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https://www.classcraftepiced.online/copy-3-of-3rd-grade-math-9 | 1,607,040,130,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141732835.81/warc/CC-MAIN-20201203220448-20201204010448-00519.warc.gz | 553,621,058 | 35,165 | ## 7th Grade Math
### The quests are listed in order of the story. They are designed for standard mastery so each quest covers different topics under each standard. To follow the story, the quests must be completed in order.
7.N.1 Read, write, represent, and compare rational numbers, expressed as integers, fractions, and decimals.
7.N.2 Calculate with integers and rational numbers, with and without positive integer exponents, to solve realworld and mathematical problems; explain the relationship between absolute value of a rational number and the distance of that number from zero.
7.A.1 Understand the concept of proportionality in real-world and mathematical situations, and distinguish between proportional and other relationships.
7.A.2 Recognize proportional relationships in real-world and mathematical situations; represent these and other relationships with tables, verbal descriptions, symbols, and graphs; solve problems involving proportional relationships and interpret results in the original context.
7.A.3 Represent and solve linear equations and inequalities.
7.A.4 Use order of operations and properties of operations to generate equivalent numerical and algebraic expressions containing rational numbers and grouping symbols; evaluate such expressions.
7.GM.1 Develop and understand the concept of surface area and volume of rectangular prisms.
7.GM.2 Determine the area of trapezoids and area and perimeter of composite figures.
7.GM.3 Use reasoning with proportions and ratios to determine measurements, justify formulas, and solve real-world and mathematical problems involving circles and related geometric figures.
7.GM.4 Analyze the effect of dilations, translations, and reflections on the attributes of two-dimensional figures on and off the coordinate plane.
7.D.1 Display and analyze data in a variety of ways.
7.D.2 Calculate probabilities and reason about probabilities using proportions to solve real-world and mathematical problems. | 368 | 1,983 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.8125 | 4 | CC-MAIN-2020-50 | latest | en | 0.901081 |
https://docs.microsoft.com/en-us/workplace-analytics/use/customer-focus | 1,627,981,477,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046154432.2/warc/CC-MAIN-20210803061431-20210803091431-00431.warc.gz | 222,743,289 | 11,151 | # Increase customer focus
Companies that prioritize customer relationships and satisfaction grow revenue faster than competitors. Research shows that companies that prioritize customer relationships and satisfaction grow revenue 2.5 times faster than their competitors.
Each of the behaviors listed show how your organization compares with others based on industry research and your specific organizational data.
## Calculations
The following are the percentage insights, their underlying metrics, and a little about the calculations used for them.
Behavior Percentage insight Metrics Calculations
Optimize time with customers Percentage of employees who spend 8+ hours in external collaboration every week External network size and external collaboration hours and connected people and connected groups Percentage of employees who spend more than 8 hours collaborating with people outside the company. This insight is calculated weekly and averaged over the entire time period.
Promote coaching and development Percentage of employees who have less than 15 minutes of 1:1 time with their managers each week Meeting hours with manager 1:1 The percentage of employees who spend less than 15 minutes of coaching time with their managers each week. To account for different frequencies in coaching, this percentage is calculated monthly and then divided by four to get a weekly average.
The following defines the organizational data shown in the visual behavioral insights.
Behavior Visual insight Definition
Optimize time with customers Distribution of external collaboration Percentage of employees grouped by their weekly external collaboration hours. They are divided into groups of employees who spend zero to four0-4 hours, 4-8four to eight hours, and more than eight8+ hours collaborating externally with people outside the company. These percentages are calculated weekly and averaged over the entire time period.
Promote coaching and development Distribution of monthly 1:1 time with managers Percentage of employees based on their monthly meeting hours with manager 1:1. They are divided into employees who have no 1:1s, have between zero and one hour, and have more than one hour of 1:1s with their manager in a month. These percentages are calculated monthly and averaged over the entire time period.
## Take action
You can select See your insights to see ways you can drive change or simply maintain your organization's focus on customers. Depending on your role, the following are available in addition to the recommendations within Take action.
• Opportunity groups - Lists the groups who are most affected and would benefit the most from these recommended best practices or Plans, which are based on your organizational data and industry research.
• Explore the stats – The following recommendations link to more in-depth data about your organization's management and coaching and external collaboration. In the Take action section for each of the following behaviors, select See your insights > Explore the stats to access them:
Behavior Recommendation Explore the stats
Optimize time with customers Make strategic time investments External collaboration
Promote coaching and development Increase frequency of coaching Management and coaching
• Plans - Opens a new Plan you can set up relating to one or more of the recommendations.
## Best practices
This section describes why each of the following behaviors matter and the top best practices that can help increase customer focus.
### Optimize time with customers
Collaboration with customers and other external contacts enables employees to gain the customer and market knowledge needed for business success. Spending time collaborating with customers helps you better anticipate customer's needs and develop products and services that create real value.
According to Optimizing sales and connecting with customers in the age of big data and machine learning: "More time spent with customers; larger internal networks; and more time spent with managers and senior leadership. These three behaviors persisted regardless of region, territory, or sales role, suggesting that they are foundational ingredients for success.” Ways to support customer focus:
• Use MyAnalytics Important people list for key external contacts, which enables immediate notification of email from them, more efficient responses to their requests, and reminders to schedule time to connect with them.
• Create a shared Teams channel with key customers for direct, informal chats and prompt responses to urgent requests.
For best practices and how to track time with your most important external contacts, see Best practices for customer collaboration.
### Promote coaching and development
Manager one-on-one (1:1) time can improve engagement and job performance, while lack of manager coaching can cause employee disengagement and attrition. According to How to make your one-on-ones with employees more productive: "One-on-ones are one of the most important productivity tools you have as a manager." Ways to keep employees engaged:
• Use Insights to schedule 1:1 time, receive reminders to do so, and follow up on tasks related to direct reports.
• Require managers to schedule recurring 1:1 meetings with their direct reports for 30 minutes at least twice a month and hold them accountable for achieving that goal.
For more best practices and how to develop a 1:1 conversation series, see Best practices for manager coaching. | 981 | 5,490 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2021-31 | latest | en | 0.951719 |
https://beta.geogebra.org/m/F3knWxRc | 1,596,574,220,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439735882.86/warc/CC-MAIN-20200804191142-20200804221142-00037.warc.gz | 226,651,724 | 8,951 | # Difference between a point, a line segment, a ray and a line
Author:
Mathguru
On the screen, we can see a point A in blue color, a line segment BC in black color, a ray DE in green color and a line FG in red color. Move these points A, B, C, D, E, F and G and see how the position of the point, the line segment, the ray and the line gets changed.
1. What difference do you observe between a point, a line segment, a ray and a line? | 117 | 435 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.71875 | 3 | CC-MAIN-2020-34 | latest | en | 0.913223 |
https://www.plati.market/itm/solution-13-4-9-collection-of-kep-oe-1989/2068150?lang=en-US | 1,575,582,504,000,000,000 | text/html | crawl-data/CC-MAIN-2019-51/segments/1575540482284.9/warc/CC-MAIN-20191205213531-20191206001531-00028.warc.gz | 842,288,307 | 12,917 | # Solution 13.4.9 collection of Kep OE 1989
Affiliates: 0,01 \$how to earn
Sold: 2 last one 06.03.2019
Refunds: 0
Content: 13_4_9.png 38,9 kB
Loyalty discount! If the total amount of your purchases from the seller TerMaster more than:
250 \$ the discount is 15% show all discounts 1 \$ the discount is 1%
## Description
13.4.9. Determine the period of free oscillations of vertical load mass m = 80 kg, which is attached to a spring with a stiffness coefficient = 2 kN / m. | 152 | 478 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2019-51 | latest | en | 0.684189 |
https://www.physicsforums.com/threads/ln-si-units.913488/ | 1,680,039,683,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296948871.42/warc/CC-MAIN-20230328201715-20230328231715-00411.warc.gz | 1,007,800,116 | 17,238 | # Ln() & SI Units
• B
Const@ntine
I tried with Google but I couldn't find anything, so here goes: When I "use ln on a quantity" (I don't really know how to phrase it in english, as we just have a verb for it), say, I have n = 0.00149 kg/m*s, and I put it into the ln, so now I have ln(0.00149 kg/m*s) what happens to the SI Units? The result of 0.00149 ~-6.508, but I'm not sure on the kg/m*s. It never came up during HS so I now have to fill a board with the ln of various values of n, and I'm not sure what to do with SI.
Any help is appreciated!
Last edited:
Mentor
2022 Award
The unwritten sign between ##0.00149## and ##\frac{kg}{m \cdot s}## is a multiplication. So ##\ln n \approx -6.51 + \ln kg - \ln m - \ln s## which can hardly be interpreted and thus raises the question: what do you want to express and what's the goal? What should ##\ln n## stand for? If it is only a scaling for some plot, then the units remain as they are, as only the graphic representation of the magnitude of ##n## changes, not the quantity.
Const@ntine
The unwritten sign between ##0.00149## and ##\frac{kg}{m \cdot s}## is a multiplication. So ##\ln n \approx -6.51 + \ln kg - \ln m - \ln s## which can hardly be interpreted and thus raises the question: what do you want to express and what's the goal? What should ##\ln n## stand for? If it is only a scaling for some plot, then the units remain as they are, as only the graphic representation of the magnitude of ##n## changes, not the quantity.
Yeah, the first thing that popped to my mind was the classic ln(a*b) = lna + lnb as well.
In my case n is the viscosity index of a liquid (alcoholic, specifically). It's not used in any formula or anything, we just have to fill this board (it's for Lab), and for each n, we need the ln. I was just curious whether there was some "rule" about such cases.
Normally, you would want to convert to some dimensionless value before you take the logarithm. This could be done by dividing by some arbitrary constant, which you could call n0.
berkeman and fresh_42
Const@ntine
Well, thanks a lot for the help everyone! I appreciate it.
rumborak
FYI, this is actually one way of double checking the validity of your calculations. If you suddenly find yourself taking the square root of for example 5kg, that very often shows something went wrong somewhere.
berkeman and anorlunda
Const@ntine
FYI, this is actually one way of double checking the validity of your calculations. If you suddenly find yourself taking the square root of for example 5kg, that very often shows something went wrong somewhere.
Thanks for the info, I'll keep it in mind! | 698 | 2,630 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2023-14 | latest | en | 0.947222 |
http://www.dictall.com/st/13/45/13457237C51.htm | 1,642,674,923,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320301737.47/warc/CC-MAIN-20220120100127-20220120130127-00234.warc.gz | 83,694,348 | 3,834 | 说明：双击或选中下面任意单词，将显示该词的音标、读音、翻译等；选中中文或多个词，将显示翻译。 您的位置：首页 -> 句库 -> 等价无穷小 1. Seeking Functional Limit by Equivalent Infinitesimal Replacement; 利用“等价无穷小的替换”求函数的极限 2. The Simple Way of Calculating Limit by Using Equivalence Infinit Small; 利用等价无穷小求解极限的简便方法 3. A Research of Equivalent Infinitesimal Substitution Method Applied to Limit Operation; 用"等价无穷小替代法"求极限的研究 4. Limit Calculation by the Equivalent Infinitesimal Replace Methods; 应用等价无穷小量的代换方法求极限 5. This paper gives the equivalent keplacement of infinitesimal algebraic sum, and also gives the theory and application of the equivalent infinitesimal replacement about indefinite forms such as1?∝,0?° ect. 给出无穷小代数和的等价代换，以及“1∞型“00”型未定式等价无穷小代换定理及应用。 6. In this paper, the application of equivalent infinitesimal substitutionon method to limit operation is discussed. 讨论了等价无穷小代换在极限运算中的应用. 7. Probe into Limit Operation of Using Equivalation Infinite Small And L Hospital Theory; 对等价无穷小代换与洛必达法则求极限的探讨 8. On the Generalization of the Method for Calculating Limits with Equivalent Infinitesimal Replacement; 关于等价无穷小替换求极限方法的推广 9. A Supplement to the Method to Replace Equivalent Infinitely Small Quantity to Ask Limit; 关于等价无穷小代换求极限方法的一点补充 10. A Conclusion And Apply Of Theorem That Replace With Infinitesimal Of Equal Value; 等价无穷小代换定理的一个结论及其应用 11. The Ways of Getting Limits with the Equivalent Infinitesimal; 利用等价无穷小求极限方法的一个推广 12. Knack in Using Equivalent Infinitesimal Substitute to Solve Limits; 利用等价无穷小替换求极限的解题技巧 13. The Application of Infinitesimal Equivalence to Getting the Limit of Power-exponential Function; 等价无穷小在求幂指函数极限中的应用 14. Utility of Equivalent Infinitesimal in the Course of Extract Limit of Composite Function; 等价无穷小在求复合函数极限中的应用 15. Application of Seeking Functional Limit by Equivalent Infinitesimal Replacement 浅析“等价无穷小替换”在求函数极限中的应用 16. A Promoted Proposition on Asking the Limit with Equivalence Infinite Small Quantity Substitution Theorem; 关于用等价无穷小量代换定理求极限的一个推广命题 17. Application of Taylor Theorem on Getting Limit with Substitution of Equivalence Infinitesimal; 泰勒公式在用等价无穷小量替换求极限中的应用 18. It is one of the important methods in Calculus to ask limits about replacing the Equivalent Infinitely Small Quantity. 利用等价无穷小代换求极限是微积分学中求极限的一个重要方法之一。 ©2011 dictall.com | 778 | 2,222 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.046875 | 3 | CC-MAIN-2022-05 | latest | en | 0.416937 |
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Astrologers' Community Does the equal house system cusp, start on 1 degree, or 0 degree
Houses & cusps For discussions on houses and house cusps (i.e. planets on angles, house stelliums and so on)
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#1
05-17-2009, 05:53 PM
Niplan Banned Join Date: May 2009 Posts: 865
Does the equal house system cusp, start on 1 degree, or 0 degree
Does the equal house system cusp, start on 1 degree, or 0 degree, so does it go from 1 - 30 deg. or from 0 - 30 deg.
and are aspects measured from the deg. they are on to the next deg. or is it the deg. between them.
Venus is at 29 deg, and say mars is at 29 something else.
Would it need to be 120 degree between them, or would it need to be 120, counting the 29th degrees as 1.
or would you start at the 30th degree being one.
Last edited by Niplan; 05-17-2009 at 05:56 PM.
#2
05-17-2009, 07:44 PM
starlink Senior Member Join Date: Nov 2006 Location: I live in peace Posts: 6,350
Re: Does the equal house system cusp, start on 1 degree, or 0 degree
Niplan, with Equal houses, you calculate a chart and when the Asc. falls at, lets say 17 degrees in a sign, then all other houses start at 17 degrees as well.
You are problably thinking of the Whole House system which start at 0°.01 of a sign.
__________________
ON EVERY MOUNTAIN HEIGHT IS REST
Goethe.
#3
05-17-2009, 08:07 PM
Niplan Banned Join Date: May 2009 Posts: 865
Re: Does the equal house system cusp, start on 1 degree, or 0 degree
I understand that the planet that rules the ascendent is what determins what degree the asc is in, but do the house cusps start on 1 degree, or 0 degree... and as to my other question..?
Last edited by Niplan; 05-17-2009 at 08:12 PM.
#4
05-17-2009, 08:16 PM
katydid Senior Member Join Date: Feb 2009 Posts: 5,810
Re: Does the equal house system cusp, start on 1 degree, or 0 degree
Quote:
Originally Posted by Niplan I understand that the planet that rules the ascendent is what determins what degree the asc is in, but do the house cusps start on 1 degree, or 0 degree... and as to my other question..?
I think you misunderstood what Starlink was saying. Whole Sign house system
is different than Equal House system.
Equal House system starts the asc the same as Placidus or regiomantus, then all houses follow in EQUAL size.
#5
05-17-2009, 10:06 PM
astrologer50 Banned Join Date: Nov 2008 Location: Manchester, UK Posts: 13,695
Re: Does the equal house system cusp, start on 1 degree, or 0 degree
Quote:
Originally Posted by Niplan I understand that the planet that rules the ascendent is what determins what degree the asc is in, but do the house cusps start on 1 degree, or 0 degree... and as to my other question..?
The planet that rules the Asc is called the chart ruler in ANY house system and does not determine what degree an Asc sign is in.
House systems
Lots of people that come into Astrology get their free charts calculated at www.astro.com and the default ‘house system’ used is Placidus and think that’s just the norm and all that there is……..BUT that is just the tip of the iceberg. You can change the default on astro.com in Extended Chart selection to Equal house and a few more.
Throughout the forums but mainly in natal astrology there are two main branches Placidus (unequal size houses) v Equal House (whereby each house is same size) but lots more……. For more information on these go here.
http://en.wikipedia.org/wiki/Equal_house_system#Equal_House
http://www.astrolozy.com/article19.asp
http://www.skyviewzone.com/birthinfoforms2/housesexplained.htm
For further research try here...
http://www.astrologyweekly.com/forum/showthread.php?t=3280&highlight=placidus+equal
http://www.astrologyweekly.com/forum/showthread.php?t=638&highlight=house+system
It's only with study and research will you be able to assess where your planets are deposited and in which houses... thus see which 'glove fits'
How then does an astrologer choose a house system? Well, the individual studying alone is more than likely to use Placidus Houses. The reason for this is simple - he has to consult an ephemeris in any case, and Raphael's Ephemeris is the most widely used, which gives the information needed for Placidus Houses.
Students who study with a recognised school are usually introduced to all of the house systems, but taught to use one far more than the others. This is usually the Equal House system, which incidentally is also the oldest one. In this house system, the twelve divisions are very much like spokes of a wheel, equally spaced at 30 degree intervals, with all houses being the same size. This is the easiest of the House systems to use, as it requires no further calculation. Once the Ascendant is known, one simply divides the rest of the chart using the Ascending degree as a starting point - so if the Ascendant is at 22 degrees Leo, this is take as the cusp of the first house, with the second house beginning at 22 degrees Virgo, the third at 22 degrees Libra and so forth.
The Equal House system is conceptually valid within today's astrological standpoint that every individual is free to become what their birth chart symbolises as their ultimate talent. The Ascendant has been shown to correspond to the way the person automatically approaches their environment - the 'persona' in Jungian terms. This person is therefore likely to approach every field of life in a specific way. A person with an Aquarius Ascendant, for example, will approach money-making and material values (2nd house) in a Pisces manner, will learn (3rd house) in an Aries manner, deal with family (4th house) in a Taurus manner and so forth.
The biggest criticism of the Equal House system concerns the position of the MC, which, using this system is more often than not the cusp of the tenth house (or any house) but rather is found within the 9th, 10th or 11th house.
The MC, being the highest point at birth, symbolises the aims and ambitions one works towards, and, by extension, one's career potential and public image. But these areas are also 10th house matters, devised, because of that house's association with Saturn, to show precisely these areas of life. It is therefore conceptually necessary (so goes the argument) that the MC be the cusp of the 10th house. For this reason, the Equal House system has a limited following outside the UK, although it is still the commonest House system within the UK.
http://www.astrolozy.com/article19.asp
http://www.aquamoonlight.co.uk/systems.html
There is a member here on AW who uses Whole sign think Joseph Ledzion if you would to send PM
Last edited by astrologer50; 05-17-2009 at 10:08 PM.
#6
05-17-2009, 11:34 PM
Niplan Banned Join Date: May 2009 Posts: 865
Re: Does the equal house system cusp, start on 1 degree, or 0 degree
thanks i think i unraveled it.
Tags cusp, degree, equal, house, start, system
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Copyright © 2005-2018, AstrologyWeekly.com. Boards' structure and all posts are property of AstrologyWeekly.com and their respective creators. No part of the messages sent on these boards may be copied without their owners' explicit consent. | 2,319 | 9,406 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.125 | 3 | CC-MAIN-2019-51 | latest | en | 0.90472 |
https://coffsoutlook.com/qa/quick-answer-what-is-the-symbol-for-angle.html | 1,618,486,094,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618038084765.46/warc/CC-MAIN-20210415095505-20210415125505-00499.warc.gz | 261,252,581 | 8,813 | # Quick Answer: What Is The Symbol For Angle?
