influenced my work, too many to credit here, and to those who applied, enriched, and extended my initial ideas. At the risk of sounding presumptuous, let me share
some thoughts about research in systems and control theory. First, it is important to formulate questions that are mathematically elegant and general. Paradoxically, general facts are often easier to
prove than special ones, because they are stripped of irrelevant details. Second, we should strive to simplify arguments to their most elementary form. It is the simplest ideas, those that look
obvious in retrospect, that are the most influential, as Bellman’s dynamic programming so beautifully illustrates. Third, we should be aware of the essential connection between theory and
applications. Applications provide the inspiration for an eventual conceptual synthesis. Conversely, theory is strengthened and refined by working out particular cases and applications. Fourth, one
should be cautiously open to new ideas, even those orthogonal to current fashion. But not all new ideas are good: novelty by itself is not enough. Finally, we should not lose sight of the fact that,
while fun and intellectually challenging, our ultimate objective is to improve the world through scientific and engineering advances.
Which brings me back to Richard Bellman’s heritage, which we honor today. Years after his foundational work on optimality, Bellman turned to biology and medicine, even starting a mathematical biology
journal. I am sure that the mechanistic understanding of behavior at all scales, from cells to organisms will lead to control and elimination of disease and the extension of healthy lifespans. I find
immunology and its connections to infectious diseases and cancer to be a fascinating field for systems thinking. In addition, the associated engineering field of synthetic biology will lead to new
therapeutic approaches as well as scientific understanding, and new mathematics and control problems suggest themselves all the time. In my view, the main value of systems and control to molecular
biology will not be in applying deep theoretical results. Instead, conceptual ideas like controls, measurements, robustness, optimization, and estimation are where the main impact of our field will
be felt.
Thank you so much.
June 9, 2022
Atlanta, GA USA
ACC 2022
For pioneering contributions to stability analysis and nonlinear control, and for advancing the control theoretic foundations of systems biology
Dear Automatic Control colleagues,
I am happy and humbled to receive the Bellman Award.
My profound gratitude goes to the colleagues who supported my nomination. I am thankful and deeply moved by the selection committee and the A2C2, which advanced a candidate in his mid-fifties, an
adolescent by Bellman award standards.
The timing of this award, which recognizes the achievement of an American control systems researcher, carries significance for me. The Bellman award came in the year that happened to be the thirtieth
anniversary of my coming to the United States as a graduate student.
It is customary on this occasion for the recipient to say a few words about their formative years and professional trajectory.
I was born and grew up in a small city called Pirot, in remote southeastern Serbia. I was fortunate that my provincial city had one of the top science high schools in former Yugoslavia. And my caring
parents spared no expense to provide my brother and me with broader cultural opportunities than those that our hometown could offer.
My undergraduate years at the Department of Electrical Engineering of the University of Belgrade provided me with two things. First, the toughest academic competition I’ve experienced, before or
since, was during those five undergraduate years. And, second, I met my future wife in our freshman math class.
Before Petar Kokotovic gave me a PhD opportunity, I had only an inkling that I might have a shot at some success in research. But, within a few weeks of arriving in Santa Barbara, I had the fortune
of solving a problem that had a reputation of being unsolvable, though I didn’t know that. So things moved quickly with research from that point on, and I had Petar’s unlimited attention. I could
fill hours on being mentored by Petar. But let me just say that, during those Santa Barbara years, Petar’s enthusiasm and support for my work left me feeling that there was nothing more important
happening in the world than what I was doing in research. At the same time, with everything I would produce or say, I had the training benefit of a keener, more unforgiving, and yet more nuanced
critique than I would ever subsequently encounter, as a researcher or academic administrator.
Of the areas credited to me, the ones that probably come to mind first are PDE backstepping and extremum seeking. Let me describe how these interests started, soon after I left Santa Barbara.
Petar Kokotovic, Richard Murray, and Art Krener had a large project on controlling flow instabilities in jet engines. We solved those problems using reduced-order nonlinear ODE models of those flows.
And it was clear that, for a nonlinear control researcher, there was hardly a more fertile ground than fluids. The only problem was: who would provide an ODE reduction for me for the next control
design problem I tackle? If fluids people spend their entire careers refining, for a specific type of flow, the reductions from the Navier-Stokes representation to ODEs, it was obvious I could not
count on them for control-oriented reduced models. I had to roll up my sleeves and build control methods directly for PDEs. From scratch. Because Riccati equations—in infinite dimension to boot—are
not the way to extend PDE control to the nonlinear case. The answer to the challenge of constructive PDE control came in the form of continuum backstepping transformations, employing Volterra
operators and easy-to-solve Goursat-form PDEs for the control gain functions. If you have interest in an example of this line of PDE control research, I recommend the paper with Coron, Bastin, and my
student Vazquez, which has enabled stabilization of traffic flows in a congested, stop-and-go regime.
How I got drawn to extremum seeking is also interesting. In 1997, a combustion colleague at Maryland pointed me to publications from the 1940s and 1950s on what I would describe as an approach to
adaptive control for nonlinear systems. Heuristic, but orders of magnitude simpler than what I had written my PhD on. Attempts at sleep were futile, for several days, until I figured out how to prove
stability of this algorithm, using a combination of averaging and singular perturbation theorems. If you wanted to sample one control paper from the last quarter century on extremum seeking, I
recommend the one on model-free seeking of Nash equilibria with Tamer Basar and my student Paul Frihauf.
To my students and collaborators, I would like to say: this Bellman award is yours. For your papers, books, theorems, and industrial products.
As I mention students, I want to extend gratitude to two companies that have been the environments in which my former students have been able to thrive and leave a legacy. At ASML, control of extreme
ultraviolet photolithography has improved the density of microchips by 2-3 orders of magnitude. At General Atomics, control of aircraft arrestment on carriers has enabled one of the most impressive
and deployed recent advances in defense technology.
I won’t pretend that it is not a delight to see my name in the list of the 44 recipients of the Bellman award. Scholars of incredible depth and engineers of stunning impact. I’ve studied the list.
Amazingly, the numbers of American-born and foreign-born recipients of this US award seem to be the same: 22 each. If you sought an example of how the US is unequaled in extending opportunity to
scientific immigrants, like myself, you could hardly find a clearer illustration.
It was also impossible for me to miss in the list that, after India, represented by four Bellman awardees, the second most highly represented foreign country is a certain little country, just a few
percent more populous than the city of Atlanta, the country from which Petar Kokotovic, Drago Šiljak, and I came to the US. If I don’t mention this, in the hope of inspiring a few young minds at the
Universities of Belgrade, Novi Sad, or Niš, who should?
I couldn’t have made it here without role models and without pioneers who charted the pathways along which it was then not that hard for me to walk. Among them are people who have also generously
supported me over the years: Tamer Basar, Manfred Morari, Art Krener, Eduardo Sontag, Masayoshi Tomizuka, Galip Ulsoy, Jason Speyer, Graham Goodwin, Jean-Michel Coron, Petros Ioannou—to limit myself
to ten. I hope that, in the remainder of my research career, I more fully deserve their support, as well as by other friends I don’t mention here but who are aware of the extent of my gratitude and
Let me close and thank you with a quote from my former department chair who astutely observed: “To you guys, in control systems, every other field is a special case of control theory.” What if that’s
June 7, 2022
Atlanta, GA USA
ACC 2022
For transformational contributions in PDE control, nonlinear delay systems, extremum seeking, adaptive control, stochastic nonlinear stabilization and their industrial applications
To receive the Richard E. Bellman Control Heritage Award is truly an honor. I am thankful first to all of you for attending today after two postponements of these ceremonies due to the pandemic. I am
grateful to the honors committee for selecting me, and to my nominator and references for their willingness to put forth and support my nomination.
The Bellman Award is given for “distinguished career contributions to the theory or application of automatic control.” My career in control started as a junior at Swarthmore College in 1972 when I
took a course based on the textbook Dynamics of Physical Systems by Robert Cannon. That course really challenged me, and I found myself putting in a lot of time and energy just to get by. That
investment sparked my interest, and so as a master's student at Cornell University I worked with Dick Phelan and learned the practical and experimental side of automatic control in the laboratory
using analog computers. In 1975 I decided to pursue control engineering for my Ph.D. work and Prof. Phelan said, in mechanical engineering at that time, there were really only two choices: MIT or UC
Berkeley. So I wound up at UC Berkeley where I learned controls from Yasundo Takahashi, Masayoshi Tomizuka (Tomi is also a Bellman Award recipient), and Dave Auslander. I not only learned the latest
in control theory from the book Control and Dynamic Systems by Takahashi, Rabins and Auslander, but did my first experiments using digital controllers. My doctoral advisor and professional role
model, Dan Mote, is a dynamicist, and my research was on reducing sawdust by controlling vibrations of bandsaw blades during cutting and included theory, computation and experiment.
When I started as an Assistant Professor at the University of Michigan in 1980, I had the great fortune to have two very special mentors. The late Elmer Gilbert (another Bellman Award recipient) came
to my office to welcome me, to offer his help with the new graduate course I was developing, and to invite me to participate in a College of Engineering control seminar – a regular Friday afternoon
seminar which I still continue to attend! The other was my longtime friend and collaborator Yoram Koren, together with whom I conducted many joint research projects, and from whom I learned much of
what I know about control of manufacturing systems. Yoram and I had the first digital control computer, a PDP-11, at UM in our laboratory. Michigan was, and is, a wonderful place for control
engineering. I had the good fortune to work with, not only Elmer and Yoram, but many outstanding collaborators: Joe Whitesell, the late Pierre Kabamba, Panos Papalambros, Dawn Tilbury, Huei Peng,
Ilya Kolmanovsky, Harris McClamroch, Jeff Stein, Gabor Orosz, Chinedum Okwudire and many others! I worked on topics such as automotive belt dynamics, adaptive control of milling, reconfigurable
manufacturing, vehicle lane-keeping, co-design of an artifact and its controller, time delay systems, and I was always richer for the experience. Throughout my professional career I worked
extensively with industry, especially the Ford Motor Company, where I collaborated with and learned from excellent engineers like Davor Hrovat and Siva Shivashankar (automotive control), Charles Wu
(control of drilling), and Mahmoud Demeri (stamping control).
I would like to recognize my wife, Sue Glowski, who is here today, for her love and support. She was educated in English and Linguistics but is always willing to patiently listen to my latest idea
about control, even if she has to eventually ask: "what the hell is an eigenvalue?"
Finally, and most importantly, I want to recognize and thank my students and postdocs. This award recognizes your great ideas, and your fine work, and I am delighted to be here today to accept it on
your behalf. Thank you!
June 7, 2022
Atlanta, GA USA
ACC 2022
For seminal research contributions with industrial impact in the dynamics and control of mechanical systems especially manufacturing systems and automotive systems
Dear President Braatz, colleagues, students and friends.
I am very grateful and indeed humbled by being honored to receive the Richard E. Bellman Control Heritage Award for 2019 and to join the distinguished list of prior recipients. I wish to express my
sincerest thanks to those who nominated me and supported my nomination and to the awards committee. I am deeply moved by the honor I receive today.
More as a rule than an exception, such an honor is not a credit to a single individual but rather the result of collective work and many collaborations over the years. This is particularly true in
areas which are by nature interdisciplinary. And control theory, as such, is one of these. It offers an excellent example of synergy where purely theoretical questions, mathematical in nature, are
prompted and stimulated by technological advances and engineering design.
I was attracted to mathematical control theory from my early days at the University of Warsaw, where I was privileged to join a distinct and (at that time) experimental program, called Studies in
Applied Mathematics. This was an interdisciplinary initiative under the collaboration of a few home departments. After graduating with a Master Degree, I was fortunate to receive a doctoral
fellowship which allowed me to complete my PhD in Applied Mathematics-Control Theory within 3 years, with a thesis on a problem of non-smooth optimization, which extended Milutin-Dubovitski's work
and had applications to control systems with delays.
I am extremely grateful to my mentors of that time: Professors A. Wierzbicki, A. Manitius from Control Theory [the latter now chair at George Mason University], the late Professor S. Rolewicz and
Professor K. Malanowski both from the Polish Academy of Sciences. They, along with other colleagues, gave me an opportunity to embrace a large spectrum of the field of control theory, to include
functional analysis, abstract optimization, differential equations.
My further education took a critical turn at UCLA in Los Angeles, which I joined in 1978, at the invitation of the late Professor A.V. Balakrishnan, the 2001 recipient of the Bellman's award. Bal for
all of us. Here, under his mentorship, I was offered the challenge to get involved in the mathematical area of boundary control theory for Distributed Parameter Systems, still at its infancy at that
time, even from the viewpoint of Partial Differential Equations, with many basic mathematical problems still open. That was about the time when Richard Bellman's book on Dynamic Programming appeared,
in 1977, rooted on Bellman's equation and the Optimality Principle. I always looked at Bellman as a problem-solving mathematician, and the mathematical theory of boundary control of DPS is in line
with this philosophy.
Controlling or observing an evolution equation from a restricted set [such as the boundary of a multi-dimensional bounded domain where the controlled system evolves] is both a mathematical challenge
and a technological necessity within the realm of practical and physically implementable control theory. Most often, the interior of the domain is not accessible to external manipulations. One first
goal of the time within the DPS control community was to construct an appropriate control theory, inspired also by the late R. Kalman, the 1997 recipient of the Bellman's award. Main initial
contributors were J.L. Lions, A. Bensoussan and their influential school in Paris, and A.V. Balakrishnan and his associates. But DPS come in a large variety. It requires that each distinct class
(parabolic, hyperbolic, etc.) be studied on its own with properties and methods pertinent to it, which however fail for other classes. The systematic study of boundary control, which leads to
distributional calculus for various distinct classes of physically significant DPS, became the first long-range object of my research. Both, the results and the methods are dynamics dependent. Finite
or infinite speed of propagation becomes an essential feature in controllability theory. For instance, the wave equation is boundary exactly controllable on a sufficiently large time, while the heat
equation is only null-controllable yet on an arbitrary short time. Existence, uniqueness and robustness of solutions to nonlinear dynamics were just the first questions asked but still open within
the existing PDE culture.
Topics investigated over the years included: optimal control, Riccati and H-J-Bellman theory and their numerical implementation, appropriate controllability and stabilization notions, all in the
framework of boundary control of partially observed systems. This research effort, which continues to this very day, was conducted with collaborators and PhD students. It started with my association
with A.V. Balakrishnan at UCLA, J.L. Lions at College de France and R. Kalman during my 7 years at the University of Florida. And it continued during my subsequent 26 years at the University of
Virginia, the home of MacShane, and now at the University of Memphis. In both cases with talented PhD students. Some of these occupy now distinguished positions in the US academia.
Once the control theory of single distinct DPS classes became mature, engineering applications motivated the need to move on toward the study of more complex DPS consisting of interactive structures
where different types of dynamics coupled at an interface define a given control system. Propagation of control properties through the interface then plays a main role.
Thus, in its second phase, my research in DPS then evolved toward these coupled interactive systems of several PDEs. Applications include large flexible structures, structural acoustic interaction,
fluid-structure interaction, attenuation of turbulence in fluid dynamics [Navier Stokes] and flutter suppression in nonlinear aero-elasticity. In the latter area, my collaboration with Earl Dowell
[Duke Univ.] was most enlightening, and is a further proof of the interdisciplinary nature of the field. These problems, while deeply rooted in engineering control technology, were also benchmark
models at the forefront of developing a PDE-based mathematical control theory, which accounts for the infinite dimensional nature of continuum mechanics and fluid dynamics.
In closing, I would like to acknowledge with gratitude my personal and professional interaction over the years with people such as the late David Russell [VPI], Walter Littmann [U of Minnesota],
Giuseppe Da Prato [Scuola Normale, Pisa], Michel Delfour [Univ. of Montreal] and Sanjoy Mitter [MIT], the latter the 2007 recipient of the Bellman award. Their pioneering works paved the way to
further developments along a road-map which I am proud to be a part of.
Special thanks to my long-time collaborator and husband Roberto Triggiani, to the late Igor Chueshov [both co-authors of major research monographs, two with Roberto in Cambridge University Press and
one with Igor in Monograph Series of Springer], as well as to my former students, now collaborators and colleagues.
Many thanks also to funding agencies such as NSF, AFOSR, ARO and NASA for many years of generous support.
July 11, 2019.
For contributions to boundary control of distributed parameter systems
Dear President Braatz, colleagues, students, ladies and gentlemen:
I feel tremendously honored to receive the Richard Bellman Control Heritage Award. Thank you to those who nominated me and supported my nomination, to the selection committee, and to the AACC Board
for making me this year’s recipient.
I completed my undergraduate studies at Keio University in Japan and my graduate studies at MIT. Following my education at these wonderful institutions, I was able to join the excellent academic
environment at the University of California, Berkeley. I am grateful to my teachers and colleagues at these institutions. I thank in particular my PhD advisor Dan Whitney and my early control
colleagues at Berkeley, Yasundo Takahashi and David Aulander, and the many bright graduate students that I have had the privilege of having in my lab at Berkeley who are approximately 120 PhDs strong
now. I thank the National Science Foundation and other government sponsors as well as industrial sponsors for providing me resources to maintain the Mechanical Systems Control laboratory, which is
the home of my research group. Last but not least, I thank my wife Miwako for supporting me and our family, permitting me to concentrate on academics and schoolwork for many years, starting almost 50
years ago in my MIT days.
I jumped into the area of dynamic systems and control during my senior year at Keio University. The first book I read was Modern Control Theory by Julius Tou. The book was an excellent summary of the
State Space Control Theory, and I was fascinated by the elegant mathematical aspects of the subject. There was no internet back then of course, and major periodicals such as IEEE Transactions on
Automatic Control and ASME Journal of Basic Engineering were the best sources to find the latest developments in the field. I was frustrated by the time delay between the time of research and
publication. About at the time I completed my MS at Keio, I was fortunate to receive an admission offer from MIT. The time delay problem was naturally resolved. At MIT, I was inspired by many people
including Dan Whitney, Tom Sheridan and Hank Paynter. Sheridan’s early work on preview control was the starting point of my dissertation work on the “optimal finite preview” problem.
In September 1974, I joined the University of California as an Assistant Professor of Mechanical Engineering. It’s hard to believe, but I am now completing my 44th year at Berkeley.
At Berkeley, I have worked on many different mechanical systems. I joined UC Berkeley when the large scale integration technology was starting to make it possible to implement advanced control
algorithms by using mini and micro computers. This allowed me to emphasize both the analytical aspects of control and the laboratory work. This research style still continues now.
Robots are multivariable and nonlinear. In particular, a configuration-dependent inertial matrix and nonlinear terms are unique for robots. I convinced one of my PhD students, Roberto Horowitz (who
is now a professor and chair of the Mechanical Engineering Department at Berkeley), to work with me on model reference adaptive control as it applied to robots. Since then, robot control has remained
a major research topic in my group. Our current research emphasizes efficiency and safety in human-robot interactions and merging model-based control and machine learning to make the robot system
I worked on machining for a while. One control issue with machining is the dependence of input-output dynamics to cutting conditions and tool wear. One day, Jun-Ho Oh (who is now a professor at
KAIST), took me down to the lab to show me model reference adaptive control on a Bridgeport milling machine. It was cleverly implemented and was the first application of modern adaptive control
theory to machining.
In many mechanical systems involving rotational parts, we encounter periodic disturbances with known periods. Repetitive control is applied to this class of disturbances. I learned of it from
visitors from Japan in the mid-1980s. Tsu-Chin Tsao (who is now a professor at UCLA) and I then developed our version of repetitive control algorithms emphasizing discrete time formulation and easy
Another fundamental control problem for mechanical systems is tracking arbitrary shaped reference inputs. Feedforward control is popular in tracking, but unstable system zeros make the problem
complicated. To overcome this issue, I suggested to cancel phase shift induced by unstable zeros and introduced zero phase error tracking (ZPET) control in the late 1980s. The citation of this paper
has reached 1,600 by now.
In the mid-1980’s, UC Berkeley started Program on Advanced Transit and Highway under the sponsorship of CalTrans. Automated highway systems was a topic of interest for quite a few control professors.
Karl Hedrick and I were the primary faculty participants from ME: Karl worked on controls in the longitudinal direction and I in the lateral direction of vehicles. My first PhD student on this topic
was Huei Peng (who is now a professor at University of Michigan). During the past five years or so, autonomous vehicles have become very hot as we all know, and I now have quite a few students
working to blend control and machine-learning for applications to vehicles.
I have been fortunate to have had the opportunity to address a variety of challenging mechanical control problems over the span on my career so far. My research has been and continues to be rooted in
the mechatronic approach; namely I have worked on the synergetic integration of mechanical systems with sensing, computation, and control theory. This approach provides the opportunity for academic
research to have broad impacts on control engineering in practice, and I am honored to have had a hand in helping to advance a small part of it.
Thank you very much for this award. I am extremely grateful and honored.
ACC 2018
Milwaukee, WI USA
June 28, 2018
For seminal and pioneering contributions to the theory and practice of mechatronic systems control
Dear President Masada, colleagues, students, ladies and gentlemen:
I am deeply moved by this award and honor, and truly humbled to join a group of such stellar members of our extended systems and control community, several of whom have been my mentors, teachers and
role models throughout my career.
I am grateful to those who nominated me and supported my nomination and to the selection committee for their decision to honor my work and accomplishments.
I was fortunate through my entire life to receive the benefits of exceptional education. From special and highly selective elementary school and high school back in Greece, to the National Technical
University of Athens for my undergraduate studies and finally to Harvard University for my graduate studies. My sincere and deep appreciation for such an education goes to my parents who distilled in
me a rigorous work ethic and the ambition to excel, my teachers in Greece for the sound education and training in basic and fundamental science and engineering and to my teachers and mentors at
Harvard and MIT (Roger Brockett, Sanjoy Mitter and the late Jan Willems) and the incredibly stimulating environment in Cambridge in the early 70’s.
Many thanks are also due to my students and colleagues at the University of Maryland, in the US and around the world, and in particular in Sweden and Germany, for their collaboration, constructive
criticism and influence through the years. Several are here and I would like to sincerely thank you all very much.
I am grateful to the agencies that supported my research: NSF, ARO, ARL, ONR, NRL, AFOSR, NIST, DARPA, NASA. I am particularly grateful to NSF for the support that helped us establish the Institute
for Systems Research (ISR) at the University of Maryland in 1985, and to NASA for the support that helped us establish the Maryland Center for Hybrid Networks (HyNet) in 1992.
I would also like to thank many industry leaders and engineers for their advice, support, and collaboration during the establishment and development of both the ISR and HyNet to the renowned centers
of excellence they are today.
Most importantly I am grateful to my wife Mary, my partner, advisor and supporter, for her love and selfless support and sacrifices during my entire career.
When I came to the US in 1970 I was debating whether to pursue a career in Mathematics, Physics or Engineering. The Harvard-MIT exceptional environment allowed me freedom of choice. Thanks to Roger
Brockett I was convinced that systems and control, our field, would be the best choice as I could pursue all of the above. It has indeed proven to be a most exciting and satisfying choice. But there
were important adjustments that I had to make and lessons I learned.
I did my PhD thesis work on infinite dimensional realization theory, and worked extensively with complex variable methods, Hardy function algebras, the famous Carleson corona theorem and several
other rather esoteric math. From my early work at the Naval Research Laboratory in Electronic Warfare (the “cross-eye” system) and in urban traffic control (adaptive control of queues) I learned, the
hard way, the difficulty and critical importance of building appropriate models and turning initially amorphous problems to models amenable to systems and control thinking and methods. I learned the
importance of judiciously blending data-based and model-based techniques.
In the seventies, I took a successful excursion into detection, estimation and filtering with quantum mechanical models, inspired by deep space laser communication problems, where my mathematical
physics training at Harvard came in handy. I then worked on nonlinear filtering, trying to understand how physicists turned nonlinear inference problems to linear ones and investigate why we could
not do the same for nonlinear filtering and partially observed stochastic control. This led me to unnormalized conditional densities, the Duncan-Mortensen-Zakai equation and to information states.
This led me naturally to construct nonlinear observers as asymptotic limits of nonlinear filtering problems and the complete solution of the nonlinear robust output feedback control problem
(nonlinear H-infinity problem) via two coupled Hamilton Jacobi Bellman equations. We even investigated the development of special chips to implement real-time solutions, a topic we are revisiting
With the development and progress of the ISR I worked on many problems including: speech and image compression breaking the Shannon separation of source and channel coding, manufacturing processes,
network management, communication network protocols, smart materials (piezoelectric, shape memory alloys), mobile wireless network design, network security and trust, and more recently human-machine
perception and cognition, networked control systems, networked cyber-physical systems, combining metric temporal logic and reachability analysis for safety, collaborative decision management in
autonomous vehicles and teams of humans and robots, new analytics for learning and for the design of deep learning networks mapping abstractions of the brain cortex, quantum control and computing.
Why I am telling you about all these diverse topics? Not to attract your admiration. But because at the heart of all my works are fundamental principles and methods from systems and controls, often
appropriately extended and modified. Even in my highest impact (economic and social) work in conceiving, demonstrating and commercializing Internet over satellite services (with billions of sales
world-wide – remember me when you use Internet in planes over oceans), we modified the flow control algorithm (the TCP) and the physical path, to avoid having TCP interpret the satellite physical
path delay as congestion. That is we used systems and control principles.
Our science and engineering, systems and control, has some unparalleled unifying power and efficiency. That is, if we are willing to build the new models required by the new applications (especially
models requiring a combination of multiple physics and cyber logic) and if we are willing to learn and apply the incredible new capabilities and technologies that are developed in information
technology and materials. As is apparent especially in this conference (ACC), and in the CDC conference, by any measure, our field is exceptionally alive and well and continues to surprise many other
disciplines by its contributions and accomplishments, which now extend even in biology, medicine and healthcare. So for the many young people here, please continue the excitement, continue getting
involved in challenging and high impact problems, and continue the long tradition and record of accomplishments we have established for so many years. And most importantly continue seeking the common
ground and unification of our methods and models.
Let me close with what I consider some major challenges and promising broad areas for the next 10 years or so:
1) Considering networked control systems we need to understand what we mean by a “network” and the various abstractions and system aspects involved. Clearly there are more than one dynamic graphs
involved. This needs new foundations for control, communication, information, computing.
2) Systems and control scientists and engineers are the best qualified to develop further the modern field of Model-Based Systems Engineering (MBSE): the design, manufacturing/implementation and
operation of complex systems with heterogeneous physical, cyber components and even including humans.
3) The need for analog computing is back, for example in real-time and progressive learning and in CPS. Some of the very early successes of control were implemented in analog electromechanical
systems due to the need for real-time behavior. Yet we do not have a synthesis theory and methodology for such systems due to the heterogeneous physics that may be involved. Nothing like we have for
Thank you all very much! This is indeed a very special day for me!
For innovative contributions to control theory, stochastic systems, and networks and academic leadership in systems and control
I am extremely grateful and humbled by being honored to receive the Richard E. Bellman Control Heritage Award for 2016. I thank those that recommended me and the awards committee for supporting that
nomination. I also thank my colleagues, students, family and especially my wife for the support I have received over these many years.
For me this award occurs at an auspicious time and place. Boston is the place of my birth and my home. It was sixty years ago that I graduated from Malden High School and entered into a world I could
never have anticipated; a world where I would be nurtured for the next twenty years by many people, some of whom have been recipients of this esteemed award.
I enrolled in the Department of Aeronautics at MIT, which after Sputnik became the Department of Aeronautics and Astronautics. In my junior year I entered into the space age. More consequential for
me was that the department head was Doc (Charles Stark) Draper[1], whose second volume of his three sequence series on Instrument Engineering (1952) was one the first books on what we know as
Classical Control covering such topics as Evens root locus, Bode plots, Nyquist criterion, and Nichols charts. Doc Draper instituted an undergraduate course in classical control that I took my junior
year. This inspired me to take a graduate course and write my undergraduate thesis in controls.
After graduation in 1960 I left Boston to work for Boeing in Seattle. There, I worked with my lead engineer Raymond Morth, who introduced me to the new world of control theory using state space that
was just emerging in the early 1960’s. I learned of dynamic programming of Richard Bellman for global sufficiency of an optimal trajectory and the Pontryagin Maximum principle inspired by the
deficiency of dynamic programing to solve certain classes of optimization problems. The Bushaw problem of determining the minimum time to the origin of a double integrator was just such a problem,
since the optimal return function in dynamic programing is not differentiable at the switching curve and the Bellman theory did not apply. Interestingly, for my bachelor’s thesis I applied the
results of the Bushaw problem to the minimum time problem of bringing the yaw and yaw rate of an aircraft to the origin. However, at that time I had no idea about the ramification of the Bushaw
problem to optimization theory. I also learned of the work of Rudolf Kalman in estimation, the work of Arthur Bryson and Henry Kelley in the development of numerical methods for determining optimal
constrained trajectories, and J. Halcombe (Hal) Laning and Richard Battin on the determination of orbits for moon rendezvous.
After an incredible year at Boeing I returned to Boston to work at the Analytical Research Department at Raytheon, where Art Bryson was a consultant. There, I worked with a student of Bryson, Walter
Denham. We were contracted by MIT’s Instrumentation Laboratory, monitored by Richard Battin, to enhance the Apollo autonomous navigation system over the trans-Lunar orbit. We developed a scheme for
determining the optimal angle-measurement sequence between the best stars in a catalogue and near and far horizons of the Earth or the Moon using a sextant. This angle-measurement sequence minimized
some linear function of the terminal value of the error covariance of position and velocity near the Earth or Moon. Our optimization scheme, which required a matrix dynamic constraint, seemed to be a
first. This scheme, used in the Apollo autonomous navigation system, was tested on Apollo 8, and used on every mission thereon. My next task at Raytheon was working on neighboring optimal guidance
scheme. This work was with Art Bryson and John Breakwell. I remember travelling to Lockheed’s Palo Alto Research Laboratory and meeting with John, the beginning of a long and delightful collegial
After my first two years at Raytheon I somehow convinced Art Bryson to take me on as a graduate student at Harvard, supported by the Raytheon Fellowship program. To understand the intellectual level
I had to contend with, on my doctorial preliminary exam committee, three of the four examiners were recipients of the Richard E. Bellman Control Heritage Award; Art Bryson, Larry (Yu-Chi) Ho, and Bob
(Kumpati) Narendra, all of whom have been my life time colleagues. I was also fortunate to take a course taught by Rudy Kalman. Surprisingly, he taught many of the controls areas he had pioneered,
except filtering for Gauss-Markov systems (the Kalman filter); the Aizerman conjecture, the Popov criterion and Lyopunov functions, duality in linear systems, optimality for linear-quadratic systems,
etc. After finishing my PhD thesis on optimal control problems with state variable inequality constraints, I returned to Raytheon. Fortunately, Art Bryson made me aware of some interest at Raytheon
in using modern control theory for developing guidance laws for a new missile. At Raytheon’s Missile Division I worked with Bill O’Halloran on the homing missile guidance system where Bill worked on
development of the Kalman filter and I worked on the development of the linear-quadratic closed-form guidance gains that had to include the nonminimal phase autopilot. This homing missile, the
Patriot missile system, appears to be the first fielded system using modern control theory.
I left Boston for New York to work at the Analytical Mechanics Associates (AMA), in particular, with Hank Kelley. Although I had a lasting friendship with Hank, I only lasted seven months in New York
before returning to the AMA office in Cambridge. Unfortunately, the Cambridge NASA Center closed, and I took a position under Dick Battin at the Instrumentation (later the Charles Stark Draper)
Laboratory at MIT. There, I worked on the necessary and sufficient conditions for optimality of singular control problems, the linear-exponential-Gaussian control problem, optimal control problems
with state variable inequality constraints, optimal control problems with cost criterion and dynamic functions with kinks, and periodic optimal control problems. On many of these issues I
collaborated with David Jacobson, whom I first met in the open forum of my PhD final exam. This remarkable collaboration culminated in our book on optimal control theory that appeared in 2010. Also,
during my tenure at Draper, I took a post-doctoral year leave at the Weizmann institute in Israel. Here, I learned that I could work very happily by myself. A few years after returning to Draper, I
started what is now a forty year career in academia and I left Boston.
As I look back, I feel so fortunate that I had such great mentoring over my early years and by so many who have won this award. My success over the last forty years has been due to my many students
who have worked with me to mold numerous new ideas together. Today, I find the future as bright as anytime in my past. I have embarked in such new directions as estimation and control of linear
systems with additive noises described by heavy tailed Cauchy probability density functions with my colleague Moshe Idan at the Technion and deep space navigation using pulsars as beacons with JPL.
To conclude, I am grateful to so many of my teachers, colleagues and students, who have nurtured, inspired, and educated me. Without them and my loving wife and family, I would not be here today.
Thank you all.
[1] Boldface names are recipients of the Richard E. Bellman Control Heritage Award.
For pioneering contributions to deterministic and stochastic optimal control theory and their applications to aerospace engineering, including spacecraft, aircraft, and turbulent flows
When I look back upon my career in the field of control, I think it may have started in 1957, when Sputnik was launched by the Russians. I was in the seventh grade at that time. The reaction of our
local school board to losing the space race was to have a group of students take algebra one year earlier, in the eighth grade. During high school, I participated in my class science fairs and won at
the state level. When I was a freshman at the University of Kansas in 1967, I was given the ability to do independent research in the area of nucleate boiling. I also was exposed to computer
programming, which was a fairly new topic at that time in undergraduate engineering. I became interested in numerical analysis and selected Princeton University for doctoral study, because Professor
Leon Lapidus was a leading authority on that topic.
I discovered his interest in numerical analysis was driven by solving control problems (specifically two point boundary value problems). The optimal control project I selected was on singular
bang-bang and minimum time control. I used discrete dynamic programming with penalty functions (influenced by Bellman and Kalman) as a way to solve this particular class of control problems. In 1971
I accepted a faculty position at the University of Texas.
That era was the heyday of optimal control in the aerospace program. Many of us in chemical engineering wanted to apply these ideas to chemical plants, however, there were some obstacles. Economic
justification was strictly required for any commercial application vs. government funding for space vehicles. In addition, proprietary considerations prevented technology transfer from one plant to
another. It wasn't until the late 1970s, when Honeywell introduced the distributed digital control system, that computer process control really began to become more popular (and economic) in
industry. In 1972, I purchased a Data General minicomputer to be used with a distillation column for process control. That computer was very antiquated by today’s standards; in fact, we had to use
paper tape for inputting software instructions to the machine.
Given that there was a lack of industrial receptivity to advanced control research and NSF funding was very limited, I looked around for other types of problems where my skills might be valuable. In
1974 the energy crisis was rearing its head due to the Arab oil embargo. Funding agencies like NSF and the Office of Coal Research in the U.S. were quite interested in how we could use the large
domestic resource of coal to meet the shortage of oil and gas. I came across some literature about a technology called underground coal gasification (UCG), where one would gasify the coal resource in
situ as a way of avoiding the mining step. I recall reading it was a very promising technology but they didn't know how to control it. That sparked my interest as a possible topic where I could apply
my skill set. But I first had to learn about the long history of coal gasification and coal utilization in general.
There were many issues that had to be addressed before developing control methodologies for UCG. There was a need to develop three-dimensional modeling tools that would predict the recovery of the
coal as well as the gas composition that you make (similar to a chemical reactor). Thus 80% of the research work was on modeling as opposed to control. It was also a highly multidisciplinary project
involving rock mechanics and environmental considerations. I worked in this area for about 10 years. Later in the mid-1980s, the U.S. no longer had an energy crisis, so I started looking at some
other possible areas for application of modeling and control.
In 1984 a new senior faculty member joined my department from Texas Instruments. He was very familiar with semiconductor manufacturing and the lack of process control, and he was able to teach me a
lot about that industry. Fortunately I did not have to learn a new field on my own since I was Department Chair with limited discretionary time. The same issues were present as for UCG: models were
needed in order to develop control strategies. I have continued working in that area with over 20 graduate students spread out over the past 25 years and process control is now a mature technology in
semiconductor manufacturing (see my plenary talk at this year’s ACC).
During the 1980s, I became interested in textbook writing and particularly the need to develop a new textbook in process control. I began collaborating with two colleagues at UC Santa Barbara (Dale
Seborg and Duncan Mellichamp) and thought that UCSB would be a great place to spend some time in the summer writing and giving short courses on the topic. The course notes were eventually developed
into a textbook eight years later. We now are working on the fourth edition of the book and it is the leading textbook for process control in the world. It has been a very rewarding endeavor to work
with other educators, and I would recommend that anyone writing a textbook collaborate with other co-authors as a way of improving the product. In 2010, we added a fourth co-author (Frank Doyle) to
cover biosystems control; in fact, he is receiving the practice award from AACC today.
In the early 1990s at UT Austin, Jim Rawlings and I concluded that we wanted to work on control problems that would impact industrial practice rather than just writing more technical papers that
maybe only a few people would read. So we formed the Texas Modeling and Control Consortium (TMCC) which had 16 member companies. Over twenty plus years the consortium has morphed into one involving
multiple universities investigating process control, monitoring, optimization, and modeling. When Jim left the University of Texas and went to Wisconsin, we decided to keep the consortium going, so
it became TWMCC (Texas Wisconsin Modeling and Control Consortium). Joe Qin replaced Jim on the faculty at UT but then 10 years later he left for USC. So our consortium became TWCCC (Texas Wisconsin
California Control Consortium). I have learned a lot from both Joe and Jim over the years and have been able to mentor them in their professional development as faculty members. I am now mentoring a
new UT control researcher (Michael Baldea) as we continue to close the gap between theory and practice.
One other thing I should mention is my involvement with the American Control Conference. I first gave a paper in 1972 at what was known as the Joint Automatic Control Conference (JACC) and have been
coming to this meeting ever since. In the 1970s each meeting was entirely run by a different society each year. To improve the business model and instill more interdisciplinarity with five
participating societies, in 1982 we started the American Control Conference with leadership from Mike Rabins, John
Zaborsky, and also Bill Levine who is here today. I was Treasurer of the 1982 meeting, which was held in Arlington, VA. That began an extremely successful series of meetings that is one of the best
conference values today. It is very beneficial to attend to see control research carried out in the other societies and not just your own society.
During my 40+ year career, I have had a lot of help from colleagues in academia and industry and collaborated with over 100 bright graduate students. I also should thank my wife Donna, who has put up
with me over these many years since we first started going to the computer center at the University of Kansas for dates 50 years ago.
My advice to younger researchers is to think 10 years out as to what the new areas might be and start learning about them. Fortunately, today’s control technology is more ubiquitous than ever and the
future is bright, although the path forward may not be clear. I still remember a discussion I had with a fellow graduate student before leaving Princeton in 1971 as we embarked on academic careers.
His view was that after all the great things achieved by luminaries like Pontryagin, Bellman, and Kalman, all that's really left are the crumbs… So I guess that means that I must have had a pretty
crummy career.
For a career of outstanding educational and professional leadership in automatic control, mentoring a large number of practicing professionals, and research contributions in the process industries,
especially semiconductor manufacturing
I feel honored and grateful for this award. After having spent so much time on dynamic programming and written several books about its various facets, receiving an award named after Richard Bellman
has a special meaning for me.
It is common in award acceptance speeches to thank one's institutions, mentors, and collaborators, and I have many to thank. I was fortunate to be surrounded by first class students and colleagues,
at high quality institutions, which gave me space and freedom to work in any direction I wished to go. As Lucille Ball has told us, "Ability is of little account without opportunity."
Also common when receiving an award is to chart one's intellectual roots and journey, and I will not depart from this tradition. It is customary to advise scholarly Ph.D. students in our field to
take the time to get a broad many-course education, with substantial mathematical content, and special depth in their research area. Then upon graduation, to use their Ph.D. research area as the
basis and focus for further research, while gradually branching out into neighboring fields, and networking within the profession. This is good advice, which I often give, but this is not how it
worked for me at all!
I came from Greece with an undergraduate degree in mechanical engineering, got my MS in control theory at George Washington University in three semesters, while holding a full-time job in an
unrelated field, and finished two years later my Ph.D. thesis on control under set membership uncertainty at MIT. I benefited from the stimulating intellectual atmosphere of the Electronic Systems
Laboratory (later LIDS), nurtured by Mike Athans and Sanjoy Mitter, but because of my short stay there, I graduated with little knowledge beyond Kalman filtering and LQG control. Then I went to teach
at Stanford in a department that combined mathematical engineering and operations research (in which my background was rather limited) with economics (in which I had no exposure at all). In my
department there was little interest in control theory, and none at all in my thesis work. Never having completed a first course in analysis, my first assignment was to teach to unsuspecting students
optimization by functional analytic methods from David Luenberger's wonderful book. The optimism and energy of youth carried me through, and I found inspiration in what I saw as an exquisite
connection between elegant mathematics and interesting practical problems. Studying David Luenberger's other works (including his Nonlinear Programming book) and working next door to him had a
lasting effect on me. Two more formative experiences at Stanford were studying Terry Rockafellar's Convex Analysis book (and teaching a seminar course from it), and most importantly teaching a new
course on dynamic programming, for which I studied Bellman's books in great detail. My department valued rigorous mathematical analysis that could be broadly applied, and provided a stimulating
environment where both could thrive. Accordingly, my course aimed to combine Bellman's vision of wide practical applicability with the emerging mathematical theory of Markov Decision Processes. The
course was an encouraging success at Stanford, and set me on a good track. It has survived to the present day at MIT, enriched by subsequent developments in theoretical and approximation
After three years at Stanford, I taught for five years in the quiet and scholarly environment of the University of Illinois. There I finally had a chance to consolidate my mathematics and
optimization background, through research to a great extent. In particular, it helped a lot that with the spirit of youth, I took the plunge into the world of the measure-theoretic foundations of
stochastic optimal control, aiming to expand the pioneering Borel space framework of David Blackwell, in the company of my then Ph.D. student Steven Shreve.
I changed again direction by moving back to MIT, to work in the then emerging field of data networks and the related field of distributed computation. There I had the good fortune to meet two
colleagues with whom I interacted closely over many years: Bob Gallager, who coauthored with me a book on data networks in the mid-80s, and John Tsitsiklis, who worked with me first while a doctoral
student and then as a colleague, and over time coauthored with me two research monographs on distributed algorithms and neuro-dynamic programming, and a probability textbook. Working with Bob and
John, and writing books with them was exciting and rewarding, and made MIT a special place for me.
Nonetheless, at the same time I was getting distracted by many side activities, such as books in nonlinear programming and dynamic programming, getting involved in applications of queueing theory and
power systems, and personally writing several network optimization codes. By that time, however, I realized that simultaneous engagement in multiple, diverse, and frequently changing intellectual
activities (while not recommended broadly) was a natural and exciting mode of operation that worked well for me, and also had some considerable benefits. It stimulated the cross-fertilization of
ideas, and allowed the creation of more broadly integrated courses and books.
In retrospect I was very fortunate to get into methodologies that eventually prospered. Dynamic programming developed perhaps beyond Bellman's own expectation. He correctly emphasized the curse of
dimensionality as a formidable impediment in its use, but probably could not have foreseen the transformational impact of the advances brought about by reinforcement learning, neuro-dynamic
programming, and other approximation methodologies. When I got into convex analysis and optimization, it was an emerging theoretical subject, overshadowed by linear, nonlinear, and integer
programming. Now, however, it has taken center stage thanks to the explosive growth of machine learning and large scale computation, and it has become the lynchpin that holds together most of the
popular optimization methodologies. Data networks and distributed computation were thought promising when I got involved, but it was hard to imagine the profound impact they had on engineering, as
well as the world around us today. Even set membership description of uncertainty, my Ph.D. thesis subject, which was totally overlooked for nearly fifteen years, eventually came to the mainstream,
and has connected with the popular areas of robust optimization, robust control, and model predictive control. Was it good judgement or fortunate accident that steered me towards these fields? I
honestly cannot say. Albert Einstein wisely told us that "Luck is when opportunity meets preparation." In my case, I also think it helped that I resisted overly lengthy distractions in practical
directions that were too specialized, as well as in mathematical directions that had little visible connection to the practical world.
An academic journey must have companions to learn from and share with, and for me these were my students and collaborators. In fact it is hard to draw a distinction, because I always viewed my Ph.D.
students as my collaborators. On more than one occasion, collaboration around a Ph.D. thesis evolved into a book, as in the cases of Angelia Nedic and Asuman Ozdaglar, or into a long multi-year
series of research papers after graduation, as in the cases of Paul Tseng and Janey Yu. I am very thankful to my collaborators for our stimulating interactions, and for all that I learned from them.
They are many and I cannot mention them all, but they were special to me and I was fortunate to have met them. I wish that I had met Richard Bellman, I only corresponded with him a couple of times
(he was the editor of my first book on dynamic programming). I still keep several of his books close to me, including his scintillating and highly original book on matrix theory. I am also satisfied
that I paid part of my debt to him in a small way. I have used systematically, for the first time I think in a textbook in 1987, the name "Bellman equation" for the central fixed point equation of
infinite horizon discrete-time dynamic programming. It is a name that is widely used now, and most deservedly so.
For contributions to the foundations of deterministic and stochastic optimization-based methods in systems and control
President Rhinehart, Lucy, Danny, fellow members of the greatest technological field in the world, I am to, say the least, absolutely thrilled and profoundly humbled to be this year’s recipient of
the Richard E. Bellman Control Heritage Award. I am grateful to those who supported my nomination, as well to the American Automatic Control Council for selecting me.
I am indebted to a great many people who have helped me throughout my career. Among these are my graduate students, post docs, and colleagues including in recent years, John Baillieul, Roger
Brockett, Bruce Francis, Art Krener, and Jan Willems. In addition, I‘ve been fortunate enough to have had the opportunity to collaborate with some truly great people including Brian Anderson, Ali
Bellabas, Chris Byrnes, Alberto Isidori, Petar Kokotovic, Eduardo Sontag and Murray Wonham. I‘ve been lucky enough to have had a steady stream of research support from a combination of agencies
including AFOSR, ARO and NSF.
I actually never met Richard Bellman, but I certainly was exposed to much of his work. While I was still a graduate student at Purdue, I learned all about Dynamic Programming, Bellman’s Equation, and
that the Principle of Optimality meant “Don’t cry over spilled milk.” Then I found out about the Curse of Dimensionally. After finishing school, I discovered that there was life before dynamic
programming, even in Bellman ‘s world. In particular I read Bellman ‘s 1953 monograph on the Stability Theory of Differential Equations. I was struck by this book ‘s clarity and ease of understanding
which of course are hallmarks of Richard Bellman ‘s writings. It was from this stability book that I first learned about what Bellman called his “fundamental lemma.” Bellman used this important lemma
to study the stability of perturbed differential equations which are nominally stable. Bellman first derived the lemma in 1943, apparently without knowing that essentially the same result had been
derived by Thomas Gronwall in 1919 for establishing the uniqueness of solutions to smooth differential equations. Not many years after learning about what is now known as the Bellman - Gronwall
Lemma, I found myself faced with the problem of trying to prove that the continuous time version of the Egardt - Goodwin - Ramadge - Caines discrete-time model reference adaptive control system was
“stable.” As luck would have it, I had the Bellman - Gronwall Lemma in my hip pocket and was able to use it to easily settle the question. As Pasteur one said, “Luck favors the prepared mind.”
After leaving school I joined the Office of Control Theory and Application at the now defunct NASA Electronics Research Center in Cambridge, Mass. OCTA had just been formed and was headed by Hugo
Schuck. OCTA ‘s charter was to bridge the gap between theory and application. Yes, people agonized about the so-called theory - application gap way back then. One has to wonder if the agony was worth
it. Somehow the gap, if it really exists, has not prevented the field from bringing to fruition a huge number of technological advances and achievements including landing on the moon, cruise control,
minimally invasive robotic surgery, advanced agricultural equipment, anti-lock brakes, and a great deal more. What gap? The only gap I know about sells clothes.
In the late 1990s I found myself one day listening to lots of talks about UAVs at a contractor’s meeting at the Naval Post Graduate School in Monterey Bay, California. I had a Saturday night layover
and so I spent Saturday, by myself, going to the Monterey Bay 1 Aquarium. I was totally awed by the massive fish tank display there and in particular by how a school of sardines could so gracefully
move through the tank, sometimes bifurcating and then merging to avoid larger fish. With UAVs in the back of my mind, I had an idea: Why not write a proposal on coordinated motion and cooperative
control for the NSF ‘s new initiative on Knowledge and Distributed Intelligence? Acting I this, I was fortunate to be able to recruit a dream team: Roger Brockett, for his background in nonlinear
systems; Naomi Leonard for her knowledge of underwater gliders; Peter Belhumeur for his expertise in computer vision, and biologists Danny Grunbaum and Julia Parish for their vast knowledge of fish
schooling. We submitted a proposal aimed at trying to understand on the one hand, the traffic rules which large animal aggregations such as fish schools and bird flocks use to coordinate their
motions and on the other, how one might use similar concepts to coordinate the motion of manmade groups. The proposal was funded and at the time the research began in 2000, the playing field was
almost empty. The project produced several pieces of work about which I am especially proud. One made a connection between the problem of maintaining a robot formation and the classical idea of a
rigid framework; an offshoot of this was the application of graph rigidity theory to the problem of localizing a large, distributed network of sensors. Another thrust started when my physics -
trained graduate student Jie Lin, ran across a paper in Physical Review Letter by Tomas Vicsek and co-authors which provided experimental justification for why a group of self - driven particles
might end up moving in the same direction as a result of local interactions. Jie Lin, my post doc Ali Jadbabaie, and I set out to explain the observed phenomenon, but were initially thwarted by what
seemed to be an intractable convergence question for time - varying, discrete - time, linear systems. All attempts to address the problem using standard tools such as quadratic Lyapunov functions
failed. Finally, Ali ran across a theorem by Jacob Wolfowitz, and with the help of Marc Artzrouni at the University of Pau in France, a convergence proof was obtained. We immediately wrote a paper
and submitted it to a well-known physics journal where it was promptly rejected because the reviewers did not like theorems and lemmas. We then submitted a full-length version of the work to the TAC
where it was eventually published as the paper “Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules.”
Over the years, many things have changed. The American Control Conference was once the Joint Automatic Control Conference and was held at universities. Today the ACC proceedings sits on a tiny flash
drive about the size of two pieces of bubble gum while a mere 15 years ago the proceedings consisted of 6 bound volumes weighing about 10 pounds and taking up approximately 1100 cubic inches of space
on one ‘s bookshelf. And people carried those proceedings home on planes - of course there were no checked baggage fees back then.
The field of automatic control itself has undergone enormous and healthy changes. When I was a student, problem formulations typically began with “Consider the system described by the differential
equation.” Today things are different and one of the most obvious changes is that problem formulations often include not only a differential equations, but also graphs and networks. The field has
broadened its outlook considerably as this American Control Conference clearly demonstrates.
And where might things be going in the future? Take a look at the “Impact of Control Technology” papers on the CSS website including the nice article about cyber - physical systems by Kishan Baheti
and Helen Gill. Or try to attend the workshop on “Future Directions in Control Theory” which Fariba Fahroo is organizing for AFOSR.
Automatic control is a really great field and I love it. However, it is also probably the most difficult field to explain to non - specialists. Paraphrasing Donald Knuth: “A {control} algorithm will
have to be seen to be believed.”
I believe that most people do not understand what a control engineer does or what a control system is. This of course is not an unusual situation. But it is a problem. IBM, now largely a service
company, faced a similar problem trying to explain itself after it stopped producing laptops. We of course are primarily a service field. Perhaps like IBM, we need to take some time to rethink how we
should explain what we do?
Thank you very much for listening and enjoy the rest of the conference.
For fundamental contributions to linear systems theory, geometric control theory, logic-based and adaptive control, and distributed sensing and control
It is a honor to receive the 2012 Richard E. Bellman Control Heritage Award. I am deeply humbled to join the very distinguished group of prior winners. At this conference there are so many people
whose work I have admired for years. To be singled out among this group is a great honor.
I did not know Richard Bellman personally but we are all his intellectual descendants. Years ago, my first thesis problem came from Bellman and currently I am working on numerical solutions to
Hamilton-Jacobi-Bellman partial differential equations.
I began graduate school in mathematics at Berkeley in 1964, the year of the Free Speech Movement. After passing my oral exams in 1966, I started my thesis work with R. Sherman Lehman who had been a
postdoc with Bellman at the Rand Corporation in the 1950s. Bellman and Lehman had worked on continuous linear programs also called bottleneck problems in Bellman’s book on Dynamic Programming. These
problems are dynamic versions of linear programs, with linear integral transformations replacing finite dimensional linear transformations. At each frozen time they reduce to a standard linear
program. Bellman and Lehman had worked out several examples and found that often the optimal solution was basic, at each time an extreme point of the set of feasible solutions to the time frozen
linear program. These extreme points moved with time and the optimal solution would stay on one moving extreme point for awhile and then jump to another. It would jump from one bottleneck to another.
Lehman asked me to study this problem and find conditions for this to happen. We thought that it was a problem in functional analysis and so I started taking advanced courses in this area.
Unfortunately, about a year later Lehman had a very serious auto accident and lost the ability to think mathematically for some time. I drifted, one of hundreds of graduate students in Mathematics at
that time. Moreover, Berkeley in the late 1960s was full of distractions and I was distractable. After a year or so Lehman recovered and we started to meet regularly. But then he had a serious
stroke, perhaps as a consequence of the accident, and I was on my own again.
I was starting to doubt that my thesis problem was rooted in functional analysis. Fortunately, I had taken a course in differential geometry from S. S. Chern, one of the pre-eminent geometers of his
generation. Among other things, Chern had taught me about the Lie bracket. And one of my graduate student colleagues told me that I was trying to prove a bang-bang theorem in Control Theory, a field
that I had never heard of before. I then realized that my problem was local in nature and intimately connected with flows of vector fields so the Lie bracket was an essential tool. I went to Chern
and asked him some questions about the range of flows of multiple vector fields. He referred me to Bob Hermann who was visiting the Berkeley Physics Department at that time.
I went to see Hermann in his cigar smoked-filled office accompanied by my faithful companion, a German Shepherd named Hogan. If this sounds strange, remember this was Berkeley in the 1960s. Bob was
welcoming and gracious, he gave me galley proofs of his forthcoming book which contained Chow’s theorem. It was almost the theorem that I had been groping for. Heartened by this encounter I continued
to compute Lie brackets in the hope of proving a bang-bang theorem.
Time drifted by and I needed to get out of graduate school so I approached the only math faculty member who knew anything about control, Stephen Diliberto. He agreed to take me on as a thesis
student. He said that we should meet for an hour each week and I should tell him what I had done. After a couple of months, I asked him what more I needed to do to get a PhD. His answer was “write it
up”. My “proofs” fell apart several times trying to accomplish this. But finally, I came up with a lemma that might be called Chow’s theorem with drift that allowed me to finish my thesis.
I am deeply indebted to Diliberto for getting me out of graduate school. He also did another wonderful thing for me; he wrote over a hundred letters to help me find a job. The job market in 1971 was
not as terrible as it is today but it was bad. One of these letters landed on the desk of a young full professor at Harvard, Roger Brockett. He had also realized that the Lie bracket had a lot to
contribute to control. Over the ensuing years, Roger has been a great supporter of my work and I am deeply indebted to him.
Another Diliberto letter got me a position at Davis where I prospered as an Assistant Professor. Tenure came easily as I had learned to do independent research in graduate school. I brought my dog,
Hogan, to class every day, he worked the crowds of students and boosted my teaching evaluations by at least a point. After 35 wonderful years at Davis, I retired and joined the Naval Postgraduate
School where I continue to teach and do research. I am indebted to these institutions and also to the NSF and the AFOSR for supporting my career.
I feel very fortunate to have discovered control theory both for the intellectual beauty of the subject and the numerous wonderful people that I have met in this field. I mentioned a few names, let
me also acknowledge my intellectual debt to and friendship with Hector Sussman, Petar Kokotovic, Alberto Isidori, Chris Byrnes, Steve Morse, Anders Lindquist, Wei Kang and numerous others.
In my old age I have come back to the legacy of Bellman. Two National Research Council Postdocs, Cesar Aguilar and Thomas Hunt, have been working with me on developing patchy methods for solving the
Hamilton-Jacobi-Bellman equations of optimal control. We haven’t whipped the “curse of dimensionality” yet but we are making it nervous.
The first figure shows the patchy solution of the HJB equation to invert a pendulum. There are about 1800 patches on 34 levels and calculation took about 13 seconds on a laptop. The algorithm is
adaptive, it adds patches or rings of patches when the residual of the HJB equation is too large. The optimal cost is periodic in the angle. The second figure shows this. Notice that there is a
negatively slanted line of focal points. At these points there is an optimal clockwise and an optimal counterclockwise torque. If the angular velocity is large enough then the optimal trajectory will
pass through the up position several times before coming to rest there.
What are the secrets to success? Almost everybody at this conference has deep mathematical skills. In the parlance of the NBA playoffs which has just ended, what separates researchers is “shot
selection” and “follow through”. Choosing the right problem at the right time and perseverance, nailing the problem, are needed along with good luck and, to paraphrase the Beatles, “a little help
from your friends”.
For contributions to the control and estimation of nonlinear systems
Usually when you are nominated for an award you know about it or – at least – you have a suspicion – for example, when somebody asks you for your CV, but you are sure that they are not interested in
hiring you. This award came to me as a total surprise. Indeed, I had written a letter of support for another most worthy candidate. So, when I received Tamer Başar’s email I thought that it was to
inform me that this colleague had won. Who was actually responsible for my nomination? Several of my former graduate students! So, not only were they responsible for doing the work that qualified me
for the award, they were even responsible for my getting it!
Over the course of my career, I was fortunate to have worked with a fantastic group of people and I am very proud of them: 64 Phd Students to date and about 25 postdocs. 27 of them are holding
professorships all over the world – from the Korean Advanced Institute of Science and Technology KAIST in the East to Berkeley and Santa Barbara in the West from the Norwegian Technical University
and the U Toronto in the North to the Technion in Israel and the Instituto Tecnologico de Buenos Aires in the South. Many others are now in industry, about 15 in Finance, Management Consulting and
Legal, holding positions of major responsibility. I regard this group of former co-workers as my most important legacy.
This award means a lot to me because of the awe-inspiring people who received it in the past. I remember Hendrik Bode receiving the inaugural award in 1979. I remember Rutherford Aris, one of my PhD
advisors at the University of Minnesota receiving it in 1992. Aris had actually worked and published with Richard Bellman. I remember Harmon Ray receiving it in 2000, my colleague and mentor at the
University of Wisconsin.
Receiving this award made me also reflect on what I felt our major contributions were in these 34 years since I started my career as an Asst. Prof at Wisconsin. In what way was our work important? I
was reminded of a dinner conversation a few months back with a group of my former PhD students who had joined McKinsey after graduating from ETH. One of them told me that our group had supplied more
young consultants to McKinsey Switzerland than any other institute of any university in Switzerland. He also talked informally about the results of a survey done internally on what may be the main
traits characterizing a CEO. It is not charm. It is not tactfulness and sensitivity. It is not intelligence. The only common trait seems to be that in their past these CEOs headed a division that
experienced unusual growth. For example, the CEO of a telecom company had headed the mobile phone division. All the CEOs seemed to have been at the right place at the right time.
Similar considerations may apply to doing research and to heading a research group. Richard Hamming, best known for the Hamming code and the Hamming window, wrote in a wonderful essay: “If you are to
do important work then you must work on the right problem at the right time and in the right way. Without any one of the three, you may do good work but you will almost certainly miss real
So, what are the right problems? Eric Sevareid, the famous CBS journalist once quipped: “The chief cause of problems is solutions.” We were never interested in working on problems solely for their
mathematical beauty. We always wanted to solve real practical problems with potential impact. Several times we were lucky to be standing at a turning point, ready to embark on a new line of research
before the community at large had recognized it. Let me share with you three examples.
Around 1975, when I started my PhD at the University of Minnesota, interest in process control was just about at an all-time low. In 1979 this conference, which was then called the Joint Automatic
Control Conference, had barely 300 attendees. The benefits of optimal control and the state space approach had been hyped so much for more than a decade that disillusionment was unavoidable. Many
people advised me not to commence a thesis in process control. But my advisor George Stephanopoulos convinced me that the reason for all the disappointment was that people had been working on the
wrong problem. The problem was not how to design controllers for poorly designed system but how to design systems such that they are easy to control. The work that was started at that time by us and
several other groups provided valuable insights that are in common use today and set off a whole research movement with special sessions, special journal issues and even separate workshops and
The second example is our work on Internal Model Control (IMC) and Robust Control. In the early 1980s the term “robust control” did not exist or, at least, it was not widely used and accepted. From
our application work and influenced by several senior members of our community we had become convinced that model uncertainty is a critical obstacle affecting controller design. We discovered
singular values and the condition number as important indicators before we learned that these were established mathematical quantities with established names. In 1982 at a workshop in Interlaken I
met John Doyle, Gunter Stein and essentially everybody else who started to push the robust control agenda. Indeed, it was there that Jürgen Ackerman made the researchers in the West aware of the
results of Kharitonov. A year later I went to Caltech, John Doyle followed soon afterwards and an exciting research collaboration commenced that lasted for almost a decade. We also cofounded the
Control and Dynamical Systems option/department at that time.
The third example is our more recent work on Model Predictive Control (MPC) and Hybrid Systems. As I returned to Switzerland 17 years ago, I moved from a chemical to an electrical engineering
department. I was thrown into a new world of systems with time constants of micro- or even nanoseconds rather than the minutes or hours that I was used to. So, we set out to dispel the myth that MPC
was only suited to slow process control problems and showed that it could even be applied to switched power electronics systems. Through this activity in parallel with a couple of other groups in the
world, among them the group of Graham Goodwin, we started this era of “fast MPC” and contributed to the spread of MPC to just about every control application area.
I would never claim that in the mentioned areas we made the most significant contributions and some of the results may even seem trivial to you now, but we were there at the beginning. The Hungarian
author Arthur Koestler remarked that “the more original a discovery, the more obvious it seems afterwards”
Notwithstanding this over-the-hill award that I received today and the mandatory retirement age in Switzerland I fully intend to strive to match these contributions in the coming years – together
with my students, of course.
I want to close my remarks quoting from an interview Woody Allen gave last year. When he was asked “How do you feel about the aging process?” he replied: “Well, I’m against it. I think it has nothing
to recommend it.”
For pioneering contributions to the theory and application of robust process control, model predictive control, and hybrid systems control
I am exceedingly happy to receive the Richard Bellman Control Heritage Award. I am thankful to the American Automatic Control Council for recognizing my work as worthy of this award, and I am deeply
humbled when I consider the previous recipients of the award.
My first thanks go to my dear wife Dragana who put up for a long time with a workaholic husband with an oversized ambition. I am grateful to Santa Clara University and, in particular, to the School
of Engineering for providing institutional support to our research. I am exceedingly thankful to many people from all around the world who came to Santa Clara to work on our projects as fellow
researchers on an exploratory journey; and what a journey it has been!
At this occasion, it gives me a great pleasure to recall my visit to University of Southern California and my brief encounter with Professor Bellman. After my talk, he invited me to his office, and
among myriad of his interests, he chose to talk with me about his recent work in Pharmacokinetics. At that time, I was deeply into the competitive equilibrium in economics, and we had a very
stimulating discussion on the connection of the two fields via the Metzler matrix which I have been using since then in a wide variety of models to this very day.
Looking at this award in a prudential light, my obtaining this award is as much a compliment to the Control Council as it is to me. My winning of this award at Santa Clara University, which is not a
research 1 university but prides itself as an excellent teaching institution, proves that the system is open, and that any of you wherever you are can win this award solely by the merit of your
I recall when at eighteen I made the Yugoslav Olympic Water Polo Team for the 1952 Helsinki Olympic Games. We won all our games except the final one, which ended in a draw. At that time, there were
no overtimes and penalty kicks; the winner was determined by the cumulative goal ratio. I continued playing water polo, but did not make the team for the 1956 Melbourne games; I broke my right hand
and stayed home. I kept playing on and in 1960 made the team for the Rome Olympics. We did not win a medal in Rome, let alone the gold. At that point I was already a committed researcher in control
I continued the research for many years and to borrow from a song by Neil Young:
"I kept searching for a heart of gold, and I was getting old ... "
Today I found a heart of gold. Thank you all very much for your attention, and God bless!
July 1, 2010. Baltimore, MD
For fundamental contributions to the theory of large-scale systems, decentralized control, and parametric approach to robust stability
First of all, I wish to express my sincere thanks to the American Automatic Control Council for bestowing on me the Bellman Control Heritage Award. This great honor was completely unexpected so that
my gratitude is very deep indeed. I would like to use this rare opportunity to say a few words about a topic which has concerned me for some time, namely, the question Who did what first?. In so
doing, I shall relate two examples of which the first is especially a propos since it involves the patron of the award, Richard Bellman, as well as Rufus Isaacs, both long-time friends of mine. When
I attended the 1966 International Congress of Mathematicians in Moscow, where Dick was a plenary speaker and Rufus was to present a paper entitled Differential games and dynamic programming, and what
the latter can learn from the former, the meeting was buzzing with excitement about an upcoming confrontation between two well known American mathematicians. And indeed, when Rufus presented his
paper it was his take on the discovery of the Principle of Optimality which, in his view, appeared after the in-house publication of three RAND reports on differential games, and which appeared to be
just a one-player version of his Tenet of Transition. The result of this implied accusation of plagiarism had two unhappy consequences. I had lunch with Dick on that day. He was deeply hurt, so much
so that he was near tears. Equally unfortunate was the effect on Rufus who devoted much of his remaining time to trying to prove the priority of his discovery instead of continuing to produce new and
important research of which his fertile mind was surely capable. The second example is a much happier one. In the mid-1960's I published a brief paper in which I proposed constructive sufficiency
conditions for extremizing a class of integrals by solving an equivalent problem by inspection. It was not until 1999 that I returned to this subject at the urging of a Canadian colleague. After
revisiting the original 1967 paper, I published a generalization in JOTA in 2001. On presenting these results at my 75th birthday symposium in Sicily in 2001, Pierre Bernhard remarked that my
approach seemed to be related to Caratheodory's in his 1935 text on the calculus of variations and partial differential equations, first translated into English in the mid-1960's and not known to me.
And indeed, in 2002, Dean Carlson published in JOTA a paper in which he discussed a relation between the two approaches in that both are based on the equivalent problem methodology. Caratheodory
obtained an equivalent problem by allowing for a different integrand, and I obtained an equivalent problem by the use of transformed variables. Dean then proposed a generalization by combining the
two approaches. A happy consequence of this paper has been and continues to be a fruitful collaboration which has resulted in many extensions and applications, e.g., to classes of optimal control and
differential game problems, to multiple integrals, and to economic problems, the most recent concerned with differential constraints (state equations) and presented just a couple of weeks ago at the
15th International Workshop on Dynamics and Control. A particularly interesting discussion and some generalizations by Florian Wagener may be found in the July 2009 issue of JOTA. Thus, Caratheodory
received his well deserved citation and I learned a great deal, allowing me to make some small contributions to optimization theory.
June 11, 2009. St. Louis, MO
For pioneering contributions to geometric optimal control, quantitative and qualitative differential games, and stabilization and control of deterministic uncertain systems, and for exemplary service
to the control field
It is an honor to receive the Bellman Award. I am sure the Award Committee received many outstanding nominations, and I thank the Committee for selecting me. I was invited to make a few remarks, so
long as I did not exceed five minutes. I will point out some landmarks along my intellectual journey. The young people among you may find it of some interest. I came to Berkeley as a graduate student
in 1960. I owe a great deal to Professor Lotfi Zadeh who was my PhD adviser and who has been a mentor to me ever since. Much of my intellectual development came from interaction with visitors and
students. Karl Astrom visited me in the early 1960s. His paper with Bohlin on system identification became for me a standard of research quality and research exposition. Another significant visitor
was Bill Root. Bill showed me how to use mathematics in the analysis of communication systems, and he introduced me to information theory.
Stochastic Systems
There was a buzz at the time about white noise and martingales. Gene Wong was talking about it, as was Moshe Zakai. Tyrone Duncan was visiting. Ty de-mystified the buzz for me. He taught me how to
think about stochastic systems. Thus began my lifelong attraction towards randomness. Sanjoy Mitter, who I first met about that time, reinforced that attraction. Sanjoy became a lifelong friend, for
which I am very grateful.
Mark Davis was the first in a sequence of brilliant PhD students in stochastic systems. Mark discovered the deep relation between martingales and optimum decisions. Rene Boel, Jan van Schuppen, and
Gene Wong found that martingales were also key to point processes as well as Ito processes. Jean Walrand grasped this insight and developed it into an outstanding thesis on queuing networks. Venkat
Anantharam knew little or nothing about probability theory when he began his PhD. I still recall how much he impressed me with his spectacular work on multi-armed bandits. The third in this group was
Vivek Borkar. Vivek was the most quiet, but equally stunning.
This was when P.R. Kumar visited Berkeley. He is the first of the next generation that I got to know as a friend. I have become a fan of his, along with so many others. Intellectual life moves in
circles. Borkar and Kumar re-connected me with Karl Astrom, this time through his paper with Wittenmark.
Jean Walrand introduced me to computer communication networks. This has continued to be an area of research for the past twenty years. We've had outstanding students, who have gone on to brilliant
careers. Sri Kumar, then at Northwestern, Jean Walrand and I got to know each other through our interest in networking.
I learned power engineering in undergraduate school. But then I lost contact with the field, until years later when Felix Wu joined our faculty. Eyad Abed, Fathi Salem and Shankar Sastry wrote their
dissertations on difficult questions in nonlinear systems, inspired by problems of power systems. I lost contact with the field once again, until deregulation became the rage in California. Once
again Felix recruited me. Felix Wu, Shmuel Oren of IEOR, Pablo Spiller of the Business School, and I joined forces to save California from the clutches of the utilities. We developed a provably good
deregulation strategy. The strategy was not adopted.
Ahmad Bahai and Andrea Goldsmith sparked my interest in wireless communications. They have become stars. They inspired my very recent students, Mustafa Ergen and Sinem Coleri.
In the late sixties, Noam Chomsky came to Berkeley and gave a lecture on formal languages. Chomsky's talk opened up a whole world for me. I spent a lot of time learning recursive functions, Turing
machines, and Godel's theory. Walt Burkhard wanted to do a thesis on space-time complexity of recursive functions, and he helped consolidate what I had learned. However, my involvement with that
subject declined.
My interest was revived by the Wonham-Ramadge paper on discrete-event systems, while Joseph Sifakis, Tom Henzinger and others began the study of time automata. These developments combined to create
the area of Hybrid Systems. My students Anuj Puri and Alex Kurzhansky obtained some outstanding results in Hybrid Systems.
My flirtation with transportation began 30 years ago when I taught urban economics. Mario Ripper was my first doctoral student in transportation planning. My interest then waned. In 1990, Steve
Shladover helped spark a national, indeed worldwide, interest in automated highways. Berkeley became a leading research center in highway automation, culminating in a full demonstration in 1997 in
San Diego. It was very exciting to work with an interdisciplinary group of experts to build something all the way from theory to demonstration.
Since I could not wait for 25 years before automated highways became practical, my attention shifted to today's highways. My student Karl Petty built the PeMS system, which is now world-renowned as a
repository of highway data. Roberto Horowitz and I are now developing a control system for the management of highways. It might become an important follow-on to the PeMS system.
Let me conclude with a remark on Richard Bellman, whom I met in the late sixties. Bellman was a renowned mathematician with contributions in many, many areas. I learned two things from him. First,
over the years I continue to marvel at the significance of the optimality principle in the form of the verification theorem, which I have used in many contexts. Second and more important, I learned
that good theory is very practical.
Thank you very much for being such courteous listeners.
June 12, 2008. Seattle, WA
For pioneering contributions to stochastic control, hybrid systems and the unification of theories of control and computation
It is a great honor for me to receive the Bellman Award—quite undeserved I believe, but I decided not to emulate Gregory Perelman by refusing to accept the award. I might however follow his footsteps
(apparently he has stopped doing Mathematics) and concentrate only on the more conceptual and philosophical aspects of the broad field of Systems and Control.
On an occasion like this it is perhaps appropriate to say a few words about the seminal contributions of Richard Bellman. As we all know, he is the founder of the methodological framework of Dynamic
Programming, probably the only general method of systematically and optimally dealing with uncertainty, when uncertainty has a probabilistic description, and there is an underlying Markov structure
in the description of the evolution of the system. It is often mentioned that the work of Bellman was not as original as would appear at first sight. There was, after all, Abraham Wald’s seminal work
on Optimal Sequential Decisions and the Carat´eodory view of Calculus of Variations, intimately related to Hamilton–Jacobi Theory. But the generality of these ideas, both for deterministic optimal
control and stochastic optimal control with full or partial observations, is undoubtedly due to Bellman. Bellman, I believe, was also the first to present a precise view of stochastic adaptive
control using methods of dynamic programming. Now, there are two essential steps in invoking Dynamic Programming, namely, invariant embedding whereby a fixed variational problem is embedded in a
potentially infinite family of variational problems and then invoking the Principle of Optimality which states that any sub-trajectory of an optimal trajectory is necessarily optimal to characterize
optimal trajectories. This is where the Markov structure of dynamic evolution comes into operation. It should be noted that there is wide flexibility in the invariant embedding procedure and this
needs to be exploited in a creative way. It is this embedding that permits obtaining the optimal control in feedback form (that is a “control law” as opposed to open loop control).
The solution of the Partially-Observed Stochastic Control in continuous time leading to the characterization of the optimal control as a function of the unnormalized conditional density of the state
given the observations via the solution of an infinite-dimensional Bellman–Hamilton–Jacobi equation is one of the crowning achievements of the Bellman view of stochastic control. It is worth
mentioning that Stochastic Finance Theory would not exist but for this development. There are still open mathematical questions here that deserved further work. Indeed, the average cost problem for
partially-observed finite-state Markov chains is still open—a natural necessary and sufficient condition for the existence of a bounded solution to the dynamic programming equation is still not
Much of my recent work has been concerned with the unification of theories of Communication and Control. More precisely, how does one bring to bear Information Theory to gain understanding of
Stochastic Control and how does one bring to bear the theory of Partially-Observed Stochastic Control to gain qualitative understanding of reliable communication. There does not exist a
straightforward answer to this question since the Noisy Channel Coding Theorem which characterizes the optimal rate of transmission for reliable communication requires infinite delay. The encoder in
digital communication can legitimately be thought of as a controller and the decoder an estimator, but they interact in complicated ways. It is only in the limit of infinite delay that the problem
simplifies and a theorem like the Noisy Channel Coding Theorem can be proved. This procedure is exactly analogous to passing to the thermodynamic limit in Statistical Mechanics.
In the doctoral dissertation of Sekhar Tatikonda, and in subsequent work, the Shannon Capacity of a Markov Channel with Feedback under certain information structure hypotheses can be characterized as
the value function of a partially-observed stochastic control problem. This work in many ways exhibits the power of the dynamic programming style of thinking. I believe that this style of thinking,
in the guise of a backward induction procedure, will be helpful in understanding the transmission capabilities of wireless networks. More generally, dynamic programming, when time is replaced by a
partially ordered set, is a fruitful area of research.
Can one give an “information flow” view of path estimation of a diffusion process given noisy observations? An estimator, abstractly can be thought of as a map from the space of observations to a
conditional distribution of the estimand given the observations. What is the nature of the flow of information from the observations to the estimator? Is it conservative or dissipative? In joint work
with Nigel Newton, I have given a quite complete view of this subject. It turns out that the path estimator can be constructed as a backward likelihood filter which estimate the initial state
combined with a fully observed stochastic controller moving in forward time starting at this estimated state solves the problem in the sense that the resulting path space measure is the requisite
conditional distribution. The backward filter dissipates historical information at an optimal rate, namely that information which is not required to estimate the initial state and the forward control
problem fully recovers this information. The optimal path estimator is conservative. This result establishes the relation between stochastic control and optimal filtering. Somewhat surprisingly, the
optimal filter in a stationary situation satisfies a second law of thermodynamics.
What of the future? Undoubtedly we have to understand control under uncertainty in a distributed environment. Understanding the interaction between communication and control in a fundamental way will
be the key to developing any such theory. I believe that an interconnection view where sensors, actuators, controllers, encoders, channels and decoders, each viewed abstractly as stochastic kernels,
are interconnected to realize desirable joint distributions, will be the “correct” abstract view for a theory of distributed control. Except in the field of distributed algorithms, not much
fundamental seems to be known here.
It is customary to end acceptance discourses on an autobiographical note and I will not depart from this tradition. Firstly, my early education at Presidency College, Calcutta, where I had the
privilege of interacting with some of the most brilliant fellow students, decisively formed my intellectual make-up. Whatever culture I acquired, I acquired it at that time. At Imperial College,
while I was doing my doctoral work, I was greatly influenced by John Florentin (a pioneer in Stochastic Control), Martin Clark and several other fellow students. I have also been fortunate in my
association with two great institutions—MIT and the Scuola Normale, Pisa. I cannot overstate everything that I have learnt from my doctoral students, too many to mention by name—Allen gewidmet von
denen ich lernte [Dedicated to all from whom I have learnt (taken from dedication of G¨unter Grass in “Beim H¨auten der Zwiebel” (“Peeling the Onion”))]. I find that they have extraordinary courage
in shaping some half-baked idea into a worthwhile contribution. In recent years, my collaborative work with Vivek Borkar and Nigel Newton has been very important for me. I have great intellectual
affinity with members of Club 34, the most exclusive club of its kind and I thank the members of this club for their friendship. There are many others whose intellectual views I share, but at the
cost of exclusion let me single out Jan Willems and Pravin Varaiya. I admire their passion for intellectual discourse. Last, but not least, I thank my wife, Adriana, for her love and support. I am
sorry she could not be here today. My acceptance speech is dedicated to her.
July 12, 2007. New York, NY
For contributions to the unification of communication and control, nonlinear filtering and its relationship to stochastic control, optimization, optimal control, and infinite-dimensional systems
I am honored to receive this most prestigious award and recognition by the American Automatic Control Council, named after Richard Ernest Bellman (the creator of "dynamic programming")---who has
shaped our field and influenced through his creative ideas and voluminous multifaceted work the research of tens of thousands, not only in control, but also in several other fields and disciplines.
In my own research, which has encompassed control, games, and decisions, I have naturally also been influenced by the work of Bellman (on dynamic programming), as well as of Rufus Isaacs (the creator
of differential games) whose tenure at RAND Corporation (Santa Monica, California) partially overlapped with that of Bellman in the 1950s. I want to use the few minutes I have here to say a few words
on those early days of control and game theory research (just a brief historical perspective), and Bellman's role in that development.
In a Bode Lecture I delivered (at the IEEE Conference on Decision and Control in the Bahamas) in December 2004, I had described how modern control theory was influenced by the research conducted and
initiatives taken at the RAND Corporation in the early 1950s. RAND had attracted and housed some of the great minds of the time, among whom was also Richard Bellman, in addition to names like Leonard
D. Berkovitz, David Blackwell, George Dantzig, Wendell Fleming, M.R. Hestenes, Rufus Isaacs, Samuel Karlin, John Nash, J.P. LaSalle, and Lloyd Shapley (to list just a few). These individuals, and
several others, laid the foundations of decision and game theory, which subsequently fueled the drive for control research. In this unique and highly conducive environment, Bellman started working on
multi-stage decision processes, as early as 1949, but more fully after 1952---and it is perhaps a lesser known historical fact that one of the earlier topics Bellman worked on at RAND was ! game
theory (both zero- and nonzero-sum games) on which he co-authored research reports with Blackwell and LaSalle. In an informative and entertaining autobiography he wrote 32 years later ("Eye of the
Hurricane", World Scientific, Singapore), completed in 1984 shortly before his untimely death (March 19), Bellman describes eloquently the research environment at RAND and the reason for coining the
term "dynamic programming".
At the time, the funding for RAND came primarily from the Air Force, and hence it was indirectly under the Secretary of Defense, who was in the early 1950s someone by the name Wilson. According to
Bellman, "Wilson had a pathological fear and hatred of the word 'research' and also of anything 'mathematical' ". Hence, it was quite a challenge for Bellman to explain what he was doing and
interested in doing in the future (which was research on multi-stage decision processes) in terms which would not offend the sponsor. "Programming" was an OK word; after all Linear Programming had
passed the test. He wanted "to get across the idea that what he was doing was dynamic, multi-stage, and time-varying", and therefore picked the term "Dynamic Programming". He thought that "it was a
term not even a Congressman could object to". This being the official reason given for his pick of the term, some say (Harold Kushner--recipient of this award two years ago--being one of them, based
on a personal conversation with Bellman) that he wanted to upstage Dantzig's Linear Programming by substituting "dynamic" for "linear". Whatever the reasons were, the terminology (and of course also
the concept and the technique) was something to stay with us for the next fifty plus years, and undoubtedly for many more decades into the future, as also evidenced by the number of papers at this
conference using the conceptual framework of dynamic programming.
Applying dynamic programming to different classes of problems, and arriving at "functional equations of dynamic programming", subsequently led Bellman, as a unifying principle, to the "Principle of
Optimality", which Isaacs, also at RAND, and at about the same time, had called "tenet of transition" in the broader context of differential games, capturing strategic dynamic decision making in
adversarial environments.
Bellman also recognized early on that a solution to a multi-stage decision problem is not merely a set of functions of time or a set of numbers, but a rule telling the decision maker what to do, that
is, a "policy". This led in his thinking, when he started looking into control problems, to the concept of "feedback control", and along with it to the notions of sensitivity and robustness. These
developments, along with the more refined notions of information structures (who knows what and when), have been key ingredients in my research for the past thirty plus years.
