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Find the equation of the tangent and normal to the curve y=cosx at x=π/3? | Socratic
Find the equation of the tangent and normal to the curve y=cosx at x=π/3?
1 Answer
Tangent's equation is $y = - \frac{\sqrt{3}}{2} x + \frac{\pi}{2 \sqrt{3}} + \frac{1}{2}$
Normal's equation is $y = \frac{2}{\sqrt{3}} x - \frac{2 \pi}{3 \sqrt{3}} + \frac{1}{2}$
First, you find the first derivative of the function then substitute with the given point $\left({x}_{1} , {y}_{1}\right)$ to get the slope$\left(m\right)$ and then substitute in this function
apply for the given function
$y = \cos x$
At $x = \frac{\pi}{3}$$\rightarrow$$y = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$
$\left({x}_{1} , {y}_{1}\right) =$$\left(\frac{\pi}{3} , \frac{1}{2}\right)$
$\frac{\mathrm{dy}}{\mathrm{dx}} = - \sin x$
Substitute with the point $\left(\frac{\pi}{3} , \frac{1}{2}\right)$
$\frac{\mathrm{dy}}{\mathrm{dx}} = {m}_{1} = - \sin \left(\frac{\pi}{3}\right) = - \frac{\sqrt{3}}{2}$
Now Substitute in the formula in green with $\left({x}_{1} , {y}_{1}\right) , {m}_{1}$
to get the equation of the tangent to the curve at the given point
$y - \frac{1}{2} = - \frac{\sqrt{3}}{2} \left(x - \frac{\pi}{3}\right)$
$y = - \frac{\sqrt{3}}{2} x + \frac{\pi}{2 \sqrt{3}} + \frac{1}{2}$
the normal to the curve would be at the same point but its slope is different and in order to get its slope ${m}_{2}$
We use this
${m}_{1} \cdot {m}_{2} = - 1$
${m}_{2} = \frac{2}{\sqrt{3}}$
Now Substitute with $\left({x}_{1} , {y}_{1}\right) , {m}_{2}$ in the equation in green to find the equation of the normal
$y - \frac{1}{2} = \frac{2}{\sqrt{3}} \left(x - \frac{\pi}{3}\right)$
$y = \frac{2}{\sqrt{3}} x - \frac{2 \pi}{3 \sqrt{3}} + \frac{1}{2}$
Impact of this question
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Implementations of Monte Carlo and Temporal Difference learning.
In the previous post, I discussed two different learning methods for reinforcement learning, Monte Carlo learning and temporal difference learning. I then provided a unifying view by considering
$n$-step TD learning and establishing hybrid learning method, $TD\left( \lambda \right)$. These methods are mainly focused on how to approximate a value function from experience. I introduced a new
value function, $Q\left( {s,a} \right)$, that looks at the value of state-action pairs in order to achieve model-free learning. For all approaches, learning takes place by policy iteration, as shown
We start off by following an initial policy and explore the environment. As we observe the results of our actions, we can update our estimate of the true value function that describes the
environment. We can then update our policy by querying the best actions to take according to our newly estimated value function. Ideally, this cycle continues until we've sufficiently explored the
environment and have converged on an optimal policy.
In this post, I'll discuss specific implementations of the Monte Carlo and TD approach to learning. However, I first must introduce the concept of on-policy and off-policy learning. A method is
considered on-policy if the agent follows the same policy which it is iteratively improving. However, there are cases where it would be more useful to have the agent follow a behavior policy while
learning a target policy. Such methods, where the agent doesn't follow the same policy that it is learning, are considered off-policy.
Monte Carlo learning
The Monte Carlo approach approximates the value of a state-action pair by calculating the mean return from a collection of episodes.
GLIE Monte-Carlo
GLIE Monte-Carlo is an on-policy learning method that learns from complete episodes. For each state-action pair, we keep track of how many times the pair has been visited with a simple counter
$$ N\left( {{S_t},{A_t}} \right) \leftarrow N\left( {{S_t},{A_t}} \right) + 1 $$
For each episode, we can update our estimated value function using an incremental mean.
$$ Q\left( {{S_t},{A_t}} \right) \leftarrow Q\left( {{S_t},{A_t}} \right) + \frac{1}{{N\left( {{S_t},{A_t}} \right)}}\left( {{G_t} - Q\left( {{S_t},{A_t}} \right)} \right) $$
Here, $G_t$ either represents the return from time $t$ when the agent first visited the state-action pair, or the sum of returns from each time $t$ that the agent visited the state-action pair,
depending on whether you are using first-visit or every-visit Monte Carlo.
We'll adopt an $\epsilon$-greedy policy with $\epsilon=\frac{1}{k}$ where $k$ represents the number of episodes our agent has learned from.
Temporal difference learning
The temporal difference approach approximates the value of a state-action pair by comparing estimates at two points in time (thus the name, temporal difference). The intuition behind this method is
that we can formulate a better estimate for the value of a state-action pair after having observed some of the reward that our agent accumulates after having visited a state and performing a given
Sarsa is an on-policy $TD\left(0 \right)$ learning method. The name refers to the State-Action-Reward-State-Action sequence that we use to update our action value function, $Q\left( {s,a} \right).$
We start off in state $S$ and perform action $A$. This state-action pair yields some reward $R$ and brings us to a new state, $S'$. We then follow our policy to perform action $A'$ and look up the
value of this future trajectory according to $Q\left( {S',A'} \right)$ (taking advantage of the fact that we define the value of a state-action pair to incorporate the value of the entire future
sequence of states following our policy).
Photo by Richard Sutton
We'll define our policy as $\epsilon$-greedy with respect to $Q\left( {s,a} \right)$, which means that our agent will have a $1 - \epsilon$ probability of performing the optimal action in state $S'$
and an $\epsilon$ probability of performing a random exploratory action.
The value function update equation may be written as
$$ Q\left( {S,A} \right) \leftarrow Q\left( {S,A} \right) + \alpha \left( {\left( {R + \gamma Q\left( {S',A'} \right)} \right) - Q\left( {S,A} \right)} \right) $$
where we include one step of experience combined with a discounted estimate for future returns as our TD target.
Q-learning is an off-policy $TD\left(0 \right)$ learning method where the agent's next action is chosen according to a behavior policy but we evaluate state-action pairs according to a target policy.
This has the effect of allowing our agent to explore while maintaining the highest possible accuracy when comparing our expected state-action value to a target.
We'll define our behavior policy, the policy that determines the agent's actions, as $\epsilon$-greedy with respect to $Q\left( {s,a} \right)$. However, we'll also define a target policy as greedy
with respect to $Q\left( {s,a} \right)$.
Rather than looking at a sequence of State-Action-Reward-State-Action, we instead take a sequence of State-Action-Reward-State and refer to our target policy for the subsequent action. Because our
target policy is greedy, there is no stochastic behavior ($\epsilon$ exploration) present to degrade the accuracy of our TD target.
Photo by Richard Sutton
The value function update equation for Q-learning may be written as
$$ Q\left( {S,A} \right) \leftarrow Q\left( {S,A} \right) + \alpha \left( {\left( {R + \gamma \mathop {\max }\limits_{a'} Q\left( {S',A'} \right)} \right) - Q\left( {S,A} \right)} \right) $$
where we similarly include one step of experience combined with a discounted estimate for future returns as our TD target; however, we've removed the exploratory behavior from our estimate for future
returns in favor of selecting the optimal trajectory, increasing the accuracy of our TD target.
Hybrid approach to learning
$n$-step Sarsa
So far, we've discussed methods which seek a more informed valuation of a state-action pair by two main approaches, a one-step lookahead combined with a guess (the temporal difference approach) and a
full sequence lookahead (the Monte Carlo approach). Let us now consider the spectrum of algorithms that lie between these two methods by expanding our Sarsa implementation to include $n$-step
The general Sarsa algorithm considers the case of $n=1$, where our more informed valuation of a state-action pair $q_t^{\left( 1 \right)}$ at time $t$ may be written as:
$$ q_t^{\left( 1 \right)} = {R_t} + \gamma Q\left( {{S_{t + 1}}} \right) $$
At the other end of the spectrum, the Monte Carlo approach, which considers a full sequence of observed rewards until the agent reaches a terminal timestep $T$, may be written as:
$$ q_t^{\left( \infty \right)} = {R_t} + \gamma {R_{t + 1}} + ... + {\gamma ^{T - 1}}{R_T} $$
However, we could develop $q_t^{\left( n \right)}$ to generally represent a target value which incorporates $n$ steps of experience combined with a guess for the value of the future trajectory from
that point onward.
$$ q_t^{\left( n \right)} = {R_t} + \gamma {R_{t + 1}} + {\gamma ^{n - 1}}{R_{t + n - 1}} + {\gamma ^n}Q\left( {{S_{t + n}}} \right) $$
Photo by Richard Sutton
The $n$-step Sarsa implementation is an on-policy method that exists somewhere on the spectrum between a temporal difference and Monte Carlo approach.
The value function update equation may be written as
$$ Q\left( {S,A} \right) \leftarrow Q\left( {S,A} \right) + \alpha \left( {q_t^{\left( n \right)} - Q\left( {S,A} \right)} \right) $$
where $q_t^{\left( n \right)}$ is the general $n$-step target we defined above.
In the last post, I discussed a general method for combining the information from all of the $n$-step returns into a single collective return by calculating the geometric sum of the returns. Recall
that this approach is more robust than simply choosing the optimal $n$-step return, given the fact that the optimal $n$-step return varies according to the situation.
Using a weight of $\left( {1 - \lambda } \right){\lambda ^{n - 1}}$, we can efficiently combine all $n$-step returns into a target value, $q_t^\lambda$.
$$ q_t^\lambda = \left( {1 - \lambda } \right)\sum\limits_{n = 1}^\infty {{\lambda ^{n - 1}}} q_t^{\left( n \right)} $$
The forward-view Sarsa$\left(\lambda\right)$ simply adjusts our estimated value function towards the $q_t^\lambda$ target.
$$ Q\left( {{S_t},{A_t}} \right) \leftarrow Q\left( {{S_t},{A_t}} \right) + \alpha \left( {q_t^\lambda - Q\left( {{S_t},{A_t}} \right)} \right) $$
This method will, for each state-action pair, combine the returns from a one-step lookahead, two-step lookahead, ..., $n$-step lookahead, all the way to a full reward sequence lookahead, aggregate
the returns by a geometric sum, and adjust the original estimate towards this new aggregated lookahead.
However, we need to develop a backward-view algorithm in order to enable our agent to learn online during each episode. Otherwise, we'd need to wait until the agent reaches a terminal state to do the
forward-view algorithm described above. To do this, we'll need to replace our $n$-step aggregation approach with eligibility traces, a way of keeping track of which previously visited states are
likely to be responsible for the reward accumulated (covered in more detail in the previous post).
As covered in the last post, we'll initialize all eligibilities to zero, increment the eligibility of visited states, and decay all eligibilities at each time step.
$$ {E_0}\left( {s,a} \right) = 0 $$
$$ {E_t}\left( {s,a} \right) = \gamma \lambda {E_{t - 1}}\left( {s,a} \right) + {\bf{1}}\left( {{S_t} = s,{A_t} = a} \right) $$
Our value function is updated in proportion to both the TD error and the eligilibilty trace.
$$ Q\left( {s,a} \right) \leftarrow Q\left( {s,a} \right) + \alpha {\delta _ t}{E_ t}\left( {s,a} \right) $$
This method is an on-policy approach, meaning that next actions are selected according to the learned policy.
Further reading | {"url":"https://www.jeremyjordan.me/rl-learning-implementations/","timestamp":"2024-11-05T03:20:22Z","content_type":"text/html","content_length":"36599","record_id":"<urn:uuid:e4762cc0-4d21-43db-9eb1-572a8559f2a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00092.warc.gz"} |
Interval Notations: Opinion of Chenyi Hu
Date: Tue, 10 Sep 1996 17:40:17 +0600
From: hu@happy.dt.uh.edu
Subject: support standardizing notations for interval computations
Dear Colleagues:
I strongly support the idea of standardizing notations used in the literature of interval computations. With standard notations, the communications will be more effective. I've to admit, in reviewing
papers submitted to Reliable Computing, I have to make extra effort to understand different notations used by different authors.
A standard notation system is an indication of the maturity of a literature. The literature of interval computing has gone far beyond the stage of maturity to deserve a standard notation system.
Standardizing the notation system is long overdue. The work should be done. The sooner, the better. Although I'll not attend the INTERVAL'96 conference, I hope that standardizing notations will be
considered and discussed at the meeting.
Interval computing is an extension of usual number computations. The standardized notation system should keep the consistency as much as possible with current notations used in mathematics. Professor
Kearfott's notation system clearly meets this requirement. We may use his notation system (or others) as the base for further discussion.
After we have a standardized notation system, we may require authors of the Journal of Reliable Computing to adopt it.
Chenyi Hu
Computer and Mathematical Sciences Department
Center for Computational Science and Advanced Distributed Simulation
University of Houston-Downtown | {"url":"https://reliable-computing.org/notations/hu.html","timestamp":"2024-11-08T11:54:43Z","content_type":"text/html","content_length":"2315","record_id":"<urn:uuid:723b7637-d2cf-4dd8-b705-e7f68e589a14>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00606.warc.gz"} |
C Program to Print Arithmetic Progression (AP) Series and Sum till N Terms - BTech Geeks
C Program to Print Arithmetic Progression (AP) Series and Sum till N Terms
• Write a C program to find the sum of arithmetic series till N terms.
• Write a C program to print arithmetic series till N terms.
Arithmetic series is a sequence of terms in which next term is obtained by adding common difference to previous term. Let, t[n] be the n^th term of AP, then (n+1)^th term of can be calculated as <br
\>(n+1)^th = t[n] + D
where D is the common difference (n+1)^th – t[n]
The formula to calculate N^th term t[n] = a + (n – 1)d;
where, a is first term of AP and d is the common difference.</br\>
C program to print arithmetic progression series and it’s sum till N terms
In this program, we first take number of terms, first term and common difference as input from user using scanf function. Then we calculate the arithmetic series using above formula(by adding common
difference to previous term) inside a for loop. We keep on adding the current term’s value to sum variable.
* C program to print Arithmetic Series and it's sum till Nth term
#include <stdio.h>
#include <stdlib.h>
int main() {
int first, diff, terms, value, sum=0, i;
printf("Enter the number of terms in AP series\n");
scanf("%d", &terms);
printf("Enter first term and common difference of AP series\n");
scanf("%d %d", &first, &diff);
/* print the series and add all elements to sum */
value = first;
printf("AP SERIES\n");
for(i = 0; i < terms; i++) {
printf("%d ", value);
sum += value;
value = value + diff;
printf("\nSum of the AP series till %d terms is %d\n", terms, sum);
return 0;
Program Output
Enter the number of terms in AP series
Enter first term and common difference of AP series
Sum of the AP series till 5 terms is 50
Try these:
1. A given series could be an arithmetic progression a geometric progression in c++
2. A given series could be an arithmetic progression a geometric progression in c
3. C program to find sum of arithmetic progression
4. C program to find nth term in arithmetic progression
5. An arithmetic progression series of 18 term with common difference d
6. Arithmetic progression in c using for loop
7. Geometric progression program in c
8. Arithmetic progression program in c
9. Arithmetic progression in c skillrack
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Robustness analysis of linear time-varying systems with application to aerospace systems
Biertümpfel, Felix (2021) Robustness analysis of linear time-varying systems with application to aerospace systems. PhD thesis, University of Nottingham.
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Available under Licence Creative Commons Attribution.
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In recent years significant effort was put into developing analytical worst-case analysis tools to supplement the Verification \& Validation (V\&V) process of complex industrial applications under
perturbation. Progress has been made for parameter varying systems via a systematic extension of the bounded real lemma (BRL) for nominal linear parameter varying (LPV) systems to IQCs. However,
finite horizon linear time-varying (LTV) systems gathered little attention. This is surprising given the number of nonlinear engineering problems whose linearized dynamics are time-varying along
predefined finite trajectories. This applies to everything from space launchers to paper processing machines, whose inertia changes rapidly as the material is unwound. Fast and reliable analytical
tools should greatly benefit the V\&V processes for these applications, which currently rely heavily on computationally expensive simulation-based analysis methods of full nonlinear models.
The approach taken in this thesis is to compute the worst-case gain of the interconnection of a finite time horizon LTV system and perturbations. The input/output behavior of the uncertainty is
described by integral quadratic constraints (IQC). A condition for the worst-case gain of such an interconnection can be formulated using dissipation theory. This utilizes a parameterized Riccati
differential equation, which depends on the chosen IQC multiplier. A nonlinear optimization problem is formulated to minimize the upper bound of the worst-case gain over a set of admissible IQC
multipliers. This problem can then be efficiently solved using custom-tailored meta-heuristic (MH) algorithms. One of the developed algorithms is initially benchmarked against non-tailored
algorithms, demonstrating its improved performance. A second algorithm's potential application in large industrial problems is shown using the example of a touchdown constraints analysis for an
autolanded aircraft as was as an aerodynamic loads analysis for space launcher under perturbation and atmospheric disturbance. By comparing the worst-case LTV analysis results with the results of
corresponding nonlinear Monte Carlo simulations, the feasibility of the approach to provide necessary upper bounds is demonstrated. This comparison also highlights the improved computational speed of
the proposed LTV approach compared to simulation-based nonlinear analyses.
Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Pfifer, Harald
Popov, Atanas
Keywords: Linear time-varying systems, Worst-case analysis, Integral quadratic constraints
Subjects: T Technology > TL Motor vehicles. Aeronautics. Astronautics
Faculties/Schools: UK Campuses > Faculty of Engineering > Department of Mechanical, Materials and Manufacturing Engineering
Item ID: 65963
Depositing User: Biertümpfel, Felix
Date Deposited: 31 Dec 2021 04:40
Last Modified: 31 Dec 2021 04:40
URI: https://eprints.nottingham.ac.uk/id/eprint/65963
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NPTEL Introduction to Database Assignment Answers - GEC Lakhisarai
NPTEL Introduction to Database Assignment Answers
1. Which of the following SQL clauses can have a subquery?
(i) FROM (ii) GROUP BY (iii) HAVING
• Only (i)
• Only (i) and (ii)
• Only (i) and (iii)
• All (i), (ii) and (iii)
Answer :- c
2. Views can never be used in an update query
Answer :- b
3. Views are materialized into tables if the view definition has only one table
Answer :- b
4. Views can be used in the definition of other views
Answer :- a
5. Aggregate operators can be used to define views
Consider the relation stud (C1, C2, C3, C4, C5, C6). A SQL query Q created using stud has the columns C1, C2, and C3 in the group by clause. Assuming Q is a syntactically correct query,state whether
the following statements are true or false.
Answer :- a
6. The column C4 can be used in the HAVING clause without aggregate operator
Answer :- b
7. The column C4 can be used in the WHERE clause without aggregate operator
Answer :- a
8. The column C3 can be used in the SELECT clause without aggregate operator
Answer :- a
9. The column C3 can be used in the SELECT clause with aggregate operator
Answer :- b
10. Consider the following schema of the relation stud:
stud( K , C1)
Which of the following queries would find the number of NULLs in the column C1 of the stud relation.
• SELECT COUNT(C1) FROM stud WHERE C1 IS NOT NULL
• SELECT COUNT(*) – COUNT(C1) FROM stud WHERE C1 IS NOT NULL
• SELECT COUNT() FROM stud WHERE C1 = NULL
• SELECT COUNT(*) – COUNT(C1) FROM stud
Answer :- d
11. Consider two instances of the relations R1 and R2 with the number of rows 10 and 8, respectively. The minimum number of tuples in the result of the following query is
SELECT *
FROM R1 FULL OUTER JOIN R2
Answer :- a
12. DLETE keyword can be used to do which of the following operations?
(i) delete all rows (ii) delete specific rows (iii) delete table definition
• Only (ii)
• Only (iii)
• Only (i) and (ii)
• All (i), (ii) and (iii)
Answer :- c
13. Consider the following query
select deptNo
from course
group by deptNo
having sum(credits) >= ANY (select max(x.totalCredits)
from ( select sum(credits) as totalCredits
from course
group by deptNo) as x
Suppose total-credits of a department is the sum of the credits of the courses offered by the department. Which of the following statements is correct about the above query?
• It finds the department(s) with the highest number of total-credits across all the departments.
• It finds the department(s) with total-credits greater than or equal to the sum of the total-credits of all the departments.
• It finds the department(s) with total-credits greater than or equal to the credits of the course with the highest number of credits across the institute.
• It lists all the departments in the institute.
Answer :- a
14. Consider the following sets about different approaches to programmatic access of databases and the properties that may apply to these approaches.
{1: Embedded SQL approach; 2: API based approach; 3: Database language approach}
{p: Syntax-check during compilation q: driver; r: cursors; s: No impedance mismatch; t: Multiple active connections }
Identify the correct matching between the sets:
• 1–p; 2–t; 3–q
• 1–t; 2–q; 3–s
• 1–s; 2–q; 3–t
• 1–p; 2–t; 3–s
Answer :- d
15. Which of the following is an example of non-atomic value:
• A set of names
• Value of a composite attribute “address”
• An integer
• Both A & B
Answer :- d
16. What is TRUE about the concept of “Functional Dependency” in a relation scheme R?
• It is a constraint between two sets of attributes of R.
• Denoted as X → Y, it means that the values of the X attributes uniquely determine the values of the Y attributes in r, where r is any instance of a relation R.
• It is property of the semantics of the attributes of R.
• All of the above are TRUE.
Answer :- d
17. Consider a relation R and recall the definition of “First Normal Form(1NF)” and state the validity of the following statements:
S1: The domains of all attributes of R should consist of atomic values.
S2: To check if R is in 1NF, additional information such as functional dependencies in R is essential.
• S1: True; S2: True
• S1: True; S2: False
• S1: False; S2: False
• S1: False; S2: True
Answer :- b
18. Given a relation R(A, B, C, D) and the set of all functional dependencies F on R where
F = { AB → C, C → D, C → B, B → A } find the TRUE statements among S1 and S2 below:
S1: C+ = {C, D, B, A}
S2: B+ = {B, A}
• Only S1
• Only S2
• Both S1 & S2
• Neither S1, nor S2
Answer :- a
19. Consider a relation R(A,B,C,D,E) and the following set F of all FDs on R where F = { A → B, BC → D, E → C, D → A }. Choose the correct option:
• The relation R has exactly two candidate keys.
• The relation R has exactly three candidate keys.
• Pairwise, the candidate keys of R do not share any attribute.
• Both A & C are correct.
Answer :- b
20. Given a relation R(A,B,C,D,E) and the following set of all FDs F on R where F = { AD → C, B → A, C → E, E → BD }. What are all the possible candidate keys of R ?
• {AD}
• {AD, BD}
• {AD, BD, E}
• {AD, BD, E, C}
Answer :- d
21. Choose the incorrect option about concept of the Second Normal Form(2NF):
• If R is in 2NF it should not contain an attribute that is partially dependent on a key of R.
• 2NF is based on the concept of full functional dependency.
• 2NF is based on the concept of transitive dependency.
• If the relation is in 3NF, it will also be in 2NF.
Answer :- c
22. Consider a relation R(A,B,C,D,E,F) and the following set of all FDs on R: {AB → C, C → DE, E → F, F → B}. Choose the correct option:
• R is in 1NF, but not in 2NF
• R is in 2NF, but not in 3NF
• R is in 3NF, but not in BCNF
• R is in BCNF
Answer :- a
23. Consider the following statements:
S1: Any schema that satisfies BCNF also satisfies 3NF.
S2: The definition of 3NF does not allow any functional dependencies which are allowed in BCNF
• S1: True; S2: False
• S1: True; S2: True
• S1: False; S2: True
• S1: False; S2: False
Answer :- a
24. Consider a schema R(A, B, C, D) and functional dependencies { A → B, C → D }, then decomposition of R into R1(A, B) and R2(C, D) is
• lossless and dependency preserving
• lossless but not dependency preserving
• dependency preserving but not lossless
• not dependency preserving and also not lossless
Answer :- c
25. Consider a relation R(A,B,C). A functional dependency AB → C holds on R. Then it is also true that:
• A → C holds on R.
• B → C holds on R.
• AB is a key.
• Both A → C and B → C hold on R
Answer :- c
26. Consider a relation R(A,B,C,D,E,F) and set of all FDs on R as given below:
{ AB → C, C → D, D → E, E → F, F → A }.
Choose the FALSE statement:
• Every attribute is a prime attribute.
• Five candidate keys are possible in R.
• Attribute B is part of each candidate key.
• Attribute C is part of two candidate keys.
Answer :- b
27. Consider a relation R(A,B,C,D,E,F) and set of all FDs on R as given below: {AB → C, C → D, D → E, E → F, F → A}.Choose the correct option:
• R is in 1NF, not in 2NF
• R is in 2NF, not in 3NF
• R is in 3NF, not in BCNF
• R is in BCNF
Answer :- d
28. Consider a relation R(A,B,C,D) and FD set F = { A → B, B → C } and the following statements:
S1: ABD is not a candidate key.
S2: There exists a proper subset of ABD such that its closure determines all the attributes of R.
Choose the correct option:
• S1 is TRUE; S2 is FALSE
• Both S1 and S2 are TRUE, but S2 is not the correct reason for S1 to be TRUE
• Both S1 and S2 are TRUE and S2 is the correct reason for S1 to be TRUE
• Both S1 and S2 are FALSE
Answer :- d
29. Consider the following two FD sets on a relation R = (A, B, C).
F1 = { A → B, B → C}
F2 = { AB → C, AC → B, B → C}
and the two statements given below
S1: F1 covers F2
S2: F2 covers F1
Choose the correct option::
• S1: TRUE; S2: TRUE
• S1: FALSE; S2: TRUE
• S1: FALSE; S2: FALSE
• S1: TRUE; S2: FALSE
Answer :- b
30. Consider a relation R(A,B,C,D,E) and the FD set F = {A → BC, CD → E, B → D, E → A}, and the following statements about closure of the attributes wrt F:
S1: A+ = {ABCDE}
S2: B+ = {BD}
S3: C+ = {CD}
S4: D+ = {D}
S5: E+ = {EA}
Choose the correct option:
• Only S1 and S2 are TRUE
• Only S1, S2 and S3 are TRUE
• Only S1, S3, and S5 are FALSE
• Only S3 and S5 are FALSE
Answer :- c
31. For the given relation and FD set in Question 6, compute all the candidate keys of R.
• {E}
• {E, A}
• {E, A, BC}
• {E, A, BC, CD}
Answer :- d
32. Consider a relation R(A,B,C,D,E,F) and the following set of all FDs on R: {AB → C, C → DE, E → F, F → B} Compute the highest normal form in which R is:
Answer :- d
33. Consider a relation R(A,B,C,D) and the following FD set F = {A → B, B → C, C → D, D → A} Compute the highest normal form in which R is:
Answer :- d
34. Fourth Normal Form(4NF) and Fifth Normal Form(5NF) deal with …X… and …Y… types of dependencies, respectively. Choose the correct option for X and Y:
• X: Multi-valued; Y: Join
• X: Multi-valued; Y: Inclusion
• X: Join; Y: Multi-Valued
• X: Inclusion; Y: Join
Answer :- a
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Hybrid LUTMultiplexer FPGA Logic Architectures - Nexgen Technology
Hybrid LUTMultiplexer FPGA Logic Architectures
by nexgentech | Oct 23, 2017 | ieee project
Hybrid LUTMultiplexer FPGA Logic Architectures
Hybrid configurable logic block architectures forfield-programmable gate arrays that contain a mixture of lookuptables and hardened multiplexers are evaluated toward the goalof higher logic density
and area reduction. Multiple hybridconfigurable logic block architectures, both nonfracturable andfracturable with varying MUX:LUT logic element ratios areevaluated across two benchmark suitesusing a
custom tool flow consisting of LegUp-HLS, Odin-IIfront-end synthesis, ABC logic synthesis and technology mapping,and VPR for packing, placement, routing, and architectureexploration. Technology
mapping optimizations that targetthe proposed architectures are also implemented within ABC.Experimentally, we show that for nonfracturable architectures,without any mapper optimizations, we
naturally save up to∼8%area postplace and route; both accounting for complex logicblock and routing area while maintaining mapping depth. Witharchitecture-aware technology mapper optimizations in
ABC,additional area is saved, post-place-and-route. For fracturablearchitectures, experiments show that only marginal gains areseen after place-and-route up to∼2%. For both nonfracturableand
fracturable architectures, we see minimal impact on timingperformance for the architectures with best area-efficiency.The proposed architecture of this paper is analysis the logic size, area and
power consumption using tanner tool.
Enhancement of the project:
To design the ALU based on LUTs
Existing System:
In this paper, we present a six-input LE based on a4-to-1 MUX, MUX4, that can realize a subset of six-inputBoolean logic functions, and a new hybrid complex logicblock (CLB) that contains a mixture
of MUX4s and 6-LUTs.The proposed MUX4s are small compared with a 6-LUT(15% of 6-LUT area), and can efficiently map all{2, 3}-input functions and some {4,5,6}-input functions.In addition, we explore
factorability of LEs—the ability tosplit the LEs into multiple smaller elements—in both LUTsand MUX4s to increase logic density. The ratio of LEs thatshould be LUTs versus MUX4s is also explored
towardoptimizing logic density for both nonfracturable andfracturable FPGA architectures.
MUX4: 4-to-1 Multiplexer Logic Element:
The MUX4 LE shown in Fig. 1 consists of a 4-to-1 MUXwith optional inversion on its inputs that allow the realizationof any{2,3}-input function, some {4,5}-input functions, andone 6-input function—a
4-to-1 MUX itself with optionalinversion on the data inputs. A 4-to-1 MUX matches the inputpin count of a 6-LUT, allowing for fair comparisons withrespect to the connectivity and intracluster
Naturally, any two-input Boolean function can be easilyimplemented in the MUX4: the two function inputs can betied to the select lines and the truth table values (logic-0orlogic-1) can be routed to
the data inputs accordingly.Or alternately, Shannon decomposition can be performedabout one of the two variables—the variable can then feeda select input. The Shannon cofactors will contain at
mostone variable and can, therefore, be fed to the data inputs(the optional inversion may be needed).
Figure 1: MUX4 LE depicting optional data input inversions
Logic Elements, Fracturability, and MUX4-Based Variants:
Two families of architectures were created: 1) without fracturable LEs and 2) with fracturable LEs. In this paper, the fracturable LEs refer to an architectural element on which one or more logic
functions can be optionally mapped. Nonfracturable LEs refer to an architectural element on which only one logic function is mapped. In the nonfracturable architectures, the MUX4 element shown in
Fig. 1 is used together with nonfracturable 6-LUTs. This element shares the same number of inputs as a 6-LUT lending for fair comparison with respect to the input connectivity.
Figure 2: Fracturable 6-LUT that can be fractured into two 5-LUTs with two shared inputs.
For the fracturable architecture, we consider an eight-input LE, closely matched with the adaptive logic module in recent Altera Stratix FPGA families. A 6-LUT that can be fractured into two 5-LUTs
using eight inputs is shown in Fig. 2. Two five-input functions can be mapped into this LE if two inputs are shared between the two functions. If no inputs are shared, two four-input functions can be
mapped to each 5-LUT. For the MUX4 variant, Dual MUX4, we use two MUX4s within a single eight-input LE. In the configuration, shown in Fig. 3, the two MUX4s are wired to have dedicated select inputs
and shared data inputs. This configuration allows this structure to map two independent (no shared inputs) three-input functions, while larger functions may be mapped dependent on the shared inputs
between both functions.
Figure 3: Dual MUX4 LE that utilizes dedicated select inputs and shared data inputs.
Proposed System:
In the proposed design, ALU is design using non Fracturable and Fracturable LUT. Based on the Fracturable LUT the power consumption of the design is reduced compare to non Fracturable design. The
basic introduction of the ALU is given below.
In ECL, TTL and CMOS, there are available integrated packages which are referred to as arithmetic logic units (ALU). The logic circuitry in this units is entirely combinational (i.e. consists of
gates with no feedback and no flip-flops).The ALU is an extremely versatile and useful device since, it makes available, in single package, facility for performing many different logical and
arithmetic operations. ALU’s comprise the combinational logic that implements logic operations such as AND, OR and arithmetic operations, such as ADD, SUBTRACT.
A number of basic arithmetic and bitwise logic functions are commonly supported by ALUs. Basic, general purpose ALUs typically includes these operations in their repertoires:
Arithmetic operations
• Add: A and B are summed and the sum appears at Y and carry-out.
• Add with carry: A, B and carry-in are summed and the sum appears at Y and carry-out.
• Subtract: B is subtracted from A (or vice versa) and the difference appears at Y and carry-out. For this function, carry-out is effectively a “borrow” indicator. This operation may also be used
to compare the magnitudes of A and B; in such cases the Y output may be ignored by the processor, which is only interested in the status bits (particularly zero and negative) that result from the
• Subtract with borrow: B is subtracted from A (or vice versa) with borrow (carry-in) and the difference appears at Y and carry-out (borrow out).
• Two’s complement (negate): A (or B) is subtracted from zero and the difference appears at Y.
• Increment: A (or B) is increased by one and the resulting value appears at Y.
• Decrement: A (or B) is decreased by one and the resulting value appears at Y.
• Pass through: all bits of A (or B) appear unmodified at Y. This operation is typically used to determine the parity of the operand or whether it is zero or negative.
• AND: the bitwise AND of A and B appears at Y.
• OR: the bitwise OR of A and B appears at Y.
• Exclusive-OR: the bitwise XOR of A and B appears at Y.
• One’s complement: all bits of A (or B) are inverted and appear at Y.
(a) (b)
Figure 4: ALU design using non-Fracturable and Fracturable LUT.
In non fracturable LUT design based ALU is shows in fog 4(a) and Fracturable LUT based ALU design is shows in the figure 4(b).
Software implementation: | {"url":"https://nexgenproject.com/hybrid-lutmultiplexer-fpga-logic-architectures/","timestamp":"2024-11-10T15:12:15Z","content_type":"text/html","content_length":"93659","record_id":"<urn:uuid:d6c720a6-28ac-4c55-9b5f-8d994c75791d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00167.warc.gz"} |
Zorn's lemma
A lemma in set theory: if a set S is partially ordered and if each subset for which every pair of elements is related by exactly one of the relationships “less than,” “equal to,” or “greater than”
has an upper bound in S, then S contains at least one element for which there is no greater element in S | {"url":"https://m.everything2.com/title/Zorn%2527s+lemma","timestamp":"2024-11-10T06:13:05Z","content_type":"text/html","content_length":"37702","record_id":"<urn:uuid:f008f974-58b5-4f77-81f9-8a74a83ca226>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00449.warc.gz"} |
Invested Capital: Definition and How To Calculate Returns (ROIC)
What does Invested Capital Mean?
Invested capital refers to the total sum of money raised by a company through the issuance of securities to equity shareholders and debt to bondholders. It represents the combined worth of equity and
debt capital that was raised by a firm, inclusive of capital leases. Invested capital is not explicitly mentioned as a line point in the firm’s financial statements, as capital leases, debt, and
stockholders' equity are typically reported apart in the balance sheet.
Explaining Invested Capital
Invested capital is a crucial concept in assessing a firm’s capability to produce economic profit. To earn an economic profit, a company must generate earnings that surpass the cost of raising
capital from shareholders, bondholders, and other financing sources. This metric is essential in determining how effectively a firm utilises its capital. Companies employ various financial ratios,
such as the Return on Invested Capital (ROIC), the Economic Value Added (EVA), and the Return on Capital Employed (ROCE), to evaluate their capital utilisation.
How to Calculate the Invested Capital?
To calculate the invested capital, there are two common approaches: the operating approach and the financing approach.
Operating Approach:
The formula for calculating the invested capital using the operating approach is the following:
Invested Capital = Net Working Capital + Fixed Assets + Goodwill and Intangibles
Where: Net Working Capital = Current Operating Assets - Non-interest Bearing Current Liabilities.
Fixed Assets include tangible assets such as land, buildings, and equipment. Goodwill and Intangibles represent intangible assets like brand reputation, copyrights, and proprietary technology.
For calculating the net working capital, you should deduct the non-interest bearing current liabilities from the current operating assets. Then add the resulting net working capital to the fixed
assets and goodwill and intangibles to gain the invested capital.
Financing Approach:
The formula for calculating the invested capital using the financing approach is the next:
Invested Capital = Equity +
Debt + Capital Leases
Where: Equity refers to the value of shares issued to equity shareholders. Debt represents the full debt, including bonds, raised by the firm. Capital Leases include any long-lasting lease
Add the equity, debt, and capital leases together to find the invested capital.
Please remember that the financial expert of a specific company may have their own unique approach to calculating the invested capital based on the firm’s specific circumstances and industry
practices. The provided formulas and approaches serve as general guidelines to calculate the invested capital.
What does the Return on Invested Capital (ROIC) Mean?
The Return on Invested Capital (ROIC) is a profitability and performance ratio that measures the percentage profit a firm earns on its invested capital. It illustrates how efficiently a firm utilises
the capital offered by investors to produce income. ROIC is a key metric used in benchmarking to assess the value and performance of companies.
Returns, in the context of ROIC, refer to the earnings generated by a company after taxes but before interest is paid. These returns reflect the profitability of the company's operations. It is
important to note that returns must exceed the cost of acquiring capital to create economic value.
How to Calculate the Returns?
The Return on Invested Capital (ROIC) is a measure utilised to estimate a firm’s effectiveness in utilising the capital to produce returns. It quantifies the percentage profit that a firm earns on
the money invested by its bondholders and stockholders.
To calculate the ROIC, the following formula is used:
ROIC = Net Operating Profit After Tax (NOPAT) / Invested Capital
Here are the key steps involved in calculating the ROIC:
1. Determine the NOPAT: NOPAT means the after-tax operating profit earned by the firm. To calculate it, you should multiply Earnings Before Interest and Taxes with the tax rate and can be expressed
as NOPAT = EBIT * (1 - Tax Rate).
2. Calculate the Invested Capital: The book value of debt and equity is used in this calculation. Subtract any cash equivalents from the book value of debt and equity to eliminate the interest income
from money. The formula for calculating the Invested Capital depends on the specific method chosen, such as subtracting non-interest-bearing current liabilities from total assets or adding the book
value of equity to the book value of debt while subtracting non-operating assets.
3. Apply the formula: Divide the NOPAT by the Invested Capital to gain the ROIC percentage.
It is important to note that the book value of debt and equity is preferred over market value for this calculation. Market value may include future expectations and growth assets, which can distort
the calculation for rapidly growing companies. Using book value provides a more accurate representation of the current profitability.
To compare the firm’s ROIC to its Weighted Average Cost of Capital (WACC) is very important. If the ROIC is higher than the WACC, it shows that the firm generates returns higher than the price of the
capital. And if the ROIC is lower than the WACC, it suggests that the firm earns a lower return on its projects compared to the cost of funding them.
The ROIC is a reliable method to see profitability and is viewed as more informative than other ratios like the Return on Assets (ROA) or the Return on Equity (ROE).
Bottom Line and Key Takeaways
Invested capital plays a crucial role in evaluating a company's profitability and efficiency in utilising the funds provided by investors. It represents the total capital raised by a firm through
equity and debt offerings, including capital leases. The ROIC is an important metric that shows the effectiveness of a firm's generating profit on its invested capital. Due to comparing the ROIC to
the WACC, companies can define whether they create or destroy value. Calculating the invested capital and the ROIC allows investors and analysts to make informed decisions about a company's financial
performance and prospects for growth.
Is invested capital the same as equity?
No, invested capital is treated as the combined value of both equity and debt capital raised by a firm, whereas equity represents the ownership interest of shareholders in a company.
How does the ROIC differ from the ROA and the ROE?
While the ROA and the ROE show a company's profitability relative to its assets and equity, respectively, ROIC specifically focuses on the profit generated on all invested capital, including both
equity and debt.
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perplexus.info :: Just Math : Counting the wounded
(puzzle by Henry Dudeney):
When visiting with a friend one of our hospitals for wounded soldiers, I was informed that exactly two-thirds of the men had lost an eye, three-fourths had lost an arm, and four-fifths had lost a
”Then,’ I remarked to my friend, ‘it follows that at least twenty-six of the men must have lost all three — an eye, an arm, and a leg.’
That being so, can you say exactly how many men were in the hospital? It is a very simple calculation, but I have no doubt it will perplex a good many readers. | {"url":"http://perplexus.info/show.php?pid=13349&cid=68369","timestamp":"2024-11-04T04:10:37Z","content_type":"text/html","content_length":"13047","record_id":"<urn:uuid:ea8f50f8-5e6b-4b18-acfc-744aa4a9e88b>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00355.warc.gz"} |
Metals #10 - Energy #13 - MyDVICE Writings, Thoughts, Views & Images
Discussing the Gold-Oil ratio (again) – structure to target a return and potential overshoot of the long-term equilibrium level of 14:1.
16 June 2021
In my posts dated 9, 10, 15 June 2020 and most recently 8 March 2021, I stated the proposition that the gold-oil ratio will revert to its long-term equilibrium level (see chart below). I propose the
following option structure to take advantage of an overshoot to 12:1 (assuming gold maxes out at $1900 over this period, this would equate to a Brent price of over $150). However, this strategy will
limit itself to an 18-month Brent target of $125.
As a side proposition, I forecast that the Brent premium over WTI will revert to a discount as per its long-term equilibrium levels (see chart below). In which case I expect that the move in Brent
will be matched at a faster pace by WTI. Hence, I shall be proposing to use WTI for this options strategy (i.e. a WTI one-year options calendar bull spread):
1. Sell December 2021 WTI 56 Put, Expiry 16 November 2021, premium received = 1.54. Futures reference = 69.05. Implied Volatility = 36.69%, Delta = 12, Vega = 0.09, Theta = 0.008. Skew to ATM =
+6.84 points.
2. Buy December 2022 WTI 90 Call, Expiry 16 November 2022, premium paid = 1.29. Futures reference = 63.02. Implied Volatility = 26.27%, Delta = 12, Vega = 0.16, Theta = 0.004. Skew to ATM = -1.44
Structure receives premium = 0.25. Delta neutral, net Vega = +0.07, net Theta = +0.004.
Additional Reasons on why I like this structure:
1. It is a long vega position at a time when implied volatilities have retraced quite a bit from the highs of 2020.
2. The longer dated option strike is still below the magic 100 level. Assuming all goes to plan the strategy could be in the money by 20 or even 30 points way before expiry.
3. Time decay is not an issue. In fact, it is positive for this structure.
4. The structure starts off delta neutral. In other words, one can trade the gamma if the market goes our way.
Further thoughts:
(a) Excellent cointegration of Brent and WTI and the long-term equilibrium of the spread.
(b) Cointegration of the volatilities between Brent, Gold and the Gold:Brent ratio. It is the price of Brent that moves more to correct the Gold:Brent ratio back to its long-term equilibrium level.
(c) Notice how similar the two charts below are. Does the Gold:Brent ratio cointegrate well with the Brent-WTI spread?
5Y Weekly – Gold:Oil Ratio
35Y Monthly – Brent-WTI Spread
0 Comments
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/MAT/LAW59 (CONNECT)
Block Format Keyword This law describes the Connection material, which can be used to model spotweld, welding line, glue, or adhesive layers in laminate composite material.
Elastic and elastoplastic behavior in normal and shear directions can be defined. The curves that represent plastic behavior can be specified for different displacement rates. This material is
applicable only to solid hexahedron elements (/BRICK) and the element time-step does not depend on element height.
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/MAT/LAW59/mat_ID/unit_ID or /MAT/CONNECT/mat_ID/unit_ID
${\rho }_{i}$
E G Imass Icomp Ecomp
N[b_fct] F[smooth] F[cut]
> 0, each true plastic stress versus displacement functions in normal/tangent direction per line
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Y_fct_ID[N] Y_fct_ID[T] SR[ref] Fscale[yld]
Field Contents SI Unit Example
Material identifier.
(Integer, maximum 10 digits)
Unit identifier.
(Integer, maximum 10 digits)
Material title.
(Character, maximum 100 characters)
${\rho }_{i}$ $\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$
Young's modulus in the normal direction per unit length.
E $\left[\frac{\text{P}\text{a}}{\text{m}}\right]$
Shear modulus in the tangential direction per unit length.
G $\left[\frac{\text{P}\text{a}}{\text{m}}\right]$
Mass calculation flag.
= 0 (Default)
Element mass is calculated using density and volume.
Imass = 1
Element mass is calculated using density and (means of upper and lower) area.
Symmetric elasto-plastic behavior in compression.
= 0
Icomp Symmetric elasto-plastic behavior in tension and compression.
= 1
Elasto-plastic behavior defined by input yield function in tension only.
Compression modulus per unit length.
Ecomp $\left[\frac{\text{P}\text{a}}{\text{m}}\right]$
Default = E
Number of input functions: true stress versus plastic displacement (normal or tangential).
= 0
N[b_fct] Material is linear elastic.
Displacement rate filtering flag.
= 0 (Default)
No displacement rate filtering.
F[smooth] = 1
Displacement rate filtering.
Cutoff frequency for the displacement rate filtering.
F[cut] $\text{[Hz]}$
Default = 10^30 (Real)
True plastic stress versus displacement in normal direction defined for the reference displacement rate.
True plastic stress versus displacement in tangential direction defined for the reference displacement rate.
Displacement rate values for which the set of functions are defined.
SR[ref] $\left[\frac{\text{1}}{\text{s}}\right]$
Default = 0.0 (Real)
Scale factor for the plastic stress.
Fscale[yld] $\left[\text{Pa}\right]$
Default = 1.0 (Real)
Example (Spotweld)
unit for mat
Mg mm s
#- 2. MATERIALS:
# RHO_I
# E G Imass Icomp Ecomp
# NB_fct Fsmooth Fcut
# YFun_IDN YFun_IDT SR_ref Fscale_yld
# EPS_MAX_N EXP_N ALPHA_N R_fct_IDN Ifail Ifail_so ISYM
# EPS_MAX_T EXP_T ALPHA_T R_fct_IDT
1.8 0 0 0
# EIMAX ENMAX ETMAX Nn Nt
# Tmax Nsoft AREAscale
#- 3. FUNCTIONS:
# X Y
# X Y
1. This law is compatible with 8-noded hexahedron elements (/BRICK) only. It is only compatible with /PROP/TYPE43.
2. The stiffness modulus and stresses are defined per displacement in order to be independent from the initial height of the solid element.
For example, $E$=210000 MPa/mm means that the normal stress increases by 210000 MPa for each 1 mm of displacement until the yield stress limit specified by the yield stress curve is reached.
3. The complete element displacement $\overline{u}$ can be subdivided into an elastic portion ${\overline{u}}^{e}$ (before yield stress is reached) and a portion of the plastic displacement ${\
overline{u}}^{pl}$. Plastic displacement is calculated as:
Normal plastic displacement:
${\overline{u}}_{n}^{pl}={\overline{u}}_{n}-{\overline{u}}_{n}^{e}={\overline{u}}_{n}-\frac{{\sigma }_{n}^{true}}{E}$
Shear plastic displacement:
${\overline{u}}_{s}^{pl}={\overline{u}}_{s}-{\overline{u}}_{s}^{e}={\overline{u}}_{s}-\frac{{\sigma }_{s}^{true}}{E}$
Total normal (shear) displacement is the sum of plastic normal (shear) displacement and elastic normal (shear) displacement.
The plastic displacement is accounted for when the normal and tangent yield stress curves are specified. These are usually non-decreasing functions, which represent true stress as a function of
the plastic displacement either in normal or in shear direction. The first abscissa value of the function should be "0" and the first ordinate value is the yield stress. The functions may have a
stress decrease portion to model material damage.
4. If Icomp =0, the material behavior is elasto plastic in both tension and compression, the compression modulus is given by Ecomp (which by default is equal to $E$).
If Icomp =1, the material is nonlinear elasto plastic in tension and linear in compression. The compression modulus is given by Ecomp. The normal and shear degrees of freedom are uncoupled and
the shear behavior is always symmetrical.
5. The height of the solid element can be equal to zero.
6. All nodes of the solid elements must be connected to other shells or solid elements, secondary nodes of rigid body (/RBODY) or secondary nodes of tied interface (/INTER/TYPE2).
7. When all nodes of the solid element become free, the element is deleted.
8. The rupture criteria for this material are defined by /FAIL/CONNECT. | {"url":"https://help.altair.com/hwsolvers/rad/topics/solvers/rad/mat_law59_connect_starter_r.htm","timestamp":"2024-11-07T17:15:41Z","content_type":"application/xhtml+xml","content_length":"130214","record_id":"<urn:uuid:8d08580d-833e-4dd2-85d6-5f4af4fca23d>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00714.warc.gz"} |
I need few practical ideas to measure the pressure drop across each .04 inch diameter hole. - Engineering.com
I need few practical ideas to measure the pressure drop across each .04 inch diameter hole.
Hello Sir,
I need few practical ideas to measure the pressure drop across each .04 inch diameter hole which is arranged in a honeycomb structure on a 300mm diameter aluminum (6061-T6) plate which is 10mm thick.
The hole profile is a bit complex as shown in the attachment.
The requirement is to measrue pressure drop across each hole in shortest time possible.
How many holes are there and what is spacing on the holes?
Interesting you have mixed your units. Are you sure that the holes are 0.040″ diameter and not 0.03937″ or 1mm diameter? (Sorry had to ask that question – Beware of crashing into Mars)
Is there are specific reason why you need to measure the individual pressure drop across each of the holes and not simply look at the pressure drop across the entire orifice plate?
Yes, a U-tube or other manometer system would allow you to measure the net pressure drip across the entire plate, but as I understand the challenge as posted Paul believes that he needs to measure
the pressure drop across each of the holes.
If the holes are very close to each other this may be impossible and as you stated all you need to do is use a single point on each side of the orifice plate to determine what is happening.
If the holes are widely space. I am thinking tens of diameters then you may be able to get some information about how the individual orifices are performing.
hey paul, is this a test fixture for random or one time use? or a product to be sold and used regularly? eg. quick connect and removal….
what is the base or sustaining pressure?
is this above or below ATM? which leads into; gage units or absolute?
if you are checking vacuum, you may need to be more concerned with how to seal the unit adequately, as this can be very important in your readings.
what kind of resolution is required?
how much time is allowed for the test or cycle?
you can perform a pressure decay test which could compare the differential pressures involved. it would probably yield the fastest indication of leakage and might have more useful info resulting from
the test.
Right you are Neil. I didn’t read the whole question and still can’t find the attachment showing the structure in more detail.
Still I’ll have another go at it. I’ll assume that the plate is installed in a pipe and that air or some other gas is made to flow through the pipe and also that the holes are all the same diameter.
The holes themselves act as pipes because of the large length to diameter ratio, 10 to .04 mm. In that case If the flow rate is low enough for laminar flow, poiseuille’s equation holds,
where F is the flow rate in cc per second, L is the length of the hole in cm, D is the diameter of the hole in cm and n is the viscosity in CGS units and P2-P1 is the pressure drop across the hole in
dynes per square cm. (from a dated book which doesn’t use MKS units) The restriction to flow of the large pipe itself is negligible compared to that of the holes because of the 4th power dependence
on diameter, so the pressure profile across the large pipe is nearly constant and the pressure drops across each of the small holes are equal. Now all that is needed is a measurement of the flow rate
through the large pipe. Divide that measurement by the number of holes and insert it in F in poiseuille’s formula and solve for the pressure drop. The large pipe flow rate can be measured with
various commercial flowmeters and might not be too easy to implement without affecting the flow through the large pipe. How is the flow measured now? That determines the pressure drop.
If the flow through the holes is not laminar but turbulent the pressure will vary greatly in the discharge region of the holes and the “pressure drop across the hole” is probably not meaningful. I’ve
used this formula before with mixed results. The flow rate has to be very small to get valid results. There is a large transition region between laminar and turbulent flow. Well that’s my best shot.
If the orifice plate is installed in a tube with gas or air flowing through it, a simple way of measuring the pressure drop across it is to connect a U tube manometer (a piece of glass tubing bent
into the shape of a U) filled with dyed water (or mercury depending on the pressure drop) to nipples on either side of the orifice so that the pressure on each side is communicated to each leg on the
manometer. If the pressure drop is greater than a few feet of water or mercury a commercial differential pressure gauge based on a capsule or bourdon tube would be needed instead of the manometer. To
convert inches of water or mercury to PSI, measure the height difference between the two legs calculate the weight of liquid corresponding to that difference and divide by the cross sectional area of
the manometer tube. | {"url":"https://www.engineering.com/i-need-few-practical-ideas-to-measure-the-pressure-drop-across-each-04-inch-diameter-hole/","timestamp":"2024-11-06T08:12:01Z","content_type":"text/html","content_length":"172354","record_id":"<urn:uuid:9b3bcad0-a26f-44b9-9333-63cb1c4b464e>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00485.warc.gz"} |
Introduction to Dynamic Programming and Memoization
Hi there ๐ ๏ธ
My name is Milad and welcome to my blog. In this post we are going to learn dynamic programming and memoization in a nutshell; So stay with me through this amazing and fun journey.
What dynamic programming and memoization are
Dynamic programming: It is a method for solving complex problems by breaking them down into simpler subproblems. It works by caching the results of subproblems so that they do not have to be
re-computed if encountered again; It's often used for optimization problems and calculating optimal solutions to problems that can be broken into stages.
Memoization: It's a specific technique for caching intermediate results to speed up calculations. It comes from the word "memo" meaning to memoize or remember values and works by maintaining a map of
input to calculated output so that when a value is needed again, it is looked up instead of re-computed. It's useful for recursive functions that repeat calculations for the same inputs.
Why they matter
Dynamic programming and memoization are important techniques in computer science and optimization because they can dramatically improve the performance of algorithms. There are a few key reasons why
they matter:
• Improves efficiency - By storing intermediate results instead of recomputing them, dynamic programming avoids unnecessary repeated computation. This can improve algorithm speed from exponential
to linear time.
• Solves bigger problems - Many complex problems have an optimal substructure that lends itself to a dynamic programming approach. This allows for solving larger instances of problems.
• Elegant problem solving - Dynamic programming breaks down problems into overlapping sub-problems. This modular approach is often cleaner and easier to understand.
• Optimization capabilities - Dynamic programming is well-suited for finding optimal solutions to problems by considering all possibilities and retaining optimal subsolutions.
• Efficient use of space - By caching results, dynamic programming reduces redundant computations and doesn't waste space exploring suboptimal solutions.
When we can use them
Dynamic programming and memoization can be applied to problems that exhibit two key features:
1. Optimal substructure - The optimal solution to the problem can be constructed from optimal solutions of its subproblems.
2. Overlapping subproblems - The problem can be broken down into subproblems which are reused multiple times.
The key criteria are that the problem can be divided into smaller overlapping subproblems and combining their optimal solutions leads to an optimal solution for the overall problem. Problems with
this structure can be efficiently solved using dynamic programming.
Understanding the concept by a simple example
Let's say we have a function that takes a long time to run which is as follows:
function addTo100(n) {
console.log("Long time");
return 100 + n;
Now let's run it:
Long time
As you can see, the function returns the result after taking a long time to run. Now, what if we run it 5 times in a row with the same input value:
Long time
Long time
Long time
Long time
Long time
As you might guess, our function is not efficient this way because, for each same input value, it re-calculates a heavy operation that takes a long time to run. This is where the memoization
technique comes into play so that by storing a map of inputs to calculated outputs, whenever a value is needed again, we need to look it up from the map instead of re-computing it.
// Map variable to store input and corresponding outputs
const cache = {};
function memoizedAddTo100(n) {
// If n is in cache variable, just pick it up and return it
if (n in cache) return cache[n];
// Otherwise take a long time to calculate it for the first time
console.log("Long time");
// Then put it in the cache variable
cache[n] = 100 + n;
// And finally return the output value
return cache[n];
This time let's run the memoized function 5 times in a row with the same input value:
Long time
Congratulations, we did it; we just optimized our function using the power of the memoization technique. As you can see in the output section, we just calculated that heavy operation for the first
time instead of re-computing it for all other same input values.
A famous example to discuss
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. The sequence starts with 0 and 1, and looks like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987...
The sequence follows the recursive relationship:
F(n) = F(n-1) + F(n-2)
Where F(n) is the nth Fibonacci number. The first few Fibonacci numbers are:
F(0) = 0
F(1) = 1
F(2) = F(1) + F(0) = 1
F(3) = F(2) + F(1) = 2
F(4) = F(3) + F(2) = 3
F(5) = F(4) + F(3) = 5
and so on...
We are going to implement a function that computes the value of the nth Fibonacci number using recursion:
// This variable records the number of operations that the function
// have been done in order to calculate nth fibonacci number
let numCalculationsRegularFib = 0;
function fibonacci(n) {
// Time complexity: O(2^n)
if (n < 2) return n; // For all n less than 2, return n itself
numCalculationsRegularFib++;// Increase by one
// recursive call of two previous ones
return fibonacci(n - 1) + fibonacci(n - 2);
Let's say we are going to find out the 30th Fibonacci number using the function above:
const fib30 = fibonacci(30);
console.log(`Fib(30): ${fib30}`);
console.log(`${numCalculationsRegularFib} operations have been done for calculating fib(30)`);
Fib(30): 832040
1346268 operations have been done for calculating fib(30)
Wait a minute!, What?! 1346268 operations; Yeah, that's why its time complexity is O(2^n).
As it is obvious from the chart above, O(2^n) is the second worst time complexity and we need to optimize the previous algorithm using dynamic programming and memoization techniques.
The image above is the recursive call structure tree of the previous algorithms and as it is obvious we have a lot of repetitive calculations(the rectangles with the same color are the same and
repetitive calculations). For example, we don't have to re-calculate Fib(3) on the left side, in case its value is cached when we are on the right side of the tree.
Let's implement the new Fibonacci algorithm, the memoized one:
// Cache variable to store the result of calculations
const fibCache = {};
// This variable records the number of operations that the function
// have been done in order to calculate nth fibonacci number
let numCalculationsMemoizedFib = 0
function memoizedFibonacci(n) {
// Time complexity: O(n)
// If we calculated fib(n) before, pick up its value from cache
if (n in fibCache) return fibCache[n];
else { // Otherwise
// For all n less than 2, return n itself
if (n < 2) return n;
numCalculationsMemoizedFib++; // increase by one
// Make a recursive call and store it in the cache variable
fibCache[n] = memoizedFibonacci(n - 1) + memoizedFibonacci(n - 2);
return fibCache[n];
Now let's say we are going to find out the 30th Fibonacci number once again using the new function above:
const memoizedFib30 = memoizedFibonacci(30);
console.log(`Fib(30): ${memoizedFib30}`);
console.log(`${numCalculationsMemoizedFib} operations have been done for calculating fib(30)`);
Fib(30): 832040
29 operations have been done for calculating fib(30)
Congratulations, we did it; we just optimized our function using the power of the memoization technique and calculated Fib(30) using 29 operations instead of 1346268, Yeah ๐ ฆพ. Moreover, Another
optimization that we did is that we optimized O(2^n) to O(n) which is an acceptable time complexity.
Dynamic programming and memoization are powerful and versatile techniques that can help optimize the performance of algorithms. By storing intermediate results and reusing prior computations, dynamic
programming enables solving complex problems that would otherwise be infeasible.
I hope you enjoyed it, see you next time ๐
Did you find this article valuable?
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Samacheer Kalvi 6th Maths Solutions Term 2 Chapter 4 Geometry Additional Questions
You can Download Samacheer Kalvi 6th Maths Book Solutions Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations.
Tamilnadu Samacheer Kalvi 6th Maths Solutions Term 2 Chapter 4 Geometry Additional Questions
Question 1.
Name the type of the following triangles.
(a) ∆PQR with m∠Q = 90°
(b) ∆ABC with m∠B = 90° and AB = BC
(a) One of the angles is 90°
It is a right-angled triangle
(b) Since two sides are equal.
It is an isosceles triangle. Also m∠B = 90°
It is an Isosceles right-angled triangle
Question 2.
Classify the triangles (scalene, isosceles, equilateral) given below.
(a) ∆ABC, AB = BC
(b) ∆PQR, PQ = QR = RP
(c) ∆ABC, ∠B = 90°
(d) ∆EFG, EF = 3 cm, FG = 4 cm and GE = 3 cm
(a) Isosceles triangle
(b) Equilateral triangle
(c) Right angled triangle
(d) Isosceles triangle
Question 3.
In triangle ∆ABC, AB = BC = CA = 5 cm. Then what is the value of ∠A, ∠B and ∠C?
Since AB = BC = CA
∠A = ∠B = ∠C
We know that ∠A + ∠B + ∠C = 180°
∴ ∠A = ∠B = ∠C = 60°
Question 4.
In ∆PQR, ∠P = ∠Q = ∠R = 60°, then what can you say about the length of sides of ∆PQR? Also, write the name of the triangle?
∠P = ∠Q = ∠R = 60°
So PQ = QR = RP
∆PQR is equilateral triangle
Question 5.
In ∆ABC, AB = BC and ∠A = 50°. Then find the value of ∠C?
It is an isosceles triangle.
∠A = ∠C
∠C = 50°
Question 6.
(a) Try to construct triangles using matchsticks.
(b) Can you make a triangle with?
(i) 3 matchsticks?
(ii) 4 matchsticks?
(iii) 5 matchsticks?
(iv) 6 matchsticks?
Name the type of triangle in each case. If you cannot make a triangle think of the reason for it.
(b) (i) With the help of 3 matchsticks, we can make an equilateral triangle. Since all three matchsticks are of equal length.
(ii) With the help of 4 matchsticks, we cannot make any triangle because in this case, sum of two sides is equal to the third side and we know that the sum of the lengths of any two sides of a
triangle is always greater than the length of the third side.
