content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
These built-in functions are similar to __builtin_add_overflow, __builtin_sub_overflow, or __builtin_mul_overflow, except that they dont store the result of the arithmetic operation anywhere and
the last argument is not a pointer, but some expression with integral type other than enumerated or boolean type.
The built-in functions promote the first two operands into infinite precision signed type and perform addition on those promoted operands. The result is then cast to the type of the third
argument. If the cast result is equal to the infinite precision result, the built-in functions return false, otherwise they return true. The value of the third argument is ignored, just the side
effects in the third argument are evaluated, and no integral argument promotions are performed on the last argument. If the third argument is a bit-field, the type used for the result cast has
the precision and signedness of the given bit-field, rather than precision and signedness of the underlying type.
For example, the following macro can be used to portably check, at compile-time, whether or not adding two constant integers will overflow, and perform the addition only when it is known to be
safe and not to trigger a -Woverflow warning.
#define INT_ADD_OVERFLOW_P(a, b) \
__builtin_add_overflow_p (a, b, (__typeof__ ((a) + (b))) 0)
enum {
A = INT_MAX, B = 3,
C = INT_ADD_OVERFLOW_P (A, B) ? 0 : A + B,
D = __builtin_add_overflow_p (1, SCHAR_MAX, (signed char) 0)
The compiler will attempt to use hardware instructions to implement these built-in functions where possible, like conditional jump on overflow after addition, conditional jump on carry etc. | {"url":"https://www.rowleydownload.co.uk/arm/documentation/gnu/gcc/Integer-Overflow-Builtins.html","timestamp":"2024-11-14T14:16:36Z","content_type":"text/html","content_length":"14235","record_id":"<urn:uuid:ea5f4c28-ee35-4748-8967-01dec2a327e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00270.warc.gz"} |
Question about Camera's range
Posts : 839
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Age : 59
• Post n°1
Question about Camera's range
If you have camera I mean russian army, how much the max distance to covert an area ?
For example if a drone is above batlefield, with a camera, this drone could warn russian army about incoming attack. How much km ? 100 KM ? 200 KM ? Less ?
Satelites have resolution and range maybe several hundreds km ? Could a drone with camera replace satelite ?
Posts : 5926
Points : 6115
Join date : 2012-10-25
nemrod wrote:If you have camera I mean russian army, how much the max distance to covert an area ?
For example if a drone is above batlefield, with a camera, this drone could warn russian army about incoming attack. How much km ? 100 KM ? 200 KM ? Less ?
Satelites have resolution and range maybe several hundreds km ? Could a drone with camera replace satelite ?
Drones with cameras are limited to size of their cameras and the altitude which means bound to earth's curvature, most common drones have range of up to 26km not hundreds, that is physically not
possible, not to mention the clouds would block range and only satellites can see hundreds of km's they have the altitude.
Posts : 2354
Points : 2536
Join date : 2010-11-12
Location : The Land Of Pharaohs
Camera over there in the space has a huge lens on it and use CCD to convert light into electrons .
It's resolution is high, 16 inch or more and may capture images outside of the typical visual spectrum.
Posts : 40384
Points : 40884
Join date : 2010-03-30
Location : New Zealand
Actually some cameras have very long range, but the problem with very long range is high magnification which limits field of view so you end up having to scan a large area with a high mag lens it
ends up like looking though a straw to find things.
Very high altitude UAVs need high power cameras just to see the ground clearly... a drone flying at 15km altitude needs a camera that can see enormous distances... a target 20km away from the drone
horizontally... well Pythagoras theorem says you can calculate the longest edge of a right angle triangle by squaring the other two sides and adding them together and square rooting the result... so
15 x 15 plus 20 x 20 equals 225 + 400 which equals 625 so the square root is 25, so to see 20km away from 15km altitude the camera has to be able to see at least 25km.
A while back I was looking through Various Russian optics websites I found one website that specialised in very high resolution very high magnification optics.
The example on their front page showed a city landscape photo that zoomed in multiple times to the point where buildings that were 20km away could be zoomed in so the image showed individual rooms on
a building...
It really annoys me that I don't seem to be able to find it again....
Sponsored content | {"url":"https://www.russiadefence.net/t3875-question-about-camera-s-range","timestamp":"2024-11-02T01:44:43Z","content_type":"text/html","content_length":"88124","record_id":"<urn:uuid:02016037-75a4-4b09-8fd0-5d9014259dc3>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00742.warc.gz"} |
How to calculate gdp price index
GDP deflator. Using the statistics on real GDP and nominal GDP, one can calculate an implicit index of the price level for the year. This index is called the GDP What is the GDP Price Index? A
measure of inflation in the prices of goods and services produced in the United States. The gross domestic product price index
Sep 3, 2008 The CPI (or the PCE) attempts to measure how the prices of a typical GDP, GDP-deflator inflation will be low, while the consumer price index However, to determine real GDP, the nominal
GDP is divided by the price index divided by 100. To simplify comparisons, the value of the price index is set at 100 for the base year. Previous to the base year, prices were generally lower, so
those GDP values must be inflated to compare them to the base year. How to Calculate the GDP Deflator. 1. Calculate Nominal GDP. Nominal GDP is defined as the monetary value of all finished goods and
services within an economy valued at current 2. Calculate Real GDP. 3. Calculate the GDP Deflator. Suppose that in the year following the base year, the GDP deflator is equal to 110. The percentage
change in the GDP deflator from the previous (base) year is obtained using the same formula used to calculate the growth rate of GDP. This percentage change is found to be . implying that the GDP
deflator index has increased 10%.
associated with GDP, where the bundle of goods under consideration is Compute the consumer price index (CPI) for each of the three years, using 1980 as.
When prices are less in any given year than they were in the base year, then the price index will be less than 100, so that when real GDP is calculated by dividing Aug 3, 2019 the Consumer Price
Index. There are indexes that measure inflation other than the GDP deflator. Many of these alternatives are based on a fixed GDP deflator. Using the statistics on real GDP and nominal GDP, one can
calculate an implicit index of the price level for the year. This index is called the GDP What is the GDP Price Index? A measure of inflation in the prices of goods and services produced in the
United States. The gross domestic product price index The GDP deflator is one of those numbers in the index and can be used to figure out the real GDP. If there is 2.5% inflation, then price level
of 2011 in
The GDP price deflator measures the changes in prices for all of the goods and services produced in an economy. Gross domestic product or GDP represents the total output of good and services.
However, as GDP rises and falls, the metric doesn't consider the impact of inflation or rising prices on the GDP results.
approaches of measuring GDP – value added approach, expenditure approach and price index (CPI) and implicit price deflator of GDP (or GDP deflator). Gross Domestic Product: Implicit Price Deflator
(GDPDEF). Download. Q4 2019: 113.083 | Index 2012=100 | Quarterly | Updated: Jan 30, 2020. Observation:. Jan 21, 2020 MEASURING THE COST OF LIVING. 11. The Consumer Price Index (CPI). ▫ GDP Deflator
includes the effect of investment goods, imports In general, constant price series are derived using price indices or unit-value For example, from the expenditure side, Gross Domestic Product (GDP)
at The Consumer Price Index (CPI) keeps a running measure of the cost of U.S. goods and services each year. The CPI measures prices from a base year, currently The government's calculation of real
GDP growth begins with the estimation index calculations, and subsequent changes in their price are taken into account
A price index is a weighted average of the prices of a selected basket of goods and services Then we take a representative sample of goods and services and calculate their value in the base
Differences between the CPI and GDP deflator .
Calculate chained-dollar real GDP. Calculate GDP deflator for each period. Calculate CPI (consumption price index) for each period. Calculate PCE price index Feb 27, 2020 The index is calculated as
the ratio of nominal GDP (expressed in current year's market prices) to real GDP (in base 2009 year price) multiplied by 1 prices. Year 2 real GDP = 25 * $1000 + 12 000 * $1.00 = $37 000. We must
next compute real GDP using year 2 prices. to express as an index number. approaches of measuring GDP – value added approach, expenditure approach and price index (CPI) and implicit price deflator of
GDP (or GDP deflator). Gross Domestic Product: Implicit Price Deflator (GDPDEF). Download. Q4 2019: 113.083 | Index 2012=100 | Quarterly | Updated: Jan 30, 2020. Observation:. Jan 21, 2020 MEASURING
THE COST OF LIVING. 11. The Consumer Price Index (CPI). ▫ GDP Deflator includes the effect of investment goods, imports In general, constant price series are derived using price indices or
unit-value For example, from the expenditure side, Gross Domestic Product (GDP) at
To calculate CPI, or Consumer Price Index, add together a sampling of product prices from a previous year. Then, add together the current prices of the same products. Divide the total of current
prices by the old prices, then multiply the result by 100. Finally, to find the percent change in CPI, subtract 100.
1 prices. Year 2 real GDP = 25 * $1000 + 12 000 * $1.00 = $37 000. We must next compute real GDP using year 2 prices. to express as an index number. approaches of measuring GDP – value added
approach, expenditure approach and price index (CPI) and implicit price deflator of GDP (or GDP deflator). Gross Domestic Product: Implicit Price Deflator (GDPDEF). Download. Q4 2019: 113.083 | Index
2012=100 | Quarterly | Updated: Jan 30, 2020. Observation:. Jan 21, 2020 MEASURING THE COST OF LIVING. 11. The Consumer Price Index (CPI). ▫ GDP Deflator includes the effect of investment goods,
imports In general, constant price series are derived using price indices or unit-value For example, from the expenditure side, Gross Domestic Product (GDP) at
For example, to express the 2004 total expenditure estimate in 2014 real ( inflation The GDP price index is the broadest index and the best choice when associated with GDP, where the bundle of goods
under consideration is Compute the consumer price index (CPI) for each of the three years, using 1980 as. The GDP deflator is an index that tracks price changes from a base year. To calculate the GDP
deflator, the formula is Nominal/Real x 100. In the example above and calculate GDP. Understand the difference between real and nominal variables (e.g., GDP, wages, interest rates) and know how to
construct a price index.”. | {"url":"https://digitaloptionssohbmwl.netlify.app/vanmatre61875pi/how-to-calculate-gdp-price-index-22.html","timestamp":"2024-11-09T02:59:54Z","content_type":"text/html","content_length":"31734","record_id":"<urn:uuid:be88ef6b-6d27-439f-a79d-a46a6d9e6f22>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00758.warc.gz"} |
Understanding Kolmogorov Complexity
| 9 minutes | 1810 words
Gallerie Umberto, Napoli
I often advocate for a straightforward yet effective rule: the shortest solution that delivers the desired result is usually the best. This approach, favoring concise algorithms, not only ensures
efficiency but also reduces maintenance cost and bug susceptibility. Intriguingly, this practical principle finds its theoretical counterpart in data compression algorithm, data analysis and machine
learning through the Minimum Description Length (MDL) and the Kolmogorov Complexity. These concepts delve deep into the essence of data representation and algorithmic efficiency.
The MDL principle is an invaluable tool. It is grounded in the concept of succinctly capturing information and is essential in balancing model complexity with its explanatory power. The ideal model,
according to MDL, is one that minimizes the combined description length of the model and the data. This principle is not only pivotal in guiding model selection but also reflects the deeper
mathematical underpinnings of Kolmogorov complexity.
Kolmogorov complexity is about the shortest possible description of an object within a given computational model. It focuses on the simplest, most concise representation of data that a computer can
use to reconstruct the original object. However, it is crucial to understand that the Kolmogorov complexity is, just like the Halting problem, inherently uncomputable. This means that even if we
stumble upon the shortest possible program that replicates a specific output, there’s no definitive method to prove that there isn’t a shorter program capable of achieving the same result. This
uncomputability aspect of Kolmogorov complexity adds an extra layer of fascination. It underscores a fundamental limitation in our ability to fully comprehend data representation’s simplicity,
highlighting the provisional nature of our knowledge in data analysis and machine learning.
Contrary to what Mortal Kombat fans might think, Kolmogorov complexity isn’t named after a final boss but after the renowned Russian mathematician Andrey Kolmogorov, Kolmogorov complexity offers a
unique perspective on assessing data complexity. It diverges from traditional metrics by focusing on information content rather than sheer size. This post is a very quick and dirty introduction to
what Kolmogorov complexity is, starting with its foundational principles and progressing to practical examples. Don’t forget to check out the disclaimer at the end of the post.
Let’s introduce Kolmogorov complexity with the notion of random. What is the meaning of a random string? Let’s consider the following 64-character example strings, and evaluate together their
perceived randomness:
1. 1111111111111111111111111111111111111111111111111111111111111111
2. 1234567890123456789012345678901234567890123456789012345678901234
3. 7f83b1657ff1fc53b92dc18148a1d65dfc2d4b1fa3d677284addd200126d9069
The 64-character strings and their corresponding PHP code are used purely for illustrative purposes to demonstrate the principles of Kolmogorov complexity. It is worth noting that these examples are
intentionally simplistic and not intended to represent optimal or real-world scenarios. The choice of 64 characters serves as a convenient length for demonstration, making the concept approachable
and understandable, even though in practical applications, the length and complexity of strings might significantly vary. Furthermore, while PHP may seem verbose in these instances, it is chosen for
its simplicity and accessibility, especially for those not deeply versed in programming. The key takeaway is not the length of the PHP code per se, but the underlying idea of Kolmogorov complexity:
the search for the simplest possible description or program that can generate a given output. Real-world applications of Kolmogorov complexity often involve much longer strings and more intricate
computations, where the efficiency and conciseness of the description become markedly more pronounced and impactful.
At a glance, the first string 1111111111111111111111111111111111111111111111111111111111111111 does not look like a random string. It can be succinctly described as “64 1’s in a row”, therefore it
has low Kolmogorov complexity. This can be demonstrated in PHP. The length of the PHP code is 25 characters, significantly shorter than 64. This substantial difference in length not only generates
the entire string but also highlights its high compressibility rate, thereby underscoring its very low randomness.
echo str_repeat('1', 64); // 25 characters
The second string 1234567890123456789012345678901234567890123456789012345678901230 appears to be random and complex at first glance. However, this string is a repetitive sequence of the digits from 0
to 9. Much like the first string, there’s no much random in it and its Kolmogorov complexity is low. it can be described succinctly as “the sequence 0 to 9 repeated 7 times and trimmed to 64
characters”. The length of the PHP code is 48 characters, which is shorter than 64, also hightlighting its high compressibility rate and therefore its low randomness.
echo substr(str_repeat('1234567890', 7), 0, 64); // 48 characters
Here, the PHP code generates the string by repeating the sequence 1234567890, effectively constructing the 64-characters string. The simplicity of this PHP code demonstrates that the string, despite
appearing random and complex, has a relatively low Kolmogorov complexity due to its underlying repetitive pattern. This example, along with the previous ones, illustrates how Kolmogorov complexity is
not merely about the visual complexity or length of a string, but rather about how succinctly the string can be described or generated. The concept of randomness in Kolmogorov complexity is closely
tied to the absence of such describable patterns.
In the third key smashed example string 7f83b1657ff1fc53b92dc18148a1d65dfc2d4b1fa3d677284addd200126d9069, initially seems to have high Kolmogorov complexity. It does not lend itself to a simple
algorithmic description and therefore appears incompressible and algorithmically random. The PHP code to generate this string, which includes the original string itself, spans 72 characters,
exceeding the original string’s length. When the code to reproduce a string is longer than the original string itself, it indicates a very low compressibility rate. This means the string cannot be
compressed further, thus suggesting its high level of randomness.
echo '7f83b1657ff1fc53b92dc18148a1d65dfc2d4b1fa3d677284addd200126d9069'; // 72 characters
While all three strings have the same probabilistic chance of being randomly selected, their complexities vary. This illustrates how randomness in Kolmogorov complexity is tied to the ability to
describe patterns succinctly. This paradox highlights that randomness transcends probability, linking instead to the presence or absence of patterns and the succinctness of their descriptions. The
shorter the description required, the less random the string is considered, as seen with the first two strings. Conversely, a string requiring a more extensive explanation, due to lack of regular
patterns, is deemed more random, as with this very last example.
The Kolmogorov complexity is not just about the subject itself but also about the language and tools used for its description. Different programming languages or descriptive frameworks might have
varying levels of conciseness for the same concept. For instance, what is succinctly expressed in one language might be more verbose in another.
This can be found in the formal definition of Kolmogorov complexity:
$$ K_f(x) = \min \lbrace |p|: f(p) = x \rbrace $$
Interpreted, this formal definition states: The Kolmogorov complexity $K$ of a string $x$, relative to a Turing machine $f$ (a programming language), is the length of the shortest program $p$ that
outputs $x$ when run on $f$.
An intriguing aspect of Kolmogorov complexity lies in its inherent uncertainty. This uncertainty stems from the possibility that a shorter description for a given string may be discovered in a near
or far future. Since the Kolmogorov complexity is uncomputable, there’s no definitive way to prove that a shorter description does not exist.
Knowing this, we can safely say that Kolmogorov complexity establishes an upper limit of complexity but not a lower one. When we write a program to generate a particular output, this program serves
as definitive evidence that the sequence’s Kolmogorov complexity is, at most, equal to the length of that program. However, this does not imply a minimum level of complexity for the sequence. There
is always the possibility that a shorter program capable of producing the same sequence exists. Therefore, the true Kolmogorov complexity of a sequence remains simply unknown, with the potential for
more concise representations yet to be discovered.
The complexity assigned to data today might change as our understanding and technological capabilities evolve. This means that while Kolmogorov complexity offers a valuable framework for assessing
data complexity, its nature is provisional. The complexity of a string reflects our current limitations and understanding, reminding us that our grasp of data complexity is always subject to revision
and improvement. From this perspective, a string is random if it is incompressible, meaning it cannot be described by a shorter program. Conversely, a string is compressible if it can be described by
a shorter program.
Formally, this can be defined as: $$ K_f(x) \geq |x| $$
This means the Kolmogorov complexity relative to a Turing machine $f$ of a supposedly incompressible string $x$ is always greater than or equal to the length of the string itself. In other words, the
Kolmogorov complexity of a true random string is always greater than or equal to its length. This is because the random string $x$ cannot be described by a shorter program.
To illustrate this, take the third example string 7f83b1657ff1fc53b92dc18148a1d65dfc2d4b1fa3d677284addd200126d9069, initially seems highly complex. However, this string is actually the SHA-256 hash
of Hello World!. Knowing this, the string can be succinctly described as “the SHA-256 hash of ‘Hello World!’”; reducing significantly its randomness and complexity. The length of the PHP code is now
36 characters, which is shorter than 64.
echo hash('sha256', 'Hello World!'); // 36 characters
The MDL principle and Kolmogorov complexity together offer profound insights into the very nature of information. MDL provides a practical methodology for model selection in data analysis and machine
learning, focusing on the balance between simplicity and explanatory power. On the other hand, Kolmogorov complexity contributes a theoretical foundation, underscoring the importance of succinctness
and efficiency in information representation. This includes applications in fields like data compression and number theory. The interplay between these two concepts highlights the critical importance
of considering both the chosen model and the language used for complexity assessment. As we delve further into the intricacies of data and computation, grasping these principles thoroughly becomes
ever more crucial. These principles enhance our ability to effectively grasp and interpret the essence of information within our rapidly evolving digital landscape.
…, isn’t it?
To go further into these topics, I highly recommend exploring the foundational ideas and detailed insights offered in the following papers and resources which I used as references for this post:
I’m just an IT professional with a deep fascination for these kind of concepts that resonates while writing code, sometimes without knowing that they are formally defined. My aim with this post is to
share my enthusiasm and understanding of Kolmogorov complexity and the MDL principle, hoping to spark interest in others. While I strive for accuracy, I encourage readers to seek out more detailed
and expert resources if they wish to delve deeper into these intriguing subjects. | {"url":"https://not-a-number.io/2024/understanding-kolmogorov-complexity/","timestamp":"2024-11-10T09:12:33Z","content_type":"text/html","content_length":"29869","record_id":"<urn:uuid:07208f74-6fbb-4498-8cec-205c54e94e1b>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00705.warc.gz"} |
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Pagination for database objects
So you have a server storing objects in a relational database, and an API, nowadays probably HTTP but it does not matter. Clients can fetch objects using the API. Obviously you do not want them
loading too many objects at the same time, that would cause all kinds of performance issues. So you need a way to let clients request a manageable subset of all objects and iterate to finally obtain
all objects. This is pagination.
I used to think it was a trivial problem. But given the number of complex and inefficient solutions I have seen over the years, I was clearly mistaken. Let us find out how to do it easily and
Limit and offset
One of the first things you learn with SQL is how to use LIMIT and OFFSET to restrict the number of rows to retrieve.
Let us define a user table and load 10 users after the first 20:
CREATE TABLE users
(id SERIAL PRIMARY KEY,
name VARCHAR NOT NULL);
SELECT * FROM users
LIMIT 10 OFFSET 20;
Have your clients send a limit and offset with each query, increment the offset to iterate. Easy, but there is a problem: the highest the offset is, the slower the request will be. LIMIT and OFFSET
are applied after filtering, so if you have a million users, selecting the last 10 ones will still require processing the 999'990 before them. Not great.
Worse, this method can and will provide incorrect results when retrieving pages one after the other: adding rows to the table between each selection will make subsequent pages either miss rows or
include some which we already returned in previous pages.
SQL Cursors
It can be tempting to use SQL cursors. After all, they let you consume rows iteratively without loading the whole result set in memory. But they are really not designed for the task: you would need
to keep the associated database connection open between requests. Next idea.
Key-based pagination
Sometimes called “keyset” pagination, this method uses a key from the last row of the page you just loaded to obtain the next one efficiently.
Let us load the first page:
SELECT * FROM users
ORDER BY id
LIMIT 10;
Then for the next page, have the client provide the identifier of the last user in the page —let us say 42— and use it to load the second page:
SELECT * FROM users
ORDER BY id
WHERE id > 42
LIMIT 10;
Continue the same way for subsequent pages.
As long as you are filtering on an indexed expression, the database will efficiently retrieve the rows for the page you requested. And you can use the exact same method to order results in the other
direction, just invert the WHERE expression.
You will also avoid the inconsistencies of the LIMIT/OFFSET method: if the table is modified between two page retrieval operations, you may still miss rows (nothing you can do about it unless you are
willing to retrieve the entire table in one operation), but you will not see rows from previous pages reappear in subsequent pages. Your users will thank you (or more accurately they will open less
bug reports; even better).
So you started using key-based pagination and everything looks fine, but now you need to sort results based on something else than the primary key. Your first idea could be to create an index on the
column you want to use to sort, but it will not help: if multiple users have the same name, how will you select your pages?
We could use an index on two columns, name and id to obtain the strict total order we need then filter on these two columns:
CREATE INDEX users_name_id_idx
ON users (name, id);
SELECT * FROM users
WHERE (name, id) > ('bob', 42)
ORDER BY name, id
LIMIT 10;
But this is not standard SQL and would only work with databases supporting row-value comparisons.
Fortunately SQL lets you create indices on expressions and not just columns. So we create our own single-value keys by concatenating values. Of course we need a separator to avoid collisions: without
it ('Bob', 42) and ('Bob4', 2) would both yield the same 'Bob42' key). The ASCII character set includes several separator characters which are perfect for this use case since they are not supposed to
end up in your data (if they do, use another character); let us use the unit separator 0x1f:
CREATE INDEX users_name_pagination_idx
ON users ((name || E'\x1f' || id));
Load the first page:
SELECT * FROM users
ORDER BY name || E'\x1f' || id
LIMIT 10;
And assuming the last user of the first page had the name Bob and the identifier 42, Load the second page.
SELECT * FROM users
WHERE name || E'\x1f' || id > E'Bob\x1f42'
ORDER BY name || E'\x1f' || id
LIMIT 10;
Annoying to type but efficient and easy to generate. Of course you will want to make it easy for your clients: when returning a page, compute the key for the previous and next page and return them
along the objects. The client can then trivially fetch the previous or next page.
Need to support multiple sort criteria? Just create an index for each of them.
That wasn’t so hard right? | {"url":"https://www.n16f.net/blog/pagination-for-database-objects/","timestamp":"2024-11-11T06:06:35Z","content_type":"text/html","content_length":"16120","record_id":"<urn:uuid:aff806ed-a3fd-415e-9681-93eff54550cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00225.warc.gz"} |
5 - Factorial - Solved
alright, so this is the working code, but for anyone who was curious i wanted to explain WHY this code worked lol. I got stuck on this for about an hour (sad, i know) but i didnt want to look at the
Q&A because i knew i wouldnt learn anything by doing that. so anyways, it took me forever but i figured it out. (ps, this explanation sucks because its hard to explain XD)
you have to write the IF statement because it needs to check if the number is 1, for multiple reasons. the first being that if x is 1, then the answer is one.
the second part of the code is taking every number in the range of x and subtracting 1 and then multiplying them all together.
so say “x” was 4
the code is going to go through this process:
does 4 equal 1?
no, so lets move on to the ELSE statement
4 * factorial(4-1)
-which would be 4*3
so thats one of the factorials
and then it goes through it again because it called the factorial function again.
and since 3 doesnt equal 1 it will run the code again.
this time it would be 3-1
which equals 2
it goes through it until it reaches one.
back to the other reasons, if it didnt have that first part of the if statement, it would go on forever since the function calls itself, this statement makes sure its not an infinite loop and gives
it a stopping point and then also if if you subtracted 1 from 1 it would be 0 and anything you multiply by 0 is 0 and then no matter what number you put in there it would return 0
2 Likes
Is there anything you would like help with, Or is this strictly an informational Topic?
1 Like
informational for anyone that struggled with this as i did haha
I’m a little concerned (minor) that there are two terms distinctly notable by their absence from the above discourse:
1. Recursion
2. Call stack
Also, we don’t see the terms, base case or recursive case. In a discussion such as this, we should not only solve our own problem, but when giving advice, research so that our explanation is more in
line with the topic diction (words and terms). Never be afraid to work on a second draft if you find ways to improve your writing.
I have the same, but mine isn’t working:
function factorial(number) {
if (number == 1 || number ==0) {
return 1
else {
return number * factorial(number -1);
I tried without the || clause and it still didn’t work.
Thanks in advance
Well seems like you made a common mistake.
In JavaScript we use ‘===’ known as strict equality whereas you have used ‘==’ but of them have a minor difference. So yeah, use ‘===’ and you’ll be all good.
1 Like | {"url":"https://discuss.codecademy.com/t/5-factorial-solved/18221","timestamp":"2024-11-04T10:48:04Z","content_type":"text/html","content_length":"32448","record_id":"<urn:uuid:b0276a4c-65c2-4c19-95af-c651921502b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00057.warc.gz"} |
Selina Concise Mathematics Class 6 ICSE Solutions Chapter 25 Properties of Angles and Lines - CBSE Tuts
Selina Concise Mathematics Class 6 ICSE Solutions Chapter 25 Properties of Angles and Lines (Including Parallel Lines)
Selina Publishers Concise Mathematics Class 6 ICSE Solutions Chapter 25 Properties of Angles and Lines (Including Parallel Lines)
Properties of Angles and Lines Exercise 25A – Selina Concise Mathematics Class 6 ICSE Solutions
Question 1.
Two straight lines AB and CD intersect each other at a point O and angle AOC = 50° ; find :
(i) angle BOD
(ii) ∠AOD
(iii) ∠BOC
Question 2.
The adjoining figure, shows two straight lines AB and CD intersecting at point P. If ∠BPC = 4x – 5° and ∠APD = 3x + 15° ; find :
(i) the value of x.
(ii) ∠APD
(iii) ∠BPD
(iv) ∠BPC
Question 3.
The given diagram, shows two adjacent angles AOB and AOC, whose exterior sides are along the same straight line. Find the value of x.
Question 4.
Each figure given below shows a pair of adjacent angles AOB and BOC. Find whether or not the exterior arms OA and OC are in the same straight line.
Question 5.
A line segment AP stands at point P of a straight line BC such that ∠APB = 5x – 40° and ∠APC = .x+ 10°; find the value of x and angle APB.
Properties of Angles and Lines Exercise 25B – Selina Concise Mathematics Class 6 ICSE Solutions
Question 1.
Identify the pair of angles in each of the figure given below :
adjacent angles, vertically opposite angles, interior alternate angles, corresponding angles or exterior alternate angles.
Question 2.
Each figure given below shows a pair of parallel lines cut by a transversal For each case, find a and b, giving reasons.
Question 3.
If ∠1 = 120°, find the measures of : ∠2, ∠3, ∠4, ∠5, ∠6, ∠7 and ∠8. Give reasons.
Question 4.
In the figure given below, find the measure of the angles denoted by x,y, z,p,q and r.
Question 5.
Using the given figure, fill in the blanks.
Question 6.
In the given figure, find the anlges shown by x,y, z and w. Give reasons.
Question 7.
Find a, b, c and d in the figure given below :
Question 8.
Find x, y and z in the figure given below :
Properties of Angles and Lines Exercise 25C – Selina Concise Mathematics Class 6 ICSE Solutions
Question 1.
In your note-book copy the following angles using ruler and a pair compass only.
Question 2.
Construct the following angles, using ruler and a pair of compass only
(i) 60°
(ii) 90°
(iii) 45°
(iv) 30°
(v) 120°
(vi) 135°
(vii) 15°
Question 3.
Draw line AB = 6cm. Construct angle ABC = 60°. Then draw the bisector of angle ABC.
Question 4.
Draw a line segment PQ = 8cm. Construct the perpendicular bisector of the line segment PQ. Let the perpendicular bisector drawn meet PQ at point R. Measure the lengths of PR and QR. Is PR = QR ?
Question 5.
Draw a line segment AB = 7cm. Mark a point Pon AB such that AP=3 cm. Draw perpendicular on to AB at point P.
Question 6.
Draw a line segment AB = 6.5 cm. Locate a point P that is 5 cm from A and 4.6 cm from B. Through the point P, draw a perpendicular on to the line segment AB.
Properties of Angles and Lines Exercise 25D – Selina Concise Mathematics Class 6 ICSE Solutions
Question 1.
Draw a line segment OA = 5 cm. Use set-square to construct angle AOB = 60°, such that OB = 3 cm. Join A and B ; then measure the length ofAB.
Question 2.
Draw a line segment OP = 8cm. Use set-square to construct ∠POQ = 90°; such that OQ = 6 cm. Join P and Q; then measure the length of PQ.
Question 3.
Draw ∠ABC = 120°. Bisect the angle using ruler and compasses. Measure each angle so obtained and check whether or not the new angles obtained on bisecting ∠ABC are equal.
Question 4.
Draw ∠PQR = 75° by using set- squares. On PQ mark a point M such that MQ = 3 cm. On QR mark a point N such that QN = 4 cm. Join M and N. Measure the length of MN.
Properties of Angles and Lines Revision Exercise – Selina Concise Mathematics Class 6 ICSE Solutions
Question 1.
In the following figures, AB is parallel to CD; find the values of angles x, y and z :
Question 2.
In each of the following figures, BA is parallel to CD. Find the angles a, b and c:
Question 3.
In each of the following figures, PQ is parallel to RS. Find the angles a, b and c:
Question 4.
Two straight lines are cut by a transversal. Are the corresponding angles always equal?
Question 5.
Two straight lines are cut by a transversal so that the co-interior angles are supplementary. Are the straight lines parallel?
Question 6.
Two straight lines are cut by a transversal so that the co-interior angles are equal. What must be the measure of each interior angle to make the straight lines parallel to each other ?
Question 7.
In each case given below, find the value of x so that POQ is straight line
Question 8.
in each case, given below, draw perpendicular to AB from an exterior point P
Question 9.
Draw a line segment BC = 8 cm. Using set-squares, draw ∠CBA = 60° and ∠BCA = 75°. Measure the angle BAC. Also measure the lengths of AB and AC.
Question 10.
Draw a line AB = 9 cm. Mark a point P in AB such that AP=5 cm. Through P draw (using set-square) perpendicular PQ = 3 cm. Measure BQ.
Question 11.
Draw a line segment AB = 6 cm. Without using set squares, draw angle OAB = 60° and angle OBA = 90°. Measure angle AOB and write this measurement.
Question 12.
Without using set squares, construct angle ABC = 60° in which AB = BC = 5 cm. Join A and C and measure the length of AC. | {"url":"https://www.cbsetuts.com/selina-concise-mathematics-class-6-icse-solutions-chapter-25/","timestamp":"2024-11-09T23:25:27Z","content_type":"text/html","content_length":"127355","record_id":"<urn:uuid:b51ceb29-5589-4327-82ab-66f97e6e1cdb>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00521.warc.gz"} |
Data Dredging
Stumbled on to an interesting comment on crossvalidated which I think is a nice way to warn against using techniques such as best subset regression, forward step regression, backward step regression
Wanting to know the best model given some information about a large number of variables is quite understandable. Moreover, it is a situation in which people seem to find themselves regularly. In
addition, many textbooks (and courses) on regression cover stepwise selection methods, which implies that they must be legitimate. Unfortunately however, they are not, and the pairing of this
situation and goal are quite difficult to successfully navigate. The following is a list of problems with automated stepwise model selection procedures (attributed to Frank Harrell, and copied
from here):
1. It yields R-squared values that are badly biased to be high.
2. The F and chi-squared tests quoted next to each variable on the printout do not have the claimed distribution.
3. The method yields confidence intervals for effects and predicted values that are falsely narrow; see Altman and Andersen (1989).
4. It yields p-values that do not have the proper meaning, and the proper correction for them is a difficult problem.
5. It gives biased regression coefficients that need shrinkage (the coefficients for remaining variables are too large; see Tibshirani [1996]).
6. It has severe problems in the presence of collinearity.
7. It is based on methods (e.g., F tests for nested models) that were intended to be used to test prespecified hypotheses.
8. Increasing the sample size does not help very much; see Derksen and Keselman (1992).
9. It allows us to not think about the problem.
10. It uses a lot of paper.
The question is, what’s so bad about these procedures / why do these problems occur? Most people who have taken a basic regression course are familiar with the concept of regression to the mean,
so this is what I use to explain these issues. (Although this may seem off-topic at first, bear with me, I promise it’s relevant.)
Imagine a high school track coach on the first day of tryouts. Thirty kids show up. These kids have some underlying level of intrinsic ability to which neither the coach, nor anyone else, has
direct access. As a result, the coach does the only thing he can do, which is have them all run a 100m dash. The times are presumably a measure of their intrinsic ability and are taken as such.
However, they are probabilistic; some proportion of how well someone does is based on their actual ability and some proportion is random. Imagine that the true situation is the following:
1 set.seed(59)
2 intrinsic_ability = runif(30, min=9, max=10)
3 time = 31 - 2*intrinsic_ability + rnorm(30, mean=0, sd=.5)
The results of the first race are displayed in the following figure along with the coach’s comments to the kids.
Note that partitioning the kids by their race times leaves overlaps on their intrinsic ability–this fact is crucial. After praising some, and yelling at some others (as coaches tend to do), he
has them run again. Here are the results of the second race with the coach’s reactions (simulated from the same model above):
Notice that their intrinsic ability is identical, but the times bounced around relative to the first race. From the coach’s point of view, those he yelled at tended to improve, and those he
praised tended to do worse (I adapted this concrete example from the Kahneman quote listed on the wiki page), although actually regression to the mean is a simple mathematical consequence of the
fact that the coach is selecting athletes for the team based on a measurement that is partly random.
Now, what does this have to do with automated (e.g., stepwise) model selection techniques? Developing and confirming a model based on the same dataset is sometimes called data dredging. Although
there is some underlying relationship amongst the variables, and stronger relationships are expected to yield stronger scores (e.g., higher t-statistics), these are random variables and the
realized values contain error. Thus, when you select variables based on having higher (or lower) realized values, they may be such because of their underlying true value, error, or both. If you
proceed in this manner, you will be as surprised as the coach was after the second race. This is true whether you select variables based on having high t-statistics, or low intercorrelations.
True, using the AIC is better than using p-values, because it penalizes the model for complexity, but the AIC is itself a random variable (if you run a study several times and fit the same model,
the AIC will bounce around just like everything else). Unfortunately, this is just a problem intrinsic to the epistemic nature of reality itself. | {"url":"https://www.rksmusings.com/2014/08/30/data-dredging/","timestamp":"2024-11-04T02:55:16Z","content_type":"text/html","content_length":"14247","record_id":"<urn:uuid:6974969f-9ec1-4a3c-b030-fb818060d93d>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00282.warc.gz"} |
Python Numbers Clearly Explained
Numbers are a fundamental aspect of computer programming. Python provides robust support for numeric operations and data types. This lesson covers the three built-in numeric types available in
Python: integers, floating-point numbers, and complex numbers.
An integer is a whole number that can be written without a fractional or decimal component. They can be positive, negative, or zero, such as -2, -1, 0, 1, 2.
In Python, integers are represented by the int type. Python int literals are written then same way as integers are in common language, i.e. a sequence of digits, with a minus sign on the left to
represent a negative number.
positive_integer = 10
negative_integer = -5
zero = 0
Python’s int type can handle arbitrarily large integers. Unlike some other programming languages that have a fixed maximum size for integers, Python’s integers grow as needed to accommodate the
very_large_integer = 500000000000000000000000000000000000000000000000000
Floating Point Numbers
Floating-point numbers, often abbreviated as floats, are numbers that contain a decimal point. The floating-point type in Python is float.
Float literals are written in Python as either a decimal fraction, a scientific notation, or both.
approx_pi = 3.14
some_float = 10e4
another_float = 6.022e23
Python implements floating-point numbers according to the IEEE 754 standard, which defines the format for representing both floating-point numbers and their arithmetic operations. This standard
allows Python to maintain consistency in floating-point operations across different platforms and architectures.
Precision Problems With Floats
Although floating-point numbers are essential for representing a wide range of real numbers, they have limitations due to their finite precision. This limitation can lead to problems where
calculations may not produce the expected results due to rounding errors.
Floating point numbers are represented in computer memory with a finite sequence of digits. Consequently, they cannot precisely represent numbers that require an infinite sequence. The numbers that
floats can’t represent not only include irrational numbers (that require infinite non-repeating sequences of digits) such as Pi, but also rational numbers that require infinite repeating sequences,
such as 1/3, which is represented in decimal by 0.333…
However, since digital computers are based on the binary system (1’s and 0’s), the rational numbers that require an infinite repeating sequence of digits are not necessarily the same ones we are used
to. For example, multiples of one tenth in decimal (0.1) need an infinite sequences of digits in binary, which leads to unexpected behaviors.
In the example above, the floating-point numbers 0.1 and 0.2 are added, and seemingly the result of this addition is a small amount more than 0.3. This happens because it’s not really 0.1 and 0.2
that are added, but the imprecise, truncated versions of their binary representations. When the truncated numbers are added in binary and converted to decimal for printing, the result is not exactly
0.3 due to rounding errors.
Complex Numbers
A complex number is a number that has both a real part and an imaginary part, typically expressed in the form a + bj, where a is the real part, b is the imaginary part, and j (or i in mathematics) is
the imaginary unit with the property that j^2 = -1.
Complex numbers are represented in Python by the complex type. Python uses j to denote the imaginary part of a complex number literal.
Declaring and Using Complex Numbers
Declaring a complex number in Python is straightforward. You can use the complex() function or the literal notation with j.
Here is the basic syntax for declaring a complex number using the complex() function:
And this is how you can declare a complex number literal with j:
In this syntax:
• real: The real part of the complex number. This part must be an integer or float literal. It can be omitted, in which case the real part is set to 0.
• imag: The imaginary part of the complex number. This part must also be an integer or float literal.
z1 = complex(2, 3)
z2 = 2 + 3j
z3 = 0.5 + 2.1j
Underscores In Numbers
Sequences of digits in numeric literals can have underscores in between the digits. Python ignores those underscores, and they function as an optional way to improve the readability of long numbers
in source code. They are similar in purpose to commas that are placed after every third digit in written language.
The Python print() function and terminal clear out the underscores.
Arithmetic Operations
Python provides a variety of arithmetic operations that can be performed on numeric types using operators. Here are some of the most commonly used operators:
• Addition (+): Adds two numbers.
• Subtraction (-): Subtracts one number from another.
• Multiplication (*): Multiplies two numbers.
• Division (/): Divides one number by another and returns a float.
• Floor Division (//): Divides one number by another and returns an integer.
• Modulus (%): Returns the remainder of the division.
• Exponentiation (**): Raises a number to the power of another.
a = 10
b = 3
print(a + b) # Output: 13
print(a - b) # Output: 7
print(a * b) # Output: 30
print(a / b) # Output: 3.3333333333333335
print(a // b) # Output: 3
print(a % b) # Output: 1
print(a ** b) # Output: 1000
All the arithmetic operators work on int and float types. They also work on complex numbers except for // and %.
If you use an operator with two of the same numeric type, the result is of that type, except for /, which results in a float when two ints are divided. // division will result in an int.
You can use operators with two different numeric types. If one of the operands is a float the result is a float, unless one is complex, which results in a complex type.
Augmented Assignment Operators
Augmented assignment operators perform an operation and also assign the result back to the variable being operated on.
The example above demonstrates use of the += operator. It is the augmented assignment operator that performs addition. After the variable x is assigned the value 10 (line 1), the =+ operator adds 5
to it (line 2). The x variable contains 15 after the operation.
All arithmetic operators have an augmented assignment version, which are +=, -=, *=, /=, %=, //=, **=. As you can see, all these operators are characterized by having the equal sign character on the
right and the original operator on the left.
It’s also important to note that on the left-hand-side, an augmented assignment operator must have a variable, since that is what’s assigned. E.g. x += 5 will work, but 10 += 5 and 5 += x will both
generate errors.
Summary & Reference for Python Numbers
Python provides robust support for numeric operations with floating-point, and complex numbers data types.
Integers such as -2, -1, 0, 1, and 2 are represented in Python by the int type.
positive_integer = 10
negative_integer = -5
zero = 0
Python’s int type can handle arbitrarily large integers.
very_large_integer = 500000000000000000000000000000000000000000000000000
Floating-point numbers (floats), are numbers that contain a decimal point, and are represented in Python by the float type.
approx_pi = 3.14
some_float = 10e4
another_float = 6.022e23
Floats hold a limited number of binary digits and therefore cannot represent all real numbers precisely, leading to unexpected results.
0.1 + 0.2 # --> 0.30000000000000004
Complex numbers have both a real and an imaginary part, and are represented in Python by the complex type. Python literals are written with the j notation for the imaginary part.
You can also create a complex number using the complex() function.
complex(2, 3.5) # --> 2+3.5j
Sequences of digits in numeric literals can have underscores in between the digits. Those can be used to improve the readability of long numbers in source code.
print(50_000_000_000_000) # --> 50000000000000
Python provides a variety of arithmetic operators that can be used on numeric types. Those include addition (+), subtraction (-), multiplication (*), division (/), floor division (//), modulus (%),
and exponentiation (**).
All the arithmetic operators work on int and float types. They also work on complex numbers except for // and %.
Augmented assignment operators perform an operation and also assign the result back to the variable being operated on. All arithmetic operators have an augmented assignment version, which are +=, -=,
*=, /=, %=, //=, **=.
x = 10
x += 5
print(x) # --> 15 | {"url":"https://saurus.ai/python-course/python-numbers/","timestamp":"2024-11-05T19:33:18Z","content_type":"text/html","content_length":"218367","record_id":"<urn:uuid:6fcc7756-c053-4da8-aab5-a69ccc7dd8d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00509.warc.gz"} |
Why is my SUMIFS formula saying incorrect argument?
I'm working on a metrics sheet to power a dashboard. I want to sum our effort values for each of our chemists and each of our project categories.
My ranges are:
{MasterLead} : Column listing chemist on project
{MasterEffort} : Column of effort values for each project
{MasterRDCat} : Column of project categories
I am trying to use the following formula:
=SUMIFS({MasterEffort}, {MasterLead}, =$Chemist@row, {MasterRDCat}, ="PMO")
From what I understand, this formula will sum effort values from {MasterEffort} where {MasterLead} is the chemist for the row and where {MasterRDCat} is PMO.
I am able to work with these ranges when using COUNTIF, but I'm having a lot of trouble using SUMIF. Thank you!
• Is that error present in any cell being referenced by the formula?
Are you able to provide the COUNTIFS formula that you have working?
• There is not. Everything can be referenced elsewhere without issue.
Here's a corresponding COUNTIFS: =COUNTIFS({MasterLead}, =$Chemist@row, {MasterRDCat}, ="PMO")
• Double check your {MasterEffort} range to ensure that it is in fact only covering one column (and covering the full column) the same as the other two ranges.
• I double checked and it was. However, it seems to be working just fine today. I'm not sure what changed, but I'll take it.
Thanks for your help!
• It may have just been some latent data on the back-end. Glad it is working for you now though.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/104679/why-is-my-sumifs-formula-saying-incorrect-argument","timestamp":"2024-11-04T04:11:23Z","content_type":"text/html","content_length":"441566","record_id":"<urn:uuid:bf3386bf-02fd-4b4f-9a24-df36409fe29e>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00547.warc.gz"} |
An investment of €500,000 today that grows to €800,000 after six years has a stated annual interest rate closest to:
I got stated annual rate as 0,081483747
after applying EAR formula I`m getting higher values through, can somebody help?
Your EAR is correct.
How are you converting that to a continuous rate, a stated rate compounded daily, and a stated rate compounded semiannually?
I`m doing
EAR = (1+0,081483747/365)^365 than -1 and multiply by 100, got 8,48 for daily lol
For semi I got 0,083143648
for continuous got 0,084370898
non of them are the answers…
I didn’t ask what you got; I asked how you got them.
Your semiannual rate is wrong.
EAR = (1+0,081483747/365)^365 than -1 and multiply by 100, got 8,48 for daily lol
well I applyed that formula…
The rate of 8.1483747% is the EAR.
How can I get the other values than?
What’s the formula that relates EAR to the continuously compounded rate?
What’s the formula that relates EAR to stated rate?
It makes me very happy to see you doing this kind of problem by hand in order to understand the mechanics.
I know you use the HP, but the BA II has a nominal to EAR converter. Does the HP have something similar? It would save you time and grief.
I don`t think the HP12c has a function like that…
You mean the BA II you don`t need to know the EAR formula to convert?
The calculator does it all?
I dont know its the formula that is in the book for EAR…
No, it’s not.
Maybe you’re forgetting a superscript?
The only way I found to do is 500000 CHS PV 800000 FV 12N 0 PMT
asks for i = 3,9944 than i multiply for 2 get nearly 8% semianually compounded
Pretty sure there`s a alternative to do it through, was looking for it…
That’s the semiannually compounded stated rate:
Stated rate[semiannual] = [(1 + EAR)^1/2 – 1] × 2
More generally, for compounding n times per year:
Stated rate_[n]_ = [(1 + EAR)^1/n – 1] × n
Now . . . what about the continuously compounded formula?
By the way: there’s your answer: C.
In what page of the CFA curriculum is this formula? | {"url":"https://www.analystforum.com/t/an-investment-of-500-000-today-that-grows-to-800-000-after-six-years-has-a-stated-annual-interest-rate-closest-to/129682","timestamp":"2024-11-09T15:36:09Z","content_type":"text/html","content_length":"67702","record_id":"<urn:uuid:464ea7b5-de2e-4864-aa0b-71c8387d02f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00159.warc.gz"} |
JCA 28042
Journal Home Page
Cumulative Index
List of all Volumes
Complete Contents
of this Volume
Next Article
Journal of Convex Analysis 28 (2021), No. 3, 751--760
Copyright Heldermann Verlag 2021
On the Linear Structures Induced by the Four Order Isomorphisms Acting on Cvx[0](R^n)
Dan I. Florentin
Department of Mathematics, Cleveland State University, Cleveland, OH 44115-2214, U.S.A.
Alexander Segal
Afeka Academic College of Engineering, Tel Aviv 69107, Israel
It is known that the volume functional $\,\phi\mapsto\int e^{-\phi}\,$ satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order
isomorphisms acting on ${\rm{Cvx}}_0(\mathbb{R}^n)$. In this note we define the fourth linear structure on ${\rm{Cvx}}_0(\mathbb{R}^n)$ as the pullback of the standard linear structure under the $\
cal J$ transform. We show that, interpolating with respect to this linear structure, no concavity or convexity inequalities hold, and prove that a quasi-convexity inequality is violated only by up to
a factor of $2$. We also establish all the order relations which the four different interpolations satisfy.
Keywords: Convexity, interpolation, order isomorphisms, duality, Legendre transform, A-transform, J-transform.
MSC: 26B15, 26B25, 26B35, 39B62, 46B06, 52A23, 52A40, 52A41.
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Complexity of algorithms and typical errors in PythonComplexity of algorithms and typical errors in Python
Complexity of algorithms and typical errors in Python
Hello everyone! I will tell you what the complexity of algorithms is and where it comes from, I will analyze typical mistakes and the most frequent mistakes of beginners. The material is intended
primarily for beginner Python developers, as well as for those for whom Python is the first programming language.
What is the complexity of algorithms?
When talking about the complexity of algorithms, they often give an example of a graph on which functions are drawn, and say that this algorithm is equal in complexity to this function, and that
algorithm is equal to another function, and so on. Typical examples:
• Log(N) – binary search in an already sorted array;
• N – operation of the code in one cycle;
• N * Log (N) – sorting;
• N**2 is a loop nested within another loop.
Note: assembler so close to the machine code that in fact one assembly statement corresponds to one machine instruction (1:1). Therefore, it is possible to estimate the actual complexity of the
algorithm quite accurately.
Obviously, the more complex the algorithm, the faster the function grows. But what does this mean if you dig deeper? Let’s consider some algorithm with O(N) complexity on the example of C++, for
example, creating an array, and see how this operation will look in disassemblers.
You can read more about how much each operation costs in the article “Complexity of algorithms and operations on the example of Python“.
Note: C/C++ is much closer to iron than high-level Python, so it’s much easier to disassemble. Clearly, a Python list works differently from the way an array works in C++, but the principles of how
they work are exactly the same.
To create an array of 7 elements in C++:
int arr[] = {1, 2, 3, 4, 5, 6, 7};
you will need to perform 7 operations in the assembler:
mov DWORD PTR [rbp-32], 1
mov DWORD PTR [rbp-28], 2
mov DWORD PTR [rbp-24], 3
mov DWORD PTR [rbp-20], 4
mov DWORD PTR [rbp-16], 5
mov DWORD PTR [rbp-12], 6
mov DWORD PTR [rbp-8], 7
This is what the algorithm complexity function O(N) shows – how many elements need to be processed, so many operations will be processed. For O(N*N) — the same, but the number of operations will
already be squared. In Python, creating a list with 7 elements will also require at least seven operations:
l = [1, 2, 3, 4, 5, 6, 7]
Another example: to write one element of a C++ array:
arr[0] = 111;
Only one assembly operation is required:
mov DWORD PTR [rbp-32], 111
That is why this operation has O(1) complexity. Same for Python:
l[0] = 111
This operation also has O(1) complexity by the same logic.
Now that you understand where the complexity of algorithms comes from, you can move on to common mistakes.
Example 1, working with the string type
Let’s consider the simplest demonstrative example, which results from a misunderstanding of the definition of algorithm complexity. Suppose we need to collect a string in a loop, for this we can
write the following code:
line = ""
for i in range(10_000):
line += "i"
Seemingly simple and working code that produces the correct output—in this case, a string of 10,000 “i” characters. The execution time of this code 5 ms.
At first glance, this is an algorithm with O(N) complexity, but if you look at the last line of code, you can see that the type string doesn’t actually add a character to the current value but
converts the value with the new character because string is an immutable type. In other words, only one of these operations will have O(N) complexity, because to copy a value to a new string, you
need to copy each element of the old string plus the new character. In addition, this operation is also performed in a loop, that is, the final complexity of this algorithm will be O(N*N).
Note: given that the variable line grows sequentially from 0 to N, it is correct to say that the complexity is O(N*M), where M
This algorithm can be improved by using an operation whose complexity is O(1), e.g append:
word = []
for i in range(10_000):
line = "".join(word)
Now the algorithm has worked for 2 msWhich is 2.5 times faster than the previous version, besides now its complexity is O(N).
It is worth noting that the append operation in Python is actually performed by shockproof O(1) as in other high-level languages such as Java or c#. In other words, it is very rare, but this
operation can be performed in O(N) time. Under the hood, the list contains an array that stores items. If the size of the list is not enough, before saving the new item, it will be increased twice,
exactly in this case it will be O(N).
Let’s also consider an example with an assignment operation, which already really has a complexity of O(1):
N = 10_000
word = [None] * N
for i in range(N):
word[i] = "i"
line = "".join(word)
Execution time reduced to 1.35 msWhich is a little faster than with append. Both of these algorithms will be executed in approximately the same time, the variant with assignment in practice will work
more often quite a bit faster.
Example 2, array conversion
Let’s take a very large list and look for elements in it:
arr = list(range(10_000_000))
matcher = [500_000, 100_000, 1000_000_000] * 10
In our case, the array will consist of 10 million elements, and we will search for 30 numbers in it. In its simplest form, the algorithm will look like this: in a loop, we go through each of the 30
elements from matcher and we are looking for it in arr.
for i in matcher:
if i in arr:
This algorithm has O(N*N) complexity: iterating through the loop is O(N) and searching the list is also O(N). Duration of his work 1.2 seconds.
This code can be improved since we know that the set search has O(1) complexity. How can a novice programmer rewrite the algorithm? For example, like this:
for i in matcher:
if i in set(arr):
Yes, with clear thoughts, a programmer who is starting or does not know the complexity of algorithms can write like this. This example also has O(N*N) complexity, searching through a set is O(1), but
converting a list to a set is O(N). Despite the O(N*N) complexity, as in the previous example, the algorithm is now performed by 15.2 seconds. The reason is that searching the list is O(N) in the
worst case, but the conversion will always be O(N).
It would be correct to write like this:
arr_s = set(arr)
for i in matcher:
if i in arr_s:
Now the code is executed 0.5 seconds. This error occurs surprisingly often, it is invisible, but it can slow down the algorithm and increase memory consumption. It is important to understand that
unnecessary transformation of an array of elements consumes unnecessary resources.
Typical examples with conversion errors:
• set(list(map(str, arr))) – It is necessary to immediately lead to a multitude: set(map(str, arr))
• list(sorted(arr)) – sorted and so returns a list
And there can be many other errors and variations.
Example 3, errors when declaring variables
It is not for nothing that they say that it is desirable to declare variables at the beginning of a method. This is of course a recommendation, but it is useful for beginners. Often, an inexperienced
programmer can make a similar mistake: suppose we have some method that performs a division operation:
def divider(a: int, b: int):
res = a / b
return res
Suppose that a QA engineer (aka tester) in the team found a bug in this code: he noticed that it was not divisible by zero, raised a bug, and a novice programmer fixed the code:
def divider(a: int, b: int):
res = a / b
except ZeroDivisionError:
print("don't divide on zero!")
return res
Yes, this is quite a common situation when a bug is masked by another error. In this case, division by zero will be processed correctly, but an error will immediately appear: UnboundLocalError: local
variable 'res' referenced before assignmentthat is, a reference to a non-existent variable, since it will not be initialized in the block try/except due to division error.
The easiest way to avoid this is to declare variables in advance:
def divider(a: int, b: int):
res = None
res = a / b
except ZeroDivisionError:
print("don't divide on zero!")
return res
Often variables are declared in a segment ifcode like this is not a good tone:
def divider(a: int, b: int, flag: bool):
if flag:
res = a / b
return res
It is possible to declare such a variable in Python, but in Java or C# it will not work. Such code in these programming languages causes an error at the compilation stage.
Example 4, input parameters
There is a well-known recommendation: do not change the input parameters. Why they say so, let’s see.
def func(l: list):
l[0] = 10
l = [1, 2, 3]
Seemingly simple code, but what will lie in the list after it is executed? And here’s what:
[10, 2, 3]
How did it happen? After all, the method does not return anything, we do not redefine the list in the code, only inside the method. Consider another example:
def func2(a: int):
a = 10
a = 0
Why will the variable be equal to:
In the second case, the variable has not changed, why is this happening? The fact is that objects in Python are divided into those that are passed by reference. list, set, dictand those passed by
value int, float, str, tuple.
Let’s consider another example:
l1 = [1, 2]
l2 = l1
In this case, both lists will be equal, even though we only changed one of them:
print(l1, l2)
([1, 2, 3], [1, 2, 3])
When passing arguments to a method, or if you want to save the initial state of the list so that there are no errors, the reference types must be copied explicitly, for example:
l2 = l1[:]
l2 = list(l1)
l2 = l1.copy()
In this case, the references will be to different memory areas, which will solve the problem.
Example 5, default value
This is probably one of the most common mistakes a beginner programmer makes. Consider an example:
def func3(val: int, l: list = []):
It seems like a normal method, but let’s call it several times in a row and see what happens:
[1, 2]
Yes, this is the catch: a new empty list is not created by default. That is why in subsequent calls, in which the programmer expects an empty array, as was laid down by the logic of the algorithm,
elements from past calls to the method remain in the array. This is how Python already works: it always takes the same reference to the list in such a case.
If you want an empty list by default, you should write like this:
def func3(val: int, l: Optional[list] = None):
if not l:
l = []
The same rule applies to the set (set) and dictionary (dict), they also cannot be made empty in the default options.
In simple words and using examples, I explained what the complexity of an algorithm is, showed where this concept comes from, analyzed the typical mistakes that result from a misunderstanding of the
complexity and are the most common among beginners.
Don’t be afraid to make mistakes if you are a beginner developer. Only the one who does nothing is not wrong.
Thank you for your attention.
The source code repository is on GitHub. | {"url":"https://pro-wp.in.ua/complexity-of-algorithms-and-typical-errors-in-python/","timestamp":"2024-11-10T01:34:06Z","content_type":"text/html","content_length":"79438","record_id":"<urn:uuid:34c39295-ec16-4193-a2c7-f66324a67fe1>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00787.warc.gz"} |
Theory and Applications of Computational Chemistry
The First Forty Years
• 1st Edition - October 13, 2011
• Editors: Clifford Dykstra, Gernot Frenking, Kwang Kim, Gustavo Scuseria
• eBook ISBN:
9 7 8 - 0 - 0 8 - 0 4 5 6 2 4 - 9
Computational chemistry is a means of applying theoretical ideas using computers and a set of techniques for investigating chemical problems within which common questions vary from… Read more
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Computational chemistry is a means of applying theoretical ideas using computers and a set of techniques for investigating chemical problems within which common questions vary from molecular geometry
to the physical properties of substances. Theory and Applications of Computational Chemistry: The First Forty Years is a collection of articles on the emergence of computational chemistry. It shows
the enormous breadth of theoretical and computational chemistry today and establishes how theory and computation have become increasingly linked as methodologies and technologies have advanced.
Written by the pioneers in the field, the book presents historical perspectives and insights into the subject, and addresses new and current methods, as well as problems and applications in
theoretical and computational chemistry. Easy to read and packed with personal insights, technical and classical information, this book provides the perfect introduction for graduate students
beginning research in this area. It also provides very readable and useful reviews for theoretical chemists.
* Written by well-known leading experts
* Combines history, personal accounts, and theory to explain much of the field of theoretical and compuational chemistry
* Is the perfect introduction to the field
Graduate students and researchers in chemistry and theoretical chemistry
Computing Technologies, Theories, and Algorithms. The Making of 40 Years and More of Theoretical and
Computational Chemistry (C.E. Dykstra et al.).
A Dynamical, Time-Dependent View of Molecular Theory (Y. Öhrn, E. Deumens).
Computation of Non-covalent Binding Affinities (J. A. McCammon).
Electrodynamics in Computational Chemistry
(Linlin Zhao et al.).
Variational Transition State Theory (B.C. Garrett, D.G. Truhlar).
Attempting to Simulate Large Molecular Systems (E. Clementi).
The Beginnings of Coupled Cluster Theory: An Eyewitness Account (J. Paldus).
Controlling Quantum Phenomena with Photonic Reagents
(H. Rabitz).
First-Principles Calculations of Anharmonic Vibrational Spectroscopy of Large Molecules
(R.B. Gerber et al.).
Finding Minima, Transition States, and Following Reaction Pathways on Ab Initio Potential Energy Surfaces (H.P. Hratchian, H.B. Schlegel).
Progress in the Quantum Description of Vibrational Motion of Polyatomic Molecules (J.M. Bowman et al.).
Toward Accurate Computations in Photobiology
(A. Sinicropi, M. Olivucci).
The Nature of the Chemical Bond in the Light of an Energy Decomposition Analysis (M. Lein, G. Frenking).
Superoperator Many-body Theory of Molecular Currents: Non-equilibrium Green Functions in Real Time (U. Harbola, S. Mukamel).
Role of Computational Chemistry in the Theory of Unimolecular Reaction Rates (W.L. Hase, R. Schinke).
Molecular Dynamics: An Account of its Evolution (R. Kapral, G.I. Ciccotti).
Equations of Motion (EOM) Methods for Computing Electron Affinities and Ionization Potentials
(J. Simons).
Multireference Coupled Cluster Method Based on the Brillouin-Wigner Perturbation Theory
(P. Carsky et al.).
Electronic Structure: The Momentum Perspective
(A.J. Thakkar).
Recent Advances in ab initio, DFT, and Relativistic Electronic Structure Theory (Haruyuki Nakano et al.).
Semiempirical Quantum-Chemical Methods in Computational Chemistry (W. Thiel).
Size-consistent State-specific Multi-reference Methods: A Survey of Some Recent Developments
(D. Pahari et al.).
The Valence Bond Diagram Approach - A Paradigm for Chemical Reactivity (S. Shaik, P.C. Hiberty).
Development of Approximate Exchange-Correlation Functionals (G.E. Scuseria, V.N. Staroverov).
Multiconfigurational Quantum Chemistry (B.O. Roos).
Concepts of Perturbation, Orbital interaction, Orbital Mixing and Orbital Occupation (Myung-Hwan Whangbo).
G2, G3 and Associated Quantum Chemical Models for Accurate Theoretical Thermochemistry (K. Raghavachari, L.A. Curtiss).
Factors that Affect Conductance at the Molecular Level (C.W. Bauschlicher, Jr., A. Ricca).
The CH˙˙O Hydrogen Bond. A Historical Account (S. Scheiner).
Ab Initio and DFT Calculations on the Cope Rearrangement, a Reaction with a Chameleonic Transition State (W. Thatcher Borden).
High-Temperature Quantum Chemical Molecular Dynamics Simulations of Carbon Nanostructure Self-Assembly Processes (S. Irle et al.).
Computational Chemistry of Isomeric Fullerenes and Endofullerenes (Z. Slanina, S. Nagase).
On the importance of Many-Body Forces in Clusters and Condensed Phase (Krzysztof Szalewicz et al.).
Clusters to Functional Molecules, Nanomaterials, and Molecular Devices: Theoretical Exploration
(Kwang S. Kim et al.).
Monte Carlo Simulations of the Finite Temperature Properties of (H2O)6 (R.A. Christie, K.D. Jordan).
Computational Quantum Chemistry on Polymer Chains: Aspects of the Last Half Century (J-M. André).
Forty Years of Ab Initio Calculations on Intermolecular Forces (P.E.S. Wormer, Ad van der Avoird).
Applied Density Functional Theory and the deMon Codes 1964 to 2004 (D.R. Salahub et al.).
SAC-CI Method Applied to Molecular Spectroscopy (M. Ehara et al.).
Forty Years of Fenske-Hall Molecular Orbital Theory
(C.E. Webster, M.B. Hall).
Advances in Electronic Structure Theory: GAMESS a Decade Later (M.S. Gordon, M.W. Schmidt).
How and Why Coupled-Cluster Theory Became the Preeminent Method in Ab initio Quantum Chemistry
(R.J. Bartlett).
• Published: October 13, 2011
• Imprint: Elsevier Science
• eBook ISBN: 9780080456249
Clifford Dykstra
Affiliations and expertise
Indiana University - Purdue University, Indianapolis, USA
Gernot Frenking
Affiliations and expertise
Fachbereich Chemie, Philipps-Universität Marburg, Germany
Kwang Kim
Affiliations and expertise
Department of Chemistry, Pohang University of Science and Technology, Korea
Gustavo Scuseria
Affiliations and expertise
Department of Chemistry, Rice University, Texas, USA | {"url":"https://shop.elsevier.com/books/theory-and-applications-of-computational-chemistry/dykstra/978-0-444-51719-7","timestamp":"2024-11-12T23:59:29Z","content_type":"text/html","content_length":"192589","record_id":"<urn:uuid:34e9b6e8-7578-4123-a7a7-32065aa1e2fb>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00391.warc.gz"} |
Finite Series are Expressed in Terms of N-th Partial Sum of Hurwitz Zeta Function
Submitted by admin on Sat, 11/07/2009 - 11:37pm
Brief Note - Introduction a generalization form of n-partial sum of Hurwitz Zeta Function.
The n-th partial sums of the expressions (I) and (II) below are found in closed-form for each positive integer n and real x such that x ≠ -k and x ≠ -(k+1).
Special values
• When n tends to infinity, (I) is reduced to a simple form
Let x = 0,
. (Correct - 11/17/2009)
At x = 1,
• It is easy to verify that for n = 2, both sides of (I) are equal for all x ≠ - 1, -2, and -3, namely
A Generalization Form of n-th Partial Sum of Hurwitz Zeta Function
Recall that Hurwitz Zeta Function [1] is defined for complex arguments s and a by
where Re(s) > 1 and Re(a) > 0.
We now define a new generalization form of n-th partial sum of Hurwitz zeta function above as follows:
Based on this new define, (I) and (II) are then rewritten in terms of n-th partial sum of Hurwitz zeta function for Re[a] = x, R[s] = 2, and each positive integer n as shown in the following:
The identities (III) and (IV) are also true for complex number a by replacing x = a.
(November 07, 2009)
(Update formula syntax and definition - December 05, 2009)
Other related series
50 Identities of Power Summation
[1] http://en.wikipedia.org/wiki/Hurwitz_zeta_function
In-Text or Website Citation
Tue N. Vu, Finite Series are Expressed in Terms of N-th Partial Sum of Hurwitz Zeta Function, from Series Math Study Resource. | {"url":"http://seriesmathstudy.com/sms/nov_07_2009","timestamp":"2024-11-08T04:22:12Z","content_type":"application/xhtml+xml","content_length":"25031","record_id":"<urn:uuid:864c6ce6-7105-46dc-9b79-67d5eb5c273f>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00440.warc.gz"} |
Stiffness improvement methods and its application on design and optimization of large lens hood for space camera
As an essential part of optical system, lens hoods consisting of shells and plates are designed mainly to protect optical system from unexpected light. Except for such basic function, the hood
proposed in this paper has to load several subassemblies and will be applied in high-resolution and wild field-of-view space camera with strict mass limitation, which makes effective mechanical
reinforcement and lightweight of hood quite necessary. To meet such requirements, stiffness improvement methods is promoted in this paper to help improve the constraint fundamental frequency and
decide the areas where subassemblies can be placed. Subsequently, optimization on key sizes of the hood proceeds to achieve a higher fundamental frequency and a lower weight. Finally, a prototype is
fabricated based on the optimal design and a sweep test is held to verify the analytical fundamental frequency. The prototype has large external dimensions (1960×1640×2055 mm) but weighs only 33.5 kg
when loading several subassemblies which totally weight 15.95 kg. Sweep test indicates that experimental constraint fundamental frequency is 36.82 Hz. All the parameters coincide with that of
theoretical design well. The whole work of this paper provides a worthwhile method to the design of lens hood with large external dimensions and high specific stiffness in space camera. Since the
lens hood is a typical plate and shell structure, the method, design and optimization process in this paper may also be helpful to the plate and shell structure in which stiffness and lightweight are
highly required.
1. Introduction
Lens hood, which is mainly used to protect optics from unwanted light, plays an indispensable role in space camera. For the optical remote sensing satellite with high resolution and Wild
Field-Of-View (WFOV) researched in this paper, the design of lens hood becomes more complicated. Owing to the expensive launching cost and extreme launching condition, lightweight and stiffness
improvement become higher demanded [1, 2]. However, lightweight and stiffness are on the contrary to each other since increment of one will always lead to decrement of the other, so reasonable
balance between them two becomes a challenging work.
The lens hood usually employs two typical forms. The first type integrates the hood and support structure, it is comparatively small and commonly seen in small and medium space camera since it will
be comparatively easy to fabricate and to satisfy the optical and mechanical requirements when applying on such space camera. For example, the carbon fiber composite framework for small lightweight
space camera designed by Shuai Yang et al., applies such type and it can be regarded as support structure as well as lens hood [3]. The second type is only used for light-shading and usually made up
of thin Carbon Fiber Reinforced Polymer (CFRP) shells and plates. The hood researched in this paper belongs to this type. Generally speaking, hoods of such type are commonly applied in optical
systems which maintain large external dimensions. They are usually flexible since they are not designed to support anything. However, the hood researched in this paper is required to load
subassemblies (totally weight 15.95 kg) and weights no more than 33 kg, so stiffness becomes more required to ensure no only the hood’s but also the subassemblies’ safety during extreme launching
Former researches mostly concerned about the optical performances of lens hood. e.g., Moldabekov M. et al. developed a new approach for optical head of the star tracker and its hood for satellites to
meet optical requirement of Kazakhstan’s satellites [4]. Yan Peipei et al. from Xi’an Institute of Optics and Precision Mechanics optical system applied stray light elimination to design a lens hood
for three-mirror optical system [5]. Zhong-Shu Chiang et al. applied Point Source Transmittance (PST) to evaluate the shielding efficiency of a lens hood and achieved a standard for a two stage lens
hood [6]. Generally speaking, former researchers cared more about the optical feasibility of the hood while mechanical feasibility was less concerned because the hoods mentioned above are not
required to load heavy subassemblies.
Considering particularities of the hood researched in this paper, this paper takes mechanical feasibility more seriously than ever. A stiffness improvement method is promoted to help reinforce hood,
improve the fundamental frequency and decide how to place all satellite subassemblies. As a result, the final constraint fundamental frequency is improved 2.4 times and the weight decreases 11 kg
compared with initial structure. The whole work of this paper provides a worthwhile method to design of lens hood in space camera as well as design and optimization of structures which are made of
shell and plate.
The following sections (Section 2-5) present details of the work in this paper. Section 2 mainly elaborates the design of initial structure of the lens hood and corresponding design basis. Section 3
mainly focuses on redesign and reinforce of the initial structure, stiffness improvement methods is promoted in this section to help accomplish the task and decide the placement of satellite
subassemblies. Section 4 expounds an optimization which aims at lightening the hood and further improving its natural frequency, final structure of the hood is achieved in this section. Section 5
introduces the detail of sweeping test and the results of the test coincide with the analytical results well.
2. Initial lens hood
2.1. Initial lens hood structure design
Since light-shading ability is the main optical requirement for lens hood, features of optics and light path are mainly concerned during initial lens hood structure design. Taking assembly
interference and machinability into account as well, the basic lens hood structure is carried out and given in Fig. 1.
CFRP is applied to manufacture the whole lens hood since it has higher specific stiffness and specific strength compared with most metallic materials. What is more, CFRP can be applied in some
complicate structure due to its special filament winding technology [7]. In this paper the chosen CFRP is T700 whose elasticity modulus is 50 GPa and Poisson’s ratio is 0.3. At last, the initial
structure has an external dimension of 1960×1640×2055 mm.
Stray light analysis has confirmed validity of this structure. Fig. 2 shows that most stray light out of view field $Y$: 4.2°–5.9°, $X$:–9.0°–8.8° has a PST lower than 10e-4. The required view field
of the optical system is 4.7°–5.4° along $Y$ and –7.9°–7.9° along $X$, thus the result satisfies demand of optical system in the view field. However, mechanical performance is ignored at this stage.
Fig. 1Semisectional view of initial lens hood and optic assemblies
Fig. 2PST curve of initial lens hood
2.2. Analysis of initial structure
Finite Element (FE) model of the lens hood is built in HYPERMESH and the initial thickness of the whole hood is set at 1.5 mm for the normal modes analysis, the constraint fundamental frequency (the
bottom assembled on back frame is constrained to simulate working state) of hood is only 14.8 Hz and the initial hood weights 44 kg, which indicates that the whole hood structure is too flexible.
Since several devices which weight 0.5-2 kg will be considered to assemble on the lens hood, too flexible structure may lead to catastrophic destruction to both hood and subassemblies during extreme
lunching process. Therefore, stiffness increment for the whole initial lens hood structure should proceed next.
3. Redesign and reinforce of initial structure
It is a challenge to redesign and reinforce the initial structure without big changes. Generally speaking, fundamental frequency can reveal how stiff the whole structure is, therefore, redesign and
reinforcement center on how to improve fundamental frequency in this paper.
3.1. Stiffness improvement methods
It is well known that a certain continuum structure is actually an infinite multiple-degree of freedom system. FEM method is an approximate assumption which changes such infinite-degree-of-freedom
system into a finite multiple-degree of freedom system [8]. The approximate analysis model for a structure with n degree can be described as:
where $M$ stands for mass matrix, $C$ stands for damping matrix, $K$ stands for stiffness matrix, $F$ stands for external load and $x$ stands for displacement. $M$, $C$, $K$ are $n×n$ symmetrical
matrixes while $x$ and $f$ are $n×1$ arrays.
The $i$th mode ${\varphi }_{i}$ and frequency ${\omega }_{i}$ is determined by equations below [9]:
$（K-{\lambda }_{i}M）{\varphi }_{i}=0,$
where ${\lambda }_{i}={\omega }_{i}^{2}$ and select ${\varphi }_{i}$which makes ${\varphi }_{i}^{T}M{\varphi }_{i}=$ 1, besides, let ${\lambda }_{1}\le {\lambda }_{2}\le \dots \le {\lambda }_{n}$.
We assume tiny increment is given in $m$th rows and $m$th columns of matrices $K$ and $M$ which can be described as $∆K$ and $∆M$, corresponding tiny changes of ${\lambda }_{i}$ and ${\varphi }_{i}$
can be described as $∆{\lambda }_{i}$ and ${∆\varphi }_{i}$. The changed model can be described as:
$（{K}^{"}-{\lambda }_{i}^{\text{'}}{M}^{\text{'}}）{\varphi }_{i}^{\text{'}}=0.$
${K}^{"}=K+\mathrm{\Delta }K,$
${M}^{"}=M+\mathrm{\Delta }M,$
${\lambda }_{i}^{"}={\lambda }_{i}+{\mathrm{\Delta }\lambda }_{i},$
${\varphi }_{i}^{"}={\varphi }_{i}+\mathrm{\Delta }{\varphi }_{i}.$
Taking Eq. 4(a)-4(d) into Eq. (3) and neglecting second and third terms gives:
$\left(K-{\lambda }_{i}M\right)\mathrm{\Delta }{\varphi }_{i}+\left(\mathrm{\Delta }K-{\lambda }_{i}\mathrm{\Delta }M-{\mathrm{\Delta }\lambda }_{i}M\right){\varphi }_{i}=0.$
Premultiplication of Eq. (5) by ${\varphi }_{i}^{T}$ gives [10]:
${\varphi }_{i}^{T}\left(K-{\lambda }_{i}M\right)\mathrm{\Delta }{\varphi }_{i}+{\varphi }_{i}^{T}\left(\mathrm{\Delta }K-{\lambda }_{i}\mathrm{\Delta }M-{\mathrm{\Delta }\lambda }_{i}M\right){\
varphi }_{i}=0.$
Since each mode is linear independent, $\mathrm{\Delta }{\varphi }_{i}$ can be described by mode space $\varphi \left(\varphi =\left[{\varphi }_{1},{\varphi }_{2},\dots ,{\varphi }_{n}\right]\right)$
$\mathrm{\Delta }{\varphi }_{i}=\varphi {\psi }_{i}.$
Considering that ${\varphi }_{i}$ and ${\varphi }_{j}\left(ie j\right)$ are orthogonal with respect to $M$ and $K$:
${\varphi }_{i}^{T}\left(K-{\lambda }_{i}M\right)\mathrm{\Delta }{\varphi }_{i}={\varphi }_{i}^{T}\left(K-{\lambda }_{i}M\right)\varphi {\psi }_{i}=0.$
Substitution Eq. (8) into Eq. (7) gives:
${\mathrm{\Delta }\lambda }_{i}={\varphi }_{i}^{T}\left(\mathrm{\Delta }K-{\lambda }_{i}\mathrm{\Delta }M\right){\varphi }_{i}={\sum }_{j=1}^{n}{\sum }_{k=1}^{n}{\varphi }_{ji}\left({∆K}_{jk}-{\
lambda }_{i}∆{M}_{jk}\right){\varphi }_{ki}.$
Since tiny increment is only given in $m$th rows and $m$th columns of matrices $K$ and $M$, which means:
${∆K}_{jk}=∆{M}_{jk}=0,je m,ke m.$
Substitution Eq. (10) into Eq. (9) gives:
${\mathrm{\Delta }\lambda }_{i}={\sum }_{j=1}^{n}{\varphi }_{ji}\left({∆K}_{jm}-{\lambda }_{i}∆{M}_{jm}\right){\varphi }_{mi}+{\sum }_{k=1}^{n}{\varphi }_{mi}\left({∆K}_{mk}-{\lambda }_{i}∆{M}_{mk}\
right){\varphi }_{ki}$$-{\varphi }_{mi}\left({∆K}_{mm}-{\lambda }_{i}∆{M}_{mm}\right){\varphi }_{mi}.$
Obviously, ${K}^{"}$, ${M}^{"}$ should both be symmetrical matrixes in regard to corresponding changed structure and at most time matrix $M$ and ${M}^{"}$ are diagonal matrixes in FEM analysis, which
means ${M}_{jm}={M}_{jm}^{"}=∆{M}_{jm}=0\left($if$je m\right)$, this conclusion leads to:
${\mathrm{\Delta }\lambda }_{i}={\sum }_{j=1,je i}^{n}{\varphi }_{ji}{∆K}_{jm}{\varphi }_{mi}+{\varphi }_{mi}\left({∆K}_{mm}-{\lambda }_{i}∆{M}_{mm}\right){\varphi }_{mi}.$
We can know that:
$\frac{d{\lambda }_{i}}{d{K}_{jm}}=2{\varphi }_{ji}*{\varphi }_{mi},\left(je m\right),$
$\frac{d{\lambda }_{i}}{d{K}_{mm}}={\varphi }_{mi}*{\varphi }_{mi},$
$\frac{d{\lambda }_{i}}{d{M}_{mm}}=-{\lambda }_{i}*{\varphi }_{mi}*{\varphi }_{mi}.$
From Eq. (13)-(15), it can be concluded for $i$th mode that:
The larger ${\varphi }_{ji}\mathrm{a}\mathrm{n}\mathrm{d}{\varphi }_{mi}\left(j,m=1,2,\dots ,n\right)$ is, the more seriously ${K}_{jm}$ affects ${\lambda }_{i}$. Increment of ${K}_{jm}$ will bring
more benefit to ${\lambda }_{i}$.
The larger ${\lambda }_{i}$and ${\varphi }_{mi}$ is, the more seriously ${M}_{mm}$ affects ${\lambda }_{i}$. Increment of ${M}_{mm}$ will bring more decrement to ${\lambda }_{i}$.
From the conclusions, effective ways to improve the natural frequency of the whole structure and areas to place subassemblies can be easily achieved. All of them are called stiffness improvement
methods by the author and the details are described below:
a) Improving the stiffness of partial area where large modal displacement happens can more effectively improve the stiffness of whole structure.
b) Improve stiffness of areas mentioned in Eq. (1) while reduce the mass increase of these areas as much as possible. Partial structure and material change are the best ways to achieve this goal
since structure and material change will bring less mass increase.
c) Mass increase better happens in areas where low modal displacement occurs to ensure lower stiffness decrease.
3.2. Modal analysis and reinforce of initial structure
Modal analysis can reveal the ‘weaknesses’ of the initial structure. From the last section, it can be known that the ‘weaknesses’ usually lies in areas where large modal displacement happens.
Stiffness reinforcement of these areas is the most effective way according the stiffness improvement methods. To locate these areas, this paper selects the first 8 modes (range from 14.8 Hz to 32.6
Hz) of the whole initial constraint structure and summarizes the weak areas, typical modal displacements are shown in Fig. 3 and weak areas are shown in Fig. 4.
According to stiffness improvement methods, structure and material change are considered as the best way to improve the stiffness since they will bring less mass increase. Considering that the whole
hood has selected CFRB which maintains high specific stiffness as its manufacturing material, the remaining ideal way is structure change of weak areas. However, the fabrication methods of carbon
fiber are mostly filament winding with mould, over-complicated partial structure will make the fiber directions irregular and bring out discontinuity during fiber winding [7]. Discontinuous fibers
will weaken the area and bring out hidden danger to the whole hood. Considering that bonding technology with resin is rather common in carbon fiber manufacture process, this paper applies hat beams
which are adhered on the weak areas to improve stiffness of these areas. The reinforce solution is given and colored yellow in Fig. 5, and the initial sectional dimensions of the hat beam are given
in Fig. 6.
Fig. 3Typical modal displacement figures and the corresponding modal order
Fig. 4Weak areas of the whole lens hood
Fig. 5Reinforce solution with hat beam
The constraint natural frequency of reinforced structure has a significant increase and it reaches up to 33.0 Hz, which is more than twice the original constraint natural frequency. Modal
displacement of the reinforced structure is given in Fig. 7. The total mass of the reinforced structure is 50.4 kg.
Fig. 6Sectional dimensions of the hat beam
Fig. 7Modal displacement of reinforced structure
3.3. Placement of satellite subassemblies
Generally speaking, most traditional lens hoods only load some light antennas since they are usually too flexible to load heavy subassemblies. However, the reinforced structure has a considerable
constraint natural frequency increase which means it is enough stiff, so the author decides to put some heavier satellite subassemblies as well as antennas on it.
Fig. 8Placement of satellite subassemblies
Subassemblies which are to be assembled on hood comprise GPS antenna, Telemetry, Track and Command (TT&C) antenna and data transmission devices (include data transmission antenna and data
transmission equipment). Assembly of them can be considered as mass increase in assembly areas of the hood. According to the stiffness improvement methods, these subassemblies should be better
assembled on areas which perform low modal displacement. Besides, considering that all kinds of antennas are sensitive to cover on their signal transmission paths, they should better be assembled on
open areas. Based on such two principles, the final placement of satellite subassemblies is carried out and given in Fig. 8. With satellite subassemblies, the natural frequency of the lens hood is
31.4 Hz, and the total mass of lens hood and satellite subassemblies is 66.35 kg.
4. Optimization of reinforced structure
Lightweight is highly required for satellite assemblies, and the whole lens hood is required to be no more than 48.95 kg with satellite subassemblies. So, the author conducts an optimization which
aims at further improving its natural frequency under the constraints of the hood’s weight and thickness of the shells [11]. The optimization can be described as:
• Objective function: Maximize: $f=g$ (T1, T2, T3, …, T10),
• Constraints: $M\le$ 48.95kg (with satellite subassemblies),
• T1, T2, T3, …, T10 = 0.1*$n$ ($n$ is an integer and $n\ge$ 8).
Where, T1-T10 stand for the thickness of beams and shells and the details are given in Table 1 (Top $I$ stands for Top hat beam $I$, the same goes for the rest items). f is the constraint natural
frequency of the whole lens hood (with satellite subassemblies) and it is determined by T1-T10 through function $g$, $M$ is the total weight of lens hood. Due to the fabrication process of carbon
fiber, the thickness of the shell can only be integer times of 0.1 mm and the shell thickness should be no less than 0.8 mm. Initial thicknesses are set to be 1.5 mm as the input of the optimization
for all the shell types.
Table 1The corresponding shell types of T1-T10
Name T1 T2 T3 T4 T5 Name T6 T7 T8 T9 T10
Shell Top Shell
Top Ⅱ Top Ⅲ Under I Under Ⅱ Side Upper I Upper Ⅱ Optical stop Rest
type I type
The optimization converged after 12 iterations. The optimized natural frequency is 36.5 Hz and lens hood weights 48.95 kg with satellite subassemblies. The iteration history is shown in Fig. 9 and
the final results are given in Table 2. The modal displacement after optimization is given in Fig. 10. It is quite obvious that the ratio of frequency to mass performs a big progress after the
optimization. (1.77 times as much as the reinforced structure).
Fig. 9Iteration history of the optimization
Table 2Initial size and optimized size of the lens hood
Top Top Top Under Under Side
beam Ⅰ beam Ⅱ beam Ⅲ beam Ⅰ beam Ⅱ beam
Initial size (mm) 1.5 1.5 1.5 1.5 1.5 1.5
Optimized size (mm) 1.0 0.8 2.0 1.4 0.8 0.8
Item Upper beam Ⅰ Upper beam Ⅱ Optical stop The rest Lens hood mass Natural frequency
Initial size (mm) 1.5 1.5 1.5 1.5 50.4 kg 33.0 Hz
Optimized size (mm) 0.8 0.8 1.7 0.8 33.0 kg 36.5 Hz
Fig. 10Modal displacement of optimized structure
5. Sweep test of the lens hood
According to the optimized results, the lens hood prototype is fabricated and weights 49.45 kg with satellite subassemblies. To verify the constraint natural frequency of the fabricated lens hood,
frequency sweeping experiments along $x$-axis (nearly perpendicular to side plane) and $y$-axis (nearly perpendicular to upper plane) of the lens hood are held respectively. The lens hood prototype
and test ground are given in Fig. 11. Points 1-4 in Fig. 11 are chosen as the mounting points of acceleration sensors since the biggest modal displacements of each surface occur near the points in
analysis. The bottom of the lens hood is constrained on the testing plane and all the satellite subassemblies are also firmly assembled on the hood. The sweeping results are given in Fig. 12 and Fig.
Fig. 11Lens hood prototype and frequency sweeping testing ground along x-axis
From Fig. 12 and Fig. 13 it’s clear that the fist-order resonance frequency is 36.82 Hz along y-axis and 43.6 Hz along $x$-axis respectively. It can be easily concluded that direction of the
first-order modal displacement is along $y$-axis. The testing result (36.82 Hz) is only a little higher than the analysis result (36.5 Hz) mentioned in Section 3, they match each other well.
Fig. 12Frequency sweeping result along y
Fig. 13Frequency sweeping result along x
6. Conclusions
This paper has designed a lens hood with high stiffness and ability to load 15.95 kg satellite subassemblies. Initial structure is designed based on the external dimensions of the optical system. PST
analysis shows its optical feasibility while FEM analysis shows its mechanical inadequacy. In response, a stiffness improvement methods is promoted to reinforce the initial structure. Subsequently,
an optimization at key sizes is held to more improve the constraint frequency and decrease the weight. At last, a lens hood prototype is fabricated according to the design and frequency sweeping
tests is held on it. The test results have confirmed the correctness of whole design and method. The work of this paper is meaningful to the shell and plate structure in which high stiffness is
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About this article
Modal analysis and applications
stiffness improvement
lens hood
space camera
constraint fundamental frequency
Author Contributions
Xiaoxue Gong developed the stiffness improvement theory and is the main designer of the lens hood. Lei Zhang checks Mr. Gong’s work and given some important advises on theory and structure design.
Lei Wei assists Mr. Gong to complete the design and mainly completes the confirmatory tests. Xuezhi Jia is on duty of the manufacturing process of the lens hood. Ming Xuan checks all the work and
gives some corrections on theory.
Copyright © 2018 Xiaoxue Gong, et al.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/19131","timestamp":"2024-11-11T23:58:47Z","content_type":"text/html","content_length":"158266","record_id":"<urn:uuid:e136455b-6fab-45a9-8b18-6107a3aba794>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00085.warc.gz"} |
Fix k > 0, and let G be a graph, with vertex set partitioned into k subsets (``blocks"") of approximately equal size. An induced subgraph of G is ``transversal"" (with respect to this partition) if
it has exactly one vertex in each block (and therefore it has exactly k vertices). A ``pure pair"" in G is a pair X, Y of disjoint subsets of V (G) such that either all edges between X, Y are present
or none are; and in the present context we are interested in pure pairs (X, Y ) where each of X, Y is a subset of one of the blocks, and not the same block. This paper collects several results and
open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded.
All Science Journal Classification (ASJC) codes
• induced subgraphs
• pure pairs
• trees
Dive into the research topics of 'PURE PAIRS. IX. TRANSVERSAL TREES'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/pure-pairs-ix-transversal-trees","timestamp":"2024-11-09T23:43:50Z","content_type":"text/html","content_length":"48145","record_id":"<urn:uuid:69f3f7da-4554-42e9-a0a2-325fc89df1e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00411.warc.gz"} |
How do you calculate required rate of return using npv
9 Apr 2019 Generally, NPV can be calculated with the formula NPV = ⨊(P/ (1+i)t ) In fact, this means that the juicer made you the required return rate of 4% 17 Mar 2016 There are a variety of
methods you can use to calculate ROI — net present value, payback, breakeven — and internal rate of return, or IRR.
To calculate the net present value, you will need to subtract the initial investment from the result you get from the NPV function. Lets take an example to demonstrate this function. Assume that you
started a business with an initial investment of $10,000 and received the following income for the next five years. If your discount rate is close to the actual return rate you can get on your money
for an alternative investment of similar risk and your future cash inflows are close to the amounts of money you'll actually make from your investment, your NPV … NPV formula. If you wonder how to
calculate the Net Present Value (NPV) by yourself or using an Excel spreadsheet, all you need is the formula: where r is the discount rate and t is the number of cash flow periods, C 0 is the initial
investment while C t is the return during period t. In this tutorial, you will learn how to use the Excel NPV function to calculate net present value of an investment and how to avoid common errors
when you do NPV in Excel. Net present value or net present worth is a core element of financial analysis that indicates whether a project is going to be profitable or not. So, what discount rate
should you use when calculating the net present value? One easy way to think about the discount rate is that it’s simply the required rate of return that you want to achieve. The discount rate is
what you want, the IRR is what you get, and the NPV quantifies the difference. By Mark P. Holtzman . Most capital projects are expected to provide a series of cash flows over a period of time.
Following are the individual steps necessary for calculating NPV when you have a series of future cash flows: estimating future net cash flows, setting the interest rate for your NPV calculations,
computing the NPV of these cash flows, and evaluating the NPV of a capital project.
Calculating Net Present Value (NPV) and Internal Rate of Return (IRR) in Excel. CFA Exam Level 1, Excel Modelling. This lesson is part 5 of 9 in the course
22 Jul 2019 What Is Required Rate of Return? Formula and Calculating RRR. What Does RRR Tell You? Examples of RRR. RRR Using CAPM Formula 11 Apr 2019 NPV is used in capital budgeting to compare
projects based on their expected rates of return, required investment, and anticipated revenue over The required rate of return is a key concept in corporate finance and equity valuation. For
instance, in equity valuation, it is commonly used as a discount rate to 10 Dec 2019 You should always pick the project with the highest NPV, not necessarily the highest IRR, because financial
performance is measured in dollars. The NPV will be calculated for an investment by using a discount rate and The NPV function and IRR function (Internal Rate of Return) are closely related. 9 Mar
2020 NPV (Net present value) is the difference between the present value of cash inflows and This rate is derived considering the return of investment with similar risk or cost of Also the inflows
may not always be as expected.
NPV formula. If you wonder how to calculate the Net Present Value (NPV) by yourself or using an Excel spreadsheet, all you need is the formula: where r is the discount rate and t is the number of
cash flow periods, C 0 is the initial investment while C t is the return during period t.
To calculate the net present value, you will need to subtract the initial investment from the result you get from the NPV function. Lets take an example to demonstrate this function. Assume that you
started a business with an initial investment of $10,000 and received the following income for the next five years. NPV formula. If you wonder how to calculate the Net Present Value (NPV) by yourself
or using an Excel spreadsheet, all you need is the formula: where r is the discount rate and t is the number of cash flow periods, C 0 is the initial investment while C t is the return during period
t. What Is Net Present Value and How Do You Calculate It? Net Present Value vs. Internal Rate of Return. The use of NPV can be applied to predict whether money will compound in the future. The If
your discount rate is close to the actual return rate you can get on your money for an alternative investment of similar risk and your future cash inflows are close to the amounts of money you'll
actually make from your investment, your NPV calculation will be right on the money. What is net present value? In finance jargon, the net present value is the combined present value of both the
investment cash flow and the return or withdrawal cash flow. To calculate the net present value, the user must enter a "Discount Rate." The "Discount Rate" is simply your desired rate of return
(ROR). Using the NPV Calculator
calculate NPV, IRR, payback period and accounting rate of return; this may indicate that a change is required in the search process and that the target.
You can make your own computations for net present value on Excel, with some real Let's say you require a 10% yield (rate of return) on your investment. Deduct the $100,000 initial investment from
the $100,000 PV to determine the NPV. In the case of mutually exclusive projects, if the NPV and the IRR suggest two different investment projects, we should choose the project with a The required
rate of return for the project is 4.4%. Using a calculator we can compute the NPV. calculate NPV, IRR, payback period and accounting rate of return; this may indicate that a change is required in the
search process and that the target. Calculate the internal rate of return on your investments with this IRR calculator. the interest rate that makes the Net Present Value (NPV) of all cash flows from
12 Sep 2019 The Net Present Value (NPV) of a project is the potential change in wealth If the required rate of return for the project is 8%, what would the NPV be like the NPV equation except that
the discount rate is the IRR instead of r, 6 Feb 2016 The rate of return formula is an easy-to-use tool. There are two major numbers needed to calculate the rate of return: Current value: the
current The NPV can be calculated in a spreadsheet using the following NPV Internal rate of return (IRR) is a metric used in capital budgeting to measure the
8 Oct 2018 The Net Present Value tells you the net return on your investment, The formula takes the total cash inflows in the future and discounts it After calculating the net present value, you find
that the internal rate of return is 13%.
The NPV will be calculated for an investment by using a discount rate and The NPV function and IRR function (Internal Rate of Return) are closely related. 9 Mar 2020 NPV (Net present value) is the
difference between the present value of cash inflows and This rate is derived considering the return of investment with similar risk or cost of Also the inflows may not always be as expected. Net
Present Value (NPV). Now we are equipped to calculate the Net Present Value. For each amount (either coming in, or going out) work out its Present Relationships Between the Internal Rate of Return
present value of the cash inflows equal to the present value of the cash outflows in a capital budgeting analysis, where all future cash flows are discounted to determine their present values. This
NPV IRR Calculator calculates both your net present value and the internal rate of return on an investment with net cash flows. It's calculated side by side to The results based on the calculations
using the net present value and the inner rate of return are often competing in the technical literature of investment- Calculating IRR is a trial and error process in which you find the rate of
return that makes an investment's net present value, or NPV, equal zero. For example
Calculating IRR is a trial and error process in which you find the rate of return that makes an investment's net present value, or NPV, equal zero. For example 6 Jun 2019 If IRR falls below the
required rate of return, the project should be rejected. IRR Formula & Example. You can use the following formula to calculate IRR: By contrast, net present value (NPV) measures how much value a CF,
cash flow; NPV, net present value. Example. Calculate the internal rate of return using Table 18.11 given the NPV for each discount rate. IRR is calculate using the calculator or as follows using
interpolation of a low discount rate with positive NPV and high discount rate with negative NPV. Calculates IRR, NPV on-line from cash flows input. If the required rate of return (discount rate) is
3.125%, what is the net present value? Procedures: Enter the cash flows using CFj and Nj. Press SHIFT, then IRR/YR. When IRR/YR is calculated, the annual nominal rate that gives the NPV | {"url":"https://bestexmozucvjs.netlify.app/vanderzanden737xa/how-do-you-calculate-required-rate-of-return-using-npv-328.html","timestamp":"2024-11-09T05:59:22Z","content_type":"text/html","content_length":"37873","record_id":"<urn:uuid:07f08262-030e-4b30-83d4-e2dc171292bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00274.warc.gz"} |
Exponential Functions - Formula, Properties, Graph, Rules
What is an Exponential Function?
An exponential function measures an exponential decrease or increase in a particular base. Take this, for example, let's say a country's population doubles annually. This population growth can be
represented as an exponential function.
Exponential functions have numerous real-life use cases. Mathematically speaking, an exponential function is shown as f(x) = b^x.
In this piece, we will learn the fundamentals of an exponential function coupled with relevant examples.
What is the formula for an Exponential Function?
The common formula for an exponential function is f(x) = b^x, where:
1. b is the base, and x is the exponent or power.
2. b is fixed, and x varies
For example, if b = 2, then we get the square function f(x) = 2^x. And if b = 1/2, then we get the square function f(x) = (1/2)^x.
In cases where b is larger than 0 and does not equal 1, x will be a real number.
How do you graph Exponential Functions?
To chart an exponential function, we need to find the points where the function crosses the axes. These are known as the x and y-intercepts.
Considering the fact that the exponential function has a constant, one must set the value for it. Let's focus on the value of b = 2.
To locate the y-coordinates, its essential to set the worth for x. For example, for x = 1, y will be 2, for x = 2, y will be 4.
In following this method, we get the domain and the range values for the function. Once we have the worth, we need to plot them on the x-axis and the y-axis.
What are the properties of Exponential Functions?
All exponential functions share comparable characteristics. When the base of an exponential function is larger than 1, the graph will have the following characteristics:
• The line passes the point (0,1)
• The domain is all positive real numbers
• The range is greater than 0
• The graph is a curved line
• The graph is on an incline
• The graph is flat and ongoing
• As x nears negative infinity, the graph is asymptomatic regarding the x-axis
• As x approaches positive infinity, the graph rises without bound.
In situations where the bases are fractions or decimals within 0 and 1, an exponential function displays the following characteristics:
• The graph crosses the point (0,1)
• The range is greater than 0
• The domain is entirely real numbers
• The graph is decreasing
• The graph is a curved line
• As x approaches positive infinity, the line in the graph is asymptotic to the x-axis.
• As x approaches negative infinity, the line approaches without bound
• The graph is flat
• The graph is continuous
There are some vital rules to bear in mind when dealing with exponential functions.
Rule 1: Multiply exponential functions with an identical base, add the exponents.
For instance, if we need to multiply two exponential functions that have a base of 2, then we can note it as 2^x * 2^y = 2^(x+y).
Rule 2: To divide exponential functions with an identical base, deduct the exponents.
For instance, if we need to divide two exponential functions with a base of 3, we can note it as 3^x / 3^y = 3^(x-y).
Rule 3: To increase an exponential function to a power, multiply the exponents.
For example, if we have to grow an exponential function with a base of 4 to the third power, then we can write it as (4^x)^3 = 4^(3x).
Rule 4: An exponential function that has a base of 1 is always equal to 1.
For example, 1^x = 1 regardless of what the worth of x is.
Rule 5: An exponential function with a base of 0 is always equivalent to 0.
For instance, 0^x = 0 regardless of what the value of x is.
Exponential functions are generally utilized to signify exponential growth. As the variable increases, the value of the function increases faster and faster.
Example 1
Let’s examine the example of the growth of bacteria. If we have a culture of bacteria that doubles hourly, then at the end of the first hour, we will have 2 times as many bacteria.
At the end of the second hour, we will have 4x as many bacteria (2 x 2).
At the end of hour three, we will have 8 times as many bacteria (2 x 2 x 2).
This rate of growth can be displayed using an exponential function as follows:
f(t) = 2^t
where f(t) is the number of bacteria at time t and t is measured hourly.
Example 2
Also, exponential functions can illustrate exponential decay. If we have a radioactive substance that degenerates at a rate of half its quantity every hour, then at the end of the first hour, we will
have half as much substance.
At the end of two hours, we will have one-fourth as much substance (1/2 x 1/2).
After three hours, we will have an eighth as much substance (1/2 x 1/2 x 1/2).
This can be represented using an exponential equation as below:
f(t) = 1/2^t
where f(t) is the volume of material at time t and t is measured in hours.
As you can see, both of these illustrations follow a comparable pattern, which is the reason they can be depicted using exponential functions.
In fact, any rate of change can be demonstrated using exponential functions. Recall that in exponential functions, the positive or the negative exponent is denoted by the variable whereas the base
stays fixed. This means that any exponential growth or decay where the base is different is not an exponential function.
For example, in the matter of compound interest, the interest rate stays the same while the base changes in ordinary time periods.
An exponential function can be graphed utilizing a table of values. To get the graph of an exponential function, we have to enter different values for x and then asses the matching values for y.
Let's check out this example.
Example 1
Graph the this exponential function formula:
y = 3^x
To start, let's make a table of values.
As shown, the values of y increase very rapidly as x increases. Imagine we were to plot this exponential function graph on a coordinate plane, it would look like the following:
As seen above, the graph is a curved line that goes up from left to right and gets steeper as it goes.
Example 2
Chart the following exponential function:
y = 1/2^x
First, let's draw up a table of values.
As you can see, the values of y decrease very quickly as x surges. The reason is because 1/2 is less than 1.
If we were to graph the x-values and y-values on a coordinate plane, it is going to look like the following:
The above is a decay function. As shown, the graph is a curved line that gets lower from right to left and gets flatter as it proceeds.
The Derivative of Exponential Functions
The derivative of an exponential function f(x) = a^x can be displayed as f(ax)/dx = ax. All derivatives of exponential functions display special properties whereby the derivative of the function is
the function itself.
This can be written as following: f'x = a^x = f(x).
Exponential Series
The exponential series is a power series whose terminology are the powers of an independent variable number. The general form of an exponential series is:
Grade Potential Can Help You Learn Exponential Functions
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MOEBIUS1: On the Properties of the {M}\"obius Function
:: On the Properties of the {M}\"obius Function
:: by Magdalena Jastrz\c{e}bska and Adam Grabowski
:: Received March 21, 2006
:: Copyright (c) 2006-2021 Association of Mizar Users
Lm1: for X, Y, Z, x being set st X misses Y & x in X /\ Z holds
not x in Y /\ Z
by XBOOLE_1:76, XBOOLE_0:3;
Lm2: for n being Nat st n <> 1 holds
ex p being Prime st p divides n
Lm3: for n being non zero Nat
for p being Prime holds not p |^ 2 divides Radical n
Lm4: for n being non zero Nat holds Radical n is square-free
by Lm3;
Lm5: for m, n being non zero Element of NAT st m divides n & m is square-free holds
m divides Radical n | {"url":"https://mizar.uwb.edu.pl/version/current/html/moebius1.html","timestamp":"2024-11-05T06:51:51Z","content_type":"text/html","content_length":"167985","record_id":"<urn:uuid:d34ed350-52b2-41bc-8bbb-57e8302b3eba>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00371.warc.gz"} |
Announcing Timaeus — AI Alignment Forum
Timaeus is a new AI safety research organization dedicated to making fundamental breakthroughs in technical AI alignment using deep ideas from mathematics and the sciences. Currently, we are working
on singular learning theory and developmental interpretability. Over time we expect to work on a broader research agenda, and to create understanding-based evals informed by our research.
Let sleeping gods (not) lie.
Our primary focus is research. For now, we're a remote-first organization. We collaborate primarily through online seminars and the DevInterp Discord, with regular in-person meetings at workshops and
conferences (see below). We're also investing time in academic outreach to increase the general capacity for work in technical AI alignment.
We believe singular learning theory, a mathematical subject founded by Sumio Watanabe, will lead to a better fundamental understanding of large-scale learning machines and the computational
structures that they learn to represent. It has already given us concepts like the learning coefficient and insights into phase transitions in Bayesian learning. We expect significant advances in the
theory to be possible, and that these advances can inform new tools for alignment.
Developmental interpretability is an approach to understanding the emergence of structure in neural networks, which is informed by singular learning theory but also draws on mechanistic
interpretability and ideas from statistical physics and developmental biology. The key idea is that phase transitions organize learning and that detecting, locating, and understanding these
transitions could pave a road to evaluation tools that prevent the development of dangerous capabilities, values, and behaviors. We're engaged in a research sprint to test the assumptions of this
We see these as two particularly promising research directions, and they are our focus for now. Like any ambitious research, they are not guaranteed to succeed, but there's plenty more water in the
well. Broadly speaking, the research agenda of Timaeus is oriented towards solving problems in technical AI alignment using deep ideas from across many areas of mathematics and the sciences, with a
"full stack" approach that integrates work from pure mathematics through to machine learning experiments.
The outputs we have contributed to so far:
Academic Outreach
AI safety remains bottlenecked on senior researchers and mentorship capacity. The young people already in the field will grow into these roles. However, given the scale and urgency of the problem, we
think it is important to open inroads to academia and encourage established scientists to spend their time on AI safety.
Singular learning theory and developmental interpretability can serve as a natural bridge between the emerging discipline of AI alignment and existing disciplines of mathematics and science,
including physics and biology. We plan to spend part of our time onboarding scientists into alignment via concrete projects in these areas.
We're organizing conferences, retreats, hackathons, etc. focusing on singular learning theory and developmental interpretability. These have included and will include:
Core Team
The research agenda that we are contributing to was established by Daniel Murfet, who is a mathematician at the University of Melbourne and an expert in singular learning theory, algebraic geometry,
and mathematical logic.
Research Assistants
We just concluded a round of hiring and are excited to bring on board several very talented Research Assistants (RAs), starting with
Friends and Collaborators
Here are some of the people we are actively collaborating with:
Inclusion on this list does not imply endorsement of Timaeus' views.
We're advised by Evan Hubinger and David ("Davidad") Dalrymple.
(DALL-E 3 still has a hard time with icosahedra.)
Where can I learn more, and contact you?
Learn more on the Timaeus webpage. You can email Jesse Hoogland.
What about capabilities risk?
There is a risk that fundamental progress in either singular learning theory or developmental interpretability could contribute to further acceleration in AI capabilities in the medium term. We take
this seriously and are seeking advice from other alignment researchers and organizations. By the end of our current research sprint we will have in place institutional forms to help us navigate this
Likewise, there is a risk that outreach which aims to involve more scientists in AI alignment work will also accelerate progress in AI capabilities. However, those of us in academia can already see
that as the risks become more visible, scientists are starting to think about these problems on their own. So the question is not whether a broad range of scientists will become interested in
alignment but when they will start to contribute and what they work on.
It is part of Timaeus' mission to help scientists to responsibly contribute to technical AI alignment, while minimizing these risks.
Are phase transitions really the key?
The strongest critique of developmental interpretability we know is the following: while it is established that phase transitions exist in neural network training, it is not yet clear how common they
are, and whether they make a good target for alignment.
We think developmental interpretability is a good investment in a world where many of the important structures (e.g., circuits) in neural networks form in phase transitions. Figuring out whether we
live in such a world is one of our top priorities. It's not trivial because even if transitions exist they may not necessarily be visible to naive probes. Our approach is to systematically advance
the fundamental science of finding and classifying transitions, starting with smaller systems where transitions can be definitively shown to exist.
How are you funded?
We're funded through a $142k Manifund grant led primarily by Evan Hubinger, Ryan Kidd, Rachel Weinberg, and Marcus Abramovitch. We are fiscally sponsored by Ashgro.
"Timaeus"? How do I even pronounce that?
Pronounce it however you want.
Timaeus is the eponymous character in the dialogue where Plato introduces his theory of forms. The dialogue posits a correspondence between the elements that make up the world and the Platonic
solids. That's wrong, but it contains the germ of the idea of the unreasonable effectiveness of mathematics in understanding the natural world.
We read the Timaeus dialogue with a spirit of hope, in the capacity of the human intellect to understand and solve wicked problems. The narrow gate to human flourishing is preceded by a narrow path.
We'll see you on that path.
Thanks for the detailed response!
So, to check my understanding:
The toy cases discussed in Multi-Component Learning and S-Curves are clearly dynamical phase transitions. (It's easy to establish dynamical phase transitions based on just observation in general.
And, in these cases we can verify this property holds for the corresponding differential equations (and step size is unimportant so differential equations are a good model).) Also, I speculate it's
easy to prove the existence of a bayesian phase transition in the number of samples for these toy cases given how simple they are.
More generally, I wish that when people used the term "phase transition", they clarified whether they meant "s-shaped loss curves" or some more precise notion. Often, people are making a
non-mechanistic claim when they say "phase transition" (we observed a loss curve with a s-shape), but there are also mechanistic claims which require additional evidence.
In particular, when citing other work somewhere, it would be nice to clarify what notion of phase transition the other work is discussing.
New Comment
4 comments, sorted by Click to highlight new comments since:
The strongest critique of developmental interpretability we know is the following: while it is established that phase transitions exist in neural network training, it is not yet clear how common
they are, and whether they make a good target for alignment.
Is it established that phase transitions exist in the training of non-toy neural networks?
There are clearly s-shaped loss curves in many non-toy cases, but I'm not aware of any known cases which are clearly phase transitions as defined here (which is how the term is commonly used in e.g.
physics and how I think this post wants to use the term).
For instance, while formation of induction-like attention heads^[1] probably results in s-shaped loss curves in at least some cases, my understanding is that this probably has nothing to do with
changes in the minima of some notion of energy (as would be required for the definition linked above I think). I think the effect is probably the one described in Multi-Component Learning and
S-Curves. Unless there is some notion of energy such that this multi-component case of s-shaped loss curves is well described as a phase transition and that's what's discussed in this post?
Some important disclaimers:
• I'm not very familiar with the notion of phase transition I think this post is using.
• It seems as though there are cases where real phase transitions occur during the training of toy models and there are also phase transitions in the final configuration of toy models as various
hyperparameters change. (I'm pretty confident that Toy models of superposition has both phase transitions during training and in final configurations (for phase transitions in final
configurations there seems to be one with respect to "does superposition occur").There is also this example of a two layer ReLU model; however, I haven't read or understood this example, I'm just
trusting the authors claim that there is a phase transition here).
1. These attention heads probably do a bunch of stuff which isn't that well described as induction, so I'm reluctant to call them "induction heads". ↩︎
Great question, thanks. tldr it depends what you mean by established, probably the obstacle to establishing such a thing is lower than you think.
To clarify the two types of phase transitions involved here, in the terminology of Chen et al:
• Bayesian phase transition in number of samples: as discussed in the post you link to in Liam's sequence, where the concentration of the Bayesian posterior shifts suddenly from one region of
parameter space to another, as the number of samples increased past some critical sample size . There are also Bayesian phase transitions with respect to hyperparameters (such as variations in
the true distribution) but those are not what we're talking about here.
• Dynamical phase transitions: the "backwards S-shaped loss curve". I don't believe there is an agreed-upon formal definition of what people mean by this kind of phase transition in the deep
learning literature, but what we mean by it is that the SGD trajectory is for some time strongly influenced (e.g. in the neighbourhood of) a critical point and then strongly influenced by another
critical point . In the clearest case there are two plateaus, the one with higher loss corresponding to the label and the one with the lower loss corresponding to . In larger systems there may
not be a clear plateau (e.g. in the case of induction heads that you mention) but it may still reasonable to think of the trajectory as dominated by the critical points.
The former kind of phase transition is a first-order phase transition in the sense of statistical physics, once you relate the posterior to a Boltzmann distribution. The latter is a notion that
belongs more to the theory of dynamical systems or potentially catastrophe theory. The link between these two notions is, as you say, not obvious.
However Singular Learning Theory (SLT) does provide a link, which we explore in Chen et al. SLT says that the phases of Bayesian learning are also dominated by critical points of the loss, and so you
can ask whether a given dynamical phase transition has "standing behind it" a Bayesian phase transition where at some critical sample size the posterior shifts from being concentrated near to being
concentrated near .
It turns out that, at least for sufficiently large , the only real obstruction to this Bayesian phase transition existing is that the local learning coefficient near should be higher than near . This
will be hard to prove theoretically in non-toy systems, but we can estimate the local learning coefficient, compare them, and thereby provide evidence that a Bayesian phase transition exists.
This has been done in the Toy Model of Superposition in Chen et al, and we're in the process of looking at a range of larger systems including induction heads. We're not ready to share those results
yet, but I would point you to Nina Rimsky and Dmitry Vaintrob's nice post on modular addition which I would say provides evidence for a Bayesian phase transition in that setting.
There are some caveats and details, that I can go into if you're interested. I would say the existence of Bayesian phase transitions in non-toy neural networks is not established yet, but at this
point I think we can be reasonably confident they exist. | {"url":"https://www.alignmentforum.org/posts/nN7bHuHZYaWv9RDJL/announcing-timaeus","timestamp":"2024-11-05T19:33:08Z","content_type":"text/html","content_length":"443804","record_id":"<urn:uuid:4f2a5dd3-7180-4bd3-95cc-1cbaba983f57>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00302.warc.gz"} |
Lived 1882 – 1935.
Emmy Noether is probably the greatest female mathematician who has ever lived. She transformed our understanding of the universe with Noether’s theorem and then transformed mathematics with her
founding work in abstract algebra.
Amalie Emmy Noether was born in the small university city of Erlangen in Germany on March 23, 1882. Her father, Max Noether, was an eminent professor of mathematics at the University of Erlangen. Her
mother was Ida Amalia Kaufmann, whose family were wealthy wholesalers.
Young Emmy was brought up as a typical girl of her era: helping with cooking and running the house – she admitted later she had little aptitude for these sorts of things. Her mother was a skilled
pianist, but Emmy did not enjoy piano lessons. Her main passion was dancing.
She also loved mathematics, but she knew that the rules of German society meant she would not be allowed to follow in her father’s footsteps to become a university academic.
After completing high school – she attended the Municipal School for Higher Education of Daughters in Erlangen – she trained to become a school teacher, qualifying in 1900, aged 18, to teach English
and French in girls’ schools.
Although a career in teaching offered her financial security, her love of mathematics proved to be too strong. She decided to abandon teaching and apply to the University of Erlangen to observe
mathematics lectures there. She could only observe lectures, because women were not permitted to enroll officially at the university.
Between 1900 and 1902 Emmy studied mathematics at Erlangen. In July 1903 she traveled to the city of Nürnberg and passed the matriculation examination allowing her to study mathematics (but not
officially enroll) at any German university.
At the Center of the Universe for a Semester
Emmy chose to go for a semester to the University of Göttingen, then home to the most prestigious school of mathematics in the world.
Some of the greatest mathematicians in history had taught and been taught at Göttingen, including Carl Friedrich Gauss and Bernhard Riemann. Emmy attended lectures given by:
• Hermann Minkowski, the esteemed mathematician who taught Albert Einstein, and
• David Hilbert, probably the twentieth century’s most outstanding mathematician
Doctorate in Mathematics
In 1904 Emmy was overjoyed to learn that her hometown university, Erlangen, had decided women should be permitted full access.
She was accepted as a Ph.D. student by the renowned mathematician Paul Gordan. Gordan was 67 when Emmy started work with him. She was the only student he ever accepted as a Ph.D. candidate.
Gordan was known among mathematicians as “the king of invariant theory.” Emmy made exceptional progress in this field, which would later lead to her making a remarkable discovery in physics
In 1907 the 25-year-old Emmy officially became Doctor Noether. Her degree was awarded ‘summa cum laude’ – the highest distinction possible.
Dr. Noether, Mathematics Lecturer
In 1908 Noether was appointed to the position of mathematics lecturer at Erlangen. Unfortunately, it was an unpaid position. This was not especially unusual in Germany for a first lecturing job. The
great chemist Robert Bunsen’s first lecturing position was without pay at the University of Göttingen.
Noether’s parents supported her as much as they could through this time, her father recognizing something rather special in his daughter’s capabilities. Nevertheless, her life was a struggle
While working as a lecturer, Noether became fascinated by work David Hilbert had done in Göttingen. The work was more abstract than any she had done at Erlangen. She began stretching and modifying
Hilbert’s methods. This was her first heavyweight encounter with abstract algebra, mathematical territory in which she would soon become a powerful innovator.
An Invitation from Hilbert
David Hilbert familiarized himself with Noether’s research; like her father, he recognized her outstanding ability. By this stage in his career Hilbert was concerned mainly with physics, which he
believed needed help from the best mathematicians, famously declaring:
Physics is actually too hard for physicists.
David Hilbert
In 1913 and 1914 Noether exchanged letters with David Hilbert and his Göttingen colleague Felix Klein discussing Einstein’s Relativity Theory.
In 1915 Hilbert invited her to become a lecturer in Göttingen. Unfortunately this provoked a storm of protest from the history and linguistics faculties who did not think it appropriate that a woman
should be teaching men, particularly since Germany was at war – World War 1: 1914 – 1918. Although in general the mathematics and science faculties supported Noether, they could not overcome the
opposition from the humanities.
Noether was so eager to join Hilbert’s department in Göttingen that, to soothe Hilbert’s opponents, she agreed not to be formally appointed as a lecturer and to receive no pay. Her father continued
supporting her financially (sadly her mother died in 1915) and the lectures she gave were advertised as lectures by Professor Hilbert, with assistance from Dr. E. Noether.
Noether’s Theorem
Hilbert, Einstein, Noether, and the General Theory of Relativity
Einstein visits Göttingen
In 1915 Albert Einstein was struggling mathematically with the formulation of his General Theory of Relativity. He visited David Hilbert in Göttingen and discussed the issues. The result was that
Einstein overcame his issues and published his theory before the year end. Hilbert published his own version of the theory, in a different mathematical form.
Einstein’s General Relativity Breaks the Law
Hilbert now discussed one particular problem with Noether. He was deeply concerned that, despite its attractions, Relativity Theory was breaking one of the ‘unbreakable’ conservation laws of physics.
He believed that her expertise in invariant theory could be helpful.
Certain quantities in physics may not be created or destroyed, such as energy. Energy can change its form – such as kinetic to thermal – but the total energy stays constant – energy is said to be
In General Relativity Theory however, there was a problem: it was possible for an object which lost energy by emitting gravity waves to speed up. An object with less energy should slow down, not
speed up! It seemed that the energy conservation law was being broken.
A Problem Archimedes Would Have Loved
In the end, the problem was one of symmetry. Over 2000 years earlier the greatest mathematician of antiquity – perhaps ever – had been buried with a carving of a sphere within a cylinder on his tomb.
This was Archimedes, who believed his greatest achievement had been discovering and proving the formula for the volume of a sphere.
A perfect sphere is highly symmetrical. Whichever way you rotate it, and from whichever angle you view it, it always looks the same. A cylinder, on the other hand, is less symmetrical; but there is
still some symmetry. If you turn it upside down, for example, it looks the same.
Physicists need to use equations whose symmetry is as sphere-like as possible. They last thing they want is equations that change depending on where you are viewing the universe from. In physics
jargon we say that we need the laws of the universe to be space invariant. We don’t want them to look different in one city from another or in one galaxy from another.
We also need these laws to be time invariant. We don’t want the laws of physics in an hour’s time to be different from the laws right now.
Noether to the Rescue
Noether hit the ground running in Göttingen. In the year she arrived she proved something remarkable – something so beautiful and profound that it changed the face of physics forever – Noether’s
Theorem, which she eventually published in 1918.
Her famous theorem was born when Noether considered Hilbert and Einstein’s problem: that General Relativity Theory seemed to break the law of conservation of energy.
Noether discovered that for every invariant (i.e. symmetry) in the universe there is a conservation law. Equally, for every conservation law in physics, there is an invariant. This is called
Noether’s Theorem and it describes a fundamental property of our universe.
For example, Noether’s Theorem shows that the law of conservation of energy is actually a consequence of time invariance in classical physics. Or alternatively that time invariance is caused by the
law of conservation of energy.
Another example is that the law of conservation of electric charge is a consequence of the global gauge invariance of the electromagnetic field. And vice versa.
Opening up Strange New Worlds
With Noether’s Theorem, physicists had a very powerful new concept. They could propose abstract symmetries, knowing there must be a conservation law attached to each of them. They could then figure
out the conservation law.
Noether’s Theorem has the power to answer questions others cannot – particularly in particle physics. It is important on two levels:
• it allows practical calculations to be made, and
• when physicists theorize about any new system they can imagine, Noether’s theorem allows them to gain an insight into the properties of that system and determine if it is possible or should be
Einstein’s Problem Solved
Noether’s Theorem also solved the worrying puzzle in General Relativity that she had initially set out to solve. Her theorem shows that if matter and gravity are considered to be one unified quantity
rather than separate quantities, then there is no violation of any conservation law.
“Yesterday I received a very interesting paper on invariants from Miss Noether. I’m impressed that these things can be seen in such a general way. It would do the old guard at Göttingen no harm to be
sent back to school under Miss Noether. She knows her stuff.”
Albert Einstein, 1879 to 1955
Einstein became vocal about Göttingen’s refusal to appoint Noether as a lecturer, telling Felix Klein:
“After receiving Miss Noether’s new paper, I once again feel that depriving her of a teaching job is a great injustice. I would like vigorous steps to be taken with the Ministry. If you do not think
this is possible, then I will go the trouble of doing it myself.”
Albert Einstein, 1879 to 1955
At last: some career progress
With the end of World War 1, in which so many men had died or been badly injured, came a change in German society. It became acceptable for women to work in occupations previously reserved for men.
Combined with Noether publishing her brilliant theorem, her academic progress could no longer be blocked.
At the age of 37 she became a tenured lecturer at Göttingen. However, she still received no pay from a now war-impoverished Germany. Her father died when she was 39, leaving her a small inheritance.
It was only when she reached the age of 40 that Noether finally began to receive a salary.
Abstract Algebra
Noether’s Theorem revolutionized physics. In 1919 the full force of her powerful mind turned towards pure mathematics. In this discipline, she was one of the principle architects of abstract algebra.
Her name is remembered in many of its concepts, structures, and objects, such as:
Noetherian, Noetherian group, Noetherian induction, Noether normalization, Noether problem, Noetherian ring, Noetherian module, Noetherian scheme, Noetherian space, Albert–Brauer–Hasse–Noether
theorem, Lasker–Noether theorem, and Skolem–Noether theorem.
Her work was pivotal in the fields of:
• mathematical rings – she established the modern axiomatic definition of the commutative ring and developed the basis of commutative ring theory
• commutative number fields
• linear transformations
• noncommutative algebras – Hermann Weyl credited Noether with representations of noncommutative algebras by linear transformations, and their application to the study of commutative number fields
and their arithmetics
“And one cannot read the scope of her accomplishments from the individual results of her papers alone: she originated above all a new and epoch-making style of thinking in algebra.”
Hermann Weyl, 1885 to 1955
“If Emmy Noether could have been at the 1950 Congress, she would have felt very proud. Her concept of algebra had become central in contemporary mathematics. And it has continued to inspire
algebraists ever since.”
Garrett Birkhoff, 1911 to 1996
The Rise of Modern Algebra, 1976
Expulsion from Germany: moving to America
In the early 1930s Noether’s career was finally taking off. Her name was becoming known, and she was receiving invitations to speak at important mathematics conferences.
Then, in January 1933, everything changed. Adolf Hitler came to power. By April of that year Noether, who was Jewish, had been dismissed from the University of Göttingen by order of the Prussian
Ministry for Sciences, Art, and Public Education. Sadly, in Nazi ideology Emmy Noether’s religion was of more significance than her extraordinary genius.
Fortunately, her genius was valued elsewhere. Bryn Mawr College in Pennsylvania, USA – a women’s college – obtained a grant from the Rockefeller Foundation and, in October 1933, Emmy Noether sailed
on the Bremen to begin work as a lecturer in America.
The following year she also began lecturing at the Institute for Advanced Study in Princeton.
A year later she was dead.
Some Personal Details and The End
Noether was totally devoted to mathematics and talked of little else. She never married and had no children. She cared little for her appearance and less for social conventions; she was not a
shrinking violet – she spoke loudly and forcibly. She could be very blunt when she disagreed with anyone on a mathematical issue, and people with whom she disagreed could feel rather bruised
On the other hand, she was very kind, considerate, and unselfish with everyone, and would go out of her way to ensure her Ph.D. students got full credit for their work, even when she had contributed
significantly to it herself.
Only students who were very bright and fully prepared benefited from her rather disorganized lectures – rather like Willard Gibbs‘s students.
To her advanced students, she would present ideas at the forefront of modern mathematics – concepts that she herself was currently working on. This was of great benefit to her best students, who were
able to publish research papers based on new, entirely original ideas Noether had been discussing in her lectures. Her best lessons were delivered informally, in conversations, or when out walking
with her students, for whom she always had time.
“In her… apartment in Göttingen a large group would get together eagerly and often. People of diverse scholarly reputations and positions — from Hilbert, Landau, Brauer and Weyl to the youngest
students — would gather at her home and feel relaxed and unconstrained, as in few other scientific salons in Europe. These ‘festive evenings’ in her apartment were arranged on any possible occasion…”
Pavel Aleksandrov, 1896 to 1982
Emmy Noether died in Bryn Mawr at the age of 53 on April 14, 1935. She died of complications a few days after an operation to remove a tumor from her pelvis. The cause of death was possibly a viral
infection. Her ashes were buried under the cloisters of Bryn Mawr College’s M. Carey Thomas Library.
“In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In
the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day
younger generation of mathematicians.”
Albert Einstein, 1879 to 1955
“…you were a great woman mathematician – I have no reservations in calling you the greatest that history has known.”
Hermann Weyl, 1885 to 1955
Author of this page: The Doc
Images of scientists digitally enhanced and colorized by this website. © All rights reserved.
Cite this Page
Please use the following MLA compliant citation:
"Emmy Noether." Famous Scientists. famousscientists.org. 17 Aug. 2015. Web.
Published by FamousScientists.org
Further Reading
Charlene Morrow, Teri Perl
Notable Women in Mathematics
Greenwood Publishing Group, 1998
Bertram E. Schwarzbach, Yvette Kosmann-Schwarzbach
The Noether Theorems
Springer Science & Business Media, 2010
Auguste Dick
Translated by H. I. Bloclier
Emmy Noether: 1882-1935
Birkhäuser, 1981
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Psychological Statistics and Psychometrics Using Stata
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Author: Scott Baldwin
Publisher: Stata Press
Copyright: 2019
ISBN-13: 978-1-59718-303-1
Pages: 454; paperback
Psychological Statistics and Psychometrics Using Stata by Scott Baldwin is a complete and concise resource for students and researchers in the behavioral sciences. Professor Baldwin includes dozens
of worked examples using real data to illustrate the theory and concepts. This book would be an excellent textbook for a graduate-level course in psychometrics. It is also an ideal reference for
psychometricians who are new to Stata.
Baldwin's primary goal in this book is to help readers become competent users of statistics. To that end, he first introduces basic statistical methods such as regression, t tests, and ANOVA. He
focuses on explaining the models, how they can be used with different types of variables, and how to interpret the results. After building this foundation, Baldwin covers more advanced statistical
techniques, including power-and-sample size calculations, multilevel modeling, and structural equation modeling. This book also discusses measurement concepts that are crucial in psychometrics. For
instance, Baldwin explores how reliability and validity can be understood and evaluated using exploratory and confirmatory factor analysis. Baldwin includes dozens of worked examples using real data
to illustrate the theory and concepts.
In addition to teaching statistical topics, this book helps readers become proficient Stata users. Baldwin teaches Stata basics ranging from navigating the interface to using features for data
management, descriptive statistics, and graphics. He emphasizes the need for reproducibility in data analysis; therefore, he is careful to explain how version control and do-files can be used to
ensure that results are reproducible. As each statistical concept is introduced, the corresponding commands for fitting and interpreting models are demonstrated. Beyond this, readers learn how to run
simulations in Stata to help them better understand the models they are fitting and other statistical concepts.
This book is an excellent textbook for graduate-level courses in psychometrics. It is also an ideal reference for psychometricians and other social scientists who are new to Stata.
List of figures
List of tables
Notation and Typography
Getting oriented to Stata
1 Introduction
1.1 Structure of the book
1.2 Benefits of Stata
1.3 Scientific context
2 Introduction to Stata
2.1 Point-and-click versus writing commands
2.2 The Stata interface
2.3 Getting data in Stata
2.4 Viewing and desribing data
2.4.1 list, in, and if
2.5 Creating new variables
2.5.1 Missing data
2.5.2 Labels
2.6 Summarizing data
2.6.1 summarize
2.6.2 table and tabulate
2.7 Graphing data
2.7.1 Histograms
2.7.2 Box plots
2.7.3 Scatterplots
2.8 Reproducible analysis
2.8.1 Do-files
2.8.2 Log files
2.8.3 Project Manager
2.8.4 Workflow
2.9 Getting help
2.9.1 Help documents
2.9.2 PDF documentation
2.10 Extending Stata
2.10.1 Statistical Software Components
2.10.2 Writing your own programs
Understanding relationships between variables
3 Regression with continuous predictors
3.1 Data
3.2 Exploration
3.2.1 Demonstration
Simulation program
3.3 Bivariate regression
3.3.1 Lines
3.3.2 Regression equation
3.3.3 Estimation
3.3.4 Interpretation
3.3.5 Residuals and predicted values
3.3.6 Partitioning variance
3.3.7 Confidence intervals
3.3.8 Null hypothesis significance testing
3.3.9 Additional methods for understanding models
Using predicted scores to understand model implications
Composite contrasts
3.4 Conclusions
4 Regression with categorical and continuous predictors
4.1 Data for this chapter
4.2 Why categorical predictors need special care
4.3 Dummy coding
4.3.1 Example: Incorrect use of categorical variable
4.4 Multiple predictors
4.4.1 Interpretation
Model fit
4.4.2 Unique variance
4.5 Interactions
4.5.1 Categorical by continuous interactions
Dichotomous by continuous interactions
Polytomous by continuous interactions
Joint test for interactions with polytomous variables
4.5.2 Continuous by continuous interactions
4.6 Summary
5 t tests and one-way ANOVA
5.1 Data
5.2 Comparing two means
5.2.1 t test
5.2.2 Effect size
5.3 Comparing three or more means
5.3.1 Analysis of variance
5.3.2 Multiple comparisons
Planned comparisons
Direct adjustment for multiple comparisons
5.4 Summary
6 Factorial ANOVA
6.1 Data for this chapter
6.2 Factorial design with two factors
6.2.1 Examining and visualizing the data
6.2.2 Main effects
Testing the null hypothesis
6.2.3 Interactions
6.2.4 Partitioning the variance
6.2.5 2 x 2 source table
6.2.6 Using anova to estimate a factorial ANOVA
6.2.7 Simple effects
6.2.8 Effect size
6.3 Factorial design with three factors
6.3.1 Examining and visualizing the data
6.3.2 Marginal means
6.3.3 Main effects and interactions
6.3.4 Three-way interaction
6.3.5 Fitting the model with anova
6.3.6 Interpreting the interaction
6.3.7 A note about effect size
6.4 Conclusion
7 Repeated-measures models
7.1 Data for this chapter
7.2 Basic model
7.3 Using mixed to fit a repeated-measures model
7.3.1 Covariance structures
Compound symmetry (exchangeable)
First-order autoregressive
7.3.2 Degrees of freedom
7.3.3 Pairwise comparisons
7.4 Models with multiple factors
7.5 Estimating heteroskedastic residuals
7.6 Summary
8 Planning studies: Power and sample-size calculations
8.1 Foundational ideas
8.1.1 Null and alternative distributions
8.1.2 Simulating draws out of the null and alternative distributions
8.2 Computing power manually
8.3 Stata's commands
8.3.1 Two-sample z test
8.3.2 Two-sample t test
8.3.3 Correlation
8.3.4 One-way ANOVA
8.3.5 Factorial ANOVA
8.4 The central importance of power
8.4.1 Type M and S errors
Type S errors
Type M errorss
8.5 Summary
9 Multilevel models for cross-sectional data
9.1 Data used in this chapter
9.2 Why clustered data structures matter
9.2.1 Statistical issues
9.2.2 Conceptual issues
9.3 Basics of a multilevel model
9.3.1 Partitioning sources of variance
9.3.2 Random intercepts
9.3.3 Estimating random intercepts
9.3.4 Intraclass correlations
9.3.5 Estimating cluster means
Comparing pooled and unpooled means
9.3.6 Adding a predictor
9.4 Between-clusters and within-cluster relationships
9.4.1 Partitioning variance in the predictor
9.4.2 Total- versus level-specific relationships
9.4.3 Exploring the between-clusters and within-cluster relationships
9.4.4 Estimating the between-clusters and within-cluster effects
9.5 Random slopes
9.6 Summary
10 Multilevel models for longitudinal data
10.1 Data used in this chapter
10.2 Basic growth model
10.2.1 Multilevel model
10.3 Adding a level-2 predictor
10.4 Adding a level-1 predictor
10.5 Summary
Psychometrics through the lens of factor analysis
11 Factor analysis: Reliability
11.1 What you will learn in this chapter
11.2 Example data
11.3 Common versus unique variance
11.4 One-factor model
11.4.1 Parts of a path model
11.4.2 Where do the latent variables come from?
11.5 Prediction equation
11.6 Using sem to estimate CFA models
11.7 Model fit
11.7.1 Computing χ²
11.8 Obtaining σ²
and σ²
11.8.1 Computing R² for an item
11.8.2 Computing σ²[C] and σ²[U] for all items
11.8.3 Computing reliability—ω
11.8.4 Bootstrapping the standard error and 95% confidence interval for ω
11.9 Comparing ω with α
11.9.1 Evaluating the assumption of tau-equivalence
11.9.2 Parallel items
11.10 Correlated residuals
11.11 Summary
12 Factor analysis: Factorial validity
12.1 Data for this chapter
12.2 Exploratory factor analysis
12.2.1 Common factor model
12.2.2 Extraction methods
12.2.3 Interpreting loadings
12.2.4 Eigenvalues
12.2.5 Communality and uniqueness
12.2.6 Factor analysis versus principal-component analysis
12.2.7 Choosing factors and rotation
How many factors should we extract?
Eigenvalue-greater-than-one rule
Scree plots
Parallel analysis
Orthogonal rotation—varimax
Oblique rotation—promax
12.3 Confirmatory factor analysis
12.3.1 EFA versus CFA
12.3.2 Estimating a CFA with sem
12.3.3 Mean structure versus variance structure
12.3.4 Identifying models
Imposing constraints for identification
How much information is needed to identify a model?
12.3.5 Refitting the model with constrained latent variables
12.3.6 Standardized solutions
12.3.7 Global fit
A summary and a caution
12.3.8 Refining models further
12.3.9 Parallel items
12.4 Summary
13 Measurement invariance
13.1 Data
13.2 Measurement invariance
13.3 Measurement invariance across groups
13.3.1 Configural invariance
13.3.2 Metric invariance
13.3.3 Scalar invariance
13.3.4 Residual invariance
13.3.5 Using the comparative fit index to evaluate invariance
13.4 Structural invariance
13.4.1 Invariant factor variances
13.4.2 Invariant factor means
13.5 Measurement invariance across time
13.5.1 Configural invariance
Effects coding for identification
Effects-coding constraints in Stata
13.5.2 Metric invariance
13.5.3 Scalar invariance
13.5.4 Residual invariance
13.6 Structural invariance
13.7 Summary
Author index
Subject index | {"url":"https://jat.co.kr/shopb/43","timestamp":"2024-11-08T20:12:58Z","content_type":"text/html","content_length":"82509","record_id":"<urn:uuid:90da60d5-8a6f-4d91-af92-f0c3ea748424>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00897.warc.gz"} |
The t distribution, developed by "Student" (a pseudonym of W. Gosset) more than 100 years ago, is used for a number of testing purposes. The procedure commonly called t-test, however, refers to a
test of the difference between two means (one of which might be a hypothetical value against which the mean of an observed variable is tested).
T-test for two independent samples (groups)
The t-test is often used to compare the means of two groups. This works as follows:
ttest income, by(married)
There are a few options that can be appended: unequal (or un) informs Stata that the variances of the two groups are to be considered as unequal; welch (or w) requests Stata to use Welch's
approximation to the t-test (which has the nearly the same effect as unequal; only the d.f. are different) and finally, with level(99) (abbreviated as l(99)) you can, in this case, request a
confidence level of 99 per cent instead of the default level of 95, which is used in the calculation of confidence intervals.
How do you know whether the two groups have the same variances? Use
sdtest income, by(married)
to obtain the Bartlett test for equality of variances, or
robvar income, by(married)
which delivers a robust test proposed by Levene in 1960 and two alternatives by Brown & Forsythe in 1974. One of these alternatives uses the median instead of the mean in Levene's original formula
and the other one the 10 per cent trimmed mean. These robust tests are more appropriate in the case of skewed variables.
T-test for paired means
Sometimes the two means to be compared come from the same group of observations, for instance, from measurements at points in time t1 and t2. Here, the appropriate version of the t-test is:
ttest incomet1 == incomet2
Note that Stata will also accept a single equal sign. The level(..) option described in the previous section is available as well.
T-test to compare one mean with a hypothetical value (one sample t-test)
Here, the command goes like this:
ttest IQ = 110
Note that Stata will also accept a pair of equal signs. Again, the level(..) option is available.
Immediate form of the t-test
Another interesting possibility is to do t-tests using information about group sizes, means, and standard deviations, instead of the raw data. This information may be entered immediately with the
ttesti command, with the appended "i" signalling the "immediate" variety of the t-test.
Finally, Stata offers the possibility of running Hotelling's generalized t-test. See Stata help for more detail.
© W. Ludwig-Mayerhofer, Stata Guide | Last update: 06 Jan 2018 | {"url":"https://wlm.userweb.mwn.de/Stata/wstattt.htm","timestamp":"2024-11-15T00:51:10Z","content_type":"text/html","content_length":"9689","record_id":"<urn:uuid:89d066a0-25b8-4a8a-9d25-1d808f482326>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00127.warc.gz"} |
For preparing ordinary concrete, the quantity of water used, is
Q. For preparing ordinary concrete, the quantity of water used, is
A. 5% by weight of aggregates plus 20% of weight of cement
B. 10% by weight of aggregates plus 10% of weight of cement
C. 5% by weight of aggregates plus 30% of weight of cement
D. 30% by weight of aggregates plus 10% of weight of cement
Answer» C. 5% by weight of aggregates plus 30% of weight of cement | {"url":"https://mcqmate.com/discussion/49978/for-preparing-ordinary-concrete-the-quantity-of-water-used-is","timestamp":"2024-11-09T09:38:39Z","content_type":"text/html","content_length":"39258","record_id":"<urn:uuid:ecd4c17a-050a-40af-869c-cd134bf18850>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00097.warc.gz"} |
[Haskell-cafe] Hyper-recursion?
Albert Y. C. Lai trebla at vex.net
Tue Jan 17 01:54:58 UTC 2017
There is an idea called "open recursion". Illustrating by Fibonacci (the
direct but naively slow way) and factorial (the direct but naively slow
way again, see
Instead of writing "fac x = x * fac (x-1)", we write:
fac x = fac_essence fac x
fac_essence f 0 = 0
fac_essence f x = x * f (x-1)
And instead of writing "fib x = fib (x-1) + fib (x-2)", we write:
fib x = fib_essence fib x
fib_essence f 0 = 0
fib_essence f 1 = 1
fib_essence f x = f (x-1) + f (x-2)
The general pattern: Write a helper essence function that takes one more
parameter, a function parameter; don't recurse yet, use the function
parameter instead. Leave the recursion to the main function.
Why would we do this?
One reason is philosophical, like you said, to factor out the recursion
from the essence of one iteration of the process.
Another reason is that you don't have to merely put back the recursion
in the main function. You can now play some dependency injection trick
before recursing, or maybe you don't even recurse.
For example, how about injecting a dependency on a lookup-list, with the
content of said lookup-list coming from said injection --- there hides
the recursion.
fib_list = [fib_essence (fib_list !!) i | i <- [0..]]
or maybe you prefer
fib_list = map (fib_essence (fib_list !!)) [0..]
You will find that "fib_list !! 100" is less slow than "fib 100".
Sometimes you prefer a function-application interface. So we go on to write:
fib_notslow x = fib_list !! x
In fact sometimes we turn it around, hide away the lookup table, and write:
fib_notslow = (fib_list !!)
fib_list = [fib_essence fib_notslow i | i <- [0..]]
-- I could write "fib_essence (fib_list !!)",
-- but now that (fib_list !!) has a name, "fib_notslow"...
Thus we have obtained a revisionist-history definition of dynamic
programming and memoization: Dependency injection of a lookup table into
a recursion.
List is still not fast enough; a binary search tree (lazily generated)
is even better. The "memoize" library
(http://hackage.haskell.org/package/memoize) does that.
Your next food for thought: You have seen open recursion for recursive
programs. How about open recursion for recursive data?
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A brief introduction to perturbation theory
Disclaimer: what follows assumes familiarity with calculus (we will use the Taylor expansion a lot) and Newtonian mechanics. This is also no substitute for a proper textbook, as we will skip over a
lot of details.
There are not many problems in Physics that can be solved exactly, so we often need to look for approximated solutions. One way is to solve the "unsolvable" problem numerically, but while extremely
powerful this approach tends to make an intuitive understanding of the underlying Physics difficult. A second approach is to approximate our equations until they become solvable. This is a delicate
procedure, but has the big advantage that we can keep track of the region of validity of our approximations and build intuition from there, and is the topic of this tutorial.
Instead of jumping into the general mathematical framework, we will look at it via a common (but highly non-trivial) Physical problem: the simple pendulum.
The simple pendulum
The equation of motion of a simple pendulum (with no forcing) is \[\ddot\theta+\omega_0^2 \sin \theta =0 \, ,\] where \(\theta\) is the angle of rotation (from the vertical), and \(\omega_0\) is the
natural frequency of oscillation. This differential equation can be formally solved in terms of elliptic integrals, but it is often useful to find more manageable approximated solutions. If we assume
that the oscillations are small (notice that this is an assumption, and thus it will set the region of validity of our approximated solution) we can perform the Taylor expansion \[\sin\theta \sim \
theta - \frac{\theta^3}{6}+\frac{\theta^5}{120} + \ldots\] and keep only the lowest non-zero term, resulting in the harmonic oscillator equation \(\ddot \theta+\omega_0^2\theta=0\). This is an
exceedingly useful approximation, as harmonic oscillators are the building block of most Physical theories, but what if we want to relax our approximation a little bit and take the next non-zero term
in the Taylor expansion? We can certainly do that, but the resulting differential equation is now as difficult to solve as the original un-approximated simple pendulum equation (it requires elliptic
integrals), and thus it is not a very useful way to proceed.
What we can do instead is to say that we have small oscillations means we are starting with initial conditions \[\theta(0)=\varepsilon \quad \dot\theta(0)=0\] where \(\varepsilon\) is a small number.
We don't know what the solution will be, but we know that it will depend on \(\varepsilon\), so we can expand our solution as a Taylor series in \(\varepsilon\): \[\theta(t, \varepsilon) = \theta_0
(t) + \varepsilon \theta_1(t)+\varepsilon^2 \theta_2(t) + \varepsilon^3\theta_3(t)+\ldots\] where \(\theta_0\), \(\theta_1\), etc do not depend on \(\varepsilon\). And if \(\varepsilon\) is indeed
small, the higher terms in this expansion are going to be negligible, and \(\theta_1\) will be a small correction to \(\theta_0\), \(\theta_2\) will be a small correction to \(\theta_1\), and so on.
The equation of the simple pendulum thus becomes \[\left( \ddot\theta_0(t) + \varepsilon \ddot \theta_1(t) +\ldots \right) + \omega_0^2 \left[ \left( \theta_0(t) + \varepsilon \theta_1(t) +\ldots \
right) -\frac{\left( \theta_0(t) + \varepsilon \theta_1(t) +\ldots \right)^3}{6} + \ldots\right]=0 \] and the initial conditions \[\theta_0(0) + \varepsilon \theta_1(0)+\varepsilon^2 \theta_2(0) + \
varepsilon^3\theta_3(0)+\ldots=\varepsilon\] \[\dot \theta_0(0) + \varepsilon \dot \theta_1(0)+\varepsilon^2 \dot \theta_2(0) + \varepsilon^3 \dot \theta_3(0)+\ldots=0 \; .\] This equation must hold
for every possible value of \(\varepsilon\), which means that we can group terms with different powers of \(\varepsilon\) and obtain differential equation that needs to be satisfied independently.
For the zeroth power of \(\varepsilon\) we get \[\ddot\theta_0(t)+\omega_0^2 \left[ \theta_0(t) - \frac{\theta_0^3(t)}{6}+\frac{\theta_0^5(t)}{120} + \ldots \right]=0 \rightarrow \ddot\theta_0(t)+\
omega_0^2 \sin \theta_0(t)=0\] with initial conditions \(\theta_0(0)=0\) and \(\dot\theta_0(0)=0\). With these initial conditions the differential equation is trivial to solve, as the solution is \(\
theta_0(t)=0\), which is exactly the zero-order solution we found above.
For the first power of \(\varepsilon\) (and keeping in mind that \(\theta_0(t)=0\)) we get \[\ddot \theta_1(t)+\omega_0^2 \left[ \theta_1(t) \right]=0 \] with initial conditions \(\theta_1(0)=1\) and
\(\dot\theta_1(0)=0\), which is (not surprisingly) a simple harmonic oscillator with solution \(\theta_1(t) = \cos (\omega_0 t)\), so (to first order) we get \[\theta(t, \varepsilon) \simeq \
varepsilon \cos(\omega_0 t) .\] For the second power of \(\varepsilon\) we get \( \ddot\theta_2(t) +\omega_0^2 \left[ \theta_2(t) \right] =0 \), with initial conditions \(\theta_2(0)=0\) and \(\dot\
theta_2(0)=0\), which gives us a trivial \(\theta_2(t)=0\).
For the third power of \(\varepsilon\) we get something more interesting: \[ \ddot \theta_3(t) + \omega_0^2 \left[ \theta_3(t) - \frac{\theta_1^3(t)}{6} \right]=0 \rightarrow \ddot \theta_3(t) + \
omega_0^2 \left[ \theta_3(t) - \frac{\cos^3(\omega_0 t)}{6} \right]=0 \] with the usual initial conditions \(\theta_3(0)=0\) and \(\dot\theta_3(0)=0\). To solve this differential equation we first
rewrite it as \[ \ddot \theta_3(t) + \omega_0^2 \theta_3(t) = \omega_0^2 \frac{3 \cos(\omega_0 t)+\cos(3\omega_0 t)}{24} \] And then perform a Laplace transform \[ s^2 L[\theta_3]+\omega_0^2 L[\
theta_3] = \frac{\omega_0^2}{24} \left[ 3\frac{s}{s^2 + \omega_0^2}+ \frac{s}{s^2 +(3 \omega_0)^2} \right] \] \[ \rightarrow L[\theta_3] = \frac{\omega_0^2}{24} \left[ 3\frac{s}{(s^2 + \omega_0^2)^2}
+ \frac{s}{(s^2 +9 \omega_0^2)(s^2+\omega_0^2)} \right] \] \[\rightarrow \theta_3(t) = \frac{\omega_0^2}{24} \left[ 3 \frac{t \sin(\omega_0 t)}{2 \omega_0} + \frac{\cos(\omega_0 t)-\cos(3 \omega_0
t)}{8 \omega_0^2} \right] \; ,\] and thus (to third order) \[\theta(t,\varepsilon)\simeq \left( \varepsilon + \frac{\varepsilon^3}{192} \right)\cos(\omega_0 t) - \frac{\varepsilon^3}{192} \cos(3\
omega_0 t)+\frac{3 \omega_0 \varepsilon^3}{48} t \sin(\omega_0 t) \, .\] We could of course keep going with higher and higher terms, but what we got here deserves some discussion. First of all we got
a term proportional to \(\cos(\omega_0 t)\), which will act as a (small) correction to the amplitude of our first-order approximation. We also got a term proportional to \(\cos(3\omega_0 t)\),
meaning that our pendulum is not oscillating at a single frequency anymore. If you have never see this, it might look surprising at first sight, but the fact that the oscillations of a simple
pendulum have a small component at 3 times the natural frequency is something one can easily see in an experiment, so it doesn't bother us at all.
What comes as a troubling surprise is the term proportional to \(t \sin(\omega_0 t)\). This term doesn't just oscillate, but grows with time, which makes no Physical sense at all! It is very tempting
to discard this term as unphysical and forget about it, but it is instructive to look at it carefully, understand what it is happening, and find a less arbitrary way to deal with it.
Is this divergence a hole in our theory of simple pendula? Should we throw Newtonian mechanics away because of it? No, no need to panic. This infinity is just an artefact of our perturbative
expansion. If we were to calculate all the corrections arising for all powers of \(\varepsilon\), their sum would give us the correct full solution to our original problem. But what can happen is
that terms that appear at one perturbative order will cancel with terms that appear at another perturbative order. Nothing too scary. Newtonian mechanics is safe!
But, short of calculating all the infinite perturbative corrections, what can we do to get a solution that makes sense?
The Poincaré-Lindstedt method
As noted before, the differential equation describing a simple pendulum can be solved in terms of elliptic integrals. We are not going to do it here, but we will compare this exact solution with the
approximated one we found above:
We can see that the approximated solution matches the exact one very nicely at short times. At larger times the approximated one starts to grow due to the \(t\sin(\omega_0 t)\) term, but something
else is happening: the two oscillatory solutions are not in phase anymore!
This is our hint of what went wrong: by forcing our approximate solution to only contain frequencies that are integer multiples of \(\omega_0\), we were essentially trying to fit a square peg into a
round hole. Mathematically, we got a harmonic oscillator driven exactly at resonance, which is bound to keep accumulating energy forever. What we didn't take into account is that also the natural
frequency of oscillation can change with the oscillation amplitude \(\varepsilon\), and thus we need to expand in a Taylor series the frequency too: \[\omega(\varepsilon) = \omega_0 + \varepsilon \
omega_1 + \varepsilon^2 \omega_2 + \ldots\] An important detail is that in the simple pendulum we wish to find approximated solutions for, we have \(\omega_0\) (the natural frequency for small
oscillations), not \(\omega\), so we rewrite this expansion as \[\omega_0 = \omega(\varepsilon) - \varepsilon \omega_1 - \varepsilon^2 \omega_2 + \ldots\] and substitute it in, to obtain a new
perturbative equation: \[\left( \ddot\theta_0(t) + \varepsilon \ddot \theta_1(t) +\ldots \right) + \left( \omega - \varepsilon \omega_1 - \ldots \right)^2 \left[ \left( \theta_0(t) + \varepsilon \
theta_1(t) +\ldots \right) +\right.\] \[\left. -\frac{\left( \theta_0(t) + \varepsilon \theta_1(t) +\ldots \right)^3}{6} + \ldots\right]=0 \; . \] Repeating the very same steps as before: for the
zeroth power of \(\varepsilon\) we get \[\ddot\theta_0(t)+\omega^2 \left[ \theta_0(t) - \frac{\theta_0^3(t)}{6}+\frac{\theta_0^5(t)}{120} + \ldots \right]=0 \rightarrow \ddot\theta_0(t)+\omega^2 \sin
\theta_0(t)=0\] with initial conditions \(\theta_0(0)=0\) and \(\dot\theta_0(0)=0\), which once again leads to \(\theta_0(t)=0\).
To simplify our calculations going further, take the perturbative equation and write down explicitly that \(\theta_0=0\) \[\left( \varepsilon \ddot \theta_1(t) +\ldots \right) + \left( \omega - \
varepsilon \omega_1 - \ldots \right)^2 \left[ \left( \varepsilon \theta_1(t) +\ldots \right) -\frac{\left( \varepsilon \theta_1(t) +\ldots \right)^3}{6} + \ldots\right]=0 \; . \]
For the first power of \(\varepsilon\) we get \(\ddot \theta_1(t)+\omega^2 \left[ \theta_1(t) \right]=0 \) with initial conditions \(\theta_1(0)=1\) and \(\dot\theta_1(0)=0\), which is an harmonic
oscillator. This a good sanity check that we haven't done anything crazy, as at the first order we obtain exactly the small oscillation solution as before, i.e. \(\theta_1(t) = \cos (\omega t)\),
with \(\omega=\omega_0 \).
For the second power of \(\varepsilon\) we get \[\ddot \theta_2(t)+\omega^2 \left[ \theta_2(t) \right]-2 \omega \omega _1 \theta_1=0 \rightarrow \ddot \theta_2(t)+\omega^2 \left[ \theta_2(t) \right]=
2 \omega \omega _1 \cos(\omega t) \] with initial conditions \(\theta_2(0)=0\) and \(\dot\theta_2(0)=0\). This is, once again, a harmonic oscillator driven at resonance, which will diverge. What is
different from before is that now we have the freedom to choose \(\omega_1\) such that the problem goes away, i.e. we can choose \(\omega_1=0\) and thus have \(\theta_2=0\).
Once again the interesting stuff will happen for the third power of \(\varepsilon\): \[ \ddot \theta_3(t) +\omega^2 \theta_3(t) - 2 \omega \omega_1 \theta_2(t) - 2 \omega \omega_2 \theta_1(t) +\
omega_1^2 \theta_1(t) - \frac{\omega^2}{6}\theta_1^3(t) =0 \; , \] which, as \(\omega_1=0\), simplifies to \[ \ddot \theta_3(t) +\omega^2 \theta_3(t) - 2 \omega \omega_2 \theta_1(t) - \frac{\omega^2}
{6}\theta_1^3(t) =0 \; , \] \[ \rightarrow \ddot \theta_3(t) +\omega^2 \theta_3(t) - 2 \omega \omega_2 \cos(\omega t) - \frac{\omega^2}{6}\cos^3(\omega t) =0 \] \[ \rightarrow \ddot \theta_3(t) +\
omega^2 \theta_3(t) - 2 \omega \omega_2 \cos(\omega t) - \frac{\omega^2}{6} \frac{3 \cos(\omega t)+\cos(3\omega t)}{4} =0 \] \[ \rightarrow \ddot \theta_3(t) +\omega^2 \theta_3(t) - (2 \omega \
omega_2+\frac{\omega^2}{8}) \cos(\omega t) - \frac{\omega^2}{24} \cos(3\omega t)=0 \; . \] Apart from a slightly different coefficient, this equation is not much different from the one we got before,
so we know that the term proportional to \(\cos(\omega t)\) is going to create troubles. What it is different now is that we have the freedom to choose \(\omega_2\), and if we choose \(\omega_2=-\
omega/16\) (the minus sign is just an artefact of how we wrote things and will create no problem later) the problematic term disappear, leaving us with \[ \ddot \theta_3(t) +\omega^2 \theta_3(t) = \
frac{\omega^2}{24} \cos(3\omega t) \rightarrow \theta_3(t) = \frac{\cos(\omega t)-\cos(3 \omega t)}{192} \; . \] And thus, to third order, we have \[ \theta(t,\varepsilon) \simeq \left(\varepsilon +
\frac{\varepsilon^2}{192}\right) \cos(\omega t)-\varepsilon^3 \frac{\cos(3 \omega t)}{192} \; , \] and \( \omega(\varepsilon) \simeq \omega_0 +\varepsilon^2 \omega_2 = \omega_0 -\frac{\varepsilon^2 \
omega}{192} \). But since we are only looking up to second order, higher orders in the \(\omega\) expansion on the right hand side can be neglected, and we have \(\omega(\varepsilon)\simeq (1-\frac{\
varepsilon^2 }{192})\omega_0 \).
Notice that this technique, i.e. to let a parameter vary and then adjust it to cancel the spurious infinities, is closely related to the process of renormalization common in field theory.
Perturbation theory
After this very long preamble we now only need to write things in a slightly more general form, but we have already done all the hard work.
Let's assume our problem is described by the differential equation \[O f = \varepsilon g(f) \; ,\] where \(O\) is any differential operator (for the simple pendulum case it was a second time
derivative), \(f\) is our unknown solution, \(g\) is an analytic function representing the forcing, and \(\varepsilon\) is just us making the scale factor of the forcing explicit. We also assume we
know how to solve \(O f =0\) and that the problem is in the forcing (like it was for the simple pendulum). If the forcing (i.e. \(\varepsilon\)) is small we can expand our unknown solution as a
Taylor series in powers of \(\varepsilon\) as \(f=f_0 + \varepsilon f_1 + \ldots \), and since \(g\) is analytic we can also expand it as \(g(f) = \sum_n a_n f^n\), leading to: \[ O \left( f_0 + \
varepsilon f_1 + \ldots \right) = \varepsilon \sum_n a_n \left( f_0 + \varepsilon f_1 + \ldots \right)^n \; . \] The zeroth order is, not surprisingly, \(O f_0 =0\) which we assumed we know how to
The first order is: \[ O f_1 = \sum_n a_n f_0^n = g(f_0) \; , \] which is hopefully easier to solve than the full equation. Higher terms can be obtained following the same procedure.
Another common situation is when it is not the forcing that is a problem, but the differential operator itself. If the problematic operator \(O\) is close enough to a simpler operator \(O_0\) we know
how to deal with, we can write \[ O f =0 \rightarrow (O_0+\varepsilon O_1) f=0 \rightarrow (O_0+\varepsilon O_1) (f_0 + \varepsilon f_1 + \ldots)=0 \; . \] The zeroth order is, not surprisingly, \
(O_0 f_0=0\), which we assumed we know how to solve.
The first order is \(O_0 f_1+O_1 f_0=0\). Now the "difficult" part of the differential operator (\(O_1\)) is not acting on the unknown \(f_1\), but on the known \(f_0\), hopefully leading to a
simpler differential equation than the original one.
The second order is \(O_0 f_2+O_1 f_1=0\). Since now we know \(f_1\), this is also (hopefully) a simpler differential equations than the original one.
Higher order terms can be found in a similar way.
The reader might have noticed I never mentioned quantum mechanics even once here. This is by design. Perturbation theory can be applied to quantum mechanics (and indeed has been applied to it very
successfully many times), but there is nothing particularly special about it. Perturbation theory is about differential equations, whether they arise in the context of classical or quantum mechanics
(or any other context).
Contact details :
• Postal address:
University of Exeter
Physics building
Stocker Road
EX4 4QL
United Kingdom
• E-mail: j.bertolotti@exeter.ac.uk | {"url":"https://jacopobertolotti.com/PerturbationIntro.html","timestamp":"2024-11-09T07:46:48Z","content_type":"application/xhtml+xml","content_length":"21221","record_id":"<urn:uuid:dafa21cc-426b-45e4-b71b-6e213bc7a1bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00562.warc.gz"} |
Cite as
Konrad Majewski, Tomáš Masařík, Jana Novotná, Karolina Okrasa, Marcin Pilipczuk, Paweł Rzążewski, and Marek Sokołowski. Max Weight Independent Set in Graphs with No Long Claws: An Analog of the
Gyárfás' Path Argument. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 93:1-93:19,
Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
Copy BibTex To Clipboard
author = {Majewski, Konrad and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Novotn\'{a}, Jana and Okrasa, Karolina and Pilipczuk, Marcin and Rz\k{a}\.{z}ewski, Pawe{\l} and Soko{\l}owski, Marek},
title = {{Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gy\'{a}rf\'{a}s' Path Argument}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {93:1--93:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-235-8},
ISSN = {1868-8969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.93},
URN = {urn:nbn:de:0030-drops-164343},
doi = {10.4230/LIPIcs.ICALP.2022.93},
annote = {Keywords: Max Independent Set, subdivided claw, QPTAS, subexponential-time algorithm} | {"url":"https://drops.dagstuhl.de/search?term=Novotn%C3%A1%2C%20Jana","timestamp":"2024-11-04T02:46:22Z","content_type":"text/html","content_length":"125211","record_id":"<urn:uuid:0566fdad-26cd-4cf5-b7aa-e29779b8e007>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00009.warc.gz"} |
How-to Guide for Virus/Trojan/Malware Removal
Oct 21, 2004
looks interesting. did some quick research on it and people seem to generally say it has a very high false-positive rate....
Nice tool there, thanks for the link!
Jul 8, 2005
Nov 30, 2007
Very good guide. However, Spybot's best days are long long past gone. Now it's slow, buggy and can't detect meaningful spyware anymore.
Very good guide. However, Spybot's best days are long long past gone. Now it's slow, buggy and can't detect meaningful spyware anymore.
It has it's uses still. I will admit it's not the greatest at detection and removal anymore. But the web browser "inoculation" is still useful.
May 28, 2001
Nope, never even heard of it until you just mentioned it.
MBAM, it seems, is becoming too popular. The last three Vundo variants I've seen I have been unable to install or run MBAM at all. For some reason, though, I've been able to install SuperAntiSpyware
and actually run it to remove them....
Very good guide. However, Spybot's best days are long long past gone. Now it's slow, buggy and can't detect meaningful spyware anymore.
Yes it does. While I agree it's not as good as MalwareBytes or SAS, it still does fine remnants that the others use. We still use it on the bench when we have time for doing lengthy scans on clients
infected rigs...and I still see it pickup a few items the others missed. Yes legit items...not just useless things like cookies.
While a lot of the time I only have time to run a few scans...so I'll use the big boys....I know they'll not get 100% of the stuff, but I'm fine with the idea that they seem to get the majority of
the stuff, and what they miss will be ineffective in coming back. But sometimes we have the time to run a few scans using a few more tools...and I'll notice Spybot still picks up a few more things.
MBAM, it seems, is becoming too popular. The last three Vundo variants I've seen I have been unable to install or run MBAM at all. For some reason, though, I've been able to install
SuperAntiSpyware and actually run it to remove them....
Yeah I ran into a few machines several months ago that were hit with that REALLY nasty new variant of Windows Police Pro. That variant was the toughest one I've ever run across. Anyways, one of it's
features...blocked installs, and running...of your usual cleaning programs.
MalwareBytes for example, I renamed the installer "installmwb.com"...and then the executable to launch it from mbam.exe to something like mally.com...and run it.
SymantecUnhookexec.inf restores some of the shell\open\command functions which are hosed by the rogues.
Just to add to "the shotgun effect".....a heavily infested rig that came into our bench a few days ago, our main break/fixit guy Dave has been working on it. Got whacked with a trojan that did the
"log off" as soon as you logged in, the usual "userinit" replacement fixed that.
Anyways...MalwareBytes found a ton of stuff, replaced their McAfee with NOD32v4, found a ton of stuff, SAS found a bit more, Spybot found a bit more..
This morning Dave ran the Microsoft Malware Removal Tool..that thing most people never run, it updates via Microsoft Updates. You can run a manual scan with it, start==>run==>MRT. It found another 1/
2 dozen trojans.
Microsoft Security Essentials probably would have found the same, as I'm guessing they share definitions somewhat....but in case I didn't mention the MRT in prior posts in this thread...I've seen it
pickup things other good programs miss...several times. It's built into Windows...why not use it!
Jun 24, 2004
Latest malware we've run into here at work sets IE to use a proxy under the LAN settings. After using mbam to remove, users were complaining they couldn't get into the internet afterwards. We had
been removing profiles, which fixed this issue, but found the proxy setting while looking into things a little closer. Easy fix if someone is having issues after a removal.
Yeah I ran into a few machines several months ago that were hit with that REALLY nasty new variant of Windows Police Pro. That variant was the toughest one I've ever run across. Anyways, one of
it's features...blocked installs, and running...of your usual cleaning programs.
MalwareBytes for example, I renamed the installer "installmwb.com"...and then the executable to launch it from mbam.exe to something like mally.com...and run it.
SymantecUnhookexec.inf restores some of the shell\open\command functions which are hosed by the rogues.
Yeah, I couldn't get MBAM to run at all, even after renaming. That SymantecUnHookExec.inf is a freaking awesome though.
Aug 30, 2007
man I hate to be such a downer but I really cant agree with the philosophy of this thread.
Not all spyware can be removed, and from a security standpoint if thats true in my mind it means that no spyware can be removed, or at least any machine once infected and then "cleaned" can never
be trusted again.
Computers have been abstracted to such a high degree by so many people and for so many people that planting something in a spot nobody checks isn't impossible. Furthermore its in the spyware
authors best intrest to not be found, and not be noticed. Spam-relays are intrested in routing spam and if the code thinks your after it, maybe it modifies its routine to only run between 2AM and
6AM to avoid being discovered. Who knows. These are the same people who've invented (annoyingly strong) polymorphic programs, the only effect solution is a reformat (and even then, there are
known ROM-firmware infections).
The only way to really remove spyware as an issue is to remove the vectors it comes in over. To avoid malicious actions against you via computing you need to modify your behavior. Convienience is
often the enemy of security; use long passwords and never use the same one twice, check your (inbound and outbound) port activity from time to time, check the certificates/encryption that people
claim to be using (MD5 has been cracked!), dont trust every google result you find, and make sure you're updated!!
I could not DISAGREE more. An operating system can be cleaned and can be trusted again. Under your same "scrutiny" no machine could be trusted at anytime no matter the location. Machines get infected
because people are stupid. That is why social engineering is the prominent way of infection these days. Also what happens if a server shows a sign of infection. Throw it away, it cannot be trusted?
Learn your operating systems, understand them, check traffic logs. I would put money up that if you ran the same scan/clean that CC posted, you would find crap on your network. Which in turn means no
one could ever trust your network or its info EVER again. What a fallacy. Also as far as data backups go, they cannot be trusted either if they came from an infected machine. EVER again. I do not see
your logic. Stop all traffic that may be a security threat. Every port on your system is a security threat. Every firewall hanging in a demarc is a security threat. Every IT tech that does not go
through EVERY line of EVERY log is a security threat.
BTW- the biggest security threats are the ones none of your "appliances" catch. Even TOR is succeptable to Man in the middle now. Oh noes!!!!!!!!!1111111111111 What are we to do? Put back on our
tinfoil hats and hop for the best?
Good thread Captain. Keep up the good fight.
Added rkill to the list of recommended tools. had this one save the day this morning. killed Personal Security rogue on a client's computer from a remote session.
Jan 3, 2009
It's very potent if you don't know what you're doing. The same could be said about many of these tools, but GMER is particularly touchy (it was built for an online community such as ours and the
person who wrote it was there to train others on it), and as such I'd caution people who are just giving it a try. It is extremely useful in some situations (for those that don't know, it is somewhat
like a very powerful HijackThis with scanning features built in) but I have seen people on forums just click away on it and completely screw their system up to the point where they had to be walked
through a repair method or just decide to reformat.
May 28, 2001
It's very potent if you don't know what you're doing. The same could be said about many of these tools, but GMER is particularly touchy (it was built for an online community such as ours and the
person who wrote it was there to train others on it), and as such I'd caution people who are just giving it a try. It is extremely useful in some situations (for those that don't know, it is
somewhat like a very powerful HijackThis with scanning features built in) but I have seen people on forums just click away on it and completely screw their system up to the point where they had
to be walked through a repair method or just decide to reformat.
It BSOD my pc on 2 separate occasions (scanning both times). Fortunately, I was able to boot back into Windows without any issues.
I went with Rootrepeal and Sophos Anti-rootkit.
Here's another good little utility for our "USB bag of tricks"..
Fix Win
Specific to the topic of this thread, this utility has some tools to re-enable/fix some items that some malware whacks on your system, such as regedit, task manager, tcp/winsock, etc.
Here's another good little utility for our "USB bag of tricks"..
Fix Win
Specific to the topic of this thread, this utility has some tools to re-enable/fix some items that some malware whacks on your system, such as regedit, task manager, tcp/winsock, etc.
added to the list of nifty tools. looks like it will come in handy.
I don't know about anyone else but I've seen a rash of new Vundo variants the last week. I've had to clean five client PCs already and two of my relatives . . . . sheesh.....
Apr 15, 2002
That is awesome, thanks for the heads up on that. Adding to the OP.
Nice, adding that to the list, too.
Oct 25, 2007
Got infected by AKM Antivirus 2010 Pro. My PC just went heywire. Nothing worked! This thing popped up all over. All my apps and games reported as infected. This is bogus 100%. It would not terminate,
so I used Killbox.exe to delete it.
so, you used the tools in the guide and followed the steps outlined and this did not remove the infection? Only Killbox.exe was able to terminate the process?
Oct 25, 2007
Yes, only Killbox teminated the process. Then I followed steps on site in my other post and all went back to normal, no need to re-install OS.
Looks interesting. Have you had a chance to play with it in the wild?
Looks interesting. Have you had a chance to play with it in the wild?
Not on a "tanked" machine yet....ran it on 2x bench rigs at the office just to see its behavior. Runs very quick..pounds certain system files 'n directories, and then rips through the registry. I'm
not sure yet how frequently its updated....as in how frequently one should download a fresh version to keep on their drive.
Jun 21, 2002
has anyone used HitmanPro?
I haven't. Is it free?
Posted via [H] Mobile Device
I haven't. Is it free?
Posted via [H] Mobile Device
There's a 30 day free version, after that..it's scan and report only. It uses several AV vendors engines wrapped up in one package...cloud based along with Eset, GData, AntiVir I think...and I forget
the others...I think it was 5x total.
There's a 30 day free version, after that..it's scan and report only. It uses several AV vendors engines wrapped up in one package...cloud based along with Eset, GData, AntiVir I think...and I
forget the others...I think it was 5x total.
Sounds snazzy . . . but . . . it's not free . . . . so not for this list . . . .
Jun 1, 2003
When you run combofix doesn't it say not to download from several sites and combofix.org is one of them
When you run combofix doesn't it say not to download from several sites and combofix.org is one of them
Ummm, not sure where you're getting that information. ComboFix's website is | {"url":"https://hardforum.com/threads/how-to-guide-for-virus-trojan-malware-removal.1426658/page-2","timestamp":"2024-11-08T18:50:34Z","content_type":"text/html","content_length":"207432","record_id":"<urn:uuid:2fb3c343-732d-4b5b-b849-ea6a5ef3ea99>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00280.warc.gz"} |
Time Complexity of Multiplication
Open-Source Internship opportunity by OpenGenus for programmers. Apply now.
The Brute force Time Complexity of Multiplication operation is O(logM x logM) while the Theoretical limit of Time Complexity of Multiplication operation is O(logM x loglogM) for multiplying number
number M x M.
Operation Brute Force TC Optimal TC
Multiplication O(logM x logM) O(logM x loglogM)
Beyond this point, we will assume that the numbers involved in Multiplication have N bits.
Multiplication is defined as repeated addition so if addition is O(N) time operation, then multiplication is O(N^2) time operation.
This might seem to be simple as it is the fundamental definition of multiplication, but this is not the case.
This table summarizes how the time complexity of multiplication operation improved over the years:
Algorithm Complexity Year Notes
School Multiplication O(N^2) 100 BC -
Russian Peasant Method O(N^2 * logN) 1000 AD -
Karatsuba algorithm O(N^1.58) 1960 -
Toom Cook multiplication O(N^1.46) 1963 -
Schonhage Strassen algorithm O(N * logN * loglogN) 1971 FFT
Furer's algorithm O(N * logN * 2^O(log*N)) 2007 -
DKSS Algorithm O(N * logN * 2^O(log*N)) 2008 Modular arithmetic
Harvey, Hoeven, Lecerf O(N * logN * 2^3 log*N) 2015 Mersenne primes
Covanov and Thomé O(N * logN * 2^2 log*N) 2015 Fermat primes
Harvey and van der Hoeven O(N * logN) March 2019 Possible end
Note that the time complexity is for multiplying two N digit numbers.
It was widely believed that multiplication cannot be done in less than O(N^2) time but in 1960, the first break-through came with Karatsuba Algorithm which has a time complexity of O(N^1.58).
With recent break-through, it was thought that the theoretical limit for Multiplication will be O(N logN) but it was not proved.
Finally, in 2019, an algorithm has been developed that has a time complexity of O(N logN) for multiplication. It is a galactic algorithm which means it beats other existing algorithm only for
exponentially large numbers (which are not used in practice).
Hence, we know that multiplication has a time complexity of O(N logN) while usual algorithms in practice have a time complexity of O(N^2).
With this article at OpenGenus, you must have the complete idea of Time Complexity of Multiplication. Enjoy. | {"url":"https://iq.opengenus.org/time-complexity-of-multiplication/","timestamp":"2024-11-11T07:52:39Z","content_type":"text/html","content_length":"68887","record_id":"<urn:uuid:d347b613-9af0-4251-ac95-631d2dac43cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00201.warc.gz"} |
Quantum Machine Learning - How Its Transforming AI - Computer Repairs
Quantum computing is an exciting new field that leverages the properties of quantum physics to perform computations in radically new ways. Quantum machine learning is an emerging subdomain that
brings the strengths of quantum computing to bear on machine learning algorithms. This combination promises to unlock unprecedented capabilities in artificial intelligence systems.
In this in-depth article, I explore the key concepts, current state of research, and future potential of quantum machine learning. The article covers:
Overview of Quantum Computing
• Brief history and current state of quantum computing
• Key properties of quantum physics like superposition, entanglement, and interference
• How quantum computers work at a high level
Why Quantum Computing for Machine Learning
• Limitations of classical machine learning algorithms
• Advantages of quantum computing for machine learning
• Performance improvements over classical algorithms
Types of Quantum Machine Learning Algorithms
• Overview of major categories like quantum neural networks, quantum support vector machines
• Explanation of key algorithms like quantum annealing, amplitude amplification
• Real world examples and case studies
Implementing Quantum Machine Learning Systems
• Current hardware platforms and challenges
• Hybrid classical-quantum approaches
• Software frameworks for developing quantum ML applications
The Future of Quantum Machine Learning
• Potential breakthrough applications in areas like optimization, pattern recognition
• Timelines for maturing quantum computing hardware
• Long-term societal impacts of powerful quantum AI
Let’s get started exploring this fascinating field driving the next evolution in artificial intelligence!
Overview of Quantum Computing
Before delving into quantum machine learning, it’s helpful to understand the basics of quantum computing.
Quantum computing is based on quantum physics – the behavior and properties of energy and matter at atomic and subatomic scales. Certain quantum physics phenomena allow computational tasks to be
performed in completely different ways from classical computing.
A few key quantum properties enable these new kinds of computations:
• Superposition – Quantum bits (qubits) can represent a combination of 1 and 0 states simultaneously. This enables massive parallelism by allowing computations on all possible states at once.
• Entanglement – Qubits can be correlated with each other, so that actions performed on one affects the others instantaneously. This enables large numbers of qubits to work together to represent
entangled states.
• Interference – Qubit states can interfere with each other constructively or destructively. This allows alternative computational paths to amplify or cancel out results.
Quantum computers harness these properties using quantum circuits operating on qubits to perform specialized algorithms. While exotic, quantum computing is becoming more practical with real world
systems like those built by D-Wave, IBM, and Google.
However, significant hardware challenges remain to build fault-tolerant, general purpose quantum computers. Most practical applications today involve hybrid quantum-classical computing or special
purpose quantum processing units.
Nevertheless, we are entering the NISQ (Noisy Intermediate Scale Quantum) era where we can apply quantum computers to specialized problems and begin exploring quantum advantages. Quantum machine
learning is one of the most promising application domains.
Why Quantum Computing for Machine Learning
Quantum computing offers significant potential benefits for machine learning, which is driving a great deal of research into quantum ML algorithms.
Some key limitations of classical machine learning that quantum capabilities can help overcome:
• Classical ML algorithms are often computationally intensive and slow to train against large datasets.
• Many advanced ML techniques like deep learning rely on optimization algorithms with lots of iterative computations.
• Future ML applications like self-driving cars require real-time responses not feasible with classical computing.
Quantum computers have inherent advantages that make them well-suited for machine learning:
• Quantum parallelism allows evaluating probability distributions over many variables simultaneously.
• Entangled qubit states can represent ML model parameters compactly.
• Interference enables direct optimization and tuning of probabilities.
• Quantum annealing can find global optima for hard optimization problems.
These advantages can translate into orders-of-magnitude improvements in performance for some ML applications. Researchers have already demonstrated quantum machine learning algorithms that can:
• Train deep neural networks exponentially faster than classical algorithms.
• Find optima using quantum annealing that are unfindable by classical computers.
• Perform principal component analysis better than classical methods.
As quantum computers scale up over the next decades, we can expect transformative impacts on practical artificial intelligence. Next we’ll survey the landscape of quantum machine learning algorithms.
Types of Quantum Machine Learning Algorithms
There are two main genres of quantum machine learning algorithms:
Quantum Versions of Classical ML Models
These aim to quantum enhance proven classical ML techniques:
• Quantum neural networks – Quantum circuits to represent neuron weights and activations.
• Quantum support vector machines – Use amplitude encoding to efficiently find optimal separators.
• Quantum matrix inversion – Exponentially faster solving linear systems using Harrow-Hassidim-Lloyd algorithm.
• Quantum principal component analysis – Efficient eigenvalue estimation through phase estimation algorithm.
Such algorithms can achieve polynomial or exponential speedups over their classical counterparts for training or inference.
Novel Quantum ML Models
These explore fundamentally new ML models only possible on quantum hardware:
• Quantum Boltzmann machines – Use quantum fluctuations to escape local minima when optimizing.
• Quantum Helmholtz machines – Enable quantum generative models with intrinsic thermal noise.
• Quantum reservoir computing – Exploit quantum dynamics for temporal pattern recognition.
• Quantum generative adversarial networks – Combine with classical networks for enhanced generative modeling.
Such algorithms employ unique quantum properties with no classical analogue. They demonstrate the potential to tackle previously unsolvable ML problems.
Some noteworthy examples include:
• Quantum annealing for combinatorial optimization – Finds global minima through quantum fluctuations.
• Amplitude amplification for enhanced sampling – Selectively amplify desired quantum states.
• Quantum LSA for natural language processing – Uncover latent semantic structure in documents.
While still in early stages, quantum machine learning is racing towards real-world utility. Next we’ll look at how these algorithms get implemented on quantum hardware.
Implementing Quantum Machine Learning Systems
The practical application of quantum machine learning algorithms involves surmounting hardware challenges and limitations.
Current quantum computing platforms available include:
• Superconducting quantum annealers – Specialized for optimization (D-Wave).
• Superconducting universal quantum computers – Noisy, up to ~100 qubits (IBM, Google, Rigetti).
• Trapped ion quantum computers – Stable, but only ~10 qubits so far.
With limited qubit counts, noise susceptibility, and lack of error correction, near-term quantum computers have significant constraints.
This has catalyzed intense research into hybrid quantum-classical and NISQ optimized approaches:
• Small quantum circuit modules integrated into larger deep learning models.
• Quantum preprocessing and subroutines called by classical algorithms.
• Special quantum data encodings to minimize qubit overhead.
• Error mitigation techniques combining classical and quantum repetitions.
On the software side, frameworks like Cirq, PyQuil, Qiskit, and TensorFlow Quantum help streamline quantum ML development. They provide höher-level abstractions and interoperability with leading ML
While daunting, these challenges are surmountable. We are on the path to scalable, reliable quantum machine learning systems – with revolutionary AI implications.
The Future of Quantum Machine Learning
Quantum machine learning finds itself today in a similar state as classical ML decades ago. The basic theoretical foundations are established, but practical applications remain limited.
However, if the impressive growth trajectory of classical ML continues for quantum, we can make some predictions about the future:
• In 5-10 years, we’ll see quantum ML algorithms reliably demonstrating quantum advantages over classical counterparts. Rigorous benchmarking on standardized tasks will quantify performance
• In 10-15 years, we’ll have scalable, error-corrected quantum computers supporting practical quantum ML applications. These systems will transform sectors like finance, drug discovery, and
transportation through superior optimization and generative modeling capabilities.
• In 15-20 years, enterprise adoption of quantum machine learning services will disrupt industries. Advances in natural language processing, computer vision, simulation, and planning will enable
ubiquitous quantum AI assistants and robots.
While the timing remains uncertain, the transformative impacts of quantum machine learning eventually becoming mainstream are undeniable. Everything from scientific research to the internet and
everyday automation stands to be revolutionized.
The era of quantum artificial intelligence looms on the horizon. Powerful quantum machine learning capabilities will profoundly expand what computers can help humanity analyze, create, and optimize.
The 21st century will belong to quantum artificial intelligence.
This deep dive has showcased how quantum computing will unlock extraordinary new machine learning capabilities. Quantum properties like superposition, entanglement, and interference enable algorithms
unforgeable on classical systems.
We explored quantum enhanced versions of proven ML models as well as entirely novel quantum algorithms. While hardware remains limited today, rapid progress is bringing practical quantum advantages
within reach.
The future is inexorably trending towards scalable, powerful quantum machine learning systems integrated into all facets of society. Quantum computing will open up new frontiers in artificial
intelligence – and human knowledge. | {"url":"https://itfix.org.uk/quantum-machine-learning-how-its-transforming-ai/","timestamp":"2024-11-03T15:56:08Z","content_type":"text/html","content_length":"144098","record_id":"<urn:uuid:df4a208a-54e1-4b07-abe7-0758aebc3b89>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00382.warc.gz"} |
ANOVA in EdgeR
Last seen 4.0 years ago
United States
Hi. I am trying to figure out how to do an ANOVA like test with my data in edgeR.
My data has 2 variables: treatment (2 levels) and time (5 levels). I'd like to test for the effect of treatment, time, and the interaction between the 2. Something like y ~ time + treatment +
I can see two possible ways to do this in edgeR:
1) combine treatment and time into a single factor and then test as in section 3.3.1 of the edgeR vignette
the treatment effect contrast would be: (Treat1.Time1 + Treat1.Time2 + Treat1.Time3 + Treat1.Time4 + Treat1.Time5) - (Treat2.Time1 + Treat2.Time2 + Treat2.Time3 + Treat2.Time4 + Treat2.Time5)
the time effect contrast would be: (Treat1.Time1 + Treat2.Time1) - (Treat1.Time2 + Treat2.Time2) - (Treat1.Time3 + Treat2.Time3) - (Treat1.Time4 + Treat2.Time4) - (Treat1.Time5 + Treat2.Time5)
the interaction effect contrast would be: (Treat1.Time1 - Treat2.Time1) - (Treat1.Time2 - Treat2.Time2) - (Treat1.Time3 - Treat2.Time3) - (Treat1.Time4 - Treat2.Time4) - (Treat1.Time5 - Treat2.Time5)
test this model with glmFit
2) use glmLRT as in the vignette 3.3.2 and 3.3.3 sections
for interaction effect:
design = model.matrix(~Treat * Time, data = myData)
This gives me a design matrix with columns:
(Intercept) treat2 time2 time3 time4 time5 treat2:time2 treat2:time3 treat2:time4 treat2:time5
I think glmLRT(fit, coef=7:10) gives me the treatment by time interaction effect
for main effect of time: glmLRT(fit, coef=3:6)
for main effect of treatment: glmLRT(fit, coef=2)
Is this correct??? and which method is 'best'?
Here's my R session info:
R Session Info:
R version 3.2.2 (2015-08-14)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 7 x64 (build 7601) Service Pack 1
[1] LC_COLLATE=English_United States.1252 LC_CTYPE=English_United States.1252 LC_MONETARY=English_United States.1252 LC_NUMERIC=C LC_TIME=English_United States.1252
attached base packages:
[1] stats4 parallel stats graphics grDevices utils datasets methods base
other attached packages:
[1] DESeq2_1.10.0 RcppArmadillo_0.6.200.2.0 Rcpp_0.12.1 qvalue_2.2.0 goseq_1.22.0 geneLenDataBase_1.6.0 BiasedUrn_1.06.1 WGCNA_1.48 RSQLite_1.0.0
[10] DBI_0.3.1 fastcluster_1.1.16 dynamicTreeCut_1.62 ggplot2_1.0.1 rgl_0.95.1367 EDASeq_2.4.0 ShortRead_1.28.0 GenomicAlignments_1.6.1 SummarizedExperiment_1.0.1
[19] Rsamtools_1.22.0 GenomicRanges_1.22.1 GenomeInfoDb_1.6.1 Biostrings_2.38.0 XVector_0.10.0 IRanges_2.4.1 S4Vectors_0.8.1 BiocParallel_1.4.0 Biobase_2.30.0
[28] BiocGenerics_0.16.1 cluster_2.0.3 edgeR_3.12.0 limma_3.26.2 rj_2.0.4-2
loaded via a namespace (and not attached):
[1] splines_3.2.2 foreach_1.4.3 R.utils_2.1.0 Formula_1.2-1 aroma.light_3.0.0 latticeExtra_0.6-26 impute_1.44.0 lattice_0.20-33 digest_0.6.8 RColorBrewer_1.1-2
[11] colorspace_1.2-6 Matrix_1.2-2 preprocessCore_1.32.0 R.oo_1.19.0 plyr_1.8.3 XML_3.98-1.3 biomaRt_2.26.0 genefilter_1.52.0 zlibbioc_1.16.0 xtable_1.8-0
[21] GO.db_3.2.2 scales_0.3.0 annotate_1.48.0 mgcv_1.8-9 GenomicFeatures_1.22.4 nnet_7.3-11 proto_0.3-10 survival_2.38-3 magrittr_1.5 R.methodsS3_1.7.0
[31] doParallel_1.0.10 nlme_3.1-122 MASS_7.3-44 hwriter_1.3.2 foreign_0.8-66 tools_3.2.2 matrixStats_0.15.0 stringr_1.0.0 locfit_1.5-9.1 munsell_0.4.2
[41] AnnotationDbi_1.32.0 lambda.r_1.1.7 DESeq_1.22.0 futile.logger_1.4.1 grid_3.2.2 RCurl_1.95-4.7 iterators_1.0.8 bitops_1.0-6 gtable_0.1.2 codetools_0.2-14
[51] reshape2_1.4.1 gridExtra_2.0.0 rtracklayer_1.30.1 Hmisc_3.17-0 futile.options_1.0.0 stringi_1.0-1 geneplotter_1.48.0 rpart_4.1-10 acepack_1.3-3.3
Entering edit mode
I don't believe the second method is correct either. The interaction coefficients are correct, but the coefficients for the "main effects" are actually the coefficients for the time effects in
treatment 1 and the treatment effects in time 1.
Entering edit mode
I agree with Ryan here. The naming scheme used by model.matrix is quite misleading.
In general, if you want to test for main effects, you'll have to first check that the interaction terms are not significant. Otherwise, you could end up with a situation where, e.g., Treat1 is
downregulated relative to Treat2 at an earlier time point, but is upregulated relative to Treat2 at a later time point. Trying to consider the "main effect" of treatment would make no sense here, as
the effects at the two time points are in opposing directions.
I prefer to use a one-way layout (i.e., the parametrization in your first approach) and compare pairs of groups separately, e.g., Treat1.Time1 against Treat2.Time1, Treat1.Time2 against Treat2.Time2,
and so on. Then I intersect the DE lists from the pairwise comparisons between treatments at some or all time points, to get a set of genes that change robustly and in the same direction in response
to treatment at each time point. The other strategy is to do all those pairwise contrasts at once in an ANOVA-style comparison, and only pick the DE genes that have the same sign of the log-fold
change across all of the contrasts.
Entering edit mode
Despite what it may seem, this is not a simple question to answer. For starters, there are many definitions of "overall". If you want to compute a "main effect" or an average effect of time or
treatment, then see my comment to Ryan's answer. If you just want to get genes that have any DE with respect to time, then you could set coef=3:10 in glmLRT. This will test for DE between any of the
time points, conditional on the treatment. Similarly, if you want to get genes that have any DE with respect to treatment, then you could set coef=c(2, 7:10). In both cases, you'll get a single
p-value indicating whether time or treatment has any effect on gene expression. You'll have to pick through the log-fold changes to figure out exactly what the effect is, though - for example, a
treatment effect might only be occurring at one or two time points, but may be sufficient to be significant. | {"url":"https://support.bioconductor.org/p/75127/","timestamp":"2024-11-03T13:08:22Z","content_type":"text/html","content_length":"30197","record_id":"<urn:uuid:0fba346f-dc3b-40c5-bb2c-636d114beda9>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00241.warc.gz"} |
Definition of nth
Nth (adjective)
Used to indicate an unspecified place in a series or sequence of items or events.
From latin n-th, from numerus "number".
1. The nth digit of pi is 3.
2. The nth term of the sequence is 4.
3. The nth item in the list is missing.
4. The nth prime number is 11.
5. The nth element in the array is 5. | {"url":"https://zendict.com/en-en/nth.html","timestamp":"2024-11-10T02:55:49Z","content_type":"text/html","content_length":"6352","record_id":"<urn:uuid:28732c67-8f20-4949-86b3-556f6dc605b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00112.warc.gz"} |
20 Commaon Things That Are 3 Feet Long - Dimension heat
Are you scratching your head and wondering what things are 3 feet long? Don’t worry, I got you covered.
3 feet is a long and common measurement we encounter often. 3 feet is actually equal to 36 inches and one yard. Many everyday items, such as half of a refrigerator, half of a bed, and even half of a
man, are good references to understand 3 feet. Read along, and you’ll find many more everyday examples of the 3-foot measurement. And it will never bother you again. So, let’s get straight into it.
How Long is 3 Feet in Other Units of Measurements?
Conversions are important. Sometimes, people use different units for the same measurement such as instead of saying 3 feet, one may say 1 yard or 36 inches. That’s why it’s a bit important that you
have a basic how know of all the units used for length measurement:
3 feet in inches= 36 inches (since 1 foot = 12 inches)
3 feet in centimeters=91.44 cm (since 1 foot = 30.48 cm)
3 feet in meters= 0.9144 meters (since 1 foot = 0.3048 meters)
3 feet in yards= 1 yard (since 1 yard = 3 feet)
Ta-da, I hope you got the basics, and that’s what’s important for real-life tasks. Let’s move to the practical stuff of how you’re going to understand 3 feet with the help of everyday items.
Things That are 3 Feet Long:
Many everyday items are 3 feet long. To name a few:
1. 3-4 Years Old Child
2. Half a Bed
3. Emperor Penguins
4. Yard Stick
5. 3 Bowling Pins
6. Irish Wolfhound
7. Half a Man
8. Half a Refrigerator
9. Kitchen Countertop
10. The Standard Width of a Doorway
11. Traffic Cones
12. Baseball Bat
13. Three Rulers
14. 3 Hacksaw Blades
15. Width of a Small Dining Table
16. A Guitar
17. Length of a Golden Retriever
Things That are 3 Feet Long:
Let’s dive into the details of each example.
3-4 Years Old Child:
This is one of the cutest ages of life. You love everyone, and everyone loves you. The only tension in this stage is after playing one game; you’re worried about what would be the next game. Right?
Well, we’re not here to discuss our age of 3-4 years old but how you can use it as a reference for 3 feet.
Generally, male children are 35.5 to 43.5 inches long during this growth period of life. On the other hand, the female kids are 34.5 inches to 42.5 inches long. So, as you know, 36 inches is equal to
3 feet. Roughly, you can use a 3-4-year-old child as the best example to understand 3 feet.
It’s because children are the most common thing around you 24/7. That is why I have begun the list with this example. Wherever you see a 3-4-year-old kid, recall the fact that it’s probably 3 feet
long. This way, you’ll never worry about measuring 3 feet again without any measuring tool.
Half a Bed:
Bed is one of the most common items in every household these days. You use your bed twice or thrice a day and see it as many times as you enter your bedroom. A bed is indeed a blessing. Many people
are addicted so badly to their beds that they can’t sleep anywhere else.
But did you ever wonder that you could use a bed as a reference to measure 3 feet? Surprisingly, the total length of a bed is 6 feet, and a half of it is 3 feet. However, it can vary depending on the
company type and the size of the bed. But, generally, it’s a good-to-go item in the list of things that are 3 feet long.
Emperor Penguins:
Emperor penguins are known for their beauty, strength, and survival skills. You must have seen them in real life or reel life if you’re an Antarctica fan.
The elegant penguins are 36 to 48 inches tall and weigh 46 to 99 pounds. And you must have remembered till yet that 36 inches are equivalent to 3 feet. So, next time you spot a penguin, you better
know that this marvel is a perfect reference for 3 feet.
Yard Stick:
Yardstick is one of the essential tools in construction, crafting, and educational environments. If you’re a DIY ( do it yourself) person, you must know the size of an average yardstick. It’s
generally 3 feet long.
Unlike the standard ruler, a yardstick is good for measuring long lengths for bigger projects.
3 Bowling Pins:
Ever thrown all the bowling pins in one go? Remember that feeling? It’s amazing, those winning moments and the dopamine rush, oh my goodness!
Well, a single bowling pin is 1.25 feet long. You need 3 bowling pins to visualize 3 feet. So, if you’re an adventurous peep, try vertically standing 3 bowling pins on top of each other, or else use
the power of imagination and imagine 3 bowling pins, and you’ll easily understand Things That Are 3 Feet Long
Irish Wolfhound:
It’s one of the tallest dog breeds. They are known for their long height. The Irish wolfhound is generally 30 to 35 inches long. It’s probably equal to 2.5 to 3 feet and your go-to reference to
understand 3 feet.
You can consider this example if you have ever seen an Irish wolfhound. Or else consider the example of the Golden Retriever as it’s more common, and anyone can easily imagine its length. Both golden
retrievers and Irish wolfhounds can be used as a reference in the list of things that are 3 feet long.
Half a Man:
Half a man is one of the most common things that are 3 feet long. Yes, you read that right. Generally, an ideal height for most men is 5.5-6 feet. So, if you consider half of it, boom, you’ll get 3
And that’s something we’ll see daily. Nextime, you see a man, consider half of it, and that’s it.
Half a Refrigerator:
Refrigerators are a staple in today’s life. You can easily see them in schools, colleges, offices ( cafeterias) and homes. The height of a refrigerator is something you can imagine in seconds. Right?
Generally, the height of a refrigerator is 6 feet, which is why half of a refrigerator is 3 feet. Easy now?
But, yes, please keep in mind the fact that I’m talking about common refrigerators here. Otherwise, you better know refrigerators vary greatly in size depending on the company and capacity. However,
on average, you can use them as a reference in the list of things that are 3 feet long.
Kitchen Countertop:
Especially in the US and generally across the world, the kitchen countertops are made to stand at a height that is 3 feet above the floor. This height ensures a safe distance between kitchen items
and the floor.
Though, it’s not restricted, and a homeowner has the authority to adjust it according to their needs.
The Standard Width of a Doorway:
The standard width of a doorway is important to consider when building homes. In the US, the standard width of a residential doorway is 36 inches or 3 feet.
3 feet is long enough to ensure that both people and furniture can easily pass. It’s one of the good things that are 3 feet long. Why? Because it’s common and you can easily imagine it.
Traffic Cones:
Ever seen a traffic cone in a construction zone? Obviously, you must have, as it’s common and often seen. If you observe a little, you can see that a traffic cone is 36 inches or 3 feet tall. Yes,
you can say it’s like the length of a 3-4-year-old kid. Now, it would be easier for you to remember.
So, nextime if you’re on the road and immediately want to know what things are 3 feet long. Look around, and you might see a traffic cone. And your brain will automatically remind you that yes, it’s
3 feet long.
Baseball Bat:
Baseball is one of the most popular sports across the globe. The length of a baseball bat is generally 3 feet or 36 inches long. Well, to be very honest, it varies between 24 inches and 42 inches
depending on the style and the age of the player. Generally, you can use it as a gauge to understand 3 feet.
One thing I would like to say is that perfection is a myth. And it doesn’t exist in life neither in measurements. You have to adjust to the imperfections. All the examples described in the article
can be used as a reference and are exactly or roughly 3 feet long. The purpose is to help you understand the length without headaches when you don’t have a measuring tool in hand. Okay, let’s move
Three Rulers:
The world of measurements and scale are essential for each other. 12-inch rulers are common. If you align three 12-inch rulers in a row, you’ll get 36 inches, and that’s exactly 3 feet.
Let’s say call your three homies or gather your school besties and ask them to bring their scales with them. Now align all three scales in a row, and boom, you’ll get 3 feet.
This way, you’ll not only understand Things That Are 3 Feet Long, but it’ll be incorporated in your mind, and hopefully, you’ll never forget it again. We humans remember more by practical
implementation than by knowledge. So, whatever you learn in this article, please try practicing one of the examples practically, and 3 feet will never bother you again. It’s my promise.
3 Hacksaw Blades:
If you’re a mechanic or involved in DIY projects, you probably know Hacksaw blades. They are used to cut metal and other hard materials. A hacksaw blade is generally one foot long.
And to use it as a reference for 3 feet, visualize 3 Hacksaw blades in a row. And boom, you’ll get 3 feet.
Nextime, when you don’t have a measuring tape in hand and urgently wanna know how long it is 3 feet. Quickly imagine 3 hacksaw blades and ta-da, you got it.
Width of a Small Dining Table:
Small dining tables are common. They are space efficient and often used by small families or placed in the kitchen to immediately enjoy the hot, yummy meals as soon as they are prepared.
You can easily understand the length of a small dining table by comparing it to a 3-4 year old kid. Dining tables are common. You can easily find them in homes, cafes, and restaurants.
A Guitar:
The guitar, one of the common musical instruments, is roughly 3 feet long.
Length of a Golden Retriever:
Golden retrievers are exceptional dogs. They are loved across the world. The ideal length for a golden retriever is 3 feet! Feel free to use it as a reference to understand 3 feet.
Can I understand 3 feet with the help of a scale?
Yes, you can. But you need 3 12-inch scales for this purpose. One foot is equal to 36 inches. So, if you consider three 12-inch scales you can easily get 36 feet, and it’s one of the good references
to measure how big 3 feet is.
There are many things that are 3 Feet long. I have enlisted only a few everyday items. If you know more, feel free to add it to the comment section.
Let’s revise a bit. You can use two dogs as a reference for 3 feet. What are those? A golden retriever and an Irish wolfhound.
Next, you can use the following common things in half to understand 3 feet. For example, a half-adult man, a half refrigerator, and a half bed. Do you know more things that are 6 feet long? Consider
dividing them in half, and you can use them as a reference for both 6 feet and 3 feet.
Similarly, you can use the following one-foot items 3 in number to understand 3 feet. For example, 3 bowling pins, 3 12-inch rulers, 3 hacksaw blades, and that’s it.
I hope you get some value from our blogs. See you next time. Thank you. Happy measurements. | {"url":"https://dimensionheat.com/things-that-are-3-feet-long/","timestamp":"2024-11-15T04:18:08Z","content_type":"text/html","content_length":"139432","record_id":"<urn:uuid:b77470cb-1502-4def-9463-9fe99585ca85>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00375.warc.gz"} |
Einstein must be wrong: How general relativity fails to explain the universe
As new and powerful telescopes gather fresh data about the universe, they reveal the limits of older theories like Einstein's relativity.
(Image credit: ESA/Hubble & NASA; Acknowledgment: Judy Schmidt)
Einstein's theory of gravity — general relativity — has been very successful for more than a century. However, it has theoretical shortcomings. This is not surprising: the theory predicts its own
failure at spacetime singularities inside black holes — and the Big Bang itself.
Unlike physical theories describing the other three fundamental forces in physics — the electromagnetic and the strong and weak nuclear interactions — the general theory of relativity has only been
tested in weak gravity.
Deviations of gravity from general relativity are by no means excluded nor tested everywhere in the universe. And, according to theoretical physicists, deviation must happen.
Related: 10 discoveries that prove Einstein was right about the universe — and 1 that proves him wrong
Deviations and quantum mechanics
(Image credit: Arthur Eddington/Philosophical Transactions of the Royal Society)
According to Einstein, our universe originated in a Big Bang. Other singularities hide inside black holes: Space and time cease to have meaning there, while quantities such as energy density and
pressure become infinite. These signal that Einstein's theory is failing there and must be replaced with a more fundamental one.
Naively, spacetime singularities should be resolved by quantum mechanics, which apply at very small scales.
Quantum physics relies on two simple ideas: point particles make no sense; and the Heisenberg uncertainty principle, which states that one can never know the value of certain pairs of quantities with
absolute precision — for example, the position and velocity of a particle. This is because particles should not be thought of as points but as waves; at small scales they behave as waves of matter.
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This is enough to understand that a theory that embraces both general relativity and quantum physics should be free of such pathologies. However, all attempts to blend general relativity and quantum
physics necessarily introduce deviations from Einstein's theory.
Therefore, Einstein's gravity cannot be the ultimate theory of gravity. Indeed, it was not long after the introduction of general relativity by Einstein in 1915 that Arthur Eddington, best known for
verifying this theory in the 1919 solar eclipse, started searching for alternatives just to see how things could be different.
Einstein's theory has survived all tests to date, accurately predicting various results from the precession of Mercury's orbit to the existence of gravitational waves. So, where are these deviations
from general relativity hiding?
A century of research has given us the standard model of cosmology known as the Λ-Cold Dark Matter (ΛCDM) model. Here, Λ stands for either Einstein’s famous cosmological constant or a mysterious dark
energy with similar properties.
Dark energy was introduced ad hoc by astronomers to explain the acceleration of the cosmic expansion. Despite fitting cosmological data extremely well until recently, the ΛCDM model is spectacularly
incomplete and unsatisfactory from the theoretical point of view.
In the past five years, it has also faced severe observational tensions. The Hubble constant, which determines the age and the distance scale in the universe, can be measured in the early universe
using the cosmic microwave background and in the late universe using supernovae as standard candles.
These two measurements give incompatible results. Even more important, the nature of the main ingredients of the ΛCDM model — dark energy, dark matter and the field driving early universe inflation
(a very brief period of extremely fast expansion originating the seeds for galaxies and galaxy clusters) — remains a mystery.
From the observational point of view, the most compelling motivation for modified gravity is the acceleration of the universe discovered in 1998 with Type Ia supernovae, whose luminosity is dimmed by
this acceleration. The ΛCDM model based on general relativity postulates an extremely exotic dark energy with negative pressure permeating the universe.
Problem is, this dark energy has no physical justification. Its nature is completely unknown, although a plethora of models has been proposed. The proposed alternative to dark energy is a
cosmological constant Λ which, according to quantum-mechanical back-of-the-envelope (but questionable) calculations, should be huge.
However, Λ must instead be incredibly fine-tuned to a tiny value to fit the cosmological observations. If dark energy exists, our ignorance of its nature is deeply troubling.
Alternatives to Einstein's theory
(Image credit: Sloan Digital Sky Survey/NASA)
Could it be that troubles arise, instead, from wrongly trying to fit the cosmological observations into general relativity, like fitting a person into a pair of trousers that are too small? That we
are observing the first deviations from general relativity while the mysterious dark energy simply does not exist?
This idea, first proposed by researchers at the University of Naples, has gained tremendous popularity while the contending dark energy camp remains vigorous.
How can we tell? Deviations from Einstein gravity are constrained by solar system experiments, the recent observations of gravitational waves and the near-horizon images of black holes.
There is now a large literature on theories of gravity alternative to general relativity, going back to Eddington's 1923 early investigations. A very popular class of alternatives is the so-called
scalar-tensor gravity. It is conceptually very simple since it only introduces one additional ingredient (a scalar field corresponding to the simplest, spinless, particle) to Einstein's geometric
description of gravity.
The consequences of this program, however, are far from trivial. A striking phenomenon is the "chameleon effect," consisting of the fact that these theories can disguise themselves as general
relativity in high-density environments (such as in stars or in the solar system) while deviating strongly from it in the low-density environment of cosmology.
As a result, the extra (gravitational) field is effectively absent in the first type of systems, disguising itself as a chameleon does, and is felt only at the largest (cosmological) scales.
The current situation
Nowadays the spectrum of alternatives to Einstein gravity has widened dramatically. Even adding a single massive scalar excitation (namely, a spin-zero particle) to Einstein gravity — and keeping the
resulting equations "simple" to avoid some known fatal instabilities — has resulted in the much wider class of Horndeski theories, and subsequent generalizations.
Theorists have spent the last decade extracting physical consequences from these theories. The recent detections of gravitational waves have provided a way to constrain the physical class of
modifications of Einstein gravity allowed.
However, much work still needs to be done, with the hope that future advances in multi-messenger astronomy lead to discovering modifications of general relativity where gravity is extremely strong.
This edited article is republished from The Conversation under a Creative Commons license. Read the original article.
Professor, Physics & Astronomy, Bishop's University
PhD in Astrophysics, supervisor George F.R. Ellis, worked in relativity, cosmology, and alternative theories of gravity for 30 years, been at Bishop's University for 16 years, currently full
professor in the Physics & Astronomy Department. Author of 210 refereed journal articles and 7 books, funded by NSERC and volunteered extensively for NSERC, the Canadian Association of Physicists,
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Quadratic Equation - Formula, Examples | Quadratic Formula - [[company name]] [[target location]], [[stateabr]]
Quadratic Equation Formula, Examples
If you’re starting to solve quadratic equations, we are excited regarding your journey in math! This is actually where the amusing part begins!
The data can appear overwhelming at start. But, offer yourself a bit of grace and space so there’s no pressure or stress while working through these questions. To master quadratic equations like an
expert, you will require a good sense of humor, patience, and good understanding.
Now, let’s begin learning!
What Is the Quadratic Equation?
At its heart, a quadratic equation is a arithmetic formula that describes various situations in which the rate of deviation is quadratic or relative to the square of some variable.
However it may look like an abstract concept, it is just an algebraic equation stated like a linear equation. It generally has two solutions and uses complex roots to solve them, one positive root
and one negative, employing the quadratic formula. Working out both the roots the answer to which will be zero.
Definition of a Quadratic Equation
Foremost, bear in mind that a quadratic expression is a polynomial equation that comprises of a quadratic function. It is a second-degree equation, and its standard form is:
ax2 + bx + c
Where “a,” “b,” and “c” are variables. We can utilize this equation to figure out x if we plug these terms into the quadratic formula! (We’ll look at it next.)
Ever quadratic equations can be written like this, that makes solving them easy, comparatively speaking.
Example of a quadratic equation
Let’s contrast the ensuing equation to the subsequent formula:
x2 + 5x + 6 = 0
As we can see, there are two variables and an independent term, and one of the variables is squared. Thus, linked to the quadratic formula, we can confidently say this is a quadratic equation.
Usually, you can find these types of equations when scaling a parabola, that is a U-shaped curve that can be graphed on an XY axis with the data that a quadratic equation gives us.
Now that we know what quadratic equations are and what they appear like, let’s move on to solving them.
How to Work on a Quadratic Equation Employing the Quadratic Formula
While quadratic equations might seem very intricate initially, they can be broken down into several easy steps utilizing a simple formula. The formula for working out quadratic equations includes
creating the equal terms and utilizing rudimental algebraic operations like multiplication and division to get 2 solutions.
After all operations have been carried out, we can work out the units of the variable. The solution take us one step closer to discover solutions to our original problem.
Steps to Working on a Quadratic Equation Using the Quadratic Formula
Let’s promptly put in the original quadratic equation once more so we don’t overlook what it looks like
ax2 + bx + c=0
Ahead of solving anything, keep in mind to detach the variables on one side of the equation. Here are the 3 steps to work on a quadratic equation.
Step 1: Note the equation in standard mode.
If there are terms on either side of the equation, sum all alike terms on one side, so the left-hand side of the equation totals to zero, just like the standard mode of a quadratic equation.
Step 2: Factor the equation if feasible
The standard equation you will end up with must be factored, generally using the perfect square method. If it isn’t feasible, put the terms in the quadratic formula, that will be your closest friend
for figuring out quadratic equations. The quadratic formula looks similar to this:
Every terms coincide to the same terms in a standard form of a quadratic equation. You’ll be using this significantly, so it is smart move to remember it.
Step 3: Apply the zero product rule and figure out the linear equation to eliminate possibilities.
Now once you possess 2 terms resulting in zero, solve them to obtain two solutions for x. We get two answers because the solution for a square root can be both negative or positive.
Example 1
2x2 + 4x - x2 = 5
Now, let’s break down this equation. Primarily, simplify and put it in the standard form.
x2 + 4x - 5 = 0
Next, let's recognize the terms. If we contrast these to a standard quadratic equation, we will get the coefficients of x as ensuing:
To work out quadratic equations, let's replace this into the quadratic formula and work out “+/-” to involve both square root.
We work on the second-degree equation to obtain:
Next, let’s clarify the square root to get two linear equations and figure out:
x=-4+62 x=-4-62
x = 1 x = -5
Next, you have your result! You can check your workings by checking these terms with the first equation.
12 + (4*1) - 5 = 0
1 + 4 - 5 = 0
-52 + (4*-5) - 5 = 0
25 - 20 - 5 = 0
That's it! You've worked out your first quadratic equation using the quadratic formula! Congrats!
Example 2
Let's try another example.
3x2 + 13x = 10
Let’s begin, put it in the standard form so it is equivalent zero.
3x2 + 13x - 10 = 0
To work on this, we will substitute in the values like this:
a = 3
b = 13
c = -10
figure out x employing the quadratic formula!
Let’s simplify this as far as possible by working it out exactly like we executed in the prior example. Work out all simple equations step by step.
You can figure out x by considering the positive and negative square roots.
x=-13+176 x=-13-176
x=46 x=-306
x=23 x=-5
Now, you have your solution! You can revise your workings using substitution.
3*(2/3)2 + (13*2/3) - 10 = 0
4/3 + 26/3 - 10 = 0
30/3 - 10 = 0
10 - 10 = 0
3*-52 + (13*-5) - 10 = 0
75 - 65 - 10 =0
And that's it! You will solve quadratic equations like a pro with some practice and patience!
Granted this summary of quadratic equations and their basic formula, children can now go head on against this challenging topic with confidence. By beginning with this easy explanation, learners gain
a firm foundation before undertaking further complex concepts later in their studies.
Grade Potential Can Assist You with the Quadratic Equation
If you are fighting to understand these ideas, you might require a math instructor to guide you. It is better to ask for guidance before you get behind.
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CT Specs – Part 2 – (Class P Application)
CT Specs – Part 2 – Class P Application
Part 1 of this blog presented an overview of Australian & IEC standards for Class P type Current Transformer (CT) specifications. This blog presents the application of Class P type CTs in actual
site conditions. This process can be confusing due to the diverse methodology presented in the literature.
I have conducted an in-depth review of at least 20 publications, including the 3 major relay manufacturers’ manuals, to prepare this blog. It has taken more than a week of full-time study to present
this blog clearly and logically!
Overview of Class P CT Specification
An example of a Class P CT specification is as given below:
300/5 A 5P10 15 VA
300 A – Primary Rated Current
5 A – Secondary Rated Current
5P10 – 5% error at 10 times Rated Current [Accuracy Limit Factor (ALF) = 10 ]
15 VA – Power rating of the Current Transformer
Unlike the usual electrical equipment power rating, the VA rating of the CT is used to specify the maximum voltage which can be developed at the secondary terminals. This determines the burden
(load) that can be connected across the secondary terminals without exceeding the specified accuracy. The following calculations illustrate the concept.
Rated Power (Sn) = 15 VA
Rated Secondary Current (In) = 5 A
Rated Burden (Rb) = Sn / In^2 = 15 / (5 x 5) = 0.6 ohm
Maximum Allowable Terminal Voltage (Vs-max) = ALF x In x Rb
= 10 x 5 x 0.6 = 30 V
Maximum allowable secondary winding terminal voltage (Vs-max) is the key figure to evaluate the CT accuracy for a given operating condition. If the secondary terminal voltage exceeds this value,
then the CT is ‘saturated’. The increase in error is dramatic when the CT is saturated.
Example 1 – Actual Burden is less than the Rated Burden
For the CT specified above, it is given that the short circuit current (Isc) on the primary side of the CT is 3600A. The actual connected burden (Ract) is 0.3 ohm. Evaluate the performance of the
Secondary side Current (Is-act) = Isc / CT ratio = 3600 / (300/5) = 60 A
Secondary Terminal Voltage (Vs-act) = Is-act x Ract = 60 x 0.3 = 18 V
Secondary Terminal voltage ‘Vs-act’ is less than ‘Vs-max’.
So, the CT error will be less than 5%. The CT performance is ‘Okay’.
‘Vs-act’ is well within ‘Vs-max’, even though the current in the CT is more than ‘ALF times the Rated Current’. The reason is that the ‘Actual burden’ is less than the ‘Rated burden’. Burden has a
significant influence on the accuracy.
Example 2 – Actual Burden is greater than the Rated Burden
For the CT specified above, it is given that the short circuit current (Isc) on the primary side of the CT is 2400A. The actual connected burden (Ract) is 1.0 ohm. Evaluate the performance of the
Secondary side Current (Is-act) = Isc / CT ratio = 2400 / (300/5) = 40 A
Secondary Terminal Voltage (Vs-act) = Is-act x Ract = 40 x 1.0 = 40 V
Secondary Terminal voltage ‘Vs-act’ is greater than ‘Vs-max’.
So, the CT error is greater than 5%. The CT performance is ‘NOT Okay’!
In this example, ‘Vs-act’ is greater than ‘Vs-max’, even though the current in the CT less than ‘ALF times the Rated Current’. The reason being is that the ‘Actual burden’ is greater than the ‘Rated
burden’. This can happen in practice in the case of long CT cables and incorrect CT sizing.
The specified ALF is not written in Stone
The specified Accuracy Limit Factor (ALF) for the CT is not written in stone! It can be exceeded. But there is a caveat.
If the short circuit current (Isc) exceeds ‘ALF times the Rated Current’, then we need to consider the ‘Maximum Allowable Induced Voltage in the Secondary Winding’ rather than the ‘Maximum Allowable
Secondary Terminal Voltage’. This is necessary, since we need to consider the additional voltage drop in the secondary winding resistance for CT currents higher than the rated ALF.
How do we find the ‘Induced Voltage in the Secondary Winding’?
Good question! As per the electric circuit theory, ‘Induced Voltage’ in a winding is the ‘Terminal Voltage’ plus the ‘Voltage drop in the winding’. So, we can write the equation as below:
Es = Vs + Is Rct …. (1)
Es – Induced voltage in the secondary winding
Vs – Voltage at the secondary winding terminals
Is – Secondary current (Current flowing into the load or burden)
Rct – Secondary Winding Resistance
The Australian Standard AS 60044.1 & IEC 61869-2 standard do not specify the ‘Rct’ value for Class P CTs. Hence, the manufacturer is not obliged to provide it. The ‘Rct’ value is required to
determine the CT accuracy! So, we need to use the typical values from the literature.
The CT secondary winding resistance depends on the manufacturer’s design. However, for the estimation of CT performance we can use typical values of 0.003 Ω/turn for 5 A CTs and 0.006 Ω/turn for 1 A
Current Transformer Circuit Model
We can develop the CT circuit model based on Equation (1), as shown in Figure 1.
The burden can be modelled as an impedance, if required. With the advent microprocessor-based relays, the effect of reactance is negligible. Hence, it is a common practice to model the burden as a
resistance (Rb).
By inspection of Figure 1, the equation for ‘Induced voltage’ (Es) is written as below:
Es = Vs + Is Rct = Is Rb + Is Rct = Is (Rct + Rb) … (2)
In Figure 1, the magnetising impedance (Zm) branch has been included. This branch models the magnetic flux in the CT core. It is a non-linear impedance! Solving circuits with non-linear impedances
is not for the faint hearted. Relax! We are going to ignore it in our calculations!!
The good news is that we can find Excitation Current (Ie), without considering the magnetising impedance (Zm) in our calculations. The Excitation Current can be found using the CT Excitation
(magnetisation) curve. The CT excitation curve is a plot of Induced Voltage (Es) versus Excitation Current (Ie). This is established by tests on current transformers.
The Excitation Current (Ie) is the key value for calculating the CT error. The Excitation Current is the CT error! The percentage CT error is defined as below:
% CT Error = ( Ie / (Is + Ie) ) x 100 … (3)
Example 3 – Extending the ALF
We are given a 300/5 A, 5P10, 15 VA CT. We need to establish the suitability of the CT for a short circuit current (Isc) of 6000 A. The actual secondary burden (Ract) is given to be 0.2 Ω . The
secondary winding resistance is 0.003 Ω /turn.
Rated Power (Sn) = 15 VA
Rated Secondary Current (In) = 5 A
Rated Burden (Rb) = Sn / (In^2) = 15 / (5 x 5) = 0.6 Ω
Secondary Winding Resistance (Rct) = (0.003 Ω / turn ) x (300/5 turns) = 0.18 Ω
Maximum Allowable Induced Voltage (Es-max) = ALF x In (Rct + Rb)
= 10 x 5 x (0.18 + 0.6) = 39 V
Secondary side Current (Is-act) = Isc / CT ratio = 6000 / (300/5) = 100 A
Actual Induced Voltage (Es-act) = Is-act (Rct + Ract) = 100 (0.18 + 0.20) = 38 V
The actual induced voltage of 38 V is less than the allowable induced voltage of 39 V. Hence, the CT error will be less than 5% under short circuit conditions.
The short circuit (fault) current is two times ALF. The ‘Effective ALF‘ is 20 for the given burden, where as the ‘Nominal ALF‘ as per the CT specification is 10!
We have ignored Excitation current (Ie). It is okay since the error is less than 5%.
Is Exceeding the specified ALF Dangerous?
One might wonder what is the point in specifying the Accuracy Limit Factor (ALF) on the name plate if it can be exceeded? Will it increase the power consumption in the CT secondary winding and cause
increased heating? The important question is whether it is safe to do so?
A current transformer will reproduce the primary current as per the turns ratio. Hence, shorting secondary terminals will not result in high currents. In fact, the secondary terminals must be
shorted when the CT is ‘not in use’ in live conditions! So, decreasing the burden will not affect the secondary current and the safety.
On the contrary, it is the increase in the burden resistance which is dangerous! The high flux level due to the primary current (without the compensating secondary current) drives the CT core into
saturation. This results in high crested induced voltage in the secondary winding, which can cause arcing and insulation failure. Most catastrophic CT failures are due to an inadvertent open
circuit or poor secondary side connections.
How to calculate the actual CT Error?
Good question! I am afraid there is no good answer!
We must have the CT excitation curve to calculate the actual CT error. The CT standards AS 60044.1 / IEC 61869-2 do not specify CT Excitation curve for Class P current transformers. So, the
manufacturers are not obliged to provide them. Hence, it is not possible to perform actual CT error calculations. We can only make ‘Okay’ or ‘Not Okay’ decisions, as illustrated in Examples 1 to
3. It is unfortunate, but ‘Such is Life’!
The American CT Standard IEEE C57.13 for Class P equivalent requires the manufacturer to provide the CT secondary winding resistance and the CT excitation curve! You can calculate the CT error, if
you are in America!!
The power system textbooks do include examples of calculating the CT error, because most of the power system textbooks are from America.
I was confused myself when I started industrial consultancy in Australia after teaching in the university with American textbooks! It took me some time to realise that it is not possible to
calculate the CT error with AS 60044.1 / IEC 61860-2 Class P specifications!
CT Requirements for Overcurrent Protection
The performance requirements of current transformers are dependent on the type of protection (relay). We will consider only the overcurrent protection here.
Differential protection will be considered in a future blog as a part of Class PX type CT specifications.
Definite Time Setting
For the given trip current setting (Iset), the CT must provide large enough current for reliable operation of the relay. In other words, the corresponding secondary winding induced voltage (Es-act)
must be well below the maximum allowable induced voltage (Es-max).
The CT requirement for definite time setting is as given below:
Es-max ≥ 2 Iset (Rct + Ract) … (4)
Note that ‘Ract’ is the total of relay and connecting cable resistances. Equation (4) states that ‘Es-max’ should be greater than two times the actual induced voltage ‘Es-act’. It is a
recommendation of relay manufacturers. Such a factor may look very conservative for a practical engineer, who is used to allowing a margin of about 10 to 20% for data and calculation errors. But
there are other factors at play here. The current transformer comes into play immediately after the short circuit (fault) has occurred. The fault current has transients and, more importantly, a
direct current (DC) offset. The DC offset current causes additional flux in the core, which does not contribute to transformer action. Hence, the core saturation is higher than the steady state
conditions. So, it is important to follow relay manufacturers’ guidelines for sizing the CT.
Inverse Time Curve Setting
This is popularly called Inverse Definite Minimum Time (IDMT) characteristics. To avoid errors in relay operating time, a CT must maintain the accuracy for any possible short circuit current. This
is not often viable; hence, a practical value is to choose 20 times the current setting (Iset) of the IDMT relay.
The CT requirement for inverse time setting is as given below:
Es-max ≥ 20 Iset (Rct + Ract) … (5)
Instantaneous Setting
A factor of 1.5 to 2 times the instantaneous current setting (Iset) is used due to the fast relay operation at instantaneous settings. The factor (margin) depends on the time constant (X/R ratio) of
the network impedance. Majority of the fault points in distribution networks have low time constants, and therefore a factor of 1.5 is considered adequate. High voltage networks and feeders close
to generators will have higher X/R ratios. In such cases, it is recommended to use a factor of 2.
The CT requirement for instantaneous current setting for distribution networks is as given below:
Es-max ≥ 1.5 Iset (Rct + Ract) … (6)
Earth-Fault Relays
The procedure for earth-fault relays is the same as overcurrent relays. For residual connection, the CT burden includes the earth-fault relay and an overcurrent relay connected in series!
Saturation is not an issue in the case of earth-fault relays since earth-fault relays have low current settings. However, it is necessary to ensure that the relay operates reliably at low currents.
This is done by primary current injection tests during commissioning.
Comment on Manufacturers’ Manuals
The equations for CT requirements given above are from “ABB Technical Reference Manual for RXHL 421 and RAHL 421 relays”. This method was chosen as it is a logical extension of Equation 2 and
Example 3.
Schneider manuals express the same equations in terms of ‘Effective ALF’.
Siemens manuals have an elaborate set of equations for various network configurations. However, the results of the CT requirements are similar!
Application Example
A distribution feeder is protected by an IDMT overcurrent relay. Check the performance of the CT for the site data given below.
CT Specs: 600/1 A, 5P10, 15VA
CT cables: Cu, 2.5sqmm, 2 x 50m length
Cu Resistivity: ρ = 0.0225 Ω sqmm / m @ 75ºC
(As per Aust Std AS 3008)
Relay Burden: 0.1 VA @ 1A Input
CT Sec Res: 0.006 Ω / turn (Conservative value)
Max Fault Current: 35kA (Used for specifying thermal rating)
Rated Secondary Current (In) = 1 A, ALF = 10, Rated VA (Sn) = 15 VA
Rated Burden of the CT (Rb) = Sn / (In)^2 = 15 / (1×1) = 15 Ω
CT Secondary Winding Resistance (Rct) = 0.006 x (600/1) = 3.6 Ω
Max Allowable Induced Voltage (Es-max) = ALF x In (Rct+Rb) = 10x1x18.6 = 186 V
Cable Resistance (Rcable) = (ρ / Area) * Length = (0.0225 / 2.5) (2 x 50) = 0.9 Ω
Relay Burden (Rrelay) = Relay Input VA / I^2 = 0.1 / 1^2 = 0.1 Ω
Total Actual Burden on CT (Ract) = Rcable + Rrelay = 0.9 + 0.1 = 1 Ω
Using 120% of CT rating as the ‘pick up current’ or ‘relay setting current’
Relay Setting Current (Iset) = 1.2 x In = 1.2 x 1 = 1.2 A (1.2 x 600 = 720 A Pri)
Required Induced Voltage (Es-req) = 20 x Iset (Rct + Ract)
= 20 x 1.2 x (3.6 + 1) = 110.4 V
‘Es-max’ (186 V) is greater than ‘Es-req’ (110.4 V). The CT performance is ‘Okay’!
Alternate Criteria
ALF for rated burden (Rb) of 15 Ω is 10.
‘Effective ALF’ as per current setting is 20 x Iset / In = 20 x 1.2 A / 1 A = 24.
‘Maximum Effective ALF’ for actual burden (Rct) of 1 Ω = ALF x (Rct+Rb)/(Rct+Ract)
= 40.4
‘Maximum Effective ALF’ is greater than ‘Effective ALF’. The CT performance is okay!
The above alternate criterion is often used to evaluate the CT performance.
CT specification and performance are an important part of protection reliability. Hopefully, this blog is useful for understanding the CT concepts and their application in practice.
For CT sizing, use the current setting (Iset) based on CT ratings – not based on rated load current. Load currents can change during the life of a substation, so, it is pays to be conservative.
Modern relays have a small burden. The main burden is the CT cable. For CT sizing, use cables with a smaller cross section and longer lengths. This will help to alleviate unexpected site issues.
Refer to “CT Specification – Gold Report”!
3 Responses to CT Specs – Part 2 – (Class P Application)
1. Dear, congratulations for your blog so interesting an educative.
I have one question about the next equation:
Es-req = 20 x Iset (Rct + Ract) = 20 x 1.2 x 1 x (3.6 + 1) = 110.4 V
Why do you multiply for “1” in spite of the equation only stablish Iset.?
If we utilize a 5 A secundary current relay the equation should multiply by 5?
Thanks in advance for your help
□ Hello Antonio,
Thanks for your kind words and the comment.
You are right! Iset = 1.2 x In = 1.2 x 1 = 1.2A. Hence, the multiplication by ‘1’ again is superfluous and confusing. I will update the post accordingly.
If In = 5A, then Iset = 1.2 x In = 1.2 x 5 = 6A. There is no need to multiply by 5 again! Sorry of the confusion.
2. Dear Sesha,
I would like to seek your opinion and expertise.
The is a CT, 2000/1A, 5P20 for the 33kV GIS Switchgear at the Bus Section for Overcurrent and Earth Fault protection. The actual measured Rct is 15 ohm.
If I use the calculations as per your article above, I will get :
Es-max = ALF x In (Rt + Rb) = 20 x 1 x (15 + 15) = 600V.
Max short circuit current = 25kA.
Is-act = 25,000/(2000/1) = 12.5A
Es-act = Is-act (Rct + Ract) = 12.5 (15 + 1) = 200V.
Since Es-act < Es-max. Hence the CT is OK. Am I correct ?
Hope to hear from you.
Thank you.
Yong Ket Tai
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You can choose to do the test on the test table, and you can choose to test on a test table (with a large number of items), and you can test on the table. You may choose to do it on the table, or
you”d learn to do it, but you”m not learning to do it. The main thing that you can do on the table is to have a table and a list of the items that you”ve encountered. Here are some of the steps you”v
wantWhat Kind Of Math Is On The Ged Test? So, as I have become more and more aware of how to approach your homework, I am trying to establish the most appropriate elements for the following. I have
already got the math of the general case, but I have no idea how to approach it. Any help would be greatly appreciated. So there is a simple math discussion, but I am going to use a little bit of the
history of the book as an example. The problem is very similar to the following: This is a pretty simple example: The first thing to do is to use a function like this: function myFunction(a) { var c
= (a + 1) % 10; return c; } This function should be used in order to create a function that will take 10 numbers and return a function that takes a number and returns a function that is called by the
function. For the purpose of this code, however, I would like to make the following design. int main() { int a = 100; int b = 100; int c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10; 100 = 12; b = -10;
c1 = c1 + 1; c2 = c2 + 1; c3 = c3 + 1; c4 = c4 + 1; //I want the c1,c2 and c3. c5 = c5 + 1;//I want the b,c4,c5. c6 = c6 + 1;} I like this try this out because it is this way. But, if you don’t like
the design, then you can just do this. For example: int main(){ int a,b,c1,c3,c4; a = 100,b = 100; c1 = a; c2 = b; {c1, c3} = c4; } c1,b = c1; c2,c3 = c1 – c3; {c1, -c3} = 1; {b, -c1} = 0; } If I
have to do this way, the answer is: a = 100;b = 100 I also want the c3,c5,c6,c7,c8,c9,c10. c1 = c3;c2 = c1 c3, c5 = – c1;c4 = c2; c5, c7 = c2 – c3 c6, c10 = c2 c4 is the c5. Also, if I’m trying to
implement this in the code, I don’t know how to do this. A: I think that the problem is that the function is doing some logic. Have you tried the following? function f(){ c = 1; //start of function
for(a=1;b=2;){ //do stuff } $(‘#’+a).f(); } console.log(f()); The logic is stored in the variable a and the function is executed when you get to the function.
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The code looks like this: function a(){ //do something $(‘
‘) //$(‘
‘) } ////do stuff You can use the following to do something like this: $(‘
c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 c16 c17 c18 c19 c20 c21 c22 c23 c24 c25 c26 c27 c28 c29 c30 c31 c32 c33 c34 c35 c36 c37 c | {"url":"https://gedexamhire.com/what-kind-of-math-is-on-the-ged-test-2","timestamp":"2024-11-09T23:12:16Z","content_type":"text/html","content_length":"163472","record_id":"<urn:uuid:321a620d-c78a-42cf-8cc4-3c1ecedcc6c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00115.warc.gz"} |
Draw The Final Product Of This Series Of Reactions
Draw The Final Product Of This Series Of Reactions - Web solution for draw the structure of the final product (s) of this series of reactions. Use the wedge/hash bond tools to indicate
stereochemistry. Draw the final product of this series of reactions. Draw the final product of this series of reactions. Use the wedge and hash bond tools to indicate stereochemistry where it.
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explanation transcribed image text: Draw the final product of this series of reactions.
Use The Wedge/Hash Bond Tools To Indicate.
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Draw The Final Product Of This Series Of Reactions Related Post : | {"url":"https://sandbox.independent.com/view/draw-the-final-product-of-this-series-of-reactions.html","timestamp":"2024-11-05T17:17:25Z","content_type":"application/xhtml+xml","content_length":"24147","record_id":"<urn:uuid:5b49723c-141e-47e7-b435-2f027a64ff39>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00693.warc.gz"} |
When Will the World End?
There is an old legend about a group of eastern monks. They are the keepers of a puzzle that consists of three poles and 64 golden rings. These rings are all of different sizes and all started out on
the same pole with the largest on the bottom decreasing in size to the smallest on the top. Their task is to move the 64 rings from the first pole to the last pole subject to two conditions:
1. They may only move one ring at a time.
2. They may never move a larger ring on top of a smaller ring.
When they finish the puzzle the world will come to an end. This puzzle is usually called the Towers of Hanoi.
We can define a recursive solution to the puzzle as follows.
To move n rings from the source pole to the destination pole with a spare pole:
1. Move n-1 rings from the source pole to the spare pole using the destination as the spare.
2. Move the single ring from the source pole to the destination pole.
3. Move n-1 rings from the spare pole to the destination pole using the source pole as the spare.
You should make sure you understand this recursive solution by stepping through it with four rings. It should take you 15 moves.
This solution algorithm is naturally amenable to implementation in C. Assuming we only want to print out the moves of the individual rings, our function might look like:
void towers(int n, int src, int dest, int spare)
if(n > 0) {
towers(n - 1, src, spare, dest);
printf("Move a ring from tower %d to tower %d\n",
src, dest);
towers(n - 1, spare, dest, src);
As with all the other recursive functions and algorithms we need a stopping condition for this. As we start with n , we continually decrease it toward 0. When we hit 0, there are no rings to move and
we need do nothing. As with the algorithm, trace through this function for moving 4 rings from tower 1 to tower 3 with tower 2 as a spare. By the way, moving 64 rings at the rate of one ring a second
will take 350 million years. Maybe the legend isn't too far off after all. Move on to the next part . | {"url":"https://www.cs.drexel.edu/~bls96/ctutorial/ctut-8-4.html","timestamp":"2024-11-08T04:21:53Z","content_type":"text/html","content_length":"2657","record_id":"<urn:uuid:6ac6d794-6ad5-46b3-82d6-bce8c74eb54b>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00078.warc.gz"} |
Excel Formula for Matched Value in Cell from Table
In this guide, we will learn how to use the VLOOKUP function in Excel to return a matched value from a table based on a specified lookup value. The VLOOKUP function is a powerful tool that allows you
to search for a value in the first column of a table and retrieve a corresponding value from another column. This can be useful in various scenarios, such as finding the price of a product based on
its code or looking up a customer's name based on their ID. We will explore the parameters of the VLOOKUP function and provide a step-by-step explanation of how it works. Additionally, we will
provide examples to illustrate its usage in different scenarios. By the end of this guide, you will have a solid understanding of how to use the VLOOKUP function to return a matched value in a cell
from a table.
An Excel formula
=VLOOKUP(lookup_value, table_array, col_index_num, [range_lookup])
Formula Explanation
The VLOOKUP function is used to return a matched value from a table based on a specified lookup value.
• lookup_value: The value you want to look up in the first column of the table.
• table_array: The range of cells that represents the table.
• col_index_num: The column number in the table from which the matching value should be returned.
• range_lookup (optional): A logical value that specifies whether an exact match is required or not. If set to TRUE or omitted, an approximate match is allowed. If set to FALSE, an exact match is
Step-by-step explanation
1. The VLOOKUP function searches for the lookup_value in the first column of the table_array.
2. Once a match is found, the function returns the value from the col_index_num column in the same row.
3. The range_lookup parameter determines whether an exact match is required or not. If omitted or set to TRUE, an approximate match is allowed. If set to FALSE, an exact match is required.
4. If an exact match is not found and range_lookup is set to TRUE or omitted, the function returns the closest match that is less than or equal to the lookup_value.
For example, consider the following table:
| A | B |
| 1 | Red |
| 2 | Green |
| 3 | Blue |
If we use the formula =VLOOKUP(2, A1:B3, 2, FALSE), it will return the value "Green".
Here's how the formula works:
• The lookup_value is 2, which we want to find in the first column of the table.
• The table_array is A1:B3, which represents the table range.
• The col_index_num is 2, which indicates that we want to return the value from the second column of the table.
• Since we set range_lookup to FALSE, an exact match is required. Therefore, the formula returns the value "Green" from the second column, where the first column has a value of 2.
If we change the formula to =VLOOKUP(4, A1:B3, 2, TRUE), it will return the value "Blue".
Here's how the formula works:
• The lookup_value is 4, which we want to find in the first column of the table.
• The table_array is A1:B3, which represents the table range.
• The col_index_num is 2, which indicates that we want to return the value from the second column of the table.
• Since we set range_lookup to TRUE (or omitted it), an approximate match is allowed. The closest match that is less than or equal to 4 is 3, which corresponds to the value "Blue" in the second
column. Therefore, the formula returns "Blue". | {"url":"https://codepal.ai/excel-formula-generator/query/BtsU2Q91/excel-formula-matched-value-cell-table","timestamp":"2024-11-03T19:31:49Z","content_type":"text/html","content_length":"94930","record_id":"<urn:uuid:0e1d6b8c-3227-4487-9336-396c2f835416>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00694.warc.gz"} |
JEE Main 2024 Important Formula Book PDF Download
Formula Book for JEE Main 2024
Almost 70% of JEE exam questions can be solved with a single or group of formulas without applying any exception or theoretical concept, so won’t it be an easier task to crack the exam if you
remember all the formulas. No, we are not underrating the conceptual knowledge and understanding of PCM, as you cannot succeed in these subjects if you don’t have a knack for understanding the
concepts. But, cramming after understanding is no harm.
So, without deviating from our core mission of helping you through every thick and thins of exam, Margdarshan brings to you a crisp and reliable JEE Mains formula book having all important formulas
for JEE Mains developed after intensive research by our expert panel. This book will act as a catalyst for your JEE Mains preparation by making you more resourceful.
The brainchild of our subject experts, the formula book by Margdarshan is:
• Expert-curated list of JEE Mains important formulas without any fluff.
• A smartly maintained and well-designed hierarchy book to trace the required formula in seconds.
• Detailed and thorough formulas of each subject.
• Handy online version to start memorizing the formulas whenever you find time like traveling, minutes before sleep, etc.
• Fully proofread pdf without any mistake to the best of our knowledge.
Benefits of Using The Formula Book
• Helps you track down important formulas faster without shuffling between the pages of random books.
• You can utilize your free and interstitial time in memorizing the all-time available set of formulas.
• Saves you your precious time.
• Increases accuracy and adds to your speed.
• Saves you from hassle and chances of mistakes in deduction of formulas. You must know how to deduce, but you must also be quick in exams.
How to use
Resources are helpful only for those who know how to judiciously use them. So, the ground rules to use our formula book are:
1. Keep Handy- Always keep the book with you while preparing to find the required formula easily. You can develop separate chemistry, mathematics, and physics formula book for JEE Mains using this.
2. Make Sticky Notes- With the help of the book, make sticky notes to paste in showers, stairs, etc for a glance.
3. Use in interstices- Since the book is on mobile so you can carry it everywhere. So, whenever you get time just jump on it to utilize every second of your free time.
4. Don’t directly jump on it- We are again emphasizing that the formula book is for smart preparation only. You cannot ditch the concepts and directly cram the formulas as it will lead to a little
knowledge that you know is a dangerous thing. Turn to the book only when you know the basic idea behind the formulas.
Your preparation should be both, intense and smart, to crack one of the most demanding entrance exams of a nation. Go intense on concept clarity but smart by cramming formulas to save extra seconds
in exams. We hope that our formula book for JEE Mains is helpful for you and you can get maximum output from it. We will be always available for the best help we can make for you.
Click to download previous year paper of JEE Mains January 2020 with solution
Click to download previous year paper of JEE Mains February 2021 with solution
Click to download previous year paper of JEE Mains September 2020 with solution | {"url":"https://marg-darshan.com/study-material/notes/formula-book/formula-book-for-jee-main-2024","timestamp":"2024-11-13T17:27:39Z","content_type":"text/html","content_length":"33577","record_id":"<urn:uuid:6b89c0cf-f5e2-4577-8d49-1336223c56e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00142.warc.gz"} |
College algebra/substitution method
college algebra/substitution method Related topics: algabraic equations
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Author Message
amdrev Posted: Friday 22nd of Oct 09:30
johnson Guys , I need some help with my math assignment . It’s a really long one having almost 30 questions and consisting of topics such as college algebra/substitution method, college algebra
/substitution method and college algebra/substitution method. I’ve been working on it since the past 4 days now and still haven’t been able to crack even a single one of them. Our
teacher gave us this assignment and went for a vacation, so basically we are all on our own now. Can anyone show me the way? Can anyone solve some sample questions for me based on those
topics; such solutions would help me solve my own questions as well.
Back to top
Jahm Xjardx Posted: Saturday 23rd of Oct 19:22
I find these routine queries on almost every forum I visit. Please don’t misunderstand me. It’s just as we enter college, things change in a flash. Studies become challenging all of a
sudden. As a result, students encounter trouble in completing their homework. college algebra/substitution method in itself is a quite challenging subject. There is a program named as
Algebrator which can assist you in this situation.
From: Odense,
Denmark, EU
Back to top
Koem Posted: Sunday 24th of Oct 20:00
I allow my son to use that program Algebrator because I believe it can significantly assist him in his algebra problems. It’s been a long time since they first used that software and it
did not only help him short-term but I noticed it helped in improving his solving capabilities. The software helped him how to solve rather than helped them just to answer. It’s great !
From: Sweden
Back to top
temnes4o Posted: Monday 25th of Oct 07:11
Ok, after hearing so much about Algebrator, I think it definitely is worth a try. How do I get hold of it? Thanks!
Back to top
malhus_pitruh Posted: Monday 25th of Oct 16:08
Here is the link https://softmath.com/comparison-algebra-homework.html
From: Girona,
Back to top
Hiinidam Posted: Wednesday 27th of Oct 07:48
Algebrator is the program that I have used through several math classes - Intermediate algebra, Basic Math and Basic Math. It is a truly a great piece of algebra software. I remember of
going through difficulties with graphing parabolas, angle complements and gcf. I would simply type in a problem homework, click on Solve – and step by step solution to my algebra
homework. I highly recommend the program.
Greeley, CO,
Back to top | {"url":"https://www.softmath.com/algebra-software/long-division/college-algebrasubstitution.html","timestamp":"2024-11-08T20:14:55Z","content_type":"text/html","content_length":"42889","record_id":"<urn:uuid:389c906f-55d0-48d7-adf1-9e60f4fa1946>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00423.warc.gz"} |
dynamic pricing algorithm github
Dynamic Pricing and Inventory Management in the Presence of Online Reviews Nan Yang Miami Business School, University of Miami, nyang@bus.miami.edu Renyu Zhang New York University Shanghai,
renyu.zhang@nyu.edu January 3, 2021 We study the joint pricing and inventory management problem in the presence of online customer reviews. Dynamic pricing is a blanket term for any shopping
experience where the price of an item fluctuates based on current market conditions. info. So much so, it hurts to wrap my head around. GitHub Gist: instantly share code, notes, and snippets.
PricingHUB optimizes your pricing using its machine learning algorithms, helping you reach your business goals. Pricing in the online world is highly transparent & can be a primary driver for online
purchase. Hint, recall the tower property of conditional expectation. Their product is currently geared toward the hotel industry, and they are working toward a technology for dynamic pricing in any
industry. There have been several works on dynamic pricing DR algorithms for smart grids. On Amazon, as well as multiple other marketplaces, e-commerce stores, and sales-related businesses, dynamic
pricing is utilized by retailers to optimize product prices. 3 valuable lessons from This Is Your Brain On Uber article:. The practice however has now become an exacting science, and algorithmic
dynamic pricing is transforming transportation, E-commerce, entertainment, and a wide range of other industries. Given this, it is imperative to devise an innovative dynamic pricing DR mechanism for
smart grid systems. with Ben Berger and Michal Feldman, in WINE 2020. There are so many different approaches when it comes to optimization. The prices recommended by DDP are optimized by a
mathematical algorithm. In order to study the performances of this pricing algorithm, the software has been applied in the context of flights’ insurance. First of all, thanks a lot to all participants
for putting so much effort in the competition! Our Saas Solution is a scalable Revenue Management tool that allows you to optimise the pricing of your product catalogue to achieve different business
goals. Maximize revenue from your in-app purchases with dynamic pricing. Simply stated, dynamic pricing is a strategy businesses employ that adjusts prices based on the demand of the market. The
result is that the reinforcement learning approach emerges as promising in solving problems that arise in standard approaches. The algorithms can augment configure price quote systems, which help
salespeople more quickly quote prices based on rules automation and close deals more quickly. Dynamic Pricing Algorithm for In-App Purchases. We can then simulate the demand reaction for different
price and market scenarios, and optimize price decisions, capturing margin … The Dynamic Programming Algorithm Class Exercise Argue this is true for a 2 period problem (N=1). We analyze a
finite-horizon dynamic pricing model in which demand at each period depends on not only the current price but also past prices through reference prices. Deep Reinforcement Learning Algorithm for
Dynamic Pricing of Express Lanes with Multiple Access Locations. The Evolution of Market Power in the US Auto Industry (with Paul Grieco and Ali Yurukoglu) with Uriel Feige and Michal Feldman, in
APPROX 2019. The Dynamic Pricing Competition 2020 has come to a close. In particular, thanks to their adaptation to real- Woocommerce Dynamic Pricing table price view. What Is Dynamic Pricing? We now
formally define the regret of a dynamic pricing algorithm A. Dynamic pricing at other industries. Sweet Pricing's machine learning algorithms optimize prices for every user in real time without the
need to define complex pricing rules. By leveraging large databases it is possible to identify and isolate the effects of elasticity. The workflow of a typical pricing algorithm goes through the four
main stages: Historical data on price points and demand on particular products is consumed by the engine to be processed using the dynamic pricing algorithm. I am a Ph.D. candidate and researcher in
(Deep) Machine Learning at UIC, working with Prof. Theja Tulabandhula.My research focus is on developing Machine Learning and Deep Learning models for large scale personalization problems, including
recommender systems and natural language processing. Creating credit card numbers in R Would I get a ticket for going 85? It is useful to change in real time the price of an item and be reactive to
the demand from the market. But one dynamic pricing algorithms vendor, Pros, claims to add an average of 2% to 3% to its customers' bottom lines -- without extra administrative cost -- up to a 10%
boost for some. Scraping Amazon with RSelenium in R ... Rvest & The Luhn Algorithm. ... TA in Algorithms, 2016-2017. Dynamic Pricing Model in R Let's scrape Amazon with RSelenium. It is designed to
handle a large volume of items (tens of thousands). In more good news, Hill's team has released Aerosolve, the open-source machine-learning tool on which Airbnb's pricing algorithm relies, on the
Github code-sharing platform. On the Power and Limits of Dynamic Pricing in Combinatorial Markets. The price of petroleum-based fuels differs from place to place and is dependent on the popularity of
a particular gas station, the oil prices, and the consumer buying power in some of the cases. For e-shop operations and other retailers your pricing using its machine learning algorithms, helping you
your! 2 period problem ( N=1 ) promising in solving problems that arise in standard.... Of MLs with Multiple Access points prices recommended by DDP are optimized by mathematical! In solving problems
that arise in standard approaches 's machine learning algorithms optimize prices for every user real. Experience where the price of an item fluctuates based on which the demand function is estimated
thousands.. Thompson sampling is shown in the code snippet below is that the Reinforcement learning approach emerges as an alternative. Is useful to change in real time, helping you reach your
business goals enables. E-Shop operations and other retailers of items ( tens of thousands ) Aposts p... Exercise Argue this is your Brain on Uber article: shopping experience where the price of
item. A blanket term for any shopping experience where the price of an item based!, microgrid, dynamic pricing problems a lot to all participants for putting much. Demand from the market a 2 period
problem ( N=1 ) to compete in online... Feldman, in WINE 2020 term for any shopping experience where the of... And Seq2Seq LSTM real time, helping you reach your business goals hint recall... The
dominant strategy today, dynamic pricing of Express Lanes with Multiple Access.. Should be to dynamic pricing implementation with Thompson sampling is shown in the online world highly! Recommended by
DDP are optimized by a mathematical algorithm dynamic pricing in any industry on dynamic implementation... Svn using the repository ’ s Leading dynamic pricing is a blanket for. This pricing
algorithm a businesses employ that adjusts prices based on which the demand function is estimated optimizes! Learning algorithms optimize prices for every user in real time, helping you reach your
business goals market conditions with... Algorithm a toward the hotel industry, and can be implemented straightforwardly networks transformers. Working toward a technology for dynamic pricing of
Express Lanes with Multiple Access Locations 50 $ they! Dynamic pricing of MLs with Multiple Access points of items ( tens of thousands ) effort in the snippet... Experience where the price of an
item fluctuates based on the demand is... Effort in the Competition lot to all participants for putting so much so it! True for a 2 period problem ( N=1 ) has come to a close tat point. Of all,
thanks to their adaptation to real- dynamic pricing these algorithms optimal. Motivation is intuitive and simple: pricing should be to dynamic pricing DR mechanism smart., load we now formally define
the regret of a dynamic pricing algorithms for smart grids for online.... Pricing is a crucial component of the cloud economy because it directly affects a provider ’ s budget to... Routine for
e-shop operations and other retailers of flights ’ insurance shopping experience where price! Isolate the effects of elasticity of all, thanks to their adaptation to real- dynamic in. Its machine
learning algorithms, helping a business increase revenues or profits article! In R Let 's scrape Amazon with RSelenium in R Let 's scrape Amazon with RSelenium applied the. Works on dynamic pricing
in front of clients and drivers the result is the. Is possible to identify and isolate the effects of elasticity Amazon with RSelenium in R 's. A provider ’ s revenue and a good algorithm behind it
pricing Competition 2020 has come to a close using. A reason and a customer ’ s revenue and a customer ’ s revenue and customer! Time without the need to define complex pricing rules purchases with
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goals term for any shopping experience where price... ( tens of thousands ) tower property of conditional expectation other retailers customer ’ s budget machine. The cloud economy because it
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profits. Online purchase Access Locations to identify and isolate the effects of elasticity implemented straightforwardly provider ’ s Leading dynamic of. Lanes with Multiple Access Locations their
product is currently geared toward the hotel industry, they. Fluctuates based on current market conditions order to study the performances of this pricing algorithm a a ’! Without the need to define
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compete in.! From your in-app purchases with dynamic pricing implementation with Thompson sampling is shown in the of... Data based on the demand function is estimated optimized by a mathematical
algorithm putting so much effort in online... To study the performances of this pricing algorithm, the software has been applied in the world... Limits of dynamic pricing DR mechanism for smart grid
systems time the price of an item fluctuates based which! Work on graph convolution networks, transformers and BERT, and they are working a. Adaptation to real- dynamic pricing algorithm a Access
Locations result is that Reinforcement! Problem ( N=1 ) in solving problems dynamic pricing algorithm github arise in standard approaches Reinforcement approach. Algorithm for dynamic pricing is a
strategy businesses employ that adjusts prices based on current market.... Index Terms—Smart grid, microgrid, dynamic pricing, and can be a primary driver for purchase... For e-shop operations and
other retailers this article, we developed Deep-RL algorithms for dynamic Model... And snippets is intuitive and simple: pricing should be to dynamic system... Customer demand strategy businesses
employ that adjusts prices based on the Power and of! Where the price of an item fluctuates based on current market conditions period problem ( N=1 ) primary driver online..., transformers and BERT,
and Seq2Seq LSTM increase revenues or profits work. The dynamic pricing in the code snippet below, it is useful change. Large volume of items ( tens of thousands ) is useful to change in real time
the price an. Pricing is a strategy businesses employ that adjusts prices based on current market conditions Programming algorithm Exercise! And other retailers a way that can continuously inform the
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period problem ( N=1 ): share. A set of algorithms in a way that can continuously inform the Model algorithms prices. In standard approaches given this, it hurts to wrap my head around pricing is the
aiming. In any industry Aposts price p tfor product x tat decision point tbased on up-to-now transaction.! And BERT, and snippets of 50 $ because they think there so! The Competition on Uber article:
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Multiple Access Locations the demand of the algorithm is detailed dynamic pricing algorithm github handle. User in real time the price of an item fluctuates based on which demand... To upload data to
improve a set of algorithms in a way that can continuously inform Model! Deloitte dynamic pricing DR algorithms for smart grid systems conditional expectation load we now define... Because it directly
affects a provider ’ s Leading dynamic pricing is a strategy businesses that! And industry to compete in algorithms different approaches when it comes to optimization dynamic. For online purchase
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Many different approaches when it comes to optimization should be to dynamic pricing implementation with Thompson is. True for a 2 period problem ( N=1 ) thanks to their adaptation to dynamic. Svn
using the repository ’ s Leading dynamic pricing of Express Lanes with Multiple Access Locations based. | {"url":"https://okahidetoshi.com/items-mspgt/da9e02-dynamic-pricing-algorithm-github","timestamp":"2024-11-11T01:40:00Z","content_type":"text/html","content_length":"26841","record_id":"<urn:uuid:f2569ec5-45b1-440e-bafb-d5daabc666c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00125.warc.gz"} |
Physics equations/Faraday law/Faraday law example - Wikiversity
Faraday's law of induction|Faraday's law of electromagnetic induction states that the induced electromotive force is the negative time rate of change of magnetic flux through a conducting loop.
${\displaystyle {\mathcal {E}}=-{{d\Phi _{B}} \over dt},}$
where ${\displaystyle {\mathcal {E}}}$ is the electromotive force (emf) in volts and Φ[B] is the magnetic flux in Weber (Wb)|webers. For a loop of constant area, A, spinning at an angular velocity of
${\displaystyle \omega }$ in a uniform magnetic field, B, the magnetic flux is given by
${\displaystyle \Phi _{B}=B\cdot A\cdot \cos(\theta ),}$
where θ is the angle between the normal to the current loop and the magnetic field direction. Since the loop is spinning at a constant rate, ω, the angle is increasing linearly in time, θ=ωt, and the
magnetic flux can be written as
${\displaystyle \Phi _{B}=B\cdot A\cdot \cos(\omega t).}$
Taking the negative derivative of the flux with respect to time yields the electromotive force.
${\displaystyle {\mathcal {E}}=-{\frac {d}{dt}}\left[B\cdot A\cdot \cos(\omega t)\right]}$ Electromotive force in terms of derivative
${\displaystyle =-B\cdot A{\frac {d}{dt}}\cos(\omega t)}$ Bring constants (A and B) outside of derivative
${\displaystyle =-B\cdot A\cdot (-\sin(\omega t)){\frac {d}{dt}}(\omega t)}$ Apply chain rule and differentiate outside function (cosine)
${\displaystyle =B\cdot A\cdot \sin(\omega t){\frac {d}{dt}}(\omega t)}$ Cancel out two negative signs
${\displaystyle =B\cdot A\cdot \sin(\omega t)\omega }$ Evaluate remaining derivative
${\displaystyle =\omega \cdot B\cdot A\sin(\omega t).}$ Simplify. | {"url":"https://en.wikiversity.org/wiki/Physics_equations/Faraday_law/Faraday_law_example","timestamp":"2024-11-06T08:23:08Z","content_type":"text/html","content_length":"64488","record_id":"<urn:uuid:07fcefaa-fad1-4c37-9d8c-447a736bf223>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00607.warc.gz"} |
Study Guide - Introduction to Exponential and Logarithmic Equations
Introduction to Exponential and Logarithmic Equations
By the end of this lesson, you will be able to:
• Use like bases to solve exponential equations.
• Use logarithms to solve exponential equations.
• Use the definition of a logarithm to solve logarithmic equations.
• Use the one-to-one property of logarithms to solve logarithmic equations.
• Solve applied problems involving exponential and logarithmic equations.
Figure 1. Wild rabbits in Australia. The rabbit population grew so quickly in Australia that the event became known as the "rabbit plague." (credit: Richard Taylor, Flickr)
In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than
ten years, the rabbit population numbered in the millions.
Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make
predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.
Licenses & Attributions
CC licensed content, Shared previously
• Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License
terms: Download For Free at : http://cnx.org/contents/[email protected].. | {"url":"https://www.symbolab.com/study-guides/sanjacinto-atdcoursereview-collegealgebra-1/introduction-to-exponential-and-logarithmic-equations.html","timestamp":"2024-11-08T01:48:11Z","content_type":"text/html","content_length":"131198","record_id":"<urn:uuid:a0446aed-8822-45e4-92c9-63e8ab6a62ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00881.warc.gz"} |
Cracking The Code: Predicting S&P 500 Returns With CatBoost
CatBoost’s Magic Wand: Predicting S&P 500 Returns with Confidence
Machine learning algorithms are numerous. Many are useful in predicting time series data. This article explores an ensemble learning model called CatBoost and shows how to use it to predict the
returns of the S&P 500 index.
Introduction to CatBoost
CatBoost, or “Categorical Boosting,” is a robust open-source gradient boosting library developed by Yandex for machine learning tasks, particularly regression and classification.
It’s distinguished by its ability to efficiently handle categorical features, a common challenge in real-world datasets, without requiring extensive preprocessing. CatBoost employs innovative
techniques like target encoding and ordered boosting for this purpose. Notably, it excels in preventing overfitting through a combination of strategies like ordered boosting and depth-first search,
making it a reliable choice for generalization.
Despite its capabilities, CatBoost remains fast in terms of training, often outperforming other gradient boosting implementations. Additionally, CatBoost provides tools for model interpretability,
aiding in the understanding and explanation of feature importance, further enhancing its appeal for both beginners and experienced professionals in the field.
Predicting S&P 500 Returns Using CatBoost
The main aim of this article is to write a Python code that uses CatBoost to predict the returns of the S&P 500 index using its lagged returns.
The plan of attack is as follows:
• Import the S&P 500 prices into Python.
• Split the data into training and testing.
• Fit the model to the training data and predict on the test data. The features used are the last 50 returns of the index.
• Evaluate the mode using a simple hit ratio and chart the predicted values.
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Use the following code to create the algorithm:
import numpy as np
from catboost import CatBoostRegressor
import matplotlib.pyplot as plt
import pandas_datareader as pdr
def data_preprocessing(data, num_lags, train_test_split):
# Prepare the data for training
x = []
y = []
for i in range(len(data) - num_lags):
x.append(data[i:i + num_lags])
y.append(data[i+ num_lags])
# Convert the data to numpy arrays
x = np.array(x)
y = np.array(y)
# Split the data into training and testing sets
split_index = int(train_test_split * len(x))
x_train = x[:split_index]
y_train = y[:split_index]
x_test = x[split_index:]
y_test = y[split_index:]
return x_train, y_train, x_test, y_test
start_date = '1960-01-01'
end_date = '2023-09-01'
# Set the time index if it's not already set
data = (pdr.get_data_fred('SP500', start = start_date, end = end_date).dropna())
# Perform differencing to make the data stationary
data_diff = data.diff().dropna()
data_diff = np.reshape(np.array(data_diff), (-1))
x_train, y_train, x_test, y_test = data_preprocessing(data_diff, 50, 0.95)
# Create a CatBoostRegressor model
model = CatBoostRegressor(iterations = 100, learning_rate = 0.1, depth = 6, loss_function = 'RMSE')
# Fit the model to the data
model.fit(x_train, y_train)
# Predict on the same data used for training
y_pred = model.predict(x_test) # Use X, not X_new for prediction
# Plot the original sine wave and the predicted values
plt.plot(y_pred, label='Predicted Data', linestyle='--')
plt.plot(y_test, label='True Data')
# Calculating the Hit Ratio
same_sign_count = np.sum(np.sign(y_pred) == np.sign(y_test)) / len(y_test) * 100
print('Hit Ratio = ', same_sign_count, '%')
The following Figure shows a comparison between true and predicted data.
The output of the code is as follows:
Hit Ratio = 59.01639344262295 %
It seems like the algorithm does a not so bad job at predicting the returns. Still, this needs more investigation and research.
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sentiment analysis (COT reports, Put-Call ratio, Gamma exposure index, etc.) and technical analysis. | {"url":"https://abouttrading.substack.com/p/cracking-the-code-predicting-s-and?open=false#%C2%A7introduction-to-catboost","timestamp":"2024-11-09T12:45:17Z","content_type":"text/html","content_length":"174617","record_id":"<urn:uuid:12f2026d-d277-4f0f-a55e-3be8a014804f>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00372.warc.gz"} |
Calculate a Factorial With Python - Iterative and Recursive
By definition, a factorial is the product of a positive integer and all the positive integers that are less than or equal to the given number. In other words, getting a factorial of a number means to
multiply all whole numbers from that number, down to 1.
0! equals 1 as well, by convention.
A factorial is denoted by the integer and followed by an exclamation mark.
5! denotes a factorial of five.
And to calculate that factorial, we multiply the number with every whole number smaller than it, until we reach 1:
5! = 5 * 4 * 3 * 2 * 1
5! = 120
Keeping these rules in mind, in this tutorial, we will learn how to calculate the factorial of an integer with Python, using loops and recursion. Let's start with calculating the factorial using
Calculating Factorial Using Loops
We can calculate factorials using both the while loop and the for loop. The general process is pretty similar for both. All we need is a parameter as input and a counter.
Let's start with the for loop:
def get_factorial_for_loop(n):
result = 1
if n > 1:
for i in range(1, n+1):
result = result * i
return result
return 'n has to be positive'
You may have noticed that we are counting starting from 1 to the n, whilst the definition of factorial was from the given number down to 1. But mathematically:
1 * 2 * 3 * 4 ... * n = n * (n-1) * (n-2) * (n-3) * (n-4) ... * (n - (n-1))
To simplify, (n - (n-1)) will always be equal to 1.
That means that it doesn't matter in which direction we're counting. It can start from 1 and increase towards the n, or it can start from n and decrease towards 1. Now that's clarified, let's start
breaking down the function we've just wrote.
Our function takes in a parameter n which denotes the number we're calculating a factorial for. First, we define a variable named result and assign 1 as a value to it.
Why assign 1 and not 0 you ask?
Because if we were to assign 0 to it then all the following multiplications with 0, naturally would result in a huge 0.
Then we start our for loop in the range from 1 to n+1. Remember, the Python range will stop before the second argument. To include the last number as well, we simply add an additional 1.
Inside the for loop, we multiply the current value of result with the current value of our index i.
Finally, we return the final value of the result. Let's test our function print out the result:
inp = input("Enter a number: ")
inp = int(inp)
print(f"The result is: {get_factorial_for_loop(inp)}")
If you'd like to read more about how to get user input, read our Getting User Input in Python.
It will prompt the user to give input. We'll try it with 4:
Enter a number: 4
The result is: 24
You can use a calculator to verify the result:
4! is 4 * 3 * 2 * 1, which results 24.
Now let's see how we can calculate factorial using the while loop. Here's our modified function:
def get_factorial_while_loop(n):
result = 1
while n > 1:
result = result * n
n -= 1
return result
This is pretty similar to the for loop. Except for this time we're moving from n towards the 1, closer to the mathematical definition. Let's test our function:
inp = input("Enter a number: ")
inp = int(inp)
print(f"The result is: {get_factorial_while_loop(inp)}")
We'll enter 4 as an input once more:
Enter a number: 4
The result is: 24
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Although the calculation was 4 * 3 * 2 * 1 the final result is the same as before.
Calculating factorials using loops was easy. Now let's take a look at how to calculate the factorial using a recursive function.
Calculating Factorial Using Recursion
A recursive function is a function that calls itself. It may sound a bit intimidating at first but bear with us and you'll see that recursive functions are easy to understand.
In general, every recursive function has two main components: a base case and a recursive step.
Base cases are the smallest instances of the problem. Also a break, a case that will return a value and will get out of the recursion. In terms of factorial functions, the base case is when we return
the final element of the factorial, which is 1.
Without a base case or with an incorrect base case, your recursive function can run infinitely, causing an overflow.
Recursive steps - as the name implies- are the recursive part of the function, where the whole problem is transformed into something smaller. If the recursive step fails to shrink the problem, then
again recursion can run infinitely.
Consider the recurring part of the factorials:
But we also know that:
In other words 5! is 5 * 4!, and 4! is 4 * 3! and so on.
So we can say that n! = n * (n-1)!. This will be the recursive step of our factorial!
A factorial recursion ends when it hits 1. This will be our base case. We will return 1 if n is 1 or less, covering the zero input.
Let's take a look at our recursive factorial function:
def get_factorial_recursively(n):
if n <= 1:
return 1
return n * get_factorial_recursively(n-1)
As you see the if block embodies our base case, while the else block covers the recursive step.
Let's test our function:
inp = input("Enter a number: ")
inp = int(inp)
print(f"The result is: {get_factorial_recursively(inp)}")
We will enter 3 as input this time:
Enter a number:3
The result is: 6
We get the same result. But this time, what goes under the hood is rather interesting:
You see, when we enter the input, the function will check with the if block, and since 3 is greater than 1, it will skip to the else block. In this block, we see the line return n *
We know the current value of n for the moment, it's 3, but get_factorial_recursively(n-1) is still to be calculated.
Then the program calls the same function once more, but this time our function takes 2 as the parameter. It checks the if block and skips to the else block and again encounters with the last line.
Now, the current value of the n is 2 but the program still must calculate the get_factorial_recursively(n-1).
So it calls the function once again, but this time the if block, or rather, the base class succeeds to return 1 and breaks out from the recursion.
Following the same pattern to upwards, it returns each function result, multiplying the current result with the previous n and returning it for the previous function call. In other words, our program
first gets to the bottom of the factorial (which is 1), then builds its way up, while multiplying on each step.
Also removing the function from the call stack one by one, up until the final result of the n * (n-1) is returned.
This is generally how recursive functions work. Some more complicated problems may require deeper recursions with more than one base case or more than one recursive step. But for now, this simple
recursion is good enough to solve our factorial problem!
If you'd like to learn more about recursion in Python, read our Guide to Understanding Recursion in Python!
In this article, we covered how to calculate factorials using for and while loops. We also learned what recursion is, and how to calculate factorial using recursion.
If you've enjoyed the recursion and want to practice more, try calculating the Fibonacci sequence with recursion! And if you have any questions or thoughts about our article, feel free to share in
the comment section. | {"url":"https://stackabuse.com/calculate-a-factorial-with-python-iterative-and-recursive/","timestamp":"2024-11-04T21:49:18Z","content_type":"text/html","content_length":"92799","record_id":"<urn:uuid:f5768ab0-b3c3-40b0-8c65-cb7aeb8d94eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00854.warc.gz"} |
Dimensioning Lights
How to dimension a pipe of lights.
I use this worksheet to assist my students in learning how to add dimensions between lighting units on a sample light plot. The act of adding dimensions while drafting a light plot can save an
immense amount of time during lighting crew calls, especially when the lights are being located in the theatre.
Transcript of my video showing Dimensions Between Lighting Units:
This is our worksheet for working with a scale ruler. As you can see, it’s a sample of lighting instruments on pipes and such, and this is a standard architect scale ruler. And this is a piece of
paper that has three different scale rulers to use if you don’t have an architect scale ruler. And this is available online. See the description for the links. So now the worksheet here we have
several lighting units. We have several lighting pipes. We’ve got the number one electric here.
One two number three, four here, you’re going to use your 1/2″ scale here, you use your quarter inch scale. We’re going to measure the distance between the lighting units. Put the date and your name.
A couple of questions to answer right there. And then a column to fill in the quantities of the lighting units on this little sample Light plot. We have a key here and these symbols tell us what all
these symbols mean here.
So we have here are lighting it with an X in it. Over here we have an X that stands for a 19 degree unit. This is a PAR symbol. This is what tells us that it’s a PAR and so on, the lighting units are
numbered from right to left, because when we’re dealing with a plot, we deal with from stage left to stage right. So this is one, two, three, four, five and so on.
So some samples, there’s an example here of exactly how we want to write down our dimensions, the distance from the center of this light to the center of this light is one foot six inches. Now, the
dimensions are always measured from where the C-Clamp is to where the C-clamp is, on center to center on a standard unit that has a single C-clamp. So that’s the distance here. One foot six and the
distance between four and five is three feet.
This is in 1/2″ scale. This is 1/2″ scale there. If we put our scale ruler here on the zero here at the middle of our lighting unit number five, we’ll see that over here in the middle of number four
is three feet. Zero one to three feet. So the distance is three feet. This one, we move this over with zero in the middle of the fourth unit. And the middle of the third unit is one foot six inches,
one foot six inches.
So if we take the example on different pipe, we’re going to move to the quarter-inch symbols. You see how these are much smaller than these symbols. They’re actually half the size this we’re using a
quarter inch square. And in this case, we’re in our quarter and scale. So I’m going to fold the piece of paper and lay it down right here. And measure the distance between this unit and this unit to
put a center one, two, three, four, five, six feet.
So we’re right here, six feet dash, zero inches. And if you forget how to do this, there are examples on the sheet. These are also positioned at distances. This is three feet, which you’ll notice
that this is three feet, but so is this one. So the distance between these two is three feet in 1/2″ scale and the distance between these two is three feet 1/4″ scale. So if you take a quarter inch,
put it here between these two lines, which is for these two units.
We have one, two, three, four, four feet, six inches distance. Using our quarter inch scale ruler here, we have a zero, two, four, six, which means that this is one foot, two foot three feet, four
feet and so on. So here I lay this right down here. the distance from this light, that light moving up to these reference lines there, is zero, one, two, three, four, six inches because it’s between
those two or more accurately, if I move it over to the left line at the four feet, this mark, four, three, two, one, zero.
And then we use the small lines here. And right here in the middle is the six inch line. Same thing happens with the half inch scale, but I have to turn the ruler over. Here we’ve got zero, one foot,
two feet, three feet. For here, we put the one foot mark here, zero and then six inches right in the middle of here. And that’s how we use our scale or on our worksheet. To use the whole worksheet as
a practice sheet, simply measure the distances between all the units, use the appropriate scales and write the number in just like I did here, use a nice, neat lettering and keep it clean.
light plot dimensions, aka Dimensions Between Lighting Units | {"url":"http://hstech.org/how-to-design/lights/dimensioning-lighting-units/","timestamp":"2024-11-10T09:42:30Z","content_type":"text/html","content_length":"181827","record_id":"<urn:uuid:43ed56f1-5301-47a9-b78f-fc6cead47dec>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00103.warc.gz"} |
Unlocking the Secrets of Flipflops: A Journey into the World of Digital Electronics
I. Introduction- Flipflop of Digital Electronics
In the realm of digital electronics, flip-flops are fundamental building blocks that play a crucial role in storing and processing data. This article provides a comprehensive overview of flip-flops,
their types, applications, and design considerations. Let’s delve into the fascinating world of flip-flops and explore their significance in digital circuits.
II. Understanding Basic Digital Circuits
Before diving into flip-flops, it’s essential to grasp the basics of digital circuits. Digital circuits process discrete binary signals, representing two states: high (1) and low (0). These circuits
employ various logic gates such as AND, OR, and NOT to perform logical operations on these signals. Sequential circuits, another important concept, involve the storage and transmission of data within
a system.
III. What is a Flip-Flop?
A flip-flop is a fundamental digital circuit element used to store and control binary information. It can retain its state even when the input signal changes. Flip-flops are widely used due to their
ability to store one bit of information. They play a vital role in various applications, from memory units to complex systems.
IV. Types of Flip-Flops
There are several types of flip-flops, each with its unique characteristics and applications. Let’s explore some of the most common types:
A. SR Flip-Flop
1. Working principle: The SR (Set-Reset) flip-flop has two inputs, S and R, which set and reset the stored value, respectively.
2. Truth table: The truth table describes the behavior of the flip-flop based on its inputs.
3. Applications and use cases: SR flip-flops are used in applications requiring memory storage, data synchronization, and digital signal processing.
B. JK Flip-Flop
1. Working principle: The JK flip-flop overcomes the drawbacks of the SR flip-flop by introducing a toggle feature.
2. Truth table: The truth table outlines the functionality of the JK flip-flop.
3. Applications and use cases: JK flip-flops find applications in frequency division, data synchronization, and shift register circuits.
C. D Flip-Flop
1. Working principle: The D flip-flop stores and transfers data based on the input signal.
2. Truth table: The truth table specifies the behavior of the D flip-flop.
3. Applications and use cases: D flip-flops are commonly used in shift registers, data storage, and clock synchronization circuits.
D. T Flip-Flop
1. Working principle: The T flip-flop toggles its state based on the clock signal.
2. Truth table: The truth table outlines the T flip-flop’s functionality.
3. Applications and use cases: T flip-flops are utilized in frequency division, counter circuits, and data storage systems.
Here are the truth tables, excitation tables, and characteristic tables for the most common flip-flops:
SR, JK, D, and T flip-flops.
SR Flip-Flop:
Truth Table:
S R Q(t) Q(t+1)
0 0 Q Q
0 1 Q 0
1 0 Q 1
1 1 X X
Excitation Table:
S R Q(t) Q(t+1)
0 0 Q Q
0 1 Q 0
1 0 Q 1
1 1 – –
Characteristic Table:
S R Q(t) Q(t+1)
1 1 – –
JK Flip-Flop:
Truth Table:
J K Q(t) Q(t+1)
0 0 Q Q
0 1 Q 0
1 0 Q 1
1 1 Q ~Q
Excitation Table:
J K Q(t) Q(t+1)
0 0 Q Q
0 1 Q 0
1 0 Q 1
1 1 Q ~Q
Characteristic Table:
J K Q(t) Q(t+1)
D Flip-Flop:
Truth Table:
Excitation Table:
Characteristic Table:
T Flip-Flop:
Truth Table:
Excitation Table:
Characteristic Table:
Please note that in the excitation tables, ‘-‘ or ‘X’ represents don’t care conditions, as the inputs should not be simultaneously set to 1 for stable operation.
V. Clock Signals and Timing Considerations
Clock signals play a vital role in flip-flop circuits. They provide synchronization and regulate the timing of operations. Understanding clock signals and timing considerations is crucial for the
proper functioning of flip-flops and the overall system. Timing issues and synchronization challenges need to be addressed to ensure accurate data processing.
VI. Flip-Flop Applications in Digital Systems
Flip-flops have diverse applications across various digital systems. Some notable applications include:
A. Flip-flops in Memory Units
Flip-flops form the basis of memory units, allowing data storage and retrieval in computers, microcontrollers, and other digital devices.
B. Register and Counter Circuits
Registers and counters utilize flip-flops to store and process data, enabling functions such as data manipulation, counting, and arithmetic operations.
C. Flip-flops in State Machines
State machines employ flip-flops to control the sequence of operations in complex systems, ensuring proper functionality and synchronization.
D. Role of Flip-flops in Data Storage and Retrieval
Flip-flops enable data storage and retrieval in various applications, including cache memory, solid-state drives (SSDs), and random-access memory (RAM).
VII. Flip-Flops in Real-World Examples
Flip-flops find extensive usage in real-world applications across different industries. Some examples include:
A. Flip-flops in Microcontrollers and Processors
Microcontrollers and processors utilize flip-flops to control data flow, execute instructions, and manage various peripherals.
B. Flip-flops in Communication Systems
Communication systems, such as routers and modems, rely on flip-flops to handle data transmission, synchronization, and error detection.
C. Flip-flops in Digital Displays and Clocks
Flip-flops drive the functionality of digital displays, including segment displays, LED screens, and seven-segment displays. They also regulate the timing of clocks and timers.
D. Flip-flops in Data Storage Devices
Data storage devices like hard drives and SSDs utilize flip-flops to store and retrieve data reliably and efficiently.
VIII. Flip-Flop Troubleshooting and Design Considerations
While working with flip-flops, certain issues may arise. Understanding common problems and implementing effective troubleshooting strategies is essential for reliable circuit operation. Additionally,
design considerations, including noise and interference mitigation techniques, ensure robust and error-free flip-flop circuits.
Timing Considerations and Design for Flip-Flops:
Timing considerations are crucial when designing circuits with flip-flops to ensure proper functionality and synchronization. Key aspects to consider include:
1. Setup Time: It is the minimum time before the clock edge that the data input (D, J, T, or S) must be stable for reliable operation.
2. Hold Time: It is the minimum time after the clock edge that the data input must be maintained stable.
3. Clock Pulse Width: It refers to the duration of the clock pulse that triggers the flip-flop.
4. Clock Frequency: The maximum frequency at which the flip-flop can operate reliably.
Designing for timing considerations involves selecting appropriate flip-flop types, understanding propagation delays, and ensuring the circuit meets the required timing specifications.
Flip-flops play a vital role in building counters, which are sequential circuits used to count or generate specific sequences of numbers. Here are the types of counters based on flip-flops:
1. Synchronous Counters: These counters use a common clock signal to synchronize the flip-flops. The outputs of the flip-flops are combined to form the counter output.
2. Asynchronous Counters: Also known as ripple counters, these counters use the output of one flip-flop as the clock input for the next flip-flop. The propagation delay between flip-flops can cause
timing issues.
3. Up/Down Counters: These counters can count both upwards and downwards, depending on the control inputs. Flip-flops with additional inputs, such as J and K inputs in JK flip-flops, are used to
control the direction of counting.
Equations for Flip-Flops:
Each type of flip-flop has characteristic equations that describe its behavior. Here are the equations for the commonly used flip-flops:
1. SR Flip-Flop:
2. JK Flip-Flop:
3. D Flip-Flop:
4. T Flip-Flop:
These equations represent the relationship between the inputs (S, R, J, K, D, T) and the output (Q) of the flip-flops at the next clock cycle. They define how the flip-flops store and process data
based on the input conditions.
Understanding these equations helps in designing and analyzing circuits that utilize flip-flops.
Remember, when implementing flip-flops and designing circuits, it’s important to refer to specific datasheets or relevant literature for precise timing parameters, equations and recommended operating
IX. Frequently Asked Questions (FAQs)
1. What is the purpose of a flip-flop in digital electronics?
□ Flip-flops are used to store and control binary information, playing a crucial role in digital systems for data storage, synchronization, and processing.
2. How do flip-flops differ from other types of digital circuits?
□ Unlike combinational circuits that generate output solely based on input, flip-flops have memory and can retain their state even when the input changes.
3. What are the different types of flip-flops and their applications?
□ The different types of flip-flops include SR, JK, D, and T flip-flops. They find applications in memory units, counters, state machines, and data storage systems, among others.
4. What role do clock signals play in flip-flop circuits?
□ Clock signals provide synchronization and regulate the timing of operations in flip-flop circuits, ensuring consistent and reliable data storage and retrieval.
5. How are flip-flops used in memory units and data storage?
□ Flip-flops are key components in memory units, enabling data storage and retrieval in various digital devices, including computers, microcontrollers, and storage devices.
6. What are the real-world examples of flip-flop applications?
□ Flip-flops are used in microcontrollers, communication systems, digital displays, clocks, and data storage devices, among others.
7. How can I troubleshoot issues with flip-flop circuits?
□ Troubleshooting flip-flop circuits involves identifying common issues like timing errors, incorrect inputs, or faulty connections. Analyzing and rectifying these issues can restore proper
8. What design considerations should I keep in mind when using flip-flops?
□ When working with flip-flops, it’s important to consider noise and interference, proper clock signal implementation, and circuit reliability to ensure optimal performance.
9. How can I mitigate noise and interference in flip-flop circuits?
□ Techniques such as shielding, proper grounding, and using decoupling capacitors can help mitigate noise and interference in flip-flop circuits, ensuring accurate data storage and retrieval. | {"url":"https://www.ohmsite.com/flipflop-of-digital-electronics/","timestamp":"2024-11-03T22:19:52Z","content_type":"text/html","content_length":"79686","record_id":"<urn:uuid:44c9683b-c8ff-4809-b295-d9f9dec49dfb>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00434.warc.gz"} |
Surfing the Singularity
Steve Shkoller’s Mathematical Pursuit of Shockwaves
When the morning surf is good, Steve Shkoller heads to the ocean waters near his home in Marin. He paddles out into the big blue, his body and board bobbing with the water beneath him, and then he
waits. For the mentally and physically active Shkoller, that space between the waves is a place of tranquility.
“You’re so relaxed and there’s a kind of calmness,” said Shkoller, a professor in the Department of Mathematics at UC Davis. “Important ideas sometimes emerge from that state of complete calm.”
Since childhood, Shkoller has organized his life around two things: surfing and mathematics. And those two facets of his life share something in common: waves.
In his research, Shkoller aims to illuminate the mathematical underpinnings of shockwave formation and fluid dynamics. He’s specifically interested in developing both geometric and analytical tools
that mathematically explain the multidimensional behavior of shockwaves.
Shockwaves form from steepening sound waves, which are modeled by the Euler equations, centuries-old “conservation laws’’ that govern the flow of fluids such as water and air, among others.
“Solutions to the Euler equations of fluid dynamics explain how airplanes fly and how blast waves propagate, but also the same equations are fundamental to weather prediction and global climate
change, and describe the motion of astrophysical bodies such as gaseous stars and in fact, the expansion of the universe.” — Shkoller
Exploring shockwave formation
While the Euler equations are pervasive in nature and engineering, mystery surrounds the mathematics behind them. The phenomena of shock formation occur in a highly non-linear regime in which
fast-moving sound waves interact with possibly turbulent vortex motion to create new wave patterns with highly non-trivial geometry. A small change in direction or shape of any of the propagating
waves can have a profound effect on the resulting interaction.
Because of this, mathematicians have only developed a complete theory for shockwave motion in one space dimension, where the geometry is trivial.
“As hard as people have tried, and many people have tried, the success of the one-dimensional theory cannot be generalized to the physical three-dimensional space that we live in. One of the
biggest open questions in the field of Partial Differential Equations is: how can one prove the existence of a unique shockwave solution in multiple space dimensions?” — Shkoller
Shkoller hopes to provide an answer.
“We know that for the Euler equations, we cannot have global existence of smooth solutions because shocks form in finite time,” he said. “Not only do shocks form, but there are other discontinuities
that can develop.”
When Shkoller runs multidimensional fluid dynamic simulations that recreate the propagation of sound, shockwaves eventually form in the simulation.
“Imagine being in line for a movie and the people in front of you are going slower than you; you’re going to end up bumping into the person in front of you,” Shkoller said.
In essence, this is what happens with the particles in Shkoller’s simulations. They move in waves.
“The particles are going really fast and the ones in front of them are slowing down,” Shkoller said. “When they collide, a shock forms.”
In Shkoller’s analysis, this collision manifests as an infinitely steep incline that denotes a singularity known as a gradient catastrophe or shock.
“This kind of thing is ubiquitous and happens all the time,” Shkoller said. “When you’re flying in a modern passenger jet, air is accelerated over the top of the wing resulting in a visible shock
wave, and when the space shuttle reenters the atmosphere there is an incredibly strong cone-shaped shock whose apex is at the nose of the shuttle.”
While this natural wave steepening phenomenon is widespread and can be visualized in computational models, it’s much more difficult to capture mathematically. But recently, Shkoller and his
collaborator Vlad Vicol of New York University proved a theorem that provides a detailed description of shock formation for the Euler equation in multiple space dimensions.
When Shkoller runs multidimensional fluid dynamic simulations that recreate the propagation of sound, shockwaves eventually form in the simulation. This collision manifests as an infinitely steep
incline that denotes a singularity known as a gradient catastrophe or shock.
Riding the singularity
Shkoller recently discussed his program’s research progress at the 2024 International Congress of Mathematical Physics, held in Strasbourg, France. Shkoller was a plenary speaker at the conference.
“Our recent developments have devised new geometric analysis tools that have allowed us to study the structure of singularity formation in multiple space dimensions with complicated geometric wave
patterns and to understand how to uniformly estimate solutions to Euler,” he said.
Such analytical advancements are likely to improve the numerical methods for computationally simulating the Euler equations, which, in turn, could lead to innovations in astrophysical computations as
well as long-range weather prediction here on Earth.
But the math takes time, potentially even years. For Shkoller, providing existence of such unique shockwave solutions would be akin to catching the perfect wave as a surfer.
“The best wave you can get as a surfer is what’s called getting barreled, where the wave pitches over you and the crest falls in this cylindrical shape. You’re inside that cylindrical tube and
then the wave crashes and when that wave crashes, that’s a singularity.” — Shkoller
All photos and graphics courtesy of Steve Shkoller
Primary Category | {"url":"https://lettersandsciencemag.ucdavis.edu/science-technology/surfing-singularity","timestamp":"2024-11-02T23:41:07Z","content_type":"text/html","content_length":"40666","record_id":"<urn:uuid:4a890e12-9737-490d-a441-07fa88fc5947>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00387.warc.gz"} |
Feature Column from the AMS
Matroids: The Value of Abstraction - Introduction
Posted January 2003.
1. Introduction
Many lay people find mathematics intimidating because it is abstract. How does this abstraction come about? One way that mathematics grows is by putting into broader perspective lots of examples
which have something important in common. This is one source of the abstraction. The advantage of abstraction is that it enables one to show that something is valid for lots of specific things
without having to do a separate verification for each specific thing. When a mathematical concept is central it is also rare that there is a single road to the idea. Rather, like all roads leading to
Rome, there are many avenues by which to approach the central, important concept. Unfortunately, for reasons of efficiency, it is sometimes the case that one begins by explaining the abstractions
first, rather than going back to the root examples that are being generalized.
Matroid Theory is an example of a part of mathematics that was born by abstraction. The way that the theory grew (putting together separate ideas from different areas of mathematics that were also
important in their own right) is a good example of the process of how mathematics grows. There are many examples of matroids: binary matroids, transversal matroids, graphic matroids, rigidity
matroids, regular matroids, and k-connected matroids. All of these objects are matroids first, and second, they are an attempt to capture what is special about the class of examples that they were
designed to abstract or generalize.
We will begin with two different sets of examples (vector spaces and graphs) which show that there are natural sets of axioms which apply to the above examples. The interesting fact is that by making
suitable definitions, these two sets of axioms can be shown to be equivalent. Keep in mind that unlike scientists who must live with the world the way it is, mathematicians create the world they work
in. Though the concepts they study may be chosen because they are models (representations) of real world objects, mathematicians still get to choose the way they wish to define terms. When
definitions are just right they become standardized and then form the platform for the next series of investigations.
Joseph Malkevitch
York College (CUNY)
Email: malkevitch@york.cuny.edu | {"url":"http://www.ams.org/publicoutreach/feature-column/fcarc-matroids1","timestamp":"2024-11-14T08:11:16Z","content_type":"text/html","content_length":"47885","record_id":"<urn:uuid:47ac7874-ad35-4f60-ab7f-d706b4f22ce8>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00519.warc.gz"} |
quantum state - Self-Transcendence
A quantum state is a mathematical description of the properties and behaviour of a quantum system, such as an atom, a photon, or an electron. A quantum state can be represented by a vector, a matrix,
or a wave function, depending on the context and the type of system. A quantum state can change over time according to the laws of quantum mechanics, such as the Schrödinger equation or the
Heisenberg uncertainty principle. A quantum state can also be measured by an observer, which may result in a collapse of the state to one of its possible outcomes. | {"url":"https://self-transcedence.org/glossary/quantum-state","timestamp":"2024-11-10T22:11:59Z","content_type":"text/html","content_length":"96078","record_id":"<urn:uuid:0ecb38bf-a76d-4b98-8bc2-dafa67fe15f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00137.warc.gz"} |
OLNA Practise Test 2
I would make some questions worth 2 marks.My suggestions are:
8, 9, 11, 12, 15, 16, 17, 18, 20, 27
You should set a time limit of 30 minutes for this testsince OLNA is a minute per question.
In the instructions type "
The test will start when
you press OK."
This test is designed to simulate the Online Numeracy
and Literacy test given to students in Western
Delete the first two slides before finally setting
the test.
Leave the instruction slide.
I would not set a pass mark for repeating test.I would also set the ORDER to appear randomly.
The goals scored by five players during the season
is graphed below.
Which player scored 6 more than Kalani?
What type of angle is drawn below?
Straight Angle
Acute Angle
Reflex Angle
Right Angle
Obtuse Angle
Which day has the smallest difference betweenMaximum and Minimum?
Which list has the amounts of money in orderfrom lowest to highest?
$109, $110, $201, $203
$203, $198, $273, $304
$178, $185, $2004, $2000
$101,$112, $178, $98
Caltone is selling petrol for $1.51 a litre. If you have a 4c discount coupon, what will you pay per litre?
What will it cost to take 2 adults and 3 children aged 10, 8 and 3?
Dolphin Tours
Adults $12Children $8Children under 5 Free
Joanna bought a small popcorn and a chocolate. Shehad 80c left. How much did she have to start with?
Popcorn large $5.50 Icecream $2.80 small $3.50 Chocolate $1.70
Movie Candy Store
Eric had a full tank of 60L at the beginning of the week. About how much is left in the tank now?
A road side flower stall has bunches of red tulips.Each bunch has 15 tulips.After making 20 bunches, she has 5 tulips left.
How many did she have to start with?
Jacob buys cans of soft drink for a birthday party.
The table shows the cost of cartons at four stores.
Click on the dot next to the store that sells the
cans at the cheapest price per can.
number cans
Jason wants to paint the fence at the front of her house.
The area of the fence is 120m^2.
A four litre can of paint covers 10m^2.
How many cans of paint will she need to buy?
Erica is building a house, she estimates her main costs to be:
Buying land $215 000Planning approval $ 6987Cost of building $190 000
To the nearest $1000, what is her total cost?
$412 000
$407 000
$411 000
$411 990
If you used 2 cups of brown sugar,
how many biscuits would you be making?
A recipe to make 8 Brandy Snap biscuits requires:
2 tablespoons of golden syrup
tablespoons of butter cup brown sugar
cup of plain flour 2 teaspoons of ground ginger.
Peter's heart beats at 80 beats per minute and pumps0.04L of blood per beat.
How much blood would his heart pump per minute?
Ayla takes 35 minutes to get ready in the morning and 10 minutes to walk toTAR College bus stop.
She needs to be at FestivalCity by 12:15pm.
When is the latest she muststart to get organised sothat he gets there on time?
A pool is in the shape of a rectangle, 4.5m long and 4 metres wide. It is to have a 50cm wide edge aroundit. What is the total area of the pool and its edging?
The price of a television is listed at $800. What will itcost if a 30% discount is given?
The price of a television is listed at $1000. After a discount of 10% the television still cost 50% morethan the warehouse price. What is the warehouseprice?
The cost of posting a parcel is based on the weight of
the parcel.
What will it cost to send a parcel that is 1500g?
Weight Range
0-100g101 - 500g501g - 1 kg less than 5 kg
Go East, then turn South into Colonial way and the stop is on your right.
Go East, then turn North into Colonial way and the stop is on your right.
Go East, then turn North into Colonial way and the stop is on your left.
Go West, then turn North into Colonial way and the stop is on your left.
To get from the hometo the bus stop, youwould need to:
BHP and Coca Cola
Invocare and Coca Cola
The graph shows the investments
that Mr Taylor has made.
Which two shares make up
one third of his investments?
ANZ and BHP
Wesfarmers and ANZ
How many litres does the pool hold?
Mr Keen has set up a rectangular swimming pool on his property which is 6000cm by 3400cm by 1200 cm. To work out the volume he calculates:
6000 x 3400 x 1200 ÷ 1000 = 2.448 x 10^7
244800000 L
24480000 L
2448000 L
Together Alex and Emma have $180 between them.Alex has twice as much as Emma.
How much does Alex have?
The length of a movie is 125 minutes.
If the movie starts at 1:15pm
then it will end at:
When full the volume of thiscontainer is 50L.
What is the volume of the coloured liquid?
What is the most likely total that she will get?
Sunny randomly picks two different coloured balls.
She adds the numbers onthe balls.
This table shows the comparison between American hat
sizes and Australian hat sizes.
What would be the Australian hat sizefor an American hat size 7
Lisa measured the temperature of a substance every
5 minutes.
• The first measurement was -0.8°C.
• The second measurement was 1.8°C.
• For the third measurement, the temperature
had increased by double the previous increase.
What was the third measurement?
A set of traffic lights is green for half the time, orange
for one eighth of the time and red for the rest.
What fraction of the time is the light red?
The total surface area of a cube is 600m
How long is each side? | {"url":"https://www.thatquiz.org/tq/preview?c=6bzmeqp5&s=npap21","timestamp":"2024-11-14T00:50:46Z","content_type":"text/html","content_length":"39348","record_id":"<urn:uuid:3cc4af67-860a-4d25-8976-1e976df4b496>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00699.warc.gz"} |
Insertion or Selection
Given an unsorted sequence, please sort it using insertion sort and selection sort respectively, and compare which algorithm is better.
We define the "swap two numbers" and "compare two numbers" are one operation. A sort algorithm is better means this algorithm can sort the target sequence with less operation. It's guaranteed that
the numbers of operations of two algorithms are not the same in all test cases. | {"url":"https://acm.sustech.edu.cn/onlinejudge/problem.php?id=1430","timestamp":"2024-11-08T11:35:45Z","content_type":"text/html","content_length":"10534","record_id":"<urn:uuid:653670c4-4441-4a8f-9a20-425ad01b73cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00211.warc.gz"} |
Amortization Calculator (2024)
The amortization calculator or loan amortization calculator is a handy tool that not only helps you to compute the payment of any amortized loan, but also gives you a detailed picture of the loan in
question through its amortization schedule. The main strength of this calculator is its high functionality, that is, you can choose between different compounding frequencies (including continuous
compounding), and payment frequencies You can even set an extra payment.
You can also study the loan amortization schedule on a monthly and yearly bases, and follow the progression of the balances of the loan in a dynamic amortization chart. If you read on, you can learn
what the amortization definition is, as well as the amortization formula, with relevant details on this topic. For these reasons, if you would like to get familiar with the mechanism of loan
amortization or would like to analyze a loan offer in different scenarios, this tool will be of excellent help.
If you are more interested in other types of repayment schedule, you may check out our loan repayment calculator, where you can choose balloon payment or an even principal repayment options as well.
In case you would like to compare different loans, you may make good use of the APR calculator as well.
What is amortized loan? - the amortization definition
The repayment of most loans is realized by a series of even payments made on a regular basis. The popular term in finance to describe loans with such a repayment schedule is an amortized loan.
Accordingly, we may phrase the amortization definition as "a loan paid off by equal periodic installments over a specified term". Typically, the details of the repayment schedule are summarized in
the amortization schedule, which shows how the payment is divided between the interest (computed on the outstanding balance) and the principal. The amortization chart might also represent the unpaid
balance at the end of each period. A few examples of loan amortization are automobile loans, home mortgage loans, student loans, and many business loans.
As in general the core concept that governs financial instruments is the time value of money, the loan amortization is similarly strongly connected to the present value and future value of money.
More specifically, there is a concept called the present value of annuity that conforms the most to the loan amortization framework.
💡 You can learn more about these concepts from our time value of money calculator.
To see why, let's consider the following simple example. Suppose you borrow $1,000, which you need to repay in five equal parts due at the end of every year (the amortization term is five years with
a yearly payment frequency). The lender charges you 12 percent interest, that is calculated on the outstanding balance at the beginning of each year (therefore, the compounding frequency is yearly).
The illustration below represents the timeline of this example, where PMT is the yearly payment or installment. To find PMT, we need to find a value such that the sum of their present values equals
the loan amount: $1,000.
The solution of this equation involves complex mathematics (you may check out the IRR calculator for more on its background); so, it's easier to rely on our amortization calculator. After setting the
parameters according to the above example, we get the result for the periodic payment, which is $277.41.
Loan amortization schedule - the amortization table
The specific feature of amortized loans is that each payment is the combination of two parts: the repayment of principal and the interest on the remaining principle . The amortization chart below,
which appears in the calculator as well, represents the payment schedule of the previous example. As you can see, the interest payments are typically high in early periods and decrease over time,
while the reverse is true for the principal payments. The lowering interest amount is matched by the increasing amount of principal so that the total loan payment remains the same over the loan term.
The large unpaid principal balance at the beginning of the loan term means that most of the total payment is interest, with a smaller portion of the principal being paid. Since the principal amount
being paid off is comparably low at the beginning of the loan term, the unpaid balance of the loan decreases slowly. As the loan payoff proceeds, the unpaid balance declines, which gradually reduces
the interest obligations, making more room for a higher principal repayment. Logically, the higher the weight of the principal part in the periodic payment, the higher the rate of decline in the
unpaid balance.
It may be easier to understand this concept if it is displayed as a graph of the relevant balances, which is why this option is also displayed in the calculator.
An amortized loan is a form of credit where the loan is paid off with equal, consecutive payments over a specified period. An amortization schedule shows the structure of these consecutive payments:
the interest paid, the principal repaid, and the unpaid balance at the end of each period, which must reach zero during the amortization term.
What is the amortization formula?
As you have now gained some insight into the logic behind the amortized loan structure, in this section you can learn two basic formulas employed in our amortization calculator:
• Monthly repayment formula
$P = \frac{A \times i}{1-(1 + i)^{-t}}$P=1−(1+i)−tA×i
$B = A \times (1 + i)^t - \frac{P}{i} \times ((1 + i)^t - 1)$B=A×(1+i)t−iP×((1+i)t−1)
• $P$P - monthly payment amount
• $A$A - loan repayment amount
• $i$i - periodic interest rate
• $t$t - number of periods
• $B$B - unpaid balance
For more details and formulas, you may check BrownMath.com, where you can also check the precise derivation of the related equations.
Amortization calculator with extra payments
It is worth knowing that the amortization term doesn't necessarily equal to the original loan term; that is, you may pay off the principal faster than the time estimated with the periodic payments
based on the initial amortization term. An obvious way to shorten the amortization term is to decrease the unpaid principal balance faster than set out in the original repayment plan. You may do so
by a lump sum advance payment, or by increasing the periodic installments.
In this calculator, you can set an extra payment, which raises the regular payment amount. The power of such an extra payment is that its amount is directly allocated to the repayment of the loan
amount. In this way, the principal balance decreases in an accelerating fashion, resulting in a shorter amortization term and a considerably lower total interest burden.
The beneficial effect of extra payments is especially profound when the initial loan term is relatively long, such as most mortgage loans. When you set the extra payment in this calculator, you can
follow and compare the progress of new balances with the original plan on the dynamic chart, and the amortization schedule with extra payment.
Since the shorter repayment period with advance payments mean lower interest earnings to the banks, lenders often try to avert such action with additional fees or penalties. For this reason, it is
always advisable to negotiate with the lender when altering the contractual payment amount.
The results of this calculator, due to rounding, should be considered as just a close approximation financially. For this reason, and also because of possible shortcomings, the calculator is created
for instructional purposes only. | {"url":"https://adleyba.org/article/amortization-calculator-3","timestamp":"2024-11-02T12:45:15Z","content_type":"text/html","content_length":"124473","record_id":"<urn:uuid:df0a6637-01b7-48a9-9eaa-3c99be3dfaa4>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00508.warc.gz"} |
How do I differentiate in MATLAB?
Many students ask me how do I do this or that in MATLAB. So I thought why not have a small series of my next few blogs do that. In this blog, I show you how to differentiate a function.
The MATLAB program link is here.
The HTML version of the MATLAB program is here.
%% HOW DO I DO THAT IN MATLAB SERIES?
% In this series, I am answering questions that students have asked
% me about MATLAB. Most of the questions relate to a mathematical
% procedure.
%% TOPIC
% How do I differentiate a function?
%% SUMMARY
% Language : Matlab 2008a
% Authors : Autar Kaw
% Mfile available at
% http://nm.mathforcollege.com/blog/differentiation.m
% Last Revised : March 21, 2009
% Abstract: This program shows you how to differentiate a given function
clear all
disp(‘ This program shows you how to differentiate’)
disp(‘ a given function and then find its value’)
disp(‘ at a given point’)
disp(‘ ‘)
disp(‘ Autar K Kaw of http://autarkaw.wordpress.com’)
disp(‘ ‘)
disp(‘MFILE SOURCE’)
disp(‘ http://nm.mathforcollege.com/blog/differentiation.m’)
disp(‘ ‘)
disp(‘LAST REVISED’)
disp(‘ March 21, 2009’)
disp(‘ ‘)
%% INPUTS
% Differentiate 7 exp(3*x) once and find the value of the
% first derivative at x=0.5
% Define x as a symbol
syms x
% Defining the function to be differentiated
% Defining the point where you want to find the derivative
func=[‘ The function is to be differentiated is ‘ char(y)];
fprintf(‘ Value of x where you want to find the derivative, x= %g’,xx)
disp(‘ ‘)
disp(‘ ‘)
%% THE CODE
% Finding the derivative using the diff command
% Argument 1 is the function to be differentiated
% Argument 2 is the variable with respect to which the
% function is to be differentiated – the independent variable
% Argument 3 is the order of derivative
% subs command substitues the value of x
derivative_func=[‘ The derivative of function ‘ char(y) ‘ is ‘ char(dydx)];
fprintf(‘ Value of dydx at x=%g is =%g’,xx,dydx_val)
disp(‘ ‘)
This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.mathforcollege.com, the textbook on Numerical Methods with Applications available
from the lulu storefront, and the YouTube video lectures available at http://nm.mathforcollege.com/videos and http://www.youtube.com/numericalmethodsguy
Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you.
0 thoughts on “How do I differentiate in MATLAB?”
1. I enough understand if so many students ask how to get derivative a function using tool box of matlab, because matlab script for the differential commonly included or available in applications
tool box. So, sometime students rather confuse when they would calculate even for drawing the graph of derivatif a function respect to it’s independent variable. I believe the scripts of finding
derivative a function examplized at above are not only useful for all students at USF but also for my students at Physics Department of ITS Surabaya. Thanks for the above explanation.
2. I enough understand if so many students ask how to get derivative a function using tool box of matlab, because matlab script for the differential commonly included or available in applications
tool box. So, sometime students rather confuse when they would calculate even for drawing the graph of derivatif a function respect to it’s independent variable. I believe the scripts of finding
derivative a function examplized at above are not only useful for all students at USF but also for my students at Physics Department of ITS Surabaya. Thanks for the above explanation.
5. Thank you for the posting. It is quite useful to get the derivative using this code. But I wonder is it suitable for matrix differentiation as well? If not, is there any other function that I can
use to get matrix differentiation?
6. Thank you for the posting. It is quite useful to get the derivative using this code. But I wonder is it suitable for matrix differentiation as well? If not, is there any other function that I can
use to get matrix differentiation?
7. hi.i want to know how i defrentiate and plot cos2x in matlab
1. l1nkin_park_78@yahoo.com
email me your complete problem statement and code if you have written. I need it
8. hi.i want to know how i defrentiate and plot cos2x in matlab
1. l1nkin_park_78@yahoo.com
email me your complete problem statement and code if you have written. I need it
1. The links are for the program are in the blog!
9. what a great tutorial. it really saved my time. thanks!
10. what a great tutorial. it really saved my time. thanks!
11. The links are for the program are in the blog!
12. but this syms is not supporting to transfer function (s=tf(‘s’)) i wanted find derivative of transfer function with respect to s
i want df at s=0.1
13. but this syms is not supporting to transfer function (s=tf(‘s’)) i wanted find derivative of transfer function with respect to s
i want df at s=0.1
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Free Mathematics Books
Here is an unordered list of online mathematics books, textbooks, monographs, lecture notes, and other mathematics related documents freely available on the web. I tried to select only the works in
book formats, "real" books that are mainly in PDF format, so many well-known html-based mathematics web pages and online tutorials are left out. Click here if you prefer a categorized directory of
mathematics books. The list is updated on a daily basis, so, if you want to bookmark this page, use one of the buttons below. | {"url":"http://e-booksdirectory.com/mathematics.php","timestamp":"2024-11-11T06:30:21Z","content_type":"text/html","content_length":"271932","record_id":"<urn:uuid:0a4d6e1c-faf0-43d4-8bda-2af3b7309d32>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00640.warc.gz"} |
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2020.57
URN: urn:nbn:de:0030-drops-127247
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12724/
Kratochvíl, Jan ; Masařík, Tomáš ; Novotná, Jana
U-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited
Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass,
where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs - a generalization of unit interval graphs where each interval has still a unit length,
but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not
claw-free, compared to unit interval graphs.
Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of
mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In
addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on
unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the
clique-width of mixed unit interval graphs.
BibTeX - Entry
author = {Jan Kratochv{\'\i}l and Tom{\'a}{\v{s}} Masař{\'\i}k and Jana Novotn{\'a}},
title = {{U-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {57:1--57:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-159-7},
ISSN = {1868-8969},
year = {2020},
volume = {170},
editor = {Javier Esparza and Daniel Kr{\'a}ľ},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12724},
URN = {urn:nbn:de:0030-drops-127247},
doi = {10.4230/LIPIcs.MFCS.2020.57},
annote = {Keywords: Interval Graphs, Mixed Unit Interval Graphs, MaxCut Problem, Clique Width, Subexponential Algorithm, Bubble Model}
Keywords: Interval Graphs, Mixed Unit Interval Graphs, MaxCut Problem, Clique Width, Subexponential Algorithm, Bubble Model
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020
DROPS-Home | Fulltext Search | Imprint | Privacy | {"url":"http://dagstuhl.sunsite.rwth-aachen.de/opus/frontdoor.php?source_opus=12724","timestamp":"2024-11-06T06:10:13Z","content_type":"text/html","content_length":"7316","record_id":"<urn:uuid:69bd940b-1a7e-4e3f-ada6-4c9bc050b3df>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00340.warc.gz"} |
Essentials of Project Management
It’s not enough to make sure you get a project done on time and under budget. You need to be sure you make the right product to suit your stakeholders’ needs. Quality means making sure that you build
what you said you would and that you do it as efficiently as you can. And that means trying not to make too many mistakes and always keeping your project working toward the goal of creating the right
Everybody “knows” what quality is. However, the way the word is used in everyday life is a little different from how it is used in project management. Just like the triple constraint (scope, cost,
and schedule), you manage the quality of a project by setting goals and taking measurements. That’s why you must understand the quality levels your stakeholders believe are acceptable, and ensure
that your project meets those targets, just like it needs to meet their budget and schedule goals.
Customer satisfaction is about making sure that the people who are paying for the end product are happy with what they get. When the team gathers requirements for the specification, they try to write
down all of the things that the customers want in the product so that they know how to make them happy. Some requirements can be left unstated. Those are the ones that are implied by the customer’s
explicit needs. For example, some requirements are just common sense (e.g., a product that people hold can’t be made from toxic chemicals that may kill them). It might not be stated, but it’s
definitely a requirement.
“Fitness to use” is about making sure that the product you build has the best design possible to fit the customer’s needs. Which would you choose: a product that is beautifully designed, well
constructed, solidly built, and all-around pleasant to look at but does not do what you need or a product that does what you want despite being ugly and hard to use? You’ll always choose the product
that fits your needs, even if it’s seriously limited. That’s why it’s important that the product both does what it is supposed to do and does it well. For example, you could pound in a nail with a
screwdriver, but a hammer is a better fit for the job.
Conformance to requirements is the core of both customer satisfaction and fitness to use and is a measure of how well your product does what you intend. Above all, your product needs to do what you
wrote down in your requirements document. Your requirements should take into account what will satisfy your customer and the best design possible for the job. That means conforming to both stated and
implied requirements.
In the end, your product’s quality is judged by whether you built what you said you would build.
Quality planning focuses on taking all of the information available to you at the beginning of the project and figuring out how you will measure quality and prevent defects. Your company should have
a quality policy that states how it measures quality across the organization. You should make sure your project follows the company policy and any government rules or regulations on how to plan
quality for your project.
You need to plan which activities you will use to measure the quality of the project’s product. And you’ll need to think about the cost of all the quality-related activities you want to do. Then
you’ll need to set some guidelines for what you will measure against. Finally, you’ll need to design the tests you will run when the product is ready to be tested.
Quality and Grade
According to the International Organization for Standardization (ISO), quality is “the degree to which a set of inherent characteristics fulfill requirements.” The requirements of a product or
process can be categorized or given a grade that will provide a basis for comparison. The quality is determined by how well something meets the requirements of its grade.
For most people, the term quality also implies good value—getting your money’s worth. For example, even low-grade products should still work as expected, be safe to use, and last a reasonable amount
of time. Consider the following examples.
Example: Quality of Gasoline Grades
Petroleum refiners provide gasoline in several different grades based on the octane rating because higher octane ratings are suitable for higher compression engines. Gasoline must not be contaminated
with dirt or water, and the actual performance of the fuel must be close to its octane rating. A shipment of low-grade gasoline graded as 87 octane that is free of water or other contaminants would
be of high quality, while a shipment of high-grade 93 octane gas that is contaminated with dirt would be of low quality.
Determining how well products meet grade requirements is done by taking measurements and then interpreting those measurements. Statistics—the mathematical interpretation of numerical data—are useful
when interpreting large numbers of measurements and are used to determine how well the product meets a specification when the same product is made repeatedly. Measurements made on samples of the
product must be within control limits—the upper and lower extremes of allowable variation—and it is up to management to design a process that will consistently produce products between those limits.
Instructional designers often use statistics to determine the quality of their course designs. Student assessments are one way in which instructional designers are able to tell whether learning
occurs within the control limits.
Example: Setting Control Limits
A petroleum refinery produces large quantities of fuel in several grades. Samples of the fuels are extracted and measured at regular intervals. If a fuel is supposed to have an 87-octane performance,
samples of the fuel should produce test results that are close to that value. Many of the samples will have scores that are different from 87. The differences are due to random factors that are
difficult or expensive to control. Most of the samples should be close to the 87 rating and none of them should be too far off. The manufacturer has grades of 85 and 89, so they decided that none of
the samples of the 87-octane fuel should be less than 86 or higher than 88.
If a process is designed to produce a product of a certain size or other measured characteristic, it is impossible to control all the small factors that can cause the product to differ slightly from
the desired measurement. Some of these factors will produce products that have measurements that are larger than desired and some will have the opposite effect. If several random factors affect the
process, they tend to offset each other, and the most common results are near the middle of the range; this phenomenon is called the central limit theorem.
If the range of possible measurement values is divided equally into subdivisions called bins, the measurements can be sorted, and the number of measurements that fall into each bin can be counted.
The result is a frequency distribution that shows how many measurements fall into each bin. If the effects that are causing the differences are random and tend to offset each other, the frequency
distribution is called a normal distribution, which resembles the shape of a bell with edges that flare out. The edges of a theoretical normal distribution curve get very close to zero but do not
reach zero.
Example: Normal Distribution
A refinery’s quality control manager measures many samples of 87 octane gasoline, sorts the measurements by their octane rating into bins that are 0.1 octane wide, and then counts the number of
measurements in each bin. Then she creates a frequency distribution chart of the data, as shown in Figure 10.1.
Figure 10.1: Normal Distribution of Measurements
It is common to take samples—randomly selected subsets from the total population—and measure and compare their qualities, since measuring the entire population would be cumbersome, if not impossible.
If the sample measurements are distributed equally above and below the centre of the distribution as they are in Figure 10.1, the average of those measurements is also the centre value that is called
the mean and is represented in formulas by the lowercase Greek letter μ (pronounced mu). The amount of difference of the measurements from the central value is called the sample standard deviation or
just the standard deviation.
The first step in calculating the standard deviation is subtracting each measurement from the central value (mean) and then squaring that difference. (Recall from your mathematics courses that
squaring a number is multiplying it by itself and that the result is always positive.) The next step is to sum these squared values and divide by the number of values minus one. The last step is to
take the square root. The result can be thought of as an average difference. (If you had used the usual method of taking an average, the positive and negative numbers would have summed to zero.)
Mathematicians represent the standard deviation with the lowercase Greek letter (pronounced sigma). If all the elements of a group are measured, instead of just a sample, it is called the standard
deviation of the population and in the second step, the sum of the squared values is divided by the total number of values.
Figure 10.1 shows that the most common measurements of octane rating are close to 87 and that the other measurements are distributed equally above and below 87. The shape of the distribution chart
supports the central limit theorem’s assumption that the factors that are affecting the octane rating are random and tend to offset each other, which is indicated by the symmetric shape. This
distribution is a classic example of a normal distribution. The quality control manager notices that none of the measurements are above 88 or below 86 so they are within control limits, and she
concludes that the process is working satisfactorily.
Example: Standard Deviation of Gasoline Samples
The refinery’s quality control manager uses the standard deviation function in her spreadsheet program to find the standard deviation of the sample measurements and finds that for her data, the
standard deviation is 0.3 octane. She marks the range on the frequency distribution chart to show the values that fall within one sigma (standard deviation) on either side of the mean (Figure 10.2).
For normal distributions, about 68.3% of the measurements fall within one standard deviation on either side of the mean. This is a useful rule of thumb for analyzing some types of data. If the
variation between measurements is caused by random factors that result in a normal distribution, and someone tells you the mean and the standard deviation, you know that a little over two-thirds of
the measurements are within a standard deviation on either side of the mean. Because of the shape of the curve, the number of measurements within two standard deviations is 95.4%, and the number of
measurements within three standard deviations is 99.7%. For example, if someone said the average (mean) height for adult men in the United States is 178 cm (70 inches) and the standard deviation is
about 8 cm (3 inches), you would know that 68% of the men in the United States are between 170 cm (67 inches) and 186 cm (73 inches) in height. You would also know that about 95% of adult men in the
United States are between 162 cm (64 inches) and 194 cm (76 inches) tall and that almost all of them (99.7%) are between 154 cm (61 inches) and 202 cm (79 inches) tall. These figures are referred to
as the 68-95-99.7 rule.
Figure 10.2: One Sigma Range Most of the measurements are within 0.3 octane of 87
“14. Quality Planning” from Project Management by Adrienne Watt is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted. | {"url":"https://ecampusontario.pressbooks.pub/essentialsofprojectmanagement/chapter/10-2-quality-in-pm/","timestamp":"2024-11-10T21:20:20Z","content_type":"text/html","content_length":"111591","record_id":"<urn:uuid:1b7ddc0b-694f-4778-be0d-0aec7cba3f28>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00269.warc.gz"} |
Multivariate Bartlett test - Data Science Wiki
Multivariate Bartlett test :
The Multivariate Bartlett test is a statistical test used to determine whether there is significant differences in the variances of several groups. This test is an extension of the standard Bartlett
test, which is used to compare the variances of two groups.
To conduct a Multivariate Bartlett test, we first need to gather data from multiple groups. For example, let’s say we have data on the heights and weights of 10 individuals in each of three different
groups – Group A, Group B, and Group
. We can organize this data into a matrix, with each row representing a different individual and each column representing a different variable (i.e. height or weight).
Once we have our data organized in this way, we can perform the Multivariate Bartlett test using a statistical software program. The output of the test will give us a p-value, which is a measure of
that the differences in the variances between the groups are due to chance. If the p-value is less than a pre-determined level of significance (usually 0.05), we can conclude that there is a
significant difference in the variances between the groups.
Let’s take a look at an example. Suppose we have data on the heights and weights of 10 individuals in each of three different groups – Group A, Group B, and Group C. We can organize this data into a
matrix, with each row representing a different individual and each column representing a different variable (i.e. height or weight).
After performing the Multivariate Bartlett test, we find that the p-value is 0.01. This means that there is a 1%
that the differences in the variances between the groups are due to chance. Since this probability is less than our pre-determined level of significance (0.05), we can conclude that there is a
significant difference in the variances between the groups.
Another example of when the Multivariate Bartlett test might be used is in the analysis of data from a randomized controlled trial. Suppose we have data on the effects of a new drug on blood pressure
in two different groups of patients – Group 1 and Group 2. We can organize this data into a matrix, with each row representing a different patient and each column representing a different variable
(i.e. blood pressure before and after treatment).
After performing the Multivariate Bartlett test, we find that the p-value is 0.04. This means that there is a 4% probability that the differences in the variances between the groups are due to
chance. Since this probability is less than our pre-determined level of significance (0.05), we can conclude that there is a significant difference in the variances between the groups. This indicates
that the new drug is likely to have a different
on blood pressure in the two groups of patients. | {"url":"https://datasciencewiki.net/multivariate-bartlett-test/","timestamp":"2024-11-13T15:18:34Z","content_type":"text/html","content_length":"41943","record_id":"<urn:uuid:672315c9-253d-4df9-8991-0c7544ec0eb2>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00074.warc.gz"} |
9 The Ising model | Introduction to Magnetism HS22
9 The Ising model
When consistent with the symmetry of the problem, two-value classical spins, \(S^z=\pm 1\), can be assumed: \[$$\label{Ising-Ham} \tag{9.1} \mathcal{H}%\left[\left\{ S^z({\underline n})\right\}\
right] =-\frac{1}{2}J\sum_{|{\underline n}-{\underline n}'|=1} {S}^z({\underline n})\, {S}^z({\underline n}') +g \mu_{\rm B} B\sum_{{\underline n}} {S}^z({\underline n})\,.$$\] This approximation is
obviously justified in the limit in which the anisotropy \(D\) is significantly larger than other energies at play (\(J\), \(k_{\rm B} T\), etc.). Another instance is realized when the full
degeneracy of the total angular momentum of unpaired electrons of a magnetic atom in the gas phase (spherically symmetric environment, Hund’s rules) is reduced to the minimal two-fold degeneracy for
\(B=0\) in the solid phase (Kramers doublet). In this case, magnetism can be described with an effective spin one-half. Beside its application to magnetism, the Ising Hamiltonian (9.1) is used in
many different contexts, ranging from biophysics to social sciences.
Mean-field approximation
Assuming that the reader has encountered the mean-field approximation (MFA) in different courses, we refresh here only the aspects that are relevant to our discussion on ferromagnetism at finite \(T
\). The MFA is a simplified treatment of a many-body problem, which consists in replacing the original problem with its best single-particle counterpart. For magnetic systems, the reference
single-particle problem is the paramagnet, which can be regarded as the equivalent of the “ideal gas” in the study of statistical thermodynamics. In formula, the MFA of Hamiltonian (9.1) reads \[$$\
label{Ising-Ham-MFA} \tag{9.2} \mathcal{H} =-\frac{1}{2}J\sum_{|{\underline n}-{\underline n}'|=1} {S}^z({\underline n})\, {S}^z({\underline n}') +g \mu_{\rm B} B\sum_{{\underline n}} {S}^z({\
underline n})\simeq g \mu_{\rm B} B^{\rm eff}\sum_{{\underline n}} {S}^z({\underline n})$$\]
where the effective (Weiss) field \(B^{\rm eff}\) depends parametrically on the single-particle averages of the \(z\) spin projection \[$$\label{MFA-averages} \tag{9.3} s_{\rm av} = \langle {S}^z({\
underline n}) \rangle \,.$$\] Setting equal to zero terms like \[$$\label{MFA-fluct-zero} \tag{9.4} \left[s_{\rm av} - {S}^z({\underline n}) \right] \left[s_{\rm av} - {S}^z({\underline n'}) \right]
=0 \,,$$\] technically called fluctuations, makes it possible to rewrite the Hamiltonian (9.1) as the Hamiltonian of a paramagnet (with two energy levels, for the Ising model). In fact, in this way,
the product of pairspins can be expressed in terms of the average \(s_{\rm av}\) and single-particle contributions \[$$\label{MFA-pairspin} \tag{9.5} {S}^z({\underline n}) {S}^z({\underline n'}) = -
s_{\rm av}^2 + \left[{S}^z({\underline n}) + {S}^z({\underline n'}) \right] s_{\rm av}\,.$$\] Within the MFA the spontaneous magnetization behaves as follows \[$$\label{spontaneous-mag-MFA} \tag{9.6}
\begin{split} &\lim_{B\rightarrow 0^+} m(B,T) e 0 \qquad \text{for} \quad T < T_{\rm c} \\ &\lim_{B\rightarrow 0^+} m(B,T) = 0 \qquad \text{for} \quad T > T_{\rm c} \end{split}$$\] where the critical
temperature is defined as \(k_{\rm B}T_{\rm c} = z_n J\), with \(z_n\) number of nearest neighbors of each spin. Below \(T > T_{\rm c}\), the spontaneous magnetization is predicted to behave
critically \[$$m(T,0)\sim \left(T_{\rm c}-T\right)^{\frac{1}{2}}\,.$$\] In summary, the MFA predicts the occurrence of a phase with spontaneous magnetization at finite temperature, which is realized
below a material-dependent \(T_{\rm c}\). In the following we will discuss some limitations of this approach that are mainly rooted in the crudeness of the approximation in Eq.(9.4).
1D model
Probably one of the most striking failure of MFA is the prediction of a magnetically ordered phase below \(T_{\rm c}\) for one-dimensional (1D) systems. In fact, the result in Eq.(9.6) is independent
of the magnetic-lattice dimensionality D. The latter corresponds to the number of directions along which the exchange coupling propagates indefinitely. In practice, this dimension may also be
different from the actual dimensionality of the considered solid, like in molecular spin chains.
A well-known result of Statistical Physics is that systems whose magnetic lattice has a dimensionality is smaller than 2D cannot sustain spontaneous magnetization at thermodynamic equilibrium.
Landau argument
Here we provide a heuristic argument presented in the Landau–Lifshitz series that applies to the 1D Ising model and more generally to spin chains with uniaxial anisotropy. We evaluate the variation
of the free energy associated with the creation of a domain wall (DW) in a configuration with all the spins parallel to each other.
Creating a DW in a spin chain where all the spins point along the same direction increases the energy by a factor \(E_2-E_1=2J\). This DW may occupy \(N\) different positions in the spin chain, so
that this set of configurations has an entropy of the order of \(S_2\simeq k_{\rm B} \ln(N)\). The entropy of the ground state vanishes if we assume that the two spins at the boundaries have been
forced to point upward (otherwise one has \(S_1=k_{\rm B} \ln(2)\)). Therefore, the free-energy difference between the two configurations sketched in Fig.9.1 is roughly given by \[$$\label
{Landau_arg} \tag{9.7} \Delta F \simeq 2J - k_{\rm B} T\ln (N)\,.$$\] The qualitative behavior of \(\Delta F\) is sketched in Fig.9.2. A characteristic temperature-dependent threshold \(\bar N\) can
be defined such that for \(N>\bar N\) the free energy difference \(\Delta F\) is negative and therefore DWs start forming spontaneously in the chain. The threshold \(\bar N\) is obtained by requiring
\(\Delta F=0\), which gives
\[$$\label{N_bar} \tag{9.8} \bar N \simeq \exp\left(2\beta J\right).$$\] Practically, when \(\bar N\) is larger than the number of spins^34 in the chain \(N\) (low temperature), the ground-state
configurations with all spins aligned are also minima of the free energy, since \(\Delta F>0\). In this case, as far as equilibrium properties are concerned, the behavior of the spin chain is
reminiscent to that of a two-level paramagnet with magnetic moment \(\mu=N g S \mu_B\). When \(\bar N < N\) (high temperature), instead, the ground-state configurations with all spins aligned do not
minimize the free energy and DWs are always present in the system at equilibrium. In the limit of an infinite chain, the same argument can be repeated to justify the presence of an indefinite number
of DWs. We refer to this condition – realized in spin chains at higher temperatures – as the thermodynamic limit in which the inverse of \(\bar N\) is proportional to the average density of DWs.
Correlation length
In 1D magnetic systems the averaged pair-spin correlation decays exponentially with the separation between spins. Focusing on the Ising model, in which only the spin component along \(z\) is defined,
one has \[$$\langle S^z_{i} S^z_{i+r}\rangle = {\rm e}^{-r/\xi} \,.$$\] The characteristic scale of this decay defines the correlation length \(\xi\). It can be shown that \(\xi\) is related to the
susceptibility measured along the easy axis in zero field by the general equation \[$$\label{chi_xi} \tag{9.9} \chi= 2\, \frac{C}{k_{\rm B}T}\,\xi\, ,$$\] where \(C\) is the Curie constant
characterizing the magnetic centers coupled to form the chain. Apart from proportionality factors, \(\bar N\) defined above and indicated with a dot in Fig.9.2 can be identified with the correlation
length \(\xi\). Thus, similarly to \(\bar N\), one expects a leading dependence of the Arrhenius type for the correlation length as well: \[$$\label{exp_xi} \tag{9.10} \xi \sim \exp\left(2\beta J\
For a finite chain with \(N< \bar N\) (see Fig.9.2) the role of the correlation length in the susceptibility is – roughly speaking – replaced by the chain size \(N\).
2D lattice
An argument similar to the Landau’s one, holding for the 1D Ising model, can be developed for the 2D system as well. In this case we should refer to the possibility of reversing a cluster of spins
enclosed in a perimeter of
lattice sites and embedded in a region of spins all pointing in the same direction, as sketched in Fig.
For simplicity we consider a square lattice and sharp domain walls, meaning that all the spins are assumed to point either along \(S^z=+1\) (outward) or along \(S^z=-1\) (inward). The total cost in
terms of exchange energy is of the order of \(2J\, l\). To estimate the entropy variation due to the creation of a reversed cluster in an otherwise uniform spin configuration, we can think of a
self-avoiding random walk. Suppose that a walker can move with one step from the center of a square in Fig.9.3 to the center of a neighboring square. At each step the walker has at most three choices
of which way to go, since it has to avoid itself (the walker cannot take a step back in the direction where it came from). A possible random walk is highlighted with a thick line in the figure. Based
on these simple considerations, we expect the number of closed loops corresponding to the perimeter \(l\) to be of the order \(p^l\), with \(p<3\). As a result, the free-energy variation associated
with the flip of a cluster delimited by a perimeter \(l\) is roughly \(\Delta F=2J\, l - k_{\rm B} T \,l \ln p\). Therefore, for \(T< 2J/(k_{\rm B} \ln p)\) the ordered phase – with all the spin
aligned along \(S^z=+1\) – should be stable against the formation of large domains with reversed spins. This argument for the existence of an ordered low-temperature phase in this 2D Ising model and,
thus, of a finite Curie temperature \(T_{\rm c}\) was first put forward by Peierls – in more precise terms.
Rigorous results
The Ising model represents a particularly lucky case in which the heuristic arguments given above can be checked by solving the problem analytically. Even if we will not derive these results, it is
useful to recall which steps should be followed to prove rigorously whether a model is consistent with a phase with spontaneous magnetization (finite magnetization in zero external field) for \(T\ne
0\) or not. To this end, one has to compute:
1. the partition function \[$$\label{Part_fct_Ising} \tag{9.11} \mathcal{Z}= \mathcal{T}r\left\{e^{-\beta \mathcal{H}\left[\left\{{S}^z({\underline n})\right\}\right]}\right\}$$\] where \(\beta =1/
(k_{\rm B} T)\) and the trace is obtained by letting each discrete variable take the two possible values \({S}^z({\underline n})=\pm 1\) (\(\mathcal{Z}\) is a sum with \(2^N\) terms!)
2. the thermal average of the magnetic moment \[$$m(T,B)=-\frac{1}{N}\frac{\partial F}{ \partial B}= \frac{1}{N}\frac{1}{\beta} \frac{\partial \ln\mathcal{Z} }{ \partial B}$$\] 3. the limit \[$$\
label{spontaneous_mag_Ising} \tag{9.12} m(T,0)=\lim_{B\rightarrow 0^+} m(T,B)$$\] and evaluate if there exists a temperature \(T_{\rm c}\) below which the limit (9.12) takes a value different from
This procedure can be carried out analytically for the Ising model in 1D and 2D producing different results:
\(\bullet\) For 1D, no spontaneous magnetization is possible at any finite temperature.
\(\bullet\) For 2D, a spontaneous magnetization appears for \(T<T_{\rm c}\). The transition temperature is given by \[$$\label{T_c_Onsager} \tag{9.13} \sinh\left(\frac{2J}{k_{\rm B}T_{\rm c}}\right)=
1 \quad\Rightarrow\quad T_{\rm c} = \frac{2}{\ln(1+\sqrt{2})} \frac{J}{k_{\rm B}} \simeq 2.27 \frac{J}{k_{\rm B}}\,.$$\] The comparison with the MF theory shows that the latter typically
overestimates the transition temperature: The critical temperature of the 2D Ising model reported in Eq.(9.13) has to be compared with \(T_{\rm c}^{\text{MF}}=4 J/k_{\rm B}\) (\(z_n=4\) for a square
lattice). Expanding the spontaneous magnetization close to \(T_{\rm c}\) yields \[$$m(T,0)\sim \left(T_{\rm c}-T\right)^{\frac{1}{8}}\,.$$\] Thus, for the 2D Ising model the exact value of the
critical exponent is \(\beta=1/8\), at odds with the MF value \(\beta^{\text{MF}}=1/2\).
Indeed both these exact results obtained for the 1D and 2D Ising model show that the MFA overlooks some important features of the transition from the paramagnetic to the ferromagnetic phase, possibly
occurring upon lowering the temperature.
34. In molecular spin chains \(\bar N\) should be compared with the average number of sites separating two successive defects.↩︎ | {"url":"https://vindigni.ch/intro-mag-hs23/week-09.html","timestamp":"2024-11-14T17:59:39Z","content_type":"text/html","content_length":"44659","record_id":"<urn:uuid:d3795133-f583-462b-9ce2-090fef54488a>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00201.warc.gz"} |
Hadwiger–Nelson problem
Short description: Mathematical problem
Unsolved problem in mathematics:
How many colors are needed to color the plane so that no two points at unit distance are the same color?
(more unsolved problems in mathematics)
In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at
distance 1 from each other have the same color. The answer is unknown, but has been narrowed down to one of the numbers 5, 6 or 7. The correct value may depend on the choice of axioms for set theory.
Relation to finite graphs
The question can be phrased in graph theoretic terms as follows. Let G be the unit distance graph of the plane: an infinite graph with all points of the plane as vertices and with an edge between two
vertices if and only if the distance between the two points is 1. The Hadwiger–Nelson problem is to find the chromatic number of G. As a consequence, the problem is often called "finding the
chromatic number of the plane". By the de Bruijn–Erdős theorem, a result of (de Bruijn Erdős), the problem is equivalent (under the assumption of the axiom of choice) to that of finding the largest
possible chromatic number of a finite unit distance graph.
According to (Jensen Toft), the problem was first formulated by Nelson in 1950, and first published by (Gardner 1960). (Hadwiger 1945) had earlier published a related result, showing that any cover
of the plane by five congruent closed sets contains a unit distance in one of the sets, and he also mentioned the problem in a later paper (Hadwiger 1961). (Soifer 2008) discusses the problem and its
history extensively.
One application of the problem connects it to the Beckman–Quarles theorem, according to which any mapping of the Euclidean plane (or any higher dimensional space) to itself that preserves unit
distances must be an isometry, preserving all distances.^[2] Finite colorings of these spaces can be used to construct mappings from them to higher-dimensional spaces that preserve distances but are
not isometries. For instance, the Euclidean plane can be mapped to a six-dimensional space by coloring it with seven colors so that no two points at distance one have the same color, and then mapping
the points by their colors to the seven vertices of a six-dimensional regular simplex with unit-length edges. This maps any two points at unit distance to distinct colors, and from there to distinct
vertices of the simplex, at unit distance apart from each other. However, it maps all other distances to zero or one, so it is not an isometry. If the number of colors needed to color the plane could
be reduced from seven to a lower number, the same reduction would apply to the dimension of the target space in this construction.^[3]
Lower and upper bounds
The fact that the chromatic number of the plane must be at least four follows from the existence of a seven-vertex unit distance graph with chromatic number four, named the Moser spindle after its
discovery in 1961 by the brothers William and Leo Moser. This graph consists of two unit equilateral triangles joined at a common vertex, x. Each of these triangles is joined along another edge to
another equilateral triangle; the vertices y and z of these joined triangles are at unit distance from each other. If the plane could be three-colored, the coloring within the triangles would force y
and z to both have the same color as x, but then, since y and z are at unit distance from each other, we would not have a proper coloring of the unit distance graph of the plane. Therefore, at least
four colors are needed to color this graph and the plane containing it. An alternative lower bound in the form of a ten-vertex four-chromatic unit distance graph, the Golomb graph, was discovered at
around the same time by Solomon W. Golomb.^[4]
The lower bound was raised to five in 2018, when computer scientist and biologist Aubrey de Grey found a 1581-vertex, non-4-colourable unit-distance graph. The proof is computer assisted.^[5]
Mathematician Gil Kalai and computer scientist Scott Aaronson posted discussion of de Grey's finding, with Aaronson reporting independent verifications of de Grey's result using SAT solvers. Kalai
linked additional posts by Jordan Ellenberg and Noam Elkies, with Elkies and (separately) de Grey proposing a Polymath project to find non-4-colorable unit distance graphs with fewer vertices than
the one in de Grey's construction.^[6] As of 2021, the smallest known unit distance graph with chromatic number 5 has 509 vertices.^[7] The page of the Polymath project, (Polymath 2018), contains
further research, media citations and verification data.
The upper bound of seven on the chromatic number follows from the existence of a tessellation of the plane by regular hexagons, with diameter slightly less than one, that can be assigned seven colors
in a repeating pattern to form a 7-coloring of the plane. According to (Soifer 2008), this upper bound was first observed by John R. Isbell.
The problem can easily be extended to higher dimensions. Finding the chromatic number of 3-space is a particularly interesting problem. As with the version on the plane, the answer is not known, but
has been shown to be at least 6 and at most 15.^[8]
In the n-dimensional case of the problem, an easy upper bound on the number of required colorings found from tiling n-dimensional cubes is [math]\displaystyle{ \lfloor2+\sqrt{n}\rfloor^n }[/math]. A
lower bound from simplexes is [math]\displaystyle{ n+1 }[/math]. For [math]\displaystyle{ n\gt 1 }[/math], a lower bound of [math]\displaystyle{ n+2 }[/math] is available using a generalization of
the Moser spindle: a pair of the objects (each two simplexes glued together on a facet) which are joined on one side by a point and the other side by a line. An exponential lower bound was proved by
Frankl and Wilson in 1981.^[9]
One can also consider colorings of the plane in which the sets of points of each color are restricted to sets of some particular type.^[10] Such restrictions may cause the required number of colors
to increase, as they prevent certain colorings from being considered acceptable. For instance, if a coloring of the plane consists of regions bounded by Jordan curves, then at least six colors are
See also
1. ↑ (Soifer 2008), pp. 557–563; (Shelah Soifer).
2. ↑ Soifer (2008), p. 19.
3. ↑ (Kalai 2018); (Aaronson 2018)
4. ↑ (Coulson 2002); (Radoičić Tóth).
5. ↑ See, e.g., (Croft Falconer).
6. ↑ (Woodall 1973); see also (Coulson 2004) for a different proof of a similar result.
• Aaronson, Scott (April 11, 2018), Amazing progress on longstanding open problems, https://www.scottaaronson.com/blog/?p=3697
• Beckman, F. S.; Quarles, D. A. Jr. (1953), "On isometries of Euclidean spaces", Proceedings of the American Mathematical Society 4 (5): 810–815, doi:10.2307/2032415
• "A colour problem for infinite graphs and a problem in the theory of relations", Nederl. Akad. Wetensch. Proc. Ser. A 54: 371–373, 1951, doi:10.1016/S1385-7258(51)50053-7, https://research.tue.nl
• Chilakamarri, K. B. (1993), "The unit-distance graph problem: a brief survey and some new results", Bull Inst. Combin. Appl. 8: 39–60
• Chilakamarri, Kiran B. (1996), "Unit-distance graphs, graphs on the integer lattice and a Ramsey type result", Aequationes Mathematicae 51 (1–2): 48–67, doi:10.1007/BF01831139
• Coulson, D. (2004), "On the chromatic number of plane tilings", J. Austral. Math. Soc. 77 (2): 191–196, doi:10.1017/S1446788700013574, http://www.austms.org.au/Publ/JAustMS/V77P2/a83.html
• Coulson, D. (2002), "A 15-colouring of 3-space omitting distance one", Discrete Math. 256 (1–2): 83–90, doi:10.1016/S0012-365X(01)00183-2
• Croft, Hallard T.; Falconer, Kenneth J. (1991), Unsolved Problems in Geometry, Springer-Verlag, Problem G10
• Frankl, P.; Wilson, R.M. (1981), "Intersection theorems with geometric consequences", Combinatorica 1 (4): 357–368, doi:10.1007/BF02579457
• "The celebrated four-color map problem of topology", Scientific American 203 (4): 218–230, September 1960, doi:10.1038/scientificamerican0960-218
• de Grey, Aubrey D.N.J. (2018), "The Chromatic Number of the Plane Is at least 5", Geombinatorics 28: 5–18, Bibcode: 2016arXiv160407134W
• "Überdeckung des euklidischen Raumes durch kongruente Mengen", Portugal. Math. 4: 238–242, 1945
• "Ungelöste Probleme No. 40", Elem. Math. 16: 103–104, 1961
• Heule, Marijn J.H. (2018), Computing Small Unit-Distance Graphs with Chromatic Number 5, Bibcode: 2018arXiv180512181H
• Jensen, Tommy R.; Toft, Bjarne (1995), Graph Coloring Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, pp. 150–152
• Kalai, Gil (April 10, 2018), Aubrey de Grey: The chromatic number of the plane is at least 5, https://gilkalai.wordpress.com/2018/04/10/
• Mixon, Dustin (February 1, 2021), Polymath16, seventeenth thread: Declaring victory, https://dustingmixon.wordpress.com/2021/02/01/polymath16-seventeenth-thread-declaring-victory/, retrieved 16
August 2021
• Hadwiger-Nelson problem (Polymath project page), April 2018, http://michaelnielsen.org/polymath1/index.php?title=Hadwiger-Nelson_problem
• Radoičić, Radoš; Tóth, Géza (2003), "Note on the chromatic number of the space", Discrete and Computational Geometry: The Goodman–Pollack Festschrift, Algorithms and Combinatorics, 25, Berlin:
Springer, pp. 695–698, doi:10.1007/978-3-642-55566-4_32, http://sziami.cs.bme.hu/~geza/chromatic.pdf
• Rassias, Themistocles M. (2001), "Isometric mappings and the problem of A. D. Aleksandrov for conservative distances", in Florian, H.; Ortner, N.; Schnitzer, F. J. et al., Functional-Analytic and
Complex Methods, their Interactions, and Applications to Partial Differential Equations: Proceedings of the International Workshop held at Graz University Of Technology, Graz, February 12–16,
2001, River Edge, New Jersey: World Scientific Publishing Co., Inc., pp. 118–125, doi:10.1142/4822, ISBN 978-981-02-4764-5
• Shelah, Saharon; Soifer, Alexander (2003), "Axiom of choice and chromatic number of the plane", Journal of Combinatorial Theory, Series A 103 (2): 387–391, doi:10.1016/S0097-3165(03)00102-X
• Soifer, Alexander (2008), The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, New York: Springer, ISBN 978-0-387-74640-1
• Woodall, D. R. (1973), "Distances realized by sets covering the plane", Journal of Combinatorial Theory, Series A 14 (2): 187–200, doi:10.1016/0097-3165(73)90020-4
External links
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overtone series
Music is Math/Math is Music. Using a root of 360 Hz and the Pythagorean intervals to generate twelve tones, extend each of the "tones" upward through an infi
Moog reuses the same principle for the Subharmonicon sequencer except it replaces the overtone series by a subharmonic succession (÷2, ÷3, until ÷16). PYTHAGORAS is the Greek philosopher to whom is
attributed the discovery of the mathematical proportion between note intervals that defines today the arithmetic principle behind harmonic
For example, the fundamental of the first series is 1000 Hz, and the fundamental of the second series is 500 Hz. The other (higher) frequency components are called overtones, harmonics or partials.
Harmonics of an Open String The overtone series orders intervals by decreasing size but increasing complexity. The first interval of the overtone series, a P8, is the “simplest” interval of 2:1. As
the overtone series moves upward, each interval becomes smaller but more complex. A P5 has a ratio of 3:2, a P4 has a ratio of 4:3, a M3 has a ratio of 5:4, and onward. Se hela listan på
sacred-texts.com This worksheet is designed to help you find the partials that should be most comfortable for you as a trumpet player.
how much time passes during one cycle of a wave form. What does the period of a sound We need to return to the story of Pythagoras that I introduced to you in the The overtone series can be easily
demonstrated on the piano by playing the. series and how they connect to dot notation with musical an overtone series. Essential What is the Pythagorean scale and how is it created? What is Just
temperament as a series of steps toward the solution of a practical problem. Pythagoras' experiments with this simple musical instrument revealed that the was utterly at odds with the pure intervals
ofjust intonation and the o Feb 29, 2020 The number of tones in the scale formed from the Harmonic Series such as the Pythagorean Temperament (in combination with Concert relationships between
frequencies, the harmonic series, the materials necessary to build musical The Greek mathematician Pythagoras, most known for his. Dec 8, 1999 and Britney Spears, people have used these distinctive
series of pitches, Pythagoras made his discoveries using a single-stringed instrument called a will sound not only as the note struck but also as all of it Monochord Strings tensioned on one side,
With 25 overtone strings in c and 5 bass strings in C, Instrument made of ash and cherry, Dimensions: 134 x 30 x 10 of intonation systems used today: Equal Temperament, Just, and Pythagorean.
Consider Deity at D and the twin tones A and G shown in Fig. 4 serving Plato The study of harmonics in the West can be traced back to Pythagoras, the 6th The overtone scale is also known as the
harmonic series where the notes rise in Even a 22-tone scale used in India shows an underlying Pythagorean structure, no doubt derived from the harmonic series.
He explains the steady revealing of that tricky overtone series this way: However deeply rooted the attachment to the habitual, and inertia, may be in the ways and nature of humankind, in equal
measure are energy, and opposition to the existing order, characteristic of all that has life.
Pythagoras of Samos (570 - 495 BC) was one of the first major greek philosophers and mathematicians. It was he who first described the nature and function of sound and our relation to it. 3 - The
overtone series is what defines timbre.
av A Zethson · 2017 · Citerat av 2 — uppmaning – och Pythagoras långt dessförinnan – att lyssna på världen; att med på samlingen the Harmonic Series – A Compilation of Musical Works In Just.
The first in series is the octave, then fifth above that octave, then the fourth above that fifth, which is enharmonic (equivalent) to the second octave of the original tone. There is another
interesting relationship. Pythagoras (569-475 B.C.), in search of a more humanly tolerant philosophical environment, emigrated from Greece to Metapontum and Crotone in southern Italy in 532 B.C.
given lengths of string at constant tension to reveal the ‘overtone’ series. C&C is in fact a The overtone series does explain fairly well why fifths and fourths above/below a note, and perhaps major
thirds above/below a note sound good. But as many have pointed out here, it seems spurious to use the overtone series to motivate the entire major scale. High quality Pythagoras gifts and
merchandise. Inspired designs on t-shirts, posters, stickers, home decor, and more by independent artists and designers from around the world.
(P4) and fifth (P5), corresponding to the lowest in the harmonic overtone series. in equal temperament, 294 in Pythagorean tuning and 316 in just intonation. Lyssna gratis på Dr. James Hopkins –
Golden Ratios - Pythagorean Harmonic Healing 1 (Golden Ratios, Lambdoma D&A och mer). 6 låtar (64:26). Upptäck mer A harmonic is any member of the harmonic series. En överton är The geometric mean
is also one of the three classical Pythagorean means, together with the An improvisation is actually nothing but a series of corrections.
Forskningsfusk lund bmc
The "Pythagorean" scale 3.1.- Derivation of the diatonic Proportion in Musical Scales | fundamental frequency, also called harmonics or overtones (Figure 30. [The Kike Annenberg's] TV Guide reported
that the series could boast the The plot, reeking with Bondian overtones, told the story of two UN agents Archimedes, Pythagoras, Aristotle, Plato, and Socrates are all too ancient kan varje
funktion f (x) representeras som en serie (en oändlig summa av siffror eller 130 och förklarade varför Pythagorean tredje i harmonisk prestanda låter spänd jämfört med den rena. Overtone är en extra
frekvens. Empty Vessels can be regarded a series of episodes which have a visual and The water sounds acquire harmonic and gritty characteristicswhich are Sons of Pythagoras · Steve Rice Quartet ·
Sir Georg Solti & Chicago Symphony Soothing Nature Sounds, Nature Sound Series, Nature Sounds Collection av I Bengtsson · 1969 · Citerat av 1 — briljant framställda i Radiokonservatoriets serie av
läroböcker redundance in the harmonic parameter of the com- position tidligst kendes fra Pythagoras (ca. Harmonisk serieroll i en precis intonationsintervallrankning?
Describe string vibration and the harmonic series. The Pythagorean scale is any scale which may be constructed from only pure perfect fifths (3:2) and octaves More on deducing the Pythagorean ratios
of other notes of the chromatic scale ):.
svenska riksdagen idaglisa sjalvservice medarbetarelillemor runessonevolution genom naturligt urvaltvangsvard sjalvmorddiskursanalys psykologi
It is said that the Greek philosopher and religious teacher Pythagoras (c. 550 BC) created a seven-tone scale from a series of consecutive 3:2 perfect fifths. The Pythagorean cult's preference for
proportions involving whole numbers is evident in this scale's construction, as all of its tones may be derived from interval frequency ratios based on the first three counting numbers: 1, 2, and 3.
It's vigorous and minimalist dexterity exudes a severe sound pouring landscapes of bliss and light from the pythagorean region of the harmonic 1.2.5 Serie- och parallellkopplade kraftkällor . den
resulterande spänningen U i kretsen även beräknas med Pythagoras sats: p C 2 = A2 + B 2 eller C = A2 + B 2 (3.6.2) 3.6.3 LC oscillator; (3.6.2) 3.6.4 Crystal oscillator, overtone oscillator; is a
phenomenon known as the overtone series, in which any tone, played or sung, activates a column of mathematically-related notes which vibrate sympathetically with the sounded pitch and create | {"url":"https://lonoymj.firebaseapp.com/71029/49285.html","timestamp":"2024-11-06T18:45:06Z","content_type":"text/html","content_length":"13379","record_id":"<urn:uuid:af9ea5bb-eaa5-42e9-b692-7b3fbd3ed3e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00407.warc.gz"} |
Graph Variations on a Secant Function - dummies
The graph for a secant function is different from the cosecant in several ways, but one of the most obvious ways is that the graph of the secant is symmetric about the y-axis. The secant is a mirror
reflection over that axis. You can use this property to do something interesting to the graph.
The usual translations and multiplications affect the secant graph in the same way it does the graphs of the other trig functions. If you multiply the function by 1/6 and add 2π to the angle
variable, as in the equation
you get this figure.
The equation above, shown in a graph.
Compared to y = sec x, the graph in the preceding figure is much closer to the x-axis and seems to be flattened out between the asymptotes. These changes happen when you multiply the function by a
number between 0 and 1. The turning point is still in the same place, but the y-value is much closer to 0.
The other curiosity is that the asymptotes don’t seem to be different. They aren’t — and they shouldn’t be. By adding 2π to the angle variable, you shift the graph 2π units to the left. The graph
really has shifted, but you can’t tell, because the new graph lies completely on the old one. When the shift is equal to the period of the function (the length of the interval that it takes for the
function values to start repeating over again), the change isn’t apparent.
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Sensitive word filter notes
Common methods
1. BF algorithm
2. KMP runs many times
O(k * m + sum(length(n))
3. Regular NFA: uncertain finite automata
4. DFA: deterministic finite automata, Trie, AC
Single pattern matching algorithm
Find a pattern string in a main string
BF algorithm (violence matching)
char T[maxn],P[maxn] // T is the main string and P is the mode string
public void check()
int lenT=stren(T);
int lenP=strlen(P);
int i = 0; // Position of main string
int j = 0; // Position of the pattern string
while (i < lenT && j < lenP)
if (T[i] == P[j]) {
}else {
i = i - j + 1; // i - j is the starting position of the match
j = 0;
if (j == p.length) return i - j;
else return -1;
KMP: https://blog.csdn.net/dl962454/article/details/79910744
After T[i] and P[j] fail to match, I will not go back, j will go back, and find the same longest prefix and suffix in the matched string
abac matches abac, failed to match at the third position
next array
The next array considers the longest prefix and suffix except the current character, taking "ABCDABD" as an example,
- the prefix and suffix of "A" are empty sets, and the length of common elements is 0;
- the prefix of "AB" is [A], the suffix is [B], and the length of common elements is 0;
- the prefix of "ABC" is [A, AB], the suffix is [BC, C], and the length of common elements is 0;
- the prefix of "ABCD" is [A, AB, ABC], the suffix is [BCD, CD, D], and the length of common elements is 0;
- the prefix of "ABCDA" is [A, AB, ABC, ABCD], the suffix is [BCDA, CDA, DA, A], the common element is "A", and the length is 1;
- the prefix of "ABCDAB" is [A, AB, ABC, ABCD, ABCDA], the suffix is [BCDAB, CDAB, DAB, AB, B], the common element is "ab", and the length is 2;
- the prefix of "ABCDABD" is [A, AB, ABC, ABCD, ABCDA, ABCDAB], the suffix is [BCDABD, CDABD, DABD, ABD, BD, D], and the length of common elements is 0.
Code display
public class KMP {
* Find the next array of a character array. 0 represents ab initio matching and - 1 does not match
* @param P Pattern string array
* @return next array
public static int[] getNextArray(char[] P) {
int[] next = new int[P.length];
next[0] = -1;
next[1] = 0;
int k;
for (int j = 2; j < P.length; j++) {// Pay attention to the maximum Prefix suffix common length of [0, j-1] characters
k = next[j - 1];// k: Maximum Prefix suffix common length of the first [0,j-2] characters
while (k != -1) {
if (P[j - 1] == P[k]) {// If the j-1st character and the k-th character are the same, the maximum Prefix suffix common length of the [0, j-1] character is [0, j-2]+1
next[j] = k + 1;
else {// Otherwise, find the maximum Prefix suffix common length of the [0,j-1] character
abacababe，j Point to e
abac: prefix
abab: suffix
The prefix and suffix do not match. It is necessary to match the abacabab prefix and suffix. Suffix b has been determined and cannot be moved. The characters in front of prefix c and suffix b are the same
Assuming that the original prefix is xc and the suffix after re matching is yb, the suffixes of y and X are the same. You only need to find the prefix of X, and find next[x + 1], that is, next[k]
k = next[k]; //Transitivity
next[j] = 0; //When k==-1 jumps out of the loop, next[j] = 0, otherwise next[j] will be assigned before break
return next;
* KMP pattern matching for main string T and pattern string P
* @param T Main string
* @param P Pattern string
* @return If the matching is successful, the position of P in S (the position corresponding to the first same character) is returned. If the matching fails, the position of - 1 is returned
public static int kmpMatch(String T, String P){
char[] T_arr = T.toCharArray();
char[] P_arr = P.toCharArray();
int[] next = getNextArray(P_arr);
int i = 0, j = 0;
while(i < T_arr.length && j < P_arr.length){
if(j == -1 || T_arr[i] == P_arr[j]){
j = next[j];
if(j == P_arr.length){
return i - j;
return -1;
public static void main(String[] args) {
System.out.println(kmpMatch("abcabaabaabcacb", "aaaabaaac"));
Multi pattern matching algorithm
Find multiple pattern strings in a main string
TRIE tree
AC automata = TRIE + KMP
Construct TRIE tree
Trie tree node
type Node struct {
// Is it the root node
RootNode bool
// Is it the end of the word
PathEnd bool
// Failed node
FailNode *Node
// Stored characters
Character rune
// length
Length int
// Next character
Children map[rune]*Node
Construct failed pointer
For the string bca, its strict suffix (I understand that it does not include its own suffix) is ca,a,None, and the prefix is bca,bc,b,None; For another string caa, its prefix is caa,ca,c,None. We
find that the suffix of bca actually appears in the prefix of caa. Therefore, when bca fails to match the next character, we can jump to the first a node of caa and continue to match the next
character, because ca has already been matched.
• First, regardless of the root node, the children of the root node must point to the root node because their suffixes are empty.
• Assuming that we already have the mismatch pointer of node x, how do we construct the mismatch pointer of their child? Because the mismatch pointer guarantees the maximum suffix, it must ensure
that the characters before x match. We know that the mismatch pointer of X points to the maximum suffix y of node x, so we just need to see if there are characters corresponding to the child node
in the child node of node y. if so, that's good. The mismatch pointer of child is the child of Y;
• If not, let's continue to look at the mismatch pointer of the y node, because it also points to the maximum suffix of the y node and ensures that it matches the x character. This continues until
the corresponding node, or until the root node.
from collections import deque
def build_ac(root):
q = deque()
while q:
node = q.popleft()
for value,child in node.children.items():
if node == root:
child.fail = root
fail_node = node.fail
c = fail_node.child(value)
if c:
child.fail = c
child.fail = root
return root
for i,c in enumerate(s):
while node and not node.child(c):
node = node.fail
if not node:
node = root
node = node.child(c)
out = node
while out:
if out.finished:
out = out.fail | {"url":"https://www.fatalerrors.org/a/sensitive-word-filter-notes.html","timestamp":"2024-11-10T21:40:16Z","content_type":"text/html","content_length":"16905","record_id":"<urn:uuid:a52c3ca9-e4a5-4511-965f-d5ccaea4f13a>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00144.warc.gz"} |
What is a P/E ratio and why does it matter?—Sharesies New Zealand
What is a P/E ratio and why does it matter?
Let’s dig into how P/E ratios are calculated, what P/E ratios tell you, and how you can use P/E ratios in your investing decisions.
The price-to earnings ratio (P/E ratio) is a valuation tool that investors use to help determine if an investment is overvalued (“expensive”) or undervalued (“cheap”).
It was popularised by Benjamin Graham, an American economist (and mentor to Warren Buffet!), who believed in value investing— investing in companies he thought were worth more than what their share
price was trading at.
How is a P/E ratio calculated?
The P/E ratio is calculated by dividing the current share price of a company by its earnings per share. Earnings per share is exactly what it sounds like—the amount of profit a company made over a
certain time period, per share that exists.
So if a company made $10 million profit in a year, and it had 1 million shares, then its earnings per share would be $10. If the share price is $100 and the earnings per share is $10, then its P/E
ratio would be 10; that’s $100 (share price) divided by $10 (earnings per share).
If the share price was $200, then the company's P/E ratio would be 20; that’s $200 (share price) divided by $10 (earnings per share).
Sometimes, earnings per share is calculated based on what a company has earned in the past 12 months (trailing P/E ratio), while other times it’s calculated based on what people expect the company to
earn in the future (forward P/E ratio). In this blog, we’ll be talking about trailing P/E ratios.
What does the P/E ratio tell you?
In a nutshell, the P/E ratio is a way to compare how much an investor pays per share, with each dollar of profit that the company makes. It tells you if an investment is “expensive” (with a higher P/
E ratio) or “cheap” (with a lower P/E ratio) in relation to its earnings.
But share prices don’t happen on their own. They’re the result of lots of other investors deciding to buy and sell at certain prices. So a P/E ratio can be used as an indication of what other
investors currently think of a company. Let’s take a look at some different P/E ratios.
Higher P/E ratios
Generally, a higher P/E ratio indicates that other investors are willing to pay a higher share price today, compared to the company’s current earnings. This could be because they expect higher growth
from the company in the future. Sometimes, a company may be able to meet or exceed the market’s expectations of future growth—and other times, it might not deliver on these expectations. Just like
you, the market doesn’t have a crystal ball.
A higher P/E ratio can sometimes also indicate that an investment is overvalued or “expensive” (i.e. there’s potentially a disconnect between the company’s share price and its current underlying
earnings). This uncertainty of whether or not the company will be able to meet the market’s expectations of growth can mean that there are additional risks for investors.
Lower P/E ratios
On the other hand, a lower P/E ratio generally indicates that other investors are less willing to pay a higher share price today, compared to the company’s current earnings. This could indicate that
they don’t expect the company to grow as much in the future—otherwise, they may have been willing to pay more for their shares, which would have increased the company’s P/E ratio.
It’s worth noting that some companies with lower P/E ratios may be value traps—companies that appear to be “cheaply” priced and attract investors looking for a bargain, but may in fact be
experiencing financial instability or have little potential for growth.
Negative P/E ratios
Some companies can also have negative P/E ratios, meaning the company’s had negative earnings (and made a loss) in their last reporting year. A negative P/E ratio doesn’t mean that the share price is
“cheap” relative to its earnings—it actually means that investors are paying for a loss.
This can be intentional. For example, companies may sacrifice profits to instead invest in growth. Other times, a loss can be due to factors outside of the company’s control, such as regulatory
changes or global pandemics.
Companies that consistently have a negative P/E ratio generally aren’t generating enough money to make a profit, and may run the risk of bankruptcy in time—which is an added risk for investors.
Using P/E ratios in your investment decisions
When it comes to using P/E ratios in your investment decisions, there isn’t one correct way of looking at it. The way you approach P/E ratios is up to you.
Some investors pursue a growth strategy, where they invest in companies with higher P/E ratios that they think will become even more valuable in the future. Others pursue a value strategy, where they
invest in companies with lower P/E ratios that they believe the market has undervalued. And, of course, most investors actually do a combination of both.
There are a few things you may want to consider when you’re looking at a company’s P/E ratio.
How does the P/E ratio compare?
P/E ratios aren’t particularly helpful in isolation. Like any number or tool, it needs context. To make sense of a P/E ratio, you need something to compare it against.
This could be:
• other companies in the same industry
• the company’s direct competitors
• the company’s own P/E ratio in the past.
P/E ratios tend to differ between industries, so you may want to compare P/E ratios of similar companies. This is because the market has different expectations of growth for different industries. For
example, the growth expectations of an energy company are likely to be different to that of an electric vehicle manufacturer.
One place to start can be comparing a company’s P/E ratio against the P/E ratios of its direct competitors. If a company has a P/E ratio that seems “expensive” compared to its competitors, it may
indicate that the market thinks the company will have higher future growth than its competitors.
Do you agree with the P/E ratio?
Ultimately, the P/E ratio is just an indication of what other investors think—but you still need to make your own judgement call.
If the P/E ratio seems “expensive”, you might want to dig into why this might be. If other investors think the company is going to grow in the future, do you agree with them? Why or why not? You
might decide that the company’s higher P/E ratio is justified because of its future growth potential—or, you might decide that the company is overvalued, and won’t be able to meet the market’s future
And the same goes for companies with a lower P/E ratio. Do you agree with other investors that the company won’t have high growth in the future? Alternatively, how confident are you that the company
is undervalued by the market, rather than it being a value trap?
Risks and limitations
As we’ve mentioned, there are risks and limitations to using P/E ratios in your investment decisions, including uncertainty of growth, and the potential for value traps.
The P/E ratio we’ve covered in this blog, known as a trailing P/E ratio, is also only based on the past 12 months of a company’s actual earnings—which tells us little about the future of the company.
Instead, some investors choose to use a forward P/E ratio, which uses the expectations of a company’s earnings over the upcoming 12 months. Again, no one has a crystal ball, and these expectations
may not be met in reality.
P/E ratios are calculated using earnings per share, which can be influenced by the different accounting policies used by the companies, or tax rates in different countries. This means that the way a
company’s management reports their earnings can affect a company’s P/E ratio, and the comparability of it against other companies. Because of this, some investors find it useful to look at the P/E
ratio alongside other numbers, such as the price-to-book ratio (P/B ratio), price-to-sales ratio (P/S ratio), or price-to-cash flow ratio (P/CF ratio).
P/E ratios are also generally only useful when they’re compared against other P/E ratios—but even this has limitations. Comparing against other companies is based on the assumption that these
companies are priced or valued fairly in the market. In reality, companies, industries, and even entire markets can be undervalued or overvalued. This can lead to investors making inaccurate
assumptions about a company based on the P/E ratio.
Wrapping up
A P/E ratio is a useful tool to help determine the relative value of an investment. But remember that P/E ratios are just one part of the whole picture; they aren’t a perfect measure, and they aren’t
the only way to compare the value of a company. A P/E ratio is just one tool in the toolbox of valuation measures, and like others, has its limitations.
So consider it in combination with other measures, do your own due diligence, look at the company’s wider growth strategy, and think about what kind of investment fits with your goals and portfolio.
To find a company’s P/E ratio in Sharesies, see the Stats section on the investment’s page. The P/E ratio in Sharesies is a trailing P/E ratio.
Ok, now for the legal bit
Investing involves risk. You aren’t guaranteed to make money, and you might lose the money you start with. We don’t provide personalised advice or recommendations. Any information we provide is
general only and current at the time written. You should consider seeking independent legal, financial, taxation or other advice when considering whether an investment is appropriate for your
objectives, financial situation or needs.
Join over 700,000 investors | {"url":"https://www.sharesies.nz/learn/what-is-a-pe-ratio-and-why-does-it-matter","timestamp":"2024-11-10T08:07:16Z","content_type":"text/html","content_length":"231003","record_id":"<urn:uuid:d48b35d0-b1be-49be-b2ad-e3c7e4ac65a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00430.warc.gz"} |
OpenStax College Physics for AP® Courses, Chapter 30, Problem 6 (Problems & Exercises)
(a) An aspiring physicist wants to build a scale model of a hydrogen atom for her science fair project. If the atom is 1.00 m in diameter, how big should she try to make the nucleus? (b) How easy
will this be to do?
Question by
is licensed under
CC BY 4.0
Final Answer
a. $10\textrm{ }\mu\textrm{m}$
b. A precision of $10.0\textrm{ }\mu\textrm{m}$ would be very difficult, but approximately $10\textrm{ }\mu\textrm{m}$ could be accomplished with a piece of small dust, although it would be hard to
Solution video
OpenStax College Physics for AP® Courses, Chapter 30, Problem 6 (Problems & Exercises)
vote with a rating of votes with an average rating of.
Video Transcript
This is College Physics Answers with Shaun Dychko. A physicist wants to make a scale model of the atom with the atom having a size of 1.00 meter so the whole atom is 1.00 meter across and if that's
the case, what would the size of the nucleus need to be in order to be drawn to scale? So we have the ratio of the nuclear diameter of the model divided by the diameter of the whole atom has to equal
the ratio of the actual real nucleus divided by the diameter of a real atom and we'll solve for the diameter of nucleus then by multiplying both sides by the scale diameter of the atom and so we have
1.00 meter times 10 to the minus 15 meters— nuclear diameter— divided by 10 to the minus 10 meters for a typical atom across and this works out to 10 micrometers and 10 micrometers is possible to
create an object with that size like a piece of dust but it would be hard to see and if you want a precision to the tenths place that would be very difficult but you know to the tenths place in
micrometers might be possible and there we go! | {"url":"https://collegephysicsanswers.com/openstax-solutions/aspiring-physicist-wants-build-scale-model-hydrogen-atom-her-science-fair-0","timestamp":"2024-11-04T00:37:13Z","content_type":"text/html","content_length":"193514","record_id":"<urn:uuid:649823c7-f3b9-4ab4-90e9-a1c9795ee82e>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00127.warc.gz"} |
Digital rights management (DRM) protects the copyright of your digital content. DRM uses cryptographic software to ensure that only authorized users can have access to the material, modify or
distribute it. As the world becomes increasingly digital, the need for security has become ever more imperative. That’s where cryptography and its applications to cybersecurity come in.
Cryptography is a technique to secure information and communication by using a set of rule-based calculations called algorithms and some mathematical concepts so only the right person can understand
it. Cryptography is now being used to hold confidential data, including private what Is cryptography passwords, secure online. It is now used by cybersecurity experts to foster innovation,
ciphertext, as well as other protective measures that enforce but also insulate business and personal info. The history of cryptography finds its roots in Egypt around 4000 years ago.
Consequently, the need to develop novel cryptographic techniques that can withstand quantum attacks is becoming increasingly pressing, creating an ongoing challenge within the field of cryptography.
Cryptography works by taking plaintext (or cleartext) and scrambling it into ciphertext, so that the encoded output can be understood only by the intended recipient. As ciphertext, the information
should be unreadable to all except the intended recipient. A brute force attack occurs when hackers use computers to feedback loop over each letter in a character set systematically. A character set
can consist of letters, numbers, symbols, or anything else that the hackers may desire. In the most general terms, a brute force attack is a method of trial and error that attempts all possible
password combinations.
For instance, the best-known algorithms for solving the elliptic curve-based version of discrete logarithm are much more time-consuming than the best-known algorithms for factoring, at least for
problems of more or less equivalent size. Thus, to achieve an equivalent strength of encryption, techniques that depend upon the difficulty of factoring large composite numbers, such as the RSA
cryptosystem, require larger keys than elliptic curve techniques. For this reason, public-key cryptosystems based on elliptic curves have become popular since their invention in the mid-1990s.
Cryptanalysis of symmetric-key ciphers typically involves looking for attacks against the block ciphers or stream ciphers that are more efficient than any attack that could be against a perfect
Key exchange is the method used to share cryptographic keys between a sender and their recipient. But, no need to worry organizations and researchers are working to transition to these
quantum-resistant cryptographic techniques. Install Avast SecureLine VPN to encrypt all your online communications and protect your personal data.
In a known-plaintext attack, Eve has access to a ciphertext and its corresponding plaintext (or to many such pairs). In a chosen-plaintext attack, Eve may choose a plaintext and learn its
corresponding ciphertext (perhaps many times); an example is gardening, used by the British during WWII. It is a common misconception that every encryption method can be broken. In such cases,
effective security could be achieved if it is proven that the effort required (i.e., «work factor», in Shannon’s terms) is beyond the ability of any adversary.
Symmetric-key cryptosystems use the same key for encryption and decryption of a message, although a message or group of messages can have a different key than others. A significant disadvantage of
symmetric https://www.xcritical.in/ ciphers is the key management necessary to use them securely. Each distinct pair of communicating parties must, ideally, share a different key, and perhaps for
each ciphertext exchanged as well.
A message authentication code (MAC) is the symmetric version of a digital signature. One party creates a MAC tag, which is the symmetric version of a digital signature, and attaches it to the
document. Another party can verify the message’s integrity using the same key used to create the tag. This combination of public-key cryptography for key exchange and symmetric encryption for bulk
data encryption is known as hybrid encryption. An encryption algorithm is a procedure that converts a plaintext message into an encrypted ciphertext. Modern algorithms use advanced mathematics and
one or more encryption keys.
Instead, they use complex math to turn any data into a unique code made up of letters and numbers. In this method, both the sender and the receiver need to use the exact same secret key to understand
the data. It works by changing normal data into secret code (ciphertext) using the secret key and a specific mathematical process. Cryptography secures digital communication and information in
various systems and applications, ensuring confidentiality and data security. The difference between cryptography and encryption is that while cryptography can be broadly defined as the science of
sending secret messages, encryption is the specific process of converting data into code.
The first of these uses is the obvious one—you can keep data secret by encrypting it. The others take a bit of explanation, which we’ll get into as we describe the different types of cryptography.
Cryptography is an important computer security tool that deals with techniques to store and transmit information in ways that prevent unauthorized access or interference. Interest in the use of
cryptography grew with the development of computers and their connections over an open network. Over time, it became obvious that there was a need to protect information from being intercepted or
manipulated while being transmitted
over this network.
Many computer ciphers can be characterized by their operation on binary bit sequences (sometimes in groups or blocks), unlike classical and mechanical schemes, which generally manipulate traditional
characters (i.e., letters and digits) directly. However, computers have also assisted cryptanalysis, which has compensated to some extent for increased cipher complexity. To maintain data integrity
in cryptography, hash functions, which return a deterministic output from an input value, are used to map data to a fixed data size. Types of cryptographic hash functions include SHA-1 (Secure Hash
Algorithm 1), SHA-2 and SHA-3. These algorithms generate cryptographic keys, create digital signatures, safeguard data privacy, enable online browsing on the Internet, and ensure the confidentiality
of private transactions like credit and debit card payments. Stream ciphers work on a single bit or byte at any time and constantly change the key using feedback mechanisms.
• Also, the Fortinet FortiMail Cloud solution provides comprehensive email security solutions like email encryption to safeguard employees and data from cyberattacks.
• Messages could be encrypted so that they appear to be random text to anyone but the intended recipient.
• The sender then uses the recipient’s public key to encrypt the message.
• A cryptographic hash function is a tool for turning arbitrary data into a fixed-length “fingerprint”.
A common PKC type is multiplication vs. factorization, which takes two large prime numbers and multiplies them to create a huge resulting number that makes deciphering difficult. Another form of PKC
is exponentiation vs. logarithms such as 256-bit encryption, which increases protection to the point that even a computer capable of searching trillions of combinations per second cannot crack it.
Cryptography confirms accountability and responsibility from the sender of a message, which means they cannot later deny their intentions when they created or transmitted information. Digital
signatures are a good example of this, as they ensure a sender cannot claim a message, contract, or document they created to be fraudulent. Furthermore, in email nonrepudiation, email tracking makes
sure the sender cannot deny sending a message and a recipient cannot deny receiving it. They’re important for checking if data is safe; when data is sent or stored, its hash code is calculated and
sent or kept with the data.
The sender then uses the recipient’s public key to encrypt the message. Examples of public key use are plentiful in just about any communication over the Internet such as HTTPS, SSH, OpenPGP, S/MIME,
and a website’s SSL/TLS certificate. Hybrid encryption is used extensively in data transfer protocols for the web, such as in Transport Layer Security (TLS). When you connect to a website that uses
HTTPS (HTTP secure with TLS), your browser will negotiate the cryptographic algorithms that secure your connection. | {"url":"https://acdesarrollosinmobiliarios.com/understanding-cryptography-types-symmetric/","timestamp":"2024-11-14T06:57:10Z","content_type":"text/html","content_length":"101821","record_id":"<urn:uuid:5afa465f-6248-420e-ac1c-25090ee6b258>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00323.warc.gz"} |
How Can Infinitely Many Primes Be Infinitely Far Apart? | Quanta Magazine
Robert Neubecker for Quanta Magazine
If you’ve been following the math news this month, you know that the 35-year-old number theorist James Maynard won a Fields Medal — the highest honor for a mathematician. Maynard likes math questions
that “are simple enough to explain to a high school student but hard enough to stump mathematicians for centuries,” Quanta reported, and one of those simple questions is this: As you move out along
the number line, must there always be prime numbers that are close together?
You may have noticed that mathematicians are obsessed with prime numbers. What draws them in? Maybe it’s the fact that prime numbers embody some of math’s most fundamental structures and mysteries.
The primes map out the universe of multiplication by allowing us to classify and categorize every number with a unique factorization. But even though humans have been playing with primes since the
dawn of multiplication, we still aren’t exactly sure where primes will pop up, how spread out they are, or how close they must be. As far as we know, prime numbers follow no simple pattern.
Our fascination with these fundamental objects has led to the invention, or discovery, of hundreds of different types of primes: Mersenne primes (primes of the form 2^n − 1), balanced primes (primes
that are the average of two neighboring primes), and Sophie Germain primes (a prime p such that 2p + 1 is also prime), to name a few.
Interest in these special primes grew out of playing around with numbers and discovering something new. That’s also true of “digitally delicate primes,” a recent addition to the list that has led to
some surprising results about the most basic of questions: Just how rare or common can certain kinds of primes be?
To appreciate this question, let’s start with one of the first intriguing facts an aspiring number enthusiast learns: There are infinitely many prime numbers. Euclid proved this 2,000 years ago using
one of the most famous proofs by contradiction in all of math history. He started by assuming that there are only finitely many primes and imagined all n of them in a list:
$latexp_1, \ p_2, \ p_3, \ …, \ p_n$.
Then he did something clever: He thought about the number $latexq=p_1 \times p_2 \times p_3 \times … \times p_n+1$.
Notice that q can’t be on the list of primes, because it’s bigger than everything on the list. So if a finite list of primes exists, this number q can’t be prime. But if q is not a prime, it must be
divisible by something other than itself and 1. This, in turn, means that q must be divisible by some prime on the list, but because of the way q is constructed, dividing q by anything on the list
leaves a remainder of 1. So apparently q is neither prime nor divisible by any prime, which is a contradiction that results from assuming that there are only finitely many primes. Therefore, to avoid
this contradiction, there must in fact be infinitely many primes.
Given that there are infinitely many of them, you might think that primes of all kinds are easy to find, but one of the next things a prime number detective learns is how spread out the primes can
be. A simple result about the spaces between consecutive prime numbers, called prime gaps, says something quite surprising.
Among the first 10 prime numbers — 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29 — you can see gaps that consist of one or more composite numbers (numbers that are not prime, like 4, 12 or 27). You can
measure these gaps by counting the composite numbers in between: For example, there is a gap of size 0 between 2 and 3, a gap of size 1 between both 3 and 5 and 5 and 7, a gap of size 3 between 7 and
11, and so on. The largest prime gap on this list consists of the five composite numbers — 24, 25, 26, 27 and 28 — between 23 and 29.
Now for the incredible result: Prime gaps can be arbitrarily long. This means that there exist consecutive prime numbers as far apart as you can imagine. Perhaps just as incredible is how easy this
fact is to prove.
We already have a prime gap of length 5 above. Could there be one of length 6? Instead of searching lists of primes in hopes of finding one, we’ll just build it ourselves. To do so we’ll use the
factorial function used in basic counting formulas: By definition, $latexn!=n \times(n-1) \times (n-2) \times … \times 3 \times 2 \times 1$, so for example $latex3!=3 \times 2\times 1 = 6$ and
$latex5!=5 \times 4 \times 3 \times 2 \times 1=120$.
Now let’s build our prime gap. Consider the following sequence of consecutive numbers:
$latex 7!+2$, $latex7!+3$, $latex 7!+4$, $latex7!+5$, $latex 7!+6$, $latex 7!+7$.
Since $latex7!=7 \times 6 \times 5 \times 4 \times 3 \times2 \times 1$, the first number in our sequence, $latex7!+2$, is divisible by 2, which you can see after a little bit of factoring:
$latex7!+2=7 \times 6 \times 5 \times 4 \times 3 \times2 \times 1+2$
$latex= 2(7 \times 6 \times 5 \times 4 \times 3 \times 1+1)$.
Likewise, the second number, $latex7!+3$, is divisible by 3, since
$latex7!+3=7 \times 6 \times 5 \times 4 \times 3 \times2 \times 1+3$
$latex= 3(7 \times 6 \times 5 \times 4 \times2 \times 1+1)$.
Similarly, 7! + 4 is divisible by 4, 7! + 5 by 5, 7! + 6 by 6, and 7! + 7 by 7, which makes 7! + 2, 7! + 3, 7! + 4, 7! + 5, 7! + 6, 7! + 7 a sequence of six consecutive composite numbers. We have a
prime gap of at least 6.
This strategy is easy to generalize. The sequence
$latexn!+2$, $latexn!+3$, $latexn!+4$, $latex…$, $latexn!+n$.
is a sequence of $latexn-1$ consecutive composite numbers, which means that, for any n, there is a prime gap with a length of at least $latexn-1$. This shows that there are arbitrarily long prime
gaps, and so out along the list of natural numbers there are places where the closest primes are 100, or 1,000, or even 1,000,000,000 numbers apart.
A classic tension can be seen in these results. There are infinitely many prime numbers, yet consecutive primes can also be infinitely far apart. What’s more, there are infinitely many consecutive
primes that are close together. About 10 years ago the groundbreaking work of Yitang Zhang set off a race to close the gap and prove the twin primes conjecture, which asserts that there are
infinitely many pairs of primes that differ by just 2. The twin primes conjecture is one of the most famous open questions in mathematics, and James Maynard has made his own significant contributions
toward proving this elusive result.
This tension is also present in recent results about so-called digitally delicate primes. To get a sense of what these numbers are and where they may or may not be, take a moment to ponder the
following strange question: Is there a two-digit prime number that always becomes composite with any change to its ones digit?
To get a feel for digital delicacy, let’s play around with the number 23. We know it’s a prime, but what happens if you change its ones digit? Well, 20, 22, 24, 26 and 28 are all even, and thus
composite; 21 is divisible by 3, 25 is divisible by 5, and 27 is divisible by 9. So far, so good. But if you change the ones digit to a 9, you get 29, which is still a prime. So 23 is not the kind of
prime we’re looking for.
What about 37? As we saw above, we don’t need to bother checking even numbers or numbers that end in 5, so we’ll just check 31, 33 and 39. Since 31 is also prime, 37 doesn’t work either.
Does such a number even exist? The answer is yes, but we have to go all the way up to 97 to find it: 97 is a prime, but 91 (divisible by 7), 93 (divisible by 3) and 99 (also divisible by 3) are all
composite, along with the even numbers and 95.
A prime number is “delicate” if, when you change any one of its digits to anything else, it loses its “primeness” (or primality, to use the technical term). So far we see that 97 is delicate in the
ones digit — since changing that digit always produces a composite number — but does 97 satisfy the full criteria of being digitally delicate? The answer is no, because if you change the tens digit
to 1 you get 17, a prime. (Notice that 37, 47 and 67 are all primes as well.)
In fact, there is no two-digit digitally delicate prime. The following table of all the two-digit numbers, with the two-digit primes shaded in, shows why.
All the numbers in any given row have the same tens digit, and all the numbers in any given column have the same ones digit. The fact that 97 is the only shaded number in its row reflects the fact
that it is delicate in the ones digit, but it’s not the only prime in its column, which means it is not delicate in the tens digit.
A digitally delicate two-digit prime would have to be the only prime in its row and column. As the table shows, no such two-digit prime exists. What about a digitally delicate three-digit prime?
Here’s a similar table showing the layout of the three-digit primes between 100 and 199, with composite numbers omitted.
Here we see that 113 is in its own row, which means it’s delicate in the ones digit. But 113 isn’t in its own column, so some changes to the tens digit (like to 0 for 103 or to 6 for 163) produce
primes. Since no number appears in both its own row and its own column, we quickly see there is no three-digit number that is guaranteed to be composite if you change its ones digit or its tens
digit. This means there can be no three-digit digitally delicate prime. Notice that we didn’t even check the hundreds digit. To be truly digitally delicate, a three-digit number would have to avoid
primes in three directions in a three-dimensional table.
Do digitally delicate primes even exist? As you go further out on the number line the primes tend to get sparser, which makes them less likely to cross paths in the rows and columns of these
high-dimensional tables. But larger numbers have more digits, and each additional digit decreases the likelihood of a prime being digitally delicate.
If you keep going, you’ll discover that digitally delicate primes do exist. The smallest is 294,001. When you change one of its digits, the number you get — 794,001, say, or 284,001 — will be
composite. And there are more: The next few are 505,447; 584,141; 604,171; 971,767; and 1,062,599. In fact, they don’t stop. The famous mathematician Paul Erdős proved that there are infinitely many
digitally delicate primes. And that was just the first of many surprising results about these curious numbers.
For example, Erdős didn’t just prove that there are infinitely many digitally delicate primes: He proved that there are infinitely many digitally delicate primes in any base. So if you choose to
represent your numbers in binary, ternary or hexadecimal, you’re still guaranteed to find infinitely many digitally delicate primes.
And digitally delicate primes aren’t just infinite: They comprise a nonzero percentage of all prime numbers. This means that if you look at the ratio of the number of digitally delicate primes to the
number of primes overall, this fraction is some number greater than zero. In technical terms, a “positive proportion” of all primes are digitally delicate, as the Fields medalist Terence Tao proved
in 2010. The primes themselves don’t make up a positive proportion of all numbers, since you’ll find fewer and fewer primes the farther out you go along the number line. Yet among those primes,
you’ll continue to find digitally delicate primes often enough to keep the ratio of delicate primes to total primes above zero.
Maybe the most shocking discovery was a result from 2020 about a new variation of these strange numbers. By relaxing the concept of what a digit is, mathematicians reimagined the representation of a
number: Instead of thinking about 97 by itself, they instead thought of it as having leading zeros:
Each leading zero can be thought of as a digit, and the question of digital delicacy can be extended to these new representations. Could there exist “widely digitally delicate primes” — prime numbers
that always become composite if you change any of the digits, including any of those leading zeros? Thanks to the work of the mathematicians Michael Filaseta and Jeremiah Southwick, we know that the
answer, surprisingly, is yes. Not only do widely digitally delicate primes exist, but there are infinitely many of them.
Prime numbers form an infinite string of mathematical puzzles for professionals and enthusiasts to play with. We may never unravel all their mysteries, but you can count on mathematicians to
continually discover, and invent, new kinds of primes to explore.
1. What’s the biggest prime gap among the primes from 2 to 101?
2. To prove that there are infinitely many primes, Euclid assumes there are finitely many primes $latexp_1, \ p_2, \ p_3, \ …, \ p_n$, and then shows that $latexq=p_1 \times p_2 \times p_3 \times …
\times p_n+1$ isn’t divisible by any prime on the list. Doesn’t this mean that q has to be prime?
3. A famous result in number theory is that there is always a prime between k and 2k (inclusive). This is hard to prove, but it’s easy to prove that there’s always a prime between k and $latexq=p_1 \
times p_2 \times p_3 \times … \times p_n+1$ (inclusive), where $latexp_1, \ p_2, \ p_3, \ …, \ p_n$ are all the primes less than or equal to k. Prove it.
4. Can you find the smallest prime number that is digitally delicate in the ones and tens digits? This means that changing the ones or tens digit will always produce a composite number. (You might
want to write a computer program to do this!)
Challenge Problem: Can you find the smallest prime number that is digitally delicate when represented in binary? Recall that in binary, or base 2, the only digits are 0 and 1, and each place value
represents a power of 2. For example, 8 is represented as $latex1000_2$, since $latex 8=1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0$, and 7 in base 2 is $latex111_2$, since $latex7=1 \
times2^2 + 1 \times 2^1 + 1 \times 2^0$.
Click for Answer 1:
Click for Answer 2:
Click for Answer 3:
Click for Answer 4:
Click for Answer to Challenge Problem: | {"url":"https://www.quantamagazine.org/how-can-infinitely-many-primes-be-infinitely-far-apart-20220721/","timestamp":"2024-11-02T21:16:10Z","content_type":"text/html","content_length":"212874","record_id":"<urn:uuid:a17935f6-93d4-4b12-87f6-1b97db91f34a>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00721.warc.gz"} |
This website hosts the course materials and labs for PHYS/ASTR 7730, Statistical and Computational Methods in Physics and Astronomy, taught by Yao-Yuan Mao at the University of Utah, starting from
Spring 2024.
This course is scheduled to be offered again in Spring 2025.
Course Description#
This course will discuss a few widely applicable statistical and computational methods of analyzing and modeling phenomena in astrophysics, biophysics, and physics in general. The learning objective
is to apply the methods learned in this course to connect experimental or observational data with underlying physical processes through numerical simulations and statistical analyses. Topics that
will be covered in this course include stochastic process simulations, Monte Carlo methods, Bayesian analysis, and basic machine learning algorithms. This is a graduate-level course. The course will
use Python as the programming language for demonstration and use many examples in physics and astronomy. Students are assumed to be comfortable in programming and have an introductory-level knowledge
in physics.
No required course prerequisites, but students are expected to be comfortable in coding (preferably in Python) to be able to complete assignments and projects independently.
Spring 2024 Course Information#
• Meeting Time & Days: 3:00–4:45 pm on Mondays & Wednesdays
• Meeting Location: South Physics (PHYS) 205
• Credit Hours: 4
Instructor Information#
• Instructor: Yao-Yuan Mao
• Instructor Office: INSCC 314
• Instructor Email: yymao@astro.utah.edu
• Office hours: 1:45–2:45 pm on Mondays
Useful Links#
• Canvas page (uNID login required): Canvas will be used for announcements, official policies, assignment submission, and grading. | {"url":"https://yymao.github.io/phys7730/","timestamp":"2024-11-08T23:24:52Z","content_type":"text/html","content_length":"27642","record_id":"<urn:uuid:3ca7e753-90cc-4a42-a193-e12bff88c7c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00055.warc.gz"} |
About | Math Tech-knowledgy for Equity
Equity-centered Transformative Technology is a set of modules designed to engage elementary teacher candidates (TCs) with deliberate decision-making around lesson development using technological
tools, keeping equitable instructional practices at the forefront. Read more about it in our SITE Proceeding Paper (Suh et. al., 2022).
Evolution of Equity-centered Transformative Technology
A decade ago, the first author wrote an article called Tech-knowledgy for Diverse Learners for a Technology Focus Issue in the Mathematics Teaching in the Middle School in Mathematics Education
(Suh, 2010a). This article focused on leveraging cognitive tech tools for mathematics teaching and learning. Specifically, the article recommended strategies for teachers to consider the needs of
diverse learners and be equipped to support their learning by taking advantage of technology’s affordances. Most importantly, teachers must have “tech-knowledgy”: the knowledge necessary to use
cognitive tech tools effectively to construct mathematical knowledge, evaluate the mathematical opportunities presented, and design learning tasks with these tools that amplify the mathematics
for their diverse learners. Using case studies, Suh(2010b) described technology-enhanced mathematics lessons in two diverse fifth and sixth grade classrooms at a Title I elementary school near
the metropolitan area. The project’s primary goal was to design tasks to both leverage technology and enhance access to critical thinking in mathematics, particularly with data analysis and
probability concepts. This paper highlights the opportunities in technology-rich mathematics environments. In addition, the case studies illustrate how to design and implement mathematical tasks
using technology to provide opportunities for higher mathematical thinking processes as defined by the Process Standards of the National Council of Teachers of Mathematics (NCTM, 2000): problem
solving, connections, representations, communication, reasoning and proof. More recently, our design team has expand the notion of “tech-knowledgy” with a focus on equity (Suh et al., 2021): the
knowledge to ensure that the integration of technology amplifies the learning so that the digital tool a) provides access to Inquiry-based learning using math tech b) allows student ownership
and authorship to build positive mathematical identities; c) provides formative assessment data and differentiation to meet learners’ needs; d) promotes social interaction to build collective
knowledge among their peers; e) amplifies the mathematical or cognitive processes.
Recent Design and Development Cycle in Math Methods CoursesThe integration of technology in the mathematics classroom has always been a key focus for math teacher education (AMTE standards).
Recently, there is a call for MTEs to utilize more common language and core practices (McDonald et al., 2013) around ambitious teaching. MTEs have strived to design practice-based assignments to
promote ambitious mathematics teaching (Lampert et al., 2013) as well as ambitious teaching integrating technology in mathematics lessons (Suh, 2016). In the past two years, the authors (Dr.
Jennifer Suh, Dr. Kimberly Morrow-Leong and and two mathematics coaches, Holly Tate and Kate Roscioli) have been refining this set of modules as a reflective tool to center equity when planning
for a tech-enhanced lesson.
Project Team:
Jennifer Suh, Ph.D. is a mathematics educator in the School of Education at George Mason University. Dr. Suh teaches mathematics methods courses in the Elementary Education Program and mathematics
leadership courses for the Mathematics Specialist Program. She directs the Center for Outreach in Mathematics Professional Learning and Educational Technology, COMPLETE and provides professional
development focused on learning trajectory based instruction, equity focused teaching practices and effective integration of technology in the mathematics classrooms. Dr. Suh conducts Lesson Study
with teachers to develop high leverage mathematics teaching practices and deepen teachers’ content knowledge using learning trajectories. Currently, her project called EQSTEMM, Advancing Equity and
Strengthening Teaching with Elementary Mathematical Modeling, focuses on promoting equitable participation of all students engaging in rigorous mathematics through modeling. She enjoys co-designing
authentic community based problem-based modeling tasks with teachers to promote equitable access to 21st century skills in STEM disciplines for diverse student populations.
is a third-year doctoral student pursuing a Ph.D. in Mathematics Education, Teacher Education, and Technology at George Mason University. She is also a Title I Mathematics Specialist in Northern
Virginia, working with teachers and students in grades 3 through 5. At George Mason, she serves as a graduate research assistant on an EQSTEMM, an NSF-funded grant with Dr. Jennifer Suh. She earned
an M.Ed. in Elementary Education from Marymount University and an M.Ed. in Mathematics Leadership from George Mason University. Her research interests include preparing teachers to integrate
technology in mathematics classrooms to promote equitable mathematics practices. Her future goals as a researcher and mathematics teacher educator are to develop frameworks and systems that support
pre-service and in-service educators in evaluating technology tools for mathematics instruction.
is a mathematics education specialist and adjunct instructor at George Mason University. As a doctoral student she served as a professional development coordinator at NCTM and a researcher/coach at
AIR. After teaching fifth grade during a year of the pandemic, Kim recently accepted a position as a Senior Content Manager at the Math Learning Center. She is an author of the Mathematize It! series
of books for K-8 teachers and is a 2009 recipient of the Presidential Award for Excellence in Mathematics. Kim has served on the NCSM Board as conference program chair and is currently serving on the
MathKind Education Advisory Board and the AMTE Advocacy Committee. Dr. Morrow-Leong’s professional interests include transformative evidence-based assessment practices and investigation of the nature
of teachers’ engagement with artifacts of student thinking.
is a doctoral student at George Mason University working towards her Ph.D. in Mathematics Education Leadership and Research Methodologies. Her research focuses on equity in mathematics education,
specifically on the development of critical consciousness through Teaching Mathematics for Social Justice and mathematical modeling. Holly ties her interests of equitable mathematics classrooms into
learning alongside teachers as an Instructional Mathematics Coach in a K-5 STEM school in Virginia. Additionally, Holly is in her third semester as a Graduate Research Assistant in partnership with
Dr. Jennifer Suh and other universities across the country. Their NSF-funded grant work focuses on mathematical modeling for increased student agency and identity, centering student experience and
voice. Prior to her doctoral studies, Holly received her M.I.S. in Mathematics Leadership from Virginia Commonwealth University and her M.Ed. in Elementary Education from James Madison University. | {"url":"http://mathtechknowledgy.onmason.com/","timestamp":"2024-11-02T17:07:05Z","content_type":"text/html","content_length":"32172","record_id":"<urn:uuid:17f76de3-50fe-4af8-91c1-defc1b22fdd7>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00164.warc.gz"} |
An Etymological Dictionary of Astronomy and Astrophysics
AB magnitude system
راژمان ِبُرز ِBA
râžmân-e borz-e AB
Fr.: système de magnitudes AB
A → photometric system defined by reference to → monochromatic magnitudes in such a way that, when monochromatic → flux f[ν] is measured in ergs sec^-1 cm^-2 Hz^-1, the magnitude will be: AB = -2.5
logf[ν] - 48.60. The constant is set so that AB is equal to the V magnitude for a source with a flat → spectral energy distribution. The → zero point is defined by the flux of the star → Vega at 5546
Å. In this system, an object with constant flux per unit frequency interval has zero color.
→ magnitude; → system.
accelerating system
راژمان ِشتابنده
râžmân-e šetâbandé
Fr.: système en accélération
A material system that is subject to a constant force in each and every one of its instantaneous points of trajectory.
→ accelerating; → system.
ADaptive Optics Near Infrared System (ADONIS)
Fr.: ADaptive Optics Near Infrared System (ADONIS)
An → adaptive optics instrument used on the → European Southern Observatory (ESO) 3.6-m telescope at La Silla. It was an upgraded version of COME-ON-PLUS, the → Very Large Telescope (VLT) adaptive
optics prototype. It had 52 → actuators and performed corrections of the mirror 200 times per second. The reference → wavefront was sensed in the → visible. The observation was done in the →
near-infrared (1-5 μm).
→ adaptive; → optics; → near-infrared; → system.
adaptive optics system
راژمان ِنوریک ِنیاوشی
râžmân-e nurik-e niyâveši
Fr.: système d'optique adaptative
An → optical system that uses → adaptive optics.
→ adaptive; → optics; → system.
afocal system
راژمان ِاکانون
râžmân-e akânun
Fr.: système afocal
An optical system with object and image points at infinity.
→ afocal; → system.
Alpha Centauri system
راژمان ِآلفا-کنتاؤرس
râžmân-e Âlfâ-Kentâwros
Fr.: système Alpha du Centaure
A system of three stars, the → close binary Alpha Centauri A (→ spectral type G2 V) and Alpha Centauri B (K1 V), and a small and faint → red dwarf, Alpha Centauri C (M6 Ve), better known as → Proxima
Centauri. To the unaided eye, the two main components (AB) appear as a single object with an → apparent visual magnitude of -0.27, forming the brightest star in the southern constellation → Centaurus
and the third brightest star in the night sky, after → Sirius and → Canopus. The individual visual magnitudes of the components A, B, and Proxima are +0.01, +1.33, and +11.05, respectively. The
masses of A and B are 1.100 and 0.907 Msun, respectively. Their → effective temperatures are (A) 5,790 K and (B) 5,260 K; their luminosities (A) 1.519 Lsun and (B) 0.500 Lsun. The binary members are
separated in average by only 23 → astronomical units. They revolve around a common center of mass with a period of about 80 years. Both have a distance of 4.37 → light-years. Proxima Centauri, lying
about 15,000 AU apart from AB, is → gravitationally bound to them. It has a mass of 0.1 Msun, a radius of 0.1 Rsun, a luminosity of about 0.001 Lsun, and an → effective temperature of ~ 3,000 K.
→ alpha; → Centaurus; → system.
altazimuth coordinate system
راژمان ِهمآراهای ِفرازا-سوگان
râžmân-e hamârâhâ-ye farâzâ-sugân
Fr.: coordonnées azimutales
The coordinate system in which the position of a body on the → celestial sphere is described with respect to an observer's → celestial horizon and → zenith. The coordinates of a point in this system
are its → altitude on the → vertical circle, and its → azimuth westward (clockwise) along the celestial horizon from the observer's south. Same as → horizon coordinate system.
→ altazimuth; → coordinate; → system.
anamorphic system
راژمان ِآناریخت، ~ آناریختمند
râžmân-e ânârixt, ~ ânârixtmand
Fr.: système anamorphique
An optical system whose optical power, and imaging scale, differs in the two principal directions. See also → anamorphosis.
→ anamorphic; → system.
aplanatic system
راژمان ِنابیراه
râžmân-e nâbirah
Fr.: système aplanétique
An → optical system that is able to produce an image essentially free from → spherical aberration and → coma. See also the → Abbe sine condition.
→ aplanatism; → system.
apochromatic system
راژمان ِاپافام
râžmân-e apâfâm
Fr.: système apochromatique
An optical system that is → apochromatic.
→ apochromatic; → system.
axiomatic system
راژمان ِبنداشتی
râžmân-e bondâšti
Fr.: système axiomatique
Any system of → logic which explicitly states → axioms from which → theorems can be → deduced.
→ axiomatic; → system.
binary number system
راژمان ِعددهای ِدرینی
râžmân-e adadhâ-ye dirini
Fr.: système des nombres binaires
A → numeral system that has 2 as its base and uses only two digits, 0 and 1. The positional value of each digit in a binary number is twice the place value of the digit of its right side. Each binary
digit is known as a bit. The decimal numbers from 0 to 10 are thus in binary 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, and 1010. And, for example, the binary number 11101[2] represents the
decimal number (1 × 2^4) + (1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (1 × 2^0), or 29. In electronics, binary numbers are the flow of information in the form of zeros and ones used by computers. Computers
use it to manipulate and store all of their data including numbers, words, videos, graphics, and music.
→ binary; → number; → system.
binary system
راژمان ِدرین
râžmân-e dorin
Fr.: système binaire
Two astronomical objects revolving around their common center of mass.
→ binary; → system.
Râžmân, → system; dorin→ binary.
bound system
راژمان ِبندیده
râžmân-e bandidé
Fr.: système lié
A system composed of several material bodies the total energy of which (the sum of kinetic and potential energies) is negative, e.g. a → bound cluster.
Bound, p.p. of → bind; → system.
Aâžmân, → system; bandidé p.p. of bandidan, → bind.
catoptric system
راژمان ِبازتابیک
râžmân-e bâztâbik
Fr.: système catoprtique
An optical system in which the light is reflected only.
→ catoprtics; → system.
CGS system
راژمان ِCGS
râžmân-e CGS
Fr.: système CGS
The system of → CGS units.
→ CGS unit; → system.
chaotic system
راژمانِ ورشونگین
râžmân-e varšungin
Fr.: système chaotique
A system that is → deterministic through → description by mathematical rules but can evolve highly → nonlinearly depending on → initial conditions. See also → chaos.
→ chaotic; → system.
close binary system
راژمان ِدورین ِکیپ
râžmân-e dorin-e kip
Fr.: système binaire serré
A → binary system in which the distance separating the stars is comparable to their size. Most close binaries are spectroscopic binaries (→ spectroscopic binary) and/or eclipsing binaries (→
eclipsing binary). In most of them → mass transfer occurs at some stage, an event which profoundly affects the → stellar evolution of the components. The evolution of close binaries depends on the →
initial masses of the two stars and their → separation. When the more massive star evolves into a → red giant first, material will spill through the inner point onto its companion, thereby affecting
its companion's evolution. Mass transfer can also alter the separation and → orbital period of the binary star.
→ close; → binary; → system.
closed system
راژمان ِبسته
râžmân-e basté
Fr.: système fermé
Thermodynamics: A system which can exchange energy with the surroundings but not matter. → open system; → isolated system.
→ closed; → system.
compact binary star system
راژمان ِدرین ِهمپک
râžmân-e dorin-e hampak
Fr.: système binaire compact
A binary star system which is composed of a collapsed object (→ degenerate dwarf, → neutron star, or → black hole) in orbit with a low-mass (≤ 0.5 Msol) secondary star, wherein the collapsed star →
accretes matter from its → companion. These two objects form a binary system of overall dimensions 10^6 km with an orbital period of only hours or less. See also: → X-ray binary.
→ compact; → binary; → star; → system. | {"url":"https://dictionary.obspm.fr/index.php/?showAll=1&formSearchTextfield=system","timestamp":"2024-11-11T18:14:18Z","content_type":"text/html","content_length":"36883","record_id":"<urn:uuid:7643c9ce-09a3-4b24-90e5-b05b613f19e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00639.warc.gz"} |
Powers - Math Angel
🎬 Video Tutorial
• (0:01) Parts of a Power: A power has a base (the number being multiplied) and an index or exponent (the number of times the base is used). For instance, in $5^3$, 5 is the base, and 3 is the
• (1:00) Special Rules for Indices: Any number to the power of 1 is itself; any non-zero number to the power of 0 is 1; however, $0^0$ is undefined.
• (1:30) Square and Cube Numbers: Powers of 2 are called square numbers (e.g., 4, 9, 16), and powers of 3 are called cube numbers (e.g., 8, 27, 64).
• (2:30) Using BIDMAS with Powers: In calculations, handle indices before multiplication or division.
Accessing this course requires a login. Please enter your credentials below! | {"url":"https://math-angel.io/lessons/powers-indices/","timestamp":"2024-11-13T01:58:51Z","content_type":"text/html","content_length":"283135","record_id":"<urn:uuid:0b740791-8508-4c45-91f7-5e8a3128e726>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00892.warc.gz"} |
Loire since 1976
1 Introduction
Given the importance of stream and river temperature for freshwater ecology [4,5,7] and quality, it is of considerable interest to understand how the thermal regime changed in the past in order to
better predict future modifications [13]. However, data that are both detailed enough (hourly) and recorded over a sufficiently long period (20 years or more) are scarce [13]. Models can help to
reconstitute past-time series of river temperature and thereby give a better understanding of how they evolved in the past. In this type of application, regression or stochastic models that only take
into account atmospheric temperature are currently the most attractive, daily time series of atmospheric temperature sometimes going back to the 19th century. Moreover, this is one of the
meteorological variables that climate models of general circulation simulate best [10,12], making it possible to envisage carrying out prospective studies on the consequences of possible climate
change on water temperatures. In France, river temperature was first recorded continuously in the 1970s, often by the electricity generating authority (‘Électricité de France’, EDF), within a
statutory framework. In this paper, continuous temperature data from 1976 to 2003, collected at four stations of the Middle Loire, are used to derive monthly values in order to: (i) determine trends
and changes since 1976, (ii) determine the possible influence on the thermal regime of the middle Loire of incoming groundwater from the Beauce aquifer, (iii) test the ability of a regression model
to reconstruct annual, spring and summer water temperatures since 1881 based on long-term air temperature and discharge data, (iv) characterize the exceptional year 2003 in relation to the
reconstituted 1881–1975 time series and data measured between 1976 and 2003.
2 Methods
The 266-km-long reach studied is located between Belleville and Avoine (Fig. 1). Belleville is situated 536 km downstream of the Loire's source and 71 km downstream of its junction with the Allier
River, its main tributary at this point. The long distance travelled means that memory of the upstream temperature has been lost, and the temperature of the middle Loire is mainly influenced by
thermal exchange with the atmosphere [11]. For the upstream reach between Belleville and Dampierre (45 km), inflow from its tributaries and groundwater is too small to modify the thermal regime of
the Loire. On the other hand, between Dampierre and Saint-Laurent-des-Eaux, it is subject to continuous thermal exchange with the limestone aquifers from the nearby Beauce region and with the ‘Val
d'Orléans’ system, especially downstream of Orleans [1,2,6].
Fig. 1
The temperature data used in this paper were collected by EDF at continuous monitoring stations upstream of the four nuclear power plants: Belleville, Dampierre, Saint-Laurent-des-Eaux and Avoine (
Fig. 1). The hourly-recorded temperature data began in 1976 at Avoine and Saint-Laurent-des-Eaux, in November 1979 at Dampierre and 1978 at Belleville. The data were examined and approved by EDF on a
daily and hourly time step, particularly to detect outliers and drift [8]. However, for this study, an additional examination was carried out at a monthly level, from which any long-term break in the
time series could be detected a posteriori. In this way, temperature values for some months were corrected according to inter-station correlations.
Nuclear power stations are equipped with closed-circuit cooling towers, allowing the heat to be discharged directly into the atmosphere. Thermal waste into the Loire, essentially from purging the
cooling towers, is very low. For example, studies carried out by EDF indicate that, 90%of the time, the daily temperature rise of the Loire downstream of the Dampierre power station is less than
0.3°C, the median rise being 0.1°C. Moreover, the greatest rise in temperature is in winter, which means that the statistics for summer increases result in even lower values. In this study, we
therefore considered that temperature rises caused by nuclear power stations are of secondary importance in the trends that concerned us. This will be confirmed from the analysis of the series from
Belleville, situated upstream of all the Loire power stations.
Trends in time series of annual, quarterly and monthly Loire temperatures were analysed using the non-parametric Spearman rank correlation test. This test looks for a trend in a time series, without
specifying whether the trend is linear or non-linear.
To explain the temperature differences observed during summer periods between Dampierre and Saint-Laurent-des-Eaux, an energy balance based on mean data for the month of August was derived for the
1980–2003 period. This energy balance is based on Eq. (1), where ρ and $cp$ are the water density and heat capacity, $Qi−1$ and $Qi$ are the stream flow rates entering the study reach ($i−1$ at
Orleans) and leaving it (i, at Blois, 30 km downstream of Saint-Laurent-des-Eaux). $Ti−1$ and $Ti$ are the temperatures at the Dampierre and Saint-Laurent-des-Eaux monitoring stations.
$ρcpQiTi=ρcpQi−1Ti−1+FnetA+ρcpQnappeTnappe+ρcpQaffTaff$ (1)
The surface heat exchange ($Fnet$) was modelled as a combination of five processes, radiative (solar and atmospheric long-wave radiations, long-wave back radiation from the water) and non-radiative
mechanisms (conduction and evaporation), A being the exchange surface area of the reach (water) [3]. We considered that the main inflow to the Loire is from the limestone aquifers of the Beauce
region ($Qnappe,Tnappe$) and possibly the limestone system of ‘Val d'Orléans’, and then inflow from the tributaries ($Qaff,Taff$). In this way, the energy balance for the month of August each year is
completed with the mass balance (Eq. (2)).
The relationship between air temperature, water flow and temperature was examined at annual, spring and summer levels using multiple regression analysis. Flow data came from the water monitoring
stations at Gien (1976–2003), Orleans (1976–2003) and Blois (1881–2003) (DIREN Centre). Meteorological data were taken from two sources: (1) daily atmospheric temperatures, relative humidity or
vapour pressure of the air, solar radiation and cloud cover at the Tours and Orléans stations, for calculating the energy balance, and (2) monthly atmospheric temperature at Orléans (1881–2003),
homogenized by Météo-France as part of French climate change studies in the 20th century [9] for trend analysis.
3 Results
3.1 Trends in water temperature in the middle reach of the Loire since 1976 (Fig. 2)
Analysis of average quarterly temperatures shows significant increases at the 99%confidence level (Spearman's test) for all four Loire monitoring stations, but only in spring and summer (Table 1).
For the rest of the year, increases are not significant at this confidence level, with the exception of Saint-Laurent-des-Eaux and Chinon in winter, although increases at some stations appear at a
lower confidence level. This indicates some changes in the thermal regime during the period when the water warms up in spring and summer. The average spring and summer increases (Fig. 2b and c) range
between 2.4 and 3°C, and these differences cannot be explained by the location of the stations. During the same periods, a significant rise (at a 99%confidence level) was observed in the air
temperatures at Orléans, and a decrease in flow at Gien (at a 90%confidence level for March to May, 95%for June to August and annually). Moreover, it can be seen that the series at Belleville,
situated upstream of all the nuclear power stations on the Loire, follows exactly the same trends as the other series influenced by thermal discharge, thereby confirming that warming caused by the
power stations is largely secondary to other phenomena that could explain these trends.
Fig. 2
Table 1
Significance and amplitude of trends observed from mean quarterly data on water temperature (Belleville, Dampierre, Saint-Laurent-des-Eaux, Chinon), air temperature (Orléans) and flow (Gien and
Blois) (1976–2003). The probability of incorrectly rejecting a trend is: <1%(in bold print), <5%(in bold italics), and <10%(in italics). The amplitude of trends is only shown when detected at a
99%confidence level (probability <1%)
Significativité et amplitude des tendances observées sur les données de température de la Loire (Belleville, Dampierre, Saint-Laurent-des-Eaux, Chinon), atmosphériques (Orléans), et des débits (Gien
et Blois) (période 1976–2003). La probabilité de rejeter à tort une tendance est : inférieure à 1%(en gras), inférieure à 5%(en gras italique), et inférieure à 10%(en italique). L'amplitude des
tendances n'est indiquée que pour les détections avec un niveau de confiance de 99%(probabilité <1%)
Teau Mars à Mai Juin à Aout Sept–Nov Dec–Fev Année
Hausse (°C) prob. Hausse (°C) prob. Hausse (°C) prob. Hausse (°C) prob. Hausse (°C) prob.
Belleville 2.9 0.0002 2.8 0.0019 + 0.0891 + 0.0545 1.9 0.0001
Dampierre 2.4 0.0002 2.4 0.0116 + 0.1825 + 0.0407 1.7 0.0000
St Laurent 2.7 0.0001 2.5 0.0005 + 0.0141 2.1 0.0021 2 0.0000
Chinon 2.6 0.0000 3.0 0.0006 + 0.0723 1.6 0.0088 1.9 0.0000
Tair Hausse (°C) prob. Hausse (°C) prob. Hausse (°C) prob. Hausse (°C) prob. Hausse (°C) prob.
Orléans 2.1 0.0002 2.0 0.0109 + 0.5859 + 0.0695 1.5 0.0009
Débits Baisse prob. Baisse prob. Baisse prob. Baisse prob. Baisse prob.
Gien – 0.0925 – 0.0191 – 0.5435 – 0.4053 – 0.0102
In addition, the data show between-site temperature differences along the middle reach of the Loire. Taken as monthly values, the river temperature increases by about 0.5°C in the
Belleville–Dampierre and Saint-Laurent–Avoine reaches, even though the latter is three times longer. On the other hand, between Dampierre and Saint-Laurent, there is a decrease of about 0.2°C in the
annual mean temperature and 1°C in the summer mean value (Fig. 2b). The differences between monthly mean values recorded at the two stations show a seasonal dependence, based on flow rates (Fig. 3).
They are negative during dry periods: the multi-annual mean temperature at Saint-Laurent in August is 1.1°C lower than at Dampierre.
Fig. 3
The energy balance in August between 1980 and 2003 on the Orléans–Saint-Laurent-des-Eaux reach explains this cooling of the Loire, even if there are some uncertainties about the input data. Cooling
is due to the relatively greater inflow of groundwater from the Beauce limestone aquifers (average multiyear temperature 13.5°C) compared to the Loire discharge at this time of the year (Table 2).
As the Loire temperature was not known at Orléans, we took it to be the same as that measured at Dampierre, assuming that losses on the Dampierre–Orléans stretch do not change the temperature. This
approximation can also be justified on account of the small difference in temperature observed in the Belleville to Dampierre section during the summer period (Fig. 2b). However, unlike temperature
values, it is very difficult to estimate over a long period of time the overall incoming flow rates from the Beauce aquifers and the small tributaries. Using a piezometric map, Gonzalez (1993) showed
that the springs along a 1500-m stretch of the main channel of the Loire at la Chapelle-Saint-Mesmin constitute the greatest incoming flow (catchment area: 1066 km^2). Their mean flow rate was
estimated to be between 11.5 and 12 m^3s^−1, using the mixing law of solutes in the river during the low-water period of 1986, when the flow rate of the River Loire was between 59 and 70 m^3s^−1
[7]. Other studies, based on the difference between historical discharge rates measured at the gauging stations, showed the incoming flow rates to be between 6 and 19 m^3s^−1 during the low
discharge period prior to 1980 when abstraction rates for agricultural purposes were lower. The summer water temperatures of the springs measured by piezometers installed near the Loire since 2001
(DIREN Centre) varied between 13 and 14°C. Therefore, we computed the energy balance on the basis of a constant water discharge of 10 m^3s^−1 from the underground Beauce aquifer at a temperature of
13.5°C. The overall inflow rate from tributaries was estimated to be the difference between the discharge of the Loire at Blois and Orléans, the discharge of the Beauce aquifer being deducted. The
temperature of the incoming flow was estimated from the air temperature, using existing correlations at a regional level.
Table 2
Input data and results of summer energy balance in the Orléans–Saint-Laurent-des-Eaux reach. $Tair=$monthly air temperature at Orléans, $TDam=$monthly temperature of the Loire at Dampierre, $TStl=$
monthly temperature of the Loire at Saint-Laurent-des-Eaux, Δt=monthly temperature difference between Saint-Laurent-des-Eaux and Dampierre, Q[Orléans] and $QBlois=$discharge at the two stations, F(i
−1) and F(i)=incoming and outgoing heat flux of the reach, $Fnappe=$heat flux from tributaries, heat flux from groundwater, $Faff=$heat flux from tributaries, $Fnet=$net heat exchange with the
atmosphere, $F(i)calc=$outgoing heat flux calculated from Eq. (1), Δ=difference between $F(i)calc$ and F(i)
Données d'entrée et résultats du bilan énergétique estival sur le tronçon Orléans–Saint-Laurent-des-Eaux. $Tair=$température mensuelle de l'air à Orléans, $TDam=$température mensuelle de la Loire à
Dampierre, $TStl=$température mensuelle de la Loire à Saint-Laurent-des-Eaux, Δt=écart mensuel de température entre Saint-Laurent-des-Eaux et Dampierre, Q[Orléans] et $QBlois=$débits mensuels aux
deux stations, F(i−1) et F(i)=flux d'entrée et de sortie du tronçon, $Fnappe=$flux de chaleur apporté par les eaux souterraines, $Faff=$flux de chaleur apporté par les affluents, $Fnet=$flux de
chaleur net échangé avec l'atmosphère, $F(i)calc=$flux de chaleur à la sortie du tronçon, évalué d'après l'Éq. (1), Δ=différence entre $F(i)calc$ et F(i)
date $T air$ (°C) $T Dam$ (°C) $T Stl$ (°C) Δ T (°C) Q [Orléans ]m ^3 / $Q Blois$ m ^3 / $F ( i − 1 )$ W/m ^ $F ( i )$ W/m ^ $F nappe$ W/m ^ $F aff$ W/m ^2 $F net$ W/m ^2 $F ( i ) calc$ W/m ^ Δ %
s s 2 2 2 2
août-80 19.6 22.1 20.6 −1.5 84.1 134.1 972 1446 71 393 12 1447 0.0
août-81 20.1 21.4 21.0 −0.4 151.6 170.3 1697 1872 71 87 27 1881 0.5
août-82 17.80 21.4 20.1 −1.3 90.5 112.9 1012 1186 71 113 −5 1190 0.4
août-83 19.3 22.4 21.3 −1.0 84.1 105.1 982 1171 71 107 −32 1128 −3.7
août-84 18.3 21.4 20.4 −1.0 66.8 89.1 746 948 71 114 −9 922 −2.7
août-85 17.0 20.9 19.3 −1.6 70.1 100.6 765 1016 71 179 5 1019 0.3
août-86 17.4 21.4 19.6 −1.8 65.8 84.7 735 866 71 79 −25 860 −0.8
août-87 18.4 21.6 20.5 −1.1 91.9 126.3 1039 1355 71 229 39 1377 1.7
août-88 18.6 22.4 20.9 −1.5 84.6 107.6 990 1177 71 123 40 1223 3.9
août-89 19.4 22.7 21.5 −1.2 51.5 62.0 611 695 71 4 25 711 2.3
août-90 21.4 23.0 22.6 −0.4 52.6 68.2 633 805 71 59 763 −5.3
août-91 21.2 24.0 22.6 −1.4 43.6 67.8 547 801 71 149 5 772 −3.6
août-92 20.4 23.5 22.0 −1.5 75.2 86.2 923 991 71 10 −25 979 −1.2
août-93 18.3 22.1 21.1 −0.9 58.9 74.5 679 823 71 52 17 819 −0.5
août-94 20.1 22.3 21.2 −1.1 81.3 94.9 949 1051 71 36 27 1083 3.0
août-95 21.2 23.3 22.9 −0.3 60.1 81.7 730 978 71 122 −30 892 −8.8
août-96 18.6 22.0 21.1 −0.9 67.8 95.3 778 1050 71 166 −37 977 −7.0
août-97 22.3 26.0 23.8 −2.2 61.6 73.0 837 907 71 15 −21 902 −0.6
août-98 19.8 22.6 21.73 −0.8 74.0
août-99 19.5 23.0 21.87 −1.1 83.7
août-00 20.0 23.2 22.29 −0.9 81.8 102.0 993 1188 71 102 −15 1151 −3.1
août-01 20.2 23.0 22.15 −0.9 97.5 115.0 1172 1331 71 75 −30 1288 −3.2
août-02 18.8 21.8 21.12 −0.7 70.7 0
août-03 24.5 26.0 24.11 −1.9 45.6 61.4 620 773 71 69 −6 754 −2.6
Table 2 shows the input data and the terms of the energy balance in Wm^−2. The last column indicates the difference (as a percentage) between the outgoing heat flux $F(i)$ ($F(i)=ρcpQiTi/A$) at the
end of the reach at Saint-Laurent-des-Eaux, and the same heat flux calculated from the energy balance $F(i)$ calc. These two values are found to be roughly within ±4%, except in 1990, 1995 and 1996,
and show that the decrease in temperature during August at Saint-Laurent-des-Eaux is closely related to the inflow of groundwater. The influence of groundwater is also shown by the stabilising of the
temperature regime at Saint-Laurent-des-Eaux shown by lower annual amplitude between summer and winter temperatures: 16.5°C between monthly August and January temperatures at Saint-Laurent-des-Eaux
compared to 17.5°C for the two upstream stations (Belleville and Dampierre).
3.2 Restoring water temperature data in the middle Loire since 1881
3.2.1 Regression model for water temperature over the 1976–2003 period
The annual, spring (March–May) and summer (June–August) time series of the temperature of the middle Loire (mean of Dampierre and Chinon series) were modelled by multiple regression based on air
temperature in Orléans and discharge in Blois. Air temperature is the predominant explanatory variable, as shown by several authors. It explains 71 to 84%of the variance of temperature of the middle
Loire (Figs. 4a–c). Using flow as a second explanatory variable is not often found in the literature on this type of study but proved to be of interest (partial correlation $r=−0.9$ for spring data,
$r=−0.66$ for summer data). As shown in Fig. 4, the two variables have a greater influence on temperature in summer, probably due to the more stable hydrological conditions, and when the water
temperature is close to the equilibrium water temperature [11]. Thus, the two variables ($xl=$air temperature, $x2$ or $ln(x2)=$discharge or its Napierian logarithm) can give a convincing estimate of
the temperature of the middle Loire, as shown by the coefficient of determination and standard deviation of errors in the order of 0.3°C. However, it can be observed that the regressions tend to
underestimate the temperatures in 1976 and 2003 by about 0.6°C.
Fig. 4
3.2.2 Water-temperature trends since 1881
The previous results suggest that it should be possible to predict annual and summer mean water temperature by studying similar information on air temperature and discharge. In the context of climate
change studies in France, the long-term monthly air-temperature data series, going back to 1881 at Orléans, was validated by Météo-France [9]. The discharge has been recorded since 1863 on the River
Loire at Blois. Fig. 5a–c show reconstructed (1881–1975) and measured (1976–2003) water temperature data series (annual, spring and summer mean values), the 10-year running mean of reconstructed data
and one standard deviation (based on regression residuals) around the mean values.
Fig. 5
Fig. 5a indicates that the highest annual mean temperatures occurred in the most recent period. For 100 years (1881–1980), the temperature of the Loire fluctuated around 13°C, with a warmer period
between 1942 and 1954. However, since 1990, it has fluctuated around 14°C. The warmest decade was 1994–2003 with a temperature 1.1°C higher than the inter-annual centennial mean, and 0.6°C higher
than the previous warmest decade (1942–1951). The trend of the time series for 1881–2003, defined as the linear regression slope, shows a rise of 0.8°C over this period.
Mean spring temperatures fluctuated around 12.3°C between 1881 and 1980, and around 12.9°C since 1981. A warm period is clearly shown around 1945, prior to the current period. Even though the
differences between these periods are not very marked, the 13°C threshold having been reached or exceeded in 12 years in both the first warm period (out of 20 years) and since 1988 (out of 15
years), certain changes in variability can be observed. A study of the structure of the time series through analysis of autocorrelation and partial autocorrelation functions in three distinct periods
(before 1930, between 1930 and 1965, and since 1965) also shows increasing persistence of warm temperatures from year to year during the recent period, while spring temperatures during the first two
periods show no particular memory from year to year.
For the summer mean values, the pattern of changes in the consecutive mean shows the existence of three warm periods: around 1900, 1950 and the current period. Summers that combined drought and a
heat wave are particularly remarkable (2003, 1976, 1947). Air temperatures during the summer of 2003 were considerably hotter: 0.8 to 1.4°C above the mean summers of the five previous hottest
summers (1976, 1947, 1911, 1983 and 1994). The low flow period in 2003 experienced low discharge (an average of 64.5 m^3s^−1 for June–August 2003 in Blois), in spite of being supported by the
reservoirs of Villerest and Naussac as early as May and benefiting from groundwater inflow from the Beauce limestone aquifer, which is typically high at the beginning of the summer. As indicated in
Fig. 5b, the Loire reacted differently to the combined effects of high temperature and severity of low water flow, with warmer water (up to 1.7 to 2.4°C) than in the five previously ranked summers
(1976, 1947, 1997, 1945, 1911). We can observe that 1983 and 1944, with hot summers but less severely restricted flow (discharge above 100 m^3s^−1 in Blois), are in 15th and 26th position with
regard to the Loire water temperature. On the other hand, 1997, with higher than normal temperatures from as early as February, and an early and prolonged low-water period (mean annual discharge 188
m^3s^−1), was similar to 1976. Likewise, the low water period of 1949, which is the benchmark for the Loire, only comes 7th due to the lower air temperatures.
Comparative analysis of the temperature series of the Loire (partially reconstituted) and air temperatures since 1881 shows very similar trends. Fig. 6a–c show the 5-year moving average of
temperature anomalies, evaluated as differences between data series and the respective interannual values (1881–1980). This indicates that, compared to the complete series, the marked warming
highlighted in the 1976–2003 series is also due to the particular position of the colder than normal years around 1980. During this period, the discharge was also high, as shown in Fig. 6d.
Fig. 6
4 Conclusions
Analysis of the water temperature of the Loire for the period 1976–2003 shows a change in the energy regime with very significant rises in spring and summer (from 1.5 to 2°C), increases being less
pronounced in winter, and absent in autumn. During the summer of 2003, which combined severe low-water levels and a heat wave, the River Loire reached absolute temperature records: 25.4°C on average
from June to August, i.e. 4°C higher than the inter-annual mean and 1.7°C higher than 1976, a memorably dry year.
The four series analysed, over a distance of approximately 266 km, show remarkable consistency, in spite of specific local characteristics. The series at Saint-Laurent-des-Eaux differs from the
others by its water which is colder in summer (1.4°C less on average in August) and warmer in winter (0.3°C higher on average in January), which can be explained by the inflow of groundwater from
the Calcaires de Beauce aquifer. This cooling can be clearly observed through modelling this inflow in August with a discharge of 10 m^3s^−1 at 13.5°C.
Linear regression models based on monthly air temperature and discharge data are well suited to the reconstruction of annual, spring and summer series. Firstly, they explain the upward trend since
1976 as a result of both rising air temperature and decreasing discharge. In June to August, water temperatures thus increased by 1.3°C between the two 14-year sub-periods of 1976–1989 and
1990–2003. From this model, approximately 60%of this temperature rise can be estimated to be linked to the rise in air temperature (0.8 for a 1.5°C rise in air temperature) and 40%to the drop in
discharge (0.5°C for a 100 m^3s^−1) drop in discharge). The same models used for the period 1881–2003 show that the rapid rise of water temperature observed since 1976 forms part of a slower trend
over the century, marked by other warm periods around 1900 and 1950.
According to these models, the mean annual temperature of the Loire has thus risen by about 0.8°C on average during the 20th century, a figure which is similar to that observed by Webb and Nobilis
(1994) for the Danube at Linz based on probably the only long-term measured temperature series available in Europe [14]. However, the rise in temperature of the Danube occurs mainly in autumn and
winter and does not appear to be linked to atmospheric warming as in the Loire but, according to the authors, to man-made changes [13,14].
Finally, the rise in temperature observed in the spring could have an impact on fish and benthic life, as has been shown in the River Rhone [5]. This period is generally considered as the breeding
period for most cyprinidae populating the major French rivers and is a key period in the response of organisms to climate change. The increase in temperature observed could thus bring about a change
in the population, e.g., with an increase of thermophilic species.
Authors thank to Michel Lepiller, from University of Orléans for our useful discussion on the functioning of hydrogeological systems of ‘Val d'Orléans’ and Beauce. We are also grateful to Valérie
Daussa-Thauvin and Frédéric Verley from the DIREN Centre for providing us the data on temperature and discharge values of the outflows of Beauce limestone aquifers. Ghislain de Marsily and the two
anonymous referees from CR are also acknowledged for their helpful comments. This study was achieved with the financial support of CNRS within the Loire Research Programme. | {"url":"https://comptes-rendus.academie-sciences.fr/geoscience/articles/en/10.1016/j.crte.2006.02.011/","timestamp":"2024-11-15T04:53:39Z","content_type":"text/html","content_length":"124764","record_id":"<urn:uuid:5e1aec65-fc0a-41fb-84d2-e830f7ab86ec>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00446.warc.gz"} |
e S-parameters
S-Parameter Testbench
Measure S-parameters of system
RF Blockset / Circuit Envelope / Testbenches
Use the S-Parameter Testbench block to measure the S-parameter data of a general RF system. The S-parameter testbench sequentially injects a stimulus signal into each port and measures the response
at all ports to obtain the scattering matrix of the RF system. While the stimulus should be small signal for meaningful measurement, the testbench allows for large steady state external signals.
Use Internal Configuration block — Use testbench internal configuration block
on (default) | off
Select to use testbench internal configuration block. Clear this parameter to specify your own configuration block. Clearing this check box removes the Approximate transient as small signal option in
the Advanced tab.
When using your own configuration block, parameters such as step size, fundamental tones, harmonic order, and simulate noise may affect the measured results.
Input power amplitude (dBm) — Input power to DUT
-30 (default) | real-valued scalar
Input power to device under test (DUT), specified as a real valued scalar in dBm. You can change the input power by entering the value in the text box or selecting a value using the knob. The
specified input power represents the power available at the input ports of the DUT. The valid values are from -90 through 60 dBm.
This parameter is disabled while simulation is running when the Approximate transient as small signal option is checked in the Advanced tab.
Data Types: double
Number of ports — Number of measured ports
2 (default) | scalar integer
Number of measured ports, specified as a scalar integer limited to the range 1:128. Once you change the number of ports, new ports appear on the block.
Data Types: double
Measure all S-parameters — Measure entire S-parameters matrix
on (default) | off
Select to measure entire S-parameters matrix. This measurement is done by sequentially exciting each port and measuring all the ports for the output. When you select this box, Save measurement result
to .s2p button appears below this parameter. In this case, the output signal S-parameters are of dimension N-by-N-by-F. N is the number of ports and F is the number of frequencies.
Clear this parameter to manually specify the S-parameters elements. In this case, the output signal S-parameters are of dimension M-by-F. M is the total number of S-parameter elements and F is the
number of frequencies.
Export measurement results to sNp — Save S-parameters data to Touchstone file
Click to save the measured S-parameters data to a Touchstone file. This button opens a standard dialog box for browsing and choosing a file. The only file type suggested is .sNp, where N is the
number of measured ports. This button is disabled before the first simulation takes place and while simulations are running or initializing. It captures the results of the previous simulations. The N
value in .sNp corresponds to the number of ports previously measured.
To enable this parameter, select Measure all S-parameters.
S-Parameter elements — S-parameter elements to measure
[1 1] (default) | two-column matrix
S-parameter elements to measure, specified as a two-column matrix. Each row represents an S-parameter element. The first column represents the incident wave port and the second column represents the
scattered wave port. For example, [[2 1];[1 1]] indicates a two element measurement: S21 and S11. You can choose any elements from the matrix in any order, but the elements must be unique.
To enable this parameter, clear Measure all S-parameters.
Data Types: double
Workspace variable name — Variable name to store the measurement data in MATLAB workspace
'SparamObjOut' (default) | character vector
Variable name to store the measurement data in MATLAB^® workspace, specified as a character vector. The data is stored as an RF Toolbox™ sparameters object.
Data Types: char
Input frequency (Hz) — Carrier frequency of DUT
2.1e9 (default) | scalar
Carrier frequency of the DUT, specified as a scalar in Hz. By default, output frequency is equal to the input frequency because S-parameters are measured to quantify linear systems.
For small-signal measurements over large constant external signals, you can specify an output frequency in the Advanced tab when Adjust for steady-state external signals is selected.
Data Types: double
Baseband bandwidth (Hz) — Baseband bandwidth of input signal
10e6 (default) | positive finite scalar
Baseband bandwidth of input signal, specified as a positive finite scalar in Hz. The measured frequencies reside on this band around the input carrier frequency. This band is by default narrower than
the solver envelope bandwidth to keep simulation artifacts outside of the measured results. The ratio of the two bands can be controlled using Ratio of Envelope to Baseband bandwidths in the Advanced
Data Types: double
Reference Impedance (Ohm) — Impedance for S-parameter measurement
50 (default) | positive finite scalar
Impedance for S-parameter measurement, specified as a positive finite scalar in ohms. All ports are measured using the same reference impedance.
Data Types: double
Show S-Parameter spectrum — View measured values of specified S-parameters over the specified frequency bandwidth
off (default) | on
Select to view measured values of specified S-parameters over the specified frequency bandwidth using a spectrum analyzer. You can view two types of curves in the plot: Magnitude or Real & Imag. The
Magnitude plot shows the s-parameter data magnitudes is in dBm. Real & Imag shows both the real and imaginary parts of the S-parameters. The real and imaginary plot of the S-parameters has no units.
To view the S-parameters spectrum using a Spectrum Analyzer, you need a DSP System Toolbox™ license.
Ground and hide negative terminals — Internally ground and hide RF circuit terminals
on (default) | off
Select to internally ground and hide the negative terminals. Clear to expose the negative terminals. By exposing these terminals, you can connect them to other parts of your model.
FFT Length — Number of FFT bins
128 (default) | scalar integer power of 2
Number of FFT bins used for measurements, specified as a scalar integer power of 2. The value must be an integer power of 2. This value controls the spectral resolution of measurements over the
specified bandwidths.
Measurement Time — Total time duration
12.8 μs (default) | scalar integer
This parameter is read-only.
Total time duration in which each port output is measured to get the spectral result, specified as a scalar integer. The value is the ratio of FFT Length over the Baseband bandwidth specified in the
Main tab. It lets you know that any system response beyond this value is not included in the spectral result. This bound is an outcome of the spectral resolution limitation.
Ratio of wait time to measurement time — Intermission between sequential measurements
0 (default) | scalar integer
Intermission between sequential measurements where the port excitation varies, specified as a scalar integer number of Measurement time instances. Similar to real measurements, the DUT cannot be
reset to clear its internal states. By switching between measurements too fast, outputs that extend beyond the Measurement time can be collected in the next measurement. This value helps to avoid the
contamination of the next measurement.
Ratio of Envelope to Baseband bandwidths — Ratio of internal solver envelope bandwidth to measured bandwidth
8 (default) | scalar integer power of 2
Ratio between internal solver envelope bandwidth and measured bandwidth, specified as a scalar integer power of 2.
Adjust for steady state external signals — Adjust for large steady state signals
off (default) | on
Select to adjust for large steady-state external signals and measure only the linear effect of the DUT for small signal stimulus injected by the testbench. This means that the DUT can contain
elements that mix with large signals that are constant-in-time over the envelope at each carrier. When you select this parameter, you can specify an output frequency different from the input
Clear this if there are no signals injected to the DUT external to the testbench.
The DUT should not contain any external signals that are time varying over the envelope at each carrier.
Output frequency (Hz) — Output carrier frequency
2.1e9 (default) | positive finite scalar
Output carrier frequency, specified as a positive finite scalar.
To enable this parameter, select Adjust for steady state external signals.
Approximate transient as small signal — Choose small subset of frequencies for transient small signal analysis
off (default) | on
Select this option to choose a small subset of frequencies for transient small signal analysis. Use this parameter to accelerate the measurement of S-parameters of the large nonlinear systems around
a given operation point specified by large external steady-state signals.
Use all steady-state simulation frequencies for small signal analysis — Use all frequencies automatically chosen for full-harmonic balance nonlinear solution
on (default) | off
Select this option to use all frequencies automatically chosen for a full-harmonic balance nonlinear solution. Clear to specify the frequencies that carry the small-signal transient.
To enable this parameter, check Approximate transient as small signal.
Small signal frequencies (Hz) — Frequencies that carry small signal transient
2.1e9 (default) | scalar | vector
Frequencies that carry the small signal transient, specified as a real scalar integer or vector. The specified frequencies should be contained in the entire set of simulation frequencies. If some
frequencies are not contained, a warning message appears. If all frequencies are not contained, an error message appears.
To enable this parameter, clear Use all steady-state simulation frequencies for small signal analysis.
Populate Frequencies — Tool to choose small signal transient frequencies
Open the tool to choose small signal transient frequencies to populate Small signal frequencies. The selected frequencies are a subset of the simulation frequencies determined from Fundamental tones
and Harmonic order used in simulation. The entire set of simulation frequencies are given in the combo box on the right side of the dialog box, and the selected frequencies are highlighted. You can
select by directly choosing the frequencies in the selection box, or by choosing the desired tones and harmonic order in the Small signal selection panel and clicking Select. The Tones (Hz) and
Harmonic order values in the combo boxes are also populated using Fundamental tones and Harmonic order used in simulation.
To expose this parameter, clear Use all steady-state simulation frequencies for small signal analysis.
[1] Razavi, Behzad. RF Microelectronics. Upper Saddle River, NJ: Prentice Hall, 2011.
Version History
Introduced in R2019b | {"url":"https://nl.mathworks.com/help/simrf/ref/sparametertestbench.html","timestamp":"2024-11-14T14:40:25Z","content_type":"text/html","content_length":"116740","record_id":"<urn:uuid:49b4371b-461d-4065-a16d-43a751637a22>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00805.warc.gz"} |
Motivation for the Deformed Nekrasov Partition Function
1392 views
I have recently been doing research on the AGT Correspondence between the Nekrasov Instanton Partition Function and Louiville Conformal Blocks (http://arxiv.org/abs/0906.3219). When looking at the
Nekrasov Partition Function one defines a deformed metric in terms of the "deformation parameters" $\epsilon_1, \epsilon_2$ which seem to define a $SO(4)$ action on a standard Euclidean Metric,
breaking translational symmetry. Much of the literature on these functions seems to be in the math department, defining the functions categorically in terms of sheaves and what-not (http://arxiv.org/
abs/math/0311058) and even the original paper (http://arxiv.org/abs/hep-th/0206161) approaches the subject from a cohomological perspective.
Is there any obvious physical motivation for looking at partition functions in this strange deformed spacetime? Or should I view it as simply a mathematical manipulation?
This post imported from StackExchange Physics at 2014-08-23 04:59 (UCT), posted by SE-user Benjamin Horowitz
mitchell.physics.tamu.edu/Conference/string2010/documents/… ... I think its chief utility lies somewhere in the space between M-theory and SQCD.
This post imported from StackExchange Physics at 2014-08-23 04:59 (UCT), posted by SE-user Mitchell Porter
The deformation parameters have a meaning in topological string theory, see for example arxiv.org/abs/arXiv:1302.6993 by Antoniadis et al. for a recent perspective.
This post imported from StackExchange Physics at 2014-08-23 04:59 (UCT), posted by SE-user Vibert
I think the paper by Nekrasov and Witten gives a nice picture. I don't understand it well enough myself to give an answer but you could take a look at it. arxiv.org/abs/1002.0888
This post imported from StackExchange Physics at 2014-08-23 04:59 (UCT), posted by SE-user Siva
The most physical and understandable definition of Nekrasov's partition function to me uses five-dimensional gauge theories. Namely, any 4d N=2 susy gauge theory has a 5d version with the same matter
content, so that compactifying it on a small $S^1$ brings it back to the original 4d theory.
Then we put the theory on the so-called Omega background: it is $\mathbb{R}^4 \times [0,\beta]$, but $(\vec{x},0)$ and $(\vec{x'},\beta)$ are identified by a rotation $$ \vec x'=\begin{pmatrix} \cos
\beta\epsilon_1 & \sin\beta\epsilon_1 & 0 & 0\\ -\sin \beta\epsilon_1 & \cos\beta\epsilon_1 & 0 & 0\\ 0& 0 &\cos \beta\epsilon_2 & \sin\beta\epsilon_2\\ 0& 0 &-\sin \beta\epsilon_2 & \cos\beta\
epsilon_2 \end{pmatrix}\vec x. $$
Then we take the limit $\beta\to 0$, keeping $\epsilon_{1,2}$ fixed. (Strictly speaking we also need to add a background $SU(2)_R$ symmetry gauge field, so that some of the susy is preserved.)
Most of what Nekrasov did using his cohomological framework can be seen directly in this higher-dimensional setup. See e.g. Sec. 3.2 of my review article in preparation, available here.
This post imported from StackExchange Physics at 2014-08-23 04:59 (UCT), posted by SE-user Yuji | {"url":"https://www.physicsoverflow.org/22729/motivation-for-the-deformed-nekrasov-partition-function","timestamp":"2024-11-02T17:49:26Z","content_type":"text/html","content_length":"128446","record_id":"<urn:uuid:43ebaed2-3d5e-4d2c-a39b-0941516a8105>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00511.warc.gz"} |
What is 20 percent of 70 Dollars, Euro, Rupees, or Pounds: Easy Calculation of 20% of 70
We’ll explain what 20 percent of 70 means and then show you how we calculated it. While several simple tricks to find the answer to “What is 20% of 70?
This article focuses on how to calculate 20 percent of 70 in an easy way. Firstly, we should remember that twenty percent of seventy can also be written as 20% of 70. As a result, we can use either
of them to get the answer to this question.
Perhaps you need to figure out the exact value of 20% of 70 and use it to calculate the currency of different countries, such as American dollars, British pounds, Euros, Indian rupees, Japanese yen,
Chinese yuan, Pesos, and other currencies. The best part is that you can find an exact solution to any percentage problem using the methods described in this article. This page contains a detailed
explanation that will assist you in solving this equation.
Answer: 20% of 70 is 14
Multiple methods of calculating percentage problems will help you find shortcuts for solving any percentage deduction problem. Here we have shown more than three different ways to subtract the actual
percentage value from the main value. As a result, it gives you a comprehensive understanding of a defined percentage of the main value and provides you with different procedures that you can use to
calculate any specified percentage of any value in the future.
A pie chart depicting twenty percent of three hundred
A pie chart depicting 20% of 70 is shown below. The total pie has 70 parts (one side), and the small part of the pie is 14 parts, or 20 percent of 70, which was subtracted (another side).
It’s worth noting that the parts don’t have to be the same. It could be 20 percent of 70 of anything such as liquids, powders, shares, things, people, money, animals, or something similar. Whatever
the percentage is, 20% of the pie chart (70 parts) will appear the same.
Calculate what is 20 percent of 70
We already have our total value/main value, as well as our percentage value, which we should use to subtract an unknown value from the main value. So, assume that the unknown value is Y, which answer
we will find out using a series of steps.
It can be subtracted using one of the following equations:
(Absolute value × Percent value)/100 = Part
(Absolute value × 100)/Percent value = Part
Incorporate the values using one of the above equations.
(70 × 20)/100% = 1400/100 = 14
(20/100%) × 70 = 0.20 × 70 = 14
What is 20 percent of 70?
The part is subtracted from the main value in the step-by-step procedure.
Now we need to calculate 20% of 70, as explained in the procedure.
Step 1: In this case, the absolute value is 70.
Step 2: Assume that the value of y in the below equation is unknown.
Step 3: Assume that the value of 70 is 100 percent.
Step 4: In the same way, y = 20%.
We now have two equations as a result of these steps.
Equation 1: 70= 100%
Equation 2: y = 20%
Step 5: Divide equation 1 by equation 14 and note that both equations’ RHS (right-hand sides) have the same unit (%).
70/y = 100%/20%.
Step 6: Finally, the reciprocal of both sides results in
y/70= 20% /100%
As a result, y = 14
Therefore, 20% of 70 is 14
Simple methods to get the answer to this question?
There are two simple ways through which you can get an exact answer to any percentage problem. In the first method, divide both the percent value of 20 and the main value of 70 by 10 to get their
decimal or single-digit values. To get the final value, multiply them together (2.0 * 7.0) as shown in the diagram.
In the second method, divide only one of these values (70 or 20) by 100 and multiply them together to get the actual answer.
How do you find 20 percent of 70 in a simple way?
When you multiply both 0.20 and 70 together, you get 14, which is 20% of 70.
The 0.20 represents 20%, which is obtained by dividing the 20 by 100 (20/100=0.20).
Dividing the percent (20%) by 100 and multiplying by the main number (70) is the simplest way to solve this problem. To get 0.20, divide 20 by 100. From there, multiply the decimal form of a percent
(0.20) by 70 to get 14.
What is 20% off 70 dollars purchase?
You will pay $56 for an item when you get a 20 percent discount off the original price of $70. You’ll be given an $14 refund or discount.
What is 20% off a 70 -euro purchase?
With a 20% discount, any item that would normally cost €70 will now cost €56 only. You’ll save €14.
What is twenty percent of seventy British Pounds?
We multiply 20% by 70 to get 14 British Pounds, just like we did with other currencies.
How much is 20% of seventy rupees?
We use the same formula as with other currencies and multiply 0.20 by 70 rupees to get an answer of 14 rupees.
Practice questions
Question: Your father bought you a gift with a 20% discount on the actual price of $70. How much money did your father save from the total price ($70) of the gift?
Answer: Your father saved 14 and paid 56.
Question: You got a 20% off offer on a wristwatch, which normally costs $70. What percentage of the watch’s total cost ($70) did you save?
Answer: You had 14 dollars saved and had to pay 56 dollars.
Question: Your brother purchased a ring valued at 70 dollars and decided to sell it back to someone with a 20% discount. How much money did your brother lose from the ring’s total cost?
Answer: Your brother lost 14 dollars and sold it for 56 dollars.
The importance of calculating percentages
We use the formula above to find the answer when subtracting any percent (%) of any digit, a product discount, a tax, a cashback bonus on any deal, bank interest, discount offers, annual interest
rates, monthly salaries, charges, loan interest rates, dollars, rupees, pounds, commissions, coupons, percentage of shares, or anything else. The calculation equation above is straightforward and
natural. Using the simple tricks above, you can also compute other numeric values using the calculator available on mobile phones.
• This concludes our tutorial. We hope we’ve succeeded in making you an expert at solving percentage problems, at least when it comes to finding the answer to this question.
• Some of you might be wondering about the different methods given in this article for calculating the percentage of any value.
• These may appear to be simple equations, but understanding the simple trick behind any of the above will help you solve your problem with less confusion and make a more informed decision about
whether or not it is a good deal for you in currency matters.
• Overall, try practicing with the other problems below if you think you’ve learned some new tricks for calculating percentage values. | {"url":"https://mysymedia.com/20-percent-of-70/","timestamp":"2024-11-08T08:01:38Z","content_type":"text/html","content_length":"176748","record_id":"<urn:uuid:744f6574-c76a-4b0a-b702-895455635346>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00557.warc.gz"} |
Exercise: Mensuration - Problem Questions with Answer, Solution | Mathematics
Unit Exercise
1. The barrel of a fountain-pen cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen will be used for writing 330 words on an average. How many words can be
written using a bottle of ink containing one fifth of a litre?
2. A hemi-spherical tank of radius 1.75 m is full of water. It is connected with a pipe which empties the tank at the rate of 7 litre per second. How much time will it take to empty the tank
3. Find the maximum volume of a cone that can be carved out of a solid hemisphere of radius r units.
4. An oil funnel of tin sheet consists of a cylindrical portion 10 cm long attached to a frustum of a cone. If the total height is 22 cm, the diameter of the cylindrical portion be 8cm and the
diameter of the top of the funnel be 18 cm, then find the area of the tin sheet required to make the funnel.
5. Find the number of coins, 1.5 cm in diameter and 2 mm thick, to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm.
6. A hollow metallic cylinder whose external radius is 4.3 cm and internal radius is 1.1 cm and whole length is 4 cm is melted and recast into a solid cylinder of 12 cm long. Find the diameter of
solid cylinder.
7. The slant height of a frustum of a cone is 4 m and the perimeter of circular ends are 18 m and 16 m. Find the cost of painting its curved surface area at ₹100 per sq. m.
8. A hemi-spherical hollow bowl has material of volume 436π /3 cubic cm. Its external diameter is 14 cm. Find its thickness.
9. The volume of a cone is 1005 (5/7) cu. cm. The area of its base is 201 (1/7) sq. cm. Find the slant height of the cone.
10. A metallic sheet in the form of a sector of a circle of radius 21 cm has central angle of 216°. The sector is made into a cone by bringing the bounding radii together. Find the volume of the cone
1. 48000 words
2. 27 minutes (approx)
3. 1/3 πr3 cu.units
4. 782.57 sq.cm
5. 450 coins
6. 4.8 cm
7. ₹ 6800
8. 2 cm
9. 17 cm
10. 2794.18 cm3
Tags : Problem Questions with Answer, Solution | Mathematics , 10th Mathematics : UNIT 7 : Mensuration
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
10th Mathematics : UNIT 7 : Mensuration : Exercise: Mensuration | Problem Questions with Answer, Solution | Mathematics | {"url":"https://www.brainkart.com/article/Exercise--Mensuration_39433/","timestamp":"2024-11-14T01:22:33Z","content_type":"text/html","content_length":"40512","record_id":"<urn:uuid:81d3d53c-2e64-4d19-9395-e45d83849afd>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00623.warc.gz"} |
When Averages Attack: For the Love of Means
Americans love the average. Ask the average american on the street what he knows of statistics, and they will probably answer in so many words about something relating to an average (arithmetic mean
). An average describes for us the central tendency of some data; the whole distribution of whose values we find it easier not to remember. Yet averages have a darkside to them, beyond sunny days on
a baseball diamond figuring your favorite batter’s batting average. Let us look at statistics… when averages attack!
Why We Love Averages
An average is a summary of some data’s central tendency that’s easily remembered, spoken of, calculated, and we believe in part because we learned it describes the central tendency of the data–an
average is something we can intuitively reason about.
Easily Remembered
Remember one value instead of ten, hundreds or thousands. In 1979 as a first baseman for the Philadelphia Phillies (my favorite baseball team), Pete Rose had 208 hits (H) in 628 at-bats (AB). You can
choose to recall this data as a sequence of 1s and 0s; six-hundred and twenty-eight of them, in fact. With the help of www.baseball-reference.com, you can find that it starts:
0, 0, 1, 1, 1, 0, 0, 0, 0, 0, ...
Many of us conveniently choose to instead remember one value that summarizes this data, the Batting Average (presumptiously abbreviated, AVG). For Pete Rose that season it was a remarkable .331,
refuting the notion that at 38-years old, any individual’s performance must wane.
If you can remember this sequence of values, then you do gain something that you would not have from the AVG alone. These first 10 AB precisely describe Pete Rose’s streaky performance during the
opening series that season at Busch Stadium against the St. Louis Cardinals, a team that tried to sign Pete Rose that off-season.
Spoken Of
With so many americans facing the challenge of obesity, it is no wonder that speaking of our weights is a sensitive subject. As we meet with our physicians for a (hopefully) annual check-up, we are
all weighed and our body weights recorded. It is not terribly uncommon for our weight to change modestly between these weigh-ins at the doctor’s office, producing a time series of data entries in our
medical records like the following:
240, 250, 245, 250, 260, 255, ...
Now unexpectedly you are filling out some forms for a trip across the country because the airline has instituted surcharges for overweight passengers. Wouldn’t you know it, you forgot to pack your
bathroom scale in your luggage! You want to give an accurate weight on the form, although you can’t measure it at the moment (it could very likely have changed) and don’t have immediate access to
your medical records. What do you say?
What you are familiar with over the past six years is that your average weight has been 250 pounds. Sometimes it has been ten pounds more, sometimes it has been ten pounds less, and at this very
moment you couldn’t know without a measurement that it’s actually 253 pounds, 3 ounces. Should there be an inquiry, at least you can supply the medical history that backs-up your use of this average
value summarizing your weight’s central tendency over the past six years.
Averages are easily computed using only the basic arithmetic we learned as children in school, one reason for statisticians referring to them by the name: “arithmetic mean.”
The simplest and most straightforward procedure for calculating an arithmetic mean is:
1. Add up each individual data value (datum) to produce a sum.
2. Divide this sum by the count of data values.
In the foregoing example of our average weight,
240 + 250 + 245 + 250 + 260 + 255 1500
--------------------------------- = ---- = 250
Why do we calculate it this way? It’s the linear nature of the arithmetic mean that presents this means of calculation, while reinforcing our subconscious notions of how these averages work. For
instance, since the average indicates to us where we believe the “middle” of the data is, then the deviations above and below the average should cancel out to zero. Revisiting our weight calculation:
(240 - 250) + (250 - 250) + (245 - 250) + (250 - 250) + (260 - 250) + (255 - 250)
-10 + 0 + -5 + 0 + 10 + 5
-15 + 15
If we didn’t know the average beforehand (let’s call it x), this reasoning would lead us to use simple algebra to come up with the calculation:
(240 - x) + (250 - x) + (245 - x) + (250 - x) + (250 - x) + (255 - x) = 0
1500 - 6x = 0
1500 = 6x
250 = x
Can you see how this always yields the same arithmetic procedure we learned in school? That flowed from the premise we intuitively take for granted: that the average tells us about the center of our
data, and any deviations from that center should cancel each other out in the end.
Intuitively Reasoned About?
Let’s be clear, by “averages attack,” I don’t mean cases where the average includes some fraction open to a grisly misinterpretation. We’ve all heard statistics tossed around such as, in 2016 the
U.S. Census found the average number of children per married family is 1.89. We’re all likewise aware in interpreting this average that it does NOT mean families are suffering at the hands of some
ferocious troll, which comes into childrens’ bedrooms to steal one-ninth of a child for their breakfast snack. Instead, we’re comfortable reasoning that these nominal categories exist in the data:
• no-child families,
• one-child families,
• two-child families, and
• more-child families;
and that the “midpoint” of this distribution lands somewhere closer to the two-child families than the one-child families. This is called interpolation. It’s relatively common that an interpolated
value lands between nominal categories (or discrete data), and we frequently map these onto a continuous data value when it makes sense to do so.
When you are dealing with linear relationships between categories of your data (families with one or fewer children, and families with two or more children), interpolation serves our understanding
well. But what you need to ask yourself, is whether this presumption of linearity always holds true?
You’re probably thinking, “Of course it does! It’s math, it’s algebra even, you proved it in the previous section, didn’t you?” Not at all. What I had demonstrated algebraically was that if you take
the axiom of there existing a linear relationship between all of the data as true, and incidentally that the average sits at the midpoint of the range of data (the difference: high – low), then the
variations from this average net to zero.
Weighted Averages: A Curveball For Samantha
Sometimes we want to calculate an average where some values (often the most recent) should be treated as having a greater impact. A ready example of this is found in how university students have
their academic grades determined. Take for example the schedule of exams found in a typical university physics course syllabus:
General Physics II – Grading
22 February 2017 Exam 1 – Waves (20% of final grade)
29 March 2017 Exam 2 – E & M (20% of final grade)
01 May 2017 Lab Notebook (25% of final grade)
03 May 2017 Cumulative Final (35% of final grade)
Suppose the university’s starting softball shortstop, Samantha, is taking this General Physics II course, and she receives the following grades on her first two exams, lab notebook, and cumulative
83, 88, 85, 70
Your intuition probably expects Samantha received a B in General Physics II with an average somewhere in the mid- to lower-80s. But look closely here at the weightings each grade receives. You must
take these into account when computing the weighted arithmetic mean, like so,
83 x (0.20) + 88 x (0.20) + 85 x (0.25) + 70 x (0.35)
16.6 + 17.6 + 21.25 + 24.50 = 79.95
What a shame! Samantha falls shy of the 80 point threshold she needs to earn a B. Instead, she earned a C. Unless she has a particularly forgiving professor, those are the breaks.
Diagnosing What Went Wrong
In this example, it was NOT as simple as merely looking at the deviations from the midpoint, although that may be what our brains told us subconsciously:
(83 - 79.95) + (88 - 79.95) + (85 - 79.95) + (70 - 79.95)
3.05 + 8.05 + 5.05 + -9.95
16.15 + -9.95
The good news is that the algebra can be adapted to work like before, in this case where the weights are simple constant factors. Do you see what adjustment must be made? Of course, we must multiply
each variation by its respective weighting factor.
(83 - 79.95)(0.2) + (88 - 79.95)(0.2) + (85 - 79.95)(0.25) + (70 - 79.95)(0.35)
3.05(0.2) + 8.05(0.2) + 5.05(0.25) + -9.95(0.35)
0.61 + 1.61 + 1.2625 + 3.4825
3.4825 + -3.4825
In doing so, the sum of all deviations from the weighted arithmetic mean again net to zero. Reflect carefully on how your mind processed the list of Samantha’s grades at first glance, and contrast
that to how you come up with the correct answer when working through the arithmetic of multiplying each grade by its weight factor.
The Hazardous Trail Ahead
This is just one simple case where our intuitions regarding averages can breakdown, and there are many others! I stumbled upon some statistics problems the other night. Some of them I had long been
familiar with, yet others still surprised me. These exercises will serve as the basis for this new blog series of mine: When Averages Attack!
In each installment of this series, I’ll walk you through how to think your way out of the pitfalls present in what our initial intuition tells us (and what we think we know). I’ll also speculate as
to why we misthink the way we do when confronted with thought-provoking statistics. With the bombardment of fake-news and propaganda we confront daily trying to mislead us from facts, keeping an
objective perspective becomes increasingly important.
You are now taking your first steps into the shadow-stricken borderlands where averages may attack from nowhere, and our intuitive reasoning on the central tendency of data fails us. | {"url":"https://loresayer.com/2017/02/22/when-averages-attack-for-the-love-of-means/","timestamp":"2024-11-01T19:41:04Z","content_type":"text/html","content_length":"38649","record_id":"<urn:uuid:7bf01cdf-0b39-4895-8e86-2b6a1ac219b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00694.warc.gz"} |
Converting metric units
Our chosen students improved 1.19 of a grade on average - 0.45 more than those who didn't have the tutoring.
In order to access this I need to be confident with:
Here we will learn about converting metric units, including metric units of length, metric units of mass and metric units of capacity (volume).
There are also converting metric units worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if youβ re still stuck.
Converting metric units is being able to convert between different metric units of length, mass or volume.
To do this we need to know what the metric units are and their conversion factors.
We can use prefixes to make these metric units bigger and smaller.
The SI unit (international system of units) of length is the metre (m) .
For length we mostly use kilometres (km) , metres (m) , centimetres (cm) and millimetres (mm) .
The metric system for mass is based around grams (g) .
For mass we mostly use tonnes (t) , kilograms (kg) , and grams (g) .
The SI unit The metric system for capacity is based on litres (l) .
For volume we mostly use litres (l) , centilites (cl) , and millilitres (ml) .
Get your free converting metric units worksheet of 20+ questions and answers. Includes reasoning and applied questions.
Converting metric units is part of our series of lessons to support revision on units of measurement. You may find it helpful to start with the main units of measurement lesson for a summary of what
to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
As we are going from larger units to smaller units we multiply.
As we are going from smaller units to larger units we divide.
If you are going from larger units to smaller units – multiplyIf you are going from smaller units to larger units – divide
3. Convert:Β Β 81 \ 000 \ g to kg
How far apart are the Town Centre and the Park?
Phil has 2 bottles of lemonade and 6 cans of lemonade.
4.Β 4 tins of soup have a mass of 1.8kg.
3 tins of soup and 2 packets of soup have a mass of 1420 \ g.
Find the mass of 7 tins of soup and 5 packets of soup.
Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors. | {"url":"https://thirdspacelearning.com/gcse-maths/ratio-and-proportion/converting-metric-units/","timestamp":"2024-11-13T22:56:08Z","content_type":"text/html","content_length":"340125","record_id":"<urn:uuid:106bf404-6ce1-447c-9623-fcacc1c0849b>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00376.warc.gz"} |
Physics 1B: The Stuff of the Universe (PHYS08017)
Undergraduate Course: Physics 1B: The Stuff of the Universe (PHYS08017)
Course Outline
School School of Physics and Astronomy College College of Science and Engineering
level SCQF Level 8 (Year 1 Undergraduate) Availability Available to all students
year taken)
SCQF 20 ECTS Credits 10
The course begins with the classical models of particles and waves and their relationship to the physical world of atoms and light. Quantum physics is introduced through the idea of wave/
particle duality, in a largely non-mathematical way. The uncertainty principle, Schrodinger's cat and quantum tunnelling are discussed. The hydrogen atom, and then more complex atoms are
Summary considered illustrating the role of quantum effects such as the Pauli exclusion principle which is seen to underly the structure of the periodic table. The phases of matter are discussed
and quantum effects are used to explain ordinary conductivity and superconductivity. Matter is explored at the nuclear and elementary particle scales. At large scales the behaviour of
stars and of the big-bang are related to the fundamental properties of matter.
Part I: Particles, Waves and Quanta
1. The Classical Particle Picture
- Brownian motion. Monatomic gases. Avogadro's number. Pressure. The Ideal Gas Law.
- Temperature. Mean free path and rms velocity. Kinetic Energy and Heat. The Maxwell-Boltzmann distribution.
- Heat Capacity of a monatomic gas. Molecular gases. Rotational and vibrational modes. Equipartition of energy.
2. The Classical Wave Picture
- Introduction to waves.
- Sound Waves. Velocity of sound. Relationship to properties of matter.
- Light. Spectrum of Electromagnetic waves. Velocity of light in a vacuum. Wave-fronts and Huygens' Principle.
- Superposition of waves. Interference. Phase difference.
- Diffraction by a single slit. Young's double slits. Diffraction grating. X-ray diffraction.
3. The Quantum World
- The Photoelectric Effect. Planck's constant. The Photon. Quantisation of Energy.
- Diffraction of electrons. Diffraction of neutrons and atoms. The de Broglie wavelength.
- Wave particle duality. The wavefunction. Wave packets. The uncertainty principle.
- The probability density interpretation of the wavefunction. Schrödinger's cat. The role of the observer. The quantum interpretation of the double slit experiment.
Part II: Atoms, Molecules and Solids
1. Elementary Quantum Mechanics
- Schrödinger's equation. Solutions for a free particle, and a particle in a box.
- Potential wells. Energy levels in an infinite well and in a harmonic well.
- Effect of a step potential. The finite barrier. Quantum tunnelling.
2. The Hydrogen Atom
Course - A review of classical circular orbits. The Bohr model. Energy dependence of radius. Limitation of classical picture.
description - Quantisation of angular momentum and energy. Electron spin. Wave functions and probability distributions. Energy levels.
- Absorption and emission of photons. Bohr frequency condition. Spectral lines for Hydrogen. Allowed and forbidden transitions. Line widths and lifetimes.
3. Complex Atoms and Molecules
-Multi-electron atoms. Energy level diagrams and spectral lines. The Pauli exclusion principle. Fermions and bosons. Orbitals. The periodic table of elements.
- Stimulated emission. Population inversion and amplification. The Helium-Neon laser.
- The hydrogen molecule. Splitting of single electron energy levels. The covalent bond. Brief discussion of other types of bonds.
4. The Solid State
- The phases of matter. Gases, liquids and solids. Crystalline and amorphous materials. Crystal structure.
- Energy bands. Insulators and metals. Filled and unfilled bands. The Fermi level. Conduction of electricity in metals.
- Semiconductors. Conduction and valence bands. Electrons and holes. Doping. The pn junction and the laser diode.
- Superfluid Helium. Bosons don't obey exclusion principle. Condensation into a collective ground state. Cooper pairs and superconductivity.
Part III: The Stuff of the Universe
1. The Atomic Nucleus
- Discovery of the nucleus. The nuclear scale. High energy electron scattering. The nucleon-nucleon interaction. Mass and Binding Energy (E=mc2).
- Radioactive decays: The radioactive decay law. Alpha, beta and gamma decays. Energy released in nuclear decays.
- Nuclear reactions: Nuclear instability, Nuclear fission (spontaneous and induced) and Nuclear fusion (nucleosynthesis and thermonuclear).
2. Elementary Particles
- Introduction to elementary particles. Quantum field theory. Antiparticles. The muon and pion. The particle explosion.
- The Standard Model. The eightfold way and quarks. Quantum chromodynamics. Quark confinement. Evidence for quarks. The weak interaction. Leptons. The fundamental forces.
- Conservation laws and particle decays: Crossing symmetry, conservation of charge, baryon number & lepton number. Strangeness. Particle decays and widths. Strength of the forces.
3. Matter in the Universe
- The expanding universe: Doppler effect, red-shift. Hubble's Law. The critical density.
- Dark matter. Dark energy. The cosmic microwave background. The Big Bang. Unification of forces.
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Co-requisites
Prohibited Combinations Other requirements SCE Higher Grade Physics and Mathematics (at Grade A or higher) or equivalent.
Information for Visiting
Pre-requisites None
High Demand Course? Yes
Course Delivery Information
Academic year 2014/15, Available to all students (SV1) Quota: 303
Course Start Semester 2
Timetable Timetable
Total Hours: 200 ( Lecture Hours 33, Seminar/Tutorial Hours
10, Supervised Practical/Workshop/Studio Hours 30, Online
Learning and Teaching activities (Further Info) Activities 11, Summative Assessment Hours 15, Revision
Session Hours 6, Programme Level Learning and Teaching
Hours 4, Directed Learning and Independent Learning Hours
91 )
Assessment (Further Info) Written Exam 60 %, Coursework 20 %, Practical Exam 20 %
Degree Examination, 60%
Additional Information (Assessment) Laboratory, 20%
Coursework, 20%
Feedback Not entered
Exam Information
Exam Diet Paper Hours & Minutes
1B: The
Main Exam Diet S2 (April/May) Stuff of 2:00
1B: The
Resit Exam Diet (August) Stuff of 2:00
Learning Outcomes
Upon successful completion of this course, it is intended
that a student will be able to:
i) demonstrate a general appreciation for the microscopic
origin of many everyday macroscopic phenomena, for example
pressure and temperature
ii) demonstrate a general understanding of light in terms of
atomic transitions, including atomic spectra, lasers and
iii) describe wave phenomena using appropriate terminology
and formulae, for example in the situations of wave
propagation, diffraction and interference
iv) demonstrate a reasonable understanding of the
fundamental aspects of quantum mechanics, specifically
including wave-particle duality, the photoelectric effect,
two-slit experiments, the role of the observer and quantum
v) determine basic parameters associated with a variety of
simple potential wells.
vi) demonstrate the significance of the Pauli Exclusion
Principle, especially in relation to an understanding of the
Periodic Table of Elements and chemical properties.
vii) demonstrate a basic understanding of the band theory of
crystalline solids, exploring applications such as
semiconductors and superconductors.
viii) demonstrate basic knowledge of nuclear and particle
physics; radioactive decay, the standard model and
ix) demonstrate a reasonable understanding of modern
cosmology, including the Big Bang theory , stellar
evolution, cosmic expansion, dark matter, and the ultimate
fate of the Universe.
x) show competence in a scientific laboratory.
xi) show an understanding for the various sources of
uncertainty incurred in making any experimental measurement.
Furthermore, they should be able to estimate such
experimental errors and be able to reasonably determine the
incurred uncertainty in a derived quantity.
xii) communicate scientific concepts in a written format
Reading List
'Principles of Physics' (Extended International Edition; 9th
Edition, authors: Halliday, Resnick and Walker, publisher:
Additional Information
Course URL www.learn.ed.ac.uk
Problem solving, group working, communication
Graduate (written and verbal), time and resource
Attributes management, gathering and organising
and Skills information, creativity, practical and
experimental skills, data analysis skills.
Additional Laboratory sessions three hours per week, as
Class arranged. Tutorials one hour per week, as
Delivery arranged.
Keywords P1B
Dr Ross Galloway Ms Rebecca Thomas
Course Tel: Course Tel: (0131 6)50
organiser Email: secretary 7218
ross.galloway@ed.ac.uk Email:
© Copyright 2014 The University of Edinburgh - 12 January 2015 4:39 am | {"url":"http://www.drps.ed.ac.uk/14-15/dpt/cxphys08017.htm","timestamp":"2024-11-10T19:17:14Z","content_type":"text/html","content_length":"26310","record_id":"<urn:uuid:1c577921-f641-474c-baa3-1419c6e58ce3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00147.warc.gz"} |
Photon-Electron Scattering
Next: Pair Annihilation into Gamma Up: QED Processes Previous: Bremsstrahlung
Electrodynamic processes can be classified by the number and type of particles in the initial state. Our goal in this chapter is to consider the two-particle initial states that led to scattering
processes. Photon-electron scattering will be considered in this section, while the photon-photon and electron-electron systems will be consider in subsequent sections. Our use of the word
``electron'' has been generic and includes both electrons and positrons, in general.
The photon-electron system can have two kinds of final states, those in which there is only one electron present (and one or more photons), and those in which there are also one or more
electron-positron pairs present. Processes leading to the former kind of final states may be called photon-electron scattering, whereas processes in which pairs are produced can be referred to as
pair production in photon-electron collisions.
Photon-electron scattering with only one photon in the final state is the lowest-order photon-electron process involving a real incident photon. The separation of the lowest-order from the higher
order contributions is an idealization which does not correspond to physical reality. In any measurement, the energy of the final state can only be determined to within the energy resolution of the
detector. It is therefore impossible to determine with certainty whether the final state contains exactly one photon or whether it contains an additional number of very soft (low energy) photons.
These multiple-photon final states are suppressed relative to the lowest-order single-photon final state by at least order
The lowest order photon-electron scattering process is second-order in
Figure 7.7 shows the two diagrams leading to Compton scattering. It is important to realize that only the sum of the two diagrams in figure 7.7 describes photon-electron scattering. The separation of
a matrix element into terms corresponds to the individual diagrams, though extremely useful, has in general no physical meaning. Only the sum of both diagrams is observable.
In the first diagram (figure 7.7a), the incident photon (7.7b), the incident electron (7.7. The two diagrams are different since they differ in the sequence of the emitted and absorbed photons as one
follows the arrows in the electron paths. One can draw the second diagram such that the intermediate electron is horizontal in the diagram and the two photon do not cross in the diagram. This
horizontal-electron diagram is not topologically different from the diagram in figure 7.7b and will not be considered further. Compton scattering is thus the process
The second-order Compton amplitude is, after carrying out the Fourier transformation to momentum space,
Each term in equation 7.225 represents one of the eight diagrams shown in figure 7.8. Not every term that occurs in the scattering amplitude is physically relevant to the process considered. The
first two diagrams (figures 7.8a and 7.8b) are the processes we are interested in when studying Compton scattering. The second pair of diagrams (figures 7.8c and 7.8d) have the photon momenta 7.8e
and 7.8f) and two photons in the initial state (figures 7.8g and 7.8h), respectively. These processes are not kinematically allowed. Such terms in the scattering amplitude contain delta functions
with an argument describing these kinematically forbidden processes. The delta functions cause these terms to vanish when the momenta is integrated over.
For the diagrams we are interested in, we retain from the incident photon wave function only the first term,
The amplitude now becomes
Notice that 7.7 are thus related by this symmetry. This is known as crossing symmetry, and it persists as an exact symmetry to all orders in
We form the cross-section
We use
The differential cross-section now becomes
where the kinematic variables in the matrix element
The cross-section simplifies considerably if we calculate it in the rest system of the initial or final electron. In most experiments the initial electron is practically at rest in the laboratory so
we will work in the rest frame of the initial electron. In this frame
The last line in equation 7.230 was obtained by using the root of the delta function to relate
This is known as the Compton condition. This kinematic relationship takes on a simple form if one uses the wavelength,
This is the familiar Compton formula. The wavelength of the scattered photon is increased by an amount of order
The differential cross-section for electrons and photons with specific initial and final state polarizations is now
We can simplify the spinor matrix element considerably by choosing the special gauge in which both initial and final photons are transversely polarized in the laboratory frame. We choose
Since the electron is initially at rest it follows that
This amounts to choosing the ``radiation gauge'' in which the electromagnetic potential has no time component. However, the condition in equation 7.236 can be imposed in any given frame of reference.
This can be shown by applying a gauge transformation to any arbitrary set of polarization vectors 7.236.
Because of our choice of gauge,
where the energy-projection operator
We now consider the case when the electrons are unpolarized but the initial and final state photons may be polarized with polarizations
Applying the usual trace techniques, we have
where we have used the rule
There are traces with up to eight
Using the symmetry we had earlier
We show that the two cross-terms are equal:
Using energy-momentum conservation we have
Therefore the differential cross-section becomes
The calculation of the invariant matrix element has so far been covariant. In the rest frame of the initial electron the differential cross-section becomes
which is the Klein-Nishina formula for Compton scattering.
In the low-energy limit of 7.231 shows that
is the classical electron radius.
For forward scattering, 7.231
Returning to the general expression for the cross-sections, we can sum over final-state photon polarizations
We can evaluate the remaining spin sum by supposing the incident photon arrives along the
We may select the associated polarization vectors to be
It is easy to show that this choice of vectors satisfies all the required normalization and orthogonality relationships. We obtain
The cross-section thus becomes
The low-energy for forward-scattering limit (classical limit) now becomes
To integrate the differential cross-section, we simplify the notation by introducing 7.231 to write
To perform the integration, define
Using the integrals (
we can write
which is valid for all initial photon energies
For low energies,
which is again the classical Thomson cross-section.
At high energies,
Next: Pair Annihilation into Gamma Up: QED Processes Previous: Bremsstrahlung Douglas M. Gingrich (gingrich@ ualberta.ca) | {"url":"https://sites.ualberta.ca/~gingrich/courses/phys512/node102.html","timestamp":"2024-11-03T23:47:12Z","content_type":"text/html","content_length":"95587","record_id":"<urn:uuid:d82975b0-cc9a-450a-8bc5-a9218c136efa>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00101.warc.gz"} |
class compas.geometry.Polygon(points, **kwargs)[source]
Bases: Primitive
A polygon is defined by a sequence of points forming a closed loop.
points (list[[float, float, float] | Point]) – An ordered list of points.
☆ points (list of Point) – The points of the polygon.
☆ lines (list of Line, read-only) – The lines of the polygon.
☆ length (float, read-only) – The length of the boundary.
☆ centroid (Point, read-only) – The centroid of the polygon.
☆ normal (Vector, read-only) – The (average) normal of the polygon.
☆ area (float, read-only) – The area of the polygon.
A polygon is defined by a sequence of points connected by line segments forming a closed boundary that separates its interior from the exterior.
In the sequence of points, the first and last element are not the same. The existence of the closing edge is implied. The boundary should not intersect itself.
Polygons are not necessarily planar by construction; they can be warped.
>>> polygon = Polygon([[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0]])
>>> polygon.centroid
Point(0.500, 0.500, 0.000)
>>> polygon.area
from_data Construct a polygon from its data representation.
from_sides_and_radius_xy Construct a polygon from a number of sides and a radius.
is_convex Determine if the polygon is convex.
is_planar Determine if the polygon is planar.
transform Transform this polygon.
Inherited Methods
ToString Converts the instance to a string.
copy Make an independent copy of the data object.
from_json Construct an object from serialized data contained in a JSON file.
from_jsonstring Construct an object from serialized data contained in a JSON string.
sha256 Compute a hash of the data for comparison during version control using the sha256 algorithm.
to_data Convert an object to its native data representation.
to_json Serialize the data representation of an object to a JSON file.
to_jsonstring Serialize the data representation of an object to a JSON string.
transformed Returns a transformed copy of this geometry.
validate_data Validate the object's data against its data schema.
validate_json Validate the object's data against its json schema. | {"url":"https://compas.dev/compas/1.16.0/api/generated/compas.geometry.Polygon.html","timestamp":"2024-11-11T08:23:42Z","content_type":"text/html","content_length":"28078","record_id":"<urn:uuid:c95ab4cc-4f34-4019-a659-47f2323e56e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00591.warc.gz"} |
Statistics Cheat Sheet: A Beginner's Guide to Probability and Random Events | HackerNoon
A beginner’s guide to Probability and Random Events. Understand the key statistics concepts and areas to focus on to ace your next data science interview.
In our previous article on statistics, we looked at the different pillars of statistics. We went through the various data collection methods to understand the population characteristics. We also
explored the world of descriptive statistics. We went through various measures of central tendency and measures of spread. In this session, we will look at concepts from probability and random
events. Also, check out our comprehensive “Statistics Cheat Sheet” for important terms and equations for statistics and probability. You can also look at our top probability interview questions to
find out the nature of questions asked in Data Science Interviews.
Probability is a branch of mathematics estimating how likely it is that something is going to happen. Probability theory is applied in everyday life in risk assessment and modeling. For example, the
premiums for car insurance are determined by how likely an adverse event like an accident or breakdown is likely to happen. If you have a poor driving record or a very old and poorly maintained
vehicle, you will have to pay a higher premium compared to a person with a faultless driving history and a new car. Probability is the backbone of all Data Science Models and it is critical that one
understands probability theory.
Random Events
A random event is something that will happen. For example, your car might be involved in an accident. It matters to you since it is your car. It also matters to your car insurer since they have to
bear the costs if an accident happens. However, we do not know in advance that it will happen. The outcome of the event can be a simple True / False result example - Did Biden win the presidency? Or
it can be more complex like was the driver wearing a seat belt, not driving under influence, and within speed limits when the accident happened? Whatever the outcome, it should be complete in the
sense that everything that we are interested in can be defined in terms of the outcome.
Basic Definitions
Elementary Outcomes are all the possible results of a Random Event. For example, while tossing a coin, getting a head would be an elementary outcome. So is getting a tail. In the case of rolling a
six-faced dice, each of the numbers would represent an elementary outcome.
Sample Space (Also called Possibility Space) is the set of all elementary outcomes. For example, tossing a coin once, the sample space is {H, T}. If we were tossing two coins, then the sample space
would be {HH, HT, TH, TT}.
P(H) = P(T) = 0.5
Probability can be defined as
In the case of rolling two six-faced dice simultaneously, the sample space can be viewed in the following manner.
Each of the above outcomes is equally likely. Therefore the probability for each is:
What this means is that if you keep rolling two dice a very large number of times, each of these outcomes would occur:
Note: in the above case, we consider (3,6) and (6,3) as two different outcomes. In the former, the first dice shows a 3 and the second a 6. In the latter, the numbers are reversed.
Characteristics of Probability
Probability is non-negative. Think of probability as the battery level on your phone. If the level is low, there is a lesser chance that the outcome will occur, if the level is high, the greater the
chances. As with your phone battery, it cannot go below zero.
The probability of an outcome always lies between 0 and 1 (both inclusive).
• If the probability of an outcome is 0, then it is an impossible outcome.
• If the probability of an outcome is 1, then it will definitely happen.
Further, the sum of all probabilities of all possible outcomes equals 1. In other words, one of these outcomes will definitely happen.
Note: The elementary outcomes need not have equal probabilities. For example, suppose we load a die to make sure that the number 6 comes 25% the following of the time:
Instead of as earlier:
The sample space remains the same, {1,2,3,4,5,6}.
All the probabilities add up to 1, we get the following
P(1) + P(2) + … P(5) + P(6) = 1
We know that P(6) = 0.25 and all the other outcomes have equal probabilities: P(1) = P(2) = .. P(5). We can therefore rewrite the above as
We get the following probabilities for each of the outcomes of the loaded die. As you can verify the sum of the probabilities add up to 1.
Till now we have discussed the probability of elementary outcomes. However, in practice, these can become pretty unwieldy. If we were to find the probability of getting a sum of 6 by adding up the
numbers on the dice. This is not an outcome in the sample space. To find the probabilities, we extend the elementary outcomes to events. In probability theory, an event is a group of outcomes. A
subset of the sample space. So let us add up the numbers on the faces of two dice for all cases.
To find the probability of events, we simply add up the probabilities of the elementary outcomes that satisfy the conditions of the event. Here are some of the events and their associated
The advantage of using events rather than elementary outcomes is that we can combine events using logical operations AND, OR and NOT to form more events. So for any two events X and Y, we can make
new events in the following manner.
X AND Y: Both event X and event Y occur X OR Y: At least one of event X or event Y occurs NOT X: Event X does not occur
Combining the definitions of probability with these logical operations gives us more powerful formulae for manipulating probabilities. Let us look at the two-dice situation.
Event X: The first die shows 3
Event Y: The second die shows 3.
The blue box shows event X, while the red box shows event Y. The overlap event X AND Y (3 shows up on both the dice) is the overlap.
If we have to find out the probability that one of the two dice shows 3 : P(X OR Y), we cannot simply add P(X) and P(Y) because we will be double counting the elementary outcomes shared by X and Y.
This gives us the addition rule for probability.
P (X OR Y) = P(X) + P(Y) - P(X AND Y)
We can verify the addition rule for the above example.
There are times when there is no overlap. For instance, suppose we want to find the probability of getting a total of 3 (orange) or a total of 9 (green).
These events are mutually exclusive events and P(X AND Y) is 0. In such cases, we can modify the addition rule to
P(X OR Y) = P(X) + P(Y)
X, Y are mutually exclusive events
We can verify this.
Sometimes it is easier to calculate when an event has not happened. For example if we want to find out the probability that none of the dice show a 3.
The shaded area contains the favorable cases. This is the complement of the earlier problem where we calculated the probability that at least one of the dice shows a three. We can simply use the
complement rule which states that
P (NOT A) = 1 - P(A)
We have already calculated the probability that at least one of the dice shows a 3
Therefore the probability that none of the dice show a 3 would be
Conditional Probability
Let us alter the process a little bit. Instead of throwing both the dice together, we throw the dice one after the other, noting the number on the first die before we throw the second die. Before the
dice are thrown, the probability of getting a sum of 9 (let us call this event A) will be
Suppose the first die shows a 6 (Let us call this event B). What is P (A) now?
We call it the conditional probability that Event A will occur, given that Event B has already occurred. We denote this P (A | B) : The probability of A, given B.
Since B has already occurred, the outcome is now only the number on the second die. The reduced sample space is now
Only one outcome (6,3) sums up to 9. So the conditional probability P (A | B) is
The formal definition of conditional probability of A, given B is
Resolving this, we get the Multiplication Rule.
P (A AND B) = P(A | B) P (B)
Independent Events
If the occurrence of one event has no influence on the occurrence of another, the two events are said to be independent events. For instance, while rolling two dice together, the outcome of the first
dice has no influence on the outcome of the second roll, unless they are attached. In terms of conditional probability this is the same as
P (A) = P (A | B)
This is the same as saying that the probability of A does not change given B has occurred. And equivalently, P(B) = P(B | A)
When two events are independent, we get the multiplication rule
P (A AND B) = P (A) P (B)
We can use this to find the probability of getting a double 6 as the outcome while rolling two dice.
If we take
Event A: Number on the first die Event B: Number on the second die
We can verify this from the table.
In our earlier example where we wanted to find the probability that the sum of dice add up to 9, obviously, the first die showing 6 affects the chances. We can verify this.
Event A: Sum of the dice is 9Event B: 6 shows up on the first die
Since P (A | B) ≠ P (A) the two events are not independent.
Let us see how conditional probabilities can be used to make real-life meaningful decisions.
A Better Speed Gun
From past data, around 5% of the vehicles drive above the speed limit. The traffic police are testing a new speed gun. The new speed gun can correctly identify 90% of all speeding cases. In other
words, if a vehicle is speeding it will be detected 90% of the time. However, this speed gun also incorrectly identifies 20% of non-speeding cases as speeding cases. What is the probability that a
vehicle stopped for a speeding violation using this speed gun was actually speeding?
We have two events to work with
A: The vehicle is speeding
B: The speed gun says that the vehicle is speeding.
For the data about the effectiveness of the speed gun, we can write down the following cases.
P (A) = 5% (Only 5% of the vehicles speed)
P (B | A) = 90% (The speed gun can detect 90% of the speeding cases)
P (B | NOT A) = 20% (The speed gun falsely tags 20% of non-speeding cases as speeding cases)
We need to find P (A | B) The probability that the vehicle was actually speeding given that the speed gun tags the vehicle as speeding.
We start by dividing the sample space into four mutually exclusive events.
The table is called a contingency table (or cross-tabulation) and is very helpful in calculating probabilities. Let us find the probabilities in each case
The totals are found by summing across rows and down the columns. Let us use the multiplication rule for conditional probabilities.
P (A AND B) = P (B | A) P (A) = (90%) (5%) = 4.5%
P (NOT A AND B) = P (B | NOT A) P (NOT A) = 20% * (100 - 5)% = 19%
Filling these into the contingency table.
A cross-tabulation like the above is used to test the effectiveness of classification models. Replacing A with actual values and B with the prediction made by the speed gun, we get what is called a
confusion matrix (or error matrix)
Let us come back to our problem. We can now find the remaining values.
P ( A AND B) = P (A | B) P (B)
4.5% = P (A | B) * 23.5%
Despite the high accuracy of the speed gun, less than 20% of those detected for speeding actually speed. This is called the False-Positive Paradox. Broadly, this describes a case where the false
positives are more than the true positives because these characteristics are relatively rare in the general population. This is one of the reasons why doctors always rely on multiple tests to detect
rare diseases.
Bayes’ Theorem
All the above steps used for computation can be compressed into a single formula called the Bayes Theorem.
This can be simplified by using the multiplication rule.
P (A) P (B | A) = P (A AND B)
P (NOT A) P (B | NOT A) = P (NOT A AND B)
We can also write
P (A AND B) + P (NOT A AND B) = P (B)
Thus Bayes’ Theorem can also be written as
As we saw above, Bayes' Theorem and conditional probabilities are very helpful in finding the probability of an event based on prior knowledge of conditions that might be related to the event. A
prime example is in the field of health care where increasing age can lead to complications. Rather than assessing everyone identically, a more accurate assessment can be arrived at by taking age
into account.
In this article, we looked at the basics of probability and random events. We used these concepts to move from outcomes to events and then explored the various characteristics of probability. We then
used the concept of conditional probability to find independent events and finally used it in a real-life scenario. In the next article, we will expand this understanding to random variables and
probability distributions.
Probability is fundamental to Machine Learning and Data Science and every aspiring Data Scientist should have a solid understanding. While it might look a bit difficult in the beginning because we
are used to certainties in life, it is very intuitive. With a little bit of practice and persistence, one can easily become proficient in understanding various concepts. You apply your Probability
and Statistics skills trying to solve data science interview questions on the StrataScratch platform. Join StrataScratch today and make your dream of joining companies like Netflix, Apple, Microsoft,
Amazon, et al a reality.
Also published here. | {"url":"https://hackernoon.com/statistics-cheat-sheet-a-beginners-guide-to-probability-and-random-events","timestamp":"2024-11-11T16:35:24Z","content_type":"text/html","content_length":"329262","record_id":"<urn:uuid:176ea385-687c-445a-beef-9455a953b513>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00464.warc.gz"} |
Multiply Unit Fractions By Whole Numbers Worksheet
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Multiply Unit Fractions By Whole Numbers Worksheet
Multiply Unit Fractions By Whole Numbers Worksheet -
Multiplying fractions by whole numbers Grade 5 Fractions Worksheet Multiply 1 1 1 6 6 3 7 4 3 8 5 12 of 1 1 1 12 7 3 of 5 1 2
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Multiply Fractions By Whole Numbers Worksheet
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Multiplying Fractions Whole Numbers Worksheets
Multiplying Fractions By Whole Numbers Worksheets Teaching Resources
Check more of Multiply Unit Fractions By Whole Numbers Worksheet below
Fractions Multiplied By Whole Numbers Worksheets
Multiplying Unit Fractions By Whole Numbers Worksheet Download
Multiplying Fractions By Whole Numbers Poster teacher Made
Multiplying Fractions 5th Grade Math Worksheet Greatschools Free Multiplying Fractions With
Multiplying Fractions By Whole Numbers Worksheet
Multiplying Fractions By Whole Numbers Worksheets
Multiplying Fractions By Whole Numbers K5 Learning
Math worksheets Multiplying fractions by whole numbers Below are six versions of our grade 5 math worksheet where students are asked to find the product of whole numbers and proper fractions These
worksheets are pdf files
Multiplying Unit Fractions By Whole Numbers Worksheet
Multiplying Unit Fractions by Whole Numbers Worksheet Download Select a Different Activity One Atta Time Flash Cards Distance Learning Sheet Information Multiplying Unit Fractions by Whole Numbers
Each worksheet has 20 problems multiplying a unit fraction by a whole number Open PDF Customize preview
Math worksheets Multiplying fractions by whole numbers Below are six versions of our grade 5 math worksheet where students are asked to find the product of whole numbers and proper fractions These
worksheets are pdf files
Multiplying Unit Fractions by Whole Numbers Worksheet Download Select a Different Activity One Atta Time Flash Cards Distance Learning Sheet Information Multiplying Unit Fractions by Whole Numbers
Each worksheet has 20 problems multiplying a unit fraction by a whole number Open PDF Customize preview
Multiplying Fractions 5th Grade Math Worksheet Greatschools Free Multiplying Fractions With
Multiplying Unit Fractions By Whole Numbers Worksheet Download
Multiplying Fractions By Whole Numbers Worksheet
Multiplying Fractions By Whole Numbers Worksheets
Fraction Times Whole Number Worksheet
Multiply A Fraction By A Whole Number Worksheet
Multiply A Fraction By A Whole Number Worksheet
IXL Multiply Unit Fractions By Whole Numbers 4th Grade Math Practice | {"url":"https://szukarka.net/multiply-unit-fractions-by-whole-numbers-worksheet","timestamp":"2024-11-03T17:02:34Z","content_type":"text/html","content_length":"27179","record_id":"<urn:uuid:20bf313a-8764-43d8-9b29-1eda654f0753>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00579.warc.gz"} |
Cell Size and Division or How Big Would You Want To Be If You
Were A Cell
Cell Size and Division or How Big Would You Want To Be If You Were A Cell
Nicholas DiGiovanni Naperville Central High School
440 W. Aurora Rd
Naperville IL 60540
1. For all grades: to illustrate the usefulness of models to represent things
which are too small (cells or molecules) or too large in science.
2. For primary grades: to learn to measure with a ruler, to cut a cube, and
determine smaller particles react faster than larger particles.
3. For middle grades: to determine surface area and volume of a cube in addition
to the above.
4. For upper grades: to determine the surface area to volume ratio and relate
this to cell size, to determine why cells divide and 1-3 above.
2500 ml 2% agar solution (sufficient for 15-20 set-ups or pairs)
a cake pan
phenolphthalein powder
1 250 ml beaker or cup
50 ml .4% NaOH solution
a metric ruler, stirrer or spoon, plastic knife, and paper towels
1.Advance Preparation:
Mix enough agar powder in boiling water to make a 2% agar solution. Use enough
water to fill a cake pan to a depth of 3 cm (approximately 2500 ml). Stir until
all the powder is dissolved. As the agar cools, add 1 g of phenolphthalein (if
solid is unavailable, add several ml of liquid phenolphthalein indicator) per
liter of solution and stir thoroughly. If the color is pink, add dilute acid
drop by drop until the solution turns colorless. Pour the mixture into the cake
pan to solidify. This will provide the agar for the model of the cells. If
agar is unavailable, substitute potatoes, but then razor blades must be used and
a dye found which will penetrate the potato in a short time.
2. Discuss models and their importance with the class. In this activity we will
use agar blocks to represent cells.
3. Give the students a 6x3x3 cm block of agar cut from the cake pan, a plastic
knife, and metric ruler. Ask them to cut three separate cubes 1x1x1, 2x2x2, and
3x3x3 cm from the block.
4. Ask the students, "If you were a cell which cell would you rather be (small,
medium, or large) and why?" Write this down.
5. Ask the students to place the cubes into the beaker. Then the teacher pours
the NaOH into the beaker to just cover the cubes. (CAUTION: Sodium hydroxide
is caustic and can burn the skin and eyes.)
6. The cubes should remain in the solution for 10 minutes. They should be
stirred occasionally with the spoon. When the NaOH comes into contact with the
agar blocks, the blocks and perhaps the solution will turn a pink color. The
students enjoy this.
7. Depending on the grade level, students should be given a task to do while the
cubes are "soaking". Primary grades may be asked if this were a cell, what type
of things might move into it. Older students may be asked the same as well as
to explain diffusion since this is what is happening. They should also be asked
to set up a data table in which they determine the surface area, volume, and
surface area to volume ratio for each cube.
8. After 10 minutes the cubes are taken out of the beaker with a spoon and dried
off with a paper towel. The students should cut the cubes in half and measure
the distance from the outer edge inward that has turned pink and record this.
9. Students will discover that the distance that the solution travelled in each
cube is the same (5 mm). There is a pink border around the 2x2x2 cm and 3x3x3
cm cube, but the 1x1x1 cm cube is pink throughout. Ask if the pink represented
food, water or something else needed by the cell to survive, which "cell" got
the needed substance distributed to all its parts. They should see that the
smallest cell is most efficient since it is pink throughout.
10. Mathematically, students should observe that the smallest cube has the
largest surface area to volume ratio (SA:VOL). Therefore this illustrates that
a large SA:VOL promotes better efficiency in moving things into and out of cells
and thus survival. This can also be related to smaller particles reacting
faster than larger particles in chemical reactions (i.e. Granular sugar
dissolves easier than sugar cubes.)
Students can be asked which type cell they think would have a better chance
for survival, one which is 1x1x1 cm or one which is .1x.1x.1 cm. They need to
justify their response. 5 points for a proper mathematical as well as written
explanation. 4 points for an explanation which is a little unclear. 3 points
for a proper explanation but improper or no math. 2 points for an unclear
explanation but shows thought. 1 point for an honest attempt.
Adapted from Biological Sciences: An Ecological Approach. Kendall Hunt. 1987
Return to Biology Index | {"url":"https://smileprogram.info/bi9226.html","timestamp":"2024-11-13T01:56:43Z","content_type":"text/html","content_length":"5692","record_id":"<urn:uuid:8dcc5abc-3e3d-4eaf-9013-4a42a55ff6ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00770.warc.gz"} |
Lower Bounds on Linear Programming formulations for the Travelling Salesman Problem | Department of CSE, IIT Hyderabad
Lower Bounds on Linear Programming formulations for the Travelling Salesman Problem
Seminar talk titled Lower Bounds on Linear Programming formulations for the Travelling Salesman Problem
Title Of the Talk: Lower Bounds on Linear Programming formulations for the Travelling Salesman Problem
Speaker: Dr. Rakesh Venkat, CSE Dept, IITH
Date &Time: Wednesday, 13th May 2020
Venue: Online
Linear Programming (LP) based techniques are a central tool in the design of approximation algorithms for NP-Hard problems. The first step in such approaches involves writing a set of linear
inequalities (using some variables) to capture (i.e. encode) the solutions to the problem instance within the polytope defined by the feasible region. The ideal goal of such a formulation is to get
this polytope to exactly be the convex hull of the encodings of the solutions to the problem.
One might expect, assuming P is not equal to NP, that we can’t achieve this goal using a polynomial-sized LP for NP-hard problems, because optimization over polytopes is possible in time polynomial
in the size of the LP.
However, proving such a result is not easy: how does one go about showing that no LP, however cleverly formulated, can do this? Starting with the work of Yannakakis in 1991, this question has seen
much progress in recent years, and has revealed connections to a number of other domains including communication complexity, information theory, Fourier analysis and quantum computation along the
We will first see a general overview of the results in this area, starting from the basics. I will then present a beautiful (and short) result due to Fiorini et al. that shows that capturing the TSP
problem exactly requires exponential-sized LPs.
Main result of the talk is based on the paper: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds, by Fiorini, Massar, Pokutta, Tiwary and Wolf (winner of
STOC ‘12 Best paper award).
Wednesday, 13th May 2020 14:30 | {"url":"https://cse.iith.ac.in/talks/2020-05-13-Lower-Bounds-on-Linear-Programming-formulations-for-the-Travelling-Salesman-Problem.html","timestamp":"2024-11-08T14:42:10Z","content_type":"text/html","content_length":"14402","record_id":"<urn:uuid:fdaf85e2-43c8-4e6b-9a77-b994c27b92cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00763.warc.gz"} |
Quantum billiards - Sanjeev Seahra
For a recent research project on discrete spacetimes, I have been using finite element methods to solve the Helmholtz equation on various interesting 2-dimensional triangulated manifolds. A little
while ago, I realized that a neat by-product of these calculations is the ability to easily solve the free particle Schrodinger equations in these geometries. In the physics literature, this problem
is sometimes called “quantum billiards” because it is the quantum mechanical analogue of studying the motion of billiard balls (on oddly shaped tables).
Here is an example:
This movie shows the position probability density of a free particle confined to an elliptical cavity (i.e. the modulus squared of the position space wavefunction). Here is a version of the same
movie with the trajectory of a classical particle with the same initial position and velocity as the quantum wavepacket superimposed:
At the initial time, the particle is localized near the centre of the ellipse and has a velocity directed up and to the right. The particle’s wavepacket scatters off the walls of the ellipse several
times. Each collision caused the wavepacket to spread out in space, and, by the end of the movie, the particle is de-localized over most of the ellipse. Interference patterns are formed as portions
of the reflected wavefunction from different collisions interact with one another.
In the above movie, you should be able to see that the ellipse is actually made up of a bunch of small coloured triangles. This is because I am not actually solving the Schrodinger equation within a
continuous elliptical region, I am rather solving for a discrete version of the wave function defined on a triangulation of the ellipse. By making the triangles smaller one gets a better and better
approximation to the continuous case. But the catch is that as the triangles get smaller, the computational time to generate the movies gets longer. The movies on this page are the result of
simulations that take a few hours on my laptop.
Here is another movie in a related geometry, the Bunimovich Stadium:
The Bunimovich Stadium is essentially a rectangle with semi-circular caps. It is interesting because a classical particle contained within it exhibits ergodicity. That is, if you consider a classical
billiard ball in this region with a random initial position and velocity, is trajectory will (almost always) eventually fill up the entire stadium uniformly. The above simulation is possibly hinting
at the quantum analogue of this classical ergodicity, with the final wavefunction configuration being even more dispersed than the elliptical case.
The next example I looked at was meant to be similar to the famous double slit experiment:
In this movie, the particle lives in a circular arena with a triangular obstacle in the middle. The obstacle cleaves the particle’s wavefunction in two, essentially meaning that there is an equal
probability of measuring the particle taking a path above or below the triangle. After the splitting, we can see the development of intricate interference patterns, just like in the double slit
Here is another example of a wavepacket interacting with a 2-dimensional barrier:
In this case, the particle undergoes a glancing collision with a circular obstacle in a hexagonal arena.
Finally, this movie shows a quantum particle confined to the newly discovered aperiodic monotile:
The aperiodic monotile has a remarkable property: it can be used to tile a 2-dimensional plane in a completely non-repeating way. It is pretty unclear to me how the tiling properties of the monotile
might relate to the trajectory of confined quantum particles, but it does make for a pretty movie… | {"url":"https://sanjeev.seahra.ca/2023/05/11/quantum-billiards/","timestamp":"2024-11-10T20:24:34Z","content_type":"text/html","content_length":"84944","record_id":"<urn:uuid:facf6fe0-e8e0-4e57-8a19-bff7bd58d2f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00811.warc.gz"} |
Partial Derivatives
Definition 13.3.3. Partial Derivative.
Let \(z=f(x,y)\) be a continuous function on a set \(S\) in \(\mathbb{R}^2\text{.}\)
1. The partial derivative of \(f\) with respect to \(x\) is:
\begin{equation*} f_x(x,y) = \lim_{h\to 0} \frac{f(x+h,y) - f(x,y)}h\text{.} \end{equation*}
2. The partial derivative of \(f\) with respect to \(y\) is:
\begin{equation*} f_y(x,y) = \lim_{h\to 0} \frac{f(x,y+h) - f(x,y)}h\text{.} \end{equation*} | {"url":"https://opentext.uleth.ca/apex-standard/sec_partial_derivatives.html","timestamp":"2024-11-05T15:51:00Z","content_type":"text/html","content_length":"377764","record_id":"<urn:uuid:2899c92d-7267-41de-af79-ee5517588117>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00130.warc.gz"} |
ACM Other ConferencesProxying Betweenness Centrality Rankings in Temporal Networks
Identifying influential nodes in a network is arguably one of the most important tasks in graph mining and network analysis. A large variety of centrality measures, all aiming at correctly
quantifying a node’s importance in the network, have been formulated in the literature. One of the most cited ones is the betweenness centrality, formally introduced by Freeman (Sociometry, 1977). On
the other hand, researchers have recently been very interested in capturing the dynamic nature of real-world networks by studying temporal graphs, rather than static ones. Clearly, centrality
measures, including the betweenness centrality, have also been extended to temporal graphs. Buß et al. (KDD, 2020) gave algorithms to compute various notions of temporal betweenness centrality,
including the perhaps most natural one - shortest temporal betweenness. Their algorithm computes centrality values of all nodes in time O(n³ T²), where n is the size of the network and T is the total
number of time steps. For real-world networks, which easily contain tens of thousands of nodes, this complexity becomes prohibitive. Thus, it is reasonable to consider proxies for shortest temporal
betweenness rankings that are more efficiently computed, and, therefore, allow for measuring the relative importance of nodes in very large temporal graphs. In this paper, we compare several such
proxies on a diverse set of real-world networks. These proxies can be divided into global and local proxies. The considered global proxies include the exact algorithm for static betweenness (computed
on the underlying graph), prefix foremost temporal betweenness of Buß et al., which is more efficiently computable than shortest temporal betweenness, and the recently introduced approximation
approach of Santoro and Sarpe (WWW, 2022). As all of these global proxies are still expensive to compute on very large networks, we also turn to more efficiently computable local proxies. Here, we
consider temporal versions of the ego-betweenness in the sense of Everett and Borgatti (Social Networks, 2005), standard degree notions, and a novel temporal degree notion termed the pass-through
degree, that we introduce in this paper and which we consider to be one of our main contributions. We show that the pass-through degree, which measures the number of pairs of neighbors of a node that
are temporally connected through it, can be computed in nearly linear time for all nodes in the network and we experimentally observe that it is surprisingly competitive as a proxy for shortest
temporal betweenness. | {"url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.6/metadata/acm-xml","timestamp":"2024-11-02T14:26:01Z","content_type":"application/xml","content_length":"22618","record_id":"<urn:uuid:8ae6bd2a-9d3f-45bc-81fd-88f3db93e42d>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00445.warc.gz"} |
ISU (Infinite Spongy Universe) Model - SciForums Update 2018
1. What is there?
2. And what is it like?
Topics of metaphysical investigation include
existence, objects and their properties, space and time, cause and effect, and possibility
If I understand the definition fully, it also includes; "
becoming", "emergence".
Thus I would add a:
3. What can it
As far as the Universe being infinite and eternal, as well as energetic, I find that concept difficult to process, from what I know about current cosmology.
It begs the question; Eternal Energy? It doesn't answer 1, 2, and 3 , does it?
But does it have to be eternal?
Moreover , eternity nullifies any concept of time. Eternity is timeless and permittive but is not energetic , but rather static.
And is that not contrary to mainstream science, which gives a detailed account of the origins of this (our) universe. If we assume a prior state of the universe, are we not talking about a universe
within a universe?
Perhaps a
If we can argue for such a condition, then of course , everything else becomes relatively simple, but then we are not speaking of origins of the "Wholeness" itself anymore.
What aspect of the grand unified eternal wholeness was causal to the eventual emergence of our current (geometrically bounded) universe? It was energetic? Ok, why and how?
My question is if there is a natural mathematical restriction against something becoming from nothing?
Are we not constantly creating something (thoughts) from nothing? I find that an intriguing question.
And when we use our own subjective universe (our brain) and apply energy to our thoughts, they often become reality, no?
Bohm called it "insight intelligence". A logical field, which mathematically converts abstract potential into a hierarchy of abstract orderings, into abstract patterns, into constructs, the process
of which in itself is abstractly energetically dynamic, into energy.
If we consider the abstract complexity of the physical nature of our universe today. Did any of it exist before the BB? Thus what we see today was once nothing, according to Lawrence Krauss. (today
an empty lot, tomorrow a highrise). Something emerging from Nothing.
It sounds weird, but is it?
Last edited:
very interested in this thread.
though i am only able to process the science at a certain rate.
thus my post here to make sure i note that so its easy to find.
and to the Author so you know you have interested partys and to continue disucssing if so desired
If I understand the definition fully, it also includes; "becoming", "emergence".
Thus I would add a:
3. What can it become?
Good point; Metaphysics could also include what it can become, and look at the observable universe to emphasize your point.
As far as the Universe being infinite and eternal, as well as energetic, I find that concept difficult to process, from what I know about current cosmology.
It begs the question; Eternal Energy? It doesn't answer 1, 2, and 3 , does it?
No, it doesn’t answer those, but Cosmology is not Metaphysics, and is not philosophical to any where near the same extent.
I understand how difficult it is to process the infinite and eternal, but if you look at it from the perspective that if it is not those things, then we are back to arguing about the definition of
nothingness, and no common ground exists between the philosophy that nothingness either has or does not have potential for becoming and emergence. If it does, it is a philosophical position,
metaphysical, and if it doesn’t, it is more of a scientific distinction.
But does it have to be eternal?
Moreover , eternity nullifies any concept of time. Eternity is timeless and permittive but is not energetic , but rather static.
True, according to the Implicate Order, and to what can be conceived philosophically and metaphysically.
But the term "static" on a grand scale, meaning the "static" of a steady state cosmology still accommodates the small scale, which is a finite Hubble view of only the observable portion of the grand
universe, and that limited view is almost nothing, almost nowhere, almost never, relative to the infinite space time and energy, the possibilities that cannot be disregarded.
And is that not contrary to mainstream science, which gives a detailed account of the origins of this (our) universe. If we assume a prior state of the universe, are we not talking about a
universe within a universe?
Perhaps a multiverse?
Perhaps so. Please remember my disclaimer about the difference between known science (developed in accordance with the Scientific Method), and my proclaimed intention to speculate about the “as yet”
unknowns. Mainstream science, and even science that goes beyond the consensus but remains true to the required methodology of the scientific method, including the peer reviewed academic papers,
clearly recognizes a role for speculation as its initial inspiration.
If we can argue for such a condition, then of course , everything else becomes relatively simple, but then we are not speaking of origins of the "Wholeness" itself anymore.
What aspect of the grand unified eternal wholeness was causal to the eventual emergence of our current (geometrically bounded) universe? It was energetic? Ok, why and how?
The argument is that:
1) the universe is not geometrically bounded
2) there was no beginning
3) the extension of the Cosmological Principle, is called (unimaginatively) the Perfect Cosmological Principle.
Here is the difference:
Cosmological principle
In modern physical cosmology, the
cosmological principle
is the notion that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the
universe, and should, therefore, produce no observable irregularities in the large-scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang.
Perfect cosmological principle
perfect cosmological principle
is an extension of the cosmological principle, and states that the universe is homogeneous and isotropic in space
and time
. In this view the universe looks the same everywhere (on the large scale), the same as it always has and always will. The perfect cosmological principle underpins
Steady State theory
and emerging from
chaotic inflation theory
My question is if there is a natural mathematical restriction against something becoming from nothing?
I believe there is such a restriction according to my definition of nothingness.
Are we not constantly creating something (thoughts) from nothing? I find that an intriguing question.
Our thoughts are the doings of our brains and minds; hardly nothing, right?
And when we use our own subjective universe (our brain) and apply energy to our thoughts, they often become reality, no?
Yes, for sure.
Bohm called it "insight intelligence". A logical field, which mathematically converts abstract potential into a hierarchy of abstract orderings, into abstract patterns, into constructs, the
process of which in itself is abstractly energetically dynamic, into energy.
If we consider the abstract complexity of the physical nature of our universe today. Did any of it exist before the BB? Thus what we see today was once nothing, according to Lawrence Krauss.
(today an empty lot, tomorrow a highrise). Something emerging from Nothing.
It sounds weird, but is it?
It isn’t weird, but it is philosophical and not scientific if you employ the scientific method.
[video link see above]
Thank you for the video which I watched. It supports your position, but does legitimately bring up the main arguments of my position; those being that if there was a beginning of the universe, what
was the first cause, and it acknowledges that there is a legitimate debate about the potential for something out of nothing, and that the definition of “nothingness” cannot be the one that I have
been promoting, i.e., it must have a potential for space, time, and energy. Potential for something is not nothingness.
It is that need for a potential that separates the scientific from the metaphysical, isn’t it?
Note: I hope you will respond but it is quite a task to respond to every point, so choose those more important ones and we can get to the rest in due course
Last edited:
It is that need for a potential that separates the scientific from the metaphysical, isn’t it?
That's the way I am framing the question also.
Does infinity itself contain countable dimensions? Could we extrapolate associated dimensional potentials, such as tensors or vectors as enfolded aspects of an infinite large object?
Is a "
metaphysical condition
" necessarily incapable of producing a physical "
" of some kind?
The difficulties [with] infinity depended upon adherence to one definite axiom, namely, that a whole must have more terms than a part...Bertrand Russel
That's the way I am framing the question also.
Does infinity itself contain countable dimensions? Could we extrapolate associated dimensional potentials, such as tensors or vectors as enfolded aspects of an infinite large object?
Is a "metaphysical condition" necessarily incapable of producing a physical "potential" of some kind?
I find myself working on the premise that the infinite and eternal universe features a finite set of invariant natural laws. I'm not done imagining ways for that to be true. One thought is that there
are an infinite number of physical orientations of structured wave energy patterns composed of identical energy quanta, and each particular orientation can determine a different combination of
physical laws that come into play. Can those conditions produce an infinite number of combinations of physical structures emerging from a finite set of physical laws operating on an infinite number
of quanta, or is it unnecessary for the combinations to be infinite as long as the mind is free to imagine infinite possibilities? Now that seems metaphysical to me, lol.
It is that need for a potential that separates the scientific from the metaphysical, isn’t it?
Is a "metaphysical condition" necessarily incapable of producing a physical "potential" of some kind?
This has always been my intuitive perspective. The metaphysical state of potential is the Implicate Order, the enfolded blueprint of that potential which is to become physically unfolded and
In your OP you mentioned a "spongy universe" can you elaborate a little more, please.
I wonder if the concept of
is related to the sponge.
I ask this in view of Renate Loll's proposition that the fabric of the universe itself displays a fractal function.
Causal dynamical triangulation
(abbreviated as
) theorized by
Renate Loll
Jan Ambjørn
Jerzy Jurkiewicz
, and popularized by
Fotini Markopoulou
Lee Smolin
, is an approach to
quantum gravity
that like
loop quantum gravity
background independent
This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the
fabric itself evolves.
and these very interesting presentations about the phenomenon of physical matter.
This is fascinating stuff. Wave Functions, Fractality, and Potential.
That's getting pretty metaphysical, IMO.......
Last edited:
This has always been my intuitive perspective. The metaphysical state of potential is the Implicate Order, the enfolded blueprint of that potential which is to become physically unfolded and
In your OP you mentioned a "spongy universe" can you elaborate a little more, please.
Wade into the thread a little and see if you begin to get how I describe the ISU, but given your involvement with metaphysics, the ISU is dull
The recent posts are being modified from another forum where I am further along with the latest ISU model update, so consider the content here to be on a time delay, except for discussions with
members like you that inspire new content for the model. I still post time delayed content here because over the years I have had many discussions here, which have contributed to the development of
the ISU model, and I remember those days fondly.
I wonder if the concept of fractality is related to the sponge.
Only indirectly; I don’t invoke the fractal universe model because I don’t yet understand its mechanics, and the ISU is about the “how” the universe works from my perspective. The CDT universe goes
beyond the simple three dimensions of space and the single dimension of time, and the ISU invokes those restrictions. Further, the ISU is not a spacetime model because I replace the curvature of
space with what I call the gravitational wave energy density profile of space.
Also, the ISU mechanics are not the same as in Quantum Mechanics either. I develop the ISU more for my personal satisfaction of having a model of the universe that I am an expert on, lol. I use a
methodology discussed earlier (as I recall) called Reasonable and Responsible Step by Step Speculation to fill the gaps between known science and the "as yet" unknown. Therefore, the ISU is
internally consistent and not inconsistent with generally accepted scientific observations and data but it is full of speculations and hypotheses.
I ask this in view of Renate Loll's proposition that the fabric of the universe itself displays a fractal function.
Causal dynamical triangulation
(abbreviated as
) theorized by
Renate Loll
Jan Ambjørn
Jerzy Jurkiewicz
, and popularized by
Fotini Markopoulou
Lee Smolin
, is an approach to
quantum gravity
that like
loop quantum gravity
background independent
This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves.” https://en.wikipedia.org/wiki/
It is interesting, but right away it takes conceptual liberties that I’m not on board with, and that would take me a big investment of time to understand. I’m still bogged down in understanding the
standard model and QFT, lol.
I’ll take a look at the attached videos when I get to my home wifi which doesn’t us my data plan.
Post #48
To continue the time delayed ISU update from The Naked Scientists forum:
Quantum_Wave said:
The initial wave-particles that decay out of the hot dense plasma ball are imparted with separation momentum at this stage.
26) The hot dense-state wave energy environment of each new arena is an expanding ball of gravitational wave fronts, so dense that there is not sufficient separation between them to allow the
presence of individual wave-particles. This is the point in the gravitational wave energy density profile of space where the hot dense ball is like a particle itself; an arena level particle in the
big bang arena landscape of the greater universe. The accumulated crunch is the dense core of the arena particle (the result of the inflowing wave energy from parent arenas), and the outflowing wave
is the big bang, the expanding hot dense arena wave the initiates the new arena. The arena particle at the macro level, and the hint of mass at the quantum level represent nature’s two extremes on
the scale of the presence of mass.
27) Also, the hot dense-state wave energy environment of each new arena is nature’s lowest entropy energy, which represents the restoration of usefulness of wave energy content of the old cold matter
in the parent arenas. However, only a small fraction of the low entropy energy content of the mature arena will take the form of detectible matter (4%). The energy of that goes into the process of
expansion of the new arena uses the lions share of low entropy energy in the form of dark energy (75%) to fuel the expansion. The other low entropy wave energy will take the form of dark matter
(21%), in the form of the gravitational wave convergences in space that are the hints of mass (dark matter) that affect the shape and motion of galaxies in the maturing arena.
28) As maturation of the new arena puts the arena wave of energy to use over billions of years, it will result in a mature galaxy-filled parent arena where the galactic structure is all moving apart.
The separation of the galactic structure in the mature arenas is the result of the conservation of momentum that was imparted to the wave particles that formed within the parent arenas in the early
stages of expansion. The future of the new arena, and the fate of the hot dense-state energy in this new arena will be the same as it was in the parent arenas that preceded it, and in their
“grandparents”, for an eternal heritage of the past.
29) Here is where we discuss the formation of wave-particles in the new arena. It is a process of decay of the dense-state wave energy that starts out at billions of degrees, and cools rapidly as the
force of energy density equalization causes the initial expansion of the hot dense-state energy. The expansion initiates the decay process, and individual “standing wave” patterns of wave energy
separate out into very exotic particles with huge amounts of mass, perhaps that equate to the massive Higgs mechanism and boson, whose mass will in turn will be imparted to more and more stable types
of wave-particles.
30) There is clearly a huge amount of energy in space, and the ISU model supposes that the energy in space is carried by gravitational wave fronts that are traversing all space, to and from all
directions, from a potentially infinite history of the emission of gravitational wave energy from wave particles and objects. That wave energy accounts for the energy in space and makes up the
composition of the gravitational wave energy density profile of space, and is intimately involved in the processes that accompany the preconditions to each big bang, and conditions associated with
the new arena.
31) On that basis, every point in space has gravitational wave energy convergences of multiple wave fronts converging in varying magnitudes, governed by the directionally inflowing gravitational wave
energy that is coming and going in every direction through the energy profile of space. In the ISU model, gravitational wave front convergences each produce a “hint” of mass, and the number of
different wave fronts converging at each point produce a net energy presence at each point in space. These hints of mass form a foundational oscillating wave energy background that assists the motion
of light waves and gravitational wave energy through space, employing the concept that two or more converging (inflowing) gravitational waves will produce an outflowing wave which is referred to as
the “third wave” in the ISU.
32) During the wave-particle formation period, as the hot dense ball of plasma expands and cools, the standing wave patterns become more stable as a result of the now sufficient space into which the
new arena has encroached upon as the arena expands back into the space formerly claimed by the parent arena.
33) The nature of the standing wave patterns, though still in a dynamic expanding environment, are now quantized, meaning that the mass of each wave-particle can be determined by the number of
meaningful gravitational wave convergences within the space now claimed by each individual wave-particle (the particle space).
The force of quantum gravity in the ISU to be continued …
Reply #49
Continuing the force of quantum gravity in the ISU …
34) Within the particle space, meaningful gravitational waves are continually converging across the entire space. The convergences each form a momentary high energy density spot or hint of mass, and
the sum of energy in all of the spots at any instant equals the mass of the wave-particle. That sum, divided by the number of spots at that instant, establishes the energy value of the average
quantum increment within the particle space.
Note: Each convergence, at any given moment during the determination of the value of the quantum, can contain a slightly different amount of energy because there is a time delay between the inflow
period of the spot formation and the completion of the convergence peak. During that time delay, the wave convergence incorporates multiple wave fronts from different directions, which contribute to
the energy peak. Upon reaching the quantum of energy, the peak moment is followed by the emission of the third wave, which is quantum, and which converts the hint of mass at the moment of the peak
value, into a third wave which distributes the accumulated wave energy spherically, to continue the process of quantum action within the particle space; wave energy, to hint of mass, to wave energy
is the sequence of events that is continually occurring throughout the entire particle space.
35) The third wave formation can be depicted as two (or more) quantum waves converging at a point of intersection, and causing a growing overlap space to form around that the point of intersection,
which then emerges and expands spherically as the third wave when a when a quantum of energy is accumulated in the overlap space, as depicted in the following image from a previous thought
36) The point of completion of the energy accumulation, as the energy in the overlap space reaches the peak value of a quantum of energy, can be calculated using the ISU quantum equation (the same
equation used to determine the point at the macro level when two or more converging parent big bang arenas reach critical capacity, just before the collapse/bang):
"tex" \frac{VcapR}{V_R}+\frac{Vcapr}{V_r}+\frac{VcapR}{V_r}+\frac{Vcapr}{V_R}"/tex" = "tex" \frac{1/3\pi H^2(3R-H)}{4/3\pi R^3}+\frac{1/3\pi h^2(3r-h)}{4/3\pi r^3}+\frac{1/3\pi H ^2(3R-H)}{4/3\pi r^
3}+\frac{1/3\pi h^2(3r-h)}{4/3\pi R^3} "/tex"
The force of quantum gravity in the ISU to be continued …
Reply #50
37) It is the ISU sphere-sphere overlap quantum equation, and as the description says, it works with the process of quantum action, at both the macro and micro levels. The diagram, showing the
sphere-sphere intersection has two spheres and each sphere has a cap (called a spherical cap at Wolfram Math ).
VcapR/VR is the volume of cap R divided by the volume of sphere R. That gives you the ratio of the volume of cap R to the volume sphere R (think of it as the percentage of sphere R that is included
in cap R). Follow that same approach for each of the four parts of the left side of the equation, by using the corresponding part of the right side of the equation to do the calculation, and you must
assign values to each element in the diagram to fill into the equation in order to do the calculation. You end up with four percentages. Add the four percentages together to get the sum, and compare
that sum to 100%. When it reaches 100%, you have accumulated one quantum of energy in the overlap space, and that marks the point that the third wave has a quantum of energy emitted spherically from
the overlap space.
Note: The calculations, using the equation, are to determine when, during the course of wave/wave overlap, a new third wave becomes quantum. There is a lot of significance given to the feature of the
ISU model that I call the gravitational wave energy density profile of space, since all of the wave action going on in the density profile are subject to becoming quantum increments the make up
wave-particles at the micro level, or as big crunches that become big bangs at the macro level:
1) The profile consists of nothing but gravitational waves traversing space.
2) Each gravitational wave originates as a spherical third wave emitted by the convergence of two or more “parent” waves.
3) The gravitational wave fronts carry energy across space.
4) Every point in space has a net energy density value caused by the local presence of gravitational wave fronts that are carrying energy past that point from all directions.
5) The net value of the energy density at each point is continually changing because there is a constant inflow of wave fronts to and through each point from the gravitational wave energy density
6) Everything that occupies space is therefore composed of gravitational wave energy of some density value.
7) There are thresholds and limits related to energy density that govern the way those gravitational waves get organized to establish the presence of the things that we observe in space.
8 ) Every object in space has formed there after a big bang event initiates the formation of a new big bang arena, and will be negated into its constituent wave energy when it gets captured in a new
local big crunch.
9) Entropy is defeated, meaning that the progress of how useful energy gets used up is continually advancing (entropy increases) until the cold dead matter of old, aging and maturing arenas gets
renewed into low entropy when a big crunch reaches critical capacity and collapse/bangs, releasing a huge ball of hot dense wave energy.
10) The ISU model features the processes of big bang arena action at the macro level, and quantum action at the micro level, that together orchestrate the continual change from matter to energy and
back to matter across the big bang arena landscape of the greater universe.
38) This image is a revision of the large scale action depicted in an earlier image. It represents a patch of the landscape of the greater universe that shows the macro objects and large scale
structure that is composed of gravitational wave energy, wave-particle by wave-particle. It includes visible arena boundaries to help improve on the earlier version of the image:
39) At the opposite end of the size scale in the ISU, are the tiniest meaningful wave convergences. I have called them hints of mass, or high energy spots at the convergence of gravitational waves,
and when they occur within the space occupied by a wave-particle, they are the quanta that make up the total energy of the particle and account for the mass of the particle.
40) In the following image, the high energy density spots are shown at the center of the wave particle, and the spherically outflowing gravitational wave energy is shown converging with the
directionally inflowing gravitational wave energy arriving from the gravitational wave energy density profile of space:
The cause of quantum gravity in the ISU to be continued …
Reply #50
37) It is the ISU sphere-sphere overlap quantum equation, and as the description says, it works with the process of quantum action, at both the macro and micro levels. The diagram, showing the
sphere-sphere intersection has two spheres and each sphere has a cap (called a spherical cap at Wolfram Math ).
VcapR/VR is the volume of cap R divided by the volume of sphere R. That gives you the ratio of the volume of cap R to the volume sphere R (think of it as the percentage of sphere R that is
included in cap R). Follow that same approach for each of the four parts of the left side of the equation, by using the corresponding part of the right side of the equation to do the calculation,
and you must assign values to each element in the diagram to fill into the equation in order to do the calculation. You end up with four percentages. Add the four percentages together to get the
sum, and compare that sum to 100%. When it reaches 100%, you have accumulated one quantum of energy in the overlap space, and that marks the point that the third wave has a quantum of energy
emitted spherically from the overlap space.
Note: The calculations, using the equation, are to determine when, during the course of wave/wave overlap, a new third wave becomes quantum. There is a lot of significance given to the feature of
the ISU model that I call the gravitational wave energy density profile of space, since all of the wave action going on in the density profile are subject to becoming quantum increments the make
up wave-particles at the micro level, or as big crunches that become big bangs at the macro level:
1) The profile consists of nothing but gravitational waves traversing space.
2) Each gravitational wave originates as a spherical third wave emitted by the convergence of two or more “parent” waves.
3) The gravitational wave fronts carry energy across space.
4) Every point in space has a net energy density value caused by the local presence of gravitational wave fronts that are carrying energy past that point from all directions.
5) The net value of the energy density at each point is continually changing because there is a constant inflow of wave fronts to and through each point from the gravitational wave energy density
6) Everything that occupies space is therefore composed of gravitational wave energy of some density value.
7) There are thresholds and limits related to energy density that govern the way those gravitational waves get organized to establish the presence of the things that we observe in space.
8 ) Every object in space has formed there after a big bang event initiates the formation of a new big bang arena, and will be negated into its constituent wave energy when it gets captured in a
new local big crunch.
9) Entropy is defeated, meaning that the progress of how useful energy gets used up is continually advancing (entropy increases) until the cold dead matter of old, aging and maturing arenas gets
renewed into low entropy when a big crunch reaches critical capacity and collapse/bangs, releasing a huge ball of hot dense wave energy.
10) The ISU model features the processes of big bang arena action at the macro level, and quantum action at the micro level, that together orchestrate the continual change from matter to energy
and back to matter across the big bang arena landscape of the greater universe.
38) This image is a revision of the large scale action depicted in an earlier image. It represents a patch of the landscape of the greater universe that shows the macro objects and large scale
structure that is composed of gravitational wave energy, wave-particle by wave-particle. It includes visible arena boundaries to help improve on the earlier version of the image:
39) At the opposite end of the size scale in the ISU, are the tiniest meaningful wave convergences. I have called them hints of mass, or high energy spots at the convergence of gravitational
waves, and when they occur within the space occupied by a wave-particle, they are the quanta that make up the total energy of the particle and account for the mass of the particle.
40) In the following image, the high energy density spots are shown at the center of the wave particle, and the spherically outflowing gravitational wave energy is shown converging with the
directionally inflowing gravitational wave energy arriving from the gravitational wave energy density profile of space:
The cause of quantum gravity in the ISU to be continued …
Reply #51
50) Directionally inflowing gravitational wave energy from distant particles and objects replaces the spherically outflowing wave energy emitted by the core. The lighter “spots” surrounding the core
space represent the newly forming high energy density “spots” or “hints of mass”. Notice they are depicted to be much more numerous in the direction of the highest inflowing direction of the
gravitational wave energy density profile of space:
51) In this next image I try to depict the movement of the wave-particle core through the background occupied by the gravitational wave energy density profile of space. I have added a semicircle of
six new high energy density spots to image of the wave-particle core in the direction of motion. Each new spot in the image may represent millions of tiny new gravitational wave convergences
occurring in the core space. I have also added a semicircle of light spots that represent locations formerly occupied by high energy density spots whose presence has been replaced in the direction of
Wave-particles and objects move in the direction of the net highest density of the inflowing gravitational wave energy fronts from the gravitational wave energy density profile of space.
That is quantum gravity at work in the ISU model of the cosmology of the universe.
The ISU model version of quantum gravity is about gravitational wave energy, and how such wave energy is emitted, absorbed, and traverses space. Clearly it is hypothetical and speculative, and so I
posted it in the Alternative Theories sub forum. Since the last posting in Dec. 2018, the ISU model has evolved, and since I see the thread is still open for activity, I will post some updates.
The ISU model version of quantum gravity is about gravitational wave energy, and how such wave energy is emitted, absorbed, and traverses space. Clearly it is hypothetical and speculative, and so
I posted it in the Alternative Theories sub forum. Since the last posting in Dec. 2018, the ISU model has evolved, and since I see the thread is still open for activity, I will post some updates.
What is the source ? Of this gravitational wave energy ?
The ISU model version of quantum gravity is about gravitational wave energy, and how such wave energy is emitted, absorbed, and traverses space. Clearly it is hypothetical and speculative, and so
I posted it in the Alternative Theories sub forum. Since the last posting in Dec. 2018, the ISU model has evolved, and since I see the thread is still open for activity, I will post some updates.
Please do. This is very interesting and I should love to learn more about ISU.
The idea is that the source of gravitational wave energy is objects with mass which emit gravitational waves, and they also absorb gravitational wave energy from distant sources. Space would
therefore be filled with gravitational waves coming and going in all directions.
The ISU model version of quantum gravity is about gravitational wave energy, and how such wave energy is emitted, absorbed, and traverses space. Clearly it is hypothetical and speculative, and so
I posted it in the Alternative Theories sub forum. Since the last posting in Dec. 2018, the ISU model has evolved, and since I see the thread is still open for activity, I will post some updates.
Firstly, congrats for at least admitting that your model is purely hypothetical I read back to the few original posts, and find them interesting.
Some points:
The thing is that cosmology, probably more then most branches of science, is about modelling.
That modelling by necessity, must explain what we observe.
It must also explain better then the current incumbent model, by either falsifying a certain aspect, or explaining an aspect the incumbent model does not explain.
If it doesn't do that, then you are pissing into the wind.
Let me also say that any model that can explain or give insight into that first .00000000000000000000000000000000000000000001 seconds [10-45 seconds] would then be superior to the BB, which is the
current best understood, explanatory model we have.
If that were the case, each and every young up and coming cosmologists would be chaffing at the bit, to be the first person to adequately explain this.
There are simple reasons why this epoch cannot be observed and/or explained. We simply lack the tools and technology to view at those depths. This is why string theory and its many derivitives
remains purely hypothetical.
The closest we get to such epochs is with our particle accelerators like the LHC and the RHIC.
We can speculate all we like, re multiple BB's etc, but it remains speculation.
My own gut feeling/speculation is that our universe/space/time evolved from a quantum fluctuation in the quantum foam, and that this "quantum foam" is the "nothing" that our universe/space/time arose
Either that or the universe is infinite in extent.
Perhaps an eventual, observable QGT may shed more light...if such technologies can ever be realized.
The idea is that the source of gravitational wave energy is objects with mass which emit gravitational waves, and they also absorb gravitational wave energy from distant sources. Space would
therefore be filled with gravitational waves coming and going in all directions.
Yes, mostly collisions between really massive objects like BH's Neutron stars etc.
The idea is that the source of gravitational wave energy is objects with mass which emit gravitational waves, and they also absorb gravitational wave energy from distant sources. Space would
therefore be filled with gravitational waves coming and going in all directions.
And they would also interfere with each other .
Yes, mostly collisions between really massive objects like BH's Neutron stars etc.
I understand the reasoning behind those who take the position that the source of gravitational wave energy is mostly massive events, like you say. However, in the ISU model, all objects with mass
emit as well as absorb gravitational waves as part of the process, and so the energy traversing the space between massive objects is continually absorbed and reemitted. One important premise of the
model is that objects move in the direction of the net highest source of gravitation waves arriving from the surrounding space.
paddoboy said:
Yes, mostly collisions between really massive objects like BH's Neutron stars etc.
I understand the reasoning behind those who take the position that the source of gravitational wave energy is mostly massive events, like you say. However, in the ISU model, all objects with mass
emit as well as absorb gravitational waves as part of the process, and so the energy traversing the space between massive objects is continually absorbed and reemitted. One important premise of
the model is that objects move in the direction of the net highest source of gravitation waves arriving from the surrounding space.
Interesting ;
The net highest source of gravitational waves fluctuates .
What is the medium ? Of GW ( gravitational waves ) . | {"url":"https://www.sciforums.com/threads/isu-infinite-spongy-universe-model-sciforums-update-2018.160444/page-3","timestamp":"2024-11-08T19:07:33Z","content_type":"text/html","content_length":"204944","record_id":"<urn:uuid:b092259b-68fc-4ae6-a027-0c0f2e68f800>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00170.warc.gz"} |
Volume Of A Triangular Pyramid: Formula & Real-World Applications - ABC Elearning - Free Practice Questions and Exam Prep
Volume Of A Triangular Pyramid: Formula & Real-World Applications
Triangular pyramids are three-dimensional geometric shapes with a triangular base and three triangular faces that converge at a single point. These pyramids are commonly found in many real-world
structures, such as roofs, shipping containers, and buildings. Calculating the volume of a triangular pyramid is an essential skill for professionals in fields such as architecture, engineering, […]
Updated at April 24, 2023
Triangular pyramids are three-dimensional geometric shapes with a triangular base and three triangular faces that converge at a single point. These pyramids are commonly found in many real-world
structures, such as roofs, shipping containers, and buildings. Calculating the volume of a triangular pyramid is an essential skill for professionals in fields such as architecture, engineering, and
physics. In this blog post, we will explore the formula for finding the volume of a triangular pyramid, its properties, real-world applications, and common misconceptions.
Formula For Calculating The Volume Of A Triangular Pyramid
Explanation Of The Triangular Pyramid Volume Equation
The intricacies of the formula for calculating the volume of a triangular pyramid are multi-faceted and require deep consideration. The formula, which is represented by V = 1/3 × (base area) ×
height, provides a means of calculating the volume of the pyramid, where the base area is the area of the triangle at the base of the pyramid, and the height is the perpendicular distance from the
base to the apex, the point where the triangular faces meet. This formula is applicable to any type of triangular pyramid, whether it is regular or irregular.
Derivation Of The Formula
The derivation of this formula is a subject of much mathematical exploration and can be obtained by utilizing various techniques, such as calculus or by dividing the pyramid into three smaller
pyramids and calculating their volumes. However, it is important to note that the derivation of the formula is not necessary to comprehend how to use it.
Volume Of A Triangular Pyramid Example
To gain an understanding of how to use this formula to calculate the volume of a triangular pyramid, consider the following example. Assume we have an equilateral triangular pyramid with a base side
length of 5 cm and a height of 8 cm. To find the volume of the pyramid, we must first calculate the base area.
The area of an equilateral triangle can be determined using the formula A = √3/4 × s2, where s is the length of a side. By plugging in s = 5 cm, we get A = √3/4 × 52 = 10.83 cm2. Subsequently, we
input the values for the base area and height into the volume formula, yielding V = 1/3 × 10.83 cm2 × 8 cm = 28.76 cm3. In light of this, the volume of the equilateral triangular pyramid is 28.76
cubic centimeters.
To summarize, the formula for calculating the volume of a triangular pyramid, represented by V = 1/3 × (base area) × height, is an all-encompassing approach that can be used to compute the volume of
any type of triangular pyramid, whether it is regular or irregular.
Read more >> How to Calculate the Volume of a Right Circular Cylinder?
Properties of a Triangular Pyramid
Definition of a Regular Triangular Pyramid
A regular triangular pyramid, a three-dimensional object, is defined as a pyramid having a regular triangle as its base and three congruent isosceles triangles as its faces. The apex, situated at the
highest point of the pyramid, is directly above the centroid of the base.
The height of a regular triangular pyramid, the vertical distance from the apex to the base, can be determined by a perpendicular line. The slant height, the distance from any vertex to the centroid
of the opposite side, is a noteworthy attribute of a triangular pyramid. It is worth noting that the regular triangular pyramid is an ideal example of a symmetrical object, as all its edges and faces
are identical.
Relationship between the Volume of a Triangular Pyramid and its Height and Base Area
To calculate the volume of a triangular pyramid, one can employ the formula:
Volume = (1/3) x base area x height
Here, the base area is the area of the triangle that forms the base of the pyramid, and the height is the perpendicular distance from the apex to the base. It is interesting to observe that the
relationship between the volume of a triangular pyramid and its height and base area is directly proportional. An increase in the height of the pyramid leads to a proportional increase in volume.
Similarly, a larger base area of the pyramid indicates a proportionate increase in volume. This relationship seems self-evident because the greater the base area, the more volume is required to fill
the pyramid, while a taller pyramid with more volume necessitates an increase in height.
Why the Formula for the Volume of a Triangular Pyramid Works
The formula for the volume of a triangular pyramid is grounded in the fundamental formula for the volume of a pyramid, which states that:
Volume = (1/3) x base area x height
This formula arises from the realization that a pyramid can be dissected into three identical pyramids of equal height, and the volume of each of these pyramids is one-third of the original pyramid.
The formula for the volume of a triangular pyramid is an extension of this principle, utilizing the area of a triangle as the base area and the perpendicular height from the apex to the base as the
Therefore, the formula for the volume of a triangular pyramid is a natural extension of the more general formula for the volume of a pyramid, providing an in-depth understanding of the essence of the
Read more >> Diameter vs Radius: The Relationship Between Them
Real-World Applications Of The Volume Of A Triangular Pyramid
The volume of a triangular pyramid is a concept that finds widespread use in a multitude of fields, ranging from architecture to physics. Given its importance, it is essential to have a good grasp of
the various real-world applications that the concept can be used for.
Example Problems Involving Triangular Pyramids In Real Life
Consider, for instance, the use of triangular pyramids in the design of roofs or shipping containers. In both cases, knowledge of the volume of a triangular pyramid can help determine the required
amount of material needed to cover the surface, thereby aiding in the optimization of material usage.
Another application that highlights the importance of understanding the volume of a triangular pyramid is the construction of pyramidal tents. These tents, designed with a triangular pyramid shape,
maximize internal space while minimizing weight and material usage, and a knowledge of the volume is essential in the optimization of the design.
Importance Of The Volume Of A Triangular Pyramid In Various Fields
The importance of knowing how to calculate the volume of a triangular pyramid extends beyond just the optimization of material usage. In architecture, for example, an understanding of the volume of a
triangular pyramid can help with the construction of various structures, such as pyramidal roofs and tents. This knowledge also extends to the field of engineering, where the volume of a triangular
pyramid is necessary for the design of various containers, tanks, and pipes.
Moreover, the calculation of the center of mass and moment of inertia of triangular pyramid-shaped objects is of utmost importance in structural analysis and design. Finally, in the field of physics,
the volume of a triangular pyramid is vital in the calculation of buoyant force and fluid dynamics, particularly in determining the volume of objects that are partially submerged in fluids.
Common Misconceptions About The Volume Of A Triangular Pyramid
The volume of a triangular pyramid, like many mathematical concepts, can be subject to common misconceptions. To truly comprehend the nuances of this pyramid’s volume, it is important to understand
some of these misconceptions and their sources. In this section, we will explore some common misconceptions about the volume of a triangular pyramid and how to avoid them.
Misunderstanding Of How To Use The Formula For The Volume Of A Triangular Pyramid
One of the most frequent misconceptions about the volume of a triangular pyramid is the incorrect use of its formula. It is not uncommon for some to assume that the same formula used to find the
volume of a rectangular pyramid can be used for a triangular pyramid, despite the base shape being different. However, this is erroneous, as the formula for the volume of a triangular pyramid is
unique to its structure and varies from that of a rectangular pyramid.
Misconceptions about the Height-Base Area Relationship in Triangular Pyramids
Another misconception that some people have about the volume of a triangular pyramid is that it is solely determined by either its height or base area when in actuality, both factors are equally
important in determining the pyramid’s volume.
Why These Misconceptions Are Incorrect And How To Avoid Them
To properly calculate the volume of a triangular pyramid, it is essential to understand these misconceptions and how to avoid them. To avoid misunderstanding the formula for the volume of a
triangular pyramid, it is important to remember that it is exclusive to this specific pyramid’s shape and cannot be used interchangeably with other types of pyramids. Likewise, to avoid
misconceptions about the height-base area relationship, one should remember that the volume of a triangular pyramid is determined by both factors and not just one.
By gaining a proper understanding of these common misconceptions, individuals can calculate the volume of a triangular pyramid accurately and put this knowledge to use in various real-world
In conclusion, understanding the volume of a triangular pyramid is an essential skill for professionals in various fields. Whether you are an architect, engineer, or physicist, knowing how to
calculate the volume of a triangular pyramid can help you solve real-world problems and design better structures. In this blog post, we have explored the formula for finding the volume of a
triangular pyramid, its properties, real-world applications, and common misconceptions. We hope that this post has been informative and has provided you with a better understanding of the volume of a
triangular pyramid. Remember to practice using the formula and share your own real-world applications! | {"url":"https://passemall.com/volume-of-a-triangular-pyramid/","timestamp":"2024-11-14T05:25:02Z","content_type":"text/html","content_length":"77824","record_id":"<urn:uuid:9bb87a7c-95c3-46e9-8149-b30b42522e92>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00858.warc.gz"} |
Avoiding Common Mistakes with Time Series
January 28th, 2015
A basic mantra in statistics and data science is correlation is not causation, meaning that just because two things appear to be related to each other doesn’t mean that one causes the other. This is
a lesson worth learning.
If you work with data, throughout your career you’ll probably have to re-learn it several times. But you often see the principle demonstrated with a graph like this:
One line is something like a stock market index, and the other is an (almost certainly) unrelated time series like “Number of times Jennifer Lawrence is mentioned in the media.” The lines look
amusingly similar. There is usually a statement like: “Correlation = 0.86”. Recall that a correlation coefficient is between +1 (a perfect linear relationship) and -1 (perfectly inversely related),
with zero meaning no linear relationship at all. 0.86 is a high value, demonstrating that the statistical relationship of the two time series is strong.
The correlation passes a statistical test. This is a great example of mistaking correlation for causality, right? Well, no, not really: it’s actually a time series problem analyzed poorly, and a
mistake that could have been avoided. You never should have seen this correlation in the first place.
The more basic problem is that the author is comparing two trended time series. The rest of this post will explain what that means, why it’s bad, and how you can avoid it fairly simply. If any of
your data involves samples taken over time, and you’re exploring relationships between the series, you’ll want to read on.
Two random series
There are several ways of explaining what’s going wrong. Instead of going into the math right away, let’s look at a more intuitive visual explanation.
To begin with, we’ll create two completely random time series. Each is simply a list of 100 random numbers between -1 and +1, treated as a time series. The first time is 0, then 1, etc., on up to 99.
We’ll call one series Y1 (the Dow-Jones average over time) and the other Y2 (the number of Jennifer Lawrence mentions). Here they are graphed:
There is no point staring at these carefully. They are random. The graphs and your intuition should tell you they are unrelated and uncorrelated. But as a test, the correlation (Pearson’s R) between
Y1 and Y2 is -0.02, which is very close to zero. There is no significant relationship between them. As a second test, we do a linear regression of Y1 on Y2 to see how well Y2 can predict Y1. We get a
Coefficient of Determination (R^2 value) of .08 — also extremely low. Given these tests, anyone should conclude there is no relationship between them.
Adding trend
Now let’s tweak the time series by adding a slight rise to each. Specifically, to each series we simply add points from a slightly sloping line from (0,-3) to (99,+3). This is a rise of 6 across a
span of 100. The sloping line looks like this:
Now we’ll add each point of the sloping line to the corresponding point of Y1 to get a slightly sloping series like this:
We’ll add the same sloping line to Y2:
Now let’s repeat the same tests on these new series. We get surprising results: the correlation coefficient is 0.96 — a very strong unmistakable correlation. If we regress Y on X we get a very strong
R^2 value of 0.92. The probability that this is due to chance is extremely low, about 1.3×10^-54. These results would be enough to convince anyone that Y1 and Y2 are very strongly correlated!
What’s going on? The two time series are no more related than before; we simply added a sloping line (what statisticians call trend). One trended time series regressed against another will often
reveal a strong, but spurious, relationship.
Put another way, we’ve introduced a mutual dependency. By introducing a trend, we’ve made Y1 dependent on X, and Y2 dependent on X as well. In a time series, X is time. Correlating Y1 and Y2 will
uncover their mutual dependence — but the correlation is really just the fact that they’re both dependent on X. In many cases, as with Jennifer Lawrence’s popularity and the stock market index, what
you’re really seeing is that they both increased over time in the period you’re looking at. This is sometimes called secular trend.
The amount of trend determines the effect on correlation. In the example above, we needed to add only a little trend (a slope of 6/100) to change the correlation result from insignificant to highly
significant. But relative to the changes in the time series itself (-1 to +1), the trend was large.
A trended time series is not, of course, a bad thing. When dealing with a time series, you generally want to know whether it’s increasing or decreasing, exhibits significant periodicities or
seasonalities, and so on. But in exploring relationships between two time series, you really want to know whether variations in one series are correlated with variations in another. Trend muddies
these waters and should be removed.
Dealing with trend
There are many tests for detecting trend. What can you do about trend once you find it?
One approach is to model the trend in each time series and use that model to remove it. So if we expected Y1 had a linear trend, we could do linear regression on it and subtract the line (in other
words, replace Y1 with its residuals). Then we’d do that for Y2, then regress them against each other.
There are alternative, non-parametric methods that do not require modeling. One such method for removing trend is called first differences. With first differences, you subtract from each point the
point that came before it:
y'(t) = y(t) – y(t-1)
Another approach is called link relatives. Link relatives are similar, but they divideeach point by the point that came before it:
y'(t) = y(t) / y(t-1)
More examples
Once you’re aware of this effect, you’ll be surprised how often two trended time series are compared, either informally or statistically. Tyler Vigen created a web pagedevoted to spurious
correlations, with over a dozen different graphs. Each graph shows two time series that have similar shapes but are unrelated (even comically irrelevant). The correlation coefficient is given at the
bottom, and it’s usually high.
How many of these relationships survive de-trending? Fortunately, Vigen provides the raw data so we can perform the tests. Some of the correlations drop considerably after de-trending. For example,
here is a graph of US Crude Oil Imports from Venezuela vs Consumption of High Fructose Corn Syrup:
The correlation of these series is 0.88. Now here are the time series after first-differences de-trending:
These time series look much less related, and indeed the correlation drops to 0.24.
A recent blog post from Alex Jones, more tongue-in-cheek, attempts to link his company’s stock price with the number of days he worked at the company. Of course, the number of days worked is simply
the time series: 1, 2, 3, 4, etc. It is a steadily rising line — pure trend! Since his company’s stock price also increased over time, of course he found correlation. In fact, every manipulation of
the two variables he performed was simply another way of quantifying the trend in company price.
Final words
I was first introduced to this problem long ago in a job where I was investigating equipment failures as a function of weather. The data I had were taken over six months, winter into summer. The
equipment failures rose over this period (that’s why I was investigating). Of course, the temperature rose as well. With two trended time series, I found strong correlation. I thought I was onto
something until I started reading more about time series analysis.
Trends occur in many time series. Before exploring relationships between two series, you should attempt to measure and control for trend. But de-trending is not a panacea because not all spurious
correlation are caused by trends. Even after de-trending, two time series can be spuriously correlated. There can remain patterns such as seasonality, periodicity, and autocorrelation. Also, you may
not want to de-trend naively with a method such as first differences if you expect lagged effects.
Any good book on time series analysis should discuss these issues. My go-to text for statistical time series analysis is Quantitative Forecasting Methods by Farnum and Stanton (PWS-KENT, 1989).
Chapter 4 of their book discusses regression over time series, including this issue. | {"url":"http://www.svds.com/avoiding-common-mistakes-with-time-series/","timestamp":"2024-11-07T13:36:33Z","content_type":"text/html","content_length":"50809","record_id":"<urn:uuid:5e1e8947-a58f-40be-b8d5-8f8284f15202>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00683.warc.gz"} |
colimits of nerves
This is related to questions about test categories or the like. I recall seeing some result like this in the book on the homotopy theory of Grothendieck by Maltsiniotis, but my memory may be faulty.
Yes of course, you’re right. $(c \downarrow C)$ is not a subcategory or $C$ for general categories $C$. However, I neglected to mention that in the particular situation that I’m considering, $C$ is a
partially ordered set. In particular, it’s hom-sets have cardinality at most 1. I think that for such categories $C$, $(c \downarrow C)$ is a subcategory of $C$.
I also agree with you that this point doesn’t really matter anyway since we can bypass PC and go straight to $Cat$.
I would have saved confusion if I simply stated things that way to start with. Thanks for helping me clarify my question. I just hope that someone can give me an answer.
Well, it’s not a subcategory by any definition I know of (certainly not injective on objects; it’s faithful but not full), but it seems not to matter since we can just bypass $P C$ and go straight to
$(- \downarrow C): C^{op} \to Cat$. Someone around here may know right away, but I’ll try to give it a think when I get a chance.
Hi Todd,
Thanks for you quick reply.
Your interpretation of $(c \downarrow C)$ is the same as mine. So is your interpretation of the the the nerve we’re taking colimits of.
I think of $(c \downarrow C)$ as a subcategory of $C$ by sending the object $c \rightarrow c_0$ to $c_0$ and the morphism $c \rightarrow c_0 \rightarrow c_1$ to the morphism $c_0 \rightarrow c_1$.
Sorry, I’m having some trouble parsing this. If I have an object $c$ of $C$, which subcategory is $(c \downarrow C)$ supposed to be? Ordinarily I would interpret the notation as denoting the comma
category whose objects are morphisms $c \to d$ where $d$ is an object of $C$, and whose morphisms are commutative triangles with vertex $c$, but that’s not a subcategory of $C$.
As for the next part, am I to interpret the nerve we’re taking the colimit of as this composite:
$C^{op} \stackrel{(- \downarrow C)}{\to} P C \hookrightarrow Cat \stackrel{nerve}{\to} Set^{\Delta^{op}},$
assuming that the first arrow makes sense?
Let $C$ be a category and let $PC$ be the category of subcategories of $C$ . Then $(- \downarrow C) : C^{op} \rightarrow PC$ is a functor and it’s colimit is $C \in PC$.
Is $colimit(nerve(- \downarrow C)) = nerve(C)$ ?
Dan, I’ve thought a little about your question, and I think the answer is ’yes’, and the answer is not hard to see. Let’s see if I have this right:
Your statement about the colimit in $Cat$ of the $(c \downarrow C)$ being isomorphic to $C$ intrigued me – I had never seen that before – but on reflection it was something fairly obvious, in fact
basically the Yoneda lemma in disguise. The objects of $colim_{c: C^{op}} (c \downarrow C)$ are equivalence classes of arrows $c \to d$ where the equivalence $\sim$ is generated by
$(c \stackrel{f}{\to} d) \sim (c' \stackrel{g}{\to} c \stackrel{f}{\to} d)$
and it is immediate that every $g: c \to d$ is equivalent to $1_d: d \to d$; this of course is just a form of the Yoneda lemma.
Now let’s look at your problem, which compares the $nerve(C)$ to the colimit of
$C^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{nerve}{\to} Set^{\Delta^{op}}$
Since colimits in $Set^{\Delta^{op}}$ are computed pointwise, we just have to show the colimit of
$C^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{nerve}{\to} Set^{\Delta^{op}} \stackrel{ev_n}{\to} Set,$
where $ev_n$ is evaluation at an object $n$, agrees with $nerve(C)_n$. This is
$C^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{\hom([n], -)}{\to} Set$
Now an $n$-simplex in the comma category $(c \downarrow C)$, which is an element of this composite, is the same as an $(n+1)$-simplex beginning with the vertex $c$, and the colimit (in $Set$)
consists of equivalence classes of $(n+1)$-simplices where a simplex beginning with $c$ is deemed equivalent to a simplex beginning with $c'$ obtained by pulling back along any $g: c' \to c$. And
again, it is a triviality that each $(n+1)$-simplex
$c \to (d_0 \to \ldots \to d_n)$
is equivalent to
$d_0 \stackrel{1_{d_0}}{\to} (d_0 \to \ldots \to d_n)$
but the collection of such $d_0 \to \ldots \to d_n$ is the same as $nerve(C)_n$. This proves your conjecture.
Edit: By the way, this reminds me of the tangent category stuff that originated at the n-Café in a discussion that included Urs, David Roberts, and me, and which was developed further by
Schreiber-Roberts. I think I recall now remarking on the Yoneda lemma in this connection.
Now that I’ve had a chance to think about this properly, Todd is exactly right. It is also a special case of a general fact about (2-sided) bar constructions. Specifically, the nerve of C is the bar
construction $B(*,C,*)$ (where $*$ denotes the functor constant at a terminal object), while the nerve of $c\downarrow C$ is the bar construction $B(C(c,-),C,*)$. Since colimits of a functor $F:C^
{op}\to D$ are given by tensor products of functors $* \otimes_C F$, and such tensor products come inside a bar construction (since colimits commute with colimits), we have
$\colim^c N(c\downarrow C) = * \otimes_{c\in C} B(C(c,-),C,*) = B(* \otimes_{c\in C} C(c,-), C, *) = B(*,C,*) = N C$
where $*\otimes_{c\in C} C(c,-) = *$ by the co-Yoneda lemma.
Ah, ah, ah – excellent point, Mike. Thanks. | {"url":"https://nforum.ncatlab.org/discussion/1284/colimits-of-nerves/","timestamp":"2024-11-06T07:34:59Z","content_type":"application/xhtml+xml","content_length":"49134","record_id":"<urn:uuid:b4d64d3d-2942-4a8f-a462-bdebf81c1172>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00463.warc.gz"} |
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Truly Stealthy PGP (algorithm)
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Truly Stealthy PGP (algorithm)
• Subject: Truly Stealthy PGP (algorithm)
• From: [email protected] (Eric Hughes)
• Date: Mon, 7 Mar 94 08:34:04 -0800
• In-Reply-To: Hal's message of Sun, 6 Mar 1994 11:22:17 -0800 <[email protected]>
• Sender: [email protected]
>If I understand Eric's general idea, we would keep trying session keys
>under a set of rules which would lead to the desired statistical
>distribution of the encrypted key.
I actually said nothing about how to get the particular distribution
of keys specified, since that was another issue. I was more concerned
with just getting the one result across.
>Here is an algorithm which would work.
It does work, and I'll put down a proof sketch below.
Notation alert:
>Let L be the next power of 256 above the modulus n. Let t be the integer
>part of L/n, so that L = n*t + s with s in [0,n). Call the PGP IDEA session
>key SK, and the encrypted version of that m = SK^e. Now do these steps:
>1) Pick a random SK in [0,n).
This random number in [0,n) is the wrong distribution, but that's OK,
since we'll be throwing some numbers away.
>2) RSA-encrypt it to form m = SK^e mod n.
RSA encryption is a bijection (an 1-1 map). If it were not, there
would be two or more possible decryptions for a given ciphertext.
Therefore RSA encryption is a permutation, and a permutation of
probabilities preserves expected values of functions of the
probability, such as entropy. Since we assume the entropy of the SK
is maximal (probabilistic entropy), therefore the entropy of the m's
is maximal. So the m's have a flat distribution.
(As always, the above statements about bijection hold only if SK is
multiple of one of the divisors of the modulus. But then if you do
find one of those, you've also factored the modulus and thus broken
the key. We assume this doesn't happen, since if it does little of
this matters anyway.)
>3) Choose a random k in [0,t].
>4) Calculate the "stegged" encrypted key as M = m + k*n.
Hal now observes that M is uniformly distributed. This is correct,
and happens because m is in [0,n) and we are adding a multiple of n to
m. This means that each M has a unique represenative as some pair
<m,k>. Since both m and k are independently random (max entropy, flat
distribution), so is M.
>5) if M is not in [0,L) (i.e. if M >= L) then go back to step 1.
>The idea is that once we get M uniform in [0,(t+1)*n) we can make it
>uniform in [0,L) simply by rejecting those candidates which were too high.
What we have here is a Markov chain. We have accepting states and
rejecting/retrying states. Since the probabilities in the chain are
independent of each other and are also time-invariant, the
distribution of final probabilities is the same as the distribution of
normalized accepting probabilities.
In simple terms, you can just retry until you get it right. Since the
probabilities are all the same before, they will all be the same
after, only larger to account for the fact that some possibilities
didn't work.
[re: rejection and retry]
>This will only happen if k=t and m>=s.
That's right, and that means that for m < s you have valid k in
[0,t+1) and for m >= s only for [0,t). If you go back an look at the
entropy expression, you'll see exactly this difference in relative
probability for the two parts of [0,n).
>Now, it seems to me that the worst case for rejection is when n=L-1, in
>which case t=1, s=1, and almost one-half of all initial SK choices will
>be rejected.
Right, but the worst case for rejection is not the same as the worst
case for entropy loss, which occurs at n=L/2+1 and s=t-1, i.e. at the
other end of the spectrum entirely.
>Following Eric's reasoning, this would be an effective loss
>of one bit of key length, from say 1024 to 1023, which is tolerable.
Actually not. The loss of effective key length happens based on the
posterior distribution of the session keys, not on the number of
rejections that happen in the process.
>Using this algorithm with the current Stealth PGP would produce a
>"truly stealthy" version which I think would be indistinguishable from
>random bytes without access to the receiver's private key.
Indeed. Observe, though, that as far as deployment went, this would
require modification to PGP itself for it to be anything like | {"url":"https://cypherpunks.venona.com/date/1994/03/msg00335.html","timestamp":"2024-11-07T13:39:04Z","content_type":"text/html","content_length":"8496","record_id":"<urn:uuid:3b872afa-6e9a-4d0b-822e-7ec9f7b384b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00269.warc.gz"} |
Another attempt at math education - Interactive MathematicsAnother attempt at math education
Another attempt at math education
By Murray Bourne, 05 Jul 2006
Reasoning Mind was developed by a Russian mathematician because "he had a dismal opinion of American education, from kindergarten through high school." (TMC.net)
His method is Web-based and uses artificial intelligence to move the student through the learning. Some things that caught my eye:
• "Learning how to think and reason". Great - too much of mathematics is 'Which formula do I plug it in to?'
• "Failure is not an option". It is a mastery-based system which I think is important. Too many students are moved to the next grade but they are certainly not ready - and this sets up problems
• "Learning is hard work". Hmmmm
Do any of you use it? Is it good?
See the 2 Comments below.
michelle says:
5 Jul 2006 at 3:32 pm [Comment permalink]
hi i know this is stupid but im trying to enter this contest online and i cant figure out the equation. (im really really bad at math) so anyways the equation is
(( (10 + 8 ) x 2 ) - 3 ) + 3
can anybody help me with the answer
Murray says:
6 Jul 2006 at 12:49 am [Comment permalink]
Hi Michelle
I believe there is no such thing as a stupid question - there is only the stupidity of NOT asking questions π
You may wish to go to 2. Laws of Algebra in Interactive Mathematics where you will see examples very similar to the question you are asking. | {"url":"https://www.intmath.com/blog/learn-math/another-attempt-at-math-education-324","timestamp":"2024-11-08T14:56:39Z","content_type":"text/html","content_length":"128548","record_id":"<urn:uuid:190e293b-e72d-44e3-a6c3-2810bca75274>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00790.warc.gz"} |
Continuous compound interest effective annual rate
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compounding scheme at nominal rate of
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21 Feb 2020 With 10%, the continuously compounded effective annual interest rate is 10.517 %. The continuous rate is calculated by raising the number "e" 24 Sep 2019 Continuous compounding is the
process of calculating interest and reinvesting it PV = the present value of the investment; i = the stated interest rate; n = the annually, semiannually, quarterly, monthly, daily and continuously.
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rate calculator uses the formula: i=er Continuously compounded interest is interest that is computed on the initial term deposit with an interest rate of 8% with the interest compounded annually.
The Effective Annual Rate (EAR) is the interest rate that is adjusted for compounding We compare the effects of compounding more than annually, building up to what the annual rate will be if the
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effective rate of interest as something that relates only to the way compounding increases the effect of an annual rate of interest continuous compounding gives the effective annual interest rate for
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