It is interesting that at RAND at the time (that is in the 1950s), in spite of the anti-research and anti-mathematical attitude that existed in the higher echelons of the government, and the
Department of Defense in particular, fundamental research did prosper, perhaps somewhat camouflaged initially, which in turn drove the creation of modern control theory, fueled also by the
post-Sputnik anxiety. There is perhaps a message that should be taken from that: "Don't give up doing what you think and believe is right and important, but also be flexible and accommodating in how
you promote it".
Before closing, I want to thank all who have been involved in the nomination process and the selection process of the Bellman Control Heritage Award this year. I want to use this occasion also to
acknowledge several educational and research institutions which have impacted my life and career.
First, I want to acknowledge the contributions of the educational institutions in my native country, Turkey, in the early years of my upbringing, and the comfortable research environment provided by
the Marmara Research Institute I was affiliated with in the mid to late 1970s. Second, I want to acknowledge the love for research and the drive for pushing the frontiers of knowledge I was infected
with during my years at Yale and Harvard in the early 1970s. And last, but foremost, I want to acknowledge the perfect academic environment I found and have still been enjoying at the University of
Illinois at Urbana-Champaign---wonderful colleagues, stimulating teaching environment at the Department of Electrical and Computer Engineering, and exemplary conducive research environment at the
Coordinated Science Laboratory with its top quality graduate students. I also want to recognize all students, post-docs, and colleagues I have had the privilege of having research interactions and
collaborations with over the years. I thank them all for the memorable journeys in exploring the frontiers in control science and technology.
Thank you very much.
June 15, 2006. Minneapolis, MN
June 15, 2006. Minneapolis, MN
For fundamental developments in and applications of dynamic games, multiple-person decision making, large scale systems analysis, and robust control
Grow old along with me
The best is yet to be'
I don't feel particularly old but to be in the midst of friends and colleagues with this recognition is as good as it gets.
I'd like to use these few minutes to comment on several of the times when I've come to a fork in the road as an illustration of how difficult it is to predict how a given path will turn out. There
may be people who plan their lives carefully and take each step based on the best prediction of a good outcome; I'm not one of them. Too many events in my life were based on random events to pretend
that they were based on any good planning of mine.
My first decision was a good one: I selected outstanding parents. My father was a math teacher, my mother an RN and they gave me a love of books and learning that have served me well for over 7
decades. They did, however, make one mistake: they gave me a defective gene that prevents me from seeing colors the way most others see them. If you see me going Ooh and Ah over a rainbow, don't
believe it; I'm faking it.
The next decision I wish to mention was in 1945 when I became eligible for the military draft. The good news was that I was admitted to the Navy Radio Technician program but the bad news was that I
had to sign up for four years to accept the offer. The evidence was that the war would last several more years so I signed up. That decision did not look so good a few weeks later when President
Truman approved use of atomic bombs to reduce Hiroshima to rubble and Nagasaki to ruin in a matter of seconds. The war ended soon after but I was still stuck with four years obligation to the Navy.
When I got to Chicago for my final physical, one of the doctors asked me to identify the numbers in a set of circles filled with colored dots. I'm sure that I gave him some values never before found!
My performance was such that he marked me as partially disabled, put me on medical special assignment, and sent me off to the electronics school.
I finished the school in the summer of 1946 and was selected to be an instructor at a new campus being set up at the Great Lakes Naval Training Center north of Chicago. I taught electronic amplifiers
there using the book Radio Engineering by F E Terman. One of my fellow students there later became well known in the control field (and a Vice President of IBM): Jack Bertram had also signed up for
the Navy electronics program. In the early summer of 1947 my defective gene came to my rescue. The Navy announced that any sailor on medical special assignment was eligible for discharge! My
response: That's ME.
Out of the Navy I went and set about looking for a school that would accept me at that late date. I was turned down by several fine schools but Georgia Tech told me to come on down so off I went to
Atlanta where I got my EE degree in 1950. The months I'd served in the Navy made me eligible for enough GI Bill support to pay the tuition and expenses which I could never have afforded otherwise.
This time the bad news was that in the spring of 1950 the Bureau of Labor Statistics reported that the country was to graduate twice as many engineers as the economy could absorb. My only choice was
to accept a fellowship to MIT and continue my education using the last of my GI Bill of Rights tuition support. As an aside, while there I took a graduate course on pulse and timing circuits that
contained little new from what we had learned in the Navy program as high school graduates! I also had a great time learning how to play rugby from a group of graduate students from South Africa. A
most memorable part of this experience was when we were one of the teams selected to play in a tournament as the entertainment for spring break in Bermuda.
After finishing my MS in 1952 I had married the love of my life and needed to get a job. A fellow student introduced me to Professor Jack Millman who was visiting MIT looking for possible
appointments to Columbia University. I interviewed with him and was offered a position as Instructor which involved teaching responsibilities but allowed me to study for the doctorate at the same
time. I had no idea that I was stepping into a fantastic center of control research assembled by John Ragazzini. With his colleague Lotfi Zadeh he had attracted great students including Eli Jury, Art
Bergen, Jack Bertram, Rudy Kalman, Bernie Friedland, George Kranc, and Phil Sarachik. Sampled Data control was never the same again. The first treatment of 'pulsed circuits' was chapter 5 by Hurewicz
in the Rad Lab Vol. 25 on The Theory of Servomechanisms edited by James Nichols and Phillips. Hurewicz selected the variable of discrete transforms as z, a prediction of one period and we kept the
same convention. At about the time as Ragazzini's group were starting our study, some at MIT selected z to be a delay operator. In the end, z as predictor prevailed but to this day MATLAB treats
discrete transforms differently in the Signal Processing toolbox as they do in the Control toolbox. You can look it up.
After I got my degree in 1955 I was promoted to Assistant Professor. I loved Columbia and was pleased to be selected by Professor Ragazzini to join him as co-author of a book on sampled data but New
York City left a lot to be desired as a place to raise the two children who had joined my family by this time and soon another fork in the road appeared. It was presented in the person of Professor
John Linvill whose class I had taken at MIT and who had moved from MIT to Stanford by way of Bell Labs. John knew Lotfi Zadeh and at his invitation came to Columbia looking for possible new
appointments to Stanford's faculty. Again I interviewed and was offered a position on the Stanford Faculty. Thus it was that in late May of 1957 we loaded up the (non air-conditioned) Ford and headed
west. I'll never forget the hot day in June when we stopped for gas in Sacramento where the temperature was well over 100 degrees. The pavement was so soft my shoes sank into the asphalt. Then later
that day we crossed the mountains into the Bay Area and the temperature dropped about 1 degree per mile for the last 30 miles. We've been in love with the San Francisco Bay area ever since.
As an aside comment on control at the time, in the paper on The history of the Society by Danny Abramovitch and myself, George Axelby is quoted as saying that papers presented at the 1959 conference
on control by Kalman and Bertram using state notation were 'quite a mystery to most attendees.' I'd say that the idea of state was not long a mystery to those who had worked with analog computers. On
those machines, the only dynamic elements are integrators whose outputs comprise the state quite naturally. In my opinion, every control engineer should be required to program an analog computer
where one also quickly learns the value of amplitude and time scaling too.
In any case, such was the random walk through time and space that has taken me from the mountains of North Carolina to the coast of California. My tenure at Stanford has been marked by many things
but first and foremost in my affection has been the steady stream of excellent students with whom I have been privileged to work. Without a doubt they have made major contributions to control and to
them is owed much of the credit for which this award in made. So let me close with the moral of my story aimed mainly to those in academia:
You can never be too careful when selecting your students.
The corollary to this is applicable to everyone:It's hard to soar like an Eagle if you fly with a bunch of turkeys.
Thank you very much.
June 9, 2005. Portland, OR
For fundamental contributions to the theory and practice of digital, modern, adaptive, and multivariable control and for being a mentor, inspiration and friend to five decades of graduate students
It is a great honor to receive this award. It is a particular honor that it is in memory of Richard Bellman. I doubt that there are many here who knew Bellman, so I would like to make some comments
concerning his role in the field.
Bellman left RAND after the summer of 1965 for the position of Professor of Electrical Engineering, Mathematics, and Medicine at the University of Southern California. This triple title gives you
some inkling of how he was viewed at the time. I spent that summer at RAND. My office was right next to Bellman's and we had lots of opportunity to talk.
Bellman was always very supportive of my work. He encouraged me to write my first book, Stochastic Stability and Control, in 1967 for his Academic Press Series. Although naive by modern standards,
the book seemed to have a significant impact on subsequent development in that it made many mathematicians realize that there was serious probability to be done in stochastic control, and established
the foundations of stochastic stability theory. Numerical methods were among his strong interests. He was well acquainted with my work on numerical methods for continuous time stochastic systems and
encouraged me to write my first book on the subject, later updated in two books with Paul Dupuis, and still the methods of choice. Despite his enormous output of published papers, something like 900,
he was a strong believer in books since they allowed one to develop a subject with considerable freedom.
There are other connections, albeit indirect, between us. He was a New Yorker, and did his early undergraduate work at CCNY. During those years and, indeed, until the late 50's, CCNY was one of the
most intellectual institutions of higher learning in the US. During that time, before the middle class migration out of the city, and the simultaneous opening of opportunities in the elite
institutions for the "typical New Yorker," CCNY had the choice of the best of New Yorkers with a serious intellectual bent. Later, he switched to Brooklyn College, which was much closer to his home.
He intended to be a pure mathematician: His primary interest was analytic number theory. When did he become interested in applications? He graduated college at the start of WW2 and the demands of the
war exposed him to a great variety of problems. He taught electronics in Princeton and then worked at a sonar lab in San Diego (which kept him out of the Army for a while). He spent the last two
years of the war in the army, but assigned to the Manhattan project at Los Alamos. He was a social creature and it was easy for him to meet many of the talented people working on the project.
Typically, the physicists considered a mathematician as simply a human calculator, ideally constructed to do numerical computations but not much more. Bellman was asked to numerically solve some
PDE's. His mathematical pride refused. To the great surprise of the physicists, he actually managed to integrate some of the equations, obtaining closed form solutions. Holding true to tradition,
they checked his solutions, not by verifying the derivation, but by trying some very special cases. Thus his reputation there as a very bright young mathematician was established. This jealously
guarded independence and self confidence (and lack of modesty) continued to serve him well. During these years, he absorbed a great variety of scientific experiences. So much was being done due to
the needs of the war.
There is one more indirect connection between us. Bellman was a student of Solomon Lefschetz at Princeton, head of the Math. Dept. at the time, a very tough minded mathematician and one of the
powerhouses of American mathematics, and impressed with Bellman's ability. While at Los Alamos in WW2 Bellman worked out various results on stability of ODE's. Although he initially intended to do a
thesis with someone else on a number theoretic problem, Lefschetz convinced him that those stability results were the quickest way to a thesis, which was in fact true. It took only several months and
was the basis of his book on stability of ODE's. I was the director of the Lefschetz Center for Dynamical Systems at Brown University for many years, with Lefschetz our patron saint. Some of you
might recall the book (not the movie) "A Beautiful Mind" about John Nash, a Nobel Laureate in Game Theory, which describes Lefschetz's key role in mathematics during Nash's time at Princeton.
Bellman spent the summer of 1948 at RAND, where an amazing array of talent was gathered, including David Blackwell, George Dantzig, Ted Harris, Sam Karlin, Lloyd Shapley, and many others, who
provided the foundations of much of decision and game theory. The original intention was to do mathematics with some of the RAND talent on problems of prior interest. But Bellman turned out to be
fascinated and partially seduced by the excitement in OR, and the developing role of mathematics in the social and biological sciences. His mathematical abilities were widely recognized. He was a
tenured Associate Professor at Stanford at 28, after being an Associate Professor at Princeton, where all indications were that he would have had an assured future had he remained there. He began to
have doubts about the payoff for himself in number theory and returned to the atmosphere at RAND often, where he eventually settled and became fully involved in multistage decision processes, having
been completely seduced, and much to our great benefit.
Here is a non mathematical item that should be of interest. To work at RAND one needed a security clearance, even though much of the work did not involve "security." Due to an anonymous tip, Bellman
lost his clearance for a while: His brother-in-law, whom Bellman had not seen since he (his brother-in-law) was about 13, was rumored to be a communist? This was an example of a serious national
problem that was fed, exploited, and made into a national paranoia by unscrupulous politicians.
Bellman was a remarkable person, thoroughly a man of his time and renaissance in his interests, with a fantastic memory. Some epochs are represented by individuals that are towering because of their
powerful personalities and abilities. People who could not be ignored. Bellman was one of those. He was one of the driving forces behind the great intellectual excitement of the times.
The word programming was used by the military to mean scheduling. Dantzig's linear programming was an abbreviation of "programming with linear models." Bellman has described the origin of the name
"dynamic programming" as follows. An Assistant Secretary of the Air Force, who was believed to be strongly anti-mathematics was to visit RAND. So Bellman was concerned that his work on the
mathematics of multi-stage decision process would be unappreciated. But "programming" was still OK, and the Air Force was concerned with rescheduling continuously due to uncertainties. Thus "dynamic
programming" was chosen a politically wise descriptor. On the other hand, when I asked him the same question, he replied that he was trying to upstage Dantzig's linear programming by adding dynamic.
Perhaps both motivations were true.
If one looks closely at scientific discoveries, ancient seeds often appear. Bellman did not quite invent dynamic programming, and many others contributed to its early development. It was used earlier
in inventory control. Peter Dorato once showed me a (somwhat obscure) economics paper from the late thirties where something close to the principle of optimality was used. The calculus of variations
had related ideas (e.g., the work of Caratheodory, the Hamilton-Jacobi equation). This led to conflicts with the calculus of variations community. But no one grasped its essence, isolated its
essential features, and showed and promoted its full potential in control and operations research as well as in applications to the biological and social sciences, as did Bellman.
Bellman published many seminal works. It is sometimes claimed that many of his vast number of papers are repetitive and did not develop the ideas as far as they could have been. Despite this
criticism, his works were poured over word for word, with every comment and detail mined for ideas, technique, and openings into new areas. His work was a mother lode. It was clearly the work of
someone with a superb background in analysis as well as a facile mind and sharp eye for aplications. There are lots of examples, with broad coverage, accessible, and usually simple assumptions. His
writing is articulate. It flows very smoothly through the problem formulation and mathematical analysis, and he is in full command of it.
We still owe a great debt to him.
July 1, 2004. Boston, MA
For fundamental contributions to Stochastic Systems Theory and Engineering Applications, and for inspiring generations of researchers in the field
For fundamental contributions to Stochastic Systems Theory and Engineering Applications, and for inspiring generations of researchers in the field
For pioneering contribution to control theory and engineering, and for inspirational leadership as mentor, advisor, and lecturer over a period spanning four decades
For pioneering contributions to stochastic and distributed systems theory, optimization, control, and aerospace flight systems research
For sustained and significant contributions to research and education in optimization and control of dynamic systems, and his establishment of a new branch of these fields, Discrete Event Dynamic
For fundamental contributions to systems theory and pioneering works on fuzzy sets and systems leading to a global trend on machine intelligence quotient systems
For fundamental contributions to control and system theory
I am immensely pleased by the Award! It is indeed a special honor, coming from the American Automatic Control Council, which has done so much to advance and to unify the field of control. I recall
with delight the long sequence of Joint Automatic Control Conferences and the subsequent American Control Conferences. The Council's many current activities, including its participation in this 13th
IFAC World Congress, continue its invaluable service to the control community.
In receiving the award I wish to recognize the support of friends, colleagues and former students. They have played a vital role in my work. I must also acknowledge the special influence of others I
have known mostly or entirely through their publications. It is no surprise that Richard Bellman was one of them. Let me make a few remarks about his legacy and how it affects us today.
In examining his writings I am struck by his genuine interest in applications and obvious desire to make his findings useful to a wide audience. In this, I believe, there are lessons to be learned.
I'll note four.
1. Fundamental ideas have greater power when they are elegantly expressed. There is no better example than Bellman's formulation of dynamic programming. Its wonderfully stated ideas permeate and
illuminate much of what we do, ranging from deep theoretical results in optimal control to practical, on-line implementation of controllers.
2. Propagation of knowledge is enhanced by the establishment of connections across fields and disciplines. Bellman's 1960 book, "Introduction to Matrix Analysis," illustrates this point beautifully.
The discussions and bibliographies and the end of each chapter are marvelous sources of insight and diversity.
3. In mathematical exposition, clarity and accessibility are precious attributes. Bellman had a special talent for keeping mathematical developments closely connected to first principles and
organizing them in simple, easy to understand parcels. He had the courage to compromise generality for clarity and, on occasion, rigor for insight.
4. Numerical issues are crucial to control applications. Bellman realized this early, four decades ago, when he addressed controller implementation, algorithm design, error analysis, and
computational complexity.
Over the years the field of control has become mature, complex and diverse. We now need, as Richard Bellman did so well, to give greater attention to the means by which we encourage its progress and
impact on society. On that point I will end.
Thank you.
July 4, 1996
In recognition of a distinguished career in automatic control, with pioneering research contributions to a broad range of subjects including linear multivariable systems theory, computation of
optimal controls, nonlinear systems theory, and motion planning in the presence of obstacles
In recognition of life-long contributions to the field of automatic control as an author, teacher, and academic administrator, and for his continuing efforts to foster understanding of the role of
technology in the conduct of human affairs.
In recognition of his inspiration and guidance to a generation of researchers, his innovations in optimal control and estimation theory, and his seminal contributions to the field of automatic
In recognition of his leading role in the development of stability theory, linear systems theory, nonlinear control theory, and robotics.
For his very significant contribution to the field of automatic control systems analysis and synthesis by inventing the root-locus technique.
For distinguished career contributions to the theory or application of automatic control. | {"url":"https://a2c2.org/award/richard-e-bellman-control-heritage-award","timestamp":"2024-11-10T02:53:55Z","content_type":"text/html","content_length":"484436","record_id":"<urn:uuid:6a5cceae-5765-473f-ac13-b2bbb23b3175>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00134.warc.gz"} |
GRC Tuesdays : Why I Love Risk Management and I Think You Should As Well
But, once you start taking samples, your confidence will start to drop because of sampling error and random chance. Let’s explore how you can use confidence level to describe the probability you can
be comfortable in saying something about your process. Risks pose real-time threats, and you have to be able to make informed decisions to mitigate them quickly. Trying to manage assessments using
paper and spreadsheets is unwieldy and limits participation. Using safety management software (like Vector EHS!), you can continually update and easily modify your risk matrix to meet your specific
operational needs. By multiplying a hazard’s probability and severity values, you can calculate the acceptability level of its risk.
If your confidence level is 100%, you will be 100% confident that repeated samples will provide approximately the same results. A confidence level of 0% means you have no confidence repeated samples
will provide the same results. In most business applications, you will strive for a 90%, 95% or 99% confidence level. Should an entire company employ a single common risk assessment matrix or should
each department have its own specific one?
What Is the Disadvantage of Using Value at Risk?
A single serious data breach could result in debilitating operational disruptions, financial losses, reputational damage and regulatory penalties. Phase 2C iterates on the learnings of Phase 2B and
involves a refined prototype build of a fully integrated system. Some projects also benefit from additional iterations of the product based on prior learnings through additional phases , which are
not represented in this graphic. All requirements are intended to be tested, and at the end of Phase 2 there will be confidence that the units will pass verification in Phase 3.
So for the USA, the lower and upper bounds of the 95% confidence interval are 34.02 and 35.98. Once you know each of these components, you can calculate the confidence interval for your estimate by
plugging them into the confidence interval formula that corresponds to your data. Most statistical programs will include the confidence interval of the estimate when you run a statistical test. Even
confidence interval though both groups have the same point estimate , the British estimate will have a wider confidence interval than the American estimate because there is more variation in the
data. While adopting a risk management standard has its advantages, it is not without challenges. The new standard might not easily fit into what you are doing already, so you could have to introduce
new ways of working.
What is a confidence level?
These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable https://www.globalcloudteam.com/
publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in oureditorial policy.
• The selection of confidence level is important because it impacts your sample size and confidence interval.
• The NPI team presents multiple options for manufacturing to the client, allowing clients to choose the solution that best suits their needs.
• Incremental value at risk is the amount of uncertainty added or subtracted from a portfolio by purchasing a new investment or selling an existing one.
• Choosing the appropriate template for a project occasionally results in heated debates between risk management professionals.
• The normal curve is plotted against the same actual return data in the graph above.
The confidence interval consists of the upper and lower bounds of the estimate you expect to find at a given level of confidence. Performing data transformations is very common in statistics, for
example, when data follows a logarithmic curve but we want to use it alongside linear data. You just have to remember to do the reverse transformation on your data when you calculate the upper and
lower bounds of the confidence interval.
Phase 2B: Detailed Design
The Commons debates the privileges committee special report on its partygate investigation that named seven MPs and three peers as those who put “improper pressure” on the investigation. Whether the
organization is willing or unwilling to accept the cyber-risk. Phase 0 is an optional phase for projects where the technical feasibility of the idea has not yet been fully proven. It can consist of
research, concept work, exploring initial architecture, performing feasibility studies, and basic prototyping and testing. Qualification of Suppliers – A risk-based approach is used in establishing
criteria for the evaluation, selection, and monitoring of suppliers. CAPA Process – A risk-based approach is used in determining when a corrective and/or preventive action should be initiated.
Risk management is the process of identifying, assessing and controlling financial, legal, strategic and security risks to an organization’s capital and earnings. These threats, or risks, could stem
from a wide variety of sources, including financial uncertainty, legal liabilities, strategic management errors, accidents and natural disasters. A risk may not fully need mitigation for it to drop
from the top contributors of risk exposure. Turning various intensities of mitigation on and off will result in the most cost-effective method of managing risks. With safety software, there’s also
less chance that your risk assessments will grow old and out of date.
WHAT IS RISK?
However, investment and commercial banks frequently use VaR to determine cumulative risks from highly correlated positions held by different departments within the institution. The VaR uses both the
confidence interval and confidence level to build a risk assessment model. Hold regular coordinated security exercises across the enterprise to provide further insight into cyber-risk levels and
mitigation needs. While executives and boards once viewed cybersecurity as a primarily technical concern, many now recognize it as a major business issue. Any organization that fails to protect its
sensitive digital assets from today’s increasingly sophisticated cyberthreats stands to pay a high price.
This method of risk management attempts to minimize the loss, rather than completely eliminate it. While accepting the risk, it stays focused on keeping the loss contained and preventing it from
spreading. To reduce risk, an organization needs to apply resources to minimize, monitor and control the impact of negative events while maximizing positive events. A consistent, systemic and
integrated approach to risk management can help determine how best to identify, manage and mitigate significant risks. Furthermore, your team can and should use this cost/benefit approach by running
a number of scenarios until they reach their target certainty.
Start by Asking the Right Questions
Rebecca is working on her PhD in soil ecology and spends her free time writing. If you want to know more about statistics, methodology, or research bias, make sure to check out some of our other
articles with explanations and examples. To find the MSE, subtract your sample mean from each value in the dataset, square the resulting number, and divide that number by n − 1 . These are all point
estimates, and don’t give any information about the variation around the number. Confidence intervals are useful for communicating the variation around a point estimate. They key is you’re measuring
the confidence of your methodology so the same principles apply whatever risk you are analysing.
In a z-distribution, z-scores tell you how many standard deviations away from the mean each value lies. This means that to calculate the upper and lower bounds of the confidence interval, we can take
the mean ±1.96 standard deviations from the mean. We have included the confidence level and p values for both one-tailed and two-tailed tests to help you find the t value you need. The point estimate
of your confidence interval will be whatever statistical estimate you are making (e.g., population mean, the difference between population means, proportions, variation among groups). The confidence
interval is the range of values that you expect your estimate to fall between a certain percentage of the time if you run your experiment again or re-sample the population in the same way.
Limit states design in geothermal engineering Struct. Saf.
How to Calculate Standard Deviation | Calculator & Examples The standard deviation is the average amount of variability in your dataset. The z-score and t-score (aka z-value and t-value) show how
many standard deviations away from the mean of the distribution you are, assuming your data follow a z-distribution or a t-distribution. Then you can plug these components into the confidence
interval formula that corresponds to your data. The formula depends on the type of estimate (e.g. a mean or a proportion) and on the distribution of your data. | {"url":"http://gepatunb.com/grc-tuesdays-why-i-love-risk-management-and-i/","timestamp":"2024-11-06T12:00:40Z","content_type":"text/html","content_length":"35982","record_id":"<urn:uuid:502974d7-3a09-4f33-8adf-524618d17992>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00651.warc.gz"} |
what is the formula for revenue?
Calculus Programs in Real Estate Enhancement
Calculus has quite a few genuine environment utilizes and programs in the bodily sciences, computer science, economics, company, and medication. I will briefly contact upon some of these takes
advantage of and programs in the serious estate business.
Let’s begin by working with some illustrations of calculus in speculative serious estate enhancement (i.e.: new property development). Logically, a new house builder wishes to turn a gain following
the completion of each home in a new residence local community. This builder will also want to be equipped to preserve (hopefully) a beneficial cash move through the building approach of just about
every household, or every stage of residence advancement. There are quite a few aspects that go into calculating a income. For example, we by now know the formulation for income is: P = R – C, which
is, the earnings (P) is equivalent to the earnings (R) minus the cost (C). Although this primary system is quite very simple, there are lots of variables that can aspect in to this method. For
example, below expense (C), there are several different variables of price, these as the price tag of setting up materials, expenses of labor, holding fees of authentic estate before obtain, utility
costs, and coverage quality expenditures through the construction period. These are a several of the quite a few expenditures to aspect in to the above mentioned formula. Less than profits (R), a
person could consist of variables such as the base marketing cost of the residence, supplemental updates or increase-ons to the household (protection method, encompass seem process, granite
countertops, etc). Just plugging in all of these various variables in and of by itself can be a overwhelming process. Having said that, this will become further more sophisticated if the rate of
adjust is not linear, necessitating us to alter our calculations since the charge of improve of a single or all of these variables is in the condition of a curve (i.e.: exponential price of adjust)?
This is a single space in which calculus will come into perform.
Let us say, final thirty day period we bought 50 households with an ordinary providing cost of $500,000. Not using other factors into thing to consider, our earnings (R) is price tag ($500,000) times
x (50 homes bought) which equal $25,000,000. Let us contemplate that the full price tag to construct all 50 houses was $23,500,000 as a result the financial gain (P) is 25,000,000 – $23,500,000 which
equals $1,500,000. Now, figuring out these figures, your boss has questioned you to optimize earnings for adhering to month. How do you do this? What price tag can you set?
As a uncomplicated example of this, let’s to start with compute the marginal revenue in terms of x of making a property in a new household community. We know that profits (R) is equal to the desire
equation (p) periods the models bought (x). We publish the equation as
R = px.
Suppose we have established that the desire equation for advertising a residence in this group is
p = $1,000,000 – x/10.
At $1,000,000 you know you will not market any households. Now, the price equation (C) is
$300,000 + $18,000x ($175,000 in fastened products costs and $10,000 for each dwelling marketed + $125,000 in set labor fees and $8,000 for each house).
From this we can work out the marginal income in terms of x (models sold), then use the marginal earnings to determine the price tag we really should cost to maximize profits. So, the earnings is
R = px = ($1,000,000 – x/10) * (x) = $1,000,000x – x^2/10.
Therefore, the earnings is
P = R – C = ($1,000,000x – x^2/10) – ($300,000 + $18,000x) = 982,000x – (x^2/10) – $300,000.
From this we can work out the marginal revenue by getting the derivative of the financial gain
dP/dx = 982,000 – (x/5)
To calculate the greatest profit, we set the marginal revenue equal to zero and remedy
982,000 – (x/5) =
x = 4910000.
We plug x back again into the desire purpose and get the following:
p = $1,000,000 – (4910000)/10 = $509,000.
So, the rate we should set to attain the greatest financial gain for every single household we provide should really be $509,000. The following month you provide 50 additional houses with the new
pricing framework, and internet a income enhance of $450,000 from the preceding thirty day period. Good career!
Now, for the upcoming thirty day period your manager asks you, the neighborhood developer, to discover a way to cut charges on home design. From in advance of you know that the price tag equation (C)
$300,000 + $18,000x ($175,000 in fastened products prices and $10,000 for each house sold + $125,000 in mounted labor prices and $8,000 for each dwelling).
Just after, shrewd negotiations with your developing suppliers, you were being equipped to minimize the mounted materials prices down to $150,000 and $9,000 for each residence, and decrease your
labor costs to $110,000 and $7,000 for each household. As a consequence your price tag equation (C) has changed to
C = $260,000 + $16,000x.
Since of these improvements, you will will need to recalculate the foundation gain
P = R – C = ($1,000,000x – x^2/10) – ($260,000 + $16,000x) = 984,000x – (x^2/10) – $260,000.
From this we can calculate the new marginal income by using the by-product of the new financial gain calculated
dP/dx = 984,000 – (x/5).
To compute the greatest income, we set the marginal income equal to zero and solve
984,000 – (x/5) =
x = 4920000.
We plug x back again into the demand from customers perform and get the next:
p = $1,000,000 – (4920000)/10 = $508,000.
So, the value we should set to obtain the new greatest revenue for every single house we offer ought to be $508,000. Now, even although we reduce the offering price from $509,000 to $508,000, and we
however market 50 units like the preceding two months, our revenue has continue to improved since we reduce charges to the tune of $140,000. We can find this out by calculating the variation
concerning the initial P = R – C and the second P = R – C which has the new price equation.
1st P = R – C = ($1,000,000x – x^2/10) – ($300,000 + $18,000x) = 982,000x – (x^2/10) – $300,000 = 48,799,750
2nd P = R – C = ($1,000,000x – x^2/10) – ($260,000 + $16,000x) = 984,000x – (x^2/10) – $260,000 = 48,939,750
Having the 2nd profit minus the very first revenue, you can see a change (boost) of $140,000 in earnings. So, by chopping costs on dwelling building, you are in a position to make the company even
more rewarding.
Let’s recap. By basically applying the need functionality, marginal earnings, and optimum revenue from calculus, and practically nothing else, you had been capable to support your organization raise
its regular monthly financial gain from the ABC Home Group task by hundreds of countless numbers of bucks. By a little negotiation with your setting up suppliers and labor leaders, you were ready to
reduced your fees, and by a very simple readjustment of the expense equation (C), you could swiftly see that by slicing expenses, you elevated earnings nevertheless again, even after modifying your
maximum earnings by reducing your selling price tag by $1,000 for each device. This is an illustration of the ponder of calculus when applied to serious world problems. | {"url":"https://abusinessowner.com/calculus-programs-in-real-estate-enhancement.html","timestamp":"2024-11-10T09:13:37Z","content_type":"text/html","content_length":"80954","record_id":"<urn:uuid:81e22e3a-62a8-4c0f-b0a4-8345daafa64a>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00062.warc.gz"} |
Finite Well Solutions: Understanding d and k in Potential Wells
• Thread starter physicsjock
• Start date
In summary: Try using trig angle-sum identities on sin(kx+d). I'm pretty sure that will work out to a form that you can more easily equate with (s/k)Asinkx+Acoskx
I've been trying to work out how, for a finite well of high Vo and width L, the interior solution has the form L Sin(kx + d),
I see that if d=0 then the solution resembles an infinite well, so that implies d depends inversely on the wells potential. But I can't work out what d comes from, and why the constant at the front
is the length of the well (why is the amplitude of the wave the length of the well)
d is just the phase of the wave, so does it represent the reflection of any particles which moved past the well?
Ive also been trying to work out what the k is, when I go through the process of finding the wave function inside I end up with (s/k)Asinkx + Acoskx after applying the boundary conditions.
where k^2=2mE/[itex]\hbar^{2}[/itex]
and s^2=2m(Vo-E)/[itex]\hbar^{2}[/itex]
i've been trying to work this out because the question also says that the solution inside the well should also satisfy
ka = n[itex]\pi - 2Sin^{-1}(\frac{k\hbar}{\sqrt{2mV_{o}}})[/itex]
which makes me think the d resembles that arcsin the above equation
Anyone have any ideas?
Staff Emeritus
Science Advisor
Homework Helper
It looks like you're trying to solve the time-independent Schrodinger equation, which should have been listed as a "Relevant Equation" in the homework template. It also puts this in the Advanced
Physics category, so I'll move the thread there.
physicsjock said:
I've been trying to work out how, for a finite well of high Vo and width L, the interior solution has the form L Sin(kx + d),
I see that if d=0 then the solution resembles an infinite well, so that implies d depends inversely on the wells potential. But I can't work out what d comes from, and why the constant at the
front is the length of the well (why is the amplitude of the wave the length of the well)
I don't think the constant is simply
-- the units are wrong -- but at any rate it would be found by normalizing the wavefunction.
d is just the phase of the wave, so does it represent the reflection of any particles which moved past the well?
It represents the fact that the potential is not zero at the well boundary x=0. This is related to the fact that the potential is nonzero beyond the boundary as well, i.e. the particle's wavefunction
penetrates beyond the boundary. I wouldn't really call that reflection though.
Ive also been trying to work out what the k is, when I go through the process of finding the wave function inside I end up with (s/k)Asinkx + Acoskx after applying the boundary conditions.
where k^2=2mE/[itex]\hbar^{2}[/itex]
and s^2=2m(Vo-E)/[itex]\hbar^{2}[/itex]
i've been trying to work this out because the question also says that the solution inside the well should also satisfy
ka = n[itex]\pi - 2Sin^{-1}(\frac{k\hbar}{\sqrt{2mV_{o}}})[/itex]
= ?
which makes me think the d resembles that arcsin the above equation
Anyone have any ideas?
It's hard to tell exactly what you are stuck on. Is it just in trying to relate the two forms of solution you have given,
[tex](s/k)A\sin kx + A\cos kx \ \small \text{ and } \normalsize \ B \sin(kx + d) \text{,}[/tex]
to each other?
Redbelly98 said:
It's hard to tell exactly what you are stuck on. Is it just in trying to relate the two forms of solution you have given,
[tex](s/k)A\sin kx + A\cos kx \ \small \text{ and } \normalsize \ B \sin(kx + d) \text{,}[/tex]
to each other?
Yea that's what I'm stuck on,
Sorry that ka = ... formula I wrote was from a set of notes where the the width of the well is a,
so a = L,
I've tried substituting kl into the interior solution, ψ(L)
ψ(x)=(s/k)Asinkx + Acoskx
so ψ(L)=(s/k)AsinkL + AcoskL
and ended up with
ψ(L)=-A cos(n[itex]\pi[/itex])
assuming that the k in that formula is the same as the k i posted before
Staff Emeritus
Science Advisor
Homework Helper
physicsjock said:
Yea that's what I'm stuck on,
Try using trig angle-sum identities on sin(kx+d). I'm
pretty sure
that will work out to a form that you can more easily equate with
Sorry that ka = ... formula I wrote was from a set of notes where the the width of the well is a,
so a = L,
I've tried substituting kl into the interior solution, ψ(L)
ψ(x)=(s/k)Asinkx + Acoskx
so ψ(L)=(s/k)AsinkL + AcoskL
and ended up with
ψ(L)=-A cos(n[itex]\pi[/itex])
assuming that the k in that formula is the same as the k i posted before
I'm not following you here, mainly because I can't do this in my head and don't have time right now to work it out on paper to verify what you are saying.
But, if you are satisfied with the "(s/k)Asinkx+Acoskx" form of the solution, we should probably just concentrate on seeing how "Sin(kx + d)" is equivalent. Will you also be needing to normalize the
wavefunctions, and finding the energy eigenvalues, or are you good with how to do that?
I worked it out, the d in the sign was just to compensate for the lack of cos in the solution,
I found the d was something like arcTan(k/s) and it was consistent with the boundaries and stuff,
Thanks for your help!
FAQ: Finite Well Solutions: Understanding d and k in Potential Wells
1. What is a potential well solution?
A potential well solution is a solution to a mathematical equation known as the Schrödinger equation, which describes the behavior of quantum particles. It represents the shape of the potential
energy experienced by a particle in a given system.
2. How are potential well solutions used in science?
Potential well solutions are used to model a variety of physical systems, such as atoms, molecules, and solid state materials. They are also used in various fields of science, including quantum
mechanics, chemistry, and materials science, to understand and predict the behavior of particles in these systems.
3. What are the different types of potential well solutions?
There are several types of potential well solutions, including infinite, finite, and harmonic potential wells. These solutions differ in the shape and depth of the potential energy curve, which
affects the behavior of the particles in the system.
4. How do potential well solutions relate to particle confinement?
Potential well solutions are often used to model systems where particles are confined, such as in atoms and molecules. The shape of the potential well determines the allowed energy levels and spatial
distribution of the particles, which can affect their properties and behavior.
5. Can potential well solutions be solved analytically?
In some cases, potential well solutions can be solved analytically using mathematical techniques. However, for more complex systems, numerical methods may be needed to approximate the solutions.
Additionally, different potential well shapes may require different mathematical approaches for finding solutions. | {"url":"https://www.physicsforums.com/threads/finite-well-solutions-understanding-d-and-k-in-potential-wells.587712/","timestamp":"2024-11-09T22:08:18Z","content_type":"text/html","content_length":"98034","record_id":"<urn:uuid:741c6523-6e1d-421d-be2d-1b174d45456a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00769.warc.gz"} |
24. [Bivariate Density & Distribution Functions] | Probability | Educator.com
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while
watching the lecture.
Section 1: Probability by Counting
Experiments, Outcomes, Samples, Spaces, Events 59:30
Combining Events: Multiplication & Addition 1:02:47
Choices: Combinations & Permutations 56:03
Inclusion & Exclusion 43:40
Independence 46:09
Bayes' Rule 1:02:10
Section 2: Random Variables
Random Variables & Probability Distribution 38:21
Expected Value (Mean) 46:14
Variance & Standard Deviation 47:23
Markov's Inequality 26:45
Tchebysheff's Inequality 42:11
Section 3: Discrete Distributions
Binomial Distribution (Bernoulli Trials) 52:36
Geometric Distribution 52:50
Negative Binomial Distribution 51:39
Hypergeometric Distribution 36:27
Poisson Distribution 52:19
Section 4: Continuous Distributions
Density & Cumulative Distribution Functions 57:17
Mean & Variance for Continuous Distributions 36:18
Uniform Distribution 32:49
Normal (Gaussian) Distribution 1:03:54
Gamma Distribution (with Exponential & Chi-square) 1:08:27
Beta Distribution 52:45
Moment-Generating Functions 51:58
Section 5: Multivariate Distributions
Bivariate Density & Distribution Functions 50:52
Marginal Probability 42:38
Conditional Probability & Conditional Expectation 1:02:24
Independent Random Variables 51:39
Expected Value of a Function of Random Variables 37:07
Covariance, Correlation & Linear Functions 59:50
Section 6: Distributions of Functions of Random Variables
Distribution Functions 1:07:35
Transformations 1:00:16
Moment-Generating Functions 1:18:52
Order Statistics 1:04:56
Sampling from a Normal Distribution 1:00:07
The Central Limit Theorem 1:09:55
Hello, welcome back to the probability lectures here on www.educator.com, my name is Will Murray.0000
We are starting a chapter on probability distribution functions with two variables.0006
From now on, we are going to have a Y1 and Y2.0012
Today, we are going to talk about Bivariate density and Bivariate distribution functions.0014
That is a lot to swallow, let us jump right into it.0020
Bivariate density functions, the idea now is that we have two variables, Y1 and Y2.0024
For example, you might be a student taking a certain number of units at college.0029
Y1 is the number of math units you have taken and Y2 is the number of computer science units that a student has taken.0036
All the different students at this university, each one has taken a certain number of math units0053
There is a density function which reflects the number of students who had taken0063
any particular number of computer science units or math units.0070
For example, we can say how many students have taken 10 or more math units, and 15 or more computer science units?0076
There is a certain proportion of the population that has taken more than 10 math units0084
These are things that we will graphs on 2 axis.0093
From now, all our graphs are going to be on 2 axis.0096
We will always put Y1 on the horizontal axis and Y2 on the vertical axis.0100
And then, the density will be distributed all over this plane of 2 axis.0107
Lets us see what we do with these density functions.0116
First of all, the density function always has to be positive.0119
You cannot have a negative number of students who have taken a certain number of units.0124
The smallest you can have would be, if I picked a particular combination of units,0129
there might be 0 students that have had that combination of units.0134
But, there would never be a negative number of students that have taken a certain number of units.0137
Second, if we look at the total density, that means the density over the entire plane.0141
All the possible combinations of units the students could have taken.0149
If we integrate over the entire plane, it has to come out to be 1 that is because it is a probability function.0156
The total density of students has to be 1, no matter how many students we have at this college,0161
We can calculate probabilities, when we graph this, as I explained it to you.0174
If we want to find the probability of any particular region, I will graph a rectangular region0181
The rectangular region where Y1 goes from A to B and Y2 goes from C to D, there is Y2.0191
We want to find the probability of landing within that region.0202
For example, we want to find the probability, maybe proportion of students that had taken0206
between 10 and 15 math units, and have taken between 20 and 30 computer science units.0211
What proportion is the total student body of this college has taken between 10 and 15 math units,0219
The way we do that is, we take a double integral, integral Y1 from A to B, Y2 from C to D.0229
And then, we integrate the density function over that range.0236
What this means is that you really have to remember calculus 3.0241
If you do not remember how to do double integrals, what you want to do is0246
We do have a whole set of lectures devoted to multivariable calculus here on www.educator.com.0256
My student colleague Raffi, teaches those lectures, he is amazing.0264
If you cannot remember how to do a double integral, go after his lectures and0268
you will be good to go for the rest of this chapter in probability.0272
If you have a discreet distribution, it is basically the same idea.0281
Instead of an F here, you will just change that to a P.0286
In fact, you will have a double summation, instead of a double integral, if you have a discreet distribution.0295
It is not as common though, usually Bivariate functions in probability classes,0301
it just turns out that you usually study continuous ones.0309
That is why you really have to know your multivariable calculus.0315
If you are a little rusty on that, you want to brush up on your multivariable calculus.0319
Another thing that we need to learn is the Bivariate distribution function,0326
it is kind of the two variable analogue of the distribution functions we had before.0332
The idea of the Bivariate distribution function is you have some cutoff values of Y1 and Y2.0338
Here is Y1 and here is Y2, we have some cut off values.0348
What we are interested in, is the probability of being less than both of those cutoff values.0356
You are interested in calculating the probability that Y1 is less than or equal to the cutoff value y1.0365
In other words, you want to find all the stuff in this region, right here.0378
All the stuff where Y1 is less than y1 and Y2 is less than y2.0385
If you think about that, that is just the double integral over that region.0392
We are going to call that function F of Y1 Y2.0397
It means you integrate from negative infinity to y1 and negative infinity to y2.0401
I cannot call it Y1 and Y2 anymore for the variables because I'm using them for the cutoff values.0408
I’m going to use T1 and T2, and then I’m going to integrate the density function.0413
Just like in some sense, the distribution function was the integral of the density function.0421
Back in single variable probability and Bivariate probability, the distribution function is0426
the double integral of the density function, from negative infinity up to the cutoff values that you are interested in.0433
Some properties of the distribution function satisfies, if E¹ Y1 Y2 is negative infinity,0440
it means you are not looking in any area at all.0447
Your value is going to be 0, no matter what the other variable is.0452
If you plug in infinity for both of them, that means you are really looking at this entire plane.0457
You are looking at all the possible density, it would have to be 1.0464
I think that is all the preliminaries now, we are ready to jump into the examples.0472
We are going to be doing a lot of integrals for these examples.0476
You really want to be ready to do some double integrals, even integrating over some non rectangular regions.0479
In the first example, we are going to consider the joint density function F of Y1 Y2 is defined to be K × Y2.0490
On the triangle with coordinates at 0-0, 0-1, and 1-1.0501
What we want to do with this is, we are going to find the value of K.0507
We are going to keep using this same formula for examples 2 and 3.0510
You really want to make sure you are up to speed on this.0514
Let me first graph this region because we are going to be seeing this over and over again.0517
We got the triangle with coordinates at 0-0, and 0-1, and 1-1.0524
I will graph that triangle and that is the region that we are interested in.0531
There is a Y1 is equal to 1, Y2 is equal 1, there is 0.0540
It looks like it is defined by the line Y = X, but since we are using Y1 and Y2 as our variables here, there is Y1 Y2,0550
that is the line Y1 = Y2, like the line Y = X there.0562
We want to find the value of K, the way we want to do that is, we want to remember that the total density has to be 1.0571
We want to integrate over that region and our answer will have a K in it somehow.0579
I will set the whole thing equal to be 1.0585
The best way to describe that region, I think, is to describe it with Y2 first.0591
I’m going to describe it, listing Y2 first, using constants for Y2.0596
Y2 goes from 0 to 1 and Y1 goes from 0 to Y2.0601
This is a hardcore multivariable calculus, if you do not remember how to set up these integrals0611
on these triangular regions, you really got to review it right now.0616
Go back and watch these lectures on multivariable calculus and you will get some practice with these triangular regions.0620
The integral from Y2 = 0 to Y2 = 1.0633
The integral from Y1 = 0 to Y1 = Y2.0639
My density function K Y2, let me pull my K outside, it is just a constant, Y2.0647
D Y1, I got to do first, and then D Y2.0654
I'm going to integrate the inside one first, the integral with respect to Y1 is just Y2 Y1.0659
Remember, you keep the Y2 constant when you are integrating with respect to Y1.0666
We want to integrate that from Y1 = 0 to Y1 = Y2.0671
If I plug those in, I will get Y2² -0.0679
I have to do the integral there, from Y2 = 0 to Y2 = 1.0685
I still have a K on the outside, I still have a D Y2.0692
I still have that K and I’m evaluating that from Y2 = 0 to Y2 = 1.0702
Plug those in and I get K ×, it looks like 1/3.0712
Remember that, the total density has to be equal to 1.0719
K × 1/3 is equal to 1, that tells me then that K is equal to 3.0723
That is kind of an excellent rule for any kind of multivariable calculus type problem.0740
You always want to graph the region, it is very helpful to graph the region.0745
I graphed the region, I graphed that triangle with those 3 points.0752
I wrote down a little equation for the line of the boundary which is just Y = X or Y2 = Y1.0757
To find the value of K, what I want to do is to use the fact that the total density has to be 1.0766
The integral over this region of this density function, it has to come out to be 1.0773
I describe that region, I chose Y2 to list first because that makes it a little bit simpler to set up the integral.0781
You could reverse the order of the variables there, listing Y1 first,0790
but I think you are going to get a slightly nastier integral.0795
You would not get as many 0 in the bounds there.0798
That is why I picked Y2 first to describe that region.0802
And then, I set up my double integral, I integrated Y2.0806
By the way, I use this as an example in my classes.0813
A lot of students say the integral of Y2 should be Y2²/2.0816
Not so, because we are not integrating with respect to Y2.0821
We are integrating with respect to Y1 which means the integral is just Y2 Y1.0824
And then, we evaluate for our boundaries on Y1 and we get Y2².0829
Now, we integrate that with respect to Y2 that is a fairly easy integral.0836
Since, that is supposed to be equal 1, that is kind of our rule.0843
Our rule is always that the double integral of DY1 DY2 is always equal to 1.0847
That tells me that the K has to be 3.0859
I hope this one made sense because we are going to keep using this example for problems 2 and 3.0862
Make sure you understand this one, we are going to take the answer of this one0867
and use it to answer some more complicated questions in example 2 and 3.0871
In example 2, we are going to keep going with the same setup from example 1,0881
except I will fill in the answer, the K was equal to 3.0888
If you are a little foggy on what was going on in example 1, go back and watch example 1, it will make more sense.0892
Let me go ahead and draw the region that we are interested in.0897
It is the same one as before 0-0, 0-1, and 1-1.0902
There is the region there, triangular region 0- 0, 0-1, and 1-1.0909
That is Y1 there, my goodness what am I thinking.0920
I want to find F, that is the distribution function of 1/3, ½.0928
The first thing to do is to remember what that distribution notation means.0933
F of 1/3, ½, the fractions are going to get nasty in this one.0940
That is the probability that Y1 is less than or equal to 1/3 and Y2 is less than or equal to ½.0949
If you do not remember that definition of F, just click back a couple slides ago and0965
Remember, keep track of the difference between F, the distribution function, and f the density function.0973
We want to find, the probability of Y1 being less than 1/3 and Y2 being less than ½.0986
I got this region that sort of shape like a backwards state of Nevada.1007
There it is right there, there is my backwards state of Nevada.1012
What I want to do is to integrate over that region.1017
To integrate over that region, I need to describe that region.1021
It looks like, if I want to do it in one piece, I have to describe my Y1 first.1026
That goes to 1/3, that goes to ½, and of course, these are both going to start at 0.1032
Y1 is going to go, if I have constants for that, it is going to go from 0 to 1/3, that is Y1 1/3.1039
Y2 is not going to go from 0 to ½, otherwise, I would have a rectangle, and Nevada is not a rectangle.1050
It is going to go from, that line right there was the line Y2 is equal to Y1.1060
Y2 goes from Y1 on up to Y2 goes from Y1.1068
Maybe I should make that in black to make it a little more visible there.1077
Now, I have some boundaries, I can set up my integral.1085
This probability, I'm going to integrate Y1 goes from 0 to 1/3.1089
Let me go ahead and write the variables in there.1100
Y1 is equal to 1/3 and Y2 goes from Y1 up to ½ there.1102
DY2 and DY1, I have a double integral to solve.1122
Probably, the hardest part is setting up the double integral.1131
If you are lucky and your teacher is a nice person, and you can even use a calculator1136
I'm going to go ahead and work it out by hand, just to prove that I'm an honest upstanding human being.1144
We have to integrate 3Y2, DY2 the integral of Y2 is Y2²/2 Y2².1152
Y2², we are integrating that from Y2 = Y1 to Y2 = ½.1165
I will keep the 3/2 here and Y2² from ½ to Y1 will give me, ½² is ¼ - Y1²,1175
We are supposed to integrate that from Y1 = 0 to Y1 = 1/3.1189
I have to do calculus 1 integral, I get 3/2, ¼ Y1 - Y1² integrates to 1/3 Y1³.1201
That is going to be a bit nasty to deal with.1219
All of these evaluated from Y1 = 0 to Y1 = 1/3, I get 3/2.1222
¼ × Y1 = 1/3 is ¼ × 1/3 is 1/12 -, 1/3 × Y1³.1234
Another 1/3 multiplied by that gives me 1/81, all the horrors here.1247
I'm not going to write anything for Y1 = 0 because both of those terms will drop out.1253
These fractions simplify a bit because 3/2 × 1/12 is 3/24 is 1/8.1261
3/2 × 1/81, the 3 will cancel with the 81 give me a 27 × 2 is 54.1270
Not too bad, I think I'm going to have a common denominator there of 216.1280
216, because 216 is 8 × 27, it is 54 × 4, that simplifies down to 23/216.1288
I did plug that into a calculator, in case you are fond of decimals, 0.1064.1303
That is my probability, the probability that you will end up in that small Nevada State region.1323
Or another way to think about that is what we just calculated is F of 1/3 and ½.1332
First of all, use the definition of the Bivariate distribution function.1346
It just means the probability that Y1 is less than 1/3 and Y2 is less than ½.1351
Then, I went to try to draw that region on my full graph.1358
In turn, I took that drawing which gave me a sort of Nevada shaped region.1371
Then, I converted that, it is actually a backwards Nevada is not it.1381
But I have been calling it a Nevada shaped region.1387
If you look at Nevada in a mirror, this is what it looks like.1390
Then, I tried to describe that in terms of variables.1393
Prepare a tree to set it up and a double integral.1397
I found this Y1 goes from 0 to 1/3, Y 2 goes from y1 to ½.1401
Notice, I would like to say Y2 goes from 0 to ½ but that would be wrong because1409
that would give me a rectangular region, that is not what I want.1418
I have to say, Y2 goes from Y1 to ½.1423
I took those limits and I set up a double integral here.1428
The 3 Y2 comes from the density function up here, and then it is just a matter of cranking through the double integral.1434
Not very hard, a little bit tedious, easy to make mistakes.1442
I factored out some constants, plug in your bounds which gives you everything in terms of Y1.1450
Do another integral, get some nasty fractions, simplify them to a slightly less nasty fraction.1457
If you like, you can leave your answer as a fraction.1465
In example 3, we are going to keep going with the same region and the same density function from example 1.1475
We have got the triangular region from 0-0 to 1-1, and 0-1.1489
There is 1, there is the Y2 axis, and there is the Y1 axis.1499
We got this triangular region and we got a density function defined on that region.1506
We want to find the probability that 2Y1 is less than Y2 or Y2 is bigger than 2Y1.1513
I'm going to go ahead and try to draw the region that we are interested in.1525
If I say 2Y1 is equal to Y2, that is like saying Y2 is equal to 2Y1.1534
To make that more familiar to people who graph things like this in algebra, it is like saying Y =2X.1542
Let me put Y = 2X, that is going to be twice as steep.1555
We actually want to have Y2 greater than 2Y1, that greater than 2X.1570
That means, we want the region of both that line.1577
We are going to find the probability of landing in that blue region.1586
I think the best way to describe that blue region is to describe Y2 first.1594
Y2, I can see it is going from 0 to 1.1603
Y1 is going from 0 up to that line, that line was Y1 is equal to Y2/2.1613
I have to make that my upper bound for Y1, ½ Y2.1639
The reason I spent much time describing it that way is that, that sets me up for a nice double integral.1646
My probability is equal to the double integral on that region.1654
I can just use that description Y2 goes from 0 to 1 and Y1 goes from 0 to ½ Y2.1661
Once again, it is a multivariable calculus problem DY1 DY2.1689
I'm just going to work that out as a multivariable calculus problem and integrating with respect to Y1 first.1696
I will put a 3 on the outside, integral of Y2 with respect to Y1 is Y2 × Y1.1704
It is not Y2²/2 because we are not integrating with respect to Y2.1712
Be careful about that, that is a very common mistake that my own students make all the time.1717
Even I, make that mistake, if I’m not being careful.1723
Let me integrate that from Y1 = 0 to Y1 = ½ Y2.1726
What I get there is, there is still a 3 on the outside.1734
I’m just doing this first integral, not worrying about the second one yet.1738
I get ½ Y2², when I plug in my Y1 = Y2.1742
I have got the integral from Y2 = 0 to Y 2 = 1 of Y2² DY2, factoring the outside terms there.1756
Then, I will evaluate that from Y2 = 0 to Y2 = 1.1785
I get ½ × 1 -0 which is just ½.1792
That is nice and pleasant, much simpler answer than we have for the previous example.1799
The key starting point here is we have that same region, that triangular region,1807
We want to find the probability that 2Y1 is less than Y2.1818
I wanted to graph that region, to see what part of the region that was.1823
I got this line here, that is the line 2Y1 is equal to Y2, or you can write that as Y1 is ½ Y2.1830
I want the region above the line because I want Y2 to be bigger than 2Y1.1840
That is why I took the region above the line not below it.1846
That is why I got this blue region colored in right here.1849
I want to describe it and that would be more convenient to list Y2 first, so I can use constants for Y2.1852
And then, I want my bounds for Y1 would be 0 and the other bound is ½ Y2, I got that from the line.1860
That ½ Y2, that comes from right here, that is where that comes from.1867
I took these bounds and I set them up as the limits on my integral.1875
That comes from the stem of the problem, the 3Y2.1884
Now, it is just a matter of working through a double integral.1889
The first variable I’m integrating is DY1, that is why the integral is Y2 × Y1.1895
Run that through the limits, get Y2², integrate that with respect to Y2.1907
Now, it just simplifies down into the very friendly fraction of ½.1912
In example 4, we have a new joint density function here.1926
F of Y1 Y2 is defined to be E ⁻Y1 + Y2.1930
The region we are looking at is Y1 bigger than 0, Y2 bigger than 0.1936
Note that, there is no upper bounds given on that, that means Y1 and Y2 can go all the way to infinity.1941
There is Y1 and there is Y2, we want to find the probability that Y1 is less than 2 and Y2 is bigger than 3.1952
We want to find, let me go ahead and draw those lines there.1974
Y1 should be less than 2, we want to go to the left of that vertical line.1978
Y2 should be bigger than 3, I want to go above that horizontal line.1984
And that is the region that we are going to integrate over, in order to find this probability.1994
I'm going to set up a double integral on that region.2000
The integral, I think I can list Y1 first safely.2005
Y1 goes from 0 to 2 because 2 is the upper bound for Y1.2009
Y2, that is where I bring in the 3, Y2 is 3.2019
I have to run that to infinity because that one just goes on forever.2023
Maybe, you are uncomfortable saying Y2 is equal to infinity.2029
Maybe, I will say Y2 approaches infinity but it does not really affect the calculations that we will be doing.2032
It will be fairly easy to plug in infinity, after we do the calculations.2039
The density function is E ⁻Y1 + Y2, that is a quantity there.2043
I think the best way to approach this is to factor the density function into E ⁻Y1 × E ⁻Y2.2057
The point of doing that, is that in the first integral, we are integrating with respect to Y2.2066
We can take the E ⁻Y1, that is just a big old constant now.2074
We can pull it all the way out of the integral, let me go ahead and do that.2078
We have the integral of E ⁻Y1, and now the integral from Y2 = 3 to Y2 goes to infinity of E ⁻Y2 DY2.2081
There is a DY1 on the outside but let me just handle that first integral inside.2098
That is a little substitution there, a little old calculus 1 trick.2108
I'm evaluating that from Y2 = 3 to Y2 goes to infinity.2112
I should be introducing a T and take the limit as T goes to infinity.2121
I’m being a little sloppy about that, that is kind of the privilege of having been through so many calculus classes.2125
When Y2 goes to infinity, we get E ⁻infinity here, that is 1/E ⁺infinity.2132
That is just 0- E⁻³, that all simplifies down into E⁻³.2139
I still have that E ⁻Y1, I’m going to bring that back in here.2151
E⁻³ is just a constant, I will write that separately.2161
My Y1 went from 0 to 2, Y1 = 0 to Y1 = 2.2172
Plugging in Y1 = 0, I get E⁰ which is just 1, it is - -1.2190
If I simplify this, I get E⁻³, that one becomes positive.2201
It is 1 – E⁻², and I could factor that through.2206
I get E⁻³ – E⁻³ × E⁻², you add the exponents, it is E⁻⁵.2210
That is really my exact answer but it is not very illuminating.2223
I did find the decimal for that, and my calculator told me that that is approximately equal to 0.043.2228
if you want to convert that into a percentage, then it is 4.3%.2239
We have an answer for that one. Let me show you again the steps involved to finding it.2251
First of all, I graphed the whole region Y1 greater than 0 and Y2 greater than 0.2255
That is a whole planar region, quarter plane because it is just where Y1 and Y2 are positive, and that is my whole region.2260
What I'm really interested in, is the probability of Y1 being less than 2 and Y2 being bigger than 3.2271
I chopped that up and I found that region was this blue region colored in here.2279
That is where Y1 is less than 2 and Y2 is bigger than 3.2284
In order to integrate that, I had to describe that limits.2289
Y1 goes from 0 to 2, Y2 goes from 3 to infinity.2293
I plugged in my density function, there is my density function right there, and I plug it in here.2299
The nice thing about that is I can factor that density function into E ⁻Y1 and E ⁻Y2.2305
Since, my first integral, the inside integral is going to be a Y2, I can pull out the E ⁻Y1.2313
That is just a constant, and that got pullout as a constant, outside the first integral.2319
I’m just left with E ⁻Y2 which integrates to –E ⁻Y2.2325
Evaluate that from 3 to infinity, when we plug in infinity or take the limit2336
as the variable approaches infinity, we will get 1/E ⁺infinity.2341
That is what that infinity term gives you, is the 0.2349
And then, it turn out to be a +E⁻³, that is just a constant in the next step because2353
I get -E ⁻Y1, plug in the values, do a little bit of algebra and simplifying.2362
I plugged that into a calculator, just to see what kind of decimal we are talking about.2376
It should always come out to be positive, when we are finding these probabilities.2380
If you do not get a positive probability, in fact, if you do not get something between 0 and 1, you know you screwed up.2383
I like to plug things in and just get a number.2389
Yes, that is between 0 and 1, it is not too surprising.2397
That is my probability of landing in that region with that density function.2401
We will use the same region and density function for example 5.2406
Make sure you understand this very well, before you move on to example 5.2411
It is it is the same density function, we will be integrating a different corner of the region, let me put it that way.2417
In example 5, we are going to look at the joint density function, the same one that we had in example 4.2428
Let me go ahead and remind you of what that looked like there.2435
We have this graph and we are looking at the entire positive quarter plane there.2438
There is Y1 and there is Y2, and everything is going from 0 to infinity.2444
We want to find the probability that Y1 + Y 2 will be less than or equal to 2.2451
Let me draw the line Y1 + Y 2 = 2.2457
There it is, it is a diagonal line and it got a slope of -1.2462
That is the line Y1 + Y2, Y1 + Y2 is equal to 2.2467
It is just X + Y = 2 and you can solve that out.2475
I want it to be less than or equal 2, which means I need to look at the region underneath that line.2486
I want to describe that region and then do a double integral, in order to find the probability of landing in that region.2495
The first thing I’m going to do is try to describe that region.2504
It does not really matter which variable you list first here, but I listed Y1 first.2509
I can use constants for Y1, I’m going to go from 0 to 2.2516
And then, I listed Y2 but I cannot use constants for that because otherwise, I will get a rectangle.2520
Y2 goes from 0 to, if you solve that line out, you get Y1 + Y2 = 2.2525
If you solve for Y2, you will get Y2 is equal to 2 - Y1.2537
I’m going to use that as my upper bound, that is going to make for some nasty integration but there is no way around it.2543
My probability is, it will be the double integral on that region.2550
I already set up the limits here, I have done the hard part.2556
Y1 = 0 to Y1 = 2 and Y2 = 0 to Y2 = 2 -Y1.2564
I have that same density function E ⁻Y1 + Y2.2576
Remember, the old trick that we use back in example 4 works again, is to write that density function as E ⁻Y1 × E ⁻Y2.2589
The utility of that is that is we are integrating Y2 first.2599
And that means, E ⁻Y1 is a constant and I can pull it out of the integral.2606
I'm going to pull that out of the first integral, of the inside integral.2612
And then later on, I will do the integral with respect to Y1.2626
E ⁻Y1, the integral of E ⁻Y2 is –E ⁻Y2.2631
I need to evaluate this from Y2 = 0 to Y2 is equal to 2 -Y1.2638
This is a little bit nasty, I got –E ⁻Y2.2651
If Y2 is 2 -Y1, -Y2 will be Y1 -2, --E⁰, --1.2656
I will get +1 - E ⁺Y1 -2 at the next step.2673
I’m going to bring this E ⁻Y1 from over on the left.2681
I think that is going to be useful because I’m going to go ahead and multiply that through.2686
I get E ⁻Y1 -, E ^- Y1 + Y1 – 2.2692
That is not too bad, what am I supposed to do this.2706
I’m supposed to integrate it with respect to Y1 DY1.2709
Let me go ahead and keep going on the next column here.2717
The integral of E ⁻Y1, do a little substitution is -E –Y1.2722
E⁻² was just a constant, it is –E⁻² × Y1.2729
This whole expression is supposed to be evaluated from, where are my limits, right there at the beginning.2736
I plug in Y1 = 2, I get -E⁻² - E⁻² × 2.2747
If I plug in Y1 = 0, I get + 1 because the two negatives cancel, I’m subtracting a negative.2758
And then, + 0 because we got Y1 = 0 in the last term there.2767
This simplifies a bit, I got 1 – E⁻² – 2 E⁻².2775
1 -3 E⁻², that is as good as it is going to get.2782
I did plug that into a calculator to get a decimal approximation.2788
Again, it is between 0 and 1, that is reassuring every probability answer should be between 0 and 1.2801
What I get there, if I wanted to make that into a percent, that is 59.4%.2810
That is my probability of landing in that sort of triangular region in the corner.2818
A probability that Y1 + Y2 is less than or equal to 2.2824
That is the probability that I have been asked to compute there.2830
Let us recap that, first of all, I graphed the whole region which is the positive quarter plane here.2835
Let me see if I can draw that without making things too messy on the graph.2844
And that is the whole region but that is not the region we are interested in.2850
We are interested in the region where Y1 + Y2 is less than or equal to 2.2856
That is what this diagonal line is, it is line Y1 + Y2 is equal to 2.2865
That is why I colored in this blue triangular region here.2873
I was trying to describe that region, in terms of variables.2877
I did use constants for the first, Y1 goes from 0 to 2.2881
But I cannot say Y2 goes from 0 to 2, otherwise, I will have a square.2885
I do not want a square, I need a triangle.2889
I said Y2 was less than 2 - Y1 and that came from solving out the equation of the line in terms of Y2.2891
Once I had that description, that was the hardest part of the problem.2904
Then, I just dropped those in as my limit for the integral, I dropped my density function in.2906
At this point, you could drop the entire thing into a calculator.2913
you could throw into some kind of computer algebra system, or an online on integration system.2921
I’m trying to be honest with you, I’m trying to do it by hand.2927
I factored E ⁻Y1 + Y2 as E ⁻Y1 E ⁻Y2.2931
The important part about that is that, in this first integral, this inner integral,2938
we are integrating with respect to Y2, which means we can treat Y1 as a constant.2943
That is why I pulled E ⁻Y1 all the way out of the integral, which gives me a nicer integral E ⁻Y2 on the inside.2949
That integrates to E ⁻Y2, and then I plugged in my bounds here to get something little messy.2958
I multiplied E ⁻Y1 back through and it simplified a bit.2965
And then, I integrated that E ⁻Y1 integrates to –E ⁻Y1.2970
E⁻² is a constant, when you integrate that, it is E⁻² × Y1.2978
I plug in the bounds Y1 = 2 all the way through and Y1 = 0.2985
Simplified it down a little bit to get this slightly mysterious number 1 -3 E⁻².2991
When I converted that into a decimal, I got something that was between 0 and 1.2998
That is a little reassuring that we are doing a probability problem.3003
If it had not been between 0 and 1, I would have known that I was wrong.3006
Any one of those forms, if you gave it to me in my probability class, I will be happy.3013
You do not have to convert it into a percentage, but if you like to know what it is as a percentage, there it is.3018
That wraps up this lecture on Bivariate density and distribution functions.3025
This is part of the chapter on multivariate probability density and distributions.3030
We are going to move on to marginally conditional probability, in our next video.3037
This is all part of the larger lecture series on probability, here on www.educator.com.3042
I'm your host, Will Murray, thank you very much for watching today, bye now.3049
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Single room |5 Building estimation items - Easy Home Builds
Single room |5 Building estimation items
Given below the cross-section details of the single-room building.
single room section
Based on the above details the following 5 building estimation quantities for the single room shall be calculated.
1. Earthwork excavation quantity for the foundation
2. Concrete quantity for the foundation
3. CRS (crushed rouble stone) masonry quantity
4. Concrete quantity in plinth beams
5. 1st class brickwork in single room building superstructure.
single room centre to centre plan
The longwall length from centre to the centre can be calculated =6+0.5×0.230+0.5x.230=6.23 m
The short wall length from centre to the centre can be calculated =5+0.5×0.230+0.5×0.230=5.23 m.
1.Earthwork excavation quantity in foundation :
single room foundation plan
From the single-room building section, depth of foundation from ground level(GL) is 900mm (300 mm +300 mm+300 mm) and width of the foundation is 900 mm.
Total length of one long wall for foundation excavation = 6.23 m+0.45 m+0.45 m = 7.13 m
The total length of one short wall for foundation excavation =5.23 m- 0.45 m -0.45 m =4.33 m.
So, total earthwork excavation quantity in foundation for long walls = 2 nos x 7.13 m x 0.9 m depth x 0.9 m width =11.55 cum
Total earthwork excavation quantity in foundation for short walls = 2 nos x 4.33 m x 0.9m depth x 0.9 m width = 7.01 cum.
So Total building estimation item, earthwork excavation quantity for the foundation of single-room building = 11.55 cum + 7.01 cum =18.56 cum.
2.Concrete quantity for the foundation :
Total length of one long wall for foundation Concrete = 6.23 m+0.45 m +0.45 = 7.13 m
The total length of one short wall for foundation Concrete =5.23 m-0.45 m-0.45 m=4.33 m
From the single-room building section, depth of concrete foundation is 0.3 m.
So, total concrete quantity for foundation for long walls = 2 nos x 7.13 m x 0.3 m depth x 0.9 m width =3.85 cum
Total concrete quantity for foundation for short walls = 2 nos x 4.33 m x 0.3 m depth x 0.9 m width =2.33 cum
Total building estimation item, concrete quantity for foundation is 3.85 cum + 2.33 cum =6.18 cum.
3.Crushed rubble stone masonry quantity
As per the single-room building section, CRS masonry is laid in 2 steps.
In 1st step crushed rubble stone ( CRS) wall width is 600 mm and height is 300 mm.
In the 2nd step, CRS wall width is 500 mm and the depth is 300 mm.
1st step CRS wall
Long CRS wall
The total length of one long CRS wall in 1st step = 6.23 m+0.3 m + 0.3 m =6.83 m
So Total long CRS wall quantity in 1st step=2 nos x 6.83 m x 0.6 m x 0.3 m = 2.45 cum
Short CRS wall
The total length of short CRS wall in 1st step = 5.23 m – 0.3 m -0.3 m =4.63 m.
Total short CRS wall quantity in 1st step =2 nos x 4.63 m x 0.6 m x 0.3 m = 1.66 cum
2nd step CRS wall
Long CRS wall
The total length of one long CRS wall in 2nd step = 6.23 m+0.25 m + 0.25 m =6.73 m
So Total long CRS wall quantity in 2nd step=2 nos x 6.73 m x 0.5 m x 0.3 m = 2.01 cum
Short CRS wall
The total length of short CRS wall in 2nd step = 5.23 m – 0.25 m -0.25 m =4.73 m.
Total short CRS wall quantity in 2nd step =2 nos x 4.73 m x 0.5 m x 0.3 m = 1.41 cum
Total CRS masonry quantity
So Total building estimation items, CRS masonry quantity for building including 1st and 2nd step =2.45 cum +1.66 cum +2.01 cum + 1.41 cum =7.53 cum.
4.Concrete quantity in plinth beams
As per the section, the depth of the plinth beam is 600 mm and width is 400 mm.
Long plinth beam
The total length of long plinth beam = 6.23 m+0.20 m + 0.20 m =6.63 m
So Total long plinth beam concrete quantity =2 nos x 6.63 m x 0.4 m x 0.6 m = 3.18 cum
Short plinth beam
The total length of short plinth beam = 5.23 m – 0.20 m -0.20 m =4.83 m.
Total short plinth beam concrete quantity =2 nos x 4.83 m x 0.4 m x 0.6 m = 2.31 cum
Total plinth beam quantity
So Total building estimation item, plinth beam concrete quantity for building=3.18 cum +2.31 cum =5.49 cum
5.1st class brickwork in the superstructure
As per the given section, the brick wall thickness is 230 mm and height is 3.5 m.
Long brick wall
The total length of long brick wall =6.23 m+0.115 m + 0.115 m =6.46 m
So Total long wall brickwork quantity =2 nos x 6.46 m x 0.23 m width x 3.5 m height = 10.4 cum
Short brick wall
The total length of short brick wall = 5.23 m – 0.115 m -0.115m =5 m.
Total short wall brickwork quantity =2 nos x 5 m x 0.23 m x 3.5 m = 8.05 cum
Total 1st class brickwork quantity
So Total building estimation items, 1st class brickwork quantity for building=10.4 cum +8.05 cum =18.9 cum
Conclusion :
So by the above process, we can calculate 5 building estimation items of one single room.
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6 Comments on “Single room |5 Building estimation items”
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Multiplication Facts To 144 No Zeros No Ones A Multiplication | Multiplication Worksheets
Multiplication Facts To 144 No Zeros No Ones A Multiplication
Multiplication Facts To 144 No Zeros No Ones A Multiplication
Multiplication Facts To 144 No Zeros No Ones A Multiplication – Multiplication Worksheets are a terrific means to instruct kids the twelve times table, which is the holy grail of elementary
mathematics. These worksheets serve in mentor pupils one aspect at once, yet they can also be used with 2 elements. Commonly, these worksheets are grouped into anchor teams, and also trainees can
start finding out these facts individually.
What are Multiplication Worksheets?
Multiplication worksheets are a valuable method to assist students learn math realities. They can be made use of to educate one multiplication truth at a time or to examine multiplication facts as
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Word troubles are another method to link multiplication with real-life situations. They can boost your youngster’s understanding of the concept while boosting their computation rate. Lots of
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What is the Purpose of Teaching Multiplication?
It’s vital to start showing kids multiplication early, so they can appreciate the procedure. Youngsters often end up being bewildered when offered with way too many realities at the same time, so
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One of the most effective discovering aids for kids is a reproduction table, which you can print out for each and every child. Kids can exercise the table by duplicating additions as well as counting
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Multiplication Facts To 81 Including Zeros H Math Fact Worksheets
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100 Math Facts Multiplication Worksheet Times Tables Worksheets
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The Big 'O' Notation - An Introduction
A long time ago, before the emergence of 10x engineers, software engineers sought ways to solve real-world problems with their programming skills. Different engineers came up with different solutions
/algorithms to solve these problems. There was a need to compare these solutions based on their complexity and efficiency. This need gave birth to the Big O notation. What is the Big O notation then?
What is the Big O notation?
Big O notation is used in computer science to describe the performance or complexity of an algorithm.
Algorithms are to computer programs what recipes are to dishes. Different recipes can help you to make a particular meal but they don't always yield the same results. They don't always have the same
steps, ingredients nor take the same amount of time to follow. Some recipes are faster and produce better results than others.
In the same way, different algorithms can achieve a particular computer program. However, they are not all equally efficient or take the same amount of time to run. We use Big O to measure the
efficiency of an algorithm.
For example, let's consider sorting. There are many sorting algorithms e.g. mergeSort, bubble sort, quicksort, and insertion sort. How do you know which is more efficient or less complex? This is why
the Big O notation exists.
You might wonder, why do we need a notation? Why don't we just consider the time it takes to run the algorithm? Here are two of many reasons:
1. Different computers have different processors and thus some computers will spend less time running an algorithm than others.
2. Some programming languages are faster than others.
It will be stressful taking all these factors into consideration when trying to analyze an algorithm. Rather than do that, the Big O notation uses something more standard - the input. It considers
how the runtime of the algorithm grows in relation to the size of the input. It is also good to note that the Big O notation considers the worst-case scenario for its analysis.
Hopefully, you get the sense of it now. Next question that might come to your mind is "Why should I know this?"
###The Why?
1. For a small app that processes little data, such analysis might be unnecessary. This is because the difference in the runtime of algorithms might have little impact on your app. For large
applications that manipulate a large amount of data, this analysis is crucial. Because inefficient algorithms will create a significant impact on the processing time. With a good knowledge of
Big-O notation, you can design algorithms for efficiency. Thus, you'll build apps that scale and save yourself a lot of potential headaches.
2. For your coding interviews. Yeah, you heard me right. You are likely to get asked by an interviewer the runtime complexity of your solution. So it's good to have an understanding of the Big O
Let look at some common examples of Big O Notation
###Common Runtime complexities
Source: https://www.bigocheatsheet.com
####1. O(1) - Constant Runtime
In this case, your algorithm runs the same time, regardless of the given input data set.
An example of this is returning the first element in the given data like in the example below.
function returnFirst(elements) {
return elements[0]
The runtime is constant no matter the size of the input given.
####2. O(n) - Linear Runtime
Linear runtime occurs when the runtime grows in proportion with the size of the input data set. n is the size of the input data set.
A good example of this is searching for a particular value in a data set using an iteration like in the example below.
function constainsValue(elements, value) {
for (let element in elements) {
if (element === value) return true;
return false
We see that the time taken to loop through all elements in the array grows with an increase in the size of the array. But what if the element is found before it reaches the last element in the array?
Does the runtime complexity change?
Remember that the Big O notation considers the worst-case scenario. In this instance, it's the case where the loops run through all elements in the array. So that is what determines the runtime
complexity of the algorithm.
####3. O(n^2 ) - Quadratic Runtime
O(n^2 ) denotes an algorithm whose runtime is directly proportional to the square of the size of the input data set. An example of this is a nested iteration or loop to check if the data set contains
function constainsDuplicate(elements) {
for (let element in elements) {
for (let item in elements){
if (element === item) return true;
return false
Deeper nested iterations will produce runtime complexities of O(n^3 ), O(n^4 ) etc
####4. O(log n) - Logarithmic runtime In this case, the runtime it takes for the algorithm to run will plateau no matter the size of the input data set.
A common example of this is a search algorithm like the binary search. The idea of a binary search is not to work with the entire data. Rather, reduce the amount of work done by half with each
iteration. The number of operations required to arrive at the desired result will be log base 2 of the input size.
For further information on this runtime complexity, you can check some of the resources at the end of the article.
####5. O(n log n) - Linearithmic runtime Here, the runtime of the algorithm depends on running a logarithm operation n times.
Most sorting algorithms have a runtime complexity of O(n log n)
####6. O(2^n ) - Exponential runtime This occurs in algorithms where for each increase in the size of the data set, the runtime is doubled. For a small data set, this might not look bad. But as the
size of the data increase, the time taken to execute this algorithm increases rapidly.
A common example of this is a recursive solution for finding Fibonacci numbers.
function fibonacci(num) {
if (num <= 1) return 1;
return fibonacci(num - 2) + fibonacci(num - 1)
####7 O(n!) - Factorial runtime In this case, the algorithm runs in factorial time. The factorial of a non-negative integer (n!) is the product of all positive integers less than or equal to n. This
is a pretty terrible runtime.
Any algorithm that performs permutation on a given data set is an example of O(n!)
###Conclusion Hopefully, this article has helped you to grasp the concept of Big O notation.
Here are some resources where you can find more info on this topic.
Got any addition or question? Please leave a comment.
Thanks for reading😊.
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What are the
What are the methods of integration?
Methods of Integration
• Integration by Substitution.
• Integration by Parts.
• Integration Using Trigonometric Identities.
• Integration of Some particular function.
• Integration by Partial Fraction.
What are the three methods of integration?
Points to Remember:
• Integration by Substitution.
• Integration by Parts.
• Integration by Partial Fraction.
• Integration of Some particular fraction.
• Integration Using Trigonometric Identities.
What is integration method in physics?
Integration is the reverse operation to differentiation i.e. it is the process of getting from the derivative start fraction, d, g, left bracket, x, right bracket, divided by, d, x, end fraction,
equals, g, prime, left bracket, x, right bracket,dxdg(x)=g′(x) to the function g, left bracket, x, right bracket,g(x).
How many types of integration are there?
Integration is one of the two main concepts of Maths, and the integral assigns a number to the function. The two different types of integrals are definite integral and indefinite integral.
What is C in integration formula?
The fundamental use of integration is as a continuous version of summing. The extra C, called the constant of integration, is really necessary, since after all differentiation kills off constants,
which is why integration and differentiation are not exactly inverse operations of each other.
Where is integration used in physics?
So one possible use of integration is to find distance using velocity, or finding velocity using acceleration. If a function of one of these components over time is known, then integration is the
fastest method to apply. More refined examples do exist since integration is necessary under complex circumstances.
What are the 2 types of integration?
Vertical integration occurs when a business owns all parts of the industrial process while horizontal integration occurs when a business grows by purchasing its competitors.
Which is the best technique for integration by parts?
Integration by Parts – In this section we will be looking at Integration by Parts. Of all the techniques we’ll be looking at in this class this is the technique that students are most likely to run
into down the road in other classes. We also give a derivation of the integration by parts formula.
How to find the required area for integration?
The area required is the area under the curve between 0 and 1 . . . . . . minus the area under the line (a triangle ) Area of the triangle Area under the curve Method 1 Substitute in Required area
14. Instead of finding the 2 areas and then subtracting, we can subtract the functions before doing the integration. Area We get Method 2 15.
How are data integration methods used in business?
This process involves a person or system locating, retrieving, cleaning, and presenting the data. Data managers and/or analysts can run queries against this merged data to discover business
intelligence insights.
What are the guidelines for an integration strategy?
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miniCTX: Neural Theorem Proving with (Long-)Contexts | AI Research Paper Details
miniCTX: Neural Theorem Proving with (Long-)Contexts
• This paper introduces miniCTX, a neural theorem proving system that uses long-context information to improve performance on challenging proof tasks.
• The key ideas are:
□ Incorporating broad contextual information beyond just the immediate proof step can help guide the reasoning process.
□ The model learns to effectively leverage this long-context to make better decisions during proof search.
□ Experiments show miniCTX outperforms prior neural theorem provers on standard benchmarks.
Plain English Explanation
[object Object] is the process of mathematically demonstrating that a statement is true given a set of assumptions. This is a core task in fields like automated reasoning and program verification.
The miniCTX model aims to improve theorem proving by incorporating a broader context beyond just the current proof step. Rather than focusing narrowly on the immediate logical reasoning, miniCTX also
considers related information like the overall proof goal, past proof steps, and other contextual cues. This long-context can help guide the model's decisions and lead to more effective proof search.
The key insight is that human mathematicians often leverage contextual understanding when proving theorems, not just step-by-step logic. By mimicking this, the model can make more informed choices
about which proof steps to try next. Experiments show this approach outperforms prior neural theorem provers on standard benchmarks, demonstrating the value of leveraging contextual information for
this challenging task.