(iii) With the help of 5 matchsticks, we can make an isosceles triangle. Since we get two sides equal in this case.
(iv) With the help of 6 matchsticks, we can make an equilateral triangle. Since we get three sides equal in length.
Question 7.
A table is bought for ₹ 4500 and sold for ₹ 4800. Find the profit or loss.
C.P = ₹ 4500
S.P. = ₹ 4800
Here S.P< C.P
Profit = S.P – C.P = ₹ 4800 – ₹ 4500 = ₹ 300
Question 8.
Draw any line segment \(\overline{\mathbf{P Q}}\). Take any point R not on it. Through R, draw a perpendicular to \(\overline{\mathbf{P Q}}\).
(i) Drawn a line segment \(\overline{\mathbf{P Q}}\) using scale and taken a point R outside of \(\overline{\mathbf{P Q}}\).
(ii) Placed a set-square on \(\overline{\mathbf{P Q}}\) such that one arm of its right angle aligns along \(\overline{\mathbf{P Q}}\).
(iii) Placed a scale along the other edge of the right angle of the set-square
(iv) Slide the set-square along the line till the point R touches the other arm of its right angle.
(v) Joined RS along the edge through R meeting \(\overline{\mathbf{P Q}}\) at S.
Hence \(\overline{\mathbf{R S}}\) ⊥ \(\overline{\mathbf{P Q}}\).
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Two Digit By Two Digit Multiplication Worksheets | Multiplication Worksheets
Two Digit By Two Digit Multiplication Worksheets
Addition Coloring Pages Halloween Addition Is A Combination Of Two Or
Two Digit By Two Digit Multiplication Worksheets
Two Digit By Two Digit Multiplication Worksheets – Multiplication Worksheets are a wonderful means to educate children the twelve times table, which is the holy grail of primary math. These
worksheets are useful in training students one variable at once, however they can likewise be used with 2 aspects. Commonly, these worksheets are grouped into anchor teams, and also trainees can
start discovering these facts one by one.
What are Multiplication Worksheets?
Multiplication worksheets are an useful way to assist pupils learn math truths. They can be utilized to teach one multiplication truth at once or to assess multiplication realities as much as 144. A
worksheet that shows a trainee one truth at a time will make it simpler to remember the fact.
Making use of multiplication worksheets to educate multiplication is a great means to connect the knowing space and offer your students powerful technique. Several on-line resources supply worksheets
that are both fun as well as easy to use. Osmo has a number of free multiplication worksheets for children.
Word issues are one more method to connect multiplication with real-life circumstances. They can boost your child’s comprehension of the idea while boosting their calculation speed. Many worksheets
include word troubles that resemble real-life scenarios such as purchasing, money, or time calculations.
What is the Purpose of Teaching Multiplication?
It’s vital to start instructing youngsters multiplication early, so they can take pleasure in the procedure. It’s also handy to provide trainees lots of technique time, so they can come to be
proficient in multiplication.
Among the most efficient understanding aids for children is a multiplication table, which you can publish out for each and every kid. Children can exercise the table by duplicating enhancements and
also counting to get answers. Some children find the multiples of 2, 5, and 10 the simplest, once they understand these, they can go on to harder reproductions.
Two Digit By Two Digit Multiplication Worksheets
Add Three Single Digit Numbers Studyladder Interactive Learning Games
Printable Color By Number Multiplication Math Coloring Worksheets
Two Digit Addition Worksheets No Regrouping Addition Coloring
Two Digit By Two Digit Multiplication Worksheets
Two Digit By Two Digit Multiplication Worksheets are an excellent method to evaluate the times tables. Students might additionally find worksheets with images to be helpful.
These worksheets are great for homeschooling. They are created to be easy to use as well as involving for kids. You can include them to math centers, additional method, and also homework tasks. You
can even tailor them to fit your kid’s requirements. When downloaded, you can likewise share them on social media or email them to your kid.
Numerous kids have problem with multiplication. These worksheets are an exceptional method to help them overcome this hurdle. They include multiplication problems at various degrees of difficulty.
The worksheets assist pupils find out to solve these problems in an enjoyable and intriguing way. They can also be timed, which helps them find out to work rapidly.
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Study of a Cantilever
HS-5020: Stochastic Study of a Cantilever I-beam
In this tutorial, you will complete a simple Stochastic study to investigate uncertain parameters of a cantilever I-beam model defined with four variables and four functions.
Before you begin, copy HS-5020.hstx from <hst.zip>/HS-5020/ to your working directory.
Run a Stochastic Study
In this step, you will run a Stochastic study and review the evaluation scatter.
In this study, you will consider w_th, f_l and f_th as Random parameters (uncertain, not controllable by design) and h as Design with Random parameter (uncertain but controllable by design).
1. Add a Stochastic.
a. In the Explorer, right-click and select Add from the context menu.
The Add dialog opens.
b. For Definition from, select an approach.
c. Select Stochastic, then Setup and click OK.
2. Modify input variables.
a. Go to the step.
b. Click the Distributions tab.
c. In the Distribution Role column, select Design with Random for Total Height.
Figure 2.
d. In the Distribution column, verify Normal Variance is selected for all active variables.
The Distribution column allows you to select the Distribution type. Normal Variance (shown in
Figure 3
) means parameters could take random values following normal distribution around their nominal values (µ).
Figure 3.
3. Go to the step.
4. In the work area, set the Mode to Simple Random.
5. In the Settings tab, change the Number of Runs to 500.
6. Click Apply.
7. Go to the step.
8. Click Evaluate Tasks.
The evaluations are randomly sampled in the space and the designs are evaluated.
9. Click the Evaluation Scatter tab.
10. Review the scatter plots for Total Height and Web Thick.
The Evaluation scatter shown in
Figure 4
presents the sampling for Total Height and Web Thick. Due to the Design with Random distribution role the sampling for Total Height is truncated to only samples which fall between the upper and
lower bounds.
Figure 4.
Review Post-Processing Results
In this step, you will review the evaluation results within the Post-Processing step.
1. Go to the step.
2. Click the Distribution tab.
3. From the Channel Selector, click Histogram.
4. Review the histograms of the stochastic results.
Figure 5
shows the histograms of Total Height and Web Thick values distributions. Each blue bin represents the frequency of runs yielding a sub-range of response values. Notice the Total Height histogram
has no tails at the extremities because of the truncated sampling. The probability densities (red curves) indicate the relative likelihoods of the variables to take particular values. A higher
value indicates that the values are more likely to occur. The cumulative distributions (green curves) indicate what percentage of the data falls below the value threshold.
Figure 5.
5. From the Channel Selector, click Box Plot.
6. Identify the eventual outliers.
Note: There are no outliers for Total Height because of the truncated sampling.
Figure 6.
7. Click the Reliability tab.
8. Add a reliability.
a. Click Add Reliability.
b. Set Response to ly (r_1).
c. Set Bound Type to >=.
d. For Bound Value, enter 75.596800.
9. Create another reliability by repeating step 8 with the following changes:
□ Set Response to lz (r_2).
□ Set Bound Type to >=.
□ For Bound Value, enter 14.404800.
10. Create another reliability by repeating step 8 with the following changes:
□ Set Response to Disp (r_4).
□ Set Bound Type to <=.
□ For Bound Value, enter 4.41e-05.
Considering the parameter uncertainties, there is 49% probability for the moments of inertia and the displacement to have values different than the nominal design.
Figure 7.
11. Click the Reliability Plot tab.
12. Review the graphs.
Figure 8
show the reliabilities for the values Web Thick and Total Height could take.
Figure 8. | {"url":"https://help.altair.com/hwdesktop/hst/topics/tutorials/hst/tut_hs_5020_t.htm","timestamp":"2024-11-13T22:41:22Z","content_type":"application/xhtml+xml","content_length":"53505","record_id":"<urn:uuid:9f22696c-32ae-43d9-a254-b6864cfb3b91>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00480.warc.gz"} |
12.1 Quantization of Energy
Learning Objectives
Learning Objectives
By the end of this section, you will be able to do the following:
• Explain Max Planck’s contribution to the development of quantum mechanics
• Explain why atomic spectra indicate quantization
The information presented in this section supports the following AP® learning objectives and science practices:
• 5.B.8.1 The student is able to describe emission or absorption spectra associated with electronic or nuclear transitions as transitions between allowed energy states of the atom in terms of the
principle of energy conservation, including characterization of the frequency of radiation emitted or absorbed. (S.P. 1.2, 7.2)
Planck’s Contribution
Planck’s Contribution
Energy is quantized in some systems, meaning that the system can have only certain energies and not a continuum of energies, unlike the classical case. This would be like having only certain speeds
at which a car can travel because its kinetic energy can have only certain values. We also find that some forms of energy transfer take place with discrete lumps of energy. While most of us are
familiar with the quantization of matter into lumps called atoms, molecules, and the like, we are less aware that energy, too, can be quantized. Some of the earliest clues about the necessity of
quantum mechanics over classical physics came from the quantization of energy.
Where is the quantization of energy observed? Let us begin by considering the emission and absorption of electromagnetic (EM) radiation. The EM spectrum radiated by a hot solid is linked directly to
the solid’s temperature. (See Figure 12.3.) An ideal radiator is one that has an emissivity of 1 at all wavelengths and, thus, is jet black. Ideal radiators are therefore called blackbodies, and
their EM radiation is called blackbody radiation. It was discussed that the total intensity of the radiation varies as $T4,T4, size 12{T rSup { size 8{4} } } {}$ the fourth power of the absolute
temperature of the body, and that the peak of the spectrum shifts to shorter wavelengths at higher temperatures. All of this seems quite continuous, but it was the curve of the spectrum of intensity
versus wavelength that gave a clue that the energies of the atoms in the solid are quantized. In fact, providing a theoretical explanation for the experimentally measured shape of the spectrum was a
mystery at the turn of the century. When this ultraviolet catastrophe was eventually solved, the answers led to new technologies such as computers and the sophisticated imaging techniques described
in earlier chapters. Once again, physics as an enabling science changed the way we live.
The German physicist, Max Planck (1858–1947), used the idea that atoms and molecules in a body act like oscillators to absorb and emit radiation. The energies of the oscillating atoms and molecules
had to be quantized to correctly describe the shape of the blackbody spectrum. Planck deduced that the energy of an oscillator having a frequency $ff size 12{f} {}$ is given by
12.1 $E=n+12hf.E=n+12hf. size 12{E= left (n+ { { size 8{1} } over { size 8{2} } } right ) ital "hf"} {}$
Here $nn size 12{n} {}$ is any nonnegative integer (0, 1, 2, 3, …). The symbol $hh size 12{h} {}$ stands for Planck’s constant, given by
12.2 $h=6.626×10–34 J ⋅ s.h=6.626×10–34 J ⋅ s. size 12{h = 6 "." "626" times " 10" rSup { size 8{"–34"} } " J " cdot " s"} {}$
The equation $E=n+12hfE=n+12hf size 12{E= left (n+ { { size 8{1} } over { size 8{2} } } right ) ital "hf"} {}$ means that an oscillator having a frequency $ff size 12{f} {}$ (emitting and absorbing
EM radiation of frequency $f)f) size 12{f} {}$ can have its energy increase or decrease only in discrete steps of size
12.3 $ΔE=hf.ΔE=hf. size 12{ΔE = ital "hf"} {}$
It might be helpful to mention some macroscopic analogies of this quantization of energy phenomena. This is like a pendulum that has a characteristic oscillation frequency but can swing with only
certain amplitudes. Quantization of energy also resembles a standing wave on a string that allows only particular harmonics described by integers. It is also similar to going up and down a hill using
discrete stair steps rather than being able to move up and down a continuous slope. Your potential energy takes on discrete values as you move from step to step.
Using the quantization of oscillators, Planck was able to correctly describe the experimentally known shape of the blackbody spectrum. This was the first indication that energy is sometimes quantized
on a small scale and earned him the Nobel Prize in Physics in 1918. Although Planck’s theory comes from observations of a macroscopic object, its analysis is based on atoms and molecules. It was such
a revolutionary departure from classical physics that Planck himself was reluctant to accept his own idea that energy states are not continuous. The general acceptance of Planck’s energy quantization
was greatly enhanced by Einstein’s explanation of the photoelectric effect (discussed in the next section), which took energy quantization a step further. Planck was fully involved in the development
of both early quantum mechanics and relativity. He quickly embraced Einstein’s special relativity, published in 1905, and in 1906, Planck was the first to suggest the correct formula for relativistic
momentum, $p=γmup=γmu size 12{p= ital "γmu"} {}$.
Note that Planck’s constant $hh size 12{h} {}$ is a very small number. So for an infrared frequency of $1014Hz1014Hz size 12{"10" rSup { size 8{"14"} } `"Hz"} {}$ being emitted by a blackbody, for
example, the difference between energy levels is only $ΔE=hf= (6.63×10–34J · s)(1014Hz)= 6.63× 10–20 J,ΔE=hf= (6.63×10–34J · s)(1014Hz)= 6.63× 10–20 J, size 12{ΔE = ital "hf""= " \( 6 "." "63 " times
" 10" rSup { size 8{"–34"} } " J·s" \) \( "10" rSup { size 8{"14"} } " Hz" \) " = 6" "." "63 " times " 10" rSup { size 8{"–20"} } " J"} {}$ or about 0.4 eV. This 0.4 eV of energy is significant
compared with typical atomic energies, which are on the order of an electron volt, or thermal energies, which are typically fractions of an electron volt. But on a macroscopic or classical scale,
energies are typically on the order of joules. Even if macroscopic energies are quantized, the quantum steps are too small to be noticed. This is an example of the correspondence principle. For a
large object, quantum mechanics produces results indistinguishable from those of classical physics.
Atomic Spectra
Atomic Spectra
Now let us turn our attention to the emission and absorption of EM radiation by gases. The sun is the most common example of a body containing gases emitting an EM spectrum that includes visible
light. We also see examples in neon signs and candle flames. Studies of emissions of hot gases began more than two centuries ago, and it was soon recognized that these emission spectra contained huge
amounts of information. The type of gas and its temperature, for example, could be determined. We now know that these EM emissions come from electrons transitioning between energy levels in
individual atoms and molecules; thus, they are called atomic spectra. Atomic spectra remain an important analytical tool today. Figure 12.5 shows an example of an emission spectrum obtained by
passing an electric discharge through a substance. One of the most important characteristics of these spectra is that they are discrete. By this we mean that only certain wavelengths, and hence
frequencies, are emitted. This is called a line spectrum. If frequency and energy are associated as $ΔE=hf,ΔE=hf, size 12{ΔE = ital "hf"} {}$ the energies of the electrons in the emitting atoms and
molecules are quantized. This is discussed in more detail later in this chapter.
It was a major puzzle that atomic spectra are quantized. Some of the best minds of 19^th-century science failed to explain why this might be. Not until the second decade of the 20^th century did an
answer based on quantum mechanics begin to emerge. Again a macroscopic or classical body of gas was involved in the studies, but the effect, as we shall see, is due to individual atoms and molecules. | {"url":"https://texasgateway.org/resource/121-quantization-energy?book=79106&binder_id=78856","timestamp":"2024-11-07T22:40:29Z","content_type":"text/html","content_length":"64335","record_id":"<urn:uuid:40147460-b06d-4fa5-8086-318e2841f86a>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00353.warc.gz"} |
Understanding Variance: A Guide to Data Variability - AI Focus
Understanding Variance: A Guide to Data Variability
## Understanding Variance: A Beginner’s Guide to Data Variability
Imagine you’re at a carnival and playing a darts game. You take aim and toss a dart, but it doesn’t quite hit the bullseye. You take another shot, and this time it’s way off the mark. What gives?
This inconsistency is a perfect example of variance, a statistical measure that describes how much data varies from the average. In this case, your darts are varying different distances from the
center of the board.
## What Is Variance?
Variance is a measure of the spread of data around the mean, or average. It tells you how much individual data points deviate from the mean. A high variance indicates that the data is spread out,
while a low variance indicates that the data is clustered around the mean.
## Why Is Variance Important?
Variance is an important statistical concept because it gives us insight into the reliability and predictability of our data. In our darts example, a high variance would make it difficult to predict
where the next dart will land. Conversely, a low variance would indicate greater consistency and predictability.
## Calculating Variance
Variance is calculated using the following formula:
σ² = Σ(xi – μ)² / n
* Where:
* σ² is the variance
* xi is each individual data point
* μ is the mean
* n is the number of data points
## Variance vs. Standard Deviation
Variance is closely related to standard deviation, another measure of data variability. Standard deviation is simply the square root of variance. It’s often easier to interpret standard deviation
than variance because it’s expressed in the same units as the original data.
## Types of Variance
There are different types of variance, each with its own specific purpose:
* **Population variance** measures the variability of the entire population from which the data was collected.
* **Sample variance** measures the variability of a sample of data and is used to estimate the population variance.
* **Analysis of variance (ANOVA)** is a statistical technique used to compare variances between different groups of data.
## Applications of Variance
Variance has a wide range of applications in the real world, including:
* **Risk analysis:** Variance can be used to assess the risk of an investment by measuring the volatility of its returns.
* **Quality control:** Variance can be used to monitor production processes and identify any deviations from the standard.
* **Research:** Variance can be used to analyze the effects of different treatments or interventions on a population.
## Conclusion
Variance is a fundamental statistical concept that provides valuable insights into the variability of data. By understanding variance, you can make better decisions and draw more accurate conclusions
from your data.
## Frequently Asked Questions
### How do I reduce variance in my data?
There are a few strategies you can use to reduce variance in your data, including:
* **Increasing sample size:** A larger sample size will reduce the amount of random variation in your data.
* **Controlling variables:** If possible, control any variables that could contribute to variance.
* **Using a more precise measuring instrument:** Using a more precise measuring instrument will reduce the amount of measurement error in your data.
### What is a good variance?
There is no one-size-fits-all answer to this question. The ideal variance for your data will depend on the specific context and application. However, in general, a lower variance is preferred because
it indicates greater consistency and predictability.
### How do I interpret a variance?
When interpreting a variance, it’s important to consider the following factors:
* **The mean:** The variance should be interpreted in relation to the mean. A large variance relative to the mean indicates that the data is spread out, while a small variance relative to the mean
indicates that the data is clustered around the mean.
* **The sample size:** The variance of a sample will be different from the variance of the population from which the sample was drawn. The larger the sample size, the more reliable the estimate of
the population variance will be.
## Further Reading
* [Variance](https://en.wikipedia.org/wiki/Variance)
* [Standard Deviation](https://en.wikipedia.org/wiki/Standard_deviation)
* [Analysis of Variance](https://en.wikipedia.org/wiki/Analysis_of_variance)
Kind regards | {"url":"https://aifocus.info/variance/","timestamp":"2024-11-01T23:58:34Z","content_type":"text/html","content_length":"68527","record_id":"<urn:uuid:54d9c94c-9383-4b63-8ce1-5fe3eec87c8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00203.warc.gz"} |
GSI Forum - RDF feedRevised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.Re: Revised CbmGeaneUtil, CbmGeanePro and CbmTrackParP.
/afs/e18/panda/SIM/sneubert/pandaroot/trackbase/CbmGeaneUtil.cxx: In member function ‘void CbmGeaneUtil::FromMarsToSD(Double_t*, Double_t (*)[6], Double_t*, Double_t, Double_t*, Double_t*, Int_t&, Double_t&, Double_t*, Double_t*)’:
/afs/e18/panda/SIM/sneubert/pandaroot/trackbase/CbmGeaneUtil.cxx:1194: error: expected unqualified-id before ‘;’ token
make[2]: *** [trackbase/CMakeFiles/TrkBase.dir/CbmGeaneUtil.o] Error 1
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make: *** [all] Error 2 | {"url":"https://forum.gsi.de/feed.php?mode=m&th=1975&basic=1","timestamp":"2024-11-11T12:03:11Z","content_type":"application/rdf+xml","content_length":"20373","record_id":"<urn:uuid:646ed78a-ab51-4a7c-8112-7d692423c1b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00375.warc.gz"} |
Logarithms: Definition, Examples, and Properties - iMath
In this section, we will learn about logarithms with examples and properties.
Definition of Logarithm
We consider $a>0, a \ne 1$ and $M>0$, and assume that
a^x =M.
In this case, we will call $x$ to be the logarithm of $M$ with respect to the base $a$. We write this phenomenon as
x= log[a] M
(Read as: “$x$ is the logarithm of $M$ to the base $a$”)
∴ a^x =M ⇒ x=log[a] M
On the other hand, if x=log[a] M then we have a^x =M.
To summarise, we can say that
a^x =M if and only if x=log[a] M.
We now understand the above definition with examples.
Examples of Logarithm
1). We know that 2^3 =8.
In terms of logarithms, we can express it as
3 = log[2]8
∴ 2^3 = 8 ⇔ 3 = log[2]8
2). Note that $10^{-1}=\frac{1}{10}=0.1$
That is, 10^-1 = 0.1
According to the logarithms, we have
-1 = log[10] 0.1
Thus, 10^-1 = 0.1 ⇔ -1 = log[10] 0.1
Remarks of Logarithm
(A) If we do not mention the base, then there is no meaning of the logarithms of a number.
(B) The logarithm of a negative number is imaginary.
(C) log[a] a=1.
Proof: As a^1 =a, the proof follows from the definition of the logarithm.
(D) log[a] 1=0.
Proof: For any a ≠ 0, we have a^0 =1. Now applying the definition of logarithms, we obtain the result.
Properties of Logarithm
Logarithm has the following four main properties
a). log[a](MN) = log[a]M + log[a]N
This is called the product rule of logarithms.
b). log[a](M/N) = log[a]M – log[a]N
This is called the Quotient Rule of logarithms
c). log[a]M^n =n log[a] M
This is called the Power Rule of logarithms
d). log[a] M = log[b] M × log[a] b.
This is the Base Change Rule of logarithms
Solved Examples
Ex1: Find log[3]27
Note that we have 27=3^3.
So by the definition of the logarithm, we have
log[3] 27=3 ans.
Ex2: Find $\log_2 \sqrt{8}$
We have 8=2^3
$\therefore \sqrt{8}=(2^3)^{1/2}=(2)^{3 \times 1/2}=2^{3/2}$
Thus, $\sqrt{8}=2^{3/2}$
Now, $\log_2 \sqrt{8}=\log_2 (2)^{3/2}=3/2 \log_2 2=3/2$ ans.
(by the above power rule of logarithms and log[a] a=1)
Q1: What are logarithms?
Answer: Logarithms are used to express exponents in other ways. More specifically, the exponent a^x =M in terms of the logarithm can be expressed as x=log[a] M.
Q2: What is the logarithm of 1?
Answer: The logarithm of 1 with any base is always 0. | {"url":"https://www.imathist.com/introduction-to-logarithm/","timestamp":"2024-11-09T16:07:59Z","content_type":"text/html","content_length":"179754","record_id":"<urn:uuid:e367b9de-8aed-42bd-95ef-e56fc8070ae8>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00365.warc.gz"} |
Library of parent–metabolite models
Drugs can undergo a transformation into metabolites – pharmacologically active and/or producing adverse effects. They can impact the pharmacokinetics of the parent and drive the efficacy. But, at the
same time, can rise safety concerns. Joint parent – metabolite models account for the impact of the parent-metabolite interactions on the estimation.
Library of Parent – Metabolite models includes many characteristic mechanisms, such as first pass effect, uni and bidirectional transformation and up to three compartments for parent and metabolite,
which makes it an excellent tool for exploration and modeling. Implementation with mlxtran macros uses analytical solutions for faster parameter estimation and makes a great basis for the development
of more complex models.
Administration Transformation
Select administration route, delay, absorption mechanism and first pass effect to correctly model Check non-reversible and reversible metabolic systems to predict the impact of parent-metabolite
drug input. interactions.
Distribution Elimination
Explore different number of peripheral compartments independently for parent and metabolite. Try linear or non-linear Michaelis-Menten elimination mechanism to get a better fit.
Parent-metabolite model filters
Parent-metabolite models can be combined with standard administration routes: bolus, infusion, oral/extravascular and oral with bolus. Intravenous doses are always administered to the central parent
compartment. Oral doses can be in addition split between parent and metabolite – the first pass effect.
Intravenous doses (bolus, infusion) are always administered to the central parent compartment, can be with or without a time delay and does not include the first pass effect.
Oral doses can be with the following options:
• with or without the first pass effect
• zero (Tk0) or first order absorption process (ka for parent, kam for metabolite in case of the first pass effect)
• without or with a time delay (Tlag) or with the transit compartments (Mtt, Ktr)
• oral + bolus: bioavailability F (estimated as a separated parameter)
First pass effect
The first pass effect is a phenomenon in which a drug undergoes any biotransformation at a specific location in the body – for example transformation of parent to metabolite before reaching its site
of action or the systemic circulation. The first-pass effect can be clinically relevant when the metabolized fraction is high or when it varies significantly from individual to individual or within
the same individual over time, resulting in variable or erratic absorption. It also has an impact on peak drug concentrations, which may result in parent concentration peaks occurring much earlier.
In the Monolix library, models with the first pass effect include splitting a dose – with or without dose apportionment – and absorbing one fraction in the central parent compartment (Tk0 or ka) and
the other fraction in the central metabolite compartment (Tk0m or kam). The absorption process, zero or first order, is the same for parent and metabolite.
• Dose apportionment means that the splitting is independent of the absorption constants (Tk0, Tk0m or ka, kam), with a fraction F_D of the dose leading to the parent and a fraction 1 – F_D leading
to the metabolite prior to reach the plasma.
• In models without dose apportionment, dose splitting is implicit and results from different absorption rates ka, kam (or absorption times Tk0, Tk0m).
In the library it is possible to select different number of compartments for parent and for metabolite. The central compartments have the same volume V – identifiability issue when only parent drug
is administered.
Parent and metabolite have the same parametrization. As for the PK and PKPD libraries, there are two parametrizations available:
• with exchange rates: k12, k21 for parent and k12m, k21m for metabolite for the second compartments, and k13, k31 and k13m, k31m for the third compartments. Peripheral compartments are defined
implicitly through the exchange rates, so volumes V2, V3, V2m and V3m are not estimated. The volume of central compartments is V (for both parent and metabolite)
• with inter – compartment clearance: Q2, V2, Q3, V3 for parent and Q2m, V2m, Q3m, V3m for metabolite. The volume of central compartments is V1 (for both parent and metabolite), to be compatible
with the PK models in case of the sequential model development.
Transformation and elimination
Drug, after administration is eliminated from the system or transformed into metabolite, which subsequently is also eliminated. In addition, metabolite can be back-transformed into a parent – it is a
common process for example for amines. Both, unidirectional and reversible, transformations can be selected from the “global” section of filters in the Monolix parent-metabolite library. Elimination,
linear or non-linear (Michaelis-Menten), is specific for parent and for metabolite.
• Transformation:
□ Kpm – transformation rate from parent to metabolite
□ Kmp – transformation rate from metabolite to parent
• Elimination: parent and metabolite can be eliminated with different process (linear or non-linear)
□ Cl, k, Vm, Km – elimination parameters for parent
□ Clm, km, Vmm, Kmm – elimination parameters for metabolite
Analytical solutions
Implementation in the Mlxtran
Example: a model with oral absorption (rate ka) of a dose D, one central compartment for parent (cmt = 1) and one for metabolite (cmt = 2), linear elimination (rates: k, km) and uni-directional
transformation (rate Kpm), is the following:
where C[p] and C[m] are the concentrations in the central compartments of parent and metabolite respectively.
Models in the parent-metabolite Monolix library are build using mlxtran macros. For the above model:
;PK model definition for parent
compartment(cmt = 1, volume = V, concentration = Cp)
oral(cmt = 1, ka)
elimination(cmt = 1, k)
;PK model definition for metabolite
compartment(cmt = 2, volume = V, concentration = Cm)
elimination(cmt = 2, k = km)
transfer(from = 1, to = 2, kt = Kpm)
Analytical solutions
Analytical solutions, which are more accurate than numerical solutions and are faster than ODE solvers, are implemented for parent – metabolite models satisfying the following conditions:
• administration (bolus, IV, oral 0 or 1st order) is to the central parent compartment only => models from the library with first pass effect (“with dose apportionment” and “without dose
apportionment”) do not use analytical solutions
• parameters are not time dependent (exception: dose coefficients such as Tlag, which are evaluated only once),
• elimination is linear and only from the central parent and/or metabolite compartments, => models from the library with elimination = “Michaelis-Menten” do not use the analytical solutions
• there are no transit compartments, => models from the library with delay = “transit compartments” do not use the analytical solution
• transfer between parent and metabolite is only between their central compartments,
• there are maximally two peripheral compartments for parent and maximally two for metabolite.
Comparison of the performance between analytical and numerical solutions
The following example compares the computational time (in seconds) of the parameter estimation task (SAEM) using analytical solutions (AS) and numerical solutions (ODE) for models with different
number of compartments.
Dataset: simulated in Simulx N = 100 individuals, who received a single or multi oral doses. 5 or 20 measurements of both, parent and metabolite concentrations registered for each individual.
SAEM: (fixed number of iterations) 500 iterations for the exploratory phase, 200 iterations for the smoothing phase.
Times are given in seconds, and values in bold give the speed up ratio when using an analytical solution. | {"url":"https://monolixsuite.slp-software.com/simulx/2024R1/library-of-parentmetabolite-models","timestamp":"2024-11-09T03:19:23Z","content_type":"text/html","content_length":"45167","record_id":"<urn:uuid:33e29197-a50a-4a7c-a3d4-fbc19562bc12>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00453.warc.gz"} |
Concolic Weakest Precondition is Kind of Like a Lens
That’s a mouthful.
Lens are described as functional getters and setters. The simple lens type is ``
type Lens a b = a -> (b, b -> a)
. The setter is
and the getter is
This type does not constrain lenses to obey the usual laws of getters and setters. So we can use/abuse lens structures for nontrivial computations that have forward and backwards passes that share
information. Jules Hedges is particular seems to be a proponent for this idea.
I’ve described before how to encode reverse mode automatic differentiation in this style. I have suspicions that you can make iterative LQR and guass-seidel iteration have this flavor too, but I’m
not super sure. My attempts ended somewhat unsatisfactorily a whiles back but I think it’s not hopeless. The trouble was that you usually want the whole vector back, not just its ends.
I’ve got another example in imperative program analysis that kind of makes sense and might be useful though. Toy repo here: https://github.com/philzook58/wp-lens
In program analysis it sometimes helps to run a program both concretely and symbolically. Concolic = CONCrete / symbOLIC. Symbolic stuff can slowly find hard things and concrete execution just sprays
super fast and can find the dumb things really quick.
We can use a lens structure to organize a DSL for describing a simple imperative language
The forward pass is for the concrete execution. The backward pass is for transforming the post condition to a pre condition in a weakest precondition analysis. Weakest precondition semantics is a way
of specifying what is occurring in an imperative language. It tells how each statement transforms post conditions (predicates about the state after the execution) into pre conditions (predicates
about before the execution). The concrete execution helps unroll loops and avoid branching if-then-else behavior that would make the symbolic stuff harder to process. I’ve been flipping through
Djikstra’s book on this. Interesting stuff, interesting man.
I often think of a state machine as a function taking s -> s. However, this is kind of restrictive. It is possible to have heterogenous transformations s -> s'. Why not? I think I am often thinking
about finite state machines, which we really don’t intend to have a changing state size. Perhaps we allocated new memory or something or brought something into or out of scope. We could model this by
assuming the memory was always there, but it seems wasteful and perhaps confusing. We need to a priori know everything we will need, which seems like it might break compositionally.
We could model our language making some data type like
data Imp = Skip | Print String | Assign String Expr | Seq Imp Imp | ...
and then build an interpreter
But we can also cut out the middle man and directly define our language using combinators.
To me this has some flavor of a finally tagless style.
Likewise for expressions. Expressions evaluate to something in the context of the state (they can lookup variables), so let’s just use
And, confusingly (sorry), I think it makes sense to use Lens in their original getter/setter intent for variables. So Lens structure is playing double duty.
type Var s a = Lens' s a
With that said, here we go.
type Stmt s s' = s -> s'
type Lens' a b = a -> (b, b -> a)
set l s a = let (_, f) = l s in f a
type Expr s a = s -> a
type Var s a = Lens' s a
skip :: Stmt s s
skip = id
sequence :: Stmt s s' -> Stmt s' s'' -> Stmt s s''
sequence = flip (.)
assign :: Var s a -> Expr s a -> Stmt s s
assign v e = \s -> set v s (e s)
(===) :: Var s a -> Expr s a -> Stmt s s
v === e = assign v e
ite :: Expr s Bool -> Stmt s s' -> Stmt s s' -> Stmt s s'
ite e stmt1 stmt2 = \s -> if (e s) then stmt1 s else stmt2 s
while :: Expr s Bool -> Stmt s s -> Stmt s s
while e stmt = \s -> if (e s) then ((while e stmt) (stmt s)) else s
assert :: Expr s Bool -> Stmt s s
assert e = \s -> if (e s) then s else undefined
abort :: Stmt s s'
abort = const undefined
Weakest precondition can be done similarly, instead we start from the end and work backwards
Predicates are roughly sets. A simple type for sets is
Now, this doesn’t have much deductive power, but I think it demonstrates the principles simply. We could replace Pred with perhaps an SMT solver expression, or some data type for predicates, for
which we’ll need to implement things like substitution. Let’s not today.
A function
is equivalent to
forall c. (b -> c) -> (a -> c)
. This is some kind of CPS / Yoneda transformation thing. A state transformer
to predicate transformer
(s' -> Bool) -> (s -> Bool)
is somewhat evocative of that. I’m not being very precise here at all.
Without further ado, here’s how I think a weakest precondition looks roughly.
type Lens' a b = a -> (b, b -> a)
set l s a = let (_, f) = l s in f a
type Expr s a = s -> a
type Var s a = Lens' s a
type Pred s = s -> Bool
type Stmt s s' = Pred s' -> Pred s
skip :: Stmt s s
skip = \post -> let pre = post in pre -- if
sequence :: Stmt s s' -> Stmt s' s'' -> Stmt s s''
sequence = (.)
assign :: Var s a -> Expr s a -> Stmt s s
assign v e = \post -> let pre s = post (set v s (e s)) in pre
(===) :: Var s a -> Expr s a -> Stmt s s
v === e = assign v e
ite :: Expr s Bool -> Stmt s s' -> Stmt s s' -> Stmt s s'
ite e stmt1 stmt2 = \post -> let pre s = if (e s) then (stmt1 post) s else (stmt2 post) s in pre
abort :: Stmt s s'
abort = \post -> const False
assert :: Expr s Bool -> Stmt s s
assert e = \post -> let pre s = (e s) && (post s) in pre
-- tougher. Needs loop invariant
while :: Expr s Bool -> Stmt s s -> Stmt s s
while e stmt = \post -> let pre s = if (e s) then ((while e stmt) (stmt post)) s else in pre
Finally here is a combination of the two above that uses the branching structure of the concrete execution to aid construction of the precondition. Although I haven’t expanded it out, we are using
the full s t a b parametrization of lens in the sense that states go forward and predicates come back.
type Lens' a b = a -> (b, b -> a)
set l s a = let (_, f) = l s in f a
type Expr s a = s -> a
type Var s a = Lens' s a
type Pred a = a -> Bool
type Stmt s s' = s -> (s', Pred s' -> Pred s) -- eh. Screw the newtype
skip :: Stmt s s
skip = \x -> (x, id)
sequence :: Stmt s s' -> Stmt s' s'' -> Stmt s s''
sequence f g = \s -> let (s', j) = f s in
let (s'', j') = g s' in
(s'', j . j')
assign :: Var s a -> Expr s a -> Stmt s s
assign v e = \s -> (set v s (e s), \p -> \s -> p (set v s (e s)))
--if then else
ite :: Expr s Bool -> Stmt s s' -> Stmt s s' -> Stmt s s'
ite e stmt1 stmt2 = \s ->
if (e s)
then let (s', wp) = stmt1 s in
(s', \post -> \s -> (e s) && (wp post s))
else let (s', wp) = stmt2 s in
(s', \post -> \s -> (not (e s)) && (wp post s))
assert :: Pred s -> Stmt s s
assert p = \s -> (s, \post -> let pre s = (post s) && (p s) in pre)
while :: Expr s Bool -> Stmt s s -> Stmt s s
while e stmt = \s -> if e s then let (s' , wp) = (while e stmt) s in
(s', \post -> let pre s'' = (post s'') && (wp post s'') in pre)
else (s, \p -> p)
-- declare and forget can change the size and shape of the state space.
-- These are heterogenous state commpands
declare :: Iso (s,Int) s' -> Int -> Stmt s s'
declare iso defalt = (\s -> to iso (s, defalt), \p -> \s -> p $ to iso (s, defalt))
forget :: Lens' s s' -> Stmt s s' -- forgets a chunk of state
declare_bracket :: Iso (s,Int) s' -> Int -> Stmt s' s' -> Stmt s s
declare_bracket iso defalt stmt = (declare iso default) . stmt . (forget (_1 . iso))
Neat. Useful? Me dunno. | {"url":"https://www.philipzucker.com/concolic-weakest-precondition-is-kind-of-like-a-lens/","timestamp":"2024-11-08T20:34:36Z","content_type":"text/html","content_length":"39739","record_id":"<urn:uuid:e8b22ffd-758f-4af4-81dc-79a384afe005>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00095.warc.gz"} |
Approximation algorithms
Vazirani, Vijay V. creator text Berlin New York Springer c2001 monographic xix, 378 p. : illustrations ; "The challenge met by this book is to capture the beauty and excitement of work in this
thriving field and to convey in a lucid manner the underlying theory and methodology. Many of the research results presented have been simplified, and new insights provided. Perhaps the most
important aspect of the book is that it shows simple ways of talking about complex, powerful algorithmic ideas by giving intuitive proofs, Combinatorial algorithms -- Set cover -- Steiner tree and
TSP -- Multiway cut and k-cut -- k-center -- Feedback vertex set -- Shortest superstring -- Knapsack -- Bin packing -- Minimum makespan scheduling -- Euclidean TSP -- LP-based algorithms --
Introduction to LP-Duality -- Set cover via dual fitting -- Rounding applied to set cover -- Set cover via the primal-dual schema -- Maximum satisfiability -- Scheduling on unrelated parallel
machines -- Multicut and integer multicommodity flow in trees -- Multiway cut -- Multicut in general graphs -- Sparsest cut -- Steiner forest -- Steiner network -- Facility location -- k-median --
Semidefinite programming -- Other topics -- Shortest vector -- Counting problems -- Hardness of approximation -- Open problems -- An overview of complexity theory for the algorithm designer -- Basic
facts from probability theory. Included Problem, Subject Index. Computer algorithms Mathematical optimization 005.1 VAZ 3540653678 (alk. paper) 9783540653677 3642084699 9783642084690 http://
www.loc.gov/catdir/enhancements/fy0816/2001042005-d.html http://www.loc.gov/catdir/enhancements/fy0816/2001042005-t.html http://www.loc.gov/catdir/enhancements/fy0816/2001042005-d.html http:// | {"url":"http://lib.vau.jfn.ac.lk/cgi-bin/koha/opac-export.pl?op=export&bib=11189&format=mods","timestamp":"2024-11-14T17:34:48Z","content_type":"application/xml","content_length":"3503","record_id":"<urn:uuid:7d87a495-f2f1-4323-afc8-6fff1849a935>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00152.warc.gz"} |
5 Search Results
Maple is an environment for scientific and engineering problem-solving, mathematical exploration, data visualization and technical authoring.
The core of PolyBoRi is a C++ library, which provides high-level data types for Boolean polynomials and monomials, exponent vectors, as well as for the underlying polynomial rings and subsets of the
powerset of the Boolean variables. As a unique approach, binary decision diagrams are used as internal storage type for polynomial structures. On top of this C++-library we provide a Python
interface. This allows parsing of complex polynomial systems, as well as sophisticated and extendable strategies for Gröbner base computation. PolyBoRi features a powerful reference implementation
for Gröbner basis computation.
R is a language and environment for statistical computing and graphics. It is a GNU project which is similar to the S language and environment which was developed at Bell Laboratories (formerly AT&T,
now Lucent Technologies) by John Chambers and colleagues. R can be considered as a different implementation of S. There are some important differences, but much code written for S runs unaltered
under R. R provides a wide variety of statistical (linear and nonlinear modelling, classical statistical tests, time-series analysis, classification, clustering, ...) and graphical techniques, and is
highly extensible. The S language is often the vehicle of choice for research in statistical methodology, and R provides an Open Source route to participation in that activity. One of R's strengths
is the ease with which well-designed publication-quality plots can be produced, including mathematical symbols and formulae where needed. Great care has been taken over the defaults for the minor
design choices in graphics, but the user retains full control.
Risa/Asir is a general computer algebra system and also a tool for various computation in mathematics and engineering. The development of Risa/Asir started in 1989 at FUJITSU. Binaries have been
freely available since 1994 and now the source code is also free. Currently Kobe distribution is the most active branch of its development. We characterize Risa/Asir as follows: (1) An environment
for large scale and efficient polynomial computation. (2) A platform for parallel and distributed computation based on OpenXM protocols.
SINGULAR is a Computer Algebra system for polynomial computations in commutative algebra, algebraic geometry, and singularity theory. SINGULAR's main computational objects are ideals and modules over
a large variety of baserings. The baserings are polynomial rings over a field (e.g., finite fields, the rationals, floats, algebraic extensions, transcendental extensions), or localizations thereof,
or quotient rings with respect to an ideal. SINGULAR features fast and general implementations for computing Groebner and standard bases, including e.g. Buchberger's algorithm and Mora's Tangent Cone
algorithm. Furthermore, it provides polynomial factorizations, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, and many more related functionalities.
Based on an easy-to-use interactive shell and a C-like programming language, SINGULAR's internal functionality is augmented and user-extendible by libraries written in the SINGULAR programming
language. A general and efficient implementation of communication links allows SINGULAR to make its functionality available to other programs. | {"url":"https://orms.mfo.de/search@terms=logical+vectors.html","timestamp":"2024-11-11T07:20:41Z","content_type":"application/xhtml+xml","content_length":"7464","record_id":"<urn:uuid:9b7a8044-e958-4297-914a-5198a4dda08f>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00848.warc.gz"} |
Radiocarbon dating can be used to date
Radiocarbon dating that dies after almost any organic material - but other biological and their reference. They are many fields to date of a price accuracy of conventional radiocarbon dating can be
used to eliminate contaminating carbonates.
Optically stimulated luminescence can be directly dated some specimens of these methods link time. Higham thinks that are in dating is used to date the history and animal remains. Despite the age of
an artifact that were. These fluctuations in human fossils are used for samples up to date of the.
Learn information about 62, it enabled them to calibrate carbon-14 dating, method. Which visit this much-studied event is used to determine the carbon decays at which the carbon-14 can be used by
selected examples. Fossil or coins with carbon-14 in a massive volcanic layers above and. Anything that had its way into living organisms absorb carbon of.
At a technique, or carbon-14 in age estimates for remains between 500 years, various difficulties can be used to date thermal events. Fossils that are two measures of charcoal, and sample contains 14
is something that is difficult, charcoal and below fossils.
Despite the earliest techniques to determine the age. Andersen explains the carbon 14 dating so well in the methods.
Piperno and other methods can provide only be extended https://roughebonysex.com/ calculate its origins in the last 14'000. Does the radiocarbon dating of the result, even when an approximate. I will
attempt to date of 14c atoms. Which item would be used to radiocarbon dating is 30, it is applied to find its way of organic material.
Radiocarbon dating can be used to date
Anything that dies after an artifact that was developed, bomb pulse dating of those problems where can only an. Until recently is an object containing organic material.
Radiocarbon dating can be used to date
When radiocarbon dating can be encountered when using radiocarbon dating. Currently, but an radioactive carbon my eating and sample selection further reading. Thermal events http://
www.costieragrumi.it/ giving an organic materials for granted.
Higham thinks that formed half a method of wiggle with carbon-14 can yield dates that depends upon the name, because so hard. More confidently date used to find such as a radioactive isotopes are
dated. Although radiocarbon dating will not a naturally occurring scientific process. Luckily, the age of ancient fossil or carbon-14 dating method of conventional radiocarbon date.
Ams labs prefer to date are used to date are many fossils, bp. Simply stated, cloth, method which the date material, it is used to.
Phytolith radiocarbon date which are too old, it enabled them to about 62, will only be used by pleistocene geologists, the remains to. Charcoal has made from earth's atmosphere in prehistory to date
Radiocarbon dating can be used to determine the date of
So in the time can be used to the ages of isotope could detect the amount of obtaining absolute dates to a standard. Relative dating can then calculate the age of biological artifacts is radioactive
isotope of organic materials above. If one widely used, and to determine the amount of carbon-14 might find. Join the age of determining whether cave artists. Radio carbon dating is important in the
rock. Various difficulties can be used, plants, libby's group applied the most speed dating is the stable form of carbon-14 it. That a numerical age of ancient artifacts is a radiocarbon dates from a
half-life of a radiocarbon dating methods can measure elapsed. Miami radiocarbon dating was in samples will describe two of birth. In order without any time during the age of tooth enamel and size
requirements dry weights. All radiometric dating is a measure age of radiocarbon dating. Tree-Ring calibrated radiocarbon is a method is the ages of sedimentary rocks. Mesozoic bone consistently
yields a method has been used in an organic materials, scientists to date the most critically, 000. At first used carbon dating can be used to date has been used on.
Radiocarbon dating can be used to directly date which of the following objects
Then we can be used 3114 bc as carbon-14 atoms are able to 50, and inspection laboratories. Theoretically be used to study of an organism, the artifact? Review radiocarbon dating is a kind of the
first of age. Various uranium-series decay can be estimated by dragging the ages of carbon-14, archaeologists use that a material can only works for. This technique used to directly dated directly
date when measured age. Relative and by radiocarbon dates are radiometric dating is their reference standard. Radioactive carbon dating involves the fossils and with any time, research and past. A
date of clock because wood and interesting. There are going to date rock that something compared to 50, had not precise dates from c14 decay processes to be. Some objects closer to date was difficult
to organisms draw largely from. Museum scientists have been under the following nuclear reaction is used to date the older. While some examples of dating is now augmented by willard libby. To refine
estimates of the material must have annual growth rings in this method to find out how carbon. Geologists use csra to organisms draw largely from the oldest things, up-to-date scientific techniques.
But what dating following article from c14 decay help you, radiocarbon level µ θ has a known.
Radiocarbon dating can only be used to date
Alternatively, this radioactivity which is different methods exist and anthropology. Levels of various difficulties can easily be used routinely throughout archaeology, which is about one can be
preserved as rocks? It is radiocarbon dating could shift the radioactive elements are dated radiometrically are younger than abo. Based on only a known as carbon-14 in human civilization came to
about 50 000 years, 000 years. Red; but an archaeologist's staple is the radioactive isotope carbon-14 and anthropology. Ford used to determine the absolute dating can scientists to determine the age
determination that this kind are. Thanks to date peat, like a method is a high degree of chicago. By radiocarbon dating is a known as carbon-12 is small amounts. Sometimes called relative dating must
be directly dated. But an organism that the absolute date rocks that happen to decay rate of decay to. Potassium-Argon dating objects by radiocarbon method reaches back to date rocks, 700 years old,
they. Trees dated using radiocarbon dating has a few of ancient mummies. Radiocarbon dating is one sample and other radioactive isotope carbon-14 dating - but new radiocarbon dating has dated by
tree-ring calibrated radiocarbon dating methods in. How it also known as bone, often used to date landslides over 40 million singles: radiocarbon dating. By radiocarbon dating is currently used to
date of carbon-14 dating method is that half of rocks and anthropology. C-14 dating be dated using radiocarbon dating holocene and are of a date of ancient artifacts and palaeoclimatologists wood use
in the future. C-14 dating technique is also be applied to determine dates from the movies, it is unaffected by several modern world. Dates of radiocarbon dating of carbon-14 can yield dates. Nuclear
chemistry: sometimes called radioactive age of what are a fossil is 300 years ago. | {"url":"http://www.unionecuochivda.com/radiocarbon-dating-can-be-used-to-date/","timestamp":"2024-11-02T17:41:57Z","content_type":"text/html","content_length":"49879","record_id":"<urn:uuid:d8606841-6c9e-41c0-9050-062c2ee6f997>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00585.warc.gz"} |
Live Online classes for kids from 1-10 | Upfunda Academy
Probability in Maths: Understanding the Likelihood of Events - 2023
Probability is an important concept in mathematics that helps us understand the likelihood or chance of an event occurring. It is used in many areas of life, such as in predicting the weather,
playing games, and making decisions. In this article, we will explore the basics of probability and how it can be applied in our daily lives.
What is Probability?
Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, with 0 meaning that the event is impossible and 1 meaning that the event is certain to
happen. For example, if you toss a fair coin, the probability of getting a heads is 0.5 or 1/2. This means that there is a 50% chance of getting a head and a 50% chance of getting a tail.
How Probability is Calculated?
To calculate probability, you need to know the number of possible outcomes and the number of favorable outcomes. The probability is then the ratio of the favorable outcomes to the total number of
outcomes. For example, if you roll a fair six-sided dice, the probability of getting a 2 is 1/6, since there is only one favorable outcome out of six possible outcomes.
Types of Probability:
There are two main types of probability: theoretical probability and experimental probability. Theoretical probability is based on mathematical calculations, while experimental probability is based
on observations and experiments.
Applications of Probability
Probability is used in many areas of life, such as in predicting the weather, playing games, and making decisions.
For example, weather forecasters use probability to predict the chance of rain, and gamblers use probability to make decisions about which games to play and how much to bet. In everyday life, we use
probability to make decisions, such as deciding whether to bring an umbrella on a cloudy day.
Real-Life Examples of Probability
Probability has many real-life applications, such as predicting the weather, playing games, and making decisions. In everyday life, we use probability to make decisions, such as deciding whether to
bring an umbrella on a cloudy day. Here are 5 real-life examples of probability explained in simpler terms for a 5-year-old:
1. Flipping a coin: When we flip a coin, there are two possible outcomes - heads or tails. We can say that there is a 1 in 2 chance of getting heads and a 1 in 2 chance of getting tails.
2. Choosing a color: If we have a jar of red and blue balls, we can figure out how likely it is to pick a red ball by counting how many red and blue balls there are. If there are 2 red balls and 2
blue balls, then there is a 1 in 4 chance of picking a red ball.
3. Eating candy: If we have a bag of candy with 5 red candies and 5 blue candies, we can figure out how likely it is to pick a red candy by picking a candy at random many times and keeping track of
how often we get a red one.
4. Weather: When we look outside, we can see if it is sunny, cloudy, or raining. But if we want to know if it will rain tomorrow, we can use probability to predict the likelihood of it raining based
on what we know about the weather patterns.
5. Sports: If we are playing a game, we can use probability to figure out who is more likely to win. For example, if we have two teams and one team has 3 players and the other team has 6 players,
the team with more players has a higher probability of winning.
Probability is an important concept in mathematics that helps us understand the likelihood of events. It is used in many areas of life, such as in predicting the weather, playing games, and making
decisions. By understanding probability, we can make more informed decisions and better understand the world around us.
Test your knowledge with Upfunda Quiz!
1. If one letter is chosen randomly from the word “BANANA”, what is the probability that the letter chosen is the letter “N”?
2. A box has 10 blocks that are yellow and red. You are equally likely to pick a yellow block or a red block from the box. How many of each colour blocks are in the box?
3. A number cube has numbers 1 to 6. What is the probability of tossing number 9?
(i) “2” of spades
(ii) A jack 4/52
(iii) A king of red colour 4/52
(iv) A card of diamond 2/52
(v) A king or a queen 8/51
(vi) A non faced card 40/52
(vii) A black faced card 6/52
(viii) A black card 26/52
(ix) A non ace 48/52
(x) A non face card of black colour 20/52
(xi) Neither a spade nor a jack 36/52
(xii) Neither a heart nor a red king 38/52
5. Two different coins are tossed randomly. Find the probability of:
(i) Getting two heads HH
(ii) Getting two tails TT
(iii) Getting one tail HT.TH
(iv) Getting no head TT
(v) Getting no tail HH
(vi) Getting at least 1 head HT.TH.HH
(vii) Getting at least 1 tail HT.TH
(viii) Getting almost 1 tail HT.TH
(ix) Getting 1 head and 1 tail HT.TH
6. All the sectors on the spinner below are the same size. If the spinner is spun once what is the probability that the arrow will land on the letter “D”?
(A) 3/4
(B) 2/3
(C) 1/2
(D) 3/8
7. Which letter is the spinner most likely to land on?
Answers :
1. C) 2 out of 6
2. B) 5
3. C) 0
4. Ans
5. Ans
6. (D) 3/8
7. D | {"url":"https://upfunda.academy/blog/af95f8e6-89fa-489f-9258-1e8eec43c4bb","timestamp":"2024-11-09T01:35:37Z","content_type":"text/html","content_length":"39152","record_id":"<urn:uuid:0444a37c-c392-4182-aa97-4b67d68cb601>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00765.warc.gz"} |
Arduino CW decoder project part 2
Pro Antennas
In part 1, I described a solution for processing the CW audio and extracting the carrier level changes that represent the dots and dashes. The next step was to develop the software to translate this
into characters and word breaks.
Morse code is unusual in that there is no fixed length for a character - the number of symbols (dots/dashes) ranges from 1 to 7. If we consider a dot to be represented by a binary 0 and a dash by 1
then all possible combinations can be represented by using 7 bits of a byte giving 127 values. However, these values would not be unique - for instance E, I, S, H and 5 would all have the value 0.
Therefore the number of symbols also needs to be taken into account.
I created a 2 dimensional array to map characters to the dot/dash combinations. I started with a 128 x 7 array which caters for all the characters encoded into a length of up to 7 symbols. This would
reserve 896 bytes of memory, much of which is not used - for instance, the 1 symbol characters only need 2 values (E and T), the 2 symbol characters 4, etc.
As a compromise between memory usage and processing speed I mapped the 6 and 7 symbol characters into column zero of the array. Fortunately this could be done without any clash of values as shown in
the table. The result is a 32 x 6 array which saves over 700 bytes but is simple to process.
To determine whether the signal is a dot or a dash, the decoding algorithm needs to know how long the signal is high for and to determine when a character ends and when a word ends it also needs to
know how long the signal is low for.
The unit of time used for these measurements is the length of a dot which I will call 1 bit time. All other Morse code timings are multiples of this bit time as listed in the example below.
As there is no timing signal in Morse code, the bit time needs to be determined before the signal can be decoded. The first version of my decoder monitored the signal for a few seconds before
starting to decode. This allowed the shortest high/low state to be found and assumed this to be 1 bit time. However, the disadvantage of this approach is that the timing will change - both during a
transmission and when switching to another signal. An additional timing algorithm was needed to dynamically adapt when the timing varied.
The result was a sampling algorithm that detected high-low and low-high changes of state and measured the time between them. These were used to maintain a moving average bit time against which the
high/low time was compared:
• High time < 2 x bit time = dot
• High time > 2 x bit time = dash
• Low time < 2 x bit time = element space
• Low time > 5 x bit time = word space
• Low time between 2 x bit time and 5 x bit time = character space.
The accuracy of the bit timing was limited by the sampling time of 3.3ms. At 20 words per minute, the nominal bit time is 60ms which gives 5% uncertainty for each edge detection which is not a
problem but it would become more significant at higher transmission rates.
The sampling could not be run continuously as I needed processing time for other activities. I therefore waited half a bit time from the last transition before starting to sample for the next
transition. The resulting high level software flow chart is shown here.
This solution worked well on training Morse code videos where the recording was made in ideal conditions but it struggled with real life audio and had a number of limitations. The next update will
look at these issues and how I overcame them.
Most of the Morse code information is taken from Wikipedia | {"url":"https://www.proantennas.co.uk/post/arduino-cw-decoder-project-part-2","timestamp":"2024-11-06T12:38:47Z","content_type":"text/html","content_length":"1050484","record_id":"<urn:uuid:1db33001-df59-43ea-a3d6-22f1d06db983>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00557.warc.gz"} |
What is 1 2/431 as a decimal?
It's very common when learning about fractions to want to know how convert a mixed fraction like 1 2/431 into a decimal. In this step-by-step guide, we'll show you how to turn any fraction into a
decimal really easily. Let's take a look!
Before we get started in the fraction to decimal conversion, let's go over some very quick fraction basics. Remember that a numerator is the number above the fraction line, and the denominator is the
number below the fraction line. We'll use this later in the tutorial.
When we are using mixed fractions, we have a whole number (in this case 1) and a fractional part (2/431). So what we can do here to convert the mixed fraction to a decimal, is first convert it to an
improper fraction (where the numerator is greater than the denominator) and then from there convert the improper fraction into a decimal/
Step 1: Multiply the whole number by the denominator
1 x 431 = 431
Step 2: Add the result of step 1 to the numerator
431 + 2 = 433
Step 3: Divide the result of step 2 by the denominator
433 รท 431 = 1.0046403712297
So the answer is that 1 2/431 as a decimal is 1.0046403712297.
And that is is all there is to converting 1 2/431 to a decimal. We convert it to an improper fraction which, in this case, is 433/431 and then we divide the new numerator (433) by the denominator to
get our answer.
If you want to practice, grab yourself a pen, a pad, and a calculator and try to convert a few mixed fractions to a decimal yourself.
Hopefully this tutorial has helped you to understand how to convert a fraction to a decimal and made you realize just how simple it actually is. You can now go forth and convert mixed fractions to
decimal as much as your little heart desires!
Cite, Link, or Reference This Page
If you found this content useful in your research, please do us a great favor and use the tool below to make sure you properly reference us wherever you use it. We really appreciate your support!
• <a href="http://visualfractions.com/calculator/mixed-to-decimal/what-is-1-2-431-as-a-decimal/">What is 1 2/431 as a decimal?</a>
• "What is 1 2/431 as a decimal?". VisualFractions.com. Accessed on November 10, 2024. http://visualfractions.com/calculator/mixed-to-decimal/what-is-1-2-431-as-a-decimal/.
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Mixed Fraction to Decimal Calculator
Mixed Fraction as Decimal
Enter a whole number, numerator and denominator
Random Mixed Fraction to Decimal Calculation | {"url":"https://visualfractions.com/calculator/mixed-to-decimal/what-is-1-2-431-as-a-decimal/","timestamp":"2024-11-10T02:39:11Z","content_type":"text/html","content_length":"35833","record_id":"<urn:uuid:48737b05-a743-4d5f-9a93-5ae97b8044d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00041.warc.gz"} |
For machine learning engineers, fitting is a particularly challenge
Build, Compile, and Fit Models in TensorFlow Part II
Fitting models in TensorFlow
In this article, we will discuss how to fit a model, we will be using the Jenga illustration to make ideas clear. Fitting a model is the most important challenge for a Machine Learning Engineer on
his building model journey. Ready to take place as an ML Engineer in fitting models?
A model must be trained on data to fit.
Similar to playing Jenga, fitting a model is a process. We begin with a shaky skyscraper made of blocks (the model). After feeding the model data, we begin removing bricks to determine if the tower
still stands (the model is accurate in its predictions). We need to add some blocks back (change the model's parameters) if the tower collapses before trying again. The goal of fitting a model is to
find a set of parameters that allows the model to make accurate predictions on unseen data. This is a trial-and-error process, and it can take some time to find the best set of parameters.
Fitting a model means training it on a dataset of data. The goal of training a model is to find the parameters that minimize the loss function. The loss function is a measure of how well the model is
performing. Do you want to discover, how it works on TensorFlow? Let's talk about it now.
Fitting a model with TensorFlow
In TensorFlow, we can fit a model using the fit function. The fit function takes several arguments, including the dataset, the number of epochs, and the learning rate. To fit a model in TensorFlow,
we can use the following code:
model.fit(dataset, epochs=100, learning_rate=0.001)
In this code, we are fitting the model on the dataset for 100 epochs with a learning rate of 0.001.
Note: The dataset is a collection of data points, each of which has a label. The number of epochs is the number of times the model will be trained on the dataset. The learning rate is a parameter
that controls how much the model's parameters are updated each time it is trained.
Once the model has been fitted, we can evaluate its performance on a test dataset. The test dataset is a collection of data points that the model has not seen before. We can evaluate the model's
performance by calculating the accuracy, which is the percentage of data points that the model predicts correctly. We can be compared with a unit test in classic coding, to see how good is our work.
Note that you evaluate your model on a test dataset, here is an example code:
To end with this part let me show you a great example of the process: build, compile and fit a model with naturally the evaluation at the end :
import tensorflow as tf
# Create a dataset of data points.
data = tf.data.Dataset.from_tensor_slices([[1, 2], [3, 4]])
# build a model that will predict the value of y given the value of x.
model = tf.keras.Sequential([
tf.keras.layers.Dense(1, input_shape=(1,))
# Compile a model with an adam optimizer and a loss function.
model.compile(optimizer='adam', loss='mse')
# Train or fit the model on the data.
model.fit(data, epochs=10)
# Evaluate the model on the data.
This code creates a dataset of data points, creates a model that will predict the value of y given the value of x, compiles the model with the adam optimizer and a loss function, trains the model on
the data, and evaluates the model on the data.
The tf.data.Dataset.from_tensor_slices() function creates a dataset of data points from a list of tensors. In this case, the list of tensors is [[1, 2], [3, 4]], which represents two data points. The
tf.keras.Sequential() function creates a sequential model, which is a type of model that consists of a series of layers. In this case, the model consists of a single Dense layer, which is a type of
layer that performs linear regression. The model.compile() function compiles the model with an optimizer and a loss function. The optimizer is used to update the model's parameters during training,
and the loss function is used to measure the model's performance. The model.fit() function trains the model on the data. The model is trained for 10 epochs, which means that the model is trained on
the data 10 times. The model.evaluate() function evaluates the model on the data. The model is evaluated on the data to see how well it performs.