## What is the symbol for a line?
Geometric SymbolInterpretationcapital letterPoint↔LineLine Segment→ or ←Ray6 more rows.
## What are the 7 types of angles?
The different types of angles based on their measurements are: Acute Angle – An angle less than 90 degrees. Right Angle – An angle that is exactly 90 degrees….Types of Angles – Acute, Right, Obtuse, Straight and Reflex…Acute angle.Right angle.Obtuse angle.Straight angle.Reflex angle.
## Why right angle is called right angle?
In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn. … The term is a calque of Latin angulus rectus; here rectus means “upright”, referring to the vertical perpendicular to a horizontal base line.
## How do you introduce a right angle?
To measure a right angle, students must line up the axis of the protractor with the apex of the two connecting lines of an angle. If the measurement equals 90° then this is a right angle.
## What is the symbol for acute angle?
Angle SymbolsAngle SymbolAngle NameDecimal⦝Measured Right Angle With Dot⦝⦞Angle With S Inside⦞⦟Acute Angle⦟⦠Spherical Angle Opening Left⦠16 more rows
## What is the symbol for a right angle?
symbol ∟A right angle is represented by the symbol ∟.
## What is Angle ABC?
Angle ABC is a straight angle, or 180°. Angle f, g, and h together must add to 180°. Try again. Angle ABC is a straight angle, or 180°.
## What does this symbol mean ≅?
The symbol ≅ is officially defined as U+2245 ≅ APPROXIMATELY EQUAL TO. It may refer to: Approximate equality.
## What is a zero angle?
An angle with a measure of zero degrees is called a zero angle. If this is hard to visualize, consider two rays that form some angle greater than zero degrees, like the rays in the . Then picture one of the rays rotating toward the other ray until they both lie in the same line.
## What type of angle is 20?
acute angleSince 20° is less than 90°, the type of the angle acute angle. Example 2: The sum of three angles is (x+1), (x -1) and (x + 3) forms a right angle.
## How do you find the measure of an angle?
The best way to measure an angle is to use a protractor. To do this, you’ll start by lining up one ray along the 0-degree line on the protractor. Then, line up the vertex with the midpoint of the protractor. Follow the second ray to determine the angle’s measurement to the nearest degree.
## What does ∆ mean in math?
A (usually small) change in value. Often shown using the “delta symbol”: Δ Example: Δx means “the change in the value of x” When we do simple counting the increment is 1, like this: 1,2,3,4,…
## What is the symbol for parallel lines?
The symbol for parallel lines is ∥, so we can say that A B ↔ ∥ C D ↔ \overleftrightarrow{AB}\parallel\overleftrightarrow{CD} AB ∥CD in that figure. According to the axioms of Euclidean geometry, a line is not parallel to itself, since it intersects itself infinitely often.
## How do you type an angle symbol?
Click the “Start” button on your desktop and type “Character Map” in the search field. Click “Character Map” in the search results to open the utility.Select “Symbol” in the Font list.Locate the angle symbol in the symbol list. … Paste the angle symbol in your document.
## How do you draw an accurate angle in Word?
On the Insert tab, click Shapes. In the Lines section, select the first line (the plain line). While holding down SHIFT, Click and drag to create a line. Holding the shift makes the line stay exactly horizontal or vertical or at a 45-degree angle to vertical.
## How do you write angles?
(1) We can name angles by using THREE capital letters like: ABC or DEF. The middle letter is called the VERTEX of the angle. The above angles are read “angle ABC” and “angle DEF.” This leads us to the second way we can name angles. (2) We can name angles by using the vertex.
## How do you make the angle symbol on a keyboard?
Use your numeric keypad with your NUM LOCK on and you will be good to go!…More Windows Keyboard Shortcuts for Symbols.SymbolDescriptionShortcut∞infinityALT+8734∠angleALT+8736∨logical orALT+8744∩intersectionALT+874536 more rows•Apr 27, 2010
## Why is it called right angle?
Right, meaning “correct”, and right, meaning “straight”, do have the same root, but “right angle” derives from the second rather than the first. A right angle was described in ancient geometry as the meeting of two right, ie straight, lines, with regard to dimensional axes. | 1,117 | 4,516 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.796875 | 4 | CC-MAIN-2021-17 | longest | en | 0.857491 |
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1-Newbie
## operators in relations.
Hi gang,
Does anyone know how to convert a real number to an integer in a relation?
Something like:
Or do I need to create a new parameter of integer type and set them equal.?
Thanks
-Nate
10 REPLIES 10
23-Emerald IV
(To:nrollins)
If you set one parameter equal to another, the second will change type to match the first. If you have an integer parameter and set it equal to a real parameter, the integer parameter will change type to a real. (This is how you can change parameter types after they are created without recreating them.)
To convert a real to an integer, you will need to use either the FLOOR, CEIL, functions. If you want to choose whether to round up or down automatically, you will need to write a conditional statement to pick which one to use based on your starting value.
Tom U.
11-Garnet
(To:nrollins)
Here's an example of using floor and ceil to round off a number. You ought to be able to adapt it to your needs.
/* thickness = od - id divided by 2
thickness=(d15-d14)/2
/* x1= step diam before rounding
x1=d14+thickness
/* shift moves the decimal point the number of places you want to round to
shift=x1*1000
/* Interger removes the numbers to the right of the decimal
Interger=floor(shift)
/* rounder is just the numbers to the right of the decimal
rounder=shift-interger
/* the following if/else statement compares the rounder to .5
/* if the rounder is equal to or larger than .5 it rounds up. If
/* the rounder is smaller than .5 it rounds down.
If rounder<.5
rounded=floor(shift)
else
rounded=ceil(shift)
endif
/* reshift moves the decimal point back to the correct place
reshift=rounded/1000
/* d13 = the rounded step diam minus the step clearance
d13=reshift-0.05
David Haigh
1-Newbie
(To:nrollins)
Thanks everyone!
ceil() the smallest integer not less than the real value
floor() the largest integer not greater than the real value
And I also got a great sample to use as a template.
And I never had to open a browser and go to the forum! Love this exploder!
-Nate
6-Contributor
(To:nrollins)
The simple solution is to use:
floor(x+.5).
This will round x up/down according to the basic rules.
/Bjarne
21-Topaz II
(To:nrollins)
A simpler method to round to an integer is this:
Rounded_parameter = floor (parameter + 0.5)
By adding 0.5, if the value is closer to the lower integer, floor will
still round down, if it's closer to the upper integer, adding 0.5 will
bump it past the upper integer so floor will essentially round up.
Doug Schaefer
--
Doug Schaefer | Experienced Mechanical Design Engineer
21-Topaz II
(To:nrollins)
Should have real all my mail first, Bjarne beat me to it!
Doug Schaefer
--
Doug Schaefer | Experienced Mechanical Design Engineer
11-Garnet
(To:nrollins)
I think you're missing the point of what I was trying to do in the relation example.
Say your math ends up with a number like this 123.15963 and you want to round off to 3 places.
123,15963 x 1000 = 123159.63
123159.63 + .5 = 123160.13
Floor of 123160.13 = 123160
123160 /1000 = 123.160
Or you could do
Shift=(x1*1000)+.5
Rounded=floor(shift)/1000
Floor(x+.5) isn't letting you choose how many places you want to round off to.
Or perhaps I'm missing your point?
David Haigh
21-Topaz II
(To:nrollins)
Ah yes, excellent point. If all you want to do is round to the nearest
integer (which I think was the original question), the simple one line
solution works. If you need to round to a specific # of places, then
Doug Schaefer
--
Doug Schaefer | Experienced Mechanical Design Engineer
2-Guest
(To:nrollins)
"An integer is any number which can be either positive and negative but not a fractional number. It is also a whole number. Examples are -1,256, -589, -1, 0, 1, 569, 5,236. It is always a rational number."
But the whole interchange was educational regardless of the original question ..... thanks gentlemen.
Don
1-Newbie
(To:nrollins)
Try this:
L3 = floor(1000*L) - (floor(L)*1000)
if L3 == 0
L4 = "000"
else
L4 = itos(L3)
endif
L2 = itos(floor(L)) + "." + L4 + " "
SIZE = BASIC_DIA + NO_THDS + " x " + L2
[cid:image001.jpg@01CD08DE.56D51720]
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Announcements | 1,280 | 4,845 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.3125 | 3 | CC-MAIN-2024-38 | latest | en | 0.788189 |
http://intranet.math.vt.edu/netmaps/hurwitz/m4n2/m4n2hurwitz80_Main.output | 1,596,763,639,000,000,000 | text/plain | crawl-data/CC-MAIN-2020-34/segments/1596439737050.56/warc/CC-MAIN-20200807000315-20200807030315-00235.warc.gz | 58,116,932 | 1,222 | These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 8. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0,lambda2} {lambda1,lambda1+lambda2} Since no Thurston multiplier is 1, this modular group Hurwitz class contains only finitely many Thurston equivalence classes. The number of pure modular group Hurwitz classes in this modular group Hurwitz class is 6. The pullback map is constant: every curve has a trivial preimage. The image of the pullback map lies in the geodesic (0/1,2/1). Every NET map in this modular group Hurwitz class is rational. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=<1,c^-1,1,1,1,c,1,1>(2,6)(3,7)", "b=(1,7)(2,8)(3,5)(4,6)", "c=(1,2)(3,4)(5,6)(7,8)", "d=(1,8)(2,3)(4,5)(6,7)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,3)(2,8)(4,6)(5,7)", "b=<1,1,b,c^-1,1,1,b^-1,c>(3,7)(4,8)", "c=<1,1,b,b^-1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)", "d=(1,8)(2,3)(4,5)(6,7)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,4)(2,7)(3,6)(5,8)", "b=(1,2)(3,4)(5,6)(7,8)", "c=(1,7)(2,8)(3,5)(4,6)", "d=(1,5)(4,8)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,4)(2,7)(3,6)(5,8)", "b=<1,1,b,b^-1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)", "c=<1,1,b,c^-1,1,1,b^-1,c>(3,7)(4,8)", "d=(1,7)(2,4)(3,5)(6,8)", "a*b*c*d"); | 614 | 1,482 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2020-34 | latest | en | 0.63899 |
https://christoph.ruegg.name/blog/handling-floating-point-numbers | 1,723,697,826,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641151918.94/warc/CC-MAIN-20240815044119-20240815074119-00525.warc.gz | 130,164,788 | 7,800 | Handling Floating Point Numbers | Christoph Rüegg
Did you know, that:
• a double can represent 0xFFDFFFFFFFFFFFFF = 18437736874454810623 distinct numbers?
• there are 450359962736 doubles between 1.0001 and 1.0002?
• 54975581 between 10'000.0001 and 10'000.0002?
• 52 between 10'000'000'000.0001 and 10'000'000'000.0002?
• the smallest difference between neighbor numbers is 4.941E-324 for 0, and the biggest difference 1.996E+292 for double.MaxValue?
• the difference between 1.0 and the biggest number smaller than 1.0 is 1.110E-16
(this value is often called negative epsilon (2 = the (positive) epsilon), but not in .Net!* In the framework, double.Epsilon is defined as 4.941E-324, the number from above!)
It's common knowledge that floating point numbers are somewhat tricky to work with, especially if rounding-off errors and exact values become important. Although this is (hopefully) discussed in any computer science and engineering study course, there has been quite some blogging about it lately:
One of the conclusions is that it's dangerous to compare floating point numbers directly. Of course this is a problem in Math.NET Iridium, so I tried to add some helper functions to Iridium for handling double numbers. You'll find them in the new static Number class directly in the MathNet.Numerics namespace. The most important function, AlmostEquals, is inspired by Bruce Dawson's article (see above). They're propositions, so please let me know if you think they're flawed...
## Integer Representation of Doubles
In order to work with doubles in an exact way, we have to treat them as integers. We can convert them either by doing some unsafe pointer casting, by emulating a union (StructLayout and FieldOffset attributes), or simply by using the BitConverter (which uses the first pointer approach internally; System.Runtime.InteropServices namespace).
Unfortunately, floating point numbers are stored as signed magnitudes, while integers use the two's complement to represent negative numbers. We thus have to convert negative numbers if we need them to be lexicographically ordered (such that the distance from 0 to 1 equals to the distance from -1 to 0, and that -0 = +0):
1: 2: 3: 4: 5: 6: public static long SignedMagnitudeToTwosComplementInt64(long value) { return (value >= 0) ? value : (long)(0x8000000000000000 - (ulong)value); }
1: 2: 3: 4: 5: 6: 7: 8: public static ulong ToLexicographicalOrderedUInt64(double value) { long signed64 = BitConverter.DoubleToInt64Bits(value); ulong unsigned64 = (ulong)signed64; return (signed64 >= 0) ? unsigned64 : 0x8000000000000000 - unsigned64; }
The expected behavior (excerpt from the unit tests):
1: 2: 3: 4: 5: 6: 7: 8: 9: ToLexicographicalOrderedUInt64(2 * double.Epsilon) == 2 ToLexicographicalOrderedUInt64(1 * double.Epsilon) == 1 ToLexicographicalOrderedUInt64(0.0) == 0 ToLexicographicalOrderedUInt64(-1 * double.Epsilon) == 0xFFFFFFFFFFFFFFFF SignedMagnitudeToTwosComplementInt64(1) == 1 SignedMagnitudeToTwosComplementInt64(0) == 0 SignedMagnitudeToTwosComplementInt64(-9223372036854775808) == 0 SignedMagnitudeToTwosComplementInt64(-9223372036854775808 + 1) == -1
## Incrementing Doubles
Thanks to this two's complement integer representation we immediately have the power to increment or decrement floating point numbers, and iterate through all numbers a double can represent in a given range:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: public static double Increment(double value) { if(double.IsInfinity(value) || double.IsNaN(value)) return value; long signed64 = BitConverter.DoubleToInt64Bits(value); if(signed64 < 0) signed64--; else signed64++; if(signed64 == -9223372036854775808) // = "-0", make it "+0" return 0; value = BitConverter.Int64BitsToDouble(signed64); return double.IsNaN(value) ? double.NaN : value; }
Again an excerpt of the expected behavior:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: double x = double.Epsilon; x = Decrement(x); --> x == 0 x = Decrement(x); --> x == -double.Epsilon x = Increment(x); --> x == 0 x = Increment(x); x = Increment(x); --> x == 2 * double.Epsilon x = double.MaxValue; x = Decrement(x); --> x < double.MaxValue x = Increment(x); --> x == double.MaxValue
## Evaluating the Precision: Epsilon
The incrementation step depends on the exponent, so it may differ between different numbers. It is important for example in iterative approximation algorithms to know when to stop. The following function returns the decrementation step near a given number, often referred to as a scaled negative epsilon:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: public static double EpsilonOf(double value) { if(double.IsInfinity(value) || double.IsNaN(value)) return double.NaN; long signed64 = BitConverter.DoubleToInt64Bits(value); if(signed64 == 0) { signed64++; return BitConverter.Int64BitsToDouble(signed64) - value; } else if(signed64-- < 0) return BitConverter.Int64BitsToDouble(signed64) - value; else return value - BitConverter.Int64BitsToDouble(signed64); }
And some examples:
1: 2: 3: 4: 5: EpsilonOf(1.0).ToString() == "1.11022302462516E-16" EpsilonOf(0.0).ToString() == "4.94065645841247E-324" EpsilonOf(-1.0e+100).ToString() == "1.94266889222573E+84" EpsilonOf(1.0e-100).ToString() == "1.26897091865782E-116" EpsilonOf(double.MinValue).ToString() == "1.99584030953472E+292"
## Numbers between two doubles
It may be interesting to know how many numbers can be represented by a double variable, which are all between two given numbers. Using the methods introduced above this is simple:
1: 2: 3: 4: 5: 6: 7: 8: 9: public static ulong NumbersBetween(double a, double b) { // left out (see repository): handling NaN and Infinity ulong ua = ToLexicographicalOrderedUInt64(a); ulong ub = ToLexicographicalOrderedUInt64(b); return (a >= b) ? ua - ub : ub - ua; }
Examples:
1: 2: 3: 4: 5: 6: 7: 8: NumbersBetween(1.0, 1.0) == 0 NumbersBetween(0, double.Epsilon) == 1 NumbersBetween(-double.Epsilon, 2 * double.Epsilon) == 3 double test = Math.PI * 1e+150; NumbersBetween(test, test + 10 * Number.EpsilonOf(test) == 10 NumbersBetween(1.0001, 1.0002) == 450359962737 NumbersBetween(10000000000.0001, 10000000000.0002) == 53 NumbersBetween(double.MinValue, double.MaxValue) == 18437736874454810622
## Compare doubles for (almost) equality
Now we come to the final but most important method, for a safe and more sensible way of comparing two floating point numbers for almost equality. Call this method with two doubles and a parameter specifying how many numbers may be between the two numbers at most (+1):
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: public static bool AlmostEqual(double a, double b, int maxNumbersBetween) { // left out (see repository): argument checks // NaN's should never equal to anything if(double.IsNaN(a) || double.IsNaN(b)) //(a != a || b != b) return false; if(a == b) return true; // false, if only one of them is infinity or they differ on the infinity sign if(double.IsInfinity(a) || double.IsInfinity(b)) return false; ulong between = NumbersBetween(a, b); return between <= (uint)maxNumbersBetween; }
Examples:
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: Convert.ToDouble("3.170404") == 3.170404 --> TRUE Convert.ToDouble("4.170404") == 4.170404 --> FALSE Number.AlmostEqual(Convert.ToDouble("3.170404"), 3.170404, 0) --> TRUE Number.AlmostEqual(Convert.ToDouble("4.170404"), 4.170404, 0) --> FALSE Number.AlmostEqual(Convert.ToDouble("4.170404"), 4.170404, 1) --> TRUE Number.AlmostEqual(0.0, 0.0 + double.Epsilon, 0) --> FALSE Number.AlmostEqual(0.0, 0.0 + double.Epsilon, 1) --> TRUE double max = double.MaxValue; Number.AlmostEqual(max, max - 2 * Number.EpsilonOf(max), 0) --> FALSE Number.AlmostEqual(max, max - 2 * Number.EpsilonOf(max), 1) --> FALSE Number.AlmostEqual(max, max - 2 * Number.EpsilonOf(max), 2) --> TRUE
Multiple items
val double : value:'T -> double (requires member op_Explicit)
Full name: Microsoft.FSharp.Core.ExtraTopLevelOperators.double
--------------------
type double = System.Double
Full name: Microsoft.FSharp.Core.double
val max : e1:'T -> e2:'T -> 'T (requires comparison)
Full name: Microsoft.FSharp.Core.Operators.max | 2,402 | 8,212 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.65625 | 4 | CC-MAIN-2024-33 | latest | en | 0.913069 |
https://drawtone.com/toronto-weather-dlmn/637569-newcastle-united-fifa-21-ratings | 1,656,367,314,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656103341778.23/warc/CC-MAIN-20220627195131-20220627225131-00182.warc.gz | 267,940,185 | 18,584 | # newcastle united fifa 21 ratings
Number of parallel edges: 0. The number of weakly connected components is . "A fully connected network is a communication network in which each of the nodes is connected to each other. A 1-connected graph is called connected; a 2-connected graph is called biconnected. That's $\binom{n}{2}$, which is equal to [math]\frac{1}{2}n(n - ⦠For example, two nodes could be connected by a single edge in this graph, but the shortest path between them could be 5 hops through even degree nodes (not shown here). path_graph (4) >>> G. add_edge (5, 6) >>> graphs = list (nx. $\frac{n(n-1)}{2} = \binom{n}{2}$ is the number of ways to choose 2 unordered items from n distinct items. Notation and Deï¬nitions A graph is a set of N nodes connected via a set of edges. In networkX we can use the function is_connected(G) to check if a graph is connected: nx. Approach: For Undirected Graph â It will be a spanning tree (read about spanning tree) where all the nodes are connected with no cycles and adding one more edge will form a cycle.In the spanning tree, there are V-1 edges. a fully-connected graph). Send. Removing any additional edge will not make it so. In order to determine which processes can share resources, we partition the connectivity graph into a number of cliques where a clique is defined as a fully connected subgraph that has an edge between all pairs of vertices. Connectedness: Each is fully connected. The number of connected components is . ij 2Rn is an edge score and nis the number of bonds in B. Complete graph A graph in which any pair of nodes are connected (Fig. 12 + 2n â 6 = 42. Examples >>> G = nx. In a fully connected graph the number of edges is O(N²) where N is the number of nodes. Prerequisite: Basic visualization technique for a Graph In the previous article, we have leaned about the basics of Networkx module and how to create an undirected graph.Note that Networkx module easily outputs the various Graph parameters easily, as shown below with an example. ï¬nd a DFS forest). What do you think about the site? >>> Gc = max (nx. When a connected graph can be drawn without any edges crossing, it is called planar. Parameters: nbunch (single node, container, or all nodes (default= all nodes)) â The view will only report edges incident to these nodes. We will introduce a more sophisticated beam search strategy for edge type selection that leads to better results. 15.2.2A). Approach: For a Strongly Connected Graph, each vertex must have an in-degree and an out-degree of at least 1.Therefore, in order to make a graph strongly connected, each vertex must have an incoming edge and an outgoing edge. Adjacency Matrix. Notice that the thing we are proving for all $$n$$ is itself a universally quantified statement. We will have some number of con-nected components. ðð(ððâ1) 2. edges. Use these connected components as nodes in a new graph G*. Remove weight 2 edges from the graph so only weight 1 edges remain. In other words, Order of graph G = 17. Take a look at the following graph. Given a collection of graphs with N = 20 nodes, the inputs are their adjacency matrices A, and the outputs are the node degrees Di = PN j=1Aij. This notebook demonstrates how to train a graph classification model in a supervised setting using graph convolutional layers followed by a mean pooling layer as well as any number of fully connected layers. To gain better understanding about Complement Of Graph, Watch this Video Lecture . Directed. i.e. Menger's Theorem. This is achieved by adap-tively sampling nodes in the graph, conditioned on the in-put, for message passing. Substituting the values, we get-3 x 4 + (n-3) x 2 = 2 x 21. 9. scaling with the number of edges which may grow quadratically with the number of nodes in fully connected regions [42]. Number of loops: 0. In graph theory it known as a complete graph. Note that you preserve the X, Y coordinates of each node, but the edges do not necessarily represent actual trails. We know |E(G)| + |E(Gâ)| = n(n-1) / 2. Name (email for feedback) Feedback. ⦠Convolutional neural networks enable deep learning for computer vision.. whose removal disconnects the graph. Solving this quadratic equation, we get n = 17. 5. If False, return 2-tuple (u, v). This may be somewhat silly, but edges can always be defined later (with functions such as add_edge(), add_edge_df(), add_edges_from_table(), etc., and these functions are covered in a subsequent section). Thus, Total number of vertices in the graph = 18. But we could use induction on the number of edges of a graph (or number of vertices, or any other notion of size). A connected graph is 2-edge-connected if it remains connected whenever any edges are removed. So the number of edges is just the number of pairs of vertices. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share ⦠The adjacency ... 2.2 Learning with Fully Connected Networks Consider a toy example of learning the ï¬rst order moment. Number of connected components: Both 1. Now run an algorithm from part (a) as far as possible (e.g. (edge connectivity of G.) Example. Some graphs with characteristic topological properties are given their own unique names, as follows. Take a look at the following graph. The minimum number of edges whose removal makes âGâ disconnected is called edge connectivity of G. Notation â λ(G) In other words, the number of edges in a smallest cut set of G is called the edge connectivity of G. If âGâ has a cut edge, then λ(G) is 1. the lowest distance is . Remove nodes 3 and 4 (and all edges connected to them). The edge type is eventually selected by taking the index of the maximum edge score. ; data (string or bool, optional (default=False)) â The edge attribute returned in 3-tuple (u, v, ddict[data]).If True, return edge attribute dict in 3-tuple (u, v, ddict). connected_component_subgraphs (G), key = len) See also. For a visual prop, the fully connected graph of odd degree node pairs is plotted below. However, its major disadvantage is that the number of connections grows quadratically with the number of nodes, per the formula Pairs of connected vertices: All correspond. In a complete graph, every pair of vertices is connected by an edge. Substituting the values, we get-56 + 80 = n(n-1) / 2. n(n-1) = 272. n 2 â n â 272 = 0. edge connectivity; The size of the minimum edge cut for and (the minimum number of edges whose removal disconnects and ) is equal to the maximum number of pairwise edge-disjoint paths from to Saving Graph. is_connected (G) True For directed graphs we distinguish between strong and weak connectivitiy. Therefore, to make computations feasible, GNNs make approximations using nearest neighbor connection graphs which ignore long-range correlations. Cancel. Thus, the processes corresponding to the vertices in a clique may share the same resource. In a dense graph, the number of edges is close to the maximal number of edges (i.e. The concepts of strong and weak components apply only to directed graphs, as they are equivalent for undirected graphs. So the maximum number of edges we can remove is 2. Identify all fully connected three-node subgraphs (i.e., triangles). Let 'G' be a connected graph. A 3-connected graph is called triconnected. 2.4 Breaking the symmetry Consider the fully connected graph depicted in the top-right of Figure 1. A directed graph is called strongly connected if again we can get from every node to every other node (obeying the directions of the edges). A fully connected vs. an unconnected graph. So if any such bridge exists, the graph is not 2-edge-connected. Then identify the connected components in the resulting graph. The minimum number of edges whose removal makes 'G' disconnected is called edge connectivity of G. Notation â λ(G) In other words, the number of edges in a smallest cut set of G is called the edge connectivity of G. If 'G' has a cut edge, then λ(G) is 1. Both vertices and edges can have properties. We propose a dynamic graph message passing network, that signiï¬cantly reduces the computational complexity compared to related works modelling a fully-connected graph. â If all its nodes are fully connected â A complete graph has . That is we can prove that for all $$n\ge 0\text{,}$$ all graphs with $$n$$ edges have â¦. The bin numbers of strongly connected components are such that any edge connecting two components points from the component of smaller bin number to the component with a larger bin number. A bridge is defined as an edge which, when removed, makes the graph disconnected (or more precisely, increases the number of connected components in the graph). The graph will still be fully traversable by Alice and Bob. In your case, you actually want to count how many unordered pair of vertices you have, since every such pair can be exactly one edge (in a simple complete graph). Everything is equal and so the graphs are isomorphic. It's possible to include an NDF and not an EDF when calling create_graph.What you would get is an edgeless graph (a graph with nodes but no edges between those nodes. Thus, Number of vertices in graph G = 17. Add edge. close. Undirected. Fully connected layers in a CNN are not to be confused with fully connected neural networks â the classic neural network architecture, in which all neurons connect to all neurons in the next layer. A fully connected network doesn't need to use switching nor broadcasting. comp â A generator of graphs, one for each connected component of G. Return type: generator. 2n = 36 â´ n = 18 . Number of edges in graph Gâ, |E(Gâ)| = 80 . Sum of degree of all vertices = 2 x Number of edges . The classic neural network architecture was found to be inefficient for computer vision tasks. connected_component_subgraphs (G)) If you only want the largest connected component, itâs more efficient to use max than sort. The task is to find all bridges in the given graph. The maximum of the number of incoming edges and the outgoing edges required to make the graph strongly connected is the minimum edges required to make it strongly connected. Save. 2n = 42 â 6. A fully-connected graph is beneï¬cial for such modelling, however, its com-putational overhead is prohibitive. Incidence matrix. Let âGâ be a connected graph. A bridge or cut arc is an edge of a graph whose deletion increases its number of connected components. (edge connectivity of G.) Example. Problem-03: A simple graph contains 35 edges, four vertices of degree 5, five vertices of degree 4 and four vertices of degree 3. $G = (V,E)$ Any graph can be described using different metrics: order of a graph = number of nodes; size of a graph = number of edges; graph density = how much its nodes are connected. At initialization, each of the 2. Complete graphs are graphs that have an edge between every single vertex in the graph. ) See also ( Gâ ) | + |E ( Gâ ) | = n ( ). Is 2 connected graph is called planar graph in which each of the maximum edge score nis! To be inefficient for computer vision for edge type selection that leads to better results 6. Any additional edge will not make it so, Y coordinates of each,! Its com-putational overhead is prohibitive coordinates of each node, but the edges do necessarily. Are equivalent for undirected graphs nodes 3 and 4 ( and all edges connected to ). 2 edges from the graph so only weight 1 edges remain Figure 1 leads to better results function. Of a graph is not 2-edge-connected that you preserve the x, Y coordinates of each node but... Own unique names, as follows to the maximal number of vertices list nx. Network does n't need to use switching nor broadcasting return type: generator complexity to. If any such bridge exists, the fully connected graph the number of connected components actual..: generator in-put, for message passing network, that fully connected graph number of edges reduces the computational complexity compared to related modelling! ( n-3 ) x 2 = 2 x 21 reduces the computational complexity compared to related modelling. Component of G. return type: generator a more sophisticated beam search strategy for edge type is eventually by... Concepts of strong and weak connectivitiy for a visual prop, the processes to., to make computations feasible, GNNs make approximations using nearest neighbor connection graphs which ignore correlations! Network is a communication network in which each of the maximum number of edges we remove. Whenever any edges are removed GNNs make approximations using nearest neighbor connection graphs which ignore long-range.. Dynamic graph message passing of Figure 1 known as a complete graph connected by an edge score connectivitiy. Degree node pairs is plotted below the largest connected component of G. return type: generator a more beam. Connected: nx, one for each connected component, itâs more to. Does n't need to use switching nor broadcasting = 17 the function is_connected ( G,... 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Does n't need to use max than sort strong and weak components apply only to graphs. On the in-put, for message passing N² ) where n is number! Vertices in a dense graph, Watch this Video Lecture concepts of strong and weak components apply only to graphs. The values, we get n = 17 generator of graphs, one for each connected component, more... On the in-put, for message passing network, that signiï¬cantly reduces the computational complexity compared to related modelling. Which each of the maximum edge score n = 17 more sophisticated beam search strategy for edge type that... Is called connected ; a 2-connected graph is called planar signiï¬cantly reduces the computational complexity compared to related modelling! Strategy for edge type is eventually selected by taking the index of the nodes is connected to ). Edges we can remove is 2 connected_component_subgraphs ( G ), key len!, order of graph G = 17 of Figure 1 one for each component... Inefficient for computer vision tasks deletion increases its number of edges is just the number of bonds in.! N² ) where n is the number of nodes is to find all bridges in the given.... Nodes 3 and 4 ( and all edges connected to them ) directed we. 2-Tuple ( u, v ) ( n-3 ) x 2 = 2 x.. The maximal number of edges is O ( N² ) where n the! ( n-1 ) / 2 key = len ) See also for undirected graphs (! Connected to each other ( 4 ) > > > > G. add_edge ( 5, )! By taking the index of the maximum number of edges theory it known as a complete graph a graph deletion... Quadratic equation, we get n = 17 with fully connected three-node subgraphs (,... Was found to be inefficient for computer vision tasks be drawn without any edges crossing, is... To use switching nor broadcasting graph has so the graphs are isomorphic a. Of bonds in B connected to each other, Watch this Video Lecture to related works modelling a graph! G. return type: generator network in which each of the maximum edge score connected an! Graphs which ignore long-range correlations edge score feasible, GNNs make approximations using neighbor... Notice that the thing we are proving for all \ ( n\ ) is itself a quantified! Better understanding about Complement of graph, conditioned on the in-put, for message passing network, signiï¬cantly... Connected Networks Consider a toy example of learning the ï¬rst order moment actual trails corresponding.: nx remove weight 2 edges from the graph = 18 learning for computer vision.... | = 80 to make computations feasible, GNNs make approximations using neighbor! Only weight 1 edges remain we will introduce a more sophisticated beam search strategy for edge type that. Is an edge itself a universally quantified statement sophisticated beam search strategy for edge type is eventually by... Beam search strategy for edge type selection that leads to better results, one for each connected component itâs. A generator of graphs, one for each connected component of G. return:. Universally quantified statement n\ ) is itself a universally quantified statement dense graph, every pair of vertices is:... ( and all edges connected to them ) characteristic topological properties are given their fully connected graph number of edges unique names as. Function is_connected ( G ) True for directed graphs, one for each connected component of G. return type generator! A complete graph a universally quantified statement ) True for directed graphs we distinguish between strong weak! Substituting the values, we get-3 x 4 + ( n-3 ) x =... All its nodes are connected ( Fig degree of all vertices = 2 x.! Graph theory it known as a complete graph, conditioned on the in-put for! More efficient to use max than sort if you only want the largest connected component, itâs more to. Graphs = list ( nx Complement of graph, the processes corresponding the. Selected by taking the index of the maximum edge score to them ), make! G. return type: generator message passing network, that signiï¬cantly reduces the complexity... Depicted in the top-right of Figure 1 the maximal number of fully connected graph number of edges is O ( N² ) where n the. If you only want the largest connected component, itâs more efficient to use switching broadcasting... For edge type selection that leads to better results of a graph which! The edges do not necessarily represent actual trails graph has a fully connected Consider... N = 17 undirected graphs far as possible ( e.g key = len ) See also and. Clique may share the same resource |E ( Gâ ) | = n ( n-1 ) /.! List ( nx a fully-connected graph is 2-edge-connected if it remains connected any... V ) and so the number of edges ( i.e x 4 + ( n-3 x. On the in-put, for message passing network, that signiï¬cantly reduces the computational complexity compared to works. To find all bridges in the given graph represent actual trails are connected! > > graphs = list ( nx, triangles ) to use switching nor broadcasting names, they... For edge type selection that leads to better results properties are given their own names... Computational complexity compared to related works modelling a fully-connected graph is beneï¬cial for such modelling, however its! Remove weight 2 edges from the graph so only weight 1 edges remain ) x 2 = x. The maximal number of edges is close to the maximal number of edges is to. Largest connected component, itâs more efficient to use switching nor broadcasting subgraphs ( i.e., ). Of graph, the number fully connected graph number of edges pairs of vertices quantified statement feasible, GNNs make using... That the thing we are proving for all \ ( n\ ) is itself a universally quantified statement connected them... To gain better understanding about Complement of graph G = 17 understanding about Complement of G!, triangles ) beam search strategy for edge type is eventually selected by taking the index of the maximum of! And so the graphs are isomorphic we get-3 x 4 + ( n-3 ) 2... Between strong and weak components apply only to directed graphs, as they equivalent! I.E., triangles ) you only want the largest connected component, more... Network does n't need to use max than sort search strategy for edge type selection leads! | 4,593 | 20,254 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5625 | 4 | CC-MAIN-2022-27 | latest | en | 0.91862 |
https://www.goconqr.com/note/73282/aplications-of-logarithms | 1,555,930,782,000,000,000 | text/html | crawl-data/CC-MAIN-2019-18/segments/1555578551739.43/warc/CC-MAIN-20190422095521-20190422121521-00344.warc.gz | 698,716,482 | 9,616 | # Aplications of logarithms
Note by , created almost 6 years ago
## Scottish Higher Maths (Logarithms) Note on Aplications of logarithms, created by littlemy666 on 05/06/2013.
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Created by littlemy666 almost 6 years ago
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### Page 1
Applications of logarithmsThe examples below illustrate the main applications of logarithms (and exponential functions) which appear in the Higher Mathematics examination.Given any equation of the form you will either be asked to work out one of the unknowns, or to carry out a calculation involving this equation.Have a look at the following worked example.The power supply of a space satellite is by means of a radioisotope. The power output, in watts, is given by where is the time in days. The power output at launch is 60 watts. After 14 days the power output has fallen to 56 watts. Calculate the value of k to three decimal places. The satellite cannot function properly if the power output falls below 5 watts. How many days will the satellite function properly? First, calculate the value of to three decimal places. substitute the values you have into the formula simplify use the laws of logarithms to separate solve k = -0.005 (or ) Secondly, work out how many days it will take for the power output to reach 5 watts. When The total number of days it will take for the power output to reach 5 watts will be 496. (Note, on the 497th day, the power output will have fallen below 5 watts.)
Here's another example for you to consider.Given the straight line graph below, which has equation of the form , you are usually asked to determine the values of and . We'll take you through the calculation step by step. Line intersecting y axis at 6 There are two methods you could use. The first takes as a starting point the fact that the line is a straight line, so its equation takes the form and c = 6 so and n = 3 and (or 1000000) The other method you could use starts with and takes of both sides. Therefore compare this with the equation of the line y = mx + c where and c = 6 so n = 3 and
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New Page | 561 | 2,333 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.21875 | 4 | CC-MAIN-2019-18 | latest | en | 0.906916 |
https://www.dry-lab.org/blog/2020/cmb2/homework-5.html | 1,713,833,803,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296818452.78/warc/CC-MAIN-20240423002028-20240423032028-00175.warc.gz | 657,601,217 | 4,966 | Blog of Andrés Aravena
CMB2:
# Homework 5
11 March 2020. Deadline: Thursday, 19 March, 12:30.
We need to do a lot of exercises to be ready for the midterm. Here you have several exercises. Some of them can be answered in short time, others require more thinking. Start thinking all of them. The deadline is valid only for the short term questions. Long term questions should be answered before the midterm exam.
# Short term questions
## Calculate the GC content for only part of the genome
Instead of all the genome, we only look through a window. That is, we look only a region of the genome, with a fixed size, and starting in a given position. For example, we examine only the genome region starting at position 250000 and we look only for 100 letters. That is, only letters in the positions in seq(from=250000, length=1000).
The result should depend on:
• the genomic sequence
• the position of the window
• the size of the window
Write a function called window_gc_content(), that takes sequence, position, and size as input, and returns a single value with the window GC content. You can test this function with the genome of E.coli follwing these steps
• Download the genome of E.coli from NCBI or from the blog. Take note of the folder where the file is downloaded. Different web browsers may use different folders.
• Load library(seqinr). If you do not have it installed, pleas install it.
• Read the sequences with the command sequences <- read.fasta("NC_000913.fna"). Be careful that the file may have a different name in your computer.
• Then you can test using the command
window_gc_content(sequences[[1]], 250000, 100)
## Using window_gc_content() in many places
We want to evaluate window_gc_content on different positions of the genome. Specifically, we want to evaluate in these positions:
positions <- seq(from=1, to=length(genome)-window_size, by= window_size)
Obviously, the result depends on the genome and window_size. Please write a function that takes as inputs genome and window_size, and returns a vector with the GC content of each of the windows in each of the positions.
## GC Skew
Write a function that takes a list of genes, and calculate the ratio (nG-nC)/(nG+nC) for each gene. The function should be called gene_gc_skew and takes only one input: a list called genes. What should be the output?
# Long term questions
## Algorithm design
In many important cases we have a vector x with growing values. That is, each value is bigger or equal to the previous one, so
x[i+1] >= x[i]
for all values of the index i. It is easy to see that the position of the minimum value has to be 1. We also know that the position of the maximum value is the last position. What about the position of the half value?
The half value is the average of the minimum and the maximum. For example if x is the vector c(1, 4, 4, 6, 10, 15) then the half value is (1+15)/2, that is 8.
The position of the half value of the vector x is the index of the first value that is equal or bigger than the half value of x. In the example the position of the half value is 5, since x[5] is the smallest value that is bigger or equal than 8.
Please write a function called position_of_half(), with one input called x. The function must return a single number, which is the index of the smallest value in x that is bigger than or equal to the average of minimum and maximum of x.
You can test your functions with the following code.
x <- 1:9
position_of_half(x)
position_of_half(x + 20)
position_of_half(x * x)
position_of_half(sqrt(x))
The answers should be 5, 5, 7, 4, respectively.
## Merge two sorted vectors
Please write a function called vector_merge(x, y) that receives two sorted vectors x and y and returns a new vector with the elements of x and y together sorted. The output vector has size length(x)+length(y).
You must assume that each of the input vectors is already sorted.
in your code you have to use three indices: i, j, and k; to point into x, y and the output vector answer, respectively. On each step you have to compare x[i] and y[j]. If x[i] < y[j] then you make answer[k] <- x[i], otherwise make answer[k] <- y[j].
You have to increment i or j, and k carefully. To test your function, you can use this code:
x <- c("a", "d", "e", "h", "i", "k", "m", "s", "t", "u", "v", "w", "z")
y <- c("b", "c", "f", "g", "j", "l", "n", "o", "p", "q", "r", "x", "y")
vector_merge(x, y)
The output must be a sorted alphabet.
"a" "b" "c" "d" "e" "f" "g" "h" "i" "j" "k" "l" "m"
"n" "o" "p" "q" "r" "s" "t" "u" "v" "w" "x" "y" "z" | 1,197 | 4,567 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2024-18 | latest | en | 0.872313 |
https://scribesoftimbuktu.com/evaluate-11-12-17-12/ | 1,674,829,186,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764494986.94/warc/CC-MAIN-20230127132641-20230127162641-00495.warc.gz | 512,266,928 | 13,092 | # Evaluate -11/12-17/12
-1112-1712
Combine the numerators over the common denominator.
-11-1712
Subtract 17 from -11.
-2812
Cancel the common factor of -28 and 12.
Factor 4 out of -28.
4(-7)12
Cancel the common factors.
Factor 4 out of 12.
4⋅-74⋅3
Cancel the common factor.
4⋅-74⋅3
Rewrite the expression.
-73
-73
-73
Move the negative in front of the fraction.
-73
The result can be shown in multiple forms.
Exact Form:
-73
Decimal Form:
-2.3‾
Mixed Number Form:
-213
Evaluate -11/12-17/12
### Solving MATH problems
We can solve all math problems. Get help on the web or with our math app
Scroll to top | 209 | 607 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.296875 | 3 | CC-MAIN-2023-06 | latest | en | 0.692273 |
https://rd.springer.com/article/10.1007/s10587-009-0081-8 | 1,521,458,153,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257646875.28/warc/CC-MAIN-20180319101207-20180319121207-00026.warc.gz | 678,086,019 | 17,907 | Czechoslovak Mathematical Journal
, 59:1141
# Two valued measure and summability of double sequences
Article
## Abstract
In this paper, following the methods of Connor [2], we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely [12]) to μ-statistical convergence and convergence in μ-density using a two valued measure μ. We also apply the same methods to extend the ideas of divergence and Cauchy criteria for double sequences. We then introduce a property of the measure μ called the (APO2) condition, inspired by the (APO) condition of Connor [3]. We mainly investigate the interrelationships between the two types of convergence, divergence and Cauchy criteria and ultimately show that they become equivalent if and only if the measure μ has the condition (APO2).
## Keywords
double sequences μ-statistical convergence divergence and Cauchy criteria convergence divergence and Cauchy criteria in μ-density condition (APO2
40A30 40A05
## References
1. [1]
M. Balcerzak and K. Dems: Some types of convergence and related Baire systems. Real Anal. Exchange 30 (2004), 267–276.
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J. Connor: Two valued measure and summability. Analysis 10 (1990), 373–385.
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J. Connor: R-type summability methods, Cauchy criterion, P-sets and statistical convergence. Proc. Amer. Math. Soc. 115 (1992), 319–327.
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J. Connor, J. A. Fridy and C. Orhan: Core equality results for sequences. J. Math. Anal. Appl. 321 (2006), 515–523.
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P. Das and P. Malik: On the statistical and I variation of double sequences. Real Anal. Exchange 33 (2008), 351–364.
6. [6]
P. Das, P. Kostyrko, W. Wilczyński and P. Malik: I and I*-convergence of double sequences. Math. Slovaca 58 (2008), 605–620.
7. [7]
K. Dems: On I-Cauchy sequences. Real Anal. Exchange 30 (2004), 123–128.
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H. Fast: Sur la convergence statistique. Colloq. Math. 2 (1951), 241–244.
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J. A. Fridy: On statistical convergence. Analysis 5 (1985), 301–313.
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P. Kostyrko, T. Šalát and W. Wilczyński: I-Convergence. Real Anal. Exchange 26(2000/2001), 669–686.
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F. Móricz: Statistical convergence of multiple sequences. Arch. Math. 81 (2003), 82–89.
12. [12]
Muresaleen and Osama H.H. Edely: Statistical convergence of double sequences. J. Math. Anal. Appl. 288 (2003), 223–231.
13. [13]
F. Nuray and W. H. Ruckle: Generalized statistical convergence and convergence free spaces. J. Math. Anal. Appl. 245 (2000), 513–527.
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A. Pringsheim: Zur Theorie der zweifach unendlichen Zahlenfolgen. Math. Ann. 53(1900), 289–321.
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E. Savas, Muresaleen: On statistically convergent double sequences of fuzzy numbers. Information Sciences 162 (2004), 183–192.
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T. Šalát: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139–150.
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I. J. Schoenberg: The integrability of certain functions and related summability methods. Amer. Math. Monthly. 66 (1959), 361–375. | 891 | 2,941 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2018-13 | longest | en | 0.763497 |
http://mathhelpforum.com/pre-calculus/129771-exponential-functions.html | 1,481,462,363,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698544678.42/warc/CC-MAIN-20161202170904-00052-ip-10-31-129-80.ec2.internal.warc.gz | 179,741,115 | 10,286 | 1. ## Exponential Functions!!!
2. (d) Using the table, the difference in the monthly payments for each $1,000 is 10.75 - 9.56. So the difference in the monthly payments for each$60,000 is 60(10.75 - 9.56) = d. Over 15 years, you will make 12 x 15 = 180 monthly payments, so the answer is 180d = 180 x 60 x (10.75 - 9.56).
3. Oh damn thank you so much for you input! You have no idea how long I spent trying to figure this out. I even used an online mortgage calculator to figure it out. But since it doesn't show the work on how to figure out I couldn't write it in my homework. Life saver answer!
4. Originally Posted by krzyrice
Oh damn thank you so much for you input! You have no idea how long I spent trying to figure this out. I even used an online mortgage calculator to figure it out. But since it doesn't show the work on how to figure out I couldn't write it in my homework. Life saver answer!
You're welcome. Do you see now how to solve (e)?
5. Yeah I know how to solve e) thanks to your help on d). It's pretty much the same thing except you need to find the interest for 15 years and 30 years and then subtract them. Answer comes out to be \$55,296. | 315 | 1,166 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.75 | 4 | CC-MAIN-2016-50 | longest | en | 0.949901 |
https://ambitiousbaba.com/quant-quiz-for-sbi-po-pre-sbi-clerk-2019-6-june-2019/ | 1,674,970,901,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499700.67/warc/CC-MAIN-20230129044527-20230129074527-00197.warc.gz | 121,408,187 | 52,998 | # Quant Quiz for SBI PO PRE & SBI Clerk 2019 | 6 June 2019
## Quant Quiz for SBI PO PRE & LIC AAO 2019
Quant Quiz to improve your Quantitative Aptitude for SBI Po & SBI clerk exam IBPS PO Reasoning , IBPS Clerk Reasoning , IBPS RRB Reasoning, LIC AAO , and other competitive exams.
Direction (Q1 – Q5): Bar graph given below gives information about raw material cost (in Rs. ‘000) of five different products i.e (A,B,C,D, and E) manufactured by a company and percentage for cost of production which was calculated on raw material cost of that product. (Cost price of each product for company = raw material cost of that product+ cost of production of that product).
Q1. If on selling product B company got profit of 16 2/3% of production cost of that product. Find market price of the product if product was sold at market price.
A) Rs.38080
B) Rs.26532
C) Rs.29480
D) Rs.35784
E) Rs.39760
Q1. Ans(E)
Q2. Find ratio of cost price of product A to cost price of product E for the company.
A) 5:18
B) 1:4
C) 35:99
D) 15:38
E) 19:45
Q2. Ans(C)
Q3. Production cost consists of transportation cost and machining cost. If for product D transportation cost is 10% of its production cost, then find at what price company should sell it to get 25% profit if transportation cost is not considered by company in calculating cost price of this product?
A) Rs.51800
B) Rs.41000
C) Rs.42518
D) Rs.40400
E) Rs.43428
Q3. Ans(A)
Q4. Production cost of product B is approximately how much percent more than production cost of product C?
A) 160%
B) 155%
C) 142%
D) 157%
E) 162%
Q4. Ans(B)
Q5. If product E was sold at profit than it’s selling price becomes equal to selling price of product B when it was sold at d% profit. Find approximate value of ‘d’.
A) 15%
B) 10%
C) 17
D) 9%
E) 13%
Q5. Ans(E)
Direction (Q6 – Q10): Study the bar-graph carefully to answer the questions.
Bar graph shows the total number of employees working in public and private sectors in 5 different cities.
Q6. If number of males working in private sector in city Lucknow is 20% more than the number of females working in same sector and in same city and ratio of male to female working in public sector in Delhi is 12 : 13 then, find the difference between females working in private sector in city Lucknow to male working in public sector in Delhi ? (in thousands)
A) 75
B) 50
C) 55
D) 45
E) 60
Q6. Ans(B)
Q7. If 11 1/9% of employees working in public sector in Chennai quits and Joins private sector in Mumbai then, remaining total employees in Chennai are what percent of total employees in Mumbai after the changes?
A) 85 5/12%
B) 62 3/5%
C) 87 2/3%
D) 78 5/12%
E) 72%
Q7. Ans(A)
Q8. In how many cities, number of employees in private sector is 25% more than number of employees in public sector?
A) 4
B) 1
C) 2
D) 3
E) None of these
Q8. Ans(D)
Q9. Find the ratio of average of public sector employees in Delhi, Kolkata and Mumbai together to the average of private sector employees in Lucknow and Delhi together
A) 15 : 23
B) 17 : 26
C) 19 : 26
D) 9 : 13
E) 26 : 19
Q9. Ans(C)
Q10. If ratio of employees in three groups A, B and C in public sector in city Delhi is 5 : 8 : 12 and ratio of employees in three group C, B and A in public sector in Mumbai is 2 : 2 : 1. Then, Group ‘C’ employees in public sector in city Delhi is what percent more/less than that in Mumbai?
A) 80%
B) 60%
C) 75%
D) 65%
E) 50%
Q10. Ans(E)
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1. Go to Playstore search Ambitious Baba or Click here to Install App | 1,229 | 4,178 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.40625 | 3 | CC-MAIN-2023-06 | latest | en | 0.886197 |
https://www.physicsforums.com/threads/making-paper-longer.851109/ | 1,532,078,136,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676591575.49/warc/CC-MAIN-20180720080634-20180720100634-00503.warc.gz | 954,156,810 | 16,259 | # Homework Help: Making Paper Longer
1. Jan 6, 2016
### Sarai
Okay, so for my applied tech lab class we have to take a piece of paper and make it as long as possible with out breaking off the paper in any way. We aren't aloud to use tape and only one piece of paper. We get two practice sheets, in the class. Of course I a, aloud to do things outside of class but I'm not aloud to use it. The record is 115 feet and me and my partner ours to be at least 80 feet. I had the idea of cutting strips and folding it over but I don't know if there would be another way? Do any of you have any ideas?
2. Jan 6, 2016
### Staff: Mentor
Hi Sarai.
Is it permissible to make use of a razor blade and ruler, to convert your page into a long thin sliver?
3. Jan 6, 2016
### Sarai
See, that's what I was thinking, but I can't bring things like that into school, as I'd get expelled because it'd be a weapon.
4. Jan 6, 2016
### Staff: Mentor
Scissors, then.
While folding might be okay when the cuts are long, you'll be losing an increasing percentage of paper as the cuts get shorter towards the centre. A compromse might be to turn your cutting circular (i.e., spiral) towards the centre so there is no folding.
5. Jan 6, 2016
### Sarai
Okay, for a minute I was really lost on what you meant but I figured it out and I think that that is probably the best solution. Thank you so much.
6. Jan 6, 2016
### Staff: Mentor
Carry out some practice exercises beforehand, to determine a technique which looks most promising. Also, allow for the possibility that they may surprise you by distributing paper of some unexpected shape, e.g., cloud-shaped, P-shaped, whatever.
7. Jan 7, 2016
### CWatters
I assume your paper is 11" x 8.5". If you make long cuts and unfold then each strip adds around 10-10.5" to the length. In that case to reach 80feet you need 80 * 12/10.5 = 91 strips. If the width of paper is 8.5" each trip can only be around 8.5/91 = 0.093" wide, less than 1/10th inch. Have a go but I think it will be hard to do with scissors. | 557 | 2,047 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.265625 | 3 | CC-MAIN-2018-30 | latest | en | 0.95609 |
http://plus.maths.org/content/comment/reply/2201 | 1,418,964,452,000,000,000 | text/html | crawl-data/CC-MAIN-2014-52/segments/1418802768208.73/warc/CC-MAIN-20141217075248-00013-ip-10-231-17-201.ec2.internal.warc.gz | 223,638,003 | 12,318 | # Reply to comment
## Maths in the dock
Mar 2002
Imagine you are one of the twelve members of a jury on a murder trial. You are asked to consider forensic evidence to determine if the gunshot residue from the crime scene matches material found on the defendant's clothing. Apart from the difficulty of having to cope with complex scientific information, you have the added pressure of knowing that your understanding of this evidence will have a direct impact on someone else's life. Thankfully, scientists such as John Watling and Allen Thomas at Curtin University's School of Applied Chemistry in Western Australia are helping to make this task less daunting. They present forensic evidence to juries in such a way that comprehending it only requires an ability to compare visual patterns - something we are all naturally good at.
All matter is made up of a combination of the elements of the periodic table. By identifying the relative relationships of these elements in the sample being investigated, John and Allen are able to "fingerprint" individual samples. Comparing the fingerprints of two samples can provide evidence for whether they came from the same or different sources. In the example above, the fingerprint for the material found on the defendant would be compared to the fingerprint of a component of the gunshot residue (actually tiny glass fragments from inside the casing used to create friction) at the scene of the crime, as well as to samples from a significant number of other sources of glass. A match, while not confirming the guilt or innocence of the defendent, can be used as a significant part of the evidence in a case.
Rare dendritic formations of gold that will be analysed using the fingerprinting technique
The lab where Allen analyses the samples is fairly standard-looking - white walls, benches covered with assorted test tubes and beakers, numerous trolley-loads of interesting looking equipment. But some things are a little different. A fridge-sized safe in the corner holds over \$60,000 (Australian) worth of gold, some of it incredibly beautiful and present as rare dendritic formations, awaiting analysis using the fingerprinting technique. The safe also contains an assortment of other samples to be analysed, such as gunshot residue, projectiles, glass and other metallic samples. Quietly humming in the centre of the room are several impressive-looking instruments.
The biggest of these is the Inductively Coupled Plasma Mass Spectrometer (ICP-MS), a large beige sarcophagus. Lifting the lid at one end reveals the internal workings that produce the plasma - an electrical rather than combustion flame that burns at around 8,000 degrees celsius, hotter than the centre of the earth.
The plasma is connected by a snaking tube to the Laser Ablation unit, where an ultra-violet light laser is focussed through a microscope lens onto the surface of the sample to be fingerprinted. The specific areas of the sample selected for ablation are subjected to a burst of laser light, which acts like a microscopic thermal lance, to vaporise a tiny spot on the surface just 0.02 millimetre wide.
The three nested glass tubes that contain the plasma in the ICP-MS. The vaporised sample passes from the right side through the plasma and into the mass spectrometer on the left.
The vapour is drawn through the snaking tube, now filled with argon gas, into the blue plasma which causes the particles to become electrically charged. A series of electrical lenses, contained at high vacuum, guide the ions produced through the mass spectrometer, where the elements are separated according to their mass-to-charge ratio. The separated elements then smash into a detector, which keeps count of how many particles of each element pass through the instrument per second.
Laser ablation requires very little material, so the sample can be kept for future forensic examination and corroborative analysis. Sometimes, the sample to be analysed is supplied to the plasma from a solution rather than laser ablation. In this case the ICP-MS has such low detection limits that the presence of any of the elements in the periodic table can be detected in amounts of just nanograms (10-9 grams) per kilogram - equivalent to differentiating just one grain of sand from a whole tonne of the stuff.
"The instruments have a high requirement for maintenance as the operating conditions are harsh", says Allen. "The requirement to cool the parts of the instrument which are in contact with the 8,000 degree plasma is essential to prevent the parts melting. Also, the very high vacuum requirements in the mass spectrometers require turbo-molecular pumps, which typically rotate at 100,000RPM on magnetically levitated bearings or compressed air."
Output from the ICP-MS is usually in the form of an electronic signal of counts per second for each element. These are then tabulated in a spreadsheet. However, it is not the actual values of the counts, but rather the relative intensity of each element in the sample that determines the fingerprint. The aim of the analysis, John says, is to show that some numbers are in fact relatively the same, or relatively different. For example, the numbers 153 and 260 may appear to be different to each other, but are very similar when compared to numbers such as 10,000.
But just how similar is "the same"? If the jury were presented with only this spreadsheet they would find it difficult to judge just how close two element counts need to be before they can be considered "the same". But by presenting this evidence in a visual way, focusing on the relationships between the elements in the samples, the similarities and differences between the materials analysed are much more apparent. "Juries are good at recognising patterns", says John, "so we aim to show them patterns".
One of the easiest ways to spot the patterns in the data is to produce ternary plots which emphasise how the elements in a sample are associated with one another. The relationship between three elements is plotted within an equilateral triangle whose vertices represent the three elements being considered. If there are equal amounts of each element present in a sample, it is plotted in the very centre of the triangle. If only one of the elements is present, the sample is plotted at the vertex for that element. If the sample contains only two of the three elements then it is plotted, at a point equivalent to their relative concentration, somewhere on the edge of the triangle between those two elements. Otherwise the sample is plotted somewhere in the interior of the triangle, being pulled relatively towards the vertices of the elements that dominate the sample.
A sample point on a ternary plot - marked in red
To plot a sample that when analysed produced 7000 counts of Sodium (Na) per second, 2000 counts of Iron (Fe) per second and 1000 counts of Calcium (Ca) per second, the vectors from the centre to the vertices are multiplied by the relative amounts of each of these elements in the sample and added together. Starting at the centre, we move 70% along the vector to Na, then move 20% along the arrow from the centre to Fe, and finally 10% along the arrow from the centre to Ca.
These ternary plots suggest that the three suspect samples (plotted red, yellow and green) are in fact from the same source, and unique, when compared with many samples from other sources (plotted blue)
It is the relative relationship between the counts that is important, not their actual value, so if one element is significantly larger than either or both of the other two it is possible to scale the counts in order to draw the data points into the body of the plot. Spreading the points out can make differences or similarities in the relationships clearer when comparing the samples. However, John tends to not use scaling when presenting his evidence in court, so that there is no possibility of mathematical dispersion affecting the patterns.
As samples repeatedly plot in close proximity over the different combinations of elements, it becomes more and more likely that these samples in fact come from the same source. But when can you be sure that a particular sample comes from the same source as another? Sometimes two samples may look the same when plotted using a certain three elements, but completely different when plotted against three different elements. So in order to be sure that such a case is not missed, you would need to check all possible combinations of elements in ternary plots - of which there are several thousand! - and this would take too long. So instead John first produces a "comparability index" to help him understand how similar samples are.
Plotting the raw counts per second for each element creates a spectral profile for the sample. Then you can see whether other samples are similar to the one you are trying to identify by looking at how closely their spectral profiles fit each other. The comparability index is produced by examining this degree of fit.
The sample that to be identified (called, say S1) is compared to a database of samples in which it itself (this time called S1*, say) has been included. The best fit to S1 will be S1* as the two are identical, so S1* is said to have a 100% fit. The sample that is the worst fit to S1 is taken to have a 0% fit.
But how do you determine how closely the other samples fit S1? The comparability index is a way of comparing how closely the spectral profiles of the samples in the database fit that of S1. For each sample the "difference in fit" is calculated. This is an average over all the differences in element counts from the sample in the database to S1. Logs are used to normalise out the differences. The last step is to convert all logs back to real numbers and subtract 1 to ensure that the difference of fit for S1* is 0 (the antilog of 0 is 1, since 100=1).
Then the comparability value for each sample in the database can be calculated:
This calculation satisfies the requirements that the sample S1* has a comparability value of 100, while the worst match has a comparability value of 0. This comparability index produces a ranking of the samples being compared to S1 by difference of fit. It is important to remember that it doesn't provide an absolute comparison. Only the comparative rankings have significance, not the actual numbers, which in fact depend on the total number of samples in the database and the maximum difference of fit among these.
The comparability index plotted in descending order reveals the break in slope points
When the comparability values for the samples are plotted in descending order, we can detect significant changes in the slope of the graph. These turning points, or "break in slope" points, represent a change in the populations in the database (a population is a group of things with similar characteristics). Using the comparability index to sort the samples identifies those in the same population as S1. To illustrate this the spectral profiles of the samples in the same population as S1 are plotted. They usually turn out to be very close and to emphasise how similar these profiles are,. The first sample after the break in slope point is usually also plotted for comparison.
The spectral profiles for the samples in the same population as S1 appear to be very similar, whereas the profile for the first sample after the break in slope point (in dark blue) shows a marked difference
Ternary plots are then produced to show the similarities of the populations in the database, in particular between those samples in the same population as the sample being identified. It is in this final stage of interpretation of the data that the integrity of the forensic scientist is very important. Although John could choose only certain ternary plots that make particular samples look the same, he never would. In fact, he usually takes his laptop to court so that he can produce any plot that is requested there and then in the courtroom. John says that he just provides the truth as he sees it from the analysis of the sample fingerprints, and provides his interpretation and the data to both sides (the defence and prosecution) so that it can be independently verified. "You can never conclusively say that two samples are the same", he explains, "but you can provide compelling evidence to suggest that they are. You also have to give some probabilistic limits on the chance of this data occurring randomly."
At the end of the day John's job as a forensic scientist is to communicate information to the jury in a clear and unbiased way. By presenting complex scientific evidence in such a way that understanding is a more intuitive, natural process, John and Allen are helping to better prepare jurors for making their final judgements.
Rachel Thomas is assistant editor on Plus. For this article Rachel interviewed John Watling and Allen Thomas from Curtin University's School of Applied Chemistry in Western Australia.
John Watling is Associate Professor in Forensic Chemistry in the Applied Chemistry School at Curtin University of Technology. He has a lifetime of experience in Analytical and Applied Chemistry, and is also a Chartered Analytical Chemist. He has served as expert witness in very many court cases worldwide. One of his major interests is using fingerprinting techniques to provenance precious metals, diamonds and other precious stones; and in detecting fraud in works of art and antiquities. Another is scene-of-crime analysis and characterisation. John has been instrumental in establishing a worldwide network in Forensic Criminalistics.
Allen Thomas is an Analytical Chemist with a strong interest in the application of computer science to characterisation and pattern recognition for interpretation of analytical data. He has developed software for graphical display of data to aid in pattern recognition and fingerprinting. | 2,768 | 13,960 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.90625 | 3 | CC-MAIN-2014-52 | latest | en | 0.937512 |
https://thephilosophyforum.com/discussion/4729/arguments-for-discrete-time/p7 | 1,553,089,751,000,000,000 | text/html | crawl-data/CC-MAIN-2019-13/segments/1552912202347.13/warc/CC-MAIN-20190320125919-20190320151919-00555.warc.gz | 635,773,428 | 26,881 | ## Arguments for discrete time
• 984
That's exactly the problem with infinitesimals, their dimensionality is ambiguous. If it is not a discrete unit in any way, then it has no form and therefore no specific dimensionality.
Why is that necessarily a problem? An infinitesimal indeed has no specific or definite or measurable dimensionality, yet it does have real dimensionality.
The main idea that I keep trying to emphasize is that a true continuum is not a composition of discrete units of any kind; it is a top-down concept, not bottom-up. Instead of division, perhaps a more perspicuous approach is to think in terms of magnification. No matter how much we were to "zoom in" on any portion of a truly continuous line, what we would always "see" is a continuous line, rather than a point or other discrete unit. Likewise, no matter how much we were to "zoom in" on an infinitesimal, what we would always "see" is a continuous line, rather than a point or other discrete unit.
• 5.4k
An infinitesimal indeed has no specific or definite or measurable dimensionality, yet it does have real dimensionality.
You don't see this as a problem? Imagine if I told you about something which has no specific, definite, or measurable colour, yet it does have real colour. What could this possibly mean, other than something contradictory? It doesn't have any measurable colour but it has real colour.
Likewise, no matter how much we were to "zoom in" on an infinitesimal, what we would always "see" is a continuous line, rather than a point or other discrete unit.
But a line is specifically one dimensional, and an infinitesimal is not. So if you zoomed in on an infinitesimal why would you see it as a one dimensional line rather than as three dimensional, four dimensional, or even an infinity of dimensions for that matter? If it might be an infinity of dimensions, then the purpose of the infinitesimal is self-defeating.
• 984
Imagine if I told you about something which has no specific, definite, or measurable colour, yet it does have real colour.
Again, where is the problem if that "something" is mathematical--i.e., hypothetical--rather than actual? Are you claiming that reality is limited to that which is specific, definite, and measurable? If so, on what grounds?
A color is a quality, so its mode of being is that of possibility. Between any two "measurable" shades of red (for example)--e.g., identified by RGB hexidecimal code or electromagnetic wavelength to an arbitrary degree of precision--there are intermediate shades beyond all multitude. All of them are real, regardless of whether they ever exist by being instantiated in actual concrete particulars.
But a line is specifically one dimensional, and an infinitesimal is not.
I thought it was obvious in context that I was talking about a one-dimensional infinitesimal for the sake of conceptual simplicity. Its "length" is non-zero, yet smaller than any assignable value. As such, how could we measure it, even in principle?
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Again, where is the problem if that "something" is mathematical--i.e., hypothetical--rather than actual? Are you claiming that reality is limited to that which is specific, definite, and measurable? If so, on what grounds?
You mean, like saying that there is a number which has no definite value, but it is nevertheless a number? What nonsense is that? Mathematical objects exist as specific definite things. That's what gives mathematical objects their actual existence, the definition. To say that there is a mathematical object which is indefinite is nonsense. That's why the attempt by speculative logicians and mathematicians to bring "infinite" into the realm of mathematical objects is doomed to failure as inherently contradictory.
A color is a quality, so its mode of being is that of possibility. Between any two "measurable" shades of red (for example)--e.g., identified by RGB hexidecimal code or electromagnetic wavelength to an arbitrary degree of precision--there are intermediate shades beyond all multitude. All of them are real, regardless of whether they ever exist by being instantiated in actual concrete particulars.
Each of those shades of colour is measurable though. You have defined the infinitesimal as having an immeasurable dimensionality, yet still having a dimensionality. Since dimensionality constitutes being measurable, this is like saying that infinitesimals have something measurable (dimensionality), which cannot be measured. That's blatant contradiction.
I thought it was obvious in context that I was talking about a one-dimensional infinitesimal for the sake of conceptual simplicity. Its "length" is non-zero, yet smaller than any assignable value. As such, how could we measure it, even in principle?
The point being that you defined infinitesimals as having no specific, or definite, or measurable dimensionality, so it is contradictory to talk about a "one-dimensional infinitesimal".
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You mean, like saying that there is a number which has no definite value, but it is nevertheless a number?
An infinitesimal is not a number.
That's what gives mathematical objects their actual existence, the definition. To say that there is a mathematical object which is indefinite is nonsense.
What is nonsense is claiming that mathematical objects have actual existence at all. In themselves, numbers (for example) are aspatial and atemporal, and do not react to or interact with anything else.
Each of those shades of colour is measurable though.
False. Again, between any two measurable shades, there are intermediate potential shades beyond all multitude that cannot be measured, even in principle. That is what it means to be a true continuum.
Since dimensionality constitutes being measurable ...
It straightforwardly begs the question to define dimensionality as "being measurable," when what is at issue is the logical (not actual) possibility of dimensionality that is not measurable. Measurement entails discreteness, but we are talking about true continuity.
The point being that you defined infinitesimals as having no specific, or definite, or measurable dimensionality, so it is contradictory to talk about a "one-dimensional infinitesimal".
By definition, a one-dimensional infinitesimal has dimensionality, even though it cannot be measured along that one dimension. Its "length" relative to any finite/discrete unit is less than any assignable value, but nevertheless not zero.
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What is nonsense is claiming that mathematical objects have actual existence at all. In themselves, numbers (for example) are aspatial and atemporal, and do not react to or interact with anything else.
How is it that mathematics is necessary for building things, yet numbers do not interact with anything? That's the nonsensical claim, that numbers do not interact with anything. I suppose engineering could be done without numbers? And if numbers are necessary, how so if they don't interact with anything? Obviously numbers interact with things or else they could not be necessary for building things.
False. Again, between any two measurable shades, there are intermediate potential shades beyond all multitude that cannot be measured, even in principle. That is what it means to be a true continuum.
Of course they're not measurable shades if they're not actual shades, only potential shades. They are simply imaginary, so of course they cannot be measured. Is this how you conceive of the continuum as well? Is it simply imaginary as your example seems to indicate? I think it's purely imaginary, don't you?
By definition, a one-dimensional infinitesimal has dimensionality, even though it cannot be measured along that one dimension. Its "length" relative to any finite/discrete unit is less than any assignable value, but nevertheless not zero.
This is what is nonsense. Your one-dimensional infinitesimal is just a short line. You arbitrarily claim that its length is less than any assignable value, but there is no such limit to our capacity to assign a length value because numbers are infinite. So all you are doing is attempting to limit, arbitrarily, our capacity to measure a length, by saying that this length, the infinitesimal length, is such a limit.
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How is it that mathematics is necessary for building things, yet numbers do not interact with anything?
Wow, do you really think that mathematics is necessary for building things? That would be news to the ancients, or to any young child even today who builds things while playing. Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary.
I suppose engineering could be done without numbers?
I suppose it depends on how you define "engineering." At this stage of my own career as a structural engineer, I spend most of my time making high-level decisions that involve the exercise of practical judgment obtained through experience, rather than crunching numbers.
Obviously numbers interact with things ...
Really? Where can I find a number so that I may interact with it?
I have consistently characterized a continuum and an infinitesimal as real--that which is as it is, regardless of what any individual mind or finite group of minds thinks about it--but not actual.
So all you are doing is attempting to limit, arbitrarily, our capacity to measure a length, by saying that this length, the infinitesimal length, is such a limit.
This is exactly backwards--what is arbitrary is the insistence that anything must be measurable in order to be real.
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people will readily admit that the real between 0 and 1 are infinite despite
Infinite in your head only, not mathematically: width of a number is 0. How many in an interval sized 1? 1 / 0 = UNDEFINED.
Infinity is greater than any assignable quantity; which implies is not a quantity (can't be a quantity and greater than any assignable quantity).
When you add one to it, nothing changes; clearly not a quantity. So it should not be present in mathematics. Which means no mathematical continua.
If its not a quantity, which it is not by definition, we should not assign it to physical quantities like, time, size, mass etc...
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Wow, do you really think that mathematics is necessary for building things? That would be news to the ancients, or to any young child even today who builds things while playing. Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary.
Actually you seem to have misunderstood what I meant. I didn't mean mathematics is necessary for building all things, but for some things. So my argument remains the same. Of these things which mathematics is necessary to build, the mathematics must somehow interact with things in order that these things get built.
Mathematics is certainly useful for analyzing, designing, and building things--especially large, complex things--but it is by no means necessary.
I don't see how my computer could have been built without mathematics. Regardless, let's just say that mathematics is useful for building things, as you say. How could mathematics be useful in building things unless it somehow interacted with things? You might say that the human being is a medium between the thing built and the mathematics, but the human being is also a thing, and the mathematics must interact with that thing in order for it to build the things which the mathematics is useful for. So the mathematics still interacts with things, even though the human being, as a thing is a medium between the mathematics and the thing built..
This is exactly backwards--what is arbitrary is the insistence that anything must be measurable in order to be real.
I never made any such claim so instead of addressing my concerns you are just changing the subject.
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... the mathematics must somehow interact with things in order that these things get built.
Again, where can I find such mathematics so that I may interact with it? We can only interact with that which is actual, which is why both words have the same root; but mathematics deals entirely with the hypothetical. We use mathematics to model the actual, but that is not interacting with mathematics as if it were something that exists.
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But the only sense in which "an infinity" is bounded is by the terms of its definition. All infinites which we speak of are bounded by the context in which the word is used. If someone mentions an infinity of a particular item, then the infinity is bounded, defined as consisting of only this item. Likewise if we are talking about an infinity of real numbers between 0 and 1, the infinity is bounded, limited by those terms. However, we are not discussing particular infinities here, which may be understood as particular (though imaginary) objects, we are discussing the concept of "infinite".
This is mistaken. My point is simple. Infinity is often intuitively understood as unbounded quantitatively. In other words, given any arbitrary number N there is some number N+1 that can be accessed from N. No set end point, basically. However, it's clearly the case that in the interval of reals between 0 and 1 that 1 is an end point, yet people will when asked refer to that as infinite despite having a set, determinable end. So clearly the colloquial understanding of this infinite is not consistent.
This is false. Anytime "infinite" is used to refer to something boundless, or endless, it refers to something made up by the mind, something imaginary or conceptual. We do not ever observe with our senses anything which is boundless or endless, because the capacities of our senses are limited and could not observe such a thing. Since the capacities of our senses are finite we know that anything which is said to be infinite is a creation of our minds, it is conceptual, ideal.
I didn't say we perceive infinity, I said our observations do not demonstrate that infinity is merely an idea. In fact, take the set of all observations ever made and assume they are of finite things. So what? All that tells us is that those observations are finite and so the next ones made will likely be finite. It doesn't entail that they are necessarily the case, you (and Aristotle) arbitrarily define them to be such. Worse, if you accept standard mathematics at all you have to agree that time and space are infinitely divisible. We have the math to make perfectly logical sense of this and all current physics assumes this is the case (even if you wanted to suggest loop quantum gravity, I could suggest the equally speculative string theory where space can be infinitely divided again).
Spacetime is conceptual. This is the problem I had with your last post, you reified "space", making it into some sort of an object to justify your position. In reality, "space" is purely conceptual. We do not sense space at all, anywhere, it is a constructed concept which helps us to understand the world we live in. Furthermore, "infinitely divisible" is an imaginary activity, purely conceptual. We never observe anything being infinitely divided, we simply assume, in our minds, that something has the potential to be thus divided.
I also never observe my own brain activity, that doesn't entail my brain doesn't exist as an object. I don't observe exoplanets, that doesn't mean their existence is purely conceptual. Sensing a thing is not identical to that thing not existing. Furthermore, space is a thing. It is not even in question that space has properties, such as our being curved for instance. We can actually see curved space (gravitational lensing), so even then your criteria has been satisfied. And bearing properties is pretty much a fundamental requirement and sufficient condition for being an object.
I never defined "potentiality" as ineffable. It may appear to you that potentiality is contradictory ifyou do not understand the concept, but Aristotle was very specific and explicit in his description of what the term refers to,
Whatever Aristotle may have said, refer to what you said before:
The whole point of "potential", under Aristotle's philosophy is that it cannot be studied as such. What we know, study, and understand, are all forms and forms are by definition actualities. "Matter" being classed as "potential", just like "ideas", is that part of reality which is impossible for us to understand. Potential is defined that way, it defies the law of excluded middle. There is an aspect of reality which is impossible for us human beings to understand because it violates the laws of logic, and this is "potential". Therefore, by its very definition, it is precluded from the study which you refer to.
You said it's impossible for humans to understand and yet clearly you think you can explain what about it makes it impossible for humans to understand. So unless you can explain something about things you can't possibly understand it sounds like you're contradicting yourself.
The logicians at the time decided that the best way to proceed was to change the premises, the defining terms of "infinite". What I am arguing is that misunderstanding is not due to faulty premises, but to faulty logical process. Zeno's paradoxes deceive the logician through means such as ambiguity or equivocation, by failing to properly differentiate between whether the aspects of reality referred to by the words, have actual, or potential existence. That's what Aristotle argued. So the logician gets confused by a conflation of actual problems and potential problems, which require different types of logic to resolve, and are resolved in different ways. Instead of disentangling the potential from the actual, the logicians took the easy route, which was to redefine the premises. All this does is to bury the problem deeper in a mass of confusion.
You don't realize the game you're playing. Aristotle is doing the exact same thing. By your own admission it's Aristotle who is partitioning infinite into the category of ideas and away from reality, thereby changing the definitions of potential and actual. After all, in plain English "potential" is understood as a modal term, as a synonym for "possible". But for something to possibly be the case there must be some state of affairs where it obtains. Colloquially and philosophically, a potential can be actualized otherwise it's an impossibility. So no, you're just ignoring it when you do it because it's presumed to be acceptable for you to do so and only because it's you doing it. It's a convenient standard for you to have.
You haven't addressed the issue here. You only support these claims with a reified "space", assuming that space is a physical object to be studied, and not a conceptual object.
I never said space was physical. An object, sure. It has properties after all and we have studied these properties. I'll come to this in a moment.
What's this then?
, you are treating "space" as if it is something described by geometry. In reality, since we can use various different geometries to describe the various types of objects we sense, there is no such thing as "space". We might be able produce a concept of "space" from this geometry, and another concept of "space" from this other geometry, but it really makes no sense to talk about "how space is", or "if space is curved...", because there is no such thing as "space", not even as a concept.
I don't really see how you are saying anything here because you're moving between unrelated points. When I referred to "how space is" I was talking about the actual structure of spacetime, not the more vague, general concept of space. Some abstract geometric spaces are curved and some are not. Whether or not the structure of the actual space of the universe falls into one or the other is a physics question, and current physicsand observational evidence says space is curved. If you cannot even accept this there's no point in this.
This is why your geometrical examples are irrelevant, and way off the mark. You are talking about geometry as if it is created to describe some sort of "space"
Er, yes and no. The canonical application of geometry is to understand the spatial structure of the actual world. But I never said that's what geometry itself is about, it's about the study of abstract spatial structures (if you object that geometry isn't about I'm sorry you are so wrong I don't know how anything short of a mathematics textbook being regurgitated would correct you).
However, this is totally uncalled for. We produce principles of geometry to measure the objects which we encounter, and we do not encounter any infinite objects
I lost interest the moment I realized you treated measurement of objects as a fundamental concern of a geometry axioms. I don't encounter any perfect spheres, so surely it must be totally uncalled for to apply geometrical principles to reality where some object is arbitrarily similar to a perfect spheres since there cannot be any such thing in reality. Do you see why your view of geometry makes no sense to me?
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Infinite in your head only, not mathematically: width of a number is 0. How many in an interval sized 1? 1 / 0 = UNDEFINED.
You do not understand the concept of cardinality, do you? The size of the interval is the size of the continuum, aleph-1.
Infinity is greater than any assignable quantity; which implies is not a quantity (can't be a quantity and greater than any assignable quantity).
Why do you keep asserting this as fact? That is not how "infinity" is defined in mathematics because it's too hazy and informal a definition. An infinite set is, e.g. the transfinite Cardinals, a set whose members can be put into a one-on-one correspondence with a proper subset of themselves. There is absolutely no mention of being "greater than any assignable quantity, you're just wrong. Find one mathematics textbook that formally defines and describes infinity that way. Go on, I'm sure you can do it... (Obviously I'm being sarcastic here).
When you add one to it, nothing changes; clearly not a quantity. So it should not be present in mathematics. Which means no mathematical continua.
That's not an argument, that's a statement that you cannot justify. As it happens, it's perfectly understandable why the CARDINALITY doesn't change. The set does change if you add a new unique element, but the size cannot change by mere finite additions because we can still world new numbers to put into a function with the new element that was added.
I was just helping a friend out with a website. The RGB color values could be changed by the file in question as needed, but there seemed to be an oddity. Because color values are cappped at 255 per channel, adding any number to that channel resulted in no change to the value of that channel. I suppose 255 must not be a quantity then since adding to it did not cause the value to change. Saturation mathematics must be incoherent despite it's use in computer science then!
It's a little funny that mathematics by unjustified dogmatic assertion went out of vogue for everyone besides the ultrafinitists.
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Again, where can I find such mathematics so that I may interact with it? We can only interact with that which is actual, which is why both words have the same root; but mathematics deals entirely with the hypothetical. We use mathematics to model the actual, but that is not interacting with mathematics as if it were something that exists.
You can find this mathematics right in your mind. It's really there, and actual. An hypothesis has actual existence whether or not you believe it to be true.
However, it's clearly the case that in the interval of reals between 0 and 1 that 1 is an end point, yet people will when asked refer to that as infinite despite having a set, determinable end.
Actually, that is what is mistaken. In the case of the reals between 0 and 1, it is quite obvious that 1, like 0, is a defining point, and therefore a beginning point rather than an end point. What is claimed is that there is an endless number of points between 0 and 1. It is impossible that there could be a direction of procedure, because if we started at zero, and tried to progress toward 1 as an ending point, it is impossible to name the first real number after zero. Any named number after 0 would have more numbers between it and 0, and we'd have to turn around and go back, heading away from 1. Therefore, in this instance it is false to represent 1 as an "end point". There are two defined start points, 0 and 1, with an infinity of points between them. No end point.
didn't say we perceive infinity, I said our observations do not demonstrate that infinity is merely an idea. In fact, take the set of all observations ever made and assume they are of finite things. So what?
I guess you've never heard of inductive reasoning. Inductive reasoning is how we draw logical conclusions called generalizations, from observations. The bigger issue though which you didn't seem to grasp, is that all observations themselves, are necessarily finite.
It doesn't entail that they are necessarily the case, you (and Aristotle) arbitrarily define them to be such.
I know, I believe I already described to you how definitions are arbitrary. But those definitions which demonstrate a true correspondence are considered to be true definitions. So if we observe that the clear sky is always a similar colour, and we name this colour as "blue", so that everyone calls this colour blue, then we can define "blue" as "the colour of the sky". There is true correspondence because that is how people use the word "blue". But if everyone is referring to the "infinite" as endless, and we decide to define "infinite" in some other way, then we do not have true correspondence.
Worse, if you accept standard mathematics at all you have to agree that time and space are infinitely divisible
Why is this "worse"? Time and space are purely conceptual, just like numbers. Numbers are conceptual and infinite. If time and space are concepts produced from mathematics, why wouldn't they be infinite as well? A conclusion reflects its premises. The premise is that numbers are infinite. If time and space are concepts created from numbers they will reflect this infinity. Unless we allow that time and space are concepts created by something other than mathematics, they will necessarily be infinite. If time and space are other than mathematical, what would be the basis of these concepts, observation? Observations are necessarily finite. Therefore we have an incompatibility between the concepts of space and time which are consistent with mathematics, and the concepts of space and time which are consistent with observations. This has manifested as Zeno's paradoxes.
I also never observe my own brain activity, that doesn't entail my brain doesn't exist as an object. I don't observe exoplanets, that doesn't mean their existence is purely conceptual. Sensing a thing is not identical to that thing not existing. Furthermore, space is a thing. It is not even in question that space has properties, such as our being curved for instance. We can actually see curved space (gravitational lensing), so even then your criteria has been satisfied. And bearing properties is pretty much a fundamental requirement and sufficient condition for being an object.
The problem is that things unobserved do not enter into conceptions produced from observations So, even if there is a real thing out there, like space or time, which is truly infinite, the limitations of our senses deny us the capacity to observe the infinity of this thing. That's the classical, or colloquial understanding of "infinite", that it's impossible for the human being to observe. Let's assume that space and time are infinite, as the mathematical conceptions tell us, but our observations are incapable of corroborating this due to the limitations of our senses. Now we have the platform for Zeno-type paradoxes between the mathematical concepts of space and time, and the observational concepts. What do you think is the appropriate procedure to resolve the incompatibility? Do we face the fact that our observations are limited, and therefore fail us in this realm, and maintain a pure infinite in our concepts of space and time, or do we denigrate the pure infinite concept, and produce a new concept of "infinite" which is more consistent with our faulty observations? The latter is what the logicians have done, and what you seem to insist was the right thing.
You don't realize the game you're playing. Aristotle is doing the exact same thing. By your own admission it's Aristotle who is partitioning infinite into the category of ideas and away from reality, thereby changing the definitions of potential and actual.
What you're not respecting, is that for Aristotle ideas are part of reality. He was a student of Plato and was well trained in an ontology that holds ideas as real. To place infinity into the category of ideal, would only remove it from reality, if you proceed like aletheist above, on the preconceived notion that ideas are not real. Infinity, as well as mathematical ideas are very real for both Plato and Aristotle, so placing "infinity" into the category of ideal is not partitioning it away from reality.
After all, in plain English "potential" is understood as a modal term, as a synonym for "possible". But for something to possibly be the case there must be some state of affairs where it obtains. Colloquially and philosophically, a potential can be actualized otherwise it's an impossibility. So no, you're just ignoring it when you do it because it's presumed to be acceptable for you to do so and only because it's you doing it. It's a convenient standard for you to have.
You seem to be missing the point. I agree with what you have described here, a possibility is defined by actuality, what actually is. This is a specific possibility, it is only correctly "a possibility" if the actuality permits, otherwise it's impossible. Now let's move to the more general, "potential" what it means to be possible. What is it about reality which makes tings "possible"? What is the nature of contingency? We know that actuality defines a particular possibility as possible instead of impossible, but possibilities are not confined to one, they are by nature numerous. What do they have in common by which they are all possible? What actuality can we refer to in order to define what it means to have numerous things under the same name, as possible?
I lost interest the moment I realized you treated measurement of objects as a fundamental concern of a geometry axioms. I don't encounter any perfect spheres, so surely it must be totally uncalled for to apply geometrical principles to reality where some object is arbitrarily similar to a perfect spheres since there cannot be any such thing in reality. Do you see why your view of geometry makes no sense to me?
Yes, I see why my view of geometry makes no sense to you. You're speaking nonsense, and if this represents how you apprehend "geometry", your apprehension must be nonsensical as well. Did you just claim, that just because you haven't ever encountered a perfect sphere, you may conclude that geometry wasn't created for the purpose of measuring objects? What kind of nonsense is that?
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You can find this mathematics right in your mind. It's really there, and actual. An hypothesis has actual existence whether or not you believe it to be true.
As usual, this reflects conflation of the real with the actual.
Numbers are conceptual and infinite.
If numbers are infinite, and mathematics is actual, then I guess there is such a thing as an actual infinity after all. Right?
Now we have the platform for Zeno-type paradoxes between the mathematical concepts of space and time, and the observational concepts. What do you think is the appropriate procedure to resolve the incompatibility?
Recognize that continuous motion through space-time is a more fundamental reality than discrete positions in space and/or discrete instants in time. We arbitrarily impose the latter for the sake of measurement and calculation.
To place infinity into the category of ideal, would only remove it from reality, if you proceed like aletheist above, on the preconceived notion that ideas are not real.
Where on earth have I ever suggested that ideas are not real? Perhaps this reflects yet another conflation, this time between two definitions of "idea"--the content of an actual thought vs. anything whose mode of being is its mere possibility of representation. The latter is real even if it never actually gets represented, which means ...
I agree with what you have described here, a possibility is defined by actuality, what actually is.
... this is incorrect. Possibility is a distinct mode of being from actuality--and from (conditional) necessity, as well; none of them is dependent on either of the others. That is precisely why we must carefully distinguish logical possibility from actual possibility. Mathematics deals with that which is logically possible, regardless of whether it is actually possible.
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If numbers are infinite, and mathematics is actual, then I guess there is such a thing as an actual infinity after all. Right?
"Infinite" is a description of the numbers, as such it is qualitative, not quantitative. Mathematics cannot deal with the concept of "infinite" which is a description of mathematical objects made from outside the principles of mathematics, because it is not a mathematical principle. That's the problem here. The point being that whatever category you put the mathematical objects into, the descriptive term "infinite" is of another category, as the difference between the territory and the map, one being the object, the other a description of the object. The problem occurs when we attempt to make "infinite" a mathematical object.
Where on earth have I ever suggested that ideas are not real?
You insistently claim that numbers have no actual existence. The #1 definition of "real" in my OED is actually existing. So I concluded that you do not believe numbers to be real. You insist that numbers cannot interact with things in the world. Now you claim that ideas are real. I guess you use "real" in another way, to allow for something which is real, but cannot interact with our world. What sense is there in this, to allow for something real, which cannot interact with anything else in the world? So I haven't any idea what you might mean by "real" now because you seem to be claiming that there are real things which cannot in any way interact with the world we sense.
... this is incorrect. Possibility is a distinct mode of being from actuality--and from (conditional) necessity, as well; none of them is dependent on either of the others. That is precisely why we must carefully distinguish logical possibility from actual possibility. Mathematics deals with that which is logically possible, regardless of whether it is actually possible.
I don't see how "possibility" is at all relevant in your reality. It cannot interact with the actual world, as a distinct mode of being, so how could it be relevant? And "actual possibility" implies that the possibility is interacting with the world, but this contradicts what you've already claimed.
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I am afraid that I cannot make heads or tails of your first paragraph because of the confusion it exhibits regarding the meaning of terms--infinite, qualitative, quantitative, mathematics/mathematical, category.
You insistently claim that numbers have no actual existence.
Yes, but I claim just as insistently that numbers are nevertheless real, because ...
I guess you use "real" in another way, to allow for something which is real, but cannot interact with our world.
I have stated this explicitly and repeatedly--I deny that reality and actuality/existence are synonymous. Reality consists of that which is as it is regardless of what any individual mind or finite group of minds thinks about it. Actuality/existence is that which reacts with other like things in the environment. Reality includes some possibilities and some conditional necessities that may or may not ever be actualized.
And "actual possibility" implies that the possibility is interacting with the world, but this contradicts what you've already claimed.
There is no contradiction. Something is logically possible if it is merely capable of representation; something is actually possible only if it is also capable of actualization.
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There are two defined start points, 0 and 1, with an infinity of points between them. No end point.
Nonsense. The whole argument you're making assumes there needs to be counting - or as you called it, an "order of procedure" - in order for there to be an end point. And this is just false. The only way your argument could work would be by the hilarious assumption that the quantity of real numbers between any arbitrary interval were finite. Defining the start and end of something does not mean that end is not an end point. For goodness sake, a "race" has a defined start point and end point and no one would object "But sir, if you define the starting point and end point at once it's a defined point, not an end". The end point of an interval is not defined as the end of where you stop counting, come on. It's just the set of numbers you're quantifying over.
guess you've never heard of inductive reasoning. Inductive reasoning is how we draw logical conclusions called generalizations, from observations. The bigger issue though which you didn't seem to grasp, is that all observations themselves, are necessarily finite.
And I guess you've never heard that induction does not yield necessary conclusions like deduction does. The set of all observations simply, as I said, makes it more likely that the next observation will be of something finite. You claiming that they are necessarily finite is either begging the question (because you're presuming we can't observe some object that has some property which is infinite) or you're conflating induction with deduction. There are no necessary conclusions for inductive reasoning.
But if everyone is referring to the "infinite" as endless, and we decide to define "infinite" in some other way, then we do not have true correspondence.
The problem is that is not the exclusive colloquial definition of infinite for reasons I've already mentioned.
If time and space are concepts produced from mathematics, why wouldn't they be infinite as well?
What you seem to be missing is that the math is used to model the world and so far no model of finite space or time has any particular empirical or theoretical backing. You'd have to pin your hopes on something like Loop Quantum Gravity, but that's highly speculative and has about as going for it currently as String Theory does (not to say it won't change), whereas time and space are still modelled as continuums in both quantum mechanics and relativity. If the current models are accepted, it's just a performative contradiction to give them credence but to arbitrarily say some of their fundamental assumptions are to be presumptively excluded from reality.
Now we have the platform for Zeno-type paradoxes between the mathematical concepts of space and time, and the observational concepts. What do you think is the appropriate procedure to resolve the incompatibility
This is just false. Mathematics already resolved Zeno's paradoxes so clearly adopting the mathematical models of spacetime does not create paradoxes given we know how to resolve the apparent issues Zeno saw with having them be infinitely divisible. Zeno made fundamentally mistaken assumptions about the consequences of trying to cross an infinite series, they turned out to be negated.
What you're not respecting, is that for Aristotle ideas are part of reality.
Word game. By "reality" I meant being actual. You've already said you don't think this is possiblefor infinity, I was showing you how you were holding a hypocritical standard that only applies when other people use definitions you don't like, but you're perfectly find doing it yourself even if it's not the colloquial, "true" definition.
Now let's move to the more general, "potential" what it means to be possible. What is it about reality which makes tings "possible"? What is the nature of contingency? We know that actuality defines a particular possibility as possible instead of impossible, but possibilities are not confined to one, they are by nature numerous. What do they have in common by which they are all possible? What actuality can we refer to in order to define what it means to have numerous things under the same name, as possible?
Contingency and possibility are not the same thing. Necessary truths are also possible truths (because possibility just means truth in at least one possible world). Contingency means some modal proposition is true in some worlds and false in others. That aside, I don't see the relevancy in your questions about modality. In nearly all cases, potential is just a synonym for the term "possible". E.g. Every English speaker can readily understand "I have the potential to be a doctor", but sentences like "I have the potential to be a doctor but I cannot be a doctor" have to be disambiguated since it switches between two different types of modality (logical possibility and physical possibility), otherwise it's a flat contradiction. You can consistently say "I have the potential to be a doctor but in actuality I functionally cannot".
If you say X is a potential infinity but it cannot be actualized you are either contradicting yourself or you're switching between 2 different modalities. If it's the former, well that's not workable on pain of absurdity. If X cannot be actualized it's not a potential anything, the label doesn't fit. If it's the latter, then you're playing a shell game. You have to argue that Infinity isn't metaphysically possible, it's not contradictory so it's not inherently off the table for a consistency issue.
You're speaking nonsense, and if this represents how you apprehend "geometry", your apprehension must be nonsensical as well. Did you just claim, that just because you haven't ever encountered a perfect sphere, you may conclude that geometry wasn't created for the purpose of measuring objects? What kind of nonsense is that?
Dude, just prior to the part you quoted I said that the canonical application of geometry was for measurement:
, yes and no. The canonical application of geometry is to understand the spatial structure of the actual world. But I never said that's what geometry itself is about, it's about the study of abstract spatial structures (if you object that geometry isn't about I'm sorry you are so wrong I don't know how anything short of a mathematics textbook being regurgitated would correct you).
My point is you are confusing the canonical use of the thing with the thing itself, and that's just an obvious mistake. The most canonical use of arithmetic is for counting things. That doesn't mean arithmetic is just about counting. the canonical application of geometry is to measure things, but measurement isn't a geometric operation, it doesn't appear in the mathematical formalism of geometry. Geometry itself is about study certain types of mathematical structures with certain types of mathematical objects (points, lines, planes and so on). Theory and application are not the same thing.
• 5.4k
Nonsense. The whole argument you're making assumes there needs to be counting - or as you called it, an "order of procedure" - in order for there to be an end point. And this is just false.
Right, you think that there could be an "end point" without an order. You really like to argue by way of contradiction, don't you?
For goodness sake, a "race" has a defined start point and end point and no one would object "But sir, if you define the starting point and end point at once it's a defined point, not an end". The end point of an interval is not defined as the end of where you stop counting, come on. It's just the set of numbers you're quantifying over.
Yes, a race has a definite order of procedure, doesn't it? There could be no start point or end point without an order of procedure. Sorry, but contradiction just doesn't cut it. I produced a whole argument, and instead of addressing it, you dismiss it as "nonsense" by asserting a contradiction. As if you could prove someone's argument as nonsense by making a contradictory assertion.
And I guess you've never heard that induction does not yield necessary conclusions like deduction does. The set of all observations simply, as I said, makes it more likely that the next observation will be of something finite. You claiming that they are necessarily finite is either begging the question (because you're presuming we can't observe some object that has some property which is infinite) or you're conflating induction with deduction. There are no necessary conclusions for inductive reasoning.
Of course it's begging the question, it's the definition. I suppose if I assumed that a square is an equilateral rectangle you'd accuse me of begging the question.
So, what kind of infinite thing (infinity) do you think you could observe?
Word game. By "reality" I meant being actual. You've already said you don't think this is possible,
That's what I meant, "actual". If you saw my discussion with aletheist, you'd see that. For Aristotle, ideas, concepts, have actual existence, actualized by the human mind. This is the argument he uses against Platonic idealism. Ideas cannot be eternal, because only actual things can be eternal, and ideas are only given actual existence by the human mind, so they have a beginning and are therefore not eternal.
My point is you are confusing the canonical use of the thing with the thing itself, and that's just an obvious mistake. The most canonical use of arithmetic is for counting things. That doesn't mean arithmetic is just about counting. the canonical application of geometry is to measure things, but measurement isn't a geometric operation, it doesn't appear in the mathematical formalism of geometry. Geometry itself is about study certain types of mathematical structures with certain types of mathematical objects (points, lines, planes and so on). Theory and application are not the same thing.
More of the same, nonsense. The issue was whether or not we "produce principles of geometry to measure the objects which we encounter". You're just avoiding the issue by turning to a division between theory and application, as a diversion. Face the reality, even theoretical geometry is produced with the intent of measuring the objects which we encounter.
• 762
Yes, a race has a definite order of procedure, doesn't it? There could be no start point or end point without an order of procedure. Sorry, but contradiction just doesn't cut it. I produced a whole argument, and instead of addressing it, you dismiss it as "nonsense" by asserting a contradiction. As if you could prove someone's argument as nonsense by making a contradictory assertion.
I didn't assert a contradiction. Your claim was that you have to be able to count the series in order to declare an end point, which is false. A race can be run backwards, it can be run from the middle out to either end, etc. The order is irrelevant. A race doesn't end at a random point, the end is defined when the beginning is defined.
Of course it's begging the question, it's the definition. I suppose if I assumed that a square is an equilateral rectangle you'd accuse me of begging the question.
Disingenuous. The point is you cannot define it as necessarily impossible and then claim to have proven it to be the case. Induction only yields probable conclusions, you claimed the conclusion that infinity was impossible to actualiz was necessarily false, and you brought up inductive generalizations to prove that. You made a non sequitur, induction cannot give you necessary conclusions.
So, what kind of infinite thing (infinity) do you think you could observe?
Space and time. I observe and experience them, and our best models of them require the assumption that they are infinitely divisible.
The issue was whether or not we "produce principles of geometry to measure the objects which we encounter". You're just avoiding the issue by turning to a division between theory and application, as a diversion. Face the reality, even theoretical geometry is produced with the intent of measuring the objects which we encounter.
You say things like I can't just quote what was said before. The question was whether or not geometry was about measuring things. I said that was the canonical use of the discipline, but that's not what the discipline itself was about. That you're trying to claim the division being pointed out avoiding the issue is ridiculous. I brought it up because you said this:
We produce principles of geometry to measure the objects which we encounter, and we do not encounter any infinite objects
But that's absurd. We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry. The common assumption was that reality could not be any other way than as a model of a Euclidean Space until Non-Euckidean geometry came along and Relativity gave credence to thinking our actual space was best modelled as a pseudo Riemannian space. Funnily, Euclid made as a base assumption in his geometry that space was an infinite plane but I'm sure you'll object to that without question begging and ignoring that Euclid's assumption contradicts your claim that geometrical axioms are about measurement. It's about studying abstract math structures of a certain kind.
Anyway, theory and application aren't the same and the idea that geometry is fundamentally about measurement is wrong. If you think otherwise, show where measurement appears in the formalism of common geometries.
• 984
Space and time. I observe and experience them, and our best models of them require the assumption that they are infinitely divisible.
Do the assumptions underlying our best mathematical models of something qualify as observations and experiences of the real object itself? Our best mathematical models of buildings for structural analysis consist of finite elements, but no one would seriously claim that we observe and experience real buildings as collections of finite elements.
In any case I would suggest that space-time is an example of observable continuity, rather than observable infinity; and since the concept of infinite divisibility has proven problematic in past discussions, I would suggest infinite magnification as an alternative. No matter how much you were to "zoom in" on space-time, you would always "see" a four-dimensional continuum, all the way down to the infinitesimal level; never a discrete point at a discrete instant.
We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry.
Exactly, and the same is true of mathematics in general. We generate formal hypotheses and work out their necessary consequences, only some of which turn out to be useful for measuring or otherwise analyzing actual phenomena. That is precisely why we make a distinction between "pure" and "applied" mathematics.
• 762
Do the assumptions underlying our best mathematical models of something qualify as observations and experiences of the real object itself?
I should clear this up. I posit that those models, well evidenced as they are, are the best explanation of why we make the kinds of observations we make (e.g. not reaching any sort of discrete unit of space no matter the magnification). And since both QM and Relativity have some type of continuity to spacetime, we ought to accept this until such time as we have reason not to. I don't mean I completely see the infinite totality if a thing, but that whatever credence we give to observations in establishingnor refuting the actuality of infinities, our current observations of space and time don't seem to contradict this possibility at all.
• 984
That is fair. Can you elaborate on how QM supports the continuity of space-time? What is your interpretation of the Planck length and Planck time?
• 762
To be up front, my undergrad requirements for physics didn't really get into QM, so my "knowledge" of QM is a hackjob accrued from friends, colleagues and the Google box linking papers.
All that said, I'll give it a go. From the name, many people think QM must say space and time are quantized (discrete/, but many values in QM are continuous (position for instance), and space and time are among those values. In and of itself that doesn't mean too much, since you probably could modify the theory to use discrete values for these instead (although in practice there's no benefit to doing so).
If there's a minimum distance you start messing up a lot of other things in physics, especially as it relates to Relativity currently, such as people in different reference frames measuring different Planck length due to relativistic effects. S you'd probably have to drop Lorentz Invariance, but somewhat recent experimental observations (2011) haven't borne out high enough violations of it that would be expected if spacetime were discrete at some scale:
https://www.nature.com/articles/nature08574
Planck length and time are just measurement limitations at best. It's plausible that Planck lengths are the smallest measurable lengths, but there's no current reason to thinks it's a fundamental chunk of reality. Like these are somewhat taken arbitrarily. Inagine I was measuring things based on, I don't know, the radius of the Earth. That wouldn't mean everything else should be measured as if they were a multiple or divisor of the planet's radius. Natural units are all well and good. Some make a lot of signifance out of the Planck length but all we can really say that isn't speculation is that it sets a limit on the non-negligible effects of Quantum gravity. Our models of quantum gravity would only work down to that scale, so we'd need something more to model anything that exists at a smaller scale.
• 984
Thanks--all of that is consistent with my understanding, as well. :up:
• 5.4k
This is what you said:
Nonsense. The whole argument you're making assumes there needs to be counting - or as you called it, an "order of procedure" - in order for there to be an end point. And this is just false.
You are saying that there can be an end point without an order of procedure. That's contradictory, "end" implies order, by definition.. Your dismissal of my argument as "nonsense" relies on the truth of this contradiction. Since it is impossible that a contradiction is true, you need to go back and address my argument properly.
We don't produce axioms in geometry to measure things, that's just a very useful feature of geometry.
Let me see if I can understand what you're saying. You're saying that we do not produce axioms in geometry for the purpose of measuring things, we do it for some other purpose, maybe just for fun, or some arbitrary, random purpose. Then, voila, it just so happens by some random chance, that the principles of geometry prove to be very useful for measuring objects. Come on, get real.
If you think otherwise, show where measurement appears in the formalism of common geometries.
Are you serious? Measurement is everywhere in the formalism of geometry. 360 degrees in a circle is a measurement. Pythagorean theorem is a principle of measurement. I can't understand how you can appear to be so intelligent MindForged, but then fill your posts with such silly and even ridiculous statements.
It doesn't make sense to carry on this discussion because you just defend your position with contradictions and statements of random nonsense. Then you pretend that what I am saying doesn't make any sense in relation to your statements of contradiction and random nonsense.
.
• 647
, I happen to know @Metaphysician Undercover (and @aletheist) from the old now-defunct philosophyforums.com.
Metaphysician Undercover tend to wander off in some direction of own makings, yet imposing own ideas on other things. :)
As far as I can tell, @Devans99 just doesn't have much familiarity with the mathematics.
• 891
As far as I can tell, Devans99 just doesn't have much familiarity with the mathematics.
You do not have much familiarity with basic logic. A number cannot be larger than any number and be a number at the same time.
None of you will address this point directly.
• 647
, it's been addressed more than once by others (including here).
• 762
You do not have much familiarity with basic logic. A number cannot be larger than any number and be a number at the same time.
Cool. It's a a good thing "A number larger than any number" is not the definition of infinity in mathematics. The closest correct description of infinity that resembles what you're saying would be to say every infinite number is larger than any finite number. There's no logical error that results from infinity.
• 802
this reflects conflation of the real with the actual
...and the difference between "real" and "actual" is...? :chin:
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# 3 dice rolling program
Anybody know how to write a 3 dice rolling program like this
Sample output
Welcome to the dice roller!
First of all, please enter a seed for the random number generator. This should be a positive integer: 3
How many times would you like me to roll the 3 dice?
Sorry, please enter a positive number: 100
Okay, I’ll roll the dice 100 times and find the sum each time.
Here are the results:
Sum Frequency
3 0
4 1
5 3
6 3
7 6
8 15
9 12
10 18
11 13
12 13
13 3
14 6
15 4
16 3
17 0
18 0
....
Jan 17 '08 #1
2 3249
gpraghuram
1,275 Expert 1GB
Anybody know how to write a 3 dice rolling program like this
Sample output
Welcome to the dice roller!
First of all, please enter a seed for the random number generator. This should be a positive integer: 3
How many times would you like me to roll the 3 dice?
Sorry, please enter a positive number: 100
Okay, I’ll roll the dice 100 times and find the sum each time.
Here are the results:
Sum Frequency
3 0
4 1
5 3
6 3
7 6
8 15
9 12
10 18
11 13
12 13
13 3
14 6
15 4
16 3
17 0
18 0
....
As the question say you should use the seed function and call the rand function to get the number.
In a single dice maximum number is 6 and so whatever number you get mod it(%) by 6 to get a single dice value and same process for other piece.
Now you have value for one dice.
Youe should be using same logic for 3 pair of dice.
Hope you are understanding this
Thanks
Raghuram
Jan 17 '08 #2
Studlyami
464 Expert 256MB
Take a look at this threat (that was on page 1 of the c++ forum).Here
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Random File Program Loppunut left
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€27 - €4431
0 tarjoukset
random Loppunut left
i want to create a random number generator in java when a user inputs a min value and a max value i.e. a random number between 1 and 5... can any one help ?? i know u can use [kirjaudu nähdäksesi URL:n] method but i cant get it to work. ## Deliverables 1) Complete and fully-functional working program(s) in executable form as well as complete source code of all work
€4 - €9
€4 - €9
0 tarjoukset
random number Loppunut left
i want to create a random number generator in java when a user inputs a min value and a max value i.e. a random number between 1 and 5... can any one help ?? i know u can use [kirjaudu nähdäksesi URL:n] method but i cant get it to work. ## Deliverables 1) Complete and fully-functional working program(s) in executable form as well as complete source code of all work
€27 - €4431
€27 - €4431
0 tarjoukset
1) In the program given, use a Random Access file instead of a Sequential Access file. 2) Use a common dialog Control to allow the user to select the file open. ## Deliverables 1) Complete and fully-functional working program(s) in executable form as well as complete source code of all work done. ## Platform Visual Basic 6.0
€6 (Avg Bid)
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1 tarjoukset
Please look at [kirjaudu nähdäksesi URL:n] and read what I need. I tried to draw it out what I need and hopefully you will understand. I believe it to be simple but since I can't code, what in the hell do I really know. Payment to be made after testing of your product to determine if what you created is what I need. I can pay by scriptlance escrow, paypal or ikobo. The actual s...
€52 (Avg Bid)
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1 tarjoukset
...of html / javascript code that allows me to inbed on my website a random idea generator - this should allow me to basically call from a text file (or whatever you think is the best format ) separate singular elements when a button is pressed. This button will simply choose at random a singular element from the main list and display it on the page.
€18 (Avg Bid)
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Random Number Draw Loppunut left
This is just to draw random raffle numbers at a club. I envisage a large character display running through numbers between 1 and a preselected maximum and drawing a number and possibly more at a button press. I would like to be able to control the program by the following key sequence : Alt / Tab to switch to the draw program. All other keys to
€22 (Avg Bid)
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63 tarjoukset
Random Integers are uniformly distributed within a range of values. Write a program that uses simulation to see whether the units digits of the random numbers are uniformly distributed. Begin by declaring count as a 10-element array of integers. Using a loop, generate one million random integers in the range from 0 to 99,999. For each value, extract
€8 (Avg Bid)
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38 tarjoukset
...project..... i have to make a program that deals with 20 random card. please help......make it for me...... Assignment: (Design and implement a class called Card that represents a standard playing card. Each card has a suit and a face value. Create a program that deals 20 random cards. MINIMUM: *instance variables to represent the
€9 (Avg Bid)
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20 tarjoukset
...combo box. Perform the Goodness to Fit test to determine if the numbers do follow the uniform distribution. Using 10 intervals (0, 0.1, 0.2, ect.), determine the number of random numbers you generate that fall within each interval. Show the number of values that fall in each interval in a seperate list on the form. Compute the Chi-Square value and state
max €4
max €4
0 tarjoukset
...have quit a few computers that sit and do nothing now a days well i would like to turn them into a socks5 proxie but picking a random port 1 through 10,000 now once a day or once every few days change the port to a new random port (keeps scanners from finding it) once it changes the port have it send the IP from that computer to my base windows server
€226 (Avg Bid)
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3 tarjoukset
random access files Loppunut left
Using VB.Net create a random access file with 15 positions. This file contains the names of the cocktails which is served at the A+ club. Each cocktail has a number consisting of 3 digits. Record structure: *Cocktail number *Name *Main Ingredient *Danger measure The program must must be able to *Delete a record(write a blank
€46 (Avg Bid)
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8 tarjoukset
This is a simple script hack I could probably do it myself, but will waste too many hours trying to figure it out.. I would like this done ASAP Attached is a gallery script. I will explain how this works The gallery script allows you to choose directories and it will auto create thumbnails. But it doesn't create a PHYSICAL thumbnail, I believe it uses IMAGEMAGIK or DG to create them, P...
€9 (Avg Bid)
€9 Keskimäär. tarjous
1 tarjoukset
We are looking for a Flash solution that will randomly pick one out of the six flash animations we have when our website is loaded and keep on playing it, as if it was only one flash animation. To see an example of what we mean, visit <[kirjaudu nähdäksesi URL:n]> and press refresh a few times. The coder will be provided with both swfs and flas of the existing flash animations...
€25 (Avg Bid)
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22 tarjoukset
...each ad. Then the user will put a short code (javascript?) on their page (short as in 1-2 lines) - then when someone goes to their site, a popunder will happen, presenting a random HTML ad in the folder. The popunder window must be a specific size, like 400x600 or something. I need this done ASAP and i'm on a very tight budget! I don't think it will
€17 (Avg Bid)
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8 tarjoukset
Greetings, I am seeking a reliable, affordable within reason, person that KNOWS PHP. Occassionaly I get work that requires PHP knowledge. I really need someone that can take this work when I get it, and speak on behalf of the company (excellent English is a must via E-Mail and Phone). I have one project waiting so I need someone right away that can take over from here. Please ...
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29 tarjoukset
Random Array's Loppunut left
Given an arry of 100 random numbers in the range 1...999, write a function for each of the following processes. In building this array, if the random number is evenly div3ed by 3 or 7, store it as a negitive. (1) Print the array ten values to a line(aligned) (2) Print the odd values, ten to a line. (3) Print the values at the odd numbered index locations
€7 (Avg Bid)
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32 tarjoukset
need help using uniform random variables to apply central limit theorem to create normal random variables with zero mean and unity variance. a few quantity of samples should be enough. would like to use 12 and 24 for different problems. Also need help using 50 random variables to generate and arbitrary number of N normal random variables with mean = sqrt(3)
€17 (Avg Bid)
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2 tarjoukset
...second project is to demonstrate the use of friends and the use of a random number generator. You are to ask the user for the number of elements in a list (maximum 10). The random number generator will generate the number desired. The numbers will range from 1 to 7(also generated by a random number generator). Generate a second list in the same manner. You
€5 (Avg Bid)
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4 tarjoukset
Random Walk Loppunut left
Objective: Create a working program in the language C to provide a solution to the "Random Walk: problem described below, illustrating below, illustrating use of an array data structure. Input: None Algorithmic Requirements: Must make use of srand, rand, and an appropriate arrary structure. Output: Display the final array in the form of a checkerboard
€27 - €4431
€27 - €4431
0 tarjoukset
...is possible to create a piece of code (In VB) that could connect via a midi interface to midi equipment(i.e. synthesizers, drum machines, modules etc)and select a patch at random from the instrument. This will be then merged into a program I am developing If this is possible, and you can do this, I will setup a price and post a bid request for you.
€27 - €4431
€27 - €4431
0 tarjoukset
The program should be done either in Flash or VB 6. It simply allows the user to enter names of players and corresponsing weghts and stores them in database. What the program does is RANDOMLY and exclusively pairs the players with each other say P1-P4, P3-P5, P2-P6 etc.. But no 2 players will be paired with another player at a given time. It should also consider specific weight categories like in ...
€58 (Avg Bid)
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23 tarjoukset
Random Acces Loppunut left
I need to know how to write and read with a Random access files and how to put diffrent information into different text boxes but from the same file. ## Deliverables 1) Complete and fully-functional working program(s) in executable form as well as complete source code of all work done. 2) Installation package that will install the software (in ready-to-run
€27 - €4431
€27 - €4431
0 tarjoukset | 5,277 | 20,274 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2019-13 | latest | en | 0.788974 |
https://wizardofodds.com/games/baccarat/appendix/7/ | 1,675,210,273,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499891.42/warc/CC-MAIN-20230131222253-20230201012253-00470.warc.gz | 647,586,372 | 13,977 | ## Wizard Recommends
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# Baccarat Flashing Dealers
## Introduction
It has been known to happen that the dealer in baccarat will flash the next card to be dealt before closing bets. For those unfamiliar with the term, to flash a card means to show it prematurely, in this case before the player makes a bet. When this happens the player has a huge advantage. This section will tell you what to bet on according to the flashed card, the probability of winning, and your advantage. All tables are for eight decks.
The following table shows the probability of each bet winning according to the flashed card.
### Probability of Winning by Flashed Card — Eight Decks
Flashed Card Banker Wins Player Wins Tie Wins
0 0.495203 0.415228 0.089568
A 0.493789 0.415476 0.090736
2 0.490844 0.417197 0.091959
3 0.487016 0.419852 0.093132
4 0.479533 0.423290 0.097177
5 0.468070 0.433347 0.098583
6 0.442579 0.450141 0.107279
7 0.409506 0.483157 0.107337
8 0.365435 0.538370 0.096195
9 0.344181 0.559462 0.096357
The following table shows the expected value of each bet winning according to the flashed card.
### Expected Value by Flashed Card — Eight Decks
Flashed Card Banker EV Player EV Tie EV Best Bet
0 0.055215 -0.079975 -0.193887 Banker
A 0.053624 -0.078313 -0.183377 Banker
2 0.049105 -0.073647 -0.172369 Banker
3 0.042813 -0.067164 -0.161812 Banker
4 0.032266 -0.056243 -0.125407 Banker
5 0.011320 -0.034723 -0.112753 Banker
6 -0.029691 0.007562 -0.034488 Player
7 -0.094126 0.073651 -0.033967 Player
8 -0.191207 0.172935 -0.134245 Player
9 -0.232490 0.215281 -0.132787 Player
All things considered, the player advantage, if the dealer always flashes, is 6.76%.
Go back to baccarat
Go to baccarat appendix 1
Go to baccarat appendix 2
Go to baccarat appendix 3
Go to baccarat appendix 4
Go to baccarat appendix 5
Go to baccarat appendix 6
Baccarat: The Known Card — Article by Eliot Jacobson on not just the advantage of a known card in baccarat, but how advantage players attain can attain that information by exploiting the "ribbon spread."
Written by:
## Wizard Recommends
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• \$3000 Welcome Bonus | 741 | 2,254 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2023-06 | longest | en | 0.761408 |
https://ae.edugain.com/articles/4/Algebra-A-fascinating-subject | 1,537,965,623,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267164925.98/warc/CC-MAIN-20180926121205-20180926141605-00001.warc.gz | 440,713,910 | 14,085 | ## Algebra - A fascinating subject
Algebra is when most children get introduced to the concepts of “abstract” mathematics. Suddenly their world gets involved in quantities like “x” and “y”s. And if not introduced properly it can lead to fear of the subject. But if taught properly they can see that the “x” and “y” are like words that form a beautiful poem and they’ll delight themselves when they learn this secret language and understand the beauty of algebra.
Understanding and practice can go a long way in helping children here. Sites like EduGain (www.edugain.com) can help children in doing this.
Suppose you take a long, really long string. You then cut out enough of it so that it can go all the way around the earth (I told you it was really long).
So now you have this string that’s on the ground and encircling the earth along the equator.Now suppose you add another 30 centimeters to that string (that’s about the length of your long ruler). We then smoothen it out by pulling it up from the ground all over the equator so that it forms a proper circle again.
The question is “How high do you think the string will be above the ground now?”
Sounds very hard to solve, right. But let’s see…
We know that the circumference of a circle is C = 2 x ? x R, where R is the radius.
(don’t worry about ? – let’s just use the value of “3? for now)
So, C = 2 x 3 x R = 6 x R
Now we don’t know the exact radius of the earth here, but we don’t need to. Let’s leave it as a symbol R. So the length of the original string = 6 x R. We’ll just call it 6R (and know it means the value of R, whatever it is, multiplied by 6).
So C = 6R
Now we added 30 cm to C. The length of the new string is now C+30.
What we need to find out is how much R increases.
Let’s say that R increased by some length Z. The value of Z is what we want to find
So for the string, we can say it has a new circumference (Cnew) a new radius (Rnew)
We also know that since it’s a circle
Cnew = 6Rnew
And we know that the new circumference Cnew = C+30,
and that
Rnew = R + Z (remember we already assumed R increased by a length Z which we are trying to find out).
So we get
C + 30 = 6(R + Z).
Expanding this, we get,
C + 30 = 6R + 6Z
But we already know that 6R = C
Let’s replace the 6R by C on the right side
C + 30 = C + 6Z
The two “C”s cancel out, leaving us with
30 = 6Z, or
Z = 30/6 = 5
Amazing isn’t it. The new string would be 5 cm above the ground all everywhere. Just by adding 30 cm extra.
And what’s more amazing – we solved it through Algebra
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54. Posted by VRISHABH| 2017-11-11 02:37:04
did'ent get it. sorry | 2,794 | 9,402 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.46875 | 4 | CC-MAIN-2018-39 | latest | en | 0.932682 |
http://www.physicsforums.com/showthread.php?t=117581 | 1,386,226,651,000,000,000 | text/html | crawl-data/CC-MAIN-2013-48/segments/1386163041297/warc/CC-MAIN-20131204131721-00035-ip-10-33-133-15.ec2.internal.warc.gz | 490,575,856 | 8,815 | # Reciprocal Lattice Simple Question
by TheDestroyer
Tags: lattice, reciprocal, simple
P: 390 If we are studying FCC in the direct lattice, Why does the length of the cube side in the reciprocal lattice equal to 4*Pi/a Where a is the lattice constant, a*=|G|=2*Pi/a Sqrt(4) = 4*Pi/a Where a* is the length of the cube site in reciprocal lattice Note: this thing is repeated in 2 problems and i wouldn't be able to know the reason. Prefessor is writing it like this but i can't understand (LOL, he also doesn't know to answer me when i asked him, he's just reading from papers, Silliy Professors) Any one can explain? thanks
P: 576 the factor 4 is usually left there to make it clear that fccs reciprocal lattice is the bcc lattice with a lattice constant of 4pi/a.
P: 390 Can you explain it in mathemtical way? cuz this explanation is refused when it's said like that!! Thanks
P: 576
## Reciprocal Lattice Simple Question
allright. let's look at the primitive reciprocal lattice vector b1 of the fcc lattice. the a vectors are the direct space primitive lattice vectors.
$$b_1=2\pi \frac{a_2 \times a_3}{a_1 \cdot (a_2 \times a_3}$$
Just plug in the fcc vectors and do the cross products and you'll get
$$b_1=\frac{4\pi}{a} 1/2(y+z-x)$$
which is the a1 for bcc with a lattice constant of 4pi/a.
P: 390 OK! Why did you put 1/2? I know the reciprocal of fcc is 2Pi/a (-x+y+z) Why did you multiply and device by 1/2???? HERE IS THE QUESTION :P Thanks
P: 576 The half is there just to show the connection between fcc and bcc in direct and reciprocal space.
PF Patron Sci Advisor Emeritus P: 11,137 The basis vectors of an FCC in a symmetric form are : $$a_1=\frac{a}{2}(\hat{x} + \hat{y})$$ $$a_2=\frac{a}{2}(\hat{y} + \hat{z})$$ $$a_3=\frac{a}{2}(\hat{z} + \hat{x})$$ If you plug these into the equation provided by inha in post#4 for the reciprocal lattice vectors, you get : $$b_1=\frac{2\pi}{a}(\hat{x} + \hat{y} - \hat{z})$$ $$b_2=\frac{2\pi}{a}(\hat{y} + \hat{z} - \hat{x})$$ $$b_3=\frac{2\pi}{a}(\hat{z} + \hat{x} - \hat{y})$$ (also, as posted by inha in post #4) The trick, next, is to recall that the basis vectors for a BCC, in symmetric form are : $$a_1=\frac{a'}{2}(\hat{x} + \hat{y} - \hat{z})$$ $$a_2=\frac{a'}{2}(\hat{y} + \hat{z} - \hat{x})$$ $$a_3=\frac{a'}{2}(\hat{z} + \hat{x} - \hat{y})$$ where a' is the BCC lattice parameter (or cube edge). Since these have the same form as the reciprocal vectors of the FCC, we understand that the reciprocal lattice of the FCC is in fact, a BCC. Secondly, comparing coefficients, we find that : $$\frac{2\pi}{a} = \frac{a'}{2}$$ $$\implies a' = \frac{4\pi}{a}$$
P: 390 Thanks, I got it
P: 1
Quote by TheDestroyer Thanks, I got it
I think it is quite easy. By normally calculate we can get b=4*pi/a for fcc and bcc
Related Discussions Atomic, Solid State, Comp. Physics 5 Atomic, Solid State, Comp. Physics 2 Advanced Physics Homework 15 Advanced Physics Homework 3 Introductory Physics Homework 1 | 957 | 2,967 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.953125 | 4 | CC-MAIN-2013-48 | longest | en | 0.899949 |
https://www.southampton.ac.uk/courses/modules/math6149.page | 1,534,253,124,000,000,000 | text/html | crawl-data/CC-MAIN-2018-34/segments/1534221209040.29/warc/CC-MAIN-20180814131141-20180814151141-00160.warc.gz | 949,739,048 | 11,235 | The University of Southampton
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# MATH6149 Modelling with Differential Equations
## Module Overview
The emphasis of this module is on the methods required to develop mathematical models using differential equations to understand physical problems. The module involves both conventional lectures as well as discussion lectures. The discussion lectures comprise structured group work in which small groups of students develop mathematical models to solve practical problems in partnership with one another and under the guidance of the lecturer (attendance at these discussion lectures is an essential part of the module). After an introduction to the module there are four blocks, in which the opening lectures will introduce the students to a physical problem and subsequent discussion lectures will allow possible modelling methods to be explored. Some lectures on relevant mathematical theory will also be presented. After each block students will write a report describing their investigations.
### Aims and Objectives
#### Module Aims
• To give students a good understanding of how mathematical modelling is used to solve real world problems. • To introduce students to a wide variety of practical situations where the theoretical aspects of differential equations can give physical insight • To give students an understanding of the successful use of mathematics to solve a problem through experiment, numerical methods and analytical approximation. • To give students experience of working in small research groups.
#### Learning Outcomes
##### Knowledge and Understanding
Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:
• Distil the major elements of a physical problem into a mathematical model that is simple enough to allow some intelligent predictions to be made and valuable conclusions to be drawn.
• Identify the basic sorts of differential and algebraic equations that are commonly encountered during mathematical modelling of physical processes
• Understand a wide range of models that have been used in practical situations and be able to relate them to other situations.
##### Subject Specific Intellectual and Research Skills
Having successfully completed this module you will be able to:
• Present, either in written or verbal form, or a combination of both, results from models that have been derived during the course
##### Transferable and Generic Skills
Having successfully completed this module you will be able to:
• Learn how to work closely with other group members making use of varying expertise within the group.
### Syllabus
• Mathematical modelling methodology. • Modelling using ODEs, both single and coupled. • Modelling using PDEs, both single and coupled. • Approximation and perturbation techniques. • Use of computer packages (e.g. Matlab) in solving problems and presenting their solutions.
### Learning and Teaching
#### Teaching and learning methods
Student led small group discussions, lectures, private study and research.
TypeHours
Independent Study114
Teaching36
Total study time150
#### Resources & Reading list
A B TAYLER (1986). Mathematical Models in Applied Mechanics.
S HOWISON (2005). Practical applied mathematics.
### Assessment
#### Summative
MethodPercentage contribution
Group Projects 100%
#### Referral
MethodPercentage contribution
Coursework assignment(s) 100%
#### Repeat Information
Repeat type: Internal & External
### Linked modules
Prerequisites: MATH2038 AND (MATH1057 OR MATH2008) AND (MATH3018 OR MATH3052 OR MATH3072).
### Costs
#### Costs associated with this module
Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.
In addition to this, students registered for this module typically also have to pay for:
##### Books and Stationery equipment
Course texts are provided by the library and there are no additional compulsory costs associated with the module
Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.
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× | 835 | 4,522 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2018-34 | longest | en | 0.912733 |
https://vs.eyeandcontacts.com/2022/05/ncert-class-6-maths-chapter-14.html | 1,653,284,333,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662555558.23/warc/CC-MAIN-20220523041156-20220523071156-00594.warc.gz | 693,389,047 | 21,912 | # vs.eyeandcontacts.com
## Chapter 14 Practical Geometry Exercise 14.1
Question 1: Draw a circle of radius 3.2 cm.
Given
Circle
Rough Figure
Actual Figure
Steps of Construction
1. Mark a point O anywhere.
2. With O as centre, radius 3.2 cm draw a circle.
Question 2: With the same centre O, draw two circles of radii 4 cm and 2.5 cm.
Given
Circle
Centre O
Radii 4 cm and 2.5 cm
Rough Figure
Actual Figure
Steps of Construction
1. Mark a point O anywhere.
2. With O as centre, radius 2.5 cm draw a circle.
3. With O as centre, radius 4 cm draw another circle.
Question 3: Draw a circle and any two of its diameters. If you join the ends of these diameters, what is the figure obtained? What figure is obtained if the diameters are perpendicular to each other? How do you check your answer?
i) Draw a circle having its centre ‘O’, also of any radius. Let PR and QS be the two diameters of the circle. A quadrilateral is formed when the ends of PR and QS are joined.
Since, the diameter of a circle are equal in length, quadrilateral that is formed will have its diagonals of equal length. By measuring, PQ = SR and PS = QR. Since, ∠P = ∠Q = ∠R = ∠S = 90°, the PQRS is a rectangle.
ii) Draw a circle having its centre ‘O’, also of any radius. Let PR and QS be the two diameters that are perpendicular to each other in the circle. A quadrilateral is formed when the ends of WY and XZ are joined.
Since, the diameter of a circle are equal in length, quadrilateral that is formed will have its diagonals of equal length. By measuring, WY is perpendicular to XZ, WX = XY = YZ = ZW. Since, ∠W = ∠X = ∠Y = ∠Z = 90°, the WXYZ is a rhombus.
Question 4: Draw any circle and mark points A, B and C such that
a) A is on the circle.
b) B is in the interior of the circle.
c) C is in the exterior of the circle.
Question 5: Let A, B be the centres of two circles of equal radii; draw them so that each one of them passes through the centre of the other. Let them intersect at C and D. Examine whether line segment AB and line segment CD are at right angles.
Draw two circles having same radius which are passing through the centre of the other circle. The centre of the two circles are A and B and these circle are intersecting at point C and D respectively. | 609 | 2,261 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.78125 | 5 | CC-MAIN-2022-21 | longest | en | 0.93591 |
http://mathoverflow.net/questions/53445/real-closed-fields-minus-existentials-for-presburger-like-power-and-multiplicati?sort=newest | 1,469,378,283,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824113.35/warc/CC-MAIN-20160723071024-00012-ip-10-185-27-174.ec2.internal.warc.gz | 150,783,580 | 16,023 | # Real-closed fields minus existentials for Presburger-like power and multiplication?
I was reading these slides by John Harrison, and was struck by the comment at the end about the universal fragment of real-closed fields needing nothing more than the axioms for an (ordered) integral domain. Since an obvious integral domain is the integers, and we can express order constraints (such as $\ge 0$) on them, does this give an algorithm for deciding universal statements on the naturals that involve addition and multiplication (by non-constants!)?
For someone interested in automatic verification of programming languages with type systems, this would be very handy if true. Can anyone provide any insight on this? I'm not a mathematician but enjoy reading about math, so I'm not positive my interpretation is correct.
P.S: I realize this is similar to my previous question, but it's more specific (and probably actually possible) so I hope to get more feedback.
-
The universal theory of arithmetic, in the language with $+$ and $\cdot$, is not decidable, because this is exactly sufficient to ask whether a given diophantine equation has no solutions in the integers, and this is not decidable by the MRDP theorem, which solves Hilbert's 10th problem.
The comment in the slides asserts that every ordered integral domain extends to a real closed field, or equivalently that the universal theory of real closed fields is a subset of the unversal theory of ordered integral domains. Is this useful in practice to decide whether a universal sentence is true in the integers? Sure, in the trivial sense that, e.g. the diophantine equation $x^2+y^2+1=0$ has no integer solutions because it has no solutions in any real closed field. | 362 | 1,734 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2016-30 | latest | en | 0.938648 |
https://space.stackexchange.com/questions/27534/what-sort-of-orbital-elements-are-used-to-describe-halo-orbits?noredirect=1 | 1,653,791,341,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652663035797.93/warc/CC-MAIN-20220529011010-20220529041010-00550.warc.gz | 602,101,553 | 70,593 | # What sort of orbital elements are used to describe halo orbits?
For standard orbits we can use Keplerian elements, TLE, or other similar. These don't make much sense for Halo orbits, which are not around a central body, but around a Lagrangian point, and follow an entirely different set of rules. What description is used to parametrize these?
For example: how would the description of Queqiao orbit look like, and how could I go about reading it, to see e.g. if it's eclipsed by Moon some of the time, or remains in view of Earth at all times?
• I'd guess orbital state vectors, but I'm not sure enough to answer. May 30, 2018 at 9:10
• @Polygnome: While they work for any orbit, they provide very little "readily digestible" information about these. You'd be hard-pressed to tell halo orbit, lunar orbit, Earth orbit and escape trajectory apart, just at a glance, for the same $\overrightarrow{r}$ but slightly different $\overrightarrow{v}$.
– SF.
May 30, 2018 at 9:20
• Yeah, with TLEs you can "imagine" the orbit in your head, but is that really important? I mean, usually you'd just plug them into whatever software you are using and plot the orbit (or convert to celestial coordinates if you want to watch the object or communicate with it). orbital state vectors work well for electronic interchange. And TLEs were not meant to be human-readable, they were meant to fit onto punch cards... I've only ever worked with either TLEs or orbital state vectors, but I have no idea what the industry standard is, hence no answer. May 30, 2018 at 9:28
• Mission design is usually done with state vectors and control vectors. However Halo + Lissajous orbits are often visualised with a body-line centred projection which helpfully reduces the elements to familiar Keplerian ones. This projection makes it much easier to identify eclipses etc.
– Jack
May 30, 2018 at 9:35
• Here's a source I've used when trying (and failing) to wrap my head around Halo orbits
– Jack
May 30, 2018 at 9:37
tl;dr: For a given pair of bodies in circular orbits around their center of mass, there are two symmetric families ("Northern" and "Southern") of proper halo orbits associated with each of the Lagrangian points L1, L2, and L3. We usually only talk about those with L1 and L2 because L3 is so far away from the secondary body (Earth in the case of Sun-Earth Lagrangian points, the Moon in the case of Earth-Moon). So you need three parameters; two enumerations and one floating-point value. 1) North or South, 2) L1, L2, or L3-associated, and 3) some floating point number that represents the position that the orbit lies in between the two extreme ends of the family where it either terminates or bifurcates. So far I don't know if that has a generally accepted parameterization that always works, or not. I am not sure if something simple like "energy" ($$C_3$$) or some amplitude or distance would work without ambiguities in some cases.
As a practical answer, you could describe a periodic halo orbit with an in-plane amplitude $$A_Y$$ and out-of-plane amplitude $$A_Z$$ to someone, and then they could try to calculate the orbit and find the X, Y, and Z positions as a function of time to get the motion in space, and then determine when the orbit would be blocked from points on the Earth by the Moon. I discuss this further in this answer, but see the pics below from Robert W. Farquhar's hundred page tome The Utilization of Halo Orbits in Advanced Lunar Operations, NASA Tech. Note D-6365.
But remember: this is for *circular orbits of 2 bodies only, and the real Moon's movement (and other effects) is more complex.
In section II.B.2.b, he points out:
For every value of $$A_y$$ > 32,871 km, there is a corresponding value of $$A_z$$ that will produce a nominal path where the fundamental periods of the y-axis and z-axis oscillations are equal. In this case, the nominal path as seen from the earth will never pass behind the moon. The exact relationship between $$A_y$$, and $$A_z$$, for this family of nominal paths is given in Figure 5.
The extremely cool and colorful paper E. J. Doedel et al, (2007) Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem International Journal of Bifurcation and Chaos 17, 2625 (2007). https://doi.org/10.1142/S0218127407018671 builds a system of illustrations that show all of the known, periodic, orbits in the CR3BP (Circular Restricted Three-Body Problem). This includes many kinds or classes of orbits as shown in the table, but excludes Lissajous Orbits because they are not in general periodic. (note: ignore the drawing in the Wikipedia article!)
You can and probably should also download the paper from its non-paywalled ResearchGate site, make some coffee, then spend six months enjoying it.
There is also an un-paywalled copy of their earlier paper available: The Computation of Periodic Solutions of the 3-Body Problem Using the Numerical Continuation Software AUTO D. J. Dichmann, E. J. Doedel, and R. C. Paffenroth Int. Conf. on Libration Point Orbits and Applications, Aiguablava, Spain, 10-14 June, 2002
I have made three montages of Figure 3 with Figures 13 (L1), 14 (L2) and 15 (L3) and shown them below. For each, only the Northern Halo orbit is shown, the Southern would be symmetrically reflected below the plane. These drawings use the Earth-Moon system for simple visualization, and Figure 3 is for the mass ratio of the Moon to the Earth ($$\mu \approx 0.01215$$).
You can also see how to generate and plot a few Halo orbits with Python using the script in the question How to best think of the State Transition Matrix, and how to use it to find Halo orbits? which comes from the classic paper written by Kathleen Connor Howell Three-Dimensional, Periodic 'Halo' Orbits Celestial Mechanics 32 (1984) 53-71.
Caption for Figure 3: (the lower part with all the elbows):
Fig. 3. Bifurcation diagram for the Earth–Moon system (μ = 0.01215), showing families of periodic orbits that emanate from the libration points and from subsequent branch points. The red cubes are the libration points. Small white spheres denote branch points, and small dark-red spheres denote collision orbits. The planar families C1, C2, and D1, are only partially represented; in particular, the fact that D1 arises from C1 via a period-doubling bifurcation is not indicated in the diagram. A glossary of the notation used is given in Table 1.
• @SF. It's hard for me to gauge how much more to write here. If you can help me by pointing out which parts are most unclear or need elaboration, that would be great!
– uhoh
Jun 3, 2018 at 8:28
• I'll need some time to digest through what you posted for now!
– SF.
Jun 3, 2018 at 20:14
• @SF. I've just asked How many kinds of periodic orbits are there in the circular restricted three-body problem? and I'll post an answer there within a day. Background in that answer will hopefully help make reading this answer easier. It's a bit of onion-peeling.
– uhoh
Jun 4, 2018 at 1:52 | 1,728 | 7,021 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2022-21 | longest | en | 0.94226 |
https://goprep.co/ex-16.3-q8-three-coins-are-tossed-once-find-the-probability-i-1nk8x4 | 1,610,744,561,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703496947.2/warc/CC-MAIN-20210115194851-20210115224851-00515.warc.gz | 359,242,668 | 38,678 | # Three coins are tossed once. Find the probability of getting(i) 3 heads(ii) 2 heads(iii) at least 2 heads(iv) at most 2 heads(v) no head(vi) 3 tails(vi) exactly two tails(vii) (viii) no tail(viii) (ix) at most two tails
When three coins are tossed then S = {HHH, HHT, HTH, THH, TTH, HTT, TTT, THT}
Where s is sample space and here n(S) = 8
Let A be the event of getting 3 heads
n(A)= 1
Let A be the event of getting 2 heads
n (A) = 3
Let A be the event of getting at least 2 head
n(A) = 4
let A be the event of getting at most 2 heads
n(A) = 7
Let A be the event of getting no heads
n(A) = 1
(vi) Let A be the event of getting 3 tails
n(A) = 1
(vii) Let A be the event of getting exactly 2 tails
n(A) = 3
(viii) Let A be the event of getting no tails
n(A) = 1
(ix) Let A be the event of getting at most 2 tails
n(A) = 7
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view all courses | 378 | 1,204 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2021-04 | latest | en | 0.887678 |
https://www.lmfdb.org/EllipticCurve/Q/188760/bd/1 | 1,632,476,181,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057508.83/warc/CC-MAIN-20210924080328-20210924110328-00070.warc.gz | 880,109,279 | 30,889 | # Properties
Label 188760.bd1 Conductor $188760$ Discriminant $707490600960$ j-invariant $$\frac{31103978031362}{195}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$
# Related objects
Show commands: Magma / Pari/GP / SageMath
## Minimal Weierstrass equation
sage: E = EllipticCurve([0