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Solution to Problem D – Delicious Cake
IPSC 2007
Solution to Problem D – Delicious Cake
For cakes up to 3×3 we can try all possible combinations of cutting/not cutting of each edge between two neigbouring unit squares. For each such combination we can verify its validity as follows. We
first compute the connected components of the cake (using the depth first search, or the breadth first search). Then we verify that there is no cut between two unit squares which belong to the same
connected component.
For cakes of the dimensions 1×N (resp. N×1) all possible combinations are valid. This gives simple formula, 2^(N-1), for the number possible cuttings.
For cakes of the dimensions 2×N (resp. N×2) we set up a recurrence as follows. Let us think of a 2×N cake as two rows of unit squares of length N each. We denote by a[N] the number of possible
cuttings of a 2×N cake in such way that the two rightmost unit squares belong to the same connected component. Similarly, we denote by b[N] the number of possible cuttings of a 2×N cake in such way
that the two rightmost unit squares belong to two different connected components. We readily check that a[1] = 1 and b[1] = 1.
The following picture demonstrates that a[N]=2a[N-1] + 3b[N-1]:
The following picture demonstrates that b[N]=3a[N-1] + 4b[N-1]:
From these recurrences we obtain a[2]=5 and b[2]=7. If we denote by s[N] the sum of a[N] and b[N], putting the two recurrences together gives: s[N] = (a[N] + b[N]) = 6(a[N-1] + b[N-1]) + b[N-1] - a
= 6s[N-1] + 3a[N-2] + 4b[N-2] - 2a[N-2] - 3b[N-2]
= 6s[N-1] + a[N-2] + b[N-2]
= 6s[N-1] + s[N-2]
and s[1] = 2 and s[1] = 12. This recurrence can be implemented in a language which supports long integer arithmetic; the unix tool bc is ideal for such job. (See bc script solving inputs 2×N.).
The solution for arbitrary dimension M×N is a generalization of the 2×N case, that is, we set up system of (linear) recurrences. However, our solution will work only for small values of M. In
practice today's PC is able to handle M ≤ 5.
(An alternate view of the same situation: we will consider one dimension M to be fixed and use dynamic programming to compute the number of cuttings for each N.)
In our solution, for every partition p of the last (i.e. N-th) column into connected components, and every combination q which determines whether any pair of the components occuring in the last
column touch each other somewhere in the cake, we compute the number a(N,p, q) – the number of different cuttings of the M×N cake obeying p and q.
This is best demonstrated on an example. We consider M=3 and the partition p=(1,2,3) of the last column:
The partition p implies that components 1,2 touch and likewise 2,3. Moreover, let us assume that the components 1,3 do not touch anywhere in the cake before the last column. This is formally captured
by the combination q={{1,2}, {2,3}}. (Formally q is a set of unordered pairs). Suppose that we have computed the value a(N-1,p,q).
We try all the possible valid combinations how to cut the last column. Several of them are shown are shown of the following picture.
Note that the last combination is valid, since we assume that the components 1,3 do not touch anywhere before the last column.
Each of the valid combinations contributes a(N-1,p,q) to some a(N,p',q') for some p', q'. For example the first two combinations in the picture together with the last one contribute 3a(N-1,p,q) to a
(N, p', q') where p'=(1,1,1) and q'={}. (Note that the partitions (4,4,4) and (1,1,1) and (1/3,1/3,1/3) are equivalent.) Likewise, the third option in the picture contributes 1a(N-1,p,q) to a(N, p'',
q''), where p''=(2,2,3) and q={{2,3}}. (Or equivalently, p''=(1,1,2) and q''=(1,1,2).)
Thus, by trying all combinations of p,q and all ways how to cut the last, yields the coefficients of the linear system of recurrences for the numbers a(N,p,q). Using these coefficients we can compute
the values a(N,p,q) for any N. For example for M=3, there are 4 options for p; ((1,1,1), (1,1,2), (1,2,2), (1,2,3)), and 8 options for q; ({}, {{1,2}}, {{2,3}}, {{1,3}}, {{1,2}, {2,3}}, {{1,2},
{1,3}}, {{1,3}, {2,3}} and , {{1,2}, {2,3}, {1,3}}). Thus the whole system of recurrences consists of 4*8=32 variables a(N,p,q) (for each N).
We omit couple of details, such as how to compute the initial values a(1,p,q), and how determine whether a particular cutting of the last column is valid for some p,q, and how to represent partions p
and sets q. | {"url":"https://ipsc.ksp.sk/2007/real/solutions/d.html","timestamp":"2024-11-04T05:56:07Z","content_type":"text/html","content_length":"9077","record_id":"<urn:uuid:3a391c24-642e-41d8-8725-02cb955949d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00139.warc.gz"} |
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Processing Symposium (IPDPS), Orlando, FL, June 2017, pp. 678-687. | {"url":"http://lpna.cs.illinois.edu/","timestamp":"2024-11-05T10:44:19Z","content_type":"text/html","content_length":"20031","record_id":"<urn:uuid:6dc612ed-b786-4efa-9667-f67b19e97432>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00788.warc.gz"} |
The following derivation is adapted from Foundations of Chemical Kinetics.^[2] This derivation assumes the reaction ${\displaystyle A+B\rightarrow C}$ . Consider a sphere of radius ${\displaystyle R_
{A}}$ , centered at a spherical molecule A, with reactant B flowing in and out of it. A reaction is considered to occur if molecules A and B touch, that is, when the distance between the two
molecules is ${\displaystyle R_{AB}}$ apart.
If we assume a local steady state, then the rate at which B reaches ${\displaystyle R_{AB}}$ is the limiting factor and balances the reaction.
Therefore, the steady state condition becomes
1. ${\displaystyle k[B]=-4\pi r^{2}J_{B}}$
${\displaystyle J_{B}}$ is the flux of B, as given by Fick's law of diffusion,
2. ${\displaystyle J_{B}=-D_{AB}({\frac {dB(r)}{dr}}+{\frac {[B]}{k_{B}T}}{\frac {dU}{dr}})}$ ,
where ${\displaystyle D_{AB}}$ is the diffusion coefficient and can be obtained by the Stokes-Einstein equation, and the second term is the gradient of the chemical potential with respect to
position. Note that [B] refers to the average concentration of B in the solution, while [B](r) is the "local concentration" of B at position r.
Inserting 2 into 1 results in
3. ${\displaystyle k[B]=4\pi r^{2}D_{AB}({\frac {dB(r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}})}$ .
It is convenient at this point to use the identity ${\displaystyle \exp(-U(r)/k_{B}T)\cdot {\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)=({\frac {dB(r)}{dr}}+{\frac {[B](r)}{k_{B}T}}{\frac {dU}{dr}})}$
allowing us to rewrite 3 as
4. ${\displaystyle k[B]=4\pi r^{2}D_{AB}\exp(-U(r)/k_{B}T)\cdot {\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)}$ .
Rearranging 4 allows us to write
5. ${\displaystyle {\frac {k[B]\exp(U(r)/k_{B}T)}{4\pi r^{2}D_{AB}}}={\frac {d}{dr}}([B](r)\exp(U(r)/k_{B}T)}$
Using the boundary conditions that ${\displaystyle [B](r)\rightarrow [B]}$ , ie the local concentration of B approaches that of the solution at large distances, and consequently ${\displaystyle U(r)\
rightarrow 0}$ , as ${\displaystyle r\rightarrow \infty }$ , we can solve 5 by separation of variables, we get
6. ${\displaystyle \int _{R_{AB}}^{\infty }dr{\frac {k[B]\exp(U(r)/k_{B}T)}{4\pi r^{2}D_{AB}}}=\int _{R_{AB}}^{\infty }d([B](r)\exp(U(r)/k_{B}T)}$ or
7. ${\displaystyle {\frac {k[B]}{4\pi D_{AB}\beta }}=[B]-[B](R_{AB})\exp(U(R_{AB})/k_{B}T)}$ (where : ${\displaystyle \beta ^{-1}=\int _{R_{AB}}^{\infty }{\frac {1}{r^{2}}}\exp({\frac {U(r)}{k_{B}T}}
dr)}$ )
For the reaction between A and B, there is an inherent reaction constant ${\displaystyle k_{r}}$ , so ${\displaystyle [B](R_{AB})=k[B]/k_{r}}$ . Substituting this into 7 and rearranging yields
8. ${\displaystyle k={\frac {4\pi D_{AB}\beta k_{r}}{k_{r}+4\pi D_{AB}\beta \exp({\frac {U(R_{AB})}{k_{B}T}})}}}$
Limiting conditions
Very fast intrinsic reaction
Suppose ${\displaystyle k_{r}}$ is very large compared to the diffusion process, so A and B react immediately. This is the classic diffusion limited reaction, and the corresponding diffusion limited
rate constant, can be obtained from 8 as ${\displaystyle k_{D}=4\pi D_{AB}\beta }$ . 8 can then be re-written as the "diffusion influenced rate constant" as
9. ${\displaystyle k={\frac {k_{D}k_{r}}{k_{r}+k_{D}\exp({\frac {U(R_{AB})}{k_{B}T}})}}}$
Weak intermolecular forces
If the forces that bind A and B together are weak, ie ${\displaystyle U(r)\approx 0}$ for all r except very small r, ${\displaystyle \beta ^{-1}\approx {\frac {1}{R_{AB}}}}$ . The reaction rate 9
simplifies even further to
10. ${\displaystyle k={\frac {k_{D}k_{r}}{k_{r}+k_{D}}}}$ This equation is true for a very large proportion of industrially relevant reactions in solution.
Viscosity dependence
The Stokes-Einstein equation describes a frictional force on a sphere of diameter ${\displaystyle R_{A}}$ as ${\displaystyle D_{A}={\frac {k_{B}T}{3\pi R_{A}\eta }}}$ where ${\displaystyle \eta }$ is
the viscosity of the solution. Inserting this into 9 gives an estimate for ${\displaystyle k_{D}}$ as ${\displaystyle {\frac {8RT}{3\eta }}}$ , where R is the gas constant, and ${\displaystyle \eta }
$ is given in centipoise. For the following molecules, an estimate for ${\displaystyle k_{D}}$ is given:
Solvents and ${\displaystyle k_{D}}$
Solvent Viscosity (centipoise) ${\displaystyle k_{D}({\frac {\times 1e9}{M\cdot s}})}$
n-Pentane 0.24 27
Hexadecane 3.34 1.9
Methanol 0.55 11.8
Water 0.89 7.42
Toluene 0.59 11 | {"url":"https://www.knowpia.com/knowpedia/Diffusion-controlled_reaction","timestamp":"2024-11-11T17:36:17Z","content_type":"text/html","content_length":"149556","record_id":"<urn:uuid:1054b4ed-9b3c-42ab-9a01-a509bc0a3e62>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00869.warc.gz"} |
Area of Parallelograms | Curious Toons
Table of Contents
Introduction to Parallelograms
Definition and Properties
A parallelogram is a special type of quadrilateral, which means it has four sides. What makes a parallelogram unique is that both pairs of opposite sides are equal in length and parallel to each
other. Additionally, the opposite angles in a parallelogram are equal, and the consecutive angles are supplementary, meaning they add up to 180 degrees. Another key property is that the diagonals of
a parallelogram bisect each other; this means that if you draw the lines connecting opposite corners, those lines will intersect at their midpoints. This beautiful geometric figure showcases symmetry
and balance, making it a fundamental shape in geometry. Whether in practical applications like architecture and engineering or in various fields of design, understanding the properties of
parallelograms helps us appreciate their presence in the world around us!
Types of Parallelograms
Parallelograms come in various shapes, each with unique properties and characteristics. The most common types include rectangles, rhombuses, and squares. A rectangle is a parallelogram with all four
angles equal to 90 degrees, making it look like a stretched-out square. A rhombus, on the other hand, has sides that are all equal in length, but the angles are not necessarily right angles; instead,
they are opposite angles that are equal. Finally, a square is a special case of both a rectangle and a rhombus—it has equal sides and all angles that measure 90 degrees. Each type of parallelogram
has its own distinct attributes, helping us tackle different mathematical problems. By recognizing the type of parallelogram we’re dealing with, we can apply the appropriate formulas to find areas,
perimeters, and understand their geometrical relationships better.
Understanding Area
What is Area?
Area is a measure of the amount of space contained within a two-dimensional shape. It tells us how much surface the shape covers. For example, when we think about a parallelogram, which looks like a
slanted rectangle, the area helps us understand how much space is inside that shape. To find the area of a parallelogram, we use the formula: Area = base × height. The base is the length of one of
the sides at the bottom, and the height is the perpendicular distance from this base to the opposite side. Understanding area is fundamental because it plays a role in real-world applications, like
determining how much paint you’ll need for a wall, how much carpet you should buy for your room, or how much land you have for a garden. So, as we explore the concept of area, remember that it’s not
just about numbers and formulas—it’s about how we can use these calculations in our everyday lives!
Units of Measure
When we talk about area, it’s important to know how we measure it, and that’s where units of measure come in. We commonly measure area in square units. This means we’re calculating how many unit
squares can fit inside a shape. For example, if we measure area in square meters (m²), we’re determining how many 1-meter by 1-meter squares can fit within the figure. Other common units include
square centimeters (cm²) for smaller areas, or acres and hectares for land. Knowing the appropriate unit is essential depending on what you’re measuring—an area of a room might be measured in square
feet, while a large field could be in acres. Additionally, converting between units is a vital skill that helps ensure accuracy, especially in practical situations. So, as we dive into the
calculations and examples, let’s keep the units of measure in mind—they’re crucial for making sure our area calculations make sense and can be applied correctly.
Formula for Area of a Parallelogram
Derivation of the Formula
To derive the formula for the area of a parallelogram, we start by understanding its properties. A parallelogram has opposite sides that are equal in length and parallel. Let’s denote the base of the
parallelogram as ( b ) and its height (the perpendicular distance from the base to the opposite side) as ( h ).
Imagine we have a parallelogram ( ABCD ). We can visualize a transformation to help derive the area: if we take triangle ( ABD ) and “slide” it across to the side ( BC ), you’ll notice that the
triangle fits perfectly against the opposite side. This rearrangement shows that the area of the parallelogram is identical to the area of the rectangle formed by the base ( b ) and height ( h ).
Calculating this area as a rectangle gives us the formula:
[ \text{Area} = \text{Base} \times \text{Height} = b \times h ]
Therefore, regardless of the specific shape of the parallelogram, as long as we know the base and the height, we can easily calculate the area using this derived formula, ( A = b \times h ).
Using Base and Height
To find the area of a parallelogram using the base and height, it’s essential to understand what we mean by these two measurements. The base refers to any one side of the parallelogram that we choose
to consider as the base. The height is the shortest distance from this base to the opposite side, measured at a right angle.
In practical terms, when identifying the base, ensure it’s a straight line along the bottom of the shape. The height must be drawn perpendicular to the base, reaching up to the upper side. This
distinction is crucial: the height is not the length of the side but rather the vertical distance measured straight up to the opposite line.
To calculate the area, simply multiply the base length ( b ) by the height ( h ):
[ A = b \times h ]
For instance, if the base of a parallelogram is 10 cm and the height is 5 cm, the area would be ( 10 \times 5 = 50 ) square centimeters. This straightforward calculation makes finding the area of a
parallelogram an accessible and effective skill in our geometry studies!
Calculating Area
Example Problems
In this section, we dive into example problems to solidify our understanding of calculating the area of parallelograms. Remember, the formula for finding the area (A) of a parallelogram is given by
(A = b \times h), where (b) is the length of the base and (h) is the height. We will start with simple problems, such as finding the area of a parallelogram with a base of 5 cm and a height of 3 cm.
Here, students will apply the formula to find that (A = 5 \times 3 = 15 \, \text{cm}^2).
Next, we’ll explore more complex examples, including parallelograms that are slanted. Understanding how to correctly identify the base and height is crucial since the height is always measured as the
perpendicular distance from the base to the opposite side. Through these examples, students will gain confidence and improve their problem-solving skills. We’ll also tackle word problems where they
must extract the necessary information from a story problem context, reinforcing their comprehension and ensuring they can apply the formula in various scenarios. This hands-on practice truly
prepares you for real-life applications of this geometry concept!
Real-World Applications
The area of parallelograms isn’t just an abstract concept; it has numerous real-world applications that highlight its importance in various fields! For instance, architects and engineers use the area
of parallelograms to calculate the surface area of different building components. When designing roofs or windows that resemble parallelograms, knowing how to measure and calculate their area
accurately is essential for material estimates and structural integrity.
In landscaping, understanding the area helps in planning garden beds or patio spaces shaped like parallelograms. By knowing the area, you can determine how much soil, grass, or materials you need.
Even in art and design, artists often incorporate geometric shapes like parallelograms into their projects, requiring precise area calculations for materials and layouts.
Moreover, in technology, computer graphics rely on understanding geometric shapes. Designers and animators calculate areas for textures and images that fit unique shapes, including parallelograms. As
you can see, mastering the area of parallelograms equips you with practical skills that extend into various careers and everyday life, making your learning relevant and meaningful!
Common Mistakes and Misconceptions
Identifying Common Errors
Understanding the area of parallelograms involves recognizing where many students often make mistakes. One common error arises from misunderstanding the formula. The area of a parallelogram is given
by the formula (A = b \times h), where (b) is the length of the base and (h) is the height. Students sometimes mistakenly think that any side can be considered the base without corresponding the
height correctly. For example, if they choose a slanted side as the base, they might forget to drop the perpendicular height to measure accurately.
Another frequent misunderstanding lies in the dimensions used; students often mix units (like centimeters and meters) if the problem is not presented clearly. Miscalculating the height—sometimes
visualizing the parallelogram incorrectly or misdrawing it—can lead to a significant error in the area. Additionally, misunderstanding the properties of a parallelogram, such as knowing that opposite
sides are equal and parallel, can affect how students approach problems. Recognizing these common pitfalls can help you avoid them and strengthen your mathematical understanding of this shape.
Tips for Avoiding Mistakes
To enhance your accuracy when calculating the area of parallelograms, several strategies can be incredibly helpful. First, always ensure you clearly label and visualize the shape. Draw the
parallelogram and mark the base and the height, which should be perpendicular to the base. This visual representation reinforces your understanding of what you’re calculating.
Next, pay careful attention to units. Always double-check that you are using the correct units and converting them when necessary. For example, if your base is measured in centimeters and height in
meters, convert one to match the other.
Practice is key! The more problems you solve, the more familiar you will become with identifying the base and height correctly. Try to find different shapes and orientations of parallelograms in
examples or worksheets. Finally, always review your calculations: after solving a problem, take a moment to reassess each step to ensure that you haven’t made any oversights. By implementing these
tips, you can confidently tackle problems related to the area of parallelograms with accuracy!
As we wrap up our exploration of the area of parallelograms, let’s take a moment to reflect on the deeper implications of what we’ve learned. The formula (A = b \times h) may seem straightforward,
but it embodies profound principles that extend far beyond the classroom. Each parallelogram, with its unique dimensions and properties, serves as a reminder of the diversity and symmetry that exist
in the world around us.
Consider how this knowledge transcends mere numbers. Architects use these principles to design buildings that can withstand the test of time, while artists adopt geometric shapes to create visually
striking compositions. Engineers rely on these concepts to innovate solutions, proving that mathematics is not just an abstract concept but a tool that shapes our reality.
As you move forward, I encourage you to see the world through a mathematical lens. Challenge yourself to find parallelograms in everyday life, from the frames of doors to the layouts of parks. Each
instance is an opportunity for discovery. Remember, math is not just about formulas—it’s about understanding our world and leveraging that understanding to foster creativity and solve real-world
problems. Keep questioning, keep exploring, and most importantly, keep embracing the beauty of mathematics! | {"url":"https://curioustoons.in/area-of-parallelograms/","timestamp":"2024-11-09T19:50:44Z","content_type":"text/html","content_length":"106929","record_id":"<urn:uuid:39ea6d17-a7d5-4699-a57c-7cf0ef1cb091>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00892.warc.gz"} |
11.4 Advanced aspects of functions | Data Science for Psychologists
11.4 Advanced aspects of functions
This section introduces four slightly more advanced topics in the context of computer programming:
These topics are of general interest when learning to program, but we will address them from the perspective of R as our current toolbox for tackling computational tasks. As computers have become
incredibly fast and contemporary programming languages provide many functions for solving problems that can be addressed by these techniques, the continued interest in these topics is partly
motivated by theoretical considerations. They help us think more carefully about data structures, the flow of control, and the contents of variables when calling functions. Ultimately, considering
the process by which functions solve their tasks enables us to create more flexible, efficient, and useful functions.
11.4.1 Recursion
Recursion is the process of defining a problem (or the solution to a problem) in terms of a simpler version of itself. More precisely, recursion solves a problem by (a) solving the problem for a
basic case, and (b) recursing more complex cases in small steps towards this solution. As this definition seems circular, people often play on the recursive nature of recursion (e.g., by titles like
To understand recursion you have to understand recursion…).
In computer science, recursion is a programming technique involving a function (or algorithm) that calls itself. To avoid getting lost in an infinite loop, the function’s call to itself typically
contains a simpler version of the original problem and re-checks a condition that can stop the process when the condition is met. Using this method results in a characteristic process in which
successive iterations are executed up to a stopping criterion. Once this criterion is met, all iterations are unwound in the opposite direction (i.e., from the last element called to the first).
A familiar instance of recursion occurs when you position yourself between two large mirrors and see your image multiplied numerous times. This phenomenon can also be observed when viewing a live
recording of yourself in a screen, as illustrated by the xkcd clip shown in Figure 11.2.
An image containing itself as part of the image will inevitably get lost in an infinite regress. In order work as an algorithm, however, a recursive function must include a stopping criterion.
Examples of recursive definitions of procedures that include such a condition include:
• The task “go home” can be implemented as:
□ If you are at home, stop moving.
□ Else, move one step towards your home; then go home.
• The task “read a book” can be implemented as:
□ If you read the last page, stop reading.
□ Else, read the current page, then read the rest of the book.
• Mathematical definitions: The factorial of a natural number \(n\) is defined as:
□ \(n! = n \cdot (n - 1)!\) for \(n > 1\), and
□ \(n! = 1\) for \(n = 1\).
These examples illustrate that a recursive solution typically contains two parts: (1) A stopping case that checks some criterion and stops the process when the criterion is met, and (2) an iterative
case that reduces the problem to a simpler version of itself. More specifically, all these examples included three elements: (a) a stopping condition, (b) a simplification step, and (c) a call of the
function (instruction, procedure, or task) to itself. However, the three elements were not always presented in the same order. For instance, the mathematical definition of the factorial (above)
provided the simplification step (b) and call to itself (c) prior to the stopping condition (a). This sequence may work for mathematical definitions, but can create problems in programming (e.g.,
when we have to test for a stopping criterion prior to the other steps).
We can easily find additional examples for recursive processes (e.g., solving a complex problem by solving its sub-steps, eating until the plate is empty, etc.). Note that recursion is closely
related to many heuristics — strategies that aim to successfully solve a problem in a fast and frugal fashion (i.e., without trying to optimize). For instance, the strategies of divide-and-conquer
and hill climbing try to solve a problem by breaking it down into many similar, but simpler ones, and aim to reach the overall goal in smaller steps.
In programming, using recursion is particularly popular when solving problems that involve linear data structures (e.g., vectors or lists, but also strings of text), as these can easily be simplified
into smaller sub-problems. For instance, we can always separate a vector v into its first element v[1] plus its rest v[-1], so that c(v[1], v[-1]) == v:
Essentials of recursion
How recursion works in practice is perhaps best explained by viewing an example. Here is a very primitive recursive function:
recurse <- function(n = 1){
if (n == 0){return("wow!")}
else {
recurse(n = n - 1)
# Check:
#> [1] "wow!"
#> [1] "wow!"
The recurse() function is recursive, as it contains a call to itself (in the else part of a conditional statement). However, note that the inner function call is not identical to the original
definition: It varies by changing its numeric argument from n to n - 1. Importantly, this small change creates an opportunity for the stopping criterion of the if-statement to become TRUE. For if n =
= 0, the then-part of the conditional {return("wow!")} is evaluated. (If the function called recurse(n = n) from its else-statement, it would call itself an infinite number of times and never stop.)
Our two calls to recurse() and recurse(10) yield the same result, but substantially differ in the way they obtained this result:
• By using the function’s default argument of n = 1, calling recurse() merely called itself once before meeting the stopping criterion.
• By contrast, recurse(10) called itself 10 times before n == 0 became TRUE.
A key insight regarding recursion concerns the question:
• What happens, when the stopping criterion of a recursive function is reached?
A tempting, but false answer would be: The process stops by evaluating the then-part of the conditional (i.e., here: {return("wow!")}). Although it is correct that this part is evaluated when the
stopping criterion is reached, the overall process does not stop there (despite the return() statement). To understand what actually happens at and after this point, consider the following variation
of the recurse() function:
recurse <- function(n = 1){
if (n == 0){return("wow!")}
else {
paste(n, recurse(n = n - 1))
# Check:
#> [1] "wow!"
#> [1] "1 wow!"
#> [1] "10 9 8 7 6 5 4 3 2 1 wow!"
In this more typical variant of a recursive function, we slightly modified the else-part of the conditional. The results of the evaluations show that the overall process was not finished when the
stopping criterion was met and return("wow!") was reached. As this part is only reached for calling recurse(n = 0), any earlier calls to recurse(n = 1), recurse(n = 2), etc., are still ongoing when
return("wow!") is evaluated. Thus, the expression paste("o", recurse(n = n - 1)) is yet to be evaluated n times, which results in adding the current value of n exactly n times to the front of “wow!”.
Thus, only the final statement returned by recurse(0) would be “wow!”, as — in this particular case of n = 0 — we would never reach the recursive call in the else-part of the conditional. By
contrast, calling recurse(n) for any n > 0 still needs to unwind all recursive loops after reaching the stopping criterion and eventually return the result of the first paste(n, recurse(n - 1))
function. Actually, there is nothing unusual or mysterious about this. For instance, when writing a nested function like:
we also know and trust that an R compiler parses it by first evaluating its innermost expression paste(1, "wow!"), before evaluating the other (and outer) two paste() functions. The result ultimately
returned by the expression is the result of its outmost function. Overall, we see that reaching the stopping criterion in a recursive function brings the recursive calls to an end, but the output of
calling the recursive function is yet to be determined by the path that led to this point.
The following Figure 11.3 illustrates a recursive process in a graphical fashion. It is implemented as a recursive plotting function that creates a new plot when n == 0. Thus, the algorithm first
prints “Stop!” and “n = 0” in the center, before exiting the recursive calls and printing n rectangles of increasing sizes for values of n > 0:
The recurse() function allows to further probe our understanding of recursion:
1. Predict the results of the following function calls, then verify your prediction by evaluating them:
2. Modify the recurse() function so that it returns "wow!" with a variable number of final exclamation marks. Specifically, the number of final exclamation marks should indicate the number of
recursive function calls to itself. For instance:
Examples of recursive functions in R
Our recurse() function may have illustrated the principle of recursion, but its uses are rather limited. To show that recursion can be a useful and powerful programming technique, here are some
examples of problems that can be solved by using recursion in R. Each example starts out with a task and is then addressed by a recursive solution.
1. Computing the factorial of a natural number n
Task: Define a function fac() that takes a natural number \(n\) as its only argument and computes the result of \(n!\).
A recursive solution to this task in R could look as follows:
fac <- function(n){
if (n == 1) {1} # stopping condition
else {n * fac(n - 1)} # simplify & recurse
# Check:
#> [1] 1
#> [1] 120
fac(10) == factorial(10)
#> [1] TRUE
The fac() function implements the above definition of the factorial of a natural number \(n\) in a recursive fashion. For any value of \(n > 1\), the function computes the product of \(n\) and the
factorial of \(n - 1\). The latter factor implies a recursive call to a simplified version of the original problem. These calls iterate to increasingly smaller values of \(n\) until the stopping
criterion of \(n=1\) is reached.
• base R provides a factorial() function that appears to do the same. However, factorial() does not use recursion in its definition, but a gamma() function that also works for non-integers.
• Our recursive definition for fac() is simple and elegant. But for practical purposes, we may want to verify its input argument before starting the recursive loop. For instance, we could add some
if statements that catch the following cases without breaking our fac() function:
A version of fac() that accommodates these additional cases could look as follows:
# Improved version:
fac_rev <- function(n){
# Verify inputs:
if (is.na(n)) {return(NA)}
if ( n < 0 | !ds4psy::is_wholenumber(n) ){
return(message("fac_rev: n must be a positive integer."))
# Main:
if (n == 1) {1} # stopping condition
else {n * fac(n - 1)} # simplify & recurse
# Check:
Note that fac_rev() still calls fac(), rather than itself. This is possible here, as the input verification based on the value of \(n\) is only needed once. However, we also could recursively invoke
fac_rev() if we wanted to check its inputs whenever the function is called.
2. Reversing a vector
Task: Define a function reverse() that reverses the elements of a vector v:
reverse <- function(v){
if (length(v) == 1) {v} # stopping condition
else {c(reverse(v[-1]), v[1])} # simplify & recurse
# Check:
reverse(c("A", "B", NA, "D"))
#> [1] "D" NA "B" "A"
#> [1] 5 4 3 2 1
#> [1] NA
The crux of the reverse() function lies in its simplification step: We take the first element v[1], and move it to the end of the reversed rest of the vector v[-1]. As reversing a scalar (i.e., a
vector of length 1) leaves v unchanged, this serves as the stopping criterion.
• base R also contains a rev() function.
3. Computing the n-th number in the Fibonacci sequence
According to Wikipedia, the series of Fibonacci numbers is characterized by the fact that the \(n\)-th number in the series is the sum of its two preceding numbers (for \(n \geq 2\)). Formally, this
requires specifiying the initial numbers and implies:
• Definition: \(F(0)=0\); \(F(1)=1\); \(F(n) = F(n-1) + F(n-2)\) for \(n>1\).
Thus, the first \(20\) values of the Fibonacci sequence are:
• \(0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, ...\)
Task: Define a function fib() that computes the \(n\)-th Fibonacci number.
Here is a recursive definition of a corresponding fib(n) function:
fib <- function(n){
# Verify n:
if (is.na(n) || n < 0 || abs(n %% 1) > 0) {
message("n must be a non-negative integer.")
# Main:
if (n <= 1) { # for (n == 0) or (n == 1):
return(n) # stop
} else {
return(fib(n - 2) + fib(n - 1)) # recurse
} # fib().
# Check: ----
#> [1] 0
#> [1] 1
#> [1] 6765
# Note:
#> [1] NA
#> [1] NA
#> [1] NA
Note that the definition of our fib() function specifies two stopping criteria (for fib(0) and fib(1)) and its final else part includes two recursive calls to itself, both of which pose a simpler
version of the original problem.
In Chapter 12, we will solve the same problem in an iterative fashion (see Exercise 1, Section 12.5.1). This will show that recursion is closely related to the concept of iteration (i.e., proceeding
in a step-wise fashion): Actually, recursion is a special type of iteration, as it employs iteration in an implicit way: By writing functions that call themselves we create loops without defining or
labeling them explicitly.
The following exercises reflect upon the above examples of recursive functions:
1. We stated above that a vector v can always be separated into its first element v[1] plus its rest v[-1], so that c(v[1], v[-1]) == v. This may be true for longer vectors, but does it also hold
for very short vectors like v <- 1 or v <- NA?
v <- 1
c(v[1], v[-1]) == v
#> [1] TRUE
v <- NA
c(v[1], v[-1]) == v
#> [1] NA
v <- NULL
c(v[1], v[-1]) == v
#> logical(0)
is.null(c(v[1], v[-1]))
#> [1] TRUE
2. Why do some of the recursive functions defined here (e.g., fib()) contain an expression if (is.na(n)) return(NA), whereas others (e.g., reverse()) do not have it? Can we reduce this expression to
if (is.na(n)) {NA}?
3. Let’s further improve and explore our fac() function (defined above):
□ Create a revised version fac_2() that does not cause errors for missing (i.e., NA) or non-integer inputs (e.g., n = .5).
□ Our fac() function included two implicit end points (in the if and else parts). Define a variant fac_2() that renders all return elements explicit by adding corresponding return() statements.
fac_2 <- function(n){
# Verify inputs:
if (is.na(n)) {return(NA)}
if (n < 0 | !ds4psy::is_wholenumber(n)){
return(message("fac: n must be a positive integer."))
# Main:
if (n==1) {return(1)} # stopping condition
return(n * fac_2(n - 1)) # simplification & recursion
Checking our function:
3. Write a recursive function rev_char() that reverses the elements (characters) of a character string s:
Note that our reverse() function and the base R function rev() work on multi-element objects (i.e., typically vectors), whereas we want to reverse the characters within objects (i.e., change the
objects, rather than just their order). We can still apply the same principles as in our recursive version of reverse(), but need to use paste0() and substr() functions (see Section 9.3) to
deconstruct and re-assemble character objects:
rev_char <- function(s){
if (is.na(s)) {return(NA)}
if (nchar(s) == 1) {s}
else {paste0(rev_char(substr(s, 2, nchar(s))), substr(s, 1, 1))}
Checking our function:
Note that our recursive versions of fib(), fac(), and rev_char() do not work well for vector inputs. We will address the issue of vectorizing functions below (in Section 11.4.3).
11.4.2 Sorting
The task of sorting elements or objects is one of the most common illustrations for the wide range of computer algorithms and comparisons between different ones. In R, we may typically want to sort
the elements of a vector, or the rows or columns of a table of data. In the previous chapters, we have seen that R and R packages provide various functions for tackling such tasks (e.g., see the base
R functions order() and sort(), or the dplyr functions arrange() and select()). However, how would we solve such tasks if we were writing our own functions?
The basic task of sorting a set of elements on some dimension can be addressed and solved in many different ways. Different approaches can be used to illustrate the efficiency of different algorithms
— and provide us with an opportunity for measuring performance (e.g., by counting steps or timing the execution of tasks).
Essentials on sorting
If we wanted to sort the elements of a vector in R, we could simply use the sort() function provided by base R:
Alternatively, we could write our own sorting function, if we saw a need to do so.^70 But now that we can write our own functions, we should be aware that not all functions are created equal.
For instance, consider the following “solution” to the sorting problem (created in this blog post, for illustrative purposes):
bogosort <- function(x) {
while(is.unsorted(x)) x <- sample(x)
# Check:
# [1] 46 57 73 81 95
If the while() loop confuses us (as we have not yet covered iteration), we can re-write the function using recursion:
bogosort_rec <- function(x) {
if (!is.unsorted(x)) {x}
else {bogosort_rec(x = sample(x))}
# Check:
# [1] 46 57 73 81 95
Note that the else part in a recursive definition typically calls the same function with a simplified version of the problem. Here, however, the “simplification” consisted merely in replacing the
argument x by sample(x). Thus, we were merely starting over again with a different random permutation of x.
As both bogosort() functions solve our task, it may seem that we have solved our problem. This is true, as long as we only want to sort the vector v containing 5 elements 73, 57, 46, 95, 81. However,
functions are usually not written for solving just one particular problem — and their performance critically depends on the type and size of problems they are solving. For the simple problem of
sorting v, these erratic versions of a sorting algorithm may have worked. But if our problem had only been slightly more complicated, the same functions may have failed. For instance, if our vector v
contains 10 or more elements, the processing times of bogosort(v) become painfully slow, and bogosort_rec(v) even throws error messages that inform us that stack usage is too close to the limit (try
this for yourself). Hence, for creating more flexible functions that can tackle a wider range of problems, efficiency considerations become important.
1. Selection sort algorithm
The basic idea of the so-called selection sort algorithm is to select the smallest element of a set and move it to the front, before sorting the remaining elements. This idea can easily be translated
into the two parts of a recursive definition:
• Stopping criterion: Sorting x is trivial when x only contains one element.
• Simplify and recurse: Move the minimum element to the front, and sort the remaining elements.
Implementing these steps in R is simple:
sort_select <- function(x) {
if (length(x) == 1) {x}
else {
ix_min <- which.min(x)
c(x[ix_min], sort_select(x[-ix_min]))
Note that which.min(x) yields the index or position of the first minimum of x (i.e., which(min(v) == v)[1]).
Let’s check our sort_select() function on a vector v:
(v <- sample(1:100, 20))
#> [1] 73 57 46 95 81 58 61 60 59 3 32 9 31 93 53 92 49 14 76 45
# [1] 73 57 46 95 81 58 61 60 59 3 32 9 31 93 53 92 49 14 76 45
(sv <- sort_select(v))
#> [1] 3 9 14 31 32 45 46 49 53 57 58 59 60 61 73 76 81 92 93 95
# Note:
sort_select(c(1, 2, 2, 1))
#> [1] 1 1 2 2
#> [1] NA
# sort_select(c("A", "C", "B")) # would yield an error.
As using which.min(x) requires a numeric argument x, our sort_select() function fails for non-numeric inputs (e.g., character vectors).
2. Quick sort algorithm
One of the most famous sorting algorithm is aptly called Quick sort. To understand its basic idea, watch its following enactment (created by Sapientia University, Tirgu Mures (Marosvásárhely),
Romania, under the direction of Kátai Zoltán and Tóth László):
As the dance illustrates, the basic idea of the Quick sort algorithm is recursive again:
• Stopping criterion: Sorting x is trivial when x only contains one element.
• Simplify and recurse: Find a pivot value pv for the unsorted set of numbers x that splits it into two similarly sized subsets. Sort all numbers that are smaller than the pivot to the left of the
pivot element, and all numbers that are larger than the pivot to the its right. Then call the same function twice — once for each subset.
Implementing these steps in R is simple again:
sort_quick <- function(x) {
if (length(x) == 1) {x}
else {
pv <- (min(x) + max(x)) / 2 # pivot to split x into 2 subsets
c(sort_quick(x[x < pv]), x[x == pv], sort_quick(x[x > pv]))
Let’s check our sort_quick() function on the vector v (from above):
Comparing algorithms
We now have created three different algorithms that seem to work fine for the task of sorting numeric vectors. This is not unusual. We can easily see that any particular algorithm can always be
tweaked in many different ways. This can be generalized into an inportant insight: There always exists an infinite number of possible algorithms for solving a computational task. Thus, the fact that
an algorithm gets a task done may be a necessary condition for being a good algorithm, but it cannot be a sufficient one.
When multiple algorithms get a particular job done, the next question to ask is: How do they differ? To learn more about them, we could vary the range of problems that they need to solve. Provided
that all functions are correct, they should always yield the same results. However, seeing how they perform on a large variety of problems may reveal some boundary cases that illustrate how the
algorithms differ in some respect. Alternatively, we could solve the same (set of) problem(s) by all algorithms and directly measure some variable that quantifies some aspect of their performance.
11.4.3 Vectorizing functions
As vectors are the main data structure of R, it is considered good practice to accommodate vector inputs when programming in R. However, as we usually tend to think in terms of a specific problem at
hand when creating a new function, we often fail to meet this requirement. For instance, note that our fib() and fac() functions — despite their recursive elegance — would not work seamlessly for
vector inputs:
# Using non-vectorized functions with vector inputs:
fib(c(6, 9))
#> Error in if (is.na(n)) return(NA): the condition has length > 1
fac(c(6, 9))
#> Error in if (n == 1) {: the condition has length > 1
We see that — when providing vector inputs to non-vectorized functions — we may be lucky to obtain the desired results. However, we sometimes may only receive partial results (here: for fib(c(6, 9)))
or get unlucky and not receive any results. Additionally, multiple warning messages indicate that we were only considering atomic inputs when using if() statements to check our input properties.
Ensuring that a new function works on vectorized inputs often requires some additional steps. Before revising a function, we should first reconsider and explicate our expectations regarding the
function (e.g., do we want to reverse the character sequence of each element, or also the sequence of elements?). Once we have clarified the task and purpose of a function, we can try to replace some
parts that only work for atomic inputs by more general expressions (e.g., replacing if() by vectorized ifelse() conditionals, see Section 11.3.4).
A convenient way for turning a non-vectorized function into a vectorized version is provided by the Vectorize() function of base R. For instance, we can easily define a vectorized version of rev_char
() as follows:
# Vectorizing functions:
fib_vec <- Vectorize(FUN = fib)
fac_vec <- Vectorize(FUN = fac)
# Checking vectorized versions:
fib_vec(c(6, 9))
#> [1] 8 34
fac_vec(c(6, 9))
#> [1] 720 362880
By specifying additional arguments of Vectorize() we can limit the arguments to be vectorized and adjust some output properties. More powerful techniques for applying functions to data structures are
introduced in Chapter 12 on Iteration (see e.g., Section 12.3 on functional programming).
11.4.4 Measuring performance
So far, our efficiency considerations in the previous sections were mostly abstract and theoretical. By contrast, waiting for some function or program to finish is a very familiar and concrete
experience for most programmers and many users, unfortunately.
A more practical approach would require methods for measuring the performance of our functions. Good questions to ask in this context are:
1. How many times is a function called?
2. How long does it take to execute it?
R provides multiple ways for answering these questions in a quantitative fashion. In this section, we merely illustrate two basic solutions.
1. Counting function calls
Counting the number of calls to a function (e.g., sort_select()) can be achieved by the trace() function. Its tracer argument can be equipped with a function that increments a variable count by one:
# Insert a counter into a function:
count <- 0 # initialize
trace(what = sort_select,
tracer = function() count <<- (count + 1),
print = FALSE)
#> [1] "sort_select"
Note that the tracer function has no argument, but changes a variable that was defined outside of the function. This is why the assignment is using the symbol <<-, rather than just <-.
To actually trace our function, we need to use it and then inspect the count variable (after initializing it to 0):
# Test:
count <- 0 # initialize
v <- sample(1:2000, 1000)
#> [1] 773 1722 652 999 548 698
sv <- sort_select(v) # apply function
count # check counter
#> [1] 1000
untrace(what = sort_select) # stops tracing this function
The value of count shows that sort_select() achieved its result by 1000 calls to the function. Actually, we could have anticipated the result: Given that sort_select(v) moves the minimum element of v
to the front on every function call, it takes length(v) function calls to sort v.
Contrast this with sort_quick():
# Insert a counter into a function:
count <- 0 # re-initialize
trace(what = sort_quick,
tracer = function() count <<- (count + 1),
print = FALSE)
#> [1] "sort_quick"
# Test:
sv <- sort_quick(v) # apply function
count # check counter
#> [1] 1597
untrace(what = sort_quick)
Thus, sort_quick() requires 1597 calls to sort the 1000 elements of v. This large value may actually be surprising — at least when knowing that quick sort is considered an efficient algorithm.
However, the fact that the vast majority of calls to sort_quick() result in two calls to the same function lets the counter increase beyond the number of elements in v.
2. Timing function calls
Beyond counting the number of function calls, we may be interested in the time it takes to evaluate an expression. Here is some code for timing the duration of a function call:
options(expressions = 50000) # Max. number of nested expressions (default = 5000).
# N <- 2000
# set.seed(111)
# v <- runif(N) # create N random numbers
# head(v)
# Measure run times: ----
system.time(sv <- sort_select(v))
#> user system elapsed
#> 0.006 0.001 0.007
system.time(sv <- sort_quick(v))
#> user system elapsed
#> 0.002 0.000 0.003
system.time(sv <- sort(v))
#> user system elapsed
#> 0 0 0
This shows that — for this particular problem (i.e., the data input v) — sort_quick(v) is considerably faster than sort_select(v). Moreover, despite its additional function calls, the time
performance of sort_quick(v)) gets pretty close to the generic base R function sort(), which is typically optimized for our system.
Here are some practice tasks on sorting algorithms and on evaluating their performance:
1. Considering ties: Do the above search algorithms also work when the vector v to be sorted includes ties (i.e., multiple identical elements)? Why or why not?
2. Reconsidering sort_select(): Write an alternative version of sort_select() that uses which.max(), rather than which.min().
3. Reconsidering sort_quick(): How important is it for the performance of sort_quick() that the problem is split into two subsets of similar sizes? Find out by writing alternative versions of
sort_quick() that use
□ (a) pv <- mean(x)
□ (b) pv <- median(x)
□ (c) pv <- (min(x) + max(x)) / 3
as alternative pivot values. How do these changes affect the performance of the search algorithm?
Hint: As such changes may substantially decrease the performance, we should consider using a simpler test vector v to avoid that our system throws an Error.
# Define alternative version: ------
sort_quick_alt <- function(x) {
if (length(x) == 1) {x}
else {
pv <- mean(x) # (a)
# pv <- median(x) # (b)
# pv <- (min(x) + max(x)) / 3 # (c)
c(sort_quick_alt(x[x < pv]), x[x == pv], sort_quick_alt(x[x > pv]))
# Check:
v <- sample(1:100, 100)
# Count performance: ------
count <- 0 # initialize
trace(what = sort_quick_alt,
tracer = function() count <<- (count + 1),
print = FALSE)
# Test (using v again):
sv <- sort_quick_alt(v) # apply function
count # check counter
untrace(what = sort_quick_alt)
# Time performance: ------
system.time(sv <- sort_quick_alt(v))
4. Vectorizing a function: Does your rev_char() function (assigned as an exercise above) also work for vectorized inputs? Verify this by evaluating:
• If it fails to work or issues multiple warnings, create a vectorized version rev_char_vec() of it and use this function to verify:
• Can you explain and control the names of the vector elements?
This concludes our foray into more advanced aspects of programming in R.
70. As the functions provided by base R are typically optimized, re-writing such functions rarely makes sense, unless we want to add or change some of its functionality. In this chapter, we are
writing alternative sort() functions for pedagogical reasons.↩︎ | {"url":"https://bookdown.org/hneth/ds4psy/11.4-functions:advanced-aspects.html","timestamp":"2024-11-06T11:44:53Z","content_type":"text/html","content_length":"234025","record_id":"<urn:uuid:ca2cc974-8b37-4373-9067-3bd802c54a28>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00343.warc.gz"} |
Which branch of statistics allows us to draw conclusions that generalize from the subjects we have studied to all the people of interest by allowing us to make inferences based on probabilities? - Answers
Still have questions?
Why do we need to understand statistics?
Statistics is the mathematical study of populations. We need statistics in order to know something about a large group of something after only studying a small group of that something. We take a
sample of a population and study it, and then we can usually draw conclusions about the rest of the population without also studying each member of the population individually. It helps us to be sure
that when we try to generalize about some pattern in the weather, behavior of certain people, or the yield of a chemical reaction, that it is objective mathematics that is doing the calculating and
not anecdotal evidence based only on human experience. We generalize about patterns and data every day, we just don't call it statistics when we do. We also count things every day, but we don't call
it math when we do. Statistics and Multivariable Calculus are both just refined versions of the skills we already use. Understanding statistics makes you a more objective person and increases your
ability to generalize about patterns and populations. | {"url":"https://math.answers.com/math-and-arithmetic/Which_branch_of_statistics_allows_us_to_draw_conclusions_that_generalize_from_the_subjects_we_have_studied_to_all_the_people_of_interest_by_allowing_us_to_make_inferences_based_on_probabilities","timestamp":"2024-11-07T09:29:56Z","content_type":"text/html","content_length":"166434","record_id":"<urn:uuid:977934ad-84ee-4f6e-975a-883a83734616>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00096.warc.gz"} |
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Series test for convergence/divergence
05-09-2018, 11:15 AM
Post: #1
DrD Posts: 1,136
Senior Member Joined: Feb 2014
Series test for convergence/divergence
How could the hp prime be used to directly determine convergence/divergence for a series, (not evaluate to the sum!)?
Example (converge/diverge)?:
∑ [n=1,∞] (2*n^2 + n) / √(4*n^7 + 3);
The idea is to use the various convergence tests, (such as done by Wolfram Alpha), for this example.
05-09-2018, 12:33 PM
Post: #2
parisse Posts: 1,337
Senior Member Joined: Dec 2013
RE: Series test for convergence/divergence
Run series((2*n^2 + n) / √(4*n^7 + 3),n=inf) and look at the series expansion main term, here it's equivalent to n^(-3/2) (const sign) hence convergent.
05-10-2018, 08:25 AM
Post: #3
DrD Posts: 1,136
Senior Member Joined: Feb 2014
RE: Series test for convergence/divergence
Thank you Parisse. While the Series expansion can help, a command which returns a result with direct indication of convergence/divergence, and possibly the confirming tests used to obtain the result,
would be a nice enhancement.
Wolfram Alpha has a nice presentation for this, as shown in the attachment. Something like this would be very useful for students studying this subject.
05-10-2018, 01:31 PM
Post: #4
parisse Posts: 1,337
Senior Member Joined: Dec 2013
RE: Series test for convergence/divergence
I don't find the wolfram alpha text very informative "By comparison test, the series converge", it's much more informative to tell that the equivalent is n^(-3/2).
Yes, it could be a nice addition, like for example tabvar.
05-10-2018, 02:09 PM
Post: #5
DrD Posts: 1,136
Senior Member Joined: Feb 2014
RE: Series test for convergence/divergence
I find Wolfram's technique to be very informative. This may be an American vs non-American teaching difference, though. During a study of series testing for convergence/divergence, a variety test
methods are taught, with problem examples provided to be solved outside of a classroom. Wolfram is showing that some tests are inconclusive, where others may reveal convergence or not. Some problems
can be mysterious, and information, (like Wolfram provides), can be useful for making progress, at such times.
I suppose a user program for this could be made, but since we have so many commands, that fill so many mathematical needs, that this one might also be included. I read, (in an older post, somewhere),
that you were thinking of this for a giac addition. It would be very nice, if included in the prime!
05-11-2018, 03:30 PM
Post: #6
parisse Posts: 1,337
Senior Member Joined: Dec 2013
RE: Series test for convergence/divergence
They just give some hints, but no proof. I mean, no math teacher would accept "the series is convergent by comparison" without details.
Yes, it would be nice to add that, but this is low priority currently.
05-11-2018, 04:17 PM
Post: #7
DrD Posts: 1,136
Senior Member Joined: Feb 2014
RE: Series test for convergence/divergence
(05-11-2018 03:30 PM)parisse Wrote: They just give some hints, but no proof. I mean, no math teacher would accept "the series is convergent by comparison" without details.
During the topic under discussion, a proof is given by the instructor. After the proof has been provided, it is expected that those methods will be used for subsequent problem solving; and once an
outcome has been reached a result is generally shown as: "convergence, (or divergence), for
∑ bn
is demonstrated, therefore
∑ an
is also convergent, (or divergent), {by the relevant series test}." So the "details" ARE part of a student's solution.
The calculator would be useful in validating intermediate steps, and if the calculator's solution was of a form similar to Wolfram's, it would be even more valuable to the student,
and the instructor as well
, since the student's answer should follow the (hopefully, correct) calculator results.
05-11-2018, 04:25 PM
Post: #8
Benjer Posts: 51
Member Joined: Apr 2017
RE: Series test for convergence/divergence
Hi Parisse,
I see that the value of the main term is n^(-3/2) but I don't know the connection between that and whether the series converges or diverges. I guess what I'm asking is, if I apply the series command
to some function, what am I looking for in the output to see if the series converges or diverges? I tried some different series that I know are convergent and divergent, but I couldn't see a pattern
in the output that helped me connect the two.
Thank you for your time.
05-11-2018, 06:28 PM
Post: #9
toshk Posts: 195
Member Joined: Feb 2015
RE: Series test for convergence/divergence
(05-11-2018 04:25 PM)Benjer Wrote: Hi Parisse,
I see that the value of the main term is n^(-3/2) but I don't know the connection between that and whether the series converges or diverges. I guess what I'm asking is, if I apply the series
command to some function, what am I looking for in the output to see if the series converges or diverges? I tried some different series that I know are convergent and divergent, but I couldn't
see a pattern in the output that helped me connect the two.
Thank you for your time.
n^(-3/2)==1/n^(3/2) therefore as n gets larger
05-11-2018, 06:56 PM
Post: #10
parisse Posts: 1,337
Senior Member Joined: Dec 2013
RE: Series test for convergence/divergence
sum(1/n^alpha,n) is convergent if alpha>1, divergent for alpha<=1 (this is easy to prove by comparing with int(1/x^alpha,x)). If f(n) is equivalent to g(n) and g(n) has constant sign then sum(f(n),n)
and sum(g(n),n) are both convergent or both divergent.
It's more complicated for non constant sign term, like sum((-1)^n/n^alpha,n) is convergent for alpha>0, but sum((-1)^n/f(n),n) may be divergent even if f(n) is equivalent to n^alpha, for this kind of
series you will usually need a series expansion of f(n) at n=inf until the remainder of the expansion is O(1/n^alpha) with alpha>1. Like for example sum((-1)^n/sqrt(n+(-1)^n*sqrt(n)),n)
05-11-2018, 11:17 PM
Post: #11
Benjer Posts: 51
Member Joined: Apr 2017
RE: Series test for convergence/divergence
Thank you for the clarification.
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Calculus 2 (Math 116 section 02)
Class times: 11:00AM - 12:15PM, MWF (BEN 207)
Office Hours: 2:30PM - 4:30PM, MWF; or by appointment
Most of the general course information can be found in the course information handout. Student grades are maintained on Blackboard.
Course Log
Jump to week: 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 -- 9 -- 10 -- 11 -- 12 -- 13 -- 14 -- 15 and beyond
Week 1
[08.28.15] - Friday - First day!
- Some of the important topics we covered. -
Sections covered: 5.2 (started)
Reading for next time: Section 5.2 (and 5.1 if you need to fill in). Focus on the following:
1. the definition of the definite integral, and
2. the content in blue boxes, e.g. statements of theorems and properties of the definite integral.
To discuss next time:
1. Where should Rich from the CarTalk Problem mark for 3/4 tank of gas? Hint: this is hard! If you don't fully solve it, that is fine, but make sure you can clearly explain your approach and where
you got stuck. Make sure to indicate if you use outside resources for parts of your solution, e.g. Desmos or WolframAlpha.
Week 2
[08.31.15] - Monday The CarTalk discussion was fantastic! Thanks for all of the participation, and extra thanks to all those that went to the board!
Sections covered: 5.2 (finished), 4.4 (started)
- CarTalking. -
Reading for next time: Section 4.4. Focus on the following:
1. L'Hospital's Rule (pg. 305, blue box)
2. Examples 1, 2, 3, and 6.
Don't forget that there's Homework (written and online) due Tuesday!
To discuss next time:
1. CarTalk again: where should Rich from the CarTalk Problem mark for 3/4 of a tank of gas? Make sure you understand the reformulation of the problem (using a definite integral) that was presented
today, and try to "solve" the CarTalk problem, i.e. make as good of an estimate as you can as to where the 20 inch stick should be marked for 3/4 of a tank of gas.
2. Try to find the error in the following argument; it is wrong!
I want to find $\lim_{x\rightarrow \infty} \frac{x+\sin x}{x}$. As $x\rightarrow \infty$, both the top and bottom of the fraction tend to $\infty$, which yields an indeterminate form of the type
$\frac{\infty}{\infty}$. Using L'Hospital's Rule, I find that $\lim_{x\rightarrow \infty} \frac{x+\sin x}{x} = \lim_{x\rightarrow \infty} \frac{1+\cos x}{1} = \lim_{x\rightarrow \infty} (1+\cos
x)$. As $x\rightarrow \infty$, $(1+\cos x)$ oscillates between $0$ and $2$, so $\lim_{x\rightarrow \infty} (1+\cos x)$ does not exist. Thus, $\lim_{x\rightarrow \infty} \frac{x+\sin x}{x}$ does
not exist.
It may be best to first figure out the correct answer to $\lim_{x\rightarrow \infty} \frac{x+\sin x}{x}$. Hint: graph it! The limit exists.
[09.02.15] - Wednesday Two more great presentations today! (But I forgot to take a pic...)
Sections covered: 4.4 (continued)
Reading for next time: End of Section 4.4 and Section 6.2. Focus on the following:
1. The final part of 4.4 about "Indeterminate Powers"
2. Example 1 in 6.2
Don't forget that there's Homework (written and online) due Friday!
To discuss next time:
1. Write down a step-by-step process to follow when trying to deal with indeterminate powers. Illustrate how to use your process as you solve \[\lim_{x\rightarrow \infty} \left(1+\frac{1}{x}\right)^
2. Given a ruler and a knife, can you find a good estimate of the volume of an eggplant? Be prepared to demonstrate.
[09.04.15] - Friday Three excellent presentations today - pics below! The eggplant never stood a chance...
- Indeterminate powers - Yay $e$! -
- We just had to know: 30.26 cubic inches -
Sections covered: 6.2 (started)
Reading for next time: End of Section 6.2 and Section 6.3. Focus on the following:
1. Example 3 and 5 in 6.2. (We did Example 7 in class today. Look in the book for a better picture than I put on the board.)
2. The intro. to 6.3 up through Example 1
To discuss next time:
1. Find the volume of Gabriel's Horn. Gabriel's Horn is the solid obtained by revolving the graph of $y=\frac{1}{x}$, from $x=1$ to $\infty$, about the $x$-axis. Hint: first find the volume of the
horn from $x=1$ to $x=a$, and then take the limit as $a\rightarrow \infty$. Be prepared to present a picture and explain the the integral that you came up with, as well as the limit.
Week 3
[09.07.15] - Monday Lots of volumes today!! ...and saw a very nice presentation that the infinitely long Gabriel's Horn has a finite volume (of $\pi$)!
Sections covered: 6.2 (finished), 6.3
Reading for next time: Section 7.1. Focus on the following:
1. the introduction and formula for integration by parts, i.e. every thing before Example 1
2. Examples 1, 2, 3
Don't forget that there's Homework (written and online) due Tuesday!
To discuss next time:
1. Find the volume of the solid obtained by revolving the region bounded by $y=e^x$, $y=0$, $x=0$, and $x=1$ about the $y$-axis. Hint: graph it first. What seems easier: shells or washers? Or maybe
they both seem hard. And now the real hint: you will likely need to use integration by parts to finish things off.
[09.09.15] - Wednesday A bit more volume today, via two nice presentations, which ended with a hook into integration by parts (and a lot more examples).
Sections covered: 7.1
Reading for next time: Section 7.2. Focus on the following:
1. Examples 1 - 4
Don't forget that there's Homework (written and online) due Friday!
To discuss next time:
1. We will start with a quiz. I will ask you set-up (but not evaluate) an integral that computes a particular volume, and I will also ask you to do a problem very similar to Example 1 in 7.2.
2. Meditate (again) on the discussion question from last time about the volume of the solid obtained by revolving the region bounded by $y=e^x$, $y=0$, $x=0$, and $x=1$ about the $y$-axis. What does
the integral look like if you use washers instead of shells? Can you comute the resulting integral?
[09.11.15] - Friday One quiz, two great, colorful presentations (one pictured below), and a whole lot of trig.
Sections covered: 7.2
Reading for next time: Section 7.3. Focus on the following:
1. The introduction
2. Examples 1 and 3
Don't forget that there's Homework (written and online) due today!
To discuss next time:
1. When trying integration by parts, describe how you choose which function is $u$ (and which is part of $dv$). In class, we decided that, first and foremost, we need to be able to integrate $dv$,
and we also hope that our choice for $u$ becomes "simpler" after differentiation. Can you be more specific? For example, suppose that one of the pieces is an inverse trig. function; would you
make it part of $u$ or $dv$? What if you encounter a logarithmic function?
2. CarTalk! Use what you read in Section 7.3 to compute $\int \sqrt{100 - x^2}\, dx$.
Week 4
[09.14.15] - Monday So...much...trig!
Sections covered: 7.3
Reading for next time: Section 7.4. Focus on the following:
1. Introduction through Example 2
Don't forget that there's Homework (written and online) due Tuesday!
To discuss next time:
1. You know how to compute $\int \frac{1}{x-6}\;dx$, right?
2. Using what you read in 7.4 (Example 1), compute $\int \frac{x^2+x+1}{x-6}\;dx$.
3. Remember how to compute $\int \frac{4x+2}{x^2+x-6}\;dx$? Hint: basic substitution.
4. Using what you read in 7.4, try to compute $\int \frac{x+7}{x^2+x-6}\;dx$.
[09.16.15] - Wednesday So...much...algebra!
- You all clearly don't need me! Nice job! -
Sections covered: 7.4 (started)
Reading for next time: End of Section 7.4. Focus on the following:
1. Examples 7-9
Don't forget that there's Homework (written and online) due Friday!
To discuss next time:
1. Be prepared to write out the form of the partial fraction decomposition (as in Example 7) for any big, nasty rational function, but you can assume that the denominator will be easy to factor.
2. Using what you read (in Example 9), try to compute $\int \frac{dx}{x^2+x\sqrt{x}}\;dx$.
[09.18.15] - Friday Very nice presentations and great discussion today. We started improper integrals at the end and had fun meditating on the fact that $\int_1^\infty \frac{1}{x^2}\,dx = 1$ while $\
int_1^\infty \frac{1}{x}\,dx = \infty$.
Sections covered: 7.4 (finished), 7.8 (started)
Reading for next time: End of Section 7.8. Focus on the following:
1. Examples 5-7
To discuss next time:
1. True or False: $\displaystyle\int_{-1}^1 \frac{1}{\sqrt[3]{x}}\,dx = 0$. Explain.
Week 5
[09.21.15] - Monday Great presentations today. I love seeing the successes and the failures!
Sections covered: 7.8 (finished), 8.1 (started)
Reading for next time: Section 8.1. Focus on the following:
1. Introduction through Example 2
Don't forget that there's Homework (written and online) due Tuesday!
To discuss next time:
1. Set up an integral to compute the length of $y=\sin(x)$, $0\le x\le 2\pi$. Can you evaluate this integral? If you can, do it; otherwise, estimate it using a Riemann sum (with not too many
2. Set up an integral to compute the length of $y=\ln|\cos(x)|$, $0\le x\le \frac{\pi}{3}$. Can you evaluate this integral? If you can, do it; otherwise, estimate it using a Riemann sum (with not
too many rectangles).
[09.23.15] - Wednesday Three excellent presentations today. The board looked gorgeous, and I totally failed to take a picture... Next time. Nice job everyone!
Sections covered: 7.8 (finished, again), 8.1 (finished), 8.2 (started)
Reading for next time: Section 8.2. Focus on the following:
1. Introduction through the blue box labeled $\fbox{4}$
2. Example 1
To discuss next time:
1. Set up an integral that computes the surface area of Gabriel's Horn. (Recall that Gabriel's Horn is the solid obtained by revolving the graph of $y=\frac{1}{x}$, from $x=1$ to $\infty$, about the
2. Show that the integral diverges. Hint: try a comparison test.
[09.25.15] - Friday Another nice set of presentations, and another failed photo op. I swera I'll remember next time...probably.
Sections covered: 8.2 (finished), 10.1 (started)
Reading for next time: Section 10.1. Focus on the following:
1. Introduction
2. Examples 1, 2, 4
To discuss next time:
1. Play around with parametric equations in Desmos. Look at the following for examples:
2. Use Desmos and what you learned in Example 4 to do problem 35 on page 647 of the textbook. Write everything down on paper, so you can reproduce it in class. Hint: each part of the picture will be
represented by a different set of parametric equations.
3. Use Desmos to make a super cool graph using parametric equations. Write everything down on paper, so you can reproduce it in class.
Week 6
[09.28.15] - Monday Two great Desmos presentations today: Smiley and Mjölnir. The picture below is a mash up of the two; smiley was never in actual danger. Many thanks to the presenters!
Sections covered: 10.1 (finished), 10.2 (started)
Reading for next time: Section 10.2. Focus on the following:
1. Introduction through Example 3
Don't forget that there's Homework (written and online) due Tuesday!
To discuss next time: Consider the curve $\mathcal{C}$ defined by the parametric equations \[\begin{align*}x=& t^3-3t\\ y=& 3t^2-9\end{align*}\]
1. Check out the graph of $\mathcal{C}$ here https://www.desmos.com/calculator/x7cia8bdpz. The graph also illustrates tangent lines.
2. Determine all places where $C$ has horizontal tangent lines. The graph may guide you, but make sure you can write down the math to support your answer.
3. Determine all places where $C$ has vertical tangent lines.
4. Find equations for all lines tangent to $C$ at $(0,0)$.
5. Find the area of the "teardrop" at the bottom of $C$. Hint: to find the area between a parametric curve $x=f(t)$, $y=g(t)$, $\alpha\le t\le \beta$, and the $x$-axis, the book gives the formula $\
int_\alpha^\beta g(t)f'(t)\;dt$. Here it might be better to find the area between the curve and the $y$-axis, and for this, the formular is $\int_\alpha^\beta f(t)g'(t)\;dt$
[09.30.15] - Wednesday Very nice presentations today! And we now know the area of the teardrop: $\frac{72\sqrt{3}}{5}$.
Sections covered: 10.2
Reading for next time: Section 10.3. Focus on the following:
1. Introduction
2. Examples 1, 4, 5, 7
To discuss next time:
1. Play around with polar equations in Desmos. Look at the following for examples:
2. Use Desmos to make a super cool graph using a polar equation. Write everything down, so you can reproduce it in class.
3. Graph the polar equation $r=\frac{\pi}{\theta}$. Make sure you zoom in enough to see the spiral and zoom out enough to see that it does not continue to spiral. Think about what $r$ and $\theta$
represent and be able to explain why the graph looks like it does.
[10.02.15] - Friday Started off with an example of computing the area of a suface obtained by revolving a parametrically defined curve, and then we launched into polar coordinates. We began with
presentations of some fun polar pictures, though parts were drawn parametrically (in $xy$-coordinates) too. To truly appreciate the polar box, one has to realize just how wild of an animal a box is
in the polar world. Nice job to both presenters! We then talked about the curve defined by $r=\frac{\pi}{\theta}$ (and I realized that I wasn't thinking about right myself). A brave voice got us
studying the behavior as $\theta \rightarrow \infty$ and we also looked at what happens as $\theta \rightarrow 0^+$. Well done!
- Some wild polar animals -
Sections covered: 10.2 (finished), (10.3 started)
Reading for next time: Section 10.3. Focus on the following:
1. The subsection "Tangents to Polar Curves" on pg 663
2. Example 9
To discuss next time: Think more about the polar equation $r=\frac{\pi}{\theta}$.
1. Graph it on Desmos, and study what happens when $\theta \rightarrow 0^+$.
2. There seems to be a horizontal asymptote. Can you confirm this? Try using our conversion formulas and studying what happens to the $y$-values as $\theta \rightarrow 0^+$.
3. Here's another approach. First compute $\frac{dy}{dx}$ for this curve. Now compute $\lim_{\theta \rightarrow 0^+} \frac{dy}{dx}$. What does this mean?
Week 7
[10.05.15] - Monday
Sections covered: 10.4
Reading for next time: None. Start reviewing for the exam, which will cover up through Section 10.2.
Don't forget that there's Homework (written and online) due Tuesday!
To discuss next time: Let's start reviewing for the exam...
1. Decide which section was the most challenging for you. Meditate on the following, and be prepared to share your answers (especially those who have not presented yet).
□ What is an example of a problem from the section that you found challenging?
□ What do you feel are the key parts of this problem that you need to understand better?
2. Decide which section was the easiest for you. Meditate on the following, and be prepared to share your answers.
□ What is an example of a problem from the section?
□ What do you feel are the key parts of this problem that your classmates should be aware of so that it becomes easy for them too?
[10.07.15] - Wednesday Wrapped-up polar today, and then started reviewing for the exam. We discussed several problems, but below is one we didn't make it to. Try it; it's a good one!
Find the volume of the solid obtained by revolving the region bounded by the curves $y=x\ln x$, $x=e$, and $y=0$ about the $y$-axis. Be aware: part of the region is below the $x$-axis, so if you use
shells, you need to split up the integral and "make that part positive."
Sections covered: 10.4 (finished), review for Midterm 1
Reading for next time: None. Focus on studying.
Don't forget that there's Homework (only written) due Friday!
To discuss next time: The exam.
[10.09.15] - Friday Started in on sequences today. Did a little group work and had a generally nice and lite day, well almost...
Sections covered: 11.1 (started)
Reading for next time: 11.1. Focus on the following:
1. Definition 10, Example 12, Definition 11
2. Theorem 12 and Figure 12 (you do not need to understand the proof of Theorem 12)
3. Example 14 (beginning and end - the middle, which proves why the sequence in bounded, can be skipped)
To discuss next time: The golden ratio and continued fractions. Specifically, we will make sense of the following continued fraction: \[1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1+\ddots}}} \] The point
is that the continued fraction really represents the limit of a particular sequence. Consider the following sequence (defined recursively - see Example 14): \[a_{n+1} = 1+\frac{1}{a_n},\quad a_1 = 1.
• Write down and do not simplify $a_1$, $a_2$, $a_3$, and $a_4$. Can you see why (and are you able to explain why) it makes sense to represent $\lim_{n\rightarrow\infty} a_n$ as the continued
fraction above?
• Explain why $\{a_n\}$ is bounded below by $0$ and above by $2$.
• Now, $\{a_n\}$ is neither increasing nor decreasing, but it still can be shown that $\{a_n\}$ has a limit $L$. That is, $\lim_{n\rightarrow\infty}a_n = L$, but this also means that $\lim_{n\
rightarrow\infty}a_{n+1} = L$. Use the method at the end of Example 14 to find $L$.
Week 8
[10.12.15] - Monday
Sections covered: 11.1
Reading for next time: 11.2. Focus on the following:
1. Intro through Example 4
2. Example 9
To discuss next time: Section 11.2 introduces our first techniques for determining if series converge; let's discuss a couple of them.
1. (Geometric series) After reading examples 2-4, try this variation: find the value of $c$ such that $\sum_{n=1}^\infty e^{nc} = 10$. How would your answer change if the equation was $\sum_{n=0}^\
infty e^{nc} = 10$?
2. (Harmonic series) Here's a different approach for showing that the series in Example 9 diverges.
□ Check out this Demos graph: https://www.desmos.com/calculator/ufyuseeewk
□ Explain why $\sum_{n=1}^\infty \frac{1}{n}$ can be thought of the blue area in the Desmos graph.
□ Now explain why the blue area is infinite by showing that the blue area is greater than a different area that we know is infinite from much earlier in the course.
[10.14.15] - Wednesday
Sections covered: 11.2
Reading for next time: End of 11.2. Focus on the following:
1. Example 8
2. Theorem 8 (on page 714) through the end of the section
To discuss next time:
1. After reading Theorem 8 and Example 11, determine if the following series is convergent or divergent: \[\sum_{n=1}^\infty \frac{2^n+e^n}{\pi^n}\]
2. Meditate on Theorem 8, and be able to explain why the following is true:
If $\sum a_n$ is convergent and $\sum b_n$ is divergent, then $\sum (a_n + b_n)$ is divergent.
3. Determine if the following series is convergent or divergent: \[\sum_{n=1}^\infty \frac{2^n+\pi^n}{e^n}\]
4. Meditate on Theorem 8 again, and determine if the following is true or false:
If $\sum a_n$ is divergent and $\sum b_n$ is divergent, then $\sum (a_n + b_n)$ is divergent.
[10.16.15] - Friday Fall Break!
Week 9
[10.19.15] - Monday
Sections covered: 11.2 (finished), 11.3 (started)
Reading for next time: 11.3. Focus on the following:
1. The statement of the Integral Test on pg 721
2. The (important) note between the statement of the Integral Test and Example 1 on pg 721
3. Example 4
To discuss next time:
1. After reading Example 4, determine if $\displaystyle\sum_{i=0}^\infty \frac{n}{e^n}$ converges or diverges.
□ You will probably want to use the integral test, but to do this, make sure to explain why $\displaystyle\frac{x}{e^x}$ is continuous, positive, and (eventually) decreasing.
2. Also, determine if $\displaystyle\sum_{i=0}^\infty \frac{-n}{e^n}$ converges or diverges.
[10.21.15] - Wednesday
Sections covered: 11.3 (finished), 11.4 (started)
Reading for next time: 11.4. Focus on the following:
1. The blue boxes, i.e. the "Comparison Test" and the "Limit Comparison Test" (LCT)
2. Note 1 and Example 2
3. Note 2 and Example 3
To discuss next time:
1. After reading the Comparison Test and LCT, think about the following questions about the series $\sum_{n=1}^\infty \frac{1}{n!}$.
□ Explain why $\frac{1}{n!} \le \frac{1}{n}$ for $n\ge 1$.
□ Explain why you cannot use the Comparison Test to compare $\sum_{n=1}^\infty\frac{1}{n!}$ to $\sum_{n=1}^\infty\frac{1}{n}$.
□ Explain why $\frac{1}{n!} \le \frac{1}{n(n-1)}$ for $n\ge 2$.
□ Show that $\sum_{n=1}^\infty\frac{1}{n(n-1)}$ converges by comparing it to an appropriate $p$-series. Were you able to use the Comparison Test or did you have to use the LCT?
□ What can you conclude about $\sum_{n=1}^\infty \frac{1}{n!}$? And why?
[10.23.15] - Friday Great day, with lots of fantastic presentations (and lots of participation from the audience)! We'll pick up with the examples we ended with next time.
Sections covered: 11.4
Reading for next time: Reread 11.4. Focus on the following:
1. The "Limit Comparison Test" (LCT)
2. Example 4
To discuss next time:
1. Think about the following questions related to $\sum_{n=1}^\infty \sin\left(\frac{1}{n}\right)$.
□ Explain why $\sin\left(\frac{1}{n}\right)$ is positive for $n\ge 1$.
□ Explain why you cannot use the Comparison Test to compare $\sum_{n=1}^\infty \sin\left(\frac{1}{n}\right)$ to $\sum_{n=1}^\infty\frac{1}{n}$. (A graph may help.)
□ Show that you can use the LCT to compare $\sum_{n=1}^\infty \sin\left(\frac{1}{n}\right)$ to $\sum_{n=1}^\infty\frac{1}{n}$.
2. Think about the following questions related to $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$.
□ Explain why you cannot use the Comparison Test to compare $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ to $\sum_{n=1}^\infty\frac{1}{n}$.
□ Explain why you cannot use the LCT to compare $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ to $\sum_{n=1}^\infty\frac{1}{n}$.
□ What's your feeling about the convergence or divergence of $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$? Write this down, with your reasoning, before going to the next part.
□ Look at the Desoms graph here: https://www.desmos.com/calculator/npezpry5vc. The orange dots represent the sequence of terms $\frac{(-1)^{n+1}}{n}$. The black dots represent the sequence of
partial sums $s_n = \sum_{i=1}^n\frac{(-1)^{i+1}}{i}$. Did this change or confirm your feeling about convergence or divergence? If you want, you can compare the previous graph to the graph of
the terms and partial sums of the harmonic series here: https://www.desmos.com/calculator/3vvosxcslp.
Week 10
[10.26.15] - Monday
Sections covered: 11.5 (started)
Reading for next time: 11.6. Focus on the following:
1. Definition 1 on page 737
2. Example 2
3. Theorem 3 on page 738
4. Example 3
To discuss next time:
1. Think about the following questions related to $\sum_{n=1}^\infty \frac{\sin\left(3^n\right)}{2^n}$.
□ Explain why you cannot use the Comparison Test or LCT to compare $\sum_{n=1}^\infty \frac{\sin\left(3^n\right)}{2^n}$ to anything.
□ Explain why the Alternating Series Test does not apply to $\sum_{n=1}^\infty \frac{\sin\left(3^n\right)}{2^n}$.
□ Show that you can use the Comparison Test to compare $\sum_{n=1}^\infty \left\lvert\frac{\sin\left(3^n\right)}{2^n}\right\rvert$ to a convergent geometric series.
□ What, if anything, does Theorem 3 on page 738 allow you to conclude about $\sum_{n=1}^\infty \frac{\sin\left(3^n\right)}{2^n}$?
[10.28.15] - Wednesday Great discussions today! Thanks to everyone that participated, at the board and from the audience.
Sections covered: 11.5 (finished), 11.6 (started)
Reading for next time: 11.6. Focus on the following:
1. The Root Test (blue box on page 741)
2. Example 6
To discuss next time:
1. Think about the following questions related to $\sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^n$ and $\sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^{n^2}$.
□ Explain why the root test does not help you with $\sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^n$.
□ Compute $\lim_{x\rightarrow \infty}\left(\frac{x}{x+1}\right)^x$. Take your time with this. It requires equal parts logarithms and L'Hospital.
□ What does the limit you found above tell you, if anything, about $\sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^n$?
□ What does the limit you found above tell you, if anything, about $\sum_{n=1}^\infty \left(\frac{n}{n+1}\right)^{n^2}$? Hint: root test.
[10.30.15] - Friday Another great day of presentations and discussions!
Sections covered: 11.6 (finished)
Reading for next time: 11.7. Focus on the following:
1. Read everything (including the examples)! The section is short and helpful.
To discuss next time:
1. Be able to crush "Converge or Diverge" questions on demand. We'll spend the entire next class working through and presenting "Converge or Diverge" problems.
2. Bonus points if you can highlight a problem (or part of a problem) that you find particularly tricky and explain what we should all be careful of. (You don't even need to be able to solve the
problem, but you do need to have thought deeply about why you find it tricky, e.g. what were all the things you tried, and why did they fail.)
Week 11
[11.02.15] - Monday Great day of group work and presentations. Every day should be like this.
Sections covered: 11.7
Reading for next time: 11.8. Focus on the following:
1. Introduction
To discuss next time: Meditate on the fact that "a power series $\sum_{n=0}^\infty c_nx^n$ is a function of $x$."
1. Let $\displaystyle f(x) = \sum_{n=0}^\infty x^n$. Writing $f(x)$ out, we get \[f(x) = 1 + x + x^2 + x^3 + x^4 + x^5 +\cdots\]
□ Is $f(0.5)$ defined? If so, what is it? How about $f(0)$? $f(1)$?
□ Explain why the domain of $f$ is $-1 < x < 1$.
□ Explain why $f(x)$ given by the formula $f(x) = \frac{1}{1-x}$ whenever $-1 < x < 1$.
2. Let $\displaystyle f(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$. Writing $f(x)$ out, we get \[f(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} +\cdots\]
□ What do you think is a formula for $f'(x)$? (You do not need to justify your answer!)
□ Go to Desmos and graph $\sum_{n=0}^{100} \frac{x^n}{n!}$.
□ Based on what you found in the previous part, hazard a guess as to a simpler formula for $f(x)$.
[11.04.15] - Wednesday
Sections covered: 11.8 (started)
Reading for next time: 11.9. Focus on the following:
1. Theorem 2 on page 754 and Notes 1, 2, and 3 that follow it.
2. Examples 4 and 5
To discuss next time: Consider the function $\displaystyle f(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}$. Writing $f(x)$ out, we get \[f(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}
{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10!} + \cdots\]
1. Find an expression for $f'(x)$.
2. Show that $f''(x) = -f(x)$.
3. Find $f(0)$ and $f'(0)$.
4. Meditate, and hazard a guess as to a simpler formula for $f(x)$.
5. Go to Desmos and graph $\sum_{n=0}^{25} (-1)^n \frac{x^{2n}}{(2n)!}$. Was your guess right?
[11.06.15] - Friday
Sections covered: 11.8 (finished), 11.9 (started)
Reading for next time: 11.9. Focus on the following:
1. Examples 5, 6, and 7
To discuss next time: Let's see what can we deduce from the geometric series \[\frac{1}{1-x} = \sum_{n=0}^\infty x^n,\quad\quad -1< x < 1 \]
1. Use the geometric series above and ideas from 11.9 to fill in the following blanks.
□ $\displaystyle \underline{\hspace{1in}} = \sum_{n=1}^\infty nx^{n-1},\quad\quad -1< x < 1$
□ $\displaystyle \underline{\hspace{1in}} = \sum_{n=1}^\infty nx^{n},\quad\quad -1< x < 1$
□ $\displaystyle \underline{\hspace{1in}} = \sum_{n=1}^\infty \frac{n}{2^{n}}$
2. Explain: \[ \ln 2 = \sum_{n=0}^\infty \frac{1}{(n+1)2^{n+1}}\]
Hint: this one does not build off of the previous parts. Try going back to the geometric series and doing something clever (but different than before)...
Week 12
[11.09.15] - Monday
Sections covered: 11.9 (finished)
Reading for next time: 11.10. Focus on the following:
1. Introduction through Example 1
2. Table 1 on page 768
To discuss next time: we now have the following power series representation of $e^x$ \[e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1+ x + \frac{x^2}{2!}+ \frac{x^3}{3!} + \cdots,\quad\quad -\infty< x <
\infty \]
1. Evaluate $\int xe^x \, dx$ by first finding a power series representation for $xe^x$ and then integrating the terms of the power series.
Hint: you can use the above series for $e^x$ to find a power series for $xe^x$; you do not need to start from scratch.
2. Of course you can also evaluate $\int xe^x \, dx$ (without series) by using integration by parts. Do it.
3. Which of the previous ways was faster? Which gives (in your opinion) a more useable answer?
4. Evaluate $\int e^{x^2} \, dx$ any way you want.
[11.11.15] - Wednesday
Sections covered: 11.10 (started)
Reading for next time: 11.10. Focus on the following:
1. Examples 3-7, 10, 11
To discuss next time: suppose the Maclaurin series for a function $f$ is \[\sum_{n=0}^\infty \frac{n+2}{2^n} x^n = 2 + \frac{3}{2}x + x^2 + \frac{5}{8}x^3 +\cdots \]
1. What is $f(0)$? How about $f'(0)$? What is an equation for the tangent line to $f$ at $x=0$?
2. What is an expression for $f^{(n)}(0)$?
[11.13.15] - Friday
Sections covered: 11.10 (finished)
Reading for next time: 12.1. Focus on the following:
1. Introduction through Example 3
To discuss next time:
1. In Example 2, you saw that the equation $x^2 + y^2 = 1$ represents a circle in $\mathbb{R}^2$, but in $\mathbb{R}^3$, the equation represents an infinite cylinder centered around the $z$-axis.
□ Describe the graph of the equation $y=sin(x)$ in $\mathbb{R}^3$.
□ How about the general case; that is, can you describe how to graph an equation $y=f(x)$ in $\mathbb{R}^3$?
□ Can you write an equation for an infinite cylinder centered around the $y$-axis?
Week 13
[11.16.15] - Monday
Sections covered: 12.1
Reading for next time: None. Start reviewing for the exam, which will cover up through Section 11.10.
Don't forget that there's Homework (written and online) due Tuesday!
To discuss next time: Let's start reviewing for the exam. Same directions as before...
1. Decide which section was the most challenging for you. Meditate on the following, and be prepared to share your answers (especially those who have not presented yet).
□ What is an example of a problem from the section that you found challenging?
□ What do you feel are the key parts of this problem that you need to understand better?
2. Decide which section was the easiest for you. Meditate on the following, and be prepared to share your answers.
□ What is an example of a problem from the section?
□ What do you feel are the key parts of this problem that your classmates should be aware of so that it becomes easy for them too?
[11.18.15] - Wednesday Started reviewing for the exam. Good luck y'all!
Sections covered: review for Midterm 2
Reading for next time: None. Focus on studying.
No Homework due Friday!
Week 14
[11.30.15] - Monday
Sections covered: 12.3 (started)
Reading for next time: 12.3. Focus on the following:
1. Theorem $\fbox{3}$ and Corollary $\fbox{6}$
2. The statement labeled $\fbox{7}$
3. The section on projections and Example 6
To discuss next time:
1. Think geometrically to find a vector that is orthogonal to $\langle 0,0,1\rangle$. How many possible answers are there? Can you describe the possible answers geometrically?
2. Use $\fbox{7}$ to explain how to find a vector that is orthogonal to $\langle 2,1,1\rangle$. How many possible answers are there? Can you describe the possible answers geometrically? Hint: assume
that $\langle x,y,z\rangle$ is orthogonal to $\langle 2,1,1\rangle$ and use $\fbox{7}$ to find possibilities for $x$, $y$, and $z$.
3. Use $\fbox{7}$ to explain how to find a vector that is orthogonal to both $\langle 2,1,1\rangle$ and $\langle 1,-1,0\rangle$. How many possible answers are there? Can you describe the possible
answers geometrically?
[12.02.15] - Wednesday
Sections covered: 12.3 (finished), 12.4 (started)
Reading for next time: 12.4. Focus on the following:
1. Theorem $\fbox{8}$, Theorem $\fbox{9}$, and Corollary $\fbox{10}$
2. The discussion following Corollary $\fbox{10}$ about area of parallelograms
3. Examples 3 and 4
To discuss next time: Describing planes
1. Consider the points $A(1,0,0)$, $B(0,2,0)$, and $C(0,0,3)$. Plot these three points and draw the plane containing them (there is only one). If you need, you can use GeoGebra.
2. Read Example 3 in 12.3, and then find a vector $\mathbf{n}$ that is perpendicular to the plane from the first part.
3. Suppose that the point $P(x,y,z)$ is in the plane. What is the relationship between $\mathbf{n}$ and the vector $\overrightarrow{AP}$? Hint: draw a picture. Can you use this to quickly determine
if $(1,-1,1)$ is in the plane? What about $(2,0,3)$?