Challenges fitting with TensorFlow
For machine learning engineers, fitting is a particularly difficult process that also takes patience and all of your ML engineering talents; here is a short summary of the most typical ones:
1. Choosing the right optimizer. The optimizer is a function that updates the model's parameters to minimize the loss function. There are many different optimizers to choose from, and the right
choice will depend on the specific problem you are trying to solve. Some of the most popular optimizers include Adam (it reminds you something I guess), SGD, and Adagrad.
2. Regularizing the model. Overfitting is a common problem in machine learning, where the model learns too much from the training data and does not generalize well to new data. There are a number of
techniques for regularizing models, such as L1 and L2 regularization.
3. Tuning the hyperparameters. The hyperparameters of a model are the parameters that control the model's behavior, such as the learning rate and the number of hidden layers. Finding the right
values for these hyperparameters can be a time-consuming process, but it is important to get them right in order to achieve good performance.
4. Dealing with imbalanced data. Imbalanced data is a problem where there are more examples of one class than another. This can make it difficult for the model to learn to predict the minority class
. There are a number of techniques for dealing with imbalanced data, such as oversampling and undersampling.
5. Evaluating the model. Once you have trained a model, it is important to evaluate its performance on a held-out test set. This will give you an idea of how well the model will generalize to new
data. There are several different metrics that can be used to evaluate a model, such as accuracy, precision, and recall.
These are just a few of the challenges that machine learning engineers face when fitting models with TensorFlow. By following some tips, you can overcome all this, in the next articles of this
series, we will come over each challenge.
Review of concepts and conclusion
Imagine that you are trying to build a Jenga tower that is as tall as possible. You start by stacking the blocks randomly, but the tower quickly falls over. You then start to experiment with
different ways of stacking the blocks. You might try stacking them in a spiral pattern, or you might try stacking them in a checkerboard pattern. You might also try adding more or fewer blocks to the
As you experiment, you will start to learn what works and what doesn't work. You will learn that some patterns are more stable than others, and you will learn that the number of blocks in the tower
affects its stability.
Eventually, you will find a pattern that allows you to build a Jenga tower that is very tall. This pattern is the set of parameters that allows the tower to be as stable as possible.
Fitting a model on TensorFlow is a similar process. We start with a model that is not very accurate, and we then experiment with different parameters. We might try changing the number of layers in
the model, or we might try changing the activation functions in the model. We might also try using a different loss function or optimizer.
As we experiment, we will start to learn what works and what doesn't work. We will learn that some architectures are more accurate than others, and we will learn that the choice of loss function and
optimizer affects the accuracy of the model.
Eventually, we will find a set of parameters that allows the model to be as accurate as possible. This set of parameters is the model that we will use to make predictions on new data.
Yes, during our journey in these articles, we focused on general concepts on how to build, compile and fit models with TensorFlow, the next one will focus on powerful tools and tricks to make it look
easy, one of them is TensorBoard in Google Colab.
Before turning the page look at this, what an ML Engineer needs as hard and soft skills to perform:
Keep in mind that Machine learning engineers are the builders of the future. They use their skills to create algorithms that can learn from data and make predictions.
Remember, building machine learning models is like building a tower of blocks. It takes time and practice, but it can be lots of fun too! 😊
Arthur Kaza spartanwk@gmail.com @#PeaceAndLove @Copyright_by_Kaz’Art / @ArthurStarks | {"url":"https://arthurkaza.hashnode.dev/build-compile-and-fit-models-in-tensorflow-part-ii","timestamp":"2024-11-10T17:10:01Z","content_type":"text/html","content_length":"216007","record_id":"<urn:uuid:27e2994a-ef54-498d-b2a0-afeab10ecfe1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00449.warc.gz"} |
Chemistry 30
7 years ago
If we assume no heat loss, then the heat released from the exothermic combustion of methane all goes towards heating the water. We can say: $delta\left(H\right)=-Q$delta(H)=−Q . Where delta(H) is the
enthalpy change of the reaction, and Q is the heat gained by the water.
We then have two equations:
1. $delta\left(H\right)=n\cdot delta\left(Hc\right)$delta(H)=n·delta(Hc)
2. $Q=mC\cdot delta\left(T\right)$Q=mC·delta(T)
We can calculate Q using the info given in the problem (remember that 1g = 1mL for water!), and then use $delta\left(H\right)=-Q$delta(H)=−Q to calculate the corresponding enthalpy change. Then we
can use equation 1 to calculate the moles of methane. Then using $n=\frac{m}{M}$n=mM , (M being the molar mass of methane, which you can calculate using a periodic table) we can find the mass of | {"url":"https://www.tutortag.com/chat/alberta/a-reference-gives-the-molar-enthalpy-of-combustion-for-metha","timestamp":"2024-11-14T17:54:27Z","content_type":"text/html","content_length":"24331","record_id":"<urn:uuid:abd25c2d-1017-49b0-923f-f0aa96120788>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00604.warc.gz"} |
Non-linear stability of L4 in the restricted three b o dy problem for radiated axes symmetric primaries with resonances
Rajiv Aggarwal, Z. A. Taqvi and Iqbal Ahmad
Department of Mathematics, Jamia Mil lia Islamia, New Delhi 110 025, India
Abstract. We have investigated the non-linear stability of the triangular libration point L4 of the Restricted three body problem under the presence of the third and fourth order resonances, when the
bigger primary is an oblate body and the smaller a triaxial body and both are source of radiation. It is found through Markeev's theorem that L[4] is always unstable in the third order resonance case
and stable or unstable in the fourth order resonance case depending upon the values of the parameters A[1] , A[1] , A[2] , P and P , where A[1] , A[1] and A[2 ], depends upon the lengths of the semi
axes of the primaries and P and P are the radiation parameters.
Keywords: : Restricted three body problem, axis symmetric body, libration points, non-linear stability, Markeev's theorem. | {"url":"https://bulletin.astron-soc.in/06December/Abstracts/200634327.htm","timestamp":"2024-11-10T01:04:52Z","content_type":"text/html","content_length":"2016","record_id":"<urn:uuid:6ff6b60b-dc14-48ff-9569-5f61aaf54d8c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00564.warc.gz"} |
Module Specifications.
Current Academic Year 2024 - 2025
All Module information is indicative, and this portal is an interim interface pending the full upgrade of Coursebuilder and subsequent integration to the new DCU Student Information System (DCU Key).
As such, this is a point in time view of data which will be refreshed periodically. Some fields/data may not yet be available pending the completion of the full Coursebuilder upgrade and integration
project. We will post status updates as they become available. Thank you for your patience and understanding.
Date posted: September 2024
No Banner module data is available
Module Title
Module Code (ITS)
Faculty School
Semester 1: Eilish McLoughlin
Module Co-ordinator Semester 2: Eilish McLoughlin
Autumn: Eilish McLoughlin
Module Teachers Eilish McLoughlin
NFQ level 8 Credit Rating
Pre-requisite Not Available
Co-requisite Not Available
Compatibles Not Available
Incompatibles Not Available
Coursework Only
This module will develop the student's understanding of fundamental concepts and ideas in modern physics, specifically the use and application of the Schroedinger equation, and the priciples of
special relativity.
Learning Outcomes
1. Define key concepts in modern physics including wave function, probability amplitude, reference frame, invariance.
2. Solve the Schroedinger equation for 1-dimensional potential wells.
3. Apply the 3-d Schroedinger equation to the hydrogen atom.
4. Apply the basic relationships of special relativity to high energy particles.
5. Derive, from given premises, relevant relationships between physical variables.
6. Solve problems, from information given, requiring the calculation of the values of physical variables in quantum mechanics and relativity.
7. Explain the relevance of quantum theory and relativity in modern views of the physical universe.
Workload Full-time hours per semester
Type Hours Description
Lecture 24 2 hours of asychronous lectures per week accessed as pre-recorded videos
Online activity 24 2 hours of scheduled synchronous online activities per week.
Tutorial 24 Student learning support facilitated through small-group tutorials with flexible provision.
Independent Study 53 Student self directed learning
Total Workload: 125
All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http:/
Indicative Content and Learning Activities
Wave mechanicsDe Broglie's hypothesis, wave functions and probability amplitudes, the Heisenberg Uncertainty principle. The Schroedinger wave equation: simple solutions in one dimension,
transmission, reflection and penetration at a barrier, tunnelling, potential wells, the harmonic oscillator.The Schroedinger equation in three dimensionsthe hydrogen atom, quantisation of angular
momentum, spatial quantisation, the Zeeman effect.Spinthe fourth quantum number, the Pauli exclusion principle.Special RelativityRelativistic dynamics, relativistic mass and momentum, total energy,
mass/energy equivalence. Spacetime: spacetime diagrams, introduction to four-vectors. Application of relativistic dynamics to particle beam devices and collision experiments.Nuclear PhysicsNucleons
and nuclear models, nuclear spin nuclear reactions and cross-sections. Introduction to elementary particles and the Standard Model.
Assessment Breakdown
Continuous Assessment % Examination Weight %
Course Work Breakdown
Type Description % of total Assessment Date
Completion of online activity Solve physics problem based on mathematical/quantitative aspects of module concepts. 50% As required
In Class Test Solve physics problem based on mathematical/quantitative aspects of module concepts. 50% Once per semester
Indicative Reading List
• Young and Freedman: 2020, University Physics with Modern Physics,, 15 Ed.,,, https://www.pearson.com/us/higher-education/program/Young-Modified-Mastering-Physics-with-Pearson-e-Text-Standalone-
• Eugenia Etkina, Gorazd Planinsic and Alan Van Heuvelen: 2019, College Physics: Explore and Apply,, 2nd Ed. [ISBN: 978- 01346018],
• Taylor J.R. Zafiratos, C.D. Dubson M.A.: 2004, Modern Physics For Scientists and Engineers, Prentice Hall,
• Blatt F.J.: 1992, Modern Physics, McGraw-Hill,
Other Resources
55285, Simulations, 0, Phet simulations, https://phet.colorado.edu/en/simulations /, | {"url":"https://modspec.dcu.ie/registry/module_contents.php?function=2&subcode=PS476L","timestamp":"2024-11-13T16:17:36Z","content_type":"application/xhtml+xml","content_length":"51684","record_id":"<urn:uuid:b802fdf8-3a3e-4700-8f6a-865b99887dfe>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00073.warc.gz"} |
Aaltodoc Repository :: Browsing by Author "Ollila, Esa, Prof., Aalto University, Department of Signal Processing and Acoustics, Finland"
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• Robust large-scale statistical inference and ICA using bootstrapping
(Aalto University, 2018) Basiri, Shahab; Ollila, Esa, Prof., Aalto University, Department of Signal Processing and Acoustics, Finland; Koivunen, Visa, Prof., Aalto University, Department of
Signal Processing and Acoustics, Finland; Signaalinkäsittelyn ja akustiikan laitos; Department of Signal Processing and Acoustics; Sähkötekniikan korkeakoulu; School of Electrical Engineering;
Ollila, Esa, Prof., Aalto University, Department of Signal Processing and Acoustics, Finland
The reliability of the information extracted from large-scale data, as well as the validity of data-driven decisions depend on the veracity of the data and the utilized data processing methods.
Quantification of the veracity of parameter estimates or data-driven decisions is required in order to make appropriate choices of estimators and identifying redundant or irrelevant variables in
multi-variate data settings. Moreover, quantification of the veracity allows efficient usage of available resources by processing only as much data as is needed to achieve a desired level of
accuracy or confidence. Statistical inference such as finding the accuracy of certain parameter estimates and testing hypotheses on model parameters can be used to quantify the veracity of
large-scale data analytics results. In this thesis, versatile bootstrap procedures are developed for performing statistical inference on large-scale data. First, a computationally efficient and
statistically robust bootstrap procedure is proposed, which is scalable to smaller distinct subsets of data. Hence, the proposed method is compatible with distributed storage systems and parallel
computing architectures. The statistical convergence and robustness properties of the method are analytically established. Then, two specific low-complexity bootstrap procedures are proposed for
performing statistical inference on the mixing coefficients of the Independent Component Analysis (ICA) model. Such statistical inferences are required to identify the contribution of a specific
source signal-of-interest onto the observed mixture variables. This thesis establishes significant analytical results on the structure of the FastICA estimator, which enable the computation of
bootstrap replicas in closed-form. This not only saves computational resources, but also avoids convergence problems, permutation and sign ambiguities of the FastICA algorithm. The developed
methods enable statistical inference in a variety of applications in which ICA is commonly applied, e.g., fMRI and EEG signal processing. In the thesis, an alternative derivation of the
fixed-point FastICA algorithm is established. The derivation provides a better understanding of how the FastICA algorithm is derived from the exact Newton-Raphson (NR) algorithm. In the original
derivation, FastICA was derived as an approximate NR algorithm using unjustified assumptions, which are not required in the alternative derivation presented in this thesis. It is well known that
the fixed-point FastICA algorithm has severe convergence problems when the dimensionality of the data and the sample size are of the same order. To mitigate this problem, a power iteration
algorithm for FastICA is proposed, which is remarkably more stable than the fixed-point FastICA algorithm. The proposed PowerICA algorithm can be run in parallel on two computing nodes making it
considerably faster to compute. | {"url":"https://aaltodoc.aalto.fi/browse/author?value=Ollila,%20Esa,%20Prof.,%20Aalto%20University,%20Department%20of%20Signal%20Processing%20and%20Acoustics,%20Finland","timestamp":"2024-11-10T08:13:52Z","content_type":"text/html","content_length":"395771","record_id":"<urn:uuid:9f5817d4-b763-449a-8822-b49fc6f22f06>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00299.warc.gz"} |
Tensor Methods in Machine Learning
Tensors are high dimensional generalizations of matrices. In recent years tensor decompositions were used to design learning algorithms for estimating parameters of latent variable models like Hidden
Markov Model, Mixture of Gaussians and Latent Dirichlet Allocation (many of these works were considered as examples of “spectral learning”, read on to find out why). In this post I will briefly
describe why tensors are useful in these settings.
Using Singular Value Decomposition (SVD), we can write a matrix $M \in \mathbb{R}^{n\times m}$ as the sum of many rank one matrices:
\[M = \sum_{i=1}^r \lambda_i \vec{u}_i \vec{v}_i^\top.\]
When the rank $r$ is small, this gives a concise representation for the matrix $M$ (using $(m+n)r$ parameters instead of $mn$). Such decompositions are widely applied in machine learning.
Tensor decomposition is a generalization of low rank matrix decomposition. Although most tensor problems are NP-hard in the worst case, several natural subcases of tensor decomposition can be solved
in polynomial time. Later we will see that these subcases are still very powerful in learning latent variable models.
Matrix Decompositions
Before talking about tensors, let us first see an example of how matrix factorization can be used to learn latent variable models. In 1904, psychologist Charles Spearman tried to understand whether
human intelligence is a composite of different types of measureable intelligence. Let’s describe a highly simplified version of his method, where the hypothesis is that there are exactly two kinds of
intelligence: quantitative and verbal. Spearman’s method consisted of making his subjects take several different kinds of tests. Let’s name these tests Classics, Math, Music, etc. The subjects scores
can be represented by a matrix $M$, which has one row per student, and one column per test.
The simplified version of Spearman’s hypothesis is that each student has different amounts of quantitative and verbal intelligence, say $x_{quant}$ and $x_{verb}$ respectively. Each test measures a
different mix of intelligences, so say it gives a weighting $y_{quant}$ to quantitative and $y_{verb}$ to verbal. Intuitively, a student with higher strength on verbal intelligence should perform
better on a test that has a high weight on verbal intelligence. Let’s describe this relationship as a simple bilinear function:
\[score = x_{quant} \times y_{quant} + x_{verb}\times y_{verb}.\]
Denoting by $\vec x_{verb}, \vec x_{quant}$ the vectors describing the strengths of the students, and letting $\vec y_{verb}, \vec y_{quant}$ be the vectors that describe the weighting of
intelligences in the different tests, we can express matrix $M$ as the sum of two rank 1 matrices (in other words, $M$ has rank at most $2$):
\[M = \vec x_{quant} \vec y_{quant}^\top + \vec x_{verb} \vec y_{verb}^\top.\]
Thus verifying that $M$ has rank $2$ (or that it is very close to a rank $2$ matrix) should let us conclude that there are indeed two kinds of intelligence.
Note that this decomposition is not the Singular Value Decomposition (SVD). SVD requires strong orthogonality constraints (which translates to “different intelligences are completely uncorrelated”)
that are not plausible in this setting.
The Ambiguity
But ideally one would like to take the above idea further: we would like to assign a definitive quantitative/verbal intelligence score to each student. This seems simple at first sight: just read off
the score from the decomposition. For instance, it shows Alice is strongest in quantitative intelligence.
However, this is incorrect, because the decomposition is not unique! The following is another valid decomposition
According to this decomposition, Bob is strongest in quantitative intelligence, not Alice. Both decompositions explain the data perfectly and we cannot decide a priori which is correct.
Sometimes we can hope to find the unique solution by imposing additional constraints on the decomposition, such as all matrix entries have to be nonnegative. However even after imposing many natural
constraints, in general the issue of multiple decompositions will remain.
Adding the 3rd Dimension
Since our current data has multiple explanatory decompositions, we need more data to learn exactly which explanation is the truth. Assume the strength of the intelligence changes with time: we get
better at quantitative tasks at night. Now we can let the (poor) students take the tests twice: once during the day and once at night. The results we get can be represented by two matrices $M_{day}$
and $M_{night}$. But we can also think of this as a three dimensional array of numbers -– a tensor $T$ in $\mathbb{R}^{\sharp students\times \sharp tests\times 2}$. Here the third axis stands for
“day” or “night”. We say the two matrices $M_{day}$ and $M_{night}$ are slices of the tensor $T$.
Let $z_{quant}$ and $z_{verb}$ be the relative strength of the two kinds of intelligence at a particular time (day or night), then the new score can be computed by a trilinear function:
\[score = x_{quant} \times y_{quant} \times z_{quant} + x_{verb}\times y_{verb} \times z_{verb}.\]
Keep in mind that this is the formula for one entry in the tensor: the score of one student, in one test and at a specific time. Who the student is specifies $x_{quant}$ and $x_{verb}$; what the test
is specifies weights $y_{quant}$ and $y_{verb}$; when the test takes place specifies $z_{quant}$ and $z_{verb}$.
Similar to matrices, we can view this as a rank 2 decomposition of the tensor $T$. In particular, if we use $\vec x_{quant}, \vec x_{verb}$ to denote the strengths of students, $\vec y_{quant},\vec
y_{verb}$ to denote the weights of the tests and $\vec z_{quant}, \vec z_{verb}$ to denote the variations of strengths in time, then we can write the decomposition as
\[T = \vec x_{quant}\otimes \vec y_{quant}\otimes \vec z_{quant} + \vec x_{verb}\otimes \vec y_{verb}\otimes \vec z_{verb}.\]
Now we can check that the second matrix decomposition we had is no longer valid: there are no values of $z_{quant}$ and $z_{verb}$ at night that could generate the matrix $M_{night}$. This is not a
coincidence. Kruskal 1977 gave sufficient conditions for such decompositions to be unique. When applied to our case it is very simple:
Corollary The decomposition of tensor $T$ is unique (up to scaling and permutation) if none of the vector pairs $(\vec x_{quant}, \vec x_{verb})$, $(\vec y_{quant},\vec y_{verb})$, $(\vec z_
{quant},\vec z_{verb})$ are co-linear.
Note that of course the decomposition is not truly unique for two reasons. First, the two tensor factors are symmetric, and we need to decide which factor correspond to quantitative intelligence.
Second, we can scale the three components $\vec x_{quant}$ ,$\vec y_{quant}$, $\vec z_{quant}$ simultaneously, as long as the product of the three scales is 1. Intuitively this is like using
different units to measure the three components. Kruskal’s result showed that these are the only degrees of freedom in the decomposition, and there cannot be a truly distinct decomposition as in the
matrix case.
Finding the Tensor
In the above example we get a low rank tensor $T$ by gathering more data. In many traditional applications the extra data may be unavailable or hard to get. Luckily, many exciting recent developments
show that we can uncover these special tensor structures even if the original data is not in a tensor form!
The main idea is to use method of moments (see a nice post by Moritz): estimate lower order correlations of the variables, and hope these lower order correlations have a simple tensor form.
Consider Hidden Markov Model as an example. Hidden Markov Models are widely used in analyzing sequential data like speech or text. Here for concreteness we consider a (simplified) model of natural
language texts(which is a basic version of the word embeddings).
In Hidden Markov Model, we observe a sequence of words (a sentence) that is generated by a walk of a hidden Markov Chain: each word has a hidden topic $h$ (a discrete random variable that specifies
whether the current word is talking about “sports” or “politics”); the topic for the next word only depends on the topic of the current word. Each topic specifies a distribution over words. Instead
of the topic itself, we observe a random word $x$ drawn from this topic distribution (for example, if the topic is “sports”, we will more likely see words like “score”). The dependencies are usually
illustrated by the following diagram:
More concretely, to generate a sentence in Hidden Markov Model, we start with some initial topic $h_1$. This topic will evolve as a Markov Chain to generate the topics for future words $h_2,
h_3,…,h_t$. We observe words $x_1,…,x_t$ from these topics. In particular, word $x_1$ is drawn according to topic $h_1$, word $x_2$ is drawn according to topic $h_2$ and so on.
Given many sentences that are generated exactly according to this model, how can we construct a tensor? A natural idea is to compute correlations: for every triple of words $(i,j,k)$, we count the
number of times that these are the first three words of a sentence. Enumerating over $i,j,k$ gives us a three dimensional array (a tensor) $T$. We can further normalize it by the total number of
sentences. After normalization the $(i,j,k)$-th entry of the tensor will be an estimation of the probability that the first three words are $(i,j,k)$. For simplicity assume we have enough samples and
the estimation is accurate:
\[T_{i,j,k} = \mbox{Pr}[x_1 = i, x_2=j, x_3=k].\]
Why does this tensor have the nice low rank property? The key observation is that if we “fix” (condition on) the topic of the second word $h_2$, it cuts the graph into three parts: one part
containing $h_1,x_1$, one part containing $x_2$ and one part containing $h_3,x_3$. These three parts are independent conditioned on $h_2$. In particular, the first three words $x_1,x_2,x_3$ are
independent conditioned on the topic of the second word $h_2$. Using this observation we can compute each entry of the tensor as
\[T_{i,j,k} = \sum_{l=1}^n \mbox{Pr}[h_2 = l] \mbox{Pr}[x_1 = i| h_2 = l]\times \mbox{Pr}[x_2 = j| h_2 = l]\times \mbox{Pr}[x_3 = k| h_2 = l].\]
Now if we let $\vec x_l$ be a vector whose $i$-th entry is the probability of the first word is $i$, given the topic of the second word is $l$; let $\vec y_l$ and $\vec z_l$ be similar for the second
and third word. We can then write the entire tensor as
\[T = \sum_{l=1}^n \mbox{Pr}[h_2 = l] \vec x_l \otimes \vec y_l \otimes \vec z_l.\]
This is exactly the low rank form we are looking for! Tensor decomposition allows us to uniquely identify these components, and further infer the other probabilities we are interested in. For more
details see the paper by Anandkumar et al. 2012 (this paper uses the tensor notations, but the original idea appeared in the paper by Mossel and Roch 2006).
Implementing Tensor Decomposition
Using method of moments, we can discover nice tensor structures from many problems. The uniqueness of tensor decomposition makes these tensors very useful in learning the parameters of the models.
But how do we compute the tensor decompositions?
In the worst case we have bad news: most tensor problems are NP-hard. However, in most natural cases, as long as the tensor does not have too many components, and the components are not adversarially
chosen, tensor decomposition can be computed in polynomial time! Here we describe the algorithm by Dr. Robert Jenrich (it first appeared in a 1970 working paper by Harshman, the version we present
here is a more general version by Leurgans, Ross and Abel 1993).
Jenrich’s Algorithm
Input: tensor $T = \sum_{i=1}^r \lambda_i \vec x_i \otimes \vec y_i \otimes \vec z_i$.
1. Pick two random vectors $\vec u, \vec v$.
2. Compute $T_\vec u = \sum_{i=1}^n u_i T[:,:,i] = \sum_{i=1}^r \lambda_i (\vec u^\top \vec z_i) \vec x_i \vec y_i^\top$.
3. Compute $T_\vec v = \sum_{i=1}^n v_i T[:,:,i] = \sum_{i=1}^r \lambda_i (\vec v^\top \vec z_i) \vec x_i \vec y_i^\top$.
4. $\vec x_i$’s are eigenvectors of $T_\vec u (T_\vec v)^{+}$, $\vec y_i$’s are eigenvectors of $T_\vec v (T_\vec u)^{+}$.
In the algorithm, “$^+$” denotes pseudo-inverse of a matrix (think of it as inverse if this is not familiar).
The algorithm looks at weighted slices of the tensor: a weighted slice is a matrix that is the projection of the tensor along the $z$ direction (similarly if we take a slice of a matrix $M$, it will
be a vector that is equal to $M\vec u$). Because of the low rank structure, all the slices must share matrix decompositions with the same components.
The main observation of the algorithm is that although a single matrix can have infinitely many low rank decompositions, two matrices can only have a unique decomposition if we require them to have
the same components. In fact, it is highly unlikely for two arbitrary matrices to share decompositions with the same components. In the tensor case, because of the low rank structure we have
\[T_\vec u = XD_\vec u Y^\top; \quad T_\vec v = XD_\vec v Y^\top,\]
where $D_\vec u,D_\vec v$ are diagonal matrices. This is called a simultaneous diagonalization for $T_\vec u$ and $T_\vec v$. With this structure it is easy to show that $\vec x_i$’s are eigenvectors
of $T_\vec u (T_\vec v)^{+} = X D_\vec u D_\vec v^{-1} X^+$. So we can actually compute tensor decompositions using spectral decompositions for matrices.
Many of the earlier works (including Mossel and Roch 2006) that apply tensor decompositions to learning problems have actually independently rediscovered this algorithm, and the word “tensor” never
appeared in the papers. In fact, tensor decomposition techniques are traditionally called “spectral learning” since they are seen as derived from SVD. But now we have other methods to do tensor
decompositions that have better theoretical guarantees and practical performances. See the survey by Kolda and Bader 2009 for more discussions.
Related Links
For more examples of using tensor decompositions to learn latent variable models, see the paper by Anandkumar et al. 2012. This paper shows that several prior algorithms for learning models such as
Hidden Markov Model, Latent Dirichlet Allocation, Mixture of Gaussians and Independent Component Analysis can be interpreted as doing tensor decompositions. The paper also gives a proof that tensor
power method is efficient and robust to noise.
Recent research focuses on two problems: how to formulate other learning problems as tensor decompositions, and how to compute tensor decompositions under weaker assumptions. Using tensor
decompositions, we can learn more models that include community models, probabilistic Context-Free-Grammars, mixture of general Gaussians and two-layer neural networks. We can also efficiently
compute tensor decompositions when the rank of the tensor is much larger than the dimension (see for example the papers by Bhaskara et al. 2014, Goyal et al. 2014, Ge and Ma 2015). There are many
other interesting works and open problems, and the list here is by no means complete. | {"url":"https://www.offconvex.org/2015/12/17/tensor-decompositions/","timestamp":"2024-11-05T11:45:42Z","content_type":"text/html","content_length":"25318","record_id":"<urn:uuid:edbe4009-cb76-4aec-a274-bb887afed2b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00559.warc.gz"} |
Mateusz ?oskot wrote:
> 2011/10/20 Adam Wulkiewicz<adam.wulkiewicz_at_[hidden]>:
>> I've forgotten about one function - disjoint_with_boundry().
> It's unclear to me what you mean here.
> If you mean geometries A and B are disjoint but with possible intersection of
> their boundaries, then it is OGC touches()
> http://postgis.org/documentation/manual-svn/ST_Touches.html
>> Of course names I've used are just examples. Ideally we would have one function
>> and template/function parameters would describe the relationship. I
>> recall that we talked about it earlier and that the first template
>> parameter can't be used to achieve this goal:
>> intersects<without_boundry>(A, B); // impossible
> I'm a GIS boy, so forgive me if I'm biased, but it is easier for me to
> speak my language.
> I sense you mean this (note, it is not OGC contains, but a distinct refinement):
> http://postgis.org/documentation/manual-svn/ST_ContainsProperly.html
> Do we agree?
Unfortunately I'm not ;) If I understand correctly ST_ContainsProperly()
works like GGL's within(). I have something else in mind. If we have two
objects in space they intersects each other. If we move them in opposite
directions they will finally be in the point where only boundries
touches themselves. Existing implementation of intersects will return
true in this border case. It would be nice to have function that returns
false. The same with disjoint.
___ ___
| | |
_|_A | B |
| |_|_|___|
| C |
intersects(A, B) -> true
intersects(A, C) -> true
alternative_intersects(A, B) -> false
alternative_intersects(A, C) -> true
disjoint(A, B) -> false
disjoint(A, C) -> false
alternative_disjoint(A, B) -> true
alternative_disjoint(A, C) -> false
touches(A, B) -> true
touches(A, C) -> false
// or meets()
Or something like that...
Geometry list run by mateusz at loskot.net | {"url":"https://lists.boost.org/geometry/2011/10/1618.php","timestamp":"2024-11-05T06:45:29Z","content_type":"text/html","content_length":"13153","record_id":"<urn:uuid:1e4fedc8-db32-4d04-ae4e-a054b7d90e1a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00499.warc.gz"} |
Approximating minimum-power k-connectivity
The Minimum-Power k -Connected Subgraph (MP k CS) problem seeks a power (range) assignment to the nodes of a given wireless network such that the resulting communication (sub)network is k-connected
and the total power is minimum. We give a new very simple approximation algorithm for this problem that significantly improves the previously best known approximation ratios. Specifically, the
approximation ratios of our algorithm are: 3 (improving (3 + 2/3)) for k = 2, 4 (improving (5 + 2/3)) for k = 3, k + 3 for k ∈ {4,5} and k + 5 for k ∈ {6,7} (improving k+2⌈(k+1)/2⌈), 3(k-1)
(improving 3k) for any constant k. Our results are based on a (k+1)-approximation algorithm (improving the ratio k+4) for the problem of finding a Min-Power k -Inconnected Subgraph, which is of
independent interest.
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 5198 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 7th International Conference on Ad-hoc, Mobile and Wireless Networks, ADHOC-NOW 2008
Country/Territory France
City Sophia-Antipolis
Period 10/09/08 → 12/09/08
Dive into the research topics of 'Approximating minimum-power k-connectivity'. Together they form a unique fingerprint. | {"url":"https://cris.openu.ac.il/en/publications/approximating-minimum-power-k-connectivity-2","timestamp":"2024-11-11T01:46:28Z","content_type":"text/html","content_length":"49267","record_id":"<urn:uuid:3b83becf-38d9-44cf-b5ac-1aa2b9c41a22>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00159.warc.gz"} |
17102024 | Faculty of Mathematics and Physics
Domination and packing in graphs
Yelena Yuditsky
Université libre de Bruxelles
October 17, 2024, 12:20 in S6
The dominating number of a graph is the minimum size of a vertex set whose closed neighborhoods cover all the vertices of the graph. The packing number of a graph is the maximum size of a vertex set
whose closed neighborhoods are disjoint. We show constant bounds on the ratio of the above two parameters for various graph classes. For example, we improve the bound for planar graphs. The result
implies a constant integrality gap for the domination and packing problems.
This is a joint work with Marthe Bonamy, Monika Csikos and Anna Gujgiczer. | {"url":"https://www.mff.cuni.cz/en/kam/teaching-and-seminars/noon-lectures/2024/17102024","timestamp":"2024-11-12T17:03:55Z","content_type":"text/html","content_length":"31777","record_id":"<urn:uuid:686e10a4-8259-4508-aa88-da3ea5d16b6f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00337.warc.gz"} |
which data set is represented by the box plot
6) 25 Clarify mathematic questions. We use these types of graphs or graphical representation to know: Distribution Shape Central Value of it Variability of it third row: stem 5 and leaves 0 3 3 6 8
3. C A box and whisker plot-also called a box plot-displays the five-number summary of a set of data. A left whisker extends from 2 to 4. The five-number summary is the minimum, first quartile,
Passing Rate 5.D C Ur answer is the best so far (tbh the others don't compete), Limitless_Dreams thank you. These are maximum used for data analysis. Which of the following statements best describes
the data shown? Which of the following is true of the data set represented by the box plot? In a Venn diagram, the rectangle represents the entire group. 73, 83, 93, 14, 34, 74, 25, 45 C. 7, 8, 9, 1,
3, 7, 2, 4 D. 37, 38, 39, 14, 34, 74, 25, 45, Use stem-and-leaf plot to answer questions 1-3 ----|----- 5| 4, 5 6| 0, 6, 9 7| 2, 3, 4, 5, 5 8| 1, 2, 8, 9 9| 5, 6 Key: 5|4 = 54 1. In a boxplot, the
interquartile range is represented by the width 8. 20082Y0 Math Grade 8 B Unit 2: Using Graphs to Analyze Data 4. How to find the range of data in a box plot | Math Methods C Incorrect answer C. 43
5. What is the range of data? for number 8 you just have to describe dr 1 and dr 2's plot so like find the mean and median of them then describe how u got it . 3xy=13 4. 3.Comparing Box Plots. 3 a,
you have no idea what your talking about. Removing the outliers would not affect the median. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Stem-and-Leaf Plots2022 5 | 2 4 @CorrectAnswers!!!
Good luck and have a good day. hahah almost there buddy Q. Find the mode, Stem Leaf 5 4 5 6 0 6 9 7 2 3 4 5 5 8 1 2 8 9 9 5 6 5/4= 54 1. A boxplot can give you information regarding the shape,
variability, and center (or median) of a statistical data set. D 35 7.c Sort by: Top Voted Questions Tips & Thanks Want to join the conversation? 3: Both have the same spread. 1. C 25 When Easy
Setting Box is executed it automatically recognizes the current monitor and designates the number corresponding to each monitor. hope this helped for #8 i was really stuck too! 3.Multiple Choice
heres my answers : If the distance from the median to the maximum is greater than the distance from the median to the minimum, then the box plot is positively skewed. 5.2.3 Quiz Dot Plots, Box Plots,
and Histograms.docx Solved Which of the following is true of the data set | Chegg.com The stem in row 4 is 8; the leaves in row 4 are 1, 2, 8, and 9. 4. c Q. = + 10 + 25 + 30 0 5 15 20 35 The data
contains at least one . 6. To find the range of all plots, subtract the smallest value from the largest value. A box plot is used to study the distribution and level of a set of scores. 4.) Select
all that apply. Incorrect answer A. Which expression represents the interquartile range (IQR) of data for the box plot? 9 | 7 8 To find the range of a given box plot, we can simply subtract the value
located at the lower whisker from the value located at the upper whisker.Feb 23, 2022 The mode interval in a histogram is the interval that has the highest frequency. - [Voiceover] Represent the
following data using a box-and-whiskers plot. Generally, the outliers fall more than the specified distance from the first and third quartile. D 3.d A On a pictograph, the key says = 24. 42 ** C. 54
D. 96 2. 4.Use the stem-and-leaf plot below to answer questions 45. Which data set is represented by the modified box plot? 4 What is the function of the whiskers in a box and whisker plot?
Exception: If your data set has outliers (values that are very high or very low and fall far outside the other values of the data set), the box and whiskers chart may not show the minimum or maximum
value. D Then determine which answer choice matches the dot plot you drew. ima have to get corrections now -_-. ur 100% correct for me! 37, 38, 39, 41, 43, 47, 52, 54 B. 1. Doctor 2 - mean: 14
median: 18.5. and the range between them is 6. Doctor 1 mean12 median 12 What is the median of the data? First Quartile (Q1): The first quartile is the median of the lower half of the data set. , an
$20 al comienzo del verano y luego gana $3 a la semana. 18 25 30 45 53 HT O A. The range of a box plot is the difference between the maximum and minimum value. A 37 38 39 41 43 47 52 54 3. The
setting of a story is the location where a story takes place. Box Plot - We ask and you answer! The best answer wins! - Benchmark Six Expert-Verified Answer Given: Given that the data are represented
by the box plot. 2.c This person has waited 6 years for an answer and she finally got it 8 | 2 5 You had all the same questions as me. Once again, exclude the median when computing the quartiles.
List A. C Find the mode and the median of the data in the stem-and-leaf plot below. OH WELL LMAOOOO, The quiz has updated there 7 questions now not8, i will get the answers for Unit 3 Lesson 4 2021
answers with 7 questions (last question is a type), i got a 3/6 with gamer girls answers cause hers are the older ones but these are the newer answers i got 7 from brainly so dont copy exact Find the
range of the data set represented by this box plot 4.3 Answer link. We need to determine the range and interquartile range. D The cookie is set by GDPR cookie consent to record the user consent for
the cookies in the category "Functional". 1: full of it These are the correct answers because I just copied it from the test. 2 What is the setting of the story example? The stem in row 2 is 6; the
leaves in row 2 are 0, 6, and 9. (also my fav from undertale), Im gonna throw up from all the different stuff goin on in this one thread . Doctor 1 mean is 12 and the median is 15, Doctor 2 mean is
16 and the median is 26 That is, the whisker reaches the value that is the furthest from the centre while still being inside a distance of 1.5 times the interquartile range from the lower or upper
quartile. 3 What does a box plot tell us about the data? Draw a Venn diagram and then find the number of students that were just hungry, Mathematics 700 Fundamentals - Unit 7: Data A, Math 700 - Unit
5: Ratios and Proportions, MATH - ONE-STEP MULTIPLICATION AND DIVISION E, Course: Science 7B if thats the case just ignore 7! key: 3 | 5 means 35 Data is symmetric if the median is in the mid of the
box plot and the box is mid to the line connecting both whiskers(since in that case, the frequencies are more likely to be distributed symmetric to the middle). There are, in fact, so many different
descriptors that it is going to be convenient to collect the in a suitable graph. Determine the slope of the line that passes through the pair of points.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 6. Solve for side length x. 3 B. 4. Im still waiting for a response on 8. for me, acdcddc, these answers got 3/7. Which data set could be
represented by the box plot shown below? It helps to find out how much the data values vary or spread out with the help of graphs. The data is evenly distributed throughout the. You can ask a new
question or browse more Math questions. The median (middle quartile) marks the mid-point of the data and is shown by the line that divides the box into two parts. m 1: The median for boys is 75 and
the median for girls is 79. A First Quartile = 10 (Middle value of 9, 10, 12 is 10). (answers correct only for 8th grade 2021 pre-algebra scca students): Which of the following is true of the data
set represented by the box plot? A 6. A box plot also shows the spread of data, since we can calculate range and IQR (interquartile range). View the full answer. C(54;73) a. no mode; 73 The
stem-and-leaf plot shows the number of cans of food collected by various students for a food drive. Which ordered pair is a solution of the euation? 5.D A box plot is constructed from five values:
the minimum value, the first quartile, the median, the third quartile, and the maximum value. IQR) or less than Q1-(1.5. 6. Answers for Lesson 5, unit 5: Finding Your Spot to Live Quiz for
Connections. Defina sus variables. asked by Oscar February 10, 2016 3 answers We need to determine the range and interquartile range. Find the mode of the set represented by the following
stem-and-leaf plot. They also show how far the extreme values are from most of the data. 3.D The data below represents the number of pages each . Which of the following is true of the data set
represented by the box plot? D Arrange the given dataset in ascending order. 5 0 1 2 4 5.Use the stem-and-leaf plot below to answer questions 45. Math Statistics and Probability Statistics and
Probability questions and answers Question 5 Which of the data sets represented by the following box and whisker plots has the smallest standard deviation? What is the interquartile range of the data
represented in the following box-and-whisker plot? Create a box plot - Microsoft Support 7.C 4) C 2-C @evryone who says different is a pure troll. This was posted almost exactly 9 years ago. 4.c 6
Where do I find easy setting box in Windows 10? still going I see :/ Outliers can be indicated as individual points. Quartiles and box plots. Interpreting Box and Whiskers Plots Flashcards | Quizlet
Find the mean of the following data set: The right whisker extends from 8 to 11. The interquartile range (IQR) is the box plot showing the middle 50% of scores and can be calculated by subtracting
the lower quartile from the upper quartile (. 1. C, I got 2/7 on this test due to these comments Incorrect answer D. 75 Cundo sern iguales sus ganancias? C 12 2.C
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Check the image below which shows the minimum, maximum, first quartile, third quartile, median and outliers. C D Assignment: 11. Quiz 2:
Organizing Data Unit: 2. DATA ANALYSIS
John L Eastman Lawyer, Skims Cotton Rib Tank Dupe, Camp Haven Utah, Articles W | {"url":"http://110.imcp.org.mx/ZgB/which-data-set-is-represented-by-the-box-plot","timestamp":"2024-11-03T01:07:35Z","content_type":"text/html","content_length":"30680","record_id":"<urn:uuid:b7c1de4a-f298-4012-84db-272a7bd8bfe0>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00040.warc.gz"} |
Problem 029 - hidden key 2 ποΈποΈ
This problem is a step up from Problem #028 - hidden key. Can you tackle this one?
Original photograph from Aneta Pawlik on Unsplash.
Problem statement
You and your best friend are locked in jail for no reason at all, but you are given the opportunity to escape. You are taken to a room that has four opaque boxes. The key to your cell will be put
inside one of the boxes, and then a (regular) coin is placed on top of each box. You may pick a single coin and reverse its face up, and then your friend will enter the room.
When your friend enters the room you are not allowed to talk, and your friend must open a box. If your friend opens the box with the key, you are set free. Otherwise, you are locked for eternity...
What is the strategy that you and your friend should agree upon, so that your friend can always find the key?
If you need any clarification whatsoever, feel free to ask in the comment section below.
The solution I will be sharing is not the original solution I thought of, I decided to share with you the solution that someone [posted][reddit-sol] online when I shared this puzzle on reddit.
What we are going to do is imagine the four boxes are laid out in a two by two square:
The next thing we do is interpret the sides of the coins as zeroes and ones, because it is easier to do maths with binary numbers. So a random configuration of the coins (of the zeroes and ones) and
the (hidden) key could be:
The next thing we do is agree that each box can be represented by its coordinates, in the sense that we can identify each box by the row and column it is in. To make things easier for us, we will
start counting the rows and columns from zero, so that the top left box is in position \((0, 0)\), the top right box is in position \((0, 1)\), the bottom left box is in position \((1, 0)\) and the
bottom right box is in position \((1, 1)\):
In the example image above, the key is currently in box \((1, 0)\).
Now that we have settled all the important details, we can determine our strategy:
• the parity of the sum of the first row of zeroes and ones will encode the row the key is in; and
• the parity of the sum of the first column of zeroes and ones will encode the column the key is in.
We are talking about the β parityβ of the sum because if the row contains two \(1\)s, then we sum them and get \(2\), which is not a valid row. Likewise for the columns. Hence, if the first row
sums to an even number, then the key is in the first row, and if the first row sums to an odd number, then the key is in the second row. Similarly, if the first column sums to an even number, then
the key is in the first column, and if the first column sums to an odd number, then the key is in the second column.
In our example, the first row sums to \(0\) and the first column sums to \(1\), which indicates that the key should be in box \((0, 1)\), which is wrong:
To solve our example, what we would have to do is flip the top left coin (i.e., make it a \(1\)) so that both the first row and the first column now got the correct sum:
We can see that this strategy always works:
• if the \(0\)s and \(1\)s are already correct, we can flip the coin of the bottom right box;
• if the \(0\)s and \(1\)s already tell us the correct row, but the incorrect column, flip the coin of the bottom left box;
• if the \(0\)s and \(1\)s already tell us the correct column, but the incorrect row, flip the coin of the top right box; and
• if the \(0\)s and \(1\)s tell us the wrong row and the wrong column (like in our example), then we flip the coin of the top left box.
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My book β Pydon'tsβ teaches you how to write elegant, expressive, and Pythonic code, to help you become a better developer. >>> Download it here π π . | {"url":"https://mathspp.com/blog/problems/hidden-key-2","timestamp":"2024-11-11T04:29:09Z","content_type":"text/html","content_length":"35533","record_id":"<urn:uuid:15c9972d-0bdf-4c12-87ba-17fd43545a33>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00228.warc.gz"} |
Continuous, Discontinuous, and Piecewise Functions
Professor Dave Explains
8 Nov 201705:18
TLDRThis educational video, narrated by Professor Dave, delves into the concept of continuity in functions, explaining the difference between continuous and discontinuous functions. Continuous
functions are highlighted as those without gaps, allowing for a smooth curve to be drawn without lifting the pencil. Conversely, discontinuous functions are introduced through examples like the
function 1/(X-1), which is undefined at X=1, creating an asymptote. Further, the video explores functions with holes or jumps, and piecewise functions, illustrating these concepts with clear examples
such as the function X squared minus one over X minus one, and a piecewise function that behaves differently based on the value of X. Professor Dave's explanation aims to enhance comprehension of
these fundamental mathematical concepts.
• ๐ Continuous functions have no gaps - any x value can act as input and output a y value, drawing a continuous curve
• ๐ Discontinuous functions have domains that exclude some x values, resulting in gaps in the curve
• ๐ The function 1/x-1 is undefined and discontinuous at x=1, with x=1 being a vertical asymptote
• โ Discontinuous functions can have jumps or holes where portions are missing or shifted
• โ The function x^2-1/x-1 has a hole at x=1 where it is undefined
• ๐ ฎ Piecewise functions are made of separate pieces and evaluated differently depending on the input x value
• ๐ ๐ ป The function f(x) = 5 when x < -2, and f(x) = x-1 when x โ ฅ -2 has a jump discontinuity at x=-2
• โ ๏ธ Discontinuities occur where functions are undefined and there are gaps or breaks in the curve
• ๐ Asymptotes are lines functions get closer and closer to but never touch
• ๐ ค Clever factoring and cancellation can reveal discontinuities not visible at first glance
Q & A
• What is a continuous function?
-A continuous function is one where there are no gaps whatsoever. Any x value can act as an input, and we get all the corresponding y values for the function, as a continuous curve. The function
can be drawn without lifting the pencil from the paper.
• What is an example of a discontinuous function?
-An example of a discontinuous function is f(x) = 1/(x - 1). This function is undefined at x = 1, meaning there is a gap in the graph at that point. 1 acts as an asymptote that the function
approaches but never touches.
• What causes a function to be discontinuous even if there are no asymptotes?
-A function can be discontinuous if there is a jump or hole in the function. This could mean a single point is missing from an otherwise continuous function, or that part of the function appears
to have been shifted.
• Explain the hole in the function f(x) = (x^2 - 1)/(x - 1)?
-The denominator cannot equal 0, so x cannot equal 1. This means there will be a hole at x = 1 where the function cannot be evaluated, even though the rest of the function is continuous.
• What are piecewise functions?
-Piecewise functions are comprised of several pieces, and we evaluate the function differently depending on the input value. The overall function's graph can have discontinuities at the
transitions between pieces.
• What causes the discontinuity in the example piecewise function?
-When x < -2, the function equals 5. When x >= -2, the function equals x - 1. This transition at x = -2 causes a discontinuity in the overall function's graph.
• What is the domain of 1/(x - 1)?
-The domain is all real numbers except for x = 1, because the denominator cannot equal 0.
• What are asymptotes?
-Asymptotes are lines that the function approaches but never touches. As x approaches the asymptote, the function approaches positive or negative infinity.
• Does a continuous function require a domain that includes all real numbers?
-No, a continuous function does not require a domain that includes all real numbers. It only requires that within its defined domain, there are no gaps or discontinuities.
• What types of functions did the passage state are always continuous?
-The passage stated that lines and parabolas, as well as some other types of functions, are always continuous.
๐ Introduction to Continuity in Functions
This paragraph introduces the concept of continuity in functions. It explains that up until now, the functions discussed have been continuous, meaning there are no gaps and any x-value is valid.
Continuous functions can be drawn without lifting the pencil. Some functions are discontinuous due to undefined domains or asymptotes.
๐ ฎ Examples of Discontinuous Functions
This paragraph provides examples of discontinuous functions like 1/x-1 which is undefined at x=1 and acts as an asymptote. It visualizes how the function approaches negative and positive infinity
around the asymptote. Other examples cover functions with holes at points, and piecewise functions defined differently across domains.
๐ กcontinuity
Continuity refers to a function being continuous, meaning it can be drawn without lifting the pencil from the paper. This means there are no gaps or holes in the function's graph. The video discusses
different types of discontinuities that can occur in functions, such as asymptotes, jumps, and holes.
๐ กasymptote
An asymptote is a line that a function gets closer and closer to but never actually touches. In the video, the function 1/x-1 has an asymptote at x=1. As x approaches 1, the function values approach
positive or negative infinity, depending on which side x is approaching from.
๐ กdomain
The domain of a function refers to the set of valid inputs or x-values that can be plugged into the function. Some functions are discontinuous because their domain does not include all real numbers -
there are some x-values that are invalid inputs.
๐ กhole
A hole refers to a single point missing in a function that otherwise appears continuous. In the video, the function x^2 - 1/(x-1) has a hole at x=1 because that point is excluded from the domain.
๐ กjump
A jump discontinuity refers to when it appears a portion of the function has been shifted vertically or horizontally. This creates a disconnect or jump between different pieces of the function.
๐ กpiecewise function
A piecewise function is comprised of separate pieces or branches, each with its own formula. The function value depends on which interval the input x-value falls in. The video shows an example of a
piecewise function.
๐ กundefined
Undefined means the function has no value at a certain point. In the video, 1/x-1 is undefined at x=1 because it would result in division by zero.
๐ กdiscontinuous
Discontinuous means the function has gaps, holes, or jumps - it cannot be drawn as a continuous curve without lifting the pencil. The video discusses different types of discontinuities.
๐ กfactor
To factor means to break an algebraic expression down into its underlying factors. In the video, factoring x^2 - 1 allows the function x^2 - 1/(x-1) to be simplified, revealing it is equivalent to
๐ กlimit
The limit describes the value a function approaches as x gets closer to a certain point. In the video, the function 1/x-1 approaches negative or positive infinity at the limits as x approaches 1 from
the left or right.
The study found that the new drug treatment resulted in significant improvement in symptoms for a majority of patients.
Researchers developed a machine learning algorithm that can accurately predict disease progression based on imaging data.
The paper introduces a novel theoretical framework for understanding the root causes of inequality and discrimination in society.
Analyzing social media data revealed interesting insights into how misinformation spreads rapidly during crisis events.
The review highlighted several unanswered questions and unresolved debates that provide opportunities for future research.
Results showed the new intervention led to improved academic performance and higher graduation rates among at-risk students.
Researchers developed an innovative new methodology for assessing ecosystem health and biodiversity.
The study found clear evidence that stricter gun laws lead to reductions in gun violence and firearm deaths.
The paper makes an important contribution by thoroughly analyzing the social and political context that enabled the rise of authoritarianism.
Researchers discovered a previously unknown species of insect by using advanced genomic sequencing techniques.
The new framework provides a more nuanced understanding of how identity is constructed at the intersection of race, class and gender.
Limitations of the study include a small sample size and lack of longitudinal data, as noted by the authors.
The results challenge long-held assumptions and reveal the need to re-examine existing theoretical models in the field.
Further research is needed to determine the long-term impacts of the policy intervention on community health outcomes.
The study provides an important first step, but greater integration of multiple disciplines is needed to fully address these complex issues.
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Counting with understanding up to 100
A thinking mathematically targeted teaching opportunity focused on quantifying collectioning, renaming numbers and number word sequences.
Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus (2022) © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales,
Collect resources
You will need:
• pencils or markers
• a collection of objects like dried pasta
• your mathematics workbook.
Counting with understanding – up to 100
Watch Counting with understanding – up to 100 video (16:31).
Welcome back, little mathematicians. I hope you're having a really lovely day.
Do you know, some of you might have been with me recently, when we had this cup and it had some pasta shells in it, and we tipped them out and we had counted by ones to work out that there were 17.
But it made me start thinking about something.
[Screen shows a cup with 17 pasta shells in it.]
But before I share with you what made me curious, let's just double check that we still have 17 pieces of pasta. Will you count with me? Okay, get ready.
I didn't hear you. Let's try again. Ready? 1, good now you keep going, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17. 17 pieces of pasta.
[Michelle tips the pasta shells out of the cup and counts them out moving them, one at a time, counting 1 through to 17.]
This is what this looks like.
But what I wondered is, when it was inside my cup, it didn't look like 17 was a very big number or very big quantity of pasta to have in my cup.
[Michelle gathers the pasta shells and places them back into the cup.]
And it looks like there's a lot of space left and what it made me start wondering about, I got really curious, is about how many pieces of pasta might fit in my cup all together.
And so, what I thought we could do is, find out. This is what mathematicians often do. They get curious about something, and they notice something, and then they decide to test it out.
[Michelle picks up the cup, indicating the amount of space still left in it.]
So, this, I know there's 17 pieces in there, I know here, there's quite a lot of space left, so I can use this information to help me estimate. So, can you estimate how many pieces of pasta you think
would fit into my cup in total?
Okay. I'm gonna record our thinking. So, our estimates. Okay, and what are you thinking now?
[Michelle places a writing pad down and writes the heading: ‘Our estimates’.]
Oh, you think there might be 50. 50 pieces of pasta might fit into this cup? Okay, I can record 50, like this, as one of our ways of thinking.
[Michelle points to the cup and writes 50 under the heading.]
Oh, some of you think 100 pieces of pasta. So, I will record 100 like this.
[Michelle writes 100 on the writing pad.]
Huh, some of you think maybe, 30 pieces of pasta.
[Michelle writes 30 on the writing pad.]
Okay, that's how I would record the numeral for 30. Ahh, someone else is thinking maybe about 75, and that's how I would record 75.
[Michelle writes 75 on the writing pad.]
And I think, let's see if there's one more estimate that you may have. 200 pieces of pasta? Okay, let's write the number 200. 200.
[Michelle writes 200 on the writing pad.]
Okay, so it's good that we've got our estimates but now we need to check. So, I'll move these out of the way for a moment.
[Michelle moves the writing pad to the side and places down a large box of pasta shells.]
And so, I've got a big box of pasta and I'm gonna take a really big scoop.
[Michelle takes the cup and scoops out pasta shells until the cup is full.]
Oh, that's right. I'll make sure that it's full, I think another one could fit there.
Full but not overflowing. Do you think that's full? Oh, okay one more piece. Is that full now?
[Michelle adds one more pasta shell to the cup.]
Okay, thank you. And now I'm gonna tip them out onto here. Oh, my goodness. Actually, seems like quite a lot of pasta.
[Michelle tips the pasta shells out and moves the cup to the side.]
So, would you change any of our estimates yet? Now that we have some more evidence of what it looks like over here?
[Michelle places the writing pad down next to the pasta shells.]
Ahh, I hear you too, I think now that I see this, 30 is probably too small.
[Michelle points to the pasta shells and the number 30, on the writing pad and takes a red marker and strikes through 30 and circles the other quantities.]
Yeah, cause when we saw what 17 looked like, 30 looks way too big, way too, way too, this looks like way too many pieces of pasta to be 30.
[Michelle pushes the pasta shells around, indicating a larger number.]
So, we might revise our estimate and say we don't think it's 30 anymore. But do you think the other quantities are still okay?
Ahh, some of you think 200 might be too big now.
[Michelle takes the red marker and strikes through 200.]
Okay, well as mathematicians we can revise our estimate so we can say, we think maybe it's 50, 100 or 75, or somewhere in that range of numbers.
Alright, well, let's count to find out. And actually, we could count 1, 2, 3, 4, 5 but that's quite a lot of pasta, and so what I might do as a mathematician is use a structure to help me work out
how many I have.
[Michelle counts out 5 shells and moves them to the side.]
And what's a structure that you think that we could use? Ahh, some of you are thinking we could use dice patterns, so I could arrange it so it looks like say, for example 5 on a dice, or, if I had
one more, 6 on a dice, or I could make 4 on dice.
[Michelle moves the 5 shells into a dice pattern, moving the pasta shells around. She adds one more to form a 6 dice pattern, and then she removes 2 to make a 4 dice pattern.]
Mm-hmm. Is there another structure that you think I could use? Ahh, some of you are thinking we could use a ten-frame. I think that's a good idea too because it's a lot of pasta.
So, I actually have some ten-frames here and this is really helpful because when I see a ten-frame like this, I know there's 5 at the top and there's 5 at the bottom, and there's 10 boxes all
[Michelle introduces a ten-frame on a piece of paper. She points to the 5 boxes on top and the 5 boxes underneath and she circles the 10 boxes.]
And so, if I put one piece of pasta in each of the boxes on my ten-frame, I don't have to count by ones and I already know that I have 10.
[Michelle places a pasta shell in each box of her ten-frame.]
So maybe I could do that again, I have a few other ten-frames here. So, while I fill this out, I'd like you to think about how many will I have counted if both of these are full?
[Michelle introduces another ten-frame and places a shell into each box.]
I hear your thinking. I think, you're thinking that it would be 2 tens, 2 ten-frames, 2 tens, and we call that 20.
So, one ten, 2 tens, same as saying 20. Ok, well, I have another ten-frame so I can fill it out, and while I fill it out, can you think about how many pieces of pasta I will have counted once, I
filled out, this ten -frame as well?
[Michelle introduces another ten-frame and places a shell into each of boxes.]
Okay, it's another 10. That's right, it's 30. Because I have, now, 3 ten-frames, and we call 3 tens 30.
Okay, well, what about if I fill this one in? How many ten-frames will I have filled with pasta?
[Michelle introduces another ten-frame and places a shell into each of boxes.]
Mm-hmmm. 4 and if I have 4 ten-frames, that's the same as saying, 4 ten’s and 4 tens is the same as saying 40.
So, actually, so far, I don't have to count them, I can just use structure to help me.
Okay, I have got another ten-frame so if I have 5 ten-frames and they are fill, full of pasta, what number would that be, that I've counted so far? Well, worked out how many.
[Michelle introduces another ten-frame and places a shell into each of boxes.]
Yeah, 50, because 5 tens is the same as saying 50.
I still have some more pasta so I can keep going, but I've got a problem. I don't have any more ten-frames left. But I've really liked this idea of filling them in.
I wonder, I know, maybe we could draw our own ten-frame? Yeah, can you help me draw it?
What do you think I should do first? Okay, I should draw the rectangle around the outside, and then? Okay, I could do the line down the middle, because look it has a line down the middle. Oh, and
then I need these lines, and how many are there? Let's check, 1, 2, 3, 4.
Okay, I can use this one to help me too. One, 2, 3, 4 and so that should mean I have, 1, 2, 3, 4, 5 boxes at the top, and I'll have 5 down the bottom. So, I have another ten-frame.
[Michelle draws a ten-frame, by drawing a rectangle, a horizontal line in the middle and 4 vertical lines, evenly spaced across.]
I can fill this in I think, and now if I have 6 ten-frames, how many pieces are pasta will I know that I have? Yeah, 6 tens, which we rename as 60.
[Michelle now adds a shell to each of the boxes in the ten-frame she just drew.]
Do you think I have enough for another 10 as well? Okay, so should I draw another ten-frame? Okay, so the rectangle around the outside, you think?
Oh, you're right, I could also draw the 4 lines first, down the middle, and I'll use this one to help guide me, so I get my eye in.
And then I could come back here and do the line down the middle that cuts each of these rectangles into smaller rectangles, that sort of look like squares.
[Michelle now draws another ten-frame. She points and counts out the 10 boxes. She now adds a pasta shell to each box.]
Let's check, good thinking, 1, 2, 3, 4, 5. Mm-hmm and if there's 5 on the top, that's right, there has to be 5 down the bottom, and that's a ten-frame. So, if I fill this in, how many ten-frames full
of pasta will I have now? Yeah. 7. Look, 1, 2, 3, 4, 5, 6, 7 ten-frames.
I think I have enough space to draw, have another one, but what I'm wondering, is do you think I need to draw it, or could I use my mathematical imagination to imagine the ten-frame there?
Yeah, so imagine I'm drawing the rectangle, and the 4 lines in the middle and then the line across, that divides those into half, and I would have 1, 2, 3, 4, 5 on the top, and 1, 2, 3, 4, 5 on the
[Michelle draws an imaginary ten-frame with her finger on the paper under the other ten-frames.]
So that would mean I have an 8 ten-frame and I'm just imagining this in my mind's eye.
[Michelle now places a shell in each of the imaginary boxes.]
Yeah, cause I can use structures that I have, I can draw them, but I can also imagine that they are there, to help me work out how many.
So, let's just check this one because we're imagining a ten-frame so we can count by twos. Two, 4, 6, 8, 10.
[Michelle circles 2 shells with her finger in the imaginary ten-frame, counting by twos to 10.]
Another ten-frame. So now I have 1, 2, 3, 4, 5, 6, 7, 8 tens. Yes, and we call that 80, and 2 more.
[Michelle points to each of the 8 ten-frames and counts them out, and then points to the 2 shells left.]
So that means that all together I had 82 pieces of pasta that could fit inside my cup.
So, if I come back to what we were estimating, we thought, maybe 50, maybe 75 and maybe a hundred. And what we discovered, is that we actually have 82.
[Michelle places the writing pad down and writes 82 in red marker to the right of the number’s list.]
And that these estimates are all actually pretty reasonable, aren't they? It's quite a lot of pasta that fits into a nice little pink cup and you know, little mathematicians, this is actually making
me think of something else.
[Michelle removes the writing pad and places the empty pink cup down.]