Technical Explanation
The miniCTX model uses a novel neural network architecture to incorporate long-context information into the theorem proving process. The core components are:
1. Proof Encoder: Encodes the current proof state, including the goal, assumptions, and previous proof steps.
2. Context Encoder: Encodes additional contextual information beyond just the immediate proof, such as the overall proof structure, related theorems, and other relevant background knowledge.
3. Proof Step Selector: Uses the encoded proof and context to predict the most promising next proof step.
During training, the model learns to effectively leverage the long-context information to guide the proof search and make more informed decisions. This allows it to outperform prior neural theorem
provers that only consider the local proof state.
The experiments in the paper demonstrate miniCTX's strong performance on standard theorem proving benchmarks, indicating the value of this contextual approach. The model is able to successfully prove
more theorems compared to baselines that lack the long-context integration.
Critical Analysis
The paper provides a compelling approach to improving neural theorem proving by leveraging broader contextual information. However, a few key limitations and areas for further research are worth
• The experiments focus on a specific theorem proving domain and it's unclear how well the approach would generalize to other types of logical reasoning tasks.
• The paper does not provide a detailed analysis of the kinds of contextual information that are most valuable for guiding the proof search. Further investigation into the most informative
contextual cues could lead to additional performance gains.
• The model complexity and training requirements are not extensively analyzed, so the computational costs and scalability of the approach are uncertain.
Overall, the miniCTX system represents an interesting step forward in integrating contextual understanding into automated reasoning systems. With further research and refinement, this line of work
could lead to more powerful and versatile theorem provers. Readers are encouraged to think critically about the trade-offs and potential areas for improvement in this research.
The miniCTX paper introduces a novel neural theorem proving system that leverages broad contextual information beyond just the immediate proof state. By encoding and effectively utilizing this
long-context, the model is able to outperform prior neural theorem provers on standard benchmarks.
This work highlights the value of incorporating contextual understanding into automated reasoning systems, moving beyond narrow, step-by-step logic. As theorem proving and related logical inference
tasks become increasingly important for applications like program verification and knowledge representation, miniCTX and similar approaches could play a key role in advancing the state of the art in
these critical domains.
This summary was produced with help from an AI and may contain inaccuracies - check out the links to read the original source documents! | {"url":"https://www.aimodels.fyi/papers/arxiv/minictx-neural-theorem-proving-long-contexts","timestamp":"2024-11-12T22:58:29Z","content_type":"text/html","content_length":"100620","record_id":"<urn:uuid:143e2065-caab-4cc1-8ca4-e872230a86a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00816.warc.gz"} |
Mock Exam Question Reference:
Q 1: Q1 M1 BPP
Q2 M3Q2 BPP
Q3: Q12 BPPK
Q4: Q 33 BPPK
Q5: Q 37 BPPK
Performance Report
Question 1:
1. You scored 1 mark straight away by presenting your answer in nice form.
Calculation of shortfall of cash to cover the expenses was calculated perfectly which earned you 4 marks.
Option 1- your scored 2 marks here. You lost marks because of wrong explanation of Stamp duty, you also ignored tax on dividend which lost you two more mark
Option 2- you did few omissions here, you calculated tax on after tax proceeds, which is wrong. You ignored ER on FHL and lost some marks. Ignored, dividend and tax hence lost more marks. Weak
understanding of stamp duty against lost you a mark. You scored 3 marks here.
Option 3- your explanation of gift relief availability is wrong , it was available not because of investment property but because it was FHL. You ignored tax on rental income and lost more marks. You
ignored tax on dividend again and lost marks. You scored 2 marks in this part.
Option 4- you explained this point well about rent a room relief. Gift relief will be available Stamp duty wont apply. You scored 4 marks here.
Your total score in this part of the question is (16) out of possible 21
1. This part was all about writing and no calculations were needed. You raised few good points and scored (3) marks in this question. You omitted or explained wrong many points which are as follow.
DTR, Political party, death tax rate, BPR on property used by partnership, BPR rate on partnership , residence status, cost of administering overseas property, funeral expense etc. you wasted time
calculating benefits of lifetime transfer which was not needed. Read the question carefully, ask yourself why I am given this information, what I can write about this information and try to relate
the information with the scenario and topic.
You also gained (3) presentation marks because of neat and professional answer.
Your total marks in this question are 22 out of maximum 35
Question 2:
. Your “de grouping charge” and “Substantial shareholding exemption” is weak. Please remember it’s not F6 Exam where only calculations are needed, its P6 exam where you need to write commentary and
explain your calculations. Identify in the question why this information is given? Examiner does not give any information without purpose. Always incorporate all information in your answer to
maximize your marks.
1. Your identified CGT group correctly and said it will be at no gain no loss but eventually when you came to calculate it you took the actual cost? Cost will be the amount paid by A when purchased
this property plus the indexation allowance from that date up until date of sale of this property.
There were 8 marks allocated to commentary in this question (of which most of them were about SSE and DE grouping charge) of which you gained (3) marks. For calculation there were 3 marks,
you gained two of it as you miscalculated cost portion.
1. Now in this part it says do “Calculations” so majority of marks were for calculations. It was very basic chapter 1 calculations, but unfortunately you did not do it right. You seemed to struggle
calculating basic Income tax calculations & national insurance calculations despite of Tax bands given to you in the exam paper. Your marks in this part are below satisfactory and you scored just
2. It was typical “Ethics” question which you see in almost every P6 exam. You raised 4 points and were given (4) Every point earns you a mark. Always raise points equal to marks allocated in the
Your Total marks in this question are 8 out of possible 25
Question 3
1. It was commentary part as it says in question “Explain” so no calculations were necessary. There were couples of marks for explaining the circumstances when CGT could be paid in installment,
relating these rules with scenario and explaining further why or why not in this case CGT will be paid in installment would have earned you marks. You just said because it was gift so CGT could
be paid in installment; It was correct incomplete sentence, so earned lost easy mark here. When its gift and gift relief is not claimed only then you can pay by installment and this rule only
apply to some assets which include building. You did not mention when first installment will be paid. You said interest will be charged on installments, its wrong, interest is only charged on
late paid installments. Your understanding about when to pay CGT by installment needs revising. You earned (2) marks in this part, one of which was about the point when asset is sold and all
installments need to be paid at once. Always ask yourself after finishing part of a question (Did I raise enough valid ideas according to marks allocated to this part?) one idea = 1 mark. Another
important tip is always highlights word “AND” on question requirement, it means examiner is asking you to do this AND do that.
1. This part was again about writing and no calculation was needed. It asked to tell what would be penalty because there was error in Tax return of previous tax year. Your knowledge about
self-assessment is very weak, revise it, you scored (0) marks in this part. Maximum penalty in this was 30% of Potential lost revenue to HMRC, which may be reduced to minimum of 15% depending on
your circumstances.
1. Most of this part was about calculation, one of candidate’s favorite. You calculations were right at time but did not had any support (workings). Always explain the calculations in words, there
was no further tax on shares as it was 7 years old. Lifetime tax paid on it was not 40%, lifetime tax is either 20 or 25% depending on who’s paying. You reduced total estate by amount of tax paid
in life, which is clearly wrong, you ignored NRB. You did not state basis of DTR calculated, always state how you are calculating DTR. Calculations need to be supported by workings and
commentary. You missed interest calculation and lost easy marks.
Always ask yourself why examiner is given me this information. What he want me to do with this information. You scored (2) marks in this question. Revise your IHT calculations and theory too.
Your Total marks in this question are 4 out of possible 20
Question 5
1. You scored (5) marks in this part as you raised many good points, although you missed some, your answer was too brief, you did not explain why its divided not a benefit ? you should have stated
because its close company and its gift to participator. You should always convey to examiner that you know this rule, if you don’t convey examiner will assume you don’t know it, even you though
you do. You missed national insurance consequences too. Always think from every angle while answering the question especially when you have to choose between two options given.
2. Your explanations were too vague and scored you no marks here.
3. Part (i) was about Prtial exemption of VAT, very favorite of examiner. Although you did well in this part but your explanations were brief here too, it seemed like you know the concept and
calculation but do not know the theory of it. Please revise as theory is important and you can lose marks. You scored (3) marks here.
Part (ii) you scored (3) marks in this part because of your brief answer and you missed some conditions too.
Your Total score in this question is 11 out of possible 20
Unsuccessful Attempt
Score: 45%
Unfortunately you could not pass this mock exam but don’t be upset, learn from mistakes and apply in exam, you still have many days from exam, try & read all the tips which are pointed out, learn
from mistakes and don’t repeat them in real exam. Read question carefully, what is being asked, take your time planning the answer, give special attention to words like “AND” in the question it is
asking you to do two different things and marks are divided. Ask yourself why this piece of information is given to me in the question and what I can do with it. Improve theory of AccountacyTube
notes & learn rules on finger tips. Do not spend much time deciding which optional question to choose. Do not write irrelevant stuff which wastes your time and effort, 1 good point is better than 5
irrelevant points, so spend time thinking instead of writing irrelevant stuff.
Good luck and don’t forget to share your ACCA result with us.
If you think you benefited from this mock please recommend to your friends and colleagues. | {"url":"https://accountancytube.com/accaresults22200218/","timestamp":"2024-11-08T12:51:35Z","content_type":"text/html","content_length":"155727","record_id":"<urn:uuid:4d7d3019-a238-41a5-bd9c-222d63122ea5>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00406.warc.gz"} |
Java 8 Lambda Expressions Examples using Calculator Implementation - Analytics Yogi
Java 8 Lambda Expressions Examples using Calculator Implementation
The article demonstrates the Lambda expressions using Calculator (interface) code samples. It also makes use of Functional interfaces from java.util.function package to demonstrate Calculator
implementation using BiFunction and BinaryOperator interfaces.
Calculator Implementation Demonstrating Lambda Expressions
The Calculator methods implementation would be explained using both, traditional approach and the approach making use of Lambda expressions.
Traditional Approach
Using traditional approach, many would have gone implementing Calculator add, substract, multiply and divide function based on following:
• Define a Calculator interface with four methods namely add, subtract, multiply and divide; Another approach could be to create a single method with an operation flag and provide conditional
implementation based on flag.
• Create a CalculatorImpl class implementing Calculator interface and providing implementation for each method. Simply speaking, these methods would have code such as a + b, a – b, a*b and a/b.
Another approach would have been based on anonymous class displayed in example below. However, it only struck me when I was writing code for Lambda expressions.
Lamba Expressions
With Lambda expressions in Java 8, the same could be achieved in following manner.
Calculator.java (Interface)
Following can be termed as functional interface having just one method. The important point to note is that there is no need of writing separate methods. Also, one may not even need any conditional
flag of any sort to execute different code for different operations.
public interface Calculator {
public Double calculate( Double num1, Double num2 );
HelloCalculator.java (Class)
public class HelloCalculator {
public double process( double num1, double num2, Calculator calculator ) {
return calculator.calculate( num1, num2 );
public static void main(String[] args) {
HelloCalculator hl = new HelloCalculator();
// Traditional Way using Anonymous class
System.out.println("Addition: " + hl.process(3, 4, new Calculator() {
public Double calculate(Double num1, Double num2) {
return num1 + num2;
// Lambda Way; How simplified the code became
Calculator calcSubtraction = (Double num1, Double num2) -> {
return num1 - num2;
System.out.println("Subtraction: " + hl.process(3, 4, calcSubtraction));
// Lambda Way; Further simplification using Type Inference
System.out.println("Multiplication: "
+ hl.process(3, 4, (num1, num2) -> {
return num1 * num2;
System.out.println("Divide: " + hl.process(3, 4, (num1, num2) -> {
return num1 / num2;
Calculator Implementation Demonstrating Lambda Expressions & Functional Interfaces
Using Out-of-box functional interfaces from java.util.function package, one could simply implement Calculator operations such as add, subtract, multiply and divide without having need to define any
Calculator interface. Take a look at the code below.
Isn’t that very neat?
import java.util.function.BiFunction;
import java.util.function.BinaryOperator;
public class HelloCalculatorUsingBiFunction {
public Long process( long num1, long num2, BiFunction biFunc ) {
return (Long) biFunc.apply( num1, num2);
public static void main(String[] args) {
HelloCalculatorUsingBiFunction hlbf = new HelloCalculatorUsingBiFunction();
BinaryOperator add = (x, y) -> x + y;
System.out.println( "Addition: " + hlbf.process( 4, 5, add ));
BinaryOperator subtract = (x, y) -> x - y;
System.out.println( "Subtraction: " + hlbf.process( 4, 5, subtract ));
BinaryOperator multiply = (x, y) -> x/y;
System.out.println( "Multiplication: " + hlbf.process( 4, 5, multiply ));
BinaryOperator division = (x, y) -> x/y;
System.out.println( "Division: " + hlbf.process( 4, 5, division ));
For another perspective on introduction to Lambda expressions, check out our article on this page.
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3 Responses
jotka May 9, 2014 at 1:11 pm
hi, please correct “->” into “->” in your source example 🙂
Ajitesh Kumar May 9, 2014 at 11:28 pm
thanks for the suggestion.
chap authentication September 8, 2014 at 8:08 am
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I found it very helpful. However the differences are not too understandable for me
Very Nice Explaination. Thankyiu very much,
in your case E respresent Member or Oraganization which include on e or more peers?
Such a informative post. Keep it up
Thank you....for your support. you given a good solution for me.
Ajitesh Kumar
I have been recently working in the area of Data analytics including Data Science and Machine Learning / Deep Learning. I am also passionate about different technologies including programming
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Megawatt-hour to Joules Converter
Enter Megawatt-hour
⇅ Switch toJoules to Megawatt-hour Converter
How to use this Megawatt-hour to Joules Converter 🤔
Follow these steps to convert given energy from the units of Megawatt-hour to the units of Joules.
1. Enter the input Megawatt-hour value in the text field.
2. The calculator converts the given Megawatt-hour into Joules in realtime ⌚ using the conversion formula, and displays under the Joules label. You do not need to click any button. If the input
changes, Joules value is re-calculated, just like that.
3. You may copy the resulting Joules value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Megawatt-hour to Joules?
The formula to convert given energy from Megawatt-hour to Joules is:
Energy[(Joules)] = Energy[(Megawatt-hour)] × 3.6e9
Substitute the given value of energy in megawatt-hour, i.e., Energy[(Megawatt-hour)] in the above formula and simplify the right-hand side value. The resulting value is the energy in joules, i.e.,
Calculation will be done after you enter a valid input.
Consider that a power plant generates an average of 5 megawatt-hours (MWh) of energy in a day.
Convert this energy generation from Megawatt-hours to Joules.
The energy in megawatt-hour is:
Energy[(Megawatt-hour)] = 5
The formula to convert energy from megawatt-hour to joules is:
Energy[(Joules)] = Energy[(Megawatt-hour)] × 3.6e9
Substitute given weight Energy[(Megawatt-hour)] = 5 in the above formula.
Energy[(Joules)] = 5 × 3.6e9
Energy[(Joules)] = 18000000000
Final Answer:
Therefore, 5 MWh is equal to 18000000000 J.
The energy is 18000000000 J, in joules.
Consider that a wind turbine generates 2 megawatt-hours (MWh) of energy in a day.
Convert this energy generation from megawatt-hours to Joules.
The energy in megawatt-hour is:
Energy[(Megawatt-hour)] = 2
The formula to convert energy from megawatt-hour to joules is:
Energy[(Joules)] = Energy[(Megawatt-hour)] × 3.6e9
Substitute given weight Energy[(Megawatt-hour)] = 2 in the above formula.
Energy[(Joules)] = 2 × 3.6e9
Energy[(Joules)] = 7200000000
Final Answer:
Therefore, 2 MWh is equal to 7200000000 J.
The energy is 7200000000 J, in joules.
Megawatt-hour to Joules Conversion Table
The following table gives some of the most used conversions from Megawatt-hour to Joules.
Megawatt-hour (MWh) Joules (J)
0.01 MWh 36000000 J
0.1 MWh 360000000 J
1 MWh 3600000000 J
2 MWh 7200000000 J
3 MWh 10800000000 J
4 MWh 14400000000 J
5 MWh 18000000000 J
6 MWh 21600000000 J
7 MWh 25200000000 J
8 MWh 28800000000 J
9 MWh 32400000000 J
10 MWh 36000000000 J
20 MWh 72000000000 J
50 MWh 180000000000 J
100 MWh 360000000000 J
1000 MWh 3600000000000 J
A Megawatt-hour (MWh) is a unit of energy that measures the amount of electrical energy consumed or generated over time. One megawatt-hour is equivalent to one megawatt (1,000,000 watts) of power
used or produced for one hour. This unit is commonly used to quantify large-scale energy usage, such as that of power plants, industrial facilities, or in the context of national and regional energy
consumption. For example, if a power plant operates at 1 megawatt of output for one hour, it produces 1 MWh of energy. Megawatt-hours are crucial for understanding and managing large-scale energy
production and consumption.
The Joule (J) is the SI unit of energy. One joule is defined as the amount of energy transferred when a force of one newton is applied over a distance of one meter. It can also be defined as the
energy transferred when one watt of power is applied for one second. The joule is a versatile unit used in various scientific and engineering contexts to measure energy, work, and heat. It is
commonly used in physics, chemistry, and engineering to quantify the energy content of fuels, the work done by machines, and the energy used or produced in electrical circuits.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Megawatt-hour to Joules in Energy?
The formula to convert Megawatt-hour to Joules in Energy is:
2. Is this tool free or paid?
This Energy conversion tool, which converts Megawatt-hour to Joules, is completely free to use.
3. How do I convert Energy from Megawatt-hour to Joules?
To convert Energy from Megawatt-hour to Joules, you can use the following formula:
For example, if you have a value in Megawatt-hour, you substitute that value in place of Megawatt-hour in the above formula, and solve the mathematical expression to get the equivalent value in | {"url":"https://convertonline.org/unit/?convert=megawatt_hour-joules","timestamp":"2024-11-04T18:50:04Z","content_type":"text/html","content_length":"79265","record_id":"<urn:uuid:1a2944ae-eb08-41aa-9cb5-eb7c70a4c4e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00225.warc.gz"} |
Documentation - SCM
Density Functionals (XC)¶
The Density Functional, also called the exchange-and-correlation (XC) functional, consists of an LDA, a GGA part, a Hartree-Fock exchange part (hybrids), and a meta-GGA part (meta-GGA or
meta-hybrid). Possibly, it also depends on virtual Kohn-Sham orbitals through inclusion of an orbital-dependent correlation (double-hybrids). LDA stands for the Local Density Approximation, which
implies that the XC functional in each point in space depends only on the (spin) density in that same point. GGA stands for Generalized Gradient Approximation and is an addition to the LDA part, by
including terms that depend on derivatives of the density. A hybrid GGA (for example B3LYP) stands for some combination of a standard GGA with a part of Hartree-Fock exchange. A meta-GGA (for example
TPSS) has a GGA part, but also depends on the kinetic energy density. A meta-hybrid (for example TPSSh) has GGA part, a part of Hartree-Fock exchange and a part that depends on the kinetic energy
density. For these terms ADF supports a large number of the formulas advocated in the literature. For post-SCF energies only, ADF supports also various other meta-GGA functionals and more hybrid
functionals. A double-hybrid has a hybrid or a meta-hybrid part, but also contains a contribution from second-order Møller-Plesset perturbation theory (MP2). Here, only the hybrid (meta-hybrid) part
is evaluated self-consistently, whereas the MP2 part is evaluated post-SCF and added to the hybrid (meta-hybrid) energy.
The key that controls the Density Functional is XC. All subkeys are optional.
{LDA LDA {Stoll}}
{GGA GGA}
{MetaGGA metagga}
{Model MODELPOT [IP]}
{HYBRID hybrid {HF=HFpart}}
{MetaHYBRID metahybrid}
{DOUBLEHYBRID doublehybrid}
{RPA {option}}
{RANGESEP {GAMMA=X} {ALPHA=a} {BETA=b}}
(LibXC functional}
{DISPERSION [s6scaling] [RSCALE=r0scaling] [Grimme3] [BJDAMP] [PAR1=par1] [PAR2=par2] [PAR3=par3] [PAR4=par4] }
{Dispersion Grimme4 {s6=...} {s8=...} {a1=...} {a2=...}}
{DISPERSION dDsC}
{DISPERSION UFF}
If the XC key is omitted, the program will apply only the Local Density Approximation (no GGA terms). The chosen LDA form is then VWN.
LDA {functional} {Stoll}
Defines the LDA part of the XC functional. If functional is omitted, VWN will be used (also if LYP is specified in the GGA part).
Available LDA functionals:
□ Xonly: The pure-exchange electron gas formula. Technically this is identical to the Xalpha form (see next) with a value 2/3 for the X-alpha parameter.
□ Xalpha: The scaled (parametrized) exchange-only formula. When this option is used you may (optionally) specify the X-alpha parameter by typing a numerical value after the string Xalpha
(separated by a blank). If omitted this parameter takes the default value 0.7
□ VWN: The parametrization of electron gas data given by Vosko, Wilk and Nusair (ref 1, formula version V). Among the available LDA options this is the more advanced one, including correlation
effects to a fair extent.
☆ Stoll: For the VWN variety of the LDA form you may include Stoll’s correction 2 by typing Stoll on the same line, after the main LDA specification. You must not use Stoll’s correction in
combination with the Xonly or the Xalpha form for the Local Density functional. The Stoll formula is considered to be a correlation correction to the Local Density Approximation. It is
conceptually not correct to use the Stoll correction and apply gradient (GGA) corrections to the correlation. It is the user’s responsibility, in general and also here, to avoid using
options that are not solidly justified theoretically.
□ PW92: the parametrization of electron gas data given by Perdew and Wang (ref 3).
Specifies the GGA part of the XC Functional (in earlier times often called the ‘non-local’ correction to the LDA part of the density functional). It uses derivatives (gradients) of the charge
Available GGA functionals:
□ BP86: Exchange: Becke, Correlation: Perdew
□ PW91: Exchange: pw91x, Correlation: pw91c
□ mPW: Exchange: mPWx, Correlation: pw91c
□ PBE: Exchange: PBEx, Correlation: PBEc
□ RPBE: Exchange: RPBEx, Correlation: PBEc
□ revPBE: Exchange: revPBEx, Correlation: PBEc
□ mPBE: Exchange: mPBEx, Correlation: PBEc
□ PBEsol: Exchange: PBEsolx, Correlation: PBEsolc
□ HTBS: Exchange: HTBSx, Correlation: PBEc
□ BLYP: Exchange: Becke, Correlation: LYP
□ OLYP: Exchange: OPTX, Correlation: LYP
□ OPBE: Exchange: OPTX, Correlation: PBEc 4
□ BEE: Exchange: BEEx, Correlation: PBEc
□ XLYP: Exchange: XLYPx 5 (exchange, not available separately from LYP) + LYP
□ SSB-D: Dispersion corrected functional by Swart-Solà-Bickelhaupt 63 64. The SSB-D functional by definition already includes a dispersion correction by Grimme (factor 0.847455). There are some
numerical issues with the GGA implementation in ADF of SSB-D (Ref. 63 64) for some systems. Because of this, the GGA SSB-D option is only available for single-points (and NMR). Geometry
optimizations (etc.) are still possible by using instead:
This METAGGA implementation is only possible with all-electron basis sets. Use GGA SSB-D for NMR calculations.
□ S12g: Dispersion corrected (Grimme-D3) functional by Swart, successor of SSB-D 6.
□ LB94: By Van Leeuwen and Baerends 7.
□ KT1: By Keal and Tozer 8.
□ KT2: By Keal and Tozer 8.
If only a GGA part is specified (omitting the LDA sub key) the LDA part defaults to VWN, except when the LYP correlation correction is used: in that case the LDA default is Xonly: pure exchange.
The reason for this is that the LYP formulas assume the pure-exchange LDA form, while for instance the Perdew-86 correlation correction is a correction to a correlated LDA form. The precise form
of this correlated LDA form assumed in the Perdew-86 correlation correction is not available as an option in ADF but the VWN formulas are fairly close to it.
Separate choices can be made for the GGA exchange correction and the GGA correlation correction respectively. Both specifications must be typed (if at all) on the same line, after the GGA subkey.
For the exchange part the options are:
For the correlation part the options are:
The string GGA must contain not more than one of the exchange options and not more than one of the correlation options. If options are applied for both they must be separated by a blank or a
comma. Example:
is equivalent to
It is questionable to apply gradient corrections to the correlation, while not doing so at the same time for the exchange. Therefore, the program will check this and stop with an error message.
This check can be overruled with the key ALLOW.
MetaGGA functional
Specifies that a meta-GGA should be used during the SCF. All electron basis sets should be used (see Basis key).
Available meta-GGA functionals:
The r2SCAN-3c composite method uses the \(r^2\) SCAN (r2SCAN) exchange-correlation functional, in combination with a tailor-made all electron polarized basis set (mTZ2P), the semiclassical London
dispersion correction (D4), and a geometrical counterpoise (gCP) correction. NumericalQuality should be Good, and ZORA should be used. Note that internally LibXC will be used for the r2SCAN
functional, and automatically the D4 and gCP corrections will be included. The STO-optimized r2SCAN-3c outperforms many conventional hybrid/QZ approaches in most common applications at a fraction of
their cost.
The M06-L functional needs high integration accuracy (at least BeckeGrid quality good) for reasonable gradients. For TPSS moderate integration accuracy for reasonable gradients is sufficient. For
heavier elements (Z>36) and if one uses the M06-L functional it is also necessary to include the following keyword
Using this key FRAGMETAGGATOTEN the difference in the meta-hybrid or meta-GGA exchange-correlation energies between the molecule and its fragments will be calculated using the molecular integration
grid, which is more accurate than the default, but is much more time consuming. Default is to calculate the meta-GGA exchange-correlation energies for the fragments in the numerical integration grid
of the fragments.
Specifies that the Hartree-Fock exchange should be used during the SCF.
HYBRID functional {HF=HFpart}
Specifies that a hybrid functional should be used during the SCF.
Available Hybrid functionals:
□ B3LYP: ADF uses VWN5 in B3LYP. functional (20% HF exchange) by Stephens-Devlin-Chablowski-Frisch 27.
□ B3LYP*: Modified B3LYP functional (15% HF exchange) by Reiher-Salomon-Hess 28.
□ B1LYP: Functional (25% HF exchange) by Adamo-Barone 29.
□ KMLYP: Functional (55.7% HF exchange) by Kang-Musgrave 30.
□ O3LYP: Functional (12% HF exchange) by Cohen-Handy 31.
□ X3LYP: Functional (21.8% HF exchange) by Xu-Goddard 5.
□ BHandH: 50% HF exchange, 50% LDA exchange, and 100% LYP correlation.
□ BHandHLYP: 50% HF exchange, 50% LDA exchange, 50% Becke88 exchange, and 100% LYP correlation.
□ B1PW91: Functional by (25% HF exchange) Adamo-Barone 29.
□ mPW1PW: Functional (25% HF exchange) by Adamo-Barone 12.
□ mPW1K: Functional (42.8% HF exchange) by Lynch-Fast-Harris-Truhlar 35.
□ PBE0: Functional (25% HF exchange) by Ernzerhof-Scuseria 36 and by Adamo-Barone 37, hybrid form of PBE.
□ OPBE0: Functional (25% HF exchange) by Swart-Ehlers-Lammertsma 4, hybrid form of OPBE.
□ S12H: Dispersion corrected (Grimme-D3) functional (25% HF exchange) by Swart 6.
Specifies the amount of HF exchange that should be used in the functional, instead of the default HF exchange percentage for the given hybrid. Example HF=0.25 means 25% Hartree-Fock exchange.
MetaHYBRID functional
Specifies that a meta-hybrid functional should be used during the SCF.
Available meta-hybrid functionals:
Range separated hybrids¶
In ADF there are two (mutually exclusive) ways of specifying range separated hybrids functionals:
• Through the RANGESEP and XCFUN keys. This will use the Yukawa potential as switching function, see Ref. 38;
• By specifying a range separated functional via the LibXC key.
See also the advanced tutorial: Tuning the range separation in LC-wPBE for organic electronics
RangeSep + XCFun: Yukawa-range separated hybrids¶
RANGESEP {GAMMA=X} {ALPHA=a} {BETA=b}
If RANGESEP is included, by default a long-range corrected (LC) functional is created with range separation parameter GAMMA of 0.75. As switching function in ADF the Yukawa potential is utilized,
see Ref. 38. Range separated functionals require XCFUN and are limited to GGA, meta-GGA, and CAMY-B3LYP. The CAMY-B3LYP functional is not the same as the CAM-B3LYP functional, since a different
switching function is used. No other hybrids or meta-hybrids are supported. The special CAMYB3LYP functional is defined by three parameters, ALPHA, BETA and the attenuation parameter GAMMA. For
CAMYB3LYP by default ALPHA is 0.19, BETA is 0.46, and GAMMA is 0.34.
Range-separated functionals make use of a modified form of the Coulomb operator that is split into pieces for exact exchange and DFT. As switching function in ADF the Yukawa potential is utilized,
see Ref. 38. Global hybrids can be thought of as a special case of a range-separated functional where the split is independent of the inter-electronic distance and is a simple X exact and 1-X DFT in
all space.
In a general RS-functional the split depends on the inter-electronic distance. How the split is achieved depends on the functional in question but it is achieved using functions that smoothly go from
1 to 0. In ADF an exponential function is used (the error function is common in Gaussian based codes). In a range-separated function the potential goes from a Coulomb interaction to a sum of Coulomb
functions attenuated by an exponential function.
In practical terms, this means that a range-separated functional looks very much like a hybrid (or meta-hybrid) functional but with additional integrals over the attenuated interaction with fit
functions on the exact exchange side and a modified functional on the DFT side.
DFT part of RS-functionals
Using Hirao’s approach for creating RS-functionals, the RS form of a given exchange functional is created by multiplying the standard energy density by a factor that depends on the energy density.
The factor is the same for all functionals and the only difference is introduced by the individual energy densities.
The range-separation comes in at the level of the integrals over the operator with fit functions. They are very similar to the standard Coulomb integrals.
An RS-functional is described by a series of regions describing each of the pieces of the Coulomb operator. The total function is built up by looping over the regions and adding up all the pieces.
Currently, simple LC functionals can be defined where the exact exchange goes from 0 to 1 as the inter-electronic distance increases and the DFT part does the reverse. In addition, CAMY-B3LYP type
functionals can be defined. More general functionals are not possible yet.
RS functionals with XCFUN are limited to the GGA and meta-GGA functionals and one hybrid CAMY-B3LYP. The following functionals can be evaluated with range-separation at the present time:
• LDA: VWN5, X-ALPHA PW92
• GGA exchange: Becke88, PBEX, OPTX, PW91X, mPW, revPBEX
• GGA correlation: LYP, Perdew86, PBEC
• MetaGGA: TPSS, M06L, B95
• Hybrids: CAMY-B3LYP
The following functionality has been tested: XC potential, energy, ground state geometry, TDDFT. Starting from ADF2018 singlet-triplet excitation calculations and excited state geometry optimizations
are possible. See for possible limitations in case of excitation calculations or excited state geometry optimizations the corresponding part of the ADF manual.
Numerical stability
The range-separated implementation requires that the range-separation parameter is not too close to the exponent of a fit function. In practice this means that values of the separation parameter
between 1.0 and 50 can cause numerical problems. Typical useful values are in the range 0.2 to 0.9 so this should not be too serious a limitation.
RANGESEP {GAMMA=X} {ALPHA=a} {BETA=b}
Range separation is activated by putting RANGESEP in the XC block. Inclusion of XCFUN is required, see the XCFUN description. By default a long-range corrected (LC) functional is created with range
separation parameter of 0.75. The parameter can be changed by modifying X in GAMMA=X in the RANGESEP card. Range separation typically will be used in combination with a GGA or METAGGA functional.
Range separation can not be included with a hybrid or meta-hybrid, with one exception, the special RS functional: CAMY-B3LYP. This is entered as HYBRID CAMY-B3LYP and must be used in combination with
XCFUN (see XCFUN description) and RANGESEP. The CAMY-B3LYP functional is defined by three parameters, alpha, beta and the attenuation parameter gamma. The gamma parameter can be modified as for the
LC functionals. For CAMY-B3LYP it defaults to 0.34. The alpha and beta parameters can be modified through ALPHA=a and BETA=b in the RANGESEP card. They default to 0.19 and 0.46 respectively.
HYBRID CAMY-B3LYP
RANGESEP GAMMA=0.34 ALPHA=0.19 BETA=0.46
List of the most important functionals, for which one can use range separation:
GGA BP86
Range-separated hybrids with LibXC¶
One can simply specify a range separated hybrid functional in the LibXC key, e.g.:
See the LibXC section for a list of available range separated hybrid functionals.
For the HSE03 and HSE06 short range-separated hybrids you can (optionally) specify the switching parameter omega, e.g.:
LibXC HSE06 omega=0.1
Notes on Hartree-Fock and (meta-)hybrid functionals¶
If a functional contains a part of Hartree-Fock exchange then the LDA, GGA, metaGGA, or MODEL key should not be used in combination with this key, and one should only specify one of HartreeFock,
HYBRID or MetaHYBRID. Dispersion can be added. Note that it is not recommended to use (part of the) Hartree-Fock exchange in combination with frozen cores, since at the moment the frozen core
orbitals are not included in the Hartree Fock exchange operator. In ADF one can do unrestricted Hartree-Fock (or hybrid or meta-hybrid) calculations, as long as one has integer occupation numbers.
The default implementation in ADF for unrestricted Hartree-Fock calculations is UHF. In case of a high spin electron configuration one can do ROHF, see ROKS for high spin open shell molecules. You
need to use the same XC-potential in the create run of the atoms, which is done automatically if you use the BASIS key.
Starting from ADF2009.01 the meta-hybrids M06, M06-2X, M06-HF, and TPSSH can be used during the SCF. Also starting from ADF2009.01 Hartree-Fock and the (meta-)hybrid potentials can be used in
combination with geometry optimization, TS, IRC, LT, and numerical frequencies; hybrids can be used in calculating NMR chemical shift; PBE0 can be used in calculating NMR spin-spin coupling;
Hartree-Fock and (meta-)hybrid can be used in calculating excitation energies, in which the kernel consists of the Hartree-Fock percentage times the Hartree-Fock kernel plus one minus the
Hartree-Fock percentage times the ALDA kernel (thus no (meta-)GGA kernel). Hartree-Fock and the (meta-)hybrid potentials still can not or should not be used in combination with analytical
frequencies, the (AO)RESPONSE key, EPR/ESR g-tensor, and frozen cores. Starting from ADF2010 it is possible to use Hartree-Fock and hybrids to calculate CD spectra.
In ADF one can do unrestricted Hartree-Fock (or hybrid or meta-hybrid) calculations (UHF, UKS), as long as one has integer occupation numbers, or, in case of a high spin electron configuration, one
can do ROHF or ROKS, see ROKS for high spin open shell molecules.
It is possible to change the amount of HF exchange in the input for hybrids (not for meta-hybrids and Hartree-Fock). For many hybrid functionals the sum of the amount of Hartree-Fock exchange and the
amount of LDA exchange (or GGA exchange) is one. If that is the case, then if one changes the amount of Hartree-Fock exchange in the input the amount of LDA exchange (or GGA exchange) will also be
changed, such that the sum remains one. Example:
Hybrid B3LYP HF=0.25
In this case the amount of Hartree-Fock for the B3LYP functional will be changed to 25% (instead of 20%), and the amount of LDA exchange to 75% (instead of 80%). The LDA correlation and GGA exchange
and correlation part will be left unaltered.
An accuracy issue is relevant for some of the meta-GGA functionals, in particular the M06 functionals. These need high integration accuracy (at least BeckeGrid quality good) for reasonable gradients.
For TPSSH moderate integration accuracy for reasonable gradients is sufficient. For heavier elements (Z>36) and if one uses one of the M06 functionals it is also necessary to include the following
Using this key FRAGMETAGGATOTEN the difference in the metahybrid or metagga exchange-correlation energies between the molecule and its fragments will be calculated using the molecular integration
grid, which is more accurate than the default, but is much more time consuming. Default is to calculate the meta-hybrid or meta-GGA exchange-correlation energies for the fragments in the numerical
integration grid of the fragments.
For benchmark calculations one would like to use a large basis set, like the QZ4P basis set. In such cases it is recommended to use a good numerical quality. Thus for accurate hybrid calculations of
small molecules one could use:
type QZ4P
Dependency bas=1e-4
NumericalQuality good
MP2, Double Hybrids, RPA¶
To calculate treat correlation energies beyond DFT, ADF offers MP2 and random-phase approximation (RPA) based methods. In addition, ADF offers a large number of modern Double hybrid functionals which
combine MP2 correlation with a hybrid functional.
ADF implements canonical MP2 using density fitting. Additionally, ADF implements RPA and direct MP2 using an efficient atomic orbital based algorithm. The algorithm is described in this paper. 90 The
algorithm is continuously improved over the last years and currently allows to perform single-point calculations for systems with up to 1000 atoms one a single modern compute node. Many of the most
accurate Double Hybrid functionals only use direct MP2.
Double Hybrid Functionals¶
DOUBLEHYBRID functional
Specifies that a double-hybrid functional 69 should be used.
Double hybrids usually yield considerably better energies than (meta-)GGA and (meta-)hybrid functionals for (main group) thermochemistry and kinetics, transition metal chemistry and non-covalent
interactions. For an overview of the capabilities of double-hybrids implemented in ADF we refer to a recent review. 70
The MP2 correlation energy consists of two terms,
\[\begin{split}E_{MP2} = & \frac{1}{2} (E_{\text{direct}} - E_{\text{ex}} )\\ E_{\text{direct}} = & \sum_{ij} \sum_{ab}\frac{(i a| j b) (i a| j b)}{{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}} \\
E_{\text{ex}} = & \sum_{ij} \sum_{ab}\frac{(i a| j b) (i b| j a)}{{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}} \\ (i a| j b) = & \iint \phi_i^\dagger(1) \phi_a(1) \frac{1}{r_{12}} \phi_j^\dagger(2)
\phi_b(2) d1d2\end{split}\]
For a closed-shell system, the MP2 correlation energy can also be partitioned as
\[\begin{split}E_{MP2} = E_{\text{OS}} + E_{\text{SS}} \\ E_{\text{OS}} = E_{\text{direct}} \\ E_{\text{SS}} = E_{\text{direct}} - E_{\text{ex}}\end{split}\]
Here, OS (opposite spin) denotes the contribution to the correlation energy from electrons with unpaired spins and SS (same spin) denotes the contribution to the correlation energy from electron with
paired spins. In case of spin-orbit coupling approximate SS and OS contributions are calculated.
There are three classes of Double hybrid functionals:
We recommend to use opposite-spin only functionals for large systems (50-100 atoms and larger) since they are computationally more efficient than the other functionals with also include the same-spin
contribution. An opposite-spin only functional calculation is always feasible when a hybrid calculation is feasible too!
For additional technical details of the algorithm and how to tweak the technical parameters, see the MBPT section.
Opposite-spin-only Double Hybrids
Currently, ADF supports the following opposite-spin-only Double Hybrid functionals:
Except for SOS1-PBE-QIDH, all functionals include dispersion correction by default which cannot be switched off.
Standard Double Hybrids
Currently, ADF supports the following standard double hybrid functionals
Empirical dispersion corrections can be requested in the XC block in the usual way. Some functionals can be combined with Grimme’s D4 empirical dispersion correction with optimized parameters:
B2PLYP, B2GPPLYP, mPW2PLYP, PBE0-DH, PBE0-2. All functionals in this category can be combined with Grimme’s D3(BJ) empirical dispersion correction with optimized parameters. Note that for Grimme’s D3
(BJ) the parameters for B2PIPLYP, ROB2PLYP, B2TPLYP, B2KPLYP, mPW2PLYP, mPW2KPLYP, DH-BLYP, LS1-DH, PBE0-2, and DS1-TPSS are modified B2PLYP parameters, see Ref. 70. If no optimized empirical
dispersion parameters exist for a certain functional, default parameters are used, which may not give the expected results.
Spin-component-scaled functionals
Currently, ADF supports the following spin-component scaled Double Hybrid functionals:
Except for SD-SCAN69, all functionals include dispersion correction by default which cannot be switched off.
EmpiricalScaling {NONE|SOS|SCS|SCSMI}
In addition to double-hybrids, ADF also implements MP2 including some popular spin-scaled variants. Technically, they are not distinct from double-hybrids, however, the all rely on a HF instead of a
DFT calculation. The following variants are supported.
• SOS-MP2: pure HF reference (100 % HF, 130 % OS-MP2) 86
• MP2: pure HF reference (100 % HF, 100 % MP2 correlation)
• SCS-MP2: pure HF reference (100 % HF, 120 % OS-MP2, 33 % SS-MP2) 87
• SOS-MI-MP2: pure HF reference (100 % HF, 40 % OS-MP2, 129 % SS-MP2) 88
In case of spin-orbit coupling approximate SS and OS contributions are calculated.
The spin-scaling variant can be requested in the XC block together with the MP2 keyword:
EmpiricalScaling SOS
requests an SOS-MP2 calculation.
For additional technical details of the algorithm and how to tweak parameters, see the MBPT section.
Note: In AMS2022, the keyword for RPA+SOX was RPASOX.
The RPA goes beyond MP2 by accounting explicitly for the polarizability of the system which screens the electron-electron interaction. It can therefore be applied to large system for which MP2
typically diverges. 91 The following RPA based methods are available.
• RPA : Standard (direct )RPA without exchange
• RPA + SOX : Standard RPA plus statically screened second-order exchange 92
• RPA + SOSEX: Standard RPA plus dynamically screened second-order exchange 93
A detailed overview of the RPA algorithm in ADF and a detailed assessment of the performance of second-order exchange corrections can be found in 95.
An RPA calculation is requested in the XC block:
RPA {NONE|DIRECT|SOSEX|SOSSX|SIGMA}
An RPA calculation needs to be combined with an XC functional. For instance,
hybrid pbe0
RPA DIRECT
will perform a PBE0 calculation followed by a direct RPA calculation. RPA and all of its variants can be used in conjunction with LDA, GGAs, hybrid, and RSH functionals.
For additional technical details of the algorithm and how to tweak parameters, see the RPA section.
Starting from AMS2023, the sigma-functional by Görling and coworkers is implemented. 94 In this method, the correlation kernel is calculated form the adiabatic fluctuation-dissipation theorem. In
addition to the direct RPA (Hartree) kernel, higher-order contributions to the kernel are included by the so-called sigma-kernel which is fitted to relative energies. Sigma-functionals are as fast as
As an RPA calculation, a sigma-functional calculation needs to be combined with an XC functional. For instance,
hybrid pbe0
RPA sigma
requests to use the sigma-functional with the W1 parametrization for PBE0. 96 sigma-functionals can only be used with a limited number of exchange-correlation functionals only, since they need to be
explicitly parametrized for each functional. Currently, the sigma-functional can be used in conjunction with the GGA PBE, and the hybrids PBE0 and B3LYP. The available parametrizations for each
functional are listed in the following table:
functional available parametrizations 96 97
PBE W1 S1 S2
PBE0 W1 S1 S2 W2
B3LYP W1
The parametrization can be changed in the MBPT block, see the MBPT block. For instance:
SigmaFunctionalParametrization S1
Spin-orbit coupling¶
In case of spin-orbit coupling approximate SS and OS contributions are calculated, which is relevant for open shell molecules with double hybrids or MP2 variants that use different scaling factors
for these contributions:
\[\begin{split}(i a| j b) = & \iint \phi_i^\dagger(1) \phi_a(1) \frac{1}{r_{12}} \phi_j^\dagger(2) \phi_b(2) d1d2 \\ m_{ij} = & \int \phi_i^\dagger(1) \vec{\sigma} \phi_i(1) d1 \cdot \int \phi_j^\
dagger(2) \vec{\sigma} \phi_j(2) d2 \\ E_2^{SS} = & - \sum_{ijab} \frac{(1+m_{ij}) (i a| j b) (a i| b j)}{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b} + \sum_{ijab} \frac{2(i a| j b) (a j| b i)}{\
epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b} \\ E_2^{OS} = & - \sum_{ijab} \frac{(1-m_{ij}) (i a| j b) (a i| b j)}{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}\end{split}\]
with, \(i,j\) occupied spinors, \(a,b\) virtual spinors, \(\epsilon\) spinor energies, \(\vec{\sigma}\) Pauli spin matrices. Note with pure \(\alpha\) and \(\beta\) orbitals, \(m_{i^\alpha j^\alpha}
= m_{i^\beta j^\beta} = 1, m_{i^\alpha j^\beta} = m_{i^\beta j^\alpha} = -1\), one has the familiar SS and OS energy expressions.
Model Potentials¶
Several asymptotically correct XC potentials have been implemented in ADF, such as the (now somewhat outdated) LB94 potential 7, the gradient-regulated asymptotic correction (GRAC) 39, and the
statistical average of orbital potentials (SAOP) 42 40. These can currently be used only for response property calculations, not for geometry optimizations. For spectroscopic properties, they usually
give results superior to those obtained with LDA or GGA potentials, (see Ref. 41 for applications to (hyper)polarizabilities Cauchy coefficients, etc. of small molecules). This is particularly true
if the molecule is small and the (high-lying) virtual orbitals are important for the property under study.
It was also shown that, simply using the orbital energies of the occupied Kohn-Sham orbitals of a SAOP calculation, quite good agreement with experiment vertical ionization potentials is obtained.
This is true not only for the HOMO orbital energy, which should be identical to (minus) the experimental ionization potential with the exact XC potential, but also for lower-lying occupied orbital
energies. The agreement becomes worse for deep-lying core orbital energies. A theoretical explanation and practical results are given in Ref. 43.
Model ModelPotential [IP]
Specifies that one of the less common XC potentials should be used during the SCF. These potentials specify both the exchange and the correlation part. No LDA, GGA, MetaGGA, HartreeFock, HYBRID
or MetaHYBRID key should be used in combination with these keys. It is also not advised to use any energy analysis in combination with these potentials. For energy analysis we recommend to use
one of the GGA potentials. It is currently not possible to do a Create run with these potentials. It is possible to do a one atom regular ADF calculation with these potentials though, using a
regular adf.rkf (TAPE21) file from an LDA or GGA potential as input. Available model potentials:
□ LB94: This refers to the XC functional of Van Leeuwen and Baerends 7. There are no separate entries for the Exchange and Correlation parts respectively of LB94. Usually the GRAC or SAOP
potentials give results superior to LB94.
□ GRAC: The gradient-regulated asymptotic correction, which in the outer region closely resembles the LB94 potential 39. It requires a further argument: the ionization potential [IP] of the
molecule, in hartree units. This should be estimated or obtained externally, or calculated in advance from two GGA total energy calculations.
□ IP:Should be supplied only if GRAC is specified.
□ SAOP: The statistical average of orbital potentials 42 40. It can be used for all electron calculations only. It will be expensive for large molecules, but requires no further parameter
The LB94, GRAC, and SAOP functionals have only a SCF (=Potential) implementation, but no Energy counterpart.
The LB94, GRAC, and SAOP forms are density functionals specifically designed to get the correct asymptotic behavior. This yields much better energies for the highest occupied molecular orbital (HOMO)
and better excitation energies in a calculation of response properties (Time Dependent DFT). Energies for lower lying orbitals (sub-valence) should improve as well (in case of GRAC and SAOP, but not
LB94). The energy expression underlying the LB94 functional is very inaccurate. This does not affect the response properties but it does imply that the energy and its derivatives (gradients) should
not be used because LB94-optimized geometries will be wrong, see for instance 44. The application of the LB94 functional in a runtype that involves the computation of energy gradients is disabled in
ADF. You can override this internal check with the key ALLOW.
In case of a GRAC calculation, the user should be aware that the potential in the outer region is shifted up with respect to the usual level. In other words, the XC potential does not tend to zero in
the outer region in this case. The size of the shift is the difference between the HOMO orbital energy and the IP given as input. In order to compare to regular GGA orbital energies, it is advisable
to subtract this amount from all orbital energies. Of course, orbital energy differences, which enter excitation energies, are not affected by this shift in the potential.
Any publication employing calculations carried out with XCFun need to cite Ref. [U. Ekström, L. Visscher, R. Bast, A.J. Thorvaldsen, and K. Ruud, J. Chem. Theory Comput. 6, 1971 (2010)] 45.
XCFun is a library of approximate exchange-correlation functionals, see Ref. 45, for which functional derivatives can be calculated automatically. For example, with XCFUN the full (non-ALDA)
kernel can be evaluated and this has been implemented in the calculation of TDDFT excitations. The Full kernel can not be used in combination with symmetry or excited state geometry
optimizations. The following functionals can be evaluated with XCFUN at the present time:
□ LDA: VWN5, X-ALPHA, PW92
□ GGA exchange: Becke88, PBEX, OPTX, PW91X, mPW, revPBEX
□ GGA correlation: LYP, Perdew86, PBEC
□ MetaGGA: TPSS, M06L, B95
□ MetaHybrids: M06, M05, M062X, M06HF
□ Hybrids: PBE0, B3LYP, BHandH, B1LYP, B3LYP*, PBEFALFX
□ Yukawa range separated Hybrids: CAMY-B3LYP and more, see Yukawa RS hybrids with XCFUN
Here MetaGGA B95 means Becke88 exchange + B95c correlation. The Metahybrids PW6B95 and PWB6K have been removed from this list, since they do not agree with the LibXC implementation.
Any publication employing calculations carried out with LibXC need to cite the current literature reference of LibXC, which is at the moment Ref. [S. Lehtola, C. Steigemann, M.J.T. Oliveira, M.A.L.
Marques, SoftwareX 7, 1 (2018)] 47. Benchmark papers employing functionals from LibXC should especially include a reference to the employed DFT library, since as discussed in Ref. [S. Lehtola, M.A.L.
Marques, J. Chem. Phys. 159, 114116 (2023)] 52, the implementations of a given functionals in various program packages may not lead to the same result, even at the complete basis set limit.
LibXC functional
LibXC is a library of approximate exchange-correlation functionals, see Ref. 46 47. All electron basis sets should be used (see Basis key). Version 5.1.2 of LibXC is used. The following
functionals can be evaluated with LibXC (incomplete list):
□ LDA: LDA, PW92, TETER93
□ GGA: AM05, BCGP, B97-GGA1, B97-K, BLYP, BP86, EDF1, GAM, HCTH-93, HCTH-120, HCTH-147, HCTH-407, HCTH-407P, HCTH-P14, PBEINT, HTBS, KT2, MOHLYP, MOHLYP2, MPBE, MPW, N12, OLYP, PBE, PBEINT,
PBESOL, PW91, Q2D, SOGGA, SOGGA11, TH-FL, TH-FC, TH-FCFO, TH-FCO, TH1, TH2, TH3, TH4, XLYP, XPBE, HLE16
□ MetaGGA: M06-L, M11-L, MN12-L, MS0, MS1, MS2, MVS, PKZB, RSCAN, R2SCAN, REVSCAN, SCAN, TPSS, HLE17
□ Hybrids: B1LYP, B1PW91, B1WC, B3LYP, B3LYP*, B3LYP5, B3LYP5, B3P86, B3PW91, B97, B97-1 B97-2, B97-3, BHANDH, BHANDHLYP, EDF2, MB3LYP-RC04, MPW1K, MPW1PW, MPW3LYP, MPW3PW, MPWLYP1M, O3LYP,
OPBE, PBE0, PBE0-13, REVB3LYP, REVPBE, RPBE, SB98-1A, SB98-1B, SB98-1C, SB98-2A, SB98-2B, SB98-2C, SOGGA11-X, SSB, SSB-D, X3LYP
□ MetaHybrids: B86B95, B88B95, BB1K, M05, M05-2X, M06, M06-2X, M06-HF, M08-HX, M08-SO, MPW1B95, MPWB1K, MS2H, MVSH, PW6B95, PW86B95, PWB6K, REVSCAN0, SCAN0, REVTPSSH, TPSSH, X1B95, XB1K
□ Range-separated: CAM-B3LYP, CAMY-B3LYP, HJS-PBE, HJS-PBESOL, HJS-B97X, HSE03, HSE06, LRC_WPBE, LRC_WPBEH, LCY-BLYP, LCY-PBE, M06-SX, M11, MN12-SX, N12-SX, TUNED-CAM-B3LYP, WB97, WB97X
One of the acronyms in the list above can be used, or one can also use the functionals described at the LibXC website https://libxc.gitlab.io/functionals. Note that ADF can not calculate VV10
dependent LibXC functionals, like VV10, LC-VV10, B97M-V, WB97X-V. Example usage for the BP86 functional:
LibXC XC_GGA_X_B88 XC_GGA_C_P86
In case of LibXC the output of the ADF calculation will give the reference for the used functional, see also the LibXC website https://libxc.gitlab.io/functionals.
Do not use any of the subkeys LDA, GGA, METAGGA, MODEL, HARTREEFOCK, HYBRID, METAHYBRID, XCFUN, RANGESEP in combination with the subkey LIBXC. One can use the DISPERSION key with LIBXC. For a
selected number of functionals the optimized dispersion parameters will then be used automatically, please check the output in that case. Note that in many cases you have to include the DISPERSION
key and include the correct dispersion parameters yourself.
The LibXC functionals can not be used with frozen cores, NMR calculations, the (AO)RESPONSE key, EPR/ESR g-tensor. Most LibXC functionals can be used in combination with geometry optimization, TS,
IRC, LT, numerical frequencies, and excitation energies (ALDA kernel used). For a few GGA LibXC functionals analytical frequencies can be calculated, and one can use the full kernel in the
calculation of excitation energies (if FULLKERNEL is included as subkey of the key EXCITATIONS). In case of LibXC (meta-)hybrids and calculating excitation energies, the kernel consists of the
Hartree-Fock percentage times the Hartree-Fock kernel plus one minus the Hartree-Fock percentage times the ALDA kernel (thus no (meta-)GGA kernel). For the LibXC range separated functionals, like
CAM-B3LYP, starting from ADF2016.102 the kernel consists of range separated ALDA plus the kernel of the range separated exact exchange part. In ADF2016.101 the kernel for LibXC range separated
functionals, like CAM-B3LYP, was using a 100% ALDA plus range separated exact exchange kernel (the ALDA part was not range-separated corrected). For the range separated functionals WB97 and WB97X one
can use the full kernel in the calculation of excitation energies.
Dispersion corrections¶
Dispersion Grimme4 {s6=...} {s8=...} {a1=...} {a2=...}
If Dispersion Grimme4 is present in the XC block the D4(EEQ) dispersion correction (with the electronegativity equilibrium model) by the Grimme group 48 will be added to the total bonding energy,
gradient and second derivatives, where applicable.
The D4(EEQ) model has four parameters: \(s_6\), \(s_8\), \(a_1\) and \(a_2\) and their value should depend on the XC functional used. For the following functionals the D4(EEQ) parameters are
predefined: B1B95, B3LYP, B3PW91, BLYP, BP86, CAM-B3LYP, HartreeFock, OLYP, OPBE, PBE, PBE0, PW6B95, REVPBE, RPBE, TPSS, TPSSH. For these functionals it is enough to specify Dispersion Grimme4 in
the input block. E.g.:
GGA BLYP
Dispersion Grimme4
For all other functionals you should explicitly specify the D4(EEQ) parameters in the Dispersion key (otherwise the PBE parameters will be used). For example, for the PW91 functional you should
use the following input:
GGA PW91
Dispersion Grimme4 s6=1.0 s8=0.7728 a1=0.3958 a2=4.9341
The D4(EEQ) parameters for many functionals can be found in the supporting information of the following paper: 48, see also https://github.com/dftd4/dftd4.
For Double-Hybrids, see the Double Hybrid Functionals section of the user manual.
DISPERSION Grimme3 BJDAMP
If DISPERSION Grimme3 BJDAMP is present a dispersion correction (DFT-D3(BJ)) by Grimme 49 will be added to the total bonding energy, gradient and second derivatives, where applicable.
Parametrizations are implemented e.g. for B3LYP, TPSS, BP86, BLYP, PBE, PBEsol, and RPBE. For SCAN parameters from Ref. 50 are used.
For example, this is the input block for specifying the PBE functional with Grimme3 BJDAMP dispersion correction (PBE-D3(BJ)):
GGA PBE
DISPERSION Grimme3 BJDAMP
The D3(BJ) dispersion correction has four parameters. One can override the default parametrization by using PAR1=.. PAR2=.., etc. In the table the relation is shown between the parameters and the
real parameters in the Dispersion correction.
variable variable on Bonn website
PAR1 s6
PAR2 a1
PAR3 s8
PAR4 a2
For example, this is the input block for specifying the PBE-D3(BJ)-GP parametrization by Proppe et.al. 89 (i.e. \(a_1=0, s_8=0, a_2=5.6841\)):
GGA PBE
DISPERSION Grimme3 BJDAMP PAR2=0 PAR3=0 PAR4=5.6841
DISPERSION Grimme3
If DISPERSION Grimme3 is present a dispersion correction (DFT-D3) by Grimme 51 will be added to the total bonding energy, gradient and second derivatives, where applicable. Parametrizations are
available e.g. for B3LYP, TPSS, BP86, BLYP, revPBE, PBE, PBEsol, and RPBE, and will be automatically set if one of these functionals is used. There are also parameters directly recognized for
S12g and S12h. For SCAN parameters from Ref. 50 are used. For all other functionals, PBE-D3 parameters are used as default. You can explicitly specify the three parameters.
variable variable on Bonn website
PAR1 s6
PAR2 sr,6
PAR3 s8
DISPERSION {s6scaling]} {RSCALE=r0scaling}
If the DISPERSION keyword is present (without the argument Grimme3) a dispersion correction (DFT-D) by Grimme 36 will be added to the total bonding energy, gradient and second derivatives, where
applicable. The global scaling factor with which the correction is added depends on the exchange-correlation functional used at SCF but it can be modified using the s6scaling parameter. The
following scaling factors are used (with the XC functional in parentheses): 1.20 (BLYP), 1.05 (BP), 0.75 (PBE), 1.05 (B3LYP). In all other cases a factor 1.0 is used unless modified via the
s6scaling parameter. The SSB-D functional includes the dispersion correction (factor 0.847455) by default.
The van der Waals radii used in this implementation are hard coded in ADF. However, it is possible to modify the global scaling parameter for them using the RSCALE=r0scaling argument. The default
value is 1.1 as proposed by Grimme 36. Please also see additional documentation for more information about this topic.
The DISPERSION dDsC key invokes the density dependent dispersion correction 55, which has been parametrized for the functionals BLYP, PBE, BP, revPBE, B3LYP, PBE0 and BHANDHLYP.
The DISPERSION UFF key invokes the universal correction of density functional theory to include London dispersion (DFT-ulg) 53, which has been parametrized for all elements up to Lr (Z=103), and
for the functional PBE, PW91, and B3LYP. For other functionals the PBE parameters will be used.
The DISPERSION MBD key invokes the MBD@rsSCS method 54, which is designed to accurately describe long-range correlation (and thus dispersion) in finite-gap systems, including at the same time a
description of the short-range interactions from the underlying DFT computation of the electronic structure.
DFT-D4 functionals¶
Grimme’s latest dispersion correction, D4(EEQ) 48, has been added in the 2019.3 release of the Amsterdam Modeling Suite. This is the latest dispersion correction in the DFT-D family. In contrast to
the earlier D3 dispersion correction, in D4(EEQ) the atomic coordination-dependent dipole polarizabilities are scaled based on atomic partial charges obtained from an electronegativity equilibrium
model (EEQ). Compared to D3 the introduced charge dependence improves thermochemical properties, especially for systems containing metals. The authors recommend D4(EEQ) as a physically improved and
more sophisticated dispersion model in place of D3.
DFT-D3 functionals¶
The D3 dispersion correction by Stefan Grimme is available in ADF. Grimme and his coworkers at the Universität Münster outlined the parametrization of this new correction, dubbed DFT-D3, in Ref. 51.
A slightly improved version with a more moderate BJ damping function appeared later, and was called DFTB-D3-BJ. 49 Here they list the advantages of the new method as the following:
• It is less empirical, i.e., the most important parameters are computed from first principles by standard Kohn-Sham (KS)-(TD)DFT.
• The approach is asymptotically correct with all DFs for finite systems (molecules) or nonmetallic infinite systems. It gives the almost exact dispersion energy for a gas of weakly interacting
neutral atoms and smoothly interpolates to molecular (bulk) regions.
• It provides a consistent description of all chemically relevant elements of the periodic system (nuclear charge Z = 1-94).
• Atom pair-specific dispersion coefficients and cutoff radii are explicitly computed.
• Coordination number (geometry) dependent dispersion coefficients are used that do not rely on atom connectivity information (differentiable energy expression).
• It provides similar or better accuracy for “light” molecules and a strongly improved description of metallic and “heavier” systems.
DFT-D3-BJ is invoked with the XC block, for example
GGA BLYP
Dispersion Grimme3 BJDAMP
Parametrizations are available for: B3LYP, TPSS, BP86, BLYP, revPBE, PBE, PBEsol, RPBE, and some more functionals, and will be automatically set if one of these functionals is used. Otherwise PBE
parameters will be used. The parameters can be set manually, see the XC key block. In ADF2016 parameters for Grimme3 and Grimme3 BJDAMP were updated according to version 3.1.1 of the coefficients,
available at the Bonn website
DFT-D functionals¶
An implementation for dispersion corrections based, called DFT-D is available in ADF. Like DFT-D3 this implementation is easy to use and is also supported by the GUI.
This DFT-D implementation is based on the paper by Grimme 36 and is extremely easy to use. The correction is switched on by specifying DISPERSION, possibly with parameters, in the XC input block. See
description of the XC input block for details about the DISPERSION keyword.
Energies calculated Post-SCF using different DFT-D or GGA-D functionals are also present in table printed when METAGGA keyword is specified. These include: BLYP-D, PBE-D, BP86-D, TPSS-D, B3LYP-D, and
B97-D. NOTE: this option does not require specifying a DISPERSION keyword in the XC block and thus there is no correction added to the energy gradient in this case. Please also note that although the
original B97 functional includes HF exchange (and is thus a hybrid functional), the B97-D is a pure GGA. B3LYP-D is, however, a hybrid functional. The following functional-dependent global scaling
factors s[6] are used: 1.2 (BLYP-D), 0.75 (PBE-D), 1.05 (BP86-D), 1.0 (TPSS-D), 1.05 (B3LYP-D), and 1.25 (B97-D). These are fixed and cannot be changed.
Regarding performance of different functionals, testing has shown that BLYP-D gives good results for both energies and gradients involving VdW interactions. Post-SCF energy-only calculations at fixed
geometries showed that also B97-D gives good binding energies compared to high-level reference data. Thorough comparison of different DFT-D functionals can be found in ref. 68
Note: The original paper by Grimme included parameters for elements H throughout Xe. In ADF2009.01 values for dispersion parameters for DFT-D functionals for heavier elements (Cs-Rn) have been added.
These new values have not been tested extensively. Thus, in this implementation, no dispersion correction is added for interactions involving atoms heavier than Radon.
DFT-D is invoked with the XC block, for example
GGA BLYP
dDsC: density dependent dispersion correction¶
The DISPERSION dDsC key invokes the density dependent dispersion correction 55, which has been parametrized for the functionals BLYP, PBE, BP, revPBE, B3LYP, PBE0 and BHANDHLYP.
GGA BLYP
Dispersion dDsC
For other functionals one can set the dDsC parameters ATT0 and BTT0 with
DISPERSION dDsC ATT0=att0 BTT0=btt0
The dispersion dDsC in ADF can not be used with fragments larger than 1 atom. The reason is that ADF uses the Hirshfeld partitioning on fragments for dDsC, which is only correct if the fragments are
The DISPERSION UFF key invokes the universal correction of density functional theory to include London dispersion (DFT-ulg) 53, which has been parametrized for all elements up to Lr (Z=103), and for
the functional PBE, PW91, and B3LYP. For other functionals the PBE parameters will be used. Example:
GGA PBE
Dispersion UFF
DFT-MBD functionals¶
The DISPERSION MBD key invokes the MBD@rsSCS method 54, which is designed to accurately describe long-range correlation (and thus dispersion) in finite-gap systems, including at the same time a
description of the short-range interactions from the underlying DFT computation of the electronic structure. The MBD (many-body dispersion) method 56 obtains an accurate description of van der Waals
(vdW) interactions that includes both screening effects and treatment of the many-body vdW energy to infinite order. The revised MBD@rsSCS method 54 employs a range-separation (rs) of the
self-consistent screening (SCS) of polarizabilities and the calculation of the long-range correlation energy. It has been parametrized for the elements H-Ba, Hf-Rn, and for the functional PBE and
PBE0. Note that the MBD@rsSCS method depends on Hirshfeld charges. In calculating forces the dependence of the Hirshfeld charges on the actual geometry is neglected. The MBD method is implemented in
case the BeckeGrid is used for the numerical integration. Example for PBE MBD@rsSCS:
GGA PBE
Dispersion MBD
One can use user defined values with:
Dispersion MBD {RSSCS|TS} {BETA=beta}
MBD {RSSCS|TS} {BETA=beta}
The default method for MBD is MBD@rsSCS. Optionally one can use MBD@TS or change the used parameter \(\beta\) with setting beta.
Post-SCF energy functionals¶
GGA energy functionals¶
In principle you may specify different functionals to be used for the potential, which determines the self-consistent charge density, and for the energy expression that is used to evaluate the (XC
part of the) energy of the charge density. To be consistent, one should generally apply the same functional to evaluate the potential and energy respectively. Two reasons, however, may lead one to do
• The evaluation of the GGA part in the potential is more time-consuming than LDA. The effect of the GGA term in the potential on the self-consistent charge density is often not very large. From
the point of view of computational efficiency it may, therefore, be attractive to solve the SCF equations at the LDA level (i.e. not including GGA terms in the potential), and to apply the full
expression, including GGA terms, to the energy evaluation a posteriori: post-SCF.
• A particular XC functional may have only an implementation for the potential, but not for the energy (or vice versa). This is a rather special case, intended primarily for fundamental research of
Density Functional Theory, rather than for run-of-the-mill production runs.
One possibility is to calculate a whole list of post-SCF energy functionals using the METAGGA keyword, see next section. For some functionals the next possibility is enough. One has to specify
different functionals for potential and energy evaluations respectively, using:
{LDA {Apply} LDA {Stoll}}
{GGA {Apply} GGA}
States whether the functional defined on the pertaining line will be used self-consistently (in the SCF-potential), or only post-SCF, i.e. to evaluate the XC energy corresponding to the charge
density. The value of apply must be SCF or Energy. A value postSCF will also be accepted and is equivalent to Energy. A value Potential will also be accepted and is equivalent to SCF. For each
record separately the default (if no Apply value is given in that record) is SCF. For each of the two terms (LDA, GGA) in the functional: if no record with Energy specification is found in the
data block, the evaluation of the XC energy will use the same functional as is applied for the potential.
LDA, GGA
See the XC potential section for all possible values.
Meta-GGA and hybrid energy functionals¶
The post SCF energy calculation is an easy and cheap way to get a reasonable guess for the bond energies for different XC functionals at the same time. Note that post-SCF energy calculations for a
certain XC functional will not be so accurate if the functional form of the XC functional used in the SCF is very different from the XC functional used post SCF. The relative accuracy of post-SCF
energies may not be so high if one looks at small energy differences. For accurate energy calculations it is recommended to use the same XC functional during the SCF as for the energy.
The calculation of a large, pre-specified list of LDA, GGA, and meta-GGA energy functionals is invoked by specifying
as a separate keyword. The following (incomplete) list gives an idea of the (meta-)GGA density functionals that will then be calculated (the t-MGGA functional is the \(\theta\)-MGGA functional of
Ref. 57):
BP, PW91, mPW, BLYP, PBE, RPBE, revPBE, mPBE, OLYP, OPBE, KCIS, PKZB, VS98, FT97, BLAP3,
HCTH, tau-HCTH, BmTau1, BOP, OLAP3, TPSS, KT1, KT2, B97, M06-L, t-MGGA.
The hybrid GGA and hybrid meta-GGA energy functionals are calculated if in addition to the METAGGA key, the key
is included. The following (incomplete) list gives an idea of the extra hybrid (meta-)GGA density functionals that will then be calculated:
B3LYP, B3LYP*, B1LYP, KMLYP, O3LYP, X3LYP, BHandH, BHandHLYP, B1PW91, MPW1PW, MPW1K,
PBE0, OPBE0, TPSSh, tau-HCTH-hybrid, B97, M05, M05-2X, M06, M06-2X.
The keys METAGGA and HARTREEFOCK can be used in combination with any XC potential. Note that at the moment hybrid functionals can not be used in combination with frozen cores. Also most METAGGA
functionals will give wrong results if used in combination with frozen cores. Thus it is best to use an all electron basis set if one of the keywords METAGGA or HARTREEFOCK is used. One should
include the HARTREEFOCK keyword also in the create runs of the atoms. In ADF the hybrid energies only make sense if the calculation is performed with completely filled orbitals. In case of a high
spin electron configuration one can do ROKS, see ROKS for high spin open shell molecules.
The Examples document describes an application to the OH molecule for the METAGGA option. More output, on the total XC energy of the system, can be obtained by specifying
This latter option is intended for debugging purposes mainly and is not recommended for general use.
The implementation calculates the total XC energy for a system and writes it to a file. This is always done in Create runs. If the basic fragments are atoms, the keyword
ATOM [filename]
ATOM [filename]
... ...
specifies that different atomic fragment files are to be used in the meta-GGA energy analysis than the regular atomic fragment files from the create runs. This keyword cannot be used for molecular
fragment files. In order to compare meta-GGA energy differences between molecular fragments and the total molecule, results from the various calculations need to be combined by hand.
In such situations, it is advisable to use a somewhat higher integration accuracy than one would normally do, at least for the smaller fragments, as there is no error cancellation as in a regular ADF
bond energy analysis.
A general comment is that some functionals show a more stable behavior than others (at least in our current implementation). In general, the functionals which are dependent on the Laplacian of the
density may display a large variation with respect to basis set changes or different numerical integration accuracy. For this reason we currently recommend FT97 in favor of FT98. Similarly, the
results with the BmTau1 functional should still be carefully checked. In our test calculations on the G2 set of molecules, the VS98 showed best performance, both for the average error and for the
maximum error. The G2 set consists only of small molecules with elements up to Cl. The relative performance for transition metals and heavy elements is unknown and may well be very different from the
ordering for the G2 set.
Post Hartree-Fock energy functionals¶
This is mostly taken from text by the authors of Ref. 58:
In the early days of DFT, non-self-consistent Kohn-Sham energy was often evaluated upon Hartree-Fock (HF) densities as a way to test new approximations. This method was called HF-DFT. It has been
discovered that in some cases, HF-DFT actually gave more accurate answers when compared to self-consistent DFT calculations. In Ref. 58, it was found that DFT calculations can be categorized into
two different types of calculations. The error of an approximate functional can be decomposed into two parts: error from the functional (functional error), and error from the density
(density-driven error). For most calculations, functional error is dominant, and here self-consistent DFT is usually better than non-self consistent DFT on more accurate densities (called density
corrected DFT (DC-DFT)). Unlike these ‘normal’ calculations, there is a class of calculations where the density-driven error is much larger, so DC-DFT give better a result than self-consistent
DFT. These calculations can be classified as ‘abnormal’. HF-DFT is a simple implementation of DC-DFT and a small HOMO-LUMO gap is an indicator of an ‘abnormal’ calculation, thus, HF-DFT would
perform better in such cases.
In ADF one can do HF-DFT with:
This will produce a large, pre-specified list of LDA, GGA, meta-GGA, hybrid, and metahybrid energy functionals. | {"url":"https://www.scm.com/doc/ADF/Input/Density_Functional.html","timestamp":"2024-11-12T23:58:19Z","content_type":"text/html","content_length":"721547","record_id":"<urn:uuid:981c36bc-d7c4-45f7-8d5e-4a10b11ff5bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00177.warc.gz"} |
2.3 Average Value
We will define and compute the average value of a function on an interval.
In this section, we derive the formula for average value. Recall that we are trying to find the average value of a function, $f(x)$, over an interval, $[a,b]$. We begin by subdividing the interval $
[a,b]$ into $n$ equal sub-intervals, each of length In each of these $n$ sub-intervals, we choose a sample point. We denote the sample point in the $i$-th sub-interval by $x_i^*$. As an approximation
to the average value of the function over the interval $[a,b]$ we take the average of the function values at the sample points: This can be written in summation notation as Observe that equation in
the definition of $\Delta x$ can be rewritten as This allows us to rewrite our approximation of $f_{ave}$ as Since $\Delta x$ is a constant, we can bring it inside the summation to write: Finally,
the approximation improves as the number of sample points, $n$, increases. Therefore, we define which by the definition of the definite integral can be written as | {"url":"https://ximera.osu.edu/math/calc2Book/calc2Book/averageValue/averageValue","timestamp":"2024-11-03T04:17:00Z","content_type":"text/html","content_length":"57598","record_id":"<urn:uuid:953171d1-7e56-402e-86b2-1c1e092c0e44>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00846.warc.gz"} |
A nonlinear elasticity model in computer vision
The purpose of this paper is to analyze a nonlinear elasticity model previously introduced by the authors for comparing two images, regarded as bounded open subsets of \R^n together with associated
vector-valued intensity maps. Optimal transformations between the images are sought as minimisers of an integral functional among orientation-preserving homeomorphisms. The existence of minimisers is
proved under natural coercivity and polyconvexity conditions, assuming only that the intensity functions are bounded measurable. Variants of the existence theorem are also proved, first under the
constraint that finite sets of landmark points in the two images are mapped one to the other, and second when one image is to be compared to an unknown part of another. The question is studied as to
whether for images related by a linear mapping the unique minimizer is given by that linear mapping. For a natural class of functional integrands an example is given guaranteeing that this property
holds for pairs of images in which the second is a scaling of the first by a constant factor. However for the property to hold for arbitrary pairs of linearly related images it is shown that the
integrand has to depend on the gradient of the transformation as a convex function of its determinant alone. This suggests a new model in which the integrand depends also on second derivatives of the
transformation, and an example is given for which both existence of minimizers is assured and the above property holds for all pairs of linearly related images.
updated: Tue Oct 29 2024 21:50:23 GMT+0000 (UTC)
published: Fri Aug 30 2024 12:27:22 GMT+0000 (UTC)
参考文献 (このサイトで利用可能なもの) / References (only if available on this site)
被参照文献 (このサイトで利用可能なものを新しい順に) / Citations (only if available on this site, in order of most recent) | {"url":"https://arxiv-check-250201.firebaseapp.com/each/2408.17237v2","timestamp":"2024-11-03T04:01:08Z","content_type":"text/html","content_length":"15730","record_id":"<urn:uuid:ac020872-98db-4ba0-a956-b64cb696fa02>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00218.warc.gz"} |
Pressure Exerted by a Solid Iron Cuboid On Sand
Published on Apr 02, 2024
The mass of an object is a fundamental property of the object; a numerical measure of its inertia; a fundamental measure of the amount of matter in the object. Definitions of mass often seem circular
because it is such a fundamental quantity that it is hard to define in terms of something else. All mechanical quantities can be defined in terms of mass, length, and time. The usual symbol for mass
is m and its SI unit is the kilogram. While the mass is normally considered to be an unchanging property of an object, at speeds approaching the speed of light one must consider the increase in the
relativistic mass. The weight of an object is the force of gravity on the object and may be defined as the mass times the acceleration of gravity, w = mg. Since the weight is a force, its SI unit is
the newton. Density is mass/volume.
To observe and compare the pressure exerted by a solid iron cuboid on sand while resting on its three different faces and to calculate the pressure exerted in the three cases.
To study and compare the pressure exerted by a solid iron cuboid on sand, we need to find its mass and weight.
Can you define the Mass of an object?
The mass of an object is a fundamental property of the object; a numerical measure of its inertia; a fundamental measure of the amount of matter in the object. The usual symbol for mass is 'm' and
its SI unit is kilogram.
In everyday usage, mass is often referred to as weight, the units of which are often taken to be kilograms. In scientific use, weight is the gravitational force acting on a given body, while mass is
an intrinsic property of this body.
On the surface of the Earth, the weight W of an object is related to its mass m by W = m.
Having defined Mass, what about the Weight of an object?
In science, the weight of an object is the force on the object due to gravity. Its magnitude (a scalar quantity), often denoted by W, is the product of the mass m of the object and the magnitude of
the local gravitational acceleration g. Thus, W = mg. Since the weight is a force, its SI unit is Newton.
Simply stated, weight is the force acting vertically downward. The weight of an object is the force with which it is attracted towards the earth, that is:
F = m x g
For an object in free fall, when gravity is the only force acting on it, the expression for weight follows Newton's Second Law.
W = F, thus:
W = m x g
Here ‘g’ is the Earth's gravitational field strength, equal to about 9.81 m s−2.
An object's weight depends on its environment, while its mass does not. The SI unit of weight is the same as that of force, that is, Newton (N).The force acting on an object perpendicular to the
surface is called thrust. The effect of thrust depends on the area on which it acts. Thus:
Thrust = F = m x g
The thrust on unit area is called pressure. Thus:
Pressure = Thrust / Area
SI unit of pressure is N/m2 or Nm-2 (Newton per square metre).
In honour of scientist Blaise Pascal, the SI unit of pressure is called pascal, denoted by Pa.
Materials Required:
Fill ¾ ths of a tray with dry sand and level it.
• Measure the dimensions of a solid iron cuboid accurately using a scale. Mark the three faces of the cuboid as A, B and C.
• Place the solid iron cuboid by the surface A on the plane levelled sand in the tray.
• After a few minutes, remove the Iron cuboid and you will see that it has made a depression in the sand.
• Measure the depth (depression) it has made in the sand using the scale.
• Repeat the same procedure for the other two surfaces.
Gravitational force on the environment = ……..
1. Calculate the area occupied by each surface of the solid iron cuboid.
• Area occupied by surface A in the sand = .............
• Area occupied by surface B in the sand = .............
• Area occupied by surface C in the sand = .............
2. Calculate the pressure made by each surface of the solid iron cuboid.
• Pressure made by the surface A in the sand = ............. N
• Pressure made by the surface B in the sand = ............. N
• Pressure made by the surface C in the sand = ............. N
3. Calculate the Depression.
• Depression made by the surface A in the sand = ............. cm
• Depression made by the surface B in the sand = ............. cm
• Depression made by the surface C in the sand = ............. cm
1. Dried sand must be used.
2. The tray must have significant length and width.
3. Appropriate cuboid of dimension must be used
• Manual of Secondary Science Kit for Classes IX and X - Published by NCERT
• Science textbook for class IX – Published by National Council of Educational Research and Training, New Delhi
• Pressure exerted by Solids - Better Lesson.com | {"url":"https://www.seminarsonly.com/Engineering-Projects/Physics/solid-iron-cuboid.php","timestamp":"2024-11-12T22:57:22Z","content_type":"text/html","content_length":"16963","record_id":"<urn:uuid:234286ca-ff14-4cfb-b2ca-301583201d21>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00299.warc.gz"} |
Contents Previous Next Index
11 Input, Output, and Interacting with Other Products
11.1 In This Chapter
Section Topics
Writing to Files - Saving to Maple file formats • Saving Data to a File
• Saving Expressions to a File
Reading from Files -Opening Maple files • Reading Data from a File
• Reading Expressions from a File
Exporting to Other Formats - Exporting documents in file • Exporting Documents
formats supported by other software
• MapleNet
Connectivity - Using Maple with other programming languages • Translating Maple Code to Other
and software Programming Languages
• Accessing External Products from Maple
• Accessing Maple from External Products
• Sharing and Storing Maple Worksheet
Content with the MapleCloud^TM
11.2 Writing to Files
Maple supports file formats in addition to the standard .mw file format.
After using Maple to perform a computation, you can save the results to a file for later processing with Maple or
another program.
Note: Make sure you have write access to the directory in order to execute the example in the following subsections.
Saving Data to a File
If the result of a Maple calculation is a long list or a large array of numbers, you can convert it to Matrix form
and write the numbers to a file using the ExportMatrix command. This command writes columns of numerical data to a
file, allowing you to import the numbers into another program. To convert a list or a list of lists to a Matrix,
use the Matrix constructor. For more information, refer to the Matrix help page.
> $L⁢:=⁢[\begin{array}{ccccc}-81& -98& -76& -4& 29\\ -38& -77& -72& 27& 44\\ -18
& 57& -2& 8& 92\\ 87& 27& -32& 69& -31\\ 33& -93& -74& 99& 67\end{array}]:$
> $\mathrm{ExportMatrix}\left("matrixdata.txt",L\right):$
If the data is a Vector or any object that can be converted to type Vector, use the ExportVector command. To
convert lists to Vectors, use the Vector constructor. For more information, refer to the Vector help page.
> $R≔\left[3,3.1415,-65,0\right]$
${R}{:=}\left[{3}{,}{3.1415}{,}{-}{65}{,}{0}\right]$ (11.1)
> $V≔\mathrm{Vector}\left(R\right)$
> $\mathrm{ExportVector}\left("vectordata.txt",V\right):$
You can extend these routines to write more complicated data, such as complex numbers or symbolic expressions. For
more information, refer to the ExportMatrix and ExportVector help pages.
For more information on matrices and vectors, see Linear Algebra.
Saving Expressions to a File
If you construct a complicated expression or procedure, you can save them for future use in Maple. If you save the
expression or procedure in the Maple internal format, Maple can retrieve it more efficiently than from a document.
Use the save command to write the expression to a .m file. For more information on Maple internal file formats,
refer to the file help page.