[12.04.15] - Friday
Sections covered: 12.4 (finished), 12.5 (started)
Reading for next time: 12.5. Focus on the following:
1. Equations $\fbox{1}$, $\fbox{2}$, $\fbox{4}$
2. Examples 1-3
To discuss next time:
1. First read Examples 2 and 3. Now, consider the following 2 lines (given parametrically) \[\begin{align*}L_1:&\quad x=8+3t,\quad y=4-t,\quad z=-3+t\quad\quad-\infty< t < \infty\\ L_2:&\quad x=2+s,
\quad y=s,\quad z=6+4s\quad\quad-\infty< s < \infty\end{align*}\]
□ Find a direction vector for $L_1$. How many possible answers are there for this? Find a direction vector for $L_2$ too.
□ By looking at the direction vectors, can you explain why $L_1$ and $L_2$ are not parallel?
□ Can you explain why $L_1$ and $L_2$ do not intersect?
□ Does $L_1$ intersect the $xy$-plane? If so, where?
Week 15
[12.07.15] - Monday
Sections covered: 12.5 (finished)
Reading for next time: 12.6. Focus on the following:
1. Examples 3-5
2. Table 1 on page 837
To discuss next time: Look over Table 1 on page 837.
1. How would you change the equation for an elliptic paraboloid so that it opens along the negative part of $x$-axis and has a vertex at $(1,2,3)$? (Of course you can check you answer with GeoGebra
2. What must be true of the equation for a hyperboliod of one sheet so that the horizontal traces are circles?
3. Consider the following equation where $k$ is a constant: \[\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2} = k\] What types of surfaces are produced when $k$ is allowed to vary between $-\infty$
and $\infty$? For example, what do you get when $k=1$? $k=4$? $k=-4$? Explain. (If you're stuck, try experimenting with GeoGebra.)
[12.09.15] - Wednesday
Sections covered: 12.6
Reading for next time: None. Start reviewing for the final exam.
Don't forget that there's Homework (the last one!) due Friday.
To discuss next time: Let's start reviewing for the final exam. Focus on the new material. Same directions as before...
1. Decide which section was the most challenging for you. Meditate on the following, and be prepared to share your answers (especially those who have not presented yet).
□ What is an example of a problem from the section that you found challenging?
□ What do you feel are the key parts of this problem that you need to understand better?
2. Decide which section was the easiest for you. Meditate on the following, and be prepared to share your answers.
□ What is an example of a problem from the section?
□ What do you feel are the key parts of this problem that your classmates should be aware of so that it becomes easy for them too? | {"url":"https://webpages.csus.edu/wiscons/teaching/math116_f15.html","timestamp":"2024-11-14T11:21:17Z","content_type":"text/html","content_length":"52163","record_id":"<urn:uuid:5c75f562-2f55-484c-8d7c-993f1a77de63>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00778.warc.gz"} |
Jamarius W.
What do you want to work on?
About Jamarius W.
Algebra, Calculus, Elementary (3-6) Math, Geometry, Trigonometry, Algebra 2, Midlevel (7-8) Math
Bachelors in Mathematics, General from Alabama State University
Math - Algebra II
Yes he was a good tutor
Math - Calculus
great tutor very patient
Math - Calculus
not the best at answering questions, could of finished in half the time.
Math - Algebra II
Thank you! :) | {"url":"https://ws.princetonreview.com/academic-tutoring/tutor/jamarius%20w--4084822","timestamp":"2024-11-12T03:46:05Z","content_type":"application/xhtml+xml","content_length":"184473","record_id":"<urn:uuid:aa7656c3-21ff-4fc6-8308-b6d76f4ff129>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00054.warc.gz"} |
Jeff's Lunchbreak
If you're not involved in a technical field, you may not be familiar with the idea of coherent or consistent units. Basically, it's the idea that you shouldn't need any fudge factors in an equation
because of the units you're using. For example, power can be calculated as a force times a speed, or P=F*V. Using consistent U.S. units gives an answer in ft-lb/sec, while using metric units gives an
answer in N-m/sec (also known as Watts). A non-consistent unit of power that most people are familiar with is horsepower. If you multiply force times speed, you then have to divide by a fudge factor
of 550 to get your answer in horsepower, or HP=F*V/550. And the fudge factors only work if the inputs are the units you're expecting. If people wanted to use mph instead of ft/s for the velocity,
then you'd need another fudge factor on top of that, HP=F*Vmph*(5280/3600)/550. Equations can get pretty messy if you're not using consistent units, having to multiply all those fudge factors
In the standard units used for engineering in the U.S., pounds are a measure of force, not mass (this is already a distinction some people are unfamiliar with, confusing weight and mass). The unit
for mass, which most non-technical people would already consider an obscure unit, is the slug. But trust me, I use slugs on a nearly daily basis as an engineer. On Earth, a slug weighs approximately
32 lbs (i.e. F=mg). Or for you metric people, it's equivalent to about 14.6 kg (which measure mass, not weight).
But the engineers who do stress calculations don't always use the normal FPS (foot-pound-second) system, because everyone's used to seeing stresses reported in lbs/in², or psi. And if you were using
the normal FPS system, your stresses would come out in lb/ft², and you'd have to do a conversion at the end of your calculations to put the results in the psi that most people are used to seeing*.
That's not a huge deal for spreadsheets or hand calcs, but it does make it more difficult for certain finite element programs. So, the stress guys sometimes use a different set of units based on
pounds and inches, with the mass unit being the slinch. A slinch is 1/12 of a slug (i.e. the ratio between feet and inches). On Earth, a slinch weighs approximately 32/12 lbs, or 2.7 lbs.
Of course, a lot of these weird units could be simplified if all the engineers in the U.S. started using the metric system like the rest of the world, but that's not the way it is right now, so I've
got to use units that other U.S. engineers are familiar with. And if Wikipedia is to be trusted, the metric stress guys have their own weird mass unit of glugs in the centimeter-gram-second system.
* Speaking of weird stress units, I remember working with a foreign engineer one time who gave me her results in Pascals, which is the normal metric way to do it. When I asked her to convert her
results to U.S. units, she gave them to me in N/ft².
Image Source: Wikimedia Commons, slightly Photoshopped to remove a dead fly
Note that this entry was adapted from a response I left on Quora. | {"url":"http://www.jefflewis.net/blog/2015/11/","timestamp":"2024-11-13T12:33:46Z","content_type":"application/xhtml+xml","content_length":"15524","record_id":"<urn:uuid:ae381129-86fd-45d5-8379-0ad6adf45c13>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00092.warc.gz"} |
rbox(1) General Commands Manual rbox(1)
rbox - generate point distributions for qhull
Command "rbox" (w/o arguments) lists the options.
rbox generates random or regular points according to the options given, and outputs the points to stdout. The points are generated in a cube, unless 's' or 'k' option is given. The format of the
output is the following: first line contains the dimension and a comment, second line contains the number of points, and the following lines contain the points, one point per line. Points are
represented by their coordinate values.
10 random points in the unit cube centered at the origin.
10 random points on a 2‐d circle.
100 random points on the surface of a cube.
1000 random points on a 4‐d sphere.
a 5‐d hypercube with one corner at the origin.
a 10‐d diamond.
100 random points on the surface of a fixed simplex
a 12‐d simplex.
10 random points along a spiral
10 regular points along a spiral plus two end points
1000 random points on the surface of a narrow lens.
a cube with coordinates +2/-2 and a diamond with coordinates +3/-3.
a rotated, {0,1,2,3} x {0,1,2,3} x {0,1,2,3} lattice (Mesh) of integer points. 'rbox 64 M1,0' is orthogonal.
5 copies of the origin in 3-d. Try 'rbox P0 P0 P0 P0 P0 | qhull QJ'.
two cospherical 100-gons plus another cospherical point.
100 s Z1
a cone of points.
100 s Z1e-7
a narrow cone of points with many precision errors.
number of points
dimension n‐d (default 3‐d)
bounding box coordinates (default 0.5)
spiral distribution, available only in 3‐d
lens distribution of radius n. May be used with 's', 'r', 'G', and 'W'.
lattice (Mesh) rotated by {[n,-m,0], [m,n,0], [0,0,r], ...}. Use 'Mm,n' for a rigid rotation with r = sqrt(n^2+m^2). 'M1,0' is an orthogonal lattice. For example, '27 M1,0' is {0,1,2} x {0,1,2} x
{0,1,2}. '27 M3,4 z' is a rotated integer lattice.
cospherical points randomly generated in a cube and projected to the unit sphere
simplicial distribution. It is fixed for option 'r'. May be used with 'W'.
simplicial distribution plus a simplex. Both 'x' and 'y' generate the same points.
restrict points to distance n of the surface of a sphere or a cube
add a unit cube to the output
c Gm
add a cube with all combinations of +m and -m to the output
add a unit diamond to the output.
d Gm
add a diamond made of 0, +m and -m to the output
add n nearly coincident points within radius r of m points
add point [n,m,r] to the output first. Pad coordinates with 0.0.
Remove the command line from the first line of output.
offset the data by adding n to each coordinate.
use time in seconds as the random number seed (default is command line).
set the random number seed to n.
generate integer coordinates. Use 'Bn' to change the range. The default is 'B1e6' for six‐digit coordinates. In R^4, seven‐digit coordinates will overflow hyperplane normalization.
restrict points to a disk about the z+ axis and the sphere (default Z1.0). Includes the opposite pole. 'Z1e-6' generates degenerate points under single precision.
same as Zn with an empty center (default G0.5).
generate a regular polygon
generate a regular cone
Some combinations of arguments generate odd results.
Report bugs to qhull_bug@qhull.org, other correspondence to qhull@qhull.org
C. Bradford Barber
August 10, 1998 Geometry Center | {"url":"https://manpages.opensuse.org/Tumbleweed/qhull/rbox.1.en.html","timestamp":"2024-11-08T21:59:59Z","content_type":"text/html","content_length":"23591","record_id":"<urn:uuid:43deda76-6b70-4c81-964a-5f1664cc34ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00079.warc.gz"} |
Wilfrid Hodges' logic page
Dr Wilfrid Hodges
Herons Brook
Devon EX20 2PY
Phone 01837 840154
E-mail my first and last names with a dot between them, at btinternet.com.
Married since 1965 to Helen M. Hodges, Emeritus Professor, Institute of Psychiatry, Kings College, University of London.
Children (all with Facebook pages):
• Sally Watson = Dr Sally Hodges, Tavistock and Portman NHS Foundation Trust;
• Gale Hodges;
• Edwin (Edd) Hodges.
Directions for reaching me by car are here, and pictures of the house and property are here. For the South Zeal Open Gardens of June 2012 we put together a guide to the mining remains at Herons
Bibliography and CV
A list of my publications is here. My CV is here.
Corrigenda to publications
1. Corrigenda to 'Model Theory'
2. Corrigenda (9 September 2014) to 'A Shorter Model Theory'
3. Corrigenda to 'Logic'
4. Corrigenda to 'Building Models by Games' are incorporated in the Dover edition.
5. Corrigenda to 'Mathematical Logic' (with Ian Chiswell)
6. Corrigenda to 'Elementary Predicate Logic' (in 'Handbook of Philosophical Logic' Vol. 1; to be added)
Several of these are due for updating.
The lectures below come with two health warnings. First, when I lecture I add explanations and comments; you will have to imagine these. Second, there are some mistakes, some of which are corrected
in the published versions indicated.
Arabic logic and semantics
Translations of Ibn Sina
1. Short Epitome sections on syllogisms.
2. Madkhal i.6. A translation of section i.6 of Ibn Sina's commentary on Porphyry's Eisagoge. This passage contains his reasons for rejecting Porphyry's notion of an inseparable accident and
replacing it by his own notion of a necessary accident. Briefly, the issue is that Porphyry took logic to be about what we can or can't imagine to be the case, whereas Ibn Sina switches attention
to what we can infer by logical rules. At first sight Ibn Sina defines logical rules in terms of the operation of the intellect, which he places in the rear cerebral ventricles. But this is
unfair; Ibn Sina is certainly not under the impression that logical rules could depend on brain function. The rules are those followed when the inference engine (the 'working mind') in the
intellect operates with 'correctness'. Ibn Sina points out towards the end of this section that 'correctness' can be understood in two ways. First, the inference engine operates correctly when it
follows correct logical inference rules. Second, the inference engine operates correctly when it doesn't tolerate things that couldn't be true in the world. These notions of correctness are
distinct because the inference engine can only handle single applications of inference rules (local formalising!) and is unable to reach conclusions that need two or more inference steps. (The
translation has not yet been checked by a native Arabic speaker.) (29 September 2013)
3. Ibara ii.1. 'Affirmative and negative in Ibn Sina', a preprint of a paper submitted for a Festschrift. It contains a translation of section ii.1 of Ibn Sina's commentary on De Interpretatione, in
which he discusses which sentences count as negative, and claims that all true affirmative predicative sentences have a nonempty subject. I comment on his discussion and argue that parts of it
should count today as linguistics rather than logic. (27 June 2012.)
4. Ibara ii.4. 'Ibn Sina on modes', a translation of section ii.4 of his commentary on De Interpretatione. Warning: the translation is complete but it has not yet been checked by a native Arabic
speaker, and the accompanying notes are incomplete. In this section Ibn Sina doesn't add much to what was in the Alexandrian commentators. But one thing seems to be original in him, namely the
distinction between 'Possibly every A is a B' and 'Every A is possibly a B'. Sadly his account of the logical differences between the two is a mess, which will leave some doubt about what he
thinks is going on in proofs that something doesn't follow from something else. The two sections Qiyas iii.4, 5 (which are now on my list to translate) develop the material here in more
interesting ways; see the opening notes to this file. Modal logic was never my thing, but I thank Zia Movahed for urging me to take an interest in this material. (9 February 2010.)
5. Ibara ii.5. 'Ibn Sina and conflict in logic', for a Festschrift. (Revised draft 18 August 2009.) This contains a commented translation of Ibn Sina's commentary on the final section of Aristotle,
Peri Hermeneias. Ibn Sina tackles some particularly incomprehensible remarks in Aristotle with the aid of some thoughts about lengths of proofs. There is also a polemical section which seems to
be intended to show that metatheorems of logic - in particular those of the kind illustrated by the more recent laws of distribution and interpolation theorems - are just as useful for
philosophical argument as the laws of logic itself. If Ibn al-Tayyib is Ibn Sina's target in this passage, it may help to explain why Ibn al-Tayyib came to dislike Ibn Sina so much.
6. Qiyas i.2. 'Ibn Sina on logic as a tool'. This section is of cardinal importance for Ibn Sina's understanding of logic as a scientific theory. Since it becomes clear that neither this nor Ibn
Sina's own logical theorising makes sense without the other, a full commentary on the section is needed, and will appear as time allows. Meanwhile I know of no logically literate discussions of
this material in the literature, and I would be grateful if anybody can point me to discussions that I missed. (11 February 2016)
7. Qiyas i.3. 'Ibn Sina on understood time and aspect', a translation of section i.3 of Qiyas. This is a very rich section containing some of his most original work. It can serve as an introduction
to his teaching that what we mean, even in careful scientific discourse, generally goes a long way beyond what we say. I put it up partly so that I can refer people to it. But like the
translation two above, the translation has not yet been checked by a native Arabic speaker and the notes are incomplete. (5 March 2010.)
8. Qiyas i.5. 'Ibn Sina on contradictories of absolutes, a commented translation of section i.5 of Qiyas. It picks up from the section i.3 translated above, and draws consequences about possible
sentence structures. The text is important for establishing Ibn Sina's links with the earlier tradition, in particular Theophrastus and Alexander. But my present interest in it is its rather
abortive discussion of quantifier scopes in terms of function representations rather than syntax. The paper is severely unfinished, but I put it up in order to be able to cite it for a talk on
scope at the September 2010 Fest for Jouko Väänänen. (10 September 2010)
9. Qiyas ii.2. 'Qiyas ii.2: Conversion of absolutes. This and the next six items are text and translation of seven of the fifteen sections covering the formal logic of modalities (and more besides)
in the book Qiyas. It will be obvious that the translations are work in progress and very much at draft stage. Also with this kind of material, raw translations without commentary have limited
value. But in view of the increasing interest in Ibn Sina's modal logic it seems sensible to make them available. More will follow as soon as time allows. (13 January 2013)
10. Qiyas ii.3. 'Qiyas ii.3: Conversion of necessaries and possibles. (4 November 2012)
11. Qiyas ii.4. 'Qiyas ii.4: Recombinant syllogisms. (13 January 2013)
12. Qiyas iii.1. 'Qiyas iii.1: On mixed syllogisms with absolute and necessary. (4 November 2012)
13. Qiyas iii.2 (part). 'Ibn Sina states and applies properties of temporal logic'. (1 May 2014)
14. Qiyas iv.1. 'Qiyas iv.1: On possibility syllogisms in the first figure. (4 November 2012)
15. Qiyas iv.2. 'Qiyas iv.2: On syllogisms that are mixtures of possible and absolute in the first figure. (4 November 2012)
16. Qiyas iv.3. 'Qiyas iv.3: On syllogisms that are mixtures of possible and necessary in the first figure. (13 January 2013)
17. Qiyas v.1. 'Ibn Sina on conditionals', work in progress, so please don't quote without checking with me. I put it up because of the centrality of the topic. (18 August 2010.)
18. Qiyas vi.4. A translation of Qiyas vi.4. This is a very boring section on propositional logic, but hidden inside it is the idea of a novel and surprisingly strong proof rule, which Ibn Sina
invokes in his treatment of reductio ad absurdum. (2 Oct 2013)
19. Qiyas viii.3. 'Ibn Sina on reductio ad absurdum. This is the revised text (submitted 5 May 2015) of a paper accepted by the Review of Symbolic Logic in March 2014. 'Ibn Sina's explanation of
reductio ad absurdum' was an expository lecture on this for a workshop in Brussels. (8 February 2016.)
20. Qiyas ix.3. 'Ibn Sina on patterns of proofs' (Revised draft 6 October 2009.) In Qiyas ix.3, Ibn Sina defines two different notions of compound syllogism and studies the relations between them.
21. Qiyas ix.6. 'Ibn Sina on analysis: 1. Proof search. Or: Abstract State Machines as a tool for history of logic', for a Festschrift for Yuri Gurevich. (Revised draft 6 September 2009.) This
contains a commented translation of a section of Ibn Sina's commentary on a couple of paragraphs of Aristotle's Prior Analytics. Ibn Sina seems to be giving his reader a string of 64 exercises to
train the reader in a proof search algorithm for syllogisms. I confirm that he is doing this, by extracting from his text all the essential ingredients of an Abstract State Machine for the
algorithm. Here is my accompanying lecture at the YuriFest in Brno where the volume was presented to Yuri. Here is a short talk on the same subject for the symposium Arabic Foundations of Science
for the International Congress of History of Science and Technology, Manchester, July 2013. (16 July 2013.)
22. Qiyas ix.18. 'Debaters', commenting on Prior Analytics ii.18, is one of the best references for Ibn Sina's notion of the correct 'order' of the premises of an argument. (7 February 2016.)
23. Raw materials for a book with Amirouche Moktefi on what skills Ibn Sina thinks he is teaching by teaching logic, under the title 'Ibn Sina on Logical Analysis'. This includes preliminary
translations of Qiyas ii.4, ix.3--4, 6--9, and it will rework the paper above on proof search. (25 January 2013.)
24. Other commented translations of Ibn Sina's logic will appear here as they become available.
Other items on Arabic logic
This section is due for a major reorganisation, because I now think (since January 2014) that I understand broadly what is going on in Ibn Sina's logic. So we move from experiment to exposition, and
the material is shaping itself into some books, unless I die first. At least two draft books are here. Also at present there is some stuff here that I now think is either badly focused or plain
wrong, and as time allows I will clear it out.
1. A draft (April 2016) of most of a monograph 'Ibn Sina's alethic modal logic' (April 2015).
2. The current draft (June 2016) of a book on Mathematical Background to the Logic of Ibn Sina. Papers in which I refer to this book for proofs of facts about two-dimensional logic have started to
appear in print, so I made it a matter of urgency to get that part of the book up on this website. Also there is some substantial material on the basics of Ibn Sina's propositional logic. It will
be clear that the presentation needs a lot of debugging, but I think the required proofs are there. This book is primarily for a mathematical audience. It has to be, because the required
mathematics - at least as far as we know it now - is quite challenging. But there are historical remarks scattered through the book. A more user-friendly book is in preparation too but will come
later. 2 May 2016.
3. Lecture on Definitions in Ibn Sina's Jadal, from conference 'The Topics in the Arabic and Latin Traditions', CRASSH, Cambridge 2006.
4. Lecture on Ibn Sina's semantics, Seminar, Department of Philosophy, St Andrews June 07.
5. Lecture on Ibn Sina's syllogistic, Oxford, November 07. This lecture had the strictly limited aim of describing in modern terms a formal system adequate to support what Ibn Sina says about the
semantics of predicate syllogisms. But the truth turns out to be rather more complicated than I realised. Ibn Sina identifies a number of sentence forms that are important in scientific discourse
and are missing from the aristotelian tradition before him; they include ∀∃ sentences (see the section above on time and aspect, and the next lecture below) and sentences where a modal operator
has a quantifier within its scope (see the section above on modes). But he shows little interest in developing a proof theory to cover sentences of these forms. I have the strong impression that
he thinks the gap can be made up by 'analysis', reducing more complicated sentences to the standard categorical forms.
6. Lecture on Ibn Sina's cyclotron, for a conference on formalising ancient logics, chiefly Indian, Hamburg, June 2010. The lecture discusses some of the sentence forms mentioned in the comments
above on the previous lecture. (Note, 8 October 2013. Michael Carter has just revealed to me that Richard Lorch, a leading authority on medieval Arabic thought in the area that Ibn Sina's
'cyclotron' sentence comes from, is the same Richard Lorch that I knew at the Cathedral Choir School in Oxford for four years in the early 1950s. I'm hoping that Richard can throw some light on
where Ibn Sina might have taken this sentence from.)
7. Draft text of my talk on `Ibn Sina's view of the practice of logic' for World Philosophy Day, Tehran, November 2010. The talk describes Ibn Sina's procedures for checking the validity of natural
language arguments, and contrasts them with those of Aristotle, Fakhr al-Din Razi and modern undergraduate logic classes. (Revised 18 November 2010.) The slides of my talk to SIHSPAI, December
2010, cover similar ground.
8. A draft paper (with accompanying talk) which attempts to collect together Ibn Sina's assumptions about sentence structure from his logical writings. This sort of exercise seems to be essential if
we are to get a realistic picture of what he understood, what he didn't understand and what he believed in semantics. For example it emerges that he had no notion of scope. (18 November 2010)
9. A draft paper on Ibn Sina and the definition of logic. (28 April 2016)
10. For a workshop on later Arabic logic and philosophy of language, a talk on the role of modality in Ibn Sina's logic. There is an accompanying handout. (17 November 2011)
11. For a seminar in Munich, a talk on whether and in what sense Ibn Sina had first order logic. There is an accompanying handout. (18 March 2012)
12. For the CNRS workshop 'Ancient and Arabic Logic', 29 March 2012, a talk on Ibn Sina's assumptions about the grammatical structure of compound meanings, and some applications in his logic. (26
March 2012)
13. For the workshop 'Modal Logic in the Middle Ages' (St Andrews, November 2012), a talk and accompanying handout on Ibn Sina's adaptation of the notions of permanence and necessity which he took
from his Aristotelian predecessors. (24 September 2012)
14. For the workshop 'Medieval Philosophy Network, Warburg Institute London 7 June 2013', a talk on Syntax and meaning in al-Sirafi and Ibn Sina based on joint work with Manuela E. B. Giolfo. There
is a handout. (6 June 2013)
15. Notes for a paper The grammar of meanings in Ibn Sina and Frege. I have to apologise for this one. A journal invited me to submit a paper comparing Ibn Sina and Frege. Since Ibn Sina and Frege
are the two main figures in the Aristotelian tradition who developed a syntax of meanings, this was a stimulating challenge. The paper has been submitted, but it contains no references and few
quotations; instead I gave a reference to the present paper for a fuller account. The paper is not yet written, though it's a good deal further down the line than these notes might suggest. I
will get the whole thing put up here as soon as I can. (1 January 2013)
16. The slides of the 2013 Lindström Lectures at Göteborg. The first discusses Ibn Sina as a logician generally, and in particular his view of the 'subject-term' of logic; his view of this broadly
puts him in the mainstream along with Pascal, Bolzano and early Tarski, but he has radical ideas about the types of entity needed for handling compound meanings. (None of this is reflected in the
standard encyclopedia accounts.) The second discusses Ibn Sina's treatment of the making and discharging of assumptions; see above. (11 November 2013)
17. These 'Notes on a remark of Street' are a preliminary announcement on Ibn Sina's system of 'modal syllogistics'. The system describes the logical relationships between some sentences with two or
three independent quantifiers, that Ibn Sina discusses in a number of places. In effect it creates a new branch of formal logic, quite unlike anything else before modern times. But Ibn Sina never
gave a good proof theory for it, probably because he wanted to do that on the pattern of Aristotle's Prior Analytics, which are not a suitable basis for a logic this advanced. Also the system is
not really modal at all, though it's an open question whether Ibn Sina expected it to have modal applications. (6 January 2014)
18. Slides of a talk at a conference in Cambridge, on Ibn Sina's use of temporal propositions to find and correct an error in an argument of Aristotle about necessary conclusions of second figure
syllogisms. The syllogisms themselves are not so interesting; Ibn Sina's strategy for dealing with the error is a masterly application of the 'new branch of formal logic' described in the
previous item. The second half of this talk was expanded into a more thorough treatment for a talk in St Andrews, and revised for a talk to SIHSPAI in Paris in October 2014. (6 October 2014)
19. Slides of a talk to the Colloquium 'Avicenna and Avicennisms', SOAS, June 2014, developing themes in the previous talk. (5 June 2014)
20. If Schoenberg can write an accompaniment to an imaginary film scene, maybe I can write slides for an imaginary lecture. This is on Ibn Sina's propositional logic. It sets out where his
propositional logic comes from, what sentences it contains and one of the main kinds of proof theory that Ibn Sina has for handling these sentences. As usual there are new features, for example
some new syllogistic moods that are not merely analogues of classical ones. But there is no major breakaway from the Aristotelian framework. There also seem to be some plain mistakes of logic,
which is rare in Ibn Sina. (16 August 2014)
21. Slides for a joint talk with Manuela E. B. Giolfo on Speaker's knowledge in Al-Sirafi and Ibn Sina for the Third International Symposium on Foundations of Arabic Linguistics, Fondation
Singer-Polignac, Paris October 2014. (20 October 2014)
22. Slides for a talk on The logic/language divide in classical Arabic semantics, for a workshop on Meaning, Concept and Conception in the Arabic Tradition, University of Göteborg, 2015. (9 April
23. This is a talk which Mohammad Maarefi was invited to give at the Arabic Logic Symposium in the International Congress of Logic, Methodology and Philosophy of Science and Technology, Helsinki
2015, on 'Could Ibn Sina's logic be undecidable?'. Unfortunately he was refused a visa, and I gave the talk on his behalf. (31 July 2015)
24. Slides of a talk 'Physical metaphors for deduction', based on Ibn Sina's analysis of formal deduction, for the closing workshop of the project Roots of Deduction, Groningen, April 2016. A paper
based on this talk has been submitted for publication. (5 April 2016)
25. Slides of a talk 'Mulla Sadra's use of Ibn Sina's logic', for a talk to the Kosovo Philosophical Association conference 'Two Philosophers of Being: Mulla Sadra and Martin Heidegger', Prishtina
May 2016. (10 May 2016)
26. Slides of a talk 'Avicenna's theory of compound meanings: language or metaphysics?', for Workshop on Sensation, Conceptualization and Language in trhe Aristotelian Tradition, Copenhagen September
2016. The handout is here. While writing this talk I became aware, through Stephen Menn's chapter in Adamson ed. Interpreting Avicenna, of Yahya b. Adi's theory of the three modes of existence.
It seems to me that this clinches what Avicenna is on about in his discussion of the subject individuals of the art of logic in Madkhal; he is using a critique of b. Adi to clarify how these
individuals can be both mental (because they are reasoned about, and the rules of reasoning are purely about mental entities), and not necessarily mental (because the rules of logic have
universal application). This will be discussed in the next draft of the paper above on the definition of logic. But this is not the main topic of the Copenhagen talk. (26 August 16)
27. Preparatory notes for an encyclopedia article on Al-Farabi's logic. (4 August 2017)
28. Slides for a talk 'Avicenna sets up a modal logic with a Kripke semantics' for Logic Colloqium 2017 in Stockholm. (11 August 2017)
29. Slides for a talk 'How far did Avicenna get with propositional logic?' for the Special Session on History of Logic at Logic Colloquium 2017 in Stockholm. The inspection of the texts that lies
behind the facts reported in that lecture is rather gruelling, but fortunately it only has to be done once. The details can be found here with further work here. (14 February 2018)
30. Slides for a talk '12th century Arabic logic diagrams: practice and theory', introducing the logical procedures of Abu al-Barakat, which are based on model-theoretic consequence and diagrams that
anticipate those of Gergonne and Venn. A paper explaining some of the background is here. (29 January 2018)
31. Slides for a talk The creation of two paradigms for modal logic: Avicenna and Razi for Workshop on the History of Arabic Logic, St Andrews, May 2019. A handout is here. I leave up a first sketch
of this talk although it is now superseded, because it has been quoted. (5 May 2019)
32. Slides for a talk Abu al-Barakat and his twelfth century logic diagrams, Arabic Logic Commission, CLMPST Prague, August 2019.
33. With Manuela E. B. Giolfo, slides for a talk Arabic grammatical treatment of 'in conditional systems: Traces pf an external influence? Panel on Formal Models in the History of Arabic Linguistic
Tradition, Henry Sweet Society, Edinburgh September 2019.
34. Slides for a talk Barakat's logical diagrams, SIHSPAI, Naples, September 2019 (same topic as 32 above, but with more on how Barakat got there as opposed to the logical content).
35. Sprenger's translation of Risala al-Shamsiyya by al-Qazwini al-Katibi. Tony Street has a new translation in preparation.
36. Slides for a talk How the teenage Avicenna planned out several new logics, for the Nordic Online Logic Seminar 24 May 2021.
37. Text 'The logical diagrams of Abu al-Barakat' to accompany YouTube talk https://www.youtube.com/watch?v=iRsnK08-_5A
38. Slides for a talk 'Abu al-Barakat's diagram method in logic' in the symposium 'Mathematical proofs and styles of reasoning: East vs. West', ICHST Prague July 2021.
39. Slides for a talk 'Avicenna motivates two new logics', Logic and Metaphysics Workshop, New York, 14 March 2022.
40. Slides for a talk 'How did Avicenna understand the Barcan formulas?', Zoom Conference in honour of John N. Crossley's 85th birthday, 14-15 June 2022.
41. Slides for a talk 'Early strands in Ibn Sina's formal logic', meeting of Avicenna Study Group IV in Aix-la-Provence, 13-15 September 2023.
General history of logic
1. The history of British logic, British Logic Colloquium, Manchester, September 2001. It would be good to have the history of the origins of the BLC written up before too many memories die. I'm not
offering to write it, but if I get a chance I will put here some of my old BLC archives. Meanwhile some corrections on this talk: (Page 3) The distinction was certainly noticed before
Lukasiewicz, but it came slowly. Already Ibn Sina studies conversions between forms like A and forms like B. The rule of arrow-introduction is explicit in the Port-Royal Logic, though only for
one-step inferences. Frege discusses the notion of making and resolving assumptions. (Page 16) As Tim Williamson pointed out, Lewis Carroll should be mentioned here. (Page 17) The word
'quantifier' appears first in De Morgan, as an abbreviation for 'quantifying phrase' - which comes from Hamilton as De Morgan acknowledges.
2. Indirect proofs and proofs from assumptions. I wrote this in 2004 to clarify in my own mind some issues raised by Paolo Mancosu, without any plans to publish it as it is. I still think it is
correct in what it covers, but the absence of medieval logicians is unbalanced. In particular Ibn Sina should be discussed alongside the Port-Royal Logic and Frege. Mancosu has seen the paper but
not responded.
3. What happened to fallacies?, Indian Logic Circle, Kolkata October 2005.
4. Logic versus theory of language in the late 19th century, Augustus De Morgan Workshop, Kings London November 2000.
5. Tutorial on Tarski and decidable theories, Mumbai 10 January 2005. A fuller version is in the Proceedings.
6. Tarski on Padoa's method, International Conference on Logic, Navya-Nyaya and Applications, Homage to Bimal Krishna Matilal, Kolkata January 2007. A fuller version is in the Proceedings.
7. The history of model theory. This was written for a handbook on the history of mathematical logic, which folded some years ago. A revised version will appear as an appendix to a forthcoming book
on the philosophy of model theory by Tim Button and Sean Walsh.
8. Bill Marsh's unpublished 1966 DPhil thesis on uncountably but not countably categorical theories (included with his permission).
9. My paper 'Detecting the logical content: Burley's "Purity of Logic"' refers to a checklist of consequentiae in Burley's book. That checklist is here.
10. Where Frege is coming from, seminar, Amsterdam March 2009. The talk compares Ibn Sina, Leibniz and Frege on some key issues in logic and semantics, and aims to illuminate Frege by putting him
into this context.
11. Why modern logic took so long to arrive, three lectures for Cameleon, March 2009. I trace the development, over the last two thousand years, of (1) relational reasoning, (2) proof rules that
discharge assumptions, (3) type-theoretic semantics. There is a collection of texts to accompany the lectures, including eight pages of translation of Ibn Sina.
12. Traditional logic, modern logic and natural language, a paper that looks for the fundamental dividing lines between traditional logic and modern logic. Comparisons between traditional and modern
logic tend to be based on an assumption that traditional logicians were aiming to do what modern logicians do, but weren't so good at it. Wouldn't it be better history, and fairer to all
concerned, to try to establish what the traditionals themselves thought they were aiming to do? This was a draft (with personal references removed) of a paper which appeared in an issue of the
Journal of Philosophical Logic in honour of Johan van Benthem. Here is a more recent lecture in the same general area but with more emphasis on deathtraps in history of logic (for the Edinburgh
University Philosophy Society, November 2010).
13. How Boole broke through the top syntactic level. For the meeting History of Modern Algebra: 19th Century and Later, in memory of Maria Panteki, Thessaloniki October 2009. The talk notes that
Boole broke with traditional logic by allowing substitutions at any depth in a formula. I discuss why this breakthrough marks a fundamental difference between traditional and modern logic, and
how far Boole understood what he was doing in making the advance. Here is a draft of the writeup for the Proceedings.
14. A talk for the 60th birthday of Oleg Belegradek, Istanbul, December 2009. The talk contains a mixture of mathematical and personal reminiscences about how the Kemerovo model theorists (Belegradek
and Zilber) came into contact with the British model theorists across the iron curtain.
15. Including history in a mathematics module - practice and theory. For the meeting History of Science in Practice, Athens, May 2010. The talk discusses the pros and cons of including history in an
undergraduate mathematics module, with some examples from my own experience.
16. Basing logic on semantics - some historical themes, a discussion of the use of dependency grammars by various traditional linguists and logicians, with particular emphasis on Ibn Sina's use of
them for describing compound thoughts. For a meeting in Cambridge, March 2011.
17. Reconciling Greek mathematics and Greek logic - Galen's question and Ibn Sina's answer, for a workshop of Catarina Dutilh's project in Groningen. Some progress on a problem I mentioned in earlier
lectures: why did Aristotelian logicians regard Euclid as the summit of logical reasoning if they couldn't represent his reasoning within their own logic? Galen at least indicated the problem,
and Alexander of Aphrodisias applied some sticking plaster to the gap pointed out by Galen. But Ibn Sina seems to have the question much better under control. I give a formal calculus consisting
of items available to Ibn Sina, which is complete for first-order logic. This shows in principle that Ibn Sina was right to suppose there was nothing in Euclid that he couldn't justify within his
own logic. But for a historian of logic an equally important question is what Ibn Sina would have counted as validating Euclid's arguments. It's clear that he would not have used any formal
calculus for the purpose. Further historical and logical details are in the handout. (24 November 2011)
18. A lecture on the history of the notion of logical scope, including some remarks on Ibn Sina's attempt to do without it, for a seminar in Amsterdam. (26 November 2011)
19. A lecture 'Tarski through a wide-angle lens' on Tarski's place in the broad history of logic, with particular reference to semantics. (3 June 2012)
20. A talk 'The influence of Augustus De Morgan' for the joint BSHM-LMS De Morgan Day marking the 150th anniversary of the founding of the London Mathematical Society. (7 May 2015)
21. A talk 'The 1950 paradigm change' for the British Postgraduate Model Theory Conference, Manchester (18 January 2016).
22. A talk 'Buridan and the Avicenna-Johnston semantics' for the Medieval Philosophy Network, Warburg Institute (1 April 2016).
23. An unauthorised translation of Chapter V of Tae-Soo Lee on the Roman Empire syllogistic. The Alexandrian school (Ammonius, Philoponus) were hugely important sources for Al-Farabi, and hence also
for Ibn Sina. I find Lee illuminating but tough going, so I translated this for my own benefit. It's not a polished translation, but I did correct a couple of printos in formulas. (16 July 2017)
24. The first of three papers on the aftermath of Aristotle's method for proving that a syllogistic premise-pair has no syllogistic conclusion, under the title 'Nonproductivity proofs from Alexander
to Abu al-Barakat'. By the time the theme has reached Barakat (12th century Baghdad), it has become a not altogether successful attempt to replace Aristotle's notion of logical consequence by a
close relative of Tarski's 'model-theoretic consequence'. This paper examines the excuses in Aristotle's text for the later developments. (15 November 2017)
25. Slides for a talk 'In pursuit of a medieval model theory', for Logical Perspectives 2018, St Petersburg, 14-18 May 2018.
26. Slides for a contributed talk talk 'Logic from Greece to Central Asia to Western Europe and then Global: a single historical development?', for Logic4Peace, a Zoom conference in aid of Ukraine,
22 and 23 April 2022.
Semantics in natural language, mathematics, engineering
Mathematical logic
1. Short model theory course, Johannesburg December 1999.
2. Three tutorials in logic, Tbilisi 6-10 October 2003.
3. Draft of lecture on non-structure, Hattingen July 1999. Chris Laskowski and I gave lectures on non-structure; I reviewed the 'classical' material and Chris introduced more recent work. There was
a plan for a joint paper, but it never materialised.
4. Classification over a predicate, Istanbul Bilgi University March 2001. The published version was devoted to the special case of theories of linear orderings. But see also the paper of Rami
Grossberg in the same volume. Istanbul is an earthquake zone, and apparently there was an earthquake during my talk, but I never noticed.
5. Model theory of pairs of abelian groups, for St Petersburg July 2005. Here is an archived preprint of the published version, which includes some nontrivial corrections. Anatoly Yakovlev was in
the audience at the talk, and the paper includes his answer to the final Question on relatively categorical pairs of finite abelian groups.
6. Relatively categorical abelian groups III. This lecture applies the analysis of the preceding item to prove that Gaifman's conjecture holds for theories of relatively categorical abelian group
pairs: Any model B of the P-theory can be extended to a model A of the whole theory so that the P-part of A is B. The occasion for the lecture was the International Conference on Fundamental
Structures of Algebra, held at Constanța on the Black Sea in April 2010 in honour of Șerban Basarab. A draft of the writeup of the proof for the Conference Proceedings is here.
7. Fully abstract valuations for subgames, De Morgan Workshop on Interactive Logic, London November 2005. A fuller version is in the Proceedings.
8. Maze games, proof games and some others, for 'Games in logic, language and computation', Amsterdam September 2005. Here is the PowerPoint. The talk contains some personal opinions about the
motivation of game-theoretic modelling.
9. Four paradigms for logical games, for the meeting 'Modelling interaction, dialog, social choice, and vagueness', Amsterdam, March 2010. Benedikt Löwe asked me to review samenesses and differences
between Obligationes, Lorenzen dialogue games, EF back-and-forth games and Hintikka game-theoretic semantics. I did this with special reference to the modelling involved. The handout contains
bibliography and some mathematical underpinning.
10. Mathematics of imperfect information, Kings London February 2000.
11. Definability versus definability-up-to-isomorphism, in groups and fields (revised 29 May 2010). One of several reports I've given on the proof (with Saharon Shelah) that there is no formula of
set theory which provably in ZFC defines an algebraic closure for each field. This report was to the Antalya Algebra Day XII in May 2010. The proof is in two papers; one of them has appeared, and
the other is in preparation (the end is in sight).
Cognitive aspects of logic
1. Some notes I put up for my students at Queen Mary: Psychologists' tips on how and how not to learn. Though I take responsibility for the contents, I checked them with my wife who is a
professional psychologist.
2. Some things a logician would like to know about human reasoning, CogSci 2001, Edinburgh August 2001.
3. Efforts to make mathematics infallible, Workshop on semantic processing, logic and cognition, Tuebingen November 2005.
4. How reasoning depends on representations, two lectures to Centre for Cognitive Science, Jadavpur University, Kolkata, October 2005. The work of Keith Stenning that I discuss here has moved on:
see Keith Stenning and Michiel van Lambalgen, Human Reasoning and Cognitive Science, MIT Press, Cambridge Mass. 2008.
5. Logical rules at deep syntactic levels, a talk to the London Reasoning Workshop, Birkbeck College London, July 2010. The talk was an appeal for help in measuring the relative difficulty or
'naturalness' of some types of inference rule. The type that particularly interest me are those which apply monotonicity to make substitutions at arbitrary syntactic depth. My belief is that most
people find these rules impossible to follow intuitively when the depth is more than 3 or so, and have to resort to explicit calculation. John of Salisbury claimed just this in the 12th century,
but without solid evidence. Several logicians have suggested in recent years that these rules are particularly 'natural', which seems to me a paradoxical description.
Mathematics and music
1. Talk on Pythagoras, QMW Oct 98. A revised version of this talk was given as the Coulter McDowell Annual Lecture at Royal Holloway in 2001. Pythagoras must have been an extraordinary person, but
modern scholarship has stripped him of most of his supposed scientific advances. One that is still sometimes attributed to him is the correlation between subdivisions of a vibrating string and
points of the musical scale. I argued, relying on the shapes of kitharas and the known techniques for playing them, that this correlation was probably clear to professionals at least a generation
before Pythagoras. It would be consistent with what we do know about Pythagoras if he took this known correlation and converted it into some kind of music therapy. (Sad footnote: I tried
unsuccessfully to discuss this with the late and much missed David Fowler, who was an expert both on Greek mathematics and on the mathematics of the musical scale. Later I learned that at the
time he was already severely ill with the brain tumour that took him not long after.)
2. Some raw material on mathematical and musical beauty, for a meeting of mathematicians and artists organised by Juliette Kennedy in Utrecht, November 2007. For copyright and technical reasons,
most of the music itself is missing here. The talk draws on an archive of geometrical patterns in music. Material from the same archive went into my Third Annual Venn Lecture in Hull in 2003, and
into two published papers (first), (second), one joint with Robin Wilson. For some years I have hoped to turn the archive into a book, but it is not high priority.
3. Time, Music and Mathematics. Slides for a talk at the Greenwich Observatory, November 2009.
4. The geometry of music, one of a group of lectures on mathematics and music arranged by Robin Wilson at Gresham College to mark his retirement from the post of Professor of Geometry at Gresham. My
handout is here. The lecturers were Robin himself and me (November 2009) and Jonathan Cross (December 2009).
1. Copper and arsenic processing at Ramsley mine, a short talk for the Industrial Archaeology group of the Devonshire Association (February 2013).
The website of the South Tawton and District Local History Group is at http://www.southtawtonhistory.org.uk. The website of the IUHPS Division of Logic, Methodology and Philosophy of Science is at
Author : Wilfrid Hodges
Last updated 10 September 2023. | {"url":"http://wilfridhodges.co.uk/","timestamp":"2024-11-13T16:28:04Z","content_type":"text/html","content_length":"50080","record_id":"<urn:uuid:ca277638-eb72-4fcc-aab2-a884828834e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00869.warc.gz"} |
2 - SAT - VietMX's Blog
SAT (Boolean satisfiability problem) is the problem of assigning Boolean values to variables to satisfy a given Boolean formula. The Boolean formula will usually be given in CNF (conjunctive normal
form), which is a conjunction of multiple clauses, where each clause is a disjunction of literals (variables or negation of variables). 2-SAT (2-satisfiability) is a restriction of the SAT problem,
in 2-SAT every clause has exactly two literals. Here is an example of such a 2-SAT problem. Find an assignment of $a, b, c$ such that the following formula is true:
$$(a \lor \lnot b) \land (\lnot a \lor b) \land (\lnot a \lor \lnot b) \land (a \lor \lnot c)$$
SAT is NP-complete, there is no known efficient solution known for it. However 2SAT can be solved efficiently in $O(n + m)$ where $n$ is the number of variables and $m$ is the number of clauses.
1. Algorithm:
First we need to convert the problem to a different form, the so-called implicative normal form. Note that the expression $a \lor b$ is equivalent to $\lnot a \Rightarrow b \land \lnot b \Rightarrow
a$ (if one of the two variables is false, then the other one must be true).
We now construct a directed graph of these implications: for each variable $x$ there will be two vertices $v_x$ and $v_{\lnot x}$. The edges will correspond to the implications.
Let’s look at the example in 2-CNF form:
$$(a \lor \lnot b) \land (\lnot a \lor b) \land (\lnot a \lor \lnot b) \land (a \lor \lnot c)$$
The oriented graph will contain the following vertices and edges:
$$\begin{array}{cccc} \lnot a \Rightarrow \lnot b & a \Rightarrow b & a \Rightarrow \lnot b & \lnot a \Rightarrow \lnot c\\ b \Rightarrow a & \lnot b \Rightarrow \lnot a & b \Rightarrow \lnot a & c \
Rightarrow a\\ \end{array}$$
You can see the implication graph in the following image:
It is worth paying attention to the property of the implication graph: if there is an edge $a \Rightarrow b$, then there also is an edge $\lnot b \Rightarrow \lnot a$.
Also note, that if $x$ is reachable from $\lnot x$, and $\lnot x$ is reachable from $x$, then the problem has no solution. Whatever value we choose for the variable $x$, it will always end in a
contradiction – if $x$ will be assigned $\text{true}$ then the implication tell us that $\lnot x$ should also be $\text{true}$ and visa versa. It turns out, that this condition is not only necessary,
but also sufficient. We will prove this in a few paragraphs below. First recall, if a vertex is reachable from a second one, and the second one is reachable from the first one, then these two
vertices are in the same strongly connected component. Therefore we can formulate the criterion for the existence of a solution as follows:
In order for this 2-SAT problem to have a solution, it is necessary and sufficient that for any variable $x$ the vertices $x$ and $\lnot x$ are in different strongly connected components of the
strong connection of the implication graph.
This criterion can be verified in $O(n + m)$ time by finding all strongly connected components.
The following image shows all strongly connected components for the example. As we can check easily, neither of the four components contain a vertex $x$ and its negation $\lnot x$, therefore the
example has a solution. We will learn in the next paragraphs how to compute a valid assignment, but just for demonstration purposes the solution $a = \text{false}$, $b = \text{false}$, $c = \text
{false}$ is given.
Now we construct the algorithm for finding the solution of the 2-SAT problem on the assumption that the solution exists.
Note that, in spite of the fact that the solution exists, it can happen that $\lnot x$ is reachable from $x$ in the implication graph, or that (but not simultaneously) $x$ is reachable from $\lnot
x$. In that case the choice of either $\text{true}$ or $\text{false}$ for $x$ will lead to a contradiction, while the choice of the other one will not. Let’s learn how to choose a value, such that we
don’t generate a contradiction.
Let us sort the strongly connected components in topological order (i.e. $\text{comp}[v] \le \text{comp}[u]$ if there is a path from $v$ to $u$) and let $\text{comp}[v]$ denote the index of strongly
connected component to which the vertex $v$ belongs. Then, if $\text{comp}[x] < \text{comp}[\lnot x]$ we assign $x$ with $\text{false}$ and $\text{true}$ otherwise.
Let us prove that with this assignment of the variables we do not arrive at a contradiction. Suppose $x$ is assigned with $\text{true}$. The other case can be proven in a similar way.
First we prove that the vertex $x$ cannot reach the vertex $\lnot x$. Because we assigned $\text{true}$ it has to hold that the index of strongly connected component of $x$ is greater than the index
of the component of $\lnot x$. This means that $\lnot x$ is located on the left of the component containing $x$, and the later vertex cannot reach the first.
Secondly we prove that there doesn’t exist a variable $y$, such that the vertices $y$ and $\lnot y$ are both reachable from $x$ in the implication graph. This would cause a contradiction, because $x
= \text{true}$ implies that $y = \text{true}$ and $\lnot y = \text{true}$. Let us prove this by contradiction. Suppose that $y$ and $\lnot y$ are both reachable from $x$, then by the property of the
implication graph $\lnot x$ is reachable from both $y$ and $\lnot y$. By transitivity this results that $\lnot x$ is reachable by $x$, which contradicts the assumption.
So we have constructed an algorithm that finds the required values of variables under the assumption that for any variable $x$ the vertices $x$ and $\lnot x$ are in different strongly connected
components. Above showed the correctness of this algorithm. Consequently we simultaneously proved the above criterion for the existence of a solution.
2. Implementation:
Now we can implement the entire algorithm. First we construct the graph of implications and find all strongly connected components. This can be accomplished with Kosaraju’s algorithm in $O(n + m)$
time. In the second traversal of the graph Kosaraju’s algorithm visits the strongly connected components in topological order, therefore it is easy to compute $\text{comp}[v]$ for each vertex $v$.
Afterwards we can choose the assignment of $x$ by comparing $\text{comp}[x]$ and $\text{comp}[\lnot x]$. If $\text{comp}[x] = \text{comp}[\lnot x]$ we return $\text{false}$ to indicate that there
doesn’t exist a valid assignment that satisfies the 2-SAT problem.
Below is the implementation of the solution of the 2-SAT problem for the already constructed graph of implication $g$ and the transpose graph $g^{\intercal}$ (in which the direction of each edge is
reversed). In the graph the vertices with indices $2k$ and $2k+1$ are the two vertices corresponding to variable $k$ with $2k+1$ corresponding to the negated variable.
int n;
vector<vector<int>> g, gt;
vector<bool> used;
vector<int> order, comp;
vector<bool> assignment;
void dfs1(int v) {
used[v] = true;
for (int u : g[v]) {
if (!used[u])
void dfs2(int v, int cl) {
comp[v] = cl;
for (int u : gt[v]) {
if (comp[u] == -1)
dfs2(u, cl);
bool solve_2SAT() {
used.assign(n, false);
for (int i = 0; i < n; ++i) {
if (!used[i])
comp.assign(n, -1);
for (int i = 0, j = 0; i < n; ++i) {
int v = order[n - i - 1];
if (comp[v] == -1)
dfs2(v, j++);
assignment.assign(n / 2, false);
for (int i = 0; i < n; i += 2) {
if (comp[i] == comp[i + 1])
return false;
assignment[i / 2] = comp[i] > comp[i + 1];
return true;
3. Practice Problems | {"url":"https://www.maixuanviet.com/2-sat.vietmx","timestamp":"2024-11-10T19:11:48Z","content_type":"text/html","content_length":"107487","record_id":"<urn:uuid:f582b626-ec32-4acb-bfd9-10db32eb91fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00226.warc.gz"} |
The Greatest Common Factor
Learning Outcomes
• Identify the greatest common factor of a polynomial
Factors are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: [latex]2[/latex] and [latex]10[/latex] are factors of [latex]20[/
latex], as are [latex]4, 5, 1, 20[/latex]. To factor a number is to rewrite it as a product. [latex]20=4\cdot{5}[/latex] or [latex]20=1\cdot{20}[/latex]. In algebra, we use the word factor as both a
noun – something being multiplied – and as a verb – the action of rewriting a sum or difference as a product. Factoring is very helpful in simplifying expressions and solving equations
involving polynomials.
The greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, [latex]4[/latex] is the GCF of [latex]16[/latex] and [latex]20[/latex]
because it is the largest number that divides evenly into both [latex]16[/latex] and [latex]20[/latex] The GCF of polynomials works the same way: [latex]4x[/latex] is the GCF of [latex]16x[/latex]
and [latex]20{x}^{2}[/latex] because it is the largest polynomial that divides evenly into both [latex]16x[/latex] and [latex]20{x}^{2}[/latex].
When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.
Greatest Common Factor
The greatest common factor (GCF) of a group of given polynomials is the largest polynomial that divides evenly into the polynomials.
Find the greatest common factor of [latex]25b^{3}[/latex] and [latex]10b^{2}[/latex].
Show Solution
In the example above, the monomials have the factors [latex]5[/latex], b, and b in common, which means their greatest common factor is [latex]5\cdot{b}\cdot{b}[/latex], or simply [latex]5b^{2}[/
The video that follows gives an example of finding the greatest common factor of two monomials with only one variable.
Sometimes you may encounter a polynomial with more than one variable, so it is important to check whether both variables are part of the GCF. In the next example, we find the GCF of two terms which
both contain two variables.
Find the greatest common factor of [latex]81c^{3}d[/latex] and [latex]45c^{2}d^{2}[/latex].
Show Solution
The video that follows shows another example of finding the greatest common factor of two monomials with more than one variable.
Now that you have practiced identifying the GCF of terms with one and two variables, we can apply this idea to factoring the GCF out of a polynomial. Notice that the instructions are now “Factor”
instead of “Find the greatest common factor.”
To factor a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property to rewrite the polynomial in factored form. Recall that the distributive
property of multiplication over addition states that a product of a number and a sum is the same as the sum of the products.
Distributive Property Forward and Backward
Forward: Product of a number and a sum: [latex]a\left(b+c\right)=a\cdot{b}+a\cdot{c}[/latex]. You can say that “[latex]a[/latex] is being distributed over [latex]b+c[/latex].”
Backward: Sum of the products: [latex]a\cdot{b}+a\cdot{c}=a\left(b+c\right)[/latex]. Here you can say that “[latex]a[/latex] is being factored out.”
We first learned that we could distribute a factor over a sum or difference, now we are learning that we can “undo” the distributive property with factoring.
Factor [latex]25b^{3}+10b^{2}[/latex].
Show Solution
The factored form of the polynomial [latex]25b^{3}+10b^{2}[/latex] is [latex]5b^{2}\left(5b+2\right)[/latex]. You can check this by doing the multiplication. [latex]5b^{2}\left(5b+2\right)=25b^{3}
Note that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over.
For example:
[latex]\begin{array}{l}25b^{3}+10b^{2}=5\left(5b^{3}+2b^{2}\right)\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }5\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=5b^{2}\left(5b+2\right) \,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }b^{2}\end{array}[/latex]
Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.
In the following video, we show two more examples of how to find and factor the GCF from binomials.
We will show one last example of finding the GCF of a polynomial with several terms and two variables. No matter how large the polynomial, you can use the same technique described below to factor out
its GCF.
How To: Given a polynomial expression, factor out the greatest common factor
1. Identify the GCF of the coefficients.
2. Identify the GCF of the variables.
3. Write together to find the GCF of the expression.
4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.
5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.
Factor [latex]6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy[/latex].
Show Solution
In the following video, you will see two more example of how to find and factor our the greatest common factor of a polynomial. | {"url":"https://courses.lumenlearning.com/intermediatealgebra/chapter/read-the-greatest-common-factor/","timestamp":"2024-11-09T13:51:17Z","content_type":"text/html","content_length":"56660","record_id":"<urn:uuid:e49c8d22-7669-4820-afe5-9dc7863c3b38>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00688.warc.gz"} |
ECE 515 - Control System Theory & Design
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This course is titled “Control System Theory & Design”. Let us start by first asking what we mean by a system; after all, it is a word used in many different contexts and with different meanings. For
our purposes, a system is a collection of interrelated processes and/or signals that, together, achieve some objective. For example, the circulatory system’s objective is to ensure oxygenated blood
from the lungs reaches all parts of the human body and, remove CO2 from different parts of the body and transfer it to the lungs. Similarly, the HVAC system’s objective is to maintain a comfortable
ambiance for the occupants of a building by controlling factors like humidity and temperature.
A control system is one in which a certain subcollection of signals and processes direct, control, command and/or regulate other processes. Recall that in classical control, we had a plant model
where a system with output \(y\) took input \(u\), which a controller determined - often by comparison against a reference signal \(r\). An example we are familiar with is cruise control in an
automobile, where a sensor measures the wheel velocity and, by comparison to the set or desired velocity, determines the appropriate amount of fuel or throttle to be provided (the control signal).
Unity feedback configuration
In this course, the object of study is control systems, which can be considered a generalization of classical dynamical systems (a control system with no inputs typically reduces to a dynamical
system). What is different from classical controls course here is the shift in focus from the frequency domain to the time domain. In prior courses, we avoided dealing with differential equations by
taking the Laplace transform and moving to the complex frequency domain. In this course, we will focus more on the ODEs and study them in detail, especially in the context of linear systems.
To illustrate what we mean, consider the following systems:
• An object in free fall
• A pendulum
• An RC circuit
and their so called equations of motion.
An object in free fall
Suppose the object has mass \(m\) and experiences acceleration \(a\). For this simplified model we will only consider the effect of gravity and aerodynamic drag (assume proportional to velocity). By
Newton’s second law, we must have the net sum of all forces acting on a free-falling object equal its mass times acceleration. Thus,
\[ mg - kv = ma \tag{1}\]
where \(kv\) is a term that retards the downward motion of the object (i.e. drag force).
A pendulum
Consider the case of a pendulum of length \(l\) and mass \(m\) (connected to its base by a massless string) subtending an angle \(\theta\). In this case, when we apply Newton’s second law seperately
to the tangential and radial axes we see that radial components cancel. On the other hand, the equation describing the tangential component gives us:
\[ F = ma \quad \implies \quad -mg \sin \theta = ma \]
However, the angle \(\theta\) is related to the arclength subtended as \(s = l \theta\) where \(s\) is the arclength. Then \(\ddot a = l \ddot \theta\) and
\[ \ddot{\theta} + \dfrac{g}{l} \sin\theta = 0 \tag{2}\]
An RC circuit
For an RC circuit with current \(I\) under an external voltage supply \(V\) we can write Kirchoff’s Voltage Law as:
\[ V = IR + V_c \]
where \(V_c\) is the voltage across the capacitor. However, we know that \(q = CV_c\) is the governing equation for the capacitor of capacitance \(C\) where \(q\) is the charge in the capacitor.
Differentiating we get \(\dot{q} = C \dfrac{dV}{dt}\) except \(\dot q = I\) the current. Thus,
\[ V = RC\dot{V_c} + V_c \tag{3}\]
At this point we have looked at an electrical system, an aerodynamical example and a mechanical system.
All of these are differential equations of the form \(\dot x = f(x)\) where \(x \in \mathbb{R}^n, n\ge 1\) and \(f: x \mapsto \mathbb{R}^n\). In other words \(f\) is some function that consumes a
vector \(x\) and returns another vector \(f(x)\).
The majority of this course is concerned with systems of the form(s):
\[ \dot x = f(x) \qquad \textrm{and} \qquad \dot x = f(x, u) \]
sometimes written with an explicit time dependence as
\[ \dot x = f(t, x(t)), \qquad \textrm{and} \qquad \dot x = f(t, x(t), u(t)) \]
The latter systems are called non-autonomous systems while the former are called autonomous systems. In either case, (autonomous or non-autonomous), the second variety (involving \(u\) or \(u(t)\))
indicates the presence of an input term \(u, u(t) \in \mathbb{R}^p\) and so signifies a control system (we usually consider the input term to be some form of control law).
Note that in \(\dot x = f(x)\) and its siblings, the left-hand-side (LHS) consists of a single (if vector valued) term that is a first derivative whereas the right-hand-side (RHS) has no derivatives
at all. This form is usually called the standard state-space formulation. Each entry of \(x\) is a state variable and \(x\) collectively captures the state of the system - information that fully
determines all aspects of a system at a particular instant of time and along with any initial conditions, sufficient to predict all future states.
Claim: Each of Equation 1, Equation 2, Equation 3 can be written in the form: \(\dot x = f(x, u)\).
For Equation 3 we have by simple re-arrangement \[ \dot V_c = \dfrac{1}{RC}V_c + \dfrac{1}{RC} V \]
where the second term involving \(V\) can be considered the input into the system. The other two require a tad bit more algebra. Let us come back to it after we discuss some course logistics.
Break to discuss: https://courses.grainger.illinois.edu/ece515/sp2024
For Equation 2, we had:
\[ mg - kv = ma \quad \implies \quad a = \dfrac{-k}{m}v + g \]
However, \(a = \dot v\) and \(v = \dot y\) where \(y\) is the downward displacement. Define auxilliary variables \(z_1 =y\) and \(z_2 = \dot y\). Then \(\dot z_2 = \ddot y\) and \(\dot z_1 = z_2\).
Thus we get,
\[ \dot z_1 &= z_2 \\ \dot z_2 &= \dfrac{-k}{m} z_2 + g \]
Now this is a two-dimensional system because \(z = \begin{bmatrix} z_1 &z_2 \end{bmatrix}^T \in \mathbb{R}^2\).
For Equation 3, similarly, we had:
\[ \ddot \theta = -\dfrac{g}{l} \sin \theta \]
Here we define, \(x_1 = \theta\) and \(x_2 = \dot x_1\). Then, we get, \[ \dot x_1 &= x_2 \\ \dot x_2 &= -\dfrac{g}{l} \sin x_1 \]
Linear vs. nonlinear system
While we have written each of Equation 1, Equation 2 and Equation 3 in the standard state space (SS) form, there is something special about Equation 3. In fact, Equation 1 and Equation 2 are linear
equations whereas Equation 3 is a nonlinear equation.
For the purposes of this class linear systems are ones in which we can write \(\dot x = f(x, u)\) or \(\dot x = f(t, x(t), u(t))\) as \[ \dot x = Ax + Bu \quad \textrm{or} \quad \dot x = A(t)x + B(t)
u \] where the matrices \(A\) and \(B\) are devoid of any state variables.
Example: We can write
\[ \dot z_1 &= z_2 \\ \dot z_2 &= \dfrac{-k}{m} z_2 + g \]
\[ \dot z = \begin{bmatrix} 0 &1 \\ 0 & -k/m \end{bmatrix}z + \begin{bmatrix} 0 \\1 \end{bmatrix} g \qquad \textrm{where} \quad z = [z_1 \; z_2]^T \]
Question: standard drag force
In the example of the free falling body we considered, the drag experienced by the falling object was directly proportional to its velocity. In reality, the drag force increases drastically with
changes in velocity and we often model it as quadratic in velocity. Is such a system linear or nonlinear?
The linear state space formulation
Generally, in a physical system that we week to influence we do not have access to all state variable. In other words, though a phenomenon may involve \(n\) variables, which together completely
determine the state vector, one may only be able to observe some \(q<n\) variables which may either be the state variables themselves or a linear combination thereof. We represent this situation by
\[ \dot x &= Ax + Bu \\ y &= Cx + Du \tag{4}\]
\[ \dot x &= A(t)x + B(t)u \\ y &= C(t)x + D(t) u \]
In the above \(A\) is a \(n\times n\) matrix, \(B\) of size \(n \times p\), and \(C\) of dimension \(q \times n\). They are typically called the system, input and output matrices. The matrix \(D\)
captures any direct influence the input \(u\) has on the output \(y\).
Note about the feed-through matrix
We will see how the \(D\) matrix arises when we discuss the relation between the state-space formulation and transfer functions.
Though we remarked that the we will focus on the time domain description in this course, and we will discuss realization theory in greater detail at a later time, it is now instructive to examine
what happens if we simply take the Laplace transform of Equation 4. For simplicity assume \(p=q=1\). We get
\[ s X(s) = AX(s) + BU(s) \qquad \textrm{and} \qquad Y(s) = CX(s) + d \cdot U(s) \] Isolating \(X(s)\) from the first, plugging it into the second and rearranging we get \[ G(s):= \dfrac{Y(s)}{U(s)}
= C \left(sI - A \right)^{-1}B + d \tag{5}\]
which provides a means to convert a given state-space model \(\left(A, B, C, D\right)\) into its transfer function model.
In the above we made the assumption \(p=q=1\). This results in a single-input and single-output (SISO) model and the \(D\) matrix necesarily reduces to a scalar entity \(d\). Equation 5 is only valid
in this case. For the general MIMO (multi-input multi-output) case, what we end up having is a transfer function matrix. In other words, a transfer function between each pair of inputs and outputs.
Question: Given a transfer function how do we find the corresponding state-space model? Is this model unique?
It turns out that there are many ways to generate a state-space model from the transfer function and they result in slightly different (but equivalent) formulations in terms of the matrices \((A, B,
C, D)\). To truly appreciate why this happens, we need to review some linear algebra material. Such material will be the focus of the next few lectures.
However, before that we review some systems concepts.
From prior courses, we are familiar with the input-output description of a linear system:
\[ y(t) = \int \limits _{t_0} ^{t} G(t, \tau) u(\tau) d \tau \]
which reduces to a convolution for time invariant systems. Let us make precise what we mean by linear and time invariant.
Definition 1 (Linearity)
A function \(f\) is linear \(f(\alpha x + y ) = \alpha f(x) + f(y)\). In terms of systems, a system is said to be linear if for any two state-input pairs
\[(x_1(t_0), u_1(t)) \quad \textrm{and} \quad (x_2(t_0), u_2(t))\]
mapping to outputs \(y_1(t)\) and \(y_2(t)\), i.e. \[ (x_1(t_0), u_1(t)) \mapsto y_1(t)) \qquad \textrm{and} \qquad (x_2(t_0), u_2(t)) \mapsto y_2(t) \] we have that, \[ (x_1(t_0) + \alpha x_2(t_0),
u_1(t) + \alpha u_2(t)) \quad \mapsto \quad y_1(t) + \alpha y_2(t) \]
In other words, if the states and inputs are scaled and added then the resultant output is also a scaled and added version of individual outputs.
Addition here is interpreted pointwise in time.
If the input to a system is zero then the system response starting from some initial condition \(x_0 = x(t_0)\) is caleld the zero-input response. If the system starts from a zero initial condition
and evolves under some input \(u(t)\), the system response is called the zero-state response.
An important property of linear systems is that the response of any system can be decomposed into the sum of a zero-state reponse and a zero-input response.
\[ \textrm{response} = \textrm{zero-state response} + \textrm{zero-input response} \]
Verify the above statement directly follows from the linearity property.
Additionally, a system is said to be lumped if the state vector necessary to describe it is finite dimensional. A system is said to be causal if the current output only depends on the past inputs to
the system and not on any future input. We will exclusively deal with lumped causal systems in this course.
If \(t \in \mathbb{R}\) we deal with continuous time systems and if \(t \in \mathbb{Z}\) then we say we deal with discrete time systems. The results we derive will primarily be for continous time
systems and occassinally for discrete time systems. LSTC does a parallel treatment of both time-invariant and time-varying systems so you are encouraged to read it as supplementary material. When we
do deal with discrete time systems, we will assume that the sampling period is uniform.
Time invariance
A special class of systems where our study can be made much deeper is the class of linear time invariant systems.
Definition 2 (Time invariance)
A system is said to be time invariant if for every initial state & input-output pair, \[ (x(t_0), u(t)) \mapsto y(t) \] and any \(T\), we have,
\[ (x(t_0 + T), u(t - T)) \mapsto y(t - T) \]
In other words, starting from the same initial condition, a time shift of the input function by an amount \(T\) results in a time shift of the output function by the same amount. Thus for linear time
invariant systems (LTI), we can assume with no loss of generality that \(t_0=0\).
Why is one argument \(t_0 + T\) and the other \(t-T\)?
Be sure to understand this point about notation. The expression: \[ (x(t_0), u(t)) \mapsto y(t) \] should be interpreted as initial condition \(x\) at time \(t_0\) (written as \(x(t_0)\)) under the
action of \(u(t)\) (for \(t>t_0\)) results in output \(y(t)\). Thus,
\[ (x(t_0 + T), u(t - T)) \mapsto y(t - T) \]
should be interpreted as initial condition \(x\) at time \(t_0 + T\) under the action of the time shifted input \(u(t-T)\) (for \(t>t_0 + T\)) gives rise to output \(y(t-T)\).
Transfer functions
Above we saw the relationship between the state-space (SS) formulation and the transfer function via the Laplace transform. Before we proceed to a review of some concepts from linear algebra, it is a
good idea to review some frequency domain concepts. Let us use \(\deg(P)\) to denote the degree of some polynomial \(P(s)\).
• For lumped systems, transfer functions are rational functions, i.e. can be written as a polynomial divided by another polynomial: \[G(s) = N(s)/D(s)\]
• If \(\deg(N) <= \deg(D)\) then a transfer function is said to be proper.
• If \(\deg(N) < \deg(D)\) then a transfer function is said to be strictly proper.
• If \(\deg(N) = \deg(D)\) then a transfer function is said to be bi proper.
Finally, a complex variable \(\lambda\) is said to be a pole of a rational transfer function \(G(s)\) if \(|G(\lambda)|= \infty\) and a zero if \(|G(\lambda)| = 0\).
If \(N(s)\) and \(D(s)\) are co-prime, i.e. do not share any common factors of degree 1 or higher, then all roots of \(N(s)\) are zeros and all roots of \(D(s)\) are poles.
Course overview
At the heart of all control systems are dynamical systems and dynamical systems essentially boil down to differential or difference equations. Our prior treatment of control systems was algebraic in
some sense because we sidestepped dealing with ODEs by applying the Laplace transform and going to the complex frequency domain. We made significant progress in studying such systems by developing
tools like the Root Locus method, Bode plots, Nyquist criterion, etc. Nevertheless, these are called classical control techniques (as opposed to modern control theory) because, in a certain sense
(the precise sense of which will hopefully become more apparent by the end of the course), the state-space formulations offer a richer characterization. In this sense, one may call the state-space
formulation a more geometric approach, and to proceed, we will make ample use of linear algebraic techniques. The rest of the course will proceed as follows:
• Linear algebra review (part 1)
• Linear systems and their solutions
• Linear algebra review (part 2)
• Stability of dynamical and control systems
• Controllability - what can we control about a system?
• Observability - what can we know about a system from observations?
• Realization theory (bridge transfer functions & SS formulation)
• Feedback control - SS perspective
• Pole placement and observer design
• Observer-based feedback
• Optimal control concepts | {"url":"https://courses.grainger.illinois.edu/ece515/sp2024/exlecs/lec01.html","timestamp":"2024-11-12T06:37:35Z","content_type":"application/xhtml+xml","content_length":"122291","record_id":"<urn:uuid:ef602d1c-fd9d-42c4-8985-412cb80dc243>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00143.warc.gz"} |
Can a ball stay on a track with a given height and friction coefficient?
• Thread starter yoni162
• Start date
In summary, the question is asking for the minimum height, h, at which a ball can be released without falling off a track with a friction coefficient of 'u' and an angle of 45 degrees. The work done
by the friction force is calculated and equated to the change in mechanical energy. The solution is derived and it is found that the condition v>0 at the top is not sufficient for the ball to stay on
the track as the reaction force must also be greater than or equal to 0. The final answer is determined to be h>(2R)/(1-u).
Homework Statement
A ball is released from height h. The friction coefficient between the straight part and the ball is 'u'. I need to find the smallest h so that the ball doesn't fall off the track.
The angle alpha=45 degrees.
Homework Equations
Work of non-conservative forces = Change in mechanical energy
The Attempt at a Solution
I calculated the work done by the friction force while the ball is going down the straight part.
X=the length of the straight part.
We get that:
since cos(alpha)/sin(alpha)=1
Now I want to say that Wf=change in mechanical energy, so:
when v=the velocity of the ball when it reaches the end of the straight part. We get:
Now to the second part, the frictionless rail. Since all the forces are conservative now:
When V1^2=g*h(1-u) -----> (the velocity we found before)
V2=the velocity at the top of the loop
So I want that V2>0, so after some work we get:
but when I put in a numerical answer I'm told that I'm wrong. Is there a mistake in my solution?
The condition v>0 at the top is not sufficient for the ball to stay on the track.
You need to have the reaction force acting on the ball at the top to be >=0.
nasu said:
The condition v>0 at the top is not sufficient for the ball to stay on the track.
You need to have the reaction force acting on the ball at the top to be >=0.
yeah you're right I forgot, thanks..
FAQ: Can a ball stay on a track with a given height and friction coefficient?
1. What is work and energy?
Work and energy are closely related concepts in physics. Work is defined as the product of force and displacement, while energy is the ability to do work. In simpler terms, work is the transfer of
energy from one object to another, and energy is the ability to cause change.
2. What is friction?
Friction is a force that resists the motion between two surfaces in contact. It is caused by the microscopic irregularities on the surfaces, which create resistance and slow down the motion. Friction
can be beneficial, such as in the case of walking or driving, but it can also be a hindrance and cause energy loss.
3. How does friction affect work and energy?
Friction is a form of resistance that opposes motion, so it can affect the amount of work and energy involved in a process. For example, when a force is applied to an object to move it, friction will
cause resistance and reduce the amount of work done. As a result, less energy will be transferred to the object, and some of it will be lost as heat due to friction.
4. How can friction be reduced?
Friction can be reduced by using lubricants, such as oil or grease, between the surfaces in contact. This creates a layer that reduces the direct contact between the surfaces, thus reducing friction.
Another way to reduce friction is by using smooth and polished surfaces, which have fewer microscopic irregularities and therefore create less resistance.
5. Can friction be completely eliminated?
No, it is not possible to completely eliminate friction. Even with the use of lubricants and smooth surfaces, there will still be some amount of friction present. However, it is possible to minimize
friction to a great extent, which can be beneficial in certain applications, such as reducing wear and tear on machinery or increasing the efficiency of engines. | {"url":"https://www.physicsforums.com/threads/can-a-ball-stay-on-a-track-with-a-given-height-and-friction-coefficient.360196/","timestamp":"2024-11-02T02:56:29Z","content_type":"text/html","content_length":"84351","record_id":"<urn:uuid:e5a108c0-993a-491f-853c-579773e067d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00512.warc.gz"} |
analysis of reactor physics processes
The aim of the deterministic method and code development within our Institute is the simulation of steady-state and transient three-dimensional reactor physics phenomena by the application of a broad
range of transport approximations and numerical techniques, as well as the realization of coupling between the developed reactor physics codes and thermal-hydraulic system codes. Our researchers have
gained considerable experience within this field since the establishment of our Institute.
Given the widely acknowledged research work of Professor Dr. Zoltán Szatmáry, a reactor dynamincs code DIMITRI (Diffusion en Milieux Tridimensionels) applying few-group diffusion theory and finite
difference spatial and time-discretization methods was developed within the NTI that is able to solve the static eigenvalue and adjoint equations and the time-dependent diffusion equation both with
considering external neutron source as well. The DIMITRI code also simulates the effect of technological uncertainties applying stochastic perturbation theory.
Since 2014, we also carry out research in finite-element-based reactor physics methods and perform related code development. One of the results is the finite-element-based diffusion code DIREMO (
Neutrondiffúziós Reaktormodellező Oktatóprogram - educational neutron diffusion code) developed within the Institute that - similarly to the DIMITRI code - is able to solve the few-group steady-state
and time-dependent diffusion equations applying continuous Galerkin spatial discretization and the theta-method for time-discretization. The DIREMO code is coupled with the GMSH code that is applied
for mesh generation and post-processing. The DIREMO code is modular, its preparation for thermal and mechanical simulations has also been started. Currently, we are working on the coupling between
the DIREMO and APROS codes that would make the simulation of coupled reactor physics/thermal-hydraulics processes possible. This research field continuously attracts students, several related theses
were and are being written.
The SPNDYN code developed in our Institute also applies finite element method, however, it is based on the so-called SP[N] theory. The code is currently able to solve the steady-state and
time-dependent SP[3] equations assuming third-order and full scattering anisotropy. Besides the application of the continuous Galerkin method, we also developed a hybrid finite element algorithm
which was implemented as a module (CPL-SP3) into the reactor physics code system C-PORCA of the Paks Nuclear Power Plant. Its testing is currently in progress.
Main publications:
B. Babcsány, I. Pós, D.P. Kis, 2021. Hybrid finite-element-based numerical solution of the multi-group SP[3] equations and its application on hexagonal reactor problems. Annals of Nuclear Energy, 155
B. Babcsány, T. Bartók, D. P. Kis, 2020. Finite element solution of the time-dependent SP[3] equations using an implicit integration scheme. Kerntechnik, 85 (2020) 4; 292-300.
B. Babcsány, T. Hajas, P. Mészáros, 2020. Finite-element-based diffusion modeling of transient reactor physics processes (in Hungarian). Nukleon, XIII. (2020) 233 | {"url":"http://www.reak.bme.hu/en/research/nuclear-and-reactor-physics-monte-carlo-methods/method-and-code-development-aiming-the-deterministic-analysis-of-reactor-physics-processes.html","timestamp":"2024-11-03T03:48:50Z","content_type":"text/html","content_length":"33741","record_id":"<urn:uuid:d411c88d-c133-4bc0-91d4-f3378c02effe>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00013.warc.gz"} |
Grid Representation#
Structured Grids: Structured grids are defined by regular connectivity and topology, typically forming a uniform pattern. They are represented using a multidimensional array, where each grid point
has a predictable neighbor. Below is an example of what a structured grid may look like.
Unstructured Grids: Unstructured grids do not follow a regular pattern, allowing for a flexible representation of complex geometries. They are composed of various elements such as triangles,
quadrilaterals, and other larger geometries, each of which are made up of nodes and edges. Below is an example of an unstructured grid used in the dynamical core of a CAM-SE model.
Node: A point within a spherical grid, representing the vertices of the elements (such as the corners of triangles or quadrilaterals)
Edge: A segment that connects two nodes within a grid.
Face: An individual polygon that is defined by nodes connected by edges.
Connectivity: Connectivity describes how nodes, edges, and faces are interconnected within a grid. It outlines the relationship between individual elements of the mesh, determining how they join
Fill Value: An arbitrary value used for representing undefined values in within connectivity variables when working with fixed-size arrays. | {"url":"https://uxarray.readthedocs.io/en/latest/user-guide/terminology.html","timestamp":"2024-11-14T03:47:09Z","content_type":"text/html","content_length":"20920","record_id":"<urn:uuid:99eb28ff-67e7-45be-8563-cfe180e8196a>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00756.warc.gz"} |
Project MUSE
• Breiman's Two Cultures:A Perspective from Econometrics
Breiman's "Two Cultures" paper painted a picture of two disciplines, data modeling, and algorithmic machine learning, both engaged in the analyses of data but talking past each other. Although that
may have been true at the time, there is now much interaction between the two. For example, in economics, machine learning algorithms have become valuable and widely appreciated tools for aiding in
the analyses of economic data, informed by causal/structural economic models.
Econometrics, Causal Inference, Instrumental Variables
1. Introduction
When we first read Breiman's "Two Cultures" paper (Breiman, 2001) in the early 2000's it baffled us. In econometrics, as in much of statistics (98% according to Breiman), the modeling culture was
dominant, and a purely prediction-based focus seemed very alien to what econometricians were doing. You can see this clearly in the history of econometrics (see for a description, e.g, Hendry and
Morgan (1997)). From the founding days of the field many researchers were more focused on identification and estimation of causal effects (e.g., the parameters of structural models) than on
prediction. This causality-based focus led econometricians to de-emphasize R^2 values as measures of success, and instead aim to build a credible case for estimation of causal effects. To illustrate
the difference between the predictive and causal approaches, we discuss two examples. Both examples are part of what Josh Angrist and Steve Pischke later called the credibility revolution (Angrist
and Pischke (2010)) that since the late 1980s has been a major influence in empirical work in economics. Then we discuss how more recently researchers in econometrics have started appreciating the
benefits of the algorithmic machine learning approaches. A rapdily growing literature attempts to combine the benefits of the prediction-focused machine learning algorithm approaches with the
traditional focus on causal model-based approaches.
The first illustration focuses on the problem of estimating supply and demand functions, an important example of a simultaneous equations problem. The study of such problems goes back to the founding
of econometrics as a separate discipline (see the example of the demand for potato flour in Tinbergen (1930), with a translation in Hendry and Morgan [End Page 127] (1997) and further discussion in
Imbens). We illustrate this with an application taken from Angrist et al. (2000). They study the demand for fish as a function of price, using daily data from the Fulton fish market in New York. The
demand function is a causal object, defined as the quantity as a function of the price that buyers would be willing to buy if the price was set exogenously. We denote the average demand function,
averaged over buyers, by the potential outcome function t indexes the markets, days in this illustration. It is of fundamental interest in economics. For example, it is of interest to policy makers
who may be interested in the effect of different market structures ( e.g., imposing a tax), or to sellers, who may be interested in the effect of charging higher prices. It is quite different from
the predictive relation between quantities and prices. The latter may be of interest for different purposes. In the Angrist et al study the researchers have observations for a number of days at the
Fulton fish market. The two main variables observed by the researcher are the quantity of fish traded, Q[t], and the average price at which it was traded on that particular day, P[t]. Figure 1 shows
the data, with each dot denoting the log quantity and log price combination for a particular day at the Fulton fish market. What should we do with such data? To estimate the demand function, one
might be tempted to try tofit a predictive model, predicting the quantity sold as a function of the price. The best predictor would be the conditional expectation E[Q[t] |P[t] = p]. But for an
economist such an exercise would make little sense as an attempt to estimate the demand function. There is little reason to believe that the conditional expectation of the quantity as a function of
price, no matter how cleverly estimated, and no matter how well it predicts (how small the residual sum of squares), would correspond to the demand function. Price are not set randomly, not in this
particular market, and not in most markets. To make sense of these data on quantities and prices and what they reveal about the demand function, one needs an economic model that explains how the
demand function relates to the data we see, and in particular why prices took on the values that were observed. In this case a standard, perhaps too simple, economic model is that in addition to the
demand function
presumably increasing in price. The final piece of the economic model is the market clearing assumption that the price and quantity we actually see on day t is the equilbrium/market-clearing price P
[t] that equates supply and demand
and that the observed quantity Q[t] is equal to the supply and demand at the equilibirum price:
Under fairly mild conditions these equilibrium prices and quanties will be unique. The relation between the observed quantities and prices combines the supply and demand function. To separate the
supply and demand functions econometricians often rely on instrumental variables methods. In this particular case ( Angrist et al., 2000) use weather conditions (wave height and wind speed) at sea as
instruments that directly affects supply, but does [End Page 128]
not directly affect demand. The traditional approach would then assume that the demand function was linear in logarithms,
lead to an instrumental variables estimator for the price elasticity, β, equal to
The solid line in Figure 1 presents the regression line from a least squares regression of Q[t] on P[t], with a slope of -0.54. The dashed line presents the instrumental variables estimate of the
demand function, with a slope of approximately -1.01. Whether the price explains much of the quantity traded here is not viewed as being of central importance. Certainly one can obtain a better fit,
as measured by the residual sum of squares, by simple linear regression of the quantity on the price. However, that would not be viewed as meaningful here by most economists for the purpose of
estimating the demand function. The stars in this figure represent the average log quantities and log prices on days where the weather was fair, mixed, or stormy. The instrumental variables estimates
are essentially trying to fit a straight line through these points.
A second example is the returns to education. There is a large literature in economics devoted to estimating the causal effect of formal education (as measured by years of education) on earnings (see
Card (2001) for a general discussion). For individual i let Y[i] be the logarithm of earnings, and X[i] be years of education, and let Y[i](x) be the potential earnings function for this individual,
measuring log earnings for this individual if this individual were to receive level of education equal to x. Initially researchers would estimate the returns to education by estimating a linear
regression of log earnings on years of education. Much of the literature has been concerned with the fact that educational choices are partly driven by unobserved individual characteristics (say,
unobserved skills) that may be related to the potential earnings outcomes. As a result the linear regression of log earnings on years of education may be biased for the causal effect of education on
log earnings, even after conditioning on observed individual characteristics. Building a better predictive model does not directly deal with this concern because it cannot adjust for unobserved
covariates. Angrist and Krueger (1991) propose a clever research strategy to estimate the causal effect of education without this omitted variable bias. They suggest using compulsory schooling laws
as an instrument. The idea is that compulsory schooling laws exogenously shift education levels without directly affecting earnings. In practice, of course compulsory schooling laws explain very
little of the variation in education levels, so little that Angrist and Krueger needed to use Census data in order to get precise estimates. So, from a predictive perspective compulsory schooling
laws appear to be largely irrelevant for modeling earnings. But, there is a reasonable argument that the compulsory schooling laws generate variation in education levels that is not associated with
the unobserved skills that create the biases in least squares regressions of log earnings on years of education. In other words, there is a reasonable argument that it is a valid instrument in the
sense of satisfying the exclusion restrictions (Imbens and Angrist (1994); Angrist et al. (1996)).
These two examples are essentially an elaboration of the point that David Cox (Cox, 2001) and Brad Efron (Efron, 2001) make in their comments on (Breiman, 2001) that much [End Page 129] of statistics
is about causal effects of interventions, rather than predictions. This distinction may often be implicit, but that does not take away from the fact that causal effects are the ultimate goal. This
view may have been part of the reason the econometrics community was initially slow in adopting the algorithmic methods that Breiman was advocating. However, although perhaps slower than one might
have hoped, many of these methods are now enthusiastically been adopted in econometrics, from deep learning methods to generative adversarial networks (Athey et al. (2019a); Kaji et al. (2019)). See
Athey and Imbens (2019) for a general discussion of the use of these methods in economics. A key insight is that although economic theory may be helpful, and in fact essential, for part of the model
(e.g., the exclusion restrictions that are the core of the instrumental variables methods, or the equilibrium assumptions that underly supply and demand models), there are parts of the model where
economic theory is silent, and where the modern machine learning methods can be extremely effective in assisting in model specification, substantially more so than traditional econometric methods.
The challenge is to incorporate the economic causal restrictions and non-prediction objectives into the algorithms.
Let me discuss two examples of this integration of machine learning methods into causal modeling. First, there is a large literature focusing on estimating average treatment effects under ignorable
treatment assignment (Rosenbaum and Rubin (1983)). Under the ignorability/unconfoundedness assumption the target (the average treatment effect) can be written as a functional of number of conditional
expectations, that of the outcome given the treatment and covariates, and that of the treatment given the covariates (the propensity score). Traditionally these conditional expectations were
estimated using nonparametric regression methods. Building on Robins et al. (1994) that introduced double robust estimation, Van der Laan and Rose (2011); Chernozhukov et al. (2017); Athey et al.
(2018) and others use algorithmic machine learning methods for recovering these conditional expectations. These methods are more effective at doing so than the traditional econometric methods,
leading to more accurate estimate of the average treatment effect.
Second, a literature has developed using machine learning techniques to estimate average treatment effects conditional on observable characteristics in a variety of settings, including those where
instrumental variables can be used to estimate treatment effects. For example, Athey et al. (2019b) develop a generalized random forest method that targets treatment effect heterogeneity. Building on
an application by Angrist and Evans (1998), Athey et al. (2019b) analyze the question of how having additional children affects the labor supply of women. The instrumental variable is an indicator
for whether a woman's first two children were of the same gender. Similarly, Hartford et al. (2017) make use of neural nets to analyze conditional average treatment effects in instrumental variables
Since the publication of the Breiman paper much progress has been made. Modellers have embraced many of the algorithms developed in the machine learning literature. The algorithm builders have
expanded beyond the original prediction problems and are now actively exploring methods for including causal objectives and restrictions into their algorithms using both graphical (Pearl (2000)) and
potential outcome perspectives (Imbens and Rubin (2015)), and going into new directions such as causal discovery (Peters et al. (2017)). The two cultures have found they have much in common and much
to learn from each other. [End Page 130]
Guido Imbens
Graduate School of Business and Department of Economics
Stanford University
Stanford, CA 94305
Susan Athey
Graduate School of Business
Stanford University
Stanford, CA 94305
Generous support from the Office of Naval Research through ONR grant N00014-17-1-2131 is gratefully acknowledged.
Joshua D Angrist and William N Evans. Children and their parents' labor supply: Evidence from exogenous variation in family size. American Economic Review , pages 450–477, 1998.
Joshua D Angrist and Alan Krueger. Does compulsory schooling affect schooling and earnings. Quarterly Journal of Economics, CVI(4):979–1014, 1991.
Joshua D Angrist and Jörn-Steffen Pischke. The credibility revolution in empirical economics: How better research design is taking the con out of econometrics. Journal of economic perspectives, 24
(2):3–30, 2010.
Joshua D Angrist, Guido W Imbens, and Donald B. Rubin. Identification of causal effects using instrumental variables. Journal of the American Statistical Association, 91:444–472, 1996.
Joshua D Angrist, Kathryn Graddy, and Guido W Imbens. The interpretation of instrumental variables estimators in simultaneous equations models with an application to the demand for fish. The Review
of Economic Studies, 67(3):499–527, 2000.
Susan Athey and Guido W Imbens. Machine learning methods that economists should know about. Annual Review of Economics, 11:685–725, 2019.
Susan Athey, Guido W Imbens, and Stefan Wager. Approximate residual balancing: debiased inference of average treatment effects in high dimensions. Journal of the Royal Statistical Society: Series B
(Statistical Methodology), 80(4):597–623, 2018.
Susan Athey, Guido W Imbens, Jonas Metzger, and Evan M Munro. Using wasserstein generative adversarial networks for the design of monte carlo simulations. Technical report, National Bureau of
Economic Research, 2019a.
Susan Athey, Julie Tibshirani, Stefan Wager, et al. Generalized random forests. The Annals of Statistics, 47(2):1148–1178, 2019b.
Leo Breiman. Statistical modeling: The two cultures (with comments and a rejoinder by the author). Statistical science, 16(3):199–231, 2001.
David Card. Estimating the return to schooling: Progress on some persistent econometric problems. Econometrica, 69(5):1127–1160, 2001.
Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, and Whitney Newey. Double/debiased/neyman machine learning of treatment effects. American Economic Review, 107
(5):261–65, 2017.
David R Cox. [statistical modeling: The two cultures]: Comment. Statistical Science, 16 (3):216–218, 2001.
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David F Hendry and Mary S Morgan. The foundations of econometric analysis. Cambridge University Press, 1997.
Guido W Imbens. Book review: The foundations of econometric analysis by hendry and morgan.
Guido W Imbens and Joshua D Angrist. Identification and estimation of local average treatment effects. Econometrica, 61:467–476, 1994.
Guido W Imbens and Donald B Rubin. Causal Inference in Statistics, Social, and Biomedical Sciences. Cambridge University Press, 2015.
T Kaji, Elena Manresa, and Guillaume Poulio. Artificial intelligence for structural estimation. Technical report, New York University, 2019.
Judea Pearl. Causality: Models, Reasoning, and Inference . Cambridge University Press, New York, NY, USA, 2000. ISBN 0-521-77362-8.
Jonas Peters, Dominik Janzing, and Bernhard Schölkopf. Elements of causal inference: foundations and learning algorithms. MIT press, 2017.
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(427):846–866, 1994.
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Jan Tinbergen. Determination and interpretation of supply curves: an example. Zeitschrift fur Nationalokonomie, 1(5):669–679, 1930.
Mark J Van der Laan and Sherri Rose. Targeted learning: causal inference for observational and experimental data. Springer Science & Business Media, 2011.
Copyright © 2021 Susan Athey and Guido Imbens | {"url":"https://muse.jhu.edu/article/799753","timestamp":"2024-11-14T02:36:36Z","content_type":"text/html","content_length":"198283","record_id":"<urn:uuid:9da0afc7-9f26-452d-82b2-7d0fc3532d14>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00087.warc.gz"} |
Central Park via Desmos
Today was a rare day where one set of classes had an "extra" day before the test. We had reviewed and I felt good about student understanding so I decided to spend part of the class period on
Our enrichment activity today was
Desmos' Central Park
. Our next unit is on systems of equations in which our focus is on problem solving. Students need to be able to write equations for problem situations. I thought Central Park would be an interesting
formative assessment to see how well my students can interpret a problem creating the necessary equation.
When I worked through Central Park on my own, I enjoyed the process. I also thought the activity might be a bit easy for my students. I guessed that only the last question might give students pause.
In class students were engaged totally in the activity. Many finished it in 15 - 20 minutes. That's what I expected for the majority since they are advanced algebra 2 students (many of them are only
in grade 9). What was more interesting to me were the few students who struggled over the last equation and their response to the struggle.
• Students would delete their whole equation instead of making adjustments.
• Students struggled with multiplying the number of dividers with the width of the dividers.
• The part they gave students the most difficulty was recognizing that they needed to divide by the number of dividers plus 1.
• Whey I suggested they go back to problems with numbers and write those down to analyze the process they were hesitant.
Before I do this activity again next year, I want to create some follow-up problems - especially for the students who struggled. One activity that I think might help are problem situations without
numbers. I used to have those for my middle school students ... I need to find some for my algebra kids! | {"url":"http://algebrasfriend.blogspot.com/2014/09/central-park-via-desmos.html","timestamp":"2024-11-14T17:53:11Z","content_type":"text/html","content_length":"45493","record_id":"<urn:uuid:f5c9dd18-8b7d-4568-892a-2eee4e8b5970>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00484.warc.gz"} |
Electric Motor Torque Types | Locked Rotor Torque, Pull-Up Torque, Breakdown Torque, Full-Load Torque | Electrical A2Z
Electric Motor Torque Types | Locked Rotor Torque, Pull-Up Torque, Breakdown Torque, Full-Load Torque
An electric motor must produce enough torque to start the load and keep it moving for the motor to operate the load connected to it. A motor connected to a load produces four types of torque. The
four types of torque are locked rotor torque (LRT), pull-up torque (PUT), breakdown torque (BDT), and full-load torque (FLT). See Figure 1.
Figure 1. A motor connected to a load produces four types of torque: locked rotor torque (LRT), pull-up torque (PUT), breakdown torque (BDT), and full-load torque (FLT).
Locked Rotor Torque
Locked rotor torque (LRT) is the torque a motor produces when its rotor is stationary and full power is applied to the motor. See Figure 2.
All motors can safely produce a higher torque output than the rated full-load torque for short periods of time. Since many loads require a higher torque to start them moving than to keep them moving,
a motor must produce a higher torque when starting the load.
LRT is also referred to as breakaway or starting torque. Starting torque is the torque required to start a motor and is normally expressed as a percentage of full-load torque.
Figure 2. Locked rotor torque (LRT) is the torque a motor produces when its rotor is stationary and full power is applied to the motor.
Pull-Up Torque
Pull-up torque (PUT) is the torque required to bring a load up to its rated speed. See Figure 3. If a motor is properly sized to the load, PUT is brief.
If a motor does not have sufficient PUT, the locked rotor torque may start the load turning but the PUT cannot bring it up to rated speed. Once the motor is up to rated speed, full-load torque keeps
the load turning. PUT is also referred to as accelerating torque.
Figure 3. Pull-up torque (PUT) is the torque required to bring a load up to its rated speed.
Breakdown Torque
Breakdown torque (BDT) is the maximum torque a motor can provide without an abrupt reduction in motor speed. See Figure 4.
As the load on a motor shaft increases, the motor produces more torque. As the load continues to increase, the point at which the motor stalls is reached. This point is the breakdown torque.
Figure 4. Breakdown torque (BDT) is the maximum torque a motor can provide without an abrupt reduction in motor speed.
Full-Load Torque
Full-load torque (FLT) is the torque required to produce the rated power at the full speed of the motor. See Figure 5.
The amount of torque a motor produces at rated power and full speed (full-load torque) can be found by using a horsepower-to-torque conversion chart.
Figure 5. Full-load torque (FLT) is the torque required to produce the rated power at full speed of the motor.
To calculate motor FLT, the following formula is applied:
\[T=\frac{HP\times 5252}{RPM}\]
T = torque (in lb-ft)
HP = horsepower 5252 = constant
RPM = revolutions per minute
Example: Calculating Full-Load Torque
What is the FLT of a 30 HP motor operating at 1725 rpm?
\[T=\frac{HP\times 5252}{RPM}=\frac{30\times 5252}{1725}=91.34lb-ft\]
If a motor is fully loaded, it produces FLT. If a motor is underloaded, it produces less than FLT. If a motor is overloaded, it must produce more than FLT to keep the load operating at the
motor’s rated speed. See Figure 6.
Figure 6. A motor may be fully loaded, underloaded, or overloaded.
For example, a 30 HP motor operating at 1725 rpm can develop 91.34 lb-ft of torque at full speed. If the load requires 91.34 lb-ft at 1725 rpm, the 30 HP motor produces an output of 30 HP.
However, if the load to which the motor is connected requires only half as much torque (45.67 lb-ft) at 1725 rpm, the 30 HP motor produces an output of 15 HP. The 30 HP motor draws less current (and
power) from the power lines and operates at a lower temperature when producing 15 HP.
However, if the 30 HP motor is connected to a load that requires twice as much torque (182.68 lb.-ft) at 1725 rpm, the motor must produce an output of 60 HP. The 30 HP motor draws more current (and
power) from the power lines and operates at a higher temperature. If the overload protection device is sized correctly, the 30 HP motor automatically disconnects from the power line before any
permanent damage is done to the motor. | {"url":"https://electricala2z.com/motors-control/electric-motor-torque-types-locked-rotor-torque-pull-torque-breakdown-torque-full-load-torque/","timestamp":"2024-11-04T15:44:48Z","content_type":"text/html","content_length":"128830","record_id":"<urn:uuid:5c112374-19f4-4c8f-964d-1993e8eae203>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00810.warc.gz"} |
map.kohonenDTW: Map data to a supervised or unsupervised SOM in e-sensing/kohonenDTW: Supervised and Unsupervised Self-Organising Maps for Satellite Image Time Series
1 ## S3 method for class 'kohonenDTW'
2 map(x, newdata, whatmap = NULL, user.weights = NULL,
3 maxNA.fraction = x$maxNA.fraction, ...)
## S3 method for class 'kohonenDTW' map(x, newdata, whatmap = NULL, user.weights = NULL, maxNA.fraction = x$maxNA.fraction, ...)
x An object of class kohonenDTW.
newdata list of data matrices (numerical) of factors, equal to the data argument of the supersom function. No data.frame objects are allowed.
whatmap, user.weights, parameters that usually will be taken from the x object, but can be supplied by the user as well. Note that it is not possible to change distance functions from the ones
maxNA.fraction used in training the map. See supersom for more information.
... Currently ignored.
list of data matrices (numerical) of factors, equal to the data argument of the supersom function. No data.frame objects are allowed.
parameters that usually will be taken from the x object, but can be supplied by the user as well. Note that it is not possible to change distance functions from the ones used in training the map. See
supersom for more information.
unit.classif a vector of units that are closest to the objects in the data matrix.
dists distances of the objects to the closest units. Distance measures are the same ones used in training the map.
whatmap,weights Values used for these arguments.
a vector of units that are closest to the objects in the data matrix.
distances of the objects to the closest units. Distance measures are the same ones used in training the map.
1 data(wines)
2 set.seed(7)
4 training <- sample(nrow(wines), 150)
5 Xtraining <- scale(wines[training, ])
6 somnet <- som(Xtraining, somgrid(5, 5, "hexagonal"))
8 map(somnet,
9 scale(wines[-training, ],
10 center=attr(Xtraining, "scaled:center"),
11 scale=attr(Xtraining, "scaled:scale")))
data(wines) set.seed(7) training <- sample(nrow(wines), 150) Xtraining <- scale(wines[training, ]) somnet <- som(Xtraining, somgrid(5, 5, "hexagonal")) map(somnet, scale(wines[-training, ], center=
attr(Xtraining, "scaled:center"), scale=attr(Xtraining, "scaled:scale")))
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/github/e-sensing/kohonenDTW/man/map.kohonenDTW.html","timestamp":"2024-11-10T08:01:41Z","content_type":"text/html","content_length":"33164","record_id":"<urn:uuid:1fa1cd55-d9a7-48f4-bc23-b97d54833f9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00430.warc.gz"} |
random variability exists because relationships between variables
random variability exists because relationships between variables
D. Positive. This rank to be added for similar values. When we consider the relationship between two variables, there are three possibilities: Both variables are categorical. D. Only the study that
measured happiness through achievement can prove that happiness iscaused by good grades. In this post I want to dig a little deeper into probability distributions and explore some of their
properties. All of these mechanisms working together result in an amazing amount of potential variation. i. The fewer years spent smoking, the fewer participants they could find. Just because we have
concluded that there is a relationship between sex and voting preference does not mean that it is a strong relationship. 46. C. non-experimental Due to the fact that environments are unstable,
populations that are genetically variable will be able to adapt to changing situations better than those that do not contain genetic variation. B. curvilinear c. Condition 3: The relationship between
variable A and Variable B must not be due to some confounding extraneous variable*. 50. Covariance is completely dependent on scales/units of numbers. Some students are told they will receive a very
painful electrical shock, others a very mildshock. Here are the prices ( $/\$ /$/ tonne) for the years 2000-2004 (Source: Holy See Country Review, 2008). A newspaper reports the results of a
correlational study suggesting that an increase in the amount ofviolence watched on TV by children may be responsible for an increase in the amount of playgroundaggressiveness they display. 43.
Random variability exists because relationships between variables are rarely perfect. A random process is a rule that maps every outcome e of an experiment to a function X(t,e). A. Which one of the
following is a situational variable? A. 51. N N is a random variable. The registrar at Central College finds that as tuition increases, the number of classes students takedecreases. We know that
linear regression is needed when we are trying to predict the value of one variable (known as dependent variable) with a bunch of independent variables (known as predictors) by establishing a linear
relationship between them. There are 3 ways to quantify such relationship. If we investigate closely we will see one of the following relationships could exist, Such relationships need to be
quantified in order to use it in statistical analysis. C. flavor of the ice cream. If we Google Random Variable we will get almost the same definition everywhere but my focus is not just on defining
the definition here but to make you understand what exactly it is with the help of relevant examples. Because we had three political parties it is 2, 3-1=2. B. variables. Specific events occurring
between the first and second recordings may affect the dependent variable. C. Curvilinear The fewer years spent smoking, the less optimistic for success. 1. Mean, median and mode imputations are
simple, but they underestimate variance and ignore the relationship with other variables. There are three 'levels' that we measure: Categorical, Ordinal or Numeric ( UCLA Statistical Consulting, Date
unknown). They then assigned the length of prison sentence they felt the woman deserved.The _____ would be a _____ variable. A. Positive c) The actual price of bananas in 2005 was 577$/577 \$ /577$/
tonne (you can find current prices at www.imf.org/external/np/ res/commod/table3.pdf.) Hence, it appears that B . c) Interval/ratio variables contain only two categories. Explain how conversion to a
new system will affect the following groups, both individually and collectively. Which of the following is true of having to operationally define a variable. An event occurs if any of its elements
occur. The researcher found that as the amount ofviolence watched on TV increased, the amount of playground aggressiveness increased. If you get the p-value that is 0.91 which means there a 91%
chance that the result you got is due to random chance or coincident. A result of zero indicates no relationship at all. Which of the following statements is accurate? A. shape of the carton. The
suppressor variable suppresses the relationship by being positively correlated with one of the variables in the relationship and negatively correlated with the other. No Multicollinearity: None of
the predictor variables are highly correlated with each other. If there were anegative relationship between these variables, what should the results of the study be like? In the above formula, PCC
can be calculated by dividing covariance between two random variables with their standard deviation. In this study Pearsons correlation coefficient formulas are used to find how strong a relationship
is between data. C. subjects which of the following in experimental method ensures that an extraneous variable just as likely to . Above scatter plot just describes which types of correlation exist
between two random variables (+ve, -ve or 0) but it does not quantify the correlation that's where the correlation coefficient comes into the picture. This is any trait or aspect from the background
of the participant that can affect the research results, even when it is not in the interest of the experiment. In order to account for this interaction, the equation of linear regression should be
changed from: Y = 0 + 1 X 1 + 2 X 2 + . The variance of a discrete random variable, denoted by V ( X ), is defined to be. C. prevents others from replicating one's results. Variance generally tells
us how far data has been spread from its mean. 5.4.1 Covariance and Properties i. This is an A/A test. 64. A random variable is any variable whose value cannot be determined beforehand meaning before
the incident. A. mediating definition These results would incorrectly suggest that experimental variability could be reduced simply by increasing the mean yield. 54. This may lead to an invalid
estimate of the true correlation coefficient because the subjects are not a random sample. Monotonic function g(x) is said to be monotonic if x increases g(x) also increases. The Spearman Rank
Correlation Coefficient (SRCC) is the nonparametric version of Pearsons Correlation Coefficient (PCC). X - the mean (average) of the X-variable. That is because Spearmans rho limits the outlier to
the value of its rank, When we quantify the relationship between two random variables using one of the techniques that we have seen above can only give a picture of samples only. C. parents'
aggression. D. Having many pets causes people to buy houses with fewer bathrooms. This relationship can best be identified as a _____ relationship. Second variable problem and third variable problem
Correlation is a statistical measure (expressed as a number) that describes the size and direction of a relationship between two or more variables. The more genetic variation that exists in a
population, the greater the opportunity for evolution to occur. The first limitation can be solved. The first is due to the fact that the original relationship between the two variables is so close
to zero that the difference in the signs simply reflects random variation around zero. A confounding variable influences the dependent variable, and also correlates with or causally affects the
independent variable. An experimenter had one group of participants eat ice cream that was packaged in a red carton,whereas another group of participants ate the same flavoured ice cream from a green
carton.Participants then indicated how much they liked the ice cream by rating the taste on a 1-5 scale. D. woman's attractiveness; response, PSYS 284 - Chapter 8: Experimental Design, Organic Chem
233 - UBC - Functional groups pr, Elliot Aronson, Robin M. Akert, Samuel R. Sommers, Timothy D. Wilson. D. paying attention to the sensitivities of the participant. This phrase used in statistics to
emphasize that a correlation between two variables does not imply that one causes the other. No relationship This is any trait or aspect from the background of the participant that can affect the
research results, even when it is not in the interest of the experiment. In our case accepting alternative hypothesis means proving that there is a significant relationship between x and y in the
population. A correlation is a statistical indicator of the relationship between variables. Participants know they are in an experiment. random variability exists because relationships between
variablesfacts corporate flight attendant training. Computationally expensive. We define there is a positive relationship between two random variables X and Y when Cov(X, Y) is positive. Since we are
considering those variables having an impact on the transaction status whether it's a fraudulent or genuine transaction. C. the drunken driver. C. Quality ratings Correlation and causes are the most
misunderstood term in the field statistics. Which one of the following represents a critical difference between the non-experimental andexperimental methods? This may be a causal relationship, but it
does not have to be. Pearson correlation ( r) is used to measure strength and direction of a linear relationship between two variables. The third variable problem is eliminated. This means that
variances add when the random variables are independent, but not necessarily in other cases. As one of the key goals of the regression model is to establish relations between the dependent and the
independent variables, multicollinearity does not let that happen as the relations described by the model (with multicollinearity) become untrustworthy (because of unreliable Beta coefficients and
p-values of multicollinear variables). Which one of the following is aparticipant variable? This process is referred to as, 11. A. say that a relationship denitely exists between X and Y,at least in
this population. increases in the values of one variable are accompanies by systematic increases and decreases in the values of the other variable--The direction of the relationship changes at least
once Sometimes referred to as a NONMONOTONIC FUNCTION INVERTED U RELATIONSHIP: looks like a U. A. always leads to equal group sizes. 2. Confounding Variables. We define there is a negative
relationship between two random variables X and Y when Cov(X, Y) is -ve. This type of variable can confound the results of an experiment and lead to unreliable findings. In our example stated above,
there is no tie between the ranks hence we will be using the first formula mentioned above. https://www.thoughtco.com/probabilities-of-rolling-two-dice-3126559, https://www.onlinemathlearning.com/
variance.html, https://www.slideshare.net/JonWatte/covariance, https://www.simplypsychology.org/correlation.html, Spearman Rank Correlation Coefficient (SRCC), IP Address:- Sets of all IP Address in
the world, Time since the last transaction:- [0, Infinity].
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Lecture Note 17
ST 361 Ch8.2 Testing Hypotheses about Means
(Part I: Testing for a Population Mean )
Topics: Hypothesis testing with population means
► One-sample problem: Testing for a population mean
1. Assume population SD is known: use a z test statistic
2. Assume population SD is not known: use a t test statistic
► A Special Case: the Paired t test
► Two-sample problem: : Testing for 2 population means 1 and 2
-------------------------------------------------------------------------------------------------------------------► Steps for Testing for a Population mean -----
Need x ~Normal !!!
Step 1. Specify H 0 and H a
H 0 : 0 vs. H a : 0 (this is referred to as _______________________)
H a : 0 (this is referred to as _______________________)
H a : 0 (this is referred to as _______________________)
Step 2. Determine the test level (also called significance level)
Step 3. Compute the test statistic
A test statistic is defined as_________________________________________________
When the population SD is known, a test statistic is ________________________
When the population SD is NOT known, a test statistic is _____________________
Step 4. Calculate the p-value
Step 5. Draw conclusions
If p-value < __________________ and draw conclusion based on _______
Otherwise we _________________________and draw conclusion based on ______
► One-sample problem: Testing for a population mean assume the population SD is known
A Working Example: (adapted from 8.14 p.355 of the textbook) Light bulbs of a certain type are
advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable,
so a potential customer has decided to go ahead with a purchase arrangement unless the true
average lifetime is smaller than what is advertised. A random sample of 50 bulbs was selected.
The sample data and result are presented below: (Assume the population SD of the bulbs lifetime
is 38.2.) What conclusion would be appropriate for a significance level of 0.05?
Mean X
SE of
Mean X
Step 1: parameter of interest =
H0 :
Ha :
Step 2: significance level =
Step 3: test statistic =
Step 4: p-value =
Step 5: Conclusion:
► The p-value
The p-value quantifies how the null hypothesis ( H 0 ) is supported by the observed data,
The smaller the p-value is, the more contradictory is the data to H 0 .
To get p-values, one begins with assuming H 0 is true (e.g., in the working example, it
means ______________________________________________________). We then
calculate that when H o is true, how likely is it to observe a sample with X = 738.44?
i.e., want to calculate P( X 738.444 | H 0 is true (i.e., μ 750) ) = ___________
To resolve this problem, one will instead calculate
the probability of observing X = 738.44 or the
more extreme cases under H 0 . Note that the
“more extreme cases” are determined by the
_____________hypothesis. Here H a : ___ 750 ,
so we calculate
Such probability is called “p-value”
Calculation of p-value (If the population SD is known….)
p-value = P( X 738.444 | μ 750 )
► In general, the calculation of p-value can be simplified in the following steps:
Let X * =
First calculate the test statistic z
X * 0
Then, the p-value = ______________
Interpretation of p-value:
(1) If the p-value, P( X 738.444 | μ 750 ), is very small (i.e., _______________), it
implies that ________________________________________________________
_______________________________________. In other words, the sample data
______________ support H0.
Reject / Do not reject (pick one) H0.
(2) If the p-value, P( X 738.444 | μ 750 ), is not very small (i.e., _____________), it implies
that the chance of observing sample with Y =738.44 or smaller when = 750 is
___________________ In other words, the sample data and H0
Reject / Do not reject (pick one) H0.
In summary,
1. The p-value describes the probability of seeing your data or more extreme IF the null hypothesis is
true. Recall that the more extreme cases are determined by_______.
If H a : 0 , p-value =
If H a : 0 , p-value =
If H a : 0 , p-value =
= ________________
2. To get p-value, all we need is to calculate the test statistic z
X * 0
X * 0
(where X * is the observed sample mean) and then find the corresponding p-value by
P( Z z* ), or P( Z z* ), or 2 P( Z z * ), depending on the alternative hypothesis.
3. To draw conclusion, compare p-value with the test level :
Reject H 0 if p-value___________, and conclude based on H a .
Otherwise we do not reject H 0 , and conclude the test based on H 0 .
Ex1. Let X = the scores on the Verbal SAT exam this year. The score X varies according to a normal
distribution with mean and variance 80. A sample of 64 students was collected, and x 580. Records
showed that the mean SAT score of two years ago is 570. Based on the data, has the average SAT score
increased over the two years? Perform a 0.05-level of test.
Step 1: parameter of interest =
H0 :
Step 2: significance level =
Step 3: test statistic =
Step 4: p-value =
Step 5: Conclusion:
Ex2. Consider the true mean stopping distances at 50 mph for cars equipped with the braking system of
brand A. It is known that the average stopping distances for braking system B is 120 inches. Result
based on 36 cars have mean 115. Assume the population SD of the stopping distance is 20. Do the
stopping distances of the two systems differ? Perform a 0.01-level of test.
Step 1: parameter of interest =
H0 :
Step 2: significance level =
Step 3: test statistic =
Step 4: p-value =
Step 5: Conclusion:
► One-sample problem: Testing for a Population mean when is unknown
If the population SD is unknown, the testing procedure is the same as what we do when the
population SD known, except that
1. the __________________ is used
2. (as a result of 1,) the test statistic is a ___________________ , which is a _________instead of a z
Need to know how to use Table VI to find p-values
EX. (From textbook Question 8.17)
1. Upper-tailed test, df=8, t=2.0
2. Lower-tailed test, df=11, t= -2.4
3. Two-tailed test, df=15, t= -1.6
4. Two tailed test, df=40, t=4.8
Ex1. Life of electric bulb: Industrial standard for the bulb life is 6000 hours. A company claims that their
bulbs are better than the industrial standard. To test their claim, a sample of 16 light bulbs was
collected and has mean 6.5 (unit=1000 hours) and SD 1 (unit = 1000 hours). (a) Perform a test at
5% level. (b) What assumption do we need to conduct a hypothesis?
(a) Step 1: parameter of interest =
H0 :
Step 2: significance level =
Step 3: test statistic:
Step 4: p-value:
Step 5: Conclusion:
(b) Assumption needed:
Ex2. A certain pen has been designed so that the true average writing lifetime is 10 hours. A random
sample of 18 pens is selected and the writing lifetime of each is determined: the mean lifetime of
the 18 pens is 10.5 hours with SD=1.2 hours. Perform a 0.01 level of test to examine if the design
specification has been satisfied.
(a) Step 1: parameter of interest =
H0 :
Step 2: significance level =
Step 3: test statistic:
Step 4: p-value:
Step 5: Conclusion:
(b) Assumption needed: (select any that apply)
_______ The sample mean lifetime follows a normal distribution
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Interesting pursuits in seismic curvature attribute analysis
Since they are second-order derivatives, seismic curvature attributes can enhance subtle information that may be difficult to see using first-order derivatives such as the dip magnitude and the
dip-azimuth attributes. As a result, these attributes form an integral part of most seismic interpretation projects. In this article we discuss some of the recent developments in curvature attributes
that we have been pursuing. These are the computation of amplitude curvature attribute and its comparison with the conventional structural curvature attribute, Euler curvature which is also termed as
azimuthal curvature, and the seismic reflection rotation and vector convergence attributes.
The conventional computation of curvature may be termed as structural curvature, as lateral second-order derivatives of structural component of seismic time or depth of reflection events are used to
generate them. Here, we explore the case of applying lateral second-order derivatives on the amplitudes of seismic data along the reflectors. We refer to this second computation as amplitude
curvature. For volumetric structural curvature we compute first-derivatives of the volumetric inline and crossline components of structural dip. For amplitude curvature we apply a similar computation
to the volumetric inline and crossline components of the energyweighted amplitude gradients, which represent the directional measures of amplitude variability. Since the amplitude and structural
position of a reflector are mathematically independent properties, application of amplitude curvature computation to real seismic data often shows different, and sometimes more detailed illumination
of geologic features than structural curvature. However, many features, such as the delineation of a fault where we encounter both a vertical shift in reflector position and a lateral change in
amplitude, will be imaged by both attributes, with images ‘coupled’ through the underlying geology.
Euler curvature is a generalization of the dip and strike components of curvature in any user-defined direction, and may be called azimuthal curvature or more simply apparent curvature (similar to
apparent dip). Euler curvature is useful for the interpretation of lineament features in desired azimuthal directions, say, perpendicular to the minimum horizontal stress. If a given azimuth is known
or hypothesized to be correlated with open fractures or if a given azimuth can be correlated to enhanced production or effective horizontal drilling, an Euler-curvature intensity volume can be
generated for that azimuth thereby high-grading potential sweet spots.
Geometric attributes such as coherence and curvature are useful for delineating a subset of seismic stratigraphic features such as shale dewatering polygons, injectites, collapse features, mass
transport complexes and overbank deposits, but have limited value in imaging classic seismic stratigraphy features such as onlap, progradation and erosional truncation. In this context, we review the
success of current geometric attribute usage and discuss the applications of newer volumetric attributes such as reflector convergence and reflector rotation about the normal to the reflector dip.
While the former attribute is useful in the interpretation of angular unconformities, the latter attribute determines the rotation of fault blocks across discontinuities such as wrench faults. Such
attributes can facilitate and quantify the use of seismic stratigraphic workflows to large 3D seismic volumes.
Structural curvature versus amplitude curvature
Since the introduction of the seismic curvature attributes by Roberts (2001), curvature has gradually become popular with interpreters, and has found its way into most commercial software packages.
Curvature is a 2D second-order derivative of time or depth structure, or a 2D first-order derivative of inline and crossline dip components. As a derivative of dip components, curvature measures
subtle lateral and vertical changes in dip that are often overpowered by stronger, regional deformation, such that a carbonate reef on a 20° dipping surface gives rise to the same curvature anomaly
as a carbonate reef on a flat surface. Such rotational invariance provides a powerful analysis tool that does not require first picking and flattening on horizons near the zone of interest. Curvature
computation can be carried out on both time surfaces as well as 3D seismic volumes. Roberts (2001) introduced curvature as a 2D second-derivative computations of picked seismic surfaces. Soon
afterwards, Al-Dossary and Marfurt (2006) showed how such computations can be computed from volumetric estimates of inline and crossline dip components. By first estimating the volumetric reflector
dip and azimuth that best represents each single sample in the volume, followed by computation of curvature from adjacent measures of dip and azimuth, a full 3D volume of curvature values is
To clarify our subsequent discussion, we denote the above calculations as structural curvature, the (explicit or implicit) lateral second derivatives of reflector time or depth. Many processing
geophysicists focused on statics and velocity analysis think of seismic data as composed of amplitude and phase components, where the phase associated with any time t and frequency f is simply φ=2πft
. Indeed, several workers have used the lateral change in phase as a means to compute reflector dip (e.g. Barnes, 2000; Marfurt and Kirlin, 2000).
We can also compute second derivatives of amplitude. Horizon-based amplitude curvature is in the hands of most interpreters. First, we generate a horizon slice through a seismic amplitude, RMS
amplitude, or impedance volume. Next, we compute the inline (∂a /∂x) and crossline (∂a /∂y) derivatives of this map. Such maps can often delineate the edges of bright spots, channels, and other
stratigraphic features at any desired direction, θ, by combining the two measures with simple trigonometry (cosθ∂a /∂x+sinθ∂a /∂y). A common edge detection algorithm is to compute the Laplacian of a
map (though more of us have probably applied this filter to digital photographs than to seismic data),
Equation 1 is the formula for the mean amplitude curvature. In Figure 1 we show a schematic diagram of an amplitude anomaly exhibiting its lateral change in one direction, x. Thereafter, we compute
the 1st and 2nd spatial derivatives of the amplitude with respect to x and show the results in Figure 1b and c. Notice, the extrema seen in Figure 1c demarcate the limits of the anomaly.
Figure 1. Effect of the first and second derivative on a one-dimensional amplitude profile. The two extrema seen in (c) shows the limits of the amplitude anomaly.
Luo et al. (1996) showed that if one were to first estimate structural inline and crossline dip, then one can generate an excellent edge detector that is approximately
where the derivatives are computed in a (-K to +K vertical sample, J-trace) analysis window oriented along the dipping plane and the derivatives are evaluated at the center of the window. Marfurt and
Kirlin (2000) and Marfurt (2006) showed how one can compute accurate estimates of reflector amplitude gradients, g, from the KL-filtered (or principal component of the data) within an analysis
where v[1] is the principal component or eigenmap of the J-trace analysis window, and λ[1] is its corresponding eigenvalue, which represents the energy of this data component.
In Figures 2a and b we show the images for 3D chair views wherein the vertical section is the seismic inline and is seen being correlated with the energy-weighted amplitude gradient in the inline and
the crossline gradients respectively. Both images express independent views of the same geology (almost N-S oriented main faults and fault related fractures) much as two orthogonal shaded
illumination maps.
Figure 2. 3D chair view showing the seismic inlines correlated with (a) inline-energy gradient and (b) crossline energy gradient strat cubes. Each strat-cube exhibits subtle information detail that
may not be so pronounced in one image or the other.
For volume computation of structural curvature, the equations applied to the components of reflector dip and azimuth in the inline and crossline directions are given by Al-Dossary and Marfurt (2006).
In the case of amplitude computation of curvature, nearly the same equations are applied to the inline and crossline components of energy-weighted amplitude gradients which represent the directional
measures of amplitude variability.
Geological structures often exhibit curvature of different wavelengths and so curvature images of different wavelengths provide different perspectives of the same geology. Al-Dossary and Marfurt
(2006) introduced the volume computation of longand short-wavelength curvature measures from seismic data. Many applications of such multispectral estimates of curvature from seismic data have been
demonstrated by Chopra and Marfurt (2006, 2007a and b, 2010). Short wavelength curvature often delineates details with intense highly localized fracture systems. Long wavelength curvature on the
other hand enhances subtle flexures on a scale of 100-200 traces that are difficult to see on conventional seismic data, but are often correlated to fracture zones that are below seismic resolution
as well as to collapse features and diagenetic alterations that result in broader bowls.
Figure 3. 3D chair views show the seismic inline and strat-cubes from (a) mostpositive amplitude curvature (long-wavelength), (b) most-positive structural curvature (long-wavelength), (c)
most-positive amplitude curvature (short-wavelength), and (d) most-positive structural curvature (short-wavelength). Notice the higher level of detail on both the amplitude curvature displays as
compared with the structural curvature displays.
Figure 4. 3D chair views show the seismic inline and strat-cubes from (a) mostnegative amplitude curvature (long-wavelength), (b) most-negative structural curvature (long-wavelength), (c)
most-negative amplitude curvature (short-wavelength), and (d) most-negative structural curvature (short-wavelength). Notice the higher level of detail on both the amplitude curvature displays as
compared with the structural curvature displays.
In Figures 3 and 4 we show a comparison of the long- and short-wavelength computation of most-positive and most-negative amplitude and structural curvature measures. In Figure 3 we notice that for
both long and short wavelength, the amplitude curvature estimates provide additional information. Structural most-positive curvature displays in Figure 3b and d show lower frequency detail as
compared with their equivalent amplitude curvature displays in Figures 3a and c. Similarly, Figures 4 a and c exhibit much greater lineament detail on the amplitude most-negative curvature displays
than what is seen on structural most-negative curvature displays in Figure 4b and d. Such fine detail is very useful when using curvature attributes for fault /fracture delineation, particularly
those that give rise to measureable amplitude changes but minimal changes in dip, such as cleats in coal beds. In Figures 5 and 6 we show a similar comparison where again we observe higher level of
lineament detail on both long and short-wavelength amplitude curvature in preference to structural curvature.
Figure 5. 3D chair views show the seismic inline and strat-cubes from (a) mostpositive structural curvature (long-wavelength) (b) most-positive amplitude curvature (long-wavelength), (c)
most-positive structural curvature (short-wavelength), and (d) most-positive amplitude curvature (short-wavelength). Notice the higher level of detail on both the amplitude curvature displays as
compared with the structural curvature displays.
Figure 6. 3D chair views show the seismic inline and strat-cubes from (a) mostnegative structural curvature (long-wavelength), (b) most-negative amplitude curvature (long-wavelength), (c)
most-negative structural curvature (short-wavelength), and (d) most-negative amplitude curvature (short-wavelength). Notice the higher level of detail on both the amplitude curvature displays as
compared with the structural curvature displays.
Euler curvature
In his seminal paper Roberts (2001) describes twelve different types of surface-based attribute measures. Many of these attributes have been extended to volume computations and implemented on
interpretation workstations. Of these different curvature attributes, the most-positive and the most-negative principal curvatures k[1] and k[2] are the most popular. Not only are k[1] and k[2]
intuitively easy to understand, they also provide more continuous maps of faults and flexures than the maximum and minimum curvatures, k[max] and k[min], which can rapidly change sign at fault and
flexure intersections. While other attributes such as the mean curvature, Gaussian curvature and shape index have also been used by a few practitioners, there are some other curvature attributes
which Roberts (2001) introduced that warrant investigation and forms the motivation for the present work.
Definition and workflow
In this paper, we describe the application of volumetric Euler curvature to 3D seismic data volumes. Euler curvature can be thought of as apparent curvature along a given strike direction. If (k[1 ],
ψ[1]) and (k[2 ], ψ[2]) represent the magnitude and strike of the most-positive and most-negative principal curvatures then the Euler curvature at an angle ψ in the dipping plane tangent to the
analysis point (where ψ[1] and ψ[2] are orthogonal) is given as
Since reflector dip magnitude and azimuth can vary considerably across a seismic survey, it is more useful to equally sample azimuths of Euler curvature on the horizontal x-y plane, project these
lines onto the local dipping plane of the reflector, and implement equation 4. The flow diagram in Figure 7 explains the method for computing Euler curvature.
Figure 7. Flow diagram showing the computation of Euler curvature.
Mapping the intensity of a given fracture set has been a major objective of reflection seismologists. The most successful work has been using attributes computed by azimuthally-limited prestack data
volumes. Chopra et al. (2000) showed how coherence attributes computed from azimuthally-restricted seismic volumes can enhance subtle features hidden or blurred in the all-azimuth volume. Vector-tile
and other migration-sorting techniques are now the method of choice for both conventional Pwave and converted wave prestack imaging (e.g. Jianming et al., 2009) allowing one to predict both fracture
strike and intensity.
Curvature, acoustic impedance, and coherence are currently the most effective attributes used to predict fractures in the post-stack world (e.g. Hunt et al., 2010). Rather than map the intensity of
the strongest attribute lineaments, Singh et al. (2008) used an image-processing (ant-tracking) algorithm to enhance curvature and coherence lineaments that were parallel to the strike of open
fractures, at an angle of some 45° to the strike of the strongest lineaments. Henning et al. (2010) use related technology to azimuthally filter lineaments in the Eagleford formation of south Texas.
They then compute RMS maps of each azimuthally-limited volume that can be correlated to production. Guo et al. (2010) hypothesize that each azimuthally-limited attribute volume computed from k[1] and
ψ[1] corresponds to open fractures. Each of these volumes is then correlated to production to either validate or reject the hypothesis.
Daber and Boe (2010) described the use of what they referred to as azimuthal curvature (and is described here as Euler curvature following Roberts (2001) definition) for reducing noise in poststack
curvature volume. They show that if the azimuthal direction is set to the inline, then the curvature computation would ignore the crossline directions and this would reduce the acquisition noise.
We describe here the application of Euler curvature to two different 3D seismic volumes from northeast British Columbia, Canada. We propose an interactive workflow, much as we do in generating a
suite of shaded relief maps where we display apparent dip rather than apparent (Euler) curvature. In Figure 8 we show 3D chair view displays for Euler curvature run at 0°, 45°, 90° and 135°. The left
column of displays shows the long-wavelength version and the right display the short wavelength displays. Notice for 00 azimuth (which would be the north), lineaments in the E-W direction seem to
stand out. For 45°s, the lineaments that are almost NW-SE are seen pronounced. Similarly for 90°s the roughly N-S events stand out and for 135°s the events slightly inclined to the vertical are more
Figure 8. 3D chair views showing the correlation of an inline with the strat-cube from Euler curvature attribute volumes run at different angles as indicated and for both long-wavelength (left
column) and the short-wavelength (right column). For each azimuth angle, the orthogonal lineaments appear more well-defined than those in other directions.
The same description applies to the short-wavelength displays that show more lineament detail and resolution than the long-wavelength display.
In Figure 9 we show a similar 3D chair view displays and again notice the lineaments becoming pronounced for particular azimuth directions.
There are obvious advantages to running Euler curvature on post stack seismic volumes in that the azimuth directions can be carefully chosen to highlight the lineaments in the directions known
through image logs or production data to better correlate to open fractures. This does not entail the processing of azimuth-restricted volumes (usually three or four) all the way to migration and
then passing them through coherence/curvature computation.
Figure 9. 3D chair views showing the correlation of an inline with the strat-cube from Euler curvature attribute volumes run at different angles as indicated and for both long-wavelength (left
column) and the short-wavelength (right column). For each azimuth angle, the orthogonal lineaments appear more well-defined than those in other directions.
Volumetric estimates of seismic reflector rotation and convergence
Seismic stratigraphic analysis refers to the analysis of the configuration and termination of seismic reflection events, packages of which are then interpreted as stratigraphic patterns. These
packages are then correlated to well-known patterns such as toplap, onlap, downlap, erosional truncation, and so forth, which in turn provide architectural elements of a depositional environment
(Mitchum et al., 1977). Through well control as well as modern and paleo analogues, we can then produce a probability map of lithofacies.
Geometric attributes such as coherence and curvature are commonly used for mapping structural deformation and depositional environment. Coherence proves useful for identification of faults, channel
edges, reef edges and collapse features while curvature images folds, flexures, sub-seismic conjugate faults that appear as drag or folds adjacent to faults, roll-over anticlines, diagenetically
altered fractures, karst and differential compaction over channels.
Although coherence and curvature are excellent at delineating a subset of seismic stratigraphic features (such as shale-dewatering polygons, injectites, collapse features, mass transport complexes,
and overbank deposits) they have only limited value in imaging classic seismic stratigraphy features such as onlap, progradation and erosional truncation. We review the success of current geometric
attribute usage and examine how the newer volumetric attributes can facilitate and quantify the use of seismic stratigraphic analysis workflows to large 3D seismic volumes.
Attribute application for stratigraphic analysis
Due to the distinct change in reflector dip and/or terminations, erosional unconformities and in particular angular unconformities are relatively easy to recognize on vertical seismic sections.
Although there will often be a low-coherence anomaly where reflectors of conflicting dip intersect, these anomalies take considerable skill to interpret. Barnes (2000) was perhaps the first to
discuss the application of attributes based on the description of seismic reflection pattern and used them to map angular unconformities amongst other features. As the first step volumetric estimates
for vector dip are computed. Next, the mean and standard deviation of the vector dip are calculated in narrow windows. Those reflections that exhibit parallelism have a smaller standard deviation and
while non-parallel events such as angular unconformities have a higher standard deviation.
Computing a vertical derivative of apparent dip at user-defined azimuth, Barnes (2000) defined the convergence/divergence of reflections. Convergent reflections would show a decreasing dip with depth
/time at constant azimuth. Marfurt and Rich (2010) built upon Barnes’ (2000) method by taking the curl of the volumetric vector dip thereby generating a 3D reflector convergence azimuth and magnitude
Compressive deformation and wrench faulting cause the fault blocks to rotate (Kim et al., 2004). Such rotation has been observed in laboratory measurements. The extent of rotation depends on the
size, the comprising lithology and the stress levels. As the individual fault blocks undergo rotation, it is expected that the edges experience higher stresses and undergo fracturing.
Natural fractures are controlled by fault block rotation and depend on how the individual fault segments intersect. Fault block rotation can also control depositional processes by providing increased
accommodation space in subsiding areas and erosional processes in uplifted areas. In view of this importance of the rotation of the fault blocks, a seismic attribute application focusing on the
rotation of the fault blocks is required. Besides the reflector convergence attribute mentioned above, Marfurt and Rich (2010) also discuss the calculation of another attribute that determines the
rotation about the normal to the reflector dip and would be a measure of the reflector rotation across a discontinuity such as a wrench fault.
As the first step, the inline and crossline components of dip are determined at every single sample in the 3D volume using semblance search or any other available method. After defining the three
components of the unit normal, n, and the rotation vector ψ, Marfurt and Rich (2010) define the rotation about the normal to the reflector dip as
which is essentially a measure of the reflector rotation across a discontinuity such as a wrench fault.
Similarly, Marfurt and Rich (2010) write reflector convergence as follows:
Note that the reflector convergence, c, is a vector consisting of a magnitude and azimuth. We use a common 2D color wheel to display such a result, where parallel reflectors (magnitude of convergence
= 0) appear as white, and the azimuth of convergence is mapped against a cyclical color bar, with the colors becoming darker for stronger convergence. In Figure 10, we try and explain the convergence
within a channel with or without levee/overbank deposits, in terms of the following cases:
Figure 10. Cartoons demonstrating convergence within a channel with or without Levee/overbank deposits. Case (a) deposition within the channel shows no significant convergence; Case (b) shows strata
within the channel where the west channel margin converging towards west and the east channel margin converging towards the east. This is displayed in color to the right with the help of a 2D color
wheel; Case (c) shows deposited sediments within the channel not converging at the margins, but the levee/overbank deposits converge towards the channel (west deposits converge towards the east and
vice-versa; Case (d) shows a combination of cases (b) and (c) where both the strata within the channel and levee/overbank deposits are converging. Notice how the convergence shows up in color as
displayed to the right in cyan and magenta colors. (Interpretation courtesy of Supratik Sarkar (OU)).
Case-1: where the deposition within the channel shows no significant convergence;
Case-2: where the deposition within the channel is such that the west channel margin is converging towards the west and the east channel margin is converging towards the east. This is displayed in
color to the right with the help of a 2D color wheel;
Case-3: where the deposited sediments within the channel are not converging at the margins, but the levee/overbank deposits converge towards the channel (west deposits converge towards the east and
Case-4: where both the strata within the channel and levee/overbank deposits are converging. This appears to be a combination of cases 2 and 3 above.
Notice how the convergence shows up in color (using the 2D color wheel) as displayed to the right in cyan and magenta colors along the channel edges.
We carried out the computation of reflector convergence and the rotation about the normal to the reflector dip attributes for a suite of 3D seismic volumes from Alberta, Canada. Figure 11 depicts a
3D chair view with a coherence time slice exhibiting a channel system, co-rendered with reflector convergence attribute using a 2D color wheel. Within the area highlighted by the ellipse in yellow
dotted line, an interpretation has been made keeping in mind the cases shown in Figure 10. Apparently, the levee/overbank deposit converging towards the channel margin generating the magenta and
green colors with respect to the reflector convergence.
Figure 11. 3D chair view with a coherence time slice as the horizontal section and showing a channel system. This slice is co-rendered with the reflector convergence attribute displayed using a 2D
color wheel. In view of the cases discussed in Figure 10, the highlighting ellipse shows a levee/overbank deposit converging towards channel margin generating magenta and green colours with respect
to the reflector convergence attribute.
In Figure 12, we show a 3D chair display with the vertical inline and crossline displays and a time slice at 1.710 s from a coherence volume showing several lineaments corresponding to faults
associated with a network of horst and grabens. This time slice is co-rendered with a multiattribute volume of the strike of the most-positive principal curvature, ψ[1], (plotted against hue)
modulated by the magnitude of the most positive principal curvature, k[1]. The fault blocks give rise to prominent North- South trending lineaments (depicted as blue) as well NE-SW trending faults
(red) and NW-SE trending faults (cyan and green). In Figure 13 we show a corresponding time slice through the reflector rotation about the average reflector normal volume. Notice the horst and graben
features show considerable contrast so as to be conveniently interpreted. Again in Figure 14 which shows a similar display, the time slice is at 1.330 s from the reflector rotation attribute about
the average reflector normal. The yellow arrow is indicative of either a rotation about antithetic faults or a suite of relay ramps. An equivalent display is shown in Figure 15, but with the time
slice from the reflector convergence attribute. The display uses a 2D color wheel (shown in Figure 10), wherein the blue color indicates reflectors pinching out to the North, red to the Southeast and
cyan to the Northwest. We co-render the slices in Figures 14 and 15 using 50 percent transparency and obtain a display as shown in Figure 16. The thickening and thinning of the reflectors appear to
be controlled by rotating fault blocks.
Figure 12. Time slice at 1.710 s through a multi-attribute volume of the strike of the most-positive principal curvature, ψ[1], (plotted against hue) modulated by the magnitude of the most positive
principal curvature, k[1], and co-rendered with coherence. The fault blocks give rise to prominent North-South trending lineaments (depicted as blue) as well NE-SW trending faults (red) and NW-SE
trending faults (cyan and green). Below the time slice we show a box probe view of the most-positive principal curvature lineaments displayed in 3D with the more planar features rendered transparent.
Figure 13. Time slice at t=1.710 through a volume of the reflector rotation about the average reflector normal. Not surprisingly, the horst and graben blocks show considerable contrast and can be
interpreted as separate units.
Figure 14. Time slice at t=1.330 s through the volume of reflector rotation about the average reflector normal. We interpret the cross hatched pattern indicated by the yellow arrow as either an
indication of rotation about antithetic faults, or a suite of relay ramps. Below the time slice, as in Figures 1 and 2, we show a box probe view of the most-positive principal curvature lineaments
displayed in 3D with the more planar features rendered transparent.
Figure 15. Time slice at t=1.330 s through a reflector convergence volume displayed using a 2D color wheel. Blue indicates reflectors pinching out to the North, red to the Southeast, and cyan to the
Northwest. Below the time slice we show a box probe view of the most-positive principal curvature lineaments displayed in 3D with the more planar features rendered transparent.
Figure 16. Time slice at t=1.330 s through a co-rendered image of reflector convergence displayed using a 2D color wheel and reflector rotation displayed using a gray scale and 50% transparency. We
interpret the thickening and thinning of the reflectors to be controlled by the rotating fault blocks. Below the time slice we show a box probe view of the most-positive principal curvature
lineaments displayed in 3D with the more planar features rendered transparent.
In Figure 17 we see a time slice at 1.330 s through the coherence attribute in gray scale co-rendered with the reflector convergence displayed against a 2D color wheel. The orange arrow indicates
sediments in the graben thinning to the Southeast. The sediments indicated by the cyan arrow show thinning to the Northwest and by the purple arrow to the North-Northeast. Sediments that have low
convergence magnitude or which are nearly parallel, has been rendered transparent. A similar display we show in Figure 18 with the time slice at 1.550s.
Figure 17. Time slice at t=1.330 s through coherence rendered against a gray-scale and reflector convergence displayed against a 2D color wheel. Sediments in the graben indicated by the orange arrow
are thinning to the Southeast, sediments indicated by the cyan arrow to the Northwest, and those by the purple arrow to the North-Northeast. Sediments that are nearly parallel (low convergence
magnitude) are rendered transparent.
Figure 18. Time slice at t=1.550 s through coherence rendered against a gray-scale and reflector convergence displayed against a 2D color wheel.
1. For data processed with an amplitude preserving sequence, amplitude variations are diagnostic of geologic information such as changes in porosity, thickness and /or lithology. Computation of
curvature on amplitude gradients furnishes higher level of lineament information that appears to be promising. The application of amplitude curvature to impedance images is particularly
interesting (Guo et al., 2010). We hope to extend this work to the generation of rose diagrams for the lineaments observed on amplitude curvature and make comparisons with similar roses obtained
from image logs. Such exercises will lend confidence in the application of amplitude curvature in seismic data interpretation.
2. Euler curvature run in desired azimuthal directions exhibit a more well-defined set of lineaments that may be of interest. Depending on the desired level of detail, the longor the
short-wavelength computations can be resorted to. For observing fracture lineaments the short-wavelength Euler curvature would be more beneficial. This work is in progress and we hope to
calibrate the observed lineaments with the image logs in terms of rose diagram matching. This would serve to enhance the interpreter’s level of confidence, should the rose-diagrams match.
3. As shown above, application of two attributes, namely reflector convergence and the rotation about the normal to the reflector dip are shown on two different 3D seismic volumes from Alberta,
Canada. These attributes have both been found to be very useful. Reflector convergence attribute gives the magnitude and direction of thickening and thinning of reflections on uninterpreted
seismic volumes. Reflector rotation about faults is clearly evident and has a valuable application in mapping of wrench faults. Such attributes would yield convincing results on datasets that
have good quality. Dip-convergence based attributes do not delineate disconformities and nonconformities exhibiting near parallel reflector patterns. Condensed sections are often seen as
stratigraphically parallel low-coherence anomalies on vertical sections. More promising solutions to mapping these features are based on changes in spectral magnitude components (Smythe et al.,
2004) or in spectral phase components (Castro de Matos et al., 2011).
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Hunt, L., S. Reynolds, T. Brown, S. Hadley, H. James, Jon Downton, and S. Chopra, 2010, Quantitative estimate of fracture density variations in the Nordegg with azimuthal AVO and curvature: a case
study: The Leading Edge, 29; 1122-1137.
Jianming, T., H. Yue, X. Xiangrong, J. Tinnin, and J. Hallin, 2009, Application of converted-wave 3D/3C data for fracture detection in a deep tight-gas reservoir: TLE, 28, 826-837.
Kim, Y. S., Peacock, D.C.P. and Sanderson, D.J., 2004. Fault damage zones, J. Struct. Geol., 26, 503–517.
Luo, Y., W. G. Higgs, and W. S. Kowalik, 1996, Edge detection and stratigraphic analysis using 3-D seismic data: 66th Annual International Meeting, SEG, Expanded Abstracts, 324–327.
Marfurt, K. J., and R. L. Kirlin, 2000, 3D broadband estimates of reflector dip and amplitude: Geophysics, 65, 304-320.
Marfurt, K. J. and J. Rich, 2010, Beyond curvature – volumetric estimates of reflector rotation and convergence, 80th Annual International Meeting, SEG, Expanded Abstracts, 1467-1472.
Mitchum, R. M. Jr., P. R. Vail, J. B. Sangree, 1977, Seismic stratigraphic and global changes of sea level: part 6. Stratigraphic interpretation of seismic reflection patterns in depositional
sequences: section 2. Application of seismic reflection configuration to stratigraphic interpretation, AAPG Special volumes, Memoir 26, 117-133.
Roberts, A., 2001, Curvature attributes and their application to 3D interpreted horizons. First Break, 19, 85-99.
Singh, S. K., H. Abu-Habbiel B. Khan, M. Akbar, A. Etchecopar and B. Montaron, 2008, Mapping fracture corridors in naturally fractured reservoirs: an example from Middle East carbonates: The First
Break, 26, 109-113.
Smythe, J., A. Gersztenkorn, B. Radovich, C.-F. Li, and, C. Liner, 2004, SPICE: Layered Gulf of Mexico shelf framework from spectral imaging: The Leading Edge, 23, 921-926. | {"url":"https://csegrecorder.com/articles/view/interesting-pursuits-in-seismic-curvature-attribute-analysis","timestamp":"2024-11-14T05:26:28Z","content_type":"text/html","content_length":"68393","record_id":"<urn:uuid:2e39f4c4-db37-4aee-a7d9-076e17092910>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00630.warc.gz"} |
Ptolemy's Theorem
This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.
This entry provides an analytic proof to Ptolemy's Theorem using polar form transformation and trigonometric identities. In this formalization, we use ideas from John Harrison's HOL Light
formalization and the proof sketch on the Wikipedia entry of Ptolemy's Theorem. This theorem is the 95th theorem of the Top 100 Theorems list.
Session Ptolemys_Theorem | {"url":"https://devel.isa-afp.org/entries/Ptolemys_Theorem.html","timestamp":"2024-11-12T03:14:51Z","content_type":"text/html","content_length":"10315","record_id":"<urn:uuid:400558bd-116d-486d-b447-1fbe2340fd67>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00144.warc.gz"} |
Working with Fractions and Rationals in Ruby
I have a confession to make. I kind of hate floating-point numbers. Sure, they're useful if you happen to be a computer, but if you're a human being your left scratching your head at situations like
129.95 * 100
# => 12994.999999999998
Not only does this fly in the face of mathematical harmony, it's also bad UX.
If a recipe told you to measure 0.37211927843 cups of flour, you would probably chuckle yourself about what an idiot the author was and proceeded to measure out a third of a cup.
Most people are able to think about fractions a lot more easily than they can think about arbitrary decimal numbers. So if your app is trying to communicate numbers to people, it might make sense to
explore ways of expressing them as fractions.
Ruby's Rational class is a great tool for working with rational numbers. It not only gives you the ability to do rational math, but it also lets you find simple fractions that approximate gnarly
floating-point numbers. Let's take a look!
What are Rationals?
For our purposes, "rational numbers" is just a fancy way of saying "fractions." They have two parts: the numerator and denominator.
1/2 # Numerator is 1. Denominator is 2.
5 # Numerator is 5. Denominator is 1.
In Ruby, rational numbers get their own data type just like integers and floating-point numbers. There are a couple of ways to create a new rational:
3/2r # This syntax was introduced in Ruby 2.1
1.5.to_r # Floats can be converted to rationals via `to_r`
"3/2".to_r # ...so can strings
Rational('3/2') # This is how we had to do things in the olden days
Rational(3, 2) # ...see how hard life was?
Simple math
When you add, subtract, multiply, or divide two rational numbers the result is also rational.
2/3r + 1/3r
# => (1/1)
2/3r - 1/3r
# => (1/3)
2/3r * 1/3r
# => (2/9)
(2/3r) / (1/3r) # We need parens here to avoid confusing the interpreter
# => (2/1)
All of the other math operators pretty much act like you would expect too: **, >, <, etc..
The general rule is that both of the inputs have to be fractions in order for the result to be a fraction. The one exception I could find is with integers. Since all integers are rational, Ruby does
the smart thing and assumes everyone a rational output:
One of the most useful things about rational numbers is that they allow us to approximate and easily do calculations in our heads. In order to take advantage of this we need to keep our fractions
simple. 3/2 instead of 3320774221237909/2251799813685248.
Fortunately, Ruby gives us an easy way to convert these precise-but-ugly numbers into approximate yet pretty numbers. I'm talking about the rationalize method.
Here's what it looks like:
# Precise but ugly
=> (3320774221237909/2251799813685248)
# Less precise, but pretty
=> (3/2)
The rationalize method has one argument. It specifies the tolerance – the amount of precision that you're willing to trade for simplicity.
The method finds a number with the lowest denominator within your tolerance. Here's what I mean:
# What's the number with the lowest denominator between 5/10 and 7/10?
# => (1/2)
# What's the number with the lowest denominator between 11/20 and 13/20?
=> (3/5)
# ..and between 1/10 and 11/10?
=> (1/1)
Cruder approximations
If all you need to do is find an integer or floating-point number that corresponds to the fraction, you've got several options.
# Return the nearest integer.
# => 1
# Round down
# => 1
Limitations of Rationals
There are a few limitations to be aware of when working with rational numbers and Ruby. Of course you can't divide by zero:
ZeroDivisionError: divided by 0
And you will run into some strange behavior if you try to treat irrational numbers as rational.
# Umm, isn't the square root of 2 irrational?
# => (6369051672525773/4503599627370496)
# And I'm pretty sure PI is irrational as well.
# => (884279719003555/281474976710656)
We might expect that asking Ruby to treat an irrational number is rational would raise some kind of exception. But unfortunately Ruby doesn't seem to be smart enough to do this. Instead, it converts
the floating-point approximations of these irrational numbers into rationals. It's not a huge problem, but something to be aware of. | {"url":"https://www.honeybadger.io/blog/ruby-rational-numbers-and-fractions/","timestamp":"2024-11-07T16:02:27Z","content_type":"text/html","content_length":"54136","record_id":"<urn:uuid:37ebc564-a72b-4a6a-be38-3c6cde5aab70>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00786.warc.gz"} |
Graphing inequalities in excel
graphing inequalities in excel Related topics: matrix operations,2
free downloadable math papers for 8th grade
polynomial exercises "middle school" download
integer division worksheets
using distributive property with fractions
define difference of squares
learning beginners algebra
factoring rational expressions calculator online
finding the value of x for an arc of a circle
ti 89 radical expressions
multiplying integers worksheets
rules to solve multi step equations
Author Message
mesareve Posted: Sunday 03rd of Oct 10:59
Hello math gurus . This is my first post in this forum. I struggle a lot with graphing inequalities in excel questions . No matter how much I try, I just am not able to crack any
question in less than an hour. If things go this way, I reckon I will not be able to get through my math exam.
From: 42° 3' N
83° 22' W
Back to top
ameich Posted: Monday 04th of Oct 07:11
I understand your situation because I had the same issues when I went to high school. I was very weak in math, especially in graphing inequalities in excel and my grades were awful . I
started using Algebrator to help me solve problems as well as with my homework and eventually I started getting A’s in math. This is an extremely good product because it explains the
problems in a step-by-step manner so we understand them well. I am absolutely certain that you will find it helpful too.
From: Prague,
Czech Republic
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MichMoxon Posted: Monday 04th of Oct 17:57
Yes I agree, Algebrator is a really useful product . I bought it a few months back and I can say that it is the reason I am passing my math class. I have recommended it to my friends
and they too find it very useful. I strongly recommend it to help you with your math homework.
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MichMoxon Posted: Tuesday 05th of Oct 07:48
A extraordinary piece of algebra software is Algebrator. Even I faced similar problems while solving algebra formulas, percentages and cramer’s rule. Just by typing in the problem from
homework and clicking on Solve – and step by step solution to my math homework would be ready. I have used it through several math classes - Pre Algebra, Pre Algebra and Pre Algebra. I
highly recommend the program.
Back to top
Varthantel Posted: Wednesday 06th of Oct 07:52
Ok, after hearing so many good things about Algebrator, I think it sure is worth a try. How do I get hold of it ?
Buenos Aires
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Svizes Posted: Thursday 07th of Oct 07:22
Check out this link https://softmath.com/news.html. I hope your math will improve and you will do a great job on the test! Good luck!
From: Slovenia
Back to top | {"url":"https://softmath.com/algebra-software-6/graphing-inequalities-in-excel.html","timestamp":"2024-11-10T17:19:35Z","content_type":"text/html","content_length":"43104","record_id":"<urn:uuid:0ac24c92-26f0-458a-aa8d-697ceb2e055a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00083.warc.gz"} |
Local and global side effects with monad transformers
Local and global side effects with monad transformers
There is an annual puzzle event that 90 or so people attend. Each designs a puzzle and manufactures 90 copies of it which are then shared with the other participants. All 90 then get to go home with
90 puzzles.
Anyway, an ex-coworker attends this event and I had a chance to play with his puzzle. It was one of those fitting blocks together types puzzles. It was ingeniously designed but it seemed pretty clear
to me that it required a lot of combinatorial searching. So I decided to write a program in Haskell to solve it using the List monad to enable simple logic programming.
But my code had a slight problem. It worked through the puzzle and found solutions, but (1) it didn't log the steps it took to achieve those solutions and (2) I couldn't count how many combinations
it searched to find the solutions. Both of these could be thought of as side-effects, so the obvious thing to do is use a monad to track these things. But there was a catch - I was already using a
monad - the List monad. When that happens there's only one thing for it - using a monad transformer to combine monads.
There are two distinct ways to combine a monad with the List monad and I needed both.
Anyway, this is literate Haskell. I'll assume you're vaguely familiar with using the List monad for logic programming. I also assume you're familiar with MonadPlus though that's not something I've
written about here. And I couldn't get this stuff to work in Hugs, so use ghc. Here's some code:
> import Control.Monad.List
> import Control.Monad.State
> import Control.Monad.Writer
I'm not going to describe the original puzzle now. Instead I'm going to look at an almost trivial logic problem so that we can concentrate on the monad transforming. The puzzle is this: find all the
possible sums of pairs of integers, one chosen from the set {1,2} and the other from the set {2,4}, where the sum is less than 5. Here's a simple implementation.
> test1 :: [Integer]
> test1 = do
> a <- [1,2]
> b <- [2,4]
> guard $ a+b<5
> return $ a+b
> go1 = test1
Run go1 and you should get the result [3,4]. But that's just the sums. What were the pairs of integers that went into those sums? We could simply return (a,b,a+b), but in more complex problems we
might want to log a complex sequence of choices and that would entail carrying all of that information around. What we'd like is to simply have some kind of log running as a side effect. For this we
need the Writer monad.
If you cast your mind back,
monad transformers
layer up monads a bit like layers of onion skin. What we want is to wrap a List monad in a Writer monad. We do this using the WriterT monad transformer. All we have to do is add a tell line to our
code, and use 'lift' to pull the items in the list monad out from one layer of onion. Here's what the code looks like:
> test2 :: WriterT [Char] [] Integer
> test2 = do
> a <- lift [1,2]
> b <- lift [2,4]
> tell ("trying "++show a++" "++show b++"\n")
> guard $ a+b<5
> return $ a+b
To get the final result we need to use runWriterT to peel the onion:
> go2 = runWriterT test2
Execute go2 and we get a list of pairs of sums and logged messages.
There's an important point to note here: we have one log per sum, so the logs are 'local'. What if we want a 'global' side effect such as a count of how many combinations were tried, regardless of
whether they succeded or failed? An obvious choice of monad to count attempts is the State monad, but to make its effects 'global' we now need to make State the inner monad and make List provide the
outer layer of skin. We're wrapping the opposite way to in the previous example. And now there's a catch. We use a line like
a <- [1,2]
to exploit the List monad. But now we no longer have a List monad, instead we have a
ListT (State Integer)
monad. This means that
is not an object in this monad. We can't use 'lift' either because the inner monad isn't List. We need to translate our lists into the
ListT (State Integer)
We can do slightly better, we can translate a list from the List monad into any other instance of MonadPlus. Remember that
return x :: [X]
is the same as
x ++ y
is the same as
x `mplus` y :: [X]
. For example
[1,2] == (return 1) `mplus` (return 2)
. The latter only uses functions from the MonadPlus interface to build the list, and hence it can be used to build the equivalent of a List in any MonadPlus. To mplus a whole list we use msum leading
to the definition:
> mlist :: MonadPlus m => [a] -> m a
> mlist = msum . map return
As a function
[a] -> [a]
, mlist is just the identity. Now we're ready to go:
> test3 :: ListT (State Integer) Integer
> test3 = do
> a <- mlist [1,2]
> b <- mlist [2,4]
> lift $ modify (+1)
> guard $ a+b<5
> return $ a+b
> go3 = runState (runListT test3) 0
Run go3 to see the result. Note we had to lift the modify line because the State monad is the inner one.
And now we have one more problem to solve: bouth logging and counting simultaneously:
> test4 :: WriterT [Char] (ListT (State Integer)) Integer
> test4 = do
> a <- lift $ mlist [1,2]
> b <- lift $ mlist [2,4]
> tell ("trying "++show a++" "++show b++"\n")
> lift $ lift $ modify (+1)
> guard $ a+b<5
> return $ a+b
> go4 = runState (runListT $ runWriterT test4) 0
That's it!
We can carry out a cute twist on this. By swapping the innermost and outermost monads we get:
> test5 :: StateT Integer (ListT (Writer [Char])) Integer
> test5 = do
> a <- lift $ mlist [1,2]
> b <- lift $ mlist [2,4]
> lift $ lift $ tell ("trying "++show a++" "++show b++"\n")
> modify (+1)
> guard $ a+b<5
> return $ a+b
> go5 = runWriter $ runListT $ runStateT test5 0
go5 returns a local count of how many combinations were required for
problem, and the Writer monad now records every 'try' in one long log.
One last thing: you don't need to explicitly 'lift' things - the monad transformers have a nice interface that automatically lifts some operations. (You may need a recent Haskell distribution for
this, it fails for older versions.)
> test6 :: WriterT [Char] (ListT (State Integer)) Integer
> test6 = do
> a <- mlist [1,2]
> b <- mlist [2,4]
> tell ("trying "++show a++" "++show b++"\n")
> modify (+1)
> guard $ a+b<5
> return $ a+b
> go6 = runState (runListT $ runWriterT test6) 0
It'd be cool to get rid of the mlist too. Maybe if the Haskell parser was hacked so that [1,2] didn't mean 1:2:[] but instead meant (return 1) `mplus` (return 2) like the way Gofer interprets list
comprehensions in any monad. (For all I know, Gofer already does exactly what I'm suggesting.)
One thing I should add - these monad transformers really kill performance. The puzzle solver I wrote no longer gives me any solutions in the few minutes that it used to...
PS I just made up the mlist thing. There may be a better way of doing this that I don't know about. I was surprised it wasn't already in the Control.Monad library somewhere. mlist is kind of a
homomorphism between MonadPlusses and I think it might make the List MonadPlus an initial object in some category or other - but that's just speculation right now.
Update: I fixed the non-htmlised <'s. I think it takes more time to convert to blogger-compatible html than to write my posts! Also take a look at
. My mlist corresponds to their liftList - and now I know I wasn't completely off the rails writing mlist.
Labels: haskell
4 Comments:
sigfpe said...
One speed-killer is the mlist function which is applied every single time a branch in the logic proram is taken. Also, most lines of code written in monadic style have implicit >>= functions in
them. As you layer up monad transformers the implementation of >>= gets more and more complex.
Hmmm...it's just become harder to proofread this. Not only do I have to read it here, but I also need to view it over at Planet Haskell. I see that the <- 'operators' are coming out differently
over there...
Great post.
I copy/pasted your code, function by function and played around a bit between each step to make sure I understood what was happening. by the time I got to test6 I run into problems. I seem to
only have a half-recent distribution of Haskell (ghc 6.6) because I could only get rid of half of the lifts :-) I still needed to lift the tell twice.
I personally wouldn't worry too much about not being able to remove the lifts. If you use implicit lifts then you can't use two state transformers of the same type simultaneosuly which is pretty
non-orthogononal. So I suggest always using the lifts. | {"url":"http://blog.sigfpe.com/2006/09/local-and-global-side-effects-with.html?m=0","timestamp":"2024-11-14T05:42:51Z","content_type":"application/xhtml+xml","content_length":"27685","record_id":"<urn:uuid:1cd5de49-32e4-4613-8e8a-8d94210b7d30>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00079.warc.gz"} |
Overview of second order effects (concrete column: EC2)
For 'isolated' columns and walls, EN1992-1-1 (EC2) allows for second order effects and member imperfections in a number of ways,
• It specifies a minimum level of member imperfection along with a conservative value - see Clause 5.2 (7).
• It provides for the additional moment due to slenderness (member buckling) using one of two methods. One method (the (Nominal) Stiffness Method) increases the first-order moments in the column
using an amplifier based on the elastic critical buckling load of the member - see Clause 5.8.7.3. The second method (the (Nominal) Curvature Method) calculates the 'second-order' moment directly
based on an adjustment to the maximum predicted curvature that the column section can achieve at failure in bending - see Clause 5.8.8.
• The impact of the slenderness is increased or decreased depending upon the effective length factor for the member. For braced members this will be ≤ 1.0 and for unbraced (bracing) members it will
be ≥ 1.0 see Clause 5.8.3.2.
• Finally (and unrelated to second order effects), EC2 also requires consideration of a minimum moment based on the likelihood that the axial load cannot be fully concentric, see Clause 6.1 (4).
The analysis moment (including the added effects of imperfection and second order effects) should not be less than this value.
Member imperfections (Clause 5.2 (7))
The imperfection moment is calculated using the eccentricity, e[i] = l[0]/400, and it is conservatively assumed that it increases the first-order moments irrespective of sign.
The imperfection moment is added to the analysis moment and if the member is slender a further second moment is added, if this total is < the 6.1(4) moment then the 6.1(4) moment applies.
In the case of the Stiffness Method the imperfection moment is added before the moment magnifier is applied. It is applied to both braced and bracing columns/walls.
Curvature Method (Clause 5.8.8)
This method is only applied to symmetrical, rectangular and circular sections and is equally applicable to columns and walls. The second-order moment, M[2] (= N[Ed] e[2]), is calculated but the
resulting design moment is only used if it is less than that calculated from the Stiffness Method. It is applied in the same manner as that for the Stiffness Method to both braced and bracing
Stiffness Method (Clause 5.8.7)
This method is applied to all columns and walls.
For braced columns the second-order moment M[2] is calculated from:
M[2] = M[e.1] x π^2 /(8 x (N[B] /N[Ed] - 1))
M[e.1] = the maximum first-order moment in the mid-fifth
N[B] = the buckling load of the column based on nominal stiffness and the effective length, hence
N[B] = π^2 EI/l[0]^2
N[Ed] = the maximum axial force in the design length
When a point of zero shear occurs inside the mid-fifth or does not exist in the member length, the value of M[2] is added algebraically to the first-order moments at the ends but only if this
increases the first-order moment. At the mid-fifth position M[2] is always "added" in such a way as to increase the first-order mid-fifth moment.
When a point of zero shear occurs within the member length and is outside the mid-fifth, the second-order moments is taken as the greater of that calculated as above and that calculated as per Clause
5.8.7.3 (4) by multiplying all first-order moments by the amplifier,
1/(1 - N[Ed]/N[B])
For bracing columns the second-order moments are calculated in the same way as braced columns except that in the determination of the amplifier, the buckling load is based on bracing effective
lengths. These are greater than 1.0L and hence produce more severe amplifiers.
Second-order analysis
When second-order analysis is selected then both braced and bracing columns are treated the same as if first-order analysis were selected. If the second-order analysis is either the amplified forces
method or the rigorous method then this approach will double count some of the global P-Δ effects in columns that are determined as having significant lateral loads. Also, when it is a rigorous
second-order analysis there is some double counting of member P-δ effects in both braced and bracing columns.
Because global second order effects are introduced when second order analysis is selected, it is logical to set all members to "braced" so that only additional effects due to member slenderness are
Minimum moment (Clause 6.1 (4))
The minimum moment about each axis, M[min] is calculated from Clause 6.1 (4).
For the specific circumstance of the moments from analysis being < the minimum moment for both axes, design codes generally require that the minimum moment need only be considered acting about one
axis at a time. In Tekla Structural Designer the behaviour in this situaton is as follows:
• For any section that is not circular or rectangular, the minimum moment is always applied about both axes, the rationale being:
□ We believe the logical intention behind the requirement is that a cross section should be able to resist a minimum moment in every direction. However, if the local X and Y axes of a member
are not aligned to the strongest and weakest axes (e.g. for an L-shape section) then applying the minimum in only one of the X or Y directions will not guarantee meeting the intention. Hence
we consider applying the moment about both axes together for this contingency is a safe and conservative approach.
• For rectangular (non-square) sections, the minimum moment is applied in the weaker direction.
• For square and circular sections, the minimum moment is applied in the direction with the smaller analysis moment. | {"url":"https://support.tekla.com/doc/tekla-structural-designer/2024/ref_overviewofsecondordereffectscolumnsec2","timestamp":"2024-11-03T18:26:46Z","content_type":"text/html","content_length":"60566","record_id":"<urn:uuid:688e18a5-1b8d-4082-b570-f1787cfc9745>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00865.warc.gz"} |
Conclusions for Research
Full text: Conclusions for Research
advocated by Roncaglia. As an example I mention the input-output
models. They assume prices as given and deal with some
problems of quantities, say a multiplier analysis, on this
bssis. Or they start from a given final output, as Sraffa
does, and deal with prices and distribution. Again, all
Kaleckian economics always work with given and constant prices
and operate on the deflated values of macro-economic
variables. In this sense, Kalecki certainly was a classicist!
The simultaneous treatment of quantities and prices involves
non-linearities: The flex price relations are typically
non-linear ( agricultural and other primary commodities)
bottle-necks generally involve non-linearities. For an
abstract equilibrium theory this involves no trouble.
But for an applied economist such simultaneous equation systems
are not very helpful. He has to start from given initial
conditions and work out the process as it evolves in time,
for example by means of difference equations, j This is
what Paul Davidson fimpjifes when he says that economic processes
are not ergodic, that is, they do not lead to a steady state
independent of initial conditions ( I should add that even
if they are ergodic they do not converge quickly so that
they never get old enough, in practice, to reach the steady
state, being again and again interrupted by disturbances from
outside). Now if you try to work with difference or other
functional equations non-linearity will not be easy to deal
with. I have no recipe to offer, unfortunately.
In the field of micro-economics ( information approach )
Streissler has haitiLy given us much encouragement to pursue | {"url":"https://viewer.wu.ac.at/viewer/fulltext/AC14446196/2/","timestamp":"2024-11-10T07:53:41Z","content_type":"application/xhtml+xml","content_length":"67495","record_id":"<urn:uuid:bceb3cd8-29d1-48e5-a910-a7d016cc0317>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00577.warc.gz"} |
One-way analysis of variance - ANOVA - in Excel data analysis add in - Deyako
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One-way analysis of variance – ANOVA – in Excel data analysis add in
Analysis of variance is a statistical method for determine if a variable has any effects on another variable. when we want to find out the relationship between two variables, we need to conduct an
experiment in which, we have to change amounts in one variable and then measuring the resulted amount in another one. If we see meaningful changes in the average of resulted amounts, then we can
conclude that the first variable has effects on the second one. The method is called, analysis of variance, in this video we show how we can conduct the one-way ANOVA in the Excel by using Data
analysis add ins
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Linear Equations
Definition of Slope The steepness of a line in the coordinate plane is called its slope. It is defined as the ratio of the rise, or vertical change in y, to the run, or horizontal change as
you move from one point to the other.
Determining Slope Given Two Given the coordinates of two points, ( x[1], y[1]) and ( x[2], y[2]), on a line, the slope m of the line can be found as follows.
What is the slope of the line that passes through (4, -6) and (-2, 3)?
Let x[1] = 4, y[1 ]= -6, x[2 ]= -2, and y[2 ]= 3.
Writing Linear Equations in Point-Slope and Standard Forms
For a given point ( x[1], y[1]) on a nonvertical line having slope of m, the point-slope form a linear equation is as follows: y - y[1]= m( x - x[1]).
Point-Slope Form of a Linear Equation
The linear equation of a vertical line, which has an undefined slope, through a point ( x[1], y[1]) is x = x[1].
Standard Form The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, A
Write the equation, first in point-slope form and then in standard form, of the line that passes through (2, 3) and has a slope of 5.
Point Slope Form y - y[1 ]= m( x - x[1])
y - 3 = 5( x - 2)
y - 3 = 5x - 10 Distribute.
5x - 10 = y - 3 Reflexive Property ( = )
5x - y = 7 Add 10 and subtract y from each side.
Standard Form: 5x - y = 7, where A = 5, B = -1 and C = 7
Writing Linear Equations in Slope-Intercept Form
The coordinates at which a graph intersects the axes are known as the x-intercept and the y-intercept.
Finding Intercepts To find the x-intercept, substitute 0 for y in the equation and solve for x. To find the y-intercept, substitute 0 for x in the equation and solve for y.
Slope-Intercept Form of a Linear Equation If a line has a slope of m and a y-intercept of b, then the slope-intercept form of an equation of the line is y = mx + b.
Find the x - and y -intercepts of the graph of 2x + 3y = 5. Then, write the equation in slope-intercept form.
2x + 3(0) = 5 Let y = 0 2(0) + 3y = 5 Let x = 0
2x = 5 3y = 5
The x-intercept is The y-intercept is
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Change Log
1. Added a new function estimate_ipwhr which enables estimation of counterfactual hazard ratios with inverse probability of treatment weighting and inverse probability of missingness weighting.
Objects produced by this function have class hr.
2. Added a new vignette called Hazard ratios describing the theory and usage of estimate_ipwhr.
3. Provided a new is.hr function to check whether an object inherits from the hr class.
4. Added make_table2, forest_plot and plot methods for the hr class.
5. Provided a new hr_data function, which takes results of estimate_ipwhr and returns a data frame with information backing the make_table2.hr and plot.hr methods.
6. Added a new inspect_ipw_weights function that produces the per-subject weights for ipw objects.
7. Exposed the boot_method formal argument to the existing make_table2, plot, and forest_plot methods.
8. estimate_ipwrisk now produces per-subject weights (with the intention that they will be accessed by the user via inspect_ipw_weights).
9. compute_cumrisk_effect_ci throws an error if an input of "log-normal" for boot_method is used with any input for effect_measure_type other than "RR" or "CR" since otherwise it can result in the
logarithm function being evaluated for a non-positive values.
10. compute_cumrisk_effect_ci throws an error if any input for boot_method is provided other than "normal" or "log-normal" rather than returning a data frame without variance information.
11. Fixed a bug in compute_cumcount_effect_ci that would result in an unintended error when using a "log-normal" input for boot_method.
12. make_table2.cumrisk now throws an error if any input for boot_method is provided other than "normal" or "log-normal".
13. knitr and forcats are each removed from the list of Imports. knitr and rmarkdown are each added to the list of Suggests. The package lower bound for gt was changed from v0.5 to v0.6.
14. Added testing for all new routines.
15. Added testing of estimate_ipwhr per-subject weights.
16. Added regression tests for estimate_ipwrisk to ensure that newer versions of the function produce the same cumulative risk results and variances as for v0.39.06.
17. Added testing of boot_method input options for make_table2.cumrisk, make_table2.cumcount, plot.cumrisk, plot.cumcount, and forest_plot.cumrisk.
18. Added an assertion that subject IDs are unique unless time-varying covariates specified. That is, across all estimators, prior to computing the estimates there is a check that there are no
duplicated subject IDs. Duplicated subject IDs in the source data are only allowed if identify_interval is used to indicate time-varying data. | {"url":"https://docs.novisci.com/causalRisk/articles/changelog.html","timestamp":"2024-11-12T02:39:31Z","content_type":"text/html","content_length":"25917","record_id":"<urn:uuid:d5293003-1f2e-4938-8d1b-02525c1bb131>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00132.warc.gz"} |
Order Of Operations Sheet 5 2 Answers 6th Grade Worksheets Pemdas | Order of Operation Worksheets
Order Of Operations Sheet 5 2 Answers 6th Grade Worksheets Pemdas
Order Of Operations Sheet 5 2 Answers 6th Grade Worksheets Pemdas
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Time Series Forecast Study with Python: Monthly Sales of French Champagne - MachineLearningMastery.comTime Series Forecast Study with Python: Monthly Sales of French Champagne - MachineLearningMastery.com
Time series forecasting is a process, and the only way to get good forecasts is to practice this process.
In this tutorial, you will discover how to forecast the monthly sales of French champagne with Python.
Working through this tutorial will provide you with a framework for the steps and the tools for working through your own time series forecasting problems.
After completing this tutorial, you will know:
• How to confirm your Python environment and carefully define a time series forecasting problem.
• How to create a test harness for evaluating models, develop a baseline forecast, and better understand your problem with the tools of time series analysis.
• How to develop an autoregressive integrated moving average model, save it to file, and later load it to make predictions for new time steps.
Kick-start your project with my new book Time Series Forecasting With Python, including step-by-step tutorials and the Python source code files for all examples.
Let’s get started.
• Updated Mar/2017: Fixed a typo in the code example.
• Updated Apr/2019: Updated the link to dataset.
• Updated Aug/2019: Updated data loading and date grouping to use new API.
• Updated Feb/2020: Updated to_csv() to remove warnings.
• Updated Feb/2020: Fixed data preparation and loading.
• Updated May/2020: Fixed small typo when making a prediction.
• Updated Dec/2020: Updated modeling for changes to the API.
In this tutorial, we will work through a time series forecasting project from end-to-end, from downloading the dataset and defining the problem to training a final model and making predictions.
This project is not exhaustive, but shows how you can get good results quickly by working through a time series forecasting problem systematically.
The steps of this project that we will through are as follows.
1. Environment.
2. Problem Description.
3. Test Harness.
4. Persistence.
5. Data Analysis.
6. ARIMA Models.
7. Model Validation.
This will provide a template for working through a time series prediction problem that you can use on your own dataset.
Stop learning Time Series Forecasting the slow way!
Take my free 7-day email course and discover how to get started (with sample code).
Click to sign-up and also get a free PDF Ebook version of the course.
1. Environment
This tutorial assumes an installed and working SciPy environment and dependencies, including:
• SciPy
• NumPy
• Matplotlib
• Pandas
• scikit-learn
• statsmodels
If you need help installing Python and the SciPy environment on your workstation, consider the Anaconda distribution that manages much of it for you.
This script will help you check your installed versions of these libraries.
1 # check the versions of key python libraries
2 # scipy
3 import scipy
4 print('scipy: %s' % scipy.__version__)
5 # numpy
6 import numpy
7 print('numpy: %s' % numpy.__version__)
8 # matplotlib
9 import matplotlib
10 print('matplotlib: %s' % matplotlib.__version__)
11 # pandas
12 import pandas
13 print('pandas: %s' % pandas.__version__)
14 # statsmodels
15 import statsmodels
16 print('statsmodels: %s' % statsmodels.__version__)
17 # scikit-learn
18 import sklearn
19 print('sklearn: %s' % sklearn.__version__)
The results on my workstation used to write this tutorial are as follows:
1 scipy: 1.5.4
2 numpy: 1.18.5
3 matplotlib: 3.3.3
4 pandas: 1.1.4
5 statsmodels: 0.12.1
6 sklearn: 0.23.2
2. Problem Description
The problem is to predict the number of monthly sales of champagne for the Perrin Freres label (named for a region in France).
The dataset provides the number of monthly sales of champagne from January 1964 to September 1972, or just under 10 years of data.
The values are a count of millions of sales and there are 105 observations.
The dataset is credited to Makridakis and Wheelwright, 1989.
Download the dataset as a CSV file and place it in your current working directory with the filename “champagne.csv“.
3. Test Harness
We must develop a test harness to investigate the data and evaluate candidate models.
This involves two steps:
1. Defining a Validation Dataset.
2. Developing a Method for Model Evaluation.
3.1 Validation Dataset
The dataset is not current. This means that we cannot easily collect updated data to validate the model.
Therefore we will pretend that it is September 1971 and withhold the last one year of data from analysis and model selection.
This final year of data will be used to validate the final model.
The code below will load the dataset as a Pandas Series and split into two, one for model development (dataset.csv) and the other for validation (validation.csv).
1 # separate out a validation dataset
2 from pandas import read_csv
3 series = read_csv('champagne.csv', header=0, index_col=0, parse_dates=True, squeeze=True)
4 split_point = len(series) - 12
5 dataset, validation = series[0:split_point], series[split_point:]
6 print('Dataset %d, Validation %d' % (len(dataset), len(validation)))
7 dataset.to_csv('dataset.csv', header=False)
8 validation.to_csv('validation.csv', header=False)
Running the example creates two files and prints the number of observations in each.
1 Dataset 93, Validation 12
The specific contents of these files are:
• dataset.csv: Observations from January 1964 to September 1971 (93 observations)
• validation.csv: Observations from October 1971 to September 1972 (12 observations)
The validation dataset is about 11% of the original dataset.
Note that the saved datasets do not have a header line, therefore we do not need to cater for this when working with these files later.
3.2. Model Evaluation
Model evaluation will only be performed on the data in dataset.csv prepared in the previous section.
Model evaluation involves two elements:
1. Performance Measure.
2. Test Strategy.
3.2.1 Performance Measure
The observations are a count of champagne sales in millions of units.
We will evaluate the performance of predictions using the root mean squared error (RMSE). This will give more weight to predictions that are grossly wrong and will have the same units as the original
Any transforms to the data must be reversed before the RMSE is calculated and reported to make the performance between different methods directly comparable.
We can calculate the RMSE using the helper function from the scikit-learn library mean_squared_error() that calculates the mean squared error between a list of expected values (the test set) and the
list of predictions. We can then take the square root of this value to give us an RMSE score.
For example:
1 from sklearn.metrics import mean_squared_error
2 from math import sqrt
3 ...
4 test = ...
5 predictions = ...
6 mse = mean_squared_error(test, predictions)
7 rmse = sqrt(mse)
8 print('RMSE: %.3f' % rmse)
3.2.2 Test Strategy
Candidate models will be evaluated using walk-forward validation.
This is because a rolling-forecast type model is required from the problem definition. This is where one-step forecasts are needed given all available data.
The walk-forward validation will work as follows:
• The first 50% of the dataset will be held back to train the model.
• The remaining 50% of the dataset will be iterated and test the model.
• For each step in the test dataset:
□ A model will be trained.
□ A one-step prediction made and the prediction stored for later evaluation.
□ The actual observation from the test dataset will be added to the training dataset for the next iteration.
• The predictions made during the iteration of the test dataset will be evaluated and an RMSE score reported.
Given the small size of the data, we will allow a model to be re-trained given all available data prior to each prediction.
We can write the code for the test harness using simple NumPy and Python code.
Firstly, we can split the dataset into train and test sets directly. We’re careful to always convert a loaded dataset to float32 in case the loaded data still has some String or Integer data types.
1 # prepare data
2 X = series.values
3 X = X.astype('float32')
4 train_size = int(len(X) * 0.50)
5 train, test = X[0:train_size], X[train_size:]
Next, we can iterate over the time steps in the test dataset. The train dataset is stored in a Python list as we need to easily append a new observation each iteration and NumPy array concatenation
feels like overkill.
The prediction made by the model is called yhat for convention, as the outcome or observation is referred to as y and yhat (a ‘y‘ with a mark above) is the mathematical notation for the prediction of
the y variable.
The prediction and observation are printed each observation for a sanity check prediction in case there are issues with the model.
1 # walk-forward validation
2 history = [x for x in train]
3 predictions = list()
4 for i in range(len(test)):
5 # predict
6 yhat = ...
7 predictions.append(yhat)
8 # observation
9 obs = test[i]
10 history.append(obs)
11 print('>Predicted=%.3f, Expected=%3.f' % (yhat, obs))
4. Persistence
The first step before getting bogged down in data analysis and modeling is to establish a baseline of performance.
This will provide both a template for evaluating models using the proposed test harness and a performance measure by which all more elaborate predictive models can be compared.
The baseline prediction for time series forecasting is called the naive forecast, or persistence.
This is where the observation from the previous time step is used as the prediction for the observation at the next time step.
We can plug this directly into the test harness defined in the previous section.
The complete code listing is provided below.
1 from pandas import read_csv
2 from sklearn.metrics import mean_squared_error
3 from math import sqrt
4 # load data
5 series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
6 # prepare data
7 X = series.values
8 X = X.astype('float32')
9 train_size = int(len(X) * 0.50)
10 train, test = X[0:train_size], X[train_size:]
11 # walk-forward validation
12 history = [x for x in train]
13 predictions = list()
14 for i in range(len(test)):
15 # predict
16 yhat = history[-1]
17 predictions.append(yhat)
18 # observation
19 obs = test[i]
20 history.append(obs)
21 print('>Predicted=%.3f, Expected=%3.f' % (yhat, obs))
22 # report performance
23 mse = mean_squared_error(test, predictions)
24 rmse = sqrt(mse)
25 print('RMSE: %.3f' % rmse)
Running the test harness prints the prediction and observation for each iteration of the test dataset.
The example ends by printing the RMSE for the model.
In this case, we can see that the persistence model achieved an RMSE of 3186.501. This means that on average, the model was wrong by about 3,186 million sales for each prediction made.
1 ...
2 >Predicted=4676.000, Expected=5010
3 >Predicted=5010.000, Expected=4874
4 >Predicted=4874.000, Expected=4633
5 >Predicted=4633.000, Expected=1659
6 >Predicted=1659.000, Expected=5951
7 RMSE: 3186.501
We now have a baseline prediction method and performance; now we can start digging into our data.
5. Data Analysis
We can use summary statistics and plots of the data to quickly learn more about the structure of the prediction problem.
In this section, we will look at the data from five perspectives:
1. Summary Statistics.
2. Line Plot.
3. Seasonal Line Plots
4. Density Plots.
5. Box and Whisker Plot.
5.1 Summary Statistics
Summary statistics provide a quick look at the limits of observed values. It can help to get a quick idea of what we are working with.
The example below calculates and prints summary statistics for the time series.
1 from pandas import read_csv
2 series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
3 print(series.describe())
Running the example provides a number of summary statistics to review.
Some observations from these statistics include:
• The number of observations (count) matches our expectation, meaning we are handling the data correctly.
• The mean is about 4,641, which we might consider our level in this series.
• The standard deviation (average spread from the mean) is relatively large at 2,486 sales.
• The percentiles along with the standard deviation do suggest a large spread to the data.
1 count 93.000000
2 mean 4641.118280
3 std 2486.403841
4 min 1573.000000
5 25% 3036.000000
6 50% 4016.000000
7 75% 5048.000000
8 max 13916.000000
5.2 Line Plot
A line plot of a time series can provide a lot of insight into the problem.
The example below creates and shows a line plot of the dataset.
1 from pandas import read_csv
2 from matplotlib import pyplot
3 series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
4 series.plot()
5 pyplot.show()
Run the example and review the plot. Note any obvious temporal structures in the series.
Some observations from the plot include:
• There may be an increasing trend of sales over time.
• There appears to be systematic seasonality to the sales for each year.
• The seasonal signal appears to be growing over time, suggesting a multiplicative relationship (increasing change).
• There do not appear to be any obvious outliers.
• The seasonality suggests that the series is almost certainly non-stationary.
There may be benefit in explicitly modeling the seasonal component and removing it. You may also explore using differencing with one or two levels in order to make the series stationary.
The increasing trend or growth in the seasonal component may suggest the use of a log or other power transform.
5.3 Seasonal Line Plots
We can confirm the assumption that the seasonality is a yearly cycle by eyeballing line plots of the dataset by year.
The example below takes the 7 full years of data as separate groups and creates one line plot for each. The line plots are aligned vertically to help spot any year-to-year pattern.
1 from pandas import read_csv
2 from pandas import DataFrame
3 from pandas import Grouper
4 from matplotlib import pyplot
5 series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
6 groups = series['1964':'1970'].groupby(Grouper(freq='A'))
7 years = DataFrame()
8 pyplot.figure()
9 i = 1
10 n_groups = len(groups)
11 for name, group in groups:
12 pyplot.subplot((n_groups*100) + 10 + i)
13 i += 1
14 pyplot.plot(group)
15 pyplot.show()
Running the example creates the stack of 7 line plots.
We can clearly see a dip each August and a rise from each August to December. This pattern appears the same each year, although at different levels.
This will help with any explicitly season-based modeling later.
It might have been easier if all season line plots were added to the one graph to help contrast the data for each year.
5.4 Density Plot
Reviewing plots of the density of observations can provide further insight into the structure of the data.
The example below creates a histogram and density plot of the observations without any temporal structure.
1 from pandas import read_csv
2 from matplotlib import pyplot
3 series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
4 pyplot.figure(1)
5 pyplot.subplot(211)
6 series.hist()
7 pyplot.subplot(212)
8 series.plot(kind='kde')
9 pyplot.show()
Run the example and review the plots.
Some observations from the plots include:
• The distribution is not Gaussian.
• The shape has a long right tail and may suggest an exponential distribution
This lends more support to exploring some power transforms of the data prior to modeling.
5.5 Box and Whisker Plots
We can group the monthly data by year and get an idea of the spread of observations for each year and how this may be changing.
We do expect to see some trend (increasing mean or median), but it may be interesting to see how the rest of the distribution may be changing.
The example below groups the observations by year and creates one box and whisker plot for each year of observations. The last year (1971) only contains 9 months and may not be a useful comparison
with the 12 months of observations for other years. Therefore, only data between 1964 and 1970 was plotted.
1 from pandas import read_csv
2 from pandas import DataFrame
3 from pandas import Grouper
4 from matplotlib import pyplot
5 series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
6 groups = series['1964':'1970'].groupby(Grouper(freq='A'))
7 years = DataFrame()
8 for name, group in groups:
9 years[name.year] = group.values
10 years.boxplot()
11 pyplot.show()
Running the example creates 7 box and whisker plots side-by-side, one for each of the 7 years of selected data.
Some observations from reviewing the plots include:
• The median values for each year (red line) may show an increasing trend.
• The spread or middle 50% of the data (blue boxes) does appear reasonably stable.
• There are outliers each year (black crosses); these may be the tops or bottoms of the seasonal cycle.
• The last year, 1970, does look different from the trend in prior years
The observations suggest perhaps some growth trend over the years and outliers that may be a part of the seasonal cycle.
This yearly view of the data is an interesting avenue and could be pursued further by looking at summary statistics from year-to-year and changes in summary stats from year-to-year.
6. ARIMA Models
In this section, we will develop Autoregressive Integrated Moving Average, or ARIMA, models for the problem.
We will approach modeling by both manual and automatic configuration of the ARIMA model. This will be followed by a third step of investigating the residual errors of the chosen model.
As such, this section is broken down into 3 steps:
1. Manually Configure the ARIMA.
2. Automatically Configure the ARIMA.
3. Review Residual Errors.
6.1 Manually Configured ARIMA
The ARIMA(p,d,q) model requires three parameters and is traditionally configured manually.
Analysis of the time series data assumes that we are working with a stationary time series.
The time series is almost certainly non-stationary. We can make it stationary this by first differencing the series and using a statistical test to confirm that the result is stationary.
The seasonality in the series is seemingly year-to-year. Seasonal data can be differenced by subtracting the observation from the same time in the previous cycle, in this case the same month in the
previous year. This does mean that we will lose the first year of observations as there is no prior year to difference with.
The example below creates a deseasonalized version of the series and saves it to file stationary.csv.
2 from pandas import read_csv
3 from pandas import Series
4 from statsmodels.tsa.stattools import adfuller
5 from matplotlib import pyplot
6 # create a differenced series
7 def difference(dataset, interval=1):
8 diff = list()
9 for i in range(interval, len(dataset)):
10 value = dataset[i] - dataset[i - interval]
11 diff.append(value)
12 return Series(diff)
13 series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
14 X = series.values
15 X = X.astype('float32')
16 # difference data
17 months_in_year = 12
18 stationary = difference(X, months_in_year)
19 stationary.index = series.index[months_in_year:]
20 # check if stationary
21 result = adfuller(stationary)
22 print('ADF Statistic: %f' % result[0])
23 print('p-value: %f' % result[1])
24 print('Critical Values:')
25 for key, value in result[4].items():
26 print('\t%s: %.3f' % (key, value))
27 # save
28 stationary.to_csv('stationary.csv', header=False)
29 # plot
30 stationary.plot()
31 pyplot.show()
Running the example outputs the result of a statistical significance test of whether the differenced series is stationary. Specifically, the augmented Dickey-Fuller test.
The results show that the test statistic value -7.134898 is smaller than the critical value at 1% of -3.515. This suggests that we can reject the null hypothesis with a significance level of less
than 1% (i.e. a low probability that the result is a statistical fluke).
Rejecting the null hypothesis means that the process has no unit root, and in turn that the time series is stationary or does not have time-dependent structure.
1 ADF Statistic: -7.134898
2 p-value: 0.000000
3 Critical Values:
4 5%: -2.898
5 1%: -3.515
6 10%: -2.586
For reference, the seasonal difference operation can be inverted by adding the observation for the same month the year before. This is needed in the case that predictions are made by a model fit on
seasonally differenced data. The function to invert the seasonal difference operation is listed below for completeness.
1 # invert differenced value
2 def inverse_difference(history, yhat, interval=1):
3 return yhat + history[-interval]
A plot of the differenced dataset is also created.
The plot does not show any obvious seasonality or trend, suggesting the seasonally differenced dataset is a good starting point for modeling.
We will use this dataset as an input to the ARIMA model. It also suggests that no further differencing may be required, and that the d parameter may be set to 0.
The next first step is to select the lag values for the Autoregression (AR) and Moving Average (MA) parameters, p and q respectively.
We can do this by reviewing Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots.
Note, we are now using the seasonally differenced stationary.csv as our dataset.
The example below creates ACF and PACF plots for the series.
1 from pandas import read_csv
2 from statsmodels.graphics.tsaplots import plot_acf
3 from statsmodels.graphics.tsaplots import plot_pacf
4 from matplotlib import pyplot
5 series = read_csv('stationary.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
6 pyplot.figure()
7 pyplot.subplot(211)
8 plot_acf(series, ax=pyplot.gca())
9 pyplot.subplot(212)
10 plot_pacf(series, ax=pyplot.gca())
11 pyplot.show()
Run the example and review the plots for insights into how to set the p and q variables for the ARIMA model.
Below are some observations from the plots.
• The ACF shows a significant lag for 1 month.
• The PACF shows a significant lag for 1 month, with perhaps some significant lag at 12 and 13 months.
• Both the ACF and PACF show a drop-off at the same point, perhaps suggesting a mix of AR and MA.
A good starting point for the p and q values is also 1.
The PACF plot also suggests that there is still some seasonality present in the differenced data.
We may consider a better model of seasonality, such as modeling it directly and explicitly removing it from the model rather than seasonal differencing.
This quick analysis suggests an ARIMA(1,0,1) on the stationary data may be a good starting point.
The historic observations will be seasonally differenced prior to the fitting of each ARIMA model. The differencing will be inverted for all predictions made to make them directly comparable to the
expected observation in the original sale count units.
Experimentation shows that this configuration of ARIMA does not converge and results in errors by the underlying library. Further experimentation showed that adding one level of differencing to the
stationary data made the model more stable. The model can be extended to ARIMA(1,1,1).
We will also disable the automatic addition of a trend constant from the model by setting the ‘trend‘ argument to ‘nc‘ for no constant in the call to fit(). From experimentation, I find that this can
result in better forecast performance on some problems.
The example below demonstrates the performance of this ARIMA model on the test harness.
# evaluate manually configured ARIMA model
from pandas import read_csv
from sklearn.metrics import mean_squared_error
from statsmodels.tsa.arima.model import ARIMA
from math import sqrt
# create a differenced series
def difference(dataset, interval=1):
diff = list()
for i in range(interval, len(dataset)):
value = dataset[i] - dataset[i - interval]
return diff
# invert differenced value
def inverse_difference(history, yhat, interval=1):
return yhat + history[-interval]
# load data
series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
# prepare data
X = series.values
X = X.astype('float32')
train_size = int(len(X) * 0.50)
train, test = X[0:train_size], X[train_size:]
# walk-forward validation
history = [x for x in train]
predictions = list()
for i in range(len(test)):
# difference data
months_in_year = 12
diff = difference(history, months_in_year)
# predict
model = ARIMA(diff, order=(1,1,1))
model_fit = model.fit()
yhat = model_fit.forecast()[0]
yhat = inverse_difference(history, yhat, months_in_year)
# observation
obs = test[i]
print('>Predicted=%.3f, Expected=%.3f' % (yhat, obs))
# report performance
rmse = sqrt(mean_squared_error(test, predictions))
print('RMSE: %.3f' % rmse)
Note, you may see a warning message from the underlying linear algebra library; this can be ignored for now.
Running this example results in an RMSE of 956.942, which is dramatically better than the persistence RMSE of 3186.501.
1 ...
2 >Predicted=3157.018, Expected=5010
3 >Predicted=4615.082, Expected=4874
4 >Predicted=4624.998, Expected=4633
5 >Predicted=2044.097, Expected=1659
6 >Predicted=5404.428, Expected=5951
7 RMSE: 956.942
This is a great start, but we may be able to get improved results with a better configured ARIMA model.
6.2 Grid Search ARIMA Hyperparameters
The ACF and PACF plots suggest that an ARIMA(1,0,1) or similar may be the best that we can do.
To confirm this analysis, we can grid search a suite of ARIMA hyperparameters and check that no models result in better out of sample RMSE performance.
In this section, we will search values of p, d, and q for combinations (skipping those that fail to converge), and find the combination that results in the best performance on the test set. We will
use a grid search to explore all combinations in a subset of integer values.
Specifically, we will search all combinations of the following parameters:
• p: 0 to 6.
• d: 0 to 2.
• q: 0 to 6.
This is (7 * 3 * 7), or 147, potential runs of the test harness and will take some time to execute.
It may be interesting to evaluate MA models with a lag of 12 or 13 as were noticed as potentially interesting from reviewing the ACF and PACF plots. Experimentation suggested that these models may
not be stable, resulting in errors in the underlying mathematical libraries.
The complete worked example with the grid search version of the test harness is listed below.
# grid search ARIMA parameters for time series
import warnings
from pandas import read_csv
from statsmodels.tsa.arima.model import ARIMA
from sklearn.metrics import mean_squared_error
from math import sqrt
import numpy
# create a differenced series
def difference(dataset, interval=1):
diff = list()
for i in range(interval, len(dataset)):
value = dataset[i] - dataset[i - interval]
return numpy.array(diff)
# invert differenced value
def inverse_difference(history, yhat, interval=1):
return yhat + history[-interval]
# evaluate an ARIMA model for a given order (p,d,q) and return RMSE
def evaluate_arima_model(X, arima_order):
# prepare training dataset
X = X.astype('float32')
train_size = int(len(X) * 0.50)
train, test = X[0:train_size], X[train_size:]
history = [x for x in train]
# make predictions
predictions = list()
for t in range(len(test)):
# difference data
months_in_year = 12
diff = difference(history, months_in_year)
model = ARIMA(diff, order=arima_order)
model_fit = model.fit()
yhat = model_fit.forecast()[0]
yhat = inverse_difference(history, yhat, months_in_year)
# calculate out of sample error
rmse = sqrt(mean_squared_error(test, predictions))
return rmse
# evaluate combinations of p, d and q values for an ARIMA model
def evaluate_models(dataset, p_values, d_values, q_values):
dataset = dataset.astype('float32')
best_score, best_cfg = float("inf"), None
for p in p_values:
for d in d_values:
for q in q_values:
order = (p,d,q)
rmse = evaluate_arima_model(dataset, order)
if rmse < best_score:
best_score, best_cfg = rmse, order
print('ARIMA%s RMSE=%.3f' % (order,rmse))
print('Best ARIMA%s RMSE=%.3f' % (best_cfg, best_score))
# load dataset
series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
# evaluate parameters
p_values = range(0, 7)
d_values = range(0, 3)
q_values = range(0, 7)
evaluate_models(series.values, p_values, d_values, q_values)
Running the example runs through all combinations and reports the results on those that converge without error. The example takes a little over 2 minutes to run on modern hardware.
The results show that the best configuration discovered was ARIMA(0, 0, 1) with an RMSE of 939.464, slightly lower than the manually configured ARIMA from the previous section. This difference may or
may not be statistically significant.
1 ...
2 ARIMA(5, 1, 2) RMSE=1003.200
3 ARIMA(5, 2, 1) RMSE=1053.728
4 ARIMA(6, 0, 0) RMSE=996.466
5 ARIMA(6, 1, 0) RMSE=1018.211
6 ARIMA(6, 1, 1) RMSE=1023.762
7 Best ARIMA(0, 0, 1) RMSE=939.464
We will select this ARIMA(0, 0, 1) model going forward.
6.3 Review Residual Errors
A good final check of a model is to review residual forecast errors.
Ideally, the distribution of residual errors should be a Gaussian with a zero mean.
We can check this by using summary statistics and plots to investigate the residual errors from the ARIMA(0, 0, 1) model. The example below calculates and summarizes the residual forecast errors.
# summarize ARIMA forecast residuals
from pandas import read_csv
from pandas import DataFrame
from statsmodels.tsa.arima.model import ARIMA
from matplotlib import pyplot
# create a differenced series
def difference(dataset, interval=1):
diff = list()
for i in range(interval, len(dataset)):
value = dataset[i] - dataset[i - interval]
return diff
# invert differenced value
def inverse_difference(history, yhat, interval=1):
return yhat + history[-interval]
# load data
series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
# prepare data
X = series.values
X = X.astype('float32')
train_size = int(len(X) * 0.50)
train, test = X[0:train_size], X[train_size:]
# walk-forward validation
history = [x for x in train]
predictions = list()
for i in range(len(test)):
# difference data
months_in_year = 12
diff = difference(history, months_in_year)
# predict
model = ARIMA(diff, order=(0,0,1))
model_fit = model.fit()
yhat = model_fit.forecast()[0]
yhat = inverse_difference(history, yhat, months_in_year)
# observation
obs = test[i]
# errors
residuals = [test[i]-predictions[i] for i in range(len(test))]
residuals = DataFrame(residuals)
# plot
residuals.plot(kind='kde', ax=pyplot.gca())
Running the example first describes the distribution of the residuals.
We can see that the distribution has a right shift and that the mean is non-zero at 165.904728.
This is perhaps a sign that the predictions are biased.
1 count 47.000000
2 mean 165.904728
3 std 934.696199
4 min -2164.247449
5 25% -289.651596
6 50% 191.759548
7 75% 732.992187
8 max 2367.304748
The distribution of residual errors is also plotted.
The graphs suggest a Gaussian-like distribution with a bumpy left tail, providing further evidence that perhaps a power transform might be worth exploring.
We could use this information to bias-correct predictions by adding the mean residual error of 165.904728 to each forecast made.
The example below performs this bias correlation.
# plots of residual errors of bias corrected forecasts
from pandas import read_csv
from pandas import DataFrame
from statsmodels.tsa.arima.model import ARIMA
from matplotlib import pyplot
from sklearn.metrics import mean_squared_error
from math import sqrt
# create a differenced series
def difference(dataset, interval=1):
diff = list()
for i in range(interval, len(dataset)):
value = dataset[i] - dataset[i - interval]
return diff
# invert differenced value
def inverse_difference(history, yhat, interval=1):
return yhat + history[-interval]
# load data
series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
# prepare data
X = series.values
X = X.astype('float32')
train_size = int(len(X) * 0.50)
train, test = X[0:train_size], X[train_size:]
# walk-forward validation
history = [x for x in train]
predictions = list()
bias = 165.904728
for i in range(len(test)):
# difference data
months_in_year = 12
diff = difference(history, months_in_year)
# predict
model = ARIMA(diff, order=(0,0,1))
model_fit = model.fit()
yhat = model_fit.forecast()[0]
yhat = bias + inverse_difference(history, yhat, months_in_year)
# observation
obs = test[i]
# report performance
rmse = sqrt(mean_squared_error(test, predictions))
print('RMSE: %.3f' % rmse)
# errors
residuals = [test[i]-predictions[i] for i in range(len(test))]
residuals = DataFrame(residuals)
# plot
residuals.plot(kind='kde', ax=pyplot.gca())
The performance of the predictions is improved very slightly from 939.464 to 924.699, which may or may not be significant.
The summary of the forecast residual errors shows that the mean was indeed moved to a value very close to zero.
RMSE: 924.699
count 4.700000e+01
mean 4.965016e-07
std 9.346962e+02
min -2.330152e+03
25% -4.555563e+02
50% 2.585482e+01
75% 5.670875e+02
max 2.201400e+03
Finally, density plots of the residual error do show a small shift towards zero.
It is debatable whether this bias correction is worth it, but we will use it for now.
It is also a good idea to check the time series of the residual errors for any type of autocorrelation. If present, it would suggest that the model has more opportunity to model the temporal
structure in the data.
The example below re-calculates the residual errors and creates ACF and PACF plots to check for any significant autocorrelation.
# ACF and PACF plots of residual errors of bias corrected forecasts
from pandas import read_csv
from pandas import DataFrame
from statsmodels.tsa.arima.model import ARIMA
from matplotlib import pyplot
from statsmodels.graphics.tsaplots import plot_acf
from statsmodels.graphics.tsaplots import plot_pacf
# create a differenced series
def difference(dataset, interval=1):
diff = list()
for i in range(interval, len(dataset)):
value = dataset[i] - dataset[i - interval]
return diff
# invert differenced value
def inverse_difference(history, yhat, interval=1):
return yhat + history[-interval]
# load data
series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
# prepare data
X = series.values
X = X.astype('float32')
train_size = int(len(X) * 0.50)
train, test = X[0:train_size], X[train_size:]
# walk-forward validation
history = [x for x in train]
predictions = list()
for i in range(len(test)):
# difference data
months_in_year = 12
diff = difference(history, months_in_year)
# predict
model = ARIMA(diff, order=(0,0,1))
model_fit = model.fit()
yhat = model_fit.forecast()[0]
yhat = inverse_difference(history, yhat, months_in_year)
# observation
obs = test[i]
# errors
residuals = [test[i]-predictions[i] for i in range(len(test))]
residuals = DataFrame(residuals)
# plot
plot_acf(residuals, ax=pyplot.gca())
plot_pacf(residuals, ax=pyplot.gca())
The results suggest that what little autocorrelation is present in the time series has been captured by the model.
7. Model Validation
After models have been developed and a final model selected, it must be validated and finalized.
Validation is an optional part of the process, but one that provides a ‘last check’ to ensure we have not fooled or misled ourselves.
This section includes the following steps:
1. Finalize Model: Train and save the final model.
2. Make Prediction: Load the finalized model and make a prediction.
3. Validate Model: Load and validate the final model.
7.1 Finalize Model
Finalizing the model involves fitting an ARIMA model on the entire dataset, in this case on a transformed version of the entire dataset.
Once fit, the model can be saved to file for later use.
The example below trains an ARIMA(0,0,1) model on the dataset and saves the whole fit object and the bias to file.
The example below saves the fit model to file in the correct state so that it can be loaded successfully later.
2 # save finalized model
3 from pandas import read_csv
4 from statsmodels.tsa.arima.model import ARIMA
5 import numpy
6 # create a differenced series
7 def difference(dataset, interval=1):
8 diff = list()
9 for i in range(interval, len(dataset)):
10 value = dataset[i] - dataset[i - interval]
11 diff.append(value)
12 return diff
13 # load data
14 series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
15 # prepare data
16 X = series.values
17 X = X.astype('float32')
18 # difference data
19 months_in_year = 12
20 diff = difference(X, months_in_year)
21 # fit model
22 model = ARIMA(diff, order=(0,0,1))
23 model_fit = model.fit()
24 # bias constant, could be calculated from in-sample mean residual
25 bias = 165.904728
26 # save model
27 model_fit.save('model.pkl')
28 numpy.save('model_bias.npy', [bias])
Running the example creates two local files:
• model.pkl This is the ARIMAResult object from the call to ARIMA.fit(). This includes the coefficients and all other internal data returned when fitting the model.
• model_bias.npy This is the bias value stored as a one-row, one-column NumPy array.
7.2 Make Prediction
A natural case may be to load the model and make a single forecast.
This is relatively straightforward and involves restoring the saved model and the bias and calling the forecast() method. To invert the seasonal differencing, the historical data must also be loaded.
The example below loads the model, makes a prediction for the next time step, and prints the prediction.
2 # load finalized model and make a prediction
3 from pandas import read_csv
4 from statsmodels.tsa.arima.model import ARIMAResults
5 import numpy
6 # invert differenced value
7 def inverse_difference(history, yhat, interval=1):
8 return yhat + history[-interval]
9 series = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
10 months_in_year = 12
11 model_fit = ARIMAResults.load('model.pkl')
12 bias = numpy.load('model_bias.npy')
13 yhat = float(model_fit.forecast()[0])
14 yhat = bias + inverse_difference(series.values, yhat, months_in_year)
15 print('Predicted: %.3f' % yhat)
Running the example prints the prediction of about 6794.
If we peek inside validation.csv, we can see that the value on the first row for the next time period is 6981.
The prediction is in the right ballpark.
7.3 Validate Model
We can load the model and use it in a pretend operational manner.
In the test harness section, we saved the final 12 months of the original dataset in a separate file to validate the final model.
We can load this validation.csv file now and use it see how well our model really is on “unseen” data.
There are two ways we might proceed:
• Load the model and use it to forecast the next 12 months. The forecast beyond the first one or two months will quickly start to degrade in skill.
• Load the model and use it in a rolling-forecast manner, updating the transform and model for each time step. This is the preferred method as it is how one would use this model in practice as it
would achieve the best performance.
As with model evaluation in previous sections, we will make predictions in a rolling-forecast manner. This means that we will step over lead times in the validation dataset and take the observations
as an update to the history.
# load and evaluate the finalized model on the validation dataset
from pandas import read_csv
from matplotlib import pyplot
from statsmodels.tsa.arima.model import ARIMA
from statsmodels.tsa.arima.model import ARIMAResults
from sklearn.metrics import mean_squared_error
from math import sqrt
import numpy
# create a differenced series
def difference(dataset, interval=1):
diff = list()
for i in range(interval, len(dataset)):
value = dataset[i] - dataset[i - interval]
return diff
# invert differenced value
def inverse_difference(history, yhat, interval=1):
return yhat + history[-interval]
# load and prepare datasets
dataset = read_csv('dataset.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
X = dataset.values.astype('float32')
history = [x for x in X]
months_in_year = 12
validation = read_csv('validation.csv', header=None, index_col=0, parse_dates=True, squeeze=True)
y = validation.values.astype('float32')
# load model
model_fit = ARIMAResults.load('model.pkl')
bias = numpy.load('model_bias.npy')
# make first prediction
predictions = list()
yhat = float(model_fit.forecast()[0])
yhat = bias + inverse_difference(history, yhat, months_in_year)
print('>Predicted=%.3f, Expected=%.3f' % (yhat, y[0]))
# rolling forecasts
for i in range(1, len(y)):
# difference data
months_in_year = 12
diff = difference(history, months_in_year)
# predict
model = ARIMA(diff, order=(0,0,1))
model_fit = model.fit()
yhat = model_fit.forecast()[0]
yhat = bias + inverse_difference(history, yhat, months_in_year)
# observation
obs = y[i]
print('>Predicted=%.3f, Expected=%.3f' % (yhat, obs))
# report performance
rmse = sqrt(mean_squared_error(y, predictions))
print('RMSE: %.3f' % rmse)
pyplot.plot(predictions, color='red')
Running the example prints each prediction and expected value for the time steps in the validation dataset.
The final RMSE for the validation period is predicted at 361.110 million sales.
This is much better than the expectation of an error of a little more than 924 million sales per month.
1 >Predicted=6794.773, Expected=6981
2 >Predicted=10101.763, Expected=9851
3 >Predicted=13219.067, Expected=12670
4 >Predicted=3996.535, Expected=4348
5 >Predicted=3465.934, Expected=3564
6 >Predicted=4522.683, Expected=4577
7 >Predicted=4901.336, Expected=4788
8 >Predicted=5190.094, Expected=4618
9 >Predicted=4930.190, Expected=5312
10 >Predicted=4944.785, Expected=4298
11 >Predicted=1699.409, Expected=1413
12 >Predicted=6085.324, Expected=5877
13 RMSE: 361.110
A plot of the predictions compared to the validation dataset is also provided.
At this scale on the plot, the 12 months of forecast sales figures look fantastic.
In this tutorial, you discovered the steps and the tools for a time series forecasting project with Python.
We have covered a lot of ground in this tutorial; specifically:
• How to develop a test harness with a performance measure and evaluation method and how to quickly develop a baseline forecast and skill.
• How to use time series analysis to raise ideas for how to best model the forecast problem.
• How to develop an ARIMA model, save it, and later load it to make predictions on new data.
How did you do? Do you have any questions about this tutorial?
Ask your questions in the comments below and I will do my best to answer.
222 Responses to Time Series Forecast Study with Python: Monthly Sales of French Champagne
1. Benson Dube February 20, 2017 at 8:46 pm #
Thank You Jason.
Would you recommend this example to a starter. I am totally new to programming and Machine Learning, and I am currently learning Python sytnax and understanding the basic terms around algorithms
□ Jason Brownlee February 21, 2017 at 9:34 am #
Hi Benson, this tutorial is an example for a beginner in time series forecasting, but does assume you know your way around Python first (or you can pick it up fast).
☆ Benson March 1, 2017 at 6:45 am #
A surely steep learning curve taking you out of your comfort zone, but that’s the way to learn. ?
○ Jason Brownlee March 1, 2017 at 8:48 am #
For sure. Dive in!
☆ laurenciatitan November 14, 2018 at 1:57 am #
helo jason will i be able to contact you and discuss, need explaination..
○ Jason Brownlee November 14, 2018 at 7:33 am #
You can use the contact page to contact me any time via email:
2. Viral Mehta February 24, 2017 at 4:56 am #
Jason, thanks for very detailed instructions on how to construct the model. When I add a few additional periods in the validation set (manually) for a short-term forecast beyond the dataset, the
model won’t run until I provide some ‘fake’ targets (expected y). However, when I provide some fake targets, I see that model quickly adjusts to those targets. I tried different levels of y
values and I see model predicted accordingly, shouldn’t the predictions be independent of what targets it sees?
□ Jason Brownlee February 24, 2017 at 10:14 am #
You should not need fake targets Viral.
You can forecast a new out of sample data point by training the model on all of the available data and predicting one-step. E.g.:
1 yhat = model_fit.forecast()[0]
☆ Viral Mehta February 25, 2017 at 2:14 am #
Jason, much appreciate your response. What I am trying to convey is that the model should predict same outputs regardless of the observations. For example, when I change y values in my
validation set (last 12 months that the model hasn’t seen), my predictions change and the model gives me a very nice RMSE every time. If the model was trained properly, it should have
same output for next twelve months regardless of my y values in the validation set. I think you will see this effect very quickly if you also change your y values from validation set.
Unless the model is only good for one period forward and needs to continuously adjust based on observed values of last period.
○ Jason Brownlee February 25, 2017 at 6:00 am #
In this specific case we are using walk-forward validation with a model trained for one-step forecasts.
This means as we walk over the validation dataset, observations are moved from validation into training and the model is refit. This is to simulate a new observation being available
after a prediction is made for a time step and us being able to update our model in response.
I think what you are talking about is a multi-step forecast, e.g. fitting the model on all of the training data, then forecasting the next n time steps. This is not the experimental
setup here, but you are welcome to try it.
☆ Benson Dube March 15, 2017 at 9:11 am #
Hello Jason,
If you don’t mind me asking, where exactly should the line below fit?:
yhat = model_fit.forecast()[0]
○ Jason Brownlee March 16, 2017 at 7:57 am #
Hi Benson,
Fit the ARIMA model on all of your data.
After that, make a forecast for the next time step by calling the forecast() function.
3. Juanlu February 27, 2017 at 10:15 am #
Time to upgrade to matplotlib 2.0, the colors are nicer 🙂
□ Jason Brownlee February 28, 2017 at 8:09 am #
I agree Juanlu! I have recently upgraded myself.
4. Hugo March 24, 2017 at 2:38 am #
Dear Jason,
Thanks for the post. Clear, concise and working.
Have you considered to forecast the TS using a SARIMA model instead of substracting the seasonality and adding it latter? As a matter of fact, statsmodel has it integrated in its dev version. (
I am wondering if it is possible to take into account hierarchy, like the forecast of monthly sales divided in sub-forecasts of french champagne destination markets, using python.
Thanks and keep posting!
□ Jason Brownlee March 24, 2017 at 7:59 am #
Hi Hugo, yes I am familiar with SARIMA, but I decided to keep the example limited to ARIMA.
Yes, as long as you have the data to train/verify the model. I would consider starting with independent models for each case, using all available data and tease out how to optimize it further
after that.
5. Luis Ashurei May 2, 2017 at 7:25 pm #
Thanks for the wonderful post Jason,
Two quick question:
When I’m doing Grid Search ARIMA Hyperparameters with your example, it’s quite slow on my machine, spent about 1 minutes. However the parameter range and data are not large at all. Is it slow
specific on my end? I’m concerning the model is too slow to use.
Does ARIMA support multi-step forecast? E.g. What if I keep use predicted value into next forecast, will it overfitting?
□ Jason Brownlee May 3, 2017 at 7:34 am #
Your timing sounds fine. If it is too slow for you, consider working with a sub-sample of your data.
You can do multi-step forecasting directly (forecast() function) or by using the ARIMA model recursively:
☆ Luis May 3, 2017 at 6:50 pm #
Thanks in advice.
6. Pramod Gupta July 3, 2017 at 7:58 pm #
Hello Jason
Thanks for this awesome hands-on on Time series. However I have a query.
I extracted year and month from date column as my features for model. Then I built a Linear Regression model and a Random Forest model. My final prediction was a weighted average of both these
models. I got an rmse of 366 (similar to yours i.e. 361 on validation data).
Can this approach be the alternative for an ARIMA model? What can be the possible drawbacks of this approach?
Appreciate your comments
□ Jason Brownlee July 6, 2017 at 10:01 am #
Nice work. ARIMA may be a simpler model if model complexity is important on a project.
7. Roberto July 5, 2017 at 7:26 am #
Hi just a silly note have you consider using itertools for your evaluate_models() function?
in python nested loops are not so readable.
□ Jason Brownlee July 6, 2017 at 10:21 am #
Thanks for the suggestion.
8. Rishi Patil July 8, 2017 at 9:15 am #
Jason, I just finished reading “Introduction to Time Series Forecasting in Python” in detail and had two questions:
* Are you planning to come out with a “Intermediate course in Time Series Forecasting in Python” soon?
* You spend the first few chapters discussing how to reframe a time-series problem into a typical Machine Learning problem using Lag features. However, in later chapters, you only use ARIMA
models, that obviate the necessity of using explicitly generated lag features. What’s the best way to use explicitly generated timeseries features with other ML algorithms (such as SVM or
(I tried out a few AR lags with XGB and got decent results using but couldn’t figure out how to incorporate the MA parts).
Would appreciate any insights.
□ Jason Brownlee July 9, 2017 at 10:50 am #
I may have a more advanced TS book in the future.
Great question. You would have to calculate the MA features manually.
9. Jan August 2, 2017 at 8:34 pm #
Hi Jason,
can it be that TimeGrouper() does not work if there are months missing in a year? Also there are no docs available for TimeGrouper(). May you can use pd.Grouper in your future examples?
□ Jason Brownlee August 3, 2017 at 6:49 am #
I think it does work with missing data, allowing you to resample.
TimeGrouper does not have docs, but Grouper does and is a good start:
10. Sambit August 17, 2017 at 5:09 am #
Thanks for this great link on time series.
I am not able to acces the dataset stored at:- https://datamarket.com/data/set/22r5/perrin-freres-monthly-champagne-sales-millions-64-72#!ds=22r5&display=line
□ Jason Brownlee August 17, 2017 at 6:49 am #
I’m sorry to hear that, here is the full dataset:
27 "Month","Sales"
28 "1964-02",2672
29 "1964-04",2721
30 "1964-06",3036
31 "1964-08",2212
32 "1964-10",4301
33 "1964-12",7312
34 "1965-02",2475
35 "1965-04",3266
36 "1965-06",3230
37 "1965-08",1759
38 "1965-10",4474
39 "1965-12",8357
40 "1966-02",3006
41 "1966-04",3523
42 "1966-06",3986
43 "1966-08",1573
44 "1966-10",5211
45 "1966-12",9254
46 "1967-02",3088
47 "1967-04",4514
48 "1967-06",4539
49 "1967-08",1643
50 "1967-10",5428
51 "1967-12",10651
52 "1968-02",4292
53 "1968-04",4121
54 "1968-06",4753
55 "1968-08",1723
56 "1968-10",6922
57 "1968-12",11331
58 "1969-02",3957
59 "1969-04",4276
60 "1969-06",4677
61 "1969-08",1821
62 "1969-10",6872
63 "1969-12",13916
64 "1970-02",2899
65 "1970-04",3740
66 "1970-06",3986
67 "1970-08",1738
68 "1970-10",6424
69 "1970-12",13076
70 "1971-02",3162
71 "1971-04",4676
72 "1971-06",4874
73 "1971-08",1659
74 "1971-10",6981
75 "1971-12",12670
76 "1972-02",3564
77 "1972-04",4788
78 "1972-06",5312
79 "1972-08",1413
11. sarrauste August 24, 2017 at 1:15 am #
i am having the following error:
X = X.astype(‘float32’)
ValueError: could not convert string to float: ‘1972-09’
can you please help me
□ Jason Brownlee August 24, 2017 at 6:44 am #
Looks like you may have missed some lines of code. Confirm you have them all.
☆ Nico October 29, 2018 at 5:02 am #
Hi Jason, I am having the same error. I checked the lines and I have them all. How can python convert ‘yyyy-mm’ to float32?
I dowload the csv from the link of datamarket you gave above.
What am i missing?
○ Jason Brownlee October 29, 2018 at 6:04 am #
I have some suggestions here:
12. sandip August 28, 2017 at 8:09 pm #
Hi Jason , Nice approach for time series forecasting.Thanks for this Article.
Just had a one small doubt,Here we are tuning our ARIMA parameter on test data using RMSE as a measure but generally we tune our parameter on the training(train+validation)data and then we check
whether those parameters are well generalized or not by using test data. is it right?
Because my concern is that if we choose our parameter on that particular test data,whether those parameter will generalized to coming new test data or not? I feel that there will be biasness
while choosing parameter because we are specifically choosing parameter those giving less RMSE for that test data.Here we are not checking whether our model is working/fitted well for our train
data or not?
If i am wrong just correct me.Let me know the logic behind this parameter tuning on the basis of test data?
□ Jason Brownlee August 29, 2017 at 5:03 pm #
Yes, that is a less biased way to prepare a model.
More here:
13. sandip August 29, 2017 at 5:58 pm #
Thank you so much Jason.
□ Jason Brownlee August 30, 2017 at 6:13 am #
You’re welcome sandip.
14. Anthy August 31, 2017 at 2:24 am #
Hi Jason,
Thank you for this detailed and clear tutorial.
I have a question concerning fiding ARIMA’s parameters. I didn’t understand why we have an imbricated “for” loop and we do consider all the p values and not all the q values.
Please ask if the question is not that clear.
Thank you in advance.
□ Jason Brownlee August 31, 2017 at 6:23 am #
Please review the example again, we grid search p, d and q values.
15. Ella Zhao September 2, 2017 at 4:26 am #
Hi Jason,
Thanks for this awesome example! It helped me a lot! However if you don’t mind, I have a question on the prediction loop.
Why did you use history.append(obs) instead of history.append(yhat) when doing the loop for prediction? Sees like you add the observation (real data in test) to history. And yhat = bias +
inverse_difference(history, yhat, months_in_year) is based on the history data. but actually we don’t have those observation data when solving practical problem.
I’ve tried to use history.append(yhat) in my model, but the result is worse than using history.append(obs).
Appreciate your comments
□ Jason Brownlee September 2, 2017 at 6:16 am #
Good question, because in this scenario, we are assuming that each new real observation is available after we make a prediction.
Pretend we are running this loop one iteration per day, and we get yesterdays observation this morning so we can accurately predict tomorrow.
☆ Ella Zhao September 2, 2017 at 11:46 pm #
Thank you so much for your explanation!!
○ Jason Brownlee September 3, 2017 at 5:45 am #
You’re welcome.
16. Alaoui September 3, 2017 at 12:27 am #
Thanks a lot for this work.
I have a point to discuss on concerning the walk forward validation.
I saw that you fill the history part by a value from the test set at each iteration :
# observation
obs = test[i]
Do we really have to do this or we have to use the new yhat computed for the next predictions?
Indeed when we have to do a future prediction on some range date we don’t have the historical value….
Regards and thanks in advance for your feedback
□ Jason Brownlee September 3, 2017 at 5:48 am #
It depends on your model and project goals.
In this case, we are assuming that the true observations are available after each modeling step and can be used in the formulation of the next prediction, a reasonable assumption, but not
required in general.
If you use outputs as inputs (e.g. recursive multi-step forecasting), error will compound and things will go more crazy, sooner (e.g. model skill will drop). See this post:
17. Alaoui September 3, 2017 at 12:49 am #
Sorry, I just read the Ella question…So for long range future prediction we have ti use the yhat…
□ Jason Brownlee September 3, 2017 at 5:48 am #
You can if you choose.
18. Sujeet September 16, 2017 at 12:22 pm #
I am getting NAN after these steps:
from pandas import Series
series = Series.from_csv(‘champagne.csv’, header=0)
split_point = len(series) – 12
dataset, validation = series[0:split_point], series[split_point:]
print(‘Dataset %d, Validation %d’ % (len(dataset), len(validation)))
□ Jason Brownlee September 17, 2017 at 5:22 am #
I’m sorry to hear that.
Ensure that you have copied all of the code and that you have removed the footer from the data file. Also confirm that your environment is up to date.
19. Jaryn September 20, 2017 at 3:24 am #
Hi Jason,
This is extremely useful. Thank you very much!
I was wondering if you could provide the full code to extend the forecast for 1 year ahead. I know you mentioned above that there is a forecast function but when I run:
yhat = model_fit.forecast()[0]
pyplot.plot(yhat, color=’green’)
pyplot.plot(predictions, color=’red’)
I don’t get any green lines in my code. I’m sure I’m missing a lot of things here, but I don’t know what.
Thank you again, much appreciated!
□ Jason Brownlee September 20, 2017 at 6:01 am #
This post will help you with making longer-term forecasts:
20. Kunal Shrivastava September 20, 2017 at 4:54 pm #
Hi Jason,
I am using this modelling steps to model my problem. When I am passing best parameter which i have got from Grid Search for doing prediction, I am getting Linear Algebra error and sometimes not
Due to this I am not able to predict the values.
My second concern is if this is happening for Grid search, how can i automate the script in production environment. Lets say i have to forecast some values for 50000 different sites then what is
best way to achieve this goal ?
□ Jason Brownlee September 21, 2017 at 5:36 am #
Sorry to hear that, perhaps you have not set all of the parameters the same as you used in the grid search?
I have some notes about models in production here that may help:
21. Tri Bien September 24, 2017 at 7:57 pm #
Hi Jason,
Thank for your post, it is really helpful for me. It is great!
I have a small question:
In the post, you use 2 data set: dataset.csv and validation.csv
+ In dataset.csv, you split it and use Walk-forward validation -> I totally agree.
+ In validation.csv, you still re-train ARIMA model with walk-forward validation (model_fit = model.fit(trend=’nc’, disp=0) – line 45 in section 7.3) -> I think it is unseen data for testing, so
we don’t train model here and only test the model’s performance. Is it correct?
□ Jason Brownlee September 25, 2017 at 5:38 am #
In the case of walk forward validation, we are assuming that the real observations are available after the time of prediction.
You can make different assumptions when evaluating your model.
22. Navid September 30, 2017 at 1:50 am #
Hi, Jason
First of all, thank you for your wonderful tutorial.
Until now (part 5.4) I just have one doubt: why you plotted a histogram/kde without removing the trend first? If the goal is to see how the distribution shape is similar to Gaussian distribution,
doesn’t a trend changes the distribution of data?
□ Jason Brownlee September 30, 2017 at 7:44 am #
Great question, at that point we are early in the analysis. Generally, I’d recommend looking at density plots before and after making the series stationary.
Trend removal generally won’t impact the form of the distribution if we are doing simple linear operations.
23. Nick October 19, 2017 at 12:46 am #
Hi Jason,
Thanks for your awesome post. Could you explain how to set the ‘interval’ in the function difference if I only have 1-year data? My dataset is recorded in half an hour, from June 2016 to June
2017. These data number are large in summer and small in winter, i.e. in winter it is between 0-2000, but in summer it is between 5000-14000.
24. Shashank Hegde October 22, 2017 at 3:46 pm #
Hi Jason,
Thank you for the post.I want to implement ‘ARIMA’ function instead of using built-in function.
Do you know where I can find the algorithm to implement ‘ARIMA’ function as well understanding that in detail manner?
□ Jason Brownlee October 23, 2017 at 5:41 am #
A good textbook on the math is this one: http://amzn.to/2zvfIcB
25. Hara October 28, 2017 at 3:38 am #
Hi Jason,
Thanks for the brief tutorial on Time Series forecasting. I am receiving the error “Given a pandas object and the index does not contain dates” when running the ARIMA model code snippet.
□ Jason Brownlee October 28, 2017 at 5:16 am #
Be sure you copied all of the sample code exactly including indenting, and also be sure you have prepared the data, including removing the footer information from the file.
26. Makis December 18, 2017 at 8:27 pm #
Dear Jason,
Thanks for your amazing tut!
I have one problem only when i run the rolling forecasts.
I use a different dataset which has temperature values for every minute of the day. I train the model with the first four days and i use the last day as a test dataset. My problem might be
because i add the test observation to the history and my arima returns the following error: raise ValueError(“The computed initial AR coefficients are not ”
ValueError: The computed initial AR coefficients are not stationary
You should induce stationarity, choose a different model order, or you can
pass your own start_params.
for i in range(0, len(x_test)):
# difference data
diff = difference(history, minutes)
# predict
model = importer.ARIMA(diff, order=(5, 0, 1))
model_fit = model.fit(trend=’nc’, disp=0)
yhat = model_fit.forecast()[0]
yhat = inverse_difference(history, yhat, minutes)
# observation
obs = x_test[i]
print(‘Predicted=%.6f, Expected=%.6f’ % (yhat, obs))
My x_test dataset contains 1440 rows and i get the error on the 1423 iteration of the loop. Until iteration 1423 the each arima model does not have issues.
Your help is precious to me.
Thanks again!
Kindest Regards,
□ Jason Brownlee December 19, 2017 at 5:17 am #
Have you confirmed that your data set is stationary? Perhaps there is still some trend or seasonal structure?
I have many posts on detrending and seasonal adjustment that may help.
☆ Makis December 20, 2017 at 12:54 am #
Dear Jason,
Thanks for your response!
I have used a different hyperparameter (Arima 5,1,1) instead, and everything worked. I don’t know if this is the right thing, since 5,0,1 and 5,1,1 had exactly the same RMSE. What is your
opinion about that? And please, one final question, my results now are extremely good, with an RMSE less than 0,01. The prediction line in the plot is almost over the the real data line.
Does this has to do with the history.append(obs)? I’m not sure i understand correctly why the test observation is added to the history. And what is the difference of doing a prediction in
a for loop with a new model for every step compared by using the steps parameter in 1 model?
Sorry for the long questions!
Your tutorials are even better than the books i’m currently reading!
Cheers from Greece.
○ Jason Brownlee December 20, 2017 at 5:47 am #
Well done. I’d recommend using the model that is skillful and stable.
■ Makis December 20, 2017 at 8:22 am #
Hey Jason! What i’m trying to say is, maybe the good predictions are because i append the y observation to the history. Why we do this? what is the purpose? If we do not append
the y observation then the results are going to be still valid?
■ Jason Brownlee December 20, 2017 at 3:49 pm #
In the test setup we are assuming that real observations are made after each prediction that we can in turn use to make the next prediction.
Your specific framing of the problem may differ and I would encourage you to design a test harness to capture that.
■ Makis December 20, 2017 at 10:42 pm #
Hey Jason, for once more thanks for your feedback!
I will follow your suggestion right now 🙂
Thanks for everything
Kind Regards,
■ Makis December 20, 2017 at 10:57 pm #
Hey Jason, one last question.
Since i do the rolling forecast manner and i introduce a new model for every iteration, why do i need to save a model and do a the first prediction based on that?
Can i simply do the rolling forecast loop to start from 0 instead of 1?
■ Jason Brownlee December 21, 2017 at 5:26 am #
27. Amir December 19, 2017 at 6:51 pm #
Hi Jason,
Thank you so much for this amazing tutorial. It really helps me to learn time series forecasting and prepare for my new job. I’m a total beginner in data analysis/science as I’m currently making
transition from engineering to data analysis.
But the problem that I’m facing with this tutorial is as I’m running the code, I’m stuck with the final step, “Validate the Model”. I’ve tried to re-copy and re-run the sample code many times but
it didn’t seem to work. Here is the error.
>Predicted=10101.763, Expected=9851
>Predicted=13219.067, Expected=12670
>Predicted=3996.535, Expected=4348
>Predicted=3465.934, Expected=3564
>Predicted=4522.683, Expected=4577
>Predicted=4901.336, Expected=4788
>Predicted=5190.094, Expected=4618
>Predicted=4930.190, Expected=5312
>Predicted=4944.785, Expected=4298
>Predicted=1699.409, Expected=1413
>Predicted=6085.324, Expected=5877
>Predicted=7135.720, Expected=nan
ValueError Traceback (most recent call last)
in ()
57 # Report performance
—> 58 mse = mean_squared_error(y, predictions)
59 rmse = sqrt(mse)
60 print(‘RMSE: %.3f’ % rmse)
/Users/amir/Library/Enthought/Canopy/edm/envs/User/lib/python3.5/site-packages/sklearn/metrics/regression.py in mean_squared_error(y_true, y_pred, sample_weight, multioutput)
229 “””
230 y_type, y_true, y_pred, multioutput = _check_reg_targets(
–> 231 y_true, y_pred, multioutput)
232 output_errors = np.average((y_true – y_pred) ** 2, axis=0,
233 weights=sample_weight)
/Users/amir/Library/Enthought/Canopy/edm/envs/User/lib/python3.5/site-packages/sklearn/metrics/regression.py in _check_reg_targets(y_true, y_pred, multioutput)
73 “””
74 check_consistent_length(y_true, y_pred)
—> 75 y_true = check_array(y_true, ensure_2d=False)
76 y_pred = check_array(y_pred, ensure_2d=False)
/Users/amir/Library/Enthought/Canopy/edm/envs/User/lib/python3.5/site-packages/sklearn/utils/validation.py in check_array(array, accept_sparse, dtype, order, copy, force_all_finite, ensure_2d,
allow_nd, ensure_min_samples, ensure_min_features, warn_on_dtype, estimator)
405 % (array.ndim, estimator_name))
406 if force_all_finite:
–> 407 _assert_all_finite(array)
409 shape_repr = _shape_repr(array.shape)
/Users/amir/Library/Enthought/Canopy/edm/envs/User/lib/python3.5/site-packages/sklearn/utils/validation.py in _assert_all_finite(X)
56 and not np.isfinite(X).all()):
57 raise ValueError(“Input contains NaN, infinity”
—> 58 ” or a value too large for %r.” % X.dtype)
ValueError: Input contains NaN, infinity or a value too large for dtype(‘float32’).
Could you please help me with this?
Thank you so much
□ Jason Brownlee December 20, 2017 at 5:41 am #
I’m not sure of the cause of your fault, sorry.
Ensure your libraries are up to date and that you have copied all of the code exactly?
□ Ankit Tripathi April 30, 2018 at 7:24 pm #
The csv file contains some jargon text which should be deleted before reading the file as list.
28. Satyajit Pattnaik December 28, 2017 at 5:03 pm #
Hi Jason,
In the prediction area, where you have added the observation to history and then running the loop to find the ARIMA results.
model = ARIMA(history, order=(4,1,2))
model_fit = model.fit(disp=0)
output = model_fit.forecast()
yhat = output[0]
obs = test[t]
Here, what if we append yhat to history, when i am appending yhat to history, my results are really bad, pls help..
Because as per my model, we need to predict the test data by using only the training data, so we cannot use obs to be appended in history, hope you get my point.
□ Jason Brownlee December 29, 2017 at 5:18 am #
Yes, that would be called a recursive multi-step forecast. It is challenging. You can learn a little more about it here:
☆ Satyajit Pattnaik December 29, 2017 at 4:06 pm #
Concept wise i understand what is Recursive Multi step forecasting, that’s the same i have used in my earlier reply code, i have appended obs in history, so that means everytime my loop
runs, it takes the existing training data + next observation, so that should work and should predict correctly, but my results are really bad..
Or do you mean to say, for this we need to plot the ACF, PACF graph in a a loop to determine the pdq values, and then run the ARIMA function on the pdq values, if that so, pls help us in
finding a way to determine pdq values in loop
○ Jason Brownlee December 30, 2017 at 5:18 am #
Generally using forecasts in a recursive model is challenging, you may need to get creative with your model/models.
29. Udi January 8, 2018 at 8:57 pm #
Hello Jason,
Thank you for your clear and straightforward post!
I have the same question as Nick above – about the choice of differentiation interval.
In your example you set it to 12 according to the expected cycle of the data.
However, in a more problematic case the data does not seem to imply a clear cycle (ACF and PACF graphs notwithstanding).
In my case, I’ve found out that setting the interval to 12 yielded better results than my default, which was 1. I can understand why choosing a small interval would be generally bad – random
noise is too dominant. I have more difficulty understanding how to calibrate the ideal interval value for my data (except brute force, that is. Maybe I shouldn’t calibrate? That could induce
overfit. Anyway, I still should find a generally decent value).
□ Jason Brownlee January 9, 2018 at 5:29 am #
If the amount of data is relatively small, consider using a grid search across difference values to see what works.
30. Udi January 8, 2018 at 9:09 pm #
And another question. I applied a grid search in order to choose the hyperparameters of my ARIMA model. I fear that bias correction could be counter productive in this case.
Do you have any insights on this subject? Would you perform both (grid search followed by bias correction)? Is the answer data dependent?
□ Jason Brownlee January 9, 2018 at 5:29 am #
I would only perform bias correction if it lifted model skill (e.g. if the model was biased).
31. Nirmal Kanti Sahoo January 29, 2018 at 11:33 am #
Hi Jason,
I have around more than 1000 of products. I have sales yearly history for last 7 years for each 1000 product. I want to develop a model by which I can predict sales amount for next 3 years.
Could you please help me how do I proceed with evaluate, prediction, validation and interpret the same.
Thanking in advance.
□ Jason Brownlee January 30, 2018 at 9:45 am #
Perhaps start here:
32. Udi February 18, 2018 at 9:32 pm #
(Noob’s parallelization question, probably)
Does anybody know whether statmodels ARIMA uses multi-threading or any other kind of parallelization?
I’m trying to run an analysis based on multiple ARIMA fits on my laptop and thread parallelization increases total runtime rather than decrease it. It seems a single ARIMA fit (part of a single
thread) uses several processors at once….
□ Jason Brownlee February 19, 2018 at 9:05 am #
I think it is single threaded.
33. Raul February 28, 2018 at 3:49 am #
Hi Jason,
This is an excellent article, and it is super helpful. I had two questions for extension that may have been asked but after reading through the comments, I’m not sure if the same advice applies
to me. My question is most similar to Nirmals.
1. I have a dataset with multiple “wines”, each with their own historical sales data
2. This dataset has other variables that aren’t just time related.
For example,
Month 1 Sales Month 2 Sales Month 3 Sales online Reviews,
Wine A
Wine B
Wine C
Wine D
I’m wondering if there’s an extension of this Time Series model that can take into account other variables and other instances of historical data. Let me know if I should clarify.
Thank you for taking the time to clarify.
□ Jason Brownlee February 28, 2018 at 6:10 am #
Yes, perhaps you could model each series separately, or groups of series or all together, or all 3 approaches and ensemble the results.
Also, I’d recommend moving on to other models, such as ML or even MLP models.
I hope to cover this topic in more detail soon.
☆ Raul February 28, 2018 at 7:01 am #
Thanks for the quick reply.
I’ll try out your first method.
But do you have any good ML/MLP models/tutorials to start with? It’s okay if you don’t! I noticed you have a nice article on multivariate time series here:
I haven’t read through it yet, but I think it only takes into account historical data of one instance of multiple variables.
I think it would be interesting to find out how to ensemble the results of my
#1. one instance of historical data (Wine A’s last 3 month sales)
#2. one instance of non-historical variables (Wine A’s online reviews, type, etc.)
To me, # 1’s output is a “variable” to be used in the ML model for #2. If that makes sense, do you think that’s the right way of going about things?
Thanks for being so responsive!
○ Jason Brownlee March 1, 2018 at 6:01 am #
I hope to cover this topic in more detail soon – maybe write a book on the topic working one dataset from many different directions.
34. Anchal April 6, 2018 at 4:20 am #
Thank you Jason! for really making ML practitioners like me being awesome in ML.
In this blog you have mentioned that the results suggest that what little autocorrelation is present in the time series has been captured by the model.
What will be the next steps if there is high autocorrelation?
Thanks in advance.
□ Jason Brownlee April 6, 2018 at 6:35 am #
Great question!
Some ideas:
– Perhaps try alternate models.
– Perhaps improve the data preparation.
– Perhaps try downstream models to further correct the forecast.
35. Ankit Tripathi May 10, 2018 at 5:55 pm #
Hey Jason,
That was a very well informed article! I am trying to forecast on a weekly data. Any tips for improving the model since weekly data is hard to forecast. Also should “months_in_year = 12” used in
differencing be changed to “weeks in month=4” to accommodate for weekly frequency?
□ Jason Brownlee May 11, 2018 at 6:34 am #
My best tips are to try lots of data preparation techniques and tune the ARIMA using a grid search.
36. Maria Dossin May 26, 2018 at 9:47 am #
Hi Jason,
Thank you for an amazing tutorial!
Just one quick question: can you please provide solution for the first scenario (not rolling forward forecast):
– Load the model and use it to forecast the next 12 months. The forecast beyond the first one or two months will quickly start to degrade in skill.
Thank you~
□ Jason Brownlee May 27, 2018 at 6:43 am #
Yes, this will help:
37. Piyasi Choudhury June 2, 2018 at 11:38 am #
Hi Jason
My time series data is non stationary as well and I tried upto 3rd degree of differentiation after which I cant proceed any more as the data is exhausted. It has 3 calculated points and on doing
Fuller test for checking stationarity, it errors out “maxlag should be < nobs". What can I do here?
□ Jason Brownlee June 3, 2018 at 6:20 am #
Sounds like you might be running out of data in your series. Perhaps you need more observations?
38. KACEM June 3, 2018 at 11:54 am #
Hello, thank you.
i wonder if we can say that python give good forecasts than other forecast logiciels like Aperia.
To forecast sales we have to base ourselves in historic and baseline but in fact, in industry there are many factors that complexify the calculation of our forecast like the marketing actions,
exceptional customer order, … How can we consider these perturbation in our forecasts to ensure good value of accuracy.
Thank you for answering.
□ Jason Brownlee June 4, 2018 at 6:21 am #
What is Aperia?
Additional elements could be used as exogenous variables in the model.
39. cathy June 23, 2018 at 10:00 am #
The dataset I have requires to double difference in order to make it stationary, so i used:
diff1 = difference(history, 12)
diff2 = difference(diff1, 12),
it worked and made it stationary with ADF test, however, how do I reverse it back please?
Thank you
□ Jason Brownlee June 24, 2018 at 7:27 am #
Add the values back. It requires that you keep the original and the first diffed data.
☆ Cathy June 24, 2018 at 10:28 am #
Thanks, so would the function look like this?
yhat = inverse_difference(diff1, yhat, interval)
yhat = inverse_difference(history, yhat, interval) ?
Thank you!
○ Jason Brownlee June 25, 2018 at 6:16 am #
40. Rui July 27, 2018 at 4:11 am #
First of all, this is a great article, well written and detailed. I have a question regarding on the use of the test set on all of the analysis. What about putting this model in production? You
wouldn’t have the ‘test[i]’ (or ‘y[i]’) for each iteration to add to the list ‘history’ to have a truly generalized prediction.
My point is that that instead of adding the values from the test set (‘y[i]’, ‘test[i]’) you’d rather add the predictions being made to the training set in order to do a true random walk.
Thanks again for the resource and all the help putting out this content.
□ Jason Brownlee July 27, 2018 at 5:57 am #
A final model will be created by training it on all available data and then using it to make predictions in the future.
You can learn more about final models here:
You can learn more about making out of sample predictions here:
41. James Lucas July 27, 2018 at 2:21 pm #
I keep getting errors
TypeError Traceback (most recent call last)
in ()
9 obs = test[i]
10 history.append(obs)
—> 11 print(‘>Predicted=%.3f, Expected=%3.f’ % (yhat, obs))
TypeError: must be real number, not ellipsis
IE: # predict
yhat = …
those 3 dots (ellipsis) are throwing errors.
The code has … throughout. Any idea how to fix this error?
□ Jason Brownlee July 28, 2018 at 6:28 am #
The code with “…” is just example code, it is not for execution. Skip it.
Perhaps a more careful read of the tutorial is in order James?
42. Ben Kesby August 3, 2018 at 2:53 am #
I’m having an issue with the PACF plot for the residuals in part 6.3 Plotting The Residuals. I have followed your code exactly with the ACF plot resulting in a perfect match of the one displayed
here tower the PACF is plotting values nearing 8 at around 36 – 38 lags. Any idea what might be causing this?
43. JP August 24, 2018 at 8:17 pm #
Hi Jason,
Thanks so much for this tutorial. One thing though. As series.from_csv is depricated, the date format gets lost when opening the dataset, for exemple when trying to generate the seasonal line
plots. Are you aware of a workaround that would keep the date format while using read_csv instead of from_csv?
□ Jason Brownlee August 25, 2018 at 5:48 am #
Yes you can use pandas.read_csv() with the same arguments.
44. Alexandra September 12, 2018 at 8:40 pm #
How can I improve the readability of Seasonal Per Year Line Plots (especially when it comes to axes) ? I would be grateful for your help.
□ Jason Brownlee September 13, 2018 at 8:01 am #
Perhaps create one plot per figure?
☆ Alexandra September 13, 2018 at 4:44 pm #
I’d rather have a comparison between all the subplots.
○ Jason Brownlee September 14, 2018 at 6:33 am #
Perhaps plot them all on the same figure?
Perhaps calculate the different and plot that?
45. Chintan September 18, 2018 at 2:16 am #
Hi Jason,
I was able to predict and chart looks nice. Just wondering how can we predict for the future, I mean to say if I wanted to see for next month prediction how are we able to do that?
□ Jason Brownlee September 18, 2018 at 6:20 am #
I show how here:
46. Pranay October 9, 2018 at 9:26 pm #
Hi Jason, I deal with daily set data. What is the issue when I try months_in_year = 364?? I mean it’s not throwing best_arima out when I set months_in_year = 364. May I know the reason?
□ Jason Brownlee October 10, 2018 at 6:09 am #
Why are you making that change? I don’t follow?
47. Patrick October 30, 2018 at 2:07 am #
Dear Jason,
thank you so much for your tutorial. I would like to apply it to a process, forecasting some process variables. The time interval is not monthly based, as in your example, but is much shorter,
like some days and the data collection is about every three seconds.
Now I would like to ask some questions.
Do you think that this model could work equally well ?
What should I put in place of ‘months_in_year’ ? For now I put 7, which is the number of different days in my dataset.
Is it normal that with 2000+ points the matrix analysis regarding the seeking of the optimal parameters is taking ages ? (e.g. one triplet after 30 min) If yes, how could I improve the code speed
Thank you again,
Best wishes,
□ Jason Brownlee October 30, 2018 at 6:08 am #
Try it and see.
2K points may be too many for the method, perhaps try reducing to a few hundred at max.
48. Markus December 25, 2018 at 12:20 am #
Where does the column ‘A’comes from by the code above which is:
groups = series[‘1964′:’1970’].groupby(TimeGrouper(‘A’))
And printing out groups contains only the first 6 months of each year, why?
□ Jason Brownlee December 25, 2018 at 7:23 am #
Right there:
49. Markus December 25, 2018 at 3:10 am #
What would be wrong if instead of defining the difference function by yourself, we would pass over d with the value of 12:
model = ARIMA(diff, order=(0,12,1))
Wouldn’t that be the same?
□ Jason Brownlee December 25, 2018 at 7:24 am #
It should be.
50. Anthony January 10, 2019 at 10:33 pm #
Hi Jeson in running your code I get this error:
raise ValueError(“The computed initial MA coefficients are not ”
ValueError: The computed initial MA coefficients are not invertible
You should induce invertibility, choose a different model order, or you can
pass your own start_params.
How can I solve it? Thank you
□ Jason Brownlee January 11, 2019 at 7:47 am #
Sorry to hear that.
Perhaps confirm your statsmodels and other libraries are up to date?
Perhaps confirm that you copied all of the code in order?
Perhaps try an alternative model configuration?
☆ Anthony January 15, 2019 at 9:50 pm #
Ok I changed ARIMA parameters and now it works thank you!
○ Jason Brownlee January 16, 2019 at 5:47 am #
Glad it hear it!
51. aravind January 12, 2019 at 9:29 pm #
This is my data: like….
Name Month Qty Unit
Wire Rods Total 2007-JAN 93798 t
Wire Rods Total 2007-FEB 86621 t
Wire Rods Total 2007-MAR 93118 t
My code is :
import pandas as pd
from sklearn.metrics import mean_squared_error
from math import sqrt
# load data
path_to_file = “C:/Users\ARAVIND\Desktop\jupyter notebook\project\datasets.csv”
data = pd.read_csv(path_to_file, encoding=’utf-8′)
# prepare data
X = data.values
X = X.astype(‘float32’)
train_size = int(len(X) * 0.50)
train, test = X[0:train_size], X[train_size:]
# walk-forward validation
history = [x for x in train]
predictions = list()
for i in range(len(test)):
# predict
yhat = history[-1]
# observation
obs = test[i]
print(‘>Predicted=%.3f, Expected=%3.f’ % (yhat, obs))
# report performance
mse = mean_squared_error(test, predictions)
rmse = sqrt(mse)
print(‘RMSE: %.3f’ % rmse)
when i run this command it shows “Value error: could not convert string to float”… so could anyone tell how to convert string to float according to my dataset.. i want to convert the columns
which is “Name, Month and Unit” to float.
□ Jason Brownlee January 13, 2019 at 5:41 am #
Perhaps remove the text and date data?
☆ aravind January 14, 2019 at 3:36 pm #
ok sir.. thank you
52. Mridul March 20, 2019 at 5:41 pm #
Hi Jason, Thanks for this amazing tutorial.
However, I get the below error when I am trying to run it on a time series:
ValueError: The computed initial MA coefficients are not invertible
You should induce invertibility, choose a different model order, or you can
pass your own start_params.
□ Mridul March 20, 2019 at 5:45 pm #
I just saw that this has already been answered and it worked when I followed that answer. Thanks!
□ Jason Brownlee March 21, 2019 at 7:59 am #
Perhaps try a different configuration of the q/d/p variables for the model?
53. Kaws March 20, 2019 at 7:42 pm #
This is really a helpful tutorial. Thank you Jason!!
And I have a small question. I got TypeError: a float is required error after i executed this code –
history = [x for x in train]
predictions = list()
for i in range(len(test)):
# predict
yhat = …
# observation
obs = test[i]
print(‘>Predicted=%.3f, Expected=%3.f’ % (yhat, obs))
Can you help me out?
□ Jason Brownlee March 21, 2019 at 8:02 am #
Perhaps confirm that you have loaded the data correctly as a float?
54. Varun April 15, 2019 at 4:23 pm #
Hi Jason,
What should be the approach when we need to provide long term forecasts ~ 12 months with exogenous variables using a technique like ARIMAX? Should we forecast the covariates and then add it in
the model?
□ Jason Brownlee April 16, 2019 at 6:45 am #
Perhaps you can frame your model to predict +12 months given only the observations available?
☆ Varun April 26, 2019 at 7:22 pm #
Hi Jason,
Just to be clear, if I add regressors and train the model, I would require future values right? E.g. xreg argument in auto.arima etc. How can I forecast +12 months without utilizing
regressors that I have used for training?
○ Jason Brownlee April 27, 2019 at 6:28 am #
You could train a new productive model that only requires t-12 data to make predictions.
55. Park1 April 24, 2019 at 8:14 pm #
It’s amazing, thank you!
Could you give all files of this project. Cannot build it on my own.
□ Jason Brownlee April 25, 2019 at 8:10 am #
You can copy them from the article, here’s how:
56. Prince Tiwari May 15, 2019 at 9:31 pm #
It’s amazing, thank you!
Actually i want develop a model which determine how many calls we can expect to come into our call center on a daily basis ?
□ Jason Brownlee May 16, 2019 at 6:31 am #
Sounds great.
Perhaps start here:
57. Prince Tiwari May 22, 2019 at 10:12 pm #
Hi Jason Brownlee ,
Thank you so much for responded, i need to one more help
Could you please explain how to use Seasonality in ARIMA with some example
□ Jason Brownlee May 23, 2019 at 6:03 am #
You can use the blog search box.
Here is an example:
Here is another:
58. Adi June 27, 2019 at 2:37 am #
Hi Jason,
That’s a very helpful article. My time series is a daily Point of sale data (eg. how many pepsi bottles get bought everyday in a walmart). There are missing dates when pepsi was not sold at all.
What I want to forecast is the bottles of pepsi sold on each day for the next 3/4 days. What might be the best approach/ algorithm?
□ Jason Brownlee June 27, 2019 at 7:59 am #
Probably a linear model like SARIMA or ETS.
I have some suggestions here:
59. Nagaraj July 9, 2019 at 4:21 pm #
Hi Jason,
I’m getting the error like this, what does this error mean?
TypeError Traceback (most recent call last)
test = …
predictions = …
mse = mean_squared_error(test, predictions)
rmse = sqrt(mse)
print(‘RMSE: %.3f’ % rmse)
TypeError: Expected sequence or array-like, got
□ Jason Brownlee July 10, 2019 at 8:03 am #
That is surprising, did you copy of all of the code exactly?
60. Rohit Singh Adhikari August 2, 2019 at 3:01 am #
Hi Jason,
I need to build a forecast having addition predictor, what should i use and it’s week forecasting i need to do?
□ Jason Brownlee August 2, 2019 at 6:54 am #
Perhaps follow the process outlined in the above tutorial?
61. Ray August 7, 2019 at 9:58 am #
Hi Jason, nice article.
I have a question in the last code sample for validation.
1) What is the purpose of making the first prediction?
# load model
#make first prediction
2) I remove the above two sections of the code and got the exact result
Does mean only order (0,0,1) and Bias(165.904728) matters and there is no need to save and load the model?
□ Jason Brownlee August 7, 2019 at 2:20 pm #
It is an example of how we might use bias correction, in general.
62. Kuba August 16, 2019 at 9:53 pm #
I got exactly same results of the Augmented Dickey-Fuller test, however my PACF plot looks much different, It has a lot of spikes from lag 50 to 80, one even reaching -120.
Any ideas or previous experience why it can look that strange?
Thank you for sharing your knowledge with us.
□ Jason Brownlee August 17, 2019 at 5:44 am #
They changed the plot recently.
Try scaling the number of time steps way down in the plot.
63. Jamie August 26, 2019 at 3:22 pm #
I have tried with earnest to work through your tutorial. Unfortunately, I am running into errors.
One being this error. This could be the whole bone to my issues. As for the life of me, I could not resolve it using your code alone. I had to tinker with it – see explanation.
series = Series.from_csv(‘dataset.csv’)
AttributeError: type object ‘Series’ has no attribute ‘from_csv’
Which I resloved by implementing
panda import pd
series = pd.read_csv(‘dataset.csv’)
Once I got over that hurdle – I then ran into this hurdle.
dataset = dataset.astype(‘float32’)
ValueError: could not convert string to float: ‘1964-01’
More than like its something I am doing wrong.
Would it be at all possible to have access to the full code? My attempts in copying and pasting obviously are not helping me.
I think I have possibly left some intrinsic part of the code out. Or I have totally confused myself.
□ Jason Brownlee August 27, 2019 at 6:27 am #
No problem, changed to:
from pandas import read_csv
2 series = read_csv('champagne.csv', header=0, index_col=0)
I have updated all examples in the tutorial.
☆ Jamie August 27, 2019 at 9:42 am #
That simple – gees… I will try that. Thanks for taking the time to answer my problem. Is your book available on amazon?
○ Jason Brownlee August 27, 2019 at 2:08 pm #
My books are not on Amazon, only on my website, I explain why here (they take a massive cut):
○ Jamie August 27, 2019 at 3:59 pm #
It worked like a charm. Thanks.
Another issue that I have found is with the code for 5.3 Seasonal Line Plots.
from pandas import TimeGrouper
from pandas import DataFrame
It seems that pandas don’t support TimeGrouper,DataFrame any more!!
Removed the previously deprecated TimeGrouper (GH16942)
Removed the previously deprecated DataFrame.reindex_axis and Series.reindex_axis (GH17842)
■ Jason Brownlee August 28, 2019 at 6:29 am #
Well done!
You can use:
1 ...
groups = series['1964':'1970'].groupby(Grouper(freq='A'))
I updated the example, thanks!
64. Arjun December 5, 2019 at 4:42 pm #
Hi jason,
I was wondering if you have used fbprophet for sales prediction. We were fetching data directly from postgresql and we seem to be running into an error
Out of bounds nanosecond timestamp: 1-08-11 00:00:00
This seems to be something related with pandas version compatibility.
Could you please look into it and try to find the problem behind it?
□ Jason Brownlee December 6, 2019 at 5:12 am #
I have not, sorry.
65. Mitra February 6, 2020 at 7:40 pm #
hey Jason,
I’m running the
series=read_csv(r’D:\industrial engineering\Thesis\monthly_champagne_sales.csv’,header=0,index_col=0)
code and what I’m getting is a data frame, not a Series what should I do?
□ Jason Brownlee February 7, 2020 at 8:13 am #
Try adding squeeze=True argument.
66. Aviral Kumar March 17, 2020 at 1:56 am #
dear sir
I am running following code in a rolling window framework however, i am not able to see results that come from the analysis. It displays only one value. Can you please let me know what and where
i need to fix so that i can get those results:
from entropy import *
import numpy as np
x = np.random.rand(3000)
n = len(x)
result = list()
block = 250
for a in range(1, n-block+1):
DATA = x[a:a+249]
results = perm_entropy(DATA, order=3, normalize=True)
return Series(result)
□ Jason Brownlee March 17, 2020 at 8:17 am | {"url":"https://machinelearningmastery.com/time-series-forecast-study-python-monthly-sales-french-champagne/","timestamp":"2024-11-08T04:22:27Z","content_type":"text/html","content_length":"1049836","record_id":"<urn:uuid:91634a9d-f883-4db5-8033-e047d6d18064>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00574.warc.gz"} |
Commutator Properties: [A,B]C+B[A,C]=[A,B](C+B)?
• I
• Thread starter Kyle Nemeth
• Start date
In summary, the commutator property states that [A,BC] = [A,B]C+B[A,C]. If B=C, then the equation becomes [A,B]C+B[A,C]=[A,B]B+B[A,B]. However, because the operators do not commute, you cannot switch
Given the property,
[A,BC] = [A,B]C+B[A,C],
is it true that, if B=C, then
I apologize if I have posted in the wrong forum.
Staff Emeritus
Science Advisor
2023 Award
Kyle Nemeth said:
Given the property,
[A,BC] = [A,B]C+B[A,C],
is it true that, if B=C, then
I apologize if I have posted in the wrong forum.
The commutator (in the assumed context above) is defined as ##[A,B]=AB-BA##.
Now you have ##[A,B]C+B[A,C]=[A,B]B+B[A,B]=AB^2-B^2A## on the left and a factor ##2## on the right.
The (presumably) operators (or linear mappings) ##A,B,C## do not commutate, so you cannot switch orders.
Thank you for your help. I understand now.
FAQ: Commutator Properties: [A,B]C+B[A,C]=[A,B](C+B)?
1. What is a commutator in mathematics?
A commutator in mathematics is an operation that measures how much two operations do not commute with each other. In other words, it is a measure of how much the order in which two operations are
performed affects the outcome.
2. How is the commutator represented mathematically?
The commutator of two operators A and B is represented as [A,B]. It is calculated by taking the product of A and B and subtracting the product of B and A, [A,B]=AB-BA.
3. What are the properties of the commutator?
One of the main properties of the commutator is that it is anti-commutative, meaning that [A,B]=-[B,A]. It also follows the Jacobi identity, [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0. Additionally, the
commutator obeys the distributive property, [A,B+C]=[A,B]+[A,C].
4. How does the commutator property [A,B]C+B[A,C]=[A,B](C+B) relate to the distributive property?
The commutator property [A,B]C+B[A,C]=[A,B](C+B) is a special case of the distributive property where C is a constant. It states that a commutator operation applied to a constant is equal to the
constant multiplied by the commutator of the operators.
5. What are some real-world applications of commutator properties?
Commutator properties have various applications in fields such as quantum mechanics, computer science, and engineering. In quantum mechanics, they are used to describe the behavior of particles and
operators. In computer science, they are used in algorithms for efficient matrix computations. In engineering, they are used in control systems to analyze the response of systems to different inputs. | {"url":"https://www.physicsforums.com/threads/commutator-properties-a-b-c-b-a-c-a-b-c-b.897510/","timestamp":"2024-11-04T23:17:51Z","content_type":"text/html","content_length":"87156","record_id":"<urn:uuid:cdc83e01-51d5-4a20-a5b5-ef1f58fe9b8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00860.warc.gz"} |
Colorado River at Lees Ferry, CO
This latest version of the reconstructed flow for the Colorado at Lees Ferry was generated as part of a project supported by the California Department of Water Resources (CDWR). This project also
includes reconstructions of southern California water year precipitation (San Gabriel Dam, Lake Arrowhead, Ojai, and Cuyamaca) and streamflow (Arroyo Seco and Santa Ana River) and the Kern River
streamflow. This version of the Lees Ferry reconstruction was developed by Dave Meko, Erica Bigio, and Connie Woodhouse in 2017, using a set of updated tree-ring collections from eight sites in the
Colorado River basin. Two reconstructions were generated: one highlighting skill, with a start date of 1416, and one emphasizing length, with a start date of 1116. The most skillful reconstruction is
comparable in skill to the reconstructions from Meko et al. 2007 and Woodhouse et al. 2006 over similar time periods.
Calibration & Validation
Average discharge of the Colorado River at Lees Ferry for the water year (October-September) was reconstructed using a two-stage regression procedure. Tree-growth at each site was first converted
into an estimate of discharge by stepwise regression of discharge using tree-ring width indices, from the current year and lagged one year, as predictors. Squared terms on the tree-ring predictors
were also included in the regression to allow for possible curvature in relationships between tree-growth and discharge. In the second step, the gage reconstruction was generated by averaging an
appropriate set of single site reconstructions. Final estimates of discharge were interpolated from a piecewise-linear smoothed scatterplot of the observed discharge values and the discharge
estimates averaged over the individual tree-ring sites. The procedure was repeated for subsets of tree-ring chronologies with different periods of common time coverage to build a “most-skillful”
reconstruction, starting in the early 1400s, and a “longest” reconstruction, starting in the early 1100s. Details of the reconstruction method can be found here.
Statistic Most Skillful: Calibration Most Skillful: Validation Longest Model: Calibration Longest Model: Validation
Explained variance (R2) 0.79 0.58
Reduction of Error (RE) 0.80 0.55
Standard Error of the Estimate 18.41 MAF 27.84 MAF
Root Mean Square Error (RMSE) 19.45 MAF 29.29 MAF
Note: The statistics listed in the table represent average accuracy, while the reconstruction method yields error bars that vary in width over time -- generally wider for wet years than for dry
years. The listed statistic R2, the decimal proportion of variance explained by the reconstruction in the calibration period, is computed directly from the reconstruction residuals. For an
explanation of these statistics, see this document.
Figure 1. Scatter plot of observed and reconstructed Colorado River at Lees Ferry annual stream flow, 1906-2014. Note that the R2 value here is slightly different than in the table. The table R2
value is the average explained variance from the three models that make up the most skillful reconstruction. The value in the scatter plot reflects the explained variance for the two models that
cover the instrumental period (more details).
Figure 2. Observed (gray) 1906-2014, and reconstructed (blue) 1900-2015, Colorado River at Lees Ferry annual stream flow. The observed mean is illustrated by the black dashed line.
Figure 3. Reconstructed annual flow for the Colorado River at Lees Ferry (1416-2015) is shown in blue. Observed flow is shown in gray and the long-term reconstructed mean is shown by the black dashed
Figure 4. The 10-year running mean (plotted on final year) of reconstructed Colorado River at Lees Ferry annual stream flow, 1416-2015. Reconstructed values are shown in blue and observed values are
shown in gray. The long-term reconstructed mean is shown by the black dashed line.
Figure 5. Scatter plot of observed and reconstructed Colorado River at Lees Ferry annual stream flow, 1906-2014 (more details).
Figure 6. Observed (gray) 1906-2014 and reconstructed (blue) 1900-2015, Colorado River at Lees Ferry annual stream flow. The observed mean is illustrated by the black dashed line.
Figure 7. Reconstructed annual flow for the Colorado River at Lees Ferry (1116-2014) is shown in blue. Observed flow is shown in gray and the long-term reconstructed mean is shown by black dashed
Figure 8. The 10-year running mean (plotted on final year) of reconstructed Colorado River at Lees Ferry annual stream flow, 1116-2014. Reconstructed values are shown in blue and observed values are
shown in gray. The long-term reconstructed mean is shown by the black dashed line. | {"url":"https://www.treeflow.info/content/upper-colorado","timestamp":"2024-11-11T23:25:54Z","content_type":"text/html","content_length":"61930","record_id":"<urn:uuid:528c8247-7c6d-47ea-8447-0f1a8d1fb253>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00132.warc.gz"} |
Monte Carlo - Computer Dictionary of Information Technology
Monte Carlo
(After Monte Carlo, Monaco - a gambling mecca) Any one of various methods involving statistical techniques for finding the solutions to mathematical or physical problems.
For example, to calculate pi: draw a square then draw the biggest circle that fits exactly inside it. Pick random points on the square. The proportion of these that lie within the circle should tend
to pi/4. | {"url":"https://www.computer-dictionary-online.org/definitions-m/monte-carlo","timestamp":"2024-11-04T05:13:00Z","content_type":"text/html","content_length":"7905","record_id":"<urn:uuid:5d696ec4-966e-4b75-aeff-5c886adfa58b>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00690.warc.gz"} |
Vertical Order Traversal of a Binary Tree in Java | Online Tutorials Library List | Tutoraspire.com
Vertical Order Traversal of a Binary Tree in Java
Vertical Order Traversal of a Binary Tree in Java
In this section, we will discuss the vertical order traversal of a binary tree in Java and the different approaches to achieve it. In the vertical order traversal, we print the node of the binary
tree vertically, i.e., from top to bottom. For example, consider the following binary tree.
The vertical order traversal is:
Approach 1: Using Horizontal Distance
In this approach, we traverse the tree only once and find the maximum and minimum horizontal distance by taking root as the reference. Assume that the root node of the binary tree is located at a
distance of 0. Also, assume that going one step in the left direction is -1, and going one step in the right direction is +1. For the above-mentioned binary tree, the minimum distance is -2 (node
with the value of 1), and the maximum distance is 3 (node with the value of 19).
After finding the minimum and maximum distances from the root, iterate across each vertical line within the range minimum to maximum. While iterating for each vertical line, print the nodes that are
present on the vertical line (see the above diagram).
Let’s see the implementation of the vertical order traversal of a binary tree using the horizontal distance approach.
FileName: VerticalTraversalExample.java
The vertical order traversal of the binary tree is: 1 2 4 3 6 5 18 7 19
Time Complexity: The time complexity of the above algorithm is O(wid * no), where “wid” is the width of the given binary tree, and “no” is the number of nodes in the binary tree. In the worst case,
the value of wid can be O(no) (for example, think of a complete tree) and, in such a case, the time complexity can become O(no^2).
Approach 2: Using TreeMap
In the previous approach, we have discussed an O(no^2) solution. In this approach, we will be using the TreeMap, which gives a better solution than the TreeMap. In this approach also, it is required
to check the horizontal distances of all the nodes using the root node as the reference. Similar to the previous approach, when we move to a node which is one unit left of the root node, the
horizontal distance is considered as -1. For the node on the right side of the root node, the horizontal distance is considered as +1. While performing the preorder traversal of the tree, we can
compute the horizontal distances. For every horizontal distance value, a list of nodes is maintained in the TreeMap.
Let’s see the implementation of the vertical order traversal of a binary tree using the TreeMap.
FileName: VerticalTraversalExample1.java
The vertical order traversal of the binary tree is: [1] [2] [4, 3, 6] [5, 18] [7] [19]
Time Complexity: The above solution is based on the technique of hashing, whose time complexity is considered as O(n), where n is the total number of nodes present in the binary tree. Note that the O
(n) time complexity is achieved when we use a good hashing method, which permits the retrieval as well as the insertion operation in O(1) time.
Approach 3: Using Level Order Traversal
One can also use the concept of the level order traversal to achieve the vertical order traversal. We take the help of a queue to do the traversal of nodes. Each element of the queue provides
information about the horizontal distance and the node of the binary tree.
Similar to other approaches, in this approach also, we find the horizontal distance by taking the root node of the tree as the reference point. Also, the leftward movement from the root node adds -1
on each of the successive nodes. Similarly, the rightward movement from the root node adds +1 on each successive node. After the level order traversal of the tree is completed, we pop out the
elements from the queue one by one.
The vertical lines, as shown in the above diagram, can be considered as a level on which the nodes are lying. To point out which node is lying on the same vertical line can be found out using the
horizontal distance (horDis). We can put these nodes in an array list, and corresponding to the list, there will be the horizontal distance. We put the list and the horizontal distance in a map.
Eventually, we iterate over the map to display the results.
Let’s see the implementation of the vertical order traversal of a binary tree using the level order traversal.
FileName: VerticalTraversalExample1.java
The vertical order traversal of the binary tree is: 1 2 4 3 6 5 18 7 19
Time Complexity: The time complexity of the above program is O(n), where n is the total number of nodes present in the tree.
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The Story of Grade 1 - IM CERTIFIED® BLOG
By Brianne Durst
Grade 1 teachers have the awesome responsibility of introducing their students to, and helping them build an understanding of, the structure of our number system. This is no small task! Just think
about how many tasks in later years depend on an understanding of place value. This understanding that begins to develop in grade 1 is the foundation of number sense.
So, how should we begin students’ journey to understanding place value? We want to build on students’ prior learning and give them sufficient time and experiences to extend and solidify their
understanding. Grade 1 does this by following the Invitation-Deep Dive-Consolidation/Application structure of the IM curriculum.
Extending an Invitation in Unit 1 (Adding, Subtracting, and Working with Data)
Unit 1 Section A is an invitational welcome for both students and teachers.
Students are invited to get to know one another and experience joy in their new math community by playing games that focus on addition and subtraction within 10, something familiar from kindergarten.
Beginning the year with fun, low-stakes activities can quell nerves that students may have as they enter a new school year. This can go a long way in creating a math community where all students feel
capable, comfortable sharing ideas, and valued as an important member of the community. Unit 1 Section A also launches the year-long work of developing fluency with addition and subtraction within
10. Giving students time to work with these smaller numbers helps set them up for success with what is to come in later units.
Section A is an invitation to teachers to learn about the students in their new class in an authentic way. As teachers, we start a school year eager to get to know our students as mathematicians.
While students play games, the teacher has time to listen to students explain their ideas and reasoning. Through observation and conversation, teachers learn a lot about their students as
mathematicians and how they are working with numbers through 10. The data work in Sections B and C allows for continued community building as students learn about each other as they create surveys
and representations of class data. They ask and answer questions about their data, continuing their work with addition and subtraction within 10. Future learning will be built on these
Building onto the Foundation in Unit 2 (Addition and Subtraction Story Problems) and Unit 3 (Adding and Subtracting Within 20)
In unit 2, the focus shifts to making sense of and solving different types of story problems. The numbers used in the problems are intentionally kept within 10 to allow students to continue to build
fluency with addition and subtraction in this range.
Then, Unit 3 Section A offers students the chance to assess their fluency within 10 and identify any facts they need to continue working with to become fluent. Through this work from the beginning of
grade 1, students develop a strong foundation on which to build.
What would be helpful to know about place value to find the value of this sum?
37 + 25
The expression above is one that students will encounter in Unit 5. Asked to find the value of this expression in the beginning of grade 1, students may count out 37 cubes and 25 cubes and count them
all to find the total. Others may start with 37 and count on 25 more. Neither strategy is very efficient when working with these numbers, but both are reflective of the work first graders have done
up to this point.
In kindergarten, students worked with teen numbers and saw that all teen numbers have 10 and some more. Without labeling them, kindergarteners were working with groups of 10 ones. In Unit 3 Section
B, first graders build on this work and learn that the group of 10 ones makes a new unit called a ten. They continue working with teen numbers and see that each teen number has 1 ten and some more
ones. Throughout this work students see teen numbers represented on double 10-frames. This representation allows students to see both the unit of ten, and the 10 ones inside that unit. This is
essential as students develop a true understanding of the unit of ten.
In grade 1, students have their first experiences using place value to add and subtract. They add and subtract teen numbers and one-digit numbers, beginning with expressions that will not require
making a new unit of ten, such as 14 + 3 or 12 + 5. They notice that the unit of ten doesn’t change, and relate the sum to adding and subtracting of ones.
Students then progress to adding and subtracting teen numbers and one-digit numbers that will require making a new unit of ten or breaking apart the unit of ten. Students rely heavily on their
previous work with numbers within 10 to support them in finding the value of these sums and differences.
How do both of these strategies rely on fluency with numbers within 10?
Diego is playing Number Card Subtraction.
He started with 15 and then picked an 8.
He started out by doing this:
What could Diego do next to find the difference?
Andre was also finding the value of 15 – 8.
He started out by doing this:
What could Andre do next to find the difference?
Deep Dive into New Learning in Unit 4 (Numbers to 99) and Unit 5 (Adding within 100)
In Unit 4, students use what they have learned about teen numbers and the unit of ten to generalize the structure of two-digit numbers, relating the two digits to the number of tens and ones. They
interpret and use multiple representations of numbers up to 99, such as connecting cubes, base-ten diagrams, words, and expressions. Connecting cubes in towers of 10 and singles are used throughout
grade 1, rather than base-ten blocks, so that units of ten can be physically composed and decomposed with the cubes.
Although students work physically with connecting cubes, they interpret base-ten diagrams, recognizing the diagram as a simplified image of the connecting cubes. This helps students make sense of a
more efficient way of drawing diagrams to match their connecting cubes. As students develop their understanding of place value and work with each of these representations, they are able to compare
any two-digit numbers by comparing the number of tens, and, if needed, the number of ones.
In Unit 5 students will add 2 two-digit numbers that require making a new unit of ten. In preparation for this, students interpret and build two digit numbers with different amounts of tens and ones.
For example, students learn to recognize 32 as both 3 tens 2 ones and 2 tens 12 ones.
Students then begin adding within 100 following a similar learning progression. They add 2 two-digit numbers that do not require making a new unit of ten, such as 42 + 35. They use their place value
understanding to add tens and tens and ones and ones, or add on adding tens first then ones.
Next, students add a two-digit and a one-digit number that requires making a new unit of ten, such as 68 + 6.
Now, we are ready to come back to the expression presented earlier: 37 + 25. Students are now equipped to add within 100 in more efficient ways. They discuss different strategies based on place
value, such as adding tens and tens and ones and ones. They might add on to make a new ten, then add the rest. Students represent their thinking using connecting cubes in towers of 10 and singles,
base-ten drawings, and equations.
Mai and her classmates volunteer to clean up the local park.
They pick up 37 plastic bottles and 25 paper wrappers.
How many pieces of litter did they pick up all together?
Consolidate and Apply Learning in Unit 6 (Length Measurements Within 120 Units)
According to standard 1.NBT.A.1, students should count to 120, read and write numbers in this range, and represent a number of objects in this range with a written numeral. Working toward this
standard allows students to apply their new understanding of place value.
While the focus on linear measurement in Unit 6 may feel unrelated to the place value work in the previous units, it is actually a wonderful opportunity to consolidate and apply this learning. After
learning proper measurement techniques, students use base-ten cubes to measure lengths up to 99 length units, then measure lengths even longer than 99 length units. In order to determine the total
number of length units, they organize the cubes into groups of 10. They see that 10 tens is 100. A hundred is not discussed as a unit in grade 1, but the written notation is introduced so students
can read and write the numbers 100–120.
10 tens 4 ones is 104.
Grade 1 teachers have the awesome responsibility of introducing their students to, and helping them build an understanding of, the structure of our number system. The IM Grade 1 materials support
teachers in doing this by taking time to build the necessary foundational skills and strategically building upon and applying these skills.
Next Steps
Consider the work you do to help students build an understanding of place value. In what ways do you support building on foundational skills to allow for a strong conceptual understanding of the
base-ten structure of numbers?
Check out the entire Stories of Grades K–5 blog post series:
Story of Kindergarten
Story of Grade 1
Story of Grade 2
Story of Grade 3
Story of Grade 4
Story of Grade 5
You can also download IM’s Stories of Grades K–5 free ebook! | {"url":"https://illustrativemathematics.blog/2022/01/27/the-story-of-grade-1/","timestamp":"2024-11-07T03:01:13Z","content_type":"text/html","content_length":"96645","record_id":"<urn:uuid:dc9b0616-e682-4dd7-a2e4-9fbcbb2b01a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00206.warc.gz"} |
EYE5: The Golden Ratio
This article is part of my series on the study of ‘The Photographer’s Eye’ by Micheal Freeman.
What is the Golden Ratio? The answer is 1.618… No, this is not the answer to the question ‘What is the meaning of Life, the Universe and Everything?‘ which is 42. If you haven’t read the Hitchhiker’s
Guide to the Galaxy by Douglas Adams, then don’t worry, read on.
History of The Golden Rule
The golden ratio appears frequently in the study of mathematics and more specifically, geometry. The ancient Greeks began studying this ratio because of it’s tie to mathematics. The Greeks have given
the credit for the discovery to Pythagoras who is most frequently known for developing the Pythagorean Theorem, or a^2 + b^2 = c^2.
Euclid was the first to record the Golden Rule. Euclid said, in his series of books called the ‘Elements‘,
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.
What Euclid is saying is that when the ratio of the total length of any line to a certain portion of that line is the same as the ratio of the two segments of the line to each other, then you have a
division equal to the Golden Rule. In mathematical formula terms, if you have a line divided into a and b like the following from the Wikpedia Article on the Golden Rule,
then, (a + b)/a is the same as a/b. If you solve this equation, you end up with a number that is approximately 1.618.
The application of this ‘magical’ number can be used by first understanding the ratio in terms of frame size. Recall we said that a 24mm x 36mm film negative is in the ratio of 1.5 to 1.0, or close
to The Golden Rule. If we have a negative that is 20mm high, then the application of the golden rule would say we want a width of 20mm x 1.618, or 32.4mm. This would provide an image Frame that
applies the Golden Rule.
Unfortunately, the size of most films and digital imagers does not follow the Golden Rule. How do we use the Golden Rule in photography? We divide the frame.
Dividing the frame into sections was the topic we began in the last essay and is the topic we continue here. First, we will round the Golden Rule to 1.6 for the remainder of our work, because we can
more easily estimate sizes.
By taking a Frame size and dividing the width and height into the Golden Rule, you can find four points, one in each quadrant of the frame, that translates to these locations.
Does this mean the Subject of an image should reside at that point? It can, especially if the subject is very small in relation to the remainder of the image. For example, a small desert plant could
be placed in one of the four locations with sand encompassing the rest of the image.
What if the Subject is large? Then we have another choice. Instead of placing the Subject at one of the four points, use the point to create a rectangle within the Frame.
Then, place a larger subject within the smaller Frame. Obviously, since there are four points within a Frame that correspond to the Golden Rule, there are four rectangles within the Frame that also
correspond to the Golden Rule. An example of using this concept would include an image of a dock at the beach in the evening. The dock could be placed in one of the two bottom divisions with the
ocean and sky filling the remaining areas.
Multiple Divisions
A concept that we will further explore as we move on is multiple divisions. Frequently our images do not have such simple subjects and backgrounds and they may include multiple subjects. A more
advanced method of applying the Golden Rule is to take a divided Frame, like in the example shown above, and further divide the remaining areas into new Golden Rule areas. This can be done several
times, although the more divisions, the more complex the image and the more time involved in placing the camera to record the image.
Multiple divisions are increasingly more complex and really only lend themselves to still images where the proper time can be taken divide the image on the Frame. In addition, practicing a simple
Golden Rule division for awhile will help make multiple divisions easier. I.e., practice makes perfect.
For this essay, take four sheets of paper and draw a Frame size that fits your photography equipment. If you have multiple Frame sizes available, pick one and stick with it. If you have access to
transparency sheets, they will work better as you can view a subject through your sheet.
After creating the Frame, divide each Frame into the Golden Rule and, using a different color than the Frame color, draw the two intersecting lines. Use a ruler and be as precise as you can. Using
the same different color, mark the point of intersection with a round dot you can see at arms length. You should end up with a Frame marked similar to the images above.
Take these four sheets and find at least four different subjects, preferably two small and two larger subjects. For each subject, hold up each frame in turn to the subject. Place the smaller subjects
at the points of intersection and the larger subjects in the largest division. Study these through the drawn frames.
Then, shoot each subject in the four locations through your camera, placing them in the proper location as best as you can. Print each of the sixteen images and study and make notes about the
different placements. Divide each image on paper, using a ruler and a marker, into the Golden Rule. Alternatively, you can take more transparency material and scale down a frame size to the printed
size. Then you can overlay the Golden Rule on each image while taking notes.
Added Challenge
As and added challenge, repeat the exercise using the same subjects as the previous exercise. Compare the locations of these subjects between simple linear divisions and the Golden Rule. Which ones
seem to represent the subject the best? the most pleasing?
Also, if you wish, try making Frames on transparencies with multiple levels of division. Shoot several complex images trying to place multiple subjects in these divisions.
Using these sheets as a guide, your eye will become accustomed to where the points lie in the Frame. This takes time to develop but will be well worth your time, especially if you shoot candid street
shots. Fast paced photography does not give you the luxury of measuring and setting up a photo opportunity. They are there and then gone.
Take these transparencies with you and use them with still images before lifting your camera. See where the placement would be. When finished, print out the image and see how well you cropped the
subject within the division.
Viewing, shooting the image, printing and reviewing after the fact will train your eye to reach the Golden Rule without thinking. Good luck!
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Theoretical chemistry
Mathematical Tools in Theoretical Chemistry: mathematical review, many electron wave functions, Born-Oppenheimer approximation, Pauli exclusion principle, Slater determinant, operators and matrix
elements, second quantization, density matrices. Hartree-Fock Theory: derivation of equations, interpretation of solutions, restricted closed-shell HF, restricted and unrestricted open-shell HF,
SCF theory, Roothaan-Hall equations, further aspects. Basis Sets: Slater and Gaussian type orbitals, classification, polarization and diffuse functions, even- and well-tempered BS, contracted BS,
Pople style, Dunning-Huzinaga, atomic natural orbital, correlation-consistent and polarization consistent BS, extrapolation, BS superposition error, effective core potentials. Configuration
Interaction: configuration expansion, conventional CI approach, diagonalization and direct CI equations, complete and approximate CI, size consistency. Multiconfigurational Self-Consistent
Theory: MCSCF wavefunction, gradient, Hessian, complete active space method, applications. Coupled Cluster Theory: coupled-cluster model, the exponential ansatz, size extensivity, CCSD model,
higher excitations, open-shell CC methods, other treatments of size-extensivity. Many-Body Perturbation Theory: applications, Rayleigh-Schrödinger, Möller-Plesset, coupled cluster and MC
perturbation theory. Density Functional Theory: electron density, Hohenberg-Kohn theorems, exchange-correlation functionals, local-density approximations, generalized gradient approximation,
hybrid functionals, Kohn-Sham theory. Local Electron Correlation Methods: localization, localized molecular orbitals, Boys and Pipek-Mezey localization, local correlation, localized
Möller-Plesset methods. Analytical Gradient Theory: properties calculated as derivatives, energy derivatives for HF wave function, molecular gradient for nonvariational wave functions. Geometry
Optimization: stationary points, local models, strategies for minimization, convergence criteria, saddle point optimizations. Accurate Quantum-Chemical Calculation: errors, calibration of
methods, choice of basis set, total electronic energy, chemical reactions, vibrational spectra, thermodynamic properties, solvent efects, relativistic terms in Hamiltonian.
LEARNING OUTCOMES:
1. Write out Schrödinger equation for molecules, u jednadžbu za molekule, explain Born-Oppenheimer approximation, Pauli exclusion principle, variational method and statement of completeness of
2. Derive Hartree-Fock equations and interpret solutions for restricted closed shell HF method, and restricted and unrestricted open shell HF method.
3. Review basis sets and pseudopotentials used in quantum chemistry. Explain on examples basis set superposition error.
4. Review equations of configuration interaction and approximation methods for solutions. Explain size consistency i size extensivity.
5. Review equations of coupled cluster theory.
6. Explain application of perturbation theory in quantum chemistry.
7. Review theory of analytical gradients and explain calculation of molecular properties.
8. Define stationary and saddle points. Review algorithms for geometry optimization and convergence criteria.
9. Review principles of accurate quantum chemical calculations along with the error estimation. Explain possible improvements.
10. Present adequate skills in technical writing and oral presentations.
1. 1. T. Hrenar: Teorijska kemija, rkp. u pripremi i dijelom dostupan putem Sveučilišnog centra za e učenje Merlin (http://merlin.srce.hr, za pristup je potreban AAI korisnički račun).
2. T. Helgaker, P. Jorgensen, J. Olsen: Molecular Electronic-Structure Theory, Wiley, Chichester, 2000.
3. I. N. Levine: Quantum Chemistry, 5th Ed., Prentice Hall, New Jersey, 2000.
4. A. Szabo and N. S. Ostlund: Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications, Inc., Mineola, New York 1996.
5. R. McWeeny: Methods of Molecular Quantum Mechanics, 2th Ed., Academic Press, San Diego, 2001.
6 L. Pauling and E. B. Wilson, Jr.: Introduction to Quantum Mechanics With Applications to Chemistry, Dover Publications, Inc., New York 1985. | {"url":"http://www.chem.pmf.hr/chem/en/course/teokem","timestamp":"2024-11-07T22:32:16Z","content_type":"text/html","content_length":"84989","record_id":"<urn:uuid:f60efa25-1f82-4429-b727-830a0095c650>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00614.warc.gz"} |
athoms (US survey) to Cubit (Greek)
Fathoms (US survey) to Cubit (Greek) Converter
Enter Fathoms (US survey)
Cubit (Greek)
β Switch toCubit (Greek) to Fathoms (US survey) Converter
How to use this Fathoms (US survey) to Cubit (Greek) Converter π €
Follow these steps to convert given length from the units of Fathoms (US survey) to the units of Cubit (Greek).
1. Enter the input Fathoms (US survey) value in the text field.
2. The calculator converts the given Fathoms (US survey) into Cubit (Greek) in realtime β using the conversion formula, and displays under the Cubit (Greek) label. You do not need to click any
button. If the input changes, Cubit (Greek) value is re-calculated, just like that.
3. You may copy the resulting Cubit (Greek) value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Fathoms (US survey) to Cubit (Greek)?
The formula to convert given length from Fathoms (US survey) to Cubit (Greek) is:
Length[(Cubit (Greek))] = Length[(Fathoms (US survey))] / 0.25305504946682206
Substitute the given value of length in fathoms (us survey), i.e., Length[(Fathoms (US survey))] in the above formula and simplify the right-hand side value. The resulting value is the length in
cubit (greek), i.e., Length[(Cubit (Greek))].
Calculation will be done after you enter a valid input.
Consider that a river's depth is measured at 5 fathoms (US survey).
Convert this depth from fathoms (US survey) to Cubit (Greek).
The length in fathoms (us survey) is:
Length[(Fathoms (US survey))] = 5
The formula to convert length from fathoms (us survey) to cubit (greek) is:
Length[(Cubit (Greek))] = Length[(Fathoms (US survey))] / 0.25305504946682206
Substitute given weight Length[(Fathoms (US survey))] = 5 in the above formula.
Length[(Cubit (Greek))] = 5 / 0.25305504946682206
Length[(Cubit (Greek))] = 19.7585
Final Answer:
Therefore, 5 fath is equal to 19.7585 cubit (Greek).
The length is 19.7585 cubit (Greek), in cubit (greek).
Consider that a dock extends into the sea for 8 fathoms (US survey).
Convert this length from fathoms (US survey) to Cubit (Greek).
The length in fathoms (us survey) is:
Length[(Fathoms (US survey))] = 8
The formula to convert length from fathoms (us survey) to cubit (greek) is:
Length[(Cubit (Greek))] = Length[(Fathoms (US survey))] / 0.25305504946682206
Substitute given weight Length[(Fathoms (US survey))] = 8 in the above formula.
Length[(Cubit (Greek))] = 8 / 0.25305504946682206
Length[(Cubit (Greek))] = 31.6137
Final Answer:
Therefore, 8 fath is equal to 31.6137 cubit (Greek).
The length is 31.6137 cubit (Greek), in cubit (greek).
Fathoms (US survey) to Cubit (Greek) Conversion Table
The following table gives some of the most used conversions from Fathoms (US survey) to Cubit (Greek).
Fathoms (US survey) (fath) Cubit (Greek) (cubit (Greek))
0 fath 0 cubit (Greek)
1 fath 3.9517 cubit (Greek)
2 fath 7.9034 cubit (Greek)
3 fath 11.8551 cubit (Greek)
4 fath 15.8068 cubit (Greek)
5 fath 19.7585 cubit (Greek)
6 fath 23.7103 cubit (Greek)
7 fath 27.662 cubit (Greek)
8 fath 31.6137 cubit (Greek)
9 fath 35.5654 cubit (Greek)
10 fath 39.5171 cubit (Greek)
20 fath 79.0342 cubit (Greek)
50 fath 197.5855 cubit (Greek)
100 fath 395.1709 cubit (Greek)
1000 fath 3951.7093 cubit (Greek)
10000 fath 39517.0933 cubit (Greek)
100000 fath 395170.933 cubit (Greek)
Fathoms (US survey)
A fathom (US survey) is a unit of length used primarily in maritime contexts in the United States to measure water depth. One US survey fathom is equivalent to exactly 6 feet or approximately 1.8288
The US survey fathom is defined as 6 feet, consistent with historical maritime measurement practices and used for depth soundings and underwater measurements.
Fathoms (US survey) are utilized in navigation, fishing, and marine activities in the United States to describe water depth. The unit provides a practical measurement for underwater distances and
ensures consistency in maritime practices.
Cubit (Greek)
A Greek cubit is an ancient unit of length used in Greece and its surrounding regions. One Greek cubit is approximately equivalent to 18.2 inches or about 0.462 meters.
The Greek cubit was used in classical Greece for various purposes, including architectural design, land measurement, and textiles. Its length was based on the distance from the elbow to the tip of
the middle finger and could vary slightly depending on the historical period and specific region.
Greek cubits are of historical interest for understanding ancient Greek construction and measurement practices. Although not in common use today, the unit provides valuable insight into the standards
and techniques of ancient Greek architecture and trade.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Fathoms (US survey) to Cubit (Greek) in Length?
The formula to convert Fathoms (US survey) to Cubit (Greek) in Length is:
Fathoms (US survey) / 0.25305504946682206
2. Is this tool free or paid?
This Length conversion tool, which converts Fathoms (US survey) to Cubit (Greek), is completely free to use.
3. How do I convert Length from Fathoms (US survey) to Cubit (Greek)?
To convert Length from Fathoms (US survey) to Cubit (Greek), you can use the following formula:
Fathoms (US survey) / 0.25305504946682206
For example, if you have a value in Fathoms (US survey), you substitute that value in place of Fathoms (US survey) in the above formula, and solve the mathematical expression to get the equivalent
value in Cubit (Greek). | {"url":"https://convertonline.org/unit/?convert=fathoms_us_survey-cubits_greek","timestamp":"2024-11-10T15:01:24Z","content_type":"text/html","content_length":"92644","record_id":"<urn:uuid:b258e76a-0229-4114-9aee-241838be63f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00301.warc.gz"} |
Search results
use scaled and digital instruments to interpret unmarked and partial units to measure and compare lengths, masses, capacities, durations and temperatures, using appropriate units
• reading the mass of objects measured with digital and analog kitchen scales and explaining what unit of mass the lines on the analog scales refer to
• deciding on which attribute, unit and measuring instrument to use to compare the length and mass of various things, such as the distance travelled by an object in a science investigation; and
explaining the use of units such as grams or millimetres to give accurate measures when needed
• using scaled instruments such as tape measures, measuring jugs, kitchen scales and thermometers to record measures using whole units (for example, 560 millimetres) or whole and part units (for
example, 5.25 metres, 1.75 litres, 2.5 kilograms, 28.5° Celsius)
• reading and interpreting the scale of an analog clock without marked minutes to estimate the time to the nearest minute and to determine the duration of time between events
• using the timer or alarm function of a clock to alert when a specified duration has elapsed from a given starting time, for example, for the different activities of an exercise routine
• making a scaled measuring instrument such as a tape measure, ruler, sand timer, sundial or measuring cup using scaled instruments and direct comparisons
• exploring the different types of scaled instruments used by Aboriginal and/or Torres Strait Islander ranger groups and other groups to make decisions about caring for Country/Place, and modelling
these in local contexts
VC2M4M01 | Mathematics | Mathematics Version 2.0 | Level 4 | Measurement | {"url":"https://victoriancurriculum.vcaa.vic.edu.au/Search?sort=type&q=The+growth+and+survival+of+living+things+are+affected+by+the+physical+conditions+of+their+environmen&f=documentType!Curriculum+content&f=levels!4&f=domain!Mathematics","timestamp":"2024-11-13T22:54:41Z","content_type":"text/html","content_length":"46238","record_id":"<urn:uuid:ef116e95-88b7-4f3f-9140-329b7eeeaefe>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00791.warc.gz"} |
ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 6 Fractions Ex 6.3
ML Aggarwal Class 6 Solutions Chapter 6 Fractions Ex 6.3 for ICSE Understanding Mathematics acts as the best resource during your learning and helps you score well in your exams.
ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 6 Fractions Ex 6.3
Question 1.
State which of the following fractions are proper, improper or mixed:
Question 2.
Convert the following improper fractions into mixed numbers:
Question 3.
Convert the following mixed number into improper fractions:
Question 4.
Write the fractions representing the shaded regions. Are all these fractions equivalent?
Question 5.
Write the fractions representing the shaded regions and pair up the equivalent fractions from each row:
Question 6.
(i) Find the equivalent fraction of \(\frac{15}{35}\) with denominator 7.
(ii) Find the equivalent fraction of \(\frac{2}{9}\) with denominator 63.
Question 7.
Find the equivalent fraction of \(\frac{3}{5}\) having
(i) denominator 30
(ii) numerator 27.
Question 8.
Replace ‘…..’ in each of the following by the correct number.
Question 9.
Check whether the given pairs of fractions are equivalent:
Question 10.
Reduce the following fractions to simplest form:
Question 11.
Convert the following fractions into equivalent like fractions: | {"url":"https://ncertmcq.com/ml-aggarwal-class-6-solutions-for-icse-maths-chapter-6-ex-6-3/","timestamp":"2024-11-02T15:42:16Z","content_type":"text/html","content_length":"69558","record_id":"<urn:uuid:437fbb91-35b2-4abc-84ec-cf4e07d7c09d>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00804.warc.gz"} |
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What is the mathematical formula to calculate GST - Tax, Accounting, and Finance Chronicle
What is the mathematical formula to calculate GST
Understanding the Calculation of GST
Calculating the Goods and Services Tax (GST) on goods or services is straightforward with a simple mathematical formula. Here’s how you can determine the GST amount and the total cost inclusive of
The GST Calculation Formula
The formula to calculate GST is:
[ \text{GST} = \left(\frac{\text{Price of goods or services} \times \text{GST rate}}{100}\right) ]
In this formula:
• Price of goods or services: The original cost of the item or service before GST.
• GST rate: The applicable GST percentage.
Example Calculations
Let’s walk through some examples to see how this formula works in practice.
Example 1: Calculating GST at 18%
Imagine you bought a product worth ₹100 and the GST rate is 18%.
1. Calculate the GST amount:
[ \text{GST} = \left(\frac{₹100 \times 18}{100}\right) = ₹18 ]
2. Add the GST amount to the original price to get the total cost:
[ \text{Total price} = ₹100 + ₹18 = ₹118 ]
Example 2: Calculating GST at 12%
Now, consider the same product priced at ₹100, but with a 12% GST rate.
1. Calculate the GST amount:
[ \text{GST} = \left(\frac{₹100 \times 12}{100}\right) = ₹12 ]
2. Add the GST amount to the original price:
[ \text{Total price} = ₹100 + ₹12 = ₹112 ]
Example 3: Calculating GST at 28%
Lastly, let’s use a 28% GST rate for the same product.
1. Calculate the GST amount:
[ \text{GST} = \left(\frac{₹100 \times 28}{100}\right) = ₹28 ]
2. Add the GST amount to the original price:
[ \text{Total price} = ₹100 + ₹28 = ₹128 ]
Applying the GST Formula to Other Rates
You can use the same formula to calculate GST for any rate. Simply substitute the Price of goods or services and the GST rate into the formula:
[ \text{GST} = \left(\frac{\text{Price of goods or services} \times \text{GST rate}}{100}\right) ]
This method ensures you accurately compute the GST amount and the final price, helping you manage your finances better. Whether you’re a consumer calculating the total cost of your purchase or a
business owner determining the correct price to charge, this straightforward formula is invaluable for ensuring compliance and transparency in pricing.
Views: 0 | {"url":"https://clerk.tax/chronicle/in-gst/what-is-the-mathematical-formula-to-calculate-gst/","timestamp":"2024-11-04T00:43:55Z","content_type":"text/html","content_length":"56150","record_id":"<urn:uuid:9214e25d-afea-4a9e-917a-64f84658ce2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00812.warc.gz"} |
Math Colloquia - Geometric Langlands theory: A bridge between number theory and physics
장소: 대면 비대면 병행
(대면) 상산관 101호
(비대면) 줌 회의실 : 889 8813 5947 (https://snu-ac-kr.zoom.us/j/88988135947)
초록: The Langlands program consists of a tantalizing collection of surprising results and conjectures which relate algebraic geometry, algebraic number theory, harmonic analysis, and representation
theory among other things. The geometric Langlands program was discovered as a geometric analogue of the Langlands program. In recent years, it has been discovered that the geometric Langlands
program has another unexpected origin in the ideas of quantum field theory, which is the best existing framework of physics in describing our universe on the micro scale. In this talk, we aim to
provide a global overview of this giant program and mention some applications of quantum field theory to the geometric Langlands program. | {"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&sort_index=room&order_type=asc&page=9&l=en&document_srl=870995","timestamp":"2024-11-14T11:16:02Z","content_type":"text/html","content_length":"45804","record_id":"<urn:uuid:2fa62eda-ee4b-438d-903b-af9ef38e4079>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00692.warc.gz"} |
Waggoner Proof Template with Table
Martha Ellen Waggoner
Creative Commons CC BY 4.0
A proof template with a truth table. The table is added as a figure, has a caption and a label, and there is a reference to the table (as a figure) in the prose. | {"url":"https://tr.overleaf.com/latex/examples/waggoner-proof-template-with-table/fmzpjvwqxndn","timestamp":"2024-11-14T05:14:08Z","content_type":"text/html","content_length":"39024","record_id":"<urn:uuid:9f1b9e29-c309-4a5f-b3c0-2e0e22ad1063>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00177.warc.gz"} |
Excel Formula: Check if A and B are Blank and Return 0
In this tutorial, we will learn how to write an Excel formula in Python that checks if both cell A and B are blank and returns 0 if they are. This formula is useful when you want to perform a
specific action if both cells are empty. We will use the IF function in combination with the AND and ISBLANK functions to achieve this. By the end of this tutorial, you will have a clear
understanding of how to implement this formula in your own Excel spreadsheets.
To check if a cell is blank, we will use the ISBLANK function. This function returns TRUE if the cell is empty and FALSE if it contains any value. We will use the ISBLANK function twice, once for
cell A1 and once for cell B1.
Next, we will use the AND function to check if both conditions are TRUE. The AND function returns TRUE only if all the conditions are TRUE. In our case, we want to check if both cell A1 and B1 are
blank, so we will pass the results of the ISBLANK functions as arguments to the AND function.
Finally, we will use the IF function to evaluate the result of the AND function. If the result is TRUE (both A1 and B1 are blank), the IF function will return 0. Otherwise, it will return an empty
To apply this formula to your own Excel spreadsheet, simply replace A1 and B1 with the cell references of the cells you want to check. You can also modify the return value to suit your needs.
Now that you have a clear understanding of how to write this formula in Python, you can confidently use it in your own Excel spreadsheets to check if both cell A and B are blank and return 0 if they
A Google Sheets formula
=IF(AND(ISBLANK(A1), ISBLANK(B1)), 0, "")
Formula Explanation
This formula uses the IF function in combination with the AND and ISBLANK functions to check if both cell A1 and B1 are blank. If they are blank, the formula returns 0. Otherwise, it returns an empty
string ("").
Step-by-step explanation
1. The ISBLANK function is used to check if cell A1 is blank. If it is blank, the function returns TRUE; otherwise, it returns FALSE.
2. The ISBLANK function is also used to check if cell B1 is blank. If it is blank, the function returns TRUE; otherwise, it returns FALSE.
3. The AND function is used to check if both the conditions (cell A1 is blank and cell B1 is blank) are TRUE. If both conditions are TRUE, the AND function returns TRUE; otherwise, it returns FALSE.
4. The IF function is used to evaluate the result of the AND function. If the result is TRUE (both A1 and B1 are blank), the IF function returns 0; otherwise, it returns an empty string ("").
For example, if we have the following data:
Applying the formula =IF(AND(ISBLANK(A1), ISBLANK(B1)), 0, "") to cell C1 would result in an empty string ("") because both A1 and B1 are not blank.
Applying the formula =IF(AND(ISBLANK(A2), ISBLANK(B2)), 0, "") to cell C2 would result in 0 because both A2 and B2 are blank.
Applying the formula =IF(AND(ISBLANK(A3), ISBLANK(B3)), 0, "") to cell C3 would result in an empty string ("") because only A3 is not blank.
Applying the formula =IF(AND(ISBLANK(A4), ISBLANK(B4)), 0, "") to cell C4 would result in an empty string ("") because only B4 is not blank.
Applying the formula =IF(AND(ISBLANK(A5), ISBLANK(B5)), 0, "") to cell C5 would result in an empty string ("") because only B5 is not blank.
Applying the formula =IF(AND(ISBLANK(A6), ISBLANK(B6)), 0, "") to cell C6 would result in an empty string ("") because only A6 is not blank.
Applying the formula =IF(AND(ISBLANK(A7), ISBLANK(B7)), 0, "") to cell C7 would result in an empty string ("") because neither A7 nor B7 is blank. | {"url":"https://codepal.ai/excel-formula-generator/query/LiHnR7HB/excel-formula-if-both-a-and-b-blank-then-0","timestamp":"2024-11-03T15:37:34Z","content_type":"text/html","content_length":"94321","record_id":"<urn:uuid:2a04cb4f-fd7c-4672-ba52-41ad8d4f53af>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00005.warc.gz"} |
Journal of Space Weather and Space Climate
Issue J. Space Weather Space Clim.
Volume 8, 2018
Article Number A53
Number of page(s) 27
DOI https://doi.org/10.1051/swsc/2018038
Published online 03 December 2018
J. Space Weather Space Clim. 2018, 8, A53
Research Article
A homogeneous aa index: 1. Secular variation
^1 Department of Meteorology, University of Reading, Whiteknights Campus Earley Gate, PO Box 243, Reading RG6 6BB, UK
^2 Institut de Physique du Globe de Strasbourg, UMR7516; Université de Strasbourg/EOST, CNRS, 5 rue René Descartes, 67084 Strasbourg Cedex, France
^3 British Geological Survey, Edinburgh EH14 4AP, UK
^4 International Service of Geomagnetic Indices, 5 rue René Descartes, 67084 Strasbourg cedex, France
^* Corresponding author: m.lockwood@reading.ac.uk
Received: 7 April 2018
Accepted: 25 September 2018
Originally complied for 1868–1967 and subsequently continued so that it now covers 150 years, the aa index has become a vital resource for studying space climate change. However, there have been
debates about the inter-calibration of data from the different stations. In addition, the effects of secular change in the geomagnetic field have not previously been allowed for. As a result, the
components of the “classical” aa index for the southern and northern hemispheres (aa [S] and aa [N]) have drifted apart. We here separately correct both aa [S] and aa [N] for both these effects using
the same method as used to generate the classic aa values but allowing δ, the minimum angular separation of each station from a nominal auroral oval, to vary as calculated using the IGRF-12 and gufm1
models of the intrinsic geomagnetic field. Our approach is to correct the quantized a [ K ]-values for each station, originally scaled on the assumption that δ values are constant, with
time-dependent scale factors that allow for the drift in δ. This requires revisiting the intercalibration of successive stations used in making the aa [S] and aa [N] composites. These
intercalibrations are defined using independent data and daily averages from 11 years before and after each station change and it is shown that they depend on the time of year. This procedure
produces new homogenized hemispheric aa indices, aa [HS] and aa [HN], which show centennial-scale changes that are in very close agreement. Calibration problems with the classic aa index are shown to
have arisen from drifts in δ combined with simpler corrections which gave an incorrect temporal variation and underestimate the rise in aa during the 20th century by about 15%.
Key words: Space climate / Space weather / Geomagnetism / Space environment / Historical records
© M. Lockwood et al., Published by EDP Sciences 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1 Introduction
1.1 The derivation of the classic aa index
In his book (Mayaud, 1980), Pierre-Noël Mayaud attributes the origins of the idea for the aa index to the 1969 IAGA (International Association of Geomagnetism and Aeronomy) meeting in Madrid, where a
request for an effort to extend geomagnetic activity indices back in time was made by Sydney Chapman on behalf of the Royal Society of London. Mayaud’s subsequent work resulted in an index somewhat
different from that which Chapman had envisaged, but which covered 100 years between 1868 and 1967 (Mayaud, 1971) and has become a key component of research into space climate change. This index,
termed aa, was adopted at the 1975 IAGA meeting in Grenoble (IAGA, 1975). It was made possible by the availability of magnetic records from two old observatories, Greenwich in southern England and
Melbourne in Australia. These two stations are almost antipodal, roughly at the same geomagnetic latitude and 10 h apart in local time. To make a full data sequence that extends from 1868 to the
present day, it is necessary to use 3 stations in each hemisphere. In England they are: Greenwich (IAGA code GRW, 1868–1925, geographic latitude 51.477°N, 0.000°E), Abinger (ABN, 1926–1956, 51.185°N,
359.613°E), and Hartland (HAD, 1957–present, 50.995°N, 355.516°E). In Australia they are: Melbourne (MEL, 1868–1919, −37.830°N, 144.975°E), Toolangi (TOO, 1920–1979, −37.533°N, 145.467°E) and
Canberra (CNB, 1980–present, −35.315°N, 149.363°E).
The aa index is based on the K values for each station, as introduced by Bartels et al. (1939). These are derived from the range of variation observed at the station in 3-hour intervals. The formal
procedure for deriving K is: the range (between minimum and maximum) of the irregular variations (that is, after elimination of the regular daily variation) observed over a 3-hour interval in either
of the horizontal components (X northward or Y eastward, whichever gives the larger value) is ranked into 1 of 10 classes (using quasi-logarithmic band limits that are specific to the observatory) to
which a K value of 0–9 is assigned. The advantage of this procedure is that the scale of threshold values used to convert the continuous range values into the quantized K values is adjusted for each
station to allow for its location and characteristics such that the K value is a standardized measure of the geomagnetic activity level, irrespective of from where it is measured. In practice, the
range limits for all K bands are all set by just one number, L, the lower limit of the K = 9 band because the same relative scale is used at all stations and so the thresholds for the K bands 1–8 are
scaled from L, the lower limit for the K = 0 band being set to zero (Menvielle & Berthelier, 1991). The derivation of the K values (and from them the a [ K ] value and aa [N] and aa [S]) is
illustrated schematically in Figure 1.
Fig. 1
Schematic illustration of the generation of K and a [K ]indices. Illustrative variations of the two orthogonal horizontal field components measured at one site are shown, X (toward geographic
north, in blue) and Y (toward geographic east, in orange). These variations are after the regular diurnal variation has been subtracted from the observations. In the fixed 3-hour UT windows (00–03
UT, or 03–06 UT, and so on up to 21–24 UT), the range of variation of both components between their maximum and minimum values is taken, ΔX and ΔY. The larger value of the two is kept and scaled
according to a standard, quasi-logarithmic scale (illustrated by the black and mauve bands to the right) for which all K-band thresholds are set for the site in question by L, the threshold range
value for the K = 9 band. The value of L for the site is assigned according to the minimum distance between the site and a nominal (fixed) auroral oval position. The K value is then converted into
the relevant quantised value of a [K ](in nT) using the standard “mid-class amplitudes” (K2aK) scale. In the schematic shown, ΔX > ΔY, thus the X component gives a K value of 8 (whereas the Y
component would have given a K of 5). Thus for this 3-hour interval, a [K ]value would be 415 nT. In the case of the classic aa indices, the hemispheric index (aa [N] or aa [S], for the observatory
in the northern or southern hemisphere, respectively) is f × a [K ], where f is a factor that is assumed constant for the observing site.
The value of L used for a station is set by its closest proximity to a nominal auroral oval. To understand this, we note that mid-latitude range indices respond most strongly to the substorm current
wedge (e.g. Saba et al., 1997; Lockwood, 2013), resulting in very high correlations with auroral electrojet indices such as AE and AL (e.g. Adebesin, 2016). For example, the correlation coefficient
between the available coincident 50 annual means of the standard auroral electrojet AE(12) index and the ap index (based on the K values from a network of stations) is 0.98 (significant at the 99.99%
level), and the correlation between the 17461 coincident daily means of AE(12) and Ap (Ap being daily means of ap) is 0.84 (significant to the same level). This means that the range response of a
station is greatest in the midnight Magnetic Local Time (MLT) sector (Clauer & McPherron, 1974). As well as the response being smaller away from midnight, the typical time variation waveform also
varies with MLT (Caan et al., 1978). The range variation in a substorm is generally greatest in the auroral oval and decreases with decreasing latitude. This is mainly because the response of
high-time-resolution geomagnetic measures (such as the H component at the ground or the equivalent currents at 1-minute resolution) show a marked decrease in amplitude with increasing distance from
the auroral oval (an example of the former is presented by Rostoker (1972) and a statistical survey of the latter during 116 substorms seen from 100 geomagnetic stations is presented by Gjerloev &
Hoffman (2014)). This means that the range in the H values in 3-hour intervals also shows a decrease with increasing distance from the auroral oval. However, we note that at lower latitudes the
variation becomes rather more complex. Ritter & Lühr (2008) surveyed the effects of 4000 substorm responses statistically at 4 stations, the most poleward of which was Niemegk. They found (their Fig.
8) that the initial response to substorm expansion phase onset in 1-minute H values is actually almost constant with latitude at these low and middle latitudes, but at the higher magnetic latitude
stations there was a faster subsequent decay in the substorm perturbation to H. The resulting effect on the values of the range in H during 3-hour intervals is again a tendency for them to decrease
with decreasing latitude, but it appears to have a different origin from that seen at higher latitudes, closer to the auroral oval.
To account for the latitude variation of the range response, the value of L used to set the K band limits is set by the minimum distance between the station and a nominal auroral oval position.
Because of the offset of the auroral oval towards the nightside, this minimum distance (quantified by the geocentric angle between the station and the point of closest approach of the nominal auroral
oval, δ) is set using a nominal oval at corrected geomagnetic latitude Λ[CG] = 69°, which is an average oval location in the midnight sector where substorm expansions occur.
A key point is that in compiling the classic aa index, the L values have been assumed to remain constant over time for a given station, which means that the effects of secular changes in the
geomagnetic field on δ have not been accounted for. Mayaud was aware of the potential for secular change in δ values but discounted it as small stating “note that the influence of the secular
variation of the field on the distances to the auroral zone is such that the resulting variations of the lower limits for K = 9 are practically negligible at a scale of some tens of years” (Mayaud,
1968). Hence, in part, his view arose because saw aa as being generated to cover the previous 100 years and did not foresee its continued extension to cover another 50 years. Being aware that the
effect of secular change in the intrinsic field could not be ignored indefinitely, Chambodut et al. (2015) proposed new α [15] indices, constructed in a way that means that the secular drift in the
magnetic latitude of the observatories used is accounted for. In addition, Mursula & Martini (2007) also noted the potential effect of the secular change on the K-values from the Sodankylä
The approach taken to generate aa is that the range data were scaled into K-values using the band limits set by assigned L values for the stations used to generate the northern and southern
hemisphere indices. The values of L used by ISGI to define the K-band scales are 500 nT for all aa stations except Canberra (CNB) for where L = 450 nT is used, because of its greater distance from
the auroral oval. These K values are then converted into a [ K ] values using a standard scale called “mid-class amplitudes”, K2aK (Mayaud, 1980), given by Figure 1. However, in order to achieve
intercalibration of the data from different stations, the a [ K ] values from each station were multiplied by a constant correction factor for that station to give aa [N] and aa [S] for the northern
and southern hemisphere, respectively. The correction factors took into account two things: a constant magnetic latitude correction and an induction effect correction. The correction factors adopted
were: 1.007 for Greenwich; 0.934 for Abinger; 1.059 for Hartland; 0.967 for Melbourne; 1.033 for Toolangi; and 0.976 for Canberra (using L = 450 nT for Canberra). Note that this has an effect on the
allowed quantization levels of the indices. Without the correction factors there would be 10 allowed levels for both aa [N] and aa [S]. Averaging them together to get aa would give 19 possible
values. Using the scaling factors means that at any one time there are still only 19 possible quantized levels, but those levels change a little with each station change (i.e. at 1920, 1925, 1957,
and 1980).
Having the two aa stations roughly 10 h of local time apart means that one of the two is on the nightside at any time. This means that we cannot expect the two stations to agree at any given time.
However, ideally there would be no systematic hemispheric asymmetries and, on average, the behavior of aa [N] and aa [S] should be the same. It has long been recognized that this is not the case for
the classic aa index. Bubenik & Fraser-Smith (1977) studied the overall distributions of aa [N] and aa [S] and found that they were different: they argued that the problem was introduced by using a
quantization scheme, a potential problem discussed by Mayaud (1980). Love (2011) investigated the difference in distributions of the K values on which aa [N] and aa [S] are based. This asymmetry will
be investigated in Paper 2 of this series (Lockwood et al., 2018b) using a model of the time-of-year and time-of-day response functions of the stations, allied to the effects of secular change in the
main field (and associated station inter-calibration issues) that are the subject of the present paper.
1.2 Hemispheric asymmetry in the centennial-scale change of the classic aa index
Figure 2a illustrates another hemispheric asymmetry in the classic aa index. It shows annual means of aa [N] (in red) and aa [S] (in blue). These are the values averaged together in the generation of
the official aa index by L’École et Observatoire des Sciences de la Terre (EOST), a joint of the University of Strasbourg and the French National Center for Scientific Research (CNRS) institute, on
behalf of the International Service of Geomagnetic Indices (ISGI). The magnetometer data are now supplied by British Geological Survey (BGS), Edinburgh for the northern hemisphere and Geoscience
Australia, Canberra for the southern hemisphere. We here refer to these aa [N], aa [S] and aa data as the “classical” values, being those that are used to derive the official aa index by EOST, as
available from ISGI (http://isgi.unistra.fr/) and data centers around the world.
Fig. 2
Variations of annual means of various forms of the aa index. (a) The published “classic” northern and southern hemisphere indices (aa [N] and aa [S] in red and blue, respectively). Also shown (in
green) is 1.5 × a [NGK], derived from the K-indices scaled from the Niemegk data. The vertical dashed lines mark aa station changes (cyan: Melbourne to Toolangi; green: Greenwich to Abinger; red:
Abinger to Hartland; and blue: Toolangi to Canberra). (b) The homogenized northern and southern hemisphere indices (aa [HN] and aa [HS] in red and blue, respectively) generated in the present
paper. The thick green and cyan line segments are, respectively, the a [NGK] and am index values used to intercalibrate segments. (c) The classic aa data series, aa = (aa [N] + aa [S])/2 (in mauve)
and the new homogeneous aa data series, aa [H] = (aa [HN] + aa [HS])/2 (in black). The orange line is the corrected aa data series aa [C] generated by Lockwood et al. (2014) by re-calibration of
the Abinger-to-Hartland join using the Ap index. (Note that before this join, aa and aa [C] are identical and the orange line is not visible as it is underneath the mauve line). The cyan line and
points show annual means of the am index. The gray-shaded area in (c) is the interval used to calibrate aa [HN] and aa [HS] (and hence aa [H]) against am.
It can be seen that although aa [N] and aa [S] agree well during solar cycles 14–16 (1900–1930), aa [N] is progressively larger than aa [S] both before and after this interval. The vertical lines
mark station changes (cyan for MEL to TOO; green for GRW to ABN; red for ABN to HAD; and blue for TOO to CNB). There has been much discussion about possible calibration errors between stations at
these times. In particular, Svalgaard et al. (2004) pointed out that the classic aa [N] values showed a major change across the ABN-HAD join. These authors argued from a comparison against their
“inter-hour variability” index, IHV, that this was responsible for an extremely large (8.1 nT) step in aa, such that all the upward drift in aa during the 20th century was entirely erroneous.
However, the early version of IHV that Svalgaard et al. had employed to draw this conclusion came from just two, nearby, Northern Hemisphere stations, Cheltenham and Fredricksburg, which were
intercalibrated using the available 0.75 yr of overlapping data in 1956. This calibration issue only influenced aa [N] and Lockwood (2003) pointed out that, as shown in Figure 2a, aa [S] also showed
the upward rise over the 20th century, albeit of slightly smaller magnitude than that in aa [N] (and hence, by definition aa). Using more stations, Mursula et al. (2004) found there was an upward
drift in IHV over the 20th century, but it depended on the station studied; nevertheless, they inferred that the upward drift in aa was probably too large. As a result, Svalgaard et al. (2003)
subsequently revised their estimates of a 1957 error in aa down to 5.2 nT (this would mean that 64% of the drift in aa was erroneous). However, Mursula & Martini (2006) showed that about half of this
difference was actually in the IHV estimates and not aa, being caused by the use of spot samples by Svalgaard et al., rather than hourly means, in constructing the early IHV data. This was corrected
by Svalgaard & Cliver (2007), who revised their estimate of the aa error further downward to 3 nT. Other studies indicated that aa needed adjusting by about 2 nT at this date (Jarvis, 2004; Martini &
Mursula, 2008). A concern about many of these comparisons is that they used hourly mean geomagnetic data which has a different dependence on different combinations of interplanetary parameters to
range data (Lockwood, 2013). Recent tests with other range indices such as Ap (Lockwood et al., 2014, Matthes et al., 2016) confirm that an upward skip of about 2 nT at 1957 is present in aa (about
one quarter of the original estimate of 8.1 nT). However, it is important to stress that this calibration arises for data which do not contain any allowance for the effects of the secular change in
the geomagnetic field (in the present paper, we will show that the rise in the classic aa between 1902 and 1987 is indeed slightly too large, but this arises more from neglecting the change in the
intrinsic geomagnetic field than from station intercalibration errors).
The argument underpinning the debate about the calibration of aa was that the minimum annual mean in 1901 (near 6 nT) was much lower than any seen in modern times (14 nT in 1965) and so, it was
argued, erroneous. This argument as shown to be specious by the low minimum of 2009 when the annual mean aa fell to 8.6 nT. Furthermore, subsequent to that sunspot minimum, solar cycle 24 in aa has
been quite similar to cycle 14 (1901–1912) and so the rise in average aa levels between cycles 14 and 22 has almost been matched by the fall over cycles 23 and 24. This does not necessarily mean that
the classic aa for cycle 14 is properly calibrated, but it does mean that the frequently-used argument that it must be in error was false.
An upward 2 nT calibration skip in aa implies a 4 nT skip in aa [N] and Figure 2a shows that after 1980 aa [N] exceeds aa [S] by approximately this amount. Hence it is tempting to ascribe this
difference between aa [N] and aa [S] to the one calibration skip. However, inspection of the figure reveals aa [N] grows relative to aa [S] before the ABN-HAD change in 1957. In Figure 2a, also
plotted (in green) are annual mean a [ K ] values based on the K-index data from Niemegk (NGK, 1880–present). These have been scaled using the same mid-class amplitudes (K2aK) to give a [NGK] and
then multiplied by a best-fit factor of 1.5 to bring it into line with aa [S]. It can be seen that 1.5 a [NGK] and aa [S] are very similar in all years, implying that the upward drift in aa [N] is
too large, even if it is not the ABN-HAD change that is solely responsible.
1.3 Studies of space climate change using the aa index
Feynman & Crooker (1978) reconstructed annual means of the solar wind speed, V [SW], from aa, using the fact that aa, like all range geomagnetic indices, has an approximately V [SW] ^2 dependence (
Lockwood, 2013). However, on annual timescales, aa also has a dependence on the IMF field strength, B, which contributes considerably to the long term drift in aa. Lockwood et al. (1999) removed the
dependence of aa on V [SW] using its 27-day recurrence (which varies with mean V [SW] on annual timescales) and derived the open solar flux (OSF, the total magnetic flux leaving the top of the solar
corona) using “the Ulysses result” that the radial component of B is largely independent of heliographic latitude (Smith & Balogh, 1995; Lockwood et al., 2004; Owens et al., 2008). This variation was
modelled using the OSF continuity equation by Solanki et al. (2000), who employed the sunspot number to quantify the OSF emergence rate. This modelling can be extended back to the start of regular
telescopic observations in 1612. Svalgaard & Cliver (2005) noted that different geomagnetic indices have different dependencies on the IMF, B and the solar wind speed, V [SW], and therefore could be
used in combination to derive both. This was exploited by Rouillard et al. (2007) who used aa in combination with indices based on hourly mean geomagnetic data to reconstruct annual means of B, V
[SW] and OSF back to 1868. Lockwood et al. (2014) used 4 different pairings of indices, including an extended aa data series (with a derived 2 nT correction for a presumed aa [N] calibration skip in
1957) to derive B, V [SW] and OSF, with a full uncertainty analysis, back to 1845. Lockwood & Owens (2014) extended the modelling to divide the OSF into that in the streamer belt and in coronal holes
and so computed the streamer belt width variation which matches well that deduced from historic eclipse images (Owens et al., 2017). The streamer belt width and OSF were used by Owens et al. (2017),
along with 30 years’ of output from a data-constrained magnetohydrodynamic model of the solar corona based on magnetograph data, to reconstruct solar wind speed V [SW] and number density N [SW] and
the IMF field strength B, based primarily on sunspot observations. Using these empirical relations, they produced the first quantitative estimate of global solar wind variations over the last 400
years and these were employed by Lockwood et al. (2017) to compute the variation in annual mean power input into the magnetosphere and by Lockwood et al. (2018a) to estimate the variation in
geomagnetic storm and substorm occurrence since before the Maunder minimum. The aa index data were also used by the CMIP-6 project (the 6th Coupled Model Intercomparison Project) to give a
comprehensive and detailed set of solar forcing reconstructions for studies of global and regional climate and of space weather (Matthes et al., 2016). Vennerstrom et al. (2016) used the aa index to
investigate the occurrence of great geomagnetic storms since 1868.
Hence the aa index has been extremely valuable in reconstructing space climate, and in taking the first steps towards a space weather climatology that covers more general conditions than do the
direct satellite observations (which were almost all recorded during the Modern Grand Maximum (Lockwood et al., 2009)). In addition, the aa data have been hugely valuable in facilitating the
exploitation of measured abundances of cosmogenic isotopes, ^14C, ^10Be and ^44Ti (Usoskin, 2017). These records of past solar variability, stored in terrestrial reservoirs such as tree trunks, ice
sheets and fallen meteorites, do not overlap much (or at all) with modern spacecraft data. For example, ^14C cannot be used after the first atomic bomb tests, and recent ^10Be data is less reliable
as it is taken from the firn rather than the compacted snow of the ice sheet, whereas ^44Ti accumulates in meteorites over very long intervals. The extension of spacecraft data by reconstructions
based on aa has given an overlap interval since 1868 which can be used to aid the interpretation of the cosmogenic data (Asvestari & Usoskin, 2016; Owens et al., 2016).
1.4 Making a homogeneous aa index
From Section 1.3, it is apparent that the aa index is very important to studies of past space climate. The issues (such as hemispheric asymmetries and calibration glitches) in the aa index discussed
here and other limitations (such as the strong artefact diurnal variation caused by the use of just 2 stations) will not invalidate the space climate work that has been done using aa, although they
may call for some corrections. However, the increasing use and importance of aa makes it timely to take a comprehensive look at these issues. In Paper 2 (Lockwood et al., 2018b) we study how the
compilation of the aa index influences its time-of-day and time-of-year response and, as far as is possible, we make corrections for this and explain and correct the north-south asymmetries in the
distributions of 3-hourly aa values. In the present paper, we study the difference in the long-term drift of the northern and southern aa indices. We show that the intercalibration glitches in aa,
particularly that between Abinger and Hartland, were actually not just errors, but were also necessary to compensate for the drifts introduced into the data by the secular change in the intrinsic
geomagnetic field. Figure 2b shows the end result of the process detailed in the present paper – a process that makes allowance for the effects of these drifts on the aa [N] and aa [S] values and
then re-calibrates the joins between data from the different stations. It can be seen from Figure 2 that the resulting “homogenized” aa [HN] and aa [HS] indices obtained from this process are much
more similar to each other than are the classic aa indices, aa [N] and aa [S].
Note that in this paper, we do just two things. Firstly, we correct Mayaud’s derivation to allow for secular drift in the main geomagnetic field – a factor which he understood but decided could be
neglected. Indeed, part of the brilliance of Mayaud’s formulation was to use the minimum distance to the auroral oval, which is less subject to secular change than the geomagnetic latitude of the
station. This is because both the geomagnetic latitude of the station and the geographic latitude of the average auroral oval drift with the secular change in and, although the two do not change in
precisely the same way, there are similarities and so part of the secular drift is cancelled out by taking the difference between the two, δ. (Of course they do not cancel completely and that is why
there is still a requirement to correct for the secular change in the main field). Secondly we revisit the inter-calibration of the stations which becomes necessary when the station data has been
corrected for the effect of the secular field change. We take the opportunity to calibrate the revised aa to modern data from the am index which is derived from a global network of 24 stations. As a
test of the validity of our approach we show that it makes the variations of the annual means of the northern and southern hemisphere aa indices, aa [N] and aa [S], much more similar although we make
no changes that were designed in advance to make them similar. The reason why this is a useful improvement to the index comes from the rationale for averaging aa [N] and aa [S] together to get an
index (aa) that is hoped to be global in its application and implications. In deriving aa, Mayaud selected the sites to be as close to antipodal as possible and give a continuous data sequence: he
did not do calculations that showed that although aa [N] and aa [S] are different, the sites are in somehow special such that the difference between aa [N] and a true global value (that would be
detected from an extensive global network) is equal and opposite to that for aa [S] – a condition that would guarantee that on averaging one gets a valid global mean. This being the case, the only
rationale for averaging aa [N] and aa [S] to get a valid representation of a global mean is that they should the same. Note that this does not alone solve the asymmetry between the distribution of
the aa [N] and aa [S] values which is investigated in Paper 2 (Lockwood et al., 2018b).
2 The effect of secular change in the magnetic field
Figure 3 shows the variation of the scale factor, s(δ), derived from the threshold range value L that defines K = 9, with the minimum geocentric angular separation of the station from a nominal
auroral oval, δ. The oval is defined to be along typical corrected geomagnetic latitude (Λ[CG]) of the nightside aurora of 69°. This empirical variation is taken from Mayaud (1968) and is the basis
of the L values used to scale K-indices from observed range for all mid-latitude stations. The scale factor s(δ) normalizes to an idealized Niemegk station (for which δ = 19° and L = L [o] = 500 nT,
the constant reference values established by Mayaud). The curve is described by the polynomial:$s ( δ ) = ( L / L o ) = 3.8309 - 0.32401 . δ + 0.01369 . δ 2 - ( 2.7711 . 10 - 4 ) . δ 3 + ( 2.1667 .
10 - 6 ) . δ 4$(1)where δ is in degrees. Equation (1) applies over the range 11° < δ < 40° which requires that the station be at mid-latitudes (the relationship not holding for either equatorial or
auroral stations).
Fig. 3
The variation of the scale factor s(δ) derived from threshold range value L that defines the K = 9 band, with the minimum angular separation of the station from a nominal auroral oval, δ. This
empirical variation is scaled from Mayaud (1968, 1972) and is the basis of the L values used to scale K-indices from observed range for all mid-latitude stations. The scale factor s(δ) normalizes
to the idealized Niemegk station for which δ = 19° and L = Lo = 500 nT (ideal static Mayaud values).
In this paper, corrected geomagnetic latitudes (Λ[CG]), and Magnetic Local Times (MLT), are computed using the IGRF-12 model (Thébault et al., 2015) for dates after 1900. For dates before this (not
covered by IGRF-12) we employ the historical gufm1 model (Jackson et al., 2000), values being scaled using linear regression of values from IGRF-12 for an overlap intercalibration interval of
1900–1920. Figure 4a shows the variations of |Λ[CG]| for the various stations used to generate aa, plus that of Niemegk (NGK, in orange). The vertical lines show the dates of transfer from one
station to the next, using the same color scheme as Figure 2. It can be seen that for much of the 20th century the geomagnetic latitude of the northern and southern hemisphere stations changed in
opposite directions, with the northern stations (GRW, ABN and HAD) drifting equatorward and southern (MEL and TOO) drifting poleward. This changed around 1984 when CNB began to drift equatorward, the
same direction as the northern hemisphere station at that time, HAD.
Fig. 4
Analysis of the effect of secular change in the geomagnetic field on the aa magnetometer stations using a spline of the IGRF-12 and the gufm1 geomagnetic field models (for after and before 1900,
respectively). (a) The modulus of the corrected geomagnetic latitude, |Λ[CG]| of the stations; (b) the angular separation of the closest approach to the station of a nominal nightside auroral oval
(at |Λ[CG]| = 69°), δ; and (c) the scale factor s(δ) = L/L [o] where L is given as a function of δ by Figure 3 and L [o] = 500 nT, the reference value for the Niemegk station (for which δ is taken
to be 19°) except for Canberra which, because of its more equatorward location, is scaled using L [o] = 450 nT. The northern hemisphere stations are Greenwich (code GRW, in mauve), Abinger (ABN, in
green) and Hartland (HAD, in red). The southern hemisphere stations are Melbourne (MEL, in black), Toolangi (TOO, in cyan) and Canberra (CNB, in blue). Also shown is Niemegk (NGK, in orange: data
available since 1890). Vertical dashed lines mark aa station changes.
These changes in the Λ[CG] of stations were accompanied by changes in the geographic latitude of the nominal aurora oval at Λ[CG] = 69°. To compute δ or a given date, we use the geomagnetic field
models to calculate the Λ[CG] = 69° contour in the relevant hemisphere in geographic coordinates and then spherical geometry to find the angular great circle distances between the station in question
and points on this contour: we then iterate the geographic longitude of the point on the contour until the minimum angular distance is found, which is δ. The variations of δ derived this way for each
station are shown in Figure 4b. Using equation (1), this gives the variation of scale factors s(δ) in Figure 4c for each station. It can be seen that the secular change in the intrinsic field has
caused a considerable drift in the threshold value for the K = 9 band, L, that should have been used. In compiling the original aa index, it was assumed that s(δ) for each station remained constant
(the scale factors given in Section 1.1 being 1/s(δ) and assumed constant). Remember also that larger s(δ) means a higher L which would give a lower aa value. We could consider reanalyzing all the
range data using K-scale band thresholds that varied according to Figure 4c: correcting the band thresholds would change many K-values, but would also leave many unchanged. However, there are now 150
years of aa data which gives 0.87 million 3-hourly intervals to analyse from the two stations, many of which are not available as digital data. Clearly this would be a massive undertaking but it
would also be a change in the construction philosophy because aa values have been scaled using constant L values (500 nT for all stations except Canberra for which 450 nT is used). The station
correction factors applied in constructing the classic aa values include an allowance for the fact that the L values used are not optimum for the station in question: however, where in the classic aa
these factors are constants over time, we here vary them to allow for the secular change in the intrinsic geomagnetic field. Therefore we divide classic aa [N] and aa [S] values by the s(δ) that
applies for that station at that date. From the above, we stress that this type of correction is already employed in the classic aa data, as it is the same principle as adopted when applying the
scale factors for the station. The only difference is that here we use the IGRF-12/gufm1 model spline to apply time-dependent scale factors, s(δ), rather than the constant ones for each station used
in deriving the classic aa.
Introducing these time-dependent scaling factors reduces the rise in aa [N] by 4.11%, over the interval of the Greenwich data (compared to a constant factor) – a rate of drift of 0.0721% p.a.; by
0.83% over the interval of the Abinger data (0.0258% p.a.) and by 5.37% over the interval of the Hartland data (0.0895% p.a.). On the other hand, they increase the rise in aa [S] by 4.77% over the
interval of the Melbourne data (0.0917% p.a.); by 5.28% over the interval of the Toolangi data (0.0880% p.a.); but decrease the rise in aa [S] by over the interval of the Canberra data by 1.84%
(0.0497% p.a.). Thus allowing for the secular change in the intrinsic magnetic field reduces the disparity in the long term-drifts in aa [N] and aa [S] that can be seen in Figure 2a.
Figure 5 summarizes the differences between the computation of the classic aa index and that of the new homogenized indices presented in this paper. The left-hand plots compare the variations in the
minimum angular distance of the stations to the auroral oval δ and compares them to the constant values used in generating the classic aa index. The right-hand plots show the corresponding scale
factors, s(δ). The (constant) correction factors used in constructing aa were derived account for several factors in addition to δ and their reciprocals are shown in the right-hand plots as dot-dash
lines. (Reciprocals are plotted because the correction factors were multiplicative whereas we divide by the s(δ) scale factors).
Fig. 5
Variations of (left) the minimum angular distance to the auroral oval, δ, and (right) the scalefactors, s(δ), for the aa stations. The colours used are as in Figure 4 (namely mauve for Greenwich,
green for Abinger, red for Hartland, black for Melbourne, cyan for Toolangi and blue for Canberra). The thin lines are the variations shown in Figure 4 and the thick lines are constant values used
in generating the classic aa. The dot-dash lines in the right-hand panels show the reciprocals of the standard multiplicative correction factors and the thick lines the factors corresponding to the
constant δ values in the left-hand panels.
The Mayaud latitude correction formulation has also been used to generate the am, an and as indices since their introduction in 1959. In generating new 15-minute indices in four local time sectors,
Chambodut et al. (2015) used a different approach employing a polynomial in the stations’ geomagnetic latitudes. Although the purpose of the two schemes is the same, a comparison cannot be made
between them because the new Chambodut et al. (2015) indices are 15-minute range values, as opposed to the 3-hour range (K index) values used by the aa, am, as and an indices. There are four separate
indices in the Chambodut et al. (2015) set, one for each of four Magnetic Local Time (MLT) sectors whereas the Mayaud formulation is designed to account predominantly for the midnight sector by
taking the minimum geomagnetic latitude offset to the auroral oval (which occurs in the midnight sector). The advantage of using geomagnetic latitude is that greater precision can be obtained
(because there is no need to employ a nominal oval location) but the station calibration factor needs considerable annual updates because of the secular drift in the station’s geomagnetic latitude.
On the other hand, the Mayaud formulation has the advantage of being less influenced by secular change in the main field, as discussed above.
We here use Mayaud’s formulation to correct for secular change via division by the s(δ) factors. However, was also taking the opportunity to re-calibrate (via linear regression) the aa index against
the am index which is based on 14 stations in the northern hemisphere and 10 stations in the south. This recalibration is carried out in Section 3.1 for the Hartland and Canberra data using linear
regression over 2002–2009 (inclusive), and then passed back (“daisy-chained”) to earlier stations (from Hartland to Abinger and then Greenwich in Sections 3.2 and 3.3 and from Canberra to Toolingi
and then Melbourne in Section 3.4). Figure 6 demonstrates how well this approach works by (top panel) comparing the results of applying this procedure to modern a [ K ] data from a range of stations
at different geographic latitudes, λ [G]: (mauve) Sodankylä, SOD, λ [G] = 67.367°N; (brown) Eskdalemuir, ESK, λ [G] = 55.314°N; (orange) Niemegk, NGK, λ [G] = 52.072°N; (red) Hartland, HAD, λ [G] =
50.995°N; (blue) Canberra, CNB, λ [G] = 35.315°S; and (green) a spline of Gangara, GNA, λ [G] = 31.780°S and nearby Gingin, GNG, λ [G] = 31.356°S, Gingin is the replacement for Gangara after January
2013 and the spline was made using the overlap data between August 2010 and January 2013: this station pair is chosen as they are in the same southern hemisphere longitude sector as Melbourne but are
at lower geomagnetic latitude (see below). The black line shows the am index data, the linear regression against which over the calibration interval (2002–2009 inclusive) gives the slope m and an
intercept i for each station. The data are means over 27-day Bartels solar rotation intervals and cover 1995 to the present day for reasons discussed in later in this section. It can be seen that the
level of agreement between the station data processed this way and the am calibration data is very close for all stations. The scalefactors s(δ) used in Figure 6 vary with time and location between a
minimum of 0.896 (for Gangara/Gingin) and maximum of 2.298 (for Sodankylä). The range covered by the aa stations is 0.940 (for Melbourne in 1875) and 1.102 (for Greenwich in 1868) – hence our test
set of stations covers all of the range of δ for the aa stations, plus a considerable amount more. The bottom panel of Figure 6 shows the root-mean-square (rms) deviation of the individual station
values from the am index, ε [rms]. For most Bartels’ rotations this is around 5%, but in the low solar minimum of 2008/2009 rises to consistently exceed 15% and in one 27-day interval reaches almost
50%. This is partly because these are percentage errors and the values of am are low, but also because by averaging 24 stations, am has much greater sensitivity at low values than a [ K ] values from
a single station. For these 27-day intervals the mean ε [rms] is 9.2% and this is reduced to 3.1% in annual mean data. Hence the procedure we deploy makes modern stations give, to a very good degree
of accuracy, highly consistent corrected a [ K ] values, even though they cover a much wider range of δ, and hence correction factors s(δ), than are covered by the aa stations since the start of the
aa data in 1868. We estimate that for the range of s(δ) involved in the historic aa data, the latitudinal correction procedure for annual means is accurate to better than 1% on average.
Fig. 6
Top: Scaled variations of modern a [K ]values from various stations using the station location correction procedure used in this paper. For all stations, the observed a [K ]values have been
corrected for any secular magnetic field change by dividing by the s(δ) factor and then scaled to the am index using the linear regression coefficients m and i obtained from the calibration
interval (2002–2009, inclusive). The plot shows 27-day Bartels rotation means for data from: (mauve) Sodankylä, SOD; (brown) Eskdalemuir, ESK; (orange) Niemegk, NGK; (red) Hartland, HAD; (blue)
Canberra, CNB; and (green) a spline of Gangara, GNA and nearby Gingin, GNG (see text for details). The black line is the am index. Bottom: the rms. fit residual of the re-scaled station a [K ]
indices compared with the am index, ε [rms], for the 27-day means. The average of ε [rms] for the whole interval shown (1995–2017), is 〈ε [rms]〉 = 9.7%
As discussed in the introduction, a major application of the aa index is in reconstructing the near-Earth interplanetary conditions of the past and so it is useful to evaluate if the errors shown in
Figure 6 are significant in this context. The data in Figure 6 are restricted after 1995 because this allows to make comparisons with near-continuous data from near-Earth interplanetary space.
Lockwood et al. (2018c) have shown that gaps in the interplanetary data series render most “coupling functions” (combinations of near-Earth interplanetary parameters used to explain or predict
geomagnetic disturbance) highly inaccurate if they are derived using data from before 1995. By introducing synthetic data gaps into near-continuous data, these authors show that in many cases
differences between derived coupling functions can arise because one is fitting to the noise introduced by the presence of many and long data gaps. After 1995 the WIND, ACE and DISCOVR satellites
give much more continuous measurements with fewer and much shorter data gaps. Because of the danger of such “overfitting”, Lockwood et al. (2018c) recommend the power input into the magnetosphere, P
[ α,] as the best coupling function. This is because P [ α ] uses the theoretical basis by Vasyliunas et al. (1982) to reduce the number of free fit variable to just one, the coupling exponent α, and
yet achieves almost as high correlations with range geomagnetic indices as coupling functions that have separate exponents for different solar wind variables which, if they do achieve a slightly
higher correlation, tend to do so by overfitting and with reduced significance because of the increased number of free fit parameters. The equation for P [ α ] shows a dependence on B ^2α V [SW] ^(7/
3-α)(m [sw] N [sw])^(2/3-α) (where B is the interplanetary magnetic field V [SW] is the solar wind speed and (m [sw] N [sw]) is the mass density on the solar wind) and so accounts for all three
near-Earth interplanetary parameters with one free fit parameter, the coupling exponent, α. This is much preferable to forms such as B ^a V [SW] ^b(m [sw] N [sw])^c which have three free fit
parameters and so are much more prone to “overfitting”.
In evaluating P [ α ], great care is here taken in handling data gaps because the often-used assumption that they have no effect on correlation studies can be a serious source of error. As pointed
out by Lockwood et al. (2018c), the much-used Omni2 interplanetary dataset gives an hourly mean value even if there is just one sample available within the hour. This is adequate for parameters such
as V [SW] that have high persistence (i.e. long autocorrelation timescales) but inadequate for parameters such as the IMF orientation factor that has and extremely short autocorrelation timescale.
Another complication is that, although coupling functions made by averaging interplanetary parameters and then combining them are valid and valuable, they are not as accurate as ones combined at high
time resolution and then averaged. Hence we here start from 1-minute Omni data (for after 1995 when data gaps are much fewer and shorter). Hourly means of a parameter are then constructed only when
there are sufficient 1-minute samples of that parameter to reduce the uncertainty in the hourly mean to 5%. The required number of samples for each parameter was obtained from the Monte-Carlo
sampling tests carried out by Lockwood et al. (2018c). From these data, hourly means of P [ α ] are constructed (for a range of α values between 0 and 1.25 in steps of 0.01). Note that a data gap in
the P [ α ] sequence is formed if any of the required parameters is unavailable. These hourly P [ α ] samples are then made into 3-hourly means (matching the 8 time-of-day intervals of the
geomagnetic range indices) only when all three of the required hourly means of P [ α ] are available. Lastly, as used by Finch & Lockwood (2007), each geomagnetic index data series is masked out at
times of the data gaps in the 3-hourly P [ α ] samples (and the P [ α ] data correspondingly masked out at the times of any gaps in the geomagnetic data it is being compared to) so that when averages
over a longer interval are taken (we here use both 27-day Bartels solar rotation intervals and 1-year intervals) only valid coincident data are included in the averages of both data sets to be
correlated. We find this rather laborious procedure improves the correlations and removes many of the apparent differences between the responses of different geomagnetic observatories.
Figure 7 shows the resulting correlograms for the Bartels rotation (27-day) means for the stations also used in Figure 6. The correlation coefficient is shown as a function of the coupling exponent,
α. The peak correlations for these 27-day means are of order 0.93 and rise to over 0.98 for annual means. Using the three separate exponents a, b and c (discussed above) causes only very small
increases in the peak correlation that are not statistically significant when one allows for the additional number of degrees of freedom. The optimum exponent for am for the 27-day means is α = 0.45
± 0.07 (see Lockwood et al. (2018c) for description of the two error estimation techniques that are used to generate these 1-σ uncertainties) giving a peak correlation of 0.93. For annual means the
peak correlation for am is 0.99 at α = 0.44 ± 0.02 (Lockwood et al., 2018c). The optimum values for all but two of the a [ K ] stations tested fall in, or close to, this range (shown by the coloured
dots and vertical dashed lines). The optimum α for Sodankylä (0.42 ± 0.10, in mauve), Niemegk (0.46 ± 0.09, in orange), Hartland (0.42 ± 0.09, in red), Canberra (0.42 ± 0.11, in blue), for Gangara/
Gingin (0.49 ± 0.12, in green) and Eskdalemuir (0.56 ± 0.16, in brown) all agree with that for am to within the estimated uncertainties and all show considerable overlap in estimated uncertainty
range with that for am. Note that the peak correlation coefficient is also considerably lower for ESK and we find, in general, that increased geomagnetic station noise, and in particular lower
instrument sensitivity, increases the optimum α (and its uncertainty range) as well as lowering the peak correlation. We find no consistent variation with magnetic latitude nor with the minimum
distance to the auroral oval, δ and effectively the same coupling exponent applies at Sodankylä (considerably closer to the auroral oval than any of the aa stations at any date) as at Gangara/Gingin
(further away from the auroral oval than any aa stations at any date). Hence this test shows that the changing magnetic latitudes of the aa stations is not introducing long-term changes into the
response of the index to interplanetary conditions.
Fig. 7
Correlogams showing the correlation between 27-day Bartels solar rotation means of power input into the magnetosphere, Pα, with the corrected a [K ]indices, a [K ]/s(δ), as a function of the
coupling exponent, α. The colours are for the same data as used in Figure 6: (mauve) Sodankylä, SOD; (brown) Eskdalemuir, ESK; (orange) Niemegk, NGK; (red) Hartland, HAD; (blue) Canberra, CNB; and
(green) a spline of Gangara, GNA and nearby Gingin, GNG (see text for details). The black line is the am index. The coloured dots and vertical dashed lines show the optimum α that gives the peak
correlation. The horizontal bars show the uncertainty in the optimum α which is the larger of the two 1-σ uncertainties computed using the two procedures described by Lockwood et al. (2018c).
3 Recalibrating the stations
The drift in the scaling factors will have influenced the intercalibration of the stations. Consider the Abinger-Hartland join in 1957, which has been the cause of much debate, as discussed in
Section 1.2. By end of the interval of the Abinger data, the use of a constant scale factor means that the classic aa was giving aa [N] values that were too high by 1.44/2 = 0.72%, compared to the
mean value for the Abinger interval. On the other hand, for the start of the Hartland data, classic aa [N] values were too low compared to the average for the Hartland interval by 4.41%. Given that
the average aa [N] value was 24.6 nT for 1956 and 31.6 nT for 1957, this makes a difference of 1.6 nT which is approximately half that required to explain the apparent calibration skip between the
Abinger and Hartland data. This throws a new light on the calibration “glitch” at the ABN-HAD join which can be regarded as being as much a necessary correction to allow for the effect of the drift
in the intrinsic magnetic field as a calibration error.
If we knew the precise dates for which the classic aa index (constant) scalefactors applied, we could generalize them using the s(δ) factors and so employ Mayaud’s original station intercalibrations.
However, these dates are not clear and so the corrected indices aa [N]/s(δ) and aa [S]/s(δ) need new intercalibrations, which is done in this section using independent data. We take the opportunity
to make calibrations that can also allow for other potential factors, such as any change in the subtraction of the regular diurnal variation associated with the change from manual to automated
scaling. For both the two northern hemisphere station changes we use data from the Niemegk (NGK) station in Germany, K indices from where are available from 1890. Figure 4c shows that the s(δ) factor
is relatively constant for NGK (orange line) but there are nevertheless some small changes (the range of variation in s(δ) for NGK in Figure 4c is 1.8%). Hence we use a [NGK]/s(δ), where a [NGK] is
scaled from the NGK K values using the standard mid-class amplitudes scale (K2aK). For the southern hemisphere we have no independent K-index record that is as long, nor as stable, as that from NGK.
For the Toolangi-Canberra join, we use the am index (compiled for a network of stations in both hemispheres, Mayaud, 1980; Chambodut et al., 2013), but find we get almost identical results if we use
the southern hemisphere component of am, as, or its northern hemisphere component, an, or even a [NGK]/s(δ). For the Melbourne-Toolangi join we have no other data of the duration and quality of
Niemegk and so we use use a [NGK]/s(δ).
The procedure used is to take 11 years’ data from each side of the join (roughly one solar cycle). For both the “before” and “after” interval we compare the aa station data with the calibration
station data. We employ daily means, thereby averaging out the diurnal variations. As discussed in the next paragraph, we carry out the calibration separately for eight independent equal-length
time-of-year (F) ranges in which we regress the corrected aa station data against the corrected calibration set (for the 11 years before and after the join, respectively). This means that each
regression is carried out on approximately 500 pairs of daily mean values (11 × 365/8). All regressions were tested to ensure problems did not arise because of lack of homoscedacity, outliers,
non-linearity, inter-dependence and using a Q-Q test to ensure the distribution of residuals was Gaussian (thereby ensuring that none of the assumptions of ordinary least squares regression, OLS, are
violated). The scatter plot was also checked in the 11 annual-mean data points because the main application of the regressions in this paper is to annual means. The “before” and “after” regressions
were then compared, as discussed below.
There are a number of reasons to be concerned about seasonal variation in magnetometer calibrations. These may be instrumental, for example early instruments were particularly temperature and
humidity sensitive. In addition, induced Earth currents can depend on the height of the water table (although their effect is predominantly in the vertical rather than the horizontal components). In
the case of Hartland, its coastal location makes ocean currents, and their seasonal variation, a potential factor. All these may differ at different sites. The conductivities of the ionosphere, and
their spatial distribution above the station, and between the station and the auroral oval, will have a strong seasonal component and again this factor may not be exactly the same at different sites.
Possibly the largest concern is the quiet-time regular variation, S [R], that must be subtracted from the data before the range is evaluated and this correction may vary with season as the S [R]
pattern moves in location over the year (Mursula et al., 2009). We note that Matthes et al. (2016) used the Ap index, derived from a wider network of mid-latitude magnetometers, to re-calibrate the
Abinger-Hartland join in the aa [N] data and found that the calibration required varied with time-of-year. For this reason, the calibrations were carried out separately in the 8 independent
time-of-year (F) bins: the number of F bins was chosen as a compromise between resolution of any annual variation and maintaining a high number of samples in each regression. Although, there was
general agreement between the results from the different F bins, there were also consistent differences at some times of year. Note that this procedure allows us to re-calibrate not only instrumental
effects but also any changes in the background subtraction and scaling practices used to derive the K-indices. Scaling has changed from manual to automated and although the latter are repeatable and
testable, the former are not; however, it helps increase homogenity that most of the classic aa data up to 1968 was scaled by Mayaud himself. Lastly, we note that Bartels recognized the need to allow
for changes during the year in the intercalibration of stations because the conversion factors that he derived (and are still used to this day to derive the Kp index) not only depend on the station
location, the Universal Time, and the activity level, but also depend on the time of year. Bartels employed 4 intervals in the year with three calibration categories (summer, winter and equinox).
By virtue of its more extensive network of stations in both hemispheres, and its use of area-weighted groupings, the am index is, by far, the best standard available to us for a global range index.
Starting in 1959, it is coincident in time with all the Canberra data and almost all of the Hartland data. It therefore makes good sense to scale both the aa [N]/s(δ) and aa [S]/s(δ) data to recent
am data, and then “daisy-chain” the calibration back to the prior two stations. As noted in the case of the sunspot number data composite (Lockwood et al., 2016), there are always concerns about
accumulating errors in daisy chaining; however, we note that the calibration is here passed across only two joins in each hemisphere and the correlations with independent data used to calibrate the
joins are exceptionally high. Furthermore, we have an additional check (of a kind not available to use when making the many joins needed for the sunspot number composite), namely that we have
independent data from other stations (and equivalent data in the IHV index) that continues through much of the sequence and across all four joins. Strictly-speaking, the Niemegk data are also a
composite, the data series coming from three nearby sites that are within 40 km of each other: Potsdam (1880–1907), Seddin (1908–1930), and Niemegk (1931–present). The site changes were made to
eliminate the influence of local electrical noise. Of these site changes, only that in 1930 falls within the 11-year calibration periods (either side of an aa station change) that are deployed here,
being 5 years after the Greenwich-Abinger join and 10 years after the Melbourne-Toolangi join. We note there are probably improvements that could be made to the Potsdam/Seddin/Niemegk a [NGK]
composite, particularly using data from relatively nearby observatories, such as Swider (SWI), Rude Skov (RSV), Lovö (LOV) and Wingst (WNG) (e.g. Kobylinski and Wysokinski, 2006). Using local
stations is preferable because the more distant they are, the larger the difference in the change in their s(δ) factors and hence the more they depend on the main field model used. Some calibration
jumps in a [NGK] have been discussed around 1932 and 1996: the latter is not in an interval used for calibration in this paper, but 1932 does fall within the 22-year spline interval used to calibrate
the Greenwich-Abinger join in 1925.
To test the suitability of the Niemegk a [ K ] index data for use as a calibration spline, we search for long-term drifts relative to independent data. Given that fluctuations within the 11-year
“before” and “after” intervals will be accommodated by the relevant regression with the aa station data, our only concern is that the mean over the before interval is consistent with that over the
after interval. One station that provides K-indices that cover all the aa calibration intervals is Sodankylä (SOD) from where K-index data is available since 1914 and the SOD data have been used to
test and re-calibrate aa in the past (Clilverd et al., 2005). The correlation between daily means of a [NGK ] and a [SOD] exceeds 0.59 for the calibration intervals and the corresponding correlation
of annual means always exceeds 0.97. However, this is not an ideal site (geographic coordinates 67.367°N, 26.633 E) in that it is closer to the auroral oval than the mid-latitude stations that we are
calibrating: its δ falls from 6.11 in 1914 to 4.69 in 2017 and these δ values are below the range over which Mayaud recommends the use of the polynomial given in Equation (1). Figure 3 highlights why
this a concern, as it shows that the effects of secular changes in the geomagnetic field on the required scaling factor are increasingly greater at smaller δ. Equation (1) predicts that s(δ) for
Sodankylä (SOD) will have risen from 2.302 to 2.586 over the interval 1914–2017, which would make the corrected SOD data more sensitive to the secular change correction than the data from
lower-latitude stations. However, at this point we must remember that in applying Equation (1) to the SOD data we are using it outside the latitude range which Mayaud intended it to be used and also
outside the latitude range of data that Mayaud used to derive it. However, Figure 6 shows that using Equation (1) with SOC data over two solar cycles has not introduced a serious error into the a
[SOD]/s(δ) and so it does supply a valuable additional test of the NGK intercalibration data (which also covers 2 solar cycles).
Nevertheless, because of these concerns over the a [SOD]/s(δ) data, we have also used data from other stations, in particular the K-indices from Lerwick (LER) and Eskdalemuir (ESK) for the 22 years
around the Abinger-Hartland join. We find it is important to correct the K-indices from these stations to allow for effect of changing δ because otherwise one finds false drifts relative to Niemegk,
where the change in δ has been much smaller (see Fig. 4). The procedure employed here is to linearly regress 〈a [NGK]/s(δ)〉[τ=1yr] and 〈a [XXX]/s(δ)〉[τ=1yr,] where XXX is a generic IAGA code of
the station used (giving regression slope α and intercept β) then compare the ratio$M = 〈 a NGK / s ( δ ) 〉 τ = 11 yr / ( α 〈 a XXX / s ( δ ) 〉 τ = 11 yr + β )$(2)for the 11-year intervals before
and after (M [B] and M [A], respectively). The ideal result would be M [A]/M [B] = 1, which would mean that any change across the join in a [NGK]/s(δ) and a [XXX]/s(δ) was the same. Because it is
highly unlikely that Neimegk and station XXX share exactly the same error at precisely the time of the join, this would give great confidence in the intercalibration.
The steps taken to generate the “homogenous” aa indices, aa [HN], aa [HS] and aa [H], are given sequentially in the following subsections. It should be noted that we are using daisy chaining of
calibrations which was partially avoided in the classic aa index only because it was assumed that the station scale factors were constant, an assumption that we here show causes its own problems.
Even then, the use of the station scale factors was, in effect, a form of daisy chaining.
3.1 Scaling of the Hartland and Canberra data to the am index
The first step is to remove the constant scale factors used in the compilation of the classic aa index to recover the 3-hourly a [ K ] indices, i.e. for Greenwich we compute a [GRW] = [aa [N]][GRW]/
1.007, and similarly we use a [ABN] = [aa [N]][ABN]/0.934, a [HAD] = [aa [N]][HAD]/1.059, a [MEL] = [aa [S]][MEL]/0.967, a [TOO] = [aa [S]][TOO]/1.033, and a [CNB] = [aa [S]][CNB]/1.084. Given that
the major application of the aa index is to map modern conditions back in time, it makes sense to scale a new corrected version to modern data. Hence we start the process of generating a new,
“homogeneous” aa data series by scaling modern a [ K ]/s(δ) data (i.e. the a [ K ] values corrected for the secular change in the geomagnetic field) against a modern standard. We use the am index as
it is by far the best range-based index in terms of reducing the false variations introduced by limited station coverage and being homogeneous over time in the distribution stations it has taken data
from. However, it contains no allowance for the effects of long-term change in the geomagnetic field and therefore we carry out scaling of a [HAD]/s(δ) and a [CNB]/s(δ) data (from Hartland and
Canberra, respectively) against am for a limited period only. We employ daily means (Am, A [CNB] and A [HAD]) to average out the strong diurnal variation in the a [ K ] indices caused by the use of
just one station and the (much smaller) residual diurnal variation in am caused by the slightly inhomogeneous longitudinal coverage (particularly in the southern hemisphere) of the am stations. We
use an interval of 7 years because we find that it is the optimum number to minimise estimated uncertainties: we employ 2002–2009 (inclusive) because that interval contains the largest annual mean aa
index in the full 150-year record (in 2003) and also the lowest in modern times (in 2009), which is only slightly larger than the minimum in the whole record. Hence this interval covers almost the
full range of classic aa values. The correlation of the daily means in this interval (23376 in number) are exceptionally high being 0.978 for Am and A [HAD]/s(δ) and 0.969 for Am and A [CNB]/s(δ).
Linear regressions (ordinary least squares) between these pairs of data series pass all tests listed above and yield the scaling factors given in Table 1. In all regressions between data series we
use both the slope (i.e. a gain term, s [c]) and the intercept (an offset term, c [c]) because, in addition to differences in instrument sensitivity, noise levels and background subtraction means
that there may, in general, also be zero-level differences. Hence we scale a [HAD]/s(δ) from Hartland using:$[ aa HN ] HAD = 0.9566 . aa HAD / s ( δ ) - 1.3448 ( for 1957 – present )$(3) and we scale
a [CNB]/s(δ) from Canberra using: $[ aa HS ] CNB = 0.9507 . a CNB / s ( δ ) + 0.4660 ( for 1980 – present )$(4)
Table 1
The correlation coefficients (r [b] and r [a] for daily means in 11 years before and after the joins, respectively) and the slope s [c] and intercept c [c] for recalibrating stations for the 8
time-of-year (F) bins employed.
Over the interval 1980–present, this gives a distribution of 3-hourly ([aa [HN]][HAD] − [aa [HS]][CNB]) values with a mode value of zero, which means there is no systematic difference between the
re-scaled indices from the two sites.
3.2 Inter-calibration of the Hartland and Abinger
Figure 8 details the method by which the Abinger data is calibrated to provide a backwards extension of the Hartland data which is as seamless as possible. As discussed above, the calibration was
separated into 8 independent, equal-duration bins of the fraction of the year, F. Bin 1 is for 0 ≤ F < 0.125; bin 2 is 0.125 ≤ F < 0.25; and so on, up to bin 8 for 0.875 ≤ F ≤ 1. The left hand column
of Figure 8 shows scatter plots between the a [ABN]/s(δ) values (i.e. the classic aa values from Abinger after removal of the original scalefactor correction and allowance for the effect of the
changing intrinsic field) against the a [NGK]/s(δ) values (the similarly-corrected values from the Niemegk K indices) for the 11-year period before the join and the middle column gives the scatter
plots of the corrected and re-scaled aa index values from Hartland, [aa [HN]][HAD], as given by Equation (2), for the 11-year period after the join, again against the simultaneous a [NGK]/s(δ)
values. In each case, the grey dots are the scatter plot for daily values and black dots are the annual means (for the range of F in question). The correlation coefficients for the daily values are
given in Table 1 (we do not give the corresponding correlations for annual means as they all between 0.99 and 0.999 but of lower statistical significance, coming from just 11 samples). The red lines
are linear least-squares regression fits to the daily values and all tests show that this is appropriate in all cases. The third column plots the best linear fit of a [NGK]/s(δ) in the interval after
the join (“fit 2”) as a function of the best linear fit of a [NGK]/s(δ) in the interval before the join (“fit 1”). The dashed line is the diagonal and would apply if the relationship of the data
before the join to a [NGK]/s(δ) were identical to that after the join. The red lines in the right-hand column have slope s [c] and intercept c [c]. Assuming that there is no discontinuity in a [NGK]/
s(δ) coincidentally at the time of the join (which means that the relationship between the calibration data and the real aa index before the join is the same as that after the join) we can calibrate
the Abinger data (corrected for secular drift) with that from Hartland (rescaled to am, as discussed in the previous section) for a given F using:$[ aa HN ] ABN ( F ) = s c ( F ) . a ABN ( F ) / s (
δ ) + c c ( F )$(5)
Fig. 8
The intercalibration of aa [N] data across the join between the Hartland (HAD) and Abinger (ABN) observations in 1957. The data are divided into eight equal-length fraction-of-year (F) bins, shown
in the 8 rows, with the bottom row being bin 1 (0 ≤ F < 0.125) and the top row being bin 8 (0.875 ≤ F < 1). The left-hand column is for an interval of duration 11-years (approximately a solar
cycle) before the join and shows scatter plots of the aa data from Abinger (after division by s(δ) to allow for secular changes in the geomagnetic field) against the similarly-corrected
simultaneous NGK data, a [NGK] /s(δ). The middle column is for an interval of duration 11-years after the join and shows the corresponding relationship between the already-homogenized aa data from
Hartland [aa [H]][HAD] and the simultaneous a [NGK] /s(δ) data. All axes are in units of nT. The grey dots are daily means to which a linear regression gives the red lines which are then checked
against the annual means (for the F bin in question) shown by the black dots. The right-hand column shows the fitted lines for the “before” interval, 1, against the corresponding fitted line for
the “after” interval, 2: the red line would lie on the dotted line if the two stations had identical responses at the F in question. The slope and intercept of these lines, giving the
intercalibration of the two stations at that F, are given in Table 1.
The first group of values in Table 1 gives the s [c] and c [c] values in each F bin for this join between the HAD and ABN data. We here ascribe these values to the centre of the respective F bin and
used PCHIP interpolation to get the value required for the F of a given [aa [N]][ABN] data point. The annual variations in both s [c] and c [c] are of quite small amplitude but are often not of a
simple form. This is not surprising considering the variety of different factors that could be influencing the variations with F, and that they are not generally the same at the two stations being
inter-calibrated nor at Niemegk.
We use the variation with F of both the scaling factor, s [c], and the offset, c [c], because at least some of the variation of the intercalibration with F will be associated with the seasonal
variation in the regular diurnal variations at the two sites and the background subtraction, which could give offset as well as gain (sensitivity) differences between the two sites.
Inspection of Figure 8 and Table 1 show that there is a variation with F in the relationship between the two sites and our procedure takes account of this. Note that the intercept values are all
small and that the red lines are actually shifted from the diagonal by the ratio of the classic aa scalefactors. This emphasizes that the data from these two stations is, after allowance had been
made for the secular geomagnetic drift through the s(δ) factor, similar. This reinforces the point that the large “calibration skip” between the Hartland and Abinger aa [N] values that has been
widely discussed in the literature was, in the main, a necessary correction step to allow for the effects of the secular changes in the intrinsic field. Hence making a correction for this apparent
calibration error, without first correcting for the temporal variation in the scaling factor s(δ), is only a first order correction and will give somewhat incorrect results in general.
As discussed above, we use Equation (2) to check the intercalibration data from Niemegk, where station XXX is SOD, LER and ESK for this join. If we do not correct for the effect of changing δ on the
scaling factor s(δ) for these stations, we obtain values of M [A]/M [B] of between 1.018 and 1.052, which implies there is drift in the average Neimegk data (to values that are slightly too low) of
between about 3% and 5% over the intercalibration interval. However, after correcting the change in the stations’ δ (in the same way as done for the aa stations and Niemegk in Fig. 4) we get an M [A]
/M [B] of 1.053, 1.022 and 0.946 for LER, ESK and SOD, respectively. Giving these 3 estimates equal weight gives an average of 1.007, which implies the Niemegk calibration is stable to within 0.7%
for our purposes. We note that this is not a test that we can repeat in such detail for all station joins. Hence we do not attempt to correct the NGK intercalibration data, beyond allowing for the
effect of the drift in δ on s(δ). However, note that we will test this approach in the level of agreement in the final full aa [HN] and aa [HS] data sequences and in section 5, we will compare the
long-term variation of these new aa indices with the equivalent IHV index as well as with a [NGK]/s(δ), a [ESK]/s(δ) and a [SOD]/s(δ).
3.3 Inter-calibration of Abinger and Greenwich
Figure 9 corresponds to Figure 8, but is for the join between the Abinger and Greenwich data. Note that because the “after” data in this case are the corrected and re-scaled Abinger data, [aa [HN]]
[ABN] given by Equation (2), the slope and intercept values (s [c] and c [c]) for this join are influenced by both the scaling of the Hartland data to am and by the Abinger-to-Hartland join. Hence
the calibration of Hartland against am is passed back to Greenwich, as is in the nature of daisy-chaining. Given the data are taken from older generations of instruments and the fact that this second
join is influenced by the first, we might have expected the plots to show more scatter than in Figure 8. In fact this is not the case and Table 1 shows the correlations are actually slightly higher
for this intercalibration than the one discussed in the last section. Because concerns have been raised about a potential skip in the calibration of the a [NGK] composite in 1932, we use an “after”
interval of 1926–1931 (inclusive, i.e. 6 years rather than the 11 years used for other joins). The correlations for all 8 F bins were indeed found to be marginally lower if the full 11 years
(1926–1936) were used but the regression coefficients were hardly influenced at all.
Fig. 9
The same as Figure 8 for the join between the Greenwich and Abinger data.
The corrected Sodankylä K-indices give M [A]/M [B] = 0.943 for this join which could imply a 6% problem with the Niemegk spline. However, we note that Sodankylä gave a lower value than the average
for the Abinger-Hartland calibration interval which is likely to be a consequence of its close proximity to the auroral oval. As for that join, we here use the Niemegk data as a calibration spline
without correction, but will test the result in Section 5.
The Greenwich data are intercalibrated using the equivalent equation to Equation (4):$[ aa HN ] GRW ( F ) = s c ( F ) . a GRW ( F ) / s ( δ ) + c c ( F )$(6)using the appropriate s [c] and c [c]
values given in Table 1 and the interpolation in F scheme described above.
3.4 Inter-calibration of the southern hemisphere stations
Figures 10 and 11 are the same as for Figure 8 for the joins between, respectively, the Canberra and Toolangi stations and between the Toolangi and Melbourne stations (note that the colours of the
regression lines matches the colours used to define the joins in Fig. 2). The Toolangi and Melbourne data are corrected using the corresponding Equations to (4) and (5) to give [aa [HS]][TOO] and [aa
Fig. 10
The same as Figure 8 for the join between the Toolangi and Canberra data.
Fig. 11
The same as Figure 8 for the join between the Melbourne and Toolangi data.
Figure 10 uses the am data to make the Canberra-Toolangi intercalibration but, as mentioned above, almost identical results were obtained if either the as index or a [NGK]/s was used. Using a [NGK]/s
did increase the scatter in the daily values slightly, but the regression fits remained almost exactly the same. In the case of the Toolangi-Melbourne join, the best comparison data available are the
Niemegk K indices, but based on the above experience of using it for the Canberra-Toolangi join, it is not a major concern that the intercalibration data are from the opposite hemisphere, although,
as expected, it does increase the scatter between the daily means.
Note that the only operation to make aa [HN] and aa [HS] similar is the scaling of both to am over the interval 2002–2009, achieved by Equations (2) and (3). Thereafter the northern and southern data
series are generated independently of each other. Therefore the degree to which the two hemispheric indices agree with each other over time becomes a test of the intercalibrations and the stability
of the datasets.
4 The homogeneous composite
We can then put together 150-year composite of aa [HN] (using [aa [HN]][GRW], [aa [HN]][ABN], and [aa [HN]][HAD]) and the red line in Figure 2b shows the resulting variations in annual means. The
blue line is the corresponding composite of aa [HS] (using [aa [HS]][MEL], [aa [HS]][TOO], and [aa [HS]][CNB]). Comparison with Figure 2a shows that the calibrations described in the previous section
have produced hemispheric data series which agree much more closely with each other than do aa [N] and aa [S.] To quantify the improvement, Figure 12 compares the distributions of the differences in
daily means of northern and southern hemisphere indices in 50-year intervals, Δ[NS]. The top row is for the classic aa indices (so Δ[NS] = aa [N] − aa [S]). The bottom row is for the homogenised aa
indices (so Δ[NS] = aa [HN] − aa [HS]). The left column is for 1868–1917 (inclusive); the middle column for 1918–1967; and the right-hand column for 1968–2017. Note that distributions are narrower
and taller for the first time interval because mean values were lower and so hemispheric differences are correspondingly lower.
Fig. 12
Distributions of the differences in daily means of northern and southern hemisphere indices, Δ[NS], for 50-year intervals. The top row is for the classic aa indices, so that Δ[NS] = aa [N] − aa
[S]. The bottom row is for the homogenised aa indices, so that Δ[NS] = aa [HN] − aa [HS]. Parts (a) and (d) are for 1868–1917 (inclusive); parts (b) and (e) are for 1918–1967; and parts (c) and (f)
are for 1968–2017. In each panel, the vertical orange line is at Δ[NS] = 0, the vertical cyan line is the median of the distribution, the vertical red line the mean (〈Δ[NS]〉), and the green lines
the upper and lower deciles. Note they are plotted in the order, orange, cyan, then red and so the mean can overplot the others (this particularly occurs in the bottom row). In each panel the
distribution mean, 〈Δ[NS]〉 and the standard deviation, σ[ΔNS], are given.
A number of improvements can be seen in the distributions for (aa [HN] − aa [HS]), compared to those for (aa [N] − aa [S]). Firstly the mean of the distributions has been reduced to zero (to within
10^−3) in all three time intervals by the homogenized index. Not only is this smaller than for the corresponding classical index, but also the upward drift in the mean value Δ[NS] has been removed.
This improvement in the mean difference quantifies the improvement that can be seen visually by comparing Figures 2a and b. Secondly, the width of the distribution in (aa [HN] − aa [HS]) is always
lower than for the corresponding distribution of (aa [N] − aa [S]): this can be seen in the given values of the standard deviation, σ[ΔNS] and in the separation of the decile values (which are given
by the vertical green lines). Thirdly the Δ[NS] distributions for the classic index show a marked asymmetry: this can be seen by the fact that the median of the distributions (vertical cyan line) is
consistently smaller than the mean and that the modulus of the lower decile value is always less than the upper decile value. This asymmetry has been removed completely in the homogenized data series
after 1917. (For 1868–1917 the 1-σ points are symmetrical but the mode is slightly lower than the mean.) Lastly the distributions for the classic index show a tendency for quantized levels
(particularly for 1868–1917) and more kurtosis in shape than for the homogenized indices. On the other hand, (aa [HN] − aa [HS]) shows very close to a Gaussian form at all times. If there is a
physical reason why the distribution should diverge from a Gaussian, it is not clear. Hence, agreement between the northern and southern hemisphere indices has been improved, in many aspects, by the
process described in this paper.
Lastly, Figure 2c compares the annual means of the homogenised aa index derived here, defined by$aa H = ( aa HN + aa HS ) / 2$(7)with the classic aa index and the corrected aa index, aa [C], that was
generated by Lockwood et al. (2014) by correcting the classic aa index for the Hartland-Abinger intercalibration using the Ap index. The black line is the aa [H] index from Equation (6) and so
contains allowance for the secular drift in the main field and for the re-calibration of stations presented in Section 3. The mauve line is the classic aa index. It can be seen that, because of the
scaling to the recent am index data, the aa [H] index values are always a bit lower than aa. The cyan line and points show annual means in the am index. It is noticeable that as we go back in time
towards the start of these data, these am means follow the classic aa rather well and so become slightly larger than the corresponding annual means in aa [H]. This indicates that the secular drift in
the intrinsic geomagnetic field is having an influence on even am over its lifetime. The orange line is the corrected aa data series, aa [C]. By definition, this is the same as aa before the
Abinger-Hartland join 1957: hence the orange line lies underneath the mauve one in this interval. Between 1957 and 1981, aa [C] is slightly larger than aa [H] most of the time, but after 1981 the
orange line can no longer be seen because it is so similar to aa [H]. Hence correcting for the Abinger-Hartland join, without correcting for the effects of the secular drift in the intrinsic field
have caused corrected indices such as aa [C] (and others like it) to underestimate the upward rise in aa [H].
Taking 11-point running means to average out the solar cycle, aa, aa [C] and aa [H] all give smoothed minima in 1902 of, respectively, 11.66 nT, 11.77 nT and 10.87 nT. The maxima for aa and aa [H]
are both in 1987, shortly after the peak of the sunspot grand maximum (Lockwood & Fröhlich, 2007), being 27.03 nT and 24.25 nT, giving a rises of 15.37 nT in aa and 13.38 nT in aa [H] over the
interval 1902–1987. The corrected index, aa [C], is somewhat different with a value of 24.51 nT in 1987, but a slightly larger peak of 24.80 nT in 1955. Over the interval 1902–1987 the rise in aa [C]
is 12.73 nT.
5 Comparison of the homogenized aa index with the IHV index and corrected a [ K ] values from Niemegk, Eskdalemuir and Sodankylä
The development of the Inter-Hour Variability (IHV) index was discussed in Section 1.2. The most recent version was published by Svalgaard & Cliver (2007). It is based on hourly means of the observed
horizontal magnetic field at each station and its compilation is considerably simpler than, and completely different to, that of the range indices such as aa. It is defined as the sum of the unsigned
differences between adjacent hourly means over a 7-hour interval centered on local midnight (in solar local time, not magnetic local time). The daytime hours are excluded to reduce the effect of the
regular diurnal variation and UT variations are removed assuming an equinoctial time-of-day/time-of-year pattern, which reduces the requirement to have a network of stations with full longitudinal
coverage. Using data from 1996–2003, Svalgaard & Cliver (2007) showed that IHV has major peaks in the auroral ovals, but equatorward of |Λ[CG]| = 55° it could be normalized to the latitude of Niemegk
using a simple ad-hoc function of Λ[CG]. Note that IHV does not allow for the changes in the stations’ Λ[CG] due to the secular change in geomagnetic field. This will be a smaller factor for IHV than
for the range indices as the latitude dependence is weaker. However, in IHV this effect will also be convolved with that of the changing distribution and number of available stations. This is because
the number of stations contributing to the annual mean IHV values tabulated by Svalgaard & Cliver (2007) varies, with just one for 1883–1889, two for 1889–1900, rising to 51 in 1979 and before
falling again to 47 in 2003. Although the removal of the diurnal variation (by assuming an equinoctial variation) and the removal of the Λ[CG] variation (by using the polynomial fit to the
latitudinal variation in the 1996–2003 data) allows the IHV index to be compiled even if only one station is available, such an index value will have a much greater uncertainty because it will not
have the noise suppression that is achieved by averaging the results from many stations in later years. It must be remembered, therefore, that the uncertainties in the IHV index increase as we go
back in time.
Lockwood et al. (2014) show that in annual mean data IHV correlates well (correlation coefficient, r = 0.952) with BV [SW] ^ n , where B is the IMF field strength, V [SW] is the solar wind speed and
n = 1.6 ± 0.8 (the uncertainty being at the 1-σ level), whereas the corrected aa index and gave r = 0.961 with n = 1.7 ± 0.8. The difference in the exponent n is small (and not statistically
significant) and so we would expect the long-term and solar cycle variations in aa and IHV to be very well correlated. Indeed, Svalgaard & Cliver (2007) found that even in Bartel’s rotation period
(27-day) means IHV and the range am index were highly correlated (r = 0.979).
Figures 13a–f compare annual means of the new homogenised indices aa [H,] aa [HN] and aa [HS] to the IHV index. The left-hand plots show the time series and the best-fit linear regression of IHV. The
right plots so scatter plots of the new indices against IHV and the least squares best-fit linear regression line in each case. For comparison, the bottom panel compares the hemispheric homogenized
indices aa [HN] and aa [HS]. The agreement is extremely good in all cases: for aa [HS] and IHV the coefficient of determination is r ^2 = 0.937; for aa [HN] and IHV r ^2 = 0.962; for aa [H] and IHV,
r ^2 = 0.958; and for aa [HN] and aa [HS], r ^2 = 0.992. This level of agreement is exceptionally high, considering IHV is constructed in an entirely different manner, and from different data and
with different assumptions (e.g. it assumes an equinoctial time-of-day/time-of-year pattern). In particular, note that IHV is not homogeneous in its construction as the number of stations
contributing decreases as we go back in time: this would increase random noise but not explain systematic differences. Also IHV only uses nightside data whereas k indices use data from all local
times; however, k indices respond primarily to substorms (see supplementary material file of Lockwood et al., 2018c) which occur in the midnight sector. Also shown in Figure 13 are the corresponding
comparisons with the corrected a [ K ] indices from Niemegk, Sodankylä, and Eskdalemuir a [NGK]/s(δ), a [SOD]/s(δ) and a [ESK]/s(δ) (parts g/h, i/j, and k/l respectively). The coefficients of
determination (r ^2) are 0.945 and 0.958 and 0.914, respectively.
Fig. 13
Comparison of the new homogenised indices aa [H,] aa [HN] and aa [HS], with the IHV index and the δ-corrected a [NGK], a [SOD] and a [ESK] values. The left-hand plots show the time series and their
respective best-fit linear regressions. The right-hand plots show scatter plots of the new indices against the test indices (a [NGK]/s(δ), a [SOD]/s(δ) or IHV) and the least-squares best-fit linear
regression line. The linear correlation coefficient r is given in each case. Parts (a) and (b) are for aa [HN] and IHV; parts (c) and (d) are for aa [HS] and IHV; parts (e) and (f) are for aa [HS]
and IHV; parts (g) and (h) are for aa [H] and the corrected a [K ]values from Niemegk, a [NGK]/s(δ); parts (i) and (j) are for aa [H] and the δ-corrected a [K ]values from Sodankylä, a [NGK]/s(δ),
parts (k) and (l) are for aa [H] and the δ-corrected a [K ]values from Eskdalemuir, a [ESK]/s(δ). For comparison, the bottom panel (parts m and n) compares the hemispheric homogenized indices aa
[HN] and aa [HS]. In each panel, the grey area defines the estimated ±1σ uncertainty in aa [H].
Hence Figure 13 is a good test of the intercalibrations used in constructing aa [HN] and aa [HS] in the context of annual mean data. There are differences between all the regressed variations but
they are small. The internal correlation between the hemispheric aa indices is now greater than that with other equivalent data series: the worst disagreements are that aa [HN] exceeds aa [HS] around
the peak of solar cycle 17 (around 1940) and aa [HS] exceeds aa [HN] around the peak of solar cycle 14 (around 1907). In both cases, the independent data in Figure 13 indicate that the error is in
both aa [HN] and aa [HS] as these data follow aa [H] more closely. In the case of the largest error (around 1940), IHV, a [NGK]/s(δ), a [SOD]/s(δ) and a [ESK]/s(δ) all also suggest that aa [HS] is an
underestimate by slightly more than aa [HN] is an overestimate and so aa [H] is very slightly underestimated, but only by less than 0.5 nT. It should be noted that this largest deviation between aa
[HN] and aa [HS] occurs when the data are supplied by the Abinger and Toolangi observatories, respectively and that Figure 2b shows that aa [HN] and aa [HS] agree more closely both earlier and later
in the interval 1925–1956 when these two stations are used. Hence the deviation is caused by relative drifts in the data from these stations and not by the inter-calibrations developed in this paper.
The grey areas in the left-hand panels of Figure 13 show the estimated ±1σ uncertainty in annual aa [H] estimates, where σ = 0.86 nT is the standard deviation of the distribution of annual (aa [HN] −
aa [HS]) values.
Figure 14 gives a more stringent test of the station joins in the aa [H] index using means over 27-day Bartels rotation intervals. The dots in the left-hand panels show the deviations of 27-means of
the aa [H] index from scaled test indices. Long-term trends in those deviations are searched for by looking at 13-point running means (just under 1 year) of those deviations, shown by the black
lines. The histograms to the right give the overall occurrence distribution of the deviation of the dots from the blacked lines and hence give an indication of the scatter around the trend. In all
cases, the test data are well correlated with aa [H], the linear correlation coefficients for Bartels rotation means being: 0.938 for IHV; 0.964 for a [NGK]/s; 0.948 for a [SOD]/s; and 0.900 for a
[ESK]/s. For comparison, the values are 0.989 and 0.987 for aa [HN] and aa [HS] (but of course they are not independent data from aa [H]). To derive the deviation, the test index is linearly
regressed with aa [H] index over a common calibration interval of 1990–2003 (for which data are available for all indices) and test data that have been scaled aa [H] using the least-squares linear
regression fit for this interval are denoted with a prime. What we are searching for are consistent step-like change in the deviations (on timescales comparable to the ~1 year smoothing time
constant) at the time of one of the aa station joins (which are marked by the vertical dashed lines using the same colour scheme as in Fig. 2). The appearance of such a step in several of the test
data series would indicate a calibration error in aa [H]. The most-tested join is that between Toolangi and Canberra in aa [HS] (blue dashed line). In generating aa [H] this was calibrated using a
spline of the am index. The only step-like feature is shortly after that join in the Eskdalemuir data and, as this is the least well correlated of the test indices and shows the latest scatter in its
27-day means, this is not good evidence for a problem with this join. The much-discussed Abinger-to-Hartland join (red dashed line) also does not produce a consistent signature in the test data, with
equal mean deviations before and after in all the corrected a [K]-index data, i.e. from Sodankyla, Neimegk, Eskdalemuir and aa [HS] (which at this time is data from Toolangi). There is a small step
in the Niemegk deviations around 1970, but this is after the calibration interval for this join (which is 1946–1967). The deviations for IHV’ show a step between 1960.5 and 1963.5, but this is after
the join and coincides with the large fall in all indices seen at the end of solar cycle 19. Hence this appears to be related to a slight non-linearity between the IHV index and range-based indices,
rather than the calibration join. For the two earlier joins, Greenwich-to-Abinger (green dashed line) and Melbourne-to-Toolangi (cyan dashed line), the independent test data available are IHV, the aa
[H] data from the opposite hemisphere and to a lesser extent a [SOD]/s (which only extends back to 1914 which is only 6 years before the MEL-TOO join). Note, however, that IHV is compiled using
hourly means from just 2 stations before 1900, rising to 11 by 1920, and one of stations is Niemegk, and so it does not provide a fully independent test of our station inter-calibrations. We note
that the data use to make the joins, a [NGK]/s, show some fluctuation over the relevant intervals, but no consistent step. The IHV data do show a step 3 years before the MEL-TOO join but this is at
the same time as the strong rise in all indices at the start of cycle 15 from the very low values during the minimum between cycles 14 and 15. Hence, as for the 1961/2 step in IHV, this appears to be
more associated with a slight non-linearity between IHV and a [K] /s values at low activity than with a calibration skip caused by an aa station change.
Fig. 14
(Left-hand plots) Bartels rotation interval (27-day) means of the deviation of various scaled indices from the new aa [H] index (points) and 13-point running means of those 27-day means (black
lines). The indices are all scaled by linear regression to the aa [H] index over the interval 1990–2003 (the interval shaded in gray). A prime on a parameter denotes that this scaling has been
carried out. (Right-hand plots) The distribution of the deviations of the 27-day means (the dots in the corresponding left-hand plot) from their simultaneous 13-point smoothed running means (the
black line in the corresponding left-hand plot). (a) and (b) are for the scaled IHV index, IHV′; (c) and (d) for the corrected and scaled a [K ]index from Niemegk, (a [NGK]/s)′; (e) and (f) for the
corrected and scaled a [K ]index from Sodankylä, (a [SOD]/s)′; (g) and (h) for the corrected and scaled a [K ]index from Eskdalemuir, (a [ESK]/s)′; (i) and (j) for the homogenized northern
hemisphere aa index, aa [HN]; and (k) and (l) for the homogenized southern hemisphere aa index, aa [HS]. The dashed lines in the left-hand plots mark the dates of aa station joins, using the same
colour scheme as used in Figure 2. Horizontal grey lines are at zero deviation.
6 Conclusions
The classic aa indices now cover 150 years, an interval long enough that there are significant effects on the indices due to the effects of secular changes in the intrinsic geomagnetic field. We here
correct for these using the standard approach to calculating K-indices, but making the scale factors employed for each station a function of time. We also show that this improves the
inter-calibration of the range-based data from other stations which had also been influenced by the assumption of constant scale factors. The intercalibrations are shown, in general, to depend on
time-of year (F), which is here accommodated using 8 equal-sized bins in F and interpolating to the date of each 3-hourly measurement. This allows us to correct for seasonal effects on both the
instrumentation and the background subtraction procedures.
We call the corrected data series that we have produced the “homogenized” aa data series, because it eliminates a number of differences between the data series from the two hemispheres. In this paper
we have concentrated on the results in annual means. In a companion paper (Paper 2), we make further allowances for the effects of the variations of each station’s sensitivity with time-of-day and
time-of-year (Lockwood et al., 2018b). These further corrections are carried out such that the annual means presented here (〈aa [H]〉[τ=1yr], 〈aa [HN]〉[τ=1yr], and 〈aa [HS]〉[τ=1yr]) remain
unchanged. In the supplementary material to the present paper we give the annual mean values of the homogenized aa [H], aa [HN], and aa [HS] data series as these will be subject to no further
corrections. We also attach a file containing the annual δ and s(δ) values used to make allowance for the secular field changes. The supplementary data attached to Paper 2 will give the daily and
3-hourly values of aa [H], aa [HN] and aa [HS]. The equations given in the text of the present paper, along with the coefficients given in Table 1, give a complete recipe for generating this first
level of the aa homogenized data series from the classical aa values.
Given the close agreement between the independently-calibrated aa [HN] and aa [HS] indices, we can be confident that the 13.38 nT rise in the 11-year averages of aa [H] is accurate. The standard
error in the difference of annual means of aa [HN] and aa [HS] is ξ[1] = 0.082 nT for 1996–2017 and ξ[2] = 0.039 nT for 1868–1889. Treating these as the uncertainties in the average levels in these
intervals gives an estimate uncertainty in the difference between them of (ξ[1] ^2 + ξ[2] ^2)^1/2 = 0.091 nT which is just ≈1% of the 13.38 nT rise in aa [H]. Also, because aa [HN] and aa [HS] are
calibrated against the am index over the last 5 years, we can also be confident that the values of 10.87 nT and 24.25 nT for 1902 and 1987 are correct to within the above accuracies. Thus the new
11-year smoothed aa [H] values reveal a rise of 123% between these two dates. In comparison, the corresponding rise in the classic aa between these dates was 15.36 nT (132%) and that in aa [C] was
12.73 nT (108%). Therefore, although the rise in the classic aa was excessive (by 9%), correcting for the Abinger-Hartland intercalibration in isolation, without allowing for the drifts caused by the
secular change in the main field, gives an over-correction and the rise in aa [C] is here found to be too small by 15%.
As a result, reconstructions of solar wind parameters that have been based on aa [C], such as the open solar flux, the solar wind speed and the near-Earth IMF (Lockwood et al., 2014) will also have
underestimated the rise that took place during the 20th century and may not have the correct temporal waveform (given that the peak aa [C] was in 1955 rather than 1987). However, it should be noted
that aa [C] was one of four geomagnetic indices (including IHV) used by Lockwood et al. (2014) which means the effect of its underestimation of the rise will be reduced in the reconstructions.
Replacing aa [C] with aa [H] should also have the effect of reducing the uncertainties. Note also that the dependencies of the various indices on the IMF, B, and the solar wind speed, V [SW], are
such that it is the V [SW] estimates that will be most affected. This will be investigated, and the reconstructions amended, in subsequent publications.
A point of general importance to geomagnetic indices is that, as shown in Figure 2c, the am index follows the classic aa series very closely, but as we go back in time towards the start of the am
index in 1959, the homogenized index aa [H] becomes progressively smaller than aa by a consistent amount. Similarly, close inspection of Figure 13 shows that IHV has fallen by slightly more than aa
[H] in this interval. These differences are the effects of changes in the intrinsic geomagnetic field on the indices. In the case of all previous indices based on K values (Kp, ap, am, as, an, and
the classic aa) this arises from drift in the L values (the threshold value of the K = 9 band) in the case of IHV it arises from the normalization to a reference latitude (and potentially also from
the use of an equinoctial pattern used to remove the diurnal variation). Hence we conclude that all geomagnetic indices should make allowance for these effects if they are to be fit for purpose in
studies of space climate change.
Supplementary Material
Supplementary files supplied by authors
Access here
Without either group knowing it, much of the work reported in this paper was carried out completely independently at University of Reading (UoR) and at École et Observatoire des Sciences de la Terre
(EOST) but when we discovered that we had independently reached almost identical results, we decided a single publication would minimise duplication and potential confusion. The authors thank all
staff of ISGI and collaborating institutes, past and present, for the long-term compilation and databasing of the aa and am indices which were downloaded from http://isgi.unistra.fr/data_download.php
. We also thank the staff of Geoscience Australia, Canberra for the southern hemisphere aa-station K-index data, and colleagues at British Geological Survey (BGS), Edinburgh for the northern
hemisphere aa-station K-index data, and the Helmholtz Centre Potsdam of GFZ, the German Research Centre for Geosciences for the Niemegk K-index data. The work at UoR and BGS is supported by the SWIGS
NERC Directed Highlight Topic Grant number NE/P016928/1 with some additional support at UoR from STFC consolidated grant number ST/M000885/1. The work at EOST is supported by CNES, France.
The editor thanks two anonymous referees for their assistance in evaluating this paper.
Cite this article as: Lockwood M, Chambodut A, Barnard L, Owens M, Clarke E, et al. 2018. A homogeneous aa index: 1. Secular variation. J. Space Weather Space Clim. 8, A53.
All Tables
Table 1
The correlation coefficients (r [b] and r [a] for daily means in 11 years before and after the joins, respectively) and the slope s [c] and intercept c [c] for recalibrating stations for the 8
time-of-year (F) bins employed.
All Figures
Fig. 1
Schematic illustration of the generation of K and a [K ]indices. Illustrative variations of the two orthogonal horizontal field components measured at one site are shown, X (toward geographic
north, in blue) and Y (toward geographic east, in orange). These variations are after the regular diurnal variation has been subtracted from the observations. In the fixed 3-hour UT windows (00–03
UT, or 03–06 UT, and so on up to 21–24 UT), the range of variation of both components between their maximum and minimum values is taken, ΔX and ΔY. The larger value of the two is kept and scaled
according to a standard, quasi-logarithmic scale (illustrated by the black and mauve bands to the right) for which all K-band thresholds are set for the site in question by L, the threshold range
value for the K = 9 band. The value of L for the site is assigned according to the minimum distance between the site and a nominal (fixed) auroral oval position. The K value is then converted into
the relevant quantised value of a [K ](in nT) using the standard “mid-class amplitudes” (K2aK) scale. In the schematic shown, ΔX > ΔY, thus the X component gives a K value of 8 (whereas the Y
component would have given a K of 5). Thus for this 3-hour interval, a [K ]value would be 415 nT. In the case of the classic aa indices, the hemispheric index (aa [N] or aa [S], for the observatory
in the northern or southern hemisphere, respectively) is f × a [K ], where f is a factor that is assumed constant for the observing site.
In the text
Fig. 2
Variations of annual means of various forms of the aa index. (a) The published “classic” northern and southern hemisphere indices (aa [N] and aa [S] in red and blue, respectively). Also shown (in
green) is 1.5 × a [NGK], derived from the K-indices scaled from the Niemegk data. The vertical dashed lines mark aa station changes (cyan: Melbourne to Toolangi; green: Greenwich to Abinger; red:
Abinger to Hartland; and blue: Toolangi to Canberra). (b) The homogenized northern and southern hemisphere indices (aa [HN] and aa [HS] in red and blue, respectively) generated in the present
paper. The thick green and cyan line segments are, respectively, the a [NGK] and am index values used to intercalibrate segments. (c) The classic aa data series, aa = (aa [N] + aa [S])/2 (in mauve)
and the new homogeneous aa data series, aa [H] = (aa [HN] + aa [HS])/2 (in black). The orange line is the corrected aa data series aa [C] generated by Lockwood et al. (2014) by re-calibration of
the Abinger-to-Hartland join using the Ap index. (Note that before this join, aa and aa [C] are identical and the orange line is not visible as it is underneath the mauve line). The cyan line and
points show annual means of the am index. The gray-shaded area in (c) is the interval used to calibrate aa [HN] and aa [HS] (and hence aa [H]) against am.
In the text
Fig. 3
The variation of the scale factor s(δ) derived from threshold range value L that defines the K = 9 band, with the minimum angular separation of the station from a nominal auroral oval, δ. This
empirical variation is scaled from Mayaud (1968, 1972) and is the basis of the L values used to scale K-indices from observed range for all mid-latitude stations. The scale factor s(δ) normalizes
to the idealized Niemegk station for which δ = 19° and L = Lo = 500 nT (ideal static Mayaud values).
In the text
Fig. 4
Analysis of the effect of secular change in the geomagnetic field on the aa magnetometer stations using a spline of the IGRF-12 and the gufm1 geomagnetic field models (for after and before 1900,
respectively). (a) The modulus of the corrected geomagnetic latitude, |Λ[CG]| of the stations; (b) the angular separation of the closest approach to the station of a nominal nightside auroral oval
(at |Λ[CG]| = 69°), δ; and (c) the scale factor s(δ) = L/L [o] where L is given as a function of δ by Figure 3 and L [o] = 500 nT, the reference value for the Niemegk station (for which δ is taken
to be 19°) except for Canberra which, because of its more equatorward location, is scaled using L [o] = 450 nT. The northern hemisphere stations are Greenwich (code GRW, in mauve), Abinger (ABN, in
green) and Hartland (HAD, in red). The southern hemisphere stations are Melbourne (MEL, in black), Toolangi (TOO, in cyan) and Canberra (CNB, in blue). Also shown is Niemegk (NGK, in orange: data
available since 1890). Vertical dashed lines mark aa station changes.
In the text
Fig. 5
Variations of (left) the minimum angular distance to the auroral oval, δ, and (right) the scalefactors, s(δ), for the aa stations. The colours used are as in Figure 4 (namely mauve for Greenwich,
green for Abinger, red for Hartland, black for Melbourne, cyan for Toolangi and blue for Canberra). The thin lines are the variations shown in Figure 4 and the thick lines are constant values used
in generating the classic aa. The dot-dash lines in the right-hand panels show the reciprocals of the standard multiplicative correction factors and the thick lines the factors corresponding to the
constant δ values in the left-hand panels.
In the text
Fig. 6
Top: Scaled variations of modern a [K ]values from various stations using the station location correction procedure used in this paper. For all stations, the observed a [K ]values have been
corrected for any secular magnetic field change by dividing by the s(δ) factor and then scaled to the am index using the linear regression coefficients m and i obtained from the calibration
interval (2002–2009, inclusive). The plot shows 27-day Bartels rotation means for data from: (mauve) Sodankylä, SOD; (brown) Eskdalemuir, ESK; (orange) Niemegk, NGK; (red) Hartland, HAD; (blue)
Canberra, CNB; and (green) a spline of Gangara, GNA and nearby Gingin, GNG (see text for details). The black line is the am index. Bottom: the rms. fit residual of the re-scaled station a [K ]
indices compared with the am index, ε [rms], for the 27-day means. The average of ε [rms] for the whole interval shown (1995–2017), is 〈ε [rms]〉 = 9.7%
In the text
Fig. 7
Correlogams showing the correlation between 27-day Bartels solar rotation means of power input into the magnetosphere, Pα, with the corrected a [K ]indices, a [K ]/s(δ), as a function of the
coupling exponent, α. The colours are for the same data as used in Figure 6: (mauve) Sodankylä, SOD; (brown) Eskdalemuir, ESK; (orange) Niemegk, NGK; (red) Hartland, HAD; (blue) Canberra, CNB; and
(green) a spline of Gangara, GNA and nearby Gingin, GNG (see text for details). The black line is the am index. The coloured dots and vertical dashed lines show the optimum α that gives the peak
correlation. The horizontal bars show the uncertainty in the optimum α which is the larger of the two 1-σ uncertainties computed using the two procedures described by Lockwood et al. (2018c).
In the text
Fig. 8
The intercalibration of aa [N] data across the join between the Hartland (HAD) and Abinger (ABN) observations in 1957. The data are divided into eight equal-length fraction-of-year (F) bins, shown
in the 8 rows, with the bottom row being bin 1 (0 ≤ F < 0.125) and the top row being bin 8 (0.875 ≤ F < 1). The left-hand column is for an interval of duration 11-years (approximately a solar
cycle) before the join and shows scatter plots of the aa data from Abinger (after division by s(δ) to allow for secular changes in the geomagnetic field) against the similarly-corrected
simultaneous NGK data, a [NGK] /s(δ). The middle column is for an interval of duration 11-years after the join and shows the corresponding relationship between the already-homogenized aa data from
Hartland [aa [H]][HAD] and the simultaneous a [NGK] /s(δ) data. All axes are in units of nT. The grey dots are daily means to which a linear regression gives the red lines which are then checked
against the annual means (for the F bin in question) shown by the black dots. The right-hand column shows the fitted lines for the “before” interval, 1, against the corresponding fitted line for
the “after” interval, 2: the red line would lie on the dotted line if the two stations had identical responses at the F in question. The slope and intercept of these lines, giving the
intercalibration of the two stations at that F, are given in Table 1.
In the text
Fig. 9
The same as Figure 8 for the join between the Greenwich and Abinger data.
In the text
Fig. 10
The same as Figure 8 for the join between the Toolangi and Canberra data.
In the text
Fig. 11
The same as Figure 8 for the join between the Melbourne and Toolangi data.
In the text
Fig. 12
Distributions of the differences in daily means of northern and southern hemisphere indices, Δ[NS], for 50-year intervals. The top row is for the classic aa indices, so that Δ[NS] = aa [N] − aa
[S]. The bottom row is for the homogenised aa indices, so that Δ[NS] = aa [HN] − aa [HS]. Parts (a) and (d) are for 1868–1917 (inclusive); parts (b) and (e) are for 1918–1967; and parts (c) and (f)
are for 1968–2017. In each panel, the vertical orange line is at Δ[NS] = 0, the vertical cyan line is the median of the distribution, the vertical red line the mean (〈Δ[NS]〉), and the green lines
the upper and lower deciles. Note they are plotted in the order, orange, cyan, then red and so the mean can overplot the others (this particularly occurs in the bottom row). In each panel the
distribution mean, 〈Δ[NS]〉 and the standard deviation, σ[ΔNS], are given.
In the text
Fig. 13
Comparison of the new homogenised indices aa [H,] aa [HN] and aa [HS], with the IHV index and the δ-corrected a [NGK], a [SOD] and a [ESK] values. The left-hand plots show the time series and their
respective best-fit linear regressions. The right-hand plots show scatter plots of the new indices against the test indices (a [NGK]/s(δ), a [SOD]/s(δ) or IHV) and the least-squares best-fit linear
regression line. The linear correlation coefficient r is given in each case. Parts (a) and (b) are for aa [HN] and IHV; parts (c) and (d) are for aa [HS] and IHV; parts (e) and (f) are for aa [HS]
and IHV; parts (g) and (h) are for aa [H] and the corrected a [K ]values from Niemegk, a [NGK]/s(δ); parts (i) and (j) are for aa [H] and the δ-corrected a [K ]values from Sodankylä, a [NGK]/s(δ),
parts (k) and (l) are for aa [H] and the δ-corrected a [K ]values from Eskdalemuir, a [ESK]/s(δ). For comparison, the bottom panel (parts m and n) compares the hemispheric homogenized indices aa
[HN] and aa [HS]. In each panel, the grey area defines the estimated ±1σ uncertainty in aa [H].
In the text
Fig. 14
(Left-hand plots) Bartels rotation interval (27-day) means of the deviation of various scaled indices from the new aa [H] index (points) and 13-point running means of those 27-day means (black
lines). The indices are all scaled by linear regression to the aa [H] index over the interval 1990–2003 (the interval shaded in gray). A prime on a parameter denotes that this scaling has been
carried out. (Right-hand plots) The distribution of the deviations of the 27-day means (the dots in the corresponding left-hand plot) from their simultaneous 13-point smoothed running means (the
black line in the corresponding left-hand plot). (a) and (b) are for the scaled IHV index, IHV′; (c) and (d) for the corrected and scaled a [K ]index from Niemegk, (a [NGK]/s)′; (e) and (f) for the
corrected and scaled a [K ]index from Sodankylä, (a [SOD]/s)′; (g) and (h) for the corrected and scaled a [K ]index from Eskdalemuir, (a [ESK]/s)′; (i) and (j) for the homogenized northern
hemisphere aa index, aa [HN]; and (k) and (l) for the homogenized southern hemisphere aa index, aa [HS]. The dashed lines in the left-hand plots mark the dates of aa station joins, using the same
colour scheme as used in Figure 2. Horizontal grey lines are at zero deviation.
In the text
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Improving the Stochastic Gradient Descent's Test Accuracy by Manipulating the ℓ<sub>∞</sub>Norm of its Gradient Approximation
The stochastic gradient descent (SGD) is a simple yet very influential algorithm used to find the minimum of a loss (cost) function which is dependent on datasets with large cardinality, such in
cases typically associated with deep learning (DL). There exists several variants/improvements over the "vanilla"SGD, which from a highlevel perspective, may be understood as using an adaptive
elementwise step-size (SS). Moreover, from an algorithmic point of view, there is a clear "incremental improvement path"which relates all of them, i.e. from simple alternative such SG Clipping (SGC)
to the well-known variance correction (Adagrad), follow by an (EMA) exponential moving average (RMSprop) to alternative furtherance such Newton (AdaDelta) or bias correction along with different EMA
options for the gradient itself (Adam, AdaMAx, AdaBelief, etc.). In this paper, inspired by previous non-stochastic results on how to avoid divergence for ill chosen SS (for the accelerated proximal
gradient algorithm), instead of directly using the standard SGD gradient s EMA g¯k, we propose to modify its entries so as to force fkg¯kk1g s moving average to be non-increasing. Our reproducible
computational results compare our proposed algorithm, called SGD- 1, with several optimizers (such Adam, AdaMax, SGC, etc.); while, as expected, SGD- 1 allows us to use larger SS without divergence
problems, (i) it also matches a well-tuned Adam s performance (superior to "default parameters"Adam), and (ii) heuristically, its convergence properties (rate, oscillations, etc.) are superior when
compared to other well-known algorithms.
Profundice en los temas de investigación de 'Improving the Stochastic Gradient Descent's Test Accuracy by Manipulating the ℓ[∞]Norm of its Gradient Approximation'. En conjunto forman una huella | {"url":"https://cris.pucp.edu.pe/es/publications/improving-the-stochastic-gradient-descents-test-accuracy-by-manip","timestamp":"2024-11-10T15:45:45Z","content_type":"text/html","content_length":"52674","record_id":"<urn:uuid:4b828452-e88d-4aa5-84f9-57f8cf1d0e95>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00389.warc.gz"} |
10 Quirky Mathematics Activities for End-of-Year Engagement10 Quirky Mathematics Activities for End-of-Year Engagement
10 Quirky Mathematics Activities for End-of-Year Engagement
By Jackson Best|2023-02-07T22:32:00+00:00October 30th, 2020|Categories: Educators|Tags: classroom activities|
Summer holidays glimmers in the near distance, but there are a few solid weeks of learning left to go yet.
And after the year we’ve had, student motivation, attention, and enthusiasm levels are harder to maintain than ever.
But before you reach for the closest DVD, try these 10 fun, end of year learning opportunities to keep your students engaged with mathematics – right up to the last day.
Mathematics Bingo
Liven things up with a game of Bingo! Here’s how you adapt it for mathematics:
1. Have students create their own Bingo cards by filling a table with numbers or unsolved equations, depending on their ability. Just make sure the values only extend to 30, so they correspond with
the numbers you draw out.
2. Draw numbers 1–30 out of a box.
3. Students circle each drawn number that appears on their card, calling “Bingo!” when they’ve circled every number.
Tip: write the numbers on ping pong balls and shake them around in a basket to get the full effect. You might even like to nominate a student as the Bingo host!
Mathematical Scavenger Hunts
If your students need a break from the classroom, send them on a hunt for mathematical items around the school. For example, you might task them with finding:
• a five-sided shape
• a tessellating pattern
• a number that contains three digits that are multiples of 3
• an acute angle
• a convex and a concave shape.
You can combine these instructions with a list of clues and put students into scavenger groups. Turn it into a competition with a reward for the first group that ticks off every clue!
Handprint Times Tables
Have students dip their hands in washable paint, and then use the handprints to visually represent times tables on a large sheet of paper.
They can start with two fingers (for 2× tables), and then move their way up to their whole hand for 5× tables.
Tip: It’s going to get messy! Give students some smocks and use the art room if you can.
Or try these multiplication tricks with your class!
Mathematical Hopscotch
Draw a hopscotch with the numbers 1–9 and mathematical symbols (plus, minus, divide) on the pavement with chalk.
Have one student jump between the squares as they call out an unfinished number sentence (e.g. 9 + 7).
The next student jumps onto the correct answer before jumping out a question of their own.
Rotate through the whole class and measure their longest streak of correct answers!
Place Value Hoops
Put three small hoops at varying distances from a throwing line, and label the furthest “hundreds”, the middle one “tens”, and the closest “ones”. Students then throw beanbags into these to score the
place values.
To ramp up the challenge, have them compete against one another in a set amount of time. When the clock counts down, they’ll have to decide whether they shoot furthest for the hundreds, or scramble
to make up points from the tens and ones!
Picture Pie
This is a fun way for students to get creative with fractions. Here’s how it works:
1. Have students draw circles with a compass or trace them onto colored card.
2. Ask them to cut the circles out with safety scissors.
3. Fold the circles and then cut along the lines to create fractional portions. Help your students label them as halves, quarters, and so on.
4. Students create an artwork or pattern by arranging and gluing the portions on a separate sheet. See what they can come up with!
Tip: cut up different craft materials, such as foil and cellophane, to make it even more colorful.
Dice Jeopardy
All it takes is a single dice, but this game will have students wide-eyed with suspense as they add their way to 100 points. The rules are simple:
1. Students take turns rolling a dice, adding up their results. They can keep rolling and racking up points for as long as they like, with the first person to reach 100 being the winner.
2. But there is a catch. If they roll a 1, they score 0 points on that turn.
3. Each turn, players have a choice. Do they take their points while they still have them, or keep rolling at the risk of losing them all?
Tip: Dice Jeopardy can be used to practice multiplication, too. Use two dice and bump the target score up to 500.
Random Number Generator
This is a quick and easy game that develops your students’ place value knowledge.
It’s also one they can do on their own.
1. Students draw five adjoining boxes and label them with their place value (tens, hundreds, thousands etc.)
2. Draw a random number from 1–9 out of a hat.
3. Students decide which box to write the number in. Once they’ve put it in a particular place value, they can’t change it.
4. Do this five times and see who can produce the biggest number.
Your students will have to make some quick decisions to maximise their chances of getting the highest number. Do they fill the highest place value with the first digit larger than 5 that comes up, or
do they wait for something bigger?
Tip: you can also turn this into a probability game if your students are older. For example, what’s the probability of the final digit being a 9?
Number Line Races
This is best done as a group activity so every student gets a turn.
1. Have each group draw a number line from 1–20 on the pavement in chalk.
2. Each group writes + and – symbols on separate pieces of paper and put them into a hat. Have them also write the numbers 1–3 on pieces of paper and place them into a separate hat.
3. The first student stands at 10 on the number line. Their peers draw a symbol from the hat, followed by a number, to determine how many steps they take forward or back.
4. Have them compete with another group to see who can reach 20 first!
Maintain the learning momentum (but give yourself a break too)
Fighting to keep your students engaged through the final weeks might sound like it requires time and energy you don’t have after the year that was.
But it doesn’t have to be hard. An online learning program can keep your students engaged with rich, curriculum-aligned mathematics activities without creating extra work for you.
In Mathletics, for example:
• Curriculum-aligned activities are automatically assigned to students as they work at their own pace
• Student work is marked automatically, so all you need to do is look at the results
• Find a library of over 700 curriculum-aligned problem-solving questions, plus printable resources for Years K–8, so you don’t have to create your own
Your students will love the engaging, gamified features of the program like Live Mathletics and Multiverse.
You’ll love the way it gives you back the time and energy you need to focus on:
• end-of-year reports
• forward planning
• yourself – you’ve earned it!
See how one school used Mathletics to help reduce teachers’ workloads.
Share this post on social media!
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Scientific program
Overview of the scientific program, list of speakers and abstracts
Program Overview
09.00-09.10 – General introduction (Garamszegi)
Part I (chair: Garamszegi)
09.10-09.30 – Talk 1: Squeezing maximum information from matrix data in behavioural ecology (speaker: Kutsukake)
09.30-09.50 – Talk 2: The tolerance interval method for assessment of agreement in behavioral ecology studies with repeated measurements (speaker: Maurer)
09.50-10.10 – Talk 3: Repeatability for binary, proportion and count data (speaker: Nakagawa)
10.10-10.30 – Talk 4: Advances in meta-analysis in Behavioural Ecology (speaker: Santos)
10.30-11.00 – coffee break
10.50-11.10 – Talk 5: rangeMapper: A package for easy generation of biodiversity (species richness) or life-history traits maps using species geographical ranges (speaker: Dale)
Part II (chair: Nakagawa)
11.10-11.30 – Talk 6: Not quite a piece of cake: problems encountered on a behavioural ecologist’s honeymoon with Akaike’s Information Criterion (speaker: Symonds)
11.30-11.50 – Talk 7: Information-theoretic approaches to statistical analysis in behavioural ecology: an introduction to a special journal issue (speaker: Garamszegi)
11.50-12.10 – Talk 8: Avoiding common pitfalls when applying ‘animal’ models to behaviour using Bayesian methods (speaker: Dugdale)
12.10-12.30 – Talk 9: Women have relatively larger brains than men: a comment on the misuse of GLM in the study of sexual dimorphism (speaker: Forstmeier)
12.30-12.50 – Talk 10: Using variance-covariance structures to incorporate data heterogeneity (speaker: Cleasby)
12.50-13.00 – Conclusion & take home message (Nakagawa)
13.00 – General discussion over lunch or beer
Overall theme: Statistical tools for Behavioral Ecologists
László Zsolt Garamszegi^1 and Shinichi Nakagawa^2
^1Department of Evolutionary Ecology, Estación Biológica de Doñana-CSIC, Seville, Spain (e-mail: laszlo.garamszegi@ebd.csic.es); ^2Department of Zoology, University of Otago, Dunedin, New Zealand
(e-mail: shinichi.nakagawa@otago.ac.nz)
We had organized a symposium on statistical topics for the previous ISBE meeting, which subsequently generated stimulating and fruitful discussions. Since then, we have been continuing to witness
that our statistical tools are developing at a high rate, and that behavioural ecologists show a huge interest in these developments and appreciate statistical dissemination. The current symposium
will revolve around various issues associated with the analysis of behavioural data. First, we will visit analytical tools that have been developed for studying the repeatability and heritability of
behaviour, which are fundamental for understanding the mechanisms that generate variation in behaviours within individuals as well as within species. Second, we will provide snapshots from intense
discussions about the usefulness of Information Theoretic (IT) approaches especially based on AIC (Akaike’s information criterion) for behavioural ecology. Third, we will provide a useful guide to
meta-analysis that has recently established itself as an essential tool for quantitative review of literature data and discuss special issues for meta-analysis in behavioral ecology. Fourth, we will
investigate some important assumptions of linear models, which often remain violated resulting in misleading conclusions. However, we will keep our forum open for statistical and methodological
discussions at a broader level. We aimed at collecting contributions that focus on any statistical and methodological matters that relate to behavioural ecological questions in general. Accordingly,
we will also host talks on social structure analysis and geographical analysis of species.
Talk 1: Squeezing maximum information from matrix data in behavioural ecology
Nobuyuki Kutsukake
Department of Evolutionary Studies of Biosystems, The Graduate University for Advanced Studies, Hayama, Kanagawa, Japan; PRESTO, Japan Science and Technology Agency, Honcho Kawaguchi, Saitama, Japan
(email: kutsu@soken.ac.jp)
Behavioural observation of social interactions or matings can be often summarized into a simple actor-receiver matrix. Here, I review a statistical toolbox for analyzing the matrix data. Association
and stable group structure can be investigated by randomization methods. Mantel test and its extended version, Hemelrijk's Kr tests, solve a problem of data non-independence and have been used to
test reciprocity and interchange at a group level. The Shannon-Weaver index which was originally developed in the information theory enables to quantify the degree of (un)evenness that one individual
allocates behaviour to other individuals. Social network analysis provides many proxies on geometric structure of a group such as density, centrality, betweeness centrality, and so on. Squeezing
maximum information from a single matrix data will help us understand complex social or mating structure in animals
Talk 2: The tolerance interval method for assessment of agreement in behavioral ecology studies with repeated measurements
Golo Maurer^1, Pankaj K. Choudhary^2, and Phillip Cassey^1
^1Centre for Ornithology and School of Biosciences, Birmingham University, UK (email: g.maurer@bham.ac.uk, p.cassey@bham.ac.uk); ^2Department of Mathematical Sciences, University of Texas at Dallas,
USA (email: pankaj@utdallas.edu)
We describe the use of a tolerance interval method for assessing agreement between repeated measurements of continuous data. Evaluation of agreement between repeated measurements is of considerable
importance in behavioral ecology. In a large number of studies, the intra-class correlation coefficient (r[I]) is cited as a measure for assessing the reliability of multiple measurements on the same
individual. A low value of r[I] is taken to indicate low repeatability. However, it is well known that a low value of r[I] may result from low variability between different individuals, not because
the repeated measurements within individuals do not agree. We show that it is important to present more data than an unaccompanied value of r[I] to address the repeatability of measurements. The
approach of the tolerance interval method is first to model the data using a linear mixed model, and then construct the relevant asymptotic tolerance interval for the distribution of appropriately
defined differences. We provide examples from two different studies: (1) laying dates in Eurasian Sparrowhawks, Accipiter nisus; and (2) claw strength in male fiddler crabs, Uca elegans. In these
examples, repeated measurements were conducted for the comparison of both intra- and inter-method agreement in behavioral ecology.
Talk 3: Repeatability for binary, proportion and count data
Shinichi Nakagawa^1 and Holger Schielzeth^2,3
^1Department of Zoology, University of Otago, New Zealand (email: shinichi.nakagawa@otago.ac.nz); ^2Department of Behavioural Ecology and Evolutionary Genetics, Max Planck Institute for Ornithology,
Germany (email: schielz@orn.mpg.de); ^3Department of Evolutionary Biology, Evolutionary Biology Centre, Uppsala University, Sweden
Repeatability (more precisely the intra-class correlation coefficient, ICC) is an important index for quantifying the accuracy of measurements and/or the constancy of phenotypes. Recently, the use of
ICC became a requirement in the area of animal personality research to measure behavioural consistency. A problem of behavioural data is that they are often binary, proportion or counts (we refer to
these as non-Gaussian data in relation to normally-distributed or Gaussian data). For non-Gaussian data, obtaining accurate ICCs is rather technical, unless one uses transformations of such data
(although the transformation of the data leads to biased estimates of ICC). Here, we explain how we can obtain unbiased ICCs using generalized linear mixed-effects models (GLMM), which have been
increasingly used by Behavioural Ecologists in recent years. We discuss a number of methods for calculating standard errors, confidence intervals and statistical significance for ICCs as well as
technical difficulties arising when we calculate ICCs from non-Gaussian data. We will also introduce the R package, named rptR, which we bundled to facilitate the accurate calculations of ICCs
(downloadable at https://r-forge.r-project.org/projects/rptr/).
Talk 4: Advances in meta-analysis in Behavioural Ecology
Eduardo S. A. Santos and Shinichi Nakagawa
Department of Zoology, University of Otago, New Zealand (e-mail: e.salves@gmail.com, shinichi.nakagawa@otago.ac.nz)
The use of meta-analysis by Behavioural Ecologists has been increasing since it was first used in the areas of Ecology and Evolution in the early 1990s. Despite its relatively recent appearance and
common usage in Ecology and Evolution, meta-analyses have been used for over a Century in the Medical and Social sciences. Due to the head start in the usage of meta-analysis, Medical and Social
scientist have been able to improve on the original method and make the meta-analytical procedure more accurate and informative. Here, we give details on some advancements that have been proposed and
are in use by both Behavioural Ecologists, and Medical and Social scientists in the field of meta-analysis. We discuss 1) the use of linear mixed-effect models (LMM) to account for non-independence
arising from multiple effect sizes per study as well as phylogenetic non-independence among different species, 2) meta-regression to relate the meta-analytic effects to covariates, such as study or
biological characteristics, and 3) a reappraisal of the methods for detecting and correcting for publication bias in meta-analysis such as Egger’s regression and the trim-and-fill method.
Talk 5: rangeMapper: A package for easy generation of biodiversity (species richness) or life-history traits maps using species geographical ranges
James Dale^1 and Mihai Valcu^2
^1Institute of Natural Sciences, Massey University, Auckland, New Zealand (e-mail: j.dale@massey.ac.nz); ^2Max Planck Institute for Ornithology, Seewiesen, Germany (e-mail: valcu@orn.mpg.de)
As species numbers and geographic ranges continue to shrink or change under the influence of human consumption of natural resources, the importance of understanding geospatial patterns of
biodiversity has never been greater. Traditionally however, geographical analyses of species have typically focused on understanding 1) what limits species ranges and 2) what drives variation in
species diversity or richness. However another important question about geographic variation in species is what determines spatial variation in phenotypes. A classic example of this is Bergmann’s
rule which states that body mass in animals tends to correlate with latitude. To facilitate the analysis of spatial variation in both biodiversity and phenotypic traits we developed a suite of R
tools called rangeMapper. rangeMapper is designed for the easy generation of biodiversity (species richness) or life-history traits maps and, in general, maps of any variable associated with a
species or population. The resulting raster maps are stored in a rangeMapper project file (a pre-customized SQLite database) and can thus be displayed and/or manipulated at any latter stage.
Talk 6: Not quite a piece of cake: problems encountered on a behavioural ecologist’s honeymoon with Akaike’s Information Criterion
Matthew R.E. Symonds
Department of Zoology, University of Melbourne, Victoria, Australia (e-mail: symondsm@unimelb.edu.au)
Increasingly, behavioural ecologists are applying novel model selection methods to the analysis of their data. Of these methods, information theory (IT) and in particular the use of Akaike’s
Information Criterion (AIC) is becoming common. AIC allows one to compare and rank multiple competing models, and to estimate which of them best approximates the “true” process underlying the
biological phenomenon under study. In theory, then, AIC provides a simple means of evaluating competing hypotheses. However, several aspects regarding the methodology and application of AIC are
currently open to much debate among statisticians. These issues include the selection of candidate models and the dangers of all-subset analyses, controlling for small sample size, and the
elimination of ‘uninformative’ models. I will discuss, from personal experience, how unsuspecting behavioural ecologists might stroll into the statistical line of fire, and suggest ways of dodging
the bullets.
Talk 7: Information-theoretic approaches to statistical analysis in behavioural ecology: an introduction to a special journal issue
László Zsolt Garamszegi
Department of Evolutionary Ecology, Estación Biológica de Doñana-CSIC, Seville, Spain (e-mail: laszlo.garamszegi@ebd.csic.es)
Scientific thinking may require the consideration of multiple hypotheses, which often call for complex statistical models at the level of data analysis. Complex models have traditionally been treated
by model selection approaches using threshold-based removal of terms, i.e. stepwise selection. A recently introduced method for model selection applies an Information Theoretic (IT) approach, which
has been increasingly propagated in the field of ecology. However, a literature survey shows that its spread in behavioural ecology has been much slower, and model simplification using stepwise
selection is still more widespread than IT-based model selection. Why has the use of IT method in behavioural ecology lagged behind other disciplines? A special issue (SI) will soon appear in
Behavioral Ecology and Sociobiology that examines the suitability of the IT method for analyzing data with multiple predictors. The volume brings together different viewpoints to aid behavioural
ecologists in understanding the method. In my talk, I will provide a brief overview on the content of the SI by emphasizing the often-misinterpreted benefits and shortcomings of the IT tool and by
pointing to avenues along which the evaluation of multiple hypotheses may develop.
Talk 8: Avoiding common pitfalls when applying ‘animal’ models to behaviour using Bayesian methods
Hannah L Dugdale^1,2, David S Richardson^3, Jan Komdeur^1 and Terry Burke^2
^1Animal Ecology, University of Groningen, Haren, Netherlands (e-mail: h.l.dugdale@rug.nl, J.Komdeur@rug.nl); ^2Department of Animal and Plant Sciences, University of Sheffield, Sheffield, UK
(e-mail: h.dugdale@sheffield.ac.uk, T.A.Burke@sheffield.ac.uk); ^3Centre for Ecology, Evolution and Conservation, School of Biological Sciences, University of East Anglia, Norwich, UK (e-mail:
For behaviour to evolve selection must act on behaviour; behaviour and variation in it must be heritable. In the past, behavioural ecologists have rarely tested whether behaviours are heritable,
assuming instead that behaviours are flexible. This is primarily because elucidating heritability in the wild is difficult, requiring long-term study of individual behaviours and a multi-generation
pedigree. As these data have become available, there has been a surge of interest in applying ‘animal’ models (mixed models that estimate how similar phenotypic traits are across related individuals)
to behaviours. However, behaviours are often binary or rate measures, requiring non-Gaussian error structures and Bayesian techniques to assess the heritability of such behaviours have only recently
become available. Furthermore, as relatives frequently not only share genes but also common environments, it is crucial to account for this in models. Using a long-term dataset and genetic pedigree
of the Seychelles warbler, we demonstrate the application of ‘animal’ models to behaviour. We highlight how interpretation of whether helping behaviour is heritable is influenced by priors (the prior
probability distribution of the unknown quantity of interest e.g. variance in helping). We therefore demonstrate the importance of using simulations to determine whether there is power to detect
Talk 9: Women have relatively larger brains than men: a comment on the misuse of GLM in the study of sexual dimorphism
Wolfgang Forstmeier
Max Planck Institute for Ornithology, Seewiesen, Germany (e-mail: forstmeier@orn.mpg.de)
General linear models (GLM) have become such universal tools of statistical inference, that their applicability to a particular data set is rarely questioned. These models are designed to minimize
residuals along the y-axis, while assuming that the predictor (x-axis) is free of statistical noise (ordinary least square regression, OLS). However, in practice, this assumption is often violated,
which can lead to erroneous conclusions, particularly when two predictors are correlated with each other (e.g. sex and body size in size dimorphic species). This is best illustrated by two examples
from the study of allometry, which have received great interest: (1) the question of whether men or women have relatively larger brains after accounting for body size differences, and (2) whether men
indeed have shorter index fingers relative to ring fingers (digit ratio) than women. These examples clearly illustrate that GLMs produce spurious sexual dimorphism in body shape where there is none
(e.g. relative brain size). Likewise, they may fail to detect existing sexual dimorphisms in which the larger sex has the lower trait values (e.g. digit ratio) and, conversely, tend to exaggerate
sexual dimorphism in which the larger sex has the relatively larger trait value (e.g. most sexually selected traits). These artifacts can be avoided with reduced major axis regression (RMA), which
simultaneously minimizes residuals along both the x and the y-axis.
Talk 10: Using variance-covariance structures to incorporate data heterogeneity
Ian R Cleasby
Department of Animal & Plant Sciences, University of Sheffield, Sheffield, UK (e-mail: bop06irc@sheffield.ac.uk)
Whenever a researcher applies a linear regression to their data they implicitly make a host of assumptions. These assumptions must be verified to ensure that we can trust the results obtained. One
particular problem that can arise when conducting a linear regression is the occurrence of heterogeneity. Heterogeneity occurs when the variance of the residuals is not constant. Although
heterogeneity does not cause coefficient estimates to be biased it does affect the standard error of these estimates. The most common method for dealing with heterogeneity is to transform the data.
However, in many cases transforming data may not be desirable as it makes results harder to interpret and because heterogeneity may provide important ecological information. Here, I briefly discuss
how we can use variance-covariance structures to successfully incorporate heterogeneity within statistical models as an alternative to data transformation. The key idea is that we can include
information on the spread of residuals within a model. To date, explicit specifications of variance-covariance structures do not appear to be widely used in behavioural ecology, which suggests that
such a technique may not be widely appreciated. However, there are a number of situations when the use of this technique may prove useful and they should be considered by researchers as a means of
dealing with heterogeneity. I also discuss extensions of the use of variance-covariance structures in regression methods in wider contexts – such as dealing with spatial, temporal and phylogenetic
covariance, which are often encountered in ecological data. | {"url":"https://bestat.ecoinformatics.org/the-2nd-isbe-post-congress-symposium-on-statistics/scientific-program.html","timestamp":"2024-11-07T09:59:49Z","content_type":"application/xhtml+xml","content_length":"45884","record_id":"<urn:uuid:902a6c94-67b4-43f4-a8ca-96f846adf771>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00885.warc.gz"} |
Recovering epipolar geometry from images of smooth surfaces
We present four methods for recovering the epipolar geometry from images of smooth surfaces. In the existing methods for recovering epipolar geometry corresponding feature points are used that cannot
be found in such images. The first method is based on finding corresponding characteristic points created by illumination (ICPM-illumination characteristic points method). The second method is based
on correspondent tangency points created by tangents from epipoles to outline of smooth bodies (OTPM-outline tangent points method). These two methods are exact and give correct results for real
images, because positions of the corresponding illumination characteristic points and corresponding outline are known with small errors. But the second method is limited either to special type of
scenes or to restricted camera motion. We also consider two more methods which are termed CCPM (curve characteristic points method, curves, denoted by word "green", are used for this method on
Figures) and CTPM (curve tangent points method, curves, denoted by word "red" are used for this method on Figures), for searching epipolar geometry for images of smooth bodies based on a set of level
curves (isophoto curves) with a constant illumination intensity. The CCPM method is based on searching correspondent points on isophoto curves with the help of correlation of curvatures between these
lines. The CTPM method is based on property of the tangential to isophoto curve epipolar line to map into the tangential to correspondent isophoto curves epipolar line. The standard method termed SM
(standard method, curves, denoted by word "blue" are used for this method on Figures) and based on knowledge of pairs of the almost exact correspondent points, has been used for testing of these two
methods. The main technical contributions of our CCPM method are following. The first of them consists of bounding the search space for epipole locations. On the face of it, this space is infinite
and unbounded. We suggest a method to partition the infinite plane into a finite number of regions. This partition is based on the desired accuracy and maintains properties that yield an efficient
search over the infinite plane. The second is an efficient method for finding correspondence between points of two closed isophoto curves and finding homography, mapping between these two isophoto
curves. Then this homography is corrected for all possible epipole positions with the help of evaluation function. A finite subset of solution is chosen from the full set given by all possible
epipole positions. This subset includes fundamental matrices giving local minimums of evaluating function close to global minimum. Epipoles of this subset lie almost on straight line directed
parallel to parallax shift. CTPM method was used to find the best solution from this subset. Our method is applicable to any pair of images of smooth objects taken under perspective projection
models, as long as assumption of the constant brightness is taken for granted. The methods have been implemented and tested on pairs of real images. Unfortunately, the last two methods give us only a
finite subset of solution that usually includes good solution, but does not allow us to find this good solution among this subset. Exception is the case of epipoles in infinity. The main reason for
such result is inaccuracy of assumption of constant brightness for smooth bodies. But outline and illumination characteristic points are not influenced by this inaccuracy. So, the first pair of
methods gives exact results.
• epipolar geometry
• homography
• isophoto curves
• level curves
• occluding contour
• smooth surfaces
Dive into the research topics of 'Recovering epipolar geometry from images of smooth surfaces'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/recovering-epipolar-geometry-from-images-of-smooth-surfaces-3","timestamp":"2024-11-08T15:33:43Z","content_type":"text/html","content_length":"64753","record_id":"<urn:uuid:39dbcbaa-d308-466b-ad27-daa049108a34>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00230.warc.gz"} |
6th Grade NYSE Math Practice Test Questions
What do you think is the best way to prepare your student for the 6th Grade NYSE Math exam? Participating in pre-tests and solving sample practice test questions can help your student prepare for the
6th Grade NYSE Math test. The more your students become familiar with the different types of test questions, the less anxious they will be on the day of the test. This type of preparation can lead to
Better answers to the various challenges and questions that your students might pose. The test takers can never predict what kind of questions await them in the exam. So it is better to be 100% ready
before the test.
Therefore, in this blog post, we provide you with a collection of 10 commonly used practice questions and their step-by-step solutions to help your student prepare for the 6th Grade NYSE Math test as
much as possible.
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10 Sample 6th Grade NYSE Math Practice Questions
1- What is the missing prime factor of number 420?
A. \(2^2×3^1×5^1×7^1\)
B. \(2^2×3^1×7^1×9^1\)
C. \(1^2×2^3×2^1×3^1\)
D. \(3^2×5^1×7^1×9^1\)
2- If the area of the following trapezoid is equal to \(A\), which equation represents \(x\)?
A. \( x = \frac{13}{A}\)
B. \( x = \frac{A}{13}\)
C. \( x=A+13\)
D. \( x=A-13\)
3- By what factor did the number below change from the first to the fourth number?
\(8, 104, 1352, 17576\)
A. 13
B. 96
C. 1456
D. 17568
4- 170 is equal to …
A. \( -20-(3×10)+(6×40)\)
B. \(((\frac{15}{8})×72 )+ (\frac{125}{5}) \)
C. \(((\frac{30}{4} + \frac{15}{2})×8) – \frac{11}{2} + \frac{222}{4}\)
D. \(\frac{481}{6} + \frac{121}{3}+50\)
5- The distance between the two cities is 3,768 feet. What is the distance of the two cities in yards?
A. 1,256 yd
B. 11,304 yd
C. 45,216 yd
D. 3,768 yd
6- Mr. Jones saves $3,400 out of his monthly family income of $74,800. What fractional part of his income does Mr. Jones save?
A. \(\frac{1}{22}\)
B. \(\frac{1}{11}\)
C. \(\frac{3}{25}\)
D. \(\frac{2}{15}\)
7- What is the lowest common multiple of 12 and 20?
A. 60
B. 40
C. 20
D. 12
8- Based on the table below, which expression represents any value of f in terms of its corresponding value of \(x\)?
A. \(f=2x-\frac{3}{10}\)
B. \(f=x+\frac{3}{10}\)
C. \(f=2x+2 \frac{2}{5}\)
D. \(2x+\frac{3}{10}\)
9- 96 kg \(=\)… ?
A. 96 mg
B. 9,600 mg
C. 960,000 mg
D. 96,000,000 mg
10- Calculate the approximate area of the following circle? (the diameter is 25)
A. 78
B. 491
C. 157
D. 1963
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1- A
2- B
The area of the trapezoid is: area= \(\frac{(base 1+base 2)}{2})×height= ((\frac{10 + 16}{2})x = A\)
\( →13x = A→x = \frac{A}{13}\)
3- A
\(\frac{104}{8}=13, \frac{1352}{104}=13, \frac{17576}{1352}=13\)
Therefore, the factor is 13
4- C
Simplify each option provided.
\( A. -20-(3×10)+(6×40)=-20-30+240=190\)
\( B. (\frac{15}{8})×72 + (\frac{125}{5}) =135+25=160\)
\(C. ((\frac{30}{4} + \frac{15}{2})×8) – \frac{11}{2} + \frac{222}{4} = ((\frac{30 + 30}{4})×8)- \frac{11}{2}+ \frac{111}{2}=(\frac{60}{4})×8) + \frac{100}{2}= 120 + 50 = 170\)this is the answer
\(D. \frac{481}{6} + \frac{121}{3}+50= \frac{481+242}{6}+50=120.5+50=170.5\)
5- A
1 yard \(= \)3 feet
Therefore, \(3,768 ft × \frac{1 \space yd }{3 \space ft}=1,256 \space yd\)
6- A
3,400 out of 74,800 equals to \(\frac{3,400}{74,800}=\frac{17}{374}=\frac{1}{22}\)
7- A
Prime factorizing of \(20=2×2×5\)
Prime factorizing of \(12=2×2×3\)
8- C
Plug in the value of \(x\) into the function f. First, plug in 3.1 for \(x\).
\(A. f=2x-\frac{3}{10}=2(3.1)-\frac{3}{10}=5.9≠8.6\)
\(B. f=x+\frac{3}{10}=3.1+\frac{3}{10}=3.4≠10.8\)
\(C. f=2x+2 \frac{2}{5}=2(3.1)+2 \frac{2}{5}=6.2+2.4=8.6 \)
This is correct!
Plug in other values of \(x. x=4.2\)
\(f=2x+2\frac{2}{5} =2(4.2)+2.4=10.8 \)
This one is also correct.
\(f=2x+2 \frac{2}{5}=2(5.9)+2.4=14.2 \)
This one works too!
\(D. 2x+\frac{3}{10}=2(3.1)+\frac{3}{10}=6.5≠8.6\)
9- D
1 kg\(=\) 1000 g and 1 g \(=\) 1000 mg
96 kg\(=\) 96 \(×\) 1000 g \(=\)96 \(×\) 1000 \(×\) 1000 \(=\)96,000,000 mg
10- B
The diameter of a circle is twice the radius. Radius of the circle is \(\frac{25}{2}\).
Area of a circle = \(πr^2=π(\frac{25}{2})^2=156.25π=156.25×3.14=490.625≅491\)
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The mere seconds pendulum is one of the ways to define the size of one meter | A Tender Love & Care FFA
The mere seconds pendulum is one of the ways to define the size of one meter
To revist this article, go to My personal visibility, after that View protected tales.
Why don’t we start off with an instant formula. You might need their calculator. Something Iˆ squared?
Really does that quantity check common? Does it appear like your local gravitational field at first glance for the environment, g? properly, no – it doesn’t as it doesn’t have any units. But the
numerical appreciate is much like the approved area value of:
Notwithstanding this, a worth of 9.81-ish N/kg is quite affordable. And yes, 9.81 N/kg contains the exact same units as 9.81 m/s repayments However, i prefer the models of N/kg since it demonstrates
the bond between area, bulk and force. Please don’t call it the ‘acceleration as a result of the law of gravity’ – that simply brings up very much conceptual trouble.
Imagine if you utilize various devices for grams? In that case, it appears to be enjoy it does not work properly. Old textbooks will list the gravitational industry with a value of 32 ft/s^2. That
plainly isn’t really pi squared.
Seconds Pendulum
Why does this g-Iˆ commitment are present? It has to manage together with the concept of the meter. Before that, let us check out the mere seconds pendulum. It is a pendulum which will take exactly 1
second commit from just one part of their movement to another (or a 2 2nd period). You may have most likely viewed such a good example – like this.
All right, that will be a grandpa time clock and never in fact a moments pendulum. In the event that you gauge the amount of the swinging supply, it’s going to be near 1 meter longer. It isn’t a
straightforward pendulum, so that it doesn’t always have are a meter very long. Straightforward pendulum of length L has the size focused in a tiny bob after the exact distance. This isn’t real for
preceding pendulum.
Go on and check it out. Get limited bulk like a nut or steel golf ball. Material is very effective since it’s body weight will likely be considerably bigger than air drag energy to be able to ignore
it. Today improve point through the middle in the size for the pivot point 1 meter and allow it to oscillate with limited position (perhaps about 10A°). If you love, you can make videos or perhaps
use a stopwatch. Regardless, it should capture about 1 second to visit from just one part to another. Is an instant exemplory instance of a seconds pendulum we make.
I am not going to derive they, it’s not as well tough to reveal that for a pendulum with limited perspective the time of oscillation is actually:
Imagine if I want a time period of 2 mere seconds?
That’s the length of your mere seconds pendulum. Guess you want to call that one meter? Therefore, I Must has grams = Iˆ – This is mindful dating for free exactly why these beliefs tend to be linked.
Definition of a Meter
The seconds pendulum ended up being one way to establish the length of one meter. However, there are some other how to establish this duration. I don’t know exactly how close of a notion this is, but
one concept of the meter was that 10 million m will be the range from North pole with the Equator driving through Paris. It simply does not appear to be this would be very easy to calculate. Exactly
what create i understand?
Really, then utilize the moments pendulum? It virtually may seem like an excellent method to establish a typical. Anybody can make one which includes simple hardware. However, it is not really
reproducible. When you move our planet, the worth of g variations (as I stated above).
Then how will you determine a meter? For a while, the theory was to a particular club of a specific length as well as a certain temperatures. Today we determine the meter while the point light
travels in a vacuum in a certain amount of opportunity.
But What About Pi?
Yes. This might be a Pi Day blog post, and so I should say one thing about Pi. Exactly why is Pi within the duration expression for a pendulum? That’s a fantastic question. Is it as the pendulum
moves in a path that follows a circle? No. The formula of motion for a oscillating mass on a spring (simple harmonic motion) comes with the same kind since the little direction pendulum and isn’t
moving in a circle. Then precisely why? I guess best answer is that means to fix simple harmonic movement are a sine or cosine features. I’m not sure what more to state apart from that gives us an
answer. Since we now have a sine function for the answer, the time scale will have to bring a pi involved.
I feel like this is an inadequate solution – but it’s the facts. They about makes Pi magical. It simply seems in places you would not expect.
Before we make you which includes extra Pi time links, I would ike to suggest one Pi Day activity predicated on this seconds pendulum. Have a meter stick. Use it to measure your local gravitational
field (which may function as the just like the vertical velocity for a totally free dropping item). Next measure the time period oscillation for a pendulum (really, i might change the duration making
a function of stage vs. size). From this duration as well as the calculated gravitational field, resolve for pi. In fact, i do believe i would do that as research. | {"url":"https://www.tlcffa.org/the-mere-seconds-pendulum-is-one-of-the-ways-to/","timestamp":"2024-11-14T02:07:04Z","content_type":"text/html","content_length":"113275","record_id":"<urn:uuid:319b5e3b-0b05-4c25-a1ab-378ed5003a35>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00110.warc.gz"} |
Let f:R→R be a function defined by f(x)=max {x,x3}. The set of ... | Filo
Let be a function defined by . The set of all points where is not differentiable is
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If , then . So, .
If , then . So, .
If , then . So, .
If , then . So, .
If , then . So, .
If , then . So, .
If , then . So, .
Clearly, is not differentiable at .
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Question Text Let be a function defined by . The set of all points where is not differentiable is
Updated On Jan 3, 2023
Topic Continuity and Differentiability
Subject Mathematics
Class Class 12
Answer Type Text solution:1 Video solution: 1
Upvotes 144
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Saturday, November 24, 2007 at 12:11AM
Asymptotic solution for the diffraction problem on a periodic array of microstrip radiators located above the surface of an infinite generalized cylinder, the curvature radius of which satisfies the
condition of slow change along the guide, is considered. By using the matrix reflectivity coefficients, the problem is solved assuming the multilayer isotropic dielectric coating is present. For
partial excitation the electric Green tensor function of Maxwell equations is written in closed form, which includes the Airy functions in the Fok definition and their first derivatives.
Friday, November 23, 2007 at 12:11AM
This work considers results of research of characteristics of rectangular printed-circuit radiators in plane phased antenna arrays. The research is conducted by using computer modeling and the
mathematical model described in [1].
Thursday, November 22, 2007 at 12:11AM
Based on the asymptotic theory, characteristics of convex cylindrical antenna array consisting of director radiators are studied. To determine the distribution of currents of the radiator vibrators,
the numeric projective method is used. Results of numeric computations are presented.
Wednesday, November 21, 2007 at 12:11AM
Based on the asymptotic diffraction theory methods, the problem of determining the near field of the convex cylindrical antenna array composed of aperture radiators is reduced to the boundary value
problem in the single cell of the array, which is solved using the Galerkin method. The asymptotic theory is further developed, which provides representation of the radiator diagram in the convex
cylindrical array as a sum of the direct wave from the excited radiator, the fast creeping waves and slow creeping waves. Results of numeric computations of the slot elliptical, particularly circular
antenna arrays are presented.
Tuesday, November 20, 2007 at 12:11AM
Results of numeric modeling of impedance and polarizational characteristics of single-channel printed-circuit radiator having circular polarization in the base of the antenna array are presented.
Possibilities for optimizing the characteristics in the frequency band and scanning angle sector. | {"url":"http://eds-soft.com/en/publications/archives/2007/index.php","timestamp":"2024-11-12T10:34:05Z","content_type":"application/xhtml+xml","content_length":"33311","record_id":"<urn:uuid:b0d00e0e-4179-4e09-a3b3-75fbf53f3134>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00830.warc.gz"} |
Sum by Color in Excel (Examples) | How To Sum By Colors in Excel?
Updated August 21, 2023
Sum by Color in Excel
In this article, we will learn about Sum By Color in Excel. In Excel, we have a function for adding numbers. But there is no direct way to add the number by their background color. By this, we don’t
need to sum the numbers separating the colored cells. We can directly consider all the cells in the formula and sum them as per their background color.
This we can do when we have many cell numbers colored, and filtering the data is not suggested there.
How to Sum by Color in Excel?
Excel Sum by Color is very simple and easy. Let’s understand how to sum by color in Excel with some examples.
Sum by Color in Excel – Example #1
Here we have data on some product and their sale. As shown below, column C has numbers with some background color.
Now go to the cell where we need to see the output and type the “=” sign (Equal). And search and select the SUBTOTAL function as shown below.
Now, as we need to sum the numbers, so from the drop-down of SUBTOTAL Function, select 9, which is for sum.
And for reference1, select the complete range of column C, which we need to total as shown below.
The Output will be as given below.
Now apply the filter in the top row by pressing Ctrl + Shift +L.
Go to Filter by Color from the drop-down menu of it. Select any color; we have selected YELLOW, as shown below.
Once we do that, we will get the Output cell filtered sum as 190, as shown below.
We can also check the correctness of the applied SUBTOTAL formula by filtering the different colors.
Sum by Color in Excel – Example #2
There is another way to sum the numbers by their colors. For this, we will consider the same data as shown in example-1. Now copy the column’s cells with numbers and paste them into a separate sheet
or in the same sheet in a different location.
Now quickly press Ctrl + T. This will enable the selected cells to convert into table format. Now click on Ok from the Create Table box.
Once we do that, selected cells will convert into the table form. And another menu will add with the name Design in the menu bar. Now Check and tick the Total Row option from the Table Style Options.
Once we do that, we will get the sum of cells at the bottom end of the column with a drop-down menu. Here we are getting a sum of 786.
Now from the drop menu of the total sum, select the Sum option as shown below.
By this, we enable the table to sum the filtered data as per colored cells. Now go to the top filter drop-down of the same column and select any color to get summed up from the Filter by Color
option. Select any color; we have selected YELLOW, as shown below.
Once we do that, we will get the YELLOW colored filtered and the sum of the YELLOW colored cells in the below cell.
Sum by Color in Excel – Example #3
There is another method of summing the numbers by their color. VBA Marcos will do this. For this, we will consider the same data we saw in example-1. And we will add separate cells for each product
name to get the sum of their quantity sold.
Now press Alt + F11 to enter Visual Basic for the Application screen.
Now go to the Insert menu and select Module.
This will open a new Module to write code. Now in the blank Module, write the code for enabling the sum by color function in Excel, as shown below. You can also use the same code to make some changes
in that.
Close the complete window of VBA. Now go to the cell reference of Mobile, where we need to see the result and type the “=” sign. Now search and select the Sum Color function we created in VBA.
And select the reference colored cell and then select the range to get summed, as shown below.
The Result will be as shown below.
Once done, drag the formula to complete respective cells to see the result as shown below.
As we can see in the above screenshot, the sum of yellow-color cells is coming at 190, which the summed value is obtained in example-1 and example-2. This means that all the formulas and functions
used in all examples are correct.
• Sum by color from the SUBTOTAL function is the easiest way to get the sum result by color in Excel.
• The process steps shown in example-2 take a little longer than in example-1, but it is still easy to apply.
• We don’t need to filter the colored cells separately to get the sum.
• Sum by color shown in example-3 by VBA coding takes time, and it doesn’t show the result if we paste the data in another file because it does carry the code with it.
Things to Remember About Sum by Color in Excel
• If you are summing colored cells by VBA Coding, it is always recommended to save in the Macro enabled Excel; this will save the coding for future use.
• These methods can use anywhere, irrespective of the data size. It is always recommended to use this method when we have a huge set of data, where if we filter the data to get the summed value may
crash the file.
Recommended Articles
This has been a guide to Sum by Color in Excel. Here we discuss how to sum by color in Excel, practical examples, and a downloadable Excel template. You may also look at the following articles to
learn more – | {"url":"https://www.educba.com/sum-by-color-in-excel/?source=leftnav","timestamp":"2024-11-03T22:10:28Z","content_type":"text/html","content_length":"300689","record_id":"<urn:uuid:eee8e12c-e328-44bd-9aea-eabe66e151e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00054.warc.gz"} |
Converting Miles to Nautical Miles
When navigating the world, whether by land, air, or sea, different units of measurement are used to calculate distances. Two such units are miles and nautical miles. While both measure distance, they
are used in different contexts and are based on different calculations.
Converting Miles to Nautical Miles
The conversion between miles and nautical miles is straightforward:
• 1 mile = 0.868976 nautical miles
• 1 nautical mile = 1.15078 miles
To convert miles to nautical miles, you simply multiply the number of miles by 0.868976.
Example Conversion:
• If you have 5114.9 miles and want to convert them to nautical miles:5114.9 miles×0.868976=4444.72534 nautical miles
So, 5114.9 miles is approximately 4444 nautical miles.
Understanding the Difference Between Miles and Nautical Miles
While miles and nautical miles both measure distance, their usage and the method of calculation differ significantly.
• Usage: Miles are used primarily on land in the United States and a few other countries for measuring road distances and land-based travel.
• Calculation: A mile is defined as exactly 1,609.344 meters or 5,280 feet.
Nautical Miles
• Usage: Nautical miles are used in aviation, maritime navigation, and in international law to define territorial waters.
• Calculation: A nautical mile is based on the Earth’s circumference and is equivalent to one minute of latitude. It is defined as exactly 1,852 meters or approximately 6,076.1 feet.
The key difference lies in their origins and applications: while miles are based on a fixed distance, nautical miles are directly related to the geometry of the Earth, making them ideal for
Why Is There a Difference?
The difference between miles and nautical miles stems from their intended purposes. Nautical miles are designed for use in navigation at sea and in the air, where it’s important to account for the
curvature of the Earth. Since the Earth is a sphere, distances over its surface are best measured using angles rather than fixed distances.
• Nautical Miles and Latitude: A nautical mile corresponds to one minute of latitude, which is 1/60th of a degree of latitude. This makes it a natural unit for navigation because it directly
relates to the Earth’s geometry. As a result, nautical miles are used universally in aviation and marine contexts.
• Miles and Land Measurement: Miles, on the other hand, were developed for land-based travel and are rooted in historical systems of measurement. They are more straightforward for measuring
distances along a relatively flat surface, such as a road.
The History of Nautical Miles
The concept of the nautical mile dates back to ancient civilizations, which needed a reliable way to measure distances over the open sea. The origins of the nautical mile are deeply tied to the need
for precise navigation across the oceans, where landmarks are few, and the curvature of the Earth plays a significant role.
• Early Navigation: Early mariners used the stars and the horizon to navigate, relying on the measurement of angles to determine their position. As navigation became more sophisticated, there was a
need for a consistent unit of measurement that could be used across different parts of the globe.
• Development of the Nautical Mile: The nautical mile was formally defined based on the Earth’s geometry. One nautical mile was set as the length of one minute of arc along a meridian, making it
directly related to the circumference of the Earth. This definition was standardized in the early 20th century as exactly 1,852 meters.
• Modern Usage: Today, the nautical mile is used universally in international navigation and aviation. It remains the standard unit for defining territorial waters and airspace boundaries. | {"url":"https://sitemap.hillhouse4design.com/miles-to-nautical-miles/","timestamp":"2024-11-01T19:08:27Z","content_type":"text/html","content_length":"49003","record_id":"<urn:uuid:7ede0a64-eced-4fbd-bbd5-bc808fc1e11a>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00176.warc.gz"} |
How to Perform Rate Analysis for Earth Excavation? [PDF]How to Perform Rate Analysis for Earth Excavation? [PDF]
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Rate analysis for earth excavation establishes the rate of labor and equipment required to excavate a site. Unlike other rate analysis works, the rate analysis of earth excavation does not undertake
any material analysis.
Manual or machine excavation method or the combination of both is used for earth excavation. The process of excavation depends on the type and complexity of excavation.
The article explains a general rate analysis of earthwork excavation performed manually as well as with the help of machines.
Features of Rate Analysis of Earthwork Excavation
The cost of excavation depends on the depth of excavation, type of soil, method of excavation to be carried out, and the distance from which the excavated soil has to be disposed of. All the costs
are added for a unit volume of excavation to get the rate of excavation.
The respective area codes of a region or a country follow a specific rate of labor productivity and a specific labor output constant.
For example, IS 7272 -1974 provides the labor output constants for building work for different country zones. It gives the coefficient value of days required by a particular type of labor to conduct
a 1-meter cube volume of excavation.
Rate Analysis of Manual Earthwork Excavation
The concept can be explained with a simple example.
Consider a work where it is required to excavate a pit of following dimensions:
Breadth (b) = 2000; Length (l) = 4000mm and Depth (d) = 1000mm;
Steps involved in rate analysis are:
Step 1: Investigate Site and Determine the tasks
Determine the type of soil of the site and associated tasks to conduct safe excavation. Sometimes, the excavation may require shoring or dewatering. If there is hard strata beneath, there is a
requirement for blasting. Also, check the provision for working space and safety features associated with this.
Listing out the tasks involved can help predetermine the labor and equipment required for the excavation.
Step 2: Determine the Volume of Excavation (m^3)
As mentioned in the example, beyond the required dimension of the pit, a minimum working space of around 150mm around the pit is provided.
Let the final dimension of the pit be:
Breadth(b) = 2000mm; Length (b) = 4000mm and Depth(d) = 1000mm;
Volume of excavation (Vp) = Length x Breadth x Height
= [4000 x 2000 x 1000 ] in mm ~ [4 x 2 x 1] in metres ~ 8m^3
Step 2: Determine the Labour time Required for 1m^3 Excavation
As per IS 7272-1974, for North Zones of India,
Labour Output Constants as per IS 7272-1974
From the figure,
The number of days required to perform 1m^3 of excavation by a mate is 0.06, and that for a mazdoor is 0.62. (Assume the case of hard dense soil).
Hence, to perform ‘V’= 8m^3 excavation:
Number of Mate = 0.06V = 0.06 x8= 0.48 nos
Number of Mazdoor = 0.62V=0.62 x 8 = 4.96 nos
This means approximately, 1 Mate and 5 numbers of Mazdoor are required to finish the excavation of the given depth in one day.
Step 3: Determine the cost of Mate and Mazdoor for Excavation
The cost of labor or machinery is obtained from Schedule of Rates (SOR) that can be taken based on PWD or Public sectors or trends of the market with respect to a particular region.
The rate of labor can be either scheduled as a rate per square feet or rate of unskilled labor per day.
Let the labor cost of mate be Rs. 370 per day, and labor cost for mazdoor be Rs. 280 per day. Then,
The cost of Mate = No. of mates x cost per mate =1 x 370= Rs. 370
The cost of Mazdoor= No. of mazdoor x cost per mazdoor= 5 x 280
= Rs. 1400
Total Cost of Labour
= The cost of mate + The cost of Mazdoor = 370 + 1400 = Rs.1770/-
Step 3: Determine the Cost of Tools & Equipment for Manual Earth Excavation
The cost of tools and equipment can be taken as a percentage of labor costs. (Assuming 5% of labor cost as the cost of tools and equipment)
Cost of tools and equipment for 8m^3 Earth Excavation
= 5% x (1770) = 0.05 x 1770= Rs. 88.5
The Basic Cost of Excavation for 8m^3 Earth Excavation
= Labour Cost + Cost of Equipment = 1770 + 88.5 = Rs. 1858.5
Step 4: Determine the Overhead Cost and Profit
The overhead cost and profit = 20% of Basic cost of excavation
= 0.20 x 1858.5 = Rs. 371.7
Total Cost Including OH and Profit= 1858.5+ 371.7= Rs. 2230.2
Rate Analysis of Machine-based Earthwork Excavation
Consider an example of excavation of soft soil for a depth of up to 1.5m and lead distance of 50m per 10m^3 of concrete. Here, the coefficient of quantity of hydraulic excavator, tractor/dumper, and
unskilled labors is based on their capacity per day (8 hours of work).
Items Units Qty. Rate Amount
Hydraulic Excavator Day 0.04125 5000 206.25
Tractor/Dumper Day 0.04125 1500 61.88
Unskilled Labor Day 1.20 311.2 373.44
Total 641.57
Water charges @1% total 6.42
Contractor Profit @15% 96.23
SUM 744.22
Gross Amt./CUM up to 1.5m depth COST 74.42
Capacity of the Equipment
From the table above, the hydraulic excavator takes 0.04125 day to perform an excavation of 10m3. It takes 0.04125 days to perform a 10m^3 excavation. Then in 1 day, it performs :
(1 x [ 10/0.04125]) m^3 of excavation = 242.24m^3/day
This is actually the capacity of the hydraulic excavator.
That means- a hydraulic excavator can excavate 242.4242 m^3 of soil in one day.
Cost of Equipment
From table-2, the cost per day, including driver and fuel for a hydraulic excavator is Rs. 5000. Then the cost of 10m^3 of excavation can be calculated as:
No. of days required for 10m^3 excavation = 10/242.4242 = 0.04125 days.
The cost of hydraulic excavator for 10m^3 excavation
= 0.04125x5000 = Rs 206.25
Likewise, based on the capacity of other equipment, laborers, etc., their cost is calculated. Contractor's profit is also added to the total cost of laborers and machineries. Then the grand total
gives the rate of excavation per 10m^3 of soil excavation.
Different mechanical equipment have different capacity per day for excavation work. Their coefficient per m^3 or per 10m^3 should be considered for calculation.
Frequently Asked Questions
How do you perform rate analysis for earth excavation?
The cost of excavation depends on the depth of excavation, type of soil, method of excavation to be carried out, and the distance where the excavated soil has to be disposed of. The cost of all these
is added for a unit volume of excavation to get the excavation rate.
The respective area codes of a region or a country follows a specific rate of labor productivity and a specific labor output constant.
For example, IS 7272 -1974 provides the labor output constants for building work for different zones of the country. It gives the coefficient value of days required by a particular type of labor to
conduct a 1-meter cube volume of excavation.
What is the rate analysis for excavation?
Rate analysis of earth excavation establishes the rate of labor and equipment required to excavate the site. Unlike other rate analysis works, the rate of earth excavation does not undertake any
material analysis.
Manual or machine excavation method or combination of both are used for earth excavation. The process of excavation depends on the type and complexity of excavation.
Read More: | {"url":"https://test.theconstructor.org/practical-guide/how-rate-analysis-earth-excavation/44936/","timestamp":"2024-11-08T18:59:17Z","content_type":"text/html","content_length":"193752","record_id":"<urn:uuid:f8af5a0a-73a2-4a05-8b3f-314c41d09f5f>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00703.warc.gz"} |
or component costs of
Compute the cost of the following:a. A bond that has...
(Individual or component costs of capital) Compute the cost of the following:a. A bond that has...
or component costs of
Compute the cost of the following:a. A bond that has
$1 comma 0001,000
par value (face value) and a contract or coupon interest rate of
percent. A new issue would have a floatation cost of
percent of the
$1 comma 1251,125
market value. The bonds mature in
years. The firm's average tax rate is 30 percent and its marginal tax rate is
percent.b. A new common stock issue that paid a
dividend last year. The par value of the stock is $15, and earnings per share have grown at a rate of
percent per year. This growth rate is expected to continue into the foreseeable future. The company maintains a constant dividend-earnings ratio of 30 percent. The price of this stock is now
percent flotation costs are anticipated.c. Internal common equity when the current market price of the common stock is
The expected dividend this coming year should be
increasing thereafter at an annual growth rate of
percent. The corporation's tax rate is
percent.d. A preferred stock paying a dividend of
percent on a
par value. If a new issue is offered, flotation costs will be
percent of the current price of
e. A bond selling to yield
percent after flotation costs, but before adjusting for the marginal corporate tax rate of
percent. In other words,
percent is the rate that equates the net proceeds from the bond with the present value of the future cash flows (principal and interest).
a. What is the firm's after-tax cost of debt on the bond?
(Round to two decimal places.)
b. What is the cost of external common equity?
(Round to two decimal places.)
c. What is the cost of internal common equity?
(Round to two decimal places.)
d. What is the cost of capital for the preferred stock?
(Round to two decimal places.)
e. What is the after-tax cost of debt on the bond?
(Round to two decimal places.)
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Biclustering allows for simultaneous clustering of the observations and variables. Martella et. al (2008) introduced biclustering in a model-based clustering framework by utilizing a structure
similar to a mixture of factor analyzer structures such that observed variables are modelled using a latent variable that is assumed to be from a MVN(0, I). In Martella et. al (2008), clustering of
variables was introduced by imposing constraints on the entries of the factor loading matrix to be 0 and 1. However, this approach restricts the non-zero off-diagonal entries of the covariance matrix
to be 1, which is very restrictive. Here, we assume the latent variable to be from a MVN(0,T) where T is a diagonal matrix and hence, the non-zero off-diagonal entires of the covariance matrix are
not restricted to be equal to 1. A family of models are developed by imposing constraints on the components of the covariance matrix. An alternating expectation conditional maximization(AECM)
algorithm is used for parameter estimation. Proposed method will be illustrated using simulated and real datasets. | {"url":"https://www2.math.binghamton.edu/p/seminars/datasci/191029","timestamp":"2024-11-11T22:36:24Z","content_type":"text/html","content_length":"17949","record_id":"<urn:uuid:a594f7bf-4a12-4d43-b535-0cf6edd670ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00183.warc.gz"} |
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2017.20
URN: urn:nbn:de:0030-drops-76687
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7668/
Cockett, Robin ; Lemay, Jean-Simon
Integral Categories and Calculus Categories
Differential categories are now an established abstract setting for differentiation. The paper presents the parallel development for integration by axiomatizing
an integral transformation in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold
(in a suitable sense): a differential category with integration which satisfies
these two theorem is called a calculus category.
Modifying an approach to antiderivatives by T. Ehrhard, it is shown how examples of calculus categories arise as differential categories with antiderivatives in this new sense. Having antiderivatives
amounts to demanding that a certain natural transformation K, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category and we
provide examples of such categories.
BibTeX - Entry
author = {Robin Cockett and Jean-Simon Lemay},
title = {{Integral Categories and Calculus Categories}},
booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)},
pages = {20:1--20:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-045-3},
ISSN = {1868-8969},
year = {2017},
volume = {82},
editor = {Valentin Goranko and Mads Dam},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7668},
URN = {urn:nbn:de:0030-drops-76687},
doi = {10.4230/LIPIcs.CSL.2017.20},
annote = {Keywords: Differential Categories, Integral Categories, Calculus Categories}
Keywords: Differential Categories, Integral Categories, Calculus Categories
Collection: 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)
Issue Date: 2017
Date of publication: 16.08.2017
DROPS-Home | Fulltext Search | Imprint | Privacy | {"url":"http://dagstuhl.sunsite.rwth-aachen.de/opus/frontdoor.php?source_opus=7668","timestamp":"2024-11-08T14:33:55Z","content_type":"text/html","content_length":"6168","record_id":"<urn:uuid:c1b4a6f7-a41c-4dda-80ae-a63a42138955>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00389.warc.gz"} |
Sloping math symbols in italic faces, is it normal?
When creating an italic face, I have always sloped the math symbols along with the figures since this makes the most sense to me. But, I recently looked at Neue Haas Unica Pro Bold Italic, a recent
release, and I was surprised to find that the math symbols were upright, as if they were upright roman! Is this odd? New thinking? Appropriate?
• Chris, this is interesting as I've had similar thoughts and questions about what other glyphs should not be italicized. There always seems to be a handful of other symbols, including math symbols
that are not always italicized (e.g. integral, product, registered, copyright, bar, currency).
• Roman is the normal convention for general math symbols. During the years that I set type I never saw italicized general math symbols until the age of desktop. To me, anyone who italicizes all
such symbols, including copyright, doesn't know anything about correct typography.
One online source that might help you is:
• General rule for book and editorial typography is that math operators are not italicised. In advertising and packaging typography they may be. Hence the design tends to depend on the nature of
the typeface design, with most having upright math operators (but spaced slightly differently in the italic fonts).
• I’ve always italicized them in italic fonts.
They look silly otherwise in text that is predominantly italic.
Now that math symbols are included in basic encoding, there’s no need to have them independent of typefaces, as they were originally, roman only as an economy.
For complex math setting, roman fonts are the default anyway, for most characters, with some letters picked out (manually selected) in italic. So it’s not as if the math typographer would ever be
in the situation of having to change italic operators to roman.
They look silly otherwise in text that is predominantly italic.
Unless, of course, one thinks they look silly — indeed, cease to look like what they are — when slanted. Especially the
• Can't even the times glyph "look like what it is" in the context of figures? Mathematicians seem to love hand writing formulas on black boards with chalk. The writing appears to be in haste and
without care for inclination. Their hands try to work as fast as their minds but they have no problem understanding the scrawl they have created?
• I’m pretty sure most people, i.e. non-mathematicians, would consider upright math symbols in an all-italic setting to be a mistake—if they were to notice.
I personally have no trouble italicizing the multiply glyph!
If a typographer is setting complex mathematics, of the sort described by Küster, they will be using a specialist, serifed math font.
Therefore, all other italic fonts should have sloped math operators, and that means all sans serifs.
If typographers simply must have “proper” upright operators, they can access them in the Roman font.
Both methods are available for Helvetica and Times.
Above, Neue Helvetica Italic and Times New Roman Italic.
Below, Helvetica Oblique and Times Italic.
And of particular note, the same distinction is present between Cambria and Cambria Math (I would have shown it, but don’t have the fonts).
• Nick, in response to your illustrations, I'd have to say that the slanted math operators are the ones that 'look wrong'. But that's a subjective answer to a subjective question. I didn't offer my
original observation
General rule for book and editorial typography is that math operators are not italicised. In advertising and packaging typography they may be. Hence the design tends to depend on the nature
of the typeface design, with most having upright math operators (but spaced slightly differently in the italic fonts).
based on what I subjectively think looks right or wrong, but with regard to the conventions of various kinds of typography and, hence, various kinds of fonts.
I'll also note that Brill's typographers were not only clear that they wanted the math operators upright in their italic fonts, but also a lot of other symbols and all parentheses, braces and
• I don't think most readers care, or even notice, if math symbols slant. But how many graphic designers will assume slanted math symbols are wrong and not buy a typeface that has them?
• I’m no mathematician, but the lines with slanted math symbols above definitely look wrong.
• My point is that you already have upright math symbols in the roman, why do you need another set of the same with the italic? You are forced to pick upright. Rather than forcing a user to do
anything I may feel is correct, in general, I would rather give the user the option to choose the version that is correct in their thinking for their express purpose rather than tie their hands
to my choice.
• Chris, the problem with that approach is that it a) requires users who want the upright math operators to selectively switch fonts within strings of numbers and operators, and b) does not provide
for optimum spacing/kerning between upright signs and slanted numbers.
I can imagine providing slanted operators as a stylistic set in an italic font, but since I mostly make book types I'm not inclined to make them the default form.
• There are situations where math symbols are used in non math situations. Like Google+ or band names like M+M, +/−. Equal signs are used in sentences. The movie title, E=mc2. Once you slant the
plus and equal sign, where do you stop slanting? I think you have to go all the way.
• As a graphic designer who flirts with type design on the side, non-italicized elements in an, otherwise, italicized block of text look mismatched to me. Consequently, I'd likely not buy or use an
italic font that didn't contain italicized figures.
In blocks of upright text, italics are used most often to draw attention to a particular word or phrase. If that italicized phrase happens to contain numbers, it would be frustrating and,
possibly, misleading to the reader to have part of that phrase revert back to upright numerals.
In numerical tabular data, I'd be unlikely to use italic figures unless, of course, I was trying to differentiate certain numbers from the others. And if this were the case, once again, I'd be
frustrated to find that no italic figures existed in the font family.
Old, obscure conventions hold no particular value in the absence of good, practical reasons supporting their continued use — especially when there are compelling reasons to the contrary.
• We often have conversations online and at conferences about typographic convention. We should have a way to ask graphic designers about this stuff. Some kind of annual typographic surveys
conducted by design organizations around the world, with the results made public like the AIGA salary survey.
We should have a way to ask graphic designers about this stuff.
And book designers, and information designers, and packaging designers....
Oh, and typographers, if you can find any.
• I don't italicize the math symbols (or copyright, registered, currency...). An italicized degree symbol looks very wrong to my eyes.
It may seem confusing for one font to share roman and italic attributes, but a mathematical symbol set includes all sorts of operators and bracketing characters that are never italicized.
We should have a way to ask graphic designers about this stuff.
And book designers, and information designers, and packaging designers....
We would probably get as many and as varied answers as here.
My experience is the same as what John observed above. And since my primary focus is also book and editorial text, I do not italicize math operators in an italic font.
Nor copyright, registered, and trademark. Nor vertical bar. And typically not degree (although I have been somewhat inconsistent on this last one).
• BTW, a specialized application of a math operator in a non-math context that might not be on everyone’s radar is the use of the multiplication sign in the botanical names of hybrids. In some
cases it is surrounded on both sides by space; in others it can occur without space directly before a genus or species name.
The rules are governed by the
International Code of Nomenclature for algae, fungi, and plants
(the current version of which is known as the Melbourne Code, adopted in 2011).
What is particularly worth noting in this context is the following recommendation from
Article H.3
in the appendix about hybrids:
H.3A.2. If the multiplication sign is not available it should be approximated by the lower-case letter “x” (not italicized).
That last part seems to indicate a clear preference, on the part of botanists anyway, not to have the multiplication symbol slanted in the context of an italic font. And this has borne out in my
experience working with gardening writers & editors.
Oh, and typographers, if you can find any.
A gem from John ;-)
That last part seems to indicate a clear preference, on the part of botanists anyway, not to have the multiplication symbol slanted in the context of an italic font. And this has borne out in
my experience working with gardening writers & editors.
The ICN considers the multiplication symbol as an operator within a name but not part of the italicized name itself. I doubt it's an indication of a broader preference for upright symbols in
italic fonts.
It's common convention for both the genus and species in binomial nomenclature to be italicized. However, botanists and zoologists differ in their approach to more complicated naming conventions.
Even within subdisciplines of these fields styles differ. The American Ornithologists’ Union, for example, places hyphens in many common species names, like sage-grouse or sea-eagle, which puts
it at odds with both other biology style guides and common punctuation conventions.
This all works fine for in-house publications produced by those specialist organizations, but when general consumer publications run into conflicting style preferences, they typically defer to
the Associated Press, Chicago Manual of Style, or their in-house guidelines.
Style guidelines shouldn't be interpreted as universal commandments. Instead, their main purpose is to ensure contextual consistency.
More to the point, publications whose styles dictate non-italicized figures will select fonts that correspond to their preferences. Publications without that requirement will likely regard an
italic font lacking italic figures as reason to use another font. The decision by a type designer to include or not include italicized figures and symbols in a font isn't a matter of right and
wrong. Instead, it seems more a matter of making a choice based on buyer preferences and, possibly, aesthetics.
• To clarify, no one is suggesting having non-italicised figures (numerals) in italic fonts. We're just talking about the typical subset of math operator symbols for basic arithmetic included in
typical text and display fonts.
• Not much, and in the early days of DTP Adobe fonts used to leave those code points glyphless, IIRC. Taking that as a cue for their irrelevance, I filled up those spots with other goodies such as
extra ligatures (before Phinney and Hudson had drummed into me the error of my ways).
It would help if Multiply and Divide, rather basic math symbols, appeared on the North American keyboard, instead of ASCII tilde and ASCII circumflex.
At least, as font producers, we can make the En dash glyph identical to Minus—although that won’t keep most people from using the Hyphen.
To clarify, no one is suggesting having non-italicised figures (numerals) in italic fonts. We're just talking about the typical subset of math operator symbols for basic arithmetic included
in typical text and display fonts.
Thanks for the clarification, John. I somehow embarrassingly missed that critical distinction. I need to pay more attention.
Still, I can think of no compelling logic for italicizing numerals in an italic font but not italicizing the various operators that accompany them. If this is just an esoteric convention born out
of economy during the metal type era, at least it was historically understandable. Today, however, I see little reason for it.
I'm not a mathematician, but I remember from a few college physics courses that mathematical conventions assign meaning to italicized glyphs in ways that differ from everyday use. In these
instances it makes sense to follow the conventions because doing so conveys information. If mathematical texts never use italicized operators, then there's no need to include them in math fonts,
but in an italic typeface designed for broad, general use, their absence might be seen as a shortcoming.
Still, I can think of no compelling logic for italicizing numerals in an italic font but not italicizing the various operators that accompany them.
As Kent and I both noted previously, this is the norm in book typography and has been for a very long time. It derives ultimately from the typography of mathematics, to which you refer, in which
operators and other symbols have normative forms that are not subject to styling of weight or slant, and in which, conversely, stylings of weight and slant of alphanumeric characters have
semantic connotations. The logic of not italicising mathematical operators is simply this: the identity of the symbols is in both their shape and their orthogonal relationship to the baseline.
And after many years working designing and setting these symbols, I really do read them that way: they just look wrong if they're slanted. I'd put italicised math symbols in the same category as
'sloped romans', a kind of contradiction in terms.
I'm no absolutist. I recognise a place for even sloped romans, and as noted above there are kinds of typography, and hence kinds of fonts, in which slanted math symbols make sense. If one is
doing that kind of work, then I can clearly see that the lack of them in a font would be 'seen as a shortcoming'. My book publishing clients were equally clear that they viewed the slanting of
math symbols in an italic font as a shortcoming.
If a font doesn't contain what you need to do the kind of work you're doing, use a different font. It's not like there's a shortage of the bloody things.
The decision by a type designer to include or not include italicized figures and symbols in a font isn't a matter of right and wrong.
To be clear, I wasn’t intending to argue right or wrong. I was just providing some perspective in response to Chris’s original questions: Odd? New thinking? Appropriate?
No, not in my opinion; certainly not; for some audiences, yes.
Instead, it seems more a matter of making a choice based on buyer preferences and, possibly, aesthetics.
Exactly. And John and I have both explained how we see our buyers’ preferences (which I also take to be general preferences). But, like John, I don’t necessarily consider this to be absolute.
Others should do as they see fit.
• I see it this way: Being symbols, they only have one appearance. Italicize them would be like slanting a logo. Now why would you do that? This applies to trademark, copyright, degree, registered,
etc. - aswell.
I see it this way: Being symbols, they only have one appearance. Italicize them would be like slanting a logo.
I see what you’re saying here, but I think the idea that math symbols are like “logos” that only have “one appearance” is too restrictive. This would be true for things like the
estimated sign
, or the way some people originally imagined the Euro symbol to work: Symbols designed exactly like logos that come with precise specs & should not even be adapted to the design of the font. I’d
think the math symbols are (nowadays) one step up the freedom-to-design ladder, certainly depending on what kind of typeface you’re making – as this thread has shown.
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BitVM and SatoshiVM Circuit | SatoshiVM
SatoshiVM adopts BitVM's approach to verifying arbitrary computations on the Bitcoin blockchain: using the primitive, non-Turing-complete Bitcoin Script code to simulate the effect of logic gate
circuits, then utilizing a massive amount of logic gate circuits to achieve the functionality of complex virtual machines.
We know that computers/processors are input-output systems composed of a large number of logic gate circuits. BitVM attempts to simulate the input-output effect of logic gate circuits using Bitcoin
Script. As long as the logic gate circuits can be simulated, in theory, it is possible to realize a Turing machine, completing all computable tasks.
In the interactive fraud proof protocol of Arbitrum, the disputing parties engage in multiple rounds of communication to continually subdivide a particular transaction instruction until they localize
a disputed opcode. Then, this opcode, along with its input and output results, is executed directly on the Ethereum blockchain for verification. This process determines which party's claim is correct
and penalizes the malicious party.
In the Bitcoin and BitVM schemes, due to the simplicity of Bitcoin Script, it's not feasible to directly verify EVM opcodes as done in Ethereum Layer2 solutions. Therefore, an alternative approach is
employed: opcodes compiled from any high-level language are decoded again into the form of logic gate circuits. Then, Bitcoin Script is used to simulate the operation of these logic gate circuits.
This allows for the indirect simulation of the operational effects of virtual machine opcodes, such as those of the EVM, on the Bitcoin blockchain. We can consider the logic gate circuits as an
Intermediate Representation (IR) between EVM opcodes and Bitcoin Script opcodes.
SatoshiVM employs the Bristol format to illustrate its logic gate circuit structure. The Bristol format is a commonly used method in the field of circuit for expressing logic gate circuits. In
essence, it provides a standardized way to describe the layout of complex logic gate circuits, including the inputs and outputs of digital circuits, the functions of logic gates (such as AND, OR, NOT
gates), and their specific connections.The Bristol format typically includes the following parts:
Circuit Size Information: Describes the basic attributes of the circuit, such as the total number of logic gates, the number of input and output signals, etc.
Input and Output Information: Provides detailed information about the assignment of each input and output signal.
Gate Description: Gives a specific description of the function of each logic gate, including the gate type (AND, OR, NOT, etc.) and its connected input and output signals.
Here is an example of Bristol format code:
1 1 0 1 INV
2 1 1 2 4 AND
1 1 4 5 INV
2 1 3 5 6 AND
The visualization of the aforementioned Bristol format circuit is as follows:
The components are as follows:
The first line's 4,7 respectively indicate that this part of the circuit has 4 logic gates and 7 signal lines.
The first number in the second line represents the bit size of the circuit's input signals, and the second number represents the quantity of input signals. For example, in the case above, it
includes a one-bit input, but the input signals comprise three separate inputs (0), (2), and (3).
The first number in the third line indicates the quantity of the circuit's output signals, and the second number represents the bit size of the outputs. In the case above, it contains one output,
which only includes a single digit (6).
Apart from the above three lines, the rest of the content specifically defines each logic gate and signal line within the circuit, listing the following details for every logic gate:
The number of input signal lines
The number of output signal lines
The identifiers for the input lines
The identifiers for the output lines
The function of the logic gate
For example, 1 1 0 1 INV indicates that the logic gate has one input line, one output line, the input line identifier is (0), the output line identifier is (1), and the logic gate operation is INV.
This effectively describes gate A in the aforementioned example circuit.
A more complex case like 2 1 3 5 6 AND indicates that the logic gate has two input lines, one output line, the input line identifiers are (3) and (5), the output line identifier is (6), and the logic
gate operation is AND. This actually defines gate D in the example circuit. We can also represent gate D with the mathematical formula $w_6 = \text{AND}(w_3, w_5)$.
In this document, we only use INV and AND logic gates. INV represents the NOT operation, and it operates according to the following rule:
The AND logic gate represents the AND operation, and it operates according to the following rule:
Input 1 Input 2 Output | {"url":"https://docs.satoshivm.io/satoshivm/bitvm-and-satoshivm-circuit","timestamp":"2024-11-08T05:22:08Z","content_type":"text/html","content_length":"371707","record_id":"<urn:uuid:76447d28-03b8-4e27-9df7-e6818d96e8a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00824.warc.gz"} |
pst-func – PSTricks package for plotting mathematical functions
The package is built for use with PSTricks. It provides macros for plotting and manipulating various mathematical functions:
• polynomials and their derivatives f(x)=an*x^n+an-1*x^(n-1)+...+a0 defined by the coefficients a0 a1 a2 ... and the derivative order;
• the Fourier sum f(x) = a0/2+a1cos(omega x)+...+b1sin(omega x)+... defined by the coefficients a0 a1 a2 ... b1 b2 b3 ...;
• the Bessel function defined by its order;
• the Gauss function defined by sigma and mu;
• Bézier curves from order 1 (two control points) to order 9 (10 control points);
• the superellipse function (the Lamé curve);
• Chebyshev polynomials of the first and second kind;
• the Thomae (or popcorn) function;
• the Weierstrass function;
• various integration-derived functions;
• normal, binomial, poisson, gamma, chi-squared, student’s t, F, beta, Cauchy and Weibull distribution functions and the Lorenz curve;
• the zeroes of a function, or the intermediate point of two functions;
• the Vasicek function for describing the evolution of interest rates; and
• implicit functions.
The plots may be generated as volumes of rotation about the X-axis, as well.
Sources /graphics/pstricks/contrib/pst-func
Home page https://tug.org/PSTricks/main.cgi/
Support https://tug.org/mailman/listinfo/pstricks
Repository https://archiv.dante.de/~herbert/TeXnik/
Version 1.02a 2024-03-31
Licenses The LaTeX Project Public License
Maintainer Herbert Voß
Contained in TeXLive as pst-func
MiKTeX as pst-func
Graphics use
Topics PSTricks
Graphics plot function
Download the contents of this package in one zip archive (3.9M).
Community Comments
Maybe you are interested in the following packages as well.
Package Links | {"url":"https://www.ctan.org/pkg/pst-func","timestamp":"2024-11-05T23:25:17Z","content_type":"text/html","content_length":"18879","record_id":"<urn:uuid:c6f815a2-05b4-4313-b0eb-fb1d0345ab6d>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00359.warc.gz"} |
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Longest Substring Without Repeating Characters in Python
In the domain of competitive programming and coding interviews, having a knack for efficiently solving complex problems is highly sought after. One prime example of such a challenge is the task of
finding the longest substring without any repeated characters. The goal of this article is to delve deeply into this problem, presenting various approaches, and shedding light on the methods for
determining the length of the longest substring without any character repetitions.
Understanding the Problem
Let's begin by establishing a clear understanding of the problem. Given a string, our goal is to identify the longest substring within that string in which no character is repeated. Essentially, we
are searching for the longest unique substring within the input string. We will also try to find a solution with the optimal time complexity.
For instance, consider the input string "FAVTUTOR". The longest substring without repeating characters, in this case, is "FAVTU," with a length of 5. Similarly, for the word "PYTHON", it’s 6.
Brute Force Approach
To gain a better understanding of the problem, we can start by examining a brute force solution. In this method, we generate all possible substrings of the input string and check each substring for
the presence of repeating characters.
Although this approach is straightforward, it proves highly inefficient, especially for longer input strings.
Here's a simplified Python code snippet to illustrate the brute force approach:
def longest_unique_substring(s):
max_length = 0
for i in range(len(s)):
for j in range(i, len(s)):
if len(set(s[i:j+1])) == j - i + 1:
max_length = max(max_length, j - i + 1)
longest_substring = s[i:j+1]
return max_length, longest_substring
def main():
input_string = "FAVTUTOR"
max_length, longest_substring = longest_unique_substring(input_string)
print(f"Input String: {input_string}")
print(f"Longest Substring without Repeating Characters: {longest_substring}")
print(f"Length of the Longest Substring: {max_length}")
if __name__ == "__main__":
Input String: FAVTUTOR
Longest Substring without Repeating Characters: FAVT
Length of the Longest Substring: 4
The brute force approach uses nested loops to consider all possible substrings of the input string. The outer loop (i) iterates through each character as a potential starting position, while the
inner loop (j) scans through characters from the starting position to the end of the string. For each substring candidate (s[i:j+1]), it creates a set to check for repeating characters. If no
repeating characters are found, the length of the candidate substring is compared to the current maximum length. The approach continues to evaluate all substrings and updates the maximum length and
longest substring as it finds longer unique substrings.
Time and Space Complexity
The time complexity of this approach is O(n^3), where n is the length of the input string. It is evident that this is not an efficient solution for practical applications. The auxiliary space
complexity of this problem is O(1).
Sliding Window Approach
To substantially improve the time complexity of our solution, we can employ the sliding window technique. The core idea is to maintain a window of characters and slide it through the input string,
ensuring that no repeating characters are present within the window.
Here's a Python implementation of the sliding window approach:
def longest_unique_substring(s):
max_length = 0
char_index = {}
start = 0
longest_substring = ""
for end in range(len(s)):
if s[end] in char_index and char_index[s[end]] >= start:
start = char_index[s[end]] + 1
char_index[s[end]] = end
if end - start + 1 > max_length:
max_length = end - start + 1
longest_substring = s[start:end+1]
return max_length, longest_substring
def main():
input_string = "FAVTUTOR"
max_length, longest_substring = longest_unique_substring(input_string)
print(f"Input String: {input_string}")
print(f"Longest Substring without Repeating Characters: {longest_substring}")
print(f"Length of the Longest Substring: {max_length}")
if __name__ == "__main__":
Input String: FAVTUTOR
Longest Substring without Repeating Characters: FAVT
Length of the Longest Substring: 4
In this code, we use a sliding window represented by the 'start' and 'end' indices, and a dictionary 'char_index' to store the last index of each character encountered.
Time and Space Complexity
This approach results in a time complexity of O(n), where n is the length of the input string. The space complexity remains the same at O(1).
Optimizing the Solution
While the sliding window approach is efficient, further optimization is possible. We can eliminate the need to track the index of every character encountered and instead directly update the 'start'
def longest_unique_substring(s):
max_length = 0
char_index = {}
start = 0
longest_substring = ""
for end in range(len(s)):
if s[end] in char_index:
start = max(start, char_index[s[end]] + 1)
char_index[s[end]] = end
if end - start + 1 > max_length:
max_length = end - start + 1
longest_substring = s[start:end+1]
return max_length, longest_substring
def main():
input_string = "FAVTUTOR"
max_length, longest_substring = longest_unique_substring(input_string)
print(f"Input String: {input_string}")
print(f"Longest Substring without Repeating Characters: {longest_substring}")
print(f"Length of the Longest Substring: {max_length}")
if __name__ == "__main__":
Input String: FAVTUTOR
Longest Substring without Repeating Characters: FAVT
Length of the Longest Substring: 4
Time and Space Complexity
By implementing this optimized approach, we can efficiently find the length of the longest substring without repeating characters in O(n) time. The space complexity for this problem is fixed at O(1).
Handling Edge Cases
While the sliding window approach is powerful, it's crucial to consider edge cases and additional scenarios.
For example, what if the input string contains spaces or special characters? How can we handle Unicode characters or non-ASCII characters?
Handling Spaces and Special Characters
When dealing with input strings that contain spaces or special characters, the sliding window approach remains effective. Spaces and special characters are treated like any other character within the
string. The algorithm will correctly identify the longest substring without repeating characters, including spaces and special characters.
Handling Unicode and Non-ASCII Characters
To handle Unicode or non-ASCII characters, you can still use the sliding window approach. Unicode characters are treated as individual characters within the input string. The algorithm works as
expected by identifying the longest substring without repeating characters, irrespective of the character encoding used.
Variations and Applications
The concept of finding the longest substring without repeating characters has various applications in computer science and programming. Here are some notable variations and use cases:
1. Longest Subarray with Distinct Elements
In an array of numbers, finding the longest subarray with distinct elements follows a similar approach. Instead of characters, you work with integers and apply the sliding window technique to achieve
the desired result.
2. Password Strength Validation
Many websites and applications use a password strength meter that evaluates the uniqueness of characters in a password. By understanding how to find the longest substring without repeating
characters, you can create more robust password strength validation algorithms.
3. DNA Sequence Analysis
In bioinformatics, analyzing DNA sequences often involves identifying unique subsequences or patterns within a given DNA sequence. The techniques used to find the longest substring without repeating
characters can be adapted for such analyses.
4. Text Processing and Parsing
Natural language processing and text analysis frequently require the identification of unique phrases or words within a text. The algorithms developed for this problem can be instrumental in these
Real-World Examples
To demonstrate the practical application of finding the longest substring without repeating characters, let's consider two real-world examples.
Example 1: Password Strength Checker
Imagine you are building a registration system for a website, and you want to ensure that users create strong and unique passwords. By employing the longest unique substring algorithm, you can check
if the password contains a sufficiently long substring without repeating characters, making it more secure.
Example 2: Language Translation
In a machine translation system, it's essential to identify unique words or phrases in the source language to improve translation accuracy. By utilizing the techniques discussed in this article, you
can preprocess the text and extract unique segments to enhance the translation process.
In this article, we dug into the intricate problem of finding the longest substring without repeating characters in Python. We began by discussing the brute force approach and highlighted its
inefficiency, paving the way for the sliding window technique. The sliding window approach, along with optimizations, provided an efficient solution to the problem, with a time complexity of O(n). | {"url":"https://favtutor.com/blogs/longest-substring-without-repeating-characters-in-python","timestamp":"2024-11-12T07:31:41Z","content_type":"text/html","content_length":"89943","record_id":"<urn:uuid:7aab7b62-3948-41f8-9bbd-31f67ea33eaa>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00198.warc.gz"} |
Logical disjunction
$(0,1)$-Category theory
Logical disjunction
In logic, logical disjunction is the join in the poset of truth values.
Assuming that (as in classical logic) the only truth values are true ($T$) and false ($F$), then the disjunction $p \vee q$ of the truth values $p$ and $q$ may be defined by a truth table:
$p$ $q$ $p \vee q$
$T$ $T$ $T$
$T$ $F$ $T$
$F$ $T$ $T$
$F$ $F$ $F$
That is, $p \vee q$ is true if and only if at least one of $p$ and $q$ is true. Disjunction also exists in nearly every non-classical logic.
More generally, if $p$ and $q$ are any two relations on the same domain, then we define their disjunction pointwise, thinking of a relation as a function to truth values. If instead we think of a
relation as a subset of its domain, then disjunction becomes union.
In natural deduction the inference rules for disjunction are given as
$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma \vdash P \vee Q \; \mathrm{prop}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm
{prop}}{\Gamma, P \; \mathrm{true} \vdash P \vee Q \; \mathrm{true}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma, Q \; \mathrm{true} \vdash P \vee Q
\; \mathrm{true}}$
$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop} \quad \Gamma, P \vee Q \; \mathrm{true} \vdash R \; \mathrm{prop} \quad \Gamma, P \; \mathrm{true} \vdash R \; \mathrm
{true} \quad \Gamma, Q \; \mathrm{true} \vdash R \; \mathrm{true}}{\Gamma, P \vee Q \; \mathrm{true} \vdash R \; \mathrm{true}}$
Disjunction as defined above is sometimes called inclusive disjunction to distinguish it from exclusive disjunction, where exactly one of $p$ and $q$ must be true.
In the context of substructural logics such as linear logic, we often have both additive disjunction $\oplus$ and multiplicative disjunction $\parr$; see the Rules of Inference below for the
distinction. In linear logic, additive disjunction is the join under the entailment relation, just like disjunction in classical logic (and intuitionistic logic), while multiplicative disjunction is
something different.
Disjunction is de Morgan dual to conjunction.
Like any join, disjunction is an associative operation, so we can take the disjunction of any finite positive whole number of truth values; the disjunction is true if and only if at least one of the
various truth values is true. Disjunction also has an identity element, which is the false truth value. Some logics allow a notion of infinitary disjunction. Indexed disjunction is existential
In dependent type theory
In dependent type theory, the disjunction of two mere propositions, $P$ and $Q$, is the bracket type of their sum type, $\| P + Q \|$. Disjunction types in general could also be regarded as a
particular sort of higher inductive type. In Coq syntax:
Inductive disjunction (P Q:Type) : Type :=
| inl : P -> disjunction P Q
| inr : Q -> disjunction P Q
| contr0 : forall (p q : disjunction P Q) p == q
If the dependent type theory has a type of propositions $\mathrm{Prop}$, such as the one derived from a type universe $U$ - $\sum_{A:U} \mathrm{isProp}(A)$, then the disjunction of two types $A$ and
$B$ is defined as the dependent function type
$A \vee B \equiv \prod_{P:\mathrm{Prop}} ((A \to P) \times (B \to P)) \to P$
By weak function extensionality, the disjunction of two types is a proposition.
The two definitions above are equivalent.
The propositional truncation of a type $A$ is equivalent to the following dependent function type
$\| A \| \simeq \prod_{P:\mathrm{Prop}} (A \to P) \to P$
Substituting the sum type $A + B$ for $A$, we have
$\| A + B \| \simeq \prod_{P:\mathrm{Prop}} ((A + B) \to P) \to P$
Given any type $C$, there is an equivalence
$((A + B) \to C) \simeq (A \to C) \times (B \to C)$
and if $A \simeq B$, then $(A \to C) \simeq (B \to C)$. In addition, for all type families $x:A \vdash B(x)$, and $x:A \vdash C(x)$, if there is a family of equivalences $e:\prod_{x:A} B(x) \simeq C
(x)$, then there is an equivalence $\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A} C(x)\right)$. All this taken together means that there are equivalences
$\| A + B \| \simeq \left(\prod_{P:\mathrm{Prop}} ((A + B) \to P) \to P\right) \simeq \left(\prod_{P:\mathrm{Prop}} ((A \to P) \times (B \to P)) \to P\right)$
If one has the boolean domain and the existential quantifier, then the disjunction of two types $A$ and $B$ is given by the following type:
$A \vee B \coloneqq \exists b:\mathrm{bool}.((b = \mathrm{true}) \to A) \times ((b = \mathrm{false}) \to B)$
The disjunction $P \vee Q$ of two mere propositions $P$ and $Q$ is also the join type of the two types $P * Q$. This is because every mere proposition is a subtype of the unit type, and the
disjunction of $P$ and $Q$ is the union of $P$ and $Q$ as two subtypes of the unit types, and the union of $P$ and $Q$ as subtypes of the unit type is defined to be the join type of $P$ and $Q$, the
pushout type of the two product projection functions from the product type $P \times Q$ to $P$ and $Q$ respectively.
Classical vs constructive
There are a variety of connectives that are distinct in intuitionistic logic but are all equivalent to disjunction in classical logic. Here is a Hasse diagram of some of them, with the strongest
statement at the bottom and the weakest at the top (so that each statement entails those above it):
$\array { & & eg(eg{P} \wedge eg{Q}) \\ & ⇗ & & ⇖ \\ eg{P} \rightarrow Q & & & & P \leftarrow eg{Q} \\ & ⇖ & & ⇗ \\ & & (eg{P} \rightarrow Q) \wedge (P \leftarrow eg{Q}) \
\ & & \Uparrow \\ & & P \vee Q }$
(A single arrow is implication in the object language; a double arrow is entailment in the metalanguage.) Note that $eg{P} \wedge eg{Q}$ is the negation of every item in this diagram.
In the double-negation interpretation? of classical logic in intuitionistic logic, $eg(eg{P} \wedge eg{Q})$ is the interpretation in intuitionistic logic of disjunction in classical logic. For this
reason, $eg(eg{P} \wedge eg{Q})$ is sometimes called classical disjunction. But this doesn't mean that it should always be used when turning classical mathematics into constructive mathematics.
Indeed, a stronger statement is almost always preferable, if one is valid; $eg(eg{P} \wedge eg{Q})$ is merely the fallback position when nothing better can be found. (And as can be seen in the
example in the paragraph after next, sometimes even this is not valid.)
In the antithesis interpretation of affine logic in intuitionistic logic, $(eg{P} \rightarrow Q) \wedge (P \leftarrow eg{Q})$ is the interpretation of the multiplicative disjunction $P \parr Q$ for
affirmative propositions. More generally, a statement $P$ in affine logic is interpreted as a pair $(P^+,P^-)$ of mutually contradictory statements in intuitionistic logic; $P^-$ is simply the
negation of $P^+$ for affirmative propositions, but in general, $P^-$ only entails $eg{P^+}$. Then $P \parr Q$ is interpreted as $\big((P^- \rightarrow Q^+) \wedge (P^+ \leftarrow Q^-), P^- \wedge Q^
-\big)$; that is, $(P \parr Q)^+$ is $(P^- \rightarrow Q^+) \wedge (P^+ \leftarrow Q^-)$, and $(P \parr Q)^-$ is $P^- \wedge Q^-$. (In contrast, the additive disjunction $P \oplus Q$ is interpreted
as $(P^+ \vee Q^+, P^- \wedge Q^-)$. Note that $P \oplus Q$ entails $P \parr Q$ in affine logic, even though they are independent in linear logic.)
For a non-affirmative example, in the arithmetic of (located) real numbers, it is not constructively valid to derive $(a = 0) \vee (b = 0)$ from $a b = 0$, and it's not even valid to derive $eg\big
(eg(a = 0) \wedge eg(b = 0)\big)$ without Markov's principle (or at least some weak version of it), but it is valid to derive $(a \# 0) \rightarrow (b = 0)$ (and conversely), where $\#$ is the usual
apartness relation between real numbers. (Here, $P^+$ is $a = 0$ and $P^-$ is $a \# 0$, and similarly for $Q$ and $b$.) Of course, it's also valid to derive $eg\big((a \# 0) \wedge (b \# 0)\big)$
(which is actually equivalent).
Rules of inference
The rules of inference for disjunction in sequent calculus are dual to those for conjunction:
$\begin {gathered} \frac { \Gamma , p , \Delta \vdash \Sigma ; \; \Gamma , q , \Delta \vdash \Sigma } { \Gamma , p \vee q , \Delta \vdash \Sigma } \; \text {left additive} \\ \frac { \Gamma \vdash \
Delta , p , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \; \text {right additive 0} \\ \frac { \Gamma \vdash \Delta , q , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \; \text
{right additive 1} \\ \end {gathered}$
Equivalently, we can use the following rules with weakened contexts:
$\begin {gathered} \frac { \Gamma , p \vdash \Delta ; \; q , \Sigma \vdash \Pi } { \Gamma , p \vee q , \Sigma \vdash \Delta , \Pi } \; \text {left multiplicative} \\ \frac { \Gamma \vdash \Delta , p
, q , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \text {right multiplicative} \\ \end {gathered}$
The rules above are written so as to remain valid in logics without the exchange rule. In linear logic, the first batch of sequent rules apply to additive disjunction (interpret $p \vee q$ in these
rules as $p \oplus q$), while the second batch of rules apply to multiplicative disjunction (interpret $p \vee q$ in those rules as $p \parr q$).
The natural deduction rules for disjunction are a little more complicated than those for conjunction:
$\begin {gathered} \frac { \Gamma , p \vdash r ; \; \Gamma , q \vdash r } { \Gamma , p \vee q \vdash r } \; \text {elimination} \\ \frac { \Gamma \vdash p } { \Gamma \vdash p \vee q } \; \text
{introduction 0} \\ \frac { \Gamma \vdash q } { \Gamma \vdash p \vee q } \; \text {introduction 1} \\ \end {gathered}$
The definition of the disjunction of two types in dependent type theory as the propositional truncation of the sum type is found in:
The definition of the disjunction of two mere propositions in dependent type theory as the join type of propositions is found in:
And the disjunction of two types defined from the type of propositions and dependent product types can be found in:
• Madeleine Birchfield, Constructing coproduct types and boolean types from universes, MathOverflow (web) | {"url":"https://ncatlab.org/nlab/show/disjunction","timestamp":"2024-11-09T04:44:50Z","content_type":"application/xhtml+xml","content_length":"107034","record_id":"<urn:uuid:b5940b34-faef-4c9d-8964-1a8869f93bb0>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00484.warc.gz"} |
Lesson 4
Understanding Decay
Lesson Narrative
This lesson continues to examine quantities that change exponentially, focusing on a quantity that decays or decreases. Students are alerted that sometimes people use the terms exponential growth and
exponential decay to distinguish between situations where the growth factor is greater than or less than 1. Additionally, students learn that when the growth factor is less than 1 (but still
positive), people sometimes refer to it as the decay factor.
The opening activity encourages students to view a quantity that decreases by a factor of itself using multiplication rather that subtraction. If a computer costs \$500 and loses \(\frac15\) of its
value each year, after one year we could write this, in dollars, as \(500 - \left(\frac15\right)\boldcdot 500\). But if we write this using multiplication, as \(500 \boldcdot \left(\frac45\right)\),
then we are in a better position to see that after 2 years its value in dollars will be \(500 \boldcdot \left(\frac45\right) \boldcdot \left(\frac45\right)\) and, after \(t\) years, the value will be
\(500 \boldcdot \left(\frac45\right)^t\). In other words, exponents are a particularly useful way to express repeated loss by a factor of the original amount. Students will carry this understanding
into future lessons that deal with repeated percentage change situations.
In the second activity, students apply this idea to write an equation for the value of a car after \(t\) years, assuming that it decreases by the same factor each year.
Using an exponent to express repeated decrease by the same factor is a good example of a generalization based on repeated calculation (MP8). Writing \(500 \boldcdot \left(\frac45\right)^t\) expresses
the computation of repeatedly decreasing by \(\frac15\), \(t\) times.
Learning Goals
Teacher Facing
• Comprehend that the term "exponential growth" describes a quantity that changes by a growth factor that is greater than 1, and the term “exponential decay” describes a quantity that changes by a
growth factor that is less than 1 but greater than 0.
• Use only multiplication to represent “decreasing a quantity by some fraction of itself.”
• Write an expression or an equation to represent a situation where a quantity decays exponentially.
Student Facing
Let’s look at exponential decay.
Student Facing
• I can use only multiplication to represent "decreasing a quantity by a fraction of itself."
• I can write an expression or equation to represent a quantity that decays exponentially.
• I know the meanings of “exponential growth” and “exponential decay.”
CCSS Standards
Building On
Building Towards
Glossary Entries
• growth factor
In an exponential function, the output is multiplied by the same factor every time the input increases by one. The multiplier is called the growth factor.
Additional Resources
Google Slides For access, consult one of our IM Certified Partners.
PowerPoint Slides For access, consult one of our IM Certified Partners. | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/1/5/4/preparation.html","timestamp":"2024-11-03T06:33:16Z","content_type":"text/html","content_length":"79667","record_id":"<urn:uuid:91183fed-dcc3-46e0-b9f2-7707d7173c10>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00281.warc.gz"} |
What are the strategies and skills of flying chess?
Flying chess is a game of strategy, and the strategy is based on probability. The more planes you have to choose from, the more strategies you can use.
Due to the rules of flying chess, the number of aircraft we can use is 0-1 - more - 1-0 process. We need to discuss the five situations in order to have a better decision.
The first and second are the opening states, in fact, there is no need to discuss, anyway, the choice is one-way, you can not say that you change the probability of 1/6.
In particular, if you throw 6 in a single machine, you should calculate the distance of the nearby enemy, if not in a critical situation, you should first take off from the base, form a multi-machine
situation, and enter the middle game.
The third situation is the middle game. The purpose of the middle game is to prepare for victory, there are two directions of combat, one is to expand the advantage, the other is to suppress the
To expand advantage is to advance and increase combat power (number of aircraft).
The strategy of going forward is nothing more than choosing which plane to go forward. In general, to ensure the safety of the aircraft after the advance, the best strategy is to keep the aircraft
away from the opposing plane after the advance (the distance is greater than 6, and not on the jump 4 grid), including the field and the opposing plane has taken off. The secondary strategy is to
choose to play with an opponent who has only one plane, and his plane, because the opponent has no choice, and whether his plane will hit you is a matter of probability.
Here is a trick, if you have two planes, after moving forward, one is one step ahead of the other plane, and the other is two steps ahead of the other plane, then the former choice will be more
favorable. Because the probability of the former getting hit is 1 in 6, and the probability of the latter getting hit is 7 in 36. The principle of this technique is to minimize the probability of
being hit, and the same is true for other point comparisons.
The decision to add power is always right, but when to add power and whether it will end after adding power need to be judged according to reality.
There is only a 1/6 chance of adding power, and for us, adding power is the opportunity to sacrifice a 6 square advance. We only need to give up the opportunity to take off if there is a high chance
that the other side will eat our pieces after advancing. The odds are just a matter of character. My recommendation is to take 1/3 as the baseline. If the probability is not less than 1/3, we must
use this opportunity to escape.
Two points should be added here: First, the above decision is based on the number of aircraft in the middle stage is greater than 2. The second is to note that the calculation of probability should
be calculated together with the number of other parties.
Whether to stand down after adding combat force, we should judge according to the number of aircraft of the other side. If only 1 aircraft can be called by the other side (this is calculated
according to a single person, taking the maximum number of aircraft can be called by the other side to judge), we can send 2 or more aircraft to enter. If the number of callable parties is 2, the
number of aircraft sent by us shall not exceed 2. If the other side has 3-4 aircraft, the single aircraft on our field is not threatened at close range, the number of our aircraft should be 1; If
threatened, we can send one more plane to increase the probability of escape and counterattack.
It's all about expanding the advantage, then we'll talk about crushing the opponent.
The crash is the fun of the game, but it's not all. For a crash plane, the ideal situation is that you are not in any danger after hitting someone. Therefore, our decision is to give priority to
advance and then consider suppressing the opponent.
The crash machine is divided into collision and collision avoidance. Personally, I understand that there are two principles in it: 1 is the priority of collision prevention over collision and 2 is
the late collision is better than early collision
Rule number one is easy to understand. Stay green, even if there's no wood to burn. The second is that the closer you get to the end, the more the dike will be hit. The plane flew so far, indicating
that a lot of points have been used in it, and the second is something we need to pay special attention to. At the same time, Article 2 can be extended to a truth: the top of the levee takes
precedence over the bottom of the levee. Because of the design of the chessboard, the starting point of the home is closer to our end, so it is more cost-effective to pay attention to the movement of
the home.
Based on the above two principles, we can distinguish several techniques:
1. When the aircraft is near the starting point of the other side, it should pay attention to countermeasures and try to quickly pass the starting point of the other side after 6 bars, which is also
the low-risk striking distance of the other side. And the closer you get to the end, the more attention you need to pay.
2 is the choice of collision, if we have two planes on the field, both in the other side of the 12 grid (12 grid is the addition of two 6, which is to give up a strike distance), then we can give
priority to the other side near the end of the aircraft.
3. Minimize stacking. Unless there are a lot of planes, don't risk it. If the superposition party is less than 6 squares ahead and the stand-alone party is blocked, the probability of our pure gain
is 2/3, the probability of two consecutive gains is 4/9, and the probability of three consecutive gains is 8/27. On the other hand, the probability of changing from the superposition state to two
machines away from the other machine is different, the probability of achieving the purpose of a roll is 1/18 (double 6 or a 6 plus a hop), if you want to wait until the roll twice, the probability
of the other party hitting us will be greatly increased. Such a situation should be avoided at all costs.
4. In the middle of the game, it is inevitable that there will be aircraft entering the final terminal. At this time, if we still have aircraft outside the terminal, then we do not need to enter the
terminal in such a hurry. The end channel can "waste" some unnecessary points for us. Ideally, use the finish line and wait until the points for avoidance and collision are available.
The middle game is basically the essence of flying chess, as long as the number of planes in the middle game has been maintained at more than 2, the probability of victory will be greatly increased.
As for going to the fourth and fifth stages, these two stages are also one-way choices, and basically can't use any strategy.
Before the arrival of these two stages, we need to pay attention to the position of our aircraft, try to avoid sudden attacks, it is best to enter the channel together, and achieve these two stages
in the channel. | {"url":"https://www.xxchessboard.com/What-are-the-strategies-and-skills-of-flying-chess-id49640756.html","timestamp":"2024-11-05T12:13:01Z","content_type":"text/html","content_length":"195004","record_id":"<urn:uuid:d08d437d-0741-4e72-86b4-7afdb6725227>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00146.warc.gz"} |
Unscramble OBELISK
How Many Words are in OBELISK Unscramble?
By unscrambling letters obelisk, our Word Unscrambler aka Scrabble Word Finder easily found 102 playable words in virtually every word scramble game!
Letter / Tile Values for OBELISK
Below are the values for each of the letters/tiles in Scrabble. The letters in obelisk combine for a total of 13 points (not including bonus squares)
• O [1]
• B [3]
• E [1]
• L [1]
• I [1]
• S [1]
• K [5]
What do the Letters obelisk Unscrambled Mean?
The unscrambled words with the most letters from OBELISK word or letters are below along with the definitions.
• obelisk (n.) - An upright, four-sided pillar, gradually tapering as it rises, and terminating in a pyramid called pyramidion. It is ordinarily monolithic. Egyptian obelisks are commonly covered
with hieroglyphic writing from top to bottom. | {"url":"https://www.scrabblewordfind.com/unscramble-obelisk","timestamp":"2024-11-13T08:35:29Z","content_type":"text/html","content_length":"54204","record_id":"<urn:uuid:e5212a39-4e55-4fdf-9128-3c2866696b20>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00742.warc.gz"} |
Course Syllabus | Unc Education Progra
top of page
Tentative Course Schedule (14 hours, one hour per week)
Lecture 1 (1/12): Welcome and Course Introduction (Zhengwu Zhang and Guorong Wu, UNC)
Lecture 2 (1/19): Neuroscience perspective of human connectome (Paul Laurienti, Wake Forest)
In this lecture, we will briefly introduce the history of brain networks in neuroscience, basic concepts in graph theory, and complex systems.
Lecture 3 (1/26): Image processing pipeline for structural neuroimages (Guorong Wu, UNC)
We will cover the basic image processing techniques, including histogram, voxel-based processing, segmentation, and registration. After that, we will introduce the major steps of T1-weighted magnetic
resonance images (MRI) and cortical surface construction.
Lecture 4 (2/2): Diffusion-weighted imaging and diffusion tensor image analysis - I (Martin Styner, UNC)
We will go through the imaging physics of diffusion-weighted imaging and the image processing pipeline for diffusion tensor images.
Lecture 5 (2/9): Diffusion-weighted imaging and diffusion tensor image analysis - II (Martin Styner, UNC)
We will demonstrate the computational pipeline for DWI/DTI image processing.
Lecture 6 (2/16): Optimizing Brain Network Acquisition (Zhengwu Zhang, UNC)
We will introduce acquisition approaches to enhance the network quality.
Lecture 7 (2/23):
Hands-on sessions for constructing your first structural and functional brain network (Jiaqi Ding and Tingting Dan, UNC)
This lecture will guide the students in practicing the tissue segmentation and registration pipelines for T1-weighted MRI, using the software developed in-house. Also, the students will practice
surface reconstruction and generate structural brain networks based on the pre-calculated tractography results.
Lecture 8 (3/1): Machine learning on human connectome data -- Session I (Zhengwu Zhang, UNC)
General introduction to machine learning, including dimension reduction, clustering, classification, evaluation metrics, and state-of-the-art deep learning techniques.
Lecture 9 (3/8): Guest lecture on Bayesian analysis for human connectomes (Heather Shappell, Wake Forest)
This lecture will talk about the Bayesian approach for modeling the brain state changes underlying functional fluctuations.
Lecture 10 (3/22): Guest lecture on functional network analysis (Martin Lindquist, John Hopkins University)
Lecture 10 (4/5): Graph theory for structural/functional network analysis (Guorong Wu, UNC)
We will explain the widely used graph measurement for brain networks. The students will practice using the web-based software to calculate the graph measures in a group comparison study. We will
study small-world properties of brain networks and computational methods to characterize network communities and hubs.
Lecture 12 (4/12): Machine learning on human connectome data -- Session II (Guorong Wu, UNC)
Graph neural networks and graph convolutional networks.
Lecture 13 (4/19): Guest lecture on connectome-genetics (Hongtu Zhu, UNC)
Lecture 14 (4/26): Course summary (Zhengwu Zhang and Guorong Wu, UNC)
We will summarize the computational techniques for human connectome analysis.
bottom of page | {"url":"https://www.unc-epic.org/course-syllabus","timestamp":"2024-11-11T21:14:44Z","content_type":"text/html","content_length":"432698","record_id":"<urn:uuid:b5f80522-2f14-42dd-9523-54b1bdaae35e>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00637.warc.gz"} |
Diving Into the Quantum Realm: Exploring the Basics and Potential of Quantum Computing for Web Developers
Hello fellow developers! Today we will embark on an exciting journey into the mysterious and intriguing world of Quantum Computing and explore its potential impact on web development. Quantum
computing is not just sci-fi; it's a real, emerging field that could revolutionize the way we approach problem-solving in computing. So put on your lab coats, and let's dive in! π §ͺπ ¨β π ¬
What is Quantum Computing?
Quantum computing is a technology that leverages the principles of quantum mechanics to process information. Unlike classical computers that use bits (0's and 1's) for computation, quantum computers
use quantum bits or 'qubits', which can exist in multiple states simultaneously thanks to properties like superposition and entanglement. This allows quantum computers to perform complex calculations
at speeds unattainable by conventional computers.
Now, before we go any further, let's setup a Python environment that can simulate a quantum computer using Qiskit, an open-source quantum computing software development framework by IBM. Python is a
popular language among web developers, so it should feel right at home!
First, ensure you have Python installed on your system, and then install Qiskit via pip:
pip install qiskit
Next, let's write our first quantum program. Create a new Python file hello_quantum.py and import Qiskit:
from qiskit import QuantumCircuit, transpile, Aer, execute
# Create a Quantum Circuit acting on a quantum register of one qubit
circuit = QuantumCircuit(1)
# Apply quantum gate
# Map the quantum measurement to the classical bits
# Execute the circuit on the qasm simulator
simulator = Aer.get_backend('aer_simulator')
compiled_circuit = transpile(circuit, simulator)
# Execute the circuit and get the result
job = execute(compiled_circuit, simulator)
result = job.result()
# Print the result
counts = result.get_counts(compiled_circuit)
print(f"Counts: {counts}")
Running this simple program will demonstrate a basic quantum computation, exhibiting the probability nature of qubits.
Quantum Computing and Web Development: A Match Made in the Quantum Realm?
You might wonder, how does this all fit with web development? The reality is that quantum computing can significantly enhance certain web applications, especially those that require complex
calculations such as cryptography, optimization problems, machine learning models, or even complex financial algorithms. Imagine a future where your web app can use quantum computing to solve in
seconds what would have taken days or even months!
However, we're still in the early days of quantum computing, and there are many challenges to be solved, such as error rate management and the creation of stable qubits. But as web developers, it's
crucial to stay informed and ready for this quantum leap.
Hereβ s a glimpse of what connecting a quantum computer to a web app might look like using Qiskit and Flask:
from flask import Flask, jsonify
from qiskit import QuantumCircuit, transpile, Aer, execute
app = Flask(__name__)
def quantum_computation():
circuit = QuantumCircuit(2)
circuit.cx(0, 1)
simulator = Aer.get_backend('aer_simulator')
compiled_circuit = transpile(circuit, simulator)
job = execute(compiled_circuit, simulator)
result = job.result()
counts = result.get_counts(compiled_circuit)
return jsonify(counts)
if __name__ == '__main__':
This sample Flask app illustrates how you could theoretically send quantum computation results to the web client in JSON format. π β ¨
Stepping into the Future
The possibilities of quantum computing are immense and can take the web to dimensions we're yet to conceive. But remember, quantum computing is a double-edged sword β with great power comes great
responsibility, especially in terms of security and cryptography.
For those who want to dive deeper and stay ahead of the curve, I highly recommend exploring more about Qiskit (Qiskit) and other quantum computing platforms. Keep learning and experimenting, but
remember that as technology evolves, so too may the resources and links we use.
It's an exciting time to be a developer, with front-row seats to the evolution of computing. Happy coding, and may the quantum force be with you! π π »
Please note that some of the links may become outdated as technology evolves rapidly. | {"url":"https://ai.fabiopacifici.com/posts/diving-into-the-quantum-realm-exploring-the-basics-and-potential-of-quantum-computing-for-web-developers","timestamp":"2024-11-14T10:41:47Z","content_type":"text/html","content_length":"16538","record_id":"<urn:uuid:cf7c5ee3-843c-43dc-9613-fc0cc12be317>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00897.warc.gz"} |
Tournament - Model comparison
Solvi Rognvaldsson, Rafael Vias, Birgir Hrafnkelsson and Axel Orn Jansson
This vignette explores the ways you can compare the fit of the different discharge rating curve models provided in the bdrc package. The package includes four different models to fit a discharge
rating curve of different complexities. These are:
plm0() - Power-law model with a constant error variance (hence the 0). This is a Bayesian hierarchical implementation of the most commonly used discharge rating curve model in hydrological practice.
plm() - Power-law model with error variance that varies with water elevation.
gplm0() - Generalized power-law model with a constant error variance (hence the 0). The generalized power law is introduced in Hrafnkelsson et al. (2022).
gplm() - Generalized power-law model with error variance that varies with water elevation. The generalized power law is introduced in Hrafnkelsson et al. (2022).
To learn more about the models, see Hrafnkelsson et al. (2022). To learn about how to run the models on your data, see the introduction vignette. The tournament is a model comparison method that uses
the Widely Applicable Information Criterion (WAIC) (see Watanabe (2010)) to select the most appropriate of the four models given the data. The WAIC consists of two terms, a measure of the
goodness-of-fit, and a penalizing term to account for model complexity (effective number of parameters). The first round of model comparisons sets up two games between model types, “gplm” vs “gplm0”
and “plm” vs. “plm0”. The two comparisons are conducted such that if the WAIC of the more complex model (“gplm” and “plm”, respectively) is smaller than the WAIC of the simpler models (“gplm0” and
“plm0”, respectively) by a pre-specified value called the winning criteria (default value = 2.2), then it wins the game and is chosen as the more appropriate model. If not, the simpler model is
chosen. The more appropriate models move on to the second round and are compared in the same way. The winner of the second round is chosen as the overall tournament winner and deemed the most
appropriate model given the data. In each match, the difference in WAIC is defined as \(\Delta\)WAIC\(=\)WAIC\(_{\text{simple}}-\)WAIC\(_{\text{complex}}\). A positive value of \(\Delta\)WAIC
indicates that the more complex model is a more appropriate model, but the more complex model only goes through to the final round if \(\Delta\)WAIC>winning_criteria.
To introduce the tournament function, we will use a dataset from the stream gauging station Krokfors in Sweden that comes with the package:
> library(bdrc)
> data(krokfors)
> krokfors
#> W Q
#> 1 9.478000 10.8211700
#> 2 8.698000 1.5010000
#> 3 9.009000 3.3190000
#> 4 8.097000 0.1595700
#> 5 9.104000 4.5462500
#> 6 8.133774 0.2121178
#> 7 8.569583 1.1580000
#> 8 9.139151 4.8110000
#> 9 9.464250 10.9960000
#> 10 8.009214 0.0984130
#> 11 8.961843 2.7847910
#> 12 8.316000 0.6631890
#> 13 8.828716 1.8911800
#> 14 9.897000 20.2600000
#> 15 7.896000 0.0190000
#> 16 9.534000 12.1000000
#> 17 9.114000 4.3560000
#> 18 8.389000 0.6200000
#> 19 8.999000 2.6800000
#> 20 9.099000 3.7310000
#> 21 8.502000 0.8930000
#> 22 8.873000 1.9000000
#> 23 8.240000 0.3200000
#> 24 9.219000 5.9000000
#> 25 9.271000 6.9000000
#> 26 8.370000 0.4420000
#> 27 9.431000 9.0000000
Running a tournament
The tournament function is easy to use. All you need are two mandatory input arguments, formula and data. The formula is of the form y~x, where y is the discharge in m\(^3/\)s, and x is the water
elevation in m (it is very important that the data is in the correct units). The data argument must be a data.frame including x and y as column names. In our case, the dataset from Krokfors has a
column named Q which includes the discharge measurements, and a column W which includes the water elevation measurements. We are ready to run our first tournament:
> set.seed(1) # set seed for reproducibility
> t_obj <- tournament(Q~W,krokfors,parallel=TRUE,num_cores=2) # by default parallel=TRUE and the number of cores is detected on the machine
#> Running tournament:
#> 25% - gplm finished
#> 50% - gplm0 finished
#> 75% - plm finished
#> 100% - plm0 finished
The function runs the four models and then the tournament. If you have already run the four different kinds of models, plm0, plm, gplm0 and gplm, and they are stored in objects, say plm0.fit,
plm.fit, gplm0.fit and gplm.fit, then you can alternatively run the tournament very efficiently in the following way:
The printing method is very simple and gives you the name of the winner
For a more detailed summary of the results of the tournament, write
> summary(t_obj)
#> round game model lppd eff_num_param WAIC Delta_WAIC winner
#> 1 1 1 gplm 6.320704 6.877144 1.112881 0.5028515 FALSE
#> 2 1 1 gplm0 5.884914 6.692781 1.615733 NA TRUE
#> 3 1 2 plm -8.903540 4.249257 26.305595 -0.3185198 FALSE
#> 4 1 2 plm0 -8.873488 4.120050 25.987075 NA TRUE
#> 5 2 3 gplm0 5.884914 6.692781 1.615733 24.3713421 TRUE
#> 6 2 3 plm0 -8.873488 4.120050 25.987075 NA FALSE
Notice here that in round 1, gplm0 is favored over gplm in the first game, and plm0 over plm in the second. In the second round, gplm0 is deemed the tournament winner, i.e., the model that provides
the best simplicity and goodness-of-fit trade-off with the data at hand.
Comparing different components of the models
There are several tools to visualize the different aspects of the model comparison. To get a visual summary of the results of the different games in the tournament, write
An informative way of comparing the goodness-of-fit of the models, is to compare their deviance posteriors. The deviance of an MCMC sample is defined as 2 times the negative log-likelihood of the
data given the values of the sampled parameters, therefore, lower values imply a better fit to the data. To plot the posterior distribution of the deviance of the different models, we write
The red diamonds on the plot denote the WAIC values for the respective models. Next, to plot the four rating curves that were estimated by the different models, write
Another useful plot is the residual plot
The differences between the four models lie in the modeling of the power-law exponent, \(f(h)\), and the error variance at the response level, \(\sigma^2_{\varepsilon}(h)\). Thus, it is insightful to
look at the posterior of the power-law exponent for the different models
and the standard deviation of the error terms at the response level
Finally, the panel option is useful to gain insight into all different model components of the winning model, which in this case is gplm0:
Customizing tournaments
There are a few ways to customize the tournament further. For example, if the parameter of zero discharge \(c\) is known, you might want to fix that parameter to the known value in the model. Assume
7.65 m is the known value of \(c\). Then you can directly run a tournament with the \(c\) parameter fixed in all the models
One can also change the winning criteria (default value = 2.2) which sets the threshold that the more complex model in each model comparison must exceed, in terms of the model comparison criteria
(default method is “WAIC”). For example, increasing the value to winning_criteria=5 raises the threshold that the more complex model must exceed to win a game, thus favoring model simplicity more
than if the default value of 2.2 were used. To re-evaluate a previously run tournament using a different winning criteria, the most efficient way is to input the list of stored model objects in the
existing tournament object. In our case we have the tournament stored in t_obj, so we can write
> t_obj_conservative <- tournament(t_obj$contestants,winning_criteria=5)
> summary(t_obj_conservative)
#> round game model lppd eff_num_param WAIC Delta_WAIC winner
#> 1 1 1 gplm 6.320704 6.877144 1.112881 0.5028515 FALSE
#> 2 1 1 gplm0 5.884914 6.692781 1.615733 NA TRUE
#> 3 1 2 plm -8.903540 4.249257 26.305595 -0.3185198 FALSE
#> 4 1 2 plm0 -8.873488 4.120050 25.987075 NA TRUE
#> 5 2 3 gplm0 5.884914 6.692781 1.615733 24.3713421 TRUE
#> 6 2 3 plm0 -8.873488 4.120050 25.987075 NA FALSE
There is also an option to change the method used to estimate the predictive performance of the models. The default method is “WAIC” (see Watanabe (2010)) which is a fully Bayesian method that uses
the full set of posterior draws to calculate the best possible estimate of the expected log pointwise predictive density. Other allowed methods are “DIC” and “Posterior_probability”. The “DIC” (see
Spiegelhalter (2002)) is similar to “WAIC” but instead of using the full set of posterior draws to compute the estimate of the expected log pointwise predictive density, it uses a point estimate of
the posterior distribution. Both “WAIC” and “DIC” have a default value of 2.2 for the winning criteria. We again run the efficient re-evaluation of the tournament
> t_obj_DIC <- tournament(t_obj$contestants,method="DIC")
> summary(t_obj_DIC)
#> round game model D_hat eff_num_param DIC Delta_DIC winner
#> 1 1 1 gplm -13.85265 6.190845 -1.4709638 0.6520799 FALSE
#> 2 1 1 gplm0 -13.53690 6.359006 -0.8188839 NA TRUE
#> 3 1 2 plm 17.59492 3.041535 23.6779944 -0.2721095 FALSE
#> 4 1 2 plm0 17.48871 2.958588 23.4058849 NA TRUE
#> 5 2 3 gplm0 -13.53690 6.359006 -0.8188839 24.2247688 TRUE
#> 6 2 3 plm0 17.48871 2.958588 23.4058849 NA FALSE
The third and final method that can be chosen is “Posterior_probability”, which uses the posterior probabilities of the models, calculated with Bayes factor (see Jeffreys (1961) and Kass and Raftery
(1995)), to compare the models, where all the models are assumed a priori to be equally likely. When using the method “Posterior_probability”, the value of the winning criteria should be a real
number between 0 and 1, since this represents the threshold value that the posterior probability of the more complex model has to surpass to be selected as the appropriate model. The default value in
this case for the winning criteria is 0.75, which again slightly favors model simplicity. The value 0.75 should give similar results to the other two methods with their respective default values of
2.2. The method “Posterior_probability” is not chosen as the default method because the Bayes factor calculations can be quite unstable. Let’s now use this method, but raise the winning criteria from
0.75 to 0.9
> t_obj_prob <- tournament(t_obj$contestants,method="Posterior_probability",winning_criteria=0.9)
> summary(t_obj_prob)
#> round game model marg_lik Post_prob winner
#> 1 1 1 gplm 3.201743e-02 1.302100e-01 FALSE
#> 2 1 1 gplm0 2.138732e-01 8.697900e-01 TRUE
#> 3 1 2 plm 1.427185e-06 4.339125e-01 FALSE
#> 4 1 2 plm0 1.861923e-06 5.660875e-01 TRUE
#> 5 2 3 gplm0 2.138732e-01 9.999913e-01 TRUE
#> 6 2 3 plm0 1.861923e-06 8.705656e-06 FALSE
We see that the results of the tournament do not change in this example, and the winner of the third and final game is still gplm0.
Hrafnkelsson, B., Sigurdarson, H., and Gardarsson, S. M. (2022). Generalization of the power-law rating curve using hydrodynamic theory and Bayesian hierarchical modeling, Environmetrics, 33
Jeffreys, H. (1961). Theory of Probability, Third Edition. Oxford University Press.
Kass, R., and A. Raftery, A. (1995). Bayes Factors. Journal of the American Statistical Association, 90, 773-795.
Spiegelhalter, D., Best, N., Carlin, B., Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 64(4),
Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research, 11, 3571–3594. | {"url":"https://pbil.univ-lyon1.fr/CRAN/web/packages/bdrc/vignettes/tournament.html","timestamp":"2024-11-01T20:30:10Z","content_type":"text/html","content_length":"571843","record_id":"<urn:uuid:b9ce2e4c-49e3-40e0-8eb3-dacb938ad158>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00238.warc.gz"} |
Graphical Representation Of Cumulative Frequency Distribution
Graphical Representation Of Cumulative Frequency Distribution
Quizizz helps math teachers design and deliver engaging lessons with great content using a variety of fun quizzes and unique resources. You can easily create, score and report on custom quizzes to
engage students and track their progress throughout the year. | {"url":"https://quizizz.com/practice/questions/en-in/curriculum/cbse/grade/10/subject/mathematics/chapter/statistics/topic/graphical-representation-of-cumulative-frequency-distribution","timestamp":"2024-11-11T10:09:18Z","content_type":"text/html","content_length":"402029","record_id":"<urn:uuid:007a07b8-0528-4718-8b34-939dcea6be74>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00527.warc.gz"} |
Quantum Computational Fluid Dynamics (QCFD)- Part 1: The Literature - Quantum Computing ReportQuantum Computational Fluid Dynamics (QCFD)- Part 1: The Literature
Quantum Computational Fluid Dynamics (QCFD)- Part 1: The Literature
Figure. SpaceX’s launch of ESA’s Hera spacecraft 7 October 2024, using its six Merlin rocket engines.
by Amara Graps
Quantum in Computational Fluid Dynamics (QCFD) has emerged as a hot research area in quantum computing in recent years. You may be familiar with Rolls Royce’s Collaborations and Use Cases, but there
are other lesser-known examples implemented in pilot projects today as well. As CFD is widely utilized in the automotive, aerospace, civil engineering, wind energy, and defense industries (Dalzell et
al., 2023), these are grouped together as ‘Advanced Industries’ in GQI’s Use Cases categories in the Figure below.
One example of the crucial role of classical CFD in SpaceX’s rocket design, testing, and optimization is the near-perfect performance of launches, such as the one that put the ESA Hera asteroid
defense mission on its interplanetary route on October 7 (header figure).
Is CFD ready for quantum?
A variety of indicators says ‘Yes’!
This 3-part series gives guidelines to:
• Part 1) the Quantum CFD Literature,
• Part 2) the QCFD Use Cases algorithm implementation: Linearization-Forward,
• Part 3) the QCFD Use Cases algorithm implementation: On the Native Hardware with Lattice-Boltzmann
Quantum CFD Scientific Literature
My primary source for algorithm research, Dalzell et al., 2023, has an excellent section on computational fluid dynamics (CFD); I’ve partially replicated the references here. Typically, CFD
implements a Navier-Stokes equation simulation. While most simulations concentrate on air or fluid movement on solid objects, it’s important to mimic other processes as well, such foaming. Large CFD
simulations, which are sometimes run at petaflop speeds, require millions of CPU cores. Some approaches to quantum algorithms are listed in the following Table. I’ve grouped the papers:
Introductions, Linear Algebra-Forward, and Native Hardware. Most are Linearized with good results; the Native Hardware algorithm is in a special class. The reason for this grouping will be clear in
Part 3.
QCFD Introductions
Table. References List for research papers implementing CFD with Quantum computers, mostly extracted from Dalzell et al., 2023 and queried with SciSpace to show their main contributions.
A New Middle Ground
In GQI’s Quantum Algorithms Outlook report, GQI writes about ‘a new middle ground’:
While VQE (low depth for NISQ applications) is often viewed in contrast to QPE (high depth requiring FTQC), interest has grown in other techniques that might offer a middle ground.
The QCFD algorithms fit into this category, for both classes: Linear Algebra-Forward and Native Quantum. See the list of algorithm summaries how they adapt to the NISQ devices, successfully.
QCFD Algorithms: Linear Algebra-Forward
QCFD Algorithms: Quantum Native
Quantum Computing Use Cases Live Tracker
From the above descriptions, many studies linearized the Navier-Stokes equation to find components that could run on the quantum device, while other components run on the classical device.
In GQI’s framework for Use Cases, quantum algorithms for ‘Linear Algebra’ would fall under the Quantum Linear System Solver (QLSS), which we described previously in QCR: Quantum Algorithms for
Solving Differential Equations. We will proceed to describe QCFD Use Cases in Parts 2 and 3.
Figure. From QC Use cases, with a focus on the technological overview. (*) .
If you are interested to learn more, please don’t hesitate to contact info@global-qi.com.
October 8, 2024
Mohamed Abdel-Kareem2024-10-10T10:59:00-07:00 | {"url":"https://quantumcomputingreport.com/quantum-computational-fluid-dynamics-qcfd-part-1-the-literature/","timestamp":"2024-11-04T17:22:16Z","content_type":"text/html","content_length":"109771","record_id":"<urn:uuid:2d3f05ed-e33c-4785-886b-0cc9e5fff04c>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00535.warc.gz"} |
Opportunity cost of studying economics
Since you are reading this, it can be safely assumed that you have made a choice to study economics. The question that arises here is – what is the opportunity cost of your choice to study economics?
Before we formally define opportunity cost, you need to do the following:
Reflect on the cost of this choice by considering the following two questions:
• How much does it cost you to study economics?
• What did you give up to study economics?
The easier question to answer is the one about how much it will cost you to study economics. This is usually interpreted as meaning how much money you would have to pay to study the course. This
would include the course fee which, in 2017, is R1 950 for ECS1501 (the course you are registered for). If you require a textbook for the course, then the cost of the book becomes part of the cost of
studying economics. This also applies to any printing costs.
If these costs are paid by your parents, family members or a bursary scheme it might cost you nothing, but the cost is borne by someone, which means that it is by no means free.
The second question is a more tricky, and goes to the heart of the concept of opportunity cost. The question is – what would you be doing if you were not studying economics?
These costs could be in the form of wages that you could have earned instead of using your time to study; or it could be those things you could be doing if you were not studying.
To determine the opportunity cost of studying for this course, you need to include not only the monetary cost of studying the course, but also those things that you have to give up in order to study.
Opportunity cost is therefore a broader concept than simply the amount you have to pay, because it also includes the best alternative that you have to give up in order to do what you are doing.
Based on the above, we can formally define opportunity cost of a choice as follows:
"Opportunity cost is the value to the decision maker of the best alternative that is given up."
Choose the correct option below:
Assume you have decided to go to see a movie instead of reading a book, taking your dogs for a walk or taking a nap. Part of your opportunity cost of going to the movies is therefore…
Since opportunity cost is the value of the best alternative that is given up is does not include the value of all the alternatives that are given up, but only the value of the best alternative. If
that best alternative for you is not reading a book the value lost of not reading a book is part of your opportunity cost of going to the movies.
Assume the move ticket cost you R50 then the opportunity cost is:
Opportunity cost is a broader concept than simply the amount you have to pay, because it also includes the best alternative that you have to give up in order to do what you are doing.
It is therefore the cost of a movie ticket plus the pleasure of not reading a book if reading a book is the best alternative that is sacrificed.
Do the following activities on opportunity cost.
You are given the following information about a student who is studying economics:
│Course fee: │R2 000│
│Internet connectivity: │R300 │
│Hardware and software cost: │R1 000│
│Cost of food: │R3 000│
│Travel cost to examination centre: │R120 │
│Wages the student could have earned if not studying economics: │R8 000│
Calculate the opportunity cost for the student to study economics.
It is R11 420. The only item that is not included in calculating the opportunity cost is the cost of food. Even if the student is not studying, he or she still has to eat.
It is Saturday, and Mpho decides to attend a soccer league match which costs her R250 and takes up two hours of her time. If she had not attended the soccer match, she would have read a couple of
Her opportunity cost to attend the soccer league match is:
It is correct.
She pays R250 for the match and loses the value of reading her magazines. The R250, plus the satisfaction of reading magazines, is her opportunity cost.
Think again.
Opportunity cost is a broader concept than simply the amount Mpho have to pay, because it also includes the best alternative that she gave you have to give up in order to do what she is doing.
She pays R250 for the match and loses the value of reading her magazines. The R250, plus the satisfaction lost of not reading magazines, is her opportunity cost.
Think again.
She pays R250 for the match and loses the value of reading her magazines. The R250, plus the satisfaction lost of reading magazines, is her opportunity cost.
Think again
Opportunity cost also apply to recreational activities. In this case Mpho had to make a decision about which recreational activities to do. Her choice is to go to the match and her opportunity cost
is the R250 she pays for the match plus the satisfaction lost of not reading her magazines.
It is Saturday, and Peter decides to attend a soccer league match which costs him R250 and takes up two hours of his time. He is supposed to mow the lawn, but decides to hire someone to do it at R50
per hour while he is at the soccer match. It will take two hours to mow the lawn.
His opportunity cost to attend the soccer league match is:
Think again.
His opportunity cost must now also includes the R100 he pays for someone to mow the lawn.
His opportunity cost is the R250 for the soccer match and the R100 he pays for someone to mow the lawn.
Think again.
His opportunity cost includes the cost of the soccer match of R250.
His opportunity cost is the R250 for the soccer match and the R100 he pays for someone to mow the lawn.
It is correct.
His opportunity cost is the R250 for the soccer match and the R100 he pays for someone to mow the lawn.
Glenda works as a consultant and earns R500 per hour. It is Saturday, and she decides to attend a soccer league match instead of working. The cost of the soccer match is R250 and it takes up two
hours of her time.
Her opportunity cost to attend the soccer match is:
Think again.
She needs to include the earning lost of R 1 000.
Her opportunity cost is therefore equal to the R250 for the soccer match and the R1 000 she loses by not working.
Think again.
She needs to include the cost of the soccer match of R250.
Her opportunity cost is therefore equal to the R250 for the soccer match and the R1 000 she loses by not working
She needs to include the cost of the soccer match of R250.
Her opportunity cost is therefore equal to the R250 for the soccer match and the R1 000 she loses by not working | {"url":"https://econdev.co.za/lesson/overview-opportunity-cost/opportunity-cost-of-studying-economics/","timestamp":"2024-11-10T05:01:52Z","content_type":"text/html","content_length":"40944","record_id":"<urn:uuid:7be7fb4a-e8ce-428c-af24-3343da078db9>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00346.warc.gz"} |
Monad . Haskell
Haskell class
-- definition
class Monad m where
return :: a -> m a
(>>=) :: m a -> (a -> m b) -> m b
-- utility functions
(>=>) :: (a -> m b) -> (b -> m c) -> (a -> m c)
(f >=> g) a = f a >>= g
fmap :: (a -> b) -> m a -> m b
fmap f ma = ma >>= (return . f)
join :: m (m a) -> m a
join mma = mma >>= id
(>>) :: m a -> m b -> m b
ma >> mb = ma >>= (const mb)
m »= return $\ \leftrightsquigarrow\ $ m
return x »= f $\ \leftrightsquigarrow\ $ f x
(m »= f) »= g $\ \leftrightsquigarrow\ $ m »= (\x → f x »= g)
The laws can be viewed as right unit, left unit and associativity.
The operation »= is also called 'bind'. Its type is somewhat ugly, especially because of a→m b. But instantiating a monad by implementing 'bind' is more concise than defining 'fmap' and 'join', which
is what mathematicians usually do.
A term 'f x y' can be viewed as an x-indexed family of functions 'f x', evaluated at y. For fixed x, we're going to use 'f x' as second argument for the bind operation 'my »=', where 'my' might also
depend on x, and then lambda abstract the x outside. The resulting function can be used argument for a second bind 'mx »='. The resulting body
mx >>= \x-> (my >>= \y -> f x y)
can be viewed as a function 'f :: a → b → m c', which has been passed two monadic values 'mx :: m a', 'mx :: m a'. Haskell has the following syntactic sugar for this expression
do {x <- mx; y <- my; f x y}
and one easily extends this to any number of arguments 'do {x ← mx; y ← my; z ←mz …'.
A 'do' block is of course still a single expression. One might say that the procedural aspect of it is that eta/beta-reduction of the inner expression possibly include reductions of the outer ones -
there is a smart and a stupid way to compute 'do { x ← return (2^2-4); y ← return (x*(x+1)^7); return (x+y) }'.
List comprehension
For the time being, this example is instructive
[1,2] >>= (\ x -> [2,3] >>= (\y -> return (x/=y) >>= (\r -> (case r of True -> return (x,y); _ -> []))))
do x <- [1,2]; y <- [2,3]; r <- return (x/=y); (case r of True -> return (x,y); _ -> [])
[(x,y) | x <- [1,2], y <- [2,3], x/=y]
Here is what makes the use of predicates possible: Note that 'join [ [1,2],[],[3],[] ]' is '[1,2,3]' and this is part of the working of the list monads »=. Since for x=2, y=2, the Bool 'x/=y' isn't
'True', the value '[]' is returned and consequently dropped from the resulting list.
Example: The List monad
The list-functors $\text{fmap}$ maps functions $f :: a \to b$ to functions $T(f)$ defined by concatenation with Haskells $\text{map}$ function, i.e. $\text{fmap}\ f\ xs\ =\ [\ f\ a\ |\ a \leftarrow
xs\ ]$, e.g. $\text{map}\ (x\mapsto x^2)\ [2,3,4]$ evaluates to $[4,9,16]$. $\text{return}$ then maps terms $x$ of type $a$ to terms $[x]$ (list with one member) of type $[a]$. $\text{join}$ is
flatten, e.g. $\text{join}$ applied to the list of lists $[[7,9,3],[2],[5,10]]$ evaluates to $[7,9,3,2,5,10]$.
Now what the functor construction does for you is generating infinite depth (lists of lists of lists of…) and once you’ve proven your system monadic, you know that your arrows are structural
throughout the system: E.g. let $f :: \text{Int}\to\text{Char}$, then
• $\text{map}\ f\ (\text{return}\ 3)$ is $\text{map}\ f\ [3]$ is $[f\ 3]$.
• $\text{return}\ (f\ 3)$ is $[f\ 3]$ as well.
• $\text{map}\ f\ (\text{join}\ [[7,3],[2]])$ is $\text{map}\ f\ [7,3,2]$ is $[f\ 7,f\ 3,f\ 2]$.
• $\text{join}\ (\text{map}\ (\text{map}\ f)\ [[7,3],[2]])$ is $\text{join}\ [(\text{map}\ f)\ [7,3],(\text{map}\ f)\ [2]]$ is $\text{join}\ [[f\ 7,f\ 3],[f\ 2]]$ is $[f\ 7,f\ 3,f\ 2]$ as well.
Example: The Maybe monad
Types $a$ get mapped to $\text{Maybe}\ a$. For each type $a$, the type $\text{Maybe}\ a$ contains a copy of all of $a$'s terms $x,y,\dots$, there written $\text{Just}\ x,\text{Just}\ y,\dots$, plus
an additional term called $\text{Nothing}$. I.e. the maybe functor effectively adds an „exception“ term to all types of $\text{Hask}$. If $f:: a\to b$ and $x::a$, then the arrow image of the functor
on $f$ is just the function which maps $\text{Just}\ x$ to $\text{Just}\ f(x)$ and
Return maps $x$ in $a$ to $\text{Just}\ x$ in $\text{Maybe}\ a$ and join maps $\text{Just}\ (\text{Just}\ x)$ to $\text{Just}\ x$ and in particular, $\text{Just}\ \text{Nothing}$ to $\text{Nothing}$.
Note that for any 'g::a→b', the function 'return.g' is of type a→m b and therefore the pattern 'mx »= \x→return (g x)' is generally common. (However not all terms of a→m b are of the form return.g.)
This is used to “lift” non-monadic functions 'f :: a → b → m c':
liftM2 f mx my = do {x <- mx; y <- my; return (f x y)}
Applicative from Monad
Since 'id f x = f x', id on functions is curried eval and so id for function spaces takes two arguments. Since id comes for every type/object, every monad gives rise to an Applicative instance via
(<*>) :: m (a -> b) -> m a -> m b
(<*>) mg ma = liftM2 id mg my
-- or = join $ fmap (\g -> fmap g mx) mg
Note that conversely, '<*>' doesn't suffice to define a '»='.
I'm going to explain what this (<*>) does in mathematical terminology: The Yoneda lemma implies that there is an isomorphism from $FA$ to $\mathrm{nat}(\mathrm{Hom}(A,B),FB)$. Consider a category
with exponential objects and arrows corresponding to components of the natural transformations (such as is the case for Hask, where the above map is $at\mapsto g\mapsto T(g)(ta):FA\to B^A\to FB$),
then we can apply the functors arrow map to the latter and obtain a map $FA\to F\,B^A\to FFB)$ to. One could say that this is provides a way to “apply” $tg:F\,B^A$ to $ta:FA$. Note, however, that $T
\,B^A$ isn't necessarily a honest function space anymore, so that's abuse of terminology. (For example, 'Noting :: Maybe (Int→Int)' isn't a function. Things still work out, however, because if you
pass 'Noting', the outer arrow mapping in the described function takes care of it. In that case the 'tg' value doesn't even matter so this is hardly an “evaluation” of 'Nothing'.). Lastly, if the
functor comes with a monadic 'join', then we can get rid of the second $F$ in $FFB$ in the type of the function, and then it's called '<*>'.
Alternative definitions
One could define a monad via 'join' and 'fmap' (mathematicians definition) and then set
f >=> g = join . (fmap g) . f
ma >>= f = (join . (fmap f)) ma
One could also define '>⇒' and set
ma >>= f = (id >=> f) ma
fmap f = id >=> (return . f)
join = id >=> id
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Search by keyword: "temperature field"
Authors: Attetkov A.V., Volkov I.K. Published: 08.06.2018
Published in issue: #3(78)/2018
DOI: 10.18698/1812-3368-2018-3-4-12
Category: Mathematics and Mechanics | Chapter: Mathematical Physics
Keywords: anisotropic half-space, heat exchange with external environment, local-heating, temperature field, integral transformations
Authors: Attetkov A.V., Volkov I.K. Published: 08.06.2018
Published in issue: #3(78)/2018
DOI: 10.18698/1812-3368-2018-3-4-12
Category: Mathematics and Mechanics | Chapter: Mathematical Physics
Keywords: anisotropic half-space, heat exchange with external environment, local-heating, temperature field, integral transformations
Authors: Attetkov A.V., Volkov I.K. Published: 10.08.2016
Published in issue: #4(67)/2016
DOI: 10.18698/1812-3368-2016-4-97-106
Category: Physics | Chapter: Thermal Physics and Theoretical Heat Engineering
Keywords: isotropic solid, spherical hot spot, thermal thin heat-absorbing coating, temperature field, selfsimilar solution
Authors: Attetkov A.V., Volkov I.K. Published: 01.09.2014
Published in issue: #1(6)/2001
Category: Physics | Chapter: Thermal Physics and Theoretical Heat Engineering
Keywords: temperature field, mathematical simulation, gas at high temperature, spherical source of heating
Authors: Attetkov A.V., Vlasova L.N., Volkov I.K. Published: 09.09.2013
Published in issue: #4(43)/2011
Category: Mathematics and Mechanics | Chapter: Mathematical Physics
Keywords: two-layer half-space, mobility of boundary, temperature field, singular integral transform
Authors: Chigiryova O.Yu. Published: 06.09.2013
Published in issue: #3(42)/2011
Category: Applied Mathematics and Methods of Mathematical Simulation
Keywords: mathematical model, nonstationary heat conduction, cylinder, local periodic heat action, temperature field, non-ideal thermal contact
Authors: Chigiryova O.Yu. Published: 05.09.2013
Published in issue: #2(41)/2011
Category: Applied Mathematics and Methods of Mathematical Simulation
Keywords: two-layer cylinder, moving heat source, process of warming up, temperature field, thermal resistance | {"url":"https://vestniken.bmstu.ru/eng/search/keyword/242/page1.html","timestamp":"2024-11-02T09:33:44Z","content_type":"application/xhtml+xml","content_length":"17556","record_id":"<urn:uuid:2fa8bd80-5afc-46bc-a1fb-842a423aec2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00724.warc.gz"} |
College Algebra Corequisite
Learning Outcome
• Identify whether an ordered pair is in the solution set of a linear inequality
The graph below shows the region of values that makes the inequality [latex]3x+2y\leq6[/latex] true (shaded red), the boundary line [latex]3x+2y=6[/latex], as well as a handful of ordered pairs. The
boundary line is solid because points on the boundary line [latex]3x+2y=6[/latex] will make the inequality [latex]3x+2y\leq6[/latex] true.
You can substitute the x and y-values of each of the [latex](x,y)[/latex] ordered pairs into the inequality to find solutions. Sometimes making a table of values makes sense for more complicated
Ordered Pair Makes the inequality [latex]3x+2y\leq6[/latex] a true statement Makes the inequality [latex]3x+2y\leq6[/latex] a false statement
[latex](−5, 5)[/ [latex]\begin{array}{r}3\left(−5\right)+2\left(5\right)\leq6\\−15+10\leq6\\−5\leq6\end
latex] {array}[/latex]
[latex](−2,−2)[/ [latex]\begin{array}{r}3\left(−2\right)+2\left(–2\right)\leq6\\−6+\left(−4\right)\leq6\\–10
latex] \leq6\end{array}[/latex]
[latex](2,3)[/ [latex]\begin{array}{r}3\left(2\right)+2\left(3\right)\leq6\\6+6\leq6\\12\leq6\end{array}
latex] [/latex]
[latex](2,0)[/ [latex]\begin{array}{r}3\left(2\right)+2\left(0\right)\leq6\\6+0\leq6\\6\leq6\end{array}[/
latex] latex]
[latex](4,−1)[/ [latex]\begin{array}{r}3\left(4\right)+2\left(−1\right)\leq6\\12+\left(−2\right)\leq6\\10
latex] \leq6\end{array}[/latex]
If substituting [latex](x,y)[/latex] into the inequality yields a true statement, then the ordered pair is a solution to the inequality, and the point will be plotted within the shaded region or the
point will be part of a solid boundary line. A false statement means that the ordered pair is not a solution, and the point will graph outside the shaded region, or the point will be part of a dotted
boundary line.
Use the graph to determine which ordered pairs plotted below are solutions of the inequality [latex]x–y<3[/latex].
Show Solution
The following video shows an example of determining whether an ordered pair is a solution to an inequality.
Is [latex](2,−3)[/latex] a solution of the inequality [latex]y<−3x+1[/latex]?
Show Solution
The following video shows another example of determining whether an ordered pair is a solution to an inequality. | {"url":"https://courses.lumenlearning.com/waymakercollegealgebracorequisite/chapter/solution-sets-of-inequalities/","timestamp":"2024-11-02T22:08:27Z","content_type":"text/html","content_length":"53853","record_id":"<urn:uuid:f3619f25-e0b3-44bf-9ff6-40a8981ac7a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00621.warc.gz"} |
The what of PCA
As discussed in the previous segment, PCA is fundamentally a dimensionality reduction technique; it helps in manipulating a data set to one with fewer variables. The following lecture will give you a
brief idea of what dimensionality reduction is and how PCA helps in achieving dimensionality reduction.
In simple terms, dimensionality reduction is the exercise of dropping the unnecessary variables, i.e., the ones that add no useful information. Now, this is something that you must have done in the
previous modules. In EDA, you dropped columns that had a lot of nulls or duplicate values, and so on. In linear and logistic regression, you dropped columns based on their p-values and VIF scores in
the feature elimination step.
Similarly, what PCA does is that it converts the data by creating new features from old ones, where it becomes easier to decide which features to consider and which not to.
Now that you have an idea of the basics of what PCA does, let’s understand its definition in the following lecture.
PCA is a statistical procedure to convert observations of possibly correlated variables to ‘principal components’ such that:
• They are uncorrelated with each other.
• They are linear combinations of the original variables.
• They help in capturing maximum information in the data set.
Now, the aforementioned definition introduces some new terms, such as ‘linear combinations’ and ‘capturing maximum information’, for which you will need some knowledge of linear algebra concepts as
well as other building blocks of PCA. In the next session, we will start our journey in the same direction with the introduction of a very basic idea: the vectorial representation of data.
Answer the following question to better understand the upcoming segments.
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MATLAB Programs | Andreas Dahlin
top of page
Matlab Scripts
Programs that may be useful for other researchers can be found here.
This is my (unexpectedly famous) algorithm for calculating the center of mass (centroid) of a resonance peak with minimal noise and high robustness. It is based on performing a high order polynomial
fit to the data, thereby generating a continuous spectrum which circumvents issues ssociated with deciding how many discrete pixels that should be included.
The polynomial fit is also very fast and the expression does not have to be integrated numerically. You can easily run the analysis while measuring in real-time unless the temporal resolution should
be extreme.
You are free to use the MATLAB code for any purpose but please cite the original reference:
A.B. Dahlin, J. Tegenfeldt, F. Höök, Analytical Chemistry 2006.
It should hopefully be clear how the calculation is done if you read through the paper.
This program calculates the transmission, reflection and absorption in an arbitrary thin film multilayer system.
When run as is, the program generates the far field angular spectrum (670 nm incident light) of a 50 nm gold film on glass in water. The surface plasmon excitation is seen as a dip in the reflection.
The simulation also includes a dielectric coating on the gold film (n = 1.4) with different thickness (hence the series of graphs). This can be thought of as a simulation of a plasmonic biosensor
The TransferMatrix program can be used to simulate transmission through and reflection from any kind of thin film multilayer - just change the parameters in the beginning of the file! You can also
change to a wavelength spectrum at a fixed angle of incidence. If a material is dispersive you should just include a new refractive index calculation in the wavelength loop.
I use the TransferMatrix program to simulate the transmission of light through thin film multilayers. Although the program naturally does not consider nanostructures it still gives a good estimate of
peaks and dips due to Fabry-Pérot interference and simplifies interpretation of experimental spectra.
You are free to use the MATLAB code for any purpose but please cite the reference:
J. Junesch, T. Sannomiya, A.B. Dahlin, ACS Nano 2012.
The supporting information for this paper describes the program. Note that this does not mean that I invented the transfer matrix algorithm! You should cite other articles for that.
This program calculates the dispersion relation for transverse magnetic surface waves in an arbitrary thin film multilayer system. When used as is, it will calculate the dispersion relation of
hybridized surface plasmon modes in a metal-insulator-metal system (20 nm Au on both sides of a 50 nm n = 2.24 dielectric in air). The figure below shows the results of solving for the higher energy
hybridized bonding mode. The plots generated are for dispersion, propagation length and fields. (The magnetic field gives a 1D plot for TM modes while the electric field is more complicated to
visualize since it has two components.)
The algorithm solves the equations by finding the real and imaginary parts of the k-vector by minimization. The program starts with generating a plot of the numerical residual for different values of
the k initial guess. You should click in the plot at a location where you see a minimum. Different minima correspond to different modes.
The program will not always converge to a solution. Sorry about that. Sometimes it gets stuck in the loop depending on what system you have simulated. Also, you need to modify the code if you want to
solve for TE modes (plasmonic modes are TM).
You are free to use the MATLAB code for any purpose but please cite a suitable reference like:
A.B. Dahlin, M. Mapar, K. Xiong, F. Mazzotta, F. Höök, T. Sannomiya, Advanced Optical Materials 2014.
The supporting information for this paper describes the calculations in detail.
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Methods That Accelerate the Modeling of Bandpass-Filter-Type Devices
When designing bandpass-filter type high-Q RF devices with the finite element method in the frequency domain, you will likely come across a situation where you need to apply many frequency samples to
more accurately describe the passband. Simulation time is directly proportional to the number of frequencies included in the simulation of a microwave device, with the time increasing as the
frequency resolution becomes finer. Two powerful simulation methods in the RF Module, an add-on to the COMSOL Multiphysics^® software, help accelerate the modeling of such devices.
Editor’s note: This blog post was originally published on July 4, 2016. It has since been updated to reflect up-to-date functionality and results representation.
A Brief Introduction to the Two RF Simulation Methods
The two simulation methods that we will discuss in today’s blog post are the asymptotic waveform evaluation (AWE) and frequency-domain modal (FDM) methods. Both methods are designed to help you
overcome the conventional issue of a longer simulation time when using a very fine frequency resolution or running an ultra-wideband simulation with a regular Frequency Domain study. The AWE method
is quite efficient when it comes to describing smooth frequency responses with a single resonance or no resonance at all. The FDM method, meanwhile, is useful for quickly analyzing multistage filters
or filters of a high number of elements that have multiple resonances in a target passband. In the next two sections, we are going to discuss their typical settings and use cases.
It’s worth mentioning that the AWE and FDM methods are both almost independent of the frequency step selected. You can decrease the value of the frequency step freely and get a well-resolved plot
without any notable slowdown or extra memory consumption. However, there’s one drawback: Decreasing the value of the frequency step may affect the amount of data saved as a final solution. Later in
this blog post, in the section dedicated to data management, you will find recommendations that allow for a significant reduction of output file size.
Note that before running either an AWE or FDM calculation with the fine resolution, it may be useful to perform a preliminary Eigenfrequency and regular Frequency Domain simulation with a coarse
frequency resolution. This would give you a fast and valuable estimation of resonance locations and a general understanding of the system’s frequency trends, including for the actual passbands and
the desired frequency resolution.
Asymptotic Waveform Evaluation Method Fosters Reduced-Order Modeling
For our purposes, it would be too technical to talk about the numerical characteristics and mathematical algorithms of AWE — an advanced reduced-order modeling technique. Instead, we will go over how
to use this method in the RF Module. As of version 6.2 of COMSOL Multiphysics, there is a dedicated Adaptive Frequency Sweep study step that implements the AWE method. When using this feature, you
should specify a target output frequency range and choose an expression to be used for error estimation by the AWE algorithm. Under the hood, the solver performs fast-frequency adaptive sweeping and
uses, by default, Padé approximations.
The Adaptive Frequency Sweep study settings. Check out the default Asymptotic waveform evaluation (AWE) expression used.
The AWE method is very useful when simulating resonant circuits, especially bandpass-filter type devices with many frequency points. For instance, the Evanescent Mode Cylindrical Cavity Filter
tutorial model, available in the Application Library, initially sweeps the simulation frequency between 3.45 GHz and 3.61 GHz with a 5 MHz frequency step via a regular Frequency Domain study.
The Evanescent Mode Cylindrical Cavity Filter tutorial model (left) and its discrete frequency sweep results (right). The S-parameter plot does not look smooth around the resonant frequency.
Say you want to run the simulation again with a much finer frequency resolution, such as a 100 kHz frequency step that is 50 times finer. You can expect that the simulation will take 50 times longer
to finish. But when using the Adaptive Frequency Sweep study in this particular example model, the simulation time is almost the same as the regular frequency sweep case, but we can obtain all of the
computed solutions on the dependent variable with the 100 kHz frequency step.
The simulation time may vary to some extent in regard the user input in the AWE expressions. Any model variable works as an AWE expression, so long as it generates a smooth resulting plot like a
Gaussian pulse or a smooth curve as a function of frequency, but the obvious and typical choice is a global expression based on S-parameters. For instance, the absolute value of S21 (abs
(comp1.emw.S21)) works great as the input for the AWE expression in the case of a two-port bandpass filter. Consider that if the frequency response of the AWE expression contains an infinite gradient
— the case for the S11 value of an antenna with excellent impedance matching at a single frequency point — the simulation will take longer to complete. If the loss from the antenna is negligible, an
alternative expression such as sqrt(1-abs(comp1.emw.S11)^2) may work better and reduce the computation time. The aforementioned expressions are the default Physics controlled choices for the
Asymptotic waveform evaluation (AWE) expression. As a sanity check, you can always run a preliminary Frequency Domain sweep with a coarse resolution, plot the expressions, and choose the smoothest
When you are ready to run the Adaptive Frequency Sweep don’t forget to use the desired finer frequency step in the study settings. Once the simulation is complete, you will notice that the simulation
time is almost the same as the discrete sweep. Let’s compare the computed S-parameters. Since the AWE solver performed a frequency sweep that was 50 times finer, its frequency response (S-parameters)
plot consequently looks much nicer. Not only do you save precious time with this approach, but as the plot below illustrates, you also still obtain accurate and good-looking results with the
resonance frequency located more accurately. As a validation, the curious reader can run a regular sweep with the same resolution and check that the results are in perfect correlation.
S-parameter plot resulting from the Adaptive Frequency Sweep (AWE) and discrete Frequency Domain simulations. The resolution of the AWE results is 50 times finer.
Frequency-Domain Modal Method Captures the Resonance of Circuits
Bandpass-frequency responses of a passive circuit result from a combination of multiple resonances, so the FDM method is the optimal choice for accelerating their modeling. Usually it contains two
subsequent steps. The Eigenfrequency analysis is key to capturing the resonance frequencies of an arbitrary shape of a device. Once we obtain all of the necessary information from the eigenfrequency
analysis, we can reuse it in the frequency-domain modal study. Doing so enables us to optimize the efficiency of the simulation when a finer frequency resolution is required to more accurately
describe the frequency response, as illustrated in the AWE method.
To perform an FDM analysis seamlessly, there are a couple of aspects to keep in mind. On one hand, you need to filter out all unwanted unphysical low-frequency residues that may be present in the
Eigenfrequency solution. On the other hand, you need to consider all physical modes that may affect the device’s performance in the target frequency range to get the correct results. Reaching both
requirements involves some tuning of the Eigenfrequency study settings (seen in the screenshot below). First, it’s beneficial to select Larger real part as the Search method around shift setting.
Next, for the Search for eigenfrequencies around setting, the lowest passband frequency works as a ballpark value. Finally, the Desired number of eigenfrequencies setting must be adjusted (based on
preliminary tests, for instance) to include the necessary amount of modes.
The two-step Frequency Domain, Modal study is added to the model. Here, the Eigenfrequency settings are highlighted.
To try out an FDM analysis, let’s take a look at the the Coupled Line Filter tutorial model, available in our Application Gallery. Initially, the simulation frequency sweeps between 3.00 GHz and 4.20
GHz with a 50 MHz frequency step within a regular Frequency Domain study.
The Coupled Line Filter tutorial model (left) and its discrete frequency sweep results (right) with the resolution of 50 MHz. The S-parameter plot does not look smooth across the target passband.
Next, you can employ the Frequency Domain, Modal study, configuring the settings for each study step as described above. Run the study with the a frequency step that is 50 times finer and check the
result enhancement. Just as with the AWE method, the S-parameter plot returned by the FDM study looks smoother and is more informative. For instance, it showcases all of the S11-parameter ripples
that were missing initially. As a validation, the curious reader can run a regular sweep with the same resolution and check that the results are in perfect correlation.
Note that the eigenfrequency analysis contains a lumped port that impacts the simulation as an extra loading factor, so the phase of the computed S-parameters is different from that of the regular
frequency sweep model. The results are compatible only with phase-independent S-parameter values such as dB-scaled, absolute value, reflectivity, or transmittivity.
S-parameter plot resulting from the Frequency Domain, Modal (FDM) and discrete Frequency Domain simulations. The resolution of the FDM results is 50 times finer.
It’s not directly related to the original topic, but in the last figure you may notice special Graph Markers that highlight all the local minima of the S11-parameter plot as well as the passband for
the S21-parameter plot. Along with the interactive result extraction from the graph plot, another recent enhancement of results evaluation functionality in COMSOL Multiphysics, it moves the
informative and interactive value of results to a new level.
Data Management When Working with Fine Frequency Resolution
As mentioned earlier, there are no real limits on how you can refine a frequency sweep with the AWE or FDM approach. However, with a really fine resolution, the solutions would contain a ton of data.
As a result, the model file size will increase tremendously when saved. In most passive RF and microwave device designs, it is a common theme that only the S-parameters are of interest, and in such
cases, it is not necessary to store all of the field solutions. By choosing the appropriate option under the Store in Output section of a study, we can control the part of the model on which the
computed solution is saved. For example, we can only add the selection, or selections, containing the boundaries where the S-parameters are calculated. These are the boundaries assigned as Ports or
Lumped Ports, and they are typically small compared to the entire modeling domain, so the total file size can be reduced dramatically.
Note that you can add such an explicit selection when setting up a port by clicking the Create Selection icon in the Boundary Selection section once the selection is specified. You can then add the
needed explicit selections created from the ports under the Store in Output section of a relevant study step.
The Store in Output section of the Frequency Domain, Modal study step with two Lumped Port selections chosen. You can check the location of these selections in the Graphics window.
Available Application Gallery Examples
The simulation methods presented in this blog post are powerful tools for enabling faster, more efficient modeling of passive RF and microwave devices. Check out the following Application Gallery
examples, which can provide further guidance in how to utilize these techniques:
Keep in mind that the methods and studies demonstrated here are universal and available not only for RF modeling. For example, you can also benefit from these methods when performing acoustic,
mechanical, MEMS, and wave optics calculations.
Next Steps
Learn about other specialized features available for your RF and microwave modeling: | {"url":"https://www.comsol.com/blogs/methods-that-accelerate-the-modeling-of-bandpass-filter-type-devices","timestamp":"2024-11-07T02:47:46Z","content_type":"text/html","content_length":"105090","record_id":"<urn:uuid:537f2a5d-b1cd-4afa-a060-03c8680b1b92>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00695.warc.gz"} |
False vacuum decay (FVD) plays a vital role in many models of the early Universe, with important implications for inflation, the multiverse, and gravitational waves. However, we still lack a
satisfying theoretical understanding of this process, with existing approaches working only in imaginary (Euclidean) time, and relying on numerous assumptions that have yet to be empirically tested.
An exciting route forward is to use laboratory experiments which undergo transitions analogous to FVD, allowing nature to simulate all of the non-perturbative quantum effects for us. | {"url":"https://apc.u-paris.fr/APC_CS/fr/taxonomy/term/135?page=5","timestamp":"2024-11-05T02:59:47Z","content_type":"text/html","content_length":"47205","record_id":"<urn:uuid:1cbe9130-66d8-4411-b1be-960bfb954866>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00437.warc.gz"} |
EcoEnsemble is an R package to set up, fit and sample from the ensemble framework described in Spence et al (2018) for time series outputs.
You can install the development version of EcoEnsemble using the devtools package:
Fitting an ensemble model in EcoEnsemble is done in three main steps:
1. Eliciting priors on discrepancy terms: This is done by using the EnsemblePrior() constructor.
2. Fitting the ensemble model: Using the fit_ensemble_model() function with simulator outputs, observations and prior information. The ensemble model can be fit, obtaining either the point estimate,
which maximises the posterior density, or running Markov chain Monte Carlo to generate a sample from the posterior denisty of the ensemble model.
3. Sampling the latent variables from the fitted model: Using the generate_sample() function with the fitted ensemble object, the discrepancy terms and the ensemble’s best guess of the truth can be
generated. Similarly to fit_ensemble_model(), this can either be a point estimate or a full sample.
We illustrate this process with datasets included with the package. EcoEnsemble comes loaded with the predicted biomasses of 4 species from 4 different mechanistic models of fish populations in the
North Sea. It also includes statistical estimates of the biomasses from single-species stock assessments, and covariances for the model outputs and assessments. The models are run for different time
periods and different species.
# Outputs from mizer. These are logs of the biomasses for each year in a simulation.
1984 10.31706 13.33601 10.80006 10.98139
1985 12.07673 13.63592 10.46646 10.87285
… … … … …
2049 12.46354 14.02923 12.27473 10.80954
2050 12.46509 14.03027 12.27422 10.81003
To encode prior beliefs about how model discrepancies are related to one another, use the EnsemblePrior() constructor. Default values are available.
priors <- EnsemblePrior(4)
or custom priors can be specified.
#Endoding prior beliefs. Details of the meanings of these terms can be found in the vignette or the documentation
num_species <- 4
priors <- EnsemblePrior(
d = num_species,
ind_st_params = IndSTPrior("lkj", list(3, 2), 3, AR_params = c(1,1)),
ind_lt_params = IndLTPrior(
list(c(10,4,8, 7),c(2,3,1, 4)),
list(matrix(5, num_species, num_species),
matrix(0.5, num_species, num_species))
sha_st_params = ShaSTPrior("inv_wishart",list(2, 1/3),list(5, diag(num_species))),
sha_lt_params = 5,
truth_params = TruthPrior(num_species, 10, list(3, 3), list(10, diag(num_species)))
This creates an EnsemblePrior object, which we can use to fit the ensemble model using the fit_ensemble_model() function and the data loaded with the package. When running a full MCMC sampling of the
posterior, this step may take some time. Samples can then be generated from the resulting object using the generate_sample() function.
fit <- fit_ensemble_model(observations = list(SSB_obs, Sigma_obs),
simulators = list(list(SSB_ewe, Sigma_ewe, "EwE"),
list(SSB_lm, Sigma_lm, "LeMans"),
list(SSB_miz, Sigma_miz, "mizer"),
list(SSB_fs, Sigma_fs, "FishSUMS")),
priors = priors)
samples <- generate_sample(fit)
This produces an EnsembleSample object containing samples of the ensemble model predictions. These can be viewed by calling the plot() function on this object. For a full MCMC sample, this includes
ribbons giving quantiles of the ensemble outputs. If only maximising the posterior density, then only the single ouput is plotted.
Spence, M. A., J. L. Blanchard, A. G. Rossberg, M. R. Heath, J. J. Heymans, S. Mackinson, N. Serpetti, D. C. Speirs, R. B. Thorpe, and P. G. Blackwell. 2018. “A General Framework for Combining
Ecosystem Models.” Fish and Fisheries 19: 1013–42. | {"url":"https://cran.case.edu/web/packages/EcoEnsemble/readme/README.html","timestamp":"2024-11-03T03:20:59Z","content_type":"application/xhtml+xml","content_length":"10023","record_id":"<urn:uuid:3391a157-2ced-4860-90a0-6edc6dbb7e28>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00161.warc.gz"} |
1.Calculate the mean angular velocity,à‰ of flywheel in earth having flywheel mass 40kg, diameter 10cm and mass of rings 500g, diameter of axle at 3.6cm, no. of chord 7.
2.Calculate the M.I of flywheel in Saturn having flywheel mass 36.5kg, diameter 10cm and mass of rings 200g, diameter of axle at 3cm, no. of chord 10.
3.Calculate the M.I of flywheel in moon having flywheel mass 31.5kg, diameter 10cm and mass of rings 800g, diameter of axle at 2.4cm, no. of chord 5 and hence find its K.E.
4.Calculate the M.I of flywheel in Jupiter having flywheel mass 11.5kg, diameter 10cm and mass of rings 100g, diameter of axle at 2.8cm, no. of chord 6.
5.Calculate the M.I of flywheel in Uranus having flywheel mass 23.5kg, diameter 10cm and mass of rings 1000g, diameter of axle at 3.6cm, no. of chord 5 and hence find its kinetic energy. | {"url":"https://vlab.amrita.edu/?sub=1&brch=74&sim=571&cnt=5","timestamp":"2024-11-08T15:41:08Z","content_type":"text/html","content_length":"15609","record_id":"<urn:uuid:533a2450-e606-46a1-bfd3-d719e0c69255>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00646.warc.gz"} |
Theorem Relating the Eigenvalue Density for Random Matrices to the Zeros of the Classical Polynomials
A theorem due to Stielties shows that the problem of locating the zeros of the classical polynomials is equivalent to finding the electrostatic equilibrium positions for a set of interacting point
charges. Defining a density function for these same charges we find that the density of zeros for the nth Hermite polynomial is the same as the density of eigenvalues for an ensemble of n‐dimensional
Hermitian matrices. Similarly the location of the zeros of the nth Laguerre polynomial determines the density of eigenvalues for an ensemble of n‐dimensional positive matrices, and the zeros of the ½
nth Tchebichef polynomial determine the density for the real part of the eigenvalues for an ensemble of n‐dimensional unitary matrices.
(a) E. P. Wigner, “Distribution Laws for Roots of a Random Hermitian Matrix” (unpublished). The pertinent results and definitions are available in
(b) N. Rosenzweig, “Brandeis Summer Institute 1962, Statistical Physics” (W. A. Benjamin, Inc., New York, 1963).
B. V. Bronk, “Exponential Ensemble for Random Matrices,” J. Math. Phys. (to be published).
F. J.
J. Math. Phys.
). Notice the results of the present paper apply to all cases of Dyson’s classification of ensembles by transformation properties. The formulas here are stated for the case
$β = 2.$
Bateman Manuscript Project, edited by A. Erdléyi (McGraw‐Hill Book Company, Inc., New York, 1953). See (a) Sec. 10.13; (b) Sec. 10.12; (c) Sec. 10.11.
From this point on the proof also applies to Rosenzweig’s “Fixed Strength” ensemble, see Ref. 1(b), p. 110.
E. P. Wigner, Proceedings of the Canadian Mathematical Congress, (1954), p. 174 (unpublished).
M. Born, Mechanics of the Atom, translated by Fisher and Hartree, (G. Bell and Sons, London, 1960), Appendix II.
We have actually proved for ensemble III a special case of a much more general theorem, which can be stated roughly, that the zeros of a set of polynomials orthogonalized with respect to an arbitrary
weight function on $[−1,1],$ are cosines of angles distributed uniformly around the circle. See Ref. 5, Theorem 12.7.2.
This content is only available via PDF.
© 1964 The American Institute of Physics.
The American Institute of Physics | {"url":"https://pubs.aip.org/aip/jmp/article-abstract/5/12/1661/379121/Theorem-Relating-the-Eigenvalue-Density-for-Random?redirectedFrom=fulltext","timestamp":"2024-11-06T00:50:36Z","content_type":"text/html","content_length":"417173","record_id":"<urn:uuid:a95ac620-f1fa-44ca-ade0-92a62490d93e>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00136.warc.gz"} |
Imaginary Numbers - Basic Introduction
The Organic Chemistry Tutor
3 Jul 202014:11
TLDRThis educational video offers an in-depth exploration of complex numbers, focusing on imaginary numbers and their properties. It explains the imaginary unit 'i', which equals the square root of
-1, and demonstrates how to simplify powers of 'i' by leveraging multiples of 4. The video proceeds to illustrate the addition, subtraction, multiplication, and division of complex numbers,
emphasizing the importance of standard form (a + bi). It also covers solving equations involving complex numbers and introduces how to plot complex numbers on a coordinate plane and calculate their
absolute value. The script is designed to provide a comprehensive understanding of complex numbers for viewers.
• π Imaginary numbers are a subset of complex numbers, with the imaginary unit 'i' defined as the square root of -1.
• π § Powers of 'i' cycle every four exponents: i^2 = -1, i^3 = -i, i^4 = 1, and this pattern repeats.
• π To simplify high powers of 'i', break down the exponent into multiples of 4 and use the cyclical nature of 'i' to simplify.
• π When adding and subtracting complex numbers, distribute and combine like terms, resulting in a standard form a + bi.
• π For multiplication of complex numbers, use the FOIL method (First, Outer, Inner, Last) and simplify by combining like terms.
• π ― To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator to simplify the division.
• π When solving equations involving complex numbers, equate real and imaginary parts separately to find the values of variables.
• π The absolute value of a complex number a + bi is calculated as β (a^2 + b^2), representing the distance from the origin in the complex plane.
• π Plotting a complex number on the complex plane involves marking the real part on the x-axis and the imaginary part on the y-axis.
• π The video provides a comprehensive introduction to operations with complex numbers, including addition, subtraction, multiplication, division, solving equations, plotting, and finding
absolute values.
Q & A
• What is the imaginary unit 'i' equal to?
-The imaginary unit 'i' is equal to the square root of negative one.
• What happens when you raise 'i' to the third power?
-Raising 'i' to the third power is equivalent to 'i' squared times 'i', which simplifies to negative 'i' because 'i' squared is negative one.
• What is the result of 'i' raised to the fourth power?
-When 'i' is raised to the fourth power, it is 'i' squared times 'i' squared, which simplifies to one because negative one times negative one is positive one.
• How can you simplify 'i' raised to a large exponent, like 'i' to the seventh power?
-To simplify 'i' raised to a large exponent, you can break up the exponent using the highest multiple of 4. For 'i' to the seventh, you can express it as 'i' to the fourth times 'i' to the third,
which simplifies to one times negative 'i', or just negative 'i'.
• How do you simplify the expression 5 times (2 + 3i) minus (4 times 7 - 2i)?
-You first distribute the multiplication across the parentheses, resulting in 10 + 15i - 28 + 8i. Then, you combine like terms to get -18 + 23i.
• What is the process for multiplying two complex numbers, such as (5 + 3i) times (8 - 2i)?
-You use the FOIL method (First, Outer, Inner, Last) to multiply the complex numbers: 5*8 + 5*(-2i) + 3i*8 + 3i*(-2i). This results in 40 - 10i + 24i - 6i^2. Since i^2 is -1, this simplifies to
46 - i.
• How do you divide a complex number by another complex number, such as (3 + 2i) divided by (4 - 3i)?
-You multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of (4 - 3i) is (4 + 3i). After multiplying and simplifying, you get (12 + 9i + 8i + 6) / (16 +
9), which simplifies to 18 + 17i / 25.
• What is the absolute value of a complex number in standard form, and how do you calculate it?
-The absolute value of a complex number in standard form a + bi is the square root of a squared plus b squared. For example, the absolute value of 4 + 3i is the square root of 4^2 + 3^2, which is
• How do you plot a complex number on the complex plane?
-On the complex plane, the x-axis represents the real part and the y-axis represents the imaginary part. For a complex number like 4 + 3i, you move 4 units to the right along the real axis and 3
units up along the imaginary axis to plot the point.
• How do you solve for x in the equation 4x + 3i = 12 - 15yi?
-You equate the real parts and the imaginary parts separately. From 4x = 12, x = 3. From 3i = -15yi, and knowing i^2 = -1, you get y = -1/5.
• What are the solutions for x in the equation x^2 + 36 = 0?
-Taking the square root of both sides gives x^2 = -36, which means x = Β±6i, as the square root of -36 is 6 times the square root of -1, which is 6i.
π Introduction to Imaginary Numbers and Simplification
This paragraph introduces the concept of imaginary numbers as complex numbers with the imaginary unit 'i', where 'i' equals the square root of negative one. It explains the powers of 'i', showing
that 'i' squared equals negative one, 'i' cubed equals negative 'i', and 'i' to the fourth equals one. The speaker then demonstrates how to simplify higher powers of 'i' by breaking down exponents
using the highest multiple of four. Examples given include simplifying 'i' to the seventh, 26th, 33rd, and 43rd power. The process involves multiplying by 'i' to a power that results in a cycle of
four, simplifying the expression, and then continuing with the remaining exponent.
π Operations with Complex Numbers and Solving Equations
The second paragraph delves into the operations of adding and subtracting complex numbers, providing an example of simplifying an expression involving multiplication and distribution. The speaker
then moves on to multiplying two imaginary numbers together, using the FOIL method and simplifying the result by combining like terms and accounting for 'i' squared being negative one. The paragraph
also covers dividing complex numbers by multiplying the denominator with the conjugate of the complex number and simplifying the result. Lastly, it discusses solving equations involving complex
numbers, showing how to isolate variables and determine the real and imaginary parts, and concludes with solving an algebraic equation that results in an imaginary number as the solution.
π Plotting Complex Numbers and Calculating Absolute Values
The final paragraph focuses on plotting complex numbers on the complex plane and calculating their absolute values. The speaker explains that the real part lies on the x-axis and the imaginary part
on the y-axis, using the complex number 4 + 3i as an example. The absolute value of a complex number is calculated as the square root of the sum of the squares of the real and imaginary parts, which
in this case results in five. The complex number is then plotted on the complex plane, with the real part represented horizontally and the imaginary part vertically, forming a right triangle with the
absolute value as the hypotenuse. The video concludes with a summary of the operations covered, including adding, subtracting, multiplying, and dividing complex numbers, solving equations, plotting
complex numbers, and finding their absolute values.
π ‘Imaginary Numbers
Imaginary numbers are a fundamental concept in complex mathematics, represented by the symbol 'i', which is defined as the square root of -1. They are integral to the video's theme as they form the
basis of complex numbers. In the script, imaginary numbers are used to explain the powers of 'i', such as i^2 = -1, i^3 = -i, and i^4 = 1, which are essential for simplifying expressions with large
π ‘Complex Numbers
Complex numbers are numbers that consist of a real part and an imaginary part, often written in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit.
They are central to the video's content as the script discusses operations such as addition, subtraction, multiplication, and division of complex numbers, as well as solving equations involving them.
π ‘Exponentiation
Exponentiation in the context of the video refers to the process of raising 'i' to various powers. The script explains how to simplify expressions with high powers of 'i' by breaking down the
exponent using multiples of 4, as seen in examples like i^7, i^26, i^33, and i^43. This concept is crucial for understanding the cyclical nature of 'i' raised to different powers.
π ‘Simplification
Simplification is the process of making complex expressions easier to understand or solve. The video script provides methods for simplifying imaginary numbers, such as breaking down exponents and
combining like terms. For instance, the script simplifies i^7 to -i and i^26 to -1, illustrating the technique of reducing complex expressions to their simplest form.
π ‘Addition and Subtraction
These operations are fundamental arithmetic actions applied to complex numbers in the video. The script demonstrates how to add and subtract complex numbers by distributing and combining like terms,
as shown in the example (5*(2 + 3i)) - (4*(7 - 2i)), which simplifies to -18 + 23i.
π ‘Multiplication
Multiplication of complex numbers is covered in the script through the FOIL method (First, Outer, Inner, Last), which is used to expand and simplify the product of two binomials. An example given is
(5 + 3i) * (8 - 2i), which results in 46 - i after combining like terms and accounting for i^2 = -1.
π ‘Division
Division of complex numbers is a key operation discussed in the video. The script explains the process of dividing complex numbers by multiplying the numerator and the denominator by the conjugate of
the denominator. An example provided is (3 + 2i) / (4 - 3i), which simplifies to (6 + 17i) / 25 after multiplying by the conjugate.
π ‘Conjugate
The conjugate of a complex number is obtained by changing the sign of the imaginary part while keeping the real part the same. In the video, the conjugate is used in division to eliminate the
imaginary part in the denominator, as demonstrated in the division example where the conjugate of (4 - 3i) is (4 + 3i).
π ‘Absolute Value
The absolute value of a complex number, also known as the modulus, is the distance of the number from the origin in the complex plane. The video script explains how to calculate the absolute value by
taking the square root of the sum of the squares of the real and imaginary parts, exemplified with the calculation of |4 + 3i|, which equals 5.
π ‘Plotting
Plotting complex numbers involves representing them as points in the complex plane, with the real part on the x-axis and the imaginary part on the y-axis. The video script describes how to plot the
complex number 4 + 3i by moving four units to the right along the real axis and three units up along the imaginary axis, forming a right triangle with sides 3, 4, and 5.
Introduction to imaginary numbers as complex numbers with the imaginary unit i, where i equals the square root of negative one.
Explanation of the powers of i: i squared equals negative one, i to the third equals negative i, and i to the fourth equals one.
Method to simplify imaginary numbers with large exponents by breaking them up using the highest multiple of 4.
Simplification example of i to the seventh power resulting in negative i.
Simplification of i to the 26th power using multiples of 4, resulting in negative one.
Breaking down i to the 33rd power and simplifying it to i.
Simplification of i to the 43rd power to negative i using the method of breaking up exponents.
Instructions on how to add and subtract complex numbers with an example involving 5 times (2 + 3i) minus 4 times (7 - 2i).
Multiplication of two imaginary numbers using the FOIL method and simplification to 46 - i.
Division of complex numbers by multiplying the denominator with the conjugate of the complex number.
Simplification of the division expression 3 + 2i divided by 4 - 3i, resulting in 6/25 + 17/25i.
Strategy for dividing when the denominator has only an imaginary part by multiplying by i.
Solving equations with complex numbers, such as finding the values of x and y in the equation 4x + 3i = 12 - 15yi.
Solving algebraic equations involving imaginary numbers, like x squared plus 36 equals zero, resulting in x being plus or minus 6i.
Plotting the complex number 4 + 3i on the complex plane and calculating its absolute value as five.
Explanation of the absolute value of a complex number as the square root of the sum of the squares of the real and imaginary parts.
Overview of the process to add, subtract, multiply, divide complex numbers, solve equations associated with them, plot them, and find their absolute value.
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LaTeX is a document preparation system for creating technical and scientific documents. It is especially useful for typesetting logic with its plethora of special symbols. The Logical Theory
textbook, as well as most of our lecture notes/slides are all produced using LaTeX.
Writing in LaTeX is much simpler than using a word processor like Word. A LaTeX file is just a plain text file containing the content of your document and a few special commands describing the
structure of your document. If you are familiar with basic html or markdown (both used extensively in web design) you will be at home with LaTeX. Feeding this text file to LaTeX will produce a pdf of
the document.
Getting started
Nowadays you can write LaTeX documents on the web without needing to download and install anything. A popular online LaTeX tool is Overleaf.
1. Start with the Learn LaTeX in 30 minutes guide.
2. Learn the interface provided by Overleaf by creating a document in Overleaf.
3. The LaTeX Wikibook is an excellant resource for learning about the more advanced features of LaTeX.
Logic in LaTeX
LaTeX is an excellent tool for typesetting mathematical notation. The basics of this are covered in the Learn LaTeX in 30 minutes guide listed above. More functionality is provided through packages
which is usually as simple as adding \usepackage{package-name} to the preamble of your document (i.e., somewhere above \begin{document}).
1. Read the documentation on typesetting mathematics in LaTeX provided by Overleaf. This guide also introduces the mathematics package amsmath that introduces the most common mathematical symbols,
provides better support for equations, and much more. See also the list of further reading provided at the end of the guide and the LaTeX wikibook on using amsmath.
2. Check out the logic related tips you can find at the LaTeX for Logicians page, ranging from logic-specific symbols to writing truth-tables and proof trees.
3. Visit the TeX StackExchange for answers to your LaTeX questions.
4. If you need to figure out how to display a particular symbol in LaTeX, detexify is your tool: you just paste or draw the symbol you want and that site will tell you the command.
5. See the LaTeX miscellany below our own recommendations.
Beam your logic with Beamer
At some point you will want to present your labour to others. Beamer is a LaTeX class for creating presentations.
1. Learn how to create a presentation in Beamer.
2. Use a beamer template to create your own presentation.
A LaTeX miscellany
There are thousands of LaTeX packages. These can be adding new symbols, new design templates, or new functionality. Here is a brief (and highly personal) recommendation. All of the following are
included in the standard LaTeX setup and their documentation is available through the Comprehensive TeX Archive Network (CTAN); simply search for the package name on that site.
Logic specific
• Alphabets You can never have too many ‘x’s. The amsmath package implements the caligraphic and fraktur scripts. Use these sparingly: your work is to be read, not framed!
• Proof trees Whether its natural deduction, tableaux or sequent calculus, we all need to typeset a formal proof at somepoint in our lifetime. The LaTeX for Logicians page has a separate entry on
typesetting natural deduction proofs with links to different packages. I recommend ebproof and bussproofs. The latter package (and its extension bussproofs-extra) is used in the Open Logic Text.
• Automata The TikZ package provides an extensive (and quite scary) toolkit for building figures and graphics within LaTeX. Search the TeXample database for what you need and adapt from there.
General packages
• Theorems, lemmas and proofs, oh my! The amsthm package is the classic. A modern alternative is provided by thmtools.
• Internal links. The hyperref package to the rescue. I recommend the overview in the LaTeX wikibook. To reduce the chance of another package conflictin with hyperref it is recommended to be the
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Understanding Mathematical Functions: How To Find The Range Of A Fract
Introduction to Mathematical Functions and the Importance of the Range
Mathematical functions play a fundamental role in various fields of study, serving as a crucial tool for solving problems and making predictions. In mathematics, functions are used to establish
relationships between different variables and describe how one quantity depends on another. One key aspect of functions is their range, which provides valuable information about the output values
they can produce. Understanding the range of a function is essential for analyzing its behavior and making informed decisions.
Understanding the basics of mathematical functions and their pivotal role in various fields of study
Functions are widely used in fields such as physics, engineering, economics, and computer science to model real-world phenomena and make accurate predictions. By defining a set of rules that map
input values to output values, functions help us understand and interpret complex relationships between variables. Whether it's calculating the trajectory of a projectile or analyzing the performance
of a financial asset, functions provide a powerful framework for problem-solving and decision-making.
Explaining the concept of the range of a function and its significance in mathematics
The range of a function refers to the set of all possible output values that the function can produce for a given set of input values. In other words, it represents the complete set of values that
the function can take on as its output. Understanding the range of a function is crucial for determining its behavior, identifying its limitations, and predicting its outcomes. By analyzing the range
of a function, mathematicians and researchers can gain valuable insights into its properties and applications.
Setting the stage for learning how to find the range of a fraction function as a skill essential for students and professionals alike
As students and professionals delve deeper into the world of mathematics, mastering the ability to find the range of functions becomes increasingly important. Fraction functions, which involve ratios
of two numbers, present a unique challenge when it comes to determining their range. By developing the skills to analyze and calculate the range of fraction functions, individuals can enhance their
problem-solving abilities, improve their analytical thinking, and gain a deeper understanding of mathematical concepts.
Key Takeaways
• Understand the concept of fraction functions.
• Identify the numerator and denominator of the fraction.
• Determine the domain of the function.
• Find the range by analyzing the behavior of the function.
• Consider restrictions on the domain for accurate range.
What is a Fraction Function?
A fraction function is a mathematical function that involves fractions or ratios of two quantities. These functions can be represented in the form f(x) = a/x, where 'a' is a constant and 'x' is the
variable. Fraction functions are commonly used in various mathematical applications and are essential in understanding the behavior of certain mathematical relationships.
A Definition and examples of fraction functions to provide a clear understanding
For example, the function f(x) = 1/x is a simple fraction function where the output value is the reciprocal of the input value. Another example is f(x) = 2/x, where the output value is twice the
reciprocal of the input value. These examples illustrate how fraction functions can vary based on the constant 'a' in the function.
The unique characteristics of fraction functions distinguishing them from other types of functions
Fraction functions have unique characteristics that distinguish them from other types of functions. One key characteristic is that fraction functions can have vertical asymptotes where the function
approaches infinity as x approaches a certain value. This is due to the denominator of the fraction becoming very small, resulting in a large output value.
Additionally, fraction functions can have horizontal asymptotes where the function approaches a constant value as x approaches infinity. This behavior is a result of the ratio between the numerator
and denominator of the fraction function.
A brief overview of how fraction functions behave graphically, helping visualize their range
Graphically, fraction functions can exhibit various behaviors depending on the values of 'a' in the function. For example, fraction functions with a positive constant 'a' will have a range of
positive and negative values, with the function approaching zero as x approaches infinity.
On the other hand, fraction functions with a negative constant 'a' will have a range of negative and positive values, with the function approaching zero as x approaches negative infinity.
Understanding the graphical behavior of fraction functions can help visualize their range and behavior in different mathematical contexts.
Understanding the Range of a Function
When working with mathematical functions, understanding the range is essential for determining the possible output values of the function. In this chapter, we will delve into the concept of range,
explore its significance in mathematical functions, and provide real-world applications where identifying the range is crucial.
Defining the range in the context of mathematical functions with examples
Range refers to the set of all possible output values of a function. It represents the vertical extent of the function's graph and helps us understand the behavior of the function in terms of its
output values. To find the range of a function, we need to determine all the possible values that the function can output.
For example, consider the function f(x) = 1/x. To find the range of this function, we need to consider all possible values that x can take. Since x cannot be equal to zero (as division by zero is
undefined), the range of this function would be all real numbers except zero. Therefore, the range of f(x) = 1/x is {y | y ≠ 0}.
The difference between range and domain to avoid common confusions
It is important to differentiate between the range and domain of a function to avoid common confusions. While the range represents the set of all possible output values of a function, the domain
refers to the set of all possible input values of a function. Understanding this distinction is crucial for accurately analyzing the behavior of a function.
For instance, consider the function g(x) = √x. The domain of this function would be all non-negative real numbers (x ≥ 0), as the square root of a negative number is undefined in the real number
system. On the other hand, the range of g(x) would be all non-negative real numbers (y ≥ 0), as the square root of any non-negative number is a non-negative number.
Real-world applications where identifying the range is crucial
Identifying the range of a function is not only important in mathematical contexts but also in real-world applications. For example, in finance, understanding the range of possible returns on an
investment can help investors make informed decisions about their portfolios. Similarly, in physics, determining the range of a projectile can aid in predicting its trajectory and impact point.
By recognizing the significance of range in various fields, we can appreciate its role in analyzing and interpreting the behavior of functions in both theoretical and practical settings.
Steps to Find the Range of a Fraction Function
Understanding mathematical functions is essential for solving complex problems in various fields. When it comes to fraction functions, finding the range can be a crucial step in analyzing their
behavior. By following specific steps, you can determine the possible values that the function can output. Let's delve into the process of finding the range of a fraction function.
Identifying the numerator and denominator of the function, crucial for determining its behavior
Before diving into finding the range of a fraction function, it is important to identify its numerator and denominator. The numerator is the top part of the fraction, while the denominator is the
bottom part. Understanding these components is crucial as they play a significant role in determining the behavior of the function.
Tip: Look for any critical points where the denominator becomes zero, as these points can affect the behavior of the function.
Applying the concept of function behavior and critical points to fraction functions
Once you have identified the numerator and denominator of the fraction function, it's time to apply the concept of function behavior and critical points. Critical points are values where the function
may exhibit unusual behavior, such as vertical asymptotes or holes in the graph.
• Identify critical points by setting the denominator equal to zero and solving for the variable.
• Examine the behavior of the function around these critical points to determine if there are any restrictions on the range.
Practical steps to calculate the range, including setting restrictions and solving inequalities
Now that you have identified the critical points and understood the behavior of the function, it's time to calculate the range. This involves setting restrictions based on the critical points and
solving any resulting inequalities.
• Step 1: Determine any restrictions on the domain based on critical points.
• Step 2: Solve any resulting inequalities to find the possible values for the range.
• Step 3: Consider any asymptotes or holes in the graph that may impact the range.
By following these practical steps and considering the behavior of the function, you can effectively find the range of a fraction function. Remember to pay attention to critical points, restrictions,
and inequalities to ensure accurate results.
Common Challenges and How to Overcome Them
When dealing with fraction functions, finding the range can sometimes be a challenging task. Here are some common pitfalls to watch out for and tips on how to overcome them:
A Addressing common pitfalls in finding the range of fraction functions, such as overlooking restrictions
• Overlooking restrictions: One of the most common mistakes when finding the range of a fraction function is overlooking any restrictions on the variables. It is important to identify any values
that the variables cannot take, as these will impact the range of the function.
• How to overcome: Carefully analyze the function and identify any restrictions on the variables. For example, if the denominator of a fraction cannot be zero, then you need to exclude that value
from the possible range of the function.
B Troubleshooting tips for complex functions that don’t fit conventional patterns
• Complex functions: Some fraction functions may not fit conventional patterns, making it difficult to determine the range using traditional methods. These functions may involve multiple variables
or non-linear relationships.
• How to overcome: Break down the function into simpler components and analyze each part separately. Look for any patterns or relationships that can help you determine the range of the function.
Consider using algebraic techniques or software tools to assist in the analysis.
C Utilizing graphing calculators and software as a tool for visualizing the function’s behavior
• Graphing calculators and software: Visualizing the behavior of a fraction function can be helpful in understanding its range. Graphing calculators and software tools can provide a graphical
representation of the function, making it easier to identify key points and trends.
• How to overcome: Use graphing calculators or software programs to plot the function and observe its behavior. Look for any asymptotes, intercepts, or other key features that can help you
determine the range. Experiment with different values to see how they affect the function's output.
Real-world Examples and Applications
Understanding the range of mathematical functions, particularly fraction functions, is not just a theoretical concept but has practical applications in various fields. Let's explore how this
knowledge is applied in engineering, economics, and data analysis.
A. Discussing how the knowledge of range is applied in fields like engineering, economics, and data analysis
In engineering, the range of a function is crucial for designing systems and structures. Engineers use mathematical functions to model physical phenomena, and knowing the range helps them determine
the possible outputs or outcomes of their designs. For example, in civil engineering, understanding the range of a function representing stress distribution in a bridge can help ensure the
structure's safety and stability.
In economics, functions are used to analyze relationships between variables such as supply and demand, cost and revenue, or production and profit. By finding the range of these functions, economists
can make informed decisions about pricing strategies, market trends, and resource allocation.
In data analysis, understanding the range of a function is essential for interpreting and visualizing data. Data scientists use mathematical functions to model patterns and trends in datasets, and
knowing the range helps them identify outliers, anomalies, or correlations that can provide valuable insights for decision-making.
B. Providing examples where solving for the range of fraction functions solves practical problems
One practical example where finding the range of a fraction function is crucial is in designing a water distribution system. By determining the range of a function representing water pressure in
pipes, engineers can ensure that all areas receive adequate water flow without exceeding the system's capacity or causing leaks.
Another example is in financial analysis, where calculating the range of a function representing investment returns helps investors assess the potential risks and rewards of different investment
options. By understanding the range of possible outcomes, investors can make informed decisions to maximize their returns while managing their risks.
C. Case studies or scenarios illustrating the importance of understanding function ranges
Consider a scenario where a company wants to optimize its production process to minimize costs and maximize efficiency. By analyzing the range of a function representing production costs in relation
to output levels, the company can identify the optimal production level that balances cost-effectiveness with productivity.
Another case study could involve a healthcare provider analyzing patient data to improve treatment outcomes. By examining the range of a function representing patient recovery rates based on
different treatment protocols, healthcare professionals can tailor their interventions to maximize the chances of successful outcomes for their patients.
Conclusion & Best Practices in Finding the Range of Fraction Functions
After delving into the intricacies of finding the range of fraction functions, it is essential to summarize the key points discussed and emphasize the importance of accurately determining the range.
A Summarizing the key points discussed and the importance of accurately finding the range
• Key Points: Understanding the range of a fraction function involves identifying all possible output values that the function can produce.
• Importance: Finding the range is crucial in understanding the behavior of the function and determining its limitations.
• Accuracy: Ensuring accuracy in finding the range helps in making informed decisions and predictions based on the function's behavior.
B Emphasizing best practices, such as always checking for undefined points and using graphical aids
• Check for Undefined Points: Always be vigilant in identifying any values that may result in division by zero, as these points are undefined in fraction functions.
• Graphical Aids: Utilize graphical representations, such as plotting the function on a graph, to visually determine the range and verify your calculations.
• Best Practices: Incorporate systematic approaches, such as simplifying the function and analyzing its behavior, to efficiently find the range.
C Encouraging continuous practice and exploration of fraction functions beyond the basics for deeper understanding
• Continuous Practice: Regularly practice finding the range of fraction functions to enhance your skills and develop a deeper understanding of their behavior.
• Exploration Beyond Basics: Challenge yourself by exploring more complex fraction functions and analyzing their ranges to broaden your mathematical knowledge.
• Deeper Understanding: By delving deeper into fraction functions and their ranges, you can gain insights into their patterns and relationships, leading to a more profound comprehension of
mathematical concepts. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-find-range-fraction-function","timestamp":"2024-11-14T23:59:53Z","content_type":"text/html","content_length":"227839","record_id":"<urn:uuid:95a15e61-9e49-431b-b9ef-c51f305e3c4e>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00018.warc.gz"} |
What I Tried-
The Question \int \csc(x-\frac{\pi}{3})\csc(x-\frac{\pi}{6})dx What I Tried- I tried dividing both numerator and
Lorraine Harvey
Answered question
The Question
$\int \mathrm{csc}\left(x-\frac{\pi }{3}\right)\mathrm{csc}\left(x-\frac{\pi }{6}\right)dx$
What I Tried- I tried dividing both numerator and denominator by $\mathrm{sin}\frac{\pi }{6}$ but couldnt
Answer & Explanation
$\mathrm{csc}\left(x-\frac{\pi }{3}\right)\mathrm{csc}\left(x-\frac{\pi }{6}\right)$
$=\frac{1}{\mathrm{sin}\left(\frac{\pi }{3}-\frac{\pi }{6}\right)}\cdot \frac{\mathrm{sin}\left(x-\frac{\pi }{6}-\left(x-\frac{\pi }{3}\right)\right)}{\mathrm{sin}\left(x-\frac{\pi }{3}\right)\mathrm
{sin}\left(x-\frac{\pi }{6}\right)}$
$\mathrm{csc}\left(x-\frac{\pi }{3}\right)\mathrm{csc}\left(x-\frac{\pi }{6}\right)=\mathrm{csc}\left(\frac{\pi }{6}-x\right)\mathrm{sec}\left(x+\frac{\pi }{6}\right)=\frac{4}{\sqrt{3}-2\mathrm{sin}\
Now, let $x={\mathrm{tan}}^{-1}\left(t\right)$ and tou face
$I=4\int \frac{dt}{\sqrt{3}{t}^{2}-4t+\sqrt{3}}$
The denominator has two simple real roots. Then partial fraction decomposition to face two simple integrals.
Substitute $t=\mathrm{tan}\left(x-\frac{\pi }{4}\right)$
$\int \mathrm{csc}\left(x-\frac{\pi }{3}\right)\mathrm{csc}\left(x-\frac{\pi }{6}\right)dx=4\int \frac{2-\sqrt{3}}{{t}^{2}-\left(2-\sqrt{3}{\right)}^{2}}dt$
$=4\mathrm{ln}|\frac{2-\sqrt{3}-t}{2-\sqrt{3}+t}|=4\mathrm{ln}|\frac{2-\sqrt{3}-\mathrm{tan}\left(x-\frac{\pi }{4}\right)}{2-\sqrt{3}+\mathrm{tan}\left(x-\frac{\pi }{4}\right)}|+C$ | {"url":"https://plainmath.org/trigonometry/50006-the-question-int-csc-frac-frac-what-tried-tried-dividing-both-numerator","timestamp":"2024-11-09T19:12:08Z","content_type":"text/html","content_length":"251168","record_id":"<urn:uuid:ba48d335-4123-477c-a40c-06435de54f92>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00721.warc.gz"} |
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Towards a representation theorem for coloring algebra
Coloring algebra (CA) captures common ideas of feature oriented programming (FOP), a general programming paradigm that provides formalisms, methods, languages, and tools for building maintainable,
customisable, as well as extensible software. FOP has widespread applications from network protocols and data structures to software product lines. It arose from the idea of level-based designs,
i.e., the idea that each program (design) can be successively built up by adding more and more levels (features). Later, this idea was generalised to the abstract concept of features. The algebra
itself is based on rings and offers simple and concise axioms for feature composition and feature interaction. From these two operations algebraic laws describing products, product lines and other
concepts of FOP can be derived.
The talk will start with a brief introduction to coloring algebra. In particular I will discuss the interplay between feature composition and interaction. I will also present different models and
discuss their relationship to and applicability for feature oriented programming.
The examples will show that the choice for defining feature composition is limited. In fact, I will prove that the composition operator is always isomorphic to symmetric-difference in a set-theoretic
model. The proof can easily be derived from Kronecker Basis Theorem. By this result we have “half” a representation theorem for CA. A detailed understanding of feature interaction is missing for the
characterisation and the proof of a full representation theorem. Therefore, I will conclude the talk with some observations and conjectures on the structure of coloring algebra, which hopefully helps
to come up with a full representation theorem.
BibTeX Entry
author = {H\"ofner, Peter},
booktitle = {Workshop on Lattices and Relations},
keywords = {fosd},
month = aug,
paperurl = {https://trustworthy.systems/publications/nicta_full_text/6567.pdf},
title = {Towards a Representation Theorem for Coloring Algebra},
year = {2012} | {"url":"https://trustworthy.systems/publications/nictaabstracts/Hoefner_12.abstract","timestamp":"2024-11-03T06:53:13Z","content_type":"text/html","content_length":"11848","record_id":"<urn:uuid:49f3134d-eeeb-4e88-88b5-7de6fa8fed88>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00114.warc.gz"} |