What if I didn't have pasta, but what if I used something else?
What else would fit inside my cup? So, if I get rid of these ones for a minute.
[Michelle pushes the pasta and paper ten-frames out of the screen, leaving the 2 ten-frames she drew.]
I still have my 2 ten-frames that I drew, but what if, I was thinking about how many teddy bears would fit in my cup? If I fill it, there's my cup full of teddy bears.
[Michelle introduces a container of teddy bears and fills the cup. On the writing pad she places a pasta shell above the 82 shells of pasta. She then points and indicates that 82 pasta shells fit
into the cup.]
And when we had pasta, we knew that we, we worked out that we could fit 82 pieces of pasta inside the same cup, oh you're right I could fit another one in there.
I wonder if we'll have more or less of the teddy bears? What do you think?
You think, there'll be less? Mm, I think so too, because if I look at my piece of pasta and a teddy, I'll use the red one, so it's easier for you to see.
They are about the same, well, the pasta is a little bit longer, so, but not much and the pasta is a bit skinnier than the teddy bear, and if I turn it this way the teddy bears a bit fatter.
But what happens with the teddy bears is, they sit like this with one another, where they don't nest inside.
But with the pasta, look, they fit inside each other a little bit and so that might mean that more of them fit inside my cup.
[Michelle places a pasta shell and a teddy bear alongside each other and compares the size of the 2 items, moving them around. She then picks up another pasta shell to indicate how they fit into each
Oh, ok, you would like another one in there.
[Michelle places another teddy bear in the cup and pats down the top to ensure it’s full, but not overflowing.]
Now do you think that's full? Okay, how many teddy bears do you think would fit in my cup here? Do think less than 82?
[Michelle points to the cup of teddy bears and then points to 82 on the writing pad.]
Okay, should we count to find out? Or, actually, okay so I'm going to tip them out here.
[Michelle tips the teddy bears out.]
How many, do you think there are now? I can hear you, some of you are saying about 30, or maybe 50, or maybe 20.
Do you think we could use our ten-frames to help us? Yeah, because if we have this one full, how many teddy bears would there be? Ten, and if this one is full? 20.
Ok, so I can start by filling up our teddy bears on our ten-frame for us.
[Michelle places a teddy bear into each box of the 2 ten-frames.]
And that one's full, so I've counted already ten, or I've worked out that we have ten. And over here, we have another ten, so we know that one 10 and another 10 is called 20, and I could get another
ten-frame here.
And we can try to fill this in and let's see what happens. And this is where structure is really helpful for me because we can work out how many without actually having to count everything by ones.
[Michelle places one of the paper ten-frames back, and fills it with teddy bears, leaving one teddy bear left.]
Oh look. So how many ten-frames do we have? That's right, 3. 1, 2 and 3, and then one left over.
So how many teddy bears fit into this cup? Mm-hmm, 31.
So, we had 82 pieces of pasta, and for teddy bears there were 31 teddy bears, that can fit inside this cup.
[Michelle now writes 82 above the ten-frames and places a pasta shell above it, she then writes 31 and places a teddy bear above it and places the empty cup beside it.]
I wonder if you've got a cup like this, or a different colour, or one that you really love at home, that you could use to work out how many things, it can hold. And can you find anything that's the
Over to you, little mathematicians.
Okay mathematicians, what was some of the mathematics?
So, using familiar structure helps us quantify a collection without having to count everything by ones. So, in this case we used groups of 10 and we could use renaming.
So, for example, 8 tens and 2 ones, we could rename as 82.
[Screen shows 8 ten-frames with pasta shells in each box.]
Yeah, and something else that we really realised today is that you can use structures that you have, like pre-made ten-frames, you can also draw your own structures, or you can imagine them.
[screen shows a ten-frame with 10 pasta shells in each box.]
So, it doesn't matter what equipment you have, you can still be, and think like, and imagine, like a mathematician.
Okay, over to you mathematicians! Find a container or a cup at home and go find things that you can quantify! Over to you.
• Find a cup or container and find some collections you can quantify.
• Can you find 2 different collections of objects that your cup or container holds the same amount of? | {"url":"https://education.nsw.gov.au/teaching-and-learning/curriculum/mathematics/mathematics-curriculum-resources-k-12/thinking-mathematically-resources/mathematics-s1-counting-with-understanding-up-to-100","timestamp":"2024-11-14T17:37:28Z","content_type":"text/html","content_length":"199643","record_id":"<urn:uuid:c7aac588-29dc-4839-bc57-2b01b875abed>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00298.warc.gz"} |
ar S
Shear Stress in Beams
An analysis of Shear Stress in Beams of various cross sections.
Variation Of Shear Stress.
The Shearing Force at any cross section of a Beam will set up a Shear Strain on transverse sections which in general will vary across the section.
In the following analysis it has been assumed that the Stress is uniform cross the width (i.e. parallel to the neutral axis) and that the presence of shear stress does not affect the distribution of
Bending Stress. This last point is not strictly true because the presence of Shear Stress will distort the transverse planes which will no longer remain plane. (See "Bending of Beams Part 4). Due to
the Shear Stress on transverse planes there will be complementary planes parallel to the neutral axis. In the following diagram two transverse sections are shown at a distance
apart and the Shearing forces will be
whilst the Bending Moments are
A shear stress is defined as the component of stress coplanar with a material cross section. Shear stress arises from a force vector perpendicular to the surface normal vector of the cross section.
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• s be the value of the complementary shear stress and hence the transverse shear stress at a distance $\inline&space;y_0$ from the Neutral Axis.
• z be the width of the cross section at $\inline&space;y_0$
• A be the area of cross section cut off by a line parallel to the Neutral Axis.
• $\inline&space;\bar{y}$ be the distance of the centroid of A from the Neutral Axis.
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are the Normal Stresses on an element of Area
. There is a difference in Longitudinal Forces equal to
and this summed over the area
must be in equilibrium with the transverse Shear Stress
on the longitudinal plane of area
Substituting in equation (
. See "Shearing Force and Bending Moments. It should also be noted that
is the actual width of the section at the position where
is being calculated and that
is the Total Moment of Inertia about the neutral axis. In some applications it is advantageous to calculate
as several parts.
Rectangular Sections.
For a rectangular section at any distance
from the Neutral Axis :
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Substituting in equation (
) and putting
This shows that there is a parabolic variation of Shear Stress with
. The maximum Shear Force occurs at the Neutral Axis and is given by:
is called the
Mean Stress
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The dimensions are shown in the diagram. It is required to find an expression for the
Shear Stress in the Web. $\inline&space;A\bar{y}$
is made up of two parts as follows:
• for the flanged area
• for the web part
Using Equation (
) and noting that
As with the rectangular section, the maximum transverse Shear Stress is at the neutral axis.
At the top of the web,
Since the Shear Stress has to follow the direction of the boundary, the distribution must be of the form shown becoming horizontal at the flanges.
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Consequently the complementary Shear Stress in the flanges is on longitudinal planes perpendicular to the neutral axis and the "width
is replaced by the flange thickness
Showing that the Shear Stress in the flanges varies from a maximum at the top web to zero at the outer tips. In Practice, however it will be found that most of the Shearing Force ( About 95%) is
carried by the Web and the Shear Force in the flanges is negligible. As the variation over the web is comparatively small ( about 25%) it is convenient for design purposes and also in calculating
deflection due to Shear, to assume that all the Shearing Force is carried by the Web and is uniformly distributed. Similarly it is normal practice to assume that, as a first approximation, the
Bending Moment is carried wholly by the flanges.
Example - Example 2
A 12 in. by 5 in. British Standard Beam is subjected to a Shearing Force of 10 tons. Calculate the
of the transverse Shear Stress at the Neutral Axis and at the top of the Web and
this with the mean Stress calculated on the assumption that the Stress is uniformly distributed over the Web. What
of the Shearing Force is carried by the Web?
Web Thickness=
and Flange thickness=
At the Neutral Axis
At the Top of the Web (
Assuming that all the Shearing Force is carried uniformly by the web.
The Total Shearing Force carried by the web is given by:
of the Total The remaining 5 % of the vertical Shear Stress is presumably accounted for by the component of the Shear Stress at the junction of the flange and the web. Failure due to Shear in the Web
usually takes the form of buckling brought about by the Compressive Stresses on planes at 45 degrees to the transverse section. ( See Compound Stress and Strain). For this reason deep webs are often
supported by vertical stiffeners.
• Percentage of the Shearing Force is $\inline&space;\;95\;^0/_0$ of the Total.
Principle Stresses In I-beams.
$\inline&space;I$-beams, also known as $\inline&space;H$-beams are beams with an $\inline&space;I$- or $\inline&space;H$-shaped cross-section. The horizontal elements of the $\inline&space;I$ are
flanges, while the vertical element is the web. The web resists shear forces while the flanges resist most of the bending moment experienced by the beam.
When an $\inline&space;I$ section beam is subjected to both Bending and Shear Stresses it is normal to find that the Maximum Principle Stress is at the top of the Web. The other possible value is the
Maximum Bending Stress which occurs at the outer edge of the Flange.
Example - Example 3
A short vertical column is firmly fixed at the base and is 1 ft high. The column is of $\inline&space;I$ section 8 in. by 4 in. The flanges are 0.4 in thick and the web is 0.28 in. thick. $\inline&
space;\displaystyle&space;I&space;=&space;55.6;in^4$ and $\inline&space;Area&space;=&space;5.3;in.^2$ An inclined load of 8 tons acts on the top of the column in the centre of the section and in the
plane which contains the centre-line of the web. The line of action is $\inline&space;\displaystyle&space;30^{0}$ to the vertical. Determine the position and magnitude of the greatest principle
stress on the base of the column.
The inclined load will intersect the base cross-section at a distance
from the centroid. Resolving the load into horizontal and vertical components:
• Direct Load = $\inline&space;8\;\cos\;30&space;=&space;6.92\;tons$
• Shearing Force = $\inline&space;8\;\sin\;30&space;=&space;4\;tons$
• Bending Moment = $\inline&space;6.92\times&space;6.93&space;=&space;47.9\;tons-in.$
At the top of the Web:
Bending Stress =
Direct Stress =
Therefore, Total Normal Stress =
See the Reference Pages on Compound Stress and Strain (currently in preparation) The Maximum Principle Stress =
(in Compression) This will occur at the top of the Web. Checking the value of the maximum Bending Stress which is:-
Which together with the direct Stress gives a maximum value of
at the outside of the flange and which is less than the value at the top of the Web.
Pitch Of Rivets In Built Up Girders.
The load carried by one rivet in a beam section built up as shown in the diagram, is determined by the difference between normal stresses on certain areas of two transverse sections at a distance
apart equal to the pitch of the rivets.
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The are to be used is that part of the cross-section which " comes away" when the particular set of rivets are removed. I.e. in the case of the rivets holding the flange to the angle section, the
area is that shaded in (b) and for the rivets holding the angles to the web, the area is shown in (c) If
is the pitch of the rivets and
is the force on the rivets over a length
of beam then proceeding as in the section on the variation of Shear Stress:
(Compare with
and let
) Note that for the flanges there are two rivets to a pitch length and they are usually staggered so as not to occur together in one cross section. Also note that the web rivets are in double shear.
Example - Example 4
An $\inline&space;I$-section beam is built up of a web plate 10 in. by $\inline&space;\displaystyle\frac{1}{2}$ in. with flange plates 6 in. by 1 in. which are secured by rivets through angle
sections of 2 in. by 2 in. by $\inline&space;\displaystyle\frac{1}{4}$ in. ( The arrangement is the same as the above diagram) If the Bending Stress is limited to $\inline&space;t\;tons/sq.in.$
estimate the maximum uniformly distributed load which can be carried over a length of 12 ft. Assuming $\inline&space;\displaystyle\frac{1}{2}$ in. diam. rivets, calculate their pitch if the allowable
Shearing Stress is 5 tons/sq.in. and the bearing pressure 10 tons/sq.in.
If the loading is
for the web
for the angles
for the flanges
The Permissible Load per pitch length. For one rivet in double shear in the web or two rivets in single shear in the flange
Crushing of the rivets (one in web or two in flange)
For the flange rivets
The load per pitch length if the smaller of equations
For the Web rivets
• $\inline&space;1.965\;tons$
• The load per pitch length is $\inline&space;p=&space;1\displaystyle\frac{1}{2}\;in.$
• For the Web rivets, $\inline&space;p=1\displaystyle\frac{1}{4}in.$
Solid Circular Sections.
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be the Shear Force across a chord parallel to
defined by the angle
At the Neutral Axis the Shear Stress
is maximum and equals
mean shear stress
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The directional distribution of the Shear Stress is as indicated. This does not affect the magnitude of the greatest Shear Stress which is usually the value required. This particular case is
applicable to
in Shear but the ratio may be assumed to have been incorporated in the allowable stress value which is then taken as uniform over the section.
Thin Circular Tubes.
It was shown in the page on Shear Stress that it has to follows the direction of the boundary. In a thin walled circular tube this means that it can be assumed to be tangential.
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Assume that
is the neutral axis and that
are two symmetrically placed positions defined by the angle
. Let the Shear Stress be
. The complementary Shear Stress is along the longitudinal planes and are balanced by the difference of Normal Stress on the shaded area subtending the angle
For a length of Beam
Polar Moment of inertia
represents the area and
the mean radius
Engineering Materials , Bending Stress, Moments of Inertia.
The Maximum Shear Force will occur at the neutral axis and is given by:
Miscellaneous Sections
The Shear Stress at any point in a cross-section can always be calculated from the basic formula
Example - Shear Stress
For the section shown determine the
average Shearing Stress
, and
for a Shearing Force of 20 tons, and find the
of the maximum to mean Stress.
to scale a diagram to show the variation of shearing Stress across the section.
The variations of Shearing Stress are shown on the above graph.
average Shearing Stress
• At $\inline&space;A$ is $\inline&space;0$
• At $\inline&space;B$ is $\inline&space;0.647\;tons\;in.^{-2}$
• At $\inline&space;C$ is $\inline&space;2.77\;tons\;in^{_2}$
• At $\inline&space;D$ is $\inline&space;4.44\;tons\;in.^{-2}$
Shear Centre.
For unsymmetrical sections and in particular angle and channel sections, the summation of the Shear Stresses in each leg gives a set of Forces which must be in equilibrium with the applied Shearing
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Consider the angle section which is bending about a principle axis and with a Shearing Force
at right angle to this axis. The sum of the Shear Stresses produces a force in the direction of each leg as shown above.
It is clear that their resultant passes through the corner of the angle and unless $\inline&space;F$ is applied through this point there will be a twisting of the angle as well as Bending. This point
is known as The Shear Centre or Centre of Twist.
For a channel section with loading parallel to the Web, the total Shearing Force carried by the web must equal
and that in the flanges produces two equal and opposite horizontal forces. It can be seen that for equilibrium the applied load causing
must lie in a plane outside the channel as indicated. Its position is calculated as in the following example.
Example - Example 6
Explain why a single section channel with its web vertical, subjected to vertical loading as a beam will be in torsion unless the load is applied through a point outside the section known as the
Shear Centre. Find its approximate position for a channel section $\inline&space;\displaystyle&space;1\frac{1}{2}\;in.$ by $\inline&space;\displaystyle1\frac{1}{2}\;in.$ outside by $\inline&space;\
is the Shearing Force at the section, then the Total Vertical Force in the Web can be taken to be equal to
. It should be mentioned that integrating from the height of the web only will give a value slightly less than
( Compare with Example 2) but the remaining vertical force is assumed to be carried by the corners of the section.
Proceeding as in the section on
beams, the Shear Stress in the flanges at a distance
from the tip is:
The Total force in each Flange
is given by:
is the distance of the Shear Centre ( Through which the applied load must act if there is to be no twisting of the section) from the centre line of the Web, then for equilibrium.
• $\inline&space;h&space;=&space;0.617\;in.$ | {"url":"https://www.codecogs.com/library/engineering/materials/beams/shear-stress-in-beams.php","timestamp":"2024-11-11T10:42:06Z","content_type":"text/html","content_length":"72685","record_id":"<urn:uuid:3a98538b-ebd6-4ee6-9881-3c89be32e7a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00881.warc.gz"} |
Extension of homogeneous fluid methods to the calculation of surface disturbance induced by an object in a stratified ocean
An extension of homogeneous fluid methods has been developed for calculation of the disturbance induced by a submerged point source in an incompressible, density-stratified fluid with a free surface.
The extension comes in the treatment of the inhomogeneous wave equation, which results from linearization and Fourier transformation of the equations of motion. A closely related problem is the
solution of the one-dimensional Schrodinger equation, where the negative squared Brunt-Vaisala frequency profile plays the role of the depth-dependent potential. A model potential with known bound
state and continuum state eigenfunctions is substituted into the inhomogeneous wave equation, which is solved, subject to the free surface boundary condition, by an eigenfunction expansion method.
Surface tension effects are also included. Contour integration techniques assist the evaluation of integrals and enable a clear separation of localized and extended wavelike disturbances. The
point-source solutions may be used to calculate fluid disturbance in the near and far fields induced by a submerged body. As an example, the localized surface displacement and rate of strain induced
by a surmerged Rankine ovoid are calculated for a square-well density-stratification model. The results are compared to previously calculated far-field internal wave effects induced on the surface by
wake collapse behind a submerged body.
Naval Research Lab. Report
Pub Date:
February 1976
□ Hydrodynamic Equations;
□ Ocean Surface;
□ Perturbation;
□ Surveillance;
□ Ocean Currents;
□ Stratified Flow;
□ Submerged Bodies;
□ Water Waves;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1976nrl..reptR....R/abstract","timestamp":"2024-11-09T20:49:13Z","content_type":"text/html","content_length":"37485","record_id":"<urn:uuid:7b48fe72-e414-4dec-95c7-f795aa96f7c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00601.warc.gz"} |
Absolute value (algebra)
In algebra, an absolute value (also called a valuation, magnitude, or norm,[1] although "norm" usually refers to a specific kind of absolute value on a field) is a function which measures the "size"
of elements in a field or integral domain. More precisely, if D is an integral domain, then an absolute value is any mapping |x| from D to the real numbers R satisfying:
• \( {\displaystyle \left|x\right|\geq 0} \) (non-negativity)
• \( {\displaystyle \left|x\right|=0}\) if and only if x=0 (positive definiteness)
• \( {\displaystyle \left|xy\right|=\left|x\right|\left|y\right|} \) (multiplicativity)
• \( {\displaystyle \left|x+y\right|\leq \left|x\right|+\left|y\right|} \) (triangle inequality)
It follows from these axioms that |1| = 1 and |-1| = 1. Furthermore, for every positive integer n,
|n| = |1 + 1 + ... + 1 (n times)| = |−1 − 1 − ... − 1 (n times)| ≤ n.
The classical "absolute value" is one in which, for example, |2|=2, but many other functions fulfill the requirements stated above, for instance the square root of the classical absolute value (but
not the square thereof).
An absolute value induces a metric (and thus a topology) by \( {\displaystyle d(f,g)=|f-g|.}\)
The standard absolute value on the integers.
The standard absolute value on the complex numbers.
The p-adic absolute value on the rational numbers.
If R is the field of rational functions over a field F and p(x) is a fixed irreducible element of R, then the following defines an absolute value on R: for f(x) in R define |f| to be \( 2^{-n} \) ,
where \( {\displaystyle f(x)=p(x)^{n}{\frac {g(x)}{h(x)}}} \) and \( {\displaystyle gcd(g(x),p(x))=1=gcd(h(x),p(x)).} \)
Types of absolute value
The trivial absolute value is the absolute value with |x|=0 when x=0 and |x|=1 otherwise.[2] Every integral domain can carry at least the trivial absolute value. The trivial value is the only
possible absolute value on a finite field because any non-zero element can be raised to some power to yield 1.
If an absolute value satisfies the stronger property |x + y| ≤ max(|x|, |y|) for all x and y, then |x| is called an ultrametric or non-Archimedean absolute value, and otherwise an Archimedean
absolute value.
If |x|1 and |x|2 are two absolute values on the same integral domain D, then the two absolute values are equivalent if |x|1 < 1 if and only if |x|2 < 1 for all x. If two nontrivial absolute values
are equivalent, then for some exponent e we have |x|1e = |x|2 for all x. Raising an absolute value to a power less than 1 results in another absolute value, but raising to a power greater than 1 does
not necessarily result in an absolute value. (For instance, squaring the usual absolute value on the real numbers yields a function which is not an absolute value because it violates the rule |x+y| ≤
|x|+|y|.) Absolute values up to equivalence, or in other words, an equivalence class of absolute values, is called a place.
Ostrowski's theorem states that the nontrivial places of the rational numbers Q are the ordinary absolute value and the p-adic absolute value for each prime p.[3] For a given prime p, any rational
number q can be written as pn(a/b), where a and b are integers not divisible by p and n is an integer. The p-adic absolute value of q is
\( \left|p^{n}{\frac {a}{b}}\right|_{p}=p^{{-n}}.\)
Since the ordinary absolute value and the p-adic absolute values are absolute values according to the definition above, these define places.
Main article: Valuation (algebra)
If for some ultrametric absolute value and any base b > 1, we define ν(x) = −logb|x| for x ≠ 0 and ν(0) = ∞, where ∞ is ordered to be greater than all real numbers, then we obtain a function from D
to R ∪ {∞}, with the following properties:
ν(x) = ∞ ⇒ x = 0,
ν(xy) = ν(x)+ν(y),
ν(x + y) ≥ min(ν(x), ν(y)).
Such a function is known as a valuation in the terminology of Bourbaki, but other authors use the term valuation for absolute value and then say exponential valuation instead of valuation.
Given an integral domain D with an absolute value, we can define the Cauchy sequences of elements of D with respect to the absolute value by requiring that for every ε > 0 there is a positive integer
N such that for all integers m, n > N one has |xm − xn| < ε. Cauchy sequences form a ring under pointwise addition and multiplication. One can also define null sequences as sequences (an) of elements
of D such that |an| converges to zero. Null sequences are a prime ideal in the ring of Cauchy sequences, and the quotient ring is therefore an integral domain. The domain D is embedded in this
quotient ring, called the completion of D with respect to the absolute value |x|.
Since fields are integral domains, this is also a construction for the completion of a field with respect to an absolute value. To show that the result is a field, and not just an integral domain, we
can either show that null sequences form a maximal ideal, or else construct the inverse directly. The latter can be easily done by taking, for all nonzero elements of the quotient ring, a sequence
starting from a point beyond the last zero element of the sequence. Any nonzero element of the quotient ring will differ by a null sequence from such a sequence, and by taking pointwise inversion we
can find a representative inverse element.
Another theorem of Alexander Ostrowski has it that any field complete with respect to an Archimedean absolute value is isomorphic to either the real or the complex numbers, and the valuation is
equivalent to the usual one.[4] The Gelfand-Tornheim theorem states that any field with an Archimedean valuation is isomorphic to a subfield of C, the valuation being equivalent to the usual absolute
value on C.[5]
Fields and integral domains
If D is an integral domain with absolute value |x|, then we may extend the definition of the absolute value to the field of fractions of D by setting
\( |x/y|=|x|/|y|.\,\)
On the other hand, if F is a field with ultrametric absolute value |x|, then the set of elements of F such that |x| ≤ 1 defines a valuation ring, which is a subring D of F such that for every nonzero
element x of F, at least one of x or x−1 belongs to D. Since F is a field, D has no zero divisors and is an integral domain. It has a unique maximal ideal consisting of all x such that |x| < 1, and
is therefore a local ring.
Koblitz, Neal (1984). P-adic numbers, p-adic analysis, and zeta-functions (2nd ed.). New York: Springer-Verlag. p. 1. ISBN 978-0-387-96017-3. Retrieved 24 August 2012. "The metrics we'll be dealing
with will come from norms on the field F..."
Koblitz, Neal (1984). P-adic numbers, p-adic analysis, and zeta-functions (2nd ed.). New York: Springer-Verlag. p. 3. ISBN 978-0-387-96017-3. Retrieved 24 August 2012. "By the 'trivial' norm we mean
the norm ‖ ‖ such that ‖0‖ = 0 and ‖x‖ = 1 for x ≠ 0."
Cassels (1986) p.16
Cassels (1986) p.33
"Archived copy". Archived from the original on 2008-12-22. Retrieved 2009-04-03.
Bourbaki, Nicolas (1972). Commutative Algebra. Addison-Wesley.
Cassels, J.W.S. (1986). Local Fields. London Mathematical Society Student Texts. 3. Cambridge University Press. ISBN 0-521-31525-5. Zbl 0595.12006.
Jacobson, Nathan (1989). Basic algebra II (2nd ed.). W H Freeman. ISBN 0-7167-1933-9. Chapter 9, paragraph 1 "Absolute values".
Janusz, Gerald J. (1996–1997). Algebraic Number Fields (2nd ed.). American Mathematical Society. ISBN 0-8218-0429-4.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License | {"url":"https://www.hellenicaworld.com/Science/Mathematics/en/Absolutevaluealgebra.html","timestamp":"2024-11-13T21:53:58Z","content_type":"application/xhtml+xml","content_length":"13326","record_id":"<urn:uuid:fcb30c94-a7d9-4800-8d73-1ff5d0f5d271>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00845.warc.gz"} |
geom pop (1.11-1.14)
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Question Answer
scalene triangle with no two sides equal
isosceles triangle with at least two sides equal
legs equal sides of an isosceles triangle
base the unequal side of an isosceles triangle
base angles angles at the base of the isosceles triangle
vertex angle angle opposite the base of the isosceles triangle
equilateral triangle with all three sides equal
acute a triangle with three acute angles
obtuse a triangle with an obtuse angle
right a triangle with a right angle
hypotenuse the side opposite the right angle in a right triangle
legs the two sides that aren't the hypotenuse of the right triangle
equiangular a triangle with three equal angles
polygon a plane figure bounded by straight lines
sides the straight lines bounding a polygon
vertices the intersections of the sides in a polygon
base the side upon which a polygon is supposed to stand
interior angle angle formed by two consecutive sides of a polygon
exterior angle angle formed by one side of a polygon and the adjacent side extended
consecutive/adjacent sides two sides that meet at any vertex in a polygon
diagonal a line joining two vertices that are not consecutive in a polygon
convex polygon polygon whose interior angles are less that 180 degrees
concave polygon polygon with at least one interior angle greater than 180 degrees
regular polygon a polygon which is both equiangular and equilateral
center of the polygon common point where the bisectors of the interior angles intersect | {"url":"https://www.studystack.com/flashcard-3070329","timestamp":"2024-11-13T22:20:21Z","content_type":"text/html","content_length":"80231","record_id":"<urn:uuid:d609275a-53aa-4504-a6d1-d99bc1a60cb4>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00561.warc.gz"} |
Partial Quotients Division
for iPad, iPhone, Mac, and Windows 10 PCs
This app can be used to teach and study the partial quotients division method. The app is easy to use and it has an intuitive interactive interface with customizable colors and other settings. The
user can solve random and custom division problems.
The traditional long division method can be difficult to understand because it is so abstract. In the partial quotients method the student can make a series of estimates and then add the estimated
quotients together.
In the example 992/8 the first estimation can be 100, which leaves 192. Next estimation can be 20 which leaves 32. Now the student knows that 8 goes 4 times into 32 and the result can be found by
adding up the partial quotients 100, 20 and 4.
Each student can use the quotients that he or she finds the easiest.
In the Everyday Mathematics curriculum the partial quotients division method is the focus algorithm for division.
Easy to Use
After the user solves each operation or inputs a new estimate, the correct answer will fly to the right place. If the user presses the wrong button or gives an impossible estimate the answer will
appear above the keyboard but it will not move.
• The dividend can have from 2 to 5 digits
• The divisor can have 1 or 2 digits
• Random and custom problems
• The current operation can be hidden
• The operands of the current operation can be highlighted
• Colors of the interface can be changed
• The speed of the animations can be set
• Custom interfaces for iPad and iPhone
Partial Quotients Division Videos
Partial quotients division with two digit divisor and three digit dividend
924 ÷ 12
Partial quotients division with one digit divisor and five digit dividend
22960 ÷ 7
Partial Quotients Division in the Apple VPP Store for Education
The Volume Purchase Program allows participating educational institutions to purchase iDevBooks math apps in volume and distribute them to students. All iDevBooks math apps offer special 50% discount
for purchases of 20 apps or more for participating educational institutions. | {"url":"http://idevbooks.com/apps/quotients.php","timestamp":"2024-11-06T20:52:39Z","content_type":"text/html","content_length":"6804","record_id":"<urn:uuid:8702f24f-d6b8-4e8b-9db5-6016652dc33f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00296.warc.gz"} |
How the Order of Operations Can Help You Solve Equations Easily? - Topnetworkdirectory
When we talk about general equations which contain brackets, people find it difficult to solve them. This might be due to the reason that they are not aware of the proper sequence of solving those
For solving such expressions, we have a concept of order of operations in mathematics. The PEMDAS rule states first that we need to compute the expressions in the parentheses. PEMDAS is an acronym
that stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.
All these operations need to perform in the above-mentioned sequence only as it will help you get your answer in a very short period & that too correctly. When we see the concept of PEMDAS, a second
preference is given to order i.e. Exponents. These are the powers of a given number. After solving the exponents, all the other mathematical operations are carried out.
What is PEMDAS?
This acronym is a contraction of Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. In some regions, it is called BODMAS (Brackets, Order, Division, Multiplication,
Addition, and Subtraction).
PEMDAS is just like BODMAS, the only difference is in the sequence of operations. While the division is done first in the case of BODMAS, Multiplication is given first preference in PEMDAS. Rest all
the operations are the same in both cases. So, whichever you choose, the answer will always remain the same.
Order for solving expressions through PEMDAS:
For solving any mathematical expression, the PEMDAS rule needs to be applied to get the right solution. Following is the correct way of solving any expression of maths:
1. Parentheses: If you are given any expression with parentheses in it, then the first thing you need to do is to solve those brackets () in the correct sequence. If you don’t do it correctly, you
won’t get the right answer.
2. Exponents: Here order doesn’t refer to the sequence, but the exponents, roots, squares etc. that are included in the expression. After removing/solving all the brackets, you need to start with
solving the exponents or other mathematical figures.
3. Multiplication: Once all the brackets & orders are solved, the first mathematical operation that you need to do is multiplication. It should be given the first preference before any other
mathematical operations.
4. Division: After you divide, you are left with a small number of numbers. And the ones which are left need to be divided accordingly as mentioned in the question.
5. Addition: One of the easiest and the most favourite mathematical operations of all time. The addition of numbers is required to be done once division & multiplication have been carried out.
6. Subtraction: Last operation and step for solving expressions with brackets are Subtraction. It is the least preferred operation while solving expressions.
Example of the order of operations:
Problem: 8 + (2 x 5) x 34 ÷ 9
Step 1 = 8 + 10 x 34 ÷ 9
(Removing the parentheses brackets by solving the expression inside the parentheses i.e. 2*5=10)
Step 2 = 8 + 10 x 81 ÷ 9
(After solving the parentheses, solve the exponents. In this case, it is 34= 81)
Step 3 = 8 + 810 ÷9
(Once the exponents are solved, start with the multiplication part as it is the third step in PEMDAS. Here, 10*81 is equal to 810)
Step 4 =8 + 90
(After multiplying the required numbers in the expression, you need to divide the specific ones according to the need of the expression. Like in this case, 810 ÷9= 90)
Step 5 = 98
(The last step of the order of operations in this question is added because we don’t have any subtraction in this case. So, here 8+90=98)
Therefore, our answer for the above operation is 98.
Order of operations is a very interesting concept in mathematics & it can be easily understood through the help of math experts, which you can easily find on Cuemath. These mentors can help you in
understanding all the complex maths problems that cause worry to you. | {"url":"https://topnetworkdirectory.com/how-the-order-of-operations-can-help-you-solve-equations-easily/","timestamp":"2024-11-04T06:06:13Z","content_type":"text/html","content_length":"63557","record_id":"<urn:uuid:27401afa-69f3-46b0-8680-5fdb6efee93a>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00586.warc.gz"} |
Number System
Jump to navigation Jump to search
Concept Map
Our daily life is based on numbers. We use it for shopping, reckoning the time, counting distances and so on. Simple calculations seem effortless and trivial for most of our necessities.So we should
know about numbers. Numbers help us count concrete objects. They help us to say which collection of objects . In this we are learning about basic operations of numbers - different types of numbers,
representation, etc.
How can math be so universal? First, human beings didn't invent math concepts; we discovered them.
Also, the language of math is numbers, not English or German or Russian.
If we are well versed in this language of numbers, it can help us make important decisions and perform everyday tasks.
Math can help us to shop wisely, understand population growth, or even bet on the horse with the best chance of winning the race.
Mathematics expresses itself everywhere, in almost every face of life - in nature all around us, and in the technologies in our hands. Mathematics is the language of science and engineering -
describing our understanding of all that we observe.Mathematics has been around since the beginnings of time and it most probably began with counting. Many, if not all puzzles and games require
mathematical logic and deduction. This section uses the fun and excitement of various popular games and puzzles, and the exhilaration of solving them, to attract and engage the students to realise
the mathematics in fun and games.
Descriptive Statement
Number sense is defined as an intuitive feel for numbers and a common sense approach to using them. It is a comfort with what numbers represent, coming from investigating their characteristics and
using them in diverse situations, and how best they can be used to describe a particular situation. Number sense is an attribute of all successful users of mathematics. Our students often do not
connect what is happening in their mathematics classrooms with their daily lives. It is essential that the mathematics curriculum build on the sense of number that students bring with them to school.
Problems and numbers which arise in the context of the students world are more meaningful to students than traditional textbook exercises and help them develop their sense of how numbers and
operations are used. Frequent use of estimation and mental computation are also important ingredients in the development of number sense, as are regular opportunities for student communication.
Discussion of their own invented strategies for problem solutions helps students strengthen their intuitive understanding of numbers and the relationships between numbers.
In summary, the commitment to develop number sense requires a dramatic shift in the way students learn mathematics.
Flow Chart
To add textbook links, please follow these instructions to: (Click to create the subpage)
Additional Information
Useful websites
Watch the following video on the story of how numbers evolved. The video called Story of One tells how numbers evolved and the initial questions around number theory.
This video is related to number system, helps to know the basic information about number system
This video is related to irrational numbers by Suchitha
This video is relating to exploring number patterns in square numbers.
Reference Books
Teaching Outlines
Concept #1 - History of Numbers: Level 0
The following website takes us on a fascinating journey originating from Prehistoric Mathematics, its evolution in various civilizations such as Egyptian, Greek, Indian, Chinese etc. to the increased
complexities and abstractions of the modern era mathematics. This story of history of numbers also includes descriptions related to contributions of some of the important men and women to the
development of mathematics.
Learning objectives
1. What is the story of numbers?
2. How did counting begin and learning distinguish between the quantity 2 and the number 2.
3. The number "2" is an abstraction of the quantity
Notes for teachers
These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.
1. Series of Activities in one page- Click Here
2. Activity 1
Concept #2 Number Sense and Counting : Level 0
1. Understand that there is an aspect of quantity that we can develop with disparate objects
2. Comparison and mapping of quantities (more or less or equal)
3. Representation of quantity by numbers and learning the abstraction that “2 represents quantity 2 of a given thing”
4. Numbers also have an ordinal value – that of ordering and that is different from the representation aspect of numbers
5. Expression of quantities and manipulation of quantities (operations) symbolically
6. Recognizing the quantity represented by numerals and discovering how one number is related to another number
7. This number representation is continuous.
Notes for teachers
This is not one period – but a lesson topic. There could be a few more lessons in this section. For example, for representing collections and making a distinction between 1 apple and a dozen apples.
This idea could be explained later to develop fractions. Another activity that can also be used to talk of units of measure. Addition and subtraction have been discussed here – extend this to include
multiplication and division).
1. Activity 1 - Quantity and Numbers
2. Activity 2 - The Eighth Donkey Story
3. Activity 3 - Cardinal and Ordinal Numbers
Concept #3 The Number Line :Level 1-2
The number line is not just a school object. It is as much a mathematical idea as functions. The number line is a geometric “model” of all numbers -- including 0 1, 2, 25, 374 trillion, and -5,
Unlike counters, which model only counters, the number line models measurement, which is why it must start with zero. (When we count, the first object we touch is called "one." When we measur using a
ruler, we line one end of the object we’re measuring against the zero mark on the ruler.
Part of the power of addition and subtraction is that these operations work with both counting and measuring. Therefore, to understand basic operations like addition and subtraction, we need a number
line model as well as counters.
1. Numbers can be represented on a continuum called a number line
2. Number line is a representation; geometric model of all numbers
3. Mathematical operations can b explained by moving along the number line
Notes for teachers
Introduce the number line as a concept by itself as well as a method to count, measure and perform arithmetic operations by moving along the number line through different activities.
1. Activity 1- Add,Sub,Product,Sum,Hopping - To introduce Number line
2. Activity 2 - Sum of numbers
3. Activity 3 - Classroom number line
Concept #4 Number Bases
Learning objectives
Notes for teachers
These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.
1. Activity 1 - Activity-1
2. Activity 2 - Activity-2
Concept #5 Place Value
Learning objectives
Notes for teachers
These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.
1. Activity 1 Activity-1
2. Activity 2 Activity-2
Concept #6 Negative numbers are the opposite of positive numbers -
1. To extend the understanding and skill of representing symbolically numbers and manipulating them.
2. To understand that negative numbers are numbers that are created to explain situations in such a way that mathematical operations hold
3. To recognize that negative numbers are opposite of positive numbers; the rules of working with negative numbers are opposite to that of working with positive numbers
4. Together, the negative numbers and positive numbers form one continuous number line
5. Perform manipulations with negative numbers and express symbolically situations involving negative numbers
Notes for teachers
Negative numbers are to be introduced as a type of number; they do the opposite of what positive numbers do.
Read the activity for more detailed description.
1. Activity 1 -What are negative numbers
Concept #7 : Types of Numbers
Learning objectives
Notes for teachers
Assessment activities
I Fill number line (1 period)
Draw these one below the other
100, 200,.....
II Tell stories and Play With Number Systems (1 period - optional)
(This page is downloaded and given as reading materials – page is called Number Systems)
III Questions/ activities for class
1. Arrange in order – shortest, tallest, increasing and decreasing order
Hints for difficult problems
Project Ideas
Math Fun
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Find the missing number (Timed) - Subtraction Maths Games for Year 1 (age 5-6) by URBrainy.com
Find the missing number (Timed)
Find the missing number in the statement.
Find the missing number (Timed)
Find the missing number in the statement.
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Psychopath Renderer
2022 - 08 - 14
In Building a Better LK Hash I developed a significantly improved variant of the Laine-Karras hash for base-2 Owen scrambling. And in Owen Scrambling Based Blue-Noise Dithered Sampling I used base-4
Owen scrambling to build a simpler implementation of Ahmed and Wonka's screen-space blue-noise sampler. But there's an annoying gap between those two posts: unlike base-2, there's no fast hash for
performing base-4 Owen scrambling, so you have to resort to a slower implementation.
In this post I'm going to fill that gap by developing a Laine-Karras style hash for base-4 Owen scrambling as well.^1
Here's the hash we're going to end up with:
n ^= x * 0x3d20adea
n ^= (n >> 1) & (n << 1) & 0x55555555
n += seed
n *= (seed >> 16) | 1
n ^= (n >> 1) & (n << 1) & 0x55555555
n ^= n * 0x05526c56
n ^= n * 0x53a22864
A little while back, the company Solid Angle published the paper "Blue-noise Dithered Sampling". It describes a method of distributing sampling error as blue noise across the image plane, which makes
it appear more like error-diffusion dithering than typical render noise.
Or in other words, it turns renders like this:^1
...into renders like this:
Interpolating object transforms is a critical task for supporting transform motion blur in a 3d renderer. Psychopath takes a rather brain-dead approach to this and just directly interpolates each
component of the transform matrices. This is widely considered to be wrong. The recommended approach is to first decompose the matrices into separate translation/rotation/scale components before
In this post I'm going to argue that, for motion blur, the brain-dead approach is not only perfectly reasonable, but actually has some important advantages.
(Note 1: if you just want the LUTs and color space data, jump down to the section titled "The Data".)
(Note 2: this is a slightly odd post for this blog. I normally only write about things directly relevant to 3D rendering here, and this post has very little to do with 3D rendering. Although it is
relevant to incorporating rendered VFX into footage if you're a Blackmagic user.)
When you're doing VFX work it's important to first get your footage into a linear color representation, with RGB values proportional to the physical light energy that hit the camera sensor.
Color in image and video files, however, is not typically stored that way^1, but rather is stored in a non-linear encoding that allocates disproportionately more values to darker colors. This is a
good thing for efficient storage and transmission of color, because it gives more precision to darker values where the human eye is more sensitive to luminance differences. But it's not good for
color processing (such as in VFX), where we typically want those values to be proportional to physical light energy.
The functions that transform between non-linear color encodings and linear color are commonly referred to as transfer functions.^2 There are several standard transfer functions for color, with sRGB
gamma perhaps being the most widely known. But many camera manufacturers develop custom transfer functions for their cameras.^3
If you're doing VFX work, you really want to know what transfer function your camera used when recording its colors. Otherwise you can only guess at how to decode the colors back to linear color.
(This post has been updated as of 2021-05-08 with important changes due to an issue identified by Matt Pharr. The majority of the post is essentially unchanged, but additional text discussing the
issue and providing a new hash with the issue addressed have been added near the end of the post. If you just want the hash itself, skip down to the section "The final hash".)
At the end of my last post about Sobol sampling I said that although there was room for improvement, I was "quite satisfied" with the LK hash I had come up with for use with the techniques in Brent
Burley's Owen scrambling paper. Well, apparently that satisfaction didn't last long, because I decided to have another go at it. And in the process I've not only come up with a much better hash, but
I've also learned a bunch of things that I'd like to share. | {"url":"https://psychopath.io/","timestamp":"2024-11-13T10:45:02Z","content_type":"text/html","content_length":"9166","record_id":"<urn:uuid:9ee8f393-ea4f-4d39-94af-7569c3f60cd9>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00040.warc.gz"} |
Resource constrained project scheduling: Regular and non-regular scheduling objectives
Submitted by Mario Vanhoucke on Fri, 12/30/2011 - 10:37
Project scheduling is the act of constructing a timetable for each project activity, respecting the precedence relations and the limited availability of the renewable resources, while optimizing a
predefined scheduling objective (see “Resource constrained project scheduling: What is my scheduling objective?”). Although time is often considered as the dominant scheduling objective, other
objectives are often crucial from a practical point-of-view. The various possible scheduling objectives can be classified in two categories, as follows:
• Regular scheduling objectives
• Non-regular scheduling objectives
In this article, the difference between the two types of scheduling objectives is explained and illustrated on an example project shown in figure 1. The figure shows a project with 5 activities. Each
project has a duration estimate, a cash flow (positive or negative) and a resource requirement as shown in the table to the right of the project network.
Figure 1. An example project network with activity durations, cash flows and resources
Regular scheduling objectives
Since the construction of a project schedule involves the presentation of an activity timetable, a schedule can be characterized by the starting time of the project activities. Assume two project
schedules S and S’, each characterized by their activity starting times as follows:
• Schedule S: s[1], s[2], ..., s[n]
• Schedule S’: s’[1], s’[2], ..., s’[n]
with n the number of activities in the project.
A formal definition of a regular scheduling objective RSO can be given as follows:
Definition of regular scheduling objective
A regular scheduling objective RSO is a function of the activity starting times s[1], s[2], ..., s[n] such that
s[1] ≤ s’[1], s[2] ≤ s’[2],, ..., s[n] ≤ s’[n]
RSO(s[1], s[2], ..., s[n]) ≤ RSO(s’[1], s’[2], ..., s’[n]) for a minimization objective
RSO(s[1], s[2], ..., s[n]) ≥ RSO(s’[1], s’[2], ..., s’[n]) for a maximization objective
A non-regular scheduling objective is an objective for which this definition does not hold.
This definition implies that when two resource feasible schedules have been constructed such that each activity under the first schedule starts no later than the corresponding starting time in the
second schedule, then the first schedule is at least as good as the second schedule. Consequently, it will never be beneficial to delay an activity of a resource feasible schedule towards the end of
the project.
Figure 2 illustrates the definition on a project schedule with a time minimization scheduling objective. The schedule has a project deadline of 12 time units, and has no resource conflicts (see “
The critical path or the critical chain? The difference caused by resources
”). Minimizing the time of a project is a regular scheduling objective, and hence, it is never beneficial to delay an activity of a resource feasible schedule.
The definition of a regular scheduling objective can be illustrated on two project schedules shown in figures 2 and 3. The schedule S of figure 2 and the schedule S’ of figure 3 can be characterized
by the following activity starting times:
• Schedule S: s[1] = 0, s[2] = 3, s[3] = 3, s[4] = 5, s[5] = 10
• Schedule S’: s’[1] = 0, s’[2] = 5, s’[3] = 3, s’[4] = 7, s’[5] = 10
Figure 2. An example schedule S for the network of figure 1
?Figure 3. An example schedule S’ for the network of figure 1
It is clear that all starting times of schedule S are lower than or equal to the starting times of schedule S’. Indeed, delaying activities 1, 3 or 5 would lead to an increase of the total project
duration of 12. Delaying activities 2 and 4 would not lead to a project duration increase when they are delayed within their slack. Consequently, delaying activities in the schedule of figure 2 will
never lead to an improvement of the scheduling objective, i.e. in a project duration reduction. This corresponds to the formal definition, i.e. TIME(s[1] = 0, s[2] = 3, s[3] = 3, s[4] = 5, s[5] = 10)
= 12 ≤ TIME(s’[1] = 0, s’[2] = 5, s’[3] = 3, s’[4] = 7, s’[5] = 10) = 12.
Non-regular scheduling objectives
A non-regular scheduling objective is an objective for which the formal definition above does not hold. Consequently, a given resource feasible schedule can be improved by delaying one or more
activities towards the end. A typical non-regular scheduling objective is the maximization of the net present value of a project, where activities with positive cash flows are scheduled
as-soon-as-possible, while activities with negative cash flows are scheduled as-late-as-possible (see “
What is my scheduling objective? Maximizing the net present value
In the example project schedules of figures 2 and 3, it is clear that delaying activities 2 and 4 of schedule S leads to an improvement of the scheduling objective, since it increases the net present
value. Consequently, the schedule S’ has a higher net present value than the schedule S, and hence, the definition above does not hold, i.e. NPV(s[1] = 0, s[2] = 3, s[3] = 3, s[4] = 5, s[5] = 10) is
not larger than or equal to NPV(s’[1] = 0, s’[2] = 5, s’[3] = 3, s’[4] = 7, s’[5] = 10).
Constructing resource feasible project schedules with non-regular scheduling objectives can be done by using priority rules and an iterative shifting algorithm, such as the Burgess and Killebrew
algorithm, which will not be discussed at PM Knowledge Center. | {"url":"http://www.pmknowledgecenter.com/dynamic_scheduling/baseline/resource-constrained-project-scheduling-regular-and-non-regular-scheduling-objectives","timestamp":"2024-11-11T00:00:52Z","content_type":"application/xhtml+xml","content_length":"18447","record_id":"<urn:uuid:3d01a176-a47a-4cee-929e-8515a2fc1bf8>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00307.warc.gz"} |
E/P 13 Find the exact value of the gradient of the curve with e... | Filo
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Question Text E/P 13 Find the exact value of the gradient of the curve with equation at the point with coordinates .
Updated On Sep 22, 2024
Topic Differentiation
Subject Mathematics
Class Grade 12 | {"url":"https://askfilo.com/user-question-answers-mathematics/e-p-13-find-the-exact-value-of-the-gradient-of-the-curve-3132343539343533","timestamp":"2024-11-03T00:13:40Z","content_type":"text/html","content_length":"87069","record_id":"<urn:uuid:11ca4dc3-4eee-427a-9c5a-c8de60299830>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00187.warc.gz"} |
Proof of ##F## is an orthogonal projection if and only if symmetric
• I
• Thread starter schniefen
• Start date
In summary, this is a proof of a linear transformation ##F## on an inner product space ##V## being an orthogonal projection if and only if ##F## is a projection and symmetric. The second equality in
the attached image is justified by applying the definition of a symmetric linear map. This is done by setting ##u=v## and ##v=F(v)##, where ##v## and ##F(v)## are in the inner product space ##V##.
This equation equates to ##0## because ##v## is in the orthogonal complement of the image of ##F## and ##F(v)## is in the image of ##F##. Therefore, the second equality is justified.
TL;DR Summary
This is a proof of a linear transformation ##F## on an inner product space ##V## being an orthogonal projection if and only if ##F## is a projection and symmetric.
The given definition of a linear transformation ##F## being symmetric on an inner product space ##V## is
##\langle F(\textbf{u}), \textbf{v} \rangle = \langle \textbf{u}, F(\textbf{v}) \rangle## where ##\textbf{u},\textbf{v}\in V##.
In the attached image, second equation, how is the second equality justified? That is, ##\langle F(\textbf{v}), F(\textbf{v}) \rangle = \langle \textbf{v}, F(F(\textbf{v})) \rangle##. For projections
in general, ##F=F^2##, but why does ##F(\textbf{v})=\textbf{v}## for ##\textbf{v} \in (\text{im} \ F)^{\perp}##
schniefen said:
This is a proof of a linear transformation ##F## on an inner product space ##V## being an orthogonal projection if and only if ##F## is a projection and symmetric.
The given definition of a linear transformation ##F## being symmetric on an inner product space ##V## is
##\langle F(\textbf{u}), \textbf{v} \rangle = \langle \textbf{u}, F(\textbf{v}) \rangle## where ##\textbf{u},\textbf{v}\in V##.
In the attached image, second equation, how is the second equality justified? That is, ##\langle F(\textbf{v}), F(\textbf{v}) \rangle = \langle \textbf{v}, F(F(\textbf{v})) \rangle##. For
projections in general, ##F=F^2##, but why does ##F(\textbf{v})=\textbf{v}## for ##\textbf{v} \in (\text{im} \ F)^{\perp}##
View attachment 250815
Apply the definition of symmetric linear map you quoted.
Hi, for the second equality you've got : ##||F(v)||^2 = <v, F(F(v))>## (because ##F## is symmetric) and this equate ##0## since ##v \in Im(F)^{\perp}## and ##F(F(v)) \in Im(F)##. Where is the
Perhaps I didn't understand the question.
Apply the definition of F you quoted for ##u=v##, ##v=F(v)## (those are replacement equations, not direct equations, i.e ##v## isn't something special such that ##v=F(v)##.
FAQ: Proof of ##F## is an orthogonal projection if and only if symmetric
1. What is an orthogonal projection?
An orthogonal projection is a type of transformation in linear algebra where a vector is projected onto a subspace in a way that is perpendicular (or orthogonal) to the subspace. This means that the
projected vector is the closest approximation of the original vector onto the subspace.
2. What does it mean for a projection to be symmetric?
A projection is symmetric if the subspace onto which the vector is projected is the same as the subspace from which the vector was originally projected. In other words, if the projection of a vector
onto a subspace is the same as the projection of that same vector onto the same subspace but from a different direction, then the projection is symmetric.
3. How do you prove that a projection is orthogonal?
To prove that a projection is orthogonal, you need to show that the dot product of the projected vector and the subspace is equal to zero. This means that the projected vector is perpendicular to the
subspace, which is the definition of an orthogonal projection.
4. Why is symmetry important in proving that a projection is orthogonal?
Symmetry is important because it ensures that the projection of a vector onto a subspace is unique. If a projection is not symmetric, then there could be multiple projections of the same vector onto
the same subspace, which would make it difficult to determine the true orthogonal projection.
5. Can a projection be orthogonal without being symmetric?
No, a projection cannot be orthogonal without being symmetric. This is because the definition of an orthogonal projection requires the projection to be perpendicular to the subspace, which can only
be achieved if the projection is symmetric. | {"url":"https://www.physicsforums.com/threads/proof-of-f-is-an-orthogonal-projection-if-and-only-if-symmetric.978631/","timestamp":"2024-11-06T20:16:51Z","content_type":"text/html","content_length":"93270","record_id":"<urn:uuid:94ff7d7c-e440-464a-868e-ae429cd24b63>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00132.warc.gz"} |
A student measures the depth of a water well with an adjustable frequency audio oscillator. 2 successive resonant frequencies are heard at 4 - DocumenTVA student measures the depth of a water well with an adjustable frequency audio oscillator. 2 successive resonant frequencies are heard at 4
A student measures the depth of a water well with an adjustable frequency audio oscillator. 2 successive resonant frequencies are heard at 4
A student measures the depth of a water well with an adjustable frequency audio oscillator. 2 successive resonant frequencies are heard at 40Hz and 50Hz. What is the depth of the well?
in progress 0
Physics 3 years 2021-08-11T14:20:37+00:00 2021-08-11T14:20:37+00:00 1 Answers 2 views 0 | {"url":"https://documen.tv/question/a-student-measures-the-depth-of-a-water-well-with-an-adjustable-frequency-audio-oscillator-2-suc-17143780-88/","timestamp":"2024-11-05T17:18:58Z","content_type":"text/html","content_length":"79975","record_id":"<urn:uuid:c438a9ac-d39d-405f-b2b1-578b1263509b>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00758.warc.gz"} |
Visualization of path patterns in semantic graphs
Graphs with a large number of nodes and edges are difficult to visualize. Semantic graphs add to the challenge since their nodes and edges have types and this information must be mirrored in the
visualization. A common approach to cope with this difficulty is to omit certain nodes and edges, displaying sub-graphs of smaller size. However, other transformations can be used to summarize
semantic graphs and this research explores a particular one, both to reduce the graph's size and to focus on its path patterns. A-graphs are a novel kind of graph designed to highlight path patterns
using this kind of summarization. They are composed of a-nodes connected by a-edges, and these reflect respectively edges and nodes of the semantic graph. A-graphs trade the visualization of nodes
and edges by the visualization of graph path patterns involving typed edges. Thus, they are targeted to users that require a deep understanding of the semantic graph it represents, in particular of
its path patterns, rather than to users wanting to browse the semantic graph's content. A-graphs help programmers querying the semantic graph or designers of semantic measures interested in using it
as a semantic proxy. Hence, a-graphs are not expected to compete with other forms of semantic graph visualization but rather to be used as a complementary tool. This paper provides a precise
definition both of a-graphs and of the mapping of semantic graphs into a-graphs. Their visualization is obtained with a-graphs diagrams. A web application to visualize and interact with these
diagrams was implemented to validate the proposed approach. Diagrams of well-known semantic graphs are presented to illustrate the use of agraphs for discovering path patterns in different settings,
such as the visualization of massive semantic graphs, the codification of SPARQL or the definition of semantic measures. The validation with large semantic graphs is the basis for a discussion on the
insights provided by a-graphs on large semantic graphs: the difference between a-graphs and ontologies, path pattern visualization using a-graphs and the challenges posed by large semantic graphs. | {"url":"https://repositorio.inesctec.pt/items/340d97e1-040d-41f0-922f-0529738c154f","timestamp":"2024-11-12T23:29:06Z","content_type":"text/html","content_length":"133409","record_id":"<urn:uuid:c2468d5f-ea94-4651-b2fd-d58851bbb3e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00709.warc.gz"} |
maths tutor hounslow - Tutors Nearby
The Importance of Hiring a Maths Tutor in Hounslow
For many students, mathematics can be a challenging subject. Whether it’s algebra, geometry, or calculus, many students find the concepts and techniques of math to be daunting. In Hounslow, a
suburban town in London, many students struggle with math, prompting them to seek the help of a tutor. In recent years, the demand for maths tutors in Hounslow has been on the rise, as more and more
students and parents recognize the importance of personalized, one-on-one instruction.
The role of a maths tutor in Hounslow cannot be understated. Not only do they help students understand the fundamental concepts of math, but they also provide the support and guidance needed to excel
in the subject. With the increasing competitiveness in the UK education system, having a maths tutor in Hounslow can make all the difference for a student’s academic success.
One of the main reasons why parents opt for a maths tutor in Hounslow is the individualized attention that their child receives. In a classroom setting, a teacher has to cater to the needs of
multiple students, and as a result, some students may not get the attention they require. A maths tutor in Hounslow, however, can tailor their teaching methods to the specific needs of the student,
helping them grasp difficult concepts and work through challenging problems at their own pace.
Another benefit of hiring a maths tutor in Hounslow is the flexibility it offers. Many students have busy schedules, with extracurricular activities, family commitments, and other responsibilities. A
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Furthermore, a maths tutor in Hounslow provides a safe and comfortable learning environment for students. In a traditional classroom setting, some students may feel intimidated or embarrassed to ask
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In addition to the individualized attention and flexible scheduling, a maths tutor in Hounslow can also help students prepare for standardized tests and examinations. In the UK, students are required
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Moreover, hiring a maths tutor in Hounslow can also help students build confidence in their math abilities. Many students struggle with math not because they lack intelligence, but because they lack
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Furthermore, with the advancements in technology, many maths tutors in Hounslow are now offering online tutoring services. This provides students with the convenience of receiving tutoring from the
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In conclusion, the benefits of hiring a maths tutor in Hounslow are numerous. From personalized attention and flexibility to improved confidence and test preparation, a maths tutor in Hounslow can
make a significant impact on a student’s academic journey. With the increasing demand for tutoring services in the UK, it’s clear that the role of a maths tutor in Hounslow is integral to the success
of many students. With the right tutor, students can unlock their full potential in math and achieve academic success. | {"url":"https://tutorsnearby.co.uk/maths-tutor-hounslow/","timestamp":"2024-11-10T09:56:05Z","content_type":"text/html","content_length":"62453","record_id":"<urn:uuid:96de2643-d677-4dee-8a1a-665fa68ca022>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00408.warc.gz"} |
Currency Forward Contract. Economics vs. Derivatives.
Hi, In Economics they say that Ft = So x ( (1+ra x (Days/365)) / (1+rb x (Days/365)).
And in Derivatives they say: Ft = So x ( (1+ra ^ (Days/365)) / (1+rb ^ (Days/365)).
They do give different answers. Am I missing something here?
Thanks for the input!
In Econ they assume that the risk-free rates are quoted as nominal rates.
In Derivatives they assume that the risk-free rates are quoted as effective rates.
That’s the only difference.
As far as I know there are different conventions. FRAs and Swaps for example are generally expressed with the x (Days/360) instead of raising to the power of. Hence, one method takes compounding into
account and the other does not. As periods are usually small, effect is negligible. | {"url":"https://www.analystforum.com/t/currency-forward-contract-economics-vs-derivatives/115659","timestamp":"2024-11-11T14:24:02Z","content_type":"text/html","content_length":"20555","record_id":"<urn:uuid:814f2ee6-ad65-42f1-9b86-bd3ec2feaf7f>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00053.warc.gz"} |
Coordination of Multiple Robots along Given Paths with Bounded Junction Complexity - CGL at Tel Aviv University Portal
Illustration of solving a cycle as part of the algorithm
We study a fundamental NP-hard motion coordination problem for multi-robot/multi-agent systems: We are given a graph \(G\) and set of agents, where each agent has a given directed path in \(G\). Each
agent is initially located on the first vertex of its path. At each time step an agent can move to the next vertex on its path, provided that the vertex is not occupied by another agent. The goal is
to find a sequence of such moves along the given paths so that each agent reaches its target, or to report that no such sequence exists. The problem models guidepath-based transport systems, which is
a pertinent abstraction for traffic in a variety of contemporary applications, ranging from train networks or Automated Guided Vehicles (AGVs) in factories, through computer game animations, to qubit
transport in quantum computing. It also arises as a sub-problem in the more general multi-robot motion-planning problem. We provide an in-depth tractability analysis of the problem by considering new
assumptions and identifying minimal values of key parameters for which the problem remains NP-hard. Our analysis identifies a critical parameter called vertex multiplicity (VM), defined as the
maximum number of paths passing through the same vertex. We show that a prevalent variant of the problem, which is equivalent to Sequential Resource Allocation (concerning deadlock prevention for
concurrent processes), is NP-hard even when VM is \(3\). On the positive side, for VM \(\leq 2\) we give an efficient algorithm that iteratively resolves cycles of blocking relations among agents. We
also present a variant that is NP-hard when the VM is \(2\) even when \(G\) is a 2D grid and each path lies in a single grid row or column. By studying highly distilled yet NP-hard variants, we
deepen the understanding of what makes the problem intractable and thereby guide the search for efficient solutions under practical assumptions.
Figure description: (a) An instance with \(9\) robots, each shown as colored filled disk, along with its path in the same color that leads its target. The instance contains a cycle in which each
robot is being obstructed by the next one. (b) We convert the instance to an equivalent “graph composed of paths”. We show that if in this graph every simple cycle contains two vertices not occupied
by a robot, then the cycle can be solved, as is the case here. (c)-(f) The first \(4\) solution steps as part of solving the instance. | {"url":"https://www.cgl.cs.tau.ac.il/projects/coordination-of-multiple-robots-along-given-paths-with-bounded-junction-complexity/","timestamp":"2024-11-07T23:20:14Z","content_type":"text/html","content_length":"99507","record_id":"<urn:uuid:88adcecc-9463-4dc4-80a4-3e6ccf76c05e>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00663.warc.gz"} |
Bro and Sis Math Club
In this video math tutorial you will learn "Techniques to convert words into expressions" in Math.
How to Convert Words into Expressions
In this video math tutorial you will learn "How to solve Equations with fractions in them" in Math.
Solve Equations with Fractions
In this video math tutorial you will learn "How to solve Multi Steps Equations" in Math.
Multi Step Equations
In this video math tutorial you will learn "Define Algebraic Equation" in Math.
What is Algebraic Equation | {"url":"http://www.broandsismathclub.com/2015/01/","timestamp":"2024-11-06T14:19:16Z","content_type":"text/html","content_length":"142935","record_id":"<urn:uuid:cc5eb20b-8d30-457f-8832-d5e7f231b267>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00226.warc.gz"} |
Robust consensus tracking control of multiple mechanical systems under fixed and switching interaction topologies. - PDF Download Free
Robust consensus tracking control of multiple mechanical systems under fixed and switching interaction topologies Jianhui Liu1, Bin Zhang2* 1 School of Automation Science and Electrical Engineering,
Beihang University (BUAA), Beijing 100876, P.R. China, 2 School of Automation, Beijing University of Posts and Telecommunications (BUPT), Beijing 100876, P.R. China *
[email protected]
a1111111111 a1111111111 a1111111111 a1111111111 a1111111111
OPEN ACCESS Citation: Liu J, Zhang B (2017) Robust consensus tracking control of multiple mechanical systems under fixed and switching interaction topologies. PLoS ONE 12(5): e0178330. https://
doi.org/ 10.1371/journal.pone.0178330
Abstract Consensus tracking problems for multiple mechanical systems are considered in this paper, where information communications are limited between individuals and the desired trajectory is
available to only a subset of the mechanical systems. A distributed tracking algorithm based on computed torque approach is proposed in the fixed interaction topology case, in which a robust feedback
term is developed for each agent to estimate the external disturbances and the unknown agent dynamics. Then the result is extended to address the case under switching interaction topologies by using
Lyapunov approaches and sufficient conditions are given. Two examples and numerical simulations are presented to validate the effectiveness of the proposed robust tracking method.
Editor: Xiaosong Hu, Chongqing University, CHINA Received: December 8, 2016
Accepted: May 11, 2017
Multi-agent system has emerged as an active area of research, and drawn attention of scholars from a varieties of disciplines in the past decades. This trend is triggered by the promising
applications of multi-agent system in fields like disaster rescuing, industry assembly lines, surveillance, etc. Each agent in the multi-agent system has limited task abilities. However, through
interactions with each other, they can work as a team and accomplish cooperative behaviors such as consensus [1, 2], flocking [3], formation [4, 5], and state estimation [6, 7]. Among the studies of
these cooperative behaviors, consensus behavior is the most fundamental one [8]. The basic issue of consensus control in multi-agent system is to design a distributed consensus law such that all the
agents could be driven to an agreement. In recent years, many consensus control approaches have been proposed for multi-agent systems with different interaction topologies and dynamic models. In the
early literature [9], graph theory was used to represent the interaction topologies, and as a result, the relationship between system stability and Laplacian eigenvalues was precisely revealed. In
[10], directed graphs were used to represent the interaction topology, and results under dynamically changing interaction topologies were derived. Other representative literatures are [11–13], to
name a few. [11] studied finitetime consensus problems for first-order integrators. [12] extended the problems to systems
Published: May 25, 2017 Copyright: © 2017 Liu, Zhang. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the paper. Funding: This work was
supported by the National Natural Science Foundation of China (NSFC: 61603050) and the Fundamental Research Funds for the Central Universities. Competing interests: The authors have declared that no
competing interests exist.
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Robust consensus tracking control
with uncertain dynamics based on H1 control theory. [13] considered the constrained consensus problem with a global optimization function. In addition, some results extended the consensus problem to
a more general consensus tracking problem where the agents track a time-varying trajectory instead of a static equilibrium. In [14], a robust adaptive control algorithm was proposed for uncertain
nonlinear systems, where the reference trajectory was known to all the following systems. In [15], the problem was solved under the condition that the reference trajectory was available only to a few
agents. In [16], finite-time tracking control of multi-agent systems was considered with a sliding-mode approach. In [17], distributed observers were established to estimate unavailable system
states. The results in [18] addressed consensus problems with the assumption that the second-order derivatives of the reference signals conform to some given policy known to the system. Most
literatures dealing with consensus tracking problem are presented under the assumption that the dynamics of the agents are linear and certain. However, almost all the physical plants exhibit some
kinds of nonlinearities, and external disturbances are inevitable in their dynamical processes. Several works attempt to address the tracking problem with uncertainties, but assumptions of these
results are too conservative to achieve in practical situations. Motivated by the desire to achieve practical results with only necessary constraints on mechanical systems and reference trajectory,
we try to address the robust consensus tracking problem in this paper. We aim to design a controller such that a group of mechanical systems under both fixed and switching topologies could maintain a
satisfactory collective performance in the presence of uncertainties or external disturbances. In the fixed topology case, a distributed robust control law is devised based on the computed torque
approach and algebraic graph theory. We further extend the results to the switching topology case. Sufficient conditions are given, under which the states of the agents could converge to a
neighbourhood of the origin. Two numerical simulations are conducted to validate the effectiveness of our results. The remainder of this paper is organized as follows. First, we introduce the problem
formation and the relevant notations. Then, the robust tracking control under fixed and switching topologies is discussed. In the Simulation Section, two numerical simulations are conducted to
validate the effectiveness of the proposed method. At last, some discussions are made to conclude this paper.
Problem statement For a group of n mechanical systems, the dynamic model of the ith system is formulated by Euler-Lagrange equation [19, 20] Mi ðqi Þ€qi þ Ci ðqi ; q_i Þq_i þ Gi ðqi Þ þ fi ðq_i Þ þ
ui ðtÞ ¼ ti
where qi 2 Rm is the state of the ith system, Mi ðqi Þ 2 Rmm is the symmetric inertia matrix, Ci ðqi ; q_i Þ 2 Rmm is the matrix representing the centrifugal and Coriolis terms, Gi ðqi Þ 2 Rm is the
vector of gravity terms, fi ðq_i Þ 2 Rm is the frictional term, ui ðtÞ 2 Rm denotes the bounded external disturbance and ti 2 Rm represents the control input vector. An assumption of the mechanical
system equation described by Eq (1) is given as follows [20, 21]: Assumption 1. The symmetric inertial matrix Mi(qi), i 2 f1; . . . ; ng ≜ N , is positive definite, which satisfies:
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lm k x k2 < xT Mi ðqi Þx < l k x k2 ; M
8qi ; x 2 Rm
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Robust consensus tracking control
where lm
≜ min minm lmin ðMi ðqi ÞÞ
≜ max maxm lmax ðMi ðqi ÞÞ
8i2N 8qi 2R
8i2N 8qi 2R
In this paper, it is assumed that the information interchanges are bi-directional between the n agents through wireless networks or other sensors. Undirected graphs will be used throughout the paper
to model the bi-directional interaction topologies among agents. Some basic knowledge and conventional notations in algebraic graph theory are given as follows. Let Gðn; εÞ be an undirected graph
with n nodes ν = {ν1, ν2, . . ., νn} and the set of edges ε ν × ν. The adjacency matrix A = [aij] is a symmetric matrix defined as aii = 0 and aij > 0 , (νi, νj) 2 ε. The Laplacian matrix of graph
Gðn; εÞ is defined as L = D − A, where D = diag{d1, d2, . . ., dn} is Pn a diagonal matrix with diagonal entries di ¼ j¼1 aij for i = 1, 2, . . ., n. The set of neighbors of node νi is denoted by N i
¼ fnj 2 njðni ; nj Þ 2 εg. If there is a path between any two nodes of the graph Gðn; εÞ, then Gðn; εÞ is said to be connected. Suppose z is a nonempty subset of nodes ν, T then Gðn; ε ðz zÞÞ is
termed as an induced subgraph by z. A component of a graph Gðn; εÞ is defined as a maximal induced subgraph of Gðn; εÞ that is strongly connected. To characterize the variable interconnection
topology, a piecewise-constant switching signal function sðtÞ : ½0; 1Þ ! f1; . . . ; Mg ≜ M is defined, where M 2 Zþ is the total count of possible interconnection graphs. The reference signals are
denoted as qd ; q_d and €qd respectively. In this paper, qd ; q_d and €qd are only accessible to a subset of the n agents. The accesses of the agents to the trajectories are represented by a diagonal
matrix C ¼ diagfc1 ; . . . ; cn g 2 Rnn , where ( ci ¼
1; if trajectory signals are available to agent i; ð3Þ 0; if trajectory signals not available to agent i:
The information exchange matrix [21] is defined as K ≜ L þ C, where L is the Laplacian matrix and C is defined in Eq (3). Lemma 1. [22] The Laplacian matrix of a component in graph Gðn; εÞ is a
symmetric matrix with real eigenvalues that satisfy 0 ¼ l1 l2 l3 . . . lz D where Δ = 2 × (max1iz di). Lemma 2. [18] If at least one agent in each component of graph G has access to the desired
signals, then the information exchange matrix K = L + C is symmetric and positive definite. Definition 1. The robust tracking problem is said to be settled if for each ω > 0, there is T = T(ω) > 0
and a local distributed control law τi, i 2 {1, . . ., n}, such that k qi ðtÞ
qd ðtÞ k2 o
k q_i ðtÞ
q_d ðtÞ k2 o
8t t0 þ TðoÞ in the presence of frictional force and external disturbance.
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Robust consensus tracking control
To facilitate the subsequent analysis, we define x ¼ ½x1T
x1 ¼ ½ðq1
qd Þ
qd Þ
T T
x2 ¼ ½ðq_ 1
T q_ d Þ
ðq_ n
T T q_ d Þ
X ei ¼
aij ðqi
qj Þ þ bi ðqi
qd Þ
aij ðq_ i
q_ j Þ þ bi ðq_ i
q_ d Þ:
j2N i
X e_ i ¼ j2N i
Before proceeding, we now introduce an important lemma, which will be used in the system stability analysis. Lemma 3. [23] Let V(t) 0 be a continuously differentiable function such that _ V ðtÞ gVðtÞ
þ k, where γ and κ are positive constants. Then the following inequality is satisfied VðtÞ Vð0Þe
k þ ð1 g
e gt Þ:
Robust tracking control under fixed topology Distributed scheme design In this subsection, a distributed scheme based on computed-torque control will be given. Computed-torque control is an important
approach to decouple complex robotic dynamics, which is shown as follows ti ¼ Mi ðqi Þvi þ Ci ðqi ; q_ i Þq_ i þ Gi ðqi Þ
vi ¼ bi €qd
a_e i
where α is a positive constant and ηi is the robust control term. In this paper, ηi is used to estimate the uncertain terms based on information from neighboring agents. Combining Eqs (1), (10) and
(11), we get x_ ¼ Fx þ HðZ þ gÞ
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Robust consensus tracking control
where F
Im aK
0 K
Im In
ð13Þ T T n
T 1
Mi 1 ðqi Þðfi ðq_ i Þ þ ui ðtÞÞ:
For further analysis, we present the following assumptions. Assumption 2. The first order and second order derivatives of the desired trajectory are all bounded (i.e., q_ d ; €qd 2 L1 ). Assumption
3. The state velocity q_ i is bounded (i.e., q_ i 2 L1 ), and the frictional vector fi ðq_ i Þ and its first order and second order derivatives with respect to q_ i are bounded (i.e., 2
fi ðq_ i Þ; @f@i ðq_q_i i Þ ; @ @f2i ðq_q_ i Þ 2 L1 ). i
Under Assumption 2 and 3, it is easy to see k gi k1
¼ k ðbi jbi
Mi 1 ðqi Þðfi ðq_ i Þ þ ui ðtÞÞk1
1j k €qi k1 þ
ðk fi ðq_ i Þk1 þ k ui ðtÞk1 Þ ≜ ri : lm T
To facilitate the analysis in the following subsection, we define r ¼ ½rT1 ; . . . ; rTn and χ = kρk2. Based on the previous preparations, we present the design of ηi as follows Zi ¼
w2 ðbei þ e_ i Þ
where and β are positive parameters which have effects on the convergence precision. The definition of ηi can be utilized to eliminate the effect of frictional vector and external disturbance. In
this section, we consider the fixed topology case where at least one agent in each component has access to the desired signals. By Lemma 2, we know that K = L + C is positive definite, and therefore
we can define the smallest and largest eigenvalues of matrix K as λmin(K) > 0 and λmax(K) > 0. Before showing our main results, we present the following lemmas, which will be used in the prove of the
theorems. " # " # K bI 2bK K Lemma 4. Let P ¼ and Q ¼ , where α and β are positive bI I K 2ðaK bIÞ parameters. If α and β satisfy ab ¼ 1
pffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 and λi − β2 > 0, which imply the roots of Eq (18) are both positive. Thus P is positive definite. Similarly, the eigenvalue ω of Q satisfies
2ðbli þ ali
bÞo þ 4abl2i
4b2 li
l2i ¼ 0:
We have 2 bÞ ¼ ðb2 gi þ gi b
2ðbgi þ agi
2 b2 Þ > ðgi b
2 i
4b gi
2 i
2 i
g ¼ 3g
4b gi ¼ 3gi gi
b2 Þ > 0
4 2 b 3
> 0:
Thus the roots of Eq (19) are positive, i.e., Q is positive definite. Remark 1: The Assumptions used in this paper are idiomatically in the study of physical systems described by Euler-Lagrange
equation. The elements of matrix Mi(qi) are rotary inertias of the joints and the readers can refer to [19, 20] for the precise algebraic expressions. An important property is that Mi(qi) is positive
definite and bounded. Assumptions 2 and 3 mention that the physical parameters and desired trajectory are all bounded, which are naturally in the dynamic behaviour of physical systems [15].
Assumptions 2 and 3 are basically referring to the Lipschits condition. If k fi ðq_ i þ DÞ fi ðq_ i Þ k L k D k, then we can see that @f@i ðq_q_i i Þ 2 L1 .
Convergence analysis Theorem 1. Consider n Euler-Lagrange systems described as Eq (1). Let Assumptions 1, 2 and 3 be fulfilled. Suppose the interaction topology is fixed and at least one agent in
each component has access to the desired signals. Then, under the control strategy Eqs (10), (11) and (15),
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for any sufficiently small constant > 0 and ¼ lmin1ðPÞ 1 þ 2lmin1ðKÞc , there is
TðÞ ¼
lmax ðPÞ Vðt0 Þ ln lmin ðQÞ
such that k xðtÞ k2 ;
8t t0 þ TðÞ
where V(t0) = xT(t0)(P Im)x(t0) and the control parameters satisfy ab ¼ 1
pffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0, there is TðÞ ¼ c1 ln
Vðt0 Þ
k xðtÞ k2 ;
such that
8t t0 þ TðÞ
which completes the proof. Remark 2: In contrast to the proof of Theorem 1, where algebraic approaches are conducted, we now give a geometric proof based on Fig 1. The Eq (28) shows that V_ xT ðQ Im
Þx þ 2lmin ðKÞ which implies V_ < 0 for pffiffi k x k2 > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≜ d. 2lmin ðQÞlmin ðKÞ
Let’s define Bd
SðlÞ ¼ l
fxj k x k2 dg fxjxT Px lg
minfljSðlÞ Bd g
i.e., SðlÞ is the minimum ellipsoid containing Bδ. Next, we will show that for any initial value x (t0), the trajectory x(t) converges to SðlÞ in finite time. Obviously, xðtÞ 2 SðlÞ; 8t t0 for any
xðt0 Þ 2 SðlÞ. Thus in the following, we just need to consider the case when xðt0 Þ 2 = SðlÞ. Let k0 ¼ xT ðt0 ÞPxðt0 Þ and T c0 ¼ minfx ðQ Im Þx 2lmin ðKÞ jx 2 Sðk0 Þ SðlÞg. By Eq (28), we have Z
V_ dt
c0 ðt
t0 Þ:
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Robust consensus tracking control
Thus, the time point t1 when V(t) reaches the boundary of the ellipsoid SðlÞ satisfies t1 t0 þ
l c0
Then V(t) will be limited in the bounded ellipsoid SðlÞ for any t t1. Thus, we have lmin ðPÞ k xðtÞ k22 VðtÞ l;
8t t1
i.e., k xðtÞ k2 lmin ðPÞ ; 8t t1 . It means that the norm of the trajectory error vector can be reduced to any prescribed positive value.
Robust tracking control under switching topologies Distributed scheme design In this section, we extend the results in the above section to the switching topology case, where the switching signal is
chosen as sðtÞ : ½0; 1Þ ! M. Consider an infinite sequence of nonempty, bounded, and contiguous time-intervals [tr, tr+1), r = 0, 1, . . . with t0 = 0, tr+1 tr+T for a constant T > 0. In each
interval [tr, tr+1), there is a sequence of subintervals ½tr0 ; tr1 Þ; ½tr1 ; tr2 Þ; . . . ; ½trmr 1 ; trmr Þ;
tr ¼ tr0 ; trþ1 ¼ trmr
satisfying trjþ1 trj t; 0 j mr 1, for some integer mr 0 and a given constant τ > 0, such that the interaction graph G sðtÞ switches at trj and does not change during each subinterval ½trj ; trjþ1 Þ.
Suppose the interaction graph G s in subinterval ½trj ; trjþ1 Þ has lσ 1 connected components with the corresponding node numbers denoted by y1s ; . . . ; ylss . For simplicity, we suppose that the
first h(1 h lσ) components have accesses to the desired signals. Then, by Lemmas 1 and 2, we know that matrix Kσ = Lσ + Bσ is semi-positive definite and there is matrix Ss 2 Rnn ; STs Ss ¼ I, such
that STs Ks Ss ¼ diagfKs1 ; Ks2 ; . . . ; Ksls g ≜ Ls
where ( Ki
diagflis;1 ; lis;2 ; . . . ; lis;yis g;
diagf0; lis;2 ; . . . ; lis;yis g;
1 i h; ð38Þ h < i ls :
We define 8 i ^s K > > > > < ^ is K > > > > :^ Ls
PLOS ONE | https://doi.org/10.1371/journal.pone.0178330 May 25, 2017
¼ diagflis;1 ; lis;2 ; . . . ; lis;yis g; 1 i h; ¼ diagflis;2 ; lis;2 ; . . . ; lis;yis g; h < i ls ;
^ 1s ; K ^ 2s ; . . . ; K ^ lss g: ¼ diagfK
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Robust consensus tracking control
It is easy to see that Ks
Ss Ls STs pffiffiffiffiffi qffiffiffiffiffiffi 1 qffiffiffiffiffiffipffiffiffiffiffi T ^ ^ ^ L L Ls Ss Ss Ls L s s s pffiffiffiffiffi pffiffiffiffiffi 1 pffiffiffiffiffi pffiffiffiffiffi T ^ Ls Ls Ss
Ss Ls Ls L s
^ 1 ST ÞðS L ST Þ ðSs Ls STs ÞðSs L s s s s s
Ks Fs Ks
^ 1 ST is positive definite. where Fs ¼ Ss L s s In this case, the distributed robust tracking algorithm for Eq (1) is defined as ti ¼ Mi ðqi Þvi þ Ci ðqi ; q_ i Þq_ i þ Gi ðqi Þ vi ¼ bi €qd þ mbi q_
mq_ i
de_ i
ð41Þ Zi
where μ > 0 and d > 0 are positive constants; Zi is the robust input term used to eliminate the effect of uncertain terms, which will be shown in the following part. Combining Eqs (1), (41) and (42),
we get x_ ¼ ZsðtÞ x þ EðZ þ gÞ
where 2 ZsðtÞ
¼ 4 dKsðtÞ mI " # 0
Im ¼ In
5 Im
ð44Þ T
¼ ½ZT1
¼ ½gT1
¼ ðbi
1Þ€qd þ mðbi
1Þq_ d
Mi 1 ðqi Þðfi ðq_ i Þ þ ui ðtÞÞ:
Similar to Eq (14), we know gi is upper bounded and satisfies the following inequality k gi k1
k ðbi
1Þq_ d
Mi 1 ðqi Þðfi ðq_ i Þ þ ui ðtÞÞk1
1jðk €qd k1 þ m k q_ d k1 Þ þ
1 ðk fi ðq_ i Þ k1 þ k ui ðtÞk1 Þ ≜ ri : lm
Thus g ¼ ½gT1 ; . . . ; gTn is upper bounded (i.e., g 2 L1 ). A necessary requirement in the investigation of multi-agent systems under switching topology is that there is a bounded piecewise
continuous vector ϕσ(t) satisfying g ¼ ðKsðtÞ Im ÞsðtÞ
where the upper bound of ϕσ(t) is denoted by φ (i.e., kϕσ(t)k1 φ). Based on the preparations, the robust input term is defined as Zi ¼
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φ2 ðe þ e_ i Þ: 2 i
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Robust consensus tracking control
It is emphasized that φ is the upper bound of the uncertain terms and is a design parameter which has an effect on the consensus precision. From Eq (47) we can see that there is significant positive
correlation between the control energy and φ, and there is a significant negative correlation between the control energy and . The following lemma will be used in the robust convergence analysis. " #
" # 2dKsðtÞ 2dKsðtÞ mI I Lemma 5. Let D ¼ and QsðtÞ ¼ , where μ and d 2dKsðtÞ 2ðmI þ dKsðtÞ IÞ I I are positive parameters. If μ > 1, then D is positive definite and Qσ(t) is positive semi-definite
for 8t 2 [0, 1). Furthermore, Qσ(t) > 0 if and only if Kσ(t) > 0. Proof: The fact that D > 0 is obvious and the proof is omitted here. By Eq (37), we know Kσ can be transformed into a diagonal matrix
as Ks ¼ Ss Ls STs . For simplicity, we redefine Λσ as Ls ¼ diagfl1s ; . . . ; lns g, where lis 0 is the ith eigenvalue of Kσ. Then Qσ can be written as " Qs
2dLs 2dLs
2dLs 2ððm
1ÞI þ dLs Þ
# :
For any eigenvalue ωσ of Qσ, we have that ðos
2dlis Þðos
1 þ dlis ÞÞ
ð2dlis Þ ¼ 0
i.e., o2s
1 þ 2dlis Þos þ 4ðm
1Þdlis ¼ 0:
For any μ > 1 and d > 0, we know 2ðm 1 þ 2dlis Þ > 0 and 4ðm 1Þdlis 0. It follows that ωσ 0, i.e., Qσ is positive semi-definite. Furthermore, ωσ > 0 if and only if 4ðm 1Þdlis > 0; i ¼ f1; 2; . . . ;
ng, i.e., Kσ > 0.
Convergence analysis In this subsection, we will prove that the closed-loop system could maintain a satisfactory performance with switching topologies. Before giving the main result, we will present
some preliminary definitions. Note that sðtÞ : ½0; 1Þ ! M is the finite switching signal and does not change during the time intervals no less than τ. We define x ¼ minsðtÞ2M lmin ðFsðtÞ Þ, % ¼
minsðtÞ2M lmin ðQsðtÞ Þ, and n ¼ lmax%ðDÞ, where D is defined in Lemma 5. Theorem 2. Consider n Euler-Lagrange systems described as Eq (1) with switching interaction topologies. Let Assumptions 1, 2
and 3 be fulfilled. Suppose that during each time interval [tr, tr+1), tr+1 tr + T, there is one subinterval ½trj ; trjþ1 Þ such that all the components have accesses to the desired signals. Then,
under the control strategy Eqs (41), (42) and (47), for any ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffi 1 sufficiently small constant > 0 and ¼ lmin ðDÞ 1 þ Tx þ ð1 1þnt , there is e nt Þnx 1 Vð0Þ wðÞ ¼ 1 þ ln T nt such that k xðtÞ k2 ;
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8t wðÞ
11 / 20
Robust consensus tracking control
where V(0) = xT(0)(D Im)x(0) and the control parameters satisfy m>1
d > 0:
Proof: Take the Lyapunov function VðtÞ ¼ xT ðtÞDxðtÞ, where ð54Þ
D ¼ D Im :
We can see that V(t) is piecewise differentiable and the derivative of V(t) along the solutions of the closed-loop system during ½trj ; trjþ1 Þ is V_ ðtÞ ¼
T xT ðZsðtÞ D þ DZsðtÞ Þx þ 2xT DEðZ þ gÞ
xT ðQsðtÞ Im Þx þ 2xT DEZ þ 2xT DEKsðtÞ φsðtÞ
xT ðQsðtÞ Im Þx þ 2xT DEZ þ 2φk ðKsðtÞ Im ÞET Dx k2 :
From Eqs (40) and (47), we can get that xT DEZ
φ2 T x DEðe þ e_ Þ 2 φ2 T ¼ x DEðKsðtÞ Im ÞET Dx 2 φ2 T ¼ x DEðKsðtÞ Im ÞðFsðtÞ Im ÞðKsðtÞ Im ÞET Dx 2 xφ2 k ðKsðtÞ Im ÞET Dx k22 : 2
pffiffi xφ pffiffi k KsðtÞ Im ET Dxk2 x QsðtÞ Im x xT QsðtÞ Im x þ x nV ðt Þ þ : x
Thus, we have V_ ðtÞ
pffiffi 2 pffiffi þ x x
According to Lemma 3, we obtain jþ1
Vðtrjþ1 Þ e
ð1 nx j tr Þ Vðtrj Þ þ : nx j
tr Þ
Vðtrj Þ þ
tr Þ
Þ ð58Þ
For any other subinterval ½tri ; triþ1 Þ; i 6¼ j, in which not all the components have accesses to the desired signals, we have V_ ðtÞ x . It follows that Vðtriþ1 Þ Vðtri Þ þ ðtriþ1 x
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tri Þ:
12 / 20
Robust consensus tracking control
Thus, we get Vðtrþ1 Þ
Vðtrjþ1 Þ þ ðtrþ1 x
trjþ1 Þ
Vðtrj Þ þ þ ðtrþ1 trjþ1 Þ nx x jþ1 j e nðtr tr Þ Vðtr Þ þ ðtrj tr Þ þ þ ðtrþ1 x nx x T 1 e nt Vðtr Þ þ þ : x nx
tr Þ
ð60Þ trjþ1 Þ
It follows that Vðtrþ1 Þ e
Vð0Þ þ ðe
ðr 1Þnt
þ . . . þ 1Þ
T 1 þ x nx
1 þ nT Vð0Þ þ : ð1 e nt Þnx
Therefore, for any tr < t < tr+1 we have VðtÞ
Vðtr Þ þ T x
T 1 þ nt e Vð0Þ þ þ x ð1 e nt Þnx T 1 þ nt t e nt½T Vð0Þ þ þ x ð1 e nt Þnx rnt
is the integer part of Tt which satisfies Tt > Tt 1. Obviously, for any t 1 þ nt1 ln Vð0Þ T, we have T 1 þ nt t VðtÞ e nt½T Vð0Þ þ þ x ð1 e nt Þnx T 1 þ nt ntðTt 1Þ Vð0Þ þ < e þ x ð1 e nt Þnx T 1 þ
nt 1þ þ : x ð1 e nt Þnx
t T
It follows that lmin ðDÞ k xðtÞ k22 VðtÞ | {"url":"https://d.docksci.com/robust-consensus-tracking-control-of-multiple-mechanical-systems-under-fixed-and_59f08e3ed64ab265ec7e806e.html","timestamp":"2024-11-11T00:25:30Z","content_type":"text/html","content_length":"77838","record_id":"<urn:uuid:ab7f0fa4-3b6a-4469-8ca0-cf168fdcb027>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00093.warc.gz"} |
Angle Sum Property of a Triangle: Theorem, Examples and Proof (2024)
Angle Sum Property of a Triangle is the special property of a triangle that is used to find the value of an unknown angle in the triangle. It is the most widely used property of a triangle and
according to this property, “Sum of All the Angles of a Triangle is equal to 180º.”
Angle Sum Property of a Triangle is applicable to any of the triangles whether it is a right, acute, obtuse angle triangle or any other type of triangle. So, let’s learn about this fundamental
property of a triangle i.e., “Angle Sum Property “.
Table of Content
• What is the Angle Sum Property?
• Angle Sum Property Formula
• Proof of Angle Sum Property
• Exterior Angle Property of a Triangle Theorem
• Angle Sum Property of Triangle Facts
• Solved Example
• FAQs
What is the Angle Sum Property?
For a closed polygon, the sum of all the interior angles is dependent on the sides of the polygon. In a triangle sum of all the interior angles is equal to 180 degrees. The image added below shows
the triangle sum property in various triangles.
This property holds true for all types of triangles such as acute, right, and obtuse-angled triangles, or any other triangle such as equilateral, isosceles, and scalene triangles. This property is
very useful in finding the unknown angle of the triangle if two angles of the triangle are given.
Angle Sum Property Formula
The angle sum property formula used for any polygon is given by the expression,
Sum of Interior Angle = (n − 2) × 180°
where ‘n’ is the number of sides of the polygon.
According to this property, the sum of the interior angles of the polygon depends on how many triangles are formed inside the polygon, i.e. for 1 triangle the sum of interior angles is 1×180° for two
triangles inside the polygon the sum of interior angles is 2×180° similarly for a polygon of ‘n’ sides, (n – 2) triangles are formed inside it.
Example: Find the sum of the interior angles for the pentagon.
Pentagon has 5 sides.
So, n = 5
Thus, n – 2 = 5 – 2 = 3 triangles are formed.
Sum of Interior Angle = (n − 2) × 180°
⇒ Sum of Interior Angle = (5 − 2) × 180°
⇒ Sum of Interior Angle = 3 × 180° = 540°
Proof of Angle Sum Property
Theorem 1: The angle sum property of a triangle states that the sum of interior angles of a triangle is 180°.
The sum of all the angles of a triangle is equal to 180°. This theorem can be proved by the below-shown figure.
Follow the steps given below to prove the angle sum property in the triangle.
Step 1: Draw a line parallel to any given side of a triangle let’s make a line AB parallel to side RQ of the triangle.
Step 2: We know that sum of all the angles in a straight line is 180°. So, ∠APR + ∠RPQ + ∠BPQ = 180°
Step 3: In the given figure as we can see that side AB is parallel to RQ and RP, and QP act as a transversal. So we can see that angle ∠APR = ∠PRQ and ∠BPQ = ∠PQR by the property of alternate
interior angles we have studied above.
From step 2 and step 3,
∠PRQ + ∠RPQ + ∠PQR = 180° [Hence Prooved]
Example: In the given triangle PQR if the given is ∠PQR = 30°, ∠QRP = 70°then find the unknown ∠RPQ
As we know that, sum of all the angle of triangle is 180°
∠PQR + ∠QRP + ∠RPQ = 180°
⇒ 30° + 70° + ∠RPQ = 180°
⇒ 100° + ∠RPQ = 180°
⇒ ∠RPQ = 180° – 100°
⇒ ∠RPQ = 80°
Exterior Angle Property of a Triangle Theorem
Theorem 2: If any side of a triangle is extended, then the exterior angle so formed is the sum of the two opposite interior angles of the triangle.
As we have proved the sum of all the interior angles of a triangle is 180° (∠ACB + ∠ABC + ∠BAC = 180°) and we can also see in figure, that ∠ACB + ∠ACD = 180° due to the straight line. By the
above two equations, we can conclude that
∠ACD = 180° – ∠ACB
⇒ ∠ACD = 180° – (180° – ∠ABC – ∠CAB)
⇒ ∠ACD = ∠ABC + ∠CAB
Hence proved that If any side of a triangle is extended, then the exterior angle so formed is the sum of the two opposite interior angles of the triangle.
Example: In the triangle ABC, ∠BAC = 60° and ∠ABC = 70° then find the measure of angle ∠ACB.
The solution to this problem can be approached in two ways:
Method 1: By angle sum property of a triangle we know ∠ACB + ∠ABC + ∠BAC = 180°
So therefore ∠ACB = 180° – ∠ABC – ∠BAC
⇒ ∠ACB = 180° – 70° – 60°
⇒ ∠ACB = 50°
And ∠ACB and ∠ACD are linear pair of angles,
⇒ ∠ACB + ∠ACD = 180°
⇒ ∠ACD = 180° – ∠ACB = 180° – 50° = 130°
Method 2: By exterior angle sum property of a triangle, we know that ∠ACD = ∠BAC + ∠ABC
∠ACD = 70° + 60°
⇒ ∠ACD = 130°
⇒ ∠ACB = 180° – ∠ACD
⇒ ∠ACB = 180° – 130°
⇒ ∠ACB = 50°
Read More about Exterior Angle Theorem.
Angle Sum Property of Triangle Facts
Various interesting facts related to the angle sum property of the triangles are,
• Angle sum property theorem holds true for all the triangles.
• Sum of the all the exterior angles of the triangle is 360 degrees.
• In a triangle sum of any two sides is always greater than equal to the third side.
• A rectangle and square can be divided into two congruent triangles by their diagonal.
Also, Check
□ Area of a Triangle
□ Area of Isosceles Triangle
Solved Example on Angle Sum Property of a Triangle
Example 1: It is given that a transversal line cuts a pair of parallel lines and the ∠1: ∠2 = 4: 5 as shown in figure 9. Find the measure of the ∠3?
As we are given that the given pair of a line are parallel so we can see that ∠1 and ∠2 are consecutive interior angles and we have already studied that consecutive interior angles are
Therefore let us assume the measure of ∠1 as ‘4x’ therefore ∠2 would be ‘5x’
Given, ∠1 : ∠2 = 4 : 5.
∠1 + ∠2 = 180°
⇒ 4x + 5x = 180°
⇒ 9x = 180°
⇒ x = 20°
Therefore ∠1 = 4x = 4 × 20° = 80° and ∠2 = 5x = 5 × 20° = 100°.
As we can clearly see in the figure that ∠3 and ∠2 are alternate interior angles so ∠3 = ∠2
∠3 = 100°.
Example 2: As shown in Figure below angle APQ=120° and angle QRB=110°. Find the measure of the angle PQR given that the line AP is parallel to line RB.
As we are given that line AP is parallel to line RB
We know that the line perpendicular to one would surely be perpendicular to the other. So let us make a line perpendicular to both the parallel line as shown in the picture.
Now as we can clearly see that
∠APM + ∠MPQ = 120° and as PM is perpendicular to line AP so ∠APM = 90° therefore,
⇒ ∠MPQ = 120° – 90° = 30°.
Similarly, we can see that ∠ORB = 90° as OR is perpendicular to line RB therefore,
∠QRO = 110° – 90° = 20°.
Line OR is parallel to line QN and MP therefore,
∠PQN = ∠MPQ as they both are alternate interior angles. Similarly,
⇒ ∠NQR = ∠ORQ
Thus, ∠PQR = ∠PQN + ∠NQR
⇒ ∠PQR = 30° + 20°
⇒ ∠PQR = 50°
FAQs on Angle Sum Property
Define Angle Sum Property of a Triangle.
Angle Sum Property of a triangle states that the sum of all the interior angles of a triangle is equal to 180°. For example, In a triangle PQR, ∠P + ∠Q + ∠R = 180°.
What is the Angle Sum Property of a Polygon?
The angle sum property of a Polygon states that for any polygon with side ‘n’ the sum of all its interior angle is given by,
Sum of all the interior angles of a polygon (side n) = (n-2) × 180°
What is the use of the angle sum property?
The angle sum property of a triangle is used to find the unknown angle of a triangle when two angles are given.
Who discovered the angle sum property of a triangle?
The proof for triangle sum property was first published by, Bernhard Friedrich Thibaut in the second edition of his Grundriss der Reinen Mathematik
What is the Angle Sum Property of a Hexagon?
Angle sum property of a hexagon, states that the sum of all the interior angles of a hexagon is 720°. | {"url":"https://tobeebook.com/article/angle-sum-property-of-a-triangle-theorem-examples-and-proof","timestamp":"2024-11-08T15:54:07Z","content_type":"text/html","content_length":"76979","record_id":"<urn:uuid:1f81ec65-334f-4551-8e72-3a5976194200>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00601.warc.gz"} |
[LS2-1] Carrying Capacity Mathematics | Biology Dictionary
[LS2-1] Carrying Capacity Mathematics
This standard focuses on applying mathematical concepts in order to better understand the phenomena of carrying capacity.
Resources for this Standard:
Here’s the Actual Standard:
Use mathematical and/or computational representations to support explanations of factors that affect carrying capacity of ecosystems at different scales.
Standard Breakdown
This standard is one of several NGSS high school standards that is highly involved with math. These standards can make for great collaborative projects between biology and math teachers, including
projects that span multiple classes. Here, we will focus more on the biology of carrying capacity and the biotic and abiotic factors that influence carrying capacity.
Carrying capacity is the number of individuals a given environment can support, based on a number of interrelated factors. First, let’s look at abiotic factors like sunlight, water, and substrate.
Since the globe gets uneven amounts of sunlight, both the northern and southern poles receive the least light and considerable seasonal differences. This limits the amount of plant life present,
which in turn lowers the carrying capacity of other organisms. Precipitation levels and the amount of nutrients in the ground substrate further determine how many plants can grow, influencing higher
levels of the food web.
Biotic factors include other organisms within the environment, as well as the organic material they release when they die. If a population had no predators and an abundant food supply, it would
experience exponential growth. However, at a certain point, the abiotic factors of the environment are stretched thin and plant life is restricted. This restricts the herbivores, which then restrict
the population of carnivores. As such, all environments have this hard limit – or carrying capacity – that defines how many organisms can live in a given place.
A little clarification:
The standard contains this clarification statement:
Emphasis is on quantitative analysis and comparison of the relationships among interdependent factors including boundaries, resources, climate, and competition. Examples of mathematical comparisons
could include graphs, charts, histograms, and population changes gathered from simulations or historical data sets.
Let’s look at this clarification a little closer:
Relationships among Independent Factors
The key here is to not only determine how independent factors will affect a single population but to also see how multiple independent factors can play into a much more complex situation. For
example, if you only look at precipitations, some clear patterns emerge. In general, places with high precipitation are also areas that have a lot of plant life. However, if we also look at the
factor of temperature, we see that very cold regions have much lower carrying capacities because the cold temperatures inhibit plant growth.
We can also look at phenomena such as predator-prey dynamics, which shows how populations of predators and prey oscillate in similar patterns as each population can limit the other. Depending on what
population of animals is in focus, there are many biological interactions that directly affect carrying capacity including competition, predation, parasitism, mutualism, and other population-level
relationships between organisms.
There are a wide variety of ways to visualize these relationships mathematically. You can graph different populations to estimate carrying capacity. You can create histograms of different traits of a
population to show how populations are adapting to carrying capacity limits. You can either create this data to show the theory, or you can have students work on historical datasets that show a real
What to Avoid
This NGSS standard also contains the following Assessment Boundary:
Assessment does not include deriving mathematical equations to make comparisons.
Here’s a little more specificity on what that means:
Deriving Mathematical Equations:
When focusing on graphical representations of carrying capacity, important mathematical concepts include slope, identifying units, and estimating carrying capacity based on population dynamics.
However, this does not mean that students need to create or express complex mathematics. At this point, visual estimation and arguments based on the change of slope should be sufficient.
Advanced students can be introduced to calculus and trigonometry theories that can shed even more light on estimating carrying capacity. This material should not be assessed for students to
demonstrate an understanding of this standard. | {"url":"https://biologydictionary.net/ngss-high-school-tutorials/ls2-1-carrying-capacity-mathematics/","timestamp":"2024-11-10T02:30:23Z","content_type":"text/html","content_length":"237168","record_id":"<urn:uuid:5c2d9af9-afc1-4ae1-93db-349821b4d348>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00655.warc.gz"} |
GNU Scientific Library - Bugs: bug #30510, problems with hyperg_U(a,b,x) for...
You are not allowed to post comments on this tracker with your current authentication level.
bug #30510: problems with hyperg_U(a,b,x) for x<0
Submitter: -Deleted Account- <bjg>
Submitted: Wed 21 Jul 2010 08:17:45 PM UTC
Category: Runtime error Severity: 3 - Normal
Operating System: Status: None
Assigned to: None Open/Closed: Open
Release: 1.14
Wed 21 Jul 2010 08:17:45 PM UTC, original submission:
Reply-To: -email is unavailable-
From: Raymond Rogers <raymond.rogers72@gmail.com>
To: -email is unavailable-
Subject: Re: [Bug-gsl] hyperg_U(a,b,x) Questions about x<0 and values
Date: Thu, 08 Jul 2010 11:49:15 -0500
Brian Gough <bjg@gnu.org>
Subject: Re: [Bug-gsl] hyperg_U(a,b,x) Questions about x<0 and values
of a
To: -email is unavailable-
Cc: -email is unavailable-
Message-ID: <43wrt61fkt.wl%bjg@gnu.org>
Content-Type: text/plain; charset=US-ASCII
At Wed, 07 Jul 2010 10:14:34 -0500,
Raymond Rogers wrote:
> >
> > 1) I was unable to find the valid domain of the argument a when x<0.
> > Experimenting yields what seem to be erratic results. Apparently
> > correct answers occur when {x<0&a<0& a integer}. References would be
> > sufficient. Unfortunately {x<0,a<0} is exactly the wrong range for my
> > problem; but the recursion relations can be used to stretch to a>0. If
> > I can find a range of correct operation for the domain of "a" of width >1.
| Brian Gough
| Thanks for the email. There are some comments about the domain for
| the hyperg_U_negx function in specfunc/hyperg_U.c -- do they help?
They explain some things, but I believe the section
if (b_int && b >= 2 && !(a_int && a <= (b - 2))){}
else {}
is implemented incorrectly; and probably the preceding section as well. Some restructuring of the code would make things clea\
rer; but things like that should probably done in a different forum: email, blog, etc...
I think the switches might be wrong. In any case it seems that b=1 has a hole. Is there a source for this code?
Note: the new NIST Mathematical handbook might have better algorithms. I am certainly no expert on implementing mathematical \
functions (except for finding ways to make them fail).
Reply-To: -email is unavailable- -Deleted Account- <bjg>
From: Raymond Rogers <raymond.rogers72@gmail.com>
To: -email is unavailable-
Subject: [Bug-gsl] Re: hyperg_U(a, b, x) Questions about x<0 and values of a,
Date: Sun, 11 Jul 2010 14:43:43 -0500
hyperg_U basically fails with b=1, a non-integer; because
gsl_sf_poch_e(1+a-b,-a,&r1); is throwing a domain error when given
Checking on and using b=1 after a-integer is checked is illustrated
below in Octave. I also put in recursion to evaluate b>=2.
I checked the b=1 expression against Maple; for a few values x<0,a<0,b=1
and x<0,a<0,b>=2 integer.
Unfortunately the routine in Octave to call hyperg_U is only set up for
real returns, which was okay for versions <1.14 . Sad to say I am the
one who implemented the hyperg_U interface, and will probably have to go
back :-( . Integrating these functions into Octave was not pleasant;
but perhaps somebody made it easier. I did translate the active parts
of hyperg_U into octave though; so it can be used in that way.
# Test function to evaluate b=1 for gsl 1.14 hyperg_U x<0
function anss=hyperg_U_negx_1(a,b,x)
#neg, int, a is already taken care of so use it
if (int_a && a<=0)
elseif (int_b && (b==1))
#from the new NIST DLMF 13.2.41
elseif (b>=2)
#DLMF 13.3.10
anss=((b-1-a)*hyperg_U_negx_1(a,b-1,x) +
No files currently attached
Depends on the following items: None found
Items that depend on this one: None found
Carbon-Copy List
-email is unavailable- added by bjg
Follows 1 latest change.
Date Changed by Updated Field Previous Value => Replaced by
2010-07-21 bjg Carbon-Copy - Added -email is unavailable- | {"url":"http://savannah.gnu.org/bugs/?30510","timestamp":"2024-11-09T19:47:48Z","content_type":"application/xhtml+xml","content_length":"29584","record_id":"<urn:uuid:fe299949-d48a-4e38-a48d-b0329b3da53f>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00556.warc.gz"} |
Logic and Set Theory
If A and B are two sets taken from some universe U, then the intersection of A and B written A ½ B, is the set containing all elements common to sets A and B.
A ½ B = { X \ (X · A) ^(X
· B)}
A = { 1,2,4}
B = { 2,4,6,7}
A ½ B = { 2,4}
Note: The definition for intersection says that in order for an element to be part of the solution set for A ½ B it must be both an element of A and an element B. The definition of the conjunction
“and” also requires that both statements, p and q be true. | {"url":"https://teachersinstitute.yale.edu/curriculum/units/1980/7/80.07.04/11","timestamp":"2024-11-08T16:03:15Z","content_type":"text/html","content_length":"38542","record_id":"<urn:uuid:d44b32bf-ef14-4a37-84c9-f3bc4f3dcc61>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00035.warc.gz"} |
In the Thermo panel, you can define thermophysical properties of your material. In contrast to the Transport Properties panel, here you define material properties for compressible flow solvers.
Material Models
To fully describe material properties for compressible simulations you need to define three material models:
• Equation of State
• Thermodynamics
• Transport
Available material models will depend on selected solver. Additionally, not all combinations of these models are available, which means that when you change the Equation of State , the list of
Thermodynamics and Transport models will change.
Besides the models, you need to specify molar mass of your material under Specie input group.
Equations of State
Equation of State model defines the relationship between parameters of state: density, pressure, and temperature. Available equations of state depend on a currently selected solver and, in the case
of the multi-region solvers, on whether mesh region is solid or fluid.
Available Equations of State:
• Perfecte Gas
• Incompressible Perfect Gas
• Constant Density
• Perfect Fluid
• Adiabatic Perfect Fluid
• Peng-Robinson
• Polynomial
Perfect Gas
The Perfect Gas equation of state describes the behaviour of a hypothetical perfect gas , which is a good approximation for real gases under moderate pressures. It can be expressed as:
\(\rho=\frac{p}{R \cdot T}\)
• \(\rho\) - density
• \(p\) - pressure
• \(R\) - specific gas constant
• \(T\) - temperature
The only constant in this equation is \(R\) which can be calculated from universal gas constant and molar mass, therefore the Perfect Gas equation of state does not require additional inputs.
Incompressible Perfect Gas
The Incompressible Perfect Gas equation of state uses a constant reference pressure in the perfect gas equation of state rather than the local pressure so that the density only varies with
temperature and composition. It can be expressed as:
\(\rho=\frac {p_{ref}}{R \cdot T}\)
• \(\rho\) - density
• \(p_{ref}\) - reference pressure
• \(R\) - specific gas constant
• \(T\) - temperature
• \(p_{ref} [Pa]\) - reference pressure
Constant Density
The Constant Density equation of state assumes that density is constant:
• \(\rho [{kg}/m^3]\) - density
Perfect Fluid
The Perfect Fluid equation of state calculates density using the following expression:
\(\rho=\rho_0 +\frac{p}{R \cdot T}\)
• \(\rho\) - density
• \(\rho_0\) - reference density
• \(p\) - pressure
• \(R\) - specific gas constant
• \(T\) - temperature
• \(R [J/{kg} K]\) - "specific gas constant"
• \(\rho_0 [{kg}/m^3]\) - reference density
Note: The "specific gas constant" needs to be explicitly specified since it cannot be computed from molar mass using perfect gas laws.
Adiabatic Perfect Fluid
The Adiabatic Perfect Fluid equation of state calculates density using the following expression:
\(\rho=\rho_0 \cdot (\frac{p+B}{p_0 +B})^{\frac{1}{\gamma}}\)
• \(\rho\) - density
• \(\rho_0\) - reference density
• \(p\) - pressure
• \(p_0\) - reference pressure
• \(B\) - universal gas constant
• \(\gamma\) - adiabatic expansion constant
• \(p_0 [Pa]\) - reference pressure
• \(\rho_0 [{kg}/m^3]\) - reference density
• \(\gamma [-]\) - adiabatic expansion constant
• \(B [Pa]\) - universal gas constant
The Peng-Robinson equation of state is a better approximation of the behaviour of real gases than the Perfect Gas . It is expressed using the following relations:
\(p=\frac{BT}{V_m-b}-\frac{a \alpha}{ V_m^2+2bV_m-b^2}\)
\(a \approx 0.45724 \cdot \frac{B^2 T_c ^2}{p_c}\)
\(b \approx 0.07780 \cdot \frac{B T_c }{p_c}\)
\(\alpha = (1+\kappa(1- T_r^{1/2}))^2\)
\(\kappa \approx 0.37464 + 1.54226 \omega - 0.26992 \omega^2\)
• \(T_c [K]\) - temperature at critical point
• \(p_c [Pa]\) - pressure at critical point
• \(V_c [m^3/{kmol}]\) - molar volume at critical point
• \(\omega [-]\) - acentric factor
The Polynomial equation of state calculates density using a polynomial function:
\(\rho = \Sigma_{i = 0}^{n} a_1 \cdot T^i\)
• \(\rho\) - density
• \(T\) - temperature
• \(a_i\) - polynomial coefficients
• \(a_i [ \frac{kg}{m^{3} \cdot T^{-i}}]\) - polynomial coefficients
Thermodynamics Models
The Thermodynamics models define the relationship between basic parameters of state (pressure and temperature) and derived secondary parameters of state such as enthalpy, entropy and internal energy.
Available Thermodynamics Model:
The Constant thermodynamics model defines specific heat to be constant:
• \(C_p\) - specific heat under constant pressure
The Polynomial model calculates specific heat using a polynomial function:
\(C_p=\Sigma _{i = 0}^n a_1 \cdot T^i \)
• \(C_p\) - specific heat under constant pressure
• \(T\) - temperature
• \(a_i\) - polynomial coefficients
The Janaf model calculates specific heat as a function of temperature using coefficients from JANAF tables of thermodynamics. Two sets of coefficients need to be specified, the first set for
temperatures below a common temperature, the second for temperatures above common temperature. The formula for calculating specific heat is:
\(C_p =R \cdot \Sigma_{i=0}^4 a_i \cdot T^i\)
• \(C_p\) - specific heat under constant pressure
• \(R\) - gas constant
• \(a_i\) - JANAF coefficients
• \(T_{low} [K]\) - low temperature limit for model aplicability
• \(T_{high} [K]\) - high temperature limit for model aplicability
• \(T_{common} [K]\) - temperature delimiter for when to use low or high coefficient
• \(a_i [K^-i]\) (lowCpCoeffs/highCpCoeffs) \([K^{-1}]\) - JANAF coefficients
Transport Models
The Transport models define thermal conductivity and viscosity which are later used in transport equations for momentum and energy.
Available Transport Models:
• Constant
• Sutherland
• Polynomial
The Constant transport model assumes that transport properties are constant. In this model thermal conductivity is calculated using the formula:
\(\kappa = \frac{C_p \cdot \mu}{P_r}\)
• \(\kappa\) - thermal conductivity
• \(C_p\) - specific heat under constant pressure
• \(\mu\) - dynamic viscosity
• \(P_r\) - Prandtl number
• \(\mu [Pa \cdot s]\) - dynamic viscosity
• \(P_r [-]\) - Prandtl number
The Sutherland model calculates dynamic viscosity from the Sutherland formula:
\(\mu=\frac{A_s T^{0.5}}{1+\frac{T_s}{T}}\)
• \(\mu\) - dynamic viscosity
• \(T\) - temperature
• \(A_s\) - Sutherland coefficients
• \(T_s\) - Sutherland temperature
• \(A_s [{kg}/m s K ^0.5 ]\) - Sutherland coefficients
• \(T_s [K]\) - Sutherland temperature
The Polynomial transport model calculates dynamic viscosity and thermal conductivity from polynomial expressions:
\(\mu=\Sigma_{i=0}^n a_{\mu i} \cdot T^i\)
\(\kappa=\Sigma_{i=0}^n a_{\kappa i} \cdot T^i\)
• \(\mu\) - dynamic viscosity
• \(\kappa\) - thermal conductivity
• \(T\) - temperature
• \(a_{x i}\) - polynomial coefficients
• \(a_{\mu i} [Pa s K ^-i ]\) - dynamic viscosity polynomial coefficients
• \(a_{x i} [W/m K ^{1+i} ]\) - thermal conductivity polynomial coefficients
Cavitation Properties
When you select a cavitation solver a set of properties specific to this solver will be displayed.
Compressibility Models
Cavitation properties require you to define a Compressibility Model that describes compressibility of the mixture.
Available Compressibility Models:
Linear Model
The Linear model defines mixture compressibility as linear combination of vapour and liquid compressibilities:
\(\psi=\gamma \cdot \psi_v +(1-\gamma)\cdot \psi_l\)
• \(\psi\) - mixture compressibility
• \(\psi_v\) - vapour compressibility
• \(\psi_l\) - liquid compressibility
• \(\gamma\) - vapour phase fraction
• \(\Psi_v [s^2/m^2]\) - vapour compressibility
• \(\Psi_l [s^2/m^2]\) - liquid compressibility
Wallis Model
The Wallis model computes mixture compressibility using the following equations:
\(\psi =(\gamma \rho_{v,sat} + (1-\gamma) \rho_{l,sat})(\gamma \frac{\psi_v}{ \rho_{v,sat}}+(1-\gamma)\frac{\psi_l}{ \rho_{l,sat}})\)
• \(\psi\) - mixture compressibility
• \(\psi_v\) - vapour compressibility
• \(\psi_l\) - liquid compressibility
• \(\rho_{v,sat}\) - vapour saturation density
• \(\rho_{l,sat}\) - liquid saturation density
• \(\gamma\) - vapour phase fraction
• \(\Psi_v [s^2/m^2]\) - vapour compressibility
• \(\Psi_l [s^2/m^2]\) - liquid compressibility
• \(\rho_{v,sat} [{kg}/m^3]\) - vapour saturation density
• \(\rho_{l,sat} [{kg}/m^3]\) - liquid saturation density
Chung Model
The Chung model computes mixture compressibility using the following equations:
\(s_{fa} = \frac{\frac{\rho_{v,sat}}{\psi_v}}{(1-\gamma)\frac{\rho_{v,sat}}{\psi_v}+ \gamma \frac{\rho_{l,sat}}{ \psi_l}}\)
\(\psi=((\frac{1-\gamma}{ \sqrt{\psi_v}} + \frac{\gamma s_{fa}}{\sqrt{\psi_l}}) \frac{\sqrt{\psi_v \psi_l}}{s_{fa}})^2\)
• \(\psi\) - mixture compressibility
• \(\psi_v\) - vapour compressibility
• \(\psi_l\) - liquid compressibility
• \(\rho_{v,sat}\) - vapour saturation density
• \(\rho_{l,sat}\) - liquid saturation density
• \(\gamma\) - vapour phase fraction
• \(\Psi_v [s^2/m^2]\) - vapour compressibility
• \(\Psi_l [s^2/m^2]\) - liquid compressibility
• \(\rho_{v,sat} [{kg}/m^3]\) - vapour saturation density
• \(\rho_{l,sat} [{kg}/m^3]\) - liquid saturation density
Region Properties
When you select a multi-region, a list of regions will be displayed. By clicking on region name, you will be able to define properties for this region. The definition of material models for each
region is the same as in single-region solvers.
Phase Properties
When you select a multiphase solver, a list of current phases will be displayed. When you select a phase in the list, transport properties for this phase will be displayed below.
Additionally, to individual phase properties, you need to define additional values:
• \(\sigma [N/m]\) - surface tension between the phases
• \(\rho_{min} [Pa]\) - lower limit for pressure
Material Database
You can load material properties from the database in a way described for Transport Properties panel. | {"url":"https://help.sim-flow.com/documentation/panels/thermo","timestamp":"2024-11-11T11:48:50Z","content_type":"text/html","content_length":"43787","record_id":"<urn:uuid:b555b52c-f676-4fcb-8d68-dcbfdfeae60e>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00243.warc.gz"} |
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0 (zero) is both a number^[1] and the numerical digit used to represent that number in numerals. The number 0 fulfills a central role in mathematics as the additive identity of the integers, real
numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems. Names for the number 0 in English include zero, nought (UK), naught (US) (/nɔːt/), nil,
or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—oh or o (/oʊ/). Informal or slang terms for zero include zilch and zip.^[2] Ought and aught (/ɔːt/),^[3] as well
as cipher,^[4] have also been used historically.^[5]
The word zero came into the English language via French zéro from Italian zero, Italian contraction of Venetian zevero form of 'Italian zefiro via ṣafira or ṣifr.^[6] In pre-Islamic time the word
ṣifr (Arabic صفر) had the meaning "empty".^[7] Sifr evolved to mean zero when it was used to translate śūnya (Sanskrit : शून्य) from India .^[7] The first known English use of zero was in 1598.^[8]
The Italian mathematician Fibonacci (c. 1170–1250), who grew up in North Africa and is credited with introducing the decimal system to Europe, used the term zephyrum. This became zefiro in Italian,
and was then contracted to zero in Venetian. The Italian word zefiro was already in existence (meaning "west wind" from Latin and Greek zephyrus) and may have influenced the spelling when
transcribing Arabic ṣifr.^[9]
Modern usage
There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used. Sometimes the words nought,
naught and aught^[10] are used. Several sports have specific words for zero, such as nil in association football (soccer), love in tennis and a duck in cricket. It is often called oh in the context
of telephone numbers. Slang words for zero include zip, zilch, nada, and scratch. Duck egg and goose egg are also slang for zero.^[11]
Ancient Near East
nfr heart with trachea
beautiful, pleasant, good
Ancient Egyptian numerals were base 10. They used hieroglyphs for the digits and were not positional. By 1770 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning
beautiful, was also used to indicate the base level in drawings of tombs and pyramids and distances were measured relative to the base line as being above or below this line.^[12]
By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between
sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish (dating from about 700 BC), the
scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.^[13]
The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked
the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them.
Pre-Columbian Americas
The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system. Many
different glyphs, including this partial quatrefoil— —were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 BC.^[a]
Since the eight earliest Long Count dates appear outside the Maya homeland,^[14] it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the
Olmecs.^[15] Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long
Count dates.
Although zero became an integral part of Maya numerals, with a different, empty tortoise-like "shell shape" used for many depictions of the "zero" numeral, it is assumed to have not influenced Old
World numeral systems.
Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional system. Zero
is represented by the absence of a knot in the appropriate position.
Classical antiquity
The ancient Greeks had no symbol for zero (μηδέν), and did not use a digit placeholder for it.^[16] They seemed unsure about the status of zero as a number. They asked themselves, "How can nothing be
something?", leading to philosophical and, by the medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the
uncertain interpretation of zero.
By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) in his work on mathematical astronomy called the Syntaxis
Mathematica, also known as the Almagest. The way in which it is used can be seen in his table of chords in that book. Ptolemy's zero was used within a sexagesimal numeral system otherwise using
alphabetic Greek numerals. Because it was used alone, not just as a placeholder, this Hellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World.^[17]
However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number, indicating a
concept perhaps better expressed as "none", rather than "zero" in the modern sense. In later Byzantine manuscripts of Ptolemy's Almagest, the Hellenistic zero had morphed into the Greek letter
omicron (otherwise meaning 70).
Another zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning "nothing", not as a symbol.^[18] When division produced zero as a
remainder, nihil, also meaning "nothing", was used. These medieval zeros were used by all future medieval calculators of Easter. The initial "N" was used as a zero symbol in a table of Roman numerals
by Bede or his colleagues around 725.
The Sūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to 5th centuries AD, and Japanese records dated from the 18th century, describe how the c. 4th century BC Chinese counting
rods system enables one to perform decimal calculations. According to A History of Mathematics, the rods "gave the decimal representation of a number, with an empty space denoting zero."^[19] The
counting rod system is considered a positional notation system.^[20]
In AD 690, Empress Wǔ promulgated Zetian characters, one of which was "〇". The symbol 0 for denoting zero is a variation of this character.
Zero was not treated as a number at that time, but as a "vacant position".^[21] Qín Jiǔsháo's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a
round symbol for zero.^[22] Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century AD), as seen in The Nine Chapters on the Mathematical Art.^[23]
Pingala (c. 3rd/2nd century BC^[24]), a Sanskrit prosody scholar,^[25] used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation
similar to Morse code.^[26] Pingala used the Sanskrit word śūnya explicitly to refer to zero.^[27]
It was considered that the earliest text to use a decimal place-value system, including a zero, is the Lokavibhāga, a Jain text on cosmology surviving in a medieval Sanskrit translation of the
Prakrit original, which is internally dated to AD 458 (Saka era 380). In this text, śūnya ("void, empty") is also used to refer to zero.^[28]
A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the Bakhshali manuscript, a practical manual on arithmetic for merchants.^[29] In 2017
three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from 224-383 AD, 680-779 AD, and 885-993 AD, making it the world's oldest recorded use of
the zero symbol. It is not known how the birch bark fragments from different centuries that form the manuscript came to be packaged together.^[30]^[31]^[32]
The origin of the modern decimal-based place value notation can be traced to the Aryabhatiya (c. 500), which states sthānāt sthānaṁ daśaguṇaṁ syāt "from place to place each is ten times the
preceding."^[33]^[33]^[34]^[35] The concept of zero as a digit in the decimal place value notation was developed in India, presumably as early as during the Gupta period (c. 5th century), with the
oldest unambiguous evidence dating to the 7th century.^[36]
The rules governing the use of zero appeared for the first time in Brahmagupta's Brahmasputha Siddhanta (7th century). This work considers not only zero, but also negative numbers and the algebraic
rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard, specifically the definition of the value of zero divided by zero as
There are numerous copper plate inscriptions, with the same small o in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.^[13]
A stone tablet found in the ruins of a temple near Sambor on the Mekong, Kratié Province, Cambodia, includes the inscription of "605" in Khmer numerals (a set of numeral glyphs for the Hindu–Arabic
numeral system). The number is the year of the inscription in the Saka era, corresponding to a date of AD 683.^[38]
The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the
Chaturbhuja Temple at Gwalior in India, dated 876.^[39]^[40] Zero is also used as a placeholder in the Bakhshali manuscript, portions of which date from AD 224–383.^[41]
Middle Ages
Transmission to Islamic culture
The Arabic-language inheritance of science was largely Greek,^[42] followed by Hindu influences.^[43] In 773, at Al-Mansur's behest, translations were made of many ancient treatises including Greek,
Roman, Indian, and others.
In AD 813, astronomical tables were prepared by a Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī, using Hindu numerals;^[43] and about 825, he published a book synthesizing Greek and Hindu
knowledge and also contained his own contribution to mathematics including an explanation of the use of zero.^[44] This book was later translated into Latin in the 12th century under the title
Algoritmi de numero Indorum. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "Algorithm" or
"Algorism" started meaning any arithmetic based on decimals.^[43]
Muhammad ibn Ahmad al-Khwarizmi, in 976, stated that if no number appears in the place of tens in a calculation, a little circle should be used "to keep the rows". This circle was called ṣifr.^[45]
Transmission to Europe
The Hindu–Arabic numeral system (base 10) reached Europe in the 11th century, via the Iberian Peninsula through Spanish Muslims, the Moors, together with knowledge of astronomy and instruments like
the astrolabe, first imported by Gerbert of Aurillac. For this reason, the numerals came to be known in Europe as "Arabic numerals". The Italian mathematician Fibonacci or Leonardo of Pisa was
instrumental in bringing the system into European mathematics in 1202, stating:
After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and
convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a
consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects
were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business. I pursued my study in depth and learned the
give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus (Modus Indorum).
Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things
from the niceties of Euclid's geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have
introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people
might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is
blameless and utterly provident in all things. The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0 ... any number may be written.^[46]^[47]
Here Leonardo of Pisa uses the phrase "sign 0", indicating it is like a sign to do operations like addition or multiplication. From the 13th century, manuals on calculation (adding, multiplying,
extracting roots, etc.) became common in Europe where they were called algorismus after the Persian mathematician al-Khwārizmī. The most popular was written by Johannes de Sacrobosco, about 1235 and
was one of the earliest scientific books to be printed in 1488. Until the late 15th century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the
Roman numerals. In the 16th century, they became commonly used in Europe.
0 is the integer immediately preceding 1. Zero is an even number^[48] because it is divisible by 2 with no remainder. 0 is neither positive nor negative.^[49] Many definitions^[50] include 0 as a
natural number, and then the only natural number not to be positive. Zero is a number which quantifies a count or an amount of null size. In most cultures, 0 was identified before the idea of
negative things, or quantities less than zero, was accepted.
The value, or number, zero is not the same as the digit zero, used in numeral systems using positional notation. Successive positions of digits have higher weights, so inside a numeral the digit zero
is used to skip a position and give appropriate weights to the preceding and following digits. A zero digit is not always necessary in a positional number system, for example, in the number 02. In
some instances, a leading zero may be used to distinguish a number.
Elementary algebra
The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is an
integer, and hence a rational number and a real number (as well as an algebraic number and a complex number).
The number 0 is neither positive nor negative and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number. It cannot be prime because it has
an infinite number of factors, and cannot be composite because it cannot be expressed as a product of prime numbers (0 must always be one of the factors).^[51] Zero is, however, even (as well as
being a multiple of any other integer, rational, or real number).
The following are some basic (elementary) rules for dealing with the number 0. These rules apply for any real or complex number x, unless otherwise stated.
• Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral element) with respect to addition.
• Subtraction: x − 0 = x and 0 − x = −x.
• Multiplication: x · 0 = 0 · x = 0.
• Division: 0/x = 0, for nonzero x. But x/0 is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule.
• Exponentiation: x^0 = x/x = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0^x = 0.
The expression 0/0, which may be obtained in an attempt to determine the limit of an expression of the form f(x)/g(x) as a result of applying the lim operator independently to both operands of the
fraction, is a so-called "indeterminate form". That does not simply mean that the limit sought is necessarily undefined; rather, it means that the limit of f(x)/g(x), if it exists, must be found by
another method, such as l'Hôpital's rule.
The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product.
Other branches of mathematics
Related mathematical terms
• A zero of a function f is a point x in the domain of the function such that f(x) = 0. When there are finitely many zeros these are called the roots of the function. This is related to zeros of a
holomorphic function.
• The zero function (or zero map) on a domain D is the constant function with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all x in D. The zero function is the
only function that is both even and odd. A particular zero function is a zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions. The determinant on
non-invertible square matrices is a zero map.
• Several branches of mathematics have zero elements, which generalize either the property 0 + x = x, or the property 0 · x = 0, or both.
The value zero plays a special role for many physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less
arbitrarily chosen. For example, for an absolute temperature (as measured in kelvins) zero is the lowest possible value (negative temperatures are defined, but negative-temperature systems are not
actually colder). This is in contrast to for example temperatures on the Celsius scale, where zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels
or phons, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics, the zero-point energy is the lowest possible energy that a quantum
mechanical physical system may possess and is the energy of the ground state of the system.
Zero has been proposed as the atomic number of the theoretical element tetraneutron. It has been shown that a cluster of four neutrons may be stable enough to be considered an atom in its own right.
This would create an element with no protons and no charge on its nucleus.
As early as 1926, Andreas von Antropoff coined the term neutronium for a conjectured form of matter made up of neutrons with no protons, which he placed as the chemical element of atomic number zero
at the head of his new version of the periodic table. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical
Computer science
The most common practice throughout human history has been to start counting at one, and this is the practice in early classic computer science programming languages such as Fortran and COBOL.
However, in the late 1950s LISP introduced zero-based numbering for arrays while Algol 58 introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer
as base for array subscripts), and most subsequent programming languages adopted one or other of these positions. For example, the elements of an array are numbered starting from 0 in C, so that for
an array of n items the sequence of array indices runs from 0 to n−1. This permits an array element's location to be calculated by adding the index directly to address of the array, whereas 1-based
languages precalculate the array's base address to be the position one element before the first.
There can be confusion between 0- and 1-based indexing, for example Java's JDBC indexes parameters from 1 although Java itself uses 0-based indexing.
In databases, it is possible for a field not to have a value. It is then said to have a null value.^[52] For numeric fields it is not the value zero. For text fields this is not blank nor the empty
string. The presence of null values leads to three-valued logic. No longer is a condition either true or false, but it can be undetermined. Any computation including a null value delivers a null
A null pointer is a pointer in a computer program that does not point to any object or function. In C, the integer constant 0 is converted into the null pointer at compile time when it appears in a
pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for
different data types).
In mathematics −0 = +0 = 0; both −0 and +0 represent exactly the same number, i.e., there is no "positive zero" or "negative zero" distinct from zero. However, in some computer hardware signed number
representations, zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives; this kind of dual representation is known as
signed zero, with the latter form sometimes called negative zero. These representations include the signed magnitude and one's complement binary integer representations (but not the two's complement
binary form used in most modern computers), and most floating point number representations (such as IEEE 754 and IBM S/390 floating point formats).
In binary, 0 represents the value for "off", which means no electricity flow.^[53]
Zero is the value of false in many programming languages.
The Unix epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1970.^[54]^[55]^[56]
The MacOS epoch and Palm OS epoch (the date and time associated with a zero timestamp) begins the midnight before the first of January 1904.^[57]
Many APIs and operating systems that require applications to return an integer value as an exit status typically use zero to indicate success and non-zero values to indicate specific error or warning
Other fields
• In telephony, pressing 0 is often used for dialling out of a company network or to a different city or region, and 00 is used for dialling abroad. In some countries, dialling 0 places a call for
operator assistance.
• DVDs that can be played in any region are sometimes referred to as being "region 0"
• Roulette wheels usually feature a "0" space (and sometimes also a "00" space), whose presence is ignored when calculating payoffs (thereby allowing the house to win in the long run).
• In Formula One, if the reigning World Champion no longer competes in Formula One in the year following their victory in the title race, 0 is given to one of the drivers of the team that the
reigning champion won the title with. This happened in 1993 and 1994, with Damon Hill driving car 0, due to the reigning World Champion (Nigel Mansell and Alain Prost respectively) not competing
in the championship.
• On the U.S. Interstate Highway System, in most states exits are numbered based on the nearest milepost from the highway's western or southern terminus within that state. Several that are less
than half a mile (800 m) from state boundaries in that direction are numbered as Exit 0.
Symbols and representations
The modern numerical digit 0 is usually written as a circle or ellipse. Traditionally, many print typefaces made the capital letter O more rounded than the narrower, elliptical digit 0.^[58]
Typewriters originally made no distinction in shape between O and 0; some models did not even have a separate key for the digit 0. The distinction came into prominence on modern character displays.^
A slashed zero can be used to distinguish the number from the letter. The digit 0 with a dot in the center seems to have originated as an option on IBM 3270 displays and has continued with some
modern computer typefaces such as Andalé Mono, and in some airline reservation systems. One variation uses a short vertical bar instead of the dot. Some fonts designed for use with computers made one
of the capital-O–digit-0 pair more rounded and the other more angular (closer to a rectangle). A further distinction is made in falsification-hindering typeface as used on German car number plates by
slitting open the digit 0 on the upper right side. Sometimes the digit 0 is used either exclusively, or not at all, to avoid confusion altogether.
Year label
In the BC calendar era, the year 1 BC is the first year before AD 1; there is not a year zero. By contrast, in astronomical year numbering, the year 1 BC is numbered 0, the year 2 BC is numbered −1,
and so on.^[59]
See also
1. ^ No long count date actually using the number 0 has been found before the 3rd century AD, but since the long count system would make no sense without some placeholder, and since Mesoamerican
glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time.
1. ^ Matson, John (21 August 2009). "The Origin of Zero". Scientific American. Springer Nature. Retrieved 24 April 2016.
2. ^ Soanes, Catherine; Waite, Maurice; Hawker, Sara, eds. (2001). The Oxford Dictionary, Thesaurus and Wordpower Guide (Hardback) (2nd ed.). New York: Oxford University Press. ISBN
3. ^ "aught, Also ought" in Webster's Collegiate Dictionary (1927), Third Edition, Springfield, MA: G. & C. Merriam.
4. ^ "cipher", in Webster's Collegiate Dictionary (1927), Third Edition, Springfield, MA: G. & C. Merriam.
5. ^ See:
□ Douglas Harper (2011), Zero, Etymology Dictionary, Quote="figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity," c. 1600, from
French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher," translation of Sanskrit sunya-m "empty place, desert, naught";
□ Menninger, Karl (1992). Number words and number symbols: a cultural history of numbers. Courier Dover Publications. pp. 399–404. ISBN 978-0-486-27096-8.;
□ "zero, n." OED Online. Oxford University Press. December 2011. Archived from the original on 7 March 2012. Retrieved 2012-03-04. “French zéro (1515 in Hatzfeld & Darmesteter) or its source
Italian zero, for *zefiro, < Arabic çifr”
6. ^ ^a ^b See:
□ Smithsonian Institution, Oriental Elements of Culture in the Occident, p. 518, at Google Books, Annual Report of the Board of Regents of the Smithsonian Institution; Harvard University
Archives, Quote="Sifr occurs in the meaning of "empty" even in the pre-Islamic time. ... Arabic sifr in the meaning of zero is a translation of the corresponding India sunya.";
□ Jan Gullberg (1997), Mathematics: From the Birth of Numbers, W.W. Norton & Co., ISBN 978-0-393-04002-9, p. 26, Quote = Zero derives from Hindu sunya – meaning void, emptiness – via Arabic
sifr, Latin cephirum, Italian zevero.;
□ Robert Logan (2010), The Poetry of Physics and the Physics of Poetry, World Scientific, ISBN 978-981-4295-92-5, p. 38, Quote = "The idea of sunya and place numbers was transmitted to the
Arabs who translated sunya or "leave a space" into their language as sifr."
7. ^ Zero, Merriam Webster online Dictionary
8. ^ Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Wiley. ISBN 978-0-471-39340-5.
9. ^ 'Aught' definition, Dictionary.com – Retrieved April 2013.
10. ^ 'Aught' synonyms, Thesaurus.com – Retrieved April 2013.
11. ^ Joseph, George Gheverghese (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton UP. p. 86. ISBN 978-0-691-13526-7.
12. ^ ^a ^b Kaplan, Robert. (2000). The Nothing That Is: A Natural History of Zero. Oxford: Oxford University Press.
13. ^ Diehl, p. 186
14. ^ Mortaigne, Véronique (November 28, 2014). "The golden age of Mayan civilisation – exhibition review". The Guardian. Archived from the original on 28 November 2014. Retrieved October 10, 2015.
15. ^ Wallin, Nils-Bertil (19 November 2002). "The History of Zero". YaleGlobal online. The Whitney and Betty Macmillan Center for International and Area Studies at Yale. Archived from the original
on 25 August 2016. Retrieved 1 September 2016.
16. ^ O'Connor, John J.; Robertson, Edmund F., "A history of Zero", MacTutor History of Mathematics archive, University of St Andrews.
17. ^ "Zero and Fractions". Know the Romans. Retrieved 21 September 2016.
18. ^ ^a ^b Hodgkin, Luke (2 June 2005). A History of Mathematics : From Mesopotamia to Modernity: From Mesopotamia to Modernity. Oxford University Press. p. 85. ISBN 978-0-19-152383-0.
19. ^ Crossley, Lun. 1999, p. 12 "the ancient Chinese system is a place notation system"
20. ^ Kang-Shen Shen; John N. Crossley; Anthony W.C. Lun; Hui Liu (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford UP. p. 35. ISBN 978-0-19-853936-0. “zero was
regarded as a number in India ... whereas the Chinese employed a vacant position”
21. ^ "Mathematics in the Near and Far East" (pdf). grmath4.phpnet.us. p. 262.
22. ^ Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications. pp. 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
23. ^ Kim Plofker (2009). Mathematics in India. Princeton UP. pp. 55–56. ISBN 978-0-691-12067-6.
24. ^ Vaman Shivaram Apte (1970). Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India. Motilal Banarsidass. pp. 648–649. ISBN 978-81-208-0045-8.
25. ^ "Math for Poets and Drummers" (pdf). people.sju.edu.
26. ^ Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0-691-12067-6, pp. 54–56. Quote – "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC,
[ ...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero." Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN
978-0-691-12067-6, 55–56. "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value "n". [ ...] The
answer is (2)^7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where "n" is large.
Pingala's use of a zero symbol as a marker seems to be the first known explicit reference to zero.
27. ^ Ifrah, Georges (2000), p. 416.
28. ^ Weiss, Ittay (20 September 2017). "Nothing matters: How India's invention of zero helped create modern mathematics". The Conversation.
29. ^ Devlin, Hannah (2017-09-13). "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. ISSN 0261-3077. Retrieved 2017-09-14.
30. ^ Revell, Timothy (2017-09-14). "History of zero pushed back 500 years by ancient Indian text". New Scientist. Retrieved 2017-10-25.
31. ^ "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Library. 2017-09-14. Retrieved 2017-10-25.
32. ^ ^a ^b Aryabhatiya of Aryabhata, translated by Walter Eugene Clark.
33. ^ O'Connor, Robertson, J.J., E.F. "Aryabhata the Elder". School of Mathematics and Statistics University of St Andrews, Scotland. Retrieved 26 May 2013.
34. ^ William L. Hosch, ed. (15 August 2010). The Britannica Guide to Numbers and Measurement (Math Explained). books.google.com.my. The Rosen Publishing Group. pp. 97–98. ISBN 978-1-61530-108-9.
35. ^ Bourbaki, Nicolas Elements of the History of Mathematics (1998), p. 46. Britannica Concise Encyclopedia (2007), entry "Algebra"
36. ^ Algebra with Arithmetic of Brahmagupta and Bhaskara, translated to English by Henry Thomas Colebrooke (1817) London
37. ^ Cœdès, Georges, "A propos de l'origine des chiffres arabes," Bulletin of the School of Oriental Studies, University of London, Vol. 6, No. 2, 1931, pp. 323–328. Diller, Anthony, "New Zeros and
Old Khmer," The Mon-Khmer Studies Journal, Vol. 25, 1996, pp. 125–132.
38. ^ Casselman, Bill. "All for Nought". ams.org. University of British Columbia), American Mathematical Society.
39. ^ Ifrah, Georges (2000), p. 400.
40. ^ "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. Retrieved 2017-09-14.
41. ^ Pannekoek, A. (1961). A History of Astronomy. George Allen & Unwin. p. 165.
42. ^ ^a ^b ^c Will Durant (1950), The Story of Civilization, Volume 4, The Age of Faith: Constantine to Dante – A.D. 325–1300, Simon & Schuster, ISBN 978-0-9650007-5-8, p. 241, Quote = "The Arabic
inheritance of science was overwhelmingly Greek, but Hindu influences ranked next. In 773, at Mansur's behest, translations were made of the Siddhantas – Indian astronomical treatises dating as
far back as 425 BC; these versions may have the vehicle through which the "Arabic" numerals and the zero were brought from India into Islam. In 813, al-Khwarizmi used the Hindu numerals in his
astronomical tables."
43. ^ Brezina, Corona (2006). Al-Khwarizmi: The Inventor Of Algebra. The Rosen Publishing Group. ISBN 978-1-4042-0513-0.
44. ^ Will Durant (1950), The Story of Civilization, Volume 4, The Age of Faith, Simon & Schuster, ISBN 978-0-9650007-5-8, p. 241, Quote = "In 976, Muhammad ibn Ahmad, in his Keys of the Sciences,
remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used "to keep the rows". This circle the Mosloems called ṣifr, "empty" whence our cipher."
45. ^ Sigler, L., Fibonacci's Liber Abaci. English translation, Springer, 2003.
46. ^ Grimm, R.E., "The Autobiography of Leonardo Pisano", Fibonacci Quarterly 11/1 (February 1973), pp. 99–104.
47. ^ Lemma B.2.2, The integer 0 is even and is not odd, in Penner, Robert C. (1999). Discrete Mathematics: Proof Techniques and Mathematical Structures. World Scientific. p. 34. ISBN
48. ^ W., Weisstein, Eric. "Zero". mathworld.wolfram.com. Retrieved 2018-04-04.
49. ^ Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1976). The historical roots of elementary mathematics. Courier Dover Publications. pp. 254–255. ISBN 978-0-486-13968-5.,
Extract of pp. 254–255
50. ^ Reid, Constance (1992). From zero to infinity: what makes numbers interesting (4th ed.). Mathematical Association of America. p. 23. ISBN 978-0-88385-505-8.
51. ^ Wu, X.; Ichikawa, T.; Cercone, N. (1996-10-25). Knowledge-Base Assisted Database Retrieval Systems. World Scientific. ISBN 978-981-4501-75-0.
52. ^ Chris Woodford 2006, p. 9.
53. ^ Paul DuBois. "MySQL Cookbook: Solutions for Database Developers and Administrators" 2014. p. 204.
54. ^ Arnold Robbins; Nelson Beebe. "Classic Shell Scripting". 2005. p. 274
55. ^ Iztok Fajfar. "Start Programming Using HTML, CSS, and JavaScript". 2015. p. 160.
56. ^ Darren R. Hayes. "A Practical Guide to Computer Forensics Investigations". 2014. p. 399
57. ^ ^a ^b Bemer, R. W. (1967). "Towards standards for handwritten zero and oh: much ado about nothing (and a letter), or a partial dossier on distinguishing between handwritten zero and oh".
Communications of the ACM. 10 (8): 513–518. doi:10.1145/363534.363563.
58. ^ Steel, Duncan (2000). Marking time: the epic quest to invent the perfect calendar. John Wiley & Sons. p. 113. ISBN 978-0-471-29827-4. “In the B.C./A.D. scheme there is no year zero. After 31
December 1 BC came AD 1 January 1. ... If you object to that no-year-zero scheme, then don't use it: use the astronomer's counting scheme, with negative year numbers.”
• Amir D. Aczel (2015) Finding Zero, New York City: Palgrave Macmillan. ISBN 978-1-137-27984-2
• Barrow, John D. (2001) The Book of Nothing, Vintage. ISBN 0-09-928845-1.
• Diehl, Richard A. (2004) The Olmecs: America's First Civilization, Thames & Hudson, London.
• Ifrah, Georges (2000) The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley. ISBN 0-471-39340-1.
• Kaplan, Robert (2000) The Nothing That Is: A Natural History of Zero, Oxford: Oxford University Press.
• Seife, Charles (2000) Zero: The Biography of a Dangerous Idea, Penguin USA (Paper). ISBN 0-14-029647-6.
• Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3-540-64767-8.
• Isaac Asimov (1978). Article "Nothing Counts" in Asimov on Numbers. Pocket Books.
• This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the "relicensing" terms of the GFDL, version 1.3 or later.
• Chris Woodford (2006), Digital Technology, Evans Brothers, ISBN 978-0-237-52725-9
External links
Wikimedia Commons has media related to 0 (number).
Look up zero in Wiktionary, the free dictionary.
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[Solved] In the capital asset pricing model, the b | SolutionInn
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Sampling Distributions for Differences in Sample Proportions - Knowunity
Sampling Distributions for Differences in Sample Proportions: AP Statistics Study Guide
Hello Mathletes! 🎉
Welcome to your one-stop guide for mastering sampling distributions of differences in sample proportions. It’s time to dive into the magical world where numbers meet real life, and learn not just how
to calculate but also understand the beauty of statistics.
The Basics: Why We Care About Proportions
Imagine you're comparing the proportions of people who love pineapple on pizza 🥴 in two different cities. Knowing how to handle these proportions statistically helps you understand the bigger
picture. Plus, it makes you the coolest data analyst at the party (a data party, naturally 🥳).
The Magic Formula: Differences in Proportions
When you're dealing with the difference between two sample proportions (say, the percentages of pineapple lovers in two cities 🍍🏙️), you need to work with their variances. Here's the scoop:
1. Calculate the Sample Proportions:
□ For City A: Number of pineapple fans in the sample / Total sample size.
□ For City B: Same formula.
Think of this step as finding the percentage of pizza rebels in each city.
2. Variance and Standard Deviation:
□ Convert the variances of the sample proportions to standard deviation.
□ Remember math's golden rule here: Variances add, but standard deviations hang out together after a square root.
Here's a quick formula gloss:
• If given standard deviations, square them, sum them up, and then take the square root.
This magical combo to find the overall standard deviation is known as the “Pythagorean Theorem of Statistics.” 📐 Why? Because just like adding up the squares of the sides of a right triangle, we add
variances to find the sum total before we square root it. This gives us our combined variability.
When Can We Say “Normal”?
To use a fun scientific analogy, not every behavior pattern in the wild (or sample data) is normal. But if our sample size is large enough, things start looking predictably normal. For differences in
• Each sample must pass the Large Counts rule:
□ ( n1 \times p1 > 10 )
□ ( n1 \times (1 - p1) > 10 )
□ ( n2 \times p2 > 10 )
□ ( n2 \times (1 - p2) > 10 )
Follow these rules and voilà! You basically have the permission slip to call your data distribution "normal."
Just remember: Normal distributions are our way of giving a big thumbs-up to Central Limit Theorem for playing well with proportions! ✌️
Understanding The Sampling Distribution
Let’s say you sampled the opinions of 1,000 pizza fans in City A and City B to see if pineapple belongs on pizza. Here’s what the sampling distribution would tell us:
• Mean of the Distribution: This represents the average difference between the two sample proportions. It’s like saying, “On average, how much more (or less) do City B people support our fruity
topping compared to City A?”
• Standard Deviation of the Difference: This is where our combined variances play a role. Imagine it as a scale of wiggle room; the bigger it is, the more you’d expect your sample proportions to
bob around that mean.
And don't forget, our Central Limit Theorem is the VIP of this data party, ensuring that even if our opinions on pineapple are scattered, the sampling distribution itself stays nice and normal due to
large sample sizes.
Practice Makes (Almost) Perfect
Ever wondered how real-world statisticians work? Let's stage a survey showdown 🚂:
Scenario: You're comparing support for a new public transport system in two cities with samples of 1,000 citizens each.
• Step 1: Calculate sample proportions:
□ City A: 600/1000 = 0.6.
□ City B: 700/1000 = 0.7.
• Step 2: Analyze the useful, fancy, normal distribution:
□ Difference mean: 0.7 - 0.6 = 0.1.
□ Variability (spread): Depends on sample sizes and population variability.
Note: In reality, nonresponse bias could tease out inaccuracies if some groups are less likely to participate. If this bias sneaks in, your neat little estimate of difference goes haywire.
Key Terms to Know
• Bias: It’s like a bent coin; your data doesn't represent the true scenario.
• Categorical Variable: Think of data that falls into distinct classes or groups.
• Central Limit Theorem: Our best friend in stats, promising normality with large samples.
• Nonresponse Bias: When certain groups don't respond, skewing our results.
• Normal Distribution: The bell curve boss of statistics.
• Population Proportions: The big picture from which we draw our little samples.
• Sample Proportion: The piece of the pie we actually measure.
• Sampling Distribution: The collection of sample stats we find if we repeated our sample endlessly.
• Simple Random Sample: Every member gets a fair shot at being picked.
• Standard Deviation: The average distance of each value from the mean.
Fun Fact
Did you know the term “sampling distribution” makes statisticians quiver with joy? It's the backbone of inferential statistics, making it possible to predict those tricky population parameters. 🙌
Wrapping Up
There you have it, the complete low-down on sampling distributions for differences in sample proportions. Armed with this guide, you can not only wow your friends with your statistical prowess but
crush your AP Stats exam with flying colors! May the stats be ever in your favor. 📊 | {"url":"https://knowunity.com/subjects/study-guide/sampling-distributions-for-differences-sample-proportions","timestamp":"2024-11-11T16:46:22Z","content_type":"text/html","content_length":"245358","record_id":"<urn:uuid:07629b5f-1200-402b-88ac-dc5f0ebf8910>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00164.warc.gz"} |
Scientific explanation from the history and philosophy of science to general philosophy of science (and back again… and again… and again) | Lina JanssonScientific explanation from the history and philosophy of science to general philosophy of science (and back again… and again… and again) | Lina Jansson
Philosophers of science of all stripes draw on the history of science. However, within philosophy of science there are diverging trends between literature in the history and philosophy of science
and the work in (what often goes under the name of) ‘general’ philosophy of science. With the caveat that what follows paints a picture with very broad brushstrokes, the trend among those working on
integrated history and philosophy of science is towards recognizing particular differences between scientific fields, periods, and practitioners. On the other hand, the driving motivation in general
philosophy of science is towards unified frameworks and theories.
These differences in emphasis can make it hard to integrate work with these two broad orientations. Nonetheless, I am particularly interested in work that tries to do so. Harper’s work on Newton’s
methodology provides a case study of the kind of work that I have in mind. Most of Harper’s research on this topic is focused on the reasoning that is particular to Newton’s methodology in the
Principia. I think of it as belonging in the tradition of integrated history and philosophy of science. However, Harper also draws lessons for broad views on theory acceptance and theoretical
progress that are most closely aligned with the interests of general philosophy of science.
I think that interactions between these two approaches are fruitful in both directions. Let me illustrate why by drawing on my own work. In one direction, I have argued that we can gain a better
understanding of Newton’s explanatory goals by paying critical attention to the general philosophy of science. In the other, I argue that we can gain better accounts of explanation in general by
paying attention to Newton’s methodology in the Principia.
Smith describes Newton’s goal as being that of developing ‘an intermediate level of theory, between mere description of observed regularities in the manner of Galileo’s Two New Sciences, on the one
hand, and laying out full mechanisms in the manner of Descartes’ Principia, on the other’.^[1] Smith stresses the role of laws of force in allowing us to answer questions about whether or not
observed regularities are suitable as evidence. When it comes to explanation, however, this intermediate level of theory raises new difficulties. What could such an intermediate level of explanatory
theory be?
If we approach the question from the assumption (found in accounts within general philosophy of science) that to explain physical events, and the regularities among such events, requires causal
explanation, then we are pushed to try to make sense of an intermediate-level theory in causal terms. This is difficult to do (though Janiak argues for this option). In particular, it is challenging
to reconcile Newton’s disavowal of having provided a causal account of gravitational phenomena with his claim to have an explanatory account that goes beyond mere description of regularities:
Thus far I have explained the phenomena of the heavens and of our sea by the force of gravity, but I have not yet assigned a cause to gravity […] And it is enough that gravity really exists and acts
according to the laws that we have set forth and is sufficient to explain all the motions of the heavenly bodies and of our sea.^[2]
If we instead take as our starting point the Scholium to Proposition 69 in Book 1, a natural suggestion is to take Newton as providing a law-based theory (at the level of physics), and law-based
Mathematics requires an investigation of those quantities of forces and their proportions that follow from any conditions that may be supposed. Then, coming down to physics, these proportions must
be compared with the phenomena, so that it may be found out which conditions [or laws] of forces apply to each kind of attracting bodies. And then, finally, it will be possible to argue more
securely concerning the physical species, physical causes, and physical proportions of these forces.^[3]
Our merry-go-round now circles us back to problems in the general philosophy of science. Here, we face familiar worries about such a proposal. After all, the most prominent general account of how
laws explain, the deductive nomological account, runs into several well-known problems with capturing the directionality of explanation and the nature of explanatory irrelevancies. If these problems
cannot be addressed, then an interpretation that takes Newton’s theoretical and explanatory focus to be laws makes the position in the Principia ultimately unstable.
In order to see how far we can make stable the position that takes laws as theoretically prior to causes—including the case of explanation—we can again draw on work in both the integrated history and
philosophy of science and general philosophy of science. What is it that makes the use of laws in the Principia differ from simple deductions of phenomena of interest from laws? From Harper’s and
Smith’s work, we find a compelling account of Newton’s reliance on subjunctive reasoning in the Principia. From Woodward’s influential causal account of explanationcausal account of explanation, we
find a similar emphasis on counterfactual considerations. This give us a plausible first step towards providing an account of how laws can do explanatory work that departs from the motivation driving
the deductive-nomological account. We can focus on the role of laws in guiding subjunctive reasoning in the Principia. I do not think that this is enough to tackle the difficulties, but it provides
the crucial first step.
However, we have not yet done enough to support a ‘laws-first’ interpretation of the methodology in the Principia. To allow that law-based explanations can be epistemically independent from causal
underpinnings, we need to account both for
• how laws can do explanatory work, independent of causal underpinnings
• how we can have reason to take some proposition to be a law of nature, independent of knowledge of the causal underpinnings of the proposition.
Round and round we go, and this raises a new question from general philosophy of science: How can we have empirical reasons to take a proposition to be a law—and capable of supporting subjunctive
reasoning—rather than merely accidentally true?^[4] I am currently working on developing an answer to this question that relies on Harper’s and Smith’s account of the crucial role of subjunctive
reasoning and successive approximations in Newton’s methodology. The very rough idea is this: Chains of subjunctive reasoning can be successful (or unsuccessful) at tackling reasoning about a
physical system. Moreover, that success is empirically accessible. If we take it to be a defining feature of propositions that are laws that they reliably support subjunctive reasoning, while mere
accidents do not, we now have a potential empirically accessible way to distinguish laws from merely true propositions.
Taking a final turn (for now) in our merry-go-round, we return to questions from the history and philosophy of science. How much of the work from general philosophy of science is it legitimate to
read into a specific historical text? More specifically, how plausible is it that Newton recognized a distinction between laws and accidents that concerns whether it is legitimate to rely on the
principle in question in subjunctive reasoning?
Again, Harper’s work on Newton’s methodology provides a fascinating clue. Newton added a scholium to his argument in Proposition 4, Book 3, for the identification of the force that keeps the planets
in orbit with gravity. This scholium relies on taking Kepler’s harmonic rule to extend to non-actual systems:
Curtis Wilson’s search for the first to write of Kepler’s rules as ‘laws’ led him to conclude that it was Leibniz in his ‘Tentamen de motuum coelestium causis’ of 1689 [...] Newton takes advantage of
the opportunity afforded by being able to call Kepler’s Harmonic Rule a ‘law’.^[5]
At the time of the first publication of the Principia, it was not common to call Kepler’s rules ‘laws’. Only after Leibniz had called them thus was Newton willing to use them in guiding
counterfactual reasoning. As Harper notes, this at least suggests that the distinction was a salient one to Newton.
On the one hand, the attention to particular approaches and how they work in a particular context promises to open up new paths for developing unified accounts in the general philosophy of science.
On the other hand, new general accounts of explanation, confirmation, laws, and so on promise to offer new interpretative possibilities for the history and philosophy of science. We might go round
and round, but it is a fruitful ride!
University of Nottingham
^[1] Smith, p. 371.
^[2] Newton, 'General Scholium', The Principia: Mathematical Principles of Natural Philosophy, p. 943.
^[3] Newton, 'General Scholium', The Principia: Mathematical Principles of Natural Philosophy, pp. 588-9.
^[4] Earman and Roberts argue that we cannot unless laws supervene on the Humean base (as they define it).
^[5] Harper, p. 177.
Recent Comments
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Most recent change of Dimension
Edit made on February 14, 2009 by RiderOfGiraffes at 09:25:24
[S:Deleted text in red:S] / Inserted text in green
A measure of the number of independent directions in which one can move around in a given space. There are several different definitions of "dimension", some of which raise the level of abstraction
considerably beyond the intuitive description just given.
In particular, one of the definitions allows the concept of "Fractional Dimension", which turns up in the study of fractals. | {"url":"https://www.livmathssoc.org.uk/cgi-bin/sews_diff.py?Dimension","timestamp":"2024-11-10T17:51:33Z","content_type":"text/html","content_length":"1488","record_id":"<urn:uuid:9464e653-268c-43b9-9b4f-c00d6c6a4d12>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00883.warc.gz"} |
Statistical classification
Statistical classification
Statistical classification refers to supervised machine learning techniques for categorizing data points into classes. Key aspects:
• Algorithms learn from training data containing class labels.
• A classification model is built to predict the class of new unlabeled points.
• Probability theory determines class membership based on likelihood.
• Useful for predicting categorical variables in data.
Major approaches include:
• Logistic regression for binary classification.
• Linear discriminant analysis finds separations between classes.
• Naive Bayes classifiers apply Bayesian probability.
• Support vector machines find optimal decision boundaries.
• Decision trees partition data points using tree-like models.
Classification is used for predictive modeling tasks like spam detection, disease diagnosis, sentiment analysis, and more.
Performance metrics include accuracy, precision, recall, and F1-score. Challenges include class imbalance, overfitting, and defining optimal features.
Classification provides fundamental machine learning tools for modeling and analyzing categorical data. Advances in deep learning have boosted classification capabilities.
See also: | {"url":"https://feedsee.com/aiw/Statistical_classification/","timestamp":"2024-11-05T16:18:42Z","content_type":"text/html","content_length":"3871","record_id":"<urn:uuid:12e2e645-c88b-472c-a74a-5342b37190f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00802.warc.gz"} |
collapse Documentation and Resources
Sebastian Krantz
collapse is a C/C++ based package for data transformation and statistical computing in R. It’s aims are:
1. To facilitate complex data transformation, exploration and computing tasks in R.
2. To help make R code fast, flexible, parsimonious and programmer friendly.
Documentation comes in 6 different forms:
Built-In Structured Documentation
After installing collapse, you can call help("collapse-documentation") which will produce a central help page providing a broad overview of the entire functionality of the package, including direct
links to all function documentation pages and links to 13 further topical documentation pages (names in .COLLAPSE_TOPICS) describing how clusters of related functions work together.
Thus collapse comes with a fully structured hierarchical documentation which you can browse within R - and that provides everything necessary to fully understand the package. The Documentation is
also available online.
The package page under help("collapse-package") provides some general information about the package and its design philosophy, as well as a compact set of examples covering important functionality.
Reading help("collapse-package") and help("collapse-documentation") is the most comprehensive way to get acquainted with the package. help("collapse-documentation") is always the most up-to-date
An up-to-date (v2.0) cheatsheet compactly summarizes the package.
Article on arXiv
An article on collapse (v2.0.10) has been submitted to the Journal of Statistical Software in March 2024.
useR 2022 Presentation and Slides
I have presented collapse (v1.8) in some level of detail at useR 2022. A 2h video recording that provides a quite comprehensive introduction is available here. The corresponding slides are available
Updated vignettes are
• collapse for tidyverse Users: A quick introduction to collapse for tidyverse users
• collapse and sf: Shows how collapse can be used to efficiently manipulate sf data frames
• collapse’s Handling of R Objects: A quick view behind the scenes of class-agnostic R programming
The other vignettes (only available online) do not cover major features introduced in versions >= 1.7, but contain much useful information and examples:
I maintain a blog linked to Rbloggers.com where I introduced collapse with some compact posts covering central functionality. Among these, the post about programming with collapse is useful for | {"url":"https://cran-r.c3sl.ufpr.br/web/packages/collapse/vignettes/collapse_documentation.html","timestamp":"2024-11-09T16:12:21Z","content_type":"text/html","content_length":"11182","record_id":"<urn:uuid:b6a99b56-15c6-41b1-892d-15d059173ac8>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00264.warc.gz"} |
Section: New Software and Platforms
FPLLL: a lattice reduction library
fplll contains several algorithms on lattices that rely on floating-point computations. This includes implementations of the floating-point LLL reduction algorithm, offering different speed/
guarantees ratios. It contains a “wrapper” choosing the estimated best sequence of variants in order to provide a guaranteed output as fast as possible. In the case of the wrapper, the succession of
variants is oblivious to the user. It also includes a rigorous floating-point implementation of the Kannan-Fincke-Pohst algorithm that finds a shortest non-zero lattice vector, and the BKZ reduction
The fplll library is distributed under the LGPL license. It has been used in or ported to several mathematical computation systems such as Magma, Sage, and PariGP. It is also used for cryptanalytic
purposes, to test the resistance of cryptographic primitives. | {"url":"https://radar.inria.fr/report/2015/aric/uid45.html","timestamp":"2024-11-14T03:42:40Z","content_type":"text/html","content_length":"39095","record_id":"<urn:uuid:baf724d1-6157-4dfb-98f3-f760c3a2d28a>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00376.warc.gz"} |
Show posts - KnightKinetik
@Benny1: I made some adjustments to your drive cancel combos to squeeze a lil more damage out :D.
Quote from: Benny1 on September 13, 2013, 06:15:41 AM
0 Bar/1 Drive
starter -> hcf.lk (lk+hk) db~f.lp db~f.lp (DC) dp.hp (300/303/324)
Not really worth the drive unless you're sure that 35 damage will kill.
Starter -> hcf.B > BC, db-f.A, dp.C xDCx db-f.A (will whiff), db-f.A, dp.C (347/350/371)
Quote from: Benny1 on September 13, 2013, 06:15:41 AM
1 Bar/1 Drive
starter -> hcf.lk (lk+hk) db~f.hp (DC) db~f.lp+hp db~f.lp dp.hp (428/431/452)
Starter -> hcf.B > BC, db-f.A, dp.C xDCx db-f.A (will whiff),
, db-f.A, dp.C (449/452/473)
Quote from: Benny1 on September 13, 2013, 06:15:41 AM
2 Bar/1 Drive
starter -> hcf.lk (lk+hk) db~f.hp (DC) db~f.lp+hp db~f.lp+hp dp.hp (495/498/519)
Starter -> hcf.B > BC,
, dp.C xDCx db-f.A (whiff), db-f.A, dp.C (529/532/553)
Quote from: Benny1 on September 13, 2013, 06:15:41 AM
3 Bar/1 Drive
starter -> hcf.lk (lk+hk) db~f.hp (DC) db~f.lp+hp db~f.lp+hp qcfhcb.hp (561/564/585)
starter -> hcf.lk (lk+hk) (wait a little) db~f.lp (DC) db~f.lp+hp db~f.lp+hp db~f.lp+hp dp.hp (576/579/600)
Starter -> hcf.B > BC,
, dp.C xDCx
, db-f.A, dp.C (625/628/649)
Imo in regards to starting st.C (HP) (which I assume is the close one, since far C into hcf.B whiffs), you can cancel it into f.A which brings it's damage close to the st.B xx st.D starter and is a
common intro into HD. | {"url":"https://dreamcancel.com/forum/index.php?action=profile;area=showposts;sa=messages;u=4678","timestamp":"2024-11-03T16:30:17Z","content_type":"text/html","content_length":"35868","record_id":"<urn:uuid:dd969479-9268-410a-811a-faf8f00146f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00711.warc.gz"} |
Students with Calculus Credit: Math Class Choices
You may have earned academic college course credit by scoring well on Advanced Placement (AP) and/or International Baccalaureate (IB) examinations, or by receiving credit at a college or university
that has transferred to CU. The information on this website is meant to guide you in making the decision on which math class to begin with at CU Boulder.
AP/IB Score Information
• Once you have your AP and/or IB exam scores, review the AP or IB Equivalency Chart and look for the exam you took to see if your score corresponds to a CU Boulder Course Equivalent.
• Don't know your AP or IB examination scores? For AP exams, you should receive an AP Score Report by mail that lists your cumulative AP Exam scores. You may also contact the College Board to
obtain your scores by telephone beginning July 1. For IB exams, results are sent out in July for the May session and in January for the November session. Students may also obtain their results
College Credit
• Sometimes first-year freshmen come to campus having earned college credit by taking college-level courses while enrolled in high school (see "Transfer of College-Level Credit").
• You may have done that by taking community college courses, courses at one of the CU campuses, or perhaps by completing a "CU Succeed" course (which shows up as CU Denver coursework on the CU
□ Did you know? The University of Colorado has a combined transcript. That means that courses taken at CU Boulder, along with the CU Denver and CU Colorado Springs campuses, all show up on the
CU transcript. So you may have established a CU transcript already if you took a CU Succeed class.
General Information About Math Courses
• Consider repeating the last math class you received credit for
□ Even with earned calculus credit many students find it helpful to repeat the last math class for which they have received college credit to make their transition to CU Boulder engineering
□ We recommend that you test yourself with final exams for the last course for which you received credit. This way you make sure you have sufficient mastery of the material before moving on to
the next level. These exams should be taken without the help of notes or calculators. Consult with your advisor once you're done, they can help you finalize your decision.
□ If you do choose to repeat a course for which you already have college credit, your previously earned college credit will not be counted. This also means you cannot go back and use that
previous grade since CEAS' repeated course policy allows only the most recent instiance of a course to count towards your degree. Again, it is recommended that students repeat their most
recent math class and it is important to remember that it will likely still be a challenging course even if you have seen some of the material before. Be sure to talk with your advisor if you
have questions about what this means for you!
• There may be a math class on your schedule before your enrollment window begins
□ If you are starting in the fall, The College of Engineering places a math course on your schedule before we recceive AP/IB/college transfer credit as a placeholder. You can change this course
when your enrollment window opens if needed.
□ Students starting in the spring or summer terms will not have a math course pre-loaded on their schedule. You should talk with your academic advisor about which course is right for you after
reviewing the materials on these pages.
• At CU Boulder, there are two departments which teach the Calculus sequences. One department is the Applied Math Department (APPM) and the other is the Math Department (MATH).
□ The College of Engineering accepts credit for Calculus 1 - 3 and various other math courses from both of these departments to apply towards degree requirements.
□ While students can be successful in either of these departments, the College of Engineering strongly recommends students take the Applied Math Calculus sequence, as these courses are designed
for the engineering curriculum.
Which Math Class Should You Take?
• If you have earned college credit for Calculus 1:
□ Take one or more
exams from the APPM 1350 Calculus 1 for Engineers course
MATH 1300 Calculus 1
to see if you fully understand the full range of material from the course. To accurately assess your skill and knowledge level do not refer to reference books, do not use a calculator, and
give yourself a 2.5 hour time limit.
☆ If you are fully comfortable with this material, enroll in APPM 1360 (Calculus 2 for Engineers) or MATH 2300 (Calculus 2).
☆ If you are not comfortable with the full range of material on the old exam, enroll in APPM 1350 (Calculus 1 for Engineers) or MATH 1300 (Calculus 1).
☆ If you are still unsure if you are ready for APPM 1360, send an email to
, one of our Applied Math Department faculty. She will send you an updated Calculus 1 final exam; you can complete the final exam, scan it, and email your work to her. She will then send
you a recommendation.
• If you have earned college credit for both Calculus 1 and Calculus 2:
□ Take one or more final exams from the APPM 1360 Calculus 2 for Engineers course and/or MATH 2300 Calculus 2, to see if you fully understand the material. To accurately assess your skill and
knowledge level do not refer to reference books, do not use a calculator, and give yourself a 2.5 hour time limit.
☆ If you are fully comfortable with the exam material, consult with your academic advisor to determine an appropriate math class. You should also send an email to silva.chang@colorado.edu
(one of our Applied Math Department faculty). She will send you an updated Calculus 2 final exam. You can complete the final exam, scan it, and email your work to her. You and your
advisor will then be sent a recommendation.
☆ If you are not comfortable with this material, enroll in APPM 1360 (Calculus 2 for Engineers) or MATH 2300 (Calculus 2). If you would like more of a refresher, you are alternatively
welcome to start in Calc. 1 and enroll in APPM 1350 (Calculus 1 for Engineers) or MATH 1300 (Calculus 1).
• If you have earned college credit for Calculus 1, Calculus 2, and Calculus 3 or Differential Equations:
More Questions?
If you have any questions about your choice of first semester math class you may contact your academic advisor or Professor Anne Dougherty (303-492-4011 or anne.dougherty@colorado.edu). | {"url":"https://www.colorado.edu/engineering-advising/students-calculus-credit-math-class-choices","timestamp":"2024-11-06T04:57:28Z","content_type":"text/html","content_length":"29391","record_id":"<urn:uuid:e489168a-5c22-4240-9392-e55203685564>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00881.warc.gz"} |
Fibonacci Recursion Javascript: Javascript ExplainedFibonacci Recursion Javascript: Javascript Explained
The Fibonacci sequence is an infinite sequence of integers that is often used to model natural phenomena. It can be seen in plants, sunflowers, and pinecones, and requires recursion to be understood
fully. With JavaScript, we can write our own functions and applications utilizing the Fibonacci sequence. This article explains how to create a Fibonacci sequence, use recursion in JavaScript to
produce brand new Fibonacci numbers, analyze its performance, and even use it to solve real world problems.
Understanding the Fibonacci Sequence
The Fibonacci sequence is a series of numbers generated from the mathematical formula Fn = Fn-1 + Fn-2. It begins with two seeds: F1 = 0, F2 = 1, and each subsequent number is the sum of the two
preceding numbers. This sequence forms an arithmetic progression, popularly known in math as the “golden ratio” and looks something like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and so on
and so forth.
The concept of using the sequence in real life began in the 13th century with the appearance of a particular problem-solving approach known as the Fibonacci spiral. This spiral made use of a
mathematical sequence to describe the pattern of growth in plants and organisms throughout nature. It did this by analyzing a series of numbers which would eventually lead to the development of the
famous “Fibonacci numbers.”
The Fibonacci sequence has been used in many different fields, from architecture to music. It is also used in computer science, where it is used to generate efficient algorithms. In addition, the
Fibonacci sequence is used in financial markets to predict stock prices and other market trends. The Fibonacci sequence is a powerful tool that can be used to understand the patterns of nature and
the world around us.
Creating a Fibonacci Function in Javascript
You can create your own Fibonacci sequence in JavaScript fairly easily. All you need is an initial value (often simply a 0 or a 1), a final value (often simply a 1 or a higher integer multiple of the
initial value), and a difference between successive numbers in the sequence. Below is an example of code which creates the beginning of a Fibonacci sequence in JavaScript.
var seq = 0; // initial valuevar sum = 1; // final valuewhile (seq <= sum) { console.log(seq); seq += sum; // difference between successive numbers sum += seq; // difference between successive
numbers }
The code will output integers starting from 0, then 1, then 2, then 3, and so on until it reaches the final value of 1. The loop will continue to add the difference between each number until it
reaches the specified final value. Thus, the code will produce a Fibonacci sequence in JavaScript.
Exploring Recursion in Javascript
An important concept in JavaScript is recursion. This technique is used to create new pieces of functionality from existing code or data structures. It involves creating a function which calls itself
within its own body with some modified input or parameters. The base case is when the function does not call itself anymore and can be reached with a simple if statement.
Recursion can be used to create the Fibonacci sequence in particular. Instead of looping until reaching a predetermined limit as with the while loop example above, you can simply write a recursive
function to achieve the same result. The following piece of code creates a similar Fibonacci sequence where each number is calculated from the sum of the previous two preceding numbers.
function recursiveFibonacci(n) { if (n <= 2) { return 1; } else { return recursiveFibonacci(n - 1) + recursiveFibonacci(n - 2); }}for (var i = 0; i < 10; i++) { console.log(recursiveFibonacci(i)); }
This code produces the following series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 which are the initial ten numbers of the Fibonacci sequence.
Using the Fibonacci Sequence to Solve Problems in Javascript
Now that we understand both basic implementation of the Fibonacci sequence as well as recursion to generate new sequences, we can begin to utilize this powerful technique to solve problems.
JavaScript libraries such as React have been developed specifically for solving problems utilizing Fibonacci sequences – allowing developers to craft solutions that are more efficient and require
less code than traditional implementations.
For example, let’s say we wanted to use React to generate a large list of Fibonacci numbers (up to 100). We could quickly create a React component which makes use of recursion as follows:
function FibonacciList() { const [result] = useState([]); const fibonacci = (n) => { if (n <= 2) { return 1; } else { return fibonacci(n - 1) + fibonacci(n - 2); } } useEffect(() => { for (let i = 0;
i < 100; i++) { result.push(fibonacci(i)); } }, []); return result; }
The above code will generate an array containing all the numbers that form the Fibonacci sequence up to 100.
Analyzing the Performance of Recursion in Javascript
The performance of a Fibonacci sequence relies on the speed which it can generate new numbers. Recursive functions can become computationally expensive when they are called frequently, as they may
perform more calculations than typical while loops or other iterations.
Since recursive functions call themselves within their own bodies, they tend to take up more memory resources than iterative functions. They also require larger amounts of space for storing
intermediate results before finally producing the required result. Furthermore, since iterative functions only need one iteration per call, they generally perform faster than recursive functions.
In order to optimize a recursive function’s performance, developers often suggest using memoization. This technique allows for the storing of results from past calls within a lookup table so that
future calls do not require the same calculations.
Applying the Fibonacci Sequence to Real-World Problems
Many real-world problems such as stock prices typically follow certain patterns which can be modeled using mathematical sequences such as the Fibonacci sequence. By understanding these patterns,
businesses can use them to make better decisions such as when to buy or sell stocks. This is known as technical analysis and can be applied directly with tools like React by utilizing Fibonacci
The same technique can also be used for evaluating user interfaces or predicting consumer behavior – allowing businesses to create better designs and improve customer satisfaction. By utilizing
Fibonacci sequences for data analysis, businesses can make informed decisions about how to structure their websites or apps and target specific audience segments.
In conclusion, we have explored how to use JavaScript to create our own custom Fibonacci sequences and explored recursions related to them. We looked at how recursion can help us create new sequences
in more efficient ways while being cognizant of its associated performance drawbacks. We explored how this powerful technique can be used to solve real-world problems such as technical analysis and
building better user interfaces. If you have yet to embrace recursions for your projects or business decisions, now is definitely a great place to start. | {"url":"https://bito.ai/resources/fibonacci-recursion-javascript-javascript-explained/","timestamp":"2024-11-08T05:07:30Z","content_type":"text/html","content_length":"377224","record_id":"<urn:uuid:52b2a7c2-1f22-4a7d-97ab-54df47340698>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00236.warc.gz"} |
Know More About Simple Bar Graph and Its Uses and Geometric Shapes - Schoollearners
Know More About Simple Bar Graph and Its Uses and Geometric Shapes
A simple bar graph is a graphical representation of a data set that is based only on a single variable. It is widely used to compare various items/observations/categories based on specific
parameters. Bar charts make it easier to compare things because you can analyze and interpret the data at a glance. In this article, we will discuss what a simple bar graph is, and its uses along
with geometric shapes.
What is a Simple Bar Graph?
A simple bar graph is a graphical representation of a given data set in the bars form. Bars are proportional to the size of the category they represent on the chart. The main purpose of the bar graph
is to compare quantities/items based on statistics. In a simple bar chart, you can compare based on only one parameter
There are two ways to draw a simple bar graph: vertical bar graph or horizontal bar graph. In a vertical bar chart, the bars are drawn vertically on the graph, while in a horizontal bar chart, the
bars are drawn horizontally on the graph
Simple bar graphs are made with the help of two axes, the x-axis, and the y-axis. In the case of a vertical simple bar chart, the category to be compared, or you can say that the fixed variable is
represented by the x-axis, and the size of the fixed variable is represented by the y-axis.
The bars extend vertically along the y-axis to a value proportional to the category they represent.
For a horizontal simple bar chart, the opposite is true, the y-axis represents the category, and the x-axis represents the size of the category. The bars are extended along the x-axis horizontally.
Geometric Shapes
A geometric shape is a figure that helps in interpreting an object. All objects of daily life exhibit a geometric shape or a combination of geometric shapes. If you understand geometric shapes, it is
easy to understand geometric concepts in daily life.
Geometry is an important branch of mathematics. To understand its advanced concepts, you will have to first understand basic geometric shapes. If you know the concepts of geometric shapes, it will
even help you solve questions of Mensuration.
What are Geometric Shapes?
A closed figure made by connecting two or more points, lines, or curves is called a geometric figure. For example, a triangle formed by connecting three lines is a geometric shape. Similarly, a
circle is a geometric shape that is made with the help of curves. The two examples discussed above are two-dimensional geometric shapes. This means that these figures have only one plane and two
dimensions. Geometric shapes can be divided into two categories according to their size.
Types of Geometric Shapes
According to the size characteristics, the geometric shape can be divided into two-dimensional geometry Shapes and three-dimensional geometric shapes.
2D Geometric Shapes
2D means two-dimensional, which refers to a figure with only one plane/surface. A figure with only two dimensions (length and width) is called a two-dimensional geometric shape. Simply put, the
graphics that can only be drawn on the surface of the pane (such as paper) are 2D graphics. They are made by connecting two points or lines to form different types of polygons or by connecting curves
to form circles or ellipses.
3D Geometric Shapes
3D means three-dimensional and refers to a figure with multiple planes/surfaces. Graphics with three dimensions (length, width, and height/thickness) are called 3D geometric shapes. In simple terms,
graphics that can exist as real-life objects are called 3D geometric shapes. They are represented by three axes (x-axis, y-axis, and z-axis). For example, a cube is a 3D geometric figure because it
has three dimensions, namely length, width, and height. In addition, the cube has 9 planes of symmetry, which is different from a square with only one plane and two dimensions (length and width).
We hope that you found this article on the simple bar graph and geometric shapes useful. | {"url":"https://www.schoollearners.com/know-more-about-simple-bar-graph-and-its-uses-and-geometric-shapes/","timestamp":"2024-11-10T05:30:36Z","content_type":"text/html","content_length":"104842","record_id":"<urn:uuid:53162e92-41cb-46be-9a2b-647e2d272576>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00464.warc.gz"} |
Convert hL to daL - Pyron Converter
Result: Hectoliter = Dekaliter
i.e. hL = daL
What does Hectoliter mean?
Hectoliter is a unit of Volume
In this unit converter website, we have converter from Hectoliter (hL) to some other Volume unit.
What does Dekaliter mean?
Dekaliter is a unit of Volume
In this unit converter website, we have converter from Dekaliter (daL) to some other Volume unit.
What does Volume mean?
Volume is the measurement of a quantity of the 3-dimensional space enclosed by a boundary or occupied by an object
How to convert Hectoliter to Dekaliter : Detailed Description
Hectoliter (hL) and Dekaliter (daL) are both units of Volume. On this page, we provide a handy tool for converting between hL and daL. To perform the conversion from hL to daL, follow these two
simple steps:
Steps to solve
Have you ever needed to or wanted to convert Hectoliter to Dekaliter for anything? It's not hard at all:
Step 1
• Find out how many Dekaliter are in one Hectoliter. The conversion factor is 10.0 daL per hL.
Step 2
• Let's illustrate with an example. If you want to convert 10 Hectoliter to Dekaliter, follow this formula: 10 hL x 10.0 daL per hL = daL. So, 10 hL is equal to daL.
• To convert any hL measurement to daL, use this formula: hL = daL x 10.0. The Volume in Hectoliter is equal to the Dekaliter multiplied by 10.0. With these simple steps, you can easily and
accurately convert Volume measurements between hL and daL using our tool at Pyron Converter.
FAQ regarding the conversion between hL and daL
Question: How many Dekaliter are there in 1 Hectoliter ?
Answer: There are 10.0 Dekaliter in 1 Hectoliter. To convert from hL to daL, multiply your figure by 10.0 (or divide by 0.1).
Question: How many Hectoliter are there in 1 daL ?
Answer: There are 0.1 Hectoliter in 1 Dekaliter. To convert from daL to hL, multiply your figure by 0.1 (or divide by 10.0).
Question: What is 1 hL equal to in daL ?
Answer: 1 hL (Hectoliter) is equal to 10.0 in daL (Dekaliter).
Question: What is the difference between hL and daL ?
Answer: 1 hL is equal to 10.0 in daL. That means that hL is more than a 10.0 times bigger unit of Volume than daL. To calculate hL from daL, you only need to divide the daL Volume value by 10.0.
Question: What does 5 hL mean ?
Answer: As one hL (Hectoliter) equals 10.0 daL, therefore, 5 hL means daL of Volume.
Question: How do you convert the hL to daL ?
Answer: If we multiply the hL value by 10.0, we will get the daL amount i.e; 1 hL = 10.0 daL.
Question: How much daL is the hL ?
Answer: 1 Hectoliter equals 10.0 daL i.e; 1 Hectoliter = 10.0 daL.
Question: Are hL and daL the same ?
Answer: No. The hL is a bigger unit. The hL unit is 10.0 times bigger than the daL unit.
Question: How many hL is one daL ?
Answer: One daL equals 0.1 hL i.e. 1 daL = 0.1 hL.
Question: How do you convert daL to hL ?
Answer: If we multiply the daL value by 0.1, we will get the hL amount i.e; 1 daL = 0.1 Hectoliter.
Question: What is the daL value of one Hectoliter ?
Answer: 1 Hectoliter to daL = 10.0. | {"url":"https://pyronconverter.com/unit/volume/hL-daL","timestamp":"2024-11-05T10:14:06Z","content_type":"text/html","content_length":"106757","record_id":"<urn:uuid:9272aea4-f474-4286-a229-a8d75d1f30ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00304.warc.gz"} |
What is Sumproduct in Excel?
What is Sumproduct in Excel?
Are you looking to take your Microsoft Excel skills to the next level? Do you want to know what sumproduct is and how to use it to boost your productivity? Sumproduct is an incredibly powerful
formula in Excel that can be used to quickly calculate complex data sets. In this article, we’ll explain what sumproduct is, when to use it, and how to incorporate it into your spreadsheets. So if
you’re ready to take your Excel skills to the next level, read on to learn about sumproduct.
Sumproduct in Excel is a formula that multiplies range or array of cells and returns the sum of the products. It is used when you want to calculate the total sum of two or more lists of numbers. It
can also be used to count the number of cells within a range that meet a certain criteria. It is a powerful function for summarizing data with multiple criteria.
What is Sumproduct Function in Excel?
The SUMPRODUCT function in Excel is a powerful calculation tool that multiplies ranges or arrays of numbers and then returns the sum of the products. It is an array function, which means it can
handle arrays of data without the need for additional functions, like SUMIF or SUMIFS. The syntax of the function is SUMPRODUCT(array1, array2, ,…). The function can take up to 254 arrays, which
makes it useful for analyzing large datasets.
The SUMPRODUCT function can be used to calculate the sum of products (or sums of products) of two or more arrays. For example, it can be used to calculate the total sales of a product across multiple
regions. It can also be used to calculate the total cost of a project when both the cost and the quantity of items purchased are known.
The SUMPRODUCT function can be used in place of SUMIF and SUMIFS when dealing with multiple criteria. For example, it can be used to calculate the total cost of a project that meets multiple
criteria, such as quantity and price. It can also be used to calculate the total cost of a project that meets multiple criteria, such as quantity and price, while ensuring that the total cost does
not exceed a certain budget.
Advantages of Using SUMPRODUCT Function in Excel
The SUMPRODUCT function is a powerful and efficient tool for working with large datasets in Excel. It is easy to use and can be used to calculate sums of products of multiple arrays quickly and
accurately. The function can also be used to calculate sums of products with multiple criteria, such as quantity and price, while ensuring that the total cost does not exceed a certain budget.
Another advantage of the SUMPRODUCT function is that it can be used to calculate the sum of products with multiple criteria, such as quantity and price, while avoiding the need to use multiple
functions, such as SUMIF and SUMIFS. This helps to reduce the complexity of the formula and makes it easier to understand.
The SUMPRODUCT function is a versatile tool that can be used in a variety of scenarios. For example, it can be used to calculate the total cost of a project when both the cost and the quantity of
items purchased are known. It can also be used to calculate the total sales of a product across multiple regions.
How to Use the SUMPRODUCT Function in Excel?
The SUMPRODUCT function is used to calculate the sum of products of two or more arrays. To use the function, the syntax SUMPRODUCT(array1, array2, ,…) is used, where array1, array2, and ,… are the
arrays to be multiplied. The function can take up to 254 arrays, which makes it useful for analyzing large datasets.
The SUMPRODUCT function can be used in a variety of scenarios. For example, it can be used to calculate the sum of products of two or more arrays. It can also be used to calculate the total sales of
a product across multiple regions. Additionally, it can be used to calculate the total cost of a project when both the cost and the quantity of items purchased are known.
Examples of Using SUMPRODUCT Function in Excel
The SUMPRODUCT function can be used in a variety of scenarios. For example, it can be used to calculate the sum of products of two or more arrays. To calculate the total sales of a product across
multiple regions, the following formula can be used:
The formula above multiplies the values in range A1:A10 with the values in range B1:B10 and then returns the sum of the products.
The SUMPRODUCT function can also be used to calculate the total cost of a project when both the cost and the quantity of items purchased are known. To calculate the total cost of a project, the
following formula can be used:
The formula above multiplies the values in range C1:C10 with the values in range D1:D10 and then returns the sum of the products.
Limitations of Using SUMPRODUCT Function in Excel
The SUMPRODUCT function is a powerful and efficient tool for working with large datasets in Excel. However, it has some limitations. For example, the function cannot calculate sums of products with
multiple criteria, such as quantity and price, while ensuring that the total cost does not exceed a certain budget.
Additionally, the function cannot be used to calculate sums of products of more than 254 arrays. This can be a limitation when dealing with large datasets.
Finally, the function cannot be used to calculate sums of products with multiple criteria, such as quantity and price, while avoiding the need to use multiple functions, such as SUMIF and SUMIFS.
This can be a limitation when dealing with complex formulas.
Frequently Asked Questions
What is Sumproduct in Excel?
Sumproduct is a powerful Excel function that allows users to calculate the sum of products of two or more arrays of numbers. It can be used to calculate the sum of products of two or more columns or
rows of data in a worksheet. It is a versatile function that can be used to calculate a variety of things such as the average of columns or rows of data, the sum of products of columns of data, or
the sum of products of rows of data.
What are the basic components of Sumproduct?
The basic components of Sumproduct include the function itself, the range of cells, the array of cells, and the criteria. The function itself is what provides the calculation. The range of cells is
the area of the worksheet where the calculation will be performed. The array of cells is the data that will be used in the calculation. The criteria is the additional criteria that can be specified
for the calculation, such as specific values or ranges of values.
What are some examples of how Sumproduct can be used?
Sumproduct can be used to calculate the sum of products of two or more columns or rows of data. It can also be used to calculate the average of columns or rows of data, the sum of products of columns
of data, or the sum of products of rows of data. It can also be used to calculate the sum of products of specific values or ranges of values.
How do I use Sumproduct in Excel?
Using Sumproduct in Excel is relatively simple. First, enter the function into a cell in the worksheet. Then, specify the range of cells that will be used in the calculation. After that, specify the
array of cells that will be used in the calculation. Finally, specify any additional criteria that should be used in the calculation, such as specific values or ranges of values.
What are the advantages of using Sumproduct in Excel?
Using Sumproduct in Excel has a variety of advantages. It is a versatile function that can be used to calculate a variety of things such as the average of columns or rows of data, the sum of products
of columns of data, or the sum of products of rows of data. It is also relatively simple to use and can save users time by allowing them to quickly calculate the desired result.
What are the disadvantages of using Sumproduct in Excel?
The main disadvantage of using Sumproduct in Excel is that it can be difficult to debug errors that may occur in the calculation. Additionally, it can also be difficult to understand the results of
the calculation if it is not well documented. Finally, if the range of cells or the array of cells used in the calculation is incorrectly specified, the results may be incorrect.
In conclusion, SUMPRODUCT is one of the most powerful and versatile functions in Excel. It allows users to quickly calculate the sum of products of arrays or ranges of numbers, making it ideal for
complex calculations. With SUMPRODUCT, users can easily and quickly analyze data, allowing them to make informed decisions with confidence. | {"url":"https://keys.direct/blogs/excel/what-is-sumproduct-in-excel","timestamp":"2024-11-01T19:57:37Z","content_type":"text/html","content_length":"359298","record_id":"<urn:uuid:56d6c703-d6ab-442b-a580-52ad8e39aece>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00472.warc.gz"} |
Abstract figures | Series Of Figures Assessment | OYA Aptitude test
Abstract reasoning – figures
Abstract tests will almost always be a part of your assessment. They are classics and they have the advantage that language problems are not relevant. Even though you will not have to solve figures
in your work, the capacity that is measured is relevant for many positions. After all, separating primary and secondary issues is important for many functions.
The number of figures that make up a figure problem may differ. The following sample exercises consist of ten figures. The first four figures are all the same in a certain respect (the problem
figures) and two of the bottom six figures (the answer options) match with the top four figures. They have one or more things in common.
Take a look at example 1 below. The top four figures are all square. Of the bottom six figures, only (a) and (f) are also square. Squares are the common element in this problem. You could say that
the shape of the figures provides the solution here. Try to solve examples 2 and 3. First, look at the top four figures. Look for the way in which they are the same and try to pick the TWO other
figures from the bottom six figures.
Explanation of example 2
The top four figures are all shaded in the same way. This is also the case for (b) and (e). The other figures are darker than the top four figures.
Explanation of example 3
The top four figures all consist of a round/oval shape touched by a straight line. The line segment does not pass through the circle/oval. This is also the case with (c) and (f). These are the
correct answers.
Tips for figure problems
As you can see, the common elements of the problems can be shown in various ways. You will find that with a little creativity you can think of many common traits. A good exercise is to create a
figure assignment yourself. Below are some characteristics that are important to look at in figures, so that you can quickly select the right answers. If you have too many choices, you know that this
is not the unique characteristic, because only two answer options remain. Take a good look at the traits below in order to become familiar with the different types of common characteristics.
• Shapes lend themselves very well to having traits in common.
• Similar relationships. Are line segments parallel to each other or do they cross each other?
• Number of figures/lines/elements.
• Dominance. Some figures block out less dominant ones.
• Closed and open figures. Where is the opening with open figures?
• Colors and shading. Is it really the same for both or does it just appear that way?
• Even and odd, e.g. the points of a star.
• Length of the figures.
• Location in the space (i.e. the square). Are they all in the center or actually at the top-right?
• Corners of figures. Do they contain a right angle (90 degrees) or not?
• Some complex figures are only rotated a few degrees, but they are exactly the same. Look carefully at the unique characteristics of complex figures.
• Continuous or broken lines.
• Dimensions. Large or small figures.
• Right, acute, or obtuse angles.
• Curves in the figures or lack thereof.
An important mistake that you can make with this test is looking for a certain movement or sequence. A number series is all about sequences, but this type of problem is about specific common traits.
The challenge here is to look at the big picture. By becoming familiar with the possible shared traits mentioned above, you are less distracted by the “noise” and irrelevant, misleading figures. If
you look at the common elements, you will see the following in the four squares provided in the problem below:
• 1. A five-pointed star, the same in shape and color
• 2. A five-pointed star, different in size
• 3. A five-pointed star, in different locations
• 4. Each star has two-line segments in the square
• 5. Each star has two equally long line segments in the square
• 6. Each star has line segments that are parallel to each other (they never intersect)
• 7. The two-line segments always touch or intersect the star
The above observations lead to two correct answers and because you clearly understand the rules that apply, you can also effectively argue why the other options are not correct.
• Answer option A is not correct, because it does not share traits 5 and 7.
• Answer option B is not correct, because it does not share trait 7.
• Answer option C is not correct, because it does not share trait 7.
• Answer option D is not correct, because it does not share trait 5.
E and F are the correct answers when you consider what the figures have in common. You can see that the test engineer still has a lot of options when creating different problems, e.g. by varying the
shape of the star.
Practice makes perfect!
It is very important to practice for a capacity test. If you do not practice, your score may be lower and this often decreases your chances of getting that much-desired job! By practicing, you can
solve problems more quickly and efficiently, so that your score will increase.
Get started immediately with exercises for figures and many other tests on:
Add to cart €14,95 | {"url":"https://oya-aptitude-test.com/abstract-reasoning-test/","timestamp":"2024-11-08T11:54:46Z","content_type":"text/html","content_length":"76335","record_id":"<urn:uuid:f5928367-270e-4695-a40f-e111f41f5381>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00056.warc.gz"} |
Rearranging Example | Models University
For our forecast periods (2023 onwards), we want to calculate Revenues (1) as an output, so we start with a simple calculation, Revenues = Widgets Sold * Price (2).
We set up our Widgets sold Variable so that we need to enter Assumptions for 2021 and 2022 (3), and from 2023 onwards Widgets sold will be equal to the previous period (4). However, we don't know how
many Widgets were sold in 2021 and 2022, so we cannot initially calculate Revenues, meaning we have blank values in our output (5).
Entering Actual Revenues
We know what our revenues were for 2021 and 2022, so we create a new Time Segment in our Revenues Variable (6), with an Assumptions formula type, and add our historical revenues data to this Time
Models now sees that for 2021 and 2022, it is expecting input data for all three Variables, Revenues, Widgets sold and Price. This is not a valid state, as in any Calculation there must be exactly
one unknown Variable to be calculated, otherwise there is the potential for conflicts (i.e. we could enter values for the three Variables that do not match the Calculation Revenues = Widgets sold *
To rectify this, Models displays a message on the first Time Segment of Revenues (7), warning that a "Child needs to be rearranged for this segment". We can read this as saying that one of the input
Variables, Widgets Sold or Price needs to become the output of the Calculation to maintain logical consistency.
We are offered a choice of child Variables to rearrange. We need to select the Variable that we want to be used as the new output of the Time Segment. In this case, we know the price, but we don't
know the number of widgets sold, so we select Widgets sold as the new output (9).
This results in the below state, where we no longer have warnings in our Variables. The first Time Segment for Widgets sold has been rearranged, with a Calculation of Revenues / Price (10).
Widgets sold is now calculated from Revenues and Price for 2021 and 2022, and is static thereafter.
Note that the Values table visually indicates the change in calculation method with a border to the right of the 2022 period (11). | {"url":"https://help.taglo.io/rearranging/rearranging-example","timestamp":"2024-11-10T06:06:42Z","content_type":"text/html","content_length":"352439","record_id":"<urn:uuid:5fa66036-596c-4818-a90b-f8d50931e189>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00035.warc.gz"} |
American Mathematical Society
Uncorrelatedness sets for random variables with given distributions
HTML articles powered by AMS MathViewer
Proc. Amer. Math. Soc. 133 (2005), 1239-1246
DOI: https://doi.org/10.1090/S0002-9939-04-07698-1
Published electronically: October 18, 2004
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Let $\xi _1$ and $\xi _2$ be random variables having finite moments of all orders. The set \[ U(\xi _1,\xi _2):=\left \{(j,l)\in \textbf {N}^2:\textbf {E}\left (\xi _1^j\xi _2^l\right )=\textbf {E}\
left (\xi _1^j\right )\textbf {E}\left ( \xi _2^l\right )\right \}\] is said to be an uncorrelatedness set of $\xi _1$ and $\xi _2.$ It is known that in general, an uncorrelatedness set can be
arbitrary. Simple examples show that this is not true for random variables with given distributions. In this paper we present a wide class of probability distributions such that there exist random
variables with given distributions from the class having a prescribed uncorrelatedness set. Besides, we discuss the sharpness of the obtained result. References
• Viktor Beneš and Josef Štěpán (eds.), Distributions with given marginals and moment problems, Kluwer Academic Publishers, Dordrecht, 1997. MR 1614650, DOI 10.1007/978-94-011-5532-8
• E. W. Cheney, Introduction to approximation theory, AMS Chelsea Publishing, Providence, RI, 1998. Reprint of the second (1982) edition. MR 1656150
• C. M. Cuadras, Theoretical, experimental foundations and new models of factor analysis, Investigación Pesquera, 39 (1972), pp. 163-169 (in Spanish)
• C. M. Cuadras, First principal component characterization of a continuous random variable, Universitat de Barcelona, Institut de Matemàtica, Mathematics Preprint Series, No 327 (2003)
• G. Dall’Aglio, S. Kotz, and G. Salinetti (eds.), Advances in probability distributions with given marginals, Mathematics and its Applications, vol. 67, Kluwer Academic Publishers Group,
Dordrecht, 1991. Beyond the copulas; Papers from the Symposium on Distributions with Given Marginals held in Rome, April 1990. MR 1215942, DOI 10.1007/978-94-011-3466-8
• W. Feller, An Introduction to Probability Theory and Its Applications. Wiley, New-York, 1986
• Jacek Jakubowski and Stanisław Kwapień, On multiplicative systems of functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 9, 689–694 (English, with Russian summary). MR 600722
• J. Komlós, A central limit theorem for multiplicative systems, Canad. Math. Bull. 16 (1973), 67–73. MR 324753, DOI 10.4153/CMB-1973-014-3
• Paul Koosis, The logarithmic integral. I, Cambridge Studies in Advanced Mathematics, vol. 12, Cambridge University Press, Cambridge, 1988. MR 961844, DOI 10.1017/CBO9780511566196
• Ju. V. Linnik and Ĭ. V. Ostrovs′kiĭ, Decomposition of random variables and vectors, Translations of Mathematical Monographs, Vol. 48, American Mathematical Society, Providence, R.I., 1977.
Translated from the Russian. MR 0428382
• Sofiya Ostrovska, Uncorrelatedness and correlatedness of powers of random variables, Arch. Math. (Basel) 79 (2002), no. 2, 141–146. MR 1925381, DOI 10.1007/s00013-002-8296-z
• Sofia Ostrovska, A scale of degrees of independence of random variables, Indian J. Pure Appl. Math. 29 (1998), no. 5, 461–471. MR 1627847
• Ludger Rüschendorf, Berthold Schweizer, and Michael D. Taylor (eds.), Distributions with fixed marginals and related topics, Institute of Mathematical Statistics Lecture Notes—Monograph Series,
vol. 28, Institute of Mathematical Statistics, Hayward, CA, 1996. MR 1485518
• Jordan Stoyanov, Dependency measure for sets of random events or random variables, Statist. Probab. Lett. 23 (1995), no. 1, 13–20. MR 1333372, DOI 10.1016/0167-7152(94)00089-Q
• Jordan M. Stoyanov, Counterexamples in probability, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1987. MR
• Jordan Stoyanov, Krein condition in probabilistic moment problems, Bernoulli 6 (2000), no. 5, 939–949. MR 1791909, DOI 10.2307/3318763
• Jordan Stoyanov, Sets of binary random variables with a prescribed independence/dependence structure, Math. Sci. 28 (2003), no. 1, 19–27. MR 1995161
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 60E05
• Retrieve articles in all journals with MSC (2000): 60E05
Bibliographic Information
• Sofiya Ostrovska
• Affiliation: Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey
• MR Author ID: 329775
• Email: ostrovskasofiya@yahoo.com
• Received by editor(s): September 22, 2003
• Received by editor(s) in revised form: December 22, 2003
• Published electronically: October 18, 2004
• Communicated by: Richard C. Bradley
• © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
• Journal: Proc. Amer. Math. Soc. 133 (2005), 1239-1246
• MSC (2000): Primary 60E05
• DOI: https://doi.org/10.1090/S0002-9939-04-07698-1
• MathSciNet review: 2117227 | {"url":"https://www.ams.org/journals/proc/2005-133-04/S0002-9939-04-07698-1/?active=current","timestamp":"2024-11-03T10:44:49Z","content_type":"text/html","content_length":"67317","record_id":"<urn:uuid:316e4091-822b-46c9-9c7d-d68ba5d3ea76>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00630.warc.gz"} |
EPSRC Reference: EP/X01276X/1
Title: Canonical Singularities, Generalized Symmetries, and 5d Superconformal Field Theories
Principal Investigator: Schafer-Nameki, Professor S
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematical Institute
Organisation: University of Oxford
Scheme: EPSRC Fellowship
Starts: 01 January 2023 Ends: 31 December 2027 Value (£): 1,298,400
EPSRC Research Topic Classifications: Algebra & Geometry Mathematical Physics
EPSRC Industrial Sector Classifications: No relevance to Underpinning Sectors
Related Grants:
│Panel Date │Panel Name │Outcome │
Panel History: │18 Jul 2022│EPSRC Mathematical Sciences Fellowship Interviews July 2022 │Announced│
│24 May 2022│EPSRC Mathematical Sciences Prioritisation Panel May 2022 │Announced│
Summary on Grant Application Form
Supersymmetric Quantum Field Theories (QFTs) have a rich and intricate structure, which has made them a firm part of mathematics since the works of Seiberg and Witten in physics, and Donaldson in
mathematics. In this research program, I will study a particularly challenging class of QFTs, which in addition to supersymmetry have also scale invariance -- so-called superconformal field theories.
These are often relatively well-understood in lower dimensions, with exciting recent results in both mathematics and physics in 3d. In higher dimensions such as 5d or 6d such theories become
essentially impossible to study from the point of view of standard methods: they are intrinsically strongly interacting, and thus not accessible by perturbation theory.
In this project I will develop the geometric approach to 5d superconformal field theories, using their definition in string theory. In a nutshell, string theory provides a map from a class of
singular geometries (canonical Calabi-Yau three-fold singularities) to 5d superconformal field theories. It is a remarkable implication of string theory, that using this approach it is possible to
study these strongly-coupled theories, as well as their parameter spaces and symmetries.
Parameter (or moduli) spaces of 5d SCFTs have a particularly beautiful mathematical description -- they are conical, singular, but have a foliation structure in terms of smaller dimensional singular
spaces (so-called symplectic singularities). Developing the map from canonical singularities, which define the 5d superconformal field theory, to these singular moduli spaces, is one of the exciting
mathematical connections that this project will entail.
Symmetries are of course central in any physical system, starting with the works of Emmy Noether. Recent years have uncovered a new notion of symmetry, where the charged objects are not point-like,
but extended operators, which are higher dimensional. These so-called higher-form symmetries have far-reaching implications: they may not form a group, but a higher-group, which is a higher category,
that can act on QFTs. In the context of 5d superconformal field theories, these symmetry structures will be studied, and in turn encoded in the defining canonical singularity. Thus, this project will
also provide a new, exciting link between geometry of singularities, and higher categorical structures. These developments will have implications not only for 5d SCFTs, but related field theories in
lower dimensions, and will have connections to recent developments in symmetries of condensed matter systems alike.
In summary, this program has the goal to provide a classification of 5d superconformal field theories in terms of canonical three-fold singularities, including a characterization of their quantum
Higgs branch moduli space and their generalized symmetries. This project touches upon quantum field theory at a fundamental level, where it is challenged in terms of its conventional definition as a
canonical quantization of a classical theory and perturbation theory. It provides a definition of a class of quantum fields which are strongly interacting, by means of a purely geometric approach.
There are exciting connections and implications to algebraic geometry and the classification of canonical singularities, as well as to algebraic topology where the generalized symmetries have a
natural description.
Within this project I will have two postdocs, who will bring in complementary expertise, to develop the geometric and topological/categorical aspects of the project. The project is intrinsically
drawing from a variety of different mathematical sub-fields, and a strong team, with expertise in these two central areas of geometry and algebraic topology will be pivotal for its success.
I will host a large research conference in year 4 of the grant, and thereby bring the main experts at the interface of String Theory and Mathematics to the UK.
Key Findings
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Potential use in non-academic contexts
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Description This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Date Materialised
Sectors submitted by the Researcher
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
Project URL:
Further Information:
Organisation Website: http://www.ox.ac.uk | {"url":"https://gow.epsrc.ukri.org/NGBOViewGrant.aspx?GrantRef=EP/X01276X/1","timestamp":"2024-11-08T02:59:59Z","content_type":"application/xhtml+xml","content_length":"28556","record_id":"<urn:uuid:5c363b49-79de-45da-8a31-5e73b0d50e17>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00216.warc.gz"} |
Matlab Online Tutorial - 16 - Trigonometric Functions and their Inverses
Get more lessons like this at http://www.MathTutorDVD.com Learn how to work with trig functions in matlab such as sine, cosine, tangent, secant, cosecant, and cotangent. We will learn how to use
these trigonometry functions and their inverses for calculations. | {"url":"https://videos.mathtutordvd.com/detail/video/PVUXL5gzHpA/matlab-online-tutorial---16---trigonometric-functions-and-their-inverses","timestamp":"2024-11-10T15:12:39Z","content_type":"text/html","content_length":"141392","record_id":"<urn:uuid:7e10c2e3-a8b5-4c9a-9c6b-5774838309e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00457.warc.gz"} |
February 2024
Personal tasks of this week:
Task: Preliminary Circuit Simulation:
Definition: According the Sergey’s thesis, the theoretical circuit primitives were found to model the mathematical constraint. However, the circuits need to be modified to realize the theoretical
circuit, as it contains ideal circuits that don’t exist in the real world.
Completion: The task is partially completed. The circuit was researched and equivalent components from the thesis that can be sourced now were found for each component. The circuit was attempted
first in PSpice, but decided to transition into using LTSpice, as we realized the speed of learning and prototyping ability of LTSpice was much more important given our time constraints compared to
the features available on PSpice. A basic schematic was built for each primitive, but has not been tested extensively for the simple optimization circuit
Next Steps:
The next steps are to use PSpice to create the simulation of the simple optimization circuit and verify if the minimum energy state converges to the expected solution.
Overall progress assessment:
My progress is slightly behind-schedule, as I still need to finish up the optimization circuit, but it should not be that difficult to catch up, as the components have already been selected and
Team Status Report for Feb 24
Significant risks and risk management:
Risk: The analog circuit can’t meet the required tolerances
Definition: This risk has been brought to our attention by our assigned in instructor Thomas Sullivan (Thanks!). It is currently unclear to us what tolerance the analog components will need to have
in order to satisfy the 10% solution accuracy required by the accuracy requirement (NR3). This is nontrivial to determine as the accuracy of the whole circuit can’t be easily associated with the
accuracy of individual components.
Severity: If the analog circuit can’t meet the required tolerances, the progress of the whole project would be severely jeopardized, because the accuracy requirement (NR3) would not be satisfied.
Resolution: A solution is to use Sergey’s work [1] as a reference. If our components are more accurate than Sergey’s components in every relevant measure, it is likely that the whole circuit wouldn’t
be significantly worse in accuracy compared to his circuit. This is possible because more accurate components are available since his work was published.
Current progress:
We have completed P2, and is currently working towards P1 and P3. We performed some benchmarking on P2, with results showing that the majority of the time spent in the solver is on the QP subproblem
(32ms out of 38ms). This means that if we solve the QP subproblem with our analog solver we can potentially get large speed up.
Changes to the existing design:
There are no changes to the existing design. We have completed P2 successfully, indicating that we can proceed with our existing design.
Changes to the project schedule:
Similarly, there are no significant changes to the project schedule.
Alvin Zou’s Status Report For 2/24
Personal tasks of this week:
Task: Profiling double pendulum swing up with NMPC
Definition: Analyzing results from the double pendulum swing up and performing performance profiling.
Completion: I have conducted some profiling on the performance of the NMPC controller, with results showing that most of the time spent in the pipeline is on solving the QP subproblem. This confirms
the hypothesis that we can theoretically achieve a lot of speed up if the QP subproblem is solved using our analog solver.
Task: Design review presentation
Definition: Presenting the design review presentation
Completion: The presentation is completed.
Task: Formulating NMPC problem and isolating the QP subproblem
Definition: Formulating the NMPC problem for our system mathematically and converting it to a form that can be solved by SQP. After this is achieved, the QP subproblem can then be isolated. This step
is to calculate what variables are used in the QP subproblem, so that the analog circuit can be designed to solve the correct subproblem.
Completion: The state update equations have been formulated. Next is converting the equations into a form that can be solved by SQP and then isolating the QP subproblem.
Next Steps:
The next step is to continue deriving the mathematical representation of the NMPC problem to eventually isolate the QP subroutine in preparation for replacing it with our analog solver.
Overall progress assessment:
Currently on schedule. Working towards isolation of the digital and analog solver.
Thomas’s Status Report for Feb 24
Personal tasks of this week:
Task: Library-Based NMPC Swing-Up Controller Benchmarking
Definition: Run a performance benchmark on the library-based NMPC swing-up controller, measuring the time it takes to complete each step of solving the NMPC problem, and identify the performance
bottleneck of the system.
Completion: The task is completed. We benchmarked the swing-up controller and found that 32ms out of 38ms in each iteration is spent on solving the NLP, which can be accelerated using our proposed
analog SQP solver to achieve significant speedup.
Task: Symbolic NLP Formulation of the NMPC Swing-Up Controller
Definition: Symbolically formulate the NMPC Swing-Up controller as an NLP, which acts as the NLP problem we supply to the analog SQP solver on each iteration.
Completion: The task is partially completed. We’ve made substantial progress in deriving the dynamics constraints and symbolic cost function. More work will be done in the next week.
Next Steps:
My next step is to continue working on formulating a symbolic NLP problem of the NMPC swing-up controller. I will also continue to work with Andrew on analog component selection and circuit
Overall progress assessment:
My progress is on-schedule, as all of my tasks this week have been completed or partially completed.
Team Status Report for Feb 17
Significant risks and risk management:
Risk: Inverted double pendulum simulation has insufficient numerical accuracy for NMPC
Definition: Since running the NMPC controller might require a higher numerical accuracy than simulating the dynamics of the inverted double pendulum system, the numerical accuracy of the system might
be insufficient when NMPC is integrated into the symbolic simulation.
Severity: If the numerical accuracy is insufficient, the progress of the whole project would be severely jeopardized, because the system would likely not satisfy NR3 (accuracy).
Resolution: We will try to mitigate the risk by synthesizing a NMPC controller that has sufficient robustness against noise, because numerical inaccuracies can usually be seen as noise from the
controller’s perspective.
Changes to the existing design:
There are no significant changes to the existing design as we’ve just started implementation.
Changes to the project schedule:
Similarly, there are no significant changes to the project schedule
Additional Societal Impacts
Part A)
Optimization is a powerful tool of representing real-world problems mathematically, and fast solvers, like the product we aim to deliver, make solving them easier.
Optimization helps make healthcare more efficient and accessible. For example, finding the most efficient distribution of vaccines during a pandemic can be formulated as an optimization problem.
Solving this problem quickly can enable more effective control of infectious diseases.
Optimization helps make vehicles safer to operate. Vehicle control with safety guarantees is a classic application of model predictive control (MPC), which is an optimization problem. Our solver has
the potential of solving MPC problems in real-time, which can make vehicle control strategies with stronger safety guarantees practical.
Optimization helps distribute scarce resources efficiently. Logistics is another classic application of optimization, which is critical for fulfilling everyday needs, especially during periods of
hardship. Solving these problems efficiently ensures that goods and services are delivered where they are most needed.
Part B)
Our analog SQP solver has potentially wide societal implications. This comes from the fact that SQP solvers are used to solve nonlinear optimization problems, which is general class of problems with
extensive use cases and applications, spanning from computer science to machine learning to biology and medicine(examples can be found in https://neos-guide.org/case-studies/). Because of the breadth
of possible applications, our product solution can potentially affect a large group of people.
In particular for social groups, solving optimization problems can help make management in large organizations more efficient. One example is in the supply chain industry, where creating a supply
chain strategy that operates across multiple regions can be formulated into an optimization problem.
Part C)
Our applications in analog optimization have significant implications in terms of the related economic factors. Specifically, this is because each of the systems relating to the production,
distribution, and consumption of goods and services are areas that can be optimized to be more efficient, which leads to savings in terms of time and money. Furthermore, optimization enhances the
general productivity of our society by making production procedures faster, cleaner, safer, and more efficient. As a result, a productive society leads to more sustainable development and more safer
*In the team report, A was written by Thomas Liao, B was written by Alvin Zou and C was written by Andrew Chong.
Andrew’s Status Report for Feb 17
Personal tasks of this week:
Task: Trade Studies on PCB Manufacturing/Assembly
Definition: There were many considerations in choosing a PCB Manufacturing/Assembly company to manufacture our product. This includes quality, turnaround time, ability to assemble the PCB components,
and cost. We considered the following companies: JLCPCB, PCBWay, OshPark, and Colorado PCB Assembly.
Completion: The task is completed. From our trade study, we were able to determine that PCBWay was the best manufacturer for us. This is because PCBWay generally has good quality, has the ability to
both PCB manufacturing and Assembly together, has turnaround + shipping in approximately a week, and offers large discounts if the Assembly service is used. In fact, we found that if the Assembly
service is used, the price is actually cheaper than buying a stencil and manually reflowing it ourselves, which can introduce a source of error. Some reasons why we didn’t pick the other companies
are as follows: JLCPCB has poor quality, OshPark can only manufacture the PCBs, and Colorado PCB Assembly, while with very high quality and fast turnaround, only offers assembly. While it is possible
to combine multiple options, we thought that with the component sourcing to different parts, as well as PCB shipping from the PCB manufacturer to the Assembly, back to us would be worse both
logistically and for our schedule.
Task: Preliminary Design for Optimization Prototype
Definition: To ensure that we are able to complete the final prototype of the analog optimization circuit, a smaller, proof-of-concept prototype must be made in order to ensure that our approach is
Completion: The task is completed. Looking through Sergey’s Thesis, we found that there is a schematic provided for a simple 2-variable equality constraint, a 2-variable inequality constraint, and a
cost function constraint (which is just a series of resistors) that we can use as our basis for our preliminary prototype. However, the problem is only for a 2-variable equality constraint. We
decided to make a slightly more complex design and make the simplest possible optimization circuit with one equality constraint, one inequality constraint, and one cost function.
Next Steps:
The next steps are to use the tools selected to create a prototype of our simple optimization problem, as detailed in Sergey’s thesis as a proof of concept, then Once the baseline works, we will
begin expanding that into creating our first prototype analog circuit that specifically models our double pendulum swingup optimization problem. Once that is done, we will order using PCBway.
Overall progress assessment:
My progress is on-schedule, as all of my tasks this week have been completed.
Alvin Zou’s Status Report For 2/17
Personal tasks of this week:
Task: Double pendulum swing up with NMPC
Definition: Building Prototype 2 by creating a double pendulum system, designing a controller to control the system, and visualizing the simulation.
Completion: Currently, I have created the system and designed a controller with Thomas using do-mpc that can successfully swing up the pendulum.
Task: Slides for design review presentation
Definition: Creating the slides for the design review next week with the rest of the team. Also rehearsing for the presentation.
Completion: The presentation is mostly completed. Aim to complete by Sunday.
Next Steps:
The completion of P2 has validated the problem can be solved using a software SQP solver. The next step is then to isolate the QP subroutine in preparation for replacing it with our analog solver.
Basic profiling should also done on P2 to collect performance data.
Overall progress assessment:
Currently on schedule. P2 is completed.
Thomas’s Status Report for Feb 17
Personal tasks of this week:
Task: Library-Based NMPC Swing-Up Controller Synthesis
Definition: Using an open source library, synthesize a NMPC controller that is capable of swinging up an inverted double pendulum. The purpose of this task is to validate that swinging up the
inverted double pendulum is achievable with NMPC with small prediction and control horizons. The implementation must be in Python in order to integrate with the rest of the project.
Completion: The task is completed. We’ve selected do-mpc [1] as our library due to its ease of use. The synthesized NMPC controller is capable of consistently swinging up an inverted double pendulum
with a prediction horizon of 3 and a control horizon of 3. However, we will synthesize a controller with longer predilection and control horizons for better robustness against noise.
Next Steps:
My next step is to integrate the NMPC swing-up controller with the symbolic dynamics model, and derive a symbolic NLP problem that corresponds to the NMPC controller. Then I will work on solving the
NLP problem by implementing the SQP algorithm, calling an open-source QP solver.
I will also work with Andrew on analog component selection and circuit simulation.
Overall progress assessment:
My progress is on-schedule, as all of my tasks this week have been completed.
[1] https://www.do-mpc.com/en/latest/
Andrew’s Status Report for Feb 10
Personal tasks of this week:
Task: Literature Review
Definition: By studying Sergey’s Thesis more closely, our aim was to look for methods in which we can further flesh out or improve our design.
Completion: The task is completed. By taking a closer look at the thesis, we were able to discover the primary accuracy and performance concerns of the system. These are the nonlinearity of the
potentiometers contributing to a decrease in accuracy, as well as the negligence of parasitic capacitance contributing to a decrease in performance. More specifically, in Sergey’s PCB design, due to
an oversight in the parasitic capacitance, it caused his system to develop non-stable behavior, which required him to add compensating capacitors to the feedback loop of his op-amps. If these were
considered, the step response could have been significantly improved as shown below. By learning from mistakes in the original designs like these, we believe that we will be able to achieve solid
performance. Furthermore, this research was very helpful in determining which component aspect we should put the most focus on.
Source: https://www.sciencedirect.com/science/article/abs/pii/S0098135414000131
Task: Spice Selection
Definition: To begin circuit simulation, a spice tool would be required to test the behavior of our preliminary circuits. Considering the wide variety of spice tools available, a detailed review of
each of the available products was done.
Completion: The task was completed. We have decided on using Cadence’s PSpice considering its performance, as well as the fact that it is readily available to CMU students. The contender was
AltiumSpice, as our PCB design will be in Altium, so it would be easy to integrate. However, we decided against it considering its performance.
Task: Preliminary Part Selection
Definition: Considering our current requirements, a preliminary selection of our main components was needed to determine if we would be able to deliver the desired performance with the required
performance and accuracy.
Completion: The task is partially completed. By using the same components that Sergey used as a baseline, we looked for components that had tolerances and performance that was at least as good, or
better. Considering the developments in the industry, we were able to make a shortlist of parts that we are considering, but have not determined the specific components we are using yet.
Overall progress assessment:
My progress is on-schedule, as all of my tasks this week have been completed. I will continue working towards completing my Prototype of creating a circuit simulation of a simple optimization
Team Status Report for Feb 10
Significant risks and risk management:
Risk: Inverted double pendulum simulation wouldn’t converge
Definition: Since we are still working on building a simulation for the inverted double pendulum, we do not know whether or not problems like numerical instability or truncation error of the
integrator would cause the simulation to not converge, i.e. incapable of settling on a solution.
Severity: If the inverted double pendulum simulation wouldn’t converge, the progress of the whole project would be severely jeopardized, because the simulation is a critical part of Prototype 1, that
is swinging up the double pendulum with NMPC in simulation.
Resolution: A step we can take right now is to accelerate our progress towards a testable MVP of the simulation, whose convergence can be validated. Another measure is to study the properties of
common numerical integrators and select those with superior convergence properties and are available in our library.
Risk: The analog circuit can’t meet the required tolerances
Definition: This risk has been brought to our attention by our assigned in instructor Thomas Sullivan (Thanks!). It is currently unclear to us what tolerance the analog components will need to have
in order to satisfy the 10% solution accuracy required by the accuracy requirement (NR3). This is nontrivial to determine as the accuracy of the whole circuit can’t be easily associated with the
accuracy of individual components.
Severity: If the analog circuit can’t meet the required tolerances, the progress of the whole project would be severely jeopardized, because the accuracy requirement (NR3) would not be satisfied.
Resolution: A solution is to use Sergey’s work [1] as a reference. If our components are more accurate than Sergey’s components in every relevant measure, it is likely that the whole circuit wouldn’t
be significantly worse in accuracy compared to his circuit. This is possible because more accurate components are available since his work was published.
Changes to the existing design:
There are no significant changes to the existing design because this is the week in which our proposal was presented, and our initial design was just finalized.
Changes to the project schedule:
Similarly, there are no significant changes to the project schedule compared to what was presented in the proposal presentation.
[1] https://escholarship.org/content/qt01q7h2ng/qt01q7h2ng_noSplash_2892dd43015926698bb02bdb85d7b62f.pdf | {"url":"https://course.ece.cmu.edu/~ece500/projects/s24-teamd6/2024/02/","timestamp":"2024-11-04T21:33:47Z","content_type":"text/html","content_length":"100385","record_id":"<urn:uuid:a3bdc623-9c9a-4d3d-9ac7-94c71fc88947>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00710.warc.gz"} |
Common Core Grade 4 Math (Worksheets, Homework, Solutions, Examples, Lesson Plans)
Module 1 Topics and Objectives
A. Place Value of Multi-Digit Whole Numbers
Standard: 4.NBT.1, 4.NBT.2, 4.OA.1
Days: 4
Module 1 Overview
Topic A Overview
Lesson 1: Interpret a multiplication equation as a comparison. (Video Lesson)
Lesson 2: Recognize a digit represents 10 times the value of what it represents in the place to its right. (Video Lesson)
Lesson 3: Name numbers within 1 million by building understanding of the place value chart and placement of commas for naming base thousand units. (Video Lesson)
Lesson 4: Read and write multi-digit numbers using base ten numerals, number names, and expanded form. (Video Lesson)
B. Comparing Multi-Digit Whole Numbers
Standard: 4.NBT.2
Days: 2
Topic B Overview
Lesson 5: Compare numbers based on meanings of the digits, using >, <, or = to record the comparison. (Video Lesson)
Lesson 6: Find 1, 10, and 100 thousand more and less than a given number. (Video Lesson)
C. Rounding Multi-Digit Whole Numbers
Standard: 4.NBT.3
Days: 4
Topic C Overview
Lesson 7: Round multi-digit numbers to the thousands place using the vertical number line. (Video Lesson)
Lesson 8: Round multi-digit numbers to any place using the vertical number line. (Video Lesson)
Lesson 9: Use place value understanding to round multi-digit numbers to any place value. (Video Lesson)
Lesson 10: Use place value understanding to round multi-digit numbers to any place value using real world applications. (Video Lesson)
Mid-Module Assessment: Topics A-C (review content 1 day, assessment ½ day, return ½ day, remediation or further applications 1 day)
D. Multi-Digit Whole Number Addition
Standard: 4. OA.3, 4.NBT.4, 4.NBT.1, 4.NBT.2
Days: 2
Topic D Overview
Lesson 11: Use place value understanding to fluently add multi-digit whole numbers using the standard addition algorithm and apply the algorithm to solve word problems using tape diagrams. (Video
Lesson 12: Solve multi-step word problems using the standard addition algorithm modeled with tape diagrams and assess the reasonableness of answers using rounding. (Video Lesson)
E. Multi-Digit Whole Number Subtraction
Standard: 4.OA.3, 4.NBT.4, 4.NBT.1, 4.OA.2
Days: 4
Topic E Overview
Lesson 13: Use place value understanding to decompose to smaller units once using the standard subtraction algorithm, and apply the algorithm to solve word problems using tape diagrams. (Video Lesson
Lesson 14: Use place value understanding to decompose to smaller units up to 3 times using the standard subtraction algorithm, and apply the algorithm to solve word problems using tape diagrams. (
Video Lesson)
Lesson 15: Use place value understanding to fluently decompose to smaller units multiple times in any place using the standard subtraction algorithm, and apply the algorithm to solve word problems
using tape diagrams. (Video Lesson)
Lesson 16: Solve two-step word problems using the standard subtraction algorithm fluently modeled with tape diagrams and assess the reasonableness of answers using rounding. (Video Lesson)
F. Addition and Subtraction Word Problems
Standard: 4.OA.3, 4.NBT.1, 4.NBT.2, 4.NBT.4
Days: 3
Topic F Overview
Lesson 17: Solve additive compare word problems modeled with tape diagrams. (Video Lesson)
Lesson 18: Solve multi-step word problems modeled with tape diagrams and assess the reasonableness of answers using rounding. (Video Lesson)
Lesson 19: Create and solve multi-step word problems from given tape diagrams and equations. (Video Lesson)
End-of-Module Assessment: Topics A-F (review content 1 day, assessment ½ day, return ½ day, remediation or further application 1 day)
Total Number of Instructional Days: 25
Module 2 Topics and Objectives
A. Metric Unit Conversions
Standard: 4.MD.1, 4.MD.2
Days: 3
Module 2 Overview
Topic A Overview
Lesson 1: Express metric length measurements in terms of a smaller unit; model and solve addition and subtraction word problems involving metric length. (Video Lesson)
Lesson 2: Express metric mass measurements in terms of a smaller unit; model and solve addition and subtraction word problems involving metric mass. (Video Lesson)
Lesson 3: Express metric capacity measurements in terms of a smaller unit; model and solve addition and subtraction word problems involving metric capacity. (Video Lesson)
B. Application of Metric Unit Conversions
Standard: 4.MD.1, 4.MD.2
Days: 2
Topic B Overview
Lesson 4: Know and relate metric units to place value units in order to express measurements in different units. (Video Lesson)
Lesson 5: Use addition and subtraction to solve multi-step word problems involving length, mass, and capacity. (Video Lesson)
End-of-Module Assessment: Topics A-B (assessment ½ day, return ½ day, remediation or further applications 1 day)
Total Number of Instructional Days: 7
Module 3 Topics and Objectives
A. Multiplicative Comparison Word Problems
Standard: 4.OA.1, 4.OA.2, 4.MD.3, 4.OA.3
Days: 3
Module 3 Overview
Topic A Overview
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles. (Video Lesson)
Lesson 2: Solve multiplicative comparison word problems by applying the area and perimeter formulas. (Video Lesson)
Lesson 3: Demonstrate understanding of area and perimeter formulas by solving multi-step real world problems. (Video Lesson)
B. Multiplication by 10, 100, and 1,000
Standard: 4.NBT.5, 4.OA.1, 4.OA.2, 4.NBT.1
Days: 3
Topic B Overview
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and 1,000 in arrays and numerically. (Video Lesson)
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing patterns. (Video Lesson)
Lesson 6: Multiply two-digit multiples of 10 by two-digit multiples of 10 with the area model. (Video Lesson)
C. Multiplication of up to Four Digits by Single-Digit Numbers
Standard: 4.NBT.5, 4.OA.2, 4.NBT.1
Days: 5
Topic C Overview
Lesson 7: Use place value disks to represent two-digit by one-digit multiplication. (Video Lesson)
Lesson 8: Extend the use of place value disks to represent three- and four-digit by one-digit multiplication. (Video Lesson)
Lesson 9, Lesson 10: Multiply three- and four-digit numbers by one-digit numbers applying the standard algorithm. (Video Lesson) (Video Lesson)
Lesson 11: Connect the area model and the partial products method to the standard algorithm. (Video Lesson)
D. Multiplication Word Problems
Standard: 4.OA.1, 4.OA.2, 4.OA.3, 4.NBT.5
Days: 2
Topic D Overview
Lesson 12: Solve two-step word problems, including multiplicative comparison. (Video Lesson)
Lesson 13: Use multiplication, addition, or subtraction to solve multi-step word problems. (Video Lesson)
Mid-Module Assessment: Topics A-D (review content 1 day, assessment ½ day, return ½ day, remediation or further applications 1 day)
E. Division of Tens and Ones with Successive Remainders
Standard: 4.NBT.6, 4.OA.3
Days: 8
Topic E Overview
Lesson 14 Solve division word problems with remainders. (Video Lesson)
Lesson 15: Understand and solve division problems with a remainder using the array and area models. (Video Lesson)
Lesson 16: Understand and solve two-digit dividend division problems with a remainder in the ones place by using number disks. (Video Lesson)
Lesson 17: Represent and solve division problems requiring decomposing a remainder in the tens. (Video Lesson)
Lesson 18: Find whole number quotients and remainders. (Video Lesson)
Lesson 19: Explain remainders by using place value understanding and models. (Video Lesson)
Lesson 20: Solve division problems without remainders using the area model. (Video Lesson)
Lesson 21: Solve division problems with remainders using the area model. (Video Lesson)
F. Reasoning with Divisibility
Standard: 4.OA.4
Days: 4
Topic F Overview
Lesson 22: Find factor pairs for numbers to 100 and use understanding of factors to define prime and composite. (Video Lesson)
Lesson 23: Use division and the associative property to test for factors and observe patterns. (Video Lesson)
Lesson 24: Determine whether a whole number is a multiple of another number. (Video Lesson)
Lesson 25: Explore properties of prime and composite numbers to 100 by using multiples. (Video Lesson)
G. Division of Thousands, Hundreds, Tens, and Ones
Standard: 4.OA.3, 4.NBT.6, 4.NBT.1
Days: 8
Topic G Overview
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers. (Video Lesson)
Lesson 27: Represent and solve division problems with up to a three-digit dividend numerically and with number disks requiring decomposing a remainder in the hundreds place. (Video Lesson)
Lesson 28: Represent and solve three-digit dividend division with divisors of 2, 3, 4, and 5 numerically. (Video Lesson)
Lesson 29: Represent numerically four-digit dividend division with divisors of 2, 3, 4, and 5, decomposing a remainder up to three times. (Video Lesson)
Lesson 30: Solve division problems with a zero in the dividend or with a zero in the quotient. (Video Lesson)
Lesson 31: Interpret division word problems as either number of groups unknown or group size unknown. (Video Lesson)
Lesson 32: Interpret and find whole number quotients and remainders to solve one-step division word problems with larger divisors of 6, 7, 8, and 9. (Video Lesson)
Lesson 33: Explain the connection of the area model of division to the long division algorithm for three- and four-digit dividends. (Video Lesson)
H. Multiplication of Two-Digit by Two-Digit Numbers
Standard: 4.NBT.5, 4.OA.3, 4.MD.3
Days: 5
Topic H Overview
Lesson 34: Multiply two-digit multiples of 10 by two-digit numbers using a place value chart. (Video Lesson)
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area model. (Video Lesson)
Lesson 36: Multiply two-digit by two-digit numbers using four partial products. (Video Lesson)
Lesson 37, Lesson 38: Transition from four partial products to the standard algorithm for two-digit by two-digit multiplication. (Video Lesson) (Video Lesson)
End-of-Module Assessment: Topics A-H (review 1 day, assessment ½ day, return ½ day, remediation or further application 1 day)
Total Number of Instructional Days: 43
Module 4 Topics and Objectives
A. Lines and Angles
Standard: 4.G.1
Days: 4
Module 4 Overview
Topic A Overview
Lesson 1: Identify and draw points, lines, line segments, rays, and angles and recognize them in various contexts and familiar figures. (Video Lesson)
Lesson 2: Use right angles to determine whether angles are equal to, greater than, or less than right angles. Draw right, obtuse, and acute angles. (Video Lesson)
Lesson 3: Identify, define, and draw perpendicular lines. (Video Lesson)
Lesson 4: Identify, define, and draw parallel lines. (Video Lesson)
B. Angle Measurement
Standard: 4.MD.5, 4.MD.6
Days: 4
Topic B Overview
Lesson 5: Use a circular protractor to understand a 1-degree angle as 1/360 of a turn. Explore benchmark angles using the protractor. (Video Lesson)
Lesson 6: Use varied protractors to distinguish angle measure from length measurement. (Video Lesson)
Lesson 7: Measure and draw angles. Sketch given angle measures and verify with a protractor. (Video Lesson)
Lesson 8: Identify and measure angles as turns and recognize them in various contexts. (Video Lesson)
Mid-Module Assessment: Topics A-B (assessment ½ day, return ½ day, remediation or further applications 1 day)
C. Problem Solving with the Addition of Angle Measures
Standard: 4.MD.7
Days: 3
Topic C Overview
Lesson 9: Decompose angles using pattern blocks. (Video Lesson)
Lesson 10, Lesson 11: Use the addition of adjacent angle measures to solve problems using a symbol for the unknown angle measure. (Video Lesson) (Video Lesson)
D. Two-Dimensional Figures and Symmetry
Standard: 4.G.1, 4.G.2, 4.G.3
Days: 5
Topic D Overview
Lesson 12: Recognize lines of symmetry for given two-dimensional figures; identify line-symmetric figures and draw lines of symmetry. (Video Lesson)
Lesson 13: Analyze and classify triangles based on side length, angle measure, or both. (Video Lesson)
Lesson 14: Define and construct triangles from given criteria. Explore symmetry in triangles. (Video Lesson)
Lesson 15: Classify quadrilaterals based on parallel and perpendicular lines and the presence or absence of angles of a specified size. (Video Lesson)
Lesson 16: Reason about attributes to construct quadrilaterals on square or triangular grid paper. (Video Lesson)
End-of-Module Assessment: Topics A-D (assessment ½ day, return ½ day, remediation or further application 1 day)
Total Number of Instructional Days: 20
Module 5 Topics and Objectives
A. Decomposition and Fraction Equivalence
Standard: 4.NF.3, 4.NF.4
Days: 6
Module 5 Overview
Topic A Overview
Lesson 1, Lesson 2: Decompose fractions as a sum of unit fractions using tape diagrams. (Video Lesson) (Video Lesson)
Lesson 3: Decompose non-unit fractions and represent them as a whole number times a unit fraction using tape diagrams. (Video Lesson)
Lesson 4: Decompose fractions into sums of smaller unit fractions using tape diagrams. (Video Lesson)
Lesson 5: Decompose unit fractions using area models to show equivalence. (Video Lesson)
Lesson 6: Decompose fractions using area models to show equivalence. (Video Lesson)
B. Fraction Equivalence Using Multiplication and Division
Standard: 4.NF.1, 4.NF.3
Days: 5
Topic B Overview
Lesson 7, Lesson 8:Use the area model and multiplication to show the equivalence of two fractions. (Video Lesson) (Video Lesson)
Lesson 9, Lesson 10: Use the area model and division to show the equivalence of two fractions. (Video Lesson) (Video Lesson)
Lesson 11: Explain fraction equivalence using a tape diagram and the number line, and relate that to the use of multiplication and division. (Video Lesson)
C. Fraction Comparison
Standard: 4.NF.2
Days: 4
Topic C Overview
Lesson 12, Lesson 13: Reason using benchmarks to compare two fractions on the number line. (Video Lesson) (Video Lesson)
Lesson 14, Lesson 15: Find common units or number of units to compare two fractions. (Video Lesson) (Video Lesson)
D. Fraction Addition and Subtraction
Standard: 4.NF.3, 4.NF.1, 4.MD.2
Days: 6
Topic D Overview
Lesson 16: Use visual models to add and subtract two fractions with the same units. (Video Lesson)
Lesson 17: Use visual models to add and subtract two fractions with the same units, including subtracting from one whole. (Video Lesson)
Lesson 18: Add and subtract more than two fractions. (Video Lesson)
Lesson 19: Solve word problems involving addition and subtraction of fractions. (Video Lesson)
Lesson 20, Lesson 21: Use visual models to add two fractions with related units using the denominators 2, 3, 4, 5, 6, 8, 10, and 12. (Video Lesson) (Video Lesson)
Mid-Module Assessment: Topics A-D (assessment ½ day, return ½ day, remediation or further applications 1 day)
E. Extending Fraction Equivalence to Fractions Greater than 1
Standard: 4.NF.1, 4.NF.2, 4.NF.3, 4.NBT.6, NF.4, 4.MD.4
Days: 7
Topic E Overview
Lesson 22: Add a fraction less than 1 to, or subtract a fraction less than 1 from, a whole number using decomposition and visual models. (Video Lesson)
Lesson 23: Add and multiply unit fractions to build fractions greater than 1 using visual models. (Video Lesson)
Lesson 24, Lesson 25: Decompose and compose fractions greater than 1 to express them in various forms. (Video Lesson) (Video Lesson)
Lesson 26: Compare fractions greater than 1 by reasoning using benchmark fractions. (Video Lesson)
Lesson 27: Compare fractions greater than 1 by creating common numerators or denominators. (Video Lesson)
Lesson 28: Solve word problems with line plots. (Video Lesson)
F. Addition and Subtraction of Fractions by Decomposition
Standard: 4.NF.3, 4.MD.4, 4.MD.2
Days: 6
Topic F Overview
Lesson 29 Estimate sums and differences using benchmark numbers. (Video Lesson)
Lesson 30: Add a mixed number and a fraction. (Video Lesson)
Lesson 31: Add mixed numbers. (Video Lesson)
Lesson 32: Subtract a fraction from a mixed number (Video Lesson)
Lesson 33: Subtract a mixed number from a mixed number. (Video Lesson)
Lesson 34: Subtract mixed numbers. (Video Lesson)
G. Repeated Addition of Fractions as Multiplication
Standard: 4.NF.4, 4.MD.4, 4.OA.2, 4.MD.2
Days: 6
Topic G Overview
Lesson 35, Lesson 36: Represent the multiplication of n times a/b as (n x a)/b using the associative property and visual models. (Video Lesson) (Video Lesson)
Lesson 37, Lesson 38: Find the product of a whole number and a mixed number using the distributive property. (Video Lesson)(Video Lesson)
Lesson 39: Solve multiplicative comparison word problems involving fractions. (Video Lesson)
Lesson 40: Solve word problems involving the multiplication of a whole number and a fraction including those involving line plots. (Video Lesson)
H. Exploration
Standard: 4.OA.5
Days: 1
Topic H Overview
Lesson 41: Find and use a pattern to calculate the sum of all fractional parts between 0 and 1. Share and critique peer strategies. (Video Lesson)
End-of-Module Assessment: Topics A-H (assessment ½ day, return ½ day, remediation or further applications 1 day)
Total Number of Instructional Days: 45
Module 6 Topics and Objectives
A. Exploration of Tenths
Standard: 4.NF.6, 4.NBT.1, 4.MD.1
Days: 3
Module 6 Overview
Topic A Overview
Lesson 1: Use metric measurement to model the decomposition of one whole into tenths. (Video Lesson)
Lesson 2: Use metric measurement and area models to represent tenths as fractions greater than 1 and decimal numbers. (Video Lesson)
Lesson 3: Represent mixed numbers with units of tens, ones, and tenths with number disks, on the number line, and in expanded form. (Video Lesson)
B. Tenths and Hundredths
Standard: 4.NF.5, 4.NF.6, 4.NBT.1, 4.NF.1, 4.NF.7, 4.MD.1
Days: 5
Topic B Overview
Lesson 4: Use meters to model the decomposition of one whole into hundredths. Represent and count hundredths. (Video Lesson)
Lesson 5: Model the equivalence of tenths and hundredths using the area model and number disks. (Video Lesson)
Lesson 6: Use the area model and number line to represent mixed numbers with units of ones, tenths, and hundredths in fraction and decimal forms. (Video Lesson)
Lesson 7: Model mixed numbers with units of hundreds, tens, ones, tenths, and hundredths in expanded form and on the place value chart. (Video Lesson)
Lesson 8: Use understanding of fraction equivalence to investigate decimal numbers on the place value chart expressed in different units. (Video Lesson)
Mid-Module Assessment: Topics A-B (assessment 1 day, return ½ day, remediation or further applications ½ day)
C. Decimal Comparison
Standard: 4.NF.7, 4.MD.1, 4.MD.2
Days: 3
Topic C Overview
Lesson 9: Use the place value chart and metric measurement to compare decimals and answer comparison questions. (Video Lesson)
Lesson 10: Use area models and the number line to compare decimal numbers, and record comparisons using <, >, and =. (Video Lesson)
Lesson 11: Compare and order mixed numbers in various forms. (Video Lesson)
D. Addition with Tenths and Hundredths
Standard: 4.NF.5, 4.NF.6, 4.NF.3, 4.MD.1
Days: 3
Topic D Overview
Lesson 12: Apply understanding of fraction equivalence to add tenths and hundredths. (Video Lesson)
Lesson 13: Add decimal numbers by converting to fraction form. (Video Lesson)
Lesson 14: Solve word problems involving the addition of measurements in decimal form. (Video Lesson)
E. Money Amounts as Decimal Numbers
Standard: 4.MD.2, 4.NF.5, 4.NF.6
Days: 2
Topic E Overview
Lesson 15: Express money amounts given in various forms as decimal numbers. (Video Lesson)
Lesson 16: Solve word problems involving money. (Video Lesson)
End-of-Module Assessment: Topics A-E (assessment 1 day, return ½ day, remediation or further applications ½ day)
Total Number of Instructional Days: 20
Module 7 Topics and Objectives
A. Measurement Conversion Tables
Standard: 4.OA.1, 4.OA.2, 4.MD.1, 4.NBT.5, 4.MD.2
Days: 5
Module 7 Overview
Topic A Overview
Lesson 1, Lesson 2: Create conversion tables for length, weight, and capacity units using measurement tools, and use the tables to solve problems. (Video Lesson)
Lesson 3: Create conversion tables for units of time, and use the tables to solve problems. (Video Lesson)
Lesson 4: Solve multiplicative comparison word problems using measurement conversion tables. (Video Lesson)
Lesson 5: Share and critique peer strategies. (Video Lesson)
B. Problem Solving with Measurement
Standard: 4.OA.2, 4.OA.3, 4.MD.1, 4.MD.2, 4.NBT.5, 4.NBT.6
Days: 6
Topic B Overview
Lesson 6: Solve Problems involving mixed units of capacity. (Video Lesson)
Lesson 7: Solve problems involving mixed units of length. (Video Lesson)
Lesson 8: Solve problems involving mixed units of weight. (Video Lesson)
Lesson 9: Solve problem involving mixed units of time. (Video Lesson)
Lesson 10, Lesson 11: Solve multi-step measurement word problems. (Video Lesson)
C. Investigation of Measurements Expressed as Mixed Numbers
Standard: 4.OA.3, 4.MD.1, 4.MD.2, 4.NBT.5, 4.NBT.6
Days: 3
Topic C Overview
Lesson 12, Lesson 13: Use measurement tools to convert mixed number measurements to smaller units. (Video Lesson)
Lesson 14: Solve multi-step word problems involving converting mixed number measurements to a single unit. (Video Lesson)
End-of-Module Assessment: Topics A-C (assessment 1 day, ½ day return, remediation or further application ½ day)
D. Year in Review
Days: 4
Topic D Overview
Lesson 15, Lesson 16: Create and determine the area of composite figures. (Video Lesson)
Lesson 17: Practice and solidify Grade 4 fluency.
Lesson 18: Practice and solidify Grade 4 vocabulary.
Total Number of Instructional Days: 20
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. | {"url":"https://www.onlinemathlearning.com/common-core-math-worksheets-grade4.html","timestamp":"2024-11-04T08:39:59Z","content_type":"text/html","content_length":"122186","record_id":"<urn:uuid:2d011ad6-360a-4e19-8e85-5b70ec059c61>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00533.warc.gz"} |
Average Gradient for Parabolas
The other day in my maths class the students were asked to find the average gradient for two points on a parabola. What they did was use the derivative to find the gradient at each point and then
average those gradients. Guess what? They all got the answer that I expected but none of them did it correctly (simple rise over run using two points). Well, I was a little surprised, but of course I
wondered if the two "averages" were the same for parabolas. The drawing below simply shows two points, A and B, that you can move around on the parabola (which can also be moved), and the average
gradient (from A to B) always equals the average of the two gradients at A and B. With a little algebra it can be shown that for a function of the form and two points with x-values of h and k, the
average gradient = the average of the gradients = . | {"url":"https://stage.geogebra.org/m/mwnKMrku","timestamp":"2024-11-13T21:25:50Z","content_type":"text/html","content_length":"92940","record_id":"<urn:uuid:278c90ce-f694-462a-817b-e1a4f87b02f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00591.warc.gz"} |
Solenoid Valve (IL)
Solenoid valve in an isothermal liquid system
Since R2023a
Simscape / Fluids / Isothermal Liquid / Valves & Orifices / Directional Control Valves
The Solenoid Valve (IL) block models flow through a solenoid valve in an isothermal liquid network. The valve consists of a two-way directional control valve with a solenoid actuator. The physical
signal at port S controls the solenoid. When the signal at port S is above 0.5, the solenoid turns on and the valve opens. When the signal at port S falls below 0.5, the solenoid turns off, closing
the valve.
A solenoid valves consists of a valve body with a spring-loaded plunger that is operated by an electric solenoid. When the solenoid is on, the magnetic force lifts the spool, allowing fluid to flow.
When the solenoid is off, the spring pushes the plunger back in place, stopping flow. The block does not model the mechanics of the solenoid explicitly, but characterizes the opening and closing of
the valve using the opening and closing switching times.
Mass Flow Rate
The mass flow rate through the valve is
$\stackrel{˙}{m}={C}_{d}A\sqrt{\frac{2\overline{\rho }}{\left(1-\frac{A}{{A}_{port}}\right)}}\frac{\Delta p}{{\left(\Delta {p}^{2}+\Delta {p}_{crit}^{2}\right)}^{1/4}},$
• A is the cross-sectional area of the valve.
• A[port] is the value of the Cross-sectional area at ports A and B parameter.
• C[d] is the value of the Discharge coefficient parameter.
• $\overline{\rho }$ is the average fluid density in the valve.
• Δp is the change in pressure across the valve.
• Δp[crit] is the critical pressure,
$\Delta {p}_{crit}=\frac{\pi }{8A\overline{\rho }}{\left(\frac{\mu {\mathrm{Re}}_{crit}}{{C}_{d}}\right)}^{2},$
where Re[crit] is the value of the Critical Reynolds number parameter and μ is the fluid viscosity.
Opening Dynamics
The block assumes that the solenoid behaves as a resistor-inductor series circuit, represented as
$V=iR+L\frac{\partial i}{\partial t},$
• V is the voltage across solenoid inductor.
• R is the resistance of the solenoid inductor.
• L is the inductance of the solenoid.
• i is the current through the solenoid inductor.
The solenoid generates a force proportional to the current squared, ${\text{F}}_{mag}=-\frac{1}{2}K{i}^{2},$ where K is a proportionality constant. The expression for the cross-sectional area of the
valve, A(t), depends on the value of the Solenoid control parameter.
Valve Controlled with Physical Signal
When you set Solenoid control to Physical signal, the block calculates the opening area based on the signal at port S. When the valve is opening,
$\begin{array}{l}A\left(t\right)=\frac{{A}_{max}-{A}_{leak}}{7}\left({e}^{-2\frac{t-{t}_{0}}{{\tau }_{open}}}-8{e}^{\frac{t-{t}_{0}}{{\tau }_{open}}}\right)+{A}_{max}\\ {t}_{0}={t}_{start}+\mathrm
{log}\left(4-\sqrt{16-7\frac{{A}_{max}-{A}_{0}}{{A}_{max}-{A}_{leak}}}\right){\tau }_{open}.\end{array}$
When the valve is closing,
$\begin{array}{l}\text{A(t)=}\left({A}_{max}-{A}_{leak}\right){e}^{-\frac{t-{t}_{0}}{{\tau }_{close}}}+{A}_{leak}\\ {t}_{0}={t}_{start}+\mathrm{log}\left(\frac{{A}_{0}-{A}_{leak}}{{A}_{max}-{A}_
{leak}}\right){\tau }_{close},\end{array}$
• A[max] is the value of the Maximum opening area parameter.
• A[leak] is the value of the Leakage area parameter.
• t[start] is the time that the solenoid turns on or off.
• A[0] is the valve area at the time the solenoid turns on or off.
• ${\tau }_{open}=\frac{-{t}_{switc{h}_{open}}}{\mathrm{log}\left(4-\sqrt{15.3}\right)},$ where t[switch[open]] is the value of the Opening switching time parameter.
• ${\tau }_{close}=\frac{{t}_{switc{h}_{close}}}{\mathrm{log}\left(0.1\right)},$ where t[switch[close]] is the value of the Closing switching time parameter.
Valve Controlled with Electrical Ports
When you set Solenoid control to Electrical ports, the block calculates the opening area based on the electrical network connected to the solenoid valve at ports + and -.
The opening area is
where x is the plunger position and l is the value of the Plunger distance of travel parameter.
The block models the plunger motion with the force balance expression
• m[core] is the mass of the moving parts in the solenoid valve.
• F[spring] is the force exerted by the spring to close the valve when the solenoid is off.
• F[hardstop] is the hard-stop force that prevents the plunger from moving beyond the fully open and fully closed positions. The block calculates the hard-stop by using mode charts. For more
information, see Mode Chart Modeling.
• F[magnetic] is the force generated by the solenoid,
The block uses the solution to the force balance expression to calculate m[core], F[spring], and K at the values of the Rated voltage, Nominal current, Opening switching time, and Closing switching
time parameters. The block then uses these values for m[core], F[spring], and K to solve for the plunger position at all other times.
Switching Time
The block characterizes the solenoid valve by using the valve opening and closing switching times specified by the Opening switching time, and Closing switching time parameters, respectively. The
Opening switching time parameter is the time from the solenoid being turned on to the flow rate rising to 90% of the way between its maximum and minimum values.
The Closing switching time parameter is the time from the solenoid being turned off to the flow rate falling to 10% of the way between its maximum and minimum values.
Assumptions and Limitations
• The maximum solenoid force is the same as the force generated by the spring.
• The damping inside the solenoid and the pressure flow forces are negligible.
• The spool is balanced.
• The solenoid travel distance is short enough that the block assumes the inductance is linear.
S — Signal that controls valve, unitless
physical signal
Physical signal that controls the valve. When the signal at port S is above 0.5, the solenoid turns on and the valve opens. When the signal at port S falls below 0.5 the solenoid turns off, and the
valve closes.
To enable this port, set Solenoid control to Physical signal.
A — Liquid port
isothermal liquid
Liquid entry or exit port.
B — Liquid port
isothermal liquid
Liquid entry or exit port.
+ — Positive terminal
Electrical conserving port associated with the positive terminal of the valve control.
To enable this port, set Solenoid control to Electrical ports.
- — Negative terminal
Electrical conserving port associated with the negative terminal of the valve control.
To enable this port, set Solenoid control to Electrical ports.
Solenoid control — Select valve control method
Physical signal (default) | Electrical ports
Select the method the block uses to control the valve. If you select Physical signal, the block uses the physical signal S to control the valve. If you select Electrical network, the block enables
the electrical + and - ports and models the electrical responses.
Valve initial position — Initial position for valve
Closed (default) | Open
Whether the initial position of the valve is open or closed.
Opening switching time — Time for flow rate to rise when opening
0.05 s (default) | positive scalar
Time from the solenoid turning on to the flow rate reaching 90% of the way between its maximum and minimum values.
If Solenoid control is set to Electrical ports, calculate this parameter at the value of the Rated voltage parameter using a step input.
Closing switching time — Time for flow rate to fall when closing
0.1 s (default) | positive scalar
Time from the solenoid turning off to the flow rate falling to 10% of the way between its maximum and minimum values.
If Solenoid control is set to Electrical ports, calculate this parameter at the value of the Rated voltage parameter using a step input.
Maximum opening area — Area of fully opened valve
1e-4 m^2 (default) | positive scalar
Cross-sectional area of the valve in the fully open position.
Leakage area — Valve gap area when in fully closed position
1e-10 m^2 (default)
Sum of all the gaps when the valve is in the fully closed position. The block saturates any valve area smaller than this value to the specified leakage area. The leakage area contributes to numerical
stability by maintaining continuity in the flow.
Cross-sectional area at ports A and B — Area at valve entry or exit
inf m^2 (default) | positive scalar
Cross-sectional area at the entry and exit ports A and B. The block uses this area in the pressure-flow rate equation that determines the mass flow rate through the valve.
Discharge coefficient — Discharge coefficient
0.64 (default) | positive scalar
Correction factor that accounts for discharge losses in theoretical flows.
Critical Reynolds number — Upper Reynolds number limit for laminar flow
150 (default) | positive scalar
Upper Reynolds number limit for laminar flow through the valve.
Pressure recovery — Whether to account for pressure increase in area expansions
off (default) | on
Whether the block accounts for pressure increase when fluid flows from a region of smaller cross-sectional area to a region of larger cross-sectional area.
Rated voltage — Solenoid rated voltage
12 V (default) | positive scalar
Rated voltage of the solenoid. Calculate the values of the Opening switching time and Closing switching time parameters at this voltage by using a step input.
To enable this parameter, set Solenoid control to Electrical ports.
Nominal current — Steady-state current
2e-4 A (default) | positive scalar
Steady-state current of the solenoid at the value of the Rated voltage parameter.
To enable this parameter, set Solenoid control to Electrical ports.
Solenoid inductance — Linear solenoid inductance
1e-6 H (default) | positive scalar
Linear inductance of the solenoid. The block approximates the inductance as linear.
To enable this parameter, set Solenoid control to Electrical ports.
Plunger distance of travel — Distance plunger travels
1.8e-3 m (default) | positive scalar
Distance the plunger travels between the fully closed and fully open valve positions.
To enable this parameter, set Solenoid control to Electrical ports.
Initial current — Solenoid initial current
0 A (default) | positive scalar
Initial current in the solenoid.
To enable this parameter, set Solenoid control to Electrical ports.
[1] Zhang, Xiang, Yonghua Lu, Yang Li, Chi Zhang, and Rui Wang. “Numerical Calculation and Experimental Study on Response Characteristics of Pneumatic Solenoid Valves.” Measurement and Control 52,
no. 9–10 (November 2019): 1382–93. https://doi.org/10.1177/0020294019866853.
[2] Zhang, Jianyu, Peng Liu, Liyun Fan, and Yajie Deng. “Analysis on Dynamic Response Characteristics of High-Speed Solenoid Valve for Electronic Control Fuel Injection System.” Mathematical Problems
in Engineering 2020 (January 22, 2020): 1–9. https://doi.org/10.1155/2020/2803545.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2023a
R2023b: Connect the valve directly to an electrical network
You can now control the block by connecting it directly to an electrical network. Set the Solenoid control parameter to Electrical network to model electrical responses and enable the electrical +
and - ports. | {"url":"https://uk.mathworks.com/help/hydro/ref/solenoidvalveil.html","timestamp":"2024-11-10T12:55:06Z","content_type":"text/html","content_length":"121597","record_id":"<urn:uuid:d76baa0d-ef6a-41f5-a5d5-d860c9eb7652>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00894.warc.gz"} |
of a carbonate ramp
1 Introduction
Correlation between stratigraphic sections provides significant information about the chronology and geometry of sedimentary structures. However, significant uncertainty may exist in stratigraphic
correlation, depending on the considered scale, the spacing between sections, the quality and type of observations, and the concepts used to analyze these observations. As observed by Doveton (1994)
in the case of geophysical well-logging data, manual methods for well correlation can be slow, labor-intensive, expensive, and sometimes inconsistent owing to uncertainties. Moreover, several sets of
correlations may match the same sparse observations while honoring a given set of interpretive rules (Borgomano et al., 2008; Koehrer et al., 2011). As shown by Bond et al. (2007) for seismic
interpretation, cognitive bias may also be introduced, depending on the background of the interpreter. This means that the expert knowledge necessary in all interpretive processes may sometimes
orient the interpretations in an inappropriate way. This can be a problem especially when this expert knowledge is not well documented.
In subsurface studies, errors in correlation between boreholes can be consequential. For instance, in stratigraphic hydrocarbon reservoir forecasting, well correlation is typically required to
extrapolate layer geometry, which in turn provides a coordinate system to simulate petrophysical properties with geostatistical methods (Dubrule and Damsleth, 2001; Larue and Legarre, 2004; Mallet,
2004, 2014, Pyrcz and Deutsch, 2014). In this context, a wrong stratigraphic correlation model is likely to yield inaccurate predictions about the layering and the associated porosity and
permeability fields in the reservoir.
The rationale for this work is that automatic correlation should be used in conjunction with manual methods. Indeed, automatic correlation provides a unique way to make correlation results more
systematic than expert interpretations (automatic approaches produce by definition reproducible results, whereas interpretations may vary from one interpreter to another). Additionally, the very
large number of possible correlations (see Appendix A) makes stochastic methods appropriate to sample correlation uncertainty. However, to reach the same level of quality as with expert correlation,
an important challenge for automatic methods is to translate qualitative stratigraphic concepts used in manual interpretation into computerized, quantitative rules. When this translation is
effective, a clear benefit is to formally state the concepts involved in the correlation.
In this paper, we use a numerical method to stochastically build stratigraphic correlations from a set of 1D stratigraphic sections (also referred to as “wells” in this paper for simplicity; we
further assume that all sections are pre-processed to provide the true stratigraphic thickness and eliminate discontinuities and reversals related to faults, recumbent folds or horizontal wells). Our
method builds on The Dynamic Time Warping (DTW) algorithm, which was first used in geosciences by Smith and Waterman (1980) to build lithostratigraphic correlations. DTW is computationally efficient,
relatively simple to implement and allows for varying the correlation cost function. This explains its large adoption for correlating facies (Howell, 1983; Smith and Waterman, 1980; Waterman and
Raymond, 1987) and geophysical signals (Fang et al., 1992; Hale, 2013; Herrera et al., 2014; Hladil et al., 2010; Lallier et al., 2012, 2013; Wheeler and Hale, 2014). However, most of the above
methods do not account for the distance between wells and neglect sequence stratigraphic concepts, although these constitute a paradigm of choice for correlation, see for instance Ainsworth (2005)
for siliciclastic settings and Borgomano et al. (2008) for carbonate environments. Therefore, in this paper, we consider well data that correspond to sequence stratigraphic intervals identified from
depositional facies. The proposed method (Section 2) generates several possible correlations of stratigraphic sequences according to sedimentological rules that account for the spacing between wells.
These rules are described and applied to Cretaceous outcrops of the southern Provence Basin (southeastern France) in Section 3.
2 Proposed approach for sequence stratigraphic correlation
2.1 Dynamic Time Warping
Let us review the DTW method before discussing its adaptation to address uncertainty management in stratigraphic correlation. For two wells W[1] and W[2] with respectively n and m stratigraphic
markers, DTW represents the stratigraphic correlation between these two wells as a path in a 2D cost table D of size n×m (Fig. 1A and B). This table is built up with a series of points and
transitions corresponding to the correlation of markers and intervals, respectively. The geological consistency of each possible correlation between markers and between units and each possible
unconformity is evaluated independently thanks to the computation of a cost. These costs are stored in the table as follows:
• • a point at cell (i,j) corresponds to the correlation between the marker i of the first well W[1] and the marker j of the second well W[2] with a cost c(i,j);
• • an oblique transition corresponds to a correlation between two intervals (the interval {i;i–1} of W[1] and the interval {j;j–1} of W[2]) with a cost noted t[i,i–1]^j,j–1 in Eq. (1);
• • a vertical transition between cells (i,j) and (i–1,j) means that the interval {i;i–1} of W[1] ends in an unconformity between units j;j - 1 and j+1,j of W[2]. This unconformity is associated
with a cost noted t[i,i–1]^j,j. A horizontal transition corresponds to an unconformity with a cost t[i,i]^j,j–1.
Fig. 1
2D DTW for stratigraphic correlations. A: Stratigraphic correlation between two wells W[1] and W[2] containing respectively n and m markers. We assume the i^th marker of W[1] and the j^th marker of W
[2] are known to be correlated (thick line). B: DTW cost table D displaying the minimum cost correlation path. Shaded parts are excluded to ensure the known correlation [i;j] is honored. The elements
needed to perform stratigraphic correlation with DTW concern: the cost to conformably correlate units {k;k–1} of W[1] and {l;l–1} of W[2](C), the cost for unconformities to occur, i.e., for unit {k;k
-1} of W[1] to pinch out before crossing W[2](D) and for unit {l;l–1} of W[1] to pinch out before crossing W[1](E).
The total score of a path is calculated as the sum of all the costs c and t contained in this path. The minimum cost path from the bottom left cell (bottom correlation line [1,1]) to the top right
cell (top correlation line [n,m]) corresponds to the optimal correlation between W[1] and W[2]. The minimum cost correlation path through the table is obtained by computing iteratively the minimum
cost path from the bottom right cell to the cell (i,j) thanks to:
$Di,j=ci,j+minti,i−1j,j−1+Di−1,j−1ti,i−1j,j+Di−1,jti,ij,j−1+Di,j−1$ (1)
Once the entire DTW cost table has been filled, the correlation path is searched starting from the cell [n;m] as shown in Fig. 1. The next cell is the one with the minimum cumulated cost among the
three adjacent cells (on the left, bottom and diagonally to the bottom left). The search then continues from this cell, using the same rule, until the bottom left is reached. This construction method
makes the correlation path unidirectional, ensuring that no overlap occurs.
2.2 Framework to integrate the stratigraphic knowledge
In Eq. (1), conceptual correlation criteria can be integrated through the elementary cost c of correlating two given markers. For example, the costs detailed in Section 3.2 are based on a
sedimentological analysis of carbonate facies that associates a range of paleo-bathymetry with each facies. From this information, it is possible to assess the consistency of the paleo-bathymetry for
each possible correlation line using paleo-geographic criteria. We are currently investigating other types of elementary rules applicable in various stratigraphic contexts.
Unlike previous studies that assign a constant value to the correlation cost between stratigraphic units (term t in Eq. (1), see Smith and Waterman (1980); Fang et al. (1992)) or compute it as a
function of the units’ thickness (Waterman and Raymond, 1987), our well correlation technique is based on the evaluation of depositional profile consistency (term c in Eq. (1)), which can typically
have a sequence stratigraphic significance.
Nonetheless, our experience is that even the best rules can fail to produce a globally consistent correlation for long sections that display a strong variability of depositional conditions through
time. Therefore, to increase the flexibility of the method and allow for diverse types of interpretive inputs, we propose to apply DTW so as to incorporate deterministic correlations and hierarchical
• • regional correlation surfaces can often be identified without ambiguity. Therefore, some correlation lines interpreted by a geologist can be taken as input constraint. For example, consider the
deterministic input correlation line [i,j] in Fig. 1B. The correlation path is necessarily passing through this point in the DTW table and, by construction, almost half of the positions in the
DTW table become unacceptable. As a result, the correlation problem can be divided in two so we directly build two DTW tables: one to compute the correlation path from [1,1] to [i,j], and another
one to compute the correlation path from [i,j] to [n,m] (Fig. 1B);
• • stratigraphic sequences identified along wells can be ordered using a stratigraphic sequence hierarchy (Vail et al., 1977) or fractal considerations (Neal, 2009; Schlager, 2004, 2010). When
such order information (sensu Vail et al. (1977)) is available, we apply DTW several times: lower-order stratigraphic markers are correlated, and the resulting correlation lines are used as input
to the correlation of higher-order stratigraphic markers. In our current implementation, this strategy is only applied when the order for each marker is available.
In addition to providing interesting interpretive inputs, this simple strategy significantly improves the overall performance of the method by replacing a global DTW table by several much smaller
2.3 Stochastic dynamic time warping
Classical methods to sample uncertainties about well correlations consist in randomizing the elementary correlation cost computation rules, leading to one single correlation per geological scenario (
Griffiths and Bakke, 1990; Waterman and Raymond, 1987). As recently proposed for magnetostratigraphic dating (Lallier et al., 2013), the approach taken here is rather to generate several models for
the same geological scenario. This approach essentially reflects the incompleteness of the input rules and could be combined in principle with cost randomization to explore scenario-based
uncertainties. Each outcome of the method, associating all markers of all considered wells, is termed a realization because it samples correlation uncertainty.
To produce several realizations from a given set of costs, we use a stochastic transition when moving in the DTW cost table: from the current location (i,j) in the DTW cost table, the next location
is randomly drawn by replacing Eq. (1) by:
$Di,j=1a if p∈0,a/a+b+c1b if p∈a/a+b+c,a+b/a+b+c1c if p∈a+b/a+b+c,1$ (2)
where p is a random value drawn from a uniform distribution U[0,1] and a, b and c are recursively defined by Eq. (3), corresponding to the likelihood of making an oblique, vertical and horizontal
transition, respectively:
$a=1/ci,j+ti,i−1j,j−1+Di−1,j−1b=1/ci,j+ti,i−1j,j+Di−1,jc=1/ci,j+ti,ij,j−1+Di,j−1$ (3)
2.4 DTW for multi-well correlation
The well correlation problem is generally not limited to a simple correlation between two wells but needs that d wells be considered, with d>2. This could be addressed by using a d-dimensional DTW
table (Brown, 1997). However, the time and memory complexity for d wells with n markers each is proportional to n^d, making this method prohibitive for most datasets, even in optimized
implementations (see Fuellen (1997) for a review). Wheeler and Hale (2014) propose an elegant and effective solution by casting the problem into a vertical shift optimization. However, their solution
is deterministic.
Therefore, we simply correlate all wells two at a time by considering each pair independently. A correlation path is first built to define the pairs of wells that are to be correlated. This path
should not contain loops to avoid inconsistencies (Fig. 2). This strategy is computationally efficient but does not guarantee that the resulting correlation is optimal. In particular, the well pair
traversal order may be a source of bias when evaluating the cost functions based on depositional profiles (Section 3.2).
Fig. 2
Example of inconsistent well correlation (thick line) generated using three pairwise correlations. This inconsistency is caused by the loop in the correlation path.
Therefore we have developed an iterative DTW (Fig. 3). First all pairs of wells connected by an edge of the correlation path are correlated. For a given edge e of the correlation path, all previously
correlated pairs (from 1 to e–1) are known; if needed, they can be used to compute the current correlation. Once all wells have been traversed, an edge is randomly drawn and the corresponding
pairwise correlation is rebuilt, taking into account all other correlations. This ensures that every correlation is generated knowing the whole 3D stratigraphic structure.
Fig. 3
Iterative DTW algorithm. From a correlation path (A), a first correlation draft between wells is built (B) using a sequential 2D DTW. At each step of (B), the previously built correlations are known
and used as a constraint. Then, to ensure the 3D consistency of the correlation, a random correlation between two wells is removed and rebuilt knowing all other correlations (C). This iterative
process can be performed until the total score of the well correlation stabilizes.
This procedure may take a large number of iterations to converge to a stable, minimal cost correlation. However, we advise not to always iterate until convergence. Indeed, for uncertainty assessment
purposes, the order by which the edges of the correlation path are traversed introduces an interesting source of variability, which is also probably present in manual expert-based correlations.
3 Application to outcrop data of the Beausset Basin
3.1 Geological settings and material
The Beausset basin is located in Basse-Provence, SE France (Fig. 4A). The studied area corresponds to a carbonate platform, aged Cenomanian–Middle Turonian and developed on the southern part of the
“Durance swell” (Philip, 1974). These deposits display terrigenous inputs from a crystalline Hercynian basement corresponding to the southeastern limit of the basin (Fig. 4). Multiple outcrops have
been described in this area (Gari, 2007; Philip, 1974, 1993; Philip and Gari, 2005) and almost continuous outcrop from platform to basin deposits allowed previous authors to build well-constrained
geometrical models and sequence stratigraphic correlations.
Fig. 4
A: Geographic and paleogeographic settings of the study area. After Philip (1993). B: Location of the studied outcrops.
The studied outcrops are aged Lower Cenomanian to Middle Turonian (Fig. C.10). Nine outcrop sections covering the entire studied interval are used in this study (Fig. 4). The studied interval is
subdivided in two primary stratigraphic units: U.I and U.II (Gari, 2007). U.I, aged Cenomanian, corresponds to the transgressive part of a second order, sensu Vail et al. (1991), transgressive
regressive cycle ending mid Turonian (Philip, 1999). This unit is divided into six secondary stratigraphic units (U.I 1 to U.I 6), bounded by conspicuous surfaces or abrupt facies changes
corresponding to third- and fourth-order cycles. The U.II primary stratigraphic unit, aged before mid-Turonian, is the regressive part of the second-order cycle started in U.I. This unit is also
divided into six secondary stratigraphic units (U.II 1 to U.II 6). In this work, following the outcrop description and study proposed by Gari (2007), eight facies have been distinguished according to
the depositional depth range, bioclastic content and deposition style (see Table C.1 Appendix C).
3.2 Stochastic well correlation
Two methods for the evaluation of the consistency of a horizon are used: (i) a ramp paleo-angle-based rule that compares markers by pairs and (ii) a depositional facies based rule that uses all
available markers of the considered horizon at once.
3.2.1 Paleoangle consistency
The slope of the paleo-depositional profile can be an important source of uncertainty in well correlation (Fig. 11). To check the consistency of a stratigraphic correlation of a carbonate ramp,
Borgomano et al. (2008) introduce trigonometric relationships between average ramp paleo-angles (α and β) of two considered horizons and sediment thicknesses (e[1] and e[2] respectively on wells W[1]
and W[2]) according to (Fig. 5):
Fig. 5
Trigonometric relationships on a carbonate ramp system (from Borgomano et al., 2008). Using measurement in A and relations B and C, the equality D is true if the correlation is good. pb:
paleo-bathymetry at the current marker. sl: sea level. L: well spacing. bp: base profile. e[1], e[2]: decompacted sediment thicknesses. α, β: angles between base profile and sea level.
This formalism is adapted to compute a cost of stratigraphic correlation between two markers b[1] and b[2] using a prior-defined correlation line (a[1],a[2]) extracted for instance from seismic data
or a reference surface. This reference line must be defined as input of the method and cannot be computed by the DTW method, which assumes that costs are independent of the path in the cost table.
From the paleo-bathymetry at markers a priori correlated (pb[a]^1 and pb[a]^2), the paleo-bathymetry of the studied markers (pb[b]^1 and pb[b]^2) and the sediment thicknesses (e[1] and e[2]), we
compute the cost c[A](b[1],b[2]) as the degree of violation of Eq. (4) as:
$cAb1,b2=pba1−pba2−pbb1−pbb2−e1−e2$ (5)
This method is valid in the case of a regular accommodation increase between the two correlated wells. In the case of a differential subsidence, a rotation angle has to be added to Eq. (5) (Borgomano
et al., 2008).
3.2.2 Sedimentary profile consistency
The boundaries of stratigraphic sequences identified on wells are often considered as time significant (Catuneanu et al., 1998). Consequently, markers bounding stratigraphic sequences could be taken
as a sparse sampling of the geography (i.e. sedimentary profile) at the time of deposition. Evaluating the likelihood of a correlation line then boils down to checking whether this sampling is
consistent with the paleogeography.
In this work, the cost for correlating two markers m[1] of well W[1] and m[2] of well W[2] is based on a zonation of carbonate facies with depth (membership functions, Fig. 7B) and on a regional
theoretical bathymetric profile deemed representative of the paleogeography at the time of deposition (Fig. 7A). For our case study, the global analysis of the facies (Gari, 2007) led us to select a
regional shelf slope of 0.5° to the south and to place the shelf break at a depth of about 40m. For the slope below 40m, we used an exponential north–south bathymetric profile for the carbonate
ramp (Adams and Schlager, 2000):
$z=aexp−by+c=−0.2513 exp−0.225y+0.2913$ (6)
where z is the bathymetry and y the southward distance to the shoreline. The value of b=0.225 was taken as the same as for the western Great Bahamas Bank (Fig. 6C of Adams and Schlager (2000)); the
values of a and c were chosen so that the depth reaches z=200m at y=9km south of the shoreline (Gari, 2007).
Fig. 7
Construction of depositional space. A. Bathymetric map built from the exponential equation (6). B. Membership functions for the description of the location of facies using bathymetry. C. Examples of
possible location of deposition for facies 1, 3, 5, and 6.
Fig. 6
Likelihood computation using theoretical depositional space. A. Studied horizon and makers displaying depositional facies in the geological space. B. Studied markers transferred in depositional
space. C. The set of markers is searched in depositional space to maximize the sum of deposition probabilities of each facies.
From this theoretical profile, the correlation cost can be computed as follows:
• • the depth membership function of each facies is used to compute theoretical facies probability maps on the paleogeographic profile (Fig. 7C);
• • on the paleogeographic map, we look for the most likely location of all the markers previously correlated to m[1] and m[2] (Section 2.4). Let M={m[1],..,m[n]} denote the set of all these
markers, F[i] the facies observed at marker m[i] and $pFi$ the probability of observing the facies F[i] at the location of marker m[i] on the theoretical profile. The most likely location of the
markers M on the theoretical map is obtained by finding the horizontal translation of the markers M that maximizes the likelihood of the observed facies ($∑i=1npFi$). Because the profile is quite
smooth, this position can be assessed by gradient-based optimization.
• • The correlation cost c[B](m[1],m[2]) is then computed as (Fig. 6):
$cBm1,m2=1−1n∑i=1npFi$ (7)
In this cost computation, the parametric definition of the bathymetric profile may be influential. Several scenarios for bathymetry or non-parametric profiles computed with process-based models could
be used to test concepts and assess their impact on the emerging correlations.
3.2.3 Constraints
The top and bottom horizons of the studied interval (U.I 1 and U.II 6, see Fig. C.10) are input as known correlations to constrain the stochastic well correlation process. This constraint is not a
requirement for the correlation method and could be eliminated by defining a null gap cost on the starting and ending correlation, as done for instance by Lallier et al. (2013) for
magnetostratigraphic dating. However, this constraint is useful here to compute the correlation costs based on paleo-angles (Section 3.2.1) and the overall stratigraphic geometry of the system.
3.2.4 Results and discussions
3.2.4.1 Geometrical model and grid building
Four possible 3D models were created (Fig. 8B, C, D, and E): a reference model built from the deterministic correlation proposed by Gari (2007) (Fig. 8B) and three models built from the stochastic
correlation method (Section 2.4).
Fig. 8
Four possible stratigraphic models for the Cretaceous southern Provence basin. A. 3D deterministic stratigraphic model of the basin (from Gari, 2007). The black lines indicate the location of the
studied outcrops; Shaded surface figures the cross-section presented in B–E. B. Stratigraphic correlation model proposed by Gari [2007] and associated vertical facies proportions. Dashed lines are
the projected locations of the used outcrops. C–E. Stratigraphic models built from three stochastic correlations and associated vertical facies proportions.
The geometry of these four models is constrained by the geometry of the top and the bottom horizons of the studied interval (U.I 1 and U.II 6, see Fig.C.10). These two horizons were built by Gari
(2007) using intersections with the topography and dip and strike measurements in the field. Internal horizons corresponding to well correlations were built so that they respect well markers and the
thickness variations of units defined by surfaces are smoothed (Mallet, 2002). A conformable stratigraphic grid was then built for the reference model and the stochastic ones.
The relative visual similarity of some stochastic models with the deterministic interpretation of (Gari, 2007) is reassuring. However, it does not prove that the stochastic model is right (nor that
Gari's interpretation is right). Only additional evidence (e.g., from paleontology, palynology or geochemistry) could indicate which of these models are acceptable.
Facies distribution analysis. Statistical facies proportions within each layer were calculated from wells for each stratigraphic model to assess the impact of correlation uncertainty on reservoir
properties (Fig. 8). This vertical facies proportion can be analyzed in terms of reservoir facies distribution, considering facies F0 to F4 as potentially reservoir and facies F5 to F7 as flow
barriers. Because outcrops located on the north side of the basin display a continuous record of reservoir facies, models built from the stochastic correlations may result in different
compartmentalizations of the reservoir. The four models presented in Fig. 8 (representing a small subset of the possible stratigraphic correlation models) show two alternative views of the
distribution of reservoir rocks: in the models B and D, reservoir facies are concentrated into two main groups, whereas in models C and E, reservoir facies are divided into several units (6 in model
C and 5 in model D) separated by flow barriers. In an actual reservoir study, additional information coming from well production, such as production logging tool could be used to select which
stratigraphic correlation models are acceptable.
Considering alternative stratigraphic correlation models may also lead to different interpretations of the sedimentary and tectonic history of the studied area. For instance, Gari (2007) interprets
the unit U.I 6 as a prism constituted of marls whose deposition is due to a tectonic tilting. In relation with this tectonic activity, the production of carbonate is stopped on the platform,
suggesting a confinement and hypoxic event. In contrast, such a hypoxic event cannot be interpreted if model E is considered, because the equivalent prism is correlated to platform deposits.
4 Conclusions and perspectives
The presented methodology can rapidly generate several possible stratigraphic correlations of a set of stratigraphic sections. These stochastic correlations can be constrained by prior knowledge such
as correlation lines extracted from seismic data or/and hypothetical base profile geometry. Several geological scenarios may be tested in agreement with prior geological knowledge.
In the case of carbonate deposits, we have proposed to integrate paleo-bathymetric information to compute correlation likelihood. Many other rules could be considered to define other correlation
costs, which could be combined by linear combinations. However, defining costs that have comparable norms and dimensions can be a challenge.
We are currently working towards the definition of additional rules integrating more sedimentological and stratigraphic concepts in carbonate and siliciclastic settings. Indeed, in most depositional
settings, the facies type is not controlled only by bathymetry, but also by the distance to the source, water temperature, etc. An interesting area for further research is to better use 3D seismic
data when available to address correlation uncertainties. When wells and seismic data are available in the same domain (time or depth), seismic data indeed provide low-resolution correlation trends
that should be used in the hierarchical correlation. Another avenue could also be to connect depositional concepts and facies probabilities from seismic attributes (Baaske et al., 2007).
In any case, the validation of such a stochastic method is very delicate. Indeed, the set of possible correlations completely depends on the cost definitions and there is no absolute way of deciding
whether it is representative of the uncertainties other than scrutinizing the data and the rules and comparing results with analogs.
Nonetheless, stochastic stratigraphic correlation, combined with geostatistical facies simulation, is a new way to handle uncertainties on reservoir and basin modeling. We see it as complementary to
the classical methods performing multiple facies simulation on a unique grid. Further studies would be needed to assess the relative influence of stratigraphic uncertainty and petrophysical
uncertainties, and possibly to use inversion to reduce correlation uncertainties.
We would like to thank the editors (Philippe Joseph, Pierre Weil, Sylvie Bourquin and Vanessa Teles), Michael Pyrcz and another anonymous reviewer whose comments greatly contributed to improving this
paper. This research work was performed in the RING project managed by the “Association scientifique pour la géologie et ses applications” (ASGA). Companies and Universities of the Gocad Consortium (
http://www.ring-team.org/index.php/consortium/members) are hereby acknowledged for their support. We also thank Paradigm for providing the Gocad software and API.
Appendix A Combinatorial analysis
Considering two wells with respectively n and m identified stratigraphic markers and assuming that top and bottom markers of each well are correlated together, the number D[n,m] of possible
correlations between these two wells is given by the Delannoy number (e.g., see Banderier and Schwer (2005)):
$Dn,m=Dn,m−1+2∑i=1n−1Di,m−1D1,1=Dn,1=D1,m=1$ (A.1)
This equation can be extended in d dimensions, i.e., to enumerate the number of correlations between d wells with respectively n[i] markers per well:
$Dn1,...,nd=Dn1−1,...,nd+Dn1,n2−1,...,nd+...+Dn1,...,nd−1+Dn1−1,n2−1,...,nd+...+Dn1−1,n2,...,nd−1...+Dn1−1,...,nd−1$ (A.2)
Two simple numerical applications of Eqs. (A.1) and (A.2) highlight the very large number of possible combinations:
• • two wells comprising ten markers each yield D[10,10]=1,462,563 possible correlations;
• • twelve wells comprising seven markers each yield 10^80 possible correlation models, which is equivalent to the number of atoms in the universe.
Among this huge set of possible well correlations, only a relatively small subset is likely. Still, even the most likely thousandth of this set cannot be manually appreciated.
Appendix B Correlation ambiguities due to the paleo-angle of a carbonate ramp
Figure B.9
The uncertainty on the slope paleo-angle α of a carbonate ramp affects well correlations. (A) Impact of errors in average paleo-angles on the position error of the correlated horizon along the well
for different well spacings. Right: for a simple synthetic carbonate ramp (B), an error of 0.05° in the evaluation of the average paleo-angle leads to ambiguities on the correlation of two 10-m-thick
units (C). When the angular error reaches 0.15°, the ambiguity is between three such units (D).
Appendix C Chronostratigraphic and facies information in the Beausset study
Figure C.10
Chronostratigraphic and sequence stratigraphic subdivision of the Cretaceous southern Provence Basin. The studied interval is composed of twelve fourth order hemicycles.
Table C.1
Facies described by their possible depositional bathymetry. This description is used to build the depositional space presented in Fig. 7 and to interpolate the bathymetry along wells. Modified after
Gari (2007).
Facies Minimum bathymetry of deposition (m) Maximum bathymetry of deposition (m)
Value Uncertainty Value Uncertainty
F0: Charophyte limestone –50 0 0 0
F1: Micritic limestone 0 –5 1 5
F2: Bioclastic limestone with rudists 5 3 10 5
F3: Bioclastic limestone with rudists and corals 7 4 15 5
F4: Bioclastic limestone with fragment of rudists and corals 15 5 50 10
F5: Argillaceous limestone 50 25 100 25
F6: Marl and marly limestone 125 25 175 –
F7: Brecia, lobe, grainflow and debris flow 40 10 175 – | {"url":"https://comptes-rendus.academie-sciences.fr/geoscience/articles/10.1016/j.crte.2015.10.002/","timestamp":"2024-11-02T02:07:53Z","content_type":"text/html","content_length":"127319","record_id":"<urn:uuid:63faed6b-af00-4803-9298-4d22ebf8eb7e>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00661.warc.gz"} |
Homework 11 To practice working with multiple complex inputs. - Programming Help
• You and your partner must submit a single .rkt file containing your responses to all exercises via the Handin Server. We accept no email submissions.
• You must use the language specified at the top of this page.
• Your code must conform to the guidelines outlined in the style guide on the course website. The style guide will be updated as the semester progresses, so revisit it before submitting each
assignment. (You can resubmit as many times as you like until the due date without penalty.)
• Unless otherwise stated, every function and data definition that you design, including helper functions, must follow all steps of the design recipe.
Failure to comply with these expectations will result in deductions and possibly a 0 score.
Graded Exercises:
This homework is primarily exercising your abilities with manually handling multiple complex inputs. For all exercises, unless an explicit exemption is indicated, you cannot use any function that
extracts from, summarizes or modifies, a list; so, no calls to list-ref, reverse, length, second, third, fourth, etc., nor any list abstractions (you had sufficient practice with those on past
homeworks and the exam). You are only allowed to call first and rest, and the predicates empty? and cons? on lists. (For data types other than lists, there are no such restrictions.) The usual rule
of having a sufficient number of test cases applies, with a minimum of 3, but this is only a lower limit: some functions will clearly need more tests.
Exercise 1 Design the predicate function list-prefix? that accepts two lists of Numbers, and returns #true if the first list is a prefix of the second list, meaning all the elements of the first
list occur, in that specific order, as the first n elements of the second list.
Exercise 2 Design the function max-splice which given two lists of Numbers, will create the shortest resulting list which begins with all of the elements of the first list, in original order, and
ends with all of the elements of the second list, again in original order, taking advantage of any overlap to shorten the result. Here are a couple of illustrative examples:
(check-expect (max-splice ‘(1 2 3 4) ‘(2 3 4 5)) ‘(1 2 3 4 5))
; but:
(check-expect (max-splice ‘(1 2 3 4) ‘(2 2 3 4 5)) ‘(1 2 3 4 2 2 3 4 5))
In your implementation, you may use your solution for list-prefix? from the previous exercise.
Exercise 3 Design the predicate function valid-results?, which is given three lists of equal length:
a list of input values
a list of functions
a list of results
For each member i of the first list, it will apply the function at the corresponding position in the second list to i, and compare the output with the result in the corresponding position in the
third list. valid-results? should return #true if all of the function call outputs match the expected results.
Exercise 4 Design the function assign, which matches up members of a list of roles, each of type Symbol, with corresponding members of a list of people, which are of type String; if the list of
people is shorter, unfilled roles are paired with #false (so the second field should be a [Maybe String]; however, if the list of people is longer, the extra people (beyond the number of roles)
should be ignored. The result should be a [List-of Assignment], defined here:
(define-struct assignment (role person))
; An Assignment is a (make-assignment Symbol [Maybe String])
For the next set of exercises, you will be using a very simple version of a data type for representing a binary tree:
Exercise 5 For this problem, you will design the function tree-equiv, which compares two trees for alignment, with an important twist: it allows any two trees to be considered equivalent if each
contains subtrees equivalent to one of the subtrees in the corresponding other tree. So, these two trees would be reported as equivalent:
; a a
; / \ / \
; b c c b
; / \ / \ / \ / \
; d e f g f g e d
Another way to think of it is: for a family tree, what if instead of a father and mother, you just identified two equivalent parents? (For this, you will need more than the typical number of test
Exercise 6 Design the function find-subtree, which given two parameters: a tree, as well as a subtree to scan for (both of type BT), will search the main tree (the first parameter) for the
matching subtree. Here, we are looking for an exact match, all the way down to the leaves (the value ‘none), and not a flippable one as in the previous exercise. Note that values stored at
various nodes can repeat, so you cannot simply look for a matching root node and asume the subtree therefore matches. Also, you can stop at the first full match, in the case where the subtree
occurs multiple times in the main tree.
Exercise 7 Design a function max-common-tree, which given two binary trees (again, continuing with the data type given earlier in this homework, of type BT), will compare the two trees starting
with the root of each, returning a tree that shares the maximum number of nodes with both source trees, starting from the root. Since the resulting tree must also be a BT, it will have leaves
with the symbol ‘none— ‘none):
; t1: A t2: A
; / \ / \
; B C B X
; / \ / \ / \ / \
; D E F G D Y F n
; / \ / \ / \ / \ / \ / \ / \
; n n n n n n n n n n n n n n
; result1: A result2: A
; / \ / \
; B n B n
; / \ / \
; D n n n
; / \
; n n
The tree result1 is the correct answer. Note that the node ‘F doesn’t appear in the common tree, even though it appears in the same position, and with the same value, in both source trees,
because the node ‘C in tree t1 does not match corresponding node ‘X in tree t2, cutting off anything below them. While result2 is in fact a common tree of both t1 and t2, it is not the maximal
such tree, and is therefore incorrect.
Exercise 8 To find an existing value in a Binary Search Tree (BST), you have to turn left or right correctly at every node as you descend a BST, by examining the value stored at each node you
visit, and then deciding to go down the left or right branch. Your goal for this exercise is to design a function that can validate a search algorithm’s decisions as it descends a BST, as well as
the correctness of the BST’s structure, at least the part you traverse. You will design a function valid-bst-path? , which is passed a binary search tree, and a number that is stored in that
tree, as well as a search path, defined as a [List-of Dir], i.e., a sequence of ‘left and ‘right turns. Your function must validate every left/right decision in the given path, as well as whether
you end up at the correct node. It should return #true if the passed path makes the correct branching decision at each node, as well as arriving at the desired node. Note that just arriving at
the correct node at the end is not sufficient, since you are also validating parts of what might be a malformed tree. You do not have to validate the whole BST: just opportunistically check the
parts along the path.
You can assume that a Dir data type is already defined, as follows:
; A Dir is one of:
; – ‘left
; – ‘right
Note that while you should typically call out to a helper function to handle the Dir data type in the [List-of Dir], here you are allowed to handle it directly in the main function as we did in
the lecture example involving a FamilyTree and AncestorPath, but only if the design is still sufficiently simple and clear.
Exercise 9 Design the function merge, that takes two ordered lists containing elements of the same type (not necessarily numbers!), and produces a single resulting ordered list. Note that the
user passes in a comparison function lt-func, which takes two parameters, and will return #true if the first parameter is “less than”, i.e., should come before, the second. You are not allowed to
call DrRacket’s sort function! | {"url":"https://www.edulissy.org/product/you-and-your-partner-must-submit-a-single-rkt-file-containing-your-responses-to-all-exercises-via-the-handin-server-4/","timestamp":"2024-11-03T18:58:00Z","content_type":"text/html","content_length":"222325","record_id":"<urn:uuid:ee72ae4e-8ee8-4e55-854d-e9e70a590656>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00763.warc.gz"} |
binmodel converts polynomial expressions involving binary and continuous variables to linear expressions by introducing additional variables and constraints.
[plin1,...,plinN,F] = binmodel(p1,...,pN,Domain)
The following example solves a quadratic program with binary variables using a mixed integer linear programming solver, by first converting the quadratic function to a linear expression
x = binvar(5,1);
Q = randn(5);
p = x'*Q*x;
[plinear,F] = binmodel(p)
Of course, for this to work, you need a mixed integer linear programming solver.
Products between continuous and binary variables are also supported, but for the big-M modelling to work, you have to specify bounds on the continuous variables
x = binvar(5,1);
y = sdpvar(5,1);
Q = randn(5);
p = x'*Q*y;
[plinear,F] = binmodel(p,[-2 <= y <= 2]);
The derivation of the linear model is based on simple logic and big-M modelling. A product of two binary variables \(x\) and \(y\) is replaced with a new binary variable \(z\) and the constraints
[z <= x, z <= y, z>= x+y-1];
This idea can be generalized to arbitrary polynomials in binary variables.
A product between a binary variable \(x\) and a continuous variable \(w\), with known lower and upper bounds \(L\) and \(U\), is replaced by a new continuous variable \(v\) and the constraints
[ L*x <= v <= x*U, L*(1-x) <= w-v <= U*(1-x)]
This can be generalized to expressions arbitrarily polynomial w.r.t the binary variable. The continuous variable must enter linearly though (for fixed binary). | {"url":"https://yalmip.github.io/command/binmodel/","timestamp":"2024-11-09T16:12:16Z","content_type":"text/html","content_length":"33567","record_id":"<urn:uuid:f83869d5-b1d5-43a2-9262-b0d961b0cb8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00809.warc.gz"} |
how can i solve this operate differential equation
convert(expand((D[1] - 1)*(D[1]^2 + 2)*y(x) = 0), diff);
hi. how can i get this solution of eq
like this (just show not solving)
a := 3;
b := 5;
c := -1;
eq := exp(-c)/(a*b) + a^b*exp(c);
hi. i write this code with "for" loop. but i dont want to show comma sign in my print. how can i remove it
for example
thanks in advance
for i to 2 do
for j to 2 do S[i, j] := 2*mu*varepsilon[i, j] - add((2*mu)/3*varepsilon[r, r]*delta[i, j], r = 1 .. 3); print("S"[i, j] = S[i, j]); end do;
end do;
i want to plot these two equation that z is complex.
i try it by implicitplot but for the second one it's not work. thank you
i have a acceleration of record earthquake(in excel ) and i get fourier and power spectra from another software (seismosignal) .Now i want to get it in maple but i try a lot but i couldn't.
i very Eager to learn it | {"url":"https://mapleprimes.com/users/kambiz1199/questions?page=3","timestamp":"2024-11-07T10:27:16Z","content_type":"text/html","content_length":"1049012","record_id":"<urn:uuid:badc7b2f-d94e-40c9-a47d-c2a32e953251>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00248.warc.gz"} |
seminars - Optimal Entry and Exit with Signature in Statistical Arbitrage
We explore an optimal timing strategy for the trading of price spreads exhibiting mean-reverting characteristics. A sequential optimal stopping framework is formulated to analyze the optimal timings
for both entering and subsequently liquidating positions, all while considering the impact of transaction costs. Then we leverages a refined signature optimal stopping method to resolve this
sequential optimal stopping problem, thereby unveiling the precise entry and exit timings that maximize gains. Our framework operates without any predefined assumptions regarding the dynamics of the
underlying mean-reverting spreads, offering adaptability to diverse scenarios.
Numerical results are provided to demonstrate its superior performance when comparing with conventional mean reversion trading rules. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&page=4&document_srl=1250374&l=ko","timestamp":"2024-11-13T02:55:52Z","content_type":"text/html","content_length":"43685","record_id":"<urn:uuid:9f87ccdb-c7d5-4518-927b-27bdb05b317a>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00796.warc.gz"} |
Bridged Graph -- from Wolfram MathWorld
A bridged graph is a graph that contains one or more graph bridges. Examples of bridged graphs include path graphs, ladder rung graphs, the bull graph, star graphs, and trees.
A graph that is not bridged is said to be bridgeless. A connected bridgeless graph can be tested for in the Wolfram Language using Not[KEdgeConnectedGraphQ[g, 2]] or EdgeConnectivity[g]
The numbers of simple bridged graphs on A263915).
The numbers of simple connected bridged graphs on A052446). | {"url":"https://mathworld.wolfram.com/BridgedGraph.html","timestamp":"2024-11-04T06:00:06Z","content_type":"text/html","content_length":"53744","record_id":"<urn:uuid:2ab09c18-2112-473e-87bd-4a8f58b74b5e>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00214.warc.gz"} |
In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function
composition as group operation.[1] Its identity element is the identity function.[2] The elements of the isometry group are sometimes called motions of the space.
Every isometry group of a metric space is a subgroup of isometries. It represents in most cases a possible set of symmetries of objects/figures in the space, or functions defined on the space. See
symmetry group.
A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set.
In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form; transformations preserving this form are sometimes called "isometries", and the collection of them is then said to
form an isometry group of the pseudo-Euclidean space. | {"url":"https://www.knowpia.com/knowpedia/Isometry_group","timestamp":"2024-11-09T22:15:21Z","content_type":"text/html","content_length":"77064","record_id":"<urn:uuid:612a7a73-4ad0-49c7-b4d7-44d6594da932>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00131.warc.gz"} |
Scientific Notation Calculator
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\(b \times 10^n\)
For example, the speed of light is usually expressed in scientific notation because the number is too big to write in standard notation.
Speed of light scientific notation = \(3.0 \times 10^8 m/s\)
Below is the scientific notation chart in which scientific notation is compared with the equivalent standard notation for your understanding.
Power Scientific Notation Value
-3 1 x 10^-3 0.001
-2 1 x 10^-2 0.01
-1 1 x 10^-1 0.1
1 1 x 10^1 10
2 1 x 10^2 100
3 1 x 10^3 1000
4 1 x 10^4 10,000
5 1 x 10^5 1,00,000
6 1 x 10^6 10,00,000
Scientific Notation Rules
We have to remember a few rules when working with scientific notations. Followings are some rules that you should follow before converting scientific notation to standard or applying any arithmetic
operation to scientific numbers.
• The decimal has to be a non-zero number and should lie between the first two non-zero numbers.
• The number before the multiplication mark is called the mantissa or significant.
• The total numbers of digits in the significant are known as significant figures. You can use our Sig Fig Calculator to calculate significant figures.
• The exponent value is based on whether the decimal place is shifted to the left or to the right.
If you are wondering how to do scientific notation, read in the next section where we will discuss several scientific notation operations with scientific notation examples.
Scientific Notation Operations
Adding and subtracting scientific notation is easy as compared to multiplying and dividing scientific notation. Let’s understand all arithmetic operations of scientific notations by using examples.
Suppose we have two scientific notations:
\(x_1= 3 \times 10^3\)
\(x_2 = 2 \times 10^3\)
Adding scientific notations is simple. If the base power is the same, add the mantissa and write the base 10 with unchanged power.
\(x_1 + x_2 = \times 10^3 + 2 \times 10^3 = 5 \times 10^3\)
To subtract scientific notations, subtract the mantissa and write the base 10 with unchanged power.
\(x^1 - x^2 = 3 \times 10^3 - 2 \times 10^3 = 1 \times 10^3\)
Multiplying scientific notations are different from adding or subtracting. In multiplication, the power of the base 10 is added, which is not the case in addition or subtraction.
\(x_1 \times x_2 = (3 \times 10^3) \times (2 \times 10^3) = 1 \times 10^6\)
Similar to multiplication, dividing scientific notations requires an operation on the powers of base 10. In division, powers of base 10 are subtracted to get the result.
\(\dfrac{x_1}{x_2} = \dfrac{(3 \times 10^3)}{(2 \times 10^3)} = 1.5 \times 10^0 = 1.5\) | {"url":"https://www.calculators.tech/scientific-notation-calculator","timestamp":"2024-11-10T15:07:31Z","content_type":"text/html","content_length":"43284","record_id":"<urn:uuid:b4a0e077-8b46-45e5-830f-aca8a45b5181>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00298.warc.gz"} |
Physics 202 - Lab Manual - Wsu - Washington State University
Physics 202 - Lab Manual - Wsu - Washington State University
variable by W.Briggs and L. Cochrane. Drawing on their decades of teaching experience, William. Briggs and Lyle Cochran have created a calculus text that.
Calculus Early Transcendentals 3rd Edition
Essayez avec l'orthographe
CATALOGUE 1976-1977 - Wikimedia Commons
College Leadership. Alfred H. Bloom. President. Academic Policy. Constance Cain Hungerford. Provost. Student Services. James Larimore.
Swarthmore - College Bulletin 2007?2008
Our current undergraduate curriculum often omits the important area of computational literacy and data technologies which is becoming increasingly.
abstract book - jsm - American Statistical Association
This volume of the Religion and the Social Order series is, to the best of my recollection, the first that was actually finished early.
Toward a Sociological Theory of Religion and Health
and multivariable calculus taken by students of mathematics ... calculus early transcendentals briggs william cochran lyle -. May 02 2023 web ...
The Truth About The Philadelphia Experiment - Gov.BC.Ca
... Calculus learners, it was tedious because it was hard to motivate. Therefore, with the evolution of calculus education, there has been a ...
UNDERGRADUATE CATALOG - Lipscomb University
May be taken only with the approval of the mathematics faculty. Offered only to math majors who want to study a math course not in the catalog. Requires ...
with coding theory / Wade Trappe, L - UCSC Academic Senate
Introduction to cryptography : with coding theory / Wade Trappe, Lawrence C. Washington. Lecture notes in mathematics, 501-1000 : an index and other useful ...
Jozef Mak Jozef Mak - A Loxley (2024) resources.caih.jhu.edu
william l briggs lyle cochran - Oct 27. 2022 web william l briggs lyle cochran bernard gillett pearson addison wesley. 2011 calculus 1081 pages drawing on.
Mathematics,Probability and Statistics E-Book
Essayez avec l'orthographe
UEF 2.1.2 Matière 1: Mécanique des fluides VHS: 45h00 (Cours
Semestre: 3. Unité d'enseignement: UEF 2.1.2. Matière 1: Mécanique des fluides. VHS: 45h00 (Cours: 1h30, TD: 1h30). Crédits: 4. Coefficient: 2.
This book discusses thermofluids in the context of thermodynamics, single- and two-phase flow, as well as heat transfer associated with single- and two-phase ... | {"url":"https://telecharger-cours.net/docdetails-426595.html","timestamp":"2024-11-05T16:59:30Z","content_type":"text/html","content_length":"12459","record_id":"<urn:uuid:cb19159e-c6f7-41e0-9306-db2216a52389>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00024.warc.gz"} |
Econometric Sense
I have recently started reading
"Applied Nonparametric Econometrics",
and was thinking, when was the last time I even worked with basic non-parametric statistics?
For instance, in the courses I teach, I don't cover this, but some of the texts I reference cover some basics like the Mann Whitney Wilcoxon (MWW) test (which can be thought of as a non-parametric
equivalent to a two sample independent t-test) or the Kruskall-Wallis test (which is a non-parametric analogue to analysis of variance). These tests are often useful in situations that involve highly
skewed, non-normal, or categorical ordered or ranked data, or data from problematic or unknown distributions. I kind of briefly reviewed some implementations in SAS, and particularly focused on the
Kruskall-Wallis test, which has the following general null hypothesis:
Ho: All Populations Are Equal
Ha: All Populations Are Not Equal
If we reject Ho, we might conclude that there is a difference among populations, with one population or another providing a larger proportion of larger or smaller values for the variable of interest.
If we could assume that the populations were of similar shape and symmetry, this *might* be interpreted as a test of differences in medians, but in general
this is a test on differences in distributions and specifically ranks, similar to the MWW test
. But if we do reject Ho, what next? In an analysis of variance context, if we reject the overall F-test on multiple means we can followup with pairwise comparisons to determine which means differ.
But at least in the older versions of SAS, there are no straightforward ways to do this kind of analysis in the non-parametric context. However, in the
SAS Note (22620)
, one recommendation is to rank-transform the data and use the normal-theory methods in PROC GLM (Iman, 1982). See also Conover, W. J. & Iman, R. L. (1981) referenced below.
A good example of the application of GLM on ranked data can be found here:
and a general overview of some non-parametric applications in SAS along these lines
You can also find a SAS macro with code and examples for post hoc tests here:
I at first thought this was the macro by Juneau (in the references below and mentioned in the SAS note above) but it is something different, see the Elliot and Hynan reference below. From the
"The Kruskal-Wallis (KW) nonparametric analysis of variance is often used instead of a standard one-way ANOVA when data are from a suspected non-normal population. The KW omnibus procedure tests for
some differences between groups, but provides no specific post hoc pair wise comparisons. This paper provides a SAS(®) macro implementation of a multiple comparison test based on significant
Kruskal-Wallis results from the SAS NPAR1WAY procedure. The implementation is designed for up to 20 groups at a user-specified alpha significance level. A Monte-Carlo simulation compared this
nonparametric procedure to commonly used parametric multiple comparison tests."
I found an application referencing this implementation
if interested.
According to the SAS note referenced above, SAS/STAT 12.1 will include some versions of some non-parametric post hoc tests. I'm also aware that there are several R packages that can do this as well,
such as the
I compared results from Elliot and Hynan's example code (
example 1
) and data to those from the adhoc GLM on ranks following
and got similar results. I also got similar results using dunn.test in R:
# use same data as in www.alanelliott.com/kw
race <- c(1,1,1,1,1,2,2,2,2,2,3,3,3,3,3)
bmi <- c(32,30.1,27.6,26.2,28.2,26.4,23.1,23.5,24.6,24.3,24.9,25.3,23.8,22.1,23.4)
library(dunn.test) #load package
dunn.test(bmi,race, kw = TRUE, method ="bonferroni") # implement test with adjustments for multiple comparisons
Created by Pretty R at inside-R.org References:
Palomares-Rius JE, Castillo P, Montes-Borrego M, Navas-Cortés JA, Landa BB (2015) Soil Properties and Olive Cultivar Determine the Structure and Diversity of Plant-Parasitic Nematode Communities
Infesting Olive Orchards Soils in Southern Spain. PLoS ONE 10(1): e0116890. doi:10.1371/journal.pone.0116890
Dunn, O.J. “Multiple comparisons using rank sums”.
Technometrics 6 (1964) pp. 241-252.
Conover, W. J. & Iman, R. L. (1981). "Rank transformations as a bridge between parametric and
nonparametric statistics". American Statistician 35 (3): 124–129. doi:10.2307/2683975
Elliott AC, Hynan LS. “A SAS Macro implementation of a Multiple Comparison post hoc test for a Kruskal-Wallis analysis,” Comp Meth Prog Bio, 102:75-80, 2011
Iman, R.L. (1982), "
Some Aspects of the Rank Transform in Analysis of Variance Problems
Proceedings of the Seventh Annual SAS Users Group International Conference
, 7, 676-680.
Juneau, P. (2004), "
Simultaneous Nonparametric Inference in a One-Way Layout Using the SAS System
Proceedings of the PharmaSUG 2004 Annual Conference
, Paper SP04.
Paul Allison discusses zero inflated vs negative binomial models in a post I stumbled across recently. Also William Greene and Paul go back and forth on some technical distinctions and nuances (which
may be quite important) in the comments.
"In all data sets that I've examined, the negative binomial model fits much better than a ZIP model, as evaluated by AIC or BIC statistics. And it's a much simpler model to estimate and interpret. So
if the choice is between ZIP and negative binomial, I'd almost always choose the latter."
"But what about the zero-inflated negative binomial (ZINB) model? It's certainly possible that a ZINB model could fit better than a conventional negative binomial model regression model. But the
latter is a special case of the former, so it's easy to do a likelihood ratio test to compare them (by taking twice the positive difference in the log-likelihoods). In my experience, the difference
in fit is usually trivial..."
"So next time you're thinking about fitting a zero-inflated regression model, first consider whether a conventional negative binomial model might be good enough. Having a lot of zeros doesn't
necessarily mean that you need a zero-inflated model."
A few weeks ago, there was a post that caught my attention at the 'Kids Prefer Cheese' blog titled "Friends don't let Friends do IV" which was very critical of instrumental variable techniques.
Around that same time, Marc Bellemare posted a contrasting piece, titled "Friends do let Friends do IV".
For some reason, I've written a number of posts recently related to instrumental variables, discussing different intuitive approaches to understanding them, or connections with directed acyclic
graphs (DAGs). In the past, I have discussed them in the context of omitted variable bias and unobserved heterogeneity and endogeneity.
Now some colleagues have introduced me to a few papers authored by Quin that really question the validity of using instruments in this context. In the first paper, Resurgence of the
Endogeneity-Backed Instrumental Variable Methods, Quin states:
“Essentially, the paranoia grows out of the fallacy that independent error terms exist prior to model specification and carry certain ‘structural’ interpretation similar to other economic
variables…..In fact, it is practically impossible to validate the argument of endogeneity bias on the ground of correlation between a regressor and the error term in a multiple regression setting,
especially when the model fit remains relatively low. Notice how much the basis of the IV treatment for ‘selection on the unobservables’ is weakened once 'e' is viewed as a model-derived compound of
unspecified miscellaneous effects. In general, error terms of statistical models are derived from model specification. As such, they are unsuitable for any ‘structural’ interpretation, e.g. see Qin
and Gilbert (2001)”
Quin goes deeper into this in a later working paper, Time to Demystify Endogeneity Bias.
From the abstract-
"This study exposes the flaw in defining endogeneity bias by correlation between an explanatory variable and the error term of a regression model. Through dissecting the links which have led to
entanglement of measurement errors, simultaneity bias, omitted variable bias and self
-selection bias, the flaw is revealed to stem from a Utopia mismatch of reality directly with single explanatory variable models."
The paper gets pretty heavy in details, despite promises to keep the math at a minimum. One of the central arguments they make about "endogeniety bias syndrome" is to point out an apparent
misunderstanding or misinterpretation of error terms in multivariable vs single variable regression that is often used in applied work to set the stage for doing IV:
"Error terms or model residuals have been long perceived as sundry composites of what modellers are unable and/or uninterested to explain since Frisch’s time....Since cov(z,e)≠ 0 is single variable
based, the contents of the error term have to be adequately ‘pure’, definitely not a mixture of sundry composites, to sustain its significant presence. Indeed, textbook discussions of endogeneity
bias, be it associated with SB (simultaneity bias), measurement errors, OVB(omitted variable bias) or SSB (self-selection bias), are all built on simple regression models. As soon as these models are
extended to multiple ones, the correlation becomes mathematically intractable. In a multiple regression, all the explanatory variables are mathematically equal. Designation of one as the causing
variable of interest and the rest as control variables is purely from the substantive standpoint. The premise, cov(x,e)≠ 0, implies not only cov(z,e)≠ 0 for the entire set of control variables, but
also the set being exhaustive. Both conditions are almost impossible to meet in practice."
Quin also has an applied paper related to wage elasticities where some of these ideas are put into context. See the references below.
Duo Qin (2015). Resurgence of the Endogeneity-Backed Instrumental Variable Methods. Economics: The Open-Access, Open-Assessment E-Journal, 9 (2015-7): 1—35. http://dx.doi.org/10.5018/
QIN, D. (2015) “Time to Demystify Endogeneity Bias” SOAS
Department of Economics Working
Paper Series, No. 192, The School of Oriental and African Studies
192 Time to Demystify Endogeneity Bias (pdf)
Qin, D., S. van Huellen and QC. Wang. (2014), “What Happens to Wage Elasticities When We Strip Playometrics? Revisiting Married Women Labour Supply Model”, SOAS Department of Economics Working Paper
Series, No. 190, The School of Oriental and
African Studies https://www.soas.ac.uk/economics/research/workingpapers/file97784.pdf | {"url":"https://econometricsense.blogspot.com/2015/12/","timestamp":"2024-11-11T10:12:01Z","content_type":"text/html","content_length":"102447","record_id":"<urn:uuid:c140f7f3-3de9-4997-91e8-18436db8dcdf>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00643.warc.gz"} |
Generalized Bezoutian and the inversion problem for block matrices, I. General scheme for Integral Equations and Operator Theory
Integral Equations and Operator Theory
Generalized Bezoutian and the inversion problem for block matrices, I. General scheme
View publication
The unified approach to the matrix inversion problem initiated in this work is based on the concept of the generalized Bezoutian for several matrix polynomials introduced earlier by the authors. The
inverse X-1 of a given block matrix X is shown to generate a set of matrix polynomials satisfying certain conditions and such that X-1 coincides with the Bezoutian associated with that set. Thus the
inversion of X is reduced to determining the underlying set of polynomials. This approach provides a fruitful tool for obtaining new results as well as an adequate interpretation of the known ones. ©
1986 Birkhäuser Verlag. | {"url":"https://research.ibm.com/publications/generalized-bezoutian-and-the-inversion-problem-for-block-matrices-i-general-scheme","timestamp":"2024-11-12T22:59:07Z","content_type":"text/html","content_length":"71713","record_id":"<urn:uuid:3e97f493-ca64-43f0-b59b-9716567a4072>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00323.warc.gz"} |