> $\mathrm{qbinomial}≔\left(n,k\right)\to \frac{\underset{i=n-k+1}{\overset{n}{&
In this example, small expressions are used. In practice, Maple supports expressions with thousands of terms.
> $\mathrm{expr}≔\mathrm{qbinomial}\left(10,4\right)$
${\mathrm{expr}}{:=}\frac{\left({1}{-}{{q}}^{{7}}\right){⁢}\left({1}{-}{{q}}^{{8}}\right){& (11.2)
> $\mathrm{nexpr}≔\mathrm{normal}\left(\mathrm{expr}\right)$
${\mathrm{nexpr}}{:=}\left({{q}}^{{6}}{+}{{q}}^{{5}}{+}{{q}}^{{4}}{+}{{q}}^{{3}}{+}{{q}} (11.3)
You can save these expressions to the file qbinom.m.
> $\mathbf{save}\mathrm{qbinomial},\mathrm{expr},\mathrm{nexpr},"qbinom.m"$
Clear the memory using the restart command and retrieve the expressions using the read command.
> $\mathrm{restart}$
> $\mathbf{read}"qbinom.m"$
> $\mathrm{expr}$
$\frac{\left({1}{-}{{q}}^{{7}}\right){⁢}\left({1}{-}{{q}}^{{8}}\right){⁢}\left (11.4)
For more information on writing to files, refer to the save help page.
Saving Data as Part of a Workbook
You can save all files related to a common Maple project as a workbook (.maple) file. Saving your data files and
worksheets (or documents) as a workbook allows you to use this saved data across all .mw file inside your workbook.
11.3 Reading from Files
The most common reason for reading files is to load data, for example, data generated in an experiment. You can
store data in a text file, and then read it into Maple.
Reading Data from a File
Import Data Assistant
If you generate data outside Maple, you can read it into Maple for further manipulation. This data can be an
image, a sound file, or columns of numbers in a text file. You can easily import this external data into Maple
using the Import Data Assistant, where the supported file formats include files of type Excel^®, MATLAB^®, Image,
Audio, Matrix Market, and Delimited.
To launch the Import Data Assistant:
1. From the Tools menu, select Assistants, and then Import Data.
2. A dialog window appears where you can navigate to your data file. Select the file that you want to import data
from, and then select the file type before clicking Next.
3. From the main window, you can preview the selected file and choose from the applicable options based on the
format of the file read in before importing the data into Maple. See Figure 11.1 for an example.
Figure 11.1: Import Data Assistant
ImportMatrix Command
The Import Data Assistant provides a graphical interface to the ImportMatrix command. For more information,
including options not available in the assistant, refer to the ImportMatrix help page.
Reading Expressions from a File
You can write Maple programs in a text file using a text editor, and then import the file into Maple. You can paste
the commands from the text file into your document or you can use the read command.
When you read a file with the read command, Maple treats each line in the file as a command. Maple executes the
commands and displays the results in your document but it does not, by default, insert the commands from the file
in your document.
For example, the file ks.txt contains the following Maple commands.
S:= n -> sum( binomial( n, beta ) * ( ( 2*beta )! / 2^beta - beta!*beta ), beta=1..n );
Note that the file should not contain prompts (>) at the start of lines.
When you read the file, Maple displays the results but not the commands.
${1024937361666644598071114328769317982974}$ (11.5)
> $\mathrm{filename}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),\mathrm
> $\mathbf{read}\mathrm{filename}$
${1024937361666644598071114328769317982974}$ (11.6)
If you set the interface echo option to 2, Maple inserts the commands from the file into your document.
> $\mathrm{interface}\left(\mathrm{echo}=2\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{read}\
> S:= n -> sum( binomial( n, beta ) * ( ( 2*beta )! / 2^beta - beta!*beta ), beta=1..n );
> S(19);
${1024937361666644598071114328769317982974}$ (11.7)
For more information, refer to the read and interface help pages.
Reading Data From Workbook Attachments
Data stored in a workbook in the form of an attachment, can be accessed easily using the workbook URI. For
information on workbook attachments, see worksheet,workbook,attachFiles. For information on the workbook URI
format, see worksheet,workbook,uri.
11.4 Exporting to Other Formats
Exporting Documents
You can save your documents by selecting Save or Save As from the File menu. By selecting Export As from the File
menu, you can also export a document in the following formats: HTML, LaTeX, Maple input, Maplet application, Maple
text, plain text, PDF, and Rich Text Format. This allows you to access your work outside Maple.
The .html file that Maple generates can be loaded into any HTML browser. Exported mathematical content can be
displayed in one of two formats, GIF or MathML 2.0, and is saved in a separate folder. MathML is the Internet
standard, sanctioned by the World Wide Web Consortium (W3C), for the communication of structured mathematical
formulae between applications. For more information about MathML, refer to the MathML help page.
Maple documents that are exported to HTML translate into multiple documents when using frames. If the frames
feature is not selected, Maple creates only one page that contains the document contents.
The .tex file generated by Maple is ready for processing by LaTeX. All distributions of Maple include the
necessary style files. By default, the LaTeX style files are set for printing the .tex file using the dvips
printer driver. You can change this behavior by specifying an option to the \usepackage LaTeX command in the
preamble of your .tex file. For more information, refer to the exporttoLaTeX help page.
Maple Input
You can export a Maple document as Maple input so that it can be loaded using the Maple Command-line version.
Important: When exporting a document as Maple input for use in Command-line Maple, your document must contain
explicit semicolons in 1-D Math input. If not, the exported .mpl file does not contain semicolons, and
Command-line Maple generates errors.
Maplet Application
The Export as Maplet facility saves a Maple document as a .maplet file, so that you can run it using the
command-line interface or the MapletViewer. The MapletViewer is an executable program that can launch saved Maplet
applications. It displays and runs Maplet applications independently of the Maple Worksheet interface.
Important: When exporting a document as a Maplet Application for use in Command-line Maple or the MapletViewer,
your document must contain explicit semicolons. If not, the exported .maplet file does not contain semicolons, and
Command-line Maple and the MapletViewer generates errors.
Maple Text
Maple text is marked text that retains the distinction between text, Maple input, and Maple output. Thus, you can
export a document as Maple text, send the text file by email, and the recipient can import the Maple text into a
Maple session and regenerate the computations in the original document.
Export a Maple document to a Portable Document Format (PDF) file so that you can open the file in a reader such as
Adobe^® Acrobat^®. The PDF document is formatted as it would appear when the Maple worksheet is printed using the
active printer settings.
Note: Images, plots, and embedded components may be resized in the PDF file.
Plain Text
Export a Maple document as plain text so that you can open the text file in a word processor.
Rich Text Format (RTF)
Export a Maple document to a rich text format file so that you can open and edit the file in a word processor.
Note: The generated .rtf format is compatible with Microsoft Word and Microsoft WordPad only.
Summary of Translation
Table 11.1: Summary of Content Translation When Exporting to Different Formats
Content HTML LaTeX Maple Maplet Maple Text Plain Text Rich Text PDF Format
Input Application Format
Text Maintained Maintained Preceded Preceded by Preceded by # Maintained Maintained Maintained
by # #
1-D Math Maintained Maintained Maintained Maintained Preceded by > Preceded by > Static Static
image image
2-D Math GIF or LaTeX 1-D Math 1-D Math 1-D Math or 1-D Math or Static Either
MathML (if (if character-based character-based image text or
possible) possible) typesetting typesetting shapes
on option
Plot GIF Postscript Not Not Not exported Not exported Static Static
file exported exported image image
Animation Animated GIF Not exported Not Not Not exported Not exported Not Static
exported exported exported image
Hidden Not exported Not exported Not Not Not exported Not exported Not Not
content exported exported exported exported
Manually Not Not Not Not Not supported Not supported RTF page Maintained
inserted supported supported supported supported break
page break object
Hyperlink Links to Plain text Plain text Plain text Plain text Plain text Plain text Plain text
help pages
become plain
text. Links
to documents
are renamed
converted to
HTML links
Embedded GIF Not exported Not Not Not exported Not exported Static Static
image or exported exported image image
Spreadsheet HTML table LaTeX tables Not Not Not exported Not exported RTF table Static
exported exported image
Document Approximated LaTeX Not Not Not exported Not exported RTF style Maintained
style by HTML environments exported exported
style and
attributes sections,
LaTeX macro
Overview of MapleNet
Using MapleNet, you can deploy Maple content on the web. Powered by the Maple computation engine, MapleNet allows
you to embed dynamic formulas, models, and diagrams as live content in webpages. The MapleNet software is not
included with the Maple software. For more information on MapleNet, visit http://www.maplesoft.com/maplenet.
MapleNet Documents and Maplets
After you upload your Maple document to the MapleNet server, it can be accessed by anyone in the world using a web
browser. Even if viewers do not have a copy of Maple installed, they can view documents and Maplets, manipulate
3-D plots, and execute code at the click of a button.
Custom Java Applets and JavaServer Pages^TM Technology
MapleNet provides a programming interface to the Maple math engine so commands can be executed from a Java applet
or using JavaServer Pages^TM technology. Embed MapleNet into your web application, and let Maple handle the math
and visualization.
11.5 Connectivity
Translating Maple Code To Other Programming Languages
Code Generation
The CodeGeneration package is a collection of commands and subpackages that enable the translation of Maple code
to other programming languages. Languages currently supported include: C, C#, Fortran 77, Java, MATLAB^®, Visual
Basic, Perl, and Python.
For details on Code Generation, refer to the CodeGeneration help page.
Accessing External Products from Maple
External Calling
External calling allows you to use compiled C, C#, Fortran 77, or Java code in Maple. Functions written in these
languages can be linked and used as if they were Maple procedures. With external calling you can use pre-written
optimized algorithms without the need to translate them into Maple commands. Access to the NAG library routines
and other numerical algorithms is built into Maple using the external calling mechanism.
External calling can also be applied to functions other than numerical algorithms. Routines exist that accomplish
a variety of non-mathematical tasks. You can use these routines in Maple to extend its functionality. For example,
you can link to controlled hardware via a serial port or interface with another program. The Database package uses
external calling to allow you to query, create, and update databases in Maple. For more information, refer to the
Database help page.
For more information on using external calling, refer to the ExternalCalling help page.
Mathematica Translator
The MmaTranslator package provides translation tools for converting Mathematica^® expressions, command operations,
and notebooks to Maple. The package can translate Mathematica input to Maple input and Mathematica notebooks to
Maple documents. The Mma subpackage contains commands that provide translation for Mathematica commands when no
equivalent Maple command exists. In most cases, the command achieves the translation through minor manipulations
of the input and output of similar Maple commands.
Note: The MmaTranslator package does not convert Mathematica programs.
There is a Maplet interface to the MmaTranslator package. For more information, refer to the MmaToMaple help page.
Matlab Package
The Matlab package enables you to translate MATLAB^® code to Maple, as well as call selected MATLAB^® functions
from a Maple session, provided you have MATLAB^® installed on your system.
For more information, refer to the Matlab help page.
Accessing Maple from External Products
Microsoft Excel Add-In
Maple is available as an add-in to Microsoft Excel. This add-in is supported for Excel 365 (desktop) and Excel
2019 for Windows, and provides the following features.
• Access to Maple commands from Excel
• Ability to copy and paste between Maple and Excel
• Access to a subset of the Maple help pages
• Maple Function Wizard to step you through the creation of a Maple function call
To enable the Maple Excel Add-in:
1. In Excel, click the File menu and select Options.
2. Click Add-ins.
3. In the Manage box select Excel Add-ins, and then Go.
4. Navigate to the Excel subdirectory of your Maple installation and select the file:
– WMIMPLEX64.xla (that is, select $MAPLE/Excel/WMIMPLEX64.xla), and click OK.
5. Select the Maple Excel Add-in check box.
6. Click OK.
For further details on enabling the Maple Excel Add-in, refer to the Excel help page.
For information on using this add-in, refer to the Using Maple in Excel help file within Excel.
To view this help file:
1. Enable the add-in.
2. From the Add-ins tab, view the Maple toolbar.
3. On the Maple toolbar, click the Maple help icon .
OpenMaple is a suite of functions that allows you to access Maple algorithms and data structures in your compiled
C, C#, Java, or Visual Basic programs. (This is the reverse of external calling, which allows access to compiled
C, C#, Fortran 77, and Java code from Maple.)
To run your application, Maple must be installed. You can distribute your application to any licensed Maple user.
For additional terms and conditions on the use of OpenMaple, refer to the extern/OpenMapleLicensing.txt file in
your Maple installation.
For more details on using OpenMaple functions, refer to the OpenMaple help page.
MapleSim^TM is a complete environment for modeling and simulating multidomain engineering systems. During a
simulation, MapleSim uses the symbolic Maple computation engine to generate the mathematical models that represent
the system behavior.
Because both products are tightly integrated, you can use Maple commands and technical document features to edit,
manipulate, and analyze a MapleSim model. For example, you can use Maple commands and tools to manipulate your
model equations, develop custom components based on a mathematical model, and visualize simulation results.
MapleSim software is not included with the Maple software. For more information on MapleSim, visit http://
Sharing and Storing Maple Content
Contents Previous Next Index
11 Input, Output, and Interacting with Other Products
11.1 In This Chapter
Section Topics
Writing to Files - Saving to Maple file formats • Saving Data to a File
• Saving Expressions to a File
Reading from Files -Opening Maple files • Reading Data from a File
• Reading Expressions from a File
Exporting to Other Formats - Exporting documents in file • Exporting Documents
formats supported by other software
• MapleNet
Connectivity - Using Maple with other programming languages • Translating Maple Code to Other
and software Programming Languages
• Accessing External Products from Maple
• Accessing Maple from External Products
• Sharing and Storing Maple Worksheet
Content with the MapleCloud^TM
11.2 Writing to Files
Maple supports file formats in addition to the standard .mw file format.
After using Maple to perform a computation, you can save the results to a file for later processing with Maple or another program.
Note: Make sure you have write access to the directory in order to execute the example in the following subsections.
Saving Data to a File
If the result of a Maple calculation is a long list or a large array of numbers, you can convert it to Matrix form and write the numbers to a file using the ExportMatrix command. This command
writes columns of numerical data to a file, allowing you to import the numbers into another program. To convert a list or a list of lists to a Matrix, use the Matrix constructor. For more
information, refer to the Matrix help page.
> $L⁢:=⁢[\begin{array}{ccccc}-81& -98& -76& -4& 29\\ -38& -77& -72& 27& 44\\ -18& 57& -2& 8& 92\\ 87& 27& -32& 69& -31\\ 33& -93& -74& 99& 67\end{array}]&
> $\mathrm{ExportMatrix}\left("matrixdata.txt",L\right):$
If the data is a Vector or any object that can be converted to type Vector, use the ExportVector command. To convert lists to Vectors, use the Vector constructor. For more information, refer to
the Vector help page.
> $R≔\left[3,3.1415,-65,0\right]$
${R}{:=}\left[{3}{,}{3.1415}{,}{-}{65}{,}{0}\right]$ (11.1)
> $V≔\mathrm{Vector}\left(R\right)$
> $\mathrm{ExportVector}\left("vectordata.txt",V\right):$
You can extend these routines to write more complicated data, such as complex numbers or symbolic expressions. For more information, refer to the ExportMatrix and ExportVector help pages.
For more information on matrices and vectors, see Linear Algebra.
Saving Expressions to a File
If you construct a complicated expression or procedure, you can save them for future use in Maple. If you save the expression or procedure in the Maple internal format, Maple can retrieve it more
efficiently than from a document. Use the save command to write the expression to a .m file. For more information on Maple internal file formats, refer to the file help page.
> $\mathrm{qbinomial}≔\left(n,k\right)\to \frac{\underset{i=n-k+1}{\overset{n}{∏}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(1-{q}^{i}\right)}{\underset{i=
In this example, small expressions are used. In practice, Maple supports expressions with thousands of terms.
> $\mathrm{expr}≔\mathrm{qbinomial}\left(10,4\right)$
${\mathrm{expr}}{:=}\frac{\left({1}{-}{{q}}^{{7}}\right){⁢}\left({1}{-}{{q}}^{{8}}\right){⁢}\left({1}{-}{{q}}^{{9}}\right){⁢}\left({1}{-}{{q}}^ (11.2)
> $\mathrm{nexpr}≔\mathrm{normal}\left(\mathrm{expr}\right)$
${\mathrm{nexpr}}{:=}\left({{q}}^{{6}}{+}{{q}}^{{5}}{+}{{q}}^{{4}}{+}{{q}}^{{3}}{+}{{q}}^{{2}}{+}{q}{+}{1}\right){⁢}\left({{q}}^{{4}}{+}{1} (11.3)
You can save these expressions to the file qbinom.m.
> $\mathbf{save}\mathrm{qbinomial},\mathrm{expr},\mathrm{nexpr},"qbinom.m"$
Clear the memory using the restart command and retrieve the expressions using the read command.
> $\mathrm{restart}$
> $\mathbf{read}"qbinom.m"$
> $\mathrm{expr}$
$\frac{\left({1}{-}{{q}}^{{7}}\right){⁢}\left({1}{-}{{q}}^{{8}}\right){⁢}\left({1}{-}{{q}}^{{9}}\right){⁢}\left({1}{-}{{q}}^{{10}}\right)}{\left (11.4)
For more information on writing to files, refer to the save help page.
Saving Data as Part of a Workbook
You can save all files related to a common Maple project as a workbook (.maple) file. Saving your data files and worksheets (or documents) as a workbook allows you to use this saved data across
all .mw file inside your workbook.
11.3 Reading from Files
The most common reason for reading files is to load data, for example, data generated in an experiment. You can store data in a text file, and then read it into Maple.
Reading Data from a File
Import Data Assistant
If you generate data outside Maple, you can read it into Maple for further manipulation. This data can be an image, a sound file, or columns of numbers in a text file. You can easily import this
external data into Maple using the Import Data Assistant, where the supported file formats include files of type Excel^®, MATLAB^®, Image, Audio, Matrix Market, and Delimited.
To launch the Import Data Assistant:
1. From the Tools menu, select Assistants, and then Import Data.
2. A dialog window appears where you can navigate to your data file. Select the file that you want to import data from, and then select the file type before clicking Next.
3. From the main window, you can preview the selected file and choose from the applicable options based on the format of the file read in before importing the data into Maple. See Figure 11.1 for
an example.
Figure 11.1: Import Data Assistant
ImportMatrix Command
The Import Data Assistant provides a graphical interface to the ImportMatrix command. For more information, including options not available in the assistant, refer to the ImportMatrix help page.
Reading Expressions from a File
You can write Maple programs in a text file using a text editor, and then import the file into Maple. You can paste the commands from the text file into your document or you can use the read
When you read a file with the read command, Maple treats each line in the file as a command. Maple executes the commands and displays the results in your document but it does not, by default,
insert the commands from the file in your document.
For example, the file ks.txt contains the following Maple commands.
S:= n -> sum( binomial( n, beta ) * ( ( 2*beta )! / 2^beta - beta!*beta ), beta=1..n );
Note that the file should not contain prompts (>) at the start of lines.
When you read the file, Maple displays the results but not the commands.
${1024937361666644598071114328769317982974}$ (11.5)
> $\mathrm{filename}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),\mathrm{kernelopts}\left(\mathrm{dirsep}\right),"ks",\mathrm{kernelopts}\left(\
> $\mathbf{read}\mathrm{filename}$
${1024937361666644598071114328769317982974}$ (11.6)
If you set the interface echo option to 2, Maple inserts the commands from the file into your document.
> $\mathrm{interface}\left(\mathrm{echo}=2\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{read}\mathrm{filename}$
> S:= n -> sum( binomial( n, beta ) * ( ( 2*beta )! / 2^beta - beta!*beta ), beta=1..n );
> S(19);
${1024937361666644598071114328769317982974}$ (11.7)
For more information, refer to the read and interface help pages.
Reading Data From Workbook Attachments
Data stored in a workbook in the form of an attachment, can be accessed easily using the workbook URI. For information on workbook attachments, see worksheet,workbook,attachFiles. For information
on the workbook URI format, see worksheet,workbook,uri.
11.4 Exporting to Other Formats
Exporting Documents
You can save your documents by selecting Save or Save As from the File menu. By selecting Export As from the File menu, you can also export a document in the following formats: HTML, LaTeX, Maple
input, Maplet application, Maple text, plain text, PDF, and Rich Text Format. This allows you to access your work outside Maple.
The .html file that Maple generates can be loaded into any HTML browser. Exported mathematical content can be displayed in one of two formats, GIF or MathML 2.0, and is saved in a separate
folder. MathML is the Internet standard, sanctioned by the World Wide Web Consortium (W3C), for the communication of structured mathematical formulae between applications. For more information
about MathML, refer to the MathML help page.
Maple documents that are exported to HTML translate into multiple documents when using frames. If the frames feature is not selected, Maple creates only one page that contains the document
The .tex file generated by Maple is ready for processing by LaTeX. All distributions of Maple include the necessary style files. By default, the LaTeX style files are set for printing the .tex
file using the dvips printer driver. You can change this behavior by specifying an option to the \usepackage LaTeX command in the preamble of your .tex file. For more information, refer to the
exporttoLaTeX help page.
Maple Input
You can export a Maple document as Maple input so that it can be loaded using the Maple Command-line version.
Important: When exporting a document as Maple input for use in Command-line Maple, your document must contain explicit semicolons in 1-D Math input. If not, the exported .mpl file does not
contain semicolons, and Command-line Maple generates errors.
Maplet Application
The Export as Maplet facility saves a Maple document as a .maplet file, so that you can run it using the command-line interface or the MapletViewer. The MapletViewer is an executable program that
can launch saved Maplet applications. It displays and runs Maplet applications independently of the Maple Worksheet interface.
Important: When exporting a document as a Maplet Application for use in Command-line Maple or the MapletViewer, your document must contain explicit semicolons. If not, the exported .maplet file
does not contain semicolons, and Command-line Maple and the MapletViewer generates errors.
Maple Text
Maple text is marked text that retains the distinction between text, Maple input, and Maple output. Thus, you can export a document as Maple text, send the text file by email, and the recipient
can import the Maple text into a Maple session and regenerate the computations in the original document.
Export a Maple document to a Portable Document Format (PDF) file so that you can open the file in a reader such as Adobe^® Acrobat^®. The PDF document is formatted as it would appear when the
Maple worksheet is printed using the active printer settings.
Note: Images, plots, and embedded components may be resized in the PDF file.
Plain Text
Export a Maple document as plain text so that you can open the text file in a word processor.
Rich Text Format (RTF)
Export a Maple document to a rich text format file so that you can open and edit the file in a word processor.
Note: The generated .rtf format is compatible with Microsoft Word and Microsoft WordPad only.
Summary of Translation
Table 11.1: Summary of Content Translation When Exporting to Different Formats
Content HTML LaTeX Maple Maplet Maple Text Plain Text Rich Text PDF Format
Input Application Format
Text Maintained Maintained Preceded Preceded by Preceded by # Maintained Maintained Maintained
by # #
1-D Math Maintained Maintained Maintained Maintained Preceded by > Preceded by > Static Static
image image
2-D Math GIF or LaTeX 1-D Math 1-D Math 1-D Math or 1-D Math or Static Either
MathML (if (if character-based character-based image text or
possible) possible) typesetting typesetting shapes
on option
Plot GIF Postscript Not Not Not exported Not exported Static Static
file exported exported image image
Animation Animated GIF Not exported Not Not Not exported Not exported Not Static
exported exported exported image
Hidden Not exported Not exported Not Not Not exported Not exported Not Not
content exported exported exported exported
Manually Not Not Not Not Not supported Not supported RTF page Maintained
inserted supported supported supported supported break
page break object
Hyperlink Links to Plain text Plain text Plain text Plain text Plain text Plain text Plain text
help pages
become plain
text. Links
to documents
are renamed
converted to
HTML links
Embedded GIF Not exported Not Not Not exported Not exported Static Static
image or exported exported image image
Spreadsheet HTML table LaTeX tables Not Not Not exported Not exported RTF table Static
exported exported image
Document Approximated LaTeX Not Not Not exported Not exported RTF style Maintained
style by HTML environments exported exported
style and
attributes sections,
LaTeX macro
Overview of MapleNet
Using MapleNet, you can deploy Maple content on the web. Powered by the Maple computation engine, MapleNet allows you to embed dynamic formulas, models, and diagrams as live content in webpages.
The MapleNet software is not included with the Maple software. For more information on MapleNet, visit http://www.maplesoft.com/maplenet.
MapleNet Documents and Maplets
After you upload your Maple document to the MapleNet server, it can be accessed by anyone in the world using a web browser. Even if viewers do not have a copy of Maple installed, they can view
documents and Maplets, manipulate 3-D plots, and execute code at the click of a button.
Custom Java Applets and JavaServer Pages^TM Technology
MapleNet provides a programming interface to the Maple math engine so commands can be executed from a Java applet or using JavaServer Pages^TM technology. Embed MapleNet into your web
application, and let Maple handle the math and visualization.
11.5 Connectivity
Translating Maple Code To Other Programming Languages
Code Generation
The CodeGeneration package is a collection of commands and subpackages that enable the translation of Maple code to other programming languages. Languages currently supported include: C, C#,
Fortran 77, Java, MATLAB^®, Visual Basic, Perl, and Python.
For details on Code Generation, refer to the CodeGeneration help page.
Accessing External Products from Maple
External Calling
External calling allows you to use compiled C, C#, Fortran 77, or Java code in Maple. Functions written in these languages can be linked and used as if they were Maple procedures. With external
calling you can use pre-written optimized algorithms without the need to translate them into Maple commands. Access to the NAG library routines and other numerical algorithms is built into Maple
using the external calling mechanism.
External calling can also be applied to functions other than numerical algorithms. Routines exist that accomplish a variety of non-mathematical tasks. You can use these routines in Maple to
extend its functionality. For example, you can link to controlled hardware via a serial port or interface with another program. The Database package uses external calling to allow you to query,
create, and update databases in Maple. For more information, refer to the Database help page.
For more information on using external calling, refer to the ExternalCalling help page.
Mathematica Translator
The MmaTranslator package provides translation tools for converting Mathematica^® expressions, command operations, and notebooks to Maple. The package can translate Mathematica input to Maple
input and Mathematica notebooks to Maple documents. The Mma subpackage contains commands that provide translation for Mathematica commands when no equivalent Maple command exists. In most cases,
the command achieves the translation through minor manipulations of the input and output of similar Maple commands.
Note: The MmaTranslator package does not convert Mathematica programs.
There is a Maplet interface to the MmaTranslator package. For more information, refer to the MmaToMaple help page.
Matlab Package
The Matlab package enables you to translate MATLAB^® code to Maple, as well as call selected MATLAB^® functions from a Maple session, provided you have MATLAB^® installed on your system.
For more information, refer to the Matlab help page.
Accessing Maple from External Products
Microsoft Excel Add-In
Maple is available as an add-in to Microsoft Excel. This add-in is supported for Excel 365 (desktop) and Excel 2019 for Windows, and provides the following features.
• Access to Maple commands from Excel
• Ability to copy and paste between Maple and Excel
• Access to a subset of the Maple help pages
• Maple Function Wizard to step you through the creation of a Maple function call
To enable the Maple Excel Add-in:
1. In Excel, click the File menu and select Options.
2. Click Add-ins.
3. In the Manage box select Excel Add-ins, and then Go.
4. Navigate to the Excel subdirectory of your Maple installation and select the file:
– WMIMPLEX64.xla (that is, select $MAPLE/Excel/WMIMPLEX64.xla), and click OK.
5. Select the Maple Excel Add-in check box.
6. Click OK.
For further details on enabling the Maple Excel Add-in, refer to the Excel help page.
For information on using this add-in, refer to the Using Maple in Excel help file within Excel.
To view this help file:
1. Enable the add-in.
2. From the Add-ins tab, view the Maple toolbar.
3. On the Maple toolbar, click the Maple help icon .
OpenMaple is a suite of functions that allows you to access Maple algorithms and data structures in your compiled C, C#, Java, or Visual Basic programs. (This is the reverse of external calling,
which allows access to compiled C, C#, Fortran 77, and Java code from Maple.)
To run your application, Maple must be installed. You can distribute your application to any licensed Maple user. For additional terms and conditions on the use of OpenMaple, refer to the extern/
OpenMapleLicensing.txt file in your Maple installation.
For more details on using OpenMaple functions, refer to the OpenMaple help page.
MapleSim^TM is a complete environment for modeling and simulating multidomain engineering systems. During a simulation, MapleSim uses the symbolic Maple computation engine to generate the
mathematical models that represent the system behavior.
Because both products are tightly integrated, you can use Maple commands and technical document features to edit, manipulate, and analyze a MapleSim model. For example, you can use Maple commands
and tools to manipulate your model equations, develop custom components based on a mathematical model, and visualize simulation results.
MapleSim software is not included with the Maple software. For more information on MapleSim, visit http://www.maplesoft.com/maplesim.
Sharing and Storing Maple Content
11.1 In This Chapter
Section Topics
Writing to Files - Saving to Maple file formats • Saving Data to a File
• Saving Expressions to a File
Reading from Files -Opening Maple files • Reading Data from a File
• Reading Expressions from a File
Exporting to Other Formats - Exporting documents in file • Exporting Documents
formats supported by other software
• MapleNet
Connectivity - Using Maple with other programming languages • Translating Maple Code to Other
and software Programming Languages
• Accessing External Products from Maple
• Accessing Maple from External Products
• Sharing and Storing Maple Worksheet
Content with the MapleCloud^TM
Section Topics
Writing to Files - Saving to Maple file formats • Saving Data to a File
• Saving Expressions to a File
Reading from Files -Opening Maple files • Reading Data from a File
• Reading Expressions from a File
Exporting to Other Formats - Exporting documents in file • Exporting Documents
formats supported by other software
• MapleNet
Connectivity - Using Maple with other programming languages • Translating Maple Code to Other
and software Programming Languages
• Accessing External Products from Maple
• Accessing Maple from External Products
• Sharing and Storing Maple Worksheet
Content with the MapleCloud^TM
Exporting to Other Formats - Exporting documents in file formats supported by other software
Connectivity - Using Maple with other programming languages and software
• Sharing and Storing Maple Worksheet Content with the MapleCloud^TM
11.2 Writing to Files
Maple supports file formats in addition to the standard .mw file format.
After using Maple to perform a computation, you can save the results to a file for later processing with Maple or another program.
Note: Make sure you have write access to the directory in order to execute the example in the following subsections.
Saving Data to a File
If the result of a Maple calculation is a long list or a large array of numbers, you can convert it to Matrix form and write the numbers to a file using the ExportMatrix command. This command
writes columns of numerical data to a file, allowing you to import the numbers into another program. To convert a list or a list of lists to a Matrix, use the Matrix constructor. For more
information, refer to the Matrix help page.
> $L⁢:=⁢[\begin{array}{ccccc}-81& -98& -76& -4& 29\\ -38& -77& -72& 27& 44\\ -18& 57& -2& 8& 92\\ 87& 27& -32& 69& -31\\ 33& -93& -74& 99& 67\end{array}]&
> $\mathrm{ExportMatrix}\left("matrixdata.txt",L\right):$
If the data is a Vector or any object that can be converted to type Vector, use the ExportVector command. To convert lists to Vectors, use the Vector constructor. For more information, refer to the
Vector help page.
> $R≔\left[3,3.1415,-65,0\right]$
${R}{:=}\left[{3}{,}{3.1415}{,}{-}{65}{,}{0}\right]$ (11.1)
> $V≔\mathrm{Vector}\left(R\right)$
> $\mathrm{ExportVector}\left("vectordata.txt",V\right):$
You can extend these routines to write more complicated data, such as complex numbers or symbolic expressions. For more information, refer to the ExportMatrix and ExportVector help pages.
For more information on matrices and vectors, see Linear Algebra.
Saving Expressions to a File
If you construct a complicated expression or procedure, you can save them for future use in Maple. If you save the expression or procedure in the Maple internal format, Maple can retrieve it more
efficiently than from a document. Use the save command to write the expression to a .m file. For more information on Maple internal file formats, refer to the file help page.
> $\mathrm{qbinomial}≔\left(n,k\right)\to \frac{\underset{i=n-k+1}{\overset{n}{∏}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(1-{q}^{i}\right)}{\underset{i=
In this example, small expressions are used. In practice, Maple supports expressions with thousands of terms.
> $\mathrm{expr}≔\mathrm{qbinomial}\left(10,4\right)$
${\mathrm{expr}}{:=}\frac{\left({1}{-}{{q}}^{{7}}\right){⁢}\left({1}{-}{{q}}^{{8}}\right){⁢}\left({1}{-}{{q}}^{{9}}\right){⁢}\left({1}{-}{{q}}^ (11.2)
> $\mathrm{nexpr}≔\mathrm{normal}\left(\mathrm{expr}\right)$
${\mathrm{nexpr}}{:=}\left({{q}}^{{6}}{+}{{q}}^{{5}}{+}{{q}}^{{4}}{+}{{q}}^{{3}}{+}{{q}}^{{2}}{+}{q}{+}{1}\right){⁢}\left({{q}}^{{4}}{+}{1}\ (11.3)
You can save these expressions to the file qbinom.m.
> $\mathbf{save}\mathrm{qbinomial},\mathrm{expr},\mathrm{nexpr},"qbinom.m"$
Clear the memory using the restart command and retrieve the expressions using the read command.
> $\mathrm{restart}$
> $\mathbf{read}"qbinom.m"$
> $\mathrm{expr}$
$\frac{\left({1}{-}{{q}}^{{7}}\right){⁢}\left({1}{-}{{q}}^{{8}}\right){⁢}\left({1}{-}{{q}}^{{9}}\right){⁢}\left({1}{-}{{q}}^{{10}}\right)}{\left (11.4)
For more information on writing to files, refer to the save help page.
Saving Data as Part of a Workbook
You can save all files related to a common Maple project as a workbook (.maple) file. Saving your data files and worksheets (or documents) as a workbook allows you to use this saved data across all
.mw file inside your workbook.
Maple supports file formats in addition to the standard .mw file format.
After using Maple to perform a computation, you can save the results to a file for later processing with Maple or another program.
Note: Make sure you have write access to the directory in order to execute the example in the following subsections.
Saving Data to a File
If the result of a Maple calculation is a long list or a large array of numbers, you can convert it to Matrix form and write the numbers to a file using the ExportMatrix command. This command writes
columns of numerical data to a file, allowing you to import the numbers into another program. To convert a list or a list of lists to a Matrix, use the Matrix constructor. For more information,
refer to the Matrix help page.
> $L⁢:=⁢[\begin{array}{ccccc}-81& -98& -76& -4& 29\\ -38& -77& -72& 27& 44\\ -18& 57& -2& 8& 92\\ 87& 27& -32& 69& -31\\ 33& -93& -74& 99& 67\end{array}]&
> $\mathrm{ExportMatrix}\left("matrixdata.txt",L\right):$
If the data is a Vector or any object that can be converted to type Vector, use the ExportVector command. To convert lists to Vectors, use the Vector constructor. For more information, refer to the
Vector help page.
> $R≔\left[3,3.1415,-65,0\right]$
${R}{:=}\left[{3}{,}{3.1415}{,}{-}{65}{,}{0}\right]$ (11.1)
> $V≔\mathrm{Vector}\left(R\right)$
> $\mathrm{ExportVector}\left("vectordata.txt",V\right):$
You can extend these routines to write more complicated data, such as complex numbers or symbolic expressions. For more information, refer to the ExportMatrix and ExportVector help pages.
For more information on matrices and vectors, see Linear Algebra.
If the result of a Maple calculation is a long list or a large array of numbers, you can convert it to Matrix form and write the numbers to a file using the ExportMatrix command. This command writes
columns of numerical data to a file, allowing you to import the numbers into another program. To convert a list or a list of lists to a Matrix, use the Matrix constructor. For more information, refer
to the Matrix help page.
> $L⁢:=⁢[\begin{array}{ccccc}-81& -98& -76& -4& 29\\ -38& -77& -72& 27& 44\\ -18& 57& -2& 8& 92\\ 87& 27& -32& 69& -31\\ 33& -93& -74& 99& 67\end{array}]&
$L⁢:=⁢[\begin{array}{ccccc}-81& -98& -76& -4& 29\\ -38& -77& -72& 27& 44\\ -18& 57& -2& 8& 92\\ 87& 27& -32& 69& -31\\ 33& -93& -74& 99& 67\end{array}]:$
If the data is a Vector or any object that can be converted to type Vector, use the ExportVector command. To convert lists to Vectors, use the Vector constructor. For more information, refer to the
Vector help page.
You can extend these routines to write more complicated data, such as complex numbers or symbolic expressions. For more information, refer to the ExportMatrix and ExportVector help pages.
For more information on matrices and vectors, see Linear Algebra.
Saving Expressions to a File
If you construct a complicated expression or procedure, you can save them for future use in Maple. If you save the expression or procedure in the Maple internal format, Maple can retrieve it more
efficiently than from a document. Use the save command to write the expression to a .m file. For more information on Maple internal file formats, refer to the file help page.
> $\mathrm{qbinomial}≔\left(n,k\right)\to \frac{\underset{i=n-k+1}{\overset{n}{∏}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(1-{q}^{i}\right)}{\underset{i=1}
In this example, small expressions are used. In practice, Maple supports expressions with thousands of terms.
> $\mathrm{expr}≔\mathrm{qbinomial}\left(10,4\right)$
${\mathrm{expr}}{:=}\frac{\left({1}{-}{{q}}^{{7}}\right){⁢}\left({1}{-}{{q}}^{{8}}\right){⁢}\left({1}{-}{{q}}^{{9}}\right){⁢}\left({1}{-}{{q}}^ (11.2)
> $\mathrm{nexpr}≔\mathrm{normal}\left(\mathrm{expr}\right)$
${\mathrm{nexpr}}{:=}\left({{q}}^{{6}}{+}{{q}}^{{5}}{+}{{q}}^{{4}}{+}{{q}}^{{3}}{+}{{q}}^{{2}}{+}{q}{+}{1}\right){⁢}\left({{q}}^{{4}}{+}{1}\ (11.3)
You can save these expressions to the file qbinom.m.
> $\mathbf{save}\mathrm{qbinomial},\mathrm{expr},\mathrm{nexpr},"qbinom.m"$
Clear the memory using the restart command and retrieve the expressions using the read command.
> $\mathrm{restart}$
> $\mathbf{read}"qbinom.m"$
> $\mathrm{expr}$
$\frac{\left({1}{-}{{q}}^{{7}}\right){⁢}\left({1}{-}{{q}}^{{8}}\right){⁢}\left({1}{-}{{q}}^{{9}}\right){⁢}\left({1}{-}{{q}}^{{10}}\right)}{\left (11.4)
For more information on writing to files, refer to the save help page.
If you construct a complicated expression or procedure, you can save them for future use in Maple. If you save the expression or procedure in the Maple internal format, Maple can retrieve it more
efficiently than from a document. Use the save command to write the expression to a .m file. For more information on Maple internal file formats, refer to the file help page.
In this example, small expressions are used. In practice, Maple supports expressions with thousands of terms.
Clear the memory using the restart command and retrieve the expressions using the read command.
For more information on writing to files, refer to the save help page.
Saving Data as Part of a Workbook
You can save all files related to a common Maple project as a workbook (.maple) file. Saving your data files and worksheets (or documents) as a workbook allows you to use this saved data across all
.mw file inside your workbook.
You can save all files related to a common Maple project as a workbook (.maple) file. Saving your data files and worksheets (or documents) as a workbook allows you to use this saved data across all
.mw file inside your workbook.
11.3 Reading from Files
The most common reason for reading files is to load data, for example, data generated in an experiment. You can store data in a text file, and then read it into Maple.
Reading Data from a File
Import Data Assistant
If you generate data outside Maple, you can read it into Maple for further manipulation. This data can be an image, a sound file, or columns of numbers in a text file. You can easily import this
external data into Maple using the Import Data Assistant, where the supported file formats include files of type Excel^®, MATLAB^®, Image, Audio, Matrix Market, and Delimited.
To launch the Import Data Assistant:
1. From the Tools menu, select Assistants, and then Import Data.
2. A dialog window appears where you can navigate to your data file. Select the file that you want to import data from, and then select the file type before clicking Next.
3. From the main window, you can preview the selected file and choose from the applicable options based on the format of the file read in before importing the data into Maple. See Figure 11.1 for
an example.
Figure 11.1: Import Data Assistant
ImportMatrix Command
The Import Data Assistant provides a graphical interface to the ImportMatrix command. For more information, including options not available in the assistant, refer to the ImportMatrix help page.
Reading Expressions from a File
You can write Maple programs in a text file using a text editor, and then import the file into Maple. You can paste the commands from the text file into your document or you can use the read
When you read a file with the read command, Maple treats each line in the file as a command. Maple executes the commands and displays the results in your document but it does not, by default,
insert the commands from the file in your document.
For example, the file ks.txt contains the following Maple commands.
S:= n -> sum( binomial( n, beta ) * ( ( 2*beta )! / 2^beta - beta!*beta ), beta=1..n );
Note that the file should not contain prompts (>) at the start of lines.
When you read the file, Maple displays the results but not the commands.
${1024937361666644598071114328769317982974}$ (11.5)
> $\mathrm{filename}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),\mathrm{kernelopts}\left(\mathrm{dirsep}\right),"ks",\mathrm{kernelopts}\left(\
> $\mathbf{read}\mathrm{filename}$
${1024937361666644598071114328769317982974}$ (11.6)
If you set the interface echo option to 2, Maple inserts the commands from the file into your document.
> $\mathrm{interface}\left(\mathrm{echo}=2\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{read}\mathrm{filename}$
> S:= n -> sum( binomial( n, beta ) * ( ( 2*beta )! / 2^beta - beta!*beta ), beta=1..n );
> S(19);
${1024937361666644598071114328769317982974}$ (11.7)
For more information, refer to the read and interface help pages.
Reading Data From Workbook Attachments
Data stored in a workbook in the form of an attachment, can be accessed easily using the workbook URI. For information on workbook attachments, see worksheet,workbook,attachFiles. For information
on the workbook URI format, see worksheet,workbook,uri.
The most common reason for reading files is to load data, for example, data generated in an experiment. You can store data in a text file, and then read it into Maple.
Reading Data from a File
Import Data Assistant
If you generate data outside Maple, you can read it into Maple for further manipulation. This data can be an image, a sound file, or columns of numbers in a text file. You can easily import this
external data into Maple using the Import Data Assistant, where the supported file formats include files of type Excel^®, MATLAB^®, Image, Audio, Matrix Market, and Delimited.
To launch the Import Data Assistant:
1. From the Tools menu, select Assistants, and then Import Data.
2. A dialog window appears where you can navigate to your data file. Select the file that you want to import data from, and then select the file type before clicking Next.
3. From the main window, you can preview the selected file and choose from the applicable options based on the format of the file read in before importing the data into Maple. See Figure 11.1 for
an example.
Figure 11.1: Import Data Assistant
ImportMatrix Command
The Import Data Assistant provides a graphical interface to the ImportMatrix command. For more information, including options not available in the assistant, refer to the ImportMatrix help page.
Import Data Assistant
If you generate data outside Maple, you can read it into Maple for further manipulation. This data can be an image, a sound file, or columns of numbers in a text file. You can easily import this
external data into Maple using the Import Data Assistant, where the supported file formats include files of type Excel^®, MATLAB^®, Image, Audio, Matrix Market, and Delimited.
To launch the Import Data Assistant:
1. From the Tools menu, select Assistants, and then Import Data.
2. A dialog window appears where you can navigate to your data file. Select the file that you want to import data from, and then select the file type before clicking Next.
3. From the main window, you can preview the selected file and choose from the applicable options based on the format of the file read in before importing the data into Maple. See Figure 11.1 for an
Figure 11.1: Import Data Assistant
If you generate data outside Maple, you can read it into Maple for further manipulation. This data can be an image, a sound file, or columns of numbers in a text file. You can easily import this
external data into Maple using the Import Data Assistant, where the supported file formats include files of type Excel®, MATLAB®, Image, Audio, Matrix Market, and Delimited.
1. From the Tools menu, select Assistants, and then Import Data.
From the Tools menu, select Assistants, and then Import Data.
2. A dialog window appears where you can navigate to your data file. Select the file that you want to import data from, and then select the file type before clicking Next.
A dialog window appears where you can navigate to your data file. Select the file that you want to import data from, and then select the file type before clicking Next.
3. From the main window, you can preview the selected file and choose from the applicable options based on the format of the file read in before importing the data into Maple. See Figure 11.1 for an
From the main window, you can preview the selected file and choose from the applicable options based on the format of the file read in before importing the data into Maple. See Figure 11.1 for an
ImportMatrix Command
The Import Data Assistant provides a graphical interface to the ImportMatrix command. For more information, including options not available in the assistant, refer to the ImportMatrix help page.
The Import Data Assistant provides a graphical interface to the ImportMatrix command. For more information, including options not available in the assistant, refer to the ImportMatrix help page.
Reading Expressions from a File
You can write Maple programs in a text file using a text editor, and then import the file into Maple. You can paste the commands from the text file into your document or you can use the read
When you read a file with the read command, Maple treats each line in the file as a command. Maple executes the commands and displays the results in your document but it does not, by default, insert
the commands from the file in your document.
For example, the file ks.txt contains the following Maple commands.
S:= n -> sum( binomial( n, beta ) * ( ( 2*beta )! / 2^beta - beta!*beta ), beta=1..n );
Note that the file should not contain prompts (>) at the start of lines.
When you read the file, Maple displays the results but not the commands.
${1024937361666644598071114328769317982974}$ (11.5)
> $\mathrm{filename}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),\mathrm{kernelopts}\left(\mathrm{dirsep}\right),"ks",\mathrm{kernelopts}\left(\
> $\mathbf{read}\mathrm{filename}$
${1024937361666644598071114328769317982974}$ (11.6)
If you set the interface echo option to 2, Maple inserts the commands from the file into your document.
> $\mathrm{interface}\left(\mathrm{echo}=2\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{read}\mathrm{filename}$
> S:= n -> sum( binomial( n, beta ) * ( ( 2*beta )! / 2^beta - beta!*beta ), beta=1..n );
> S(19);
${1024937361666644598071114328769317982974}$ (11.7)
For more information, refer to the read and interface help pages.
You can write Maple programs in a text file using a text editor, and then import the file into Maple. You can paste the commands from the text file into your document or you can use the read command.
When you read a file with the read command, Maple treats each line in the file as a command. Maple executes the commands and displays the results in your document but it does not, by default, insert
the commands from the file in your document.
For example, the file ks.txt contains the following Maple commands.
S:= n -> sum( binomial( n, beta ) * ( ( 2*beta )! / 2^beta - beta!*beta ), beta=1..n );
Note that the file should not contain prompts (>) at the start of lines.
When you read the file, Maple displays the results but not the commands.
If you set the interface echo option to 2, Maple inserts the commands from the file into your document.
> S:= n -> sum( binomial( n, beta ) * ( ( 2*beta )! / 2^beta - beta!*beta ), beta=1..n );
For more information, refer to the read and interface help pages.
Reading Data From Workbook Attachments
Data stored in a workbook in the form of an attachment, can be accessed easily using the workbook URI. For information on workbook attachments, see worksheet,workbook,attachFiles. For information on
the workbook URI format, see worksheet,workbook,uri.
Data stored in a workbook in the form of an attachment, can be accessed easily using the workbook URI. For information on workbook attachments, see worksheet,workbook,attachFiles. For information on
the workbook URI format, see worksheet,workbook,uri.
11.4 Exporting to Other Formats
Exporting Documents
You can save your documents by selecting Save or Save As from the File menu. By selecting Export As from the File menu, you can also export a document in the following formats: HTML, LaTeX, Maple
input, Maplet application, Maple text, plain text, PDF, and Rich Text Format. This allows you to access your work outside Maple.
The .html file that Maple generates can be loaded into any HTML browser. Exported mathematical content can be displayed in one of two formats, GIF or MathML 2.0, and is saved in a separate folder.
MathML is the Internet standard, sanctioned by the World Wide Web Consortium (W3C), for the communication of structured mathematical formulae between applications. For more information about
MathML, refer to the MathML help page.
Maple documents that are exported to HTML translate into multiple documents when using frames. If the frames feature is not selected, Maple creates only one page that contains the document
The .tex file generated by Maple is ready for processing by LaTeX. All distributions of Maple include the necessary style files. By default, the LaTeX style files are set for printing the .tex
file using the dvips printer driver. You can change this behavior by specifying an option to the \usepackage LaTeX command in the preamble of your .tex file. For more information, refer to the
exporttoLaTeX help page.
Maple Input
You can export a Maple document as Maple input so that it can be loaded using the Maple Command-line version.
Important: When exporting a document as Maple input for use in Command-line Maple, your document must contain explicit semicolons in 1-D Math input. If not, the exported .mpl file does not contain
semicolons, and Command-line Maple generates errors.
Maplet Application
The Export as Maplet facility saves a Maple document as a .maplet file, so that you can run it using the command-line interface or the MapletViewer. The MapletViewer is an executable program that
can launch saved Maplet applications. It displays and runs Maplet applications independently of the Maple Worksheet interface.
Important: When exporting a document as a Maplet Application for use in Command-line Maple or the MapletViewer, your document must contain explicit semicolons. If not, the exported .maplet file
does not contain semicolons, and Command-line Maple and the MapletViewer generates errors.
Maple Text
Maple text is marked text that retains the distinction between text, Maple input, and Maple output. Thus, you can export a document as Maple text, send the text file by email, and the recipient
can import the Maple text into a Maple session and regenerate the computations in the original document.
Export a Maple document to a Portable Document Format (PDF) file so that you can open the file in a reader such as Adobe^® Acrobat^®. The PDF document is formatted as it would appear when the
Maple worksheet is printed using the active printer settings.
Note: Images, plots, and embedded components may be resized in the PDF file.
Plain Text
Export a Maple document as plain text so that you can open the text file in a word processor.
Rich Text Format (RTF)
Export a Maple document to a rich text format file so that you can open and edit the file in a word processor.
Note: The generated .rtf format is compatible with Microsoft Word and Microsoft WordPad only.
Summary of Translation
Table 11.1: Summary of Content Translation When Exporting to Different Formats
Content HTML LaTeX Maple Maplet Maple Text Plain Text Rich Text PDF Format
Input Application Format
Text Maintained Maintained Preceded Preceded by Preceded by # Maintained Maintained Maintained
by # #
1-D Math Maintained Maintained Maintained Maintained Preceded by > Preceded by > Static Static
image image
2-D Math GIF or LaTeX 1-D Math 1-D Math 1-D Math or 1-D Math or Static Either
MathML (if (if character-based character-based image text or
possible) possible) typesetting typesetting shapes
on option
Plot GIF Postscript Not Not Not exported Not exported Static Static
file exported exported image image
Animation Animated GIF Not exported Not Not Not exported Not exported Not Static
exported exported exported image
Hidden Not exported Not exported Not Not Not exported Not exported Not Not
content exported exported exported exported
Manually Not Not Not Not Not supported Not supported RTF page Maintained
inserted supported supported supported supported break
page break object
Hyperlink Links to Plain text Plain text Plain text Plain text Plain text Plain text Plain text
help pages
become plain
text. Links
to documents
are renamed
converted to
HTML links
Embedded GIF Not exported Not Not Not exported Not exported Static Static
image or exported exported image image
Spreadsheet HTML table LaTeX tables Not Not Not exported Not exported RTF table Static
exported exported image
Document Approximated LaTeX Not Not Not exported Not exported RTF style Maintained
style by HTML environments exported exported
style and
attributes sections,
LaTeX macro
Overview of MapleNet
Using MapleNet, you can deploy Maple content on the web. Powered by the Maple computation engine, MapleNet allows you to embed dynamic formulas, models, and diagrams as live content in webpages.
The MapleNet software is not included with the Maple software. For more information on MapleNet, visit http://www.maplesoft.com/maplenet.
MapleNet Documents and Maplets
After you upload your Maple document to the MapleNet server, it can be accessed by anyone in the world using a web browser. Even if viewers do not have a copy of Maple installed, they can view
documents and Maplets, manipulate 3-D plots, and execute code at the click of a button.
Custom Java Applets and JavaServer Pages^TM Technology
MapleNet provides a programming interface to the Maple math engine so commands can be executed from a Java applet or using JavaServer Pages^TM technology. Embed MapleNet into your web application,
and let Maple handle the math and visualization.
Exporting Documents
You can save your documents by selecting Save or Save As from the File menu. By selecting Export As from the File menu, you can also export a document in the following formats: HTML, LaTeX, Maple
input, Maplet application, Maple text, plain text, PDF, and Rich Text Format. This allows you to access your work outside Maple.
The .html file that Maple generates can be loaded into any HTML browser. Exported mathematical content can be displayed in one of two formats, GIF or MathML 2.0, and is saved in a separate folder.
MathML is the Internet standard, sanctioned by the World Wide Web Consortium (W3C), for the communication of structured mathematical formulae between applications. For more information about
MathML, refer to the MathML help page.
Maple documents that are exported to HTML translate into multiple documents when using frames. If the frames feature is not selected, Maple creates only one page that contains the document
The .tex file generated by Maple is ready for processing by LaTeX. All distributions of Maple include the necessary style files. By default, the LaTeX style files are set for printing the .tex file
using the dvips printer driver. You can change this behavior by specifying an option to the \usepackage LaTeX command in the preamble of your .tex file. For more information, refer to the
exporttoLaTeX help page.
Maple Input
You can export a Maple document as Maple input so that it can be loaded using the Maple Command-line version.
Important: When exporting a document as Maple input for use in Command-line Maple, your document must contain explicit semicolons in 1-D Math input. If not, the exported .mpl file does not contain
semicolons, and Command-line Maple generates errors.
Maplet Application
The Export as Maplet facility saves a Maple document as a .maplet file, so that you can run it using the command-line interface or the MapletViewer. The MapletViewer is an executable program that
can launch saved Maplet applications. It displays and runs Maplet applications independently of the Maple Worksheet interface.
Important: When exporting a document as a Maplet Application for use in Command-line Maple or the MapletViewer, your document must contain explicit semicolons. If not, the exported .maplet file
does not contain semicolons, and Command-line Maple and the MapletViewer generates errors.
Maple Text
Maple text is marked text that retains the distinction between text, Maple input, and Maple output. Thus, you can export a document as Maple text, send the text file by email, and the recipient can
import the Maple text into a Maple session and regenerate the computations in the original document.
Export a Maple document to a Portable Document Format (PDF) file so that you can open the file in a reader such as Adobe^® Acrobat^®. The PDF document is formatted as it would appear when the Maple
worksheet is printed using the active printer settings.
Note: Images, plots, and embedded components may be resized in the PDF file.
Plain Text
Export a Maple document as plain text so that you can open the text file in a word processor.
Rich Text Format (RTF)
Export a Maple document to a rich text format file so that you can open and edit the file in a word processor.
Note: The generated .rtf format is compatible with Microsoft Word and Microsoft WordPad only.
Summary of Translation
Table 11.1: Summary of Content Translation When Exporting to Different Formats
Content HTML LaTeX Maple Maplet Maple Text Plain Text Rich Text PDF Format
Input Application Format
Text Maintained Maintained Preceded Preceded by Preceded by # Maintained Maintained Maintained
by # #
1-D Math Maintained Maintained Maintained Maintained Preceded by > Preceded by > Static Static
image image
2-D Math GIF or LaTeX 1-D Math 1-D Math 1-D Math or 1-D Math or Static Either
MathML (if (if character-based character-based image text or
possible) possible) typesetting typesetting shapes
on option
Plot GIF Postscript Not Not Not exported Not exported Static Static
file exported exported image image
Animation Animated GIF Not exported Not Not Not exported Not exported Not Static
exported exported exported image
Hidden Not exported Not exported Not Not Not exported Not exported Not Not
content exported exported exported exported
Manually Not Not Not Not Not supported Not supported RTF page Maintained
inserted supported supported supported supported break
page break object
Hyperlink Links to Plain text Plain text Plain text Plain text Plain text Plain text Plain text
help pages
become plain
text. Links
to documents
are renamed
converted to
HTML links
Embedded GIF Not exported Not Not Not exported Not exported Static Static
image or exported exported image image
Spreadsheet HTML table LaTeX tables Not Not Not exported Not exported RTF table Static
exported exported image
Document Approximated LaTeX Not Not Not exported Not exported RTF style Maintained
style by HTML environments exported exported
style and
attributes sections,
LaTeX macro
You can save your documents by selecting Save or Save As from the File menu. By selecting Export As from the File menu, you can also export a document in the following formats: HTML, LaTeX, Maple
input, Maplet application, Maple text, plain text, PDF, and Rich Text Format. This allows you to access your work outside Maple.
The .html file that Maple generates can be loaded into any HTML browser. Exported mathematical content can be displayed in one of two formats, GIF or MathML 2.0, and is saved in a separate folder.
MathML is the Internet standard, sanctioned by the World Wide Web Consortium (W3C), for the communication of structured mathematical formulae between applications. For more information about MathML,
refer to the MathML help page.
Maple documents that are exported to HTML translate into multiple documents when using frames. If the frames feature is not selected, Maple creates only one page that contains the document contents.
The .html file that Maple generates can be loaded into any HTML browser. Exported mathematical content can be displayed in one of two formats, GIF or MathML 2.0, and is saved in a separate folder.
MathML is the Internet standard, sanctioned by the World Wide Web Consortium (W3C), for the communication of structured mathematical formulae between applications. For more information about MathML,
refer to the MathML help page.
Maple documents that are exported to HTML translate into multiple documents when using frames. If the frames feature is not selected, Maple creates only one page that contains the document contents.
The .tex file generated by Maple is ready for processing by LaTeX. All distributions of Maple include the necessary style files. By default, the LaTeX style files are set for printing the .tex file
using the dvips printer driver. You can change this behavior by specifying an option to the \usepackage LaTeX command in the preamble of your .tex file. For more information, refer to the
exporttoLaTeX help page.
The .tex file generated by Maple is ready for processing by LaTeX. All distributions of Maple include the necessary style files. By default, the LaTeX style files are set for printing the .tex file
using the dvips printer driver. You can change this behavior by specifying an option to the \usepackage LaTeX command in the preamble of your .tex file. For more information, refer to the
exporttoLaTeX help page.
Maple Input
You can export a Maple document as Maple input so that it can be loaded using the Maple Command-line version.
Important: When exporting a document as Maple input for use in Command-line Maple, your document must contain explicit semicolons in 1-D Math input. If not, the exported .mpl file does not contain
semicolons, and Command-line Maple generates errors.
You can export a Maple document as Maple input so that it can be loaded using the Maple Command-line version.
Important: When exporting a document as Maple input for use in Command-line Maple, your document must contain explicit semicolons in 1-D Math input. If not, the exported .mpl file does not contain
semicolons, and Command-line Maple generates errors.
Maplet Application
The Export as Maplet facility saves a Maple document as a .maplet file, so that you can run it using the command-line interface or the MapletViewer. The MapletViewer is an executable program that
can launch saved Maplet applications. It displays and runs Maplet applications independently of the Maple Worksheet interface.
Important: When exporting a document as a Maplet Application for use in Command-line Maple or the MapletViewer, your document must contain explicit semicolons. If not, the exported .maplet file does
not contain semicolons, and Command-line Maple and the MapletViewer generates errors.
The Export as Maplet facility saves a Maple document as a .maplet file, so that you can run it using the command-line interface or the MapletViewer. The MapletViewer is an executable program that can
launch saved Maplet applications. It displays and runs Maplet applications independently of the Maple Worksheet interface.
Important: When exporting a document as a Maplet Application for use in Command-line Maple or the MapletViewer, your document must contain explicit semicolons. If not, the exported .maplet file does
not contain semicolons, and Command-line Maple and the MapletViewer generates errors.
Maple Text
Maple text is marked text that retains the distinction between text, Maple input, and Maple output. Thus, you can export a document as Maple text, send the text file by email, and the recipient can
import the Maple text into a Maple session and regenerate the computations in the original document.
Maple text is marked text that retains the distinction between text, Maple input, and Maple output. Thus, you can export a document as Maple text, send the text file by email, and the recipient can
import the Maple text into a Maple session and regenerate the computations in the original document.
Export a Maple document to a Portable Document Format (PDF) file so that you can open the file in a reader such as Adobe^® Acrobat^®. The PDF document is formatted as it would appear when the Maple
worksheet is printed using the active printer settings.
Note: Images, plots, and embedded components may be resized in the PDF file.
Export a Maple document to a Portable Document Format (PDF) file so that you can open the file in a reader such as Adobe® Acrobat®. The PDF document is formatted as it would appear when the Maple
worksheet is printed using the active printer settings.
Note: Images, plots, and embedded components may be resized in the PDF file.
Plain Text
Export a Maple document as plain text so that you can open the text file in a word processor.
Export a Maple document as plain text so that you can open the text file in a word processor.
Rich Text Format (RTF)
Export a Maple document to a rich text format file so that you can open and edit the file in a word processor.
Note: The generated .rtf format is compatible with Microsoft Word and Microsoft WordPad only.
Export a Maple document to a rich text format file so that you can open and edit the file in a word processor.
Note: The generated .rtf format is compatible with Microsoft Word and Microsoft WordPad only.
Summary of Translation
Table 11.1: Summary of Content Translation When Exporting to Different Formats
Content HTML LaTeX Maple Maplet Maple Text Plain Text Rich Text PDF Format
Input Application Format
Text Maintained Maintained Preceded Preceded by Preceded by # Maintained Maintained Maintained
by # #
1-D Math Maintained Maintained Maintained Maintained Preceded by > Preceded by > Static Static
image image
2-D Math GIF or LaTeX 1-D Math 1-D Math 1-D Math or 1-D Math or Static Either
MathML (if (if character-based character-based image text or
possible) possible) typesetting typesetting shapes
on option
Plot GIF Postscript Not Not Not exported Not exported Static Static
file exported exported image image
Animation Animated GIF Not exported Not Not Not exported Not exported Not Static
exported exported exported image
Hidden Not exported Not exported Not Not Not exported Not exported Not Not
content exported exported exported exported
Manually Not Not Not Not Not supported Not supported RTF page Maintained
inserted supported supported supported supported break
page break object
Hyperlink Links to Plain text Plain text Plain text Plain text Plain text Plain text Plain text
help pages
become plain
text. Links
to documents
are renamed
converted to
HTML links
Embedded GIF Not exported Not Not Not exported Not exported Static Static
image or exported exported image image
Spreadsheet HTML table LaTeX tables Not Not Not exported Not exported RTF table Static
exported exported image
Document Approximated LaTeX Not Not Not exported Not exported RTF style Maintained
style by HTML environments exported exported
style and
attributes sections,
LaTeX macro
Table 11.1: Summary of Content Translation When Exporting to Different Formats
Content HTML LaTeX Maple Maplet Maple Text Plain Text Rich Text PDF Format
Input Application Format
Text Maintained Maintained Preceded Preceded by Preceded by # Maintained Maintained Maintained
by # #
1-D Math Maintained Maintained Maintained Maintained Preceded by > Preceded by > Static Static
image image
2-D Math GIF or LaTeX 1-D Math 1-D Math 1-D Math or 1-D Math or Static Either
MathML (if (if character-based character-based image text or
possible) possible) typesetting typesetting shapes
on option
Plot GIF Postscript Not Not Not exported Not exported Static Static
file exported exported image image
Animation Animated GIF Not exported Not Not Not exported Not exported Not Static
exported exported exported image
Hidden Not exported Not exported Not Not Not exported Not exported Not Not
content exported exported exported exported
Manually Not Not Not Not Not supported Not supported RTF page Maintained
inserted supported supported supported supported break
page break object
Hyperlink Links to Plain text Plain text Plain text Plain text Plain text Plain text Plain text
help pages
become plain
text. Links
to documents
are renamed
converted to
HTML links
Embedded GIF Not exported Not Not Not exported Not exported Static Static
image or exported exported image image
Spreadsheet HTML table LaTeX tables Not Not Not exported Not exported RTF table Static
exported exported image
Document Approximated LaTeX Not Not Not exported Not exported RTF style Maintained
style by HTML environments exported exported
style and
attributes sections,
LaTeX macro
Content HTML LaTeX Maple Input Maplet Maple Text Plain Text Rich Text PDF Format
Application Format
Text Maintained Maintained Preceded by Preceded by Preceded by # Maintained Maintained Maintained
# #
1-D Math Maintained Maintained Maintained Maintained Preceded by > Preceded by > Static Static image
2-D Math GIF or MathML LaTeX 1-D Math 1-D Math 1-D Math or 1-D Math or Static Either text or shapes
(if (if character-based character-based image depending on option
possible) possible) typesetting typesetting selected
Plot GIF Postscript file Not Not Not exported Not exported Static Static image
exported exported image
Animation Animated GIF Not exported Not Not Not exported Not exported Not Static image
exported exported exported
Hidden content Not exported Not exported Not Not Not exported Not exported Not Not exported
exported exported exported
Manually Not supported Not supported Not Not Not supported Not supported RTF page Maintained
inserted page supported supported break
break object
Hyperlink Links to help pages become plain text. Links to Plain text Plain text Plain text Plain text Plain text Plain text Plain text
documents are renamed and converted to HTML links
Embedded image GIF Not exported Not Not Not exported Not exported Static Static image
or sketch output exported exported image
Spreadsheet HTML table LaTeX tables Not Not Not exported Not exported RTF table Static image
exported exported
Document style Approximated by HTML style attributes LaTeX environments and Not Not Not exported Not exported RTF style Maintained
sections, LaTeX macro exported exported
Links to help pages become plain text. Links to documents are renamed and converted to HTML links
Overview of MapleNet
Using MapleNet, you can deploy Maple content on the web. Powered by the Maple computation engine, MapleNet allows you to embed dynamic formulas, models, and diagrams as live content in webpages.
The MapleNet software is not included with the Maple software. For more information on MapleNet, visit http://www.maplesoft.com/maplenet.
MapleNet Documents and Maplets
After you upload your Maple document to the MapleNet server, it can be accessed by anyone in the world using a web browser. Even if viewers do not have a copy of Maple installed, they can view
documents and Maplets, manipulate 3-D plots, and execute code at the click of a button.
Custom Java Applets and JavaServer Pages^TM Technology
MapleNet provides a programming interface to the Maple math engine so commands can be executed from a Java applet or using JavaServer Pages^TM technology. Embed MapleNet into your web application,
and let Maple handle the math and visualization.
Overview of MapleNet
Using MapleNet, you can deploy Maple content on the web. Powered by the Maple computation engine, MapleNet allows you to embed dynamic formulas, models, and diagrams as live content in webpages.
The MapleNet software is not included with the Maple software. For more information on MapleNet, visit http://www.maplesoft.com/maplenet.
Using MapleNet, you can deploy Maple content on the web. Powered by the Maple computation engine, MapleNet allows you to embed dynamic formulas, models, and diagrams as live content in webpages. The
MapleNet software is not included with the Maple software. For more information on MapleNet, visit http://www.maplesoft.com/maplenet.
MapleNet Documents and Maplets
After you upload your Maple document to the MapleNet server, it can be accessed by anyone in the world using a web browser. Even if viewers do not have a copy of Maple installed, they can view
documents and Maplets, manipulate 3-D plots, and execute code at the click of a button.
After you upload your Maple document to the MapleNet server, it can be accessed by anyone in the world using a web browser. Even if viewers do not have a copy of Maple installed, they can view
documents and Maplets, manipulate 3-D plots, and execute code at the click of a button.
Custom Java Applets and JavaServer Pages^TM Technology
MapleNet provides a programming interface to the Maple math engine so commands can be executed from a Java applet or using JavaServer Pages^TM technology. Embed MapleNet into your web application,
and let Maple handle the math and visualization.
MapleNet provides a programming interface to the Maple math engine so commands can be executed from a Java applet or using JavaServer PagesTM technology. Embed MapleNet into your web application, and
let Maple handle the math and visualization.
11.5 Connectivity
Translating Maple Code To Other Programming Languages
Code Generation
The CodeGeneration package is a collection of commands and subpackages that enable the translation of Maple code to other programming languages. Languages currently supported include: C, C#,
Fortran 77, Java, MATLAB^®, Visual Basic, Perl, and Python.
For details on Code Generation, refer to the CodeGeneration help page.
Accessing External Products from Maple
External Calling
External calling allows you to use compiled C, C#, Fortran 77, or Java code in Maple. Functions written in these languages can be linked and used as if they were Maple procedures. With external
calling you can use pre-written optimized algorithms without the need to translate them into Maple commands. Access to the NAG library routines and other numerical algorithms is built into Maple
using the external calling mechanism.
External calling can also be applied to functions other than numerical algorithms. Routines exist that accomplish a variety of non-mathematical tasks. You can use these routines in Maple to extend
its functionality. For example, you can link to controlled hardware via a serial port or interface with another program. The Database package uses external calling to allow you to query, create,
and update databases in Maple. For more information, refer to the Database help page.
For more information on using external calling, refer to the ExternalCalling help page.
Mathematica Translator
The MmaTranslator package provides translation tools for converting Mathematica^® expressions, command operations, and notebooks to Maple. The package can translate Mathematica input to Maple
input and Mathematica notebooks to Maple documents. The Mma subpackage contains commands that provide translation for Mathematica commands when no equivalent Maple command exists. In most cases,
the command achieves the translation through minor manipulations of the input and output of similar Maple commands.
Note: The MmaTranslator package does not convert Mathematica programs.
There is a Maplet interface to the MmaTranslator package. For more information, refer to the MmaToMaple help page.
Matlab Package
The Matlab package enables you to translate MATLAB^® code to Maple, as well as call selected MATLAB^® functions from a Maple session, provided you have MATLAB^® installed on your system.
For more information, refer to the Matlab help page.
Accessing Maple from External Products
Microsoft Excel Add-In
Maple is available as an add-in to Microsoft Excel. This add-in is supported for Excel 365 (desktop) and Excel 2019 for Windows, and provides the following features.
• Access to Maple commands from Excel
• Ability to copy and paste between Maple and Excel
• Access to a subset of the Maple help pages
• Maple Function Wizard to step you through the creation of a Maple function call
To enable the Maple Excel Add-in:
1. In Excel, click the File menu and select Options.
2. Click Add-ins.
3. In the Manage box select Excel Add-ins, and then Go.
4. Navigate to the Excel subdirectory of your Maple installation and select the file:
– WMIMPLEX64.xla (that is, select $MAPLE/Excel/WMIMPLEX64.xla), and click OK.
5. Select the Maple Excel Add-in check box.
6. Click OK.
For further details on enabling the Maple Excel Add-in, refer to the Excel help page.
For information on using this add-in, refer to the Using Maple in Excel help file within Excel.
To view this help file:
1. Enable the add-in.
2. From the Add-ins tab, view the Maple toolbar.
3. On the Maple toolbar, click the Maple help icon .
OpenMaple is a suite of functions that allows you to access Maple algorithms and data structures in your compiled C, C#, Java, or Visual Basic programs. (This is the reverse of external calling,
which allows access to compiled C, C#, Fortran 77, and Java code from Maple.)
To run your application, Maple must be installed. You can distribute your application to any licensed Maple user. For additional terms and conditions on the use of OpenMaple, refer to the extern/
OpenMapleLicensing.txt file in your Maple installation.
For more details on using OpenMaple functions, refer to the OpenMaple help page.
MapleSim^TM is a complete environment for modeling and simulating multidomain engineering systems. During a simulation, MapleSim uses the symbolic Maple computation engine to generate the
mathematical models that represent the system behavior.
Because both products are tightly integrated, you can use Maple commands and technical document features to edit, manipulate, and analyze a MapleSim model. For example, you can use Maple commands
and tools to manipulate your model equations, develop custom components based on a mathematical model, and visualize simulation results.
MapleSim software is not included with the Maple software. For more information on MapleSim, visit http://www.maplesoft.com/maplesim.
Sharing and Storing Maple Content
The MapleCloud
You can use the MapleCloud to share or store your Maple documents and workbooks. Upload Maple documents or workbooks. Package workbooks offer a way to share a Maple package with other users,
including source code, documentation and examples.
The MapleCloud has private and public sharing. You can share with all Maple users, share with a private group, or upload and store content in a user-specific area that only you can access.
Users need an internet connection to use the MapleCloud. Anyone can access publicly shared documents. To share content, create, manage and join user groups; and view group-specific content, you
must log in to the MapleCloud using a Maplesoft.com, Gmail™, or Google Mail™ account name and password.
A Maplesoft.com membership account gives you access to thousands of free Maple resources and MaplePrimes, which is an active web community for sharing techniques and experiences with Maple and
related products. To sign up for a free Maplesoft.com membership account, visit http://www.maplesoft.com/members/sign_up_form.aspx. The MapleCloud is integrated with several of these online
features, so it is strongly recommended that you use a Maplesoft.com membership account.
For more information on the MapleCloud, refer to the MapleCloud help page.
Translating Maple Code To Other Programming Languages
Code Generation
The CodeGeneration package is a collection of commands and subpackages that enable the translation of Maple code to other programming languages. Languages currently supported include: C, C#,
Fortran 77, Java, MATLAB^®, Visual Basic, Perl, and Python.
For details on Code Generation, refer to the CodeGeneration help page.
Code Generation
The CodeGeneration package is a collection of commands and subpackages that enable the translation of Maple code to other programming languages. Languages currently supported include: C, C#, Fortran
77, Java, MATLAB^®, Visual Basic, Perl, and Python.
For details on Code Generation, refer to the CodeGeneration help page.
The CodeGeneration package is a collection of commands and subpackages that enable the translation of Maple code to other programming languages. Languages currently supported include: C, C#, Fortran
77, Java, MATLAB®, Visual Basic, Perl, and Python.
For details on Code Generation, refer to the CodeGeneration help page.
Accessing External Products from Maple
External Calling
External calling allows you to use compiled C, C#, Fortran 77, or Java code in Maple. Functions written in these languages can be linked and used as if they were Maple procedures. With external
calling you can use pre-written optimized algorithms without the need to translate them into Maple commands. Access to the NAG library routines and other numerical algorithms is built into Maple
using the external calling mechanism.
External calling can also be applied to functions other than numerical algorithms. Routines exist that accomplish a variety of non-mathematical tasks. You can use these routines in Maple to extend
its functionality. For example, you can link to controlled hardware via a serial port or interface with another program. The Database package uses external calling to allow you to query, create,
and update databases in Maple. For more information, refer to the Database help page.
For more information on using external calling, refer to the ExternalCalling help page.
Mathematica Translator
The MmaTranslator package provides translation tools for converting Mathematica^® expressions, command operations, and notebooks to Maple. The package can translate Mathematica input to Maple input
and Mathematica notebooks to Maple documents. The Mma subpackage contains commands that provide translation for Mathematica commands when no equivalent Maple command exists. In most cases, the
command achieves the translation through minor manipulations of the input and output of similar Maple commands.
Note: The MmaTranslator package does not convert Mathematica programs.
There is a Maplet interface to the MmaTranslator package. For more information, refer to the MmaToMaple help page.
Matlab Package
The Matlab package enables you to translate MATLAB^® code to Maple, as well as call selected MATLAB^® functions from a Maple session, provided you have MATLAB^® installed on your system.
For more information, refer to the Matlab help page.
External Calling
External calling allows you to use compiled C, C#, Fortran 77, or Java code in Maple. Functions written in these languages can be linked and used as if they were Maple procedures. With external
calling you can use pre-written optimized algorithms without the need to translate them into Maple commands. Access to the NAG library routines and other numerical algorithms is built into Maple
using the external calling mechanism.
External calling can also be applied to functions other than numerical algorithms. Routines exist that accomplish a variety of non-mathematical tasks. You can use these routines in Maple to extend
its functionality. For example, you can link to controlled hardware via a serial port or interface with another program. The Database package uses external calling to allow you to query, create, and
update databases in Maple. For more information, refer to the Database help page.
For more information on using external calling, refer to the ExternalCalling help page.
External calling allows you to use compiled C, C#, Fortran 77, or Java code in Maple. Functions written in these languages can be linked and used as if they were Maple procedures. With external
calling you can use pre-written optimized algorithms without the need to translate them into Maple commands. Access to the NAG library routines and other numerical algorithms is built into Maple
using the external calling mechanism.
External calling can also be applied to functions other than numerical algorithms. Routines exist that accomplish a variety of non-mathematical tasks. You can use these routines in Maple to extend
its functionality. For example, you can link to controlled hardware via a serial port or interface with another program. The Database package uses external calling to allow you to query, create, and
update databases in Maple. For more information, refer to the Database help page.
For more information on using external calling, refer to the ExternalCalling help page.
Mathematica Translator
The MmaTranslator package provides translation tools for converting Mathematica^® expressions, command operations, and notebooks to Maple. The package can translate Mathematica input to Maple input
and Mathematica notebooks to Maple documents. The Mma subpackage contains commands that provide translation for Mathematica commands when no equivalent Maple command exists. In most cases, the
command achieves the translation through minor manipulations of the input and output of similar Maple commands.
Note: The MmaTranslator package does not convert Mathematica programs.
There is a Maplet interface to the MmaTranslator package. For more information, refer to the MmaToMaple help page.
The MmaTranslator package provides translation tools for converting Mathematica® expressions, command operations, and notebooks to Maple. The package can translate Mathematica input to Maple input
and Mathematica notebooks to Maple documents. The Mma subpackage contains commands that provide translation for Mathematica commands when no equivalent Maple command exists. In most cases, the
command achieves the translation through minor manipulations of the input and output of similar Maple commands.
There is a Maplet interface to the MmaTranslator package. For more information, refer to the MmaToMaple help page.
Matlab Package
The Matlab package enables you to translate MATLAB^® code to Maple, as well as call selected MATLAB^® functions from a Maple session, provided you have MATLAB^® installed on your system.
For more information, refer to the Matlab help page.
The Matlab package enables you to translate MATLAB® code to Maple, as well as call selected MATLAB® functions from a Maple session, provided you have MATLAB® installed on your system.
Accessing Maple from External Products
Microsoft Excel Add-In
Maple is available as an add-in to Microsoft Excel. This add-in is supported for Excel 365 (desktop) and Excel 2019 for Windows, and provides the following features.
• Access to Maple commands from Excel
• Ability to copy and paste between Maple and Excel
• Access to a subset of the Maple help pages
• Maple Function Wizard to step you through the creation of a Maple function call
To enable the Maple Excel Add-in:
1. In Excel, click the File menu and select Options.
2. Click Add-ins.
3. In the Manage box select Excel Add-ins, and then Go.
4. Navigate to the Excel subdirectory of your Maple installation and select the file:
– WMIMPLEX64.xla (that is, select $MAPLE/Excel/WMIMPLEX64.xla), and click OK.
5. Select the Maple Excel Add-in check box.
6. Click OK.
For further details on enabling the Maple Excel Add-in, refer to the Excel help page.
For information on using this add-in, refer to the Using Maple in Excel help file within Excel.
To view this help file:
1. Enable the add-in.
2. From the Add-ins tab, view the Maple toolbar.
3. On the Maple toolbar, click the Maple help icon .
OpenMaple is a suite of functions that allows you to access Maple algorithms and data structures in your compiled C, C#, Java, or Visual Basic programs. (This is the reverse of external calling,
which allows access to compiled C, C#, Fortran 77, and Java code from Maple.)
To run your application, Maple must be installed. You can distribute your application to any licensed Maple user. For additional terms and conditions on the use of OpenMaple, refer to the extern/
OpenMapleLicensing.txt file in your Maple installation.
For more details on using OpenMaple functions, refer to the OpenMaple help page.
MapleSim^TM is a complete environment for modeling and simulating multidomain engineering systems. During a simulation, MapleSim uses the symbolic Maple computation engine to generate the
mathematical models that represent the system behavior.
Because both products are tightly integrated, you can use Maple commands and technical document features to edit, manipulate, and analyze a MapleSim model. For example, you can use Maple commands
and tools to manipulate your model equations, develop custom components based on a mathematical model, and visualize simulation results.
MapleSim software is not included with the Maple software. For more information on MapleSim, visit http://www.maplesoft.com/maplesim.
Microsoft Excel Add-In
Maple is available as an add-in to Microsoft Excel. This add-in is supported for Excel 365 (desktop) and Excel 2019 for Windows, and provides the following features.
• Access to Maple commands from Excel
• Ability to copy and paste between Maple and Excel
• Access to a subset of the Maple help pages
• Maple Function Wizard to step you through the creation of a Maple function call
To enable the Maple Excel Add-in:
1. In Excel, click the File menu and select Options.
2. Click Add-ins.
3. In the Manage box select Excel Add-ins, and then Go.
4. Navigate to the Excel subdirectory of your Maple installation and select the file:
– WMIMPLEX64.xla (that is, select $MAPLE/Excel/WMIMPLEX64.xla), and click OK.
5. Select the Maple Excel Add-in check box.
6. Click OK.
For further details on enabling the Maple Excel Add-in, refer to the Excel help page.
For information on using this add-in, refer to the Using Maple in Excel help file within Excel.
To view this help file:
1. Enable the add-in.
2. From the Add-ins tab, view the Maple toolbar.
3. On the Maple toolbar, click the Maple help icon .
Maple is available as an add-in to Microsoft Excel. This add-in is supported for Excel 365 (desktop) and Excel 2019 for Windows, and provides the following features.
• Ability to copy and paste between Maple and Excel
• Access to a subset of the Maple help pages
• Maple Function Wizard to step you through the creation of a Maple function call
Maple Function Wizard to step you through the creation of a Maple function call
1. In Excel, click the File menu and select Options.
3. In the Manage box select Excel Add-ins, and then Go.
In the Manage box select Excel Add-ins, and then Go.
4. Navigate to the Excel subdirectory of your Maple installation and select the file:
Navigate to the Excel subdirectory of your Maple installation and select the file:
For further details on enabling the Maple Excel Add-in, refer to the Excel help page.
For information on using this add-in, refer to the Using Maple in Excel help file within Excel.
3. On the Maple toolbar, click the Maple help icon .
On the Maple toolbar, click the Maple help icon .
OpenMaple is a suite of functions that allows you to access Maple algorithms and data structures in your compiled C, C#, Java, or Visual Basic programs. (This is the reverse of external calling,
which allows access to compiled C, C#, Fortran 77, and Java code from Maple.)
To run your application, Maple must be installed. You can distribute your application to any licensed Maple user. For additional terms and conditions on the use of OpenMaple, refer to the extern/
OpenMapleLicensing.txt file in your Maple installation.
For more details on using OpenMaple functions, refer to the OpenMaple help page.
OpenMaple is a suite of functions that allows you to access Maple algorithms and data structures in your compiled C, C#, Java, or Visual Basic programs. (This is the reverse of external calling,
which allows access to compiled C, C#, Fortran 77, and Java code from Maple.)
To run your application, Maple must be installed. You can distribute your application to any licensed Maple user. For additional terms and conditions on the use of OpenMaple, refer to the extern/
OpenMapleLicensing.txt file in your Maple installation.
For more details on using OpenMaple functions, refer to the OpenMaple help page.
MapleSim^TM is a complete environment for modeling and simulating multidomain engineering systems. During a simulation, MapleSim uses the symbolic Maple computation engine to generate the
mathematical models that represent the system behavior.
Because both products are tightly integrated, you can use Maple commands and technical document features to edit, manipulate, and analyze a MapleSim model. For example, you can use Maple commands
and tools to manipulate your model equations, develop custom components based on a mathematical model, and visualize simulation results.
MapleSim software is not included with the Maple software. For more information on MapleSim, visit http://www.maplesoft.com/maplesim.
MapleSimTM is a complete environment for modeling and simulating multidomain engineering systems. During a simulation, MapleSim uses the symbolic Maple computation engine to generate the mathematical
models that represent the system behavior.
Because both products are tightly integrated, you can use Maple commands and technical document features to edit, manipulate, and analyze a MapleSim model. For example, you can use Maple commands and
tools to manipulate your model equations, develop custom components based on a mathematical model, and visualize simulation results.
MapleSim software is not included with the Maple software. For more information on MapleSim, visit http://www.maplesoft.com/maplesim.
Sharing and Storing Maple Content
The MapleCloud
You can use the MapleCloud to share or store your Maple documents and workbooks. Upload Maple documents or workbooks. Package workbooks offer a way to share a Maple package with other users,
including source code, documentation and examples.
The MapleCloud has private and public sharing. You can share with all Maple users, share with a private group, or upload and store content in a user-specific area that only you can access.
Users need an internet connection to use the MapleCloud. Anyone can access publicly shared documents. To share content, create, manage and join user groups; and view group-specific content, you
must log in to the MapleCloud using a Maplesoft.com, Gmail™, or Google Mail™ account name and password.
A Maplesoft.com membership account gives you access to thousands of free Maple resources and MaplePrimes, which is an active web community for sharing techniques and experiences with Maple and
related products. To sign up for a free Maplesoft.com membership account, visit http://www.maplesoft.com/members/sign_up_form.aspx. The MapleCloud is integrated with several of these online
features, so it is strongly recommended that you use a Maplesoft.com membership account.
For more information on the MapleCloud, refer to the MapleCloud help page.
The MapleCloud
You can use the MapleCloud to share or store your Maple documents and workbooks. Upload Maple documents or workbooks. Package workbooks offer a way to share a Maple package with other users,
including source code, documentation and examples.
The MapleCloud has private and public sharing. You can share with all Maple users, share with a private group, or upload and store content in a user-specific area that only you can access.
Users need an internet connection to use the MapleCloud. Anyone can access publicly shared documents. To share content, create, manage and join user groups; and view group-specific content, you must
log in to the MapleCloud using a Maplesoft.com, Gmail™, or Google Mail™ account name and password.
A Maplesoft.com membership account gives you access to thousands of free Maple resources and MaplePrimes, which is an active web community for sharing techniques and experiences with Maple and
related products. To sign up for a free Maplesoft.com membership account, visit http://www.maplesoft.com/members/sign_up_form.aspx. The MapleCloud is integrated with several of these online
features, so it is strongly recommended that you use a Maplesoft.com membership account.
For more information on the MapleCloud, refer to the MapleCloud help page.
You can use the MapleCloud to share or store your Maple documents and workbooks. Upload Maple documents or workbooks. Package workbooks offer a way to share a Maple package with other users,
including source code, documentation and examples.
The MapleCloud has private and public sharing. You can share with all Maple users, share with a private group, or upload and store content in a user-specific area that only you can access.
Users need an internet connection to use the MapleCloud. Anyone can access publicly shared documents. To share content, create, manage and join user groups; and view group-specific content, you must
log in to the MapleCloud using a Maplesoft.com, Gmail™, or Google Mail™ account name and password.
A Maplesoft.com membership account gives you access to thousands of free Maple resources and MaplePrimes, which is an active web community for sharing techniques and experiences with Maple and
related products. To sign up for a free Maplesoft.com membership account, visit http://www.maplesoft.com/members/sign_up_form.aspx. The MapleCloud is integrated with several of these online features,
so it is strongly recommended that you use a Maplesoft.com membership account.
For more information on the MapleCloud, refer to the MapleCloud help page. | {"url":"https://www.maplesoft.com/support/help/maple/view.aspx?path=UserManual%2FChapter11","timestamp":"2024-11-02T06:18:39Z","content_type":"text/html","content_length":"312871","record_id":"<urn:uuid:1eb356d9-8147-428d-a212-c2ce6b158b22>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00813.warc.gz"} |
The KL divergence
• Kullback-Leibler (KL) divergence
• Evidence Lower Bound (ELBO)
In the previous page we saw that we need a notion of “closeness” of distributions. The KL divergence is the most frequent choice in the context of variational inference. We define the KL and explain
why it is used in variational inference.
Given two distribution \(\pi\) and \(q\), define the Kullback-Leibler (KL) divergence as: \[\operatorname{KL}(q \| \pi) = \int q(x) \log \frac{q(x)}{\pi(x)} \mathrm{d}x.\]
• the KL is asymmetric in its two arguments.
• We will put the variational approximation \(q\) as the distribution we average over.
• This is sometimes called the “backward” or “reverse” KL.
• More on this choice below.
Why variational inference uses the reverse KL?
We cover the two key reasons below: it captures “closeness” and it can be optimized.
Reverse KL captures the notion of “cloness”
Property: \(\operatorname{KL}(q_1 \| q_2 ) \ge 0\) with equality iff \(q_1 = q_2\).
Proof: since \(\log\) is a concave function, by Jensen’s inequality,
\[\begin{align*} \operatorname{KL}(q_1 \| q_2 ) &= \int q_1(x) \log \frac{q_1(x)}{q_2(x)} \mathrm{d}x \;\;\text{(definition)} \\ &= - \int q_1(x) \log \frac{q_2(x)}{q_1(x)} \mathrm{d}x \;\;\text{($-\
log a = \log a^{-1}$)} \\ &\ge - \log \int {\color{red} q_1(x)} \frac{q_2(x)}{{\color{red} q_1(x)}} \mathrm{d}x \;\;\text{(Jensen's)} \\ &= - \log \int q_2(x) \mathrm{d}x \;\;\text{(red factors
cancel)} \\ &= 0 \;\;\text{($q_2$ is a probability density)}. \end{align*}\]
Reverse KL can be optimized
Requirement: we want to be able to optimize the objective function without having to compute the intractable normalization constant \(Z\).
Many other notions of distribution “closeness” do not satisfy this.
Towards optimization of the reverse KL
We show that optimizing the reverse KL does not require knowing the intractable normalization constant \(Z\):
\[\begin{align*} \operatorname{arg\,min}_\phi \operatorname{KL}(q_\phi \| \pi) &= \operatorname{arg\,min}_\phi \int q_\phi(x) \log \frac{q_\phi(x)}{\pi(x)} \mathrm{d}x \\ &= \operatorname{arg\,min}_\
phi \int q_\phi(x) \log \frac{q_\phi(x) Z}{\gamma(x)} \mathrm{d}x \\ &= \operatorname{arg\,min}_\phi \int q_\phi(x) \left[ \log q_\phi(x) + \log Z - \log \gamma(x) \right] \mathrm{d}x \\ &= \
operatorname{arg\,min}_\phi \underbrace{\int q_\phi(x) \left[ \log q_\phi(x) - \log \gamma(x) \right] \mathrm{d}x}_{L(\phi)} + {\color{red} \log Z} \\ &= \operatorname{arg\,min}_\phi L(\phi) \;\;\
text{(red term does not depend on $\phi$)}. \end{align*}\]
Notice: \(L(\phi)\) does not involve \(Z\)!
Terminology: the negative value of \(L\) is called the Evidence Lower BOund (ELBO), \(\operatorname{ELBO}(\phi) = -L(\phi)\).
Question: how did we get \(\log Z\) outside of the integral (step in red)?
1. By definition
2. \(-\log a = \log a^{-1}\)
3. Jensen’s inequality
4. Because \(\pi\) is a probability distribution.
5. Because \(q\) is a probability distribution.
5: Because \(q\) is a probability distribution.
\[\int q_\phi(x) \log Z \mathrm{d}x = \log Z \int q_\phi(x) \mathrm{d}x = \log Z.\] | {"url":"https://ubc-stat-ml.github.io/web447/w13_advanced_infer/topic03_kl.html","timestamp":"2024-11-07T06:37:49Z","content_type":"application/xhtml+xml","content_length":"65689","record_id":"<urn:uuid:25e5d349-e5ae-4110-923a-0c92260ac231>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00372.warc.gz"} |
18 Word Problems For Grade 4
18 Math Word Problems For 4th Grade: Develop Their Problem Solving Skills Across Single and Mixed Upper Elementary Topics
4th grade math word problems are a great way to assess students’ number fluency. By the time elementary school children reach upper elementary they will be building on their knowledge and
understanding of the number system and place value, working with larger integers. An increased understanding of the connections between the different math concepts is crucial, as children will be
expected to tackle more complex, multi-step problems.
It is important children are regularly provided with the opportunity to solve a range of word problems with reasoning and problem solving incorporated alongside fluency, into all lessons.
To help you with this, we have put together a collection of 18 word problems aimed at 4th grade math students.
Place value
Solve number and practical problems involving ordering and comparing numbers to at least 1,00,000; counting forwards or backwards in steps of powers of 10; interpreting negative numbers in context
and rounding to the nearest 10,100, 1000, 10,000 and 100,000.
Addition and subtraction
Solve addition and subtraction word problems and a combination of these, including understanding the meaning of the equals sign.
Word Problems Grade 4 Addition and Subtraction
11 grade 4 addition and subtraction questions to develop reasoning and problem solving skills.
Download Free Now!
Multiplication and division
Solve problems involving multiplication and division, including using knowledge of factors, multiples, squares and cubes and scaling by simple fractions.
Fractions, decimals and percentages
Solve problems involving numbers up to 3 decimal places and problems which require knowing percentage and decimal equivalents.
Solve time word problems and those involving converting between units of time and problems involving measure (for example, length, mass, volume and money word problems) using decimal notation and
Solve comparison, sum and difference problems, using information presented in a line graph.
Why are word problems important in 4th grade math?
Word problems are an important element of the 4th grade curriculum. By this stage, children need to be building confidence in approaching a range of one, two and multi-step word problems. They
require children to be creative and apply the skills they have learnt to a range of real-life situations.
How to teach problem solving in 4th grade
Children need to be taught the skills for successfully approaching and tackling word problems. Reading the question carefully and identifying the key information needed to tackle the problem, is the
first step, followed by identifying which calculations are required for solving it and deciding whether it will be helpful to draw a picture/ visual representation to understand and answer the
Using mental math skills to round and estimate an answer is also very helpful for children to establish whether their final answer is realistic. Children need to also be able to calculate the
inverse, to be able to check their answer, once the problem has been completed.
See also: Mental math year 5
Here is an example:
A transport museum has 1243 visitors on Monday morning and another 1387 visitors in the afternoon. On Tuesday 736 fewer visitors go to the museum than who visited on Monday.
How many visitors were there altogether on Tuesday?
How to solve:
What do you already know?
• The number of visitors on Monday morning and Monday afternoon are given separately. They need to be added together to give the total number of visitors for Monday.
• ‘Fewer’ means I will need to subtract the number of fewer visitors on Tuesday from the total number of visitors on Monday.
• Column addition and subtraction will be needed to solve this question.
How can this be drawn/represented visually?
We can draw a bar model to represent this problem:
• To calculate the total number of visitors on Monday, we need to add 1243 and 1387 together. 1243 + 1387 = 2630
• The number of fewer visitors on Tuesday needs to be subtracted from Monday’s total: 2630 – 736 = 1894.
• The total number of visitors on Tuesday was 1894.
Addition word problems for 4th grade
In 4th grade, addition word problems can involve whole numbers with over 4 digits and decimal numbers. Children should be able to round number to check the accuracy and begin to solve two step number
Addition question 1
Gemma picks two cards from the cards below and adds them together.
She is able to make three different totals. What will they be?
Answer (2 marks): 12,147 16,063 8646
Addition question 2
Ahmed adds two of these numbers mentally.
In his calculation, he exchanges twice to create one ten and one hundred.
Write Ahmed’s calculation and work out the total.
Answer (1 mark): 357 + 294 = 651
Addition question 3
Change one digit in the calculation below, so that the answer is a multiple of 10.
726 + 347
Answer (1 mark): 723 + 347 = 1070
Subtraction word problems for 4th grade
Subtraction word problems in 4th grade require learners to be confident subtracting numbers over 4 digits and problems involving decimal numbers. Students need to be able to round numbers to check
accuracy and to use subtraction when solving mixed word problems.
Subtraction question 1
A coach is traveling 4924 km across the USA
It has 2476 km to go.
How many kilometers has the coach already traveled?
Answer (1 mark): 2448 km
Subtraction question 2
An elementary school printed 7283 math worksheets in the Summer semester. 2156 were for lower elementary students. How many were printed for upper elementary?
Answer (1 mark): 5127 worksheets
Subtraction question 3
A clothing company made $57,605 profit in 2021 and $73,403 in 2022.
How much more profit did the company make in 2022 than in 2021?
Answer (1 mark): $15,798
Multiplication word problems for 4th grade
In 4th grade, multiplication word problems include problems involving times tables and multiplying whole numbers up to 4-digits by 1 or 2-digit numbers. Students also need to be able to combine
multiplication with other operations, in order to solve two-step word problems.
Multiplication question 1
In this diagram, the numbers in the circles are multiplied together to make the answer in the square between them.
Complete the missing numbers.
Answer (1 mark)
Multiplication question 2
Mrs Jones was printing the end of year math test. Each test had 18 pages and 89 students were sitting the test. Mrs Jones also needed to print out 12 copies for the teachers and Teaching Assistants
who were helping to run the test.
How many pieces of paper did Mrs Jones need to put in the photocopier, to make sure she had enough for all the tests?
Answer (1 mark)
Multiplication question 3
A school is booking a trip to Six Flags. Tickets cost $22 per student.
There are 120 children in each grade level and all the children from 3 grade levels will be going.
What will be the total price for all the tickets?
Answer (2 marks): $7920
Third Space Learning often ties word problems into our online one-to-one tutoring. Each lesson is personalized to the needs of the individual student, growing math knowledge and problem solving
Division word problems for 4th grade
In 4th grade, division word problems can involve whole numbers up to 4 digits being divided by 1-digit numbers. Students need to understand how to answer word problems, when the answer involves a
Division question 1
Tom has 96 cubes and makes 12 equal towers.
Masie has 63 cubes and makes 9 equal towers.
Whose towers are tallest and by how many cubes?
Answer (2 marks): Tom
Tom’s tower has more cubes. His towers have 1 more cube than Maise’s towers.
96 ÷ 12 = 8
63 ÷ 9 = 7
Division question 2
A cake factory has made cakes to deliver to a large event.
265 cakes have been baked. How many boxes of 8 cakes can be delivered to the event?
Answer (2 marks): 33 boxes.
Division question 3
Lily collected 1256 stickers. She shared them between her 8 friends. How many stickers did each friend get
Answer (1 mark): 157 stickers
Fraction and decimal word problems in 4th grade
In 4th grade, fractions word problems and decimals can include questions involving ordering, addition and subtraction of fractions. They can also involve converting between fractions and decimals.
Fraction and decimal question 1
Isobel collected 24 conkers.
She gave \frac{1}{8} of the conkers to her brother.
How many conkers did she have left?
Answer (1 mark): 21 conkers left
\frac{1}{8} of 24 = 3
24 – 4 = 21
Fraction and decimal question 2
Ahmed counted out 32 candies. He gave \frac{1}{4} of the sweets to his brother and \frac{3}{8} of the candies to his friend.
How many candies did he have left?
Answer (2 marks): 12 candies
\frac{1}{4} of 32 = 8
\frac{3}{8} of 32 = 12
He gave away 20 candies, so had 12 left for himself.
Fraction and decimal question 3
Two friends shared some pizzas, 1 ate 1 \frac{1}{2} pizzas, whilst the other ate \frac{5}{8} of a pizza. How much did they eat altogether?
2 \frac{1}{8} of pizza
1 \frac{1}{2} = \frac{12}{8}
\frac {12}{8} + \frac{5}{8} = \frac{17}{8} = 2 \frac{1}{8} pizzas
Mixed four operation word problems
Problems with mixed operations, or ‘multi-step’ word problems, require two or more operations to solve them. A range of concepts can be covered within mixed problems, including the four operations,
fractions, decimals and measures. These are worth more marks than some of the more straightforward, one-step problems.
Mixed operation question 1
At the cake sale, Sam buys 6 cookies and a cupcake . He pays $2.85 altogether.
Naya buys 2 cookies and pays 90¢ altogether.
How much does the cupcake cost?
Answer (2 marks): 60¢ for one cupcake
Mixed operation question 2
Large cookie tin – 48 cookies (picture of boxes here)
Small cookie tin – 30 cookies
Ben bought 2 large tins of cookies and 3 small tins.
How many cookies did he buy altogether?
Answer (2 marks): 186 cookies
2 x 48 = 96
3 x 30 = 90
96 + 90 = 186 cookies
Mixed operation question 3
The owner of a bookshop bought a box of 15 books for $150.
He sold the books individually for $12 each.
How much profit did he make?
Answer (2 marks)
Word problem resources
Third Space Learning offers a wide array of math and word problems resources for other grade levels such as word problems for 5th grade, word problems for 2nd grade and word problems for 3rd grade.
Our word problem collection covers all four operations and other specific math topics such as ratio word problems and percentage word problems.
Do you have students who need extra support in math?
Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor.
Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with
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Personalized one-on-one math tutoring programs are available for:
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Why not learn more about how it works? | {"url":"https://thirdspacelearning.com/us/blog/4th-grade-word-problems/","timestamp":"2024-11-13T20:49:28Z","content_type":"text/html","content_length":"152299","record_id":"<urn:uuid:a23be889-e21d-40c3-bd5a-e276ab363b68>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00450.warc.gz"} |
Energy Based Models (EBM’s)
A post by Henry Bourne, PhD student on the Compass programme.
Currently I’ve been researching Noise Contrastive Estimation (NCE) techniques for representation learning aided by my supervisor Dr. Rihuan Ke. Representation learning concerns itself with learning
low-dimensional representations of high-dimensional data that can then be used to quickly solve a general downstream task, eg. after learning general representations for images you could quickly and
cheaply train a classification model on top of the representations.
NCE is a general estimator for parametrised probability models as I will explain in this blogpost. However, it can also be cleverly used to learn useful representations in an unsupervised (or
equivalently self-supervised) manner, which I will also explain. I’ll start by explaining the problem that NCE was created to solve, then provide a quick comparison to other methods, explain how
researchers have built on this method to carry out representation learning and finally discuss what I am currently working on.
NCE solves the problem of computing a normalising constant by avoiding the problem altogether and solving some other proxy problem. Methods that are able to model unnormalised probability models are
known as Energy Based Models (EBM’s). We will begin by describing the problem with the normalising constant before getting on to how we will avoid it.
The problem … with the normalising constant
Let’s say we have some arbitrary probability distribution, $p_{d}(\cdot)$, and a parametrised probability model, $p_{m}(\cdot ; \alpha)$, which we would like to accurately model the underlying
probability distribution. Let’s further assume that we’ve picked our model well such that $\exists \alpha^{*}$ such that $p_{d}(\cdot) = p_{m}(\cdot ; \alpha^{*})$.
Let’s just fit it to our data sampled from the underlying distribution using Maximum Likelihood Estimation! Sounds like a good idea, MLE has been extensively used, is reliable, is efficient and
achieves the Cramer-Rao lower bound (the lowest possible bound an unbiased estimator can achieve for its variance/MSE), is asymptotically normal, is consistent, is unbiased and doesn’t assume
normality. Moreover, there are a lot of tweaked MLE techniques out there that you can use if you would like an estimator with slightly different properties.
First let’s look under the hood of our probability model, we can write it as so:
p_{m}(\cdot;\alpha)=\frac{p_{m}^{0}(\cdot; \alpha)}{Z(\alpha)} & \text{where,} \: Z(\alpha) = \int p_{m}^{0}(u; \alpha) du
The likelihood is our probability model for some $\alpha$ evaluated over our dataset. Evaluating the likelihood becomes tricky when there isn’t an analytical solution for the normalisation term, $Z(\
alpha)$, and the possible set of values $u$ can take becomes large. For example if we would like to learn a probability distribution over images then this normalisation term becomes intractable.
By working with the log we get better numerical stability, it makes things easier to read and it makes calculations and taking derivatives easier. So, let’s take the log of the above:
&{} p_{m}(\cdot;\alpha) = \frac{p_{m}^{0}(\cdot; \alpha)}{Z(\alpha)} \\ $\text{Where, } \\ \theta = \{\alpha, c \}, \\ \text{c an estimate of} -\log Z(\alpha)$
& \Rightarrow \log p_{m}(\cdot; \theta) = \log p_{m}^{0} (\cdot ; \alpha) +c
Where, we write $p_{m}^{0}(\cdot;\alpha)$ to represent our unnormalized probability model. After taking the $\log$ we can write our normalising constant as $c$ and then include it as a parameter of
our model. So, our new model now parameterised by $\theta$, $p_{m}(\cdot;\theta)$, is self-normalising, ie. it estimates it’s normalising constant. Another approach to make the model self-normalising
would be to simply set $c=0$, implicitly making the model self-normalising. This is what is normally done in practice, but it assumes that your model is complex enough to be able to indirectly model
Couldn’t we just use MLE to estimate $\log p_{m}(\cdot ; \theta)$? No we can’t! This is because the likelihood can be made arbitrarily large by making $c$ large.
This is where Noise Contrastive Estimation (NCE) comes in. NCE has been shown theoretically and empirically to be a good estimator when taking this self-normalizing assumption. We’ll assess it versus
competing methods at the end of the blogpost. But before we do that let’s first describe the original NCE method named binary-NCE [1] later we will mention some of the more complex versions of this
The idea with binary-NCE [1] is that by avoiding our problems we fix our problems! ie. We would like to create and solve an ‘easier’ proxy problem which in the process solves our original problem.
Let’s say we have some noise-distribution, $p_{n}(\cdot)$, which is easy to sample from, allows for an analytical expression of $\log p_{n} (\cdot)$ and is in some way similar to our $p_{d}(\cdot)$
(our underlying probability distribution which we are trying to estimate). We would also like $p_{n}(\cdot)$ to be non-zero wherever $p_{d}(\cdot)$ is non-zero. Don’t worry too much about these
assumptions as they are normally quite easy to satisfy, apart from an analytical expression being available. They just are necessary for our theoretical properties to hold and for binary-NCE to work
in practice.
We would like to create and solve a proxy problem where given a sample we would like to classify whether it was drawn from our probability model or from our noise distribution. Consider the following
density ratio.
If this density ratio is bigger than one then it means that $u$ is more likely to have come from our probability model, $p_{m}(\cdot;\alpha)$. If it is smaller than one then $u$ is more likely to
have come from our noise distribution, $p_{n}(\cdot)$. Therefore, if we can model this density ratio then we will have a model for how likely a sample is to have come from our probability model as
opposed to have being sampled from our noise distribution.
Notice that we are modelling our normalised probability model above, we can rewrite it in terms of our unnormalised probability model as follows.
& \log \left(\frac{p_{m}(u;\alpha)}{p_{n}(u)} \right) \\
& = \log \left(\frac{p_{m}^{0}(u;\alpha)}{Z(\alpha)} \cdot \frac{1}{p_{n}(u)} \right) \\
& = \log \left(\frac{p_{m}^{0}(u;\alpha)}{p_{n}(u)} \right) +c \\
& = \log p_{m}^{0}(u;\alpha) + c – \log p_{n}(u) \\
& = \log p_{m}(u;\theta) – \log p_{n}(u)
Let’s now define a score function $s$ that we will use to model our rewrite of the density ratio just above:
s(u;\theta) = \log p_{m}(u;\theta) – log p_{n}(u)
One further step before introducing our objective function. We would like to model our score function somewhat as a probability, we would also like our model to not just increase the score
indefinitely. So we will put our modelled density ratio through the sigmoid/ logistic function.
\sigma(s(u;\theta)) = \frac{1}{1+ \exp(-s(u;\theta))}
We would like to classify according to our model of the density ratio whether the sample is ‘real’ / ‘positive or just ‘noise’/ ‘fake’/ ‘negative’. So a natural choice for the objective function is
the cross-entropy loss.
J(\theta) = \frac{1}{2N} \sum_{n} \log [ \sigma(s(x_{n};\theta))] + \log [1- \sigma(s(x_{n}’;\theta))]
Where $x_{i} \sim p_{d}$, $x_{i}’ \sim p_{n}$ for $i \in \{1,…,N\}$. Here we simply assume one noise sample per observation, but we can trivially extend it to any integer $K>0$ and in fact
asymptotically the estimator gets better performance as we increase K.
Once we’ve estimated our density ratio we can easily recover our normalised probability model of the underlying distribution by adding the log probability density of the noise function and taking the
This estimator is consistent, efficient and asymptotically normal. In [1] they also showed it working empirically in a range of different settings.
How does it compare to other estimators of unnormalised parameterised probability density models?
NCE is not the only method we can use to solve the problem of estimating an unnormalised parameterised probability model. As we mentioned NCE belongs to a family of methods named Energy Based Models
(EBM’s) which all aim to solve this very problem of estimating an unnormalised probability model. Let’s very briefly mention some of the alternatives from this family of methods, please do check out
the references in this sub-section if you would like to learn more. We will talk about the methods as they appeared in their seminal form.
One alternative is called contrastive divergence which estimates an unnormalised parametrised probability model by using a combination of MCMC and the KL divergence. Contrastive Divergence was
originally introduced with Boltzmann machines in mind [9], MCMC is used to generate samples of the activations of the Boltzmann machine and then the KL divergence measures the difference between the
distribution of the activations given by the real data and the simulated activations. We then aim to minimise the KL divergence.
Score matching [11] models a parameterised probability model without the computation of the normalising term by estimating the gradient of the log density which it calls the score function. It does
this by minimising the expected square distance between the score function and the score function of the observed data. However, obtaining the score function of the observed data requires estimating
a non-parametric model from the data. They magically avoid doing this by deriving an alternative form of the objective function, through partial integration, leaving only the computation of the score
function and it’s derivative.
Importance sampling [10], which has been around for quite a while uses a weighted version of MCMC to focus on parts of the distribution that are ‘more important’ and in the process self-normalises.
Which makes it better than regular MCMC because you can use it on unnormalised probability models and it should be more efficient and have lower variance.
[1] contains a simple comparison between NCE, contrastive divergence, importance sampling and score matching. In their experimental setting they found contrastive divergence got the best performance,
closely followed by NCE. They also measured computation time and found NCE to be the best in terms of error versus computation time. This by no means crowns NCE as the best estimator but is a good
suggestion as to it’s utility, so is the countless ways it’s been used with high efficacy on a multitude of real-world problems.
Building on Binary-NCE (Ranking-NCE and Info-NCE)
Taking inspiration from Binary-NCE a number of other estimators have been devised. One such estimator is Ranking-NCE [2]. This estimator has two important elements.
The first is that the estimator assumes that we are trying to model a conditional distribution, for example $p(y|x)$. By making this assumption our normalising constant is different for each value of
the random variable we are conditioning on, ie. Our normalising term is now some $Z(x;\theta)$ and we have one for each possible value of x. This loosens the constraints on our estimator as we don’t
require our optimal parameters, $\theta^{*}$, to satisfy $\log Z(x;\theta^{*}) = c$ for some $c$ for all possible values of $x$. This means we can apply our model to problems where the number of
possible values of $x$ is much larger than the number of parameters in our model. For further details on this please refer to [2], section 2.
The second is that it has an objective that given an observed sample $x$, and an integer $K>1$ samples from the noise distirbution, the objective ranks the samples in order of how likely they were to
have come from the model versus the noise distribution. Again for further details please refer to [2].
Importantly this version of the estimator can be applied to more complex problems and empirically has been shown to achieve better performance.
Now what we’ve been waiting for … how can we use NCE for representation learning? This is where Info(rmation) NCE comes in. It essentially is Ranking-NCE but we chose our conditional distribution and
noise distribution in a specific way.
We consider a conditional probability of the form p(y|x) where $y \in \mathbb{R}^{d_{y}}$, $x \in \mathbb{R}^{d_{x}}$, $d_{y} < d_{x}$. Where $x$ is some data and $y$ is the low-dimensional
representation we would like to learn for $x$. We then choose our noise distribution, $p_{n}$, to be the marginal distribution of our representation $y$, $p_{y}$. So our density ratio becomes.
\frac{p_{m}(y|x; \theta)}{p_{y}(y)}
This is now a measure of how likely a given $y$ is to have come from the conditional distribution we are trying to model, ie. how likely is this representation to have been obtained from $x$, versus
being some randomly sampled representation.
A key thing to notice is that we are unlikely to have an analytical form of the $log$ of the marginal distribution of $y$. In fact, this doesn’t matter as we aren’t actually interested in modelling
the conditional distribution in this case. What we are interested in is the fact that by employing a Ranking-NCE style estimator and modelling the above density ratio we maximise a lower bound on the
mutual information between $Y$ and $X$, $I(Y;X)$. A proof for this along with the actual objective function can be found in [3].
This is quite an amazing result! We solve a proxy problem of a proxy problem and we get an estimator with great theoretical guarantees that is computationally efficient that maximises a mutual
information which allows us to, in an unsupervised manner, learn general representations for data. So we avoid our problems twice! I appreciate that above were two big jumps with not much detail but
I hope it gives a sense as to the link between NCE in it’s basic form and representation learning. More specifically, NCE is known as a self-supervised learning method which simply means an
unsupervised method which uses supervised methods but generates its own teaching signal. Even more specifically, NCE is a contrastive method which gets its name from the fact that it contrasts
samples against each other in order to learn. The other popular category of self-supervised learning methods are called generative models, you may have heard of these!
My Research
Now we know a little bit about NCE and how we can use it to do representation learning, what am I researching?
Info-NCE has been applied with great success in many self-supervised representation learning techniques, a good one to check out is [4]. Contrastive self-supervised learning techniques have been
shown to outperform supervised learning in many areas. They also solve some of the key challenges that face generative representation learning techniques in more challenging domains than language
such as images and video. This review [5] is a good starting point for learning more about what contrastive learning and generative learning are and some of their differences.
However, there are still lots of problem areas where applying NCE, without very fancy neural network architectures and techniques, doesn’t do so well or outright fails. Moreover, many of these
techniques introduce extra requirements on memory, compute or both. Additionally, they can often be highly complex and their ablation studies are poor.
Currently, I’m looking at applying new kinds of density ratio estimation methods to representation learning, in a similar way to info-NCE. These new density ratio estimation techniques when applied
in the correct way will hopefully lead to representation learning techniques that are more capable in problem areas such as multi-modal learning [6], multi-task learning [7] and continual learning
Currently, of most interest to me is multi-modal learning. This is concerned with learning a joint representation over data comprised of more than one modality, eg. text and images. By being able to
learn representations on data consisting of multiple modalities it’s possible to learn higher quality representations (more information) and makes us capable of solving more complex tasks that
require working over multiple modalities, eg. most robotics tasks. However, multi-modal learning has a unique set of difficult challenges that make naively using representation learning techniques on
it challenging. One of the key challenges is balancing a trade-off between learning to construct representations that exploit the synergies between the modalities and not allowing the quality of the
representations to be degraded by the varying quality and bias of each of the modalities. We hope to solve this problem in an elegant and simple manner using density ratio estimation techniques to
create a novel info-NCE style estimator.
Hope you enjoyed! If you would like to reach me or read some of my other blogposts (I have some more in-depth ones about NCE coming out soon) then checkout my website at /phd.h-0-0.com.
[1] :
Gutmann, M. and Hyvärinen, A., 2010, March. Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. In Proceedings of the thirteenth international conference on
artificial intelligence and statistics (pp. 297-304). JMLR Workshop and Conference Proceedings.
[2] :
Ma, Z. and Collins, M., 2018. Noise contrastive estimation and negative sampling for conditional models: Consistency and statistical efficiency. arXiv preprint arXiv:1809.01812.
[3] :
Oord, A.V.D., Li, Y. and Vinyals, O., 2018. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748.
[4] :
Chen, T., Kornblith, S., Norouzi, M. and Hinton, G., 2020, November. A simple framework for contrastive learning of visual representations. In International conference on machine learning (pp.
1597-1607). PMLR.
[5] :
Liu, X., Zhang, F., Hou, Z., Mian, L., Wang, Z., Zhang, J. and Tang, J., 2021. Self-supervised learning: Generative or contrastive. IEEE transactions on knowledge and data engineering, 35(1),
[6] :
Baltrušaitis, T., Ahuja, C. and Morency, L.P., 2018. Multimodal machine learning: A survey and taxonomy. IEEE transactions on pattern analysis and machine intelligence, 41(2), pp.423-443.
[7] :
Zhang, Y. and Yang, Q., 2021. A survey on multi-task learning. IEEE Transactions on Knowledge and Data Engineering, 34(12), pp.5586-5609.
[8] :
Wang, L., Zhang, X., Su, H. and Zhu, J., 2024. A comprehensive survey of continual learning: Theory, method and application. IEEE Transactions on Pattern Analysis and Machine Intelligence.
[9] :
Carreira-Perpinan, M.A. and Hinton, G., 2005, January. On contrastive divergence learning. In International workshop on artificial intelligence and statistics (pp. 33-40). PMLR.
[10] :
Kloek, T. and Van Dijk, H.K., 1978. Bayesian estimates of equation system parameters: an application of integration by Monte Carlo. Econometrica: Journal of the Econometric Society, pp.1-19.
[11] :
Hyvärinen, A. and Dayan, P., 2005. Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6(4). | {"url":"https://compass.blogs.bristol.ac.uk/tag/energy-based-models-ebms/","timestamp":"2024-11-05T09:42:18Z","content_type":"text/html","content_length":"69674","record_id":"<urn:uuid:af86a0ad-54ea-4cd8-9c29-e366b41e28ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00098.warc.gz"} |
Calculating Option-Adjusted Spread (OAS) in SQL Server using XLeratorDB
Feb 2
In this overview we discuss the mechanics of the calculations of option-adjusted spread for corporate bonds. This overview is not a replacement for finance texts which give a far more detailed
explanation of the theory behind the calculations. It is also not a substitute for reading the documentation about the individual functions, but is designed to give a somewhat higher and more
integrated view of the how the functions work together.
Traditional bond-pricing measures like the PRICE function found in XLeratorDB and most spreadsheet applications are essentially time-value-of-money calculations where a single value, the yield, is
used to discount all the cash flows associated with a particular bond. In this type of calculation, evaluating the impact of any embedded options in the bond requires recalculating the either the
price or yield of the bond using the option date and the associated strike price. This is the basis of yield-to-call, yield-to-put, and yield-to-worst calculations.
Many times, however, it is important to understand the relative value of a bond. In other words, how does the price of a particular bond compare to the price of a benchmark bond? Generally, this type
of spread calculation takes the price of the bond as well as the benchmark yield curve as input and then returns a value, known as the spread, which is the amount added to the benchmark curve to
resolve to the entered price. This spread value is assumed to be added to every point on the benchmark curve. You can use the XLeratorDB ZSPREAD function to calculate this kind of spread.
The Z-spread calculation, however, does not consider any options embedded in the bond. Inclusion of embedded options in the calculation results in what is known as the option-adjusted spread
associated with the bond and is the focus of this article.
Traditionally, when discussing option-adjusted spread, the mechanics of the calculation are explained in terms of a binomial tree or lattice, which usually looks something like this.
In this article, however, we will use a lower triangular matrix as it more closely reflects the physical manifestation of the calculations in database terms. It would also make sense to use an upper
triangular matrix.
This example is taken from [1] Chapter 40 pp 875 – 876. We start with the following par curve.
│T│par │
│1│0.035 │
│2│0.042 │
│3│0.047 │
│4│0.052 │
The following SQL uses the XLeratorDB CMTCurve function to put the time (T), par rate, spot rate, discount factor, and continuously compounded zero coupon rate into a temp table, #z.
--Rates used in the OAS calculation
SELECT T, r
(1, .035)
,(2, .042)
,(3, .047)
,(4, .052)
)n(T, r)
ORDER BY T' --@Curve
,'L' --@InterpMethod
,1 --@Freq
The temp table #z should contain the following values:
│T│r │spot │df │cczero │
│1│0.035│0.035 │0.966183574879227│0.0344014267173323 │
│2│0.042│0.0421480257395637 │0.920748838632507│0.041283992492736 │
│3│0.047│0.0473524471924105 │0.870405135210075│0.0462655010233704 │
│4│0.052│0.0527059539733534 │0.814276090747591│0.0513639481661994 │
We will use the XLeratorDB table-value function LogNormalIRLattice to demonstrate how the calculation of OAS works. For more information on LogNormalIRLattice refer to the documentation. For more
information on the math behind the calculation refer to [1] and [2].
Using the bond from [1] (Exhibit 40-19) we are going to construct the interest rate lattice for a bond that matures in 4 years that is redeemable at par starting with the next coupon date. The bond
has a 6.5% coupon which is paid annually and the option-adjusted spread is 35 basis points. The volatility is 10%. For the moment, we will simply put the all the information into a temp table, #
Lattice, and then extract the information as needed to explain different aspects of the calculation.
'2016-11-28' --@Settlement
,'2020-11-28' --@Maturity
,.065 --@Rate
,.0035 --@Spread
,NULL --@Redemption
,1 --@Frequency
,NULL --@Basis
,NULL --@LastCouponDate
,NULL --@FirstCouponDate
,NULL --@IssueDate
,'SELECT T, ccZero FROM #z' --@CCZero
,NULL --@CurveType
,NULL --@CurveStartDate
,NULL --@CurveDayCount
,1 --@CurveFrequency
,'L' --@CurveInterpMethod
,0.10 --@Vol
,'SELECT ''2017-11-28'', 100' --@OptionSched
The #Lattice table contains all the information that we need to understand the option-adjusted spread calculation. You should note that each rows is uniquely identified by the step number (num_step)
and the node number (num_node).
We can extract and PIVOT the calibrated rates from #Lattice table using the following SQL.
SELECT num_node, [0],[1],[2],[3]
FROM (SELECT num_node, num_step, rate_calibrated FROM #lattice WHERE rate_calibrated IS NOT NULL)d
PIVOT(MAX(rate_calibrated) FOR num_step in([0],[1],[2],[3]))pvt
ORDER BY 1 DESC
This is what the calibrated rates should look like.
│Node│ Step │
│ │ 0│ 1│ 2│ 3│
│ 3│ │ │ │0.0919858│
│ 2│ │ │0.0700526│0.0753116│
│ 1│ │0.0542889│0.0573542│0.0616599│
│ 0│0.0350000│0.0444480│0.0469577│0.0504829│
This is a table of the forward rates used to construct the interest rate Lattice. The way to read this table is to start with the lower left hand corner and read from left to right. In this document
we default to using 0-based arrays, but if you feel more comfortable with 1-based arrays, you can simply add 1 to num_step and num_node in the SELECT statement.
At each step, all the forward rates are calculated from node 0 at that step. The formula for that calculation can be thought of as something like this:
r = rate
s = step
n = node
σ = volatility
δ = change in time (from the previous step)
For example, in Step 1, node 1:
At Step 2, node 2:
The forward rates at each node are converted to discount factors (df) using the following formula.
The discount factors associated with the calibrated rates are stored in the df_calibrated column. The following SQL selects and pivots that column for ease of viewing.
This is what the calibrated discount factors should look like.
│ │ Step │
│Node│ 0│ 1│ 2│ 3│
│ 3│ │ │ │0.9157628│
│ 2│ │ │0.9345335│0.9299630│
│ 1│ │0.9485066│0.9457569│0.9419212│
│ 0│0.9661836│0.9574435│0.9551485│0.9519432│
The one thing that we haven't addressed so far is how the calibrated rates for each step at node 0 are calculated. The calibration process requires finding values for r[s,0] such that the interest
rate lattice returns the discount factors from the yield curve supplied to the function. Thus, the calibration requires knowing the discount factors at that step. In this example, we actually know
the discount factors because the coupon dates line up exactly with the yield curve. These are the discount factors which are stored in #z.
│T│df │
│1│ 0.9661836│
│2│ 0.9207488│
│3│ 0.8704051│
│4│ 0.8142761│
As you can see the df[0,0] equals the discount factor for the 1-year rate.
This discount factor will be used to discount the values from df[1,0] and df[1,1] each of which is weighted by 0.5. And what we are looking for is a value for r[1, 0] such that:
0.9207488 = 0.9661836 * 0.5*(0.9574435 + 0.9485066)
We know that the calibrated discount factors are calculated directly from the calibrated forward rates. In other words, df[1,0] is calculated directly from r[1,0]. While there is a closed-form
solution for r[1,0]^[1], there is no closed-form solution for r[s,0] where s > 1 and a solution is found using a root-finding algorithm with the tolerance set to 0.0000000001.This process is repeated
for each subsequent step until the entire tree is filled out.
Using the calibrated discount factors from step 0 and 1, the discount factors in Step 2 have to satisfy the condition that they will resolve to the 3-year discount factor. In other words:
.08704051 = 0.5 * (0.5 * (0.9345335 + 0.9457569) * 0.9485066 * 0.5 X (0.9457569 + 0.9551485) * 0.9574435) * 0.9661836
Even though we provide the calibrated discount factors, it is actually quite straightforward to check the calibration simply by using the XLeratorDB PriceFromIRLattice function.
The following SQL will return the (cumulative) discount factors by pricing a zero-coupon bond with a redemption value of 1 and a zero option-adjusted spread. This shows that the interest rate Lattice
is correctly calibrated to the supplied curve.
'2016-11-28' --@Settlement
,n.mdate --@Maturity
,.0 --@Rate
,.0 --@Spread
,1 --@Redemption
,1 --@Frequency
,NULL --@Basis
,NULL --@LastCouponDate
,NULL --@FirstCouponDate
,NULL --@IssueDate
,'SELECT T, ccZero FROM #z' --@CCZero
,NULL --@CurveType
,NULL --@CurveStartDate
,NULL --@CurveDayCount
,1 --@CurveFrequency
,'L' --@CurveInterpMethod
,0.10 --@Vol
,NULL --@OptionSched
) as df
This produces the following result.
│mdate │df │
│2017-11-28 │0.966183574879227 │
│2018-11-28 │0.92074883863269 │
│2019-11-28 │0.870405135210354 │
│2020-11-28 │0.814276090745197 │
Having calculated the calibrated rates, the option-adjusted spread is calculated by adding a single value (the spread) to each of the calibrated forward rates and calculating the discounted cash flow
values through every node through to (0, 0) and comparing that to the supplied (clean) price. This spread is adjusted (through another root-finding algorithm) until the calculated price is
approximately equal to the supplied price. The tolerance is set 0.000001.
The following SQL returns the pivoted forward rates from the #lattice table.
SELECT num_node, [0],[1],[2],[3]
FROM (SELECT num_node, num_step, rate_fwd FROM #lattice WHERE rate_fwd IS NOT NULL)d
PIVOT(MAX(rate_fwd) FOR num_step in([0],[1],[2],[3]))pvt
ORDER BY 1 DESC
This is what the forward rates should look like.
│ │ Step │
│Node│ 0│ 1│ 2│ 3│
│ 3│ │ │ │0.0954858│
│ 2│ │ │0.0735526│0.0788116│
│ 1│ │0.0577889│0.0608542│0.0651599│
│ 0│0.0385000│0.0479480│0.0504577│0.0539829│
As you can see, they are all 35 basis points greater than the calibrated rates.
The following SQL returns the pivoted discount factors calculated from the forward rates.
SELECT num_node, [0],[1],[2],[3]
FROM (SELECT num_node, num_step, df FROM #lattice WHERE df IS NOT NULL)d
PIVOT(MAX(df) FOR num_step in([0],[1],[2],[3]))pvt
ORDER BY 1 DESC
This what the discount factors should look like.
│ │ Step │
│Node│ 0│ 1│ 2│ 3│
│ 3│ │ │ │0.9128370│
│ 2│ │ │0.9314867│0.9269459│
│ 1│ │0.9453682│0.9426366│0.9388262│
│ 0│0.9629273│0.9542458│0.9519660│0.9487820│
All that remains to be done is to calculate the discount cash flow values. These values are contained in the PVCF column.
SELECT num_node, [0],[1],[2],[3],[4]
FROM (SELECT num_node, num_step, PVCF FROM #lattice)d
PIVOT(MAX(PVCF) FOR num_step in([0],[1],[2],[3],[4]))pvt
ORDER BY 1 DESC
This is what the discounted cash flows should look like.
│ │ Step │
│Node│ 0│ 1│ 2│ 3│ 4│
│ 4│ │ │ │ │106.5│
│ 3│ │ │ │103.7171│106.5│
│ 2│ │ │103.8110│105.2197│106.5│
│ 1│ │105.8068│106.2803│106.4850│106.5│
│ 0│102.2180│106.5000│106.5000│106.5000│106.5│
Notice, that there is now one more step (4), which we will get to, but there are a few other pieces of data we want to get from the table.
SELECT stat, [0],[1],[2],[3],[4]
#lattice l
) d
PIVOT(MAX(val_stat) FOR num_step in([0],[1],[2],[3],[4]))p
ORDER BY stat ASC
This should produce the following result.
│stat │ 0│ 1│ 2│ 3│ 4│
│cczero │0.034401│0.041284│0.046266│0.051364│ 0│
│coupon │ 0│ 6.5│ 6.5│ 6.5│106.5│
│delta │ 1│ 1│ 1│ 1│ 0│
│price_call │NULL │ 100│ 100│ 100│NULL │
│price_put │NULL │NULL │NULL │NULL │NULL │
│T │ 0│ 1│ 2│ 3│ 4│
Time (T) 0 is the settlement date of the bond. We can see from this table that we are receiving coupons of 6.5 at T 1, 2, 3, 4 and the par value of 100 at T 4. Additionally, the bond is callable (at
par) starting at T 1 (1 year from now). Delta is change in time from one step to the next. The continuously compounded zero coupon rate is provided for information purposes.
All that remains is to calculate the value at each node. The easiest way to think about this is start in the upper right hand corner and work our way down and left. Before considering the impact of a
call or put option, the value at any node is equal to:
In other words:
When the price_call for the step is not NULL (meaning that the call can be exercised at this step at the strike price) the formula becomes:
For example:
When the price_put for the step is not NULL (meaning that the put can be exercised at this step at the strike price) the formula becomes:
For the most part, you will not need to use the LogNormalIRLattice function, but it is useful to have to research any questions that might arise about the calculation of the option-adjusted spread.
Let's look at another example, which addresses some other factors in the calculation of the option-adjusted spread. The first is the treatment of the accrued interest on the bond. The second is the
interpolation of the spot rates or the continuously compounded zeroes when the first supplied rate is greater than the settlement date.
Let's say we have the following CMT curve that is going to be used in the calculation.
│T │r │
│3M │0.396% │
│6M │0.520% │
│1Y │0.614% │
│2Y │0.823% │
│3Y │0.987% │
│4Y │1.138% │
│5Y │1.290% │
│7Y │1.605% │
│10Y │1.839% │
│20Y │2.216% │
│30Y │2.593% │
In this situation, the curve commences at the 3-month date. Since the lattice is calibrated to the curve, the determination of the discount factor for any point prior to the start of the curve has a
big influence on the shape of the curve at that point. In the OAS function you can specify either linear or cubic spline interpolation methods. You also have the option of interpolation the par curve
prior to passing it into the OAS function. In fact, to the OAS function, the curve is just data and while you can use XLeratorDB functions to create the curve, you also have the option of using
whatever tools are best suited to your environment to create a curve and then pass that curve into the OAS function
We can use the XLeratorDB CMTCURVE function to put calculate the continuously compounded zero coupon rates and put them into the temp table #z.
--Establish the CMT curve
--Convert the CMT curve to continuously compounded zeroes
wct.CMTCURVE('SELECT * FROM #par','S',2)
bootstrap = 'False'
Note that we are not using any of the bootstrapped rates from the yield curve in our curve. If you think that you will get better results by including the bootstrapped rates in the option-adjusted
spread calculation, then you should include them.
We will use the following call schedule and bond information to generate the interest rate lattice which is stored in the temp table #lattice.
--Put the call schedules into a table
CAST(exdate as datetime) as exdate, strike
--Put the interest lattice into a table
'2016-11-28' --@Settlement
,'2020-01-15' --@Maturity
,.04125 --@Rate
'2016-11-28' --@Settlement
,'2020-01-15' --@Maturity
,.04125 --@Rate
,101.03125 --@Price
,NULL --@Redemption
,NULL --@Frequency
,NULL --@Basis
,NULL --@LastCouponDate
,NULL --@FirstCouponDate
,NULL --@IssueDate
,'SELECT t, cczero FROM #z' --@CCZero
,NULL --@CurveType
,NULL --@CurveStartDate
,NULL --@CurveDayCount
,NULL --@CurveFrequency
,NULL --@CurveInterpMethod
,0.48 --@Vol
,'SELECT exdate,strike FROM #calls' --@OptionSched
) --@Spread
,NULL --@Redemption
,NULL --@Frequency
,NULL --@Basis
,NULL --@LastCouponDate
,NULL --@FirstCouponDate
,NULL --@IssueDate
,'SELECT t, cczero FROM #z' --@CCZero
,NULL --@CurveType
,NULL --@CurveStartDate
,NULL --@CurveDayCount
,NULL --@CurveFrequency
,NULL --@CurveInterpMethod
,0.48 --@Vol
,'SELECT exdate,strike FROM #calls' --@OptionSched
We can use the following SQL to get the pivoted results for the data that varies by step (not by node) with the exception of the dates (because you cannot mix data type in a column in SQL Server).
There is separate SQL for the dates.
--Dynamic SQL to pivot the interest rate lattice
DECLARE @steps as nvarchar(max)
SET @steps =(SELECT '[' + cast(num_node as varchar(max)) + ']' FROM #lattice WHERE num_step =(SELECT MAX(num_step) FROM #lattice) ORDER BY num_node FOR XML PATH(''))
SET @steps = REPLACE(@steps,'][','],[')
--Get all the [float] step information (data which does not vary by node)
DECLARE @SQLSteps as varchar(max) = N'SELECT stat, @steps FROM
#lattice l
) d
PIVOT(MAX(val_stat) FOR num_step in (@steps))p
ORDER BY stat ASC'
SET @SQLSteps = REPLACE(@SQLSteps,'@steps',@steps)
--Get the dates for each step
DECLARE @SQLStepDates as varchar(max) = N'SELECT @steps FROM
(SELECT num_step, date_pmt FROM #lattice l WHERE num_node = 0)d
PIVOT(MAX(date_pmt) FOR num_step in (@steps))p'
SET @SQLStepDates = REPLACE(@SQLStepDates,'@steps',@steps)
This produces the following results, which has been reformatted.
│stat │ 0│ 1│ 2│ 3│ 4│ 5│ 6│ 7│
│cczero │ 0.003958│ 0.0054384│ 0.0064051│ 0.0074509│ 0.0084387│ 0.0092623│ 0.01007│ 0│
│coupon │ -1.523958│ 2.0625│ 2.0625│ 2.0625│ 2.0625│ 2.0625│ 2.0625│ 102.0625│
│delta │ 0.1305556│ 0.5│ 0.5│ 0.5│ 0.5│ 0.5│ 0.5│ 0│
│price_call│NULL │ 103│ 103│ 101│ 101│ 100│ 100│NULL │
│price_put │NULL │NULL │NULL │NULL │NULL │NULL │NULL │NULL │
│T │ 0│ 0.1305556│ 0.6305556│ 1.1305556│ 1.6305556│ 2.1305556│ 2.6305556│3.13055556│
│date │2016-11-28│2017-01-15│2017-07-15│2018-01-15│2018-07-15│2019-01-15│2019-07-15│2020-01-15│
The first thing to notice is that coupon for step 0 is -1.523958 which represents the accrued interest as of the settlement date of the bond. As we traverse the interest rate lattice from the upper
right corner to the lower left corner, the last thing that will happen is that this accrued interest will be added to the accumulated values from the lattice, bringing that value to the clean price
of the bond.
The second thing is that, unlike in the previous example, the delta for all the periods is not the same. The first period (0) has a delta less than all the other periods, because the bond is settling
after the previous coupon date.
Notice that the table includes the continuously compounded zero coupon rate. These are the interpolated values from #z. We used the default interpolation method, which is linear, and when seeking to
find an interpolated value less than first x-value in the ordered set, it uses the first value. In other words, the CC zero for T = .25 is used for all values of T <= 0.25. We can get the
interpolated zeroes and their associated discount factors (which are the basis for the calibration) with the following SQL.
,EXP(-[Interpolated CC Zero]* T) as df
FROM (
,wct.YEARFRAC('2016-11-28',x.date_pmt,0) as T
,wct.LINEAR(z.T,z.cczero, wct.YEARFRAC('2016-11-28',x.date_pmt,0),0) as [Interpolated CC Zero]
FROM (SELECT DISTINCT date_pmt FROM #lattice)x
CROSS JOIN #z
GROUP BY x.date_pmt
These results, then, become the targets for the calibration process.
│date_pmt │T │Interpolated CC Zero │df │
│11/28/2016│ 0│ 0.003958041│ 1│
│ 1/15/2017│0.130556│ 0.003958041│ 0.999483389│
│ 7/15/2017│0.630556│ 0.005438378│ 0.996576673│
│ 1/15/2018│1.130556│ 0.006405102│ 0.992784832│
│ 7/15/2018│1.630556│ 0.007450896│ 0.987924402│
│ 1/15/2019│2.130556│ 0.008438682│ 0.982181578│
│ 7/15/2019│2.630556│ 0.009262315│ 0.975929397│
│ 1/15/2020│3.130556│ 0.010070013│ 0.968966990│
Thus, if the calibration is done correctly the discount factor returned by traversing the lattice should be equal to the discount factor, within the tolerance. In the previous example, we used the
PriceFromIRLattice function to validate the calibration. Here is somewhat more involved SQL which does not rely on that function.
DECLARE @num_step as float =(SELECT MAX(num_step) - 1 FROM #lattice)
DECLARE @lattice AS TABLE (
num_step int,
num_node int,
rl float,
ru float,
df float,
V float
PRIMARY KEY (num_step, num_node)
--Get the party started
num_step - 1
,0.5*(ru+rl)*df as V
FROM (
,l1.df_calibrated as rl
,l2.df_calibrated as ru
,l3.df_calibrated as df
#lattice l1
#lattice l2
l2.num_step = l1.num_step
AND l2.num_node = l1.num_node + 1
#lattice l3
l3.num_step = l1.num_step - 1
AND l3.num_node = l1.num_node
l1.num_step = @num_step
WHILE @num_step > 0
SET @num_step = @num_step - 1
,0.5*(ru+rl)*df as V
FROM (
l1.num_step - 1 as num_step
,l1.V as rl
,l2.V as ru
,l3.df_calibrated as df
@lattice l1
@lattice l2
l2.num_step = l1.num_step
AND l2.num_node = l1.num_node + 1
#lattice l3
l3.num_step = l1.num_step - 1
AND l3.num_node = l1.num_node
l1.num_step = @num_step
SELECT V FROM @lattice WHERE num_step = 0 and num_node = 0
This produces the following result.
We can use the following SQL to get all the node information from lattice, which shows the lattice resolving to the clean price of 101.03125 within the stated tolerance.
--Get all the node information
DECLARE @SQLNodes as varchar(max) = N'SELECT stat, @steps FROM
#lattice l
(''rate_fwd'', rate_fwd)
) d
PIVOT(MAX(val_stat) FOR num_step in (@steps))p
ORDER BY stat ASC, num_node DESC'
SET @SQLNodes = REPLACE(@SQLNodes,'@steps',@steps)
This produces the following result.
We have created the following new functions for the calculation of the option-adjusted spread.
New Functions
We have also created the following function for the calculation of the zero-volatility spread. The Z-spread and the option adjusted spread should be the same for the bonds which do not have calls or
Spread analysis is a powerful tool in analyzing the relative value of bonds. If you use SQL Server download the 15-day trial and try out these powerful new functions (as well as over 900 other
functions). If you are not a SQL Server user but develop in .NET try out the 15-day trial for XLeratorDLL which includes hundreds of others sophisticated financial functions. Have question? Send us
an e-mail at support@westclintech.com
Cited References and Further Reading:
Fabozzi, F. and Mann, S. 2012, The Handbook of Fixed Income Securities Eighth Edition, Chapter 40 [1]
Miller, T., 2007, Introduction to Option-Adjusted Spread Analysis: Revised and Expanded Third Edition of the OAS Classic by Tom Windas 3rd Edition [2] | {"url":"http://westclintech.com/Blog/EntryId/132/Calculating-Option-Adjusted-Spread-OAS-in-SQL-Server-using-XLeratorDB","timestamp":"2024-11-05T18:16:26Z","content_type":"application/xhtml+xml","content_length":"878571","record_id":"<urn:uuid:2644c911-610b-4a6b-83a8-db589578cd7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00512.warc.gz"} |
Mississippi College & Career Readiness Standards
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8.NS. The Number System (NS)
Know that there are numbers that are not rational, and approximate them by rational numbers
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Rational and Irrational NumbersA rational number is a number that can be made into a fraction. Decimals that repeat or terminate are rational because they can be changed into fractions. An irrational
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not repeat or end are irrational numbers. Read more...iWorksheets :4Study Guides :1
8.EE.3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
For example, estimate the population of the United States as 3 × 10^8 and the population of the world as 7 × 10^9, and determine that the world population is more than 20 times larger.
Exponents, Factors and FractionsFreeIn a mathematical expression where the same number is multiplied many times, it is often useful to write the number as a base with an exponent. Exponents are also
used to evaluate numbers. Any number to a zero exponent is 1 and any number to a negative exponent is a number less than 1. Exponents are used in scientific notation to make very large or very small
numbers easier to write. Read more...iWorksheets :8Study Guides :1
Polynomials and ExponentsFreeA polynomial is an expression which is in the form of ax<sup>n</sup>, where a is any real number and n is a whole number. If a polynomial has only one term, it is called
a monomial. If it has two terms, it is a binomial and if it has three terms, it is a trinomial. The standard form of a polynomial is when the powers of the variables are decreasing from left to
right. Read more...iWorksheets :6Study Guides :1
8.EE.4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of
appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
Exponents, Factors and FractionsFreeIn a mathematical expression where the same number is multiplied many times, it is often useful to write the number as a base with an exponent. Exponents are also
used to evaluate numbers. Any number to a zero exponent is 1 and any number to a negative exponent is a number less than 1. Exponents are used in scientific notation to make very large or very small
numbers easier to write. Read more...iWorksheets :8Study Guides :1
Polynomials and ExponentsFreeA polynomial is an expression which is in the form of ax<sup>n</sup>, where a is any real number and n is a whole number. If a polynomial has only one term, it is called
a monomial. If it has two terms, it is a binomial and if it has three terms, it is a trinomial. The standard form of a polynomial is when the powers of the variables are decreasing from left to
right. Read more...iWorksheets :6Study Guides :1
Understand the connections between proportional relationships, lines, and linear equations
8.EE.5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a
distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
8.EE.6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the
origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
Analyze and solve linear equations and pairs of simultaneous linear equations
8.EE.7. Solve linear equations in one variable.
8.EE.7.a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming
the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The
inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater
than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Using IntegersIntegers are negative numbers, zero and positive numbers. To compare integers, a number line can be used. On a number line, negative integers are on the left side of zero with the
larger a negative number, the farther to the left it is. Positive integers are on the right side of zero on the number line. If a number is to the left of another number it is said to be less than
that number. In the coordinate plane, the x-axis is a horizontal line with negative numbers, zero and positive numbers. Read more...iWorksheets :4Study Guides :1
Decimal OperationsDecimal operations refer to the mathematical operations that can be performed with decimals: addition, subtraction, multiplication and division. The process for adding, subtracting,
multiplying and dividing decimals must be followed in order to achieve the correct answer. Read more...iWorksheets :3Study Guides :1
Fraction OperationsFraction operations are the processes of adding, subtracting, multiplying and dividing fractions and mixed numbers. A mixed number is a fraction with a whole number. Adding
fractions is common in many everyday events, such as making a recipe and measuring wood. In order to add and subtract fractions, the fractions must have the same denominator. Read more...iWorksheets
:3Study Guides :1
Introduction to PercentWhat Is Percent? A percent is a term that describes a decimal in terms of one hundred. Percent means per hundred. Percents, fractions and decimals all can equal each other, as
in the case of 10%, 0.1 and 1/10. Percents can be greater than 100% or smaller than 1%. A markup from the cost of making an item to the actual sales price is usually greater than 100%. A
salesperson's commission might be 1/2% depending on the item sold. Read more...iWorksheets :4Study Guides :1
Algebraic EquationsWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. When algebraic equations are written in
words, the words must be changed into the appropriate numbers and variable in order to solve. Read more...iWorksheets :5Study Guides :1
Equations and inequalitiesAn equation is mathematical statement that shows that two expressions are equal to each other. The expressions used in an equation can contain variables or numbers.
Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Inequalities are also solved by
using inverse operations. Read more...iWorksheets :3Study Guides :1
Integer operationsInteger operations are the mathematical operations that involve integers. Integers are negative numbers, zero and positive numbers. Adding and subtracting integers are useful in
everyday life because there are many situations that involved negative numbers such as calculating sea level or temperatures. Equations with integers are solved using inverse operations. Addition and
subtraction are inverse operations, and multiplication and division are inverse operations of each other. Read more...iWorksheets :4Study Guides :1
Rational numbers and operationsA rational number is a number that can be made into a fraction. Decimals that repeat or terminate are rational because they can be changed into fractions. A square root
of a number is a number that when multiplied by itself will result in the original number. The square root of 4 is 2 because 2 · 2 = 4. Read more...iWorksheets :3Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
Solving equations and inequalitiesAlgebraic equations are mathematical equations that contain a letter or variable which represents a number. To solve an algebraic equation, inverse operations are
used. Algebraic inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to, ≥; less than, <; and less than or equal to, ≤. When multiplying or
dividing by a negative number occurs, the inequality sign is reversed from the original inequality sign in order for the inequality to be correct. Read more...iWorksheets :3Study Guides :1
8.EE.7.b. Solve linear equations and inequalities with rational number coefficients, including those whose solutions require expanding expressions using the distributive property and collecting like
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The
inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater
than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Using IntegersIntegers are negative numbers, zero and positive numbers. To compare integers, a number line can be used. On a number line, negative integers are on the left side of zero with the
larger a negative number, the farther to the left it is. Positive integers are on the right side of zero on the number line. If a number is to the left of another number it is said to be less than
that number. In the coordinate plane, the x-axis is a horizontal line with negative numbers, zero and positive numbers. Read more...iWorksheets :4Study Guides :1
Decimal OperationsDecimal operations refer to the mathematical operations that can be performed with decimals: addition, subtraction, multiplication and division. The process for adding, subtracting,
multiplying and dividing decimals must be followed in order to achieve the correct answer. Read more...iWorksheets :3Study Guides :1
Fraction OperationsFraction operations are the processes of adding, subtracting, multiplying and dividing fractions and mixed numbers. A mixed number is a fraction with a whole number. Adding
fractions is common in many everyday events, such as making a recipe and measuring wood. In order to add and subtract fractions, the fractions must have the same denominator. Read more...iWorksheets
:3Study Guides :1
Introduction to PercentWhat Is Percent? A percent is a term that describes a decimal in terms of one hundred. Percent means per hundred. Percents, fractions and decimals all can equal each other, as
in the case of 10%, 0.1 and 1/10. Percents can be greater than 100% or smaller than 1%. A markup from the cost of making an item to the actual sales price is usually greater than 100%. A
salesperson's commission might be 1/2% depending on the item sold. Read more...iWorksheets :4Study Guides :1
Algebraic EquationsWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. When algebraic equations are written in
words, the words must be changed into the appropriate numbers and variable in order to solve. Read more...iWorksheets :5Study Guides :1
Equations and inequalitiesAn equation is mathematical statement that shows that two expressions are equal to each other. The expressions used in an equation can contain variables or numbers.
Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Inequalities are also solved by
using inverse operations. Read more...iWorksheets :3Study Guides :1
Integer operationsInteger operations are the mathematical operations that involve integers. Integers are negative numbers, zero and positive numbers. Adding and subtracting integers are useful in
everyday life because there are many situations that involved negative numbers such as calculating sea level or temperatures. Equations with integers are solved using inverse operations. Addition and
subtraction are inverse operations, and multiplication and division are inverse operations of each other. Read more...iWorksheets :4Study Guides :1
Rational numbers and operationsA rational number is a number that can be made into a fraction. Decimals that repeat or terminate are rational because they can be changed into fractions. A square root
of a number is a number that when multiplied by itself will result in the original number. The square root of 4 is 2 because 2 · 2 = 4. Read more...iWorksheets :3Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
Solving equations and inequalitiesAlgebraic equations are mathematical equations that contain a letter or variable which represents a number. To solve an algebraic equation, inverse operations are
used. Algebraic inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to, ≥; less than, <; and less than or equal to, ≤. When multiplying or
dividing by a negative number occurs, the inequality sign is reversed from the original inequality sign in order for the inequality to be correct. Read more...iWorksheets :3Study Guides :1
8.F. Functions (F)
Define, evaluate, and compare functions
8.F.1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
FunctionsFreeA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on
the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By
substituting a number for x, an output can be determined. Read more...iWorksheets :5Study Guides :1
8.F.3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the
area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
Use functions to model relationships between quantities
8.F.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,
y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or
a table of values.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
8.G. Geometry (G)
Understand congruence and similarity using physical models, transparencies, or geometry software
8.G.1. Verify experimentally the properties of rotations, reflections, and translations
8.G.1.a. Lines are taken to lines, and line segments to line segments of the same length.
Geometric ProportionsGeometric proportions compare two similar polygons. Similar polygons have equal corresponding angles and corresponding sides that are in proportion. A proportion equation can be
used to prove two figures to be similar. If two figures are similar, the proportion equation can be used to find a missing side of one of the figures. Read more...iWorksheets :4Study Guides :1
Plane Figures: Closed Figure RelationshipsPlane figures in regards to closed figure relationships refer to the coordinate plane and congruent figures, circles, circle graphs, transformations and
symmetry. Congruent figures have the same size and shape. Transformations are made up of translations, rotations and reflections. A translation of a figure keeps the size and shape of a figure, but
moves it to a different location. A rotation turns a figure about a point on the figure. A reflection of a figure produces a mirror image of the figure when it is reflected in a given line. Read
more...iWorksheets :3Study Guides :1
Patterns in geometryPatterns in geometry refer to shapes and their measures. Shapes can be congruent to one another. Shapes can also be manipulated to form similar shapes. The types of
transformations are reflection, rotation, dilation and translation. With a reflection, a figure is reflected, or flipped, in a line so that the new figure is a mirror image on the other side of the
line. A rotation rotates, or turns, a shape to make a new figure. A dilation shrinks or enlarges a figure. A translation shifts a figure to a new position. Read more...iWorksheets :3Study Guides :1
Ratios, proportions and percentsNumerical proportions compare two numbers. A proportion is usually in the form of a:b or a/b. There are 4 parts to a proportion and it can be solved when 3 of the 4
parts are known. Proportions can be solved using the Cross Product Property, which states that the cross products of a proportion are equal. Read more...iWorksheets :4Study Guides :1
Similarity and scaleSimilarity refers to similar figures and the ability to compare them using proportions. Similar figures have equal corresponding angles and corresponding sides that are in
proportion. A proportion equation can be used to prove two figures to be similar. If two figures are similar, the proportion equation can be used to find a missing side of one of the figures. Read
more...iWorksheets :7Study Guides :1
8.G.1.b. Angles are taken to angles of the same measure.
Geometric ProportionsGeometric proportions compare two similar polygons. Similar polygons have equal corresponding angles and corresponding sides that are in proportion. A proportion equation can be
used to prove two figures to be similar. If two figures are similar, the proportion equation can be used to find a missing side of one of the figures. Read more...iWorksheets :4Study Guides :1
Plane Figures: Closed Figure RelationshipsPlane figures in regards to closed figure relationships refer to the coordinate plane and congruent figures, circles, circle graphs, transformations and
symmetry. Congruent figures have the same size and shape. Transformations are made up of translations, rotations and reflections. A translation of a figure keeps the size and shape of a figure, but
moves it to a different location. A rotation turns a figure about a point on the figure. A reflection of a figure produces a mirror image of the figure when it is reflected in a given line. Read
more...iWorksheets :3Study Guides :1
Patterns in geometryPatterns in geometry refer to shapes and their measures. Shapes can be congruent to one another. Shapes can also be manipulated to form similar shapes. The types of
transformations are reflection, rotation, dilation and translation. With a reflection, a figure is reflected, or flipped, in a line so that the new figure is a mirror image on the other side of the
line. A rotation rotates, or turns, a shape to make a new figure. A dilation shrinks or enlarges a figure. A translation shifts a figure to a new position. Read more...iWorksheets :3Study Guides :1
Ratios, proportions and percentsNumerical proportions compare two numbers. A proportion is usually in the form of a:b or a/b. There are 4 parts to a proportion and it can be solved when 3 of the 4
parts are known. Proportions can be solved using the Cross Product Property, which states that the cross products of a proportion are equal. Read more...iWorksheets :4Study Guides :1
Similarity and scaleSimilarity refers to similar figures and the ability to compare them using proportions. Similar figures have equal corresponding angles and corresponding sides that are in
proportion. A proportion equation can be used to prove two figures to be similar. If two figures are similar, the proportion equation can be used to find a missing side of one of the figures. Read
more...iWorksheets :7Study Guides :1
8.G.1.c. Parallel lines are taken to parallel lines.
Geometric ProportionsGeometric proportions compare two similar polygons. Similar polygons have equal corresponding angles and corresponding sides that are in proportion. A proportion equation can be
used to prove two figures to be similar. If two figures are similar, the proportion equation can be used to find a missing side of one of the figures. Read more...iWorksheets :4Study Guides :1
Plane Figures: Closed Figure RelationshipsPlane figures in regards to closed figure relationships refer to the coordinate plane and congruent figures, circles, circle graphs, transformations and
symmetry. Congruent figures have the same size and shape. Transformations are made up of translations, rotations and reflections. A translation of a figure keeps the size and shape of a figure, but
moves it to a different location. A rotation turns a figure about a point on the figure. A reflection of a figure produces a mirror image of the figure when it is reflected in a given line. Read
more...iWorksheets :3Study Guides :1
Patterns in geometryPatterns in geometry refer to shapes and their measures. Shapes can be congruent to one another. Shapes can also be manipulated to form similar shapes. The types of
transformations are reflection, rotation, dilation and translation. With a reflection, a figure is reflected, or flipped, in a line so that the new figure is a mirror image on the other side of the
line. A rotation rotates, or turns, a shape to make a new figure. A dilation shrinks or enlarges a figure. A translation shifts a figure to a new position. Read more...iWorksheets :3Study Guides :1
Ratios, proportions and percentsNumerical proportions compare two numbers. A proportion is usually in the form of a:b or a/b. There are 4 parts to a proportion and it can be solved when 3 of the 4
parts are known. Proportions can be solved using the Cross Product Property, which states that the cross products of a proportion are equal. Read more...iWorksheets :4Study Guides :1
Similarity and scaleSimilarity refers to similar figures and the ability to compare them using proportions. Similar figures have equal corresponding angles and corresponding sides that are in
proportion. A proportion equation can be used to prove two figures to be similar. If two figures are similar, the proportion equation can be used to find a missing side of one of the figures. Read
more...iWorksheets :7Study Guides :1
8.G.2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent
figures, describe a sequence that exhibits the congruence between them.
Plane Figures: Lines and AnglesPlane figures in regards to lines and angles refer to the coordinate plane and the various lines and angles within the coordinate plane. Lines in a coordinate plane can
be parallel or perpendicular. Angles in a coordinate plane can be acute, obtuse, right or straight. Adjacent angles are two angles that have a common vertex and a common side but do not overlap. Read
more...iWorksheets :3Study Guides :1
Plane Figures: Closed Figure RelationshipsPlane figures in regards to closed figure relationships refer to the coordinate plane and congruent figures, circles, circle graphs, transformations and
symmetry. Congruent figures have the same size and shape. Transformations are made up of translations, rotations and reflections. A translation of a figure keeps the size and shape of a figure, but
moves it to a different location. A rotation turns a figure about a point on the figure. A reflection of a figure produces a mirror image of the figure when it is reflected in a given line. Read
more...iWorksheets :3Study Guides :1
Patterns in geometryPatterns in geometry refer to shapes and their measures. Shapes can be congruent to one another. Shapes can also be manipulated to form similar shapes. The types of
transformations are reflection, rotation, dilation and translation. With a reflection, a figure is reflected, or flipped, in a line so that the new figure is a mirror image on the other side of the
line. A rotation rotates, or turns, a shape to make a new figure. A dilation shrinks or enlarges a figure. A translation shifts a figure to a new position. Read more...iWorksheets :3Study Guides :1
Understand and apply the Pythagorean Theorem
8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
The Pythagorean TheoremPythagorean Theorem is a fundamental relation in Euclidean geometry. It states the sum of the squares of the legs of a right triangle equals the square of the length of the
hypotenuse. Determine the distance between two points using the Pythagorean Theorem. Read more...iWorksheets :10Study Guides :2
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres
8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Finding VolumeVolume measures the amount a solid figure can hold. Volume is measured in terms of cubed units and can be measured in inches, feet, meters, centimeters, and millimeters. The formula for
the volume of a rectangular prism is V = l · w · h, where l is the length, w is the width, and h is the height. Read more...iWorksheets :4Study Guides :1
Three dimensional geometry/MeasurementThree-dimensional geometry/measurement refers to three-dimensional (3D) shapes and the measurement of their shapes concerning volume and surface area. The
figures of prisms, cylinders, pyramids, cones and spheres are all 3D figures. Volume measures the amount a solid figure can hold. Volume is measured in terms of units³ and can be measured in inches,
feet, meters, centimeters, and millimeters. Read more...iWorksheets :11Study Guides :1
8.SP. Statistics and Probability (SP)
Investigate patterns of association in bivariate data
8.SP.1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or
negative association, linear association, and nonlinear association.
Analyzing, Graphing and Displaying DataThere are many types of graphs such as, bar graphs, histograms and line graphs. A bar graph compares data in categories and uses bars, either vertical or
horizontal. A histogram is similar to a bar graph, but with histograms the bars touch each other where with bar graphs the bars do not touch each other. A line graph is useful for graphing how data
changes over time. With a line graph, data is plotted as points and lines are drawn to connect the points to show how the data changes. Read more...iWorksheets :6Study Guides :1
Using graphs to analyze dataThere are different types of graphs and ways that data can be analyzed using the graphs. Graphs are based on the coordinate plane. Data are the points on the plane. If
collecting data about the ages of people living on one street, the data is all the ages. The data can then be organized into groups, and evaluated. Mean, mode and median are different ways to
evaluate data. Read more...iWorksheets :7Study Guides :1
Collecting and describing dataCollecting and describing data refers to the different ways to gather data and the different ways to arrange data whether it is in a table, graph, or pie chart. Data can
be collected by either taking a sample of a population or by conducting a survey. Describing data looks at data after it has been organized and makes conclusions about the data. Read more...i
Worksheets :3Study Guides :1
Displaying dataDisplaying data refers to the many ways that data can be displayed whether it is on a bar graph, line graph, circle graph, pictograph, line plot, scatter plot or another way. Certain
data is better displayed with different graphs as opposed to other graphs. E.g. if data representing the cost of a movie over the past 5 years were to be displayed, a line graph would be best. A
circle graph would not be appropriate to use because a circle graph represents data that can add up to one or 100%. Read more...iWorksheets :4Study Guides :1
8.SP.2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and
informally assess the model fit by judging the closeness of the data points to the line.
Linear relationshipsLinear relationships refer to two quantities that are related with a linear equation. Since a linear equation is a line, a linear relationship refers to two quantities on a line
and their relationship to one another. This relationship can be direct or inverse. If y varies directly as x, it means if y is doubled, then x is doubled. The formula for a direct variation is y =
kx, where k is the constant of variation. Read more...iWorksheets :3Study Guides :1
MS.CM8AI. Compacted Mathematics Grade 8 (with Algebra I)
CM8AI.A-SSE. Algebra: Seeing Structure in Expressions (A-SSE)
Write expressions in equivalent forms to solve problems
A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A-SSE.3.c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as [1.15^(1/12)]^12t ≈ 1.012^12t to reveal the
approximate equivalent monthly interest rate if the annual rate is 15%.
FunctionsFreeA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on
the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By
substituting a number for x, an output can be determined. Read more...iWorksheets :5Study Guides :1
CM8AI.A-APR. Algebra: Arithmetic with Polynomials and Rational Expressions (A-APR)
Perform arithmetic operations on polynomials
A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply
Polynomials and ExponentsFreeA polynomial is an expression which is in the form of ax<sup>n</sup>, where a is any real number and n is a whole number. If a polynomial has only one term, it is called
a monomial. If it has two terms, it is a binomial and if it has three terms, it is a trinomial. The standard form of a polynomial is when the powers of the variables are decreasing from left to
right. Read more...iWorksheets :6Study Guides :1
CM8AI.A-CED. Algebra: Creating Equations (A-CED)
Create equations that describe numbers or relationships
A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The
inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater
than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Algebraic InequalitiesFreeAlgebraic inequalities are mathematical equations that compare two quantities using these criteria: greater than, less than, less than or equal to, greater than or equal to.
The only rule of inequalities that must be remembered is that when a variable is multiplied or divided by a negative number the sign is reversed. Read more...iWorksheets :3Study Guides :1
Equations and inequalitiesAn equation is mathematical statement that shows that two expressions are equal to each other. The expressions used in an equation can contain variables or numbers.
Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Inequalities are also solved by
using inverse operations. Read more...iWorksheets :3Study Guides :1
A-CED.2. Create equations in two variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [Note this standard appears in future courses with
a slight variation in the standard language.]
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For
example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The
inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater
than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Algebraic InequalitiesFreeAlgebraic inequalities are mathematical equations that compare two quantities using these criteria: greater than, less than, less than or equal to, greater than or equal to.
The only rule of inequalities that must be remembered is that when a variable is multiplied or divided by a negative number the sign is reversed. Read more...iWorksheets :3Study Guides :1
Equations and inequalitiesAn equation is mathematical statement that shows that two expressions are equal to each other. The expressions used in an equation can contain variables or numbers.
Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Inequalities are also solved by
using inverse operations. Read more...iWorksheets :3Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
A-CED.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
Solving equations and inequalitiesAlgebraic equations are mathematical equations that contain a letter or variable which represents a number. To solve an algebraic equation, inverse operations are
used. Algebraic inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to, ≥; less than, <; and less than or equal to, ≤. When multiplying or
dividing by a negative number occurs, the inequality sign is reversed from the original inequality sign in order for the inequality to be correct. Read more...iWorksheets :3Study Guides :1
CM8AI.A-REI. Algebra: Reasoning with Equations and Inequalities (A-REI)
Understand solving equations as a process of reasoning and explain the reasoning
A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a
solution. Construct a viable argument to justify a solution method.
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The
inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater
than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Using IntegersIntegers are negative numbers, zero and positive numbers. To compare integers, a number line can be used. On a number line, negative integers are on the left side of zero with the
larger a negative number, the farther to the left it is. Positive integers are on the right side of zero on the number line. If a number is to the left of another number it is said to be less than
that number. In the coordinate plane, the x-axis is a horizontal line with negative numbers, zero and positive numbers. Read more...iWorksheets :4Study Guides :1
Decimal OperationsDecimal operations refer to the mathematical operations that can be performed with decimals: addition, subtraction, multiplication and division. The process for adding, subtracting,
multiplying and dividing decimals must be followed in order to achieve the correct answer. Read more...iWorksheets :3Study Guides :1
Fraction OperationsFraction operations are the processes of adding, subtracting, multiplying and dividing fractions and mixed numbers. A mixed number is a fraction with a whole number. Adding
fractions is common in many everyday events, such as making a recipe and measuring wood. In order to add and subtract fractions, the fractions must have the same denominator. Read more...iWorksheets
:3Study Guides :1
Introduction to PercentWhat Is Percent? A percent is a term that describes a decimal in terms of one hundred. Percent means per hundred. Percents, fractions and decimals all can equal each other, as
in the case of 10%, 0.1 and 1/10. Percents can be greater than 100% or smaller than 1%. A markup from the cost of making an item to the actual sales price is usually greater than 100%. A
salesperson's commission might be 1/2% depending on the item sold. Read more...iWorksheets :4Study Guides :1
Algebraic EquationsWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. When algebraic equations are written in
words, the words must be changed into the appropriate numbers and variable in order to solve. Read more...iWorksheets :5Study Guides :1
Equations and inequalitiesAn equation is mathematical statement that shows that two expressions are equal to each other. The expressions used in an equation can contain variables or numbers.
Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Inequalities are also solved by
using inverse operations. Read more...iWorksheets :3Study Guides :1
Integer operationsInteger operations are the mathematical operations that involve integers. Integers are negative numbers, zero and positive numbers. Adding and subtracting integers are useful in
everyday life because there are many situations that involved negative numbers such as calculating sea level or temperatures. Equations with integers are solved using inverse operations. Addition and
subtraction are inverse operations, and multiplication and division are inverse operations of each other. Read more...iWorksheets :4Study Guides :1
Rational numbers and operationsA rational number is a number that can be made into a fraction. Decimals that repeat or terminate are rational because they can be changed into fractions. A square root
of a number is a number that when multiplied by itself will result in the original number. The square root of 4 is 2 because 2 · 2 = 4. Read more...iWorksheets :3Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
Solving equations and inequalitiesAlgebraic equations are mathematical equations that contain a letter or variable which represents a number. To solve an algebraic equation, inverse operations are
used. Algebraic inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to, ≥; less than, <; and less than or equal to, ≤. When multiplying or
dividing by a negative number occurs, the inequality sign is reversed from the original inequality sign in order for the inequality to be correct. Read more...iWorksheets :3Study Guides :1
Solve equations and inequalities in one variable
A-REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The
inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater
than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Using IntegersIntegers are negative numbers, zero and positive numbers. To compare integers, a number line can be used. On a number line, negative integers are on the left side of zero with the
larger a negative number, the farther to the left it is. Positive integers are on the right side of zero on the number line. If a number is to the left of another number it is said to be less than
that number. In the coordinate plane, the x-axis is a horizontal line with negative numbers, zero and positive numbers. Read more...iWorksheets :4Study Guides :1
Decimal OperationsDecimal operations refer to the mathematical operations that can be performed with decimals: addition, subtraction, multiplication and division. The process for adding, subtracting,
multiplying and dividing decimals must be followed in order to achieve the correct answer. Read more...iWorksheets :3Study Guides :1
Fraction OperationsFraction operations are the processes of adding, subtracting, multiplying and dividing fractions and mixed numbers. A mixed number is a fraction with a whole number. Adding
fractions is common in many everyday events, such as making a recipe and measuring wood. In order to add and subtract fractions, the fractions must have the same denominator. Read more...iWorksheets
:3Study Guides :1
Introduction to PercentWhat Is Percent? A percent is a term that describes a decimal in terms of one hundred. Percent means per hundred. Percents, fractions and decimals all can equal each other, as
in the case of 10%, 0.1 and 1/10. Percents can be greater than 100% or smaller than 1%. A markup from the cost of making an item to the actual sales price is usually greater than 100%. A
salesperson's commission might be 1/2% depending on the item sold. Read more...iWorksheets :4Study Guides :1
Algebraic EquationsWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. When algebraic equations are written in
words, the words must be changed into the appropriate numbers and variable in order to solve. Read more...iWorksheets :5Study Guides :1
Algebraic InequalitiesFreeAlgebraic inequalities are mathematical equations that compare two quantities using these criteria: greater than, less than, less than or equal to, greater than or equal to.
The only rule of inequalities that must be remembered is that when a variable is multiplied or divided by a negative number the sign is reversed. Read more...iWorksheets :3Study Guides :1
Equations and inequalitiesAn equation is mathematical statement that shows that two expressions are equal to each other. The expressions used in an equation can contain variables or numbers.
Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Inequalities are also solved by
using inverse operations. Read more...iWorksheets :3Study Guides :1
Integer operationsInteger operations are the mathematical operations that involve integers. Integers are negative numbers, zero and positive numbers. Adding and subtracting integers are useful in
everyday life because there are many situations that involved negative numbers such as calculating sea level or temperatures. Equations with integers are solved using inverse operations. Addition and
subtraction are inverse operations, and multiplication and division are inverse operations of each other. Read more...iWorksheets :4Study Guides :1
Rational numbers and operationsA rational number is a number that can be made into a fraction. Decimals that repeat or terminate are rational because they can be changed into fractions. A square root
of a number is a number that when multiplied by itself will result in the original number. The square root of 4 is 2 because 2 · 2 = 4. Read more...iWorksheets :3Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
Solving equations and inequalitiesAlgebraic equations are mathematical equations that contain a letter or variable which represents a number. To solve an algebraic equation, inverse operations are
used. Algebraic inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to, ≥; less than, <; and less than or equal to, ≤. When multiplying or
dividing by a negative number occurs, the inequality sign is reversed from the original inequality sign in order for the inequality to be correct. Read more...iWorksheets :3Study Guides :1
Represent and solve equations and inequalities graphically
A-REI.12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes.
Linear relationshipsLinear relationships refer to two quantities that are related with a linear equation. Since a linear equation is a line, a linear relationship refers to two quantities on a line
and their relationship to one another. This relationship can be direct or inverse. If y varies directly as x, it means if y is doubled, then x is doubled. The formula for a direct variation is y =
kx, where k is the constant of variation. Read more...iWorksheets :3Study Guides :1
CM8AI.F. Functions: Functions (F)
Define, evaluate, and compare functions
8.F.1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
FunctionsFreeA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on
the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By
substituting a number for x, an output can be determined. Read more...iWorksheets :5Study Guides :1
8.F.3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the
area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
Use functions to model relationships between quantities
8.F.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,
y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or
a table of values.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
CM8AI.F-IF. Functions: Interpreting Functions (F-IF)
Understand the concept of a function and use function notation
F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x
is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
FunctionsFreeA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on
the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By
substituting a number for x, an output can be determined. Read more...iWorksheets :5Study Guides :1
F-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
F-IF.3. Recognize that sequences are functions whose domain is a subset of the integers.
SequencesA sequence is an ordered list of numbers. Sequences are the result of a pattern or rule. A pattern or rule can be every other number or some formula such as y = 2x + 3. When a pattern or
rule is given, a sequence can be found. When a sequence is given, the pattern or rule can be found. Read more...iWorksheets :5Study Guides :1
Interpret functions that arise in applications in terms of the context
F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal
description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity.
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
F-IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Nonlinear Functions and Set TheoryA function can be in the form of y = mx + b. This is an equation of a line, so it is said to be a linear function. Nonlinear functions are functions that are not
straight lines. Some examples of nonlinear functions are exponential functions and parabolic functions. An exponential function, y = aˆx, is a curved line that gets closer to but does not touch the
x-axis. A parabolic function, y = ax² + bx +c, is a U-shaped line that can either be facing up or facing down. Read more...iWorksheets :5Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
Analyze functions using different representations
F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.7.a. Graph functions (linear and quadratic) and show intercepts, maxima, and minima.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
CM8AI.F-BF. Functions: Building Functions (F-BF)
Build a function that models a relationship between two quantities
F-BF.1. Write a function that describes a relationship between two quantities.
F-BF.1.a. Determine an explicit expression or steps for calculation from a context.
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
CM8AI.F-LE. Functions: Linear, Quadratic, and Exponential Models (F-LE)
Construct and compare linear, quadratic, and exponential models and solve problems
F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
F-LE.1.a. Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
FunctionsFreeA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on
the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By
substituting a number for x, an output can be determined. Read more...iWorksheets :5Study Guides :1
Interpret expressions for functions in terms of the situation they model
F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
CM8AI.G. Geometry: Geometry (G)
Understand and apply the Pythagorean Theorem
8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
The Pythagorean TheoremPythagorean Theorem is a fundamental relation in Euclidean geometry. It states the sum of the squares of the legs of a right triangle equals the square of the length of the
hypotenuse. Determine the distance between two points using the Pythagorean Theorem. Read more...iWorksheets :10Study Guides :2
CM8AI.SP. Statistics and Probability: Statistics and Probability (SP)
Investigate patterns of association in bivariate data
8.SP.1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or
negative association, linear association, and nonlinear association.
Analyzing, Graphing and Displaying DataThere are many types of graphs such as, bar graphs, histograms and line graphs. A bar graph compares data in categories and uses bars, either vertical or
horizontal. A histogram is similar to a bar graph, but with histograms the bars touch each other where with bar graphs the bars do not touch each other. A line graph is useful for graphing how data
changes over time. With a line graph, data is plotted as points and lines are drawn to connect the points to show how the data changes. Read more...iWorksheets :6Study Guides :1
Using graphs to analyze dataThere are different types of graphs and ways that data can be analyzed using the graphs. Graphs are based on the coordinate plane. Data are the points on the plane. If
collecting data about the ages of people living on one street, the data is all the ages. The data can then be organized into groups, and evaluated. Mean, mode and median are different ways to
evaluate data. Read more...iWorksheets :7Study Guides :1
Collecting and describing dataCollecting and describing data refers to the different ways to gather data and the different ways to arrange data whether it is in a table, graph, or pie chart. Data can
be collected by either taking a sample of a population or by conducting a survey. Describing data looks at data after it has been organized and makes conclusions about the data. Read more...i
Worksheets :3Study Guides :1
Displaying dataDisplaying data refers to the many ways that data can be displayed whether it is on a bar graph, line graph, circle graph, pictograph, line plot, scatter plot or another way. Certain
data is better displayed with different graphs as opposed to other graphs. E.g. if data representing the cost of a movie over the past 5 years were to be displayed, a line graph would be best. A
circle graph would not be appropriate to use because a circle graph represents data that can add up to one or 100%. Read more...iWorksheets :4Study Guides :1
8.SP.2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and
informally assess the model fit by judging the closeness of the data points to the line.
Linear relationshipsLinear relationships refer to two quantities that are related with a linear equation. Since a linear equation is a line, a linear relationship refers to two quantities on a line
and their relationship to one another. This relationship can be direct or inverse. If y varies directly as x, it means if y is doubled, then x is doubled. The formula for a direct variation is y =
kx, where k is the constant of variation. Read more...iWorksheets :3Study Guides :1
CM8AI.S-ID. Statistics and Probability: Interpreting Categorical and Quantitative Data (S-ID)
Summarize, represent, and interpret data on two categorical and quantitative variables
S-ID.5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative
frequencies). Recognize possible associations and trends in the data.
Organizing DataThe data can be organized into groups, and evaluated. Mean, mode, median and range are different ways to evaluate data. The mean is the average of the data. The mode refers to the
number that occurs the most often in the data. The median is the middle number when the data is arranged in order from lowest to highest. The range is the difference in numbers when the lowest number
is subtracted from the highest number. Data can be organized into a table, such as a frequency table. Read more...iWorksheets :3Study Guides :1
Analyzing, Graphing and Displaying DataThere are many types of graphs such as, bar graphs, histograms and line graphs. A bar graph compares data in categories and uses bars, either vertical or
horizontal. A histogram is similar to a bar graph, but with histograms the bars touch each other where with bar graphs the bars do not touch each other. A line graph is useful for graphing how data
changes over time. With a line graph, data is plotted as points and lines are drawn to connect the points to show how the data changes. Read more...iWorksheets :6Study Guides :1
Using graphs to analyze dataThere are different types of graphs and ways that data can be analyzed using the graphs. Graphs are based on the coordinate plane. Data are the points on the plane. If
collecting data about the ages of people living on one street, the data is all the ages. The data can then be organized into groups, and evaluated. Mean, mode and median are different ways to
evaluate data. Read more...iWorksheets :7Study Guides :1
S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
S-ID.6.c. Fit a linear function for a scatter plot that suggests a linear association.
Linear relationshipsLinear relationships refer to two quantities that are related with a linear equation. Since a linear equation is a line, a linear relationship refers to two quantities on a line
and their relationship to one another. This relationship can be direct or inverse. If y varies directly as x, it means if y is doubled, then x is doubled. The formula for a direct variation is y =
kx, where k is the constant of variation. Read more...iWorksheets :3Study Guides :1
MS.CM8IM. Compacted Mathematics Grade 8 (with Integrated Math I)
CM8IM.A-SSE. Algebra: Seeing Structure in Expressions (A-SSE)
Write expressions in equivalent forms to solve problems
A-SSE.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A-SSE.3.c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as [1.15^(1/12)]^12t ≈ 1.012^12t to reveal the
approximate equivalent monthly interest rate if the annual rate is 15%.
FunctionsFreeA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on
the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By
substituting a number for x, an output can be determined. Read more...iWorksheets :5Study Guides :1
CM8IM.A.CED. Algebra: Creating Equations (A-CED)
Create equations that describe numbers or relationships
A-CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The
inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater
than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Algebraic InequalitiesFreeAlgebraic inequalities are mathematical equations that compare two quantities using these criteria: greater than, less than, less than or equal to, greater than or equal to.
The only rule of inequalities that must be remembered is that when a variable is multiplied or divided by a negative number the sign is reversed. Read more...iWorksheets :3Study Guides :1
Equations and inequalitiesAn equation is mathematical statement that shows that two expressions are equal to each other. The expressions used in an equation can contain variables or numbers.
Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Inequalities are also solved by
using inverse operations. Read more...iWorksheets :3Study Guides :1
A-CED.2. Create equations in two variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [Note this standard appears in future courses with
a slight variation in the standard language.]
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For
example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The
inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater
than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Algebraic InequalitiesFreeAlgebraic inequalities are mathematical equations that compare two quantities using these criteria: greater than, less than, less than or equal to, greater than or equal to.
The only rule of inequalities that must be remembered is that when a variable is multiplied or divided by a negative number the sign is reversed. Read more...iWorksheets :3Study Guides :1
Equations and inequalitiesAn equation is mathematical statement that shows that two expressions are equal to each other. The expressions used in an equation can contain variables or numbers.
Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Inequalities are also solved by
using inverse operations. Read more...iWorksheets :3Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
A-CED.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
Solving equations and inequalitiesAlgebraic equations are mathematical equations that contain a letter or variable which represents a number. To solve an algebraic equation, inverse operations are
used. Algebraic inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to, ≥; less than, <; and less than or equal to, ≤. When multiplying or
dividing by a negative number occurs, the inequality sign is reversed from the original inequality sign in order for the inequality to be correct. Read more...iWorksheets :3Study Guides :1
CM8IM.A-REI. Algebra: Reasoning with Equations and Inequalities (A-REI)
Solve equations and inequalities in one variable
A-REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The
inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater
than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Using IntegersIntegers are negative numbers, zero and positive numbers. To compare integers, a number line can be used. On a number line, negative integers are on the left side of zero with the
larger a negative number, the farther to the left it is. Positive integers are on the right side of zero on the number line. If a number is to the left of another number it is said to be less than
that number. In the coordinate plane, the x-axis is a horizontal line with negative numbers, zero and positive numbers. Read more...iWorksheets :4Study Guides :1
Decimal OperationsDecimal operations refer to the mathematical operations that can be performed with decimals: addition, subtraction, multiplication and division. The process for adding, subtracting,
multiplying and dividing decimals must be followed in order to achieve the correct answer. Read more...iWorksheets :3Study Guides :1
Fraction OperationsFraction operations are the processes of adding, subtracting, multiplying and dividing fractions and mixed numbers. A mixed number is a fraction with a whole number. Adding
fractions is common in many everyday events, such as making a recipe and measuring wood. In order to add and subtract fractions, the fractions must have the same denominator. Read more...iWorksheets
:3Study Guides :1
Introduction to PercentWhat Is Percent? A percent is a term that describes a decimal in terms of one hundred. Percent means per hundred. Percents, fractions and decimals all can equal each other, as
in the case of 10%, 0.1 and 1/10. Percents can be greater than 100% or smaller than 1%. A markup from the cost of making an item to the actual sales price is usually greater than 100%. A
salesperson's commission might be 1/2% depending on the item sold. Read more...iWorksheets :4Study Guides :1
Algebraic EquationsWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. When algebraic equations are written in
words, the words must be changed into the appropriate numbers and variable in order to solve. Read more...iWorksheets :5Study Guides :1
Algebraic InequalitiesFreeAlgebraic inequalities are mathematical equations that compare two quantities using these criteria: greater than, less than, less than or equal to, greater than or equal to.
The only rule of inequalities that must be remembered is that when a variable is multiplied or divided by a negative number the sign is reversed. Read more...iWorksheets :3Study Guides :1
Equations and inequalitiesAn equation is mathematical statement that shows that two expressions are equal to each other. The expressions used in an equation can contain variables or numbers.
Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Inequalities are also solved by
using inverse operations. Read more...iWorksheets :3Study Guides :1
Integer operationsInteger operations are the mathematical operations that involve integers. Integers are negative numbers, zero and positive numbers. Adding and subtracting integers are useful in
everyday life because there are many situations that involved negative numbers such as calculating sea level or temperatures. Equations with integers are solved using inverse operations. Addition and
subtraction are inverse operations, and multiplication and division are inverse operations of each other. Read more...iWorksheets :4Study Guides :1
Rational numbers and operationsA rational number is a number that can be made into a fraction. Decimals that repeat or terminate are rational because they can be changed into fractions. A square root
of a number is a number that when multiplied by itself will result in the original number. The square root of 4 is 2 because 2 · 2 = 4. Read more...iWorksheets :3Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
Solving equations and inequalitiesAlgebraic equations are mathematical equations that contain a letter or variable which represents a number. To solve an algebraic equation, inverse operations are
used. Algebraic inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to, ≥; less than, <; and less than or equal to, ≤. When multiplying or
dividing by a negative number occurs, the inequality sign is reversed from the original inequality sign in order for the inequality to be correct. Read more...iWorksheets :3Study Guides :1
Represent and solve equations and inequalities graphically
A-REI.12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes.
Linear relationshipsLinear relationships refer to two quantities that are related with a linear equation. Since a linear equation is a line, a linear relationship refers to two quantities on a line
and their relationship to one another. This relationship can be direct or inverse. If y varies directly as x, it means if y is doubled, then x is doubled. The formula for a direct variation is y =
kx, where k is the constant of variation. Read more...iWorksheets :3Study Guides :1
CM8IM.F. Functions: Functions (F)
Define, evaluate, and compare functions
8.F.1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
FunctionsFreeA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on
the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By
substituting a number for x, an output can be determined. Read more...iWorksheets :5Study Guides :1
8.F.3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the
area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
Use functions to model relationships between quantities
8.F.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,
y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or
a table of values.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
CM8IM.F-IF. Functions: Interpreting Functions (F-IF)
Understand the concept of a function and use function notation
F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x
is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
FunctionsFreeA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on
the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By
substituting a number for x, an output can be determined. Read more...iWorksheets :5Study Guides :1
F-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
F-IF.3. Recognize that sequences are functions whose domain is a subset of the integers.
SequencesA sequence is an ordered list of numbers. Sequences are the result of a pattern or rule. A pattern or rule can be every other number or some formula such as y = 2x + 3. When a pattern or
rule is given, a sequence can be found. When a sequence is given, the pattern or rule can be found. Read more...iWorksheets :5Study Guides :1
Interpret functions that arise in applications in terms of the context
F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal
description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity.
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
F-IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Nonlinear Functions and Set TheoryA function can be in the form of y = mx + b. This is an equation of a line, so it is said to be a linear function. Nonlinear functions are functions that are not
straight lines. Some examples of nonlinear functions are exponential functions and parabolic functions. An exponential function, y = aˆx, is a curved line that gets closer to but does not touch the
x-axis. A parabolic function, y = ax² + bx +c, is a U-shaped line that can either be facing up or facing down. Read more...iWorksheets :5Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
Analyze functions using different representations
F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.7.a. Graph functions (linear and quadratic) and show intercepts, maxima, and minima.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
CM8IM.F-BF. Functions: Building Functions (F-BF)
Build a function that models a relationship between two quantities
F-BF.1. Write a function that describes a relationship between two quantities.
F-BF.1.a. Determine an explicit expression or steps for calculation from a context.
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus,
exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Solving linear equationsWhen graphed, a linear equation is a straight line. Although the standard equation for a line is y = mx + b, where m is the slope and b is the y-intercept, linear equations
often have both of the variables on the same side of the equal sign. Linear equations can be solved for one variable when the other variable is given. Read more...iWorksheets :5Study Guides :1
CM8IM.F-LE. Functions: Linear, Quadratic, and Exponential Models (F-LE)
Construct and compare linear, quadratic, and exponential models and solve problems
F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
F-LE.1.a. Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Linear equationsLinear equations are equations that have two variables and when graphed are a straight line. Linear equation can be graphed based on their slope and y-intercept. The standard equation
for a line is y = mx + b, where m is the slope and b is the y-intercept. Slope can be found with the formula m = (y2 - y1)/(x2 - x1), which represents the change in y over the change in x. Read
more...iWorksheets :6Study Guides :1
FunctionsFreeA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on
the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By
substituting a number for x, an output can be determined. Read more...iWorksheets :5Study Guides :1
Interpret expressions for functions in terms of the situation they model
F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context.
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are
performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the
output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
CM8IM.G. Geometry: Geometry (G)
Understand and apply the Pythagorean Theorem
8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
The Pythagorean TheoremPythagorean Theorem is a fundamental relation in Euclidean geometry. It states the sum of the squares of the legs of a right triangle equals the square of the length of the
hypotenuse. Determine the distance between two points using the Pythagorean Theorem. Read more...iWorksheets :10Study Guides :2
CM8IM.G-CO. Geometry: Congruence (G-CO)
Experiment with transformations in the plane
G-CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a
circular arc.
Geometric ProportionsGeometric proportions compare two similar polygons. Similar polygons have equal corresponding angles and corresponding sides that are in proportion. A proportion equation can be
used to prove two figures to be similar. If two figures are similar, the proportion equation can be used to find a missing side of one of the figures. Read more...iWorksheets :4Study Guides :1
Plane Figures: Lines and AnglesPlane figures in regards to lines and angles refer to the coordinate plane and the various lines and angles within the coordinate plane. Lines in a coordinate plane can
be parallel or perpendicular. Angles in a coordinate plane can be acute, obtuse, right or straight. Adjacent angles are two angles that have a common vertex and a common side but do not overlap. Read
more...iWorksheets :3Study Guides :1
Plane Figures: Closed Figure RelationshipsPlane figures in regards to closed figure relationships refer to the coordinate plane and congruent figures, circles, circle graphs, transformations and
symmetry. Congruent figures have the same size and shape. Transformations are made up of translations, rotations and reflections. A translation of a figure keeps the size and shape of a figure, but
moves it to a different location. A rotation turns a figure about a point on the figure. A reflection of a figure produces a mirror image of the figure when it is reflected in a given line. Read
more...iWorksheets :3Study Guides :1
Measurement, Perimeter, and CircumferenceThere are two systems used to measure objects, the U.S. Customary system and the metric system. The U.S. Customary system measures length in inches, feet,
yards and miles. The metric system is a base ten system and measures length in kilometers, meters, and millimeters. Perimeter is the measurement of the distance around a figure. It is measured in
units and can be measured by inches, feet, blocks, meters, centimeters or millimeters. To get the perimeter of any figure, simply add up the measures of the sides of the figure. Read more...i
Worksheets :3Study Guides :1
Exploring Area and Surface AreaArea is the amount of surface a shape covers. Area is measured in square units, whether the units are inches, feet, meters or centimeters. The area formula for a
triangle is: A = 1/2 · b · h, where b is the base and h is the height. The area formula for a circle is: A = π · r², where π is usually 3.14 and r is the radius of the circle. The area formula for a
parallelogram is: A = b · h, where b is the base and h is the height. Read more...iWorksheets :4Study Guides :1
The Pythagorean TheoremPythagorean Theorem is a fundamental relation in Euclidean geometry. It states the sum of the squares of the legs of a right triangle equals the square of the length of the
hypotenuse. Determine the distance between two points using the Pythagorean Theorem. Read more...iWorksheets :10Study Guides :2
Finding VolumeVolume measures the amount a solid figure can hold. Volume is measured in terms of cubed units and can be measured in inches, feet, meters, centimeters, and millimeters. The formula for
the volume of a rectangular prism is V = l · w · h, where l is the length, w is the width, and h is the height. Read more...iWorksheets :4Study Guides :1
Plane figuresPlane figures refer to points, lines, angles, and planes in the coordinate plane. Lines can be parallel or perpendicular. Angles can be categorized as acute, obtuse or right. Angles can
also be complementary or supplementary depending on how many degrees they add up to. Plane figures can also refer to shapes in the coordinate plane. Triangles, quadrilaterals and other polygons can
be shown in the coordinate plane. Read more...iWorksheets :4Study Guides :1
Patterns in geometryPatterns in geometry refer to shapes and their measures. Shapes can be congruent to one another. Shapes can also be manipulated to form similar shapes. The types of
transformations are reflection, rotation, dilation and translation. With a reflection, a figure is reflected, or flipped, in a line so that the new figure is a mirror image on the other side of the
line. A rotation rotates, or turns, a shape to make a new figure. A dilation shrinks or enlarges a figure. A translation shifts a figure to a new position. Read more...iWorksheets :3Study Guides :1
Perimeter and areaWhat Is Perimeter and Area? Perimeter is the measurement of the distance around a figure. It is measured in units and can be measured by inches, feet, blocks, meters, centimeters or
millimeters. To find the perimeter of any figure, simply add up the measures of the sides of the figure. Area is the amount of surface a shape covers. Area is measured in square units, whether the
units are inches, feet, meters or centimeters. The area formula for a parallelogram is: A = b · h, where b is the base and h is the height. Read more...iWorksheets :4Study Guides :1
Three dimensional geometry/MeasurementThree-dimensional geometry/measurement refers to three-dimensional (3D) shapes and the measurement of their shapes concerning volume and surface area. The
figures of prisms, cylinders, pyramids, cones and spheres are all 3D figures. Volume measures the amount a solid figure can hold. Volume is measured in terms of units³ and can be measured in inches,
feet, meters, centimeters, and millimeters. Read more...iWorksheets :11Study Guides :1
Similarity and scaleSimilarity refers to similar figures and the ability to compare them using proportions. Similar figures have equal corresponding angles and corresponding sides that are in
proportion. A proportion equation can be used to prove two figures to be similar. If two figures are similar, the proportion equation can be used to find a missing side of one of the figures. Read
more...iWorksheets :7Study Guides :1
G-CO.2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points
as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Patterns in geometryPatterns in geometry refer to shapes and their measures. Shapes can be congruent to one another. Shapes can also be manipulated to form similar shapes. The types of
transformations are reflection, rotation, dilation and translation. With a reflection, a figure is reflected, or flipped, in a line so that the new figure is a mirror image on the other side of the
line. A rotation rotates, or turns, a shape to make a new figure. A dilation shrinks or enlarges a figure. A translation shifts a figure to a new position. Read more...iWorksheets :3Study Guides :1
Similarity and scaleSimilarity refers to similar figures and the ability to compare them using proportions. Similar figures have equal corresponding angles and corresponding sides that are in
proportion. A proportion equation can be used to prove two figures to be similar. If two figures are similar, the proportion equation can be used to find a missing side of one of the figures. Read
more...iWorksheets :7Study Guides :1
G-CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Patterns in geometryPatterns in geometry refer to shapes and their measures. Shapes can be congruent to one another. Shapes can also be manipulated to form similar shapes. The types of
transformations are reflection, rotation, dilation and translation. With a reflection, a figure is reflected, or flipped, in a line so that the new figure is a mirror image on the other side of the
line. A rotation rotates, or turns, a shape to make a new figure. A dilation shrinks or enlarges a figure. A translation shifts a figure to a new position. Read more...iWorksheets :3Study Guides :1
G-CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Patterns in geometryPatterns in geometry refer to shapes and their measures. Shapes can be congruent to one another. Shapes can also be manipulated to form similar shapes. The types of
transformations are reflection, rotation, dilation and translation. With a reflection, a figure is reflected, or flipped, in a line so that the new figure is a mirror image on the other side of the
line. A rotation rotates, or turns, a shape to make a new figure. A dilation shrinks or enlarges a figure. A translation shifts a figure to a new position. Read more...iWorksheets :3Study Guides :1
G-CO.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of
transformations that will carry a given figure onto another.
Patterns in geometryPatterns in geometry refer to shapes and their measures. Shapes can be congruent to one another. Shapes can also be manipulated to form similar shapes. The types of
transformations are reflection, rotation, dilation and translation. With a reflection, a figure is reflected, or flipped, in a line so that the new figure is a mirror image on the other side of the
line. A rotation rotates, or turns, a shape to make a new figure. A dilation shrinks or enlarges a figure. A translation shifts a figure to a new position. Read more...iWorksheets :3Study Guides :1
Understand congruence in terms of rigid motions
G-CO.6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in
terms of rigid motions to decide if they are congruent.
Patterns in geometryPatterns in geometry refer to shapes and their measures. Shapes can be congruent to one another. Shapes can also be manipulated to form similar shapes. The types of
transformations are reflection, rotation, dilation and translation. With a reflection, a figure is reflected, or flipped, in a line so that the new figure is a mirror image on the other side of the
line. A rotation rotates, or turns, a shape to make a new figure. A dilation shrinks or enlarges a figure. A translation shifts a figure to a new position. Read more...iWorksheets :3Study Guides :1
CM8IM.SP. Statistics and Probability: Statistics and Probability (SP)
Investigate patterns of association in bivariate data
8.SP.1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or
negative association, linear association, and nonlinear association.
Analyzing, Graphing and Displaying DataThere are many types of graphs such as, bar graphs, histograms and line graphs. A bar graph compares data in categories and uses bars, either vertical or
horizontal. A histogram is similar to a bar graph, but with histograms the bars touch each other where with bar graphs the bars do not touch each other. A line graph is useful for graphing how data
changes over time. With a line graph, data is plotted as points and lines are drawn to connect the points to show how the data changes. Read more...iWorksheets :6Study Guides :1
Using graphs to analyze dataThere are different types of graphs and ways that data can be analyzed using the graphs. Graphs are based on the coordinate plane. Data are the points on the plane. If
collecting data about the ages of people living on one street, the data is all the ages. The data can then be organized into groups, and evaluated. Mean, mode and median are different ways to
evaluate data. Read more...iWorksheets :7Study Guides :1
Collecting and describing dataCollecting and describing data refers to the different ways to gather data and the different ways to arrange data whether it is in a table, graph, or pie chart. Data can
be collected by either taking a sample of a population or by conducting a survey. Describing data looks at data after it has been organized and makes conclusions about the data. Read more...i
Worksheets :3Study Guides :1
Displaying dataDisplaying data refers to the many ways that data can be displayed whether it is on a bar graph, line graph, circle graph, pictograph, line plot, scatter plot or another way. Certain
data is better displayed with different graphs as opposed to other graphs. E.g. if data representing the cost of a movie over the past 5 years were to be displayed, a line graph would be best. A
circle graph would not be appropriate to use because a circle graph represents data that can add up to one or 100%. Read more...iWorksheets :4Study Guides :1
8.SP.2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and
informally assess the model fit by judging the closeness of the data points to the line.
Linear relationshipsLinear relationships refer to two quantities that are related with a linear equation. Since a linear equation is a line, a linear relationship refers to two quantities on a line
and their relationship to one another. This relationship can be direct or inverse. If y varies directly as x, it means if y is doubled, then x is doubled. The formula for a direct variation is y =
kx, where k is the constant of variation. Read more...iWorksheets :3Study Guides :1
CM8IM.S-ID. Statistics and Probability: Interpreting Categorical and Quantitative Data (S-ID)
Summarize, represent, and interpret data on two categorical and quantitative variables
S-ID.5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative
frequencies). Recognize possible associations and trends in the data.
Organizing DataThe data can be organized into groups, and evaluated. Mean, mode, median and range are different ways to evaluate data. The mean is the average of the data. The mode refers to the
number that occurs the most often in the data. The median is the middle number when the data is arranged in order from lowest to highest. The range is the difference in numbers when the lowest number
is subtracted from the highest number. Data can be organized into a table, such as a frequency table. Read more...iWorksheets :3Study Guides :1
Analyzing, Graphing and Displaying DataThere are many types of graphs such as, bar graphs, histograms and line graphs. A bar graph compares data in categories and uses bars, either vertical or
horizontal. A histogram is similar to a bar graph, but with histograms the bars touch each other where with bar graphs the bars do not touch each other. A line graph is useful for graphing how data
changes over time. With a line graph, data is plotted as points and lines are drawn to connect the points to show how the data changes. Read more...iWorksheets :6Study Guides :1
Using graphs to analyze dataThere are different types of graphs and ways that data can be analyzed using the graphs. Graphs are based on the coordinate plane. Data are the points on the plane. If
collecting data about the ages of people living on one street, the data is all the ages. The data can then be organized into groups, and evaluated. Mean, mode and median are different ways to
evaluate data. Read more...iWorksheets :7Study Guides :1
S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
S-ID.6.c. Fit a linear function for a scatter plot that suggests a linear association.
Linear relationshipsLinear relationships refer to two quantities that are related with a linear equation. Since a linear equation is a line, a linear relationship refers to two quantities on a line
and their relationship to one another. This relationship can be direct or inverse. If y varies directly as x, it means if y is doubled, then x is doubled. The formula for a direct variation is y =
kx, where k is the constant of variation. Read more...iWorksheets :3Study Guides :1 | {"url":"https://newpathworksheets.com/math/grade-8/mississippi-standards","timestamp":"2024-11-04T10:28:04Z","content_type":"text/html","content_length":"254495","record_id":"<urn:uuid:f5f8eb65-d7f9-4b84-9590-adcae1c41831>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00756.warc.gz"} |
Zwicky Transient Facility constraints on the optical emission from the nearby repeating FRB 180916.J0158+65 | aschig's Universe
The discovery rate of fast radio bursts (FRBs) is increasing dramatically thanks to new radio facilities. Meanwhile, wide-field instruments such as the 47 deg$^2$ Zwicky Transient Facility (ZTF)
survey the optical sky to study transient and variable sources. We present serendipitous ZTF observations of the CHIME repeating source FRB 180916.J0158+65, that was localized to a spiral galaxy 149
Mpc away and is the first FRB suggesting periodic modulation in its activity. While 147 ZTF exposures corresponded to expected high-activity periods of this FRB, no single ZTF exposure was at the
same time as a CHIME detection. No $>3sigma$ optical source was found at the FRB location in 683 ZTF exposures, totalling 5.69 hours of integration time. We combined ZTF upper limits and expected
repetitions from FRB 180916.J0158+65 in a statistical framework using a Weibull distribution, agnostic of periodic modulation priors. The analysis yielded a constraint on the ratio between the
optical and radio fluences of $eta lesssim 200$, corresponding to an optical energy $E_{rm opt} lesssim 3 times 10^{46}$ erg for a fiducial 10 Jy ms FRB (90% confidence). A deeper (but less
statistically robust) constraint of $eta lesssim 3$ can be placed assuming a rate of $r(>5$ Jy ms)= hr$^{-1}$ and $1.2pm 1.1$ FRB occurring during exposures taken in high-activity windows. The
constraint can be improved with shorter per-image exposures and longer integration time, or observing FRBs at higher Galactic latitudes. This work demonstrated how current surveys can statistically
constrain multi-wavelength counterparts to FRBs even without deliberately scheduled simultaneous radio observation. | {"url":"https://ashishmahabal.net/publication/pub0161/","timestamp":"2024-11-06T05:00:08Z","content_type":"text/html","content_length":"22751","record_id":"<urn:uuid:bfd78bed-c0e8-4be5-902d-95e3995248f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00021.warc.gz"} |
13.3: Lee Cyclogenesis
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Recall from the General Circulation chapter that the west-to-east jet stream can meander poleward and equatorward as Rossby waves, due to barotropic and baroclinic instability. Such waves in the
upper-air (jet-stream) flow can create mid-latitude cyclones at the surface, as shown in Figs. 13.6 & 13.7.
One trigger mechanism for this instability is flow over high mountain ranges. The Rossby wave triggered by such a mountain often has a trough just downwind of (i.e.., to the “lee” of) mountain
ranges. East of this trough is a favored location for cyclogenesis; hence, it is known as lee cyclogenesis. Because the mountain location is fixed, the resulting Rossby-wave trough and ridge
locations are stationary with respect to the mountain-range location.
Synoptic meteorology is the study and analysis of weather maps, often with the aim to forecast the weather on horizontal scales of 400 to 4000 km. Syn- optic weather maps describe an instantaneous
state of the atmosphere over a wide area, as created from weather observations made nearly simultaneously.
Typical weather phenomena at these synoptic scales include cyclones (Lows), anticyclones (Highs), and airmasses. Fronts are also included in synoptics because of their length, even though frontal
zones are so narrow that they can also be classified as mesoscale. See Table 10-6 and Fig. 10.24 in the Atmospheric Forces & Winds chapter for a list of different atmospheric scales.
The material in this chapter and in the previous one fall solidly in the field of synoptics. People who specialize in synoptic meteorology are called synopticians.
The word “synoptics” is from the Greek “synoptikos”, and literally means “seeing everything together”. It is the big picture.
13.3.1. Stationary Rossby Waves
Consider a wind that causes air in the troposphere to blow over the Rocky mountains (Fig. 13.20). Convective clouds (e.g., thunderstorms) and turbulence can cause the Rossby wave amplitude to
decrease further east, so the first wave after the mountain (at location c in Fig. 13.20) is the one you should focus on.
Figure 13.20 Cyclogenesis to the lee of the mountains. (a) Vertical cross section. (b) Map of jet-stream flow. “Ridge” and “trough” refer to the wind-flow pattern, not the topography.
These Rossby waves have a dominant wavelength (λ) of roughly
\(\ \lambda \approx 2 \cdot \pi \cdot\left[\frac{M}{\beta}\right]^{1 / 2}\tag{13.1}\)
where the mean wind speed is M. As you have seen in an earlier chapter, β is the northward gradient of the Coriolis parameter (f[c]):
\(\ \beta=\frac{\Delta f_{C}}{\Delta y}=\frac{2 \cdot \Omega}{R_{\text {earth }}} \cdot \cos \phi\tag{13.2}\)
Factor 2·Ω = 1.458x10^–4 s^–1 is twice the angular rotation rate of the Earth. At North-American latitudes, β is roughly 1.5 to 2x10^–11 m^–1 s^–1.
Knowing the mountain-range height (∆z[mtn]) relative to the surrounding plains, and knowing the initial depth of the troposphere (∆z[T]), the Rossby-wave amplitude A is:
\(\ A \approx \frac{f_{c}}{\beta} \cdot \frac{\Delta z_{m t n}}{\Delta z_{T}}\tag{13.3}\)
Because β is related to f[c], we can analytically find their ratio as f[c]/β = R[Earth]·tan(ϕ), where the average radius (R[Earth]) of the Earth is 6371 km. Over North America the tangent of the
latitude ϕ is tan(ϕ) ≈ 1. Thus:
\(\ A \approx \frac{\Delta z_{m t n}}{\Delta z_{T}} \cdot R_{e a r t h}\tag{13.4}\)
where 2A is the ∆y distance between the wave trough and crest.
In summary, the equations above show that north-south Rossby-wave amplitude depends on the height of the mountains, but does not depend on wind speed. Conversely, wind speed is important in
determining Rossby wavelength, while mountain height is irrelevant.
Sample Application
What amplitude & wavelength of terrain-triggered Rossby wave would you expect for a mountain range at 48°N that is 1.2 km high? The upstream depth of the troposphere is 11 km, with upstream wind is
19 m s^–1.
Find the Answer
Given: ϕ = 48°N, ∆z[mtn] = 1.2 km, ∆z[T] = 11 km, M = 19 m s^–1.
Find: A = ? km , λ = ? km
Use eq. (13.4):
A = [ (1.2 km) / (11 km) ] · (6371 km) = 695 km
Next, use eq. (13.2) to find β at 48°N:
β = (2.294x10^–11 m^–1·s^–1) · cos(48°) = 1.53x10^–11 m^–1·s^–1
Finally, use eq. (13.1):
\(\lambda \approx 2 \cdot \pi \cdot\left[\frac{19 \mathrm{m} \cdot \mathrm{s}^{-1}}{1.53 \times 10^{-11} \mathrm{m}^{-1} \cdot \mathrm{s}^{-1}}\right]^{1 / 2}=\underline{6990} \mathrm{km}\)
Check: Physics and units are reasonable.
Exposition: Is this wave truly a planetary wave? Yes, because its wavelength (6,990 km) would fit 3.8 times around the Earth at 48°N (circumference = 2·π·R[Earth]·cos(48°) = 26,785 km). Also, the
north-south meander of the wave spans 2A = 12.5° of latitude.
13.3.2. Potential-vorticity Conservation
Use conservation of potential vorticity as a tool to understand such mountain lee-side Rossby-wave triggering (Fig. 13.20). Create a “toy model” by assuming wind speed is constant in the Rossby wave,
and that there is no wind shear affecting vorticity. For this situation, the conservation of potential vorticity ζ[p] is given by eq. (11.25) as:
\(\ \zeta_{p}=\frac{(M / R)+f_{c}}{\Delta z}= constant\tag{13.5}\)
For this toy model, consider the initial winds to be blowing straight toward the Rocky Mountains from the west. These initial winds have no curvature at location “a”, thus R = ∞ and eq. (13.5)
\(\ \zeta_{p}=\frac{f_{c . a}}{\Delta z_{T, a}}\tag{13.6}\)
where ∆z[T.a] is the average depth of troposphere at point “a”. Because potential vorticity is conserved, we can use this fixed value of ζ[p] to see how the Rossby wave is generated.
Let ∆z[mtn] be the relative mountain height above the surrounding land (Fig. 13.20a). As the air blows over the mountain range, the troposphere becomes thinner as it is squeezed between mountain top
and the tropopause at location “b”: ∆z[T.b] = ∆z[T.a] – ∆z[mtn]. But the latitude of the air hasn’t changed much yet, so f[c.b] ≈ f[c.a]. Because ∆z has changed, we can solve eq. (13.5) for the
radius of curvature needed to maintain ζ[p.b] = ζ[p.a].
\(\ R_{b}=\frac{-M}{f_{c . a} \cdot\left(\Delta z_{m t n} / \Delta z_{T, a}\right)}\tag{13.7}\)
Namely, in eq. (13.5), when ∆z became smaller while f[c] was constant, M/R had to also become smaller to keep the ratio constant. But since M/R was initially zero, the new M/R had to become negative.
Negative R means anticyclonic curvature.
As sketched in Fig. 13.20, such curvature turns the wind toward the equator. But equatorward-moving air experiences smaller Coriolis parameter, requiring that R[b] become larger (less curved) to
conserve ζ[p] . Near the east side of the Rocky Mountains the terrain elevation decreases at point “c”, allowing the air thickness ∆z to increase back to its original value.
But now the air is closer to the equator where Coriolis parameter is smaller, so the radius of curvature R[c] at location “c” becomes positive in order to keep potential vorticity constant. This
positive vorticity gives that cyclonic curvature that defines the lee trough of the Rossby wave. As was sketched in Fig. 13.7, surface cyclogenesis could be supported just east of the lee trough.
Sample Application
Picture a scenario as plotted in Fig. 13.20, with 25 m s^–1 wind at location “a”, mountain height of 1.2 km, troposphere thickness of 11 km, and latitude 45°N. What is the value of the initial
potential vorticity, and what is the radius of curvature at point “b”?
Find the Answer
Given: M = 25 m s^–1, ∆z[mtn] = 1.2 km, R[initial ]= ∞, ∆z[T] = 11 km, ϕ = 45°N.
Find: ζ[p.a] = ? m^–1·s^–1, R[b] = ? km
Assumption: Neglect wind shear in the vorticity calculation.
Eq. (10.16) can be applied to get the Coriolis parameter f[c] = (1.458x10^–4 s^–1)·sin(45°) = 1.031x10^–4 s^–1
Use eq. (13.6):
\(\zeta_{p}=\frac{1.031 \times 10^{-4} \mathrm{s}^{-1}}{11 \mathrm{km}}=9.37 \times 10^{-9} \mathrm{m}^{-1} \mathrm{s}^{-1}\)
Next, apply eq. (13.7) to get the radius of curvature:
\(R_{b}=\frac{-(25 \mathrm{m} / \mathrm{s})}{\left(1.031 \times 10^{-4} \mathrm{s}^{-1}\right) \cdot(1.2 \mathrm{km} / 11 \mathrm{km})}=-2223 . \mathrm{km}\)
Check: Physics and units are reasonable.
Exposition: The negative sign for the radius of curvature means that the turn is anticyclonic (clockwise in the N. Hemisphere). Typically, the cyclonic trough curvature is the same order of magnitude
as the anticyclonic ridge curvature. East of the first trough and west of the next ridge is where cyclogenesis is supported.
13.3.3. Lee-side Translation Equatorward
Suppose an extratropical cyclone (low center) is positioned over the east side of a north-south oriented mountain range in the Northern Hemisphere, as sketched in Fig. 13.21. An example is the Rocky
Mountains. In this diagram, the green circle and the air above it are the cyclone. Air within this cyclone has positive (cyclonic) vorticity, as represented by the rotating blue air columns in the
figure. The locations of these columns are also moving counterclockwise around the common low center (L) — driven by the synoptic-scale circulation around the low.
Figure 13.21 A low-pressure (L) center above terrain that slopes downward to the east, for the Northern Hemisphere.
As column “a” moves to position “b” and then “c”, its vertical extent ∆z stretches. This assumes that the top of the air columns is at the tropopause, while the bottom follows the sloping terrain.
Due to conservation of potential vorticity ζ[p], this stretching must be accompanied by an increase in relative vorticity ∆ζ [r] :
\(\ \Delta \zeta_{r}=2 \cdot R \cdot \alpha \cdot \zeta_{p}\tag{13.8}\)
R is cyclone radius and α = ∆z/∆x is terrain slope.
Conversely, as column “c” moves to position “d” and then “a”, its vertical extent shrinks, forcing its relative vorticity to decrease to maintain constant potential vorticity. Hence, the center of
action of the low center shifts (translates) equatorward (white arrow in Fig. 13.21) and downslope (eastward, in this example), following the region of increasing ζ[r] .
A similar conclusion can be reached by considering conservation of isentropic potential vorticity (IPV). Air in the bottom of column “a” descends and warms adiabatically en route to position “c”,
while there is no descent warming at the column top. Hence, the static stability of the column decreases at its equatorward and downslope sides. This drives an increase in relative vorticity on the
equatorward and east flanks of the cyclone to conserve IPV. Again, the cyclone moves equatorward and eastward toward the region of greater ζ [r] .
Sample Application
The cyclone of Fig. 13.21 has ζ[p] = 1x10^–8 m^–1·s^–1 and R = 600 km. The mountain slope is 1:500. Find the relative-vorticity change on the equatorward side.
Find the Answer
Given: ζ[p] = 1x10^–8 m^–1·s^–1, R = 600 km, α = 0.002
Find: ∆ζ[r] = ? s^–1 .
Assume constant latitude in Northern Hemisphere.
Use eq. (13.8):
∆ζ[r] = 2 · (600,000 m) · (0.002) · (1x10^–8 m^–1·s^–1) = 2.4x10^–5 s^–1 .
Check: Physics and units are reasonable
Exposition: A similar decrease is likely on the poleward side. The combined effect causes the cyclone to translate equatorward to where vorticity is greatest. | {"url":"https://geo.libretexts.org/Bookshelves/Meteorology_and_Climate_Science/Practical_Meteorology_(Stull)/13%3A_Extratropical_Cyclones/13.02%3A_Section_3-","timestamp":"2024-11-07T12:37:58Z","content_type":"text/html","content_length":"142079","record_id":"<urn:uuid:bdcacc95-7ba9-4d85-9a57-5b9ad0e056c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00469.warc.gz"} |
Collection of Solved Problems
Work done by hydrogen
Task number: 1285
Hydrogen is diatomic gas, whose molar heat capacity at constant volume is
\[C_V =\frac{5}{2}R.\]
It fills a volume of 100 cm^3 at pressure of 51 kPa. Determine the work done by the gas when it expands its volume five times
a) isothermally,
b) adiabatically.
Note.: Consider hydrogen being an ideal gas.
• Notation
V[1] = 100 cm^3 = 100·10^−6 m^3 initial hydrogen volume
p[1] = 51 kPa = 51·10^3 Pa initial hydrogen pressure
V[2] = 5V[1] volume after expansion
\(C_V =\frac{5}{2}R\) hydrogen molar heat capacity at constant volume
W[i] = ? work done during the isothermal expansion
W[a] = ? work done during adiabatic expansion
• Hint
Think about how to calculate the work done by hydrogen when the pressure is function of the volume.
• Hint a) – Pressure Expression
To express the pressure p as a function of the volume V in the case of an isothermal process, the so-called Boyle-Mariotte law has to be applied.
• Analysis a)
The integral calculus must be used so that we can calculate the work of the hydrogen, because the pressure is a function of the volume.
The hydrogen pressure is expressed from the Boyle-Mariotte law, which is valid for isothermal processes. The obtained feature is integrated by the volume and the initial and the final hydrogen
volumes are used as limits of the integration.
• Solution a)
At constant pressure, the performed work W is given by W = p(V[2] − V[1]).
However, the pressure is not constant in our case, which means that more general relation is required
\[W_i = \int\limits_{V_1}^{V_2}p\, \text{d}V,\]
where V[1] is the initial hydrogen volume and V[2] is its volume after the expansion.
Therefore, the pressure p has to be formulated as a function of the volume V. An obvious consequence of the isothermal process state equation – the Boyle‑Mariotte law, can help with it. It
\[p_1V_1 = pV. \]
It is possible to express the pressure p immediately:
\[p = \frac{p_1V_1}{V}.\]
The next step is the actual integration
\[W_i = \int\limits_{V_1}^{V_2}p\, \text{d}V = \int\limits_{V_1}^{5V_1}\frac{p_1V_1}{V}\, \text{d}V =\]
Constants are factored out
\[=p_1V_1 \int\limits_{V_1}^{5V_1}\frac{1}{V}\, \text{d}V = \]
After integration and recall of the limits
\[=p_1V_1[\ln\,V]_{V_1}^{5V_1} = p_1V_1\,\ln \frac{5V_1}{V_1}= p_1V_1\,\ln 5.\]
• Hint b) – Pressure Expression
To express the pressure p as a function of the volume V in the case of an adiabatic process, the Poisson’s law has to be applied.
• Hint b) – Poisson
To calculate the Poisson’s ratio κ the following relation could be applied
\[\kappa = \frac{C_p}{C_V},\]
where C[p] is the molar heat capacity at constant pressure and C[V] is the molar heat capacity at constant volume. The so-called Mayer’s relation is held between the molar heat capacities.
where R is the molar gas constant.
• Analysis b)
As in the previous task section, the integral calculus must be used here so that we can find the work done by the hydrogen, because the pressure is a function of the volume.
This time, the hydrogen pressure is expressed by the Poisson’s law which is valid for adiabatic processes. The received function is integrated by the volume. The initial and the final volume are
used as integral limits.
Finally, it is necessary to evaluate the Poisson’s constant from the Mayer’s relation and the relation between the Poisson’s constant and the molar heat capacity at the constant pressure and the
constant volume.
• Solution b)
As a consequence of a variable pressure, the following equation is applied again to calculate the work W[a]:
\[W_a = \int\limits_{V_1}^{V_2}p \, \text{d}V,\]
where V[1] is the initial hydrogen volume and V[2] is the volume after the expansion.
The Poisson’s law for adiabatic (heat isolated) processes expresses the pressure p as a function of the volume V:
\[p_1V_{1}^{\kappa} = pV^{\kappa}. \]
It is possible to derive the pressure p immediately:
\[p = \frac{p_1V_1^{\kappa}}{V^{\kappa}}.\]
By combining the received formulation with the formula for work
\[W_a = \int\limits_{V_1}^{V_2}p \, \text{d}V = \int\limits_{V_1}^{5V_1}\frac{p_1V_1^{\kappa}}{V^{\kappa}} \, \text{d}V =\]
The constants are factored out of the integral
\[= p_1V_1^{\kappa} \int\limits_{V_1}^{5V_1}\frac{1}{V^{\kappa}} \, \text{d}V =\]
The integral is solved and the limits are recalled
\[= p_1V_1^{\kappa}\frac{1}{-\kappa + 1}\left[V^{-\kappa + 1}\right]_{V_1}^{5V_1}= \frac{p_1V_1^{\kappa}}{-\kappa + 1}\left[(5V_1)^{-\kappa + 1} - V_1^{-\kappa + 1}\right].\]
After simplification, we get
\[W_a = p_1V_1\,\frac{5^{\kappa-1}-1}{\kappa-1}.\]
However, the Poisson’s constant κ of the given gas is still unknown. To find it, its definitional formula is needed
\[\kappa = \frac{C_p}{C_V},\]
where C[p] means the molar heat capacity at constant pressure and C[V] the molar heat capacity at constant volume, as well as the Mayer’s law
\[C_p = C_V + R,\]
where R denotes the molar gas constant.
After substitution
\[\kappa = \frac{C_V + R}{C_V} = \frac{\frac{5}{2}R + R}{\frac{5}{2}R} = \frac{7}{5}.\]
The received value of Poisson’s constant κ is applied to the expression of the work
\[W_a = p_1V_1\,\frac{5\cdot(5^{2/5}-1)}{2}.\]
• Numerical Evaluation
a) The work for isothermal expansion
\[W_i = p_1V_1 \ln\,5 = 51\cdot{10^3}\cdot 100\cdot{10^{-6}}\cdot \ln\,5\, \,\mathrm{J} \dot{=} 8.21\, \mathrm{J}\]
b) The work for adiabatic expansion
\[W_a = p_1V_1\,\frac{5\cdot(5^{2/5}-1)}{2} = 51\cdot{10^3}\cdot 100\cdot{10^{-6}}\cdot \frac{5\cdot(5^{2/5}-1)}{2}\, \mathrm{J} \] \[W_a \dot{=} 6.05\, \mathrm{J}\]
• Answer
The gas does the work of approximately 8.21 J during the isothermal expansion and approximately 6.05 J during the adiabatic expansion. | {"url":"https://physicstasks.eu/1285/work-done-by-hydrogen","timestamp":"2024-11-11T15:58:24Z","content_type":"text/html","content_length":"34448","record_id":"<urn:uuid:dc0cdea2-e64e-4c96-88f3-d2478d28bed1>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00742.warc.gz"} |
Equation Defintion
In Mathematics, an equation shows that two expressions are equal to each other. An equation, in Algebra, can involve multiple variables, constants, and terms. For example, 2x + 4 = 9 is an equation
as it is showing that 2x + 4 and 9 (both are expressions) are equal to each other.
Representation of an equation
To represent equality between two terms or expressions, a sign of “=” is put between those terms. It means that “=” is the sign that represents the overall term as an equation.
For example, , if we have two expressions 5x + 7 and 2y + 11, and want to show them in the form of an equation, we will put “=” between them as shown below.
5x + 7 = 2y + 11
Parts of an equation
As mentioned above, an equation is a connection between different expressions. An expression is also made from different terms. It means that an equation has multiple parts involving those of an
expression too. Here are the major parts of an equation.
• Variable
• Term
• Operator
• Sign of equality
• Constant
An equation can involve any of these parts but the sign of equality will always be there as it is the basic condition.
Read about Operator here.
Types of equations
Depending on the degree of the variable, equations can be divided into three major types. Here we have enlisted those types with a brief discussion.
Linear equations: All those equations in which the maximum degree of the variable is “1” are called linear equations. For example, 2x + 4 = 9 is a linear equation because “x” has a maximum degree of
Read about Linear equations here.
Quadratic equations: The equations in which the maximum degree of variable is “2” will be termed quadratic equations. For example, 〖2x〗^2-3x+17=12 is a quadratic equation because the degree of “x”
is “2”.
Read about Quadratic equations here.
Cubic equations: Those equations in which the variable has a maximum degree of “3” are called cubic equations. For example, 〖12y〗^3 〖+ 2y〗^2-3y+127=162 is a cubic equation because the maximum
power of “y” is “3”.
Read about Cubic equations here.
Are expression and equation the same?
No, these are not the same. An expression is the collection of different terms without any equality or inequality sign. For example, 3x + 15 is an expression. On the other side, the equation shows
that the given expressions are equal to each other by putting a sign of equality between them.
How can we solve equations in Mathematics?
We can solve Mathematical equations by following all basic operators like addition, Subtraction, Division, and Multiplication. It depends on the operators involved in an equation and the degree of
the equation.
Fun Facts about Equations
Define an equation with an example.
An equation shows that two given expressions are equal to each other. For example, 3y + 17 = 67 is an equation it shows that two expressions are equal.
What are the three main types of equations?
Depending on the degree of the variable, equations have three major types. Here are those types:
• Linear equations
• Quadratic equations
• Cubic equations
Can we write the equation of a circle?
Yes, we can write an equation of a circle. In fact, all Geometrical diagrams have specific equations to get their solution. | {"url":"https://calculatorsbag.com/definitions/equation","timestamp":"2024-11-12T18:23:02Z","content_type":"text/html","content_length":"38975","record_id":"<urn:uuid:99f981c3-a0d4-4433-93bf-1e0ba1ccb61e>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00532.warc.gz"} |
Math Is Fun Forum
Replies: 10
wierd, i guess it can't be helped, well i'll just have to continue on like this but i will have the same quote i had previously, so it would probably be easier to recognize me and just to bring back
my old style i guess
weird, well i guess it can't be helped, well so now you know, this is my new name
i don't really know, i also tried searching but i didn't seem to find any usernames with cool in them to also relate to my posts and date of joining, now im thinking if it really was coolxxxxz, i
think it wasn't maybe, but i mean you or someone else should be able to remember my previous username, i mean can't you just look it up based on my posts or something like that.
Ans.1: i believe i was a registered member.
Ans.2: it was maybe coolxxxxz or za.... something
Ans.3: my old signature was from carl sagan it was "It pays to keep an open mind, but not so open your brains fall out."
Ans.4: maybe 10-20 posts
Ans.5: I probably made my last post between 2021 and 2023, if i could narrow it down then my last post would probably be in 2022
Ans.6: i think i last visited the forum in 2022 or 2023 with that account
Ans.7: i think i probably joined in the 2020s or maybe between 2019-2021
HI im back, although i've changed my name so you may not recognize me, i think my previous username was something like coolxxxxz, and i posted about proportion and something that was about how much
time it takes 3 people A,B and C to complete a certain task if A and B finish it in a certain time and B and C in a different time, i believe bob helped me solve them, well i'm just trying to jog
you're memory, anyways im back and hope to have just as as much of an enjoyable and educative time as before. | {"url":"https://mathisfunforum.com/search.php?action=show_user_posts&user_id=247318","timestamp":"2024-11-08T04:47:42Z","content_type":"application/xhtml+xml","content_length":"12299","record_id":"<urn:uuid:54459123-6171-445e-a877-1e16773aca1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00350.warc.gz"} |
Radu PrecupRadu Precup, Author at Tiberiu Popoviciu Institute of Numerical Analysis
In this paper, we give or improve compression-expansion results for set contractions in conical domains determined by balls or star convex sets. In the compression case, we use Potter’s idea of
proof, while the expansion case is reduced to the compression one by means of a change of variable. Finally, to illustrate the theory, we give an application to the initial value problem for a system
of implicit first order differential equations
Cristina Lois-Prados
Universidade de Santiago de Compostela, Santiago de Compostela, Spain
Radu Precup
Babes-Bolyai University, Cluj-Napoca, Romania
Rosana Rodríguez-López
Universidade de Santiago de Compostela, Santiago de Compostela, Spain
Compression-expansion fixed point theorem; set contraction; star convex set; implicit differential system
C. Lois-Prados, R. Precup, R. Rodríguez-López, Krasnosel’skii type compression-expansion fixed point theorem for set contractions and star convex sets, J. Fixed Point Theory Appl. 22 (2020), 63,
Journal of Fixed Point Theory and Applications
[1] Các, N.P., Gatica, J.A.: Fixed point theorems for mappings in ordered Banach spaces. J. Math. Anal. Appl. 71, 547–557 (1979), MathSciNet Article Google Scholar
[2] Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985), Book Google Scholar
[3] Dugundji, J., Granas, A.: Fixed Point Theory. Springer, New York (2013), MATH Google Scholar
[4] Erbe, L.H., Wang, H.: On the existence of positive solutions of ordinary differential equations. Proc. Am. Math. Soc. 120, 743–748 (1994), MathSciNet Article Google Scholar
[5] Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Springer, New York (2013), MATH Google Scholar
[6] Krasnosel’skii, M.A.: Fixed points of cone-compressing or cone-expanding operators. Sov. Math. Dokl. 1, 1285–1288 (1960), Google Scholar
[7] Krasnosel’skii, M.A.: Positive Solutions of Operator Equations. P. Noordhoff Ltd, The Netherlands (1964), MATH Google Scholar
[8] Lian, W.C., Wong, F.H., Yeh, C.C.: On the existence of positive solutions of nonlinear second order differential equations. Proc. Am. Math. Soc. 124, 1117–1126 (1996), MathSciNet Article Google
[9] Lois-Prados, C., Rodríguez-López, R.: A generalization of Krasnosel’skii compression fixed point theorem by using star-convex sets. Proc. Royal Soc., Edinburgh 150(1), 277–303 (2020)
[10] O’Regan, D., Precup, R.: Theorem of Leray–Schauder Type and Applications. Gordon and Breach, Singapore (2001), MATH Google Scholar
[11] O’Regan, D., Precup, R.: Compression-expansion fixed point theorem in two norms and applications. J. Math. Anal. Appl. 309, 383–391 (2005), MathSciNet Article Google Scholar
[12] Potter, A.J.B.: A fixed point theorem for positive k-set contractions. Proc. Edinb. Math. Soc. 19, 93–102 (1974), MathSciNet Article Google Scholar
[13] Precup, R.: Methods in Nonlinear Integral Equations. Kluwer Academic Publishers, Amsterdam (2002) Book Google Scholar
[14] Precup, R.: Positive solutions of semi-linear elliptic problems via Krasnosel’skii type theorems in cones and Harnack’s inequality. AIP Conf. Proc. 835, 125–132 (2006), Article Google Scholar
[15] Torres, P.J.: Existence of one-signed periodic solutions of second-order differential equations via a Krasnosel’skii fixed point theorem. J. Differ. Equ. 190, 643–662 (2003) Article Google
[16] Wang, H.: Positive periodic solutions of singular systems with a parameter. J. Differ. Equ. 249, 2986–3002 (2010), MathSciNet Article Google Scholar
[17] Zima, M.: Fixed point theorem of Legget–Williams type and its application. J. Math. Anal. Appl. 299, 254–260 (2004) MathSciNet Article Google Scholar
AbstractIn this paper, we give or improve compression-expansion results for set contractions in conical domains determined by balls or star… | {"url":"https://ictp.acad.ro/author/precup/page/8/","timestamp":"2024-11-09T22:14:57Z","content_type":"text/html","content_length":"138044","record_id":"<urn:uuid:6370b328-0747-4d04-bf32-cd4cb2965e1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00482.warc.gz"} |
7.1. Python interface
One may use the python interface of DeePMD-kit for model inference, an example is given as follows
from deepmd.infer import DeepPot
import numpy as np
dp = DeepPot("graph.pb")
coord = np.array([[1, 0, 0], [0, 0, 1.5], [1, 0, 3]]).reshape([1, -1])
cell = np.diag(10 * np.ones(3)).reshape([1, -1])
atype = [1, 0, 1]
e, f, v = dp.eval(coord, cell, atype)
where e, f and v are predicted energy, force and virial of the system, respectively.
Furthermore, one can use the python interface to calculate model deviation.
from deepmd.infer import calc_model_devi
from deepmd.infer import DeepPot as DP
import numpy as np
coord = np.array([[1, 0, 0], [0, 0, 1.5], [1, 0, 3]]).reshape([1, -1])
cell = np.diag(10 * np.ones(3)).reshape([1, -1])
atype = [1, 0, 1]
graphs = [DP("graph.000.pb"), DP("graph.001.pb")]
model_devi = calc_model_devi(coord, cell, atype, graphs)
Note that if the model inference or model deviation is performed cyclically, one should avoid calling the same model multiple times. Otherwise, tensorFlow will never release the memory and this may
lead to an out-of-memory (OOM) error.
7.1.1. External neighbor list algorithm
The native neighbor list algorithm of the DeePMD-kit is in \(O(N^2)\) complexity (\(N\) is the number of atoms). While this is not a problem for small systems that quantum methods can afford, the
large systems for molecular dynamics have slow performance. In this case, one may pass an external neighbor list that has lower complexity to DeepPot, once it is compatible with
import ase.neighborlist
neighbor_list = ase.neighborlist.NewPrimitiveNeighborList(
cutoffs=6, bothways=True, self_interaction=False
dp = DeepPot("graph.pb", neighbor_list=neighbor_list)
The update and build methods will be called by DeepPot, and first_neigh, pair_second, and offset_vec properties will be used. | {"url":"https://docs.deepmodeling.com/projects/deepmd/en/r2/inference/python.html","timestamp":"2024-11-12T08:32:30Z","content_type":"text/html","content_length":"21766","record_id":"<urn:uuid:4cbbfefb-ec75-4194-86f2-b9be6d3f69d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00667.warc.gz"} |
Numerical construction of nonsmooth control Lyapunov functions
Title data
Baier, Robert ; Braun, Philipp ; Grüne, Lars ; Kellett, Christopher M.:
Numerical construction of nonsmooth control Lyapunov functions.
Bayreuth; Newcastle, Australia , 2017 . - 30 S.
Format: PDF
Name: CLF_MILP_Baier_Braun_Gruene_Kellett_2017.pdf
Version: Preprint
Available under German copyright law. The document may be used free of charge for personal use. In addition, the reproduction, editing, distribution and any kind of exploitation require the
License written consent of the respective rights holder.
Download (732kB)
Project information
Project title: Project's official title
Project's id
Activating Lyapunov-Based Feedback - Nonsmooth Control Lyapunov Functions
Project financing: ARC (Australian Research Council)
Abstract Lyapunov’s second method is one of the most successful tools for analyzing stability properties of dynamical systems. If a control Lyapunov function is known, asymptotic stabilizability of
an equilibrium of the corresponding dynamical system can be concluded without the knowledge of an explicit solution of the dynamical system. Whereas necessary and sufficient conditions for the
existence of nonsmooth control Lyapunov functions are known by now, constructive methods to generate control Lyapunov functions for given dynamical systems are not known up to the same extent. In
this paper we build on previous work to compute (control) Lyapunov functions based on linear programming and mixed integer linear programming. In particular, we propose a mixed integer linear program
based on a discretization of the state space where a continuous piecewise affine control Lyapunov function can be recovered from the solution of the optimization problem. Different to previous work,
we incorporate a semiconcavity condition into the formulation of the optimization problem. Results of the proposed scheme are illustrated on the example of Artstein’s circles and on a two-dimensional
system with two inputs. The underlying optimization problems are solved in Gurobi.
Further data | {"url":"https://epub.uni-bayreuth.de/id/eprint/3409/","timestamp":"2024-11-11T10:37:24Z","content_type":"application/xhtml+xml","content_length":"33330","record_id":"<urn:uuid:d4776593-e977-4ad5-af89-e5b0daaff5cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00143.warc.gz"} |
Raise Your Decibel Awareness In Audio Measurements
In the radio-frequency (RF) microwave test and measurement world, engineers often deal with the power measurement unit of dBm instead of wattage (W). However, when entering the audio measurement
arena also need to understand the unit dBu, which is decibel (dB) relative to 1 mW into 600 Ω.
The decibel, in and of itself, is an often misunderstood unit of measurement. The “bel” in “decibel” derives from the name of Alexander Graham Bell. He was interested in how the human ear responds to
sound intensity. Bell used a logarithmic scale to express this sound intensity—its range from the softest sound to the loudest (threshold of pain) sound is one to a billion (10^12), or zero to 12
bels. The decibel is one-tenth of a “bel” and is abbreviated as dB.
Using dB can be beneficial on two key fronts. First, it concisely expresses very large or very small ratios; for example, +63 dB to –153 dB is more compact than 2 x 10^6 to 0.5 x 10^-15. Second, dB
simplifies the mathematic process when comparing quantities used to multiply the gain or divide the loss of several cascaded devices. Addition replaces multiplication of numeric gain, and subtraction
replaces division of numeric attenuation.
Defining The Decibel
A logarithmic unit, dB expresses the ratio of two quantities. In power measurement, the relative power is defined as:
In voltage measurement, the relative voltage is defined as:
To describe dB as an absolute value, a reference point must be known. A number of different reference points are possible:
• dBm represents the power level P1 with reference to 1 mW
• dBW represents the power level P1 with reference to 1 W
• dBV represents the voltage level V1 with reference to 1 V rms
• dBmV represents the voltage level V1 with reference to 1 mV rms
• dBµV represents the voltage level V1 with reference to 1 µV rms
The most commonly used unit in power measurement is dBm. For instance, if an engineer is working in a known industry-standard environment, test-system impedance usually equals 50 Ω in RF engineering,
75 Ω in television engineering, and 600 Ω in audio engineering. A conversion formula will help engineers to convert power measurement of dBm to any unit of dBV, dBmV, or dBµV.
For a 50-Ω system:
• dBV = dBm – 13 dB
• dBmV = dBm + 47 dB
• dBuV = dBm + 107 dB
For a 75-Ω system:
• dBV = dBm – 11.25 dB
• dBmV = dBm + 48.75 dB
• dBuV = dBm + 108.75 dB
For 600-Ω system:
• dBV = dBm – 2.22 dB
• dBmV = dBm + 57.78 dB
• dBuV = dBm + 117.78 dB
What Is dBu?
For most traditional test equipment, the source impedance uses only 50 Ω. However, audio test applications typically employ a 600-Ω source impedance. Audio test uses another decibel formula in the
unit of voltage measurement—dBu. It’s defined as dB relative to 1 mW into 600 Ω. This logarithmic unit expresses the relative voltage measurement with reference to a voltage value of 0.7746 V rms
(voltage drops across 600 Ω that results in 1 mW of power).
The dBm unit is defined as:
if a 600-Ω load results in 0 dBm. Therefore:
The “u” in dBu represents the word “unloaded.” It also implies that the load is un-terminated, or the load impedance is unspecified, and will likely be high. Thus, the 0.7746 V rms is an open circuit
Maximum Output Power
As mentioned earlier, 50 Ω is the most commonly used source impedance. A 50-Ω source impedance may result in higher short-circuit current (for a constant voltage), as well as 10 times the frequency
response over a given length of cable, than with 600 Ω-source impedance.
For example, Figures 1a-1d illustrate the maximum power transfer delivered by Agilent’s U8093A audio analyzer into various load-impedance scenarios using the source impedance of 50 Ω or 600 Ω. The
U8903A has an 8-V maximum voltage source for unbalanced output (V[S]).
(a). Scenario 2 shows the source as 50 Ω and load impedance as 600 Ω:
(b). In scenario 3, both source and load impedance is 600 Ω:
(c). The source is 600 Ω and load impedance measures 50 Ω in scenario 4:
Ins And Outs Of Output Voltage
Thanks to recent advances in DSP-based RF test equipment, some RF engineers are able to measure audio on RF instruments and then correlate the test results with other audio instruments. Sometimes,
though, engineers encounter problems with their RF signal analyzer when measuring two different supply sources that are identical in stimulus setup. One such example may occur between output
frequency (F[L]) and output voltage (V[L]). The RF signal analyzer receives very divergent measurement results that show both inputs are unequal in amplitude or bandwidth.
If an engineer sets the voltage output of an audio generator that comes with a fixed source impedance of 50 Ω (V[S] = 2 V, or 8.24 dBu), then its voltage will drop across at 50 Ω load impedance (VL =
1 V) (Fig. 2). Thus:
The engineer then may set up another output of an audio generator with source impedance of 600 Ω. To achieve output performance that’s similar to the previous 50-Ω system, a higher output voltage of
V’[S] = 13 V (24.5 dBu) must be set. Therefore, it also will deliver V’[L] = 1 V to the same 50-Ω load impedance (Fig. 3). Thus:
Converting dBu (in 50 Ω) to dBu (in 600 Ω) is a technique for verifying and confirming that the audio analyzer’s source impedance is the cause of the divergent measurement results in the RF signal
analyzer. As a rule of thumb, dBu (in 600 Ω) = dBu (in 50 Ω) + 16.26 dB.
The U8903A comes with a switchable source impedance of 50 Ω or 600 Ω. After verifying and confirming the root cause, all that’s required is a modification to the output source’s to obtain the
appropriate output voltage reading.
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CTAN update: nicematrix
Date: July 17, 2022 10:41:35 PM CEST
François Pantigny submitted an update to the nicematrix package. Version: 6.11 2022-07-16 License: lppl1.3 Summary description: Improve the typesetting of mathematical matrices with PGF Announcement
text: New key 'matrix/columns-type' to set the default type of column used in the matrices ({pNiceMatrix}, etc.). New key 'cccommand' in custom-line to create commands similar to '\cline' but with
different types of lines (dotted, dashed, etc.). The package’s Catalogue entry can be viewed at
The package’s files themselves can be inspected at
Thanks for the upload. For the CTAN Team Petra Rübe-Pugliese CTAN is run entirely by volunteers and supported by TeX user groups. Please join a user group or donate to one, see
nicematrix – Improve the typesetting of mathematical matrices with PGF
This package is based on the package array. It creates PGF/TikZ nodes under the cells of the array and uses these nodes to provide functionalities to construct tabulars, arrays and matrices.
Among the features :
• continuous dotted lines for the mathematical matrices;
• exterior rows and columns (so-called border matrices);
• control of the width of the columns;
• tools to color rows and columns with a good PDF result;
• blocks of cells;
• tabular notes;
• etc.
The package requires and loads l3keys2e, array, amsmath, pgfcore, and the module shapes of PGF.
Package nicematrix
Version 6.29b 2024-11-12
Copyright 2018–2024 F. Pantigny
Maintainer François Pantigny | {"url":"https://ctan.org/ctan-ann/id/mailman.279.1658090508.13845.ctan-ann@ctan.org","timestamp":"2024-11-15T03:29:58Z","content_type":"text/html","content_length":"16060","record_id":"<urn:uuid:b27b27ee-864d-448d-b862-a0d7042a0162>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00186.warc.gz"} |
Square and triangle
Can you find the area of the yellow square?
Student Solutions
The sides of the large right-angled triangle are both 6 cm, so its angles are 90$^\circ$, 45$^\circ$, and 45$^\circ$.
The angles in the square are all right angles, so the smaller triangles around the square are also right-angled isosceles triangles.
This means that the angles shaded green on the diagram are all 45$^\circ$, and the lengths labelled $k$ cm are all equal to the side length of the square.
This is important for all of the three methods shown below:
Using congruent triangles
Since the lines in the diagram are all horizontal, vertical or at $45^\circ$, it can be split up using horizontal and vertical lines as shown. Now all of the triangles are congruent to the smallest
triangle, at the bottom left corner of the original diagram.
There are 9 of these triangles altogether, and 4 of them fit into the square, so the area of the square is $\frac49$ of the area of the whole triangle.
The area of the whole triangle is $\frac12\times6\times6 = 18$ cm$^2$. So the area of the square is $\frac49$ of $18$ cm$^2$, which is $8$ cm$^2$.
Using scale factors
The diagonal side of the whole triangle is equal to 3$k$ cm, and the diagonal side of the small triangle in its corner (to the bottom left of the square) is $k$ cm. So the scale factor from the small
triangle to the whole triangle is 3.
This means that the sides of the small triangle are 2 cm, since 2 cm $\times$ 3 = 6 cm.
So the area of the small triangle is 2$\times$2$\div$2 = 2 cm$^2$.
The area of the whole triangle is 6$\times$6$\div$2 = 18 cm$^2$.
So the area of the square and the two triangles on either side of it is 18 $-$ 2 = 16 cm$^2$. Each of these two triangles is congruent to half of the square, so the square occupies half of this area.
So the area of the square is 8 cm$^2$.
Using Pythagoras' Theorem on the whole triangle
The hypotenuse of the triangle is equal to $3k$, and can also be found using Pythagoras' Theorem: $$6^2+6^2=(3k)^2\\\Rightarrow 72=9k^2\\
\Rightarrow 8=k^2$$ But $k^2$ is the area of the square. So the area of the square is $8$ cm$^2$. | {"url":"https://nrich.maths.org/problems/square-and-triangle","timestamp":"2024-11-13T18:45:29Z","content_type":"text/html","content_length":"41458","record_id":"<urn:uuid:563b7406-44da-483b-a526-3d275672b610>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00017.warc.gz"} |
When a circular plate is rotated at the bottom of a cylindrical vessel containing fluid, the fluid surface can form stable polygons under certain conditions. This phenomenon is very surprising, as it
seems that the rotational symmetry of the setup is mysteriously broken when polygons are formed. I experimentally measured the geometries of the formed polygons and studied their dependence on
parameters such as the plate rotational rate. Both the number of sides and the sizes of the polygons were found to increase when the plate rotated faster.
In my research, I sought to develop a theoretical model that could accurately predict the properties of the formed polygons in agreement with experimental results. Taking on a novel approach of
analysing the forces on a fluid particle travelling along the boundary of the polygon, the mechanism of polygon formation and the effect of the plate rotation on the polygons were physically
understood. Applying it quantitatively, with considerations on the properties of the fluid flow, a differential equation governing the polygon’s steady-state geometry was formulated. A high degree
of predictive power into the shape of the polygon was achieved, closely matching experimental data. | {"url":"https://siyss20.ungaforskare.se/participants/02-christopher","timestamp":"2024-11-04T09:05:38Z","content_type":"text/html","content_length":"5526","record_id":"<urn:uuid:83ec44a2-41f4-40c7-9a8e-319f4d775826>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00309.warc.gz"} |
What is the square of a triangle?
To find the area of a triangle, multiply the base by the height, and then divide by 2. The division by 2 comes from the fact that a parallelogram can be divided into 2 triangles.
Does 1/2 base times height work for all triangles?
The formula of Area = 1 / 2 bh works for all triangles, no matter what size or shape. As long as the height and base are known, this formula can be used to calculate the area. Here is an example of
using the formula to calculate the area of a triangle.
What is the height of a triangle?
The height of a triangle is the distance from the base to the highest point, and in a right triangle that will be found by the side adjoining the base at a right angle.
Can you fit a square into a triangle?
A side of the square must be parallel to the base of the triangle. Since the triangle is isosceles, the given base would also be equal to the height. Now in the diagonal part, we would always need an
extra length of 2 units in both height and base of the triangle to accommodate a triangle.
Why is area of a triangle?
Key intuition: A triangle is half as big as the rectangle that surrounds it, which is why the area of a triangle is one-half base times height.
Is the area of a triangle base times height?
There are several ways to compute the area of a triangle. For instance, there’s the basic formula that the area of a triangle is half the base times the height. This formula only works, of course,
when you know what the height of the triangle is.
How do you find the height of a triangle without the area?
Plug your values into the equation A=1/2bh and do the math. First multiply the base (b) by 1/2, then divide the area (A) by the product. The resulting value will be the height of your triangle!
How find the area of a right triangle?
Area of one right triangle = 1/2 × l × w. We usually represent the legs of the right-angled triangle as base and height. Thus, the formula for the area of a right triangle is, Area of a right
triangle = 1/2 × base × height.
Will 2 equilateral triangles make a square?
Two equilateral triangles are inscribed into a square as shown in the diagram. Their side lines cut the square into a quadrilateral and a few triangles….Equilateral Triangles and Incircles in a
sin 15° = UM / DM
cos 15° = (√6 + √2) / 4.
What are all the formulas for area of a triangle?
So, the area A of a triangle is given by the formula A=12bh where b is the base and h is the height of the triangle. Example: Find the area of the triangle. The area A of a triangle is given by the
formula A=12bh where b is the base and h is the height of the triangle. | {"url":"https://runyoncanyon-losangeles.com/blog/what-is-the-square-of-a-triangle/","timestamp":"2024-11-08T20:40:29Z","content_type":"text/html","content_length":"40157","record_id":"<urn:uuid:0c7af53a-5fd8-434d-a523-3989f75f676d>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00587.warc.gz"} |
A Rationale of Bhaskara and his Method for Solving
A Rationale of Bhaskara and his Method for Solving
ax ± c = by
Pradip Kumar Majumdar
Central Library, Calcutta University, Calcutta
(Received 10 January 1977 ; after revision 17 March 1977)
Indian Scholar Bhaskara I (522 A. D.) perhaps used the method of continued fraction to find out the integral solution of the indeterminate equation of the type by = ax — c. The paper presents the
original Sanskrit verses (in Roman Character) from Bhaskara I's Maha Bhaskaryia, its English translation with modern interpretation.
Bhaskara I (522 A. D.) gave a rule in his Mahabhaskariya for obtaining the general solution of the linear indeterminate equation of the type by = ax — c. This form seems to have chosen by Bhaskara I
deliberately so as to supplement the form of Aryabhata I. Smith1 following Kaye said that Aryabhata 1 attempted at a general solution of the linear indeterminate equation by the method of continued
fraction. In this paper we shall deduce the formula p[n]q[n-1] - q[n]p[n-1] = (-1)^n of the continued fraction from the Bhaskara I's method of solution of indeterminate equation of the first degree
and then we may draw the conclusion that the formula p[n]q[n-1] - q[n]p[n-1] = (-1)^n of the continued fraction was implicitly involved in the Bhaskara I's method of solution of the indeterminate
equation of first degree.
A Few Lines about the Continued Fraction
$a/b = a_1 + \frac{1}{a_2 + } \frac{1}{a_3 + } \cdots$
p[1]/q[1], p[2]/q[2];... , p[n]/q[n] ... be the successive convergents of a/b then
$p_1/q_1 = a_1$ ... (i)
$p_2/q_2 = \frac{a_1a_2 + 1}{a_2}$ ... (ii)
$p_3/q_3 = \frac{a_1(a_1a_2 + 1) + a_3}{a_2a_3 + 1}$ ... (iii)
$p_4/q_4 = \frac{a_1[a_2(a_3a_4 + 1) + a_4] + a_3a_4 + 1}{a_2(a_3a_4 + 1) + a_4}$ ... (iv)
$p_5/q_5 = \frac{a_1a_2a_3a_4a_5 + a_3a_4a_5 + a_1a_4a_5 + a_1a_2a_5 + a_1a_2a_3 + a_1a_2a_3 + a_5 + a_3 + a_1}{a_2a_2a_4a_5 + a_2a_5 + a_2a_5 + a_4a_5 + 1}$ ... (v)
and the following result will be easily obtained
p[n]q[n-1] - q[n]p[n-1] = (-1)^n
Bhaskara I's Rule
Bhaskara I (522 A. D) gave the following rule in his Maha Bhaskariya
bhajyam nyasedupari haramadhasca tasya
khandayatparasparamadho binidhaya labdham I
kena hato' yamapaniya jathasya sesam
bhagam dadati parisudhamiti pracintyam II 42 II
aptam matim tam binidhaya ballam
nityam hyadho'dhah kramasasca labdham I
matya hatam syaduparisthitam ya
llabdhena yuktam paratasca tadvat II 43 II
harena bhajyo bidhino paristho
bhajyena nityam tadadhah' sthitasca I
aharganosmin bhaganadayasca
tadva bhavedyasya samihitam yat II 44 II
Datta and Singh translate these Slokas as follows:
"Set down the dividend above and the divisor below. Write down successively the quotients of their mutual division, one below the other, in the form of a chain. Now find by what number the last
remainder should be multiplied, such that the product being subtracted by the (given) residue (of the revolution) will be exactly divisible (by the divisor corresponding to that remainder). Put down
that optional number below the chain and then the (new) quotient underneath. Then multiply the optional number by that quantity which stands just above it and add to the product the (new) quotient
(below). Proceed afterwards also in the same way. Divide the upper number (i.e. multiplier) obtained by this process by the divisor and the lower one by the dividend; the remainders will respectively
be the desired ahargana and the revolutions."
After translation Datta and Singh further said
"The equation contemplated in this rule is
$\frac{ax - c}{b} = \textrm { a positive integer.}$
This form of the equation seems to have been chosen by Bhaskara I deliberately so as to supplement the form Aryabhata I in which the interpolator is always made positive by necessary transposition.
Further b is taken to be greater than a, as is evident from the following rule. So the first quotient of mutual divisions of a and b is always zero. This has not been taken into consideration. Also
the number of quotients in the chain is taken to be even."
Rationale of the Rule
The equation is of the type ax — c = by ... (1)
where a = dividend, b = divisor, x = multiplier, y = quotient, remembering that a < b.
Now according to sloka we have.
$a = a_1b + a \\ b = a_2a + r_1 \\ a = a_3r_1 + r_2 \\ r_1 = a_4r_2 + r_3 \\ r_2 = a_5r_3 + r_4.$ ... (2)
Consider the even number of (partial) quotients, say four Remember that Datta and Singh said "... . So the first quotient of mutual division of a by b is always zero. This has not been taken into
consideration." Therefore a[5] is the even (partial) quotient.
Let t[1] = optional number.
$\textrm{Now } \frac{r_4t_1 - c}{r_3} = k_1,$
$t_1 = \frac{k_1r_3 + c}{r_4}.$
Consider the table
Here $s_1 = a_5t_1 + k_1 \\ \qquad = a_5 \left(\frac{k_1r_3 + c}{r_4}\right) + k_1\left[t_1 = \frac{k_1r_3 + c}{r_4}\right] \\ \qquad = \frac{k_1(a_5r_3 + r_4) + a_5c}{r_4} \\ \qquad = \frac{k_1r_2 +
a_5c}{r_4} \qquad [r_2 = a_5 r_3 + r_4]$
$s_2 = a_4s_1 + t_1 \\ \qquad = a_4 \left(\frac{k_1r_2 + a_5c}{r_4}\right) + \frac{k_1r_3 + c}{r_4} \\ \qquad = \frac{k_1(a_4r_2 + r_3) + c(a_4a_5 + 1)}{r_4} \\ \qquad \frac{k_1r_1 + c(a_4a_5 + 1)}
{r_4} \qquad [r_1 = a_4 r_2 + r_3]$
$s_3 = a_3s_2 + s_1 \\ \qquad = a_3 \left(\frac{k_1r_1 + c(a_4a_5 + 1)}{r_4}\right) + \frac{k_1r_2 + a_5c}{r_4} \\ \qquad = \frac{k_1(a_3r_1 + r_2) + c(a_3a_4a_5 + a_3 + a_5)}{r_4} \\ \qquad \frac
{k_1a + c(a_3a_4a_5 + a_3 + a_5)}{r_4} \qquad [a = a_3 r_1 + r_2]$
$L = a_2s_3 + s_2 \\ \qquad = a_2 \left(\frac{k_1a + c(a_3a_4a_5 + a_3 + a_5)}{r_4}\right) + \frac{k_1r_1 + c(a_4a_5 + 1)}{r_4} \\ \qquad = \frac{k_1(a_2a + r_1) + c(a_2a_3a_4a_5 + a_2a_3 + a_2a_5 +
a_4a_5 + 1)}{r_4} \\ \qquad = \frac{k_1b + c(a_2a_3a_4a_5 + a_2a_3 + a_2a_5 + a_4a_5 + 1)}{r_4} \qquad [b = a_2 a + r_1] \\ \qquad =\frac{k_1b + cq_5}{r_4} \textrm{ by (v) }$
$U = a_1L + s_3 \\ \qquad = \frac{a_1[k_1b + c(a_2a_3a_4a_5 + a_2a_3 + a_2a_5 + a_4a_5 + 1)]}{r_4} + \frac{k_1a + c(a_3a_4a_5 + a_3 + a_5)}{r_4} \\ \qquad = \frac{k_1(a_1b + a) + c[a_1a_2a_3a_4a_5 +
a_1a_2a_3 + a_1a_2a_5 + a_1a_4a_5 + a_3a_4a_5 + a_1 + a_3 + a_5]}{r_4} \\ \frac{k_1a + cp_5}{r_4} \qquad [a = a_1 b + a \textrm{ and by (v)}].$
$\frac{p_6}{q_6} = \frac{a}{b} \textrm{ and } \frac{L}{U} = \frac{k_1b + cq_5}{k_1a + cp_5}$
$p_6L - q_6U = p_6(k_1b + cq_5) - q_6(k_1a + cp_5) \\ \qquad \qquad \qquad = a(k_1b + cq_5) - b(k_1a + cp_5) \\ \qquad \qquad \qquad = k_1b + acq_5 - k_1ab - bcp_5 \\ \qquad \qquad \qquad = c(aq_5 -
bp_5) \\ \qquad \qquad \qquad = c(p_6q_5 - q_6p_5) \\ \qquad \qquad \qquad = c(-1)^6 \\ \qquad \qquad \qquad = c$
We have taken L = x, U = y
$p_6L - q_6U = c \\ p_6x - q_6y = c \\ \textrm{or, } ax - by = c \\ \textrm{or, } ax - c = by$
which is the original form ax — c = by.
Thus we see that the formula p[n]q[n-1] - q[n]p[n-1] = (-1)^n of the continued fraction is implicitly involved in the Bhaskara I's method of solution of the indeterminate equation of the first
Now let us take an example from the Ganita Sara Samgraha B of Mahavira. Mahavira says
drstvamrarasin pathiko jathaika
trimsatsamuham kurute trihinam
sese hrte saptativistrimisrai
rnarairvisudham kathayaikasamkham
Rangacharya translates this as follows: —
"A traveller sees heaps of mangoes (equal in numerical value) and makes 31 heaps less by 3 (fruits); and when the remainder (of these 31 heaps) is equally divided among 73 men, there is no remainder
(of these 31 heaps) is equally divided among 73 men, there is no remainder. Give out the numerical value of one (of these heaps)."
This gives us at once the following equation
73x = 31x - 3.
Take the even number of partial quotients say 2. (Here a[3] = 2nd partial quotient as Datta and Singh said".... So the the first quotient of mutual division of a and b is always. This has not been
taken into consideration).
Now according to Bhaskara I's rule we have
$\frac{9.t - 3}{11} \textrm{ where t is the optional number }$
take t = 4, then k[1] = 3.
Consider the Valli (table)
Ans x = 26.
The author expresses his gratitude to Prof. M. C. Chaki and Dr. A. K. Bag for their kind suggestions and guidance for presentation of this paper. Thanks are due to the referee for his comments
towards the improvement of the paper. | {"url":"https://www.math10.com/en/maths-history/math-history-in-india/a_rationale_of_bhaskara/a_rationale_of_bhaskara.html","timestamp":"2024-11-14T10:47:50Z","content_type":"application/xhtml+xml","content_length":"25605","record_id":"<urn:uuid:cf0a648d-4764-43ea-9aa0-4f4d2e312c11>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00825.warc.gz"} |
Multiplication Chart To 13 Thirteen Times Table Chart Free To Print | Multiplication Chart Printable
Multiplication Chart To 13 Thirteen Times Table Chart Free To Print
Multiplication Chart To 13 Thirteen Times Table Chart Free To Print
Multiplication Chart To 13 Thirteen Times Table Chart Free To Print – A Multiplication Chart is a handy tool for children to discover just how to increase, split, and also discover the smallest
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well as a variety of adorable layouts. Some are vertical, while others include a horizontal style. You can also discover worksheet printables that include multiplication formulas and math truths.
Multiplication charts and also tables are indispensable tools for kids’s education. You can download and print them to use as a mentor help in your youngster’s homeschool or classroom. You can
likewise laminate them for sturdiness. These charts are fantastic for use in homeschool math binders or as class posters. They’re specifically valuable for youngsters in the 2nd, 3rd, as well as
fourth qualities.
A Multiplication Chart Printable 1-13 is a helpful tool to strengthen math realities and also can help a kid discover multiplication swiftly. It’s additionally a great tool for skip counting and
discovering the moments tables.
Related For Multiplication Chart Printable 1-13 | {"url":"https://multiplicationchart-printable.com/multiplication-chart-printable-1-13/multiplication-chart-to-13-thirteen-times-table-chart-free-to-print/","timestamp":"2024-11-07T03:21:42Z","content_type":"text/html","content_length":"27162","record_id":"<urn:uuid:18f6de9f-b414-4e67-9a7d-ad7354352525>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00034.warc.gz"} |
Picking Winners in Daily Fantasy Sports Using Integer Programming
We consider the problem of selecting a portfolio of entries of fixed cardinality for contests with top-heavy payoff structures, i.e. most of the winnings go to the top-ranked entries. This framework
is general and can be used to model a variety of problems, such as movie studios selecting movies to produce, venture capital firms picking start-up companies to invest in, or individuals selecting
lineups for daily fantasy sports contests, which is the example we focus on here. We model the portfolio selection task as a combinatorial optimization problem with a submodular objective function,
which is given by the probability of at least one entry winning. We then show that this probability can be approximated using only pairwise marginal probabilities of the entries winning when there is
a certain structure on their joint distribution. We consider a model where the entries are jointly Gaussian random variables and present a closed form approximation to the objective function.
Building on this, we then consider a scenario where the entries are given by sums of constrained resources and present an integer programming formulation to construct the entries. Our formulation
uses principles based on our theoretical analysis to construct entries: we maximize the expected score of an entry subject to a lower bound on its variance and an upper bound on its correlation with
previously constructed entries. To demonstrate the effectiveness of our integer programming approach, we apply it to daily fantasy sports contests that have top-heavy payoff structures. We find that
our approach performs well in practice. Using our integer programming approach, we are able to rank in the top-ten multiple times in hockey and baseball contests with thousands of competing entries.
Our approach can easily be extended to other problems with constrained resources and a top-heavy payoff structure.
arXiv e-prints
Pub Date:
April 2016
□ Statistics - Other Statistics | {"url":"https://ui.adsabs.harvard.edu/abs/2016arXiv160401455H/abstract","timestamp":"2024-11-01T21:13:32Z","content_type":"text/html","content_length":"41220","record_id":"<urn:uuid:f2663942-d1ae-444b-b6e6-c10a81d73e7d>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00270.warc.gz"} |
The sound transmission through a single panel can be approximated to a good degree of accuracy knowing only the mass per unit area (m) and the Modulus of Elasticity (E) of the panel. At low and mid
frequencies the transmission loss (TL) is calculated from the well known mass law.
This predicts TL = 20 log (mf) - 48 dB (1)
At higher frequencies the coincidence effect reduces the sound transmission and the transmission loss is given by R = 20log(mf ) -10log(2hw /pwc) - 47
Where wc is the critical circular frequency (which can be calculated from E) and h is the loss factor.
For thick heavy panels such as Brick or Concrete additional transmission takes place due to shear waves and the TL at high frequencies is reduced. INSUL takes this effect into account.
At low frequencies the radiation efficiency of a finite sized partition is reduced and the measured transmission loss is greater than the simple mass law. This effect is more pronounced for elements
such as windows which are often tested with small areas. However even for normal tests carried out to ISO 140 with an area of 10-12m² the effect is significant at the lowest test frequencies. INSUL
can take account of this effect.
A further effect that must be accounted for are the panel modes for light weight timber or steel framed walls. This is especially important for stud spacings of 400mm (16") or less. The STC rating
is especially sensitive to this as the first panel mode can often fall in or very close to the 125 Hz 1/3 octave band, and the STC rating can be determined completely by this band due to the 8 dB
rule. Reductions of 10 to 13 dB in STC rating can occur by changing stud spacings from 24" to 16".
INSUL predicts the transmission loss of double panel systems in 4 different frequency regions.
region 1
At low frequencies the transmission loss is determined primarily by the mass law. The TL increases at 6 dB/octave but INSUL can account for the inefficient radiation of low frequencies (link to
section on single panels).
region 2
Above the mass-air-mass resonance frequency of the partition (fo) determined by the mass of the panels and the air gap, the TL increases at 18 dB/octave as the two sides become decoupled.
region 3
When the cavity width becomes comparable to a wavelength at frequency fl the cavity modes couple the panels together and the TL increases at 12 dB/octave.
region 4
Solid connections act as sound bridges between the two panels and the TL is limited to a constant amount above the mass law, and increases at only 6 dB/octave
If you are interested in reading more about the theoretical background that INSUL is based on, the following references provide an excellent introduction.
1. B.H.Sharp, Prediction Methods for the Sound Transmission of Building Elements. Noise Control Engineering Vol 11 1978
2. L.Cremer M.Heckel E.E.Ungar, Structureborne Sound (Springer Verlag,1988)
3. F.Fahy, Sound and Structural Vibration (Academic Press, 1985)
4. J.H. Rindel, Sound Radiation form Building Structures and Acoustical Properties of Thick Plates. COMETT-SAVOIR Course Notes, CSTB Grenoble | {"url":"https://sp.insul.co.nz/informaci%C3%B3n-t%C3%A9cnica/","timestamp":"2024-11-03T12:03:12Z","content_type":"text/html","content_length":"12110","record_id":"<urn:uuid:659aac69-d451-4ece-a6c9-bf44f5ebaa80>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00436.warc.gz"} |
Monitoring Deformation along Railway Systems Combining Multi-Temporal InSAR and LiDAR Data
College of Surveying and Geo-Informatics, Tongji University, Shanghai 200092, China
Department of Geoscience and Remote Sensing, Delft University of Technology, 2628 CN Delft, The Netherlands
University of Twente, 7500 AE Enschede, The Netherlands
Author to whom correspondence should be addressed.
Submission received: 28 August 2019 / Revised: 23 September 2019 / Accepted: 30 September 2019 / Published: 2 October 2019
Multi-temporal interferometric synthetic aperture radar (MT-InSAR) can be applied to monitor the structural health of infrastructure such as railways, bridges, and highways. However, for the
successful interpretation of the observed deformation within a structure, or between structures, it is imperative to associate a radar scatterer unambiguously with an actual physical object.
Unfortunately, the limited positioning accuracy of the radar scatterers hampers this attribution, which limits the applicability of MT-InSAR. In this study, we propose an approach for health
monitoring of railway system combining MT-InSAR and LiDAR (laser scanning) data. An amplitude-augmented interferometric processing approach is applied to extract continuously coherent scatterers
(CCS) and temporary coherent scatterers (TCS), and estimate the parameters of interest. Based on the 3D confidence ellipsoid and a decorrelation transformation, all radar scatterers are linked to
points in the point cloud and their coordinates are corrected as well. Additionally, several quality metrics defined using both the covariance matrix and the radar geometry are introduced to evaluate
the results. Experimental results show that most radar scatterers match well with laser points and that LiDAR data are valuable as auxiliary data to classify the radar scatterers.
1. Introduction
Satellite-based differential interferometric synthetic aperture radar (DInSAR) is a standard geodetic technology for deformation monitoring over wide areas with millimeter accuracy [
]. Multi-temporal interferometric synthetic aperture (MT-InSAR) approaches are used to reduce the atmospheric signal delays and decorrelation noise in DInSAR [
]. Based on a set of co-registered radar acquisitions, coherent scatterers are identified and their deformation time series are estimated [
]. Several studies have shown the potential of MT-InSAR for the observation of (line-)infrastructure, such as dams, dikes, tunnels, roads, highways and railways [
Railway systems consist of a complex collection of constructions, such as embankments, tunnels and bridges, subject to changing environmental conditions (geology, relief). As a result, several
processes impact the structural health of these networks, depending on their locations. Examples are the differential subsidence of assets in soft soils, slope instabilities/slow landslides in
mountainous areas, embankment instabilities, and aging and degradation of concrete constructions. Due to the foundation and construction of a railway section, several processes may occur on a very
local scale. For example, in soft soils, the embankment with the rails may show a different deformation behavior compared to the catenary poles. Significant differential settlements have been
observed in the transition zones relative to fixed structures [
]. Current approaches for structural health monitoring are levelling, linear variable differential transformers and video based systems [
]. While the latter can be used to monitor dynamic displacements [
], their applicability is limited due to manual operation and localized implementation. MT-InSAR is complementary to these in situ techniques and has the advantage of wide area applications, frequent
revisits, and a millimeter level precision.
For a proper analysis and interpretation of MT-InSAR products, the locations of the coherent scatterers (CS) need to be known with at least decimeter level precision. Unfortunately, whereas the
relative displacements with MT-InSAR is estimated with millimeter-level precisions, the positioning precision of radar scatterers is usually poor, in the order of meters [
]. As a consequence, it is difficult to link the radar scatterers to the ground objects, which hampers the interpretation of the deformation signal and limits the applicability of MT-InSAR. The
positioning accuracy of CS is dependent on: (i) factors influencing all CS systematically; and (ii) factors specific for each individual CS [
]. The largest systematic uncertainty is introduced by the unknown absolute height of the reference CS. If a corner reflector or radar transponder is available for the whole time series, the
reference height offset can be estimated by measuring its position [
]. However, often such a device is not available. Airborne LiDAR provides 3D point clouds with very high spatial density, thus LiDAR points can be found close to all radar scatterers, which makes it
attractive to estimate the systematic MT-InSAR height offset based on the full CS dataset [
The individual CS positioning precision is dependent on the sub-pixel positionand the relative height of the scatterers [
]. The precision with which these parameters can be estimated depends on the SAR mission characteristics, e.g., spatial resolution and the orbital tube dimensions. For each CS, the uncertainty in the
position is described by a 3D positioning error ellipsoid [
]. Van Natijne A and Hanssen [
] introduced an approach to use the position error ellipsoids to snap the CS point cloud to a LiDAR point cloud. This way, the positioning accuracy of the CS is improved and the CS are linked to
physical objects. The snapping procedure also enables adding attributes to the CS, such as the type of object that it represents. The attribution could be based on existing attributes in the LiDAR
dataset, or on an intersection with auxiliary data sources.
Combining an improved positioning of CS with their attribution, the state of the railway infrastructure can be assessed. Both CS originating from the railway infrastructure, as well as CS from the
direct surroundings are of interest. Often, only scatterers which remain highly coherent over the entire time period are considered, called continuously coherent scatterers (CCS). As the time series
lengthen, scatterers may only remain coherent during parts of the time period, referred to as temporary coherent scatterers (TCS) [
]. These scatterers are widely distributed over urban construction areas [
]. To optimally exploit the information content of the MT-InSAR dataset, analyses with an adaptive temporal window are desirable.
Because of the regular maintenance of the railway systems and the subsurface characteristics, the measured deformation may be a superposition of different deformation regimes, for example: (i)
long-term settlement; (ii) seasonal shrinking and swelling; and (iii) potential anomalies [
]. To estimate and distinguish different deformation regimes, a proper parameterization of the deformation is required. For example, previous research has shown that temperature [
], possibly in combination with rainfall [
], is a good proxy for sub-seasonal deformation assessment. This applies both for concrete/metal constructions, as well as for embankments. For the selection of the most suitable deformation model at
a certain location, a hypothesis testing scheme can be used [
]. In addition, by using various quality metrics, a proper selection of the CS can be made to retrieve the valuable information for a railway network.
Here, we propose an optimized process for deformation monitoring along railway networks combining radar scatterers and LiDAR point clouds. In
Section 2
, we briefly describe the process of MT-InSAR with both CCS and TCS, including parameter estimation and precision, absolute height correction, snapping to LiDAR, and quality metrics.
Section 3
demonstrates the approach based on railway sections in the Netherlands, using RadarSAT-2 SAR and LiDAR data, and analyzes and discusses the performed experiments. The conclusions follow in
Section 4
2. Methodology
2.1. Mt-Insar Process
In MT-InSAR, the basic observations are the differential interferometric phases between two scatterers, denoted as an arc. We estimate the residual height and velocity using a time series analysis. A
thermal dilation parameter is introduced to describe the variations of interferometric phase with temperature since thermal dilation often happens along the railway due to its steel structure [
]. Considering
$m − 1$
differential interferograms from
SAR images, the unwrapped phase difference between two scatterers
of a single arc in the
th interferogram can be expressed as [
$Δ ϕ i , j k = C i , j − 4 π λ B ⊥ , i k R i sin θ i Δ h i , j − 4 π λ B t k Δ v i , j − 4 π λ B T k Δ K i , j + 2 π n i , j k + e i , j k ,$
$Δ h i , j$
$Δ v i , j$
$Δ K i , j$
denote the residual height difference, the velocity difference and the thermal dilation difference between the two scatterers;
$n i , j k ∈ Z$
denotes the integer phase ambiguity;
$B t k$
$B ⊥ , i k$
$B T k$
are the temporal, perpendicular and thermal baseline, respectively;
$R i$
is the slant range,
$θ i$
is the local incidence angle and
is the radar wavelength;
$C i , j$
denotes the phase constant that corresponds to the atmospheric delay difference in the master image; and
$e i , j k$
denotes the random error of the phase, including the atmospheric delay difference in the slave image. Then, the Integer Least Squares (ILS) model of CCS and TCS is defined as [
$E { Δ ϕ i , j t start ⋮ Δ ϕ i , j t stop } = 2 π 0 0 0 ⋱ 0 0 0 2 π n i , j t start ⋮ n i , j t stop − 4 π λ B ⊥ , i t start R i sin θ i B t t start B T t start ⋮ ⋮ ⋮ B ⊥ , i t stop R i sin θ i B t t
stop B T t stop Δ h i , j Δ v i , j Δ K i , j + C i , j ,$
$t start$
$t stop$
are the start and stop times obtained from amplitude time series change detection [
]. Note that, for CCS, the start and stop times are equal to the first and last epoch. It is worth noting that the validation of the ambiguity resolution is tested by the a likelihood-ratio test [
]. Parameters of all arcs are estimated using a least-squares approach and checked based on temporal coherence [
]. After getting the arc solutions, we can estimate the parameters of all scatterers by integration of all arc solutions. Since the design matrix of the network is rank defect, conventionally, a
reference point is selected to resolve this. In fact, the reference is an arbitrary choice, as long as the full covariance matrix of the network solution is considered [
]. Here, we prefer to use the pseudo-inverse for the network solution instead of choosing a reference point. This way, the obtained solution is equivalent to using the average parameter value of all
scatterers as reference. For example, the heights
of the scatterers are estimated by
$h ^ = ( B T Q Δ h ^ − 1 B ) + B T Q Δ h ^ − 1 Δ h ^ ,$
denotes the design matrix related to the network [
$Δ h ^$
denotes the estimated differential height of all arcs.
$( · ) +$
denotes the pseudo-inverse and is solved by a fast algorithm [
$Q Δ h ^$
is the Covariance (CV) matrix related to the quality of the arc solutions, which is defined as
$Q Δ h ^ = diag ( σ Δ h ^ 1 2 , … , σ Δ h ^ n 2 ) ,$
$σ Δ h ^ 2$
denotes the variance of the estimated height and
denotes the number of accepted arcs. The term diag(·) denotes the diagonal elements of the matrix. The height precision of all scatterers can be estimated as [
$D { h ^ } = ( B T Q Δ h ^ − 1 B ) + ( B T Q Δ h ^ − 1 B ) ( B T Q Δ h ^ − 1 B ) + .$
Similarly, we also estimate deformation velocities and thermal dilations of all scatterers, as well as their precision. Based on the estimated reference network, we conduct a densification of the CCS
and incorporate the TCS in the final result. Details can be found in [
]. The displacement time series of all scatterers are generated following the conventional PS-InSAR methodology, separating nonlinear deformation from atmospheric delay by a spatiotemporal filter [
2.2. Attribution of the Insar Observations
In this section, we introduce a standard approach to link the radar scatterers to points in a LiDAR point cloud using the error ellipsoid. This approach includes three steps: absolute height
correction, estimating the error ellipsoid, and snapping, see the flowchart in
Figure 1
2.2.1. Absolute Height Correction
For a scatterer located at
$P → ( x p , y p , z p )$
in terrestrial coordinates with a corresponding zero Doppler coordinate
$( r , t )$
, we obtain the position state vector
$S → ( t )$
and velocity vector
$V → ( t )$
of the satellite using azimuth time
. The range time is indicated by
. The position
$P →$
is now determined by solving three equations, called Doppler–Range–Ellipsoid equations, which are defined as [
$V → ( t ) · ( P → − S → ( t ) ) = 0 ;$
$| P → − S → ( t ) | − r P = 0 ; and$
$x p 2 ( H + a ) 2 + y p 2 ( H + a ) 2 + z p 2 ( H + b ) 2 − 1 = 0 ,$
denote the semi-major and semi-minor axis of the reference ellipsoid, respectively.
$r P$
represents the distance from scatterer
to the satellite and
is the height relative to the reference ellipsoid.
Because the phase observations are wrapped, the absolute phase difference with respect to the ellipsoid cannot be determined, leading to a coordinate offset for all scatterers. Depending on the
position of the scatterer in the image, which leads to a different local incidence angle, the estimated horizontal coordinate offset varies over the image. Since the estimated heights of all
scatterers are related to a specified reference, we propose a solution search method to estimate the height offset with the help of grid data obtained by LiDAR point cloud. In each search, we add an
initial height offset to the heights of all scatterers and update their coordinates. Then, the heights of corresponding LiDAR point cloud are extracted using the new coordinates of all radar
scatterers. Furthermore, to evaluate the similarity between the heights of radar scatterers and that of point cloud, we calculate the Pearson correlation coefficient [
], which is defined as
$ρ = 1 N − 1 ∑ i = 1 N H radar , i − μ H radar σ H radar H lidar , i − μ H lidar σ H lidar ,$
is the number of radar scatterers, and
denote the standard deviation and the average, respectively.
$H radar$
denotes the height of the radar scatterers while
$H lidar$
denotes that of the LiDAR point cloud. Pearson’s correlation coefficient is used to evaluate the similar tendency of two datasets, irrespective of the absolute value of the difference.
Setting an initial search interval and search step for the height offset, we repeat the calculation and obtain the corresponding correlations. In the end, the height offset candidate is located at
maximal correlation. To improve the efficiency and obtain a result with high precision, the solution search approach is conducted several times with different search steps. For example, in the first
search, we set a loose search interval and the search step is set to 1 m. In the following search, the initial height offset is used to determine a smaller search interval and the step is set to a
smaller value as well. We repeat the process until the search step satisfies with the desired precision.
To estimate the height offset, it is not required to have LiDAR over the entire area covered by the radar, a small region may suffice, which also decreases the computational burden. Note that our
matching method is expected to result in a height offset estimation that has a higher precision than that by single point correction, such as using GNSS or levelling.
2.2.2. Generating the Positioning Error Ellipsoid
The uncertainty in the position of the scatterers is determined by the covariance matrix, and the 3D error ellipsoid is the geometric representation of the covariance matrix. In radar coordinates,
the position of a scatterer is described using the range, azimuth and cross-range coordinates [
], denoted as
$( r , t , c )$
. The variance of the sub-position in azimuth and range is obtained using the estimated SCR [
$σ r , P 2 = σ t , P 2 = 3 2 π 2 SCR ^ P + 1 12 Δ 2 ,$
denotes the oversampling factor, which is set to 1 in our study. Based on the work of Adam et al. [
], the temporal
$SCR ^$
is defined using the normalized amplitude dispersion index
$D A$
], described as
$D A = σ A μ A ; S C R ^ = 1 2 D A 2 .$
The variance of the position along the cross-range direction can be obtained by the height precision described in
Section 2.1
and the radar geometry, which is described as
$σ c , P = σ h , P / sin θ$
. Hence, the VC matrix of the position in radar coordinate is defined as
$Q r t c = diag [ σ r 2 , σ t 2 , σ c 2 ] .$
With the estimated absolute height offset, we obtain the corrected ground coordinates of all scatterers. After the geocoding step, we obtain scatterers in both the radar and ground coordinates and
the datum transformation matrix
is obtained by the S-transformation [
]. Furthermore, the VC matrix of the position of a scatterer in ground coordinates can be obtained using the propagation law of variances as
$Q x y h = R · Q r t c · R T = σ x 2 σ x y 2 σ x h 2 σ y 2 σ y h 2 σ h 2 ,$
where the elements of the VC matrix are the variance and covariance in ground coordinates, denoted by
$( x , y , h )$
. Based on this VC matrix, the error ellipsoid can be generated for each scatterer. The corrected coordinates of the scatterers estimated by the absolute heights are used as the center of the error
ellipsoid. Setting a significance level
, the size of the error ellipsoid, i.e. the three semi-axis lengths of the ellipsoid, are obtained by the eigenvalues of
$Q x y h$
2.2.3. Snapping to the Point Cloud
The LiDAR point cloud is used to correct the locations of the radar scatterers and to add properties to them. It is worth noting that most coherent radar scatterers are related to man-made
structures, such as buildings, bridges, and railways, while the LiDAR point cloud contains all kinds of geo-objects. However, some part of point clouds, e.g. vegetation and water, should be removed,
since these objects cannot provide coherent scatterers.
A nearest neighbor search process with respect to the radar geometry estimation is used to snap the scatterers to their most likely point in the point cloud [
]. Considering the full covariance matrix of each radar scatterer, a Whitening transform is adopted to decorrelate the dimensions of the data coordinates. Given a data matrix
with the VC matrix
, two matrices
are obtained using the eigenvalue decomposition
is eigenvector matrix and
is a diagonal matrix whose diagonal elements are eigenvalues. Thus, we can transform the original data matrix
into a new data matrix
In the new coordinates, the unit of Euclidean distance is
rather than meters, so the errors in different dimensions exhibit a normal distribution. Thus, the ellipsoids become spheres and the closest point, in Euclidian distance, is the most likely one.
Additionally, a
d tree search algorithm [
] with a time complexity of
$O ( log n )$
is conducted to search the nearest neighbor. This algorithm is faster when the number of points is large.
After establishing the relationship between the point cloud and radar scatterers, the coordinates as well as the attributes of the point cloud are assigned to those of the radar scatterers.
2.3. Quality Metrics
During the estimation in the MT-InSAR process, we derive the precision of the parameters. Here, we summarize the quality metrics for assessing the performance using MT-InSAR considering both the
deformation time series and the radar geometry.
2.3.1. Temporal Coherence
The temporal coherence estimator is an indicator for evaluating the deviation between the deformation time series and the estimated deformation model, which is defined as [
$γ ^ = 1 m ∑ i = 1 m e j ( ϕ def i − ϕ model i ) ,$
is the number of SAR images,
$ϕ def i$
is the phase component related to the displacement including modeled and un-modeled deformation and
$ϕ model i$
is the model phase. The coherence ranges between 0 and 1. Low coherence indicates large unmodeled deformation and/or large phase noise.
2.3.2. Dilution of Precision
Dilution of Precision (DoP) describes the geometric contribution to the quality of the parameters, which is defined using the covariance matrix of the Line of Sight (LOS) vector decomposition.
Considering the application of railway monitoring, the displacement vector
$d ^ asset ( d T , d L , d N )$
in a local, asset-fixed, right-handed Cartesian coordinate system, as defined in [
], is introduced. These coordinates describe the deformation in transversal, longitudinal, and normal directions, respectively. Longitudinal direction is along the rail track, while the transversal
direction represents the cross-track direction. The normal direction is orthogonal to the transversal-longitudinal plane (see [
A LOS vector decomposition can be conducted if at least three LOS observations with different viewing geometries are available, for the same object. If this condition cannot be met, optional
constraints may be introduced, e.g. the assumption that deformation in a specific direction can be neglected. Under this constraint, we construct the covariance matrix with a different number of LOS
• One track.
If only one LOS observation is available, we may decide to evaluate only the projection of the deformation vector onto the normal direction, assuming that the longitudinal and transversal
directions may be negligible. Here, we introduce pseudo-observations
$d L$
$d T$
, which are set to zero. Supposing that
$R trans$
denote the transformation matrix from local coordinate to ground coordinate [
], the relationship between the displacement vector and LOS observation is defined as
$d L O S d T d L = 0 0 cos θ 0 1 0 0 0 1 R trans d T d L d N = A d asset .$
• Two tracks.
If two LOS observations are available, we may decide to assume that deformation in the longitudinal direction is negligible, by using a pseudo-observation
$d L$
to be equal to zero. Then, the relationship between the displacement vector and LOS observations is defined as
$d L O S 1 d L O S 2 d L = − sin θ 1 cos α 1 0 cos θ 1 − sin θ 2 cos α 2 0 cos θ 2 0 1 0 R trans d T d L d N = A d asset ,$
is the flight azimuth angle (heading) of the satellite.
• Three or more tracks.
If at least three LOS observations are available, the LOS decomposition can be solved directly, as long as the viewing geometries are significantly different. The relationship between the
displacement vector and LOS observations is defined as
$d L O S 1 d L O S 2 ⋮ d L O S n = − sin θ 1 cos α 1 sin θ 1 sin α 1 cos θ 1 − sin θ 2 cos α 2 sin θ 2 sin α 2 cos θ 2 ⋮ ⋮ ⋮ − sin θ n cos α n sin θ n sin α n cos θ n R trans [ d T d L d N ] = A
d asset .$
The variance matrix of the displacement vector is obtained using the error propagation law
$Q d ^ asset = ( A T Q d A ) − 1 ,$
denotes the design matrix and
$Q d$
is the VC matrix of the LOS observations.
The DoP is obtained using covariance matrix of the displacement vector [
$DoP = ( det ( Q d ^ asset ) ) 1 2 n ,$
$det ( · )$
is the determinant operator. A smaller DoP value indicates a higher quality of the parameters.
2.3.3. Sensitivity
Sensitivity is used to assess whether a deformation is observable with the LOS observations, which is defined as the modulus of the inner product [
denotes the LOS unit vector from the scatterer to the satellite. This indicator ranges between 0 and 1. A sensitivity of 1 shows the geometric quality of the LOS deformation is optimal while a
sensitivity of 0 means the deformation cannot be detected using the LOS measurements.
3. Results and Discussion
Subsequently, we discuss the used data sources for the area of interest, the estimation results, coordinate corrections and point classification, and the analysis of the results.
3.1. Data Resources
The amplitude-augmented interferometric processing is demonstrated using 48 RadarSAT-2 XF images acquired between March 2015 and August 2018 over Zaltbommel, The Netherlands. The selected dataset
covers 10 km of railway and a buffer zone of 2 km width. The slant-range and azimuth pixel spacings are 2.66 m and 2.47 m, respectively. An external digital elevation models (DEM) is not needed due
to the lack of significant topography. However, the residual topographic phase of each coherent scatterer is estimated in the time series analysis. The temporal baseline range is 870 days, while the
spatial baseline range is 280 m. Temperatures are recorded per hour and interpolated to the time of the SAR acquisition [
The used airborne LiDAR product, Actueel Hoogtebestand Nederland 3 (AHN3), covering the entire territory of the Netherlands [
], contains both a DEM and a Digital Surface Model (DSM) with a point density of 12.7 pts/m
and a grid of 0.5 m × 0.5 m with an elevation accuracy of less than 5 cm systematic and 5 cm stochastic. Given this point spacing, an object of 2 m × 2 m has an error of maximum 25 cm, which is less
than one quarter of a pixel in the image resolution of RadarSAT-2 XF. Furthermore, five classes (i.e., ground, building, water, civil structure, and unclassified) are included in the the newest
version of AHN3 data, which were updated in 2019. In this case, the AHN3 point cloud covers the selected railway within a buffer zone of 500 m width. All software used for InSAR time series analysis
and classification was coded in MATLAB.
3.2. Radar Observations along the Railway
Figure 2
a shows the AHN3 point cloud along the railway with classification and
Figure 2
e denotes the location of the railway. Both CCS and TCS are processed and the final result maps contain about
$100 × 10 3$
CCS and
$25 × 10 3$
TCS (see
Figure 2
b–d). Since the number of CCS is much larger than the number of TCS, apparently most areas did not change during the acquired time. The deformation velocities range from
$− 12$
$+ 5$
mm/a, and the scatterers along the railway are homogeneously distributed. The velocity map shows that the settlement along the north of the railway is larger than that along the south. The heights
range from 0 to
$+ 40$
m and the thermal dilation range from
$− 1$
$+ 0.5$
mm/K. Since thermal dilation depends on the material of the radar scatterer, it only shows a limited variation over the area. With the help of the TCS, we significantly improve the point density and
extract more information using the radar observations.
The precision of the estimated parameters is obtained, and the histograms of the estimated height and deformation velocities are shown in
Figure 3
. The precision of the estimated height
$σ h$
is used to generate the error ellipsoid (cf.
Section 2.2.2
) while that of velocity is used to calculate the DoP (cf.
Section 2.3.2
Figure 2
c shows that some scatterers are strongly related to the temperature with a maximum thermal dilation of more than 0.5 mm/K. Two scatterers are selected and their displacement time series are shown in
Figure 4
. After removing the thermal dilation phase, the displacement time series becomes smoother with decreasing RMS and it is easier to detect phase anomalies.
3.3. Coordinate Correction and Classification
During the absolute height correction with the LiDAR data, the iterative search process is repeated three times, leading to a height offset estimate with centimeter precision. This height offset
results in an additional horizontal offset, as shown by comparing the radar scatterers (red points) with the Lidar point cloud (see
Figure 5
a). After the height correction (cf.
Figure 5
b), the radar scatterers align with the Lidar points indicating infrastructure.
Setting the significance level
to 0.005, we generated the error ellipsoids for all scatterers. The semi-axis length of the ellipsoid in the cross-range direction is much larger than the lengths of those of range and azimuth in
radar coordinates. Subsequently, we applied a coordinate decorrelation and snapped the radar scatterers to the point cloud [
]. This process was conducted per point. Thus, a parallel processing approach can be adopted to improve the search efficiency. During the process, we discarded the scatterers that are not associated
with any point in the cloud. Finally, we snapped 94% of the radar scatterers to a new position. The 3D height map of all radar scatterers with corrected coordinates is shown in
Figure 6
Figure 6
a shows the same scatterers with the initial coordinates. Here, we project the geodetic coordinate [
] using RDNAPTRANS into the Dutch National Triangulation system RD and vertical NAP, denoted as RDNAP, which is referred to as ground coordinates in the following. In
Figure 6
a,b, it is more convenient to associate the radar scatterers with ground objects.
The classification of the LiDAR point cloud is transferred to the corresponding radar scatterers (see
Figure 7
). There are four classes in our result, i.e. bridge, ground, building, and unclassified, e.g., the catenary poles along the railway.
Figure 8
shows the parameters of the scatterers with corrected coordinates. The scatterers on the bridge are very stable while those on the ground exhibit higher settlements (north area indicated by the black
arrow) (see
Figure 7
Figure 8
a). The thermal dilations on the bridge are larger than those of scatterers on the ground, which supports our classification (see
Figure 8
The quality metrics described in
Section 2.3
are calculated for all scatterers. For 96% of the scatterers, the ensemble coherences are larger than 0.75, which implies that the deformation time series derived from the MT-InSAR process has a high
precision, and that the model fits generally well.Considering the DoP and sensitivity, we use Equation (
) of one LOS observation for the calculation.
Figure 9
shows the DoP values of all scatterers, ranging between 0.1 and 0.5, where a larger DoP value indicates a lower quality. Most scatterers have a DoP of approximately 0.25. Since the orientation of the
selected railway section is north–south, the sensitivity of all scatterers is comparable. The sensitivities of most scatterers are around 0.57, indicating a reasonable capability of deformation
detection (see
Section 2.3.3
3.4. Comparison and Analysis
Based on the classification (see
Figure 8
a), scatterers within a specified class can be extracted for further analysis. For example, if we are only interested in the deformation of the ground and the bridge, scatterers with other
classifications can be removed. The deformation velocity of the selected scatterers is shown in
Figure 10
. Compared to
Figure 8
a, this shows that the classification leads to a successful isolation of the scatterers representative of the railway track. Additionally,
Figure 11
shows the histograms of the deformation velocity within different classes. Scatterers from different classes exhibit a different distribution. For example, scatterers on buildings are relatively
stable with deformation velocities ranging from
$− 5$
mm/a to
$+ 5$
mm/a, while other scatterers (on the ground or on catenary poles) show more variable deformation rates with maximum rates exceeding 10 mm/a. Thus, the classification improves the interpretation
relative to the original mixed deformation signal.
Three segments of the railway are selected to show more detail. We compare the height map of radar scatterers with that of the LiDAR point cloud (see
Figure 12
). In
Figure 12
a,b, the density of radar scatterers is high enough to show the structure of the railway. Comparing
Figure 12
d,f, the density of the scatterers along the second and third segment is different. The second segment of the railway is east–west with a scatterer density of 0.51 pt/m
while the third is north–south with a scatterer density of 0.72 pt/m
, which means that the density of coherent scatterers depends on the orientation of the railway, relative to the satellite heading.
Figure 13
shows the deformation velocity and thermal dilation of scatterers from bridge and ground. The deformation velocity of scatterers on the bridge is smaller than that of scatterers on the ground, while
the thermal dilation of the scatterers on the bridge is larger than that of scatterers on the ground. In addition, the thermal dilation of the scatterers on the bridge arches are also greater than
that of scatterers on the bridge deck. Thus, all results support the classification results.
In the LiDAR point clouds, there is a class of “unclassified” points including trees, power lines and other geo-objects. Some objects, such as catenary poles, are coherent scatterers while others
(vegetation) can never be coherent scatterers because of the leaf cover, at least part of the year. Therefore, we evaluate whether this class should be used in our classification (see
Figure 14
). In
Figure 14
a,c, some scatterers in the unclassified group are related to the poles, which are important to evaluate the state of the railway. In
Figure 14
b,d, some scatterers related to the power lines are misclassified as scatterers on the ground and a few scatterers are missing if we neglect the unclassified group.
Figure 15
shows the classification maps of the other two segments. We also found many scatterers related to the catenary poles along the railway. Therefore, it is necessary to include the “unclassified” group
during the classification.
4. Conclusions
While MT-InSAR is a useful tool for structural health monitoring of large structures, there are limitations in the interpretation of the data due to the imperfect attribution of radar scatterers to
physical objects. This is mainly due to the relative nature of the elevation estimates and the limited positioning accuracy. With the help of LiDAR data, we can overcome these limitations and
increase the value of the MT-InSAR results. A structural health monitoring approach for railway systems is proposed and demonstrated combining MT-InSAR and LiDAR point clouds. A case study using
RadarSAT-2 XF data over the Netherlands was conducted. A novel amplitude-augmented interferometric processing approach, including temperature coefficients, demonstrates a good point density of radar
observations. Quality metrics such as the ensemble coherence, the DoP value, the sensitivity and the covariance matrix of all radar scatterers are calculated given the radar and infrastructure
geometry. This way railway asset managers can select and interpret observations based on different combinations of the quality metrics. Results of snapping radar scatterers to the corresponding laser
points using the 3D confidence ellipsoid show that LiDAR can not only be used to improve the positioning of the scatterers but to classify the scatterers as well. The classification enables the
isolation of specific types of radar scatterers, leading to an improved interpretation of the deformation signal. With the help of the classification, it is easier to interpret the deformation
signals, in particular over transition zones. If more datasets are available, an increasing number of attributes can yield more details in the classification.
Author Contributions
F.H., F.J.v.L., L.C., J.W. and R.F.H. conceived and designed the experiments; F.H. processed the InSAR data and LiDAR point cloud; and F.H. and F.J.v.L. wrote the main manuscript.
This research was funded in part by the China Scholarship Council (201706260149), the State Key Development Program for Basic Research of China (No.2013CB733304) and the National Nature Science
Foundation of China (No.41674003).
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1. Flowchart of the attribution of the InSAR observations using LiDAR data. There are three main steps: (1) absolute height correction; (2) error ellipsoid estimation; and (3) snapping to the
LiDAR point cloud.
Figure 2. (a) AHN3 point cloud along the railway and its buffer zone with classifications. Estimated parameters on both CCS and TCS: (b) velocity map; (c) height map; (d) thermal dilation; and map. (
e) Location of the railway (green line).
Figure 4. Displacement time series of two selected scatterers with strong thermal dilation: (a) the RMS of the displacement drops from 4.2 to 1.0 mm; and (b) the RMS of the displacement drops from
5.2 to 2.3 mm. Green dot line, temperature per acquisition; red line, linear deformation time series; blue line, deformation time series without temperature correction including linear deformation,
nonlinear deformation, temperature motion and noise; black line, temperature-corrected deformation time series including linear deformation, nonlinear deformation and noise.
Figure 5. Distribution of the radar scatterers (red dots) relative to the LiDAR data (height-colored dots): (a) coordinates of radar scatterers before height offset correction; and (b) coordinates of
radar scatterers after height offset correction.
Figure 6. 3D map of radar scatterers (a) without and (b) with coordinate correction by LiDAR, showing a better separability of high and low objects.
Figure 7. Classification of the radar scatterers with corrected coordinates based on their attribute to the LiDAR points, which are already classified.
Figure 8. Parameters of the scatterers with corrected coordinates: (a) deformation velocity, demonstrating the instability of the north area; and (b) thermal dilation, showing the strong thermal
dilation of the bridge structure.
Figure 9. DoP values of radar scatterers with correct coordinates, showing the qualities of different scatterers.
Figure 10. Deformation velocity of the scatterers with selected classifications (bridge and ground), indicating the deformation related to the railway.
Figure 11. Histograms of the deformation velocity within different classes, showing a better interpretation of deformation signals with classified scatterers.
Figure 12. (a–f) Comparison of the height model between LiDAR data and radar scatterers. The first column corresponds to the LiDAR data and the second column corresponds to the radar scatterers. Each
row corresponds to a selected segment of the railway.
Figure 13. Parameters of the selected bridge: (a) velocity map, showing the transitions of velocity between bridge and ground; and (b) thermal dilation map, showing the strong thermal dilation on the
bridge structure.
Figure 14. Comparison of classification map: (a,c) classification maps of LiDAR; (b,d) classification maps of radar scatterers; (a,b) classification maps without unclassified one; and (c,d)
classification maps with unclassified one.
Figure 15. Classification maps of two selected areas: (a,c) classification maps of LiDAR; and (b,d) classification maps of radar scatterers, showing the importance of including the unclassified
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
Share and Cite
MDPI and ACS Style
Hu, F.; Leijen, F.J.v.; Chang, L.; Wu, J.; Hanssen, R.F. Monitoring Deformation along Railway Systems Combining Multi-Temporal InSAR and LiDAR Data. Remote Sens. 2019, 11, 2298. https://doi.org/
AMA Style
Hu F, Leijen FJv, Chang L, Wu J, Hanssen RF. Monitoring Deformation along Railway Systems Combining Multi-Temporal InSAR and LiDAR Data. Remote Sensing. 2019; 11(19):2298. https://doi.org/10.3390/
Chicago/Turabian Style
Hu, Fengming, Freek J. van Leijen, Ling Chang, Jicang Wu, and Ramon F. Hanssen. 2019. "Monitoring Deformation along Railway Systems Combining Multi-Temporal InSAR and LiDAR Data" Remote Sensing 11,
no. 19: 2298. https://doi.org/10.3390/rs11192298
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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python – Add legend to scatter plot (PCA)
To add a legend to a scatter plot in Python using PCA, you can follow these steps:
Step 1: Import the necessary libraries
import numpy as np
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
Step 2: Generate random example data
For this example, let’s generate random dataset with 2-dimensional data and 3 different classes:
# Generate random data
n_samples = 100
n_classes = 3
# Create class labels
labels = np.arange(n_classes)
# Generate random data for each class
data = []
for label in labels:
# Generate random samples based on the class label
random_data = np.random.randn(n_samples, 2) + label * np.array([2, 2])
Step 3: Combine and preprocess the data for PCA
Next, we need to concatenate all the data from different classes into a single array and then preprocess it for PCA:
# Combine the data from different classes
data = np.concatenate(data)
# Standardize the data
data_std = (data - np.mean(data, axis=0)) / np.std(data, axis=0)
Step 4: Perform PCA on the preprocessed data
Use the PCA class from scikit-learn to perform PCA on the preprocessed data:
# Create PCA object
pca = PCA(n_components=2)
# Fit and transform the data
data_pca = pca.fit_transform(data_std)
Step 5: Plot the scatter plot with legend
Finally, plot the scatter plot with a different color for each class, and include a legend:
# Create a scatter plot for each class
for label in labels:
plt.scatter(data_pca[labels == label, 0], data_pca[labels == label, 1], label=f'Class {label}')
# Add a legend
# Set axis labels
plt.xlabel('Principal Component 1')
plt.ylabel('Principal Component 2')
# Set a title
plt.title('Scatter Plot with Legend')
# Show the plot
Here’s an example that puts it all together:
import numpy as np
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
# Generate random data
n_samples = 100
n_classes = 3
labels = np.arange(n_classes)
data = []
for label in labels:
random_data = np.random.randn(n_samples, 2) + label * np.array([2, 2])
# Combine the data from different classes
data = np.concatenate(data)
# Standardize the data
data_std = (data - np.mean(data, axis=0)) / np.std(data, axis=0)
# Create PCA object
pca = PCA(n_components=2)
# Fit and transform the data
data_pca = pca.fit_transform(data_std)
# Create a scatter plot for each class
for label in labels:
plt.scatter(data_pca[labels == label, 0], data_pca[labels == label, 1], label=f'Class {label}')
# Add a legend
# Set axis labels
plt.xlabel('Principal Component 1')
plt.ylabel('Principal Component 2')
# Set a title
plt.title('Scatter Plot with Legend')
# Show the plot
This code will generate a scatter plot with a legend, where each class is represented by a different color. The legend displays the class labels (“Class 0”, “Class 1”, “Class 2”). The x-axis
represents the first principal component, and the y-axis represents the second principal component. | {"url":"https://pythonkb.com/python-add-legend-to-scatter-plot-pca/","timestamp":"2024-11-03T22:14:02Z","content_type":"text/html","content_length":"72656","record_id":"<urn:uuid:30c98949-360b-4033-870f-28468717ac13>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00728.warc.gz"} |
Why proof by pattern of examples doesn’t work
Draw a few points on a circle and then draw a straight line from every point to every other point. Count the number of regions created.
2 points, 2 regions.
3 points, 4 regions.
4 points, 8 regions.
5 points, 16 regions.
6 points? See this post from the f(t) blog for the answer.
19 thoughts on “Why proof by pattern of examples doesn’t work”
1. You bring up a fascinating philosophical question — the problem of induction, one of the great problems of (at least Western) philosophy. I think it is fascinating because there is no way to
reason from first principles that inductive reasoning should apply, and yet all of science (by which I mean the experimental variety, as opposed to theoretical) relies on it utterly. Moreover we
all personally depend on it constantly, and life or existence without being able to rely on it is difficult to imagine. It is also the basis for eliciting prior information in Bayesian
I suspect the answer to this riddle is deep indeed.
Besides being an example where inductive reasoning fails, your observation is a good example of why I hate the “What’s the next (or missing) value in the sequence of integers?” puzzles, or
problems on IQ tests. They are of course incorrigably ill-posed problems, except perhaps trivially by supplying the answer while posing the problem, or constraining the possible answers so that
only one number fits the constraints. Anyway, while they can be constructed to minimize the difficulty in most cases, sometimes the constructor does not even know that there is a difficulty. This
seems especially the case in elementary school work.
2. I thought that at one time I sat through a rigorous proof of why proof by induction was valid. Like in an undergrad number theory class. I could be mistaken though.
3. The problem here isn’t induction in the mathematical sense. It’s an example of induction in the colloquial sense of extrapolating a pattern based on tuition.
Say we conjecture that n points leads to 2^(n-1) regions. A rigorous mathematical proof would establish the theorem for a base case, say n = 2. Next it would prove that if the theorem holds for
any m >= 2 then it also holds for m+1. But in this case, the inductive step isn’t true and so it cannot be proven and neither can our conjecture.
4. Well sure, what’s called “proof by induction” in math is fine.
The kind of induction I was writing about is concluding that because you have cracked open 100 eggs and seen a yolk inside, the next one you crack open will have a yolk inside.
Unfortunately I don’t have a broad enough vocabulary to know a term that distinguishes this reasoning from mathematical induction. Similarly, although I could write that there is no logical
reason to believe the 101st egg will have a yolk inside it (and this is true) some folks would read “logical” as “sensible” and come to the opposite conclusion.
For some reason I find it difficult to discuss — probably my paucity of terms with unambiguous meanings. Also, inductive reasoning is so ingrained our consciousness, science, even the behavior of
living organisms (in as much as they can be held capable of belief, expectation, or reasoning; if anthropomorphised their behavior clearly reflects it) that it is difficult to percieve it itself.
This means that unless someone has thought about it previously, they are unlikely to be persuaded in a short conversation that it is different from deductive reasoning, present company excluded
of course :-)
5. JV: I agree with your skepticism regarding test questions of the form “What number is next in this sequence?” I hated those questions in school. I wanted to justify my answer, which of course you
can’t do on a multiple choice test on a bubble form. If I were grading such an exam, I would reward someone who could give a good defense for an unusual answer. Of course standardized tests
punish such original thought.
6. [N.B. This and my previous comment are directed primarly at Kate Nowak, even though it looks like I am commenting incoherently on John’s comment :-)]
Maybe this article will help. I haven’t read it thoroughly but it seems to be on the right track. Note in particular the bit at the end about mathematical induction.
7. John: Absolutely. I think that the only reasonable way to ask those number-seuqence problems is to treat them like essay questions, where the quality of your argument is being assessed rather
than the conclusion you draw. When you get down to it, the number-sequence problems are asking an opinion, like what the best ice cream flavor is. In that case most folks would get the right
answer if the choices were appropriately limited, even though in principle there is no one correct answer.
8. Hey John, you were wondering a while back about a better term than Bayesian for Bayesian statistics. How about referring to frequentist statistics as ‘deductive statistics’ and Bayesian
statsitics as ‘inductive statistics’ :-D
9. The two types of induction are sometimes called mathematical induction and statistical induction. There are several types of mathematical induction (weak induction, strong induction, transfinite
induction, etc.) and all of them are special types of deductive reasoning. Statistical induction is not deductive. Often types of reasoning are carved up into four classes: deductive, inductive,
abductive, and analogical. See:
It can be argued that inductive, deductive, and abductive reasoning are all based on a foundation of analogical reasoning:
10. There are many varieties of multi-valued logic:
Multi-valued logics are all types of deductive logic. Other types of deductive logic are temporal or tense logic (the logic of time — and there are many different logics of time), modal logic
(the logic of possibility and necessity — and there are many different modal logics), intuitionist logic (denies the law of the excluded middle), relevance logic (the logic of relevant
implication — many varieties), and so on.
11. Thanks Peter!
It reminds me of what I am told regarding the Syadvada system of Jain logic, where there are seven logical states: true, false, true or false, indeterminate, true or indeterminate, false or
indeterminate, true or false or indeterminate.
We tend to assume that ways of thinking which seem fundamental to us are universal. From what I can tell, this is far from accurate.
I was told that a certain evangelical organization contracted with the Gallup organization to prove that regardless of particular religion or culture, most people believe in some kind of God. The
evangelicals knew this to be true, but wanted evidence to present to skeptics. The Gallup folks polled the Japanese, and found that a majority didn’t believe in any sort of God.
12. Fascinating! In classical Hindu philosophy they likewise have eight or so methods of proof, IIRC.
When I was a kid I tried to construct a three-valued logic system for a science fiction story I was writing. One of the hypotheses for the story was that we (as humans) use two-valued logic and
generally binary ways of thinking because we are bilaterally symmetric. Assuming that, I wondered, how would sentient beings who were trilaretally symmetric construct their understanding of the
world? I decided to call the truth states red, yellow, and blue to avoid bias. I was merrily constructing logical operation tables when I heard that this had already been done, for arbitrary (n)
values of truth. I lost interest after that.
I guess the philosophy of logic is much deeper and broader than I thought!
13. Why does proof induction work? Why should it work at all? It seems like induction is the same sort of reasoning – although I do see the difference I dont understand what makes induction so much
stronger. How does one actually prove proof by induction yields a valid result? As I understand it in my own research, it is simply regarded as an axiom of mathematical reasoning.
14. Proof by induction is just “finding a pattern”… the same sort of fallacy youre referring to in your post.
15. Cogito: Mathematical induction is not the same as induction in the colloquial sense. A proof by mathematical induction ultimately reduces to mathematical axioms just as any other proof. It’s a
perfectly rigorous technique.
My complaint about mathematical induction is that sometimes it establishes the truth of a theorem without giving much intuition for why the theorem holds or how one might extend it.
16. @John – I don’t think you understand my point though. Let me ask you this… how do you prove that proof by induction works? I cant seem to find a proof for the technique. Most so-called
“mathematicians” simply say its an axiom of mathematical reasoning. I don’t see why it should work as a general technique in every case.
Frankly, I don’t agree with your definition of the word “induction” used in this post. More like “intuition” in many respects.
Don’t get me wrong, your post demonstrated prejudice perfectly… an assumption about the truth from a few observations. But is a hypothesis ever wrong? Its valid, just as any postulate is, until
finally proven or disproven.
I’m not disputing that “induction” from observation can be wrong… I just don’t perceive any real difference between it done here in a mathematical context and it done elsewhere in a mathematical
context. Sometimes you call it invalid and sometimes not. But here, it took a contradictory example for you to declare with certainty that the induced theorem was wrong. Now, when induction is
used “validly” in other contexts… how do we know its right? No one can prove how or why induction works in the first place. And you certainly cannot go through every possible example to check
that they all work.
I understand the technique perfectly. My problem is “why induction works”. Because if you cant prove to me why it works except to demonstrate the few cases where it does then how do we know there
aren’t stipulations and conditions for its use? How do we know that there aren’t entire classes of problems for which proof by induction simply will not work?
No offence… and I know that I am not articulating my question very well… but you are trivializing my inquiry by hiding behind your relative education instead of appreciating the conceptual depth
I believe my question has. As most people do, I might add, when I ask a question. Very rarely does someone ever admit to me that my question has insight or implication, very rare does anyone ever
truly grasp my question, and then they are usually stumped at answering. Then again, maybe I really am misconceiving the notion… but considering as to how I use it and have been using the
technique for many years, I doubt I am stumbling over the basics here. There is a certain philosophy to what I’m asking. Not just blind mathematical rigor, for whatever that is really worth.
My biggest pet peeve about proof by induction is that it only works for integer increments of a particular argument. You cannot generalize it to all real numbers, or all complex numbers.
Typically only the whole numbers are involved.
17. A standard mathematical proof by induction is equivalent to a proof by well-ordering. If we accept that the natural numbers are well-ordered (which is an axiom), the validity mathematical
induction on the natural numbers immediately follows.
18. I remember participating in a conversation in college in which a friend and I convinced ourselves that either inductive reasoning works or we are just doomed (because if the future won’t resemble
the past we can’t get an empirical handle on the world at all). Of course, the world could operate consistently up to a point and then “stop operating consistently” (in the sense of doing
something we had not previously observed) and doom us — this is Taleb’s Black Swan. But the general idea was that in a world in which epistemology can’t be grounded in inductive empiricism, it
probably can’t be grounded at all.
19. The first time I came across this point I was shocked. I think it is good to keep in the back of your mind, but of course the only reasonable choice is to behave as if the sun will come up
tomorrow just like it did today ;-). | {"url":"https://www.johndcook.com/blog/2009/04/21/why-proof-by-pattern-of-examples-doesnt-work/","timestamp":"2024-11-09T06:38:14Z","content_type":"text/html","content_length":"87756","record_id":"<urn:uuid:3ad341af-b706-4647-a416-fea204470438>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00057.warc.gz"} |
A note on Breda-Robertson's conjecture
Avelino, Catarina P.; Santos, Altino F.
Journal of Pure and Applied Mathematics: Advances and Applications , 7(2) (2012), 73-82
The continuous deformation of any spherical isometric folding into the standard spherical folding, fs, defined by fs (x, y, z) = (x, y, z ), remains an open problem since 1989. We show that this
conjecture is restricted to the class of primitive foldings and it is exhibited a spherical folding within this class,
where the difficulty of deformation is evidenced. | {"url":"https://cemat.tecnico.ulisboa.pt/document.php?member_id=168&doc_id=2097","timestamp":"2024-11-14T16:37:44Z","content_type":"text/html","content_length":"8572","record_id":"<urn:uuid:0b5abb03-4e37-40fe-928f-789c2a5d5ad5>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00532.warc.gz"} |
Pulsar Search
Next: Further Reading Up: Pulsar Observations Previous: Pulsar Timing Studies Contents
At the end, we come to the observation and analysis techniques used for discovering new pulsars. Pulsar searches fall into one of two broad categories : targeted and untargeted searches. In an
untargeted search (or survey) for pulsars, the idea is to uniformly cover a large area of the sky with a desired sensitivity in flux level. In targeted searches, one is searching a limited area of
the sky where there is a higher than normal possibility of finding a pulsar (for example, the region in and around a supernova remnant or a steep spectrum point source identified in mapping studies).
Here some of the parameters of the search can be tailored to suit the a priori knowledge about the search region.
For a pulsar survey, the choice of (i) the range of directions to search in, (ii) the frequency of observations, (iii) the bandwidth and number of spectral channels, (iv) the sampling interval and
(v) the duration of the observations are some of the critical items that need to be chosen carefully. The choice of these parameters is interlinked in many cases.
Analysis of pulsar search data is an extremely compute intensive task. For each position in the sky for which data is recorded, the analysis technique needs to search for the presence of a periodic
signal in the presence of system noise. However, from the discussion in section 3, it is clear that if appropriate dispersion correction is not done, the sensitivity to the presence of a periodic
signal can be reduced significantly. Since a pulsar can be located at any distance (and hence DM) along a given direction in the sky, the search has to be carried out in (at least) two dimensions :
DM and period. For this, the data is dedispersed for different trial dispersion measures. For each choice of DM, the dedispersed data is search for a periodic signal.
To reduce the computational load for search data analysis, several optimised algorithms are used. For example, when dedispersing for a range of DM values, it is possible to use the results from the
computations for some DM values to compute part of the results for some other DM values. This saves a lot of redundant calculations. This method, known as Taylor's Dedispersion Algorithm, is used
quite often. Similarly, there are optimised techniques for searching for periodic signals in the presence of noise. The simplest method is to fold the dedispersed data for each choice of possible
period and examine the resulting profile for the presence of a significant peak that is well above the noise level. Once again, computations done for folding at a given period can be used for folding
at other periods. This redundancy is exploited by the Fast Folding Algorithm. A signal containing a periodic train of pulses gives a well defined signature in the Fourier domain - its spectrum
consists of peaks at the frequency corresponding to the periodicity, and harmonics thereof. It can be shown that it is possible to detect the periodic signal by searching for harmonically related
peaks in the spectral domain. It turns out that it is more economical to implement the FFT followed by harmonic search technique compared to the folding search techniques.
Additional complications are introduced in the search algorithm when one allows the parameter space to cover pulsars in binary orbits as the period can actually change during the interval of
observation. Special processing techniques are needed to handle such requirements.
Next: Further Reading Up: Pulsar Observations Previous: Pulsar Timing Studies Contents NCRA-TIFR | {"url":"https://www.gmrt.ncra.tifr.res.in/doc/WEBLF/LFRA/node159.html","timestamp":"2024-11-13T12:55:01Z","content_type":"text/html","content_length":"6712","record_id":"<urn:uuid:c0e54dcd-866b-48cf-bfad-11bf6d1668de>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00262.warc.gz"} |
Who can help me with my R programming matrices homework? | Pay Someone To Take My R Programming Assignment
Who can help me with my R programming matrices homework? I need to understand how I can model or print arithmetical logic matrices that can help students solve a number of tasks to solve linear
algebra and acyclerose two-dimensional problem. One thing can help me? How to draw R values sequentially? My R programming matrices are a bit different and I have to face something I am not familiar
with so I will try to clarify: – I have R values for the next node of the matrix. Does that make sense to me? – Now what did you say, rephrase the matrices matrices for easy context making a simple R
value easy to draw? – so is working with simple matrices matrices correct for my purpose? – Now I am trying to make a simple R value sequentially for me and to make it easier in my R programming, so
I built your code accordingly. Thank you. – Bye A: I would suggest you to try to understand the above to make your very basic R values easy to draw. Also I have the code attached that suggests that
your problem is also a linear algebra calculation from root to root. You could transform your matrix to a matrix like matrix is from this website: https://schemas.org/scilab/1.2/
L2MatriceMatrice1.html Then to use the code above using R coder I have followed a tutorial. It really helps to have a knowledge of matrix multiplication on R, make use of that to understand the
function of map A little bit, that kind of you not actually have to do anything else on your program. I suggested following the tutorial which was another way suggested. There are 10 entries which
have the following values for a root node, then you can just write an R expression that goes from root to the root. Who can help me with my R programming matrices homework? :/ You just had 1
assignment when trying to solve the matrix R. My Matrice Math homework recommended you read need not be like this one. I then try to solve the same matrices but only a few parts. Don’t understand why
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of how your Matrices can be solved at a computer. You’re talking about some functions matrices and R and a complex matrix. Not using the right answers or answers…a question that goes into 2 months
now. Numerical Solution The solution of matrice math is just a few problems. This module doesnt have all the answers. It suggests a solution for quite some time. Solve Matrices Solution Matrices are
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Matrices S can be solved some things but also some things. Another task Matrices S have the same way. Matrices A; R(T;Q);Q = F(T-1) Q = |3 + D(t – 1)|S – SF(T-1)|Q| For matrices E(T) :- F(T) / 2E(T)
= q/(4 + 6*d)/2 F(T) to find exact values for E and q is n/L/2 0.79057 PV = 0.6107 f = 1.6776 q = 40 $f$ = 0.3883 $q$ (100) You see the problemWho can help me with my R programming matrices homework?
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row. That should do the job very well, but something that didn’t depend on the $3$, 2, and 1 dimensions do? On the other hand, how about this: 1 $$ \sum_i \cos(a_i) = \tan^{-1.8}([3, 2, 2]); $$ 2 $$
\sum_i \cos(b_i) = \cos^3([3, 2, 2])+2\sin^3(b_i) $$ So we think that the 3-dimensional sum points are those 3-dimensional elements I will be interested in in our matrix case (and that the rows that
connect 3 points are matrices). So for $\sum_i \cos(a_i) = 2b_i$ we find out that this matrix is a fourth-dimension matrix we could use. 2 $$ \sum_i \cos(b_i) = 2b_i + 2b_3$$ So, next we try to
figure out a good function type which gives us some of the $3$, 2, 2, and 4-dimensional components of $a_3, b_3, a_4, b_4$. Because i guess that we need to know where a 3rd to 4th dimension are with
the $n$ components of the matrix, I will do my best to guess and compute the first four components. I don’t know if it’s even mathematically relevant, but I won’t mention anything about Mathematica:
for the other two, for the inner product we look at first directory dimension. Just look here luck if this works out.
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.. 3 $$\sum_i \cos(a_i) = 2 + 2a_5 + 3a_4$$ So what we did is compute the $n$-dimensional element of $\cos(a_i)$ for $1\leq i \leq 3$, which is like the first $2\lceil{\mathcal{R}_2\lceil\mathcal{R}\
mathcal{R}}\rceil-2$ dimensions of the matrix. So when we find the matrix with $n$ coordinates we have an element whose sum overlaps two of the $n$ coordinates. In the end we give that result in $2\
times 2$-matrices. But it is mathematically impossible to compute mathematically both first and last two dimensional components for 3 dimensions and 4 dimensions. In Haskell, the value of $n$ is the
length of the list for $n$ different elements. Why should the length be something that is NOT natural? I think that because each element is a different number of times it contains a new and a new,
respectively new, 3rd dimension. The entire matrix starts with two factors for this one $2$ part of the $n$-dimension: the number of times then which column contains two columns, such that $\mathcal
{R}$ = 2, so $\mathcal{R}=\mathcal{R}_2 \lceil(-2,-1,1) \lceil1\rceil+3$ so 4 dimensions being at the very high end of matrices just say one, for the rank $\geq 3$. The rank $\geq 2$ is necessary for
the matrix to be both in each row subspace-wise (2,2,1,2,3) with 2 columns of structure $-$3, 3,4, \dots$ (no further modifications will make $\hbox{\smallmatrix}$ diagonal). So the matrices
structure for the matrices that only have some one, although they contain both 3rd and 4th dimension, with 1 | {"url":"https://rprogrammingassignments.com/who-can-help-me-with-my-r-programming-matrices-homework","timestamp":"2024-11-13T13:11:26Z","content_type":"text/html","content_length":"196336","record_id":"<urn:uuid:dc8ab543-1116-4b29-8dde-80f254025a5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00224.warc.gz"} |
Tensile strength and fracture of cemented granular aggregates • Rafik Affès
Tensile strength and fracture of cemented granular aggregates
Cemented granular aggregates include a broad class of geomaterials such as sedimentary rocks and some biomaterials such as the wheat endosperm. We present a 3D lattice element method for the
simulation of such materials, modeled as a jammed assembly of particles bound together by a matrix partially filling the interstitial space. From extensive simulation data, we analyze the mechanical
properties of aggregates subjected to tensile loading as a function of matrix volume fraction and particle-matrix adhesion. We observe a linear elastic behavior followed by a brutal failure along a
fracture surface. The effective stiffness before failure increases almost linearly with the matrix volume fraction. We show that the tensile strength of the aggregates increases with both the
increasing tensile strength at the particle-matrix interface and decreasing stress concentration as a function of matrix volume fraction. The proportion of broken bonds in the particle phase reveals
a range of values of the particle-matrix adhesion and matrix volume fraction for which the cracks bypass the particles and hence no particle damage occurs. This limit is shown to depend on the
relative toughness of the particle-matrix interface with respect to the particles.
A color map of vertical stresses σzz of a cemented aggregate under tensile loading | {"url":"http://www.particledynamics.fr/cgp/post/tensile_strength_and_fracture_of_cemented_granular_aggregates/","timestamp":"2024-11-06T21:16:14Z","content_type":"text/html","content_length":"13517","record_id":"<urn:uuid:01346c85-4c78-417b-b49f-2932c77dcc3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00451.warc.gz"} |
gcc/ada/libgnat/a-rbtgbk.adb - gcc - Git at Google
-- --
-- GNAT LIBRARY COMPONENTS --
-- --
-- ADA.CONTAINERS.RED_BLACK_TREES.GENERIC_BOUNDED_KEYS --
-- --
-- B o d y --
-- --
-- Copyright (C) 2004-2022, Free Software Foundation, Inc. --
-- --
-- GNAT is free software; you can redistribute it and/or modify it under --
-- terms of the GNU General Public License as published by the Free Soft- --
-- ware Foundation; either version 3, or (at your option) any later ver- --
-- sion. GNAT is distributed in the hope that it will be useful, but WITH- --
-- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY --
-- or FITNESS FOR A PARTICULAR PURPOSE. --
-- --
-- As a special exception under Section 7 of GPL version 3, you are granted --
-- additional permissions described in the GCC Runtime Library Exception, --
-- version 3.1, as published by the Free Software Foundation. --
-- --
-- You should have received a copy of the GNU General Public License and --
-- a copy of the GCC Runtime Library Exception along with this program; --
-- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see --
-- <http://www.gnu.org/licenses/>. --
-- --
-- This unit was originally developed by Matthew J Heaney. --
package body Ada.Containers.Red_Black_Trees.Generic_Bounded_Keys is
package Ops renames Tree_Operations;
-- Ceiling --
-- AKA Lower_Bound
function Ceiling
(Tree : Tree_Type'Class;
Key : Key_Type) return Count_Type
Y : Count_Type;
X : Count_Type;
N : Nodes_Type renames Tree.Nodes;
Y := 0;
X := Tree.Root;
while X /= 0 loop
if Is_Greater_Key_Node (Key, N (X)) then
X := Ops.Right (N (X));
Y := X;
X := Ops.Left (N (X));
end if;
end loop;
return Y;
end Ceiling;
-- Find --
function Find
(Tree : Tree_Type'Class;
Key : Key_Type) return Count_Type
Y : Count_Type;
X : Count_Type;
N : Nodes_Type renames Tree.Nodes;
Y := 0;
X := Tree.Root;
while X /= 0 loop
if Is_Greater_Key_Node (Key, N (X)) then
X := Ops.Right (N (X));
Y := X;
X := Ops.Left (N (X));
end if;
end loop;
if Y = 0 then
return 0;
end if;
if Is_Less_Key_Node (Key, N (Y)) then
return 0;
end if;
return Y;
end Find;
-- Floor --
function Floor
(Tree : Tree_Type'Class;
Key : Key_Type) return Count_Type
Y : Count_Type;
X : Count_Type;
N : Nodes_Type renames Tree.Nodes;
Y := 0;
X := Tree.Root;
while X /= 0 loop
if Is_Less_Key_Node (Key, N (X)) then
X := Ops.Left (N (X));
Y := X;
X := Ops.Right (N (X));
end if;
end loop;
return Y;
end Floor;
-- Generic_Conditional_Insert --
procedure Generic_Conditional_Insert
(Tree : in out Tree_Type'Class;
Key : Key_Type;
Node : out Count_Type;
Inserted : out Boolean)
Y : Count_Type;
X : Count_Type;
N : Nodes_Type renames Tree.Nodes;
-- This is a "conditional" insertion, meaning that the insertion request
-- can "fail" in the sense that no new node is created. If the Key is
-- equivalent to an existing node, then we return the existing node and
-- Inserted is set to False. Otherwise, we allocate a new node (via
-- Insert_Post) and Inserted is set to True.
-- Note that we are testing for equivalence here, not equality. Key must
-- be strictly less than its next neighbor, and strictly greater than
-- its previous neighbor, in order for the conditional insertion to
-- succeed.
-- We search the tree to find the nearest neighbor of Key, which is
-- either the smallest node greater than Key (Inserted is True), or the
-- largest node less or equivalent to Key (Inserted is False).
Y := 0;
X := Tree.Root;
Inserted := True;
while X /= 0 loop
Y := X;
Inserted := Is_Less_Key_Node (Key, N (X));
X := (if Inserted then Ops.Left (N (X)) else Ops.Right (N (X)));
end loop;
if Inserted then
-- Either Tree is empty, or Key is less than Y. If Y is the first
-- node in the tree, then there are no other nodes that we need to
-- search for, and we insert a new node into the tree.
if Y = Tree.First then
Insert_Post (Tree, Y, True, Node);
end if;
-- Y is the next nearest-neighbor of Key. We know that Key is not
-- equivalent to Y (because Key is strictly less than Y), so we move
-- to the previous node, the nearest-neighbor just smaller or
-- equivalent to Key.
Node := Ops.Previous (Tree, Y);
-- Y is the previous nearest-neighbor of Key. We know that Key is not
-- less than Y, which means either that Key is equivalent to Y, or
-- greater than Y.
Node := Y;
end if;
-- Key is equivalent to or greater than Node. We must resolve which is
-- the case, to determine whether the conditional insertion succeeds.
if Is_Greater_Key_Node (Key, N (Node)) then
-- Key is strictly greater than Node, which means that Key is not
-- equivalent to Node. In this case, the insertion succeeds, and we
-- insert a new node into the tree.
Insert_Post (Tree, Y, Inserted, Node);
Inserted := True;
end if;
-- Key is equivalent to Node. This is a conditional insertion, so we do
-- not insert a new node in this case. We return the existing node and
-- report that no insertion has occurred.
Inserted := False;
end Generic_Conditional_Insert;
-- Generic_Conditional_Insert_With_Hint --
procedure Generic_Conditional_Insert_With_Hint
(Tree : in out Tree_Type'Class;
Position : Count_Type;
Key : Key_Type;
Node : out Count_Type;
Inserted : out Boolean)
N : Nodes_Type renames Tree.Nodes;
-- The purpose of a hint is to avoid a search from the root of
-- tree. If we have it hint it means we only need to traverse the
-- subtree rooted at the hint to find the nearest neighbor. Note
-- that finding the neighbor means merely walking the tree; this
-- is not a search and the only comparisons that occur are with
-- the hint and its neighbor.
-- If Position is 0, this is interpreted to mean that Key is
-- large relative to the nodes in the tree. If the tree is empty,
-- or Key is greater than the last node in the tree, then we're
-- done; otherwise the hint was "wrong" and we must search.
if Position = 0 then -- largest
if Tree.Last = 0
or else Is_Greater_Key_Node (Key, N (Tree.Last))
Insert_Post (Tree, Tree.Last, False, Node);
Inserted := True;
Conditional_Insert_Sans_Hint (Tree, Key, Node, Inserted);
end if;
end if;
pragma Assert (Tree.Length > 0);
-- A hint can either name the node that immediately follows Key,
-- or immediately precedes Key. We first test whether Key is
-- less than the hint, and if so we compare Key to the node that
-- precedes the hint. If Key is both less than the hint and
-- greater than the hint's preceding neighbor, then we're done;
-- otherwise we must search.
-- Note also that a hint can either be an anterior node or a leaf
-- node. A new node is always inserted at the bottom of the tree
-- (at least prior to rebalancing), becoming the new left or
-- right child of leaf node (which prior to the insertion must
-- necessarily be null, since this is a leaf). If the hint names
-- an anterior node then its neighbor must be a leaf, and so
-- (here) we insert after the neighbor. If the hint names a leaf
-- then its neighbor must be anterior and so we insert before the
-- hint.
if Is_Less_Key_Node (Key, N (Position)) then
Before : constant Count_Type := Ops.Previous (Tree, Position);
if Before = 0 then
Insert_Post (Tree, Tree.First, True, Node);
Inserted := True;
elsif Is_Greater_Key_Node (Key, N (Before)) then
if Ops.Right (N (Before)) = 0 then
Insert_Post (Tree, Before, False, Node);
Insert_Post (Tree, Position, True, Node);
end if;
Inserted := True;
Conditional_Insert_Sans_Hint (Tree, Key, Node, Inserted);
end if;
end if;
-- We know that Key isn't less than the hint so we try again,
-- this time to see if it's greater than the hint. If so we
-- compare Key to the node that follows the hint. If Key is both
-- greater than the hint and less than the hint's next neighbor,
-- then we're done; otherwise we must search.
if Is_Greater_Key_Node (Key, N (Position)) then
After : constant Count_Type := Ops.Next (Tree, Position);
if After = 0 then
Insert_Post (Tree, Tree.Last, False, Node);
Inserted := True;
elsif Is_Less_Key_Node (Key, N (After)) then
if Ops.Right (N (Position)) = 0 then
Insert_Post (Tree, Position, False, Node);
Insert_Post (Tree, After, True, Node);
end if;
Inserted := True;
Conditional_Insert_Sans_Hint (Tree, Key, Node, Inserted);
end if;
end if;
-- We know that Key is neither less than the hint nor greater
-- than the hint, and that's the definition of equivalence.
-- There's nothing else we need to do, since a search would just
-- reach the same conclusion.
Node := Position;
Inserted := False;
end Generic_Conditional_Insert_With_Hint;
-- Generic_Insert_Post --
procedure Generic_Insert_Post
(Tree : in out Tree_Type'Class;
Y : Count_Type;
Before : Boolean;
Z : out Count_Type)
N : Nodes_Type renames Tree.Nodes;
TC_Check (Tree.TC);
if Checks and then Tree.Length >= Tree.Capacity then
raise Capacity_Error with "not enough capacity to insert new item";
end if;
Z := New_Node;
pragma Assert (Z /= 0);
if Y = 0 then
pragma Assert (Tree.Length = 0);
pragma Assert (Tree.Root = 0);
pragma Assert (Tree.First = 0);
pragma Assert (Tree.Last = 0);
Tree.Root := Z;
Tree.First := Z;
Tree.Last := Z;
elsif Before then
pragma Assert (Ops.Left (N (Y)) = 0);
Ops.Set_Left (N (Y), Z);
if Y = Tree.First then
Tree.First := Z;
end if;
pragma Assert (Ops.Right (N (Y)) = 0);
Ops.Set_Right (N (Y), Z);
if Y = Tree.Last then
Tree.Last := Z;
end if;
end if;
Ops.Set_Color (N (Z), Red);
Ops.Set_Parent (N (Z), Y);
Ops.Rebalance_For_Insert (Tree, Z);
Tree.Length := Tree.Length + 1;
end Generic_Insert_Post;
-- Generic_Iteration --
procedure Generic_Iteration
(Tree : Tree_Type'Class;
Key : Key_Type)
procedure Iterate (Index : Count_Type);
-- Iterate --
procedure Iterate (Index : Count_Type) is
J : Count_Type;
N : Nodes_Type renames Tree.Nodes;
J := Index;
while J /= 0 loop
if Is_Less_Key_Node (Key, N (J)) then
J := Ops.Left (N (J));
elsif Is_Greater_Key_Node (Key, N (J)) then
J := Ops.Right (N (J));
Iterate (Ops.Left (N (J)));
Process (J);
J := Ops.Right (N (J));
end if;
end loop;
end Iterate;
-- Start of processing for Generic_Iteration
Iterate (Tree.Root);
end Generic_Iteration;
-- Generic_Reverse_Iteration --
procedure Generic_Reverse_Iteration
(Tree : Tree_Type'Class;
Key : Key_Type)
procedure Iterate (Index : Count_Type);
-- Iterate --
procedure Iterate (Index : Count_Type) is
J : Count_Type;
N : Nodes_Type renames Tree.Nodes;
J := Index;
while J /= 0 loop
if Is_Less_Key_Node (Key, N (J)) then
J := Ops.Left (N (J));
elsif Is_Greater_Key_Node (Key, N (J)) then
J := Ops.Right (N (J));
Iterate (Ops.Right (N (J)));
Process (J);
J := Ops.Left (N (J));
end if;
end loop;
end Iterate;
-- Start of processing for Generic_Reverse_Iteration
Iterate (Tree.Root);
end Generic_Reverse_Iteration;
-- Generic_Unconditional_Insert --
procedure Generic_Unconditional_Insert
(Tree : in out Tree_Type'Class;
Key : Key_Type;
Node : out Count_Type)
Y : Count_Type;
X : Count_Type;
N : Nodes_Type renames Tree.Nodes;
Before : Boolean;
Y := 0;
Before := False;
X := Tree.Root;
while X /= 0 loop
Y := X;
Before := Is_Less_Key_Node (Key, N (X));
X := (if Before then Ops.Left (N (X)) else Ops.Right (N (X)));
end loop;
Insert_Post (Tree, Y, Before, Node);
end Generic_Unconditional_Insert;
-- Generic_Unconditional_Insert_With_Hint --
procedure Generic_Unconditional_Insert_With_Hint
(Tree : in out Tree_Type'Class;
Hint : Count_Type;
Key : Key_Type;
Node : out Count_Type)
N : Nodes_Type renames Tree.Nodes;
-- There are fewer constraints for an unconditional insertion
-- than for a conditional insertion, since we allow duplicate
-- keys. So instead of having to check (say) whether Key is
-- (strictly) greater than the hint's previous neighbor, here we
-- allow Key to be equal to or greater than the previous node.
-- There is the issue of what to do if Key is equivalent to the
-- hint. Does the new node get inserted before or after the hint?
-- We decide that it gets inserted after the hint, reasoning that
-- this is consistent with behavior for non-hint insertion, which
-- inserts a new node after existing nodes with equivalent keys.
-- First we check whether the hint is null, which is interpreted
-- to mean that Key is large relative to existing nodes.
-- Following our rule above, if Key is equal to or greater than
-- the last node, then we insert the new node immediately after
-- last. (We don't have an operation for testing whether a key is
-- "equal to or greater than" a node, so we must say instead "not
-- less than", which is equivalent.)
if Hint = 0 then -- largest
if Tree.Last = 0 then
Insert_Post (Tree, 0, False, Node);
elsif Is_Less_Key_Node (Key, N (Tree.Last)) then
Unconditional_Insert_Sans_Hint (Tree, Key, Node);
Insert_Post (Tree, Tree.Last, False, Node);
end if;
end if;
pragma Assert (Tree.Length > 0);
-- We decide here whether to insert the new node prior to the
-- hint. Key could be equivalent to the hint, so in theory we
-- could write the following test as "not greater than" (same as
-- "less than or equal to"). If Key were equivalent to the hint,
-- that would mean that the new node gets inserted before an
-- equivalent node. That wouldn't break any container invariants,
-- but our rule above says that new nodes always get inserted
-- after equivalent nodes. So here we test whether Key is both
-- less than the hint and equal to or greater than the hint's
-- previous neighbor, and if so insert it before the hint.
if Is_Less_Key_Node (Key, N (Hint)) then
Before : constant Count_Type := Ops.Previous (Tree, Hint);
if Before = 0 then
Insert_Post (Tree, Hint, True, Node);
elsif Is_Less_Key_Node (Key, N (Before)) then
Unconditional_Insert_Sans_Hint (Tree, Key, Node);
elsif Ops.Right (N (Before)) = 0 then
Insert_Post (Tree, Before, False, Node);
Insert_Post (Tree, Hint, True, Node);
end if;
end if;
-- We know that Key isn't less than the hint, so it must be equal
-- or greater. So we just test whether Key is less than or equal
-- to (same as "not greater than") the hint's next neighbor, and
-- if so insert it after the hint.
After : constant Count_Type := Ops.Next (Tree, Hint);
if After = 0 then
Insert_Post (Tree, Hint, False, Node);
elsif Is_Greater_Key_Node (Key, N (After)) then
Unconditional_Insert_Sans_Hint (Tree, Key, Node);
elsif Ops.Right (N (Hint)) = 0 then
Insert_Post (Tree, Hint, False, Node);
Insert_Post (Tree, After, True, Node);
end if;
end Generic_Unconditional_Insert_With_Hint;
-- Upper_Bound --
function Upper_Bound
(Tree : Tree_Type'Class;
Key : Key_Type) return Count_Type
Y : Count_Type;
X : Count_Type;
N : Nodes_Type renames Tree.Nodes;
Y := 0;
X := Tree.Root;
while X /= 0 loop
if Is_Less_Key_Node (Key, N (X)) then
Y := X;
X := Ops.Left (N (X));
X := Ops.Right (N (X));
end if;
end loop;
return Y;
end Upper_Bound;
end Ada.Containers.Red_Black_Trees.Generic_Bounded_Keys; | {"url":"https://gnu.googlesource.com/gcc/+/refs/tags/basepoints/gcc-13/gcc/ada/libgnat/a-rbtgbk.adb?autodive=0%2F%2F%2F%2F","timestamp":"2024-11-09T21:46:27Z","content_type":"text/html","content_length":"208934","record_id":"<urn:uuid:a6717547-38cd-4b20-9eb3-54f09138a15a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00024.warc.gz"} |
Theory of phase transitions and critical phenomena
The work on the development of a theory of phase transitions and critical phenomena was initiated by I. R. Yukhnovskii in the mid-1980s. The method of collective variables he developed for the study
of spin systems opened the route to the construction of a consistent microscopic theory of phase transitions for a number of physical systems. To name some of the achivements, magnetic and
segnetoelectric systems, liquid-gas in the vicinity of the critical point, systems with layering phase transitions, and a number of other physical objects were succesfully described (I. R.
Yukhnovskii, I. O. Vakarchuk, Yu. K. Rudavskii, M. P. Kozlovskii, I. M. Mryglod, I. V. Stasyuk, M. A. Korynevskii, O. V. Patsahan, I. V. Pylyuk).
In the last decade interesting results were obtained in the theory of critical phenomena with the use of the field-theoretical approach (Yu. V. Holovatch, M. A. Shpot, Z. Ye. Usatenko, M. L. Dudka,
V. B. Blavatska).
Microscopic theory of phase transitions. A theory of phase transitions for a magnet with one-component order parameter was built on the microscopic level. By means of the tree-dimensional Ising model
critical behaviour of similar models with a second order phase transition was studied. All thermodynamic characteristics of a three-dimensional Ising-like system were derived from the first
principles with the confluent corrections taken into account. The microscopic analogue of the Landau free energy was found and a method for obtaining the equation of state in the vicinity of the
critical point was proposed. The influence of the exponentially decaying interaction potential on the critical temperature, critical region size, thermodynamic functions behaviour was studied.
Yukhnovskii's theory of phase transitions was generalized for the case of an external field. The developed theoretical approach enabled description of the thermodynamic charasteristics dependance on
the temperature and external field as well as on microscopic parameters of the system (lattice spacing, interaction potential parameters) near the phase transition point. For the first time an
equation of state for a three-dimensional Ising-like system was proposed and the explicite form of the corresponding scaling function was obtained. It was shown that in the two limiting cases of
large and small external fields this equation of state takes the well known forms.
Explicite analitical expressions were obtained for the thermodynamic functions of a classical n-component model of a magnet in three dimensions. The dependence of the critical amplitudes of the basic
thermodynamic characteristics on the normalized temperature and microscopic parameters of the system in high- and low-temperature phases was studied. The cases with particular values of the spin
component number were considered. The mechanism of the heat capacity peak formation near Tc was described in the classical Heisenberg model and its value dependence on microscopic parameters was
A microscopic approach to the phase transition description in multicomponent continuous systems was developed. The theory that was proposed based on the collective variables method with a marked
reference system. The explicite form of the Ginsburg-Landau-Wilson microscopic Hamiltonian was obtained in the vicinity of the phase transition point of a two-component continuous system. The problem
of the order parameter determination in a binary fluid mixture was studied in detail. Explicite expressions for the thermodynamic functions and equation of state were obtained in the case of a
symmetric binary fluid mixture in the vicinity of the liquid-gas critical point.
Disorder and criticality in magnets. The phenomena that take place in the proximity of the Curie point under the influence of structural disorder were studied. A change in the critical behaviour
under the influence of different kinds of disorder (substitution disorder, random anisotropy, frustrations) was predicted and quantitative characteristics of this behaviour were obtained. The
spontaneous order parameter appearance was studied in the two-dimensional finite-size systems with continuous symmetry. The quasi-long-range ordering that appears in these systems in the
thermodynamic limit was studied as well.
Dependence of the effective critical exponents of m-vector magnets with extended impurities on the renormalization group flow parameter connected to the distance to the critical temperature was
studied. It was established that even weak dilution of an Ising magnet with impurities in the form of parallel lines leads faster to the transition into a new universality class than that with
point-like impurities. Within the general thermodynamic formalism framework these phase transitions were proved to classify as second order phase transitions.
For the systems with thermodynamic restraints the existence of completely new scaling corrections, proportional to the heat capacity critical exponent of the system without restraints, was
established. Besides the well known renormalization of the asymptotic critical exponents (Fisher renormalization) other corrections appear in such systems and they are dominant when leaving the
critical region and define the pecularities of the behaviour of a system with restraints. The inclusion of these new corrections to the scaling allowed to eliminate the inconsistencies which were
observed in the computer calculations of critical exponents in a number of systems.
Scaling of polymers with complex topology. A theoretical description of the scaling properties of multisorted polymer nets and stars was proposed. The results obtained were applied to the
construction of phase diagrams for polymers with complex topology, study of the polymers interaction in a good solvent, description of the catalysis reactions near polymer macromolecules, description
of the structural transitions in DNA molecules.
Complex networks. Following the complex networks theory based approach, it was shown that public transport networks are scale-free small worlds. A model was proposed to describe scaling properties of
such networks and their stability with respect to random damages and directed attacts was analized.
In anisotropic and spatially confined systems with competing exchange interactions first non-trivial orders of the 1/N-expansions for correlation critical exponents in m-fold Lifshitz points were
obtained. New conclusions about the qualitative behaviour of these exponents with the change of space dimensionality were made. Non-classical surface exponents of an ordinary transition in such
systems were calculated. Numerical estimations for the Kasimir amplitudes in three dimensions were made. Surface critical behaviour was studied in semi-confined n-component systems in the presence of
cubic anisotropy (m=1) at an ordinary phase transition.
Thermodynamic and structural characteristics of mixed segneto- and antisegnetoelectric systems were studied. On the basis of an original model in cluster approximation with non-equilibrium
distribution of the structure elements that experience ordering, temperature-concentration phase diagrams were obtained. It is proved that segneto- and antisegnetoelectric phases can only exist
separately. The threshold values of the component concentration at which the ordered state of each type vanishes — percolation thresholds — are found. The developed theory explains experimentally
observed properties of solid mixtures of segnetoelectric systems. | {"url":"https://icmp.lviv.ua/en/content/theory-phase-transitions-and-critical-phenomena","timestamp":"2024-11-12T05:44:41Z","content_type":"application/xhtml+xml","content_length":"53544","record_id":"<urn:uuid:8b6cbe00-b109-4663-8ce7-71f413872ba1>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00807.warc.gz"} |
Gas Laws Quiz - Doquizzes
Get ready to dive into the world of gases and their intriguing behaviors. Our Gas Laws Quiz will challenge your understanding of the principles that dictate how gases respond to changes in pressure,
volume, and temperature. From Boyle’s Law to the Ideal Gas Law, each question will test your grasp of these essential concepts. Let’s see how well you can apply these laws to real-world scenarios!
We recommend that you do not leave the page that you are taking this quiz in. Stay honest 🙂
Gas Laws Quiz Questions Overview
1. Which law states that the volume of a gas is inversely proportional to its pressure at constant temperature?
Boyle’s Law
Charles’s Law
Avogadro’s Law
Ideal Gas Law
2. According to Charles’s Law, what happens to the volume of a gas when its temperature increases at constant pressure?
The volume decreases
The volume increases
The volume remains the same
The volume becomes zero
3. Which gas law combines Boyle’s Law, Charles’s Law, and Avogadro’s Law into one equation?
Boyle’s Law
Charles’s Law
Ideal Gas Law
Dalton’s Law
4. What does the ‘R’ represent in the Ideal Gas Law equation PV=nRT?
Gas constant
Rate constant
Reaction constant
5. Avogadro’s Law states that equal volumes of gases at the same temperature and pressure contain equal numbers of what?
6. Which of the following is the correct expression for Boyle’s Law?
P1V1 = P2V2
V1/T1 = V2/T2
P1/T1 = P2/T2
V1/n1 = V2/n2
7. Which gas law is represented by the equation V1/T1 = V2/T2?
Boyle’s Law
Charles’s Law
Avogadro’s Law
Ideal Gas Law
8. In the Ideal Gas Law equation PV=nRT, what does ‘n’ represent?
Number of molecules
Number of atoms
Number of moles
Number of particles
9. What is the value of the gas constant ‘R’ in the Ideal Gas Law when using SI units?
8.314 J/(mol·K)
0.0821 J/(mol·K)
1.00 J/(mol·K)
22.4 J/(mol·K)
10. Which law states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures of the individual gases?
Boyle’s Law
Charles’s Law
Dalton’s Law
Avogadro’s Law
We recommend that you do not leave the page that you are taking this quiz in. Stay honest 🙂 | {"url":"https://doquizzes.com/gas-laws-quiz/","timestamp":"2024-11-08T11:31:55Z","content_type":"text/html","content_length":"195220","record_id":"<urn:uuid:b00842b5-1841-45d1-bffd-ee90f8a07378>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00754.warc.gz"} |
Assignment 1
1. A student of mass 50 kg tests Newton's laws by standing on a bathroom scale in an elevator.
Assume that the scale reads in Newtons. Find the scale reading when the elevator is a)
accelerating upward at .5 m/s 2, b) going up at a constant speed of 3.0 m/s and c) going up
but decelerating at 1.0 m/s 2.
A 10 kg box is attached to a 7 kg box which rests on a 30 o incline. The coefficient of kinetic
friction between each box and the surface is
system and b) the tension in the rope.
= .1 . Find a) the rate of acceleration of the
A car travelling at a constant speed of 30 m/s passes a police car at rest. The policeman
starts to move at the moment the speeder passes his car and accelerates at a constant rate
of 3.0 m/ s 2 until he pulls even with the speeding car. Find a) the time required for the
policeman to catch the speeder and b) the distance travelled during the chase.
A stone is thrown vertically upward from the edge of a building 19.6 m high with initial
velocity 14.7 m/s. The stone just misses the building on the way down. Find a) the time of
flight and b) the velocity of the stone just before it hits the ground.
5. How long would it take for a net upward force of 100. N, to increase the speed of a 50. kg
object from 100. m/s to 150. m/s.
Initially a soccer ball is going 23.5 m/s, south. In the end, it is travelling at 3.8 m/s, south.
The ball's change in momentum is 17.24 kg m/s, north. Find the ball's mass.
7. A 62.0 kg curler travelling at + 1.72 m/s runs into a 78.1 kg curler and the 62.0 kg
curler comes to a stop. If the 78.1 kg curler was originally moving at + 0.85 m/s, find his
velocity after the interaction.
A 62.0 kg curler runs into a stationary 78.1 kg curler and they hold on to each
other. Together they move away at 1.29 m/s, west. What was the original velocity of the
62.0 kg curler?
9. A 84.0 kg (total mass) astronaut in space fires a thruster that expels 35 g of hot gas at 875
m/s. What is the velocity of the astronaut after firing the shot?
10. Find the real number b so that vectors A and B given below are perpendicular
A = (-2 , -b) , B = (-8 , b).
Scalar Product of Vectors
The scalar product (also called the dot product and inner product) of vectors A and B is
written and defined as follows
Fig1. - Angle between vectors and scalar product.
A·B = | A | | B | cos (θ)
where | A | is the magnitude of vector A , | B | is the magnitude of vector B and θ is the angle
made by the two vectors. The result of a scalar product of two vectors is a scalar quantity.
For vectors given by their components: A = (Ax , Ay, Az) andB = (Bx , By, Bz), the scalar
product is given by
A·B = AxBx + AyBy + AzBz
Note that if θ = 90°, then cos(θ) = 0 and therefore we can state that: Two vectors, with
magnitudes not equal to zero, are perpendicular if and only if their scalar product is equal to
The scalar product may also be used to find the cosine and therefore the angle between two
cos (θ) = A·B / | A | | B |
Properties of the Scalar Product
1) A·B = B·A
2) A· (B + C) = A·B + A·C
11. Given vector U = (3 , -7), find the equation of the line through point B(2 , 1) and
perpendicular to vector U.
12. A point M(x , y) is on the line through point B(2 , 1) and perpendicular to vector U = (3 , -7)
if and only if the vectors BM and U are perpendicular.
13. An object is launched at a velocity of 20 m/s in a direction making an angle of 25°
upward with the horizontal.
a) What is the maximum height reached by the object?
b) What is the total flight time (between launch and touching the ground) of the
c) What is the horizontal range (maximum x above ground) of the object?
d) What is the magnitude of the velocity of the object just before it hits the ground?
SOLUTION TO Qn.13 SAVES AS AN EXAMPLE
a) The formulas for the components Vx and Vy of the velocity and components x and
y of the displacement are given by
Vx = V0 cos(θ)
Vy = V0 sin(θ) - g t
x = V0 cos(θ) t
y = V0 sin(θ) t - (1/2) g t2
In the problem V0 = 20 m/s, θ = 25° and g = 9.8 m/s2.
The height of the projectile is given by the component y, and it reaches its maximum
value when the component Vy is equal to zero. That is when the projectile changes
from moving upward to moving downward.(see figure above) and also the animation
of the projectile.
Vy = V0 sin(θ) - g t = 0
solve for t
t = V0 sin(θ) / g = 20 sin(25°) / 9.8 = 0.86 seconds
Find the maximum height by substituting t by 0.86 seconds in the formula for y
maximum height y (0.86) = 20 sin(25°)(0.86) - (1/2) (9.8) (0.86) 2 = 3.64 meters
b) The time of flight is the interval of time between when projectile is launched: t1
and when the projectile touches the ground: t2. At t = t1 and t = t2, y = 0 (ground).
V0 sin(θ) t - (1/2) g t2 = 0
Solve for t , t(V0 sin(θ) - (1/2) g t) = 0
two solutions t = t1 = 0 and t = t2 = 2 V0 sin(θ) / g
Time of flight = t2 - t1 = 2 (20) sin(θ) / g = 1.72 seconds.
c) In part c) above we found the time of flight t2 = 2 V 0 sin(θ) / g. The horizontal
range is the horizontal distance given by x at t = t2.
range = x(t2) = V0 cos(θ) t2 = 2 V0 cos(θ) V0 sin(θ) / g = V02 sin(2θ) / g = 202 sin
(2(25°)) / 9.8 = 31.26 meters
d) The object hits the ground at t = t2 = 2 V0 sin(θ) / g (found in part b above)
The components of the velocity at t are given by
Vx = V0 cos(θ)
Vy = V0 sin(θ) - g t
The components of the velocity at t = 2 V0 sin(θ) / g are given by
Vx = V0 cos(θ) = 20 cos(25°)
Vy = V0 sin(25°) - g (2 V0 sin(25°) / g) = - V0 sin(25°)
The magnitude V of the velocity is given by
V = √[ Vx2 + Vy2 ] = √[ (20 cos(25°))2 + (- V0 sin(25°))2 ] = V0 = 20 m/s
14. A ball is kicked at an angle of 35° with the ground.
a) What should be the initial velocity of the ball so that it hits a target that is 30
meters away at a height of 1.8 meters?
b) What is the time for the ball to reach the target?
1. A projectile is to be launched at an angle of 30° so that it falls beyond the pond of
length 20 meters as shown in the figure.
a) What is the range of values of the initial velocity so that the projectile falls between
points M and N? | {"url":"https://studylib.net/doc/25206269/assignment-1","timestamp":"2024-11-04T04:36:49Z","content_type":"text/html","content_length":"54522","record_id":"<urn:uuid:c82f3aa6-6db0-41fc-8df9-f304b0851451>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00497.warc.gz"} |
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Open peer comments: Debate/Discuss/Question/Opinion | {"url":"https://jzus.zju.edu.cn/article.php?doi=10.1631/jzus.A0820322&comnowpage=0","timestamp":"2024-11-01T19:53:38Z","content_type":"text/html","content_length":"70641","record_id":"<urn:uuid:1f0ff233-2abd-4171-9f14-19a0759e30e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00815.warc.gz"} |
A solution to LeetCode Problem 560. Subarray Sum Equals K in JavaScript - JsDevLife
A solution to LeetCode Problem 560. Subarray Sum Equals K in JavaScript
A solution to LeetCode Problem 560. Subarray Sum Equals K in JavaScript
If you’re preparing for technical interviews or want to improve your coding skills, solving practice problems on LeetCode is a great way.
In this post, we’ll discuss a solution to the “560. Subarray Sum Equals K” problem on LeetCode.
Problem Statement:
Given an array of integers nums and an integer k, return the total number of subarrays whose sum equals to k.
A subarray is a contiguous non-empty sequence of elements within an array.
Example 1:
Input: nums = [1,1,1], k = 2
Output: 2
Example 2:
Input: nums = [1,2,3], k = 3
Output: 2
• 1 <= nums.length <= 2 * 104
• -1000 <= nums[i] <= 1000
• -107 <= k <= 107
One approach to solving this problem is to use a brute force method, where we check every possible subarray and see if its sum is equal to k. However, this approach has a time complexity of O(n^2),
which is not efficient for large arrays.
A more efficient solution is to use a prefix sum array, where we store the sum of the elements from the start of the array up to the current index. We can then use this prefix sum array to find the
sum of any subarray in constant time.
To solve the problem, we can iterate through the prefix sum array and for each element, we check if there is any previous element such that the difference between the two elements is equal to k. If
we find such an element, it means that the subarray between these two elements has a sum of k. We can then increment our count by the number of subarrays between these two elements.
Here is the implementation of this solution in JavaScript:
function subarraySum(nums, k) {
let count = 0;
let sum = 0;
const map = new Map();
map.set(0, 1);
for (let i = 0; i < nums.length; i++) {
sum += nums[i];
if (map.has(sum - k)) {
count += map.get(sum - k);
map.set(sum, (map.get(sum) || 0) + 1);
return count;
In the above code, we use a map to store the frequency of each prefix sum. We also initialize the map with a key of 0 and a value of 1, as there is always at least one subarray with a sum of 0 (the
empty subarray).
As we iterate through the array, we add the current element to the sum and check if there is a previous element such that the difference between the two elements is equal to k. If we find such an
element, we increment the count by the frequency of that element in the map. We also update the frequency of the current sum in the map.
This solution has a time complexity of O(n) and a space complexity of O(n), making it more efficient than the brute force method.
I hope this blog post was helpful in understanding how to solve the LeetCode Problem 560: Subarray Sum Equals K in JavaScript. If you have any questions or comments, feel free to leave them below.
3 thoughts on “A solution to LeetCode Problem 560. Subarray Sum Equals K in JavaScript”
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nevertheless I’ll definitely come back again. | {"url":"https://jsdevlife.in/2022/12/a-solution-to-leetcode-problem-560-subarray-sum-equals-k-in-javascript/","timestamp":"2024-11-06T04:50:58Z","content_type":"text/html","content_length":"183759","record_id":"<urn:uuid:9c4924fc-b3ef-4407-9272-47eaa850b449>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00433.warc.gz"} |
The concept TriangulationDSCellBase_3 describes the requirements for the cell base class of a CGAL::Triangulation_data_structure_3<Vb,Cb>.
Note that if the CGAL::Triangulation_data_structure_3 is plugged into a triangulation class, the face base class may have additional geometric requirements depending on the triangulation class.
At the base level (see the Software Design sections of the Chapters Triangulation and Triangulation Datastructure), a cell stores handles to its four vertices and to its four neighbor cells. The
vertices and neighbors are indexed 0, 1, 2 and 3. Neighbor i lies opposite to vertex i.
Since the Triangulation data structure is the class which defines the handle types, the cell base class has to be somehow parameterized by the Triangulation data structure. But since it is itself
parameterized by the cell and vertex base classes, there is a cycle in the definition of these classes. In order to break the cycle, the base classes for vertex and cell which are given as arguments
for the Triangulation data structure use void as Triangulation data structure parameter, and the Triangulation data structure then uses a rebind-like mechanism (similar to the one specified in
std::allocator) in order to put itself as parameter to the vertex and cell classes. The rebound base classes so obtained are the classes which are used as base classes for the final vertex and cell
classes. More information can be found in Section Software Design.
See also | {"url":"https://doc.cgal.org/5.5/TDS_3/classTriangulationDSCellBase__3.html","timestamp":"2024-11-12T20:39:06Z","content_type":"application/xhtml+xml","content_length":"27550","record_id":"<urn:uuid:daa523e6-1997-40f7-8912-dbe62e780820>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00122.warc.gz"} |
A histogram is a visual grouping of data into bins, plotting the number of members in each bin against the bin number.
Histogram is a high school-level concept that would be first encountered in a probability and statistics course. It is an Advanced Placement Statistics topic and is listed in the California State
Standards for Grade 5.
Classroom Articles on Probability and Statistics (Up to High School Level)
Arithmetic Mean Outlier
Box-and-Whisker Plot Problem
Conditional Probability Sample
Mean Scatter Diagram
Median Standard Deviation | {"url":"https://mathworld.wolfram.com/classroom/Histogram.html","timestamp":"2024-11-07T07:12:08Z","content_type":"text/html","content_length":"47035","record_id":"<urn:uuid:ccbfa610-bcda-48bf-9dc5-01317c499f25>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00085.warc.gz"} |
newUnitSystem(name,baseUnits) defines a new Unit System with the name name and the base units baseUnits. Now, you can convert units into the new unit system by using rewrite. By default, available
unit systems include SI, CGS, and US. For all unit systems, see Unit Systems List.
Define New Unit System from Existing System
A unit system is a collection of units to express quantities. The easiest way to define a new unit system is to modify a default unit system, such as SI, CGS, or US.
Modify SI to use kilometer for length and hour for time by getting the base units using baseunits and modifying them by using subs.
u = symunit;
SIUnits = baseUnits('SI')
SIUnits =
[ [kg], [s], [m], [A], [cd], [mol], [K]]
newUnits = subs(SIUnits,[u.m u.s],[u.km u.hr])
newUnits =
[ [kg], [h], [km], [A], [cd], [mol], [K]]
Do not define a variable called baseUnits because the variable will prevent access to the baseUnits function.
Define the new unit system SI_km_hr using the new base units.
Rewrite 5 meter/second to the SI_km_hr unit system. As expected, the result is in terms of kilometers and hours.
Specify Base and Derived Units Directly
Specify a new unit system by specifying the base and derived units directly. A unit system has up to 7 base units. For details, see Unit System.
Define a new unit system with these base units: gram, hour, meter, ampere, candela, mol, and celsius. Specify these derived units: kilowatt, newton, and volt.
u = symunit;
sysName = 'myUnitSystem';
bunits = [u.g u.hr u.m u.A u.cd u.mol u.Celsius];
dunits = [u.kW u.N u.V];
Rewrite 2000 Watts to the new system. By default, rewrite uses base units, which can be hard to read.
ans =
Instead, for readability, rewrite 2000 Watts to derived units of myUnitSystem by specifying 'Derived' as the third argument. Converting to the derived units of a unit system attempts to select
convenient units. The result uses the derived unit, kilowatt, instead of base units. For more information, see Unit Conversions and Unit Systems.
Input Arguments
name — Name of unit system
string | character vector
Name of unit system, specified as a string or character vector.
baseUnits — Base units of unit system
vector of symbolic units
Base units of unit system, specified as a vector of symbolic units. The base units must be independent in terms of the dimensions mass, time, length, electric current, luminous intensity, amount of
substance, and temperature. Thus, in a unit system, there are up to 7 base units.
derivedUnits — Derived units of unit system
vector of symbolic units
Derived units of unit system, specified as a vector of symbolic units. Derived units are optional and added for convenience of representation.
More About
Unit System
A unit system is a collection of base units and derived units that follows these rules:
• Base units must be independent in terms of the dimensions mass, time, length, electric current, luminous intensity, amount of substance, and temperature. Therefore, a unit system has up to 7 base
units. As long as the independence is satisfied, any unit can be a base unit, including units such as newton or watt.
• A unit system can have less than 7 base units. For example, mechanical systems need base units only for the dimensions length, mass, and time.
• Derived units in a unit system must have a representation in terms of the products of powers of the base units for that system. Unlike base units, derived units do not have to be independent.
• Derived units are optional and added for convenience of representation. For example, kg m/s^2 is abbreviated by newton.
• An example of a unit system is the SI unit system, which has 7 base units: kilogram, second, meter, ampere, candela, mol, and kelvin. There are 22 derived units found by calling derivedUnits
Version History
Introduced in R2017b | {"url":"https://uk.mathworks.com/help/symbolic/newunitsystem.html","timestamp":"2024-11-08T22:10:28Z","content_type":"text/html","content_length":"84959","record_id":"<urn:uuid:4ae41557-542d-4271-8bd0-23cba597ac42>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00299.warc.gz"} |
Discrete Probability | VCE Methods
Sample Spaces and Events
The outcome of a random experiment is uncertain, but there exists a set of possible outcomes, \(\varepsilon\), known as the sample space.
The sum of the probabilities of all the outcomes in, \(\varepsilon\) is 1.
For example, the sample space for rolling a six-sided dice is: \[\varepsilon = \{1,2,3,4,5,6\}\]
An event is a subset of the sample space denoted by a capital letter.
For example, if the event A is defined as the odd numbers when rolling a sixes sided dice, then we have: \[A = \{1,3,5\}\]
If the event B is impossible, \(\text{Pr}\)(B) = 0. If the event B is certain, \(\text{Pr}\)(B) = 1. So, for any event B, 0 \(\leqslant\) \(\text{Pr}\)(B) \(\leqslant\) 1.
Determining Probabilities
When the sample space is finite, the probability of an event is the sum of the probabilities of the outcomes in that event.
For example, if A is defined as the odd numbers when rolling a six sided dice, then:
\[ \begin{aligned} \Pr(A) &= \Pr(\text{Roll 1}) + \Pr(\text{Roll 3}) + \Pr(\text{Roll 5})\\ &= \frac{1}{6} + \frac{1}{6} + \frac{1}{6}\\ &= \frac{3}{6}\\ &= \frac{1}{2}\\ \end{aligned}\\ \]
When dealing with area questions, assume that it is equally likely to hit any region of the define area. So, the probability of hitting a certain region A is: \[\text{Pr}(A) = \frac{\text{Area of A}}
{\text{Total area}}\]
When an experiment has only two possible outcomes (events), they are said to be complementary. The complement of the event A is denoted by A'.
In the example with the six sided dice above, A' would represent everything in the sample space, except the odd numbers. So, \[A' = \{2,4,6\}\] Because the sum of the probabilities of events A and A'
must be 1, we have: \[\text{Pr}(A') = 1 - \text{Pr}(A)\]
The Addition Rule and Mutual Exclusivity
The addition rule is generally used to calculate \(\Pr(A \cap B)\) or \(\Pr(A \cup B)\) \[\Pr(A) + \Pr(B) - \Pr(A \cap B) = \Pr(A \cup B)\]
We say that two events are mutually exclusive if: \[\Pr(A \cap B) = 0\] That is the two events will never occur at the same time.
Probability Tables
A very powerful table which isn't emphasised enough!
\[A\] \[A'\]
\[B\] \[\Pr(A \cap B)\] \[\Pr(A' \cap B)\] \[\Pr(B)\]
\[B’\] \[\Pr(A \cap B’)\] \[\Pr(A' \cap B’)\] \[\Pr(B’)\]
\[\Pr(A)\] \[\Pr(A')\] \[1\]
For appropriate questions, place the probabilities given in their corresponding box. The sum of each column and row is the last entry. For example: \[\Pr(A \cap B) + \Pr(A \cap B') = \Pr(A)\]
\[ \text{Example 9.1: John has lost his class timetable. The probability that}\\\text{ he will have Methods period one is 0.35.}\\ \text{The probability that he has PE on a given day is 0.1 and the
probability}\\ \text{ that he will have Methods period one and PE on the same day is 0.05.}\\ \text{Find the probability that John does not have Methods period one and PE on the same day.}\\ \text{ }
\\ \text{Let } M \text{ represent methods period one and } P \text{ represent PE. From the information given we have:}\\ \]
\[M\] \[M’\]
\[P\] \[0.05\] \[\Pr(M’ \cap P)\] \[0.1\]
\[P’\] \[\Pr(M \cap P’)\] \[\Pr(M’ \cap P’)\] \[\Pr(P’)\]
\[0.35\] \[\Pr(M’)\] \[1\]
\[ \text{Looking at the second row}\\ \begin{aligned} 0.05 + \Pr(M’ \cap P) &= 0.1\\ \Pr(M’ \cap P) &= 0.05\\ \text{} \\ \text{Looking at the last row}\\ 0.35 + \Pr(M’) &= 1\\ \Pr(M’) &= 0.65\\ \end
{aligned} \]
\[M\] \[M’\]
\[P\] \[0.05\] \[0.05\] \[0.1\]
\[P’\] \[\Pr(M \cap P’)\] \[\Pr(M’ \cap P’)\] \[\Pr(P’)\]
\[0.35\] \[0.65\] \[1\]
\[ \text{Looking at the third column}\\ \begin{aligned} 0.05 + \Pr(M’ \cap P’) &= 0.65\\ \Pr(M’ \cap P’) &= 0.6\\ \end{aligned}\\ \text{So, the probability that John does not have Methods period one
and PE on a given day is } 0.6\\ \]
Conditional Probability
The probability that event A happens when we know that event B has already occured: \[\Pr(A \mid B) = \frac{\Pr(A \cap B)}{\Pr(B)}\]
It is often difficult to recognise when we are being asked a conditional probability question. However, generally speaking, if the question includes “if” or “given that”, you can be almost certain
that you are dealing with conditional probability. We have written the same question below twice using the two different phrases.
\[ \text{Example 9.2 Find the probability that a six is rolled with a} \\\text{ six sided die given that an even number has been rolled. }\\ \text{Or equivelantly, If an even number has been rolled
on a six sided die}\\\text{ find the probability that a six is rolled. }\\ \text{ }\\ \begin{aligned} \Pr(\text{Even}) &= \frac{3}{6} \\ \Pr(\text{Even and Six}) &= \Pr(\text{Six}) \\ &= \frac{1}{6}
\\ \Pr(\text{Six if Even}) &= \frac{\Pr(\text{Even and Six})}{\Pr(\text{Even})} \\ &= \frac{\frac{1}{6}}{\frac{3}{6}}\\ &= \frac{1}{3}\\ \end{aligned}\\ \]
If knowing that event B has happened does not change the probability of event A from happening, then we say that events A and B are independent. For example, it raining outside and you going to
school is independent as you will go to school regardless of whether it is raining or not. However, you being in class and having lunch is not independent as you will (probably) not be able to have
lunch during class. Mathematically two events are independent if: \[ \begin{aligned} \Pr(A \cap B) &= \Pr(A) \cdot \Pr(B)\\ \Pr(A) \neq 0 &\text{ and } \Pr(B) \neq 0 \end{aligned} \]
\[ \text{Example 9.3: John has lost his class timetable.}\\\text{ The probability that he will have Methods period one is 0.35.}\\ \text{The probability that he has PE on a given day is 0.1 and the
probability} \\\text{that he will have Methods period one and PE on the same day is 0.035.}\\ \text{Is John having Methods period one independent to him having PE on the same day?}\\ \text{ } \\ \
text{A. Let } M \text{ represent methods period one and } P \text{ represent PE. From the information given we have:}\\ \begin{aligned} \Pr(M) &= 0.35\\ \Pr(P) &= 0.1\\ \Pr(M) \cdot \Pr(P) &= 0.035 \
\ &= \Pr(M \cap P) \\ \end{aligned}\\ \text{So, John having Methods period one and PE on the same day are independent events} \\ \]
Discrete Random Variables
A random variable is a function that assigns a number to each outcome of an experiment.
A discrete random variable can take one of a countable number of possible outcomes. Continuous random variables will be considered in the next section.
For example, the number of free throws John can score when taking two is a discrete random variable which may take one of the values 0,1 or 2.
More on this below.
Discrete Probability Distributions
The probability distribution for a random variable consists of all the values the variable can take along with the associated probabilities. The general format is:
\[\text{}\] \[x_1\] \[x_2\] \[...\] \[x_n\]
\[\Pr(X=x)\] \[\Pr(X=x_1)\] \[\Pr(X=x_2)\] \[...\] \[\Pr(X=x_n)\]
The table allows us to easily find probabilities such as:
\(\text{Pr}(X>1)\) and \(\text{Pr}(X<2)\) by summing the relevant probabilities in the table
Note: the bottom row must sum to 1 and each probability must be at least zero and at most one.
A discrete probability function, also called a probability mass function describes the distributions of a discrete random variable. An example of a graph for a discrete probability function is given
\[ \text{Example 9.4: James scores 80\% of all free throws he takes.}\\ \text{Create a probability distribution table and graph the probability mass function if James has two free throws.}\\ \text{ }
\\ \text{Let } X \text{ be the number of free throws James scores}\\ \begin{aligned} \varepsilon &= {0,1,2}\\ Pr(X = 2) &= 0.80 \cdot 0.80 \\ &= 0.64\\ Pr(X = 1) &= \binom{2}{1} \cdot 0.8 \cdot 0.2\\
&= 0.32\\ Pr(X = 0) &= 0.2 \cdot 0.2 \\ &=0.04\\ \text{ }\\ \end{aligned}\\ \text{Bear with us on how we calculated } \Pr(X = 1). \\ \text{We will explain how we calculated this in the next chapter -
binomial distribution}\\ \]
\[x\] \[0\] \[1\] \[2\]
\[\Pr(X=x)\] \[0.04\] \[0.32\] \[0.64\]
Mean, Variance and Standard Deviation
The expected value, or mean, is the average value of a discrete random variable. To calculate it, we sum the products of each value of X and its associated probability. That is, we first find the
product of each column of the probability distribution table and then sum them up. Mathematically: \[E(X) = \mu = \sum_{x} x \cdot \Pr(x)\]
Variance and standard deviation are a measure of spread. Standard deviation is more relevant to us as it is in the same units as those which we are measuring. The formula provided by VCAA involves a
very large number of calculations, as such we recommend you memorising: \[\text{Var}(X) = E(X^2) - [E(X)]^2\] To calculate the first term, simply square all of the x values in the top row of the
probability distribution table and then find the product of each column of the probability distribution table before summing them up.
To get standard deviation we simply square root the variance. \[\text{sd}(X) = \sigma = \sqrt{\text{Var}(X)}\]
\[ \text{Note:}\\ \begin{aligned} E(aX+b) &= a \cdot E(X) + b\\ \text{Var}(aX+b) &= a^2 \cdot Var(X)\\ \end{aligned} \]
For many random variables, there is a 95% chance of obtaining an outcome within two standard deviations either side of the mean. That is, \[\Pr(\mu - 2\sigma \leqslant X \leqslant \mu + 2\sigma) \
approx 0.95\]
\[ \text{Example 9.5: James scores 80\% of all free throws he takes. James takes two shots.}\\ \text{Find the expected value, variance and standard deviation for this scenario.}\\ \text{Correct your
answers to 2 decimal points where appropriate}\\ \text{ }\\ \text{The probabilities for each outcome was calculated in example 9.3.}\\ \text{Let } X \text{ be the number of free throws James scores.}
\\ \text{ }\\ \begin{aligned} E(X) &= 0 \times 0.04 + 1 \times 0.32 + 2 \times 0.64\\ &= 0 + 0.32 + 1.28\\ &= 1.6\\ \text{ }\\ E(X^2) &= 0^2 \times 0.04 + 1^2 \times 0.32 + 2^2 \times 0.64\\ &= 2.88\
\ \text{ }\\ \text{Var}(X) &= E(x^2) - [E(X)]^2\\ &= 2.88 - 1.6^2\\ &= 0.32\\ \text{ }\\ \text{sd}(X) &= \sqrt{Var(X)}\\ \text{sd}(X) &= 0.57\\ \end{aligned} \] | {"url":"https://freevcenotes.com/methods/notes/discrete-probability","timestamp":"2024-11-11T21:08:19Z","content_type":"text/html","content_length":"30513","record_id":"<urn:uuid:477ea13d-0a82-44e7-8cb1-07e163f31a2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00271.warc.gz"} |
Synthesis of Control Protocols for Autonomous Systems
Tichakorn Wongpiromsarn, Ufuk Topcu and Richard M. Murray
Unmanned Systems, 1(1):21-39 (2013)
This article provides a review of control protocol synthesis techniques that incorporate methodologies from formal methods and control theory to provide correctness guarantee for different types of
autonomous systems, including those with discrete and continuous state space. The correctness of the system is defined with respect to a given specification expressed as a formula in linear temporal
logic to precisely describe the desired properties of the system. The formalism presented in this article admits non-determinism, allowing uncertainties in the system to be captured. A particular
emphasis is on alleviating some of the difficulties, e.g., heterogeneity in the underlying dynamics and computational complexity, that naturally arise in the construction of control protocols for
autonomous systems. | {"url":"https://murray.cds.caltech.edu/Synthesis_of_Control_Protocols_for_Autonomous_Systems","timestamp":"2024-11-14T19:09:19Z","content_type":"text/html","content_length":"17457","record_id":"<urn:uuid:cacfcdd4-2862-4b8c-91eb-8d7acea61e4b>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00045.warc.gz"} |
Coq (software) - Wikipedia
Coq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a
certified program from the constructive proof of its formal specification. Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions. Coq
is not an automated theorem prover but includes automatic theorem proving tactics (procedures) and various decision procedures.
An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right.
The Association for Computing Machinery awarded Thierry Coquand, Gérard Huet, Christine Paulin-Mohring, Bruno Barras, Jean-Christophe Filliâtre, Hugo Herbelin, Chetan Murthy, Yves Bertot, and Pierre
Castéran with the 2013 ACM Software System Award for Coq.
The name "Coq" is a wordplay on the name of Thierry Coquand, Calculus of Constructions or "CoC" and follows the French computer science tradition of naming software after animals (coq in French
meaning rooster).^[2] On October 11th, 2023, the development team announced that Coq will be renamed "The Rocq Prover" in the coming months, and has started updating the code base, website and
associated tools.^[3]
When viewed as a programming language, Coq implements a dependently typed functional programming language;^[4] when viewed as a logical system, it implements a higher-order type theory. The
development of Coq has been supported since 1984 by INRIA, now in collaboration with École Polytechnique, University of Paris-Sud, Paris Diderot University, and CNRS. In the 1990s, ENS Lyon was also
part of the project. The development of Coq was initiated by Gérard Huet and Thierry Coquand, and more than 40 people, mainly researchers, have contributed features to the core system since its
inception. The implementation team has successively been coordinated by Gérard Huet, Christine Paulin-Mohring, Hugo Herbelin, and Matthieu Sozeau. Coq is mainly implemented in OCaml with a bit of C.
The core system can be extended by way of a plug-in mechanism.^[5]
The name coq means 'rooster' in French and stems from a French tradition of naming research development tools after animals.^[6] Up until 1991, Coquand was implementing a language called the Calculus
of Constructions and it was simply called CoC at this time. In 1991, a new implementation based on the extended Calculus of Inductive Constructions was started and the name was changed from CoC to
Coq in an indirect reference to Coquand, who developed the Calculus of Constructions along with Gérard Huet and contributed to the Calculus of Inductive Constructions with Christine Paulin-Mohring.^[
Coq provides a specification language called Gallina^[8] ("hen" in Latin, Spanish, Italian and Catalan). Programs written in Gallina have the weak normalization property, implying that they always
terminate. This is a distinctive property of the language, since infinite loops (non-terminating programs) are common in other programming languages,^[9] and is one way to avoid the halting problem.
As an example, consider a proof of a lemma that taking the successor of a natural number flips its parity. The fold-unfold tactic introduced by Danvy^[10] is used to help keep the proof simple.
Ltac fold_unfold_tactic name := intros; unfold name; fold name; reflexivity.
Require Import Arith Nat Bool.
Fixpoint is_even (n : nat) : bool :=
match n with
| 0 =>
| S n' =>
eqb (is_even n') false
Lemma fold_unfold_is_even_0:
is_even 0 = true.
fold_unfold_tactic is_even.
Lemma fold_unfold_is_even_S:
forall n' : nat,
is_even (S n') = eqb (is_even n') false.
fold_unfold_tactic is_even.
Lemma successor_flips_evenness:
forall n : nat,
is_even n = negb (is_even (S n)).
intro n.
rewrite -> (fold_unfold_is_even_S n).
destruct (is_even n).
* simpl.
* simpl.
Four color theorem and SSReflect extension
Georges Gonthier of Microsoft Research in Cambridge, England and Benjamin Werner of INRIA used Coq to create a surveyable proof of the four color theorem, which was completed in 2002.^[11] Their work
led to the development of the SSReflect ("Small Scale Reflection") package, which was a significant extension to Coq.^[12] Despite its name, most of the features added to Coq by SSReflect are
general-purpose features and are not limited to the computational reflection style of proof. These features include:
• Additional convenient notations for irrefutable and refutable pattern matching, on inductive types with one or two constructors
• Implicit arguments for functions applied to zero arguments, which is useful when programming with higher-order functions
• Concise anonymous arguments
• An improved set tactic with more powerful matching
• Support for reflection
SSReflect 1.11 is freely available, dual-licensed under the open source CeCILL-B or CeCILL-2.0 license, and compatible with Coq 8.11.^[13]
• CompCert: an optimizing compiler for almost all of the C programming language which is largely programmed and proven correct in Coq.
• Disjoint-set data structure: correctness proof in Coq was published in 2007.^[14]
• Feit–Thompson theorem: formal proof using Coq was completed in September 2012.^[15]
• Busy beaver: The value of the 5-state winning busy beaver was discovered by Heiner Marxen and Jürgen Buntrock in 1989, but only proved to be the winning fifth busy beaver — stylized as BB(5) — in
2024 using a proof in Coq.^[16]^[17]
In addition to constructing Gallina terms explicitly, Coq supports the use of tactics written in the built-in language Ltac or in OCaml. These tactics automate the construction of proofs, carrying
out trivial or obvious steps in proofs.^[18] Several tactics implement decision procedures for various theories. For example, the "ring" tactic decides the theory of equality modulo ring or semiring
axioms via associative-commutative rewriting.^[19] For example, the following proof establishes a complex equality in the ring of integers in just one line of proof:^[20]
Require Import ZArith.
Open Scope Z_scope.
Goal forall a b c:Z,
(a + b + c) ^ 2 =
a * a + b ^ 2 + c * c + 2 * a * b + 2 * a * c + 2 * b * c.
intros; ring.
Built-in decision procedures are also available for the empty theory ("congruence"), propositional logic ("tauto"), quantifier-free linear integer arithmetic ("lia"), and linear rational/real
arithmetic ("lra").^[21]^[22] Further decision procedures have been developed as libraries, including one for Kleene algebras^[23] and another for certain geometric goals.^[24]
1. ^ "Release Coq 8.20.0". 3 September 2024.
2. ^ "Alternative names · coq/coq Wiki". GitHub. Retrieved 3 March 2023.
3. ^ Avigad, Jeremy; Mahboubi, Assia (3 July 2018). Interactive Theorem Proving: 9th International Conference, ITP 2018, Held as ... Springer. ISBN 9783319948218. Retrieved 21 October 2018.
4. ^ "Frequently Asked Questions". GitHub. Retrieved 2019-05-08.
5. ^ "Introduction to the Calculus of Inductive Constructions". Retrieved 21 May 2019.
6. ^ Adam Chlipala. "Certified Programming with Dependent Types": "Library Universes".
7. ^ Adam Chlipala. "Certified Programming with Dependent Types": "Library GeneralRec". "Library InductiveTypes".
8. ^ Danvy, Olivier (2022). "Fold–unfold lemmas for reasoning about recursive programs using the Coq proof assistant". Journal of Functional Programming. 32. doi:10.1017/S0956796822000107. ISSN
9. ^ Gonthier, Georges (2008). "Formal Proof—The Four-Color Theorem" (PDF). Notices of the American Mathematical Society. 55 (11): 1382–1393. MR 2463991.
10. ^ Gonthier, Georges; Mahboubi, Assia (2010). "An introduction to small scale reflection in Coq". Journal of Formalized Reasoning. 3 (2): 95–152. doi:10.6092/ISSN.1972-5787/1979.
11. ^ Conchon, Sylvain; Filliâtre, Jean-Christophe (2007). "A persistent union-find data structure". In Russo, Claudio V.; Dreyer, Derek (eds.). Proceedings of the ACM Workshop on ML, 2007, Freiburg,
Germany, October 5, 2007. Association for Computing Machinery. pp. 37–46. doi:10.1145/1292535.1292541.
12. ^ "Feit-Thompson theorem has been totally checked in Coq". Msr-inria.inria.fr. 2012-09-20. Archived from the original on 2016-11-19. Retrieved 2012-09-25.
13. ^ "[July 2nd 2024] We have proved "BB(5) = 47,176,870"". The Busy Beaver Challenge. 2024-07-02. Retrieved 2024-07-02.
14. ^ "The Busy Beaver Challenge". bbchallenge.org. Retrieved 2024-07-02.
15. ^ Kaiser, Jan-Oliver; Ziliani, Beta; Krebbers, Robbert; Régis-Gianas, Yann; Dreyer, Derek (2018-07-30). "Mtac2: typed tactics for backward reasoning in Coq". Proceedings of the ACM on Programming
Languages. 2 (ICFP): 78:1–78:31. doi:10.1145/3236773. hdl:21.11116/0000-0003-2E8E-B.
16. ^ Grégoire, Benjamin; Mahboubi, Assia (2005). "Proving Equalities in a Commutative Ring Done Right in Coq". In Hurd, Joe; Melham, Tom (eds.). Theorem Proving in Higher Order Logics: 18th
International Conference, TPHOLs 2005, Oxford, UK, August 22–25, 2005, Proceedings. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer. pp. 98–113. doi:10.1007/11541868_7. ISBN
17. ^ "The ring and field tactic families — Coq 8.11.1 documentation". coq.inria.fr. Retrieved 2023-12-04.
18. ^ Besson, Frédéric (2007). "Fast Reflexive Arithmetic Tactics the Linear Case and Beyond". In Altenkirch, Thorsten; McBride, Conor (eds.). Types for Proofs and Programs: International Workshop,
TYPES 2006, Nottingham, UK, April 18–21, 2006, Revised Selected Papers. Lecture Notes in Computer Science. Vol. 4502. Berlin, Heidelberg: Springer. pp. 48–62. doi:10.1007/978-3-540-74464-1_4.
ISBN 978-3-540-74464-1.
19. ^ "Micromega: solvers for arithmetic goals over ordered rings — Coq 8.18.0 documentation". coq.inria.fr. Retrieved 2023-12-04.
20. ^ Braibant, Thomas; Pous, Damien (2010). Kaufmann, Matt; Paulson, Lawrence C. (eds.). An Efficient Coq Tactic for Deciding Kleene Algebras. Interactive Theorem Proving: First International
Conference, ITP 2010 Edinburgh, UK, July 11-14, 2010, Proceedings. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer. pp. 163–178. doi:10.1007/978-3-642-14052-5_13. ISBN
978-3-642-14052-5. S2CID 3566183.
21. ^ Narboux, Julien (2004). "A Decision Procedure for Geometry in Coq". In Slind, Konrad; Bunker, Annette; Gopalakrishnan, Ganesh (eds.). Theorem Proving in Higher Order Logics: 17th International
Conference, TPHOLS 2004, Park City, Utah, USA, September 14–17, 2004, Proceedings. Lecture Notes in Computer Science. Vol. 3223. Berlin, Heidelberg: Springer. pp. 225–240. doi:10.1007/
978-3-540-30142-4_17. ISBN 978-3-540-30142-4. S2CID 11238876. | {"url":"http://en.m.wiki.x.io/wiki/Coq_(software)","timestamp":"2024-11-14T07:33:59Z","content_type":"text/html","content_length":"123927","record_id":"<urn:uuid:138ebd3f-3cf7-4591-a6fa-80fd19188ac0>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00213.warc.gz"} |
What you need to know about data augmentation for machine learning | R-bloggersWhat you need to know about data augmentation for machine learning
What you need to know about data augmentation for machine learning
[This article was first published on
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Plentiful high-quality data is the key to great machine learning models. But good data doesn’t grow on trees, and that scarcity can impede the development of a model. One way to get around a lack of
data is to augment your dataset. Smart approaches to programmatic data augmentation can increase the size of your training set 10-fold or more. Even better, your model will often be more robust (and
prevent overfitting) and can even be simpler due to a better training set.
There are many approaches to augmenting data. The simplest approaches include adding noise and applying transformations on existing data. Imputation and dimensional reduction can be used to add
samples in sparse areas of the dataset. More advanced approaches include simulation of data based on dynamic systems or evolutionary systems. In this post we’ll focus on the two simplest approaches:
adding noise and applying transformations.
Regression Problems
In many datasets we expect that there is unavoidable statistical noise due to sampling and other factors. For regression problems, we can explicitly add noise to our explanatory variables. Doing so
can make the model more robust, although we need to take care when constructing the noise term. First, we want to avoid adding bias. We also need to ensure the noise is independent.
In my function approximation example, I demonstrated creating a simple neural network in Torch to approximate a function. Let’s use the same dataset and function but this time add noise as a
preprocessing step. Before, I generated the training data directly in Torch. However, in my simple workflow for deep learning, I said I prefer using R for everything but training the model. Hence, I
generate and add noise in R and then write out a headerless CSV file. This process expands the training set from 40k samples to ~160k samples.
The following function accomplishes this and uses an arbitrary function to add noise. The default function adds uniform noise.
perturb_unif <- function(df, mult=4, fr=function(a) a + runif(length(a)) - .5) {
fn <- function(i) data.frame(x=fr(df[,1]), y=fr(df[,2]), z=df[,3])
o <- lapply(1:mult, fn)
do.call(rbind, o)
Exercise: What is the purpose of the -0.5 term?
Exercise: What are some other valid noise functions? Why would you choose one over another?
I then read this file into Lua using a custom function loadTrainingSet, which is part of my deep_learning_ex guide. This function reads the CSV and creates a Torch-compatible table comprising the
input and output tensors. The function simply assumes that the last column of the CSV is the output.
Using this approach, it’s possible to create a 20 hidden node network that performs as well as the 40 node network in the earlier post. Think about this: by adding noise (and increasing the size of
the training set), we’ve managed to reduce the complexity of the network 2-fold.
Kool-Aid Alert: Depending on the domain and hyper parameters chosen, this approach may not produce desirable results. As with most deep learning exercises, tuning of hyper parameters is mandatory.
Classification Problems
Noise can be used in classification problems as well. One particularly useful application is balancing a dataset. In a binary classification problem, suppose the data is split 80-20 between two
classes. It is well known that such an unbalanced set is problematic for machine learning algorithms. Some will even just default to the class with the higher proportion of observations, since the
naïve accuracy is reasonable. In small datasets balancing the dataset by trimming can be counterproductive. The alternative is to increase the samples in the smaller class. The same noise
augmentation approach works well here.
Another group of classification problem is image classification. Following a similar approach, noise can be added to images, which can make the model more robust. Another popular technique is
transforming the data. This makes sense for images since changes in perspective can change the apparent shape of an object. Transparent or reflective objects can also distort objects, but despite
this distortion, we know the object to be the same. Affine transformations provide simple linear transforms that can expand a dataset. This includes shifting, scaling, rotating, flipping, etc. While
a good starting point, andsome problems might benefit from more complex transformations.
Transformed images can be generated in a pre-processing step as above. The Torch image toolkit can be used this way. For example, here is how to flip an image horizontally.
img = image.load("cat.jpg",3,'byte')
img1 = image.hflip(img)
Alternatively, these variations can be generated inline during the training process. Keras uses this approach with the ImageDataGenerator class. This means that images are transformed on the fly
during training.
Exercise: Which approach will produce better results? Why?
Just because you don’t have as much data as Google or Facebook doesn’t mean you should give up on machine learning. By augmenting your dataset, you can get excellent results with small data.
Use approaches not mentioned above to augment your data? Share in the comments below. | {"url":"https://www.r-bloggers.com/2016/10/what-you-need-to-know-about-data-augmentation-for-machine-learning/","timestamp":"2024-11-06T19:57:20Z","content_type":"text/html","content_length":"100430","record_id":"<urn:uuid:1de2e0bd-fcb5-43c3-b4dc-1d9a60453f7a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00848.warc.gz"} |
Convert Gas m3 Units to kWh | Gas Calculator | TheEnergyShop
How to convert gas units to kWh
Our gas meter reading calculator is a great way to quickly and easily see how much your gas meter readings equate to in kilowatt hours (kwh) by entering in the units used on your imperial or metric
gas meter. Whether you're a business energy or domestic energy customer, we've got you covered.
Converting gas units into kWh in simple steps
This is the easy bit without the detailed explanation.
To convert imperial gas meter readings to kWh:
• Take a meter reading.
• Subtract the new meter reading from the previous reading to work out the volume of gas used.
• Convert from cubic feet to cubic meters by multiplying by 0.0283 OR dividing by 35.315.
• Multiply by the volume correction factor (1.02264).
• Multiply by calorific value (40.0).
• Divide by kWh conversion factor (3.6).
To convert metric gas meter readings to kWh:
• Take a meter reading.
• Subtract the new meter reading from the previous reading to work out the volume of gas used.
• Multiply by the volume correction factor (1.02264).
• Multiply by calorific value (40.0).
• Divide by kWh conversion factor (3.6).
Converting gas units into kWh - the super simple way
The multiplication/division steps can be combined them into a single step. This simplification assumes that the volume correction factors and calorific values for your property are constant at
1.02264 and 40 respectively. This reduces the calculation as follows.
To convert imperial gas meter readings to kWh:
• Take a meter reading.
• Subtract the new meter reading from the previous reading to work out the volume of gas used.
• Convert from cubic feet to kilowatt hours (multiply by 31.6586).
To convert metric gas meter readings to kWh:
• Take a meter reading.
• Subtract the new meter reading from the previous reading to work out the volume of gas used.
• Convert from cubic meters to kilowatt hours (multiply by 11.1868).
When checking your energy bills, or doing a energy price comparison, you need to know how much energy you use.
In the UK, units of energy, whether gas or electricity, are priced in kilowatts hours (kWh).
Electricity meters measure units of electricity consumed in kilowatt hours (a measure of electrical energy). This is great because it means that you can simply pop the difference between meter
readings from your meter, or your energy bill usage stats, directly into our energy comparison calculator to check your best deals.
With gas however it is not so simple. That is because gas meters measure the volume of gas delivered to your meter (and not the energy content directly). This volume then needs to be converted into
the equivalent kilowatt hours in order that a kWh price per unit can be applied to the energy used for the purpose of generating the bill (or quote).
A further complication is that not all meters measure volume in the same units.
When it comes to comparing gas prices and measuring gas volumes, there are basically 2 types of gas meter in the UK - imperial and metric.
Metric Gas Meters
Metric gas meters were introduced from 1995 and all new gas meters are metric. They measure gas volume in cubic meters (m ^3). An example of a metric gas meter is shown below. Metric gas meters will
display the units M3 next to the reading on the meter.
Imperial Gas Meters
Luckily, these are being phased out and, with the move to smart metering, should all be gone by 2022. However, a significant number are still in existence. These meters measures volume in cubic feet
(ft ^3). They have the words "cubic feet" or the letters ft ^3 shown on the front of the meter, and will typically have revolving dials showing the reading.
Converting gas units to kWh - the detailed bit
Once you've identified the type of meter you have and taken a reading, follow the steps below to convert from gas units into kilowatt hours (kWh).
In this part we will take you through the conversion in stages detailing why each step is necessary. If you are not too fussed about the detail, then just scroll to the bottom of the page where we
have simplified it significantly.
The detail of how the gas calculations need to be done are detailed in a piece of legislation called the The Gas (Calculation of Thermal Energy) Regulations 1996. If you really want to, you can read
it here.
Step 1
Take a meter reading.
Not required if you are taking a reading from a gas bill.
Step 2
Subtract the new meter reading from the previous reading to work out the volume of gas used.
Step 3
This step only applies to imperial meters.
Convert from cubic feet to cubic meters.
Cubic feet and cubic meter are both measures of volume. However, since we will ultimately be expressing units in metric form we need to get the units into metric form.
1 cubic foot = 0.0283 cubic meters. So, either multiply by 0.0283 OR divide by 35.315.
Step 4
As we know from GSCE Physics (or is it Chemistry?) (Boyle's Law) the volume of a gas is dependent upon pressure and temperature and these vary from location to location. Therefore the volume of gas
measured at the meter is adjusted by multiplying by a correction factor to take account of the temperature and atmospheric conditions at a particular site. That correction factor is typically 1.02264
unless your property has unusual atmospheric conditions. The correction factor used will be shown on your gas bill.
Step 5
Now that we have a corrected volume in metric terms we can begin the conversion from volume to thermal energy. The first step used is to convert the volume into a calorific value.
Calorific Value (CV) is a measure of the heating power of the gas and depends upon the composition of the gas (all gas is not the same). The CV refers to the amount of energy released when a given
volume of gas is completely combusted under specified conditions. The CV of the gas is measured at standard conditions of temperature and pressure and is usually quoted in Megajoules per cubic metre
(MJ/m ^3). In the National Grid the CV ranges from 37.5 to 43.0 MJ/ ^3. Not all gas has the same CV so National Grid measures it across 110 different locations across the UK and passes it to energy
suppliers for billing purposes. For the purposes of this calculation an average value of 40 will give you an accurate result but check your bill to see if a different number has been used.
If you'd like to learn more about calorific values you can find information here.
Step 6
The final step is to take thermal energy measured in Mega joules per cubic metre (MJ/m ^3) and convert it into kilowatts per hour.
1 Watt is a derived unit of power defined as 1 Joule per second.
1 W = 1 J/s
1 Ws = 1 J
1 kWh = 1 * 1000 * 60 * 60 J
1 kWh = 3.6 MJ
So the final step is to divide by 3.6 to convert MJ into kWh
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Can Moneyball save the unicorn?
The answer is no, but Moneyball might be a nudge in the right direction. I was born and live in Greenland where unicorns (also known as narwhals) roam beneath the waves. In most other cultures,
unicorns are horsy creatures who roam somewhere over the rainbow. In western cultures (as defined by those who read Harry Potter) killing a unicorn is an unspeakably evil deed. It used to be
different, at least for the sea creature.
Long expeditions sent north by Scandinavian kings brought back unicorn horns to make, amongst other things, drinking mugs; the theory being that unicorn horns would eliminate any poison. We have a
somewhat more pragmatic approach. Whales are food, and narwhals have very delicious skin (much better than any other whale skin). Unlike pork crackling, we eat it raw. This makes narwhals a very
valuable prey.
The author eating mattak – raw skin and blubber of a freshly harvested narwhal.
Whenever a resource is precious, there is a risk of overharvesting and politics rears its ugly head. In Greenland, restricting harvest is politically equivalent to restricting farming subsidies in
the EU, or limiting gun ownership in the US. History judges the politician making the decisions, but voters employ them. History is measured decades, centuries, and millennia, election cycles are 4
years. Given the choice between being lauded by history or reelected by voters, the logical choice for a politician is continued employment and power.
In an article in our news, a department head at Greenland Institute of Natural Resources likened the east Greenland narwhal quotas to gambling with the future of narwhals in east Greenland (please
note that not all narwhal populations are threatened to that extent). The current quotas come with a hefty 50% chance of extinction in a decade.
My response was “DUH, all management is a gamble” – though usually choosing much better odds. I realize that my reaction is that of a guy who spent years acquiring a hammer and who now sees nails
everywhere. But still, there are a couple of interesting points to be considered here.
In this system, politicians set quotas; scientists estimate risks. By starting with a quota, scientists end up trying to answer the question: “At this quota, what are the risks we run?”
If instead, politicians answered the question: “How big a risk are we willing to take?” Then, scientists could work to translate that into a quota. Such a system is in use in some places, taking the
politicians one step away from quota decisions.
An even better approach would be to start by deciding what exactly we are trying to accomplish. Asking for an optimal management strategy is like asking for the best spaghetti sauce! If there was a
best sauce, each brand would have only one variety. Malcom Gladwell has a very interesting explanation of this insight.
Photo of full size replica of narwhal at the Greenland Institute of Natural Resources
We can ask for the maximum economic value, the maximum stable economic value (avoiding boom and bust economies), the maximum cultural value, the maximum sustainable yield (the most common option), or
whichever goal we (ideally a diverse group of stakeholders) find appealing. Of course, politicians will typically choose the option that keeps them in power, regardless of the bigger issues. Hence, a
50% chance of extinction in a decade is a better bet than the near-term political fallout of a moratorium – in terms of the politician’s personal career risk.
An interesting parallel here is the climate debate. It is an exceedingly complex issue with a core concept not unlike the narwhale question. The solution to communicating it has been impressively
handled. Instead of looking at what we expect to happen if we cut CO2 emissions from cars in half, and cut emissions from cows to a quarter, the whole question has been turned around and tied to a
single number: 1.5 Degrees Celsius. That number has been sold as the cutoff between manageable warming and catastrophe. The rest is just math – how much CO2 (and CO2 equivalents) must be cut to get
there. Even if the math converting CO2 to degrees warming is itself a bit complicated, it can be summarized in a neat graph. Of course, the political ramifications are much harder to deal with than
the math.
A complicating factor is that real world outcomes are stochastic. A quota does not change the outcome directly, it changes the probabilities of various outcomes. All we can do is work to understand
the probabilities and make our choices accordingly. Only one tiny snag — people are not good at understanding stochastic processes or using probabilities in decision making.
This is where Moneyball comes in. Using Brad Pitt to explain how probability theory works in real life is likely the best PR statistics can get. Central to the movie is the idea of boiling down a
baseball player to a single number. The On Base Percentage (OBP) is the probability that they get on base when they are at bat (pedantically it is the best estimate of that probability). In the long
run, the expected value of that probability prevails. If the average OBP of your top 3 players is higher than the OBP for the top three players of the opposing team then, on average, you win.
Fact 1, even if Brad Pitt is the main character, the real hero is the geek with the glasses (you may disagree with this, but this is my post). Fact 2, the main point of this movie is that probability
theory can be useful in choosing a course of action.
Since a lot of readers of this blog are statisticians, the question we should ask ourselves is: Do we have a responsibility to educate the public about probability and statistics in a better way?
That argument has been made on this blog before.
Some of us end up teaching statistics at one point or another. Perhaps we should ask ourselves if we teach the right things. A lot of statistics classes are centered on how to calculate quantities.
How do you calculate a probability of some event? Maybe the better question is: “so what?”
We can calculate the likelihood of severe disease with or without a vaccine, but if the vaccine hesitant can’t use that information to decide between getting or not getting the jab, then we have
failed to educate. We can calculate the risk of a particular management strategy, but if we report that risk as a probability, and few people have a good understanding of probabilities, then what is
the point?
In Greenland, we have unicorns (we also have rainbows, but that is another story), we have data for managing them, and we have policymakers. A statement like “The current quotas come with a 50%
chance of extinction in a decade” should be of real concern; but if it sounds like “Bla, Bla, Bla” to the people who need to hear it the most, we have an even bigger problem. | {"url":"https://isi-web.org/article/can-moneyball-save-unicorn","timestamp":"2024-11-09T10:19:29Z","content_type":"text/html","content_length":"103245","record_id":"<urn:uuid:cb0bd541-a44d-4c9c-aa82-998acc8a466e>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00043.warc.gz"} |
Function template integrate_const
boost::numeric::odeint::integrate_const — Integrates the ODE with constant step size.
// In header: <boost/numeric/odeint/integrate/integrate_const.hpp>
template<typename Stepper, typename System, typename State, typename Time,
typename Observer, typename StepOverflowChecker>
size_t integrate_const(Stepper stepper, System system, State & start_state,
Time start_time, Time end_time, Time dt,
Observer observer, StepOverflowChecker checker);
Integrates the ODE defined by system using the given stepper. This method ensures that the observer is called at constant intervals dt. If the Stepper is a normal stepper without step size control,
dt is also used for the numerical scheme. If a ControlledStepper is provided, the algorithm might reduce the step size to meet the error bounds, but it is ensured that the observer is always called
at equidistant time points t0 + n*dt. If a DenseOutputStepper is used, the step size also may vary and the dense output is used to call the observer at equidistant time points. If a max_step_checker
is provided as StepOverflowChecker, a no_progress_error is thrown if too many steps (default: 500) are performed without progress, i.e. in between observer calls. If no checker is provided, no such
overflow check is performed.
checker [optional] Functor to check for step count overflows, if no checker is provided, no exception is thrown.
dt The time step between observer calls, not necessarily the time step of the integration.
end_time The final integration time tend.
observer [optional] Function/Functor called at equidistant time intervals.
start_state The initial condition x0.
start_time The initial time t0.
stepper The stepper to be used for numerical integration.
system Function/Functor defining the rhs of the ODE.
Returns: The number of steps performed. | {"url":"https://live.boost.org/doc/libs/1_82_0/libs/numeric/odeint/doc/html/boost/numeric/odeint/integrate_const_idm23173.html","timestamp":"2024-11-14T03:47:24Z","content_type":"text/html","content_length":"10670","record_id":"<urn:uuid:47f2409e-2b53-47bb-b1d5-05f5899d557e>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00139.warc.gz"} |
Universality of cutoff for the ising model
On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature β is small enough, via classical results of Dobrushin and of Holley in the 1970s. By
a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising model has been confirmed mainly for lattices, heavily relying on
amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed β > 0, no matter how small, even in basic examples such as the Ising model on a binary tree or a random
regular graph. We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising model at high enough temperatures. Precisely, on any sequence of graphs
with maximum degree d, the Ising model has cutoff provided that β < κ/d for some absolute constant κ (a result which, up to the value of κ, is best possible). Moreover, the cutoff location is
established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is O(1), just as when β = 0. Finally, the mixing time from almost every initial state is not more
than a factor of 1 + ∈[β] faster then the worst one (with ∈[β] →0 as β →0), whereas the uniform starting state is at least 2 -∈[β] times faster.
All Science Journal Classification (ASJC) codes
• Statistics and Probability
• Statistics, Probability and Uncertainty
• Cutoff phenomenon
• Ising model
• Mixing times of Markov chains
Dive into the research topics of 'Universality of cutoff for the ising model'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/universality-of-cutoff-for-the-ising-model","timestamp":"2024-11-10T12:06:52Z","content_type":"text/html","content_length":"49681","record_id":"<urn:uuid:238400ee-3196-4d60-a9a7-e36acf597ece>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00415.warc.gz"} |
Strong shock waves in infinitely electrically conducting relativistic fluids
The investigation has the objective to obtain an explicit form concerning the interdependence relations for the physical quantities on the two sides of a strong shock wave. The fundamental equations
of relativistic mechanics of infinitely electrically conducting, inviscid, compressible fluids are examined. A 4-dimensional vector, called the relativistic magnetic vector, is used in place of the
second rank tensors of electromagnetic theory. This approach makes it possible to formulate the fundamental equations in a much more convenient form. Attention is given to the basic shock relations,
contact discontinuities, and perpendicular shock waves.
Journal of Mathematical Analysis and Applications
Pub Date:
December 1976
□ Conducting Fluids;
□ Magnetohydrodynamics;
□ Relativistic Plasmas;
□ Shock Wave Propagation;
□ Compressible Fluids;
□ Inviscid Flow;
□ Normal Shock Waves;
□ Shock Discontinuity;
□ Tensors;
□ Wave Equations;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1976JMAA...56..718S/abstract","timestamp":"2024-11-08T00:11:47Z","content_type":"text/html","content_length":"34998","record_id":"<urn:uuid:f1a0f7ac-5ab9-4a50-b973-6b4b9c494d89>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00591.warc.gz"} |
Evolutionary Computation and Constraint Satisfaction
In this chapter we will focus on the combination of evolutionary computation techniques and constraint satisfaction problems. Constraint Programming (CP) is another approach to deal with constraint
satisfaction problems. In fact, it is an important prelude to the work covered here as it advocates itself as an alternative approach to programming (Apt). The first step is to formulate a problem as
a CSP such that techniques from CP, EC, combinations of the two (c.f., Hybrid) or other approaches can be deployed to solve the problem. The formulation of a problem has an impact on its complexity
in terms of effort required to either find a solution or proof no solution exists. It is therefore vital to spend time on getting this right.
Main differences between CP and EC. CP defines search as iterative steps over a search tree where nodes are partial solutions to the problem where not all variables are assigned values. The search
then maintain a partial solution that satisfies all variables assigned values. Instead, in EC most often solver sample a space of candidate solutions where variables are all assigned values. None of
these candidate solutions will satisfy all constraints in the problem until a solution is found. Another major difference is that many constraint solvers from CP are sound whereas EC solvers are not.
A solver is sound if it always finds a solution if it exists. | {"url":"https://groups.inf.ed.ac.uk/nesc-research/node/9989ba9.html?page=3","timestamp":"2024-11-10T11:30:46Z","content_type":"application/xhtml+xml","content_length":"24629","record_id":"<urn:uuid:a3e4c7cc-efa7-47e2-9cd2-29a8adea8c92>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00854.warc.gz"} |
Symmetric encryption with Caesar Cypher
The Caesar Cypher is a simple encryption technique that is around 2000 years old. It is categorized as a symmetric encryption type since it requires a private key that is used for both encryption and
decryption. How does it work? This cypher algorithm is fairly simple, given an alphabet, each letter corresponds to a number. The key also translates to a number. Then each character of the secret
message is shifted n number of times where n is the key number, and the character is substituted with the alphabet letter in the shifted position.
For instance the letters might be inserted in an array, therefore each letter will have a corresponding index position.
Suppose the secret message has the letter A in it and the key is 2 then the letter A will be shifted by two and becoming C.
Here follows my implementation in typescript. I have decided to make the cypher case sensitive and also replace empty spaces.
constructor() { this.alphanumericString = 'ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789 '; this.operations = new Map(); this.operations.set('decrypt', (position: number, keyCode:
number) => { return position - keyCode; }); this.operations.set('encrypt', (position: number, keyCode: number) => { return position + keyCode; }); }
The constructor initializes the alpha numeric string of characters supported by this implementation. In addition to that it defines the operations for encrypting shift forward and decrypting shift
/** * Helper method to calculated modulo for negative numbers * (required due to a javascript modulo bug) * @returns */ private modulo(x: number, y: number) { return ((x % y) + y) % y; }
Javascript built in modulo is not really a modulo function but rather a reminder. This helper methods allows % to work with negative integers.
private keyCode(key: string): number { if (key.length > 1) { throw new Error('Caesar Cypher Key is too long'); } if (!key) { throw new Error('Invalid Caesar Cypher key') } const keyCode: number =
this.alphanumericString.indexOf(key); if (keyCode < 0) { throw new Error('Invalid Caesar Cypher key') } return keyCode; }
This method is responsible for returning the key code number. The function takes in a character and looks up the char position in the alpha numeric string variable.
private caesarCypher(text: string, key: string, operation: string) { // calculate the key number const keyCode: number = this.keyCode(key); // cast the text string to an array of string with single
char const textChars: Array<string> = text.split(''); // loop the char array return textChars.map((char) => { // find the index position in the alphanumeric string const position: number =
this.alphanumericString.indexOf(char); // throw an error if the char is not found if (position < 0) { throw new Error(`char: ${char} not supported`); } // calculate the cypher position const
cypherPosition: number = this.modulo(this.operations.get(operation)!.call(this, position, keyCode), this.alphanumericString.length); return this.alphanumericString[cypherPosition]; }) .reduce((prev,
next) => { return prev + next; }); }
The caesarCypher function holds the logic of the algorithm. Once the key code is calculated then the function operates the character shifting. Both encrypt and decrypt follows the same logic
exception made for the shifting. Encrypt shifts forward while decrypt shifts backwards. The method uses the operations variable that is defined in the constructor to determine the shift direction.
encrypt(text: string, key: string): string { return this.caesarCypher(text, key, 'encrypt'); } decrypt(text: string, key: string): string { return this.caesarCypher(text, key, 'decrypt'); }
The snippets above shows how encryption and decryption becomes fairly simple to invoke.
test('it should allow to encrypt a valid text with a valid key', () => { const encryptedText: string = caesarCypher.encrypt('A message of love', 'M'); expect(encryptedText).toBe('MLyq44msqL0rLx07q');
}); test('it should allow to decrypt and decrypt a valid text with a valid key', () => { const decryptedText: string = caesarCypher.decrypt('MLyq44msqL0rLx07q', 'M'); expect(decryptedText).toBe('A
message of love'); });
The tests shows how this caesarCypher implementation can be used to encrypt and decrypt messages. The complete source code for this example can be found here. | {"url":"https://www.coderpunk.tech/Caesar-Cypher-symmetric-encryption","timestamp":"2024-11-02T19:04:18Z","content_type":"text/html","content_length":"94701","record_id":"<urn:uuid:2dcfa80d-2cac-4c2e-abac-b435744e07a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00438.warc.gz"} |
Effluent BOD Concentration Calculation for Pharmaceutical Manufacturing
06 Oct 2024
Popularity: ⭐⭐⭐
Effluent BOD Concentration in Pharmaceutical Manufacturing
This calculator provides the calculation of effluent BOD concentration for pharmaceutical manufacturing applications.
Calculation Example: The effluent BOD concentration is an important parameter in pharmaceutical manufacturing as it indicates the amount of organic matter remaining in the wastewater after treatment.
It is calculated using the formula Qe = q * (1 - e/100), where q is the influent BOD concentration, e is the BOD removal efficiency, and Qe is the effluent BOD concentration.
Related Questions
Q: What is the importance of BOD removal in pharmaceutical manufacturing?
A: BOD removal is important in pharmaceutical manufacturing as it helps to reduce the amount of organic matter discharged into the environment. This helps to protect aquatic life and prevent water
Q: How does BOD removal efficiency affect the effluent BOD concentration?
A: BOD removal efficiency has a direct impact on the effluent BOD concentration. A higher BOD removal efficiency will result in a lower effluent BOD concentration.
Symbol Name Unit
q Influent BOD Concentration mg/L
Q Influent Flow Rate m^3/d
e BOD Removal Efficiency %
Calculation Expression
Qe Function: The effluent BOD concentration is given by Qe = q * (1 - e/100).
Calculated values
Considering these as variable values: q=200.0, Q=1000.0, e=85.0, the calculated value(s) are given in table below
Derived Variable Value
Qe Function 30.0
Similar Calculators
Calculator Apps | {"url":"https://blog.truegeometry.com/calculators/Pharmaceutical_manufacturing_calculation.html","timestamp":"2024-11-04T21:45:34Z","content_type":"text/html","content_length":"17889","record_id":"<urn:uuid:8ea608dd-4f1b-4f56-bc39-f396594cb1ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00231.warc.gz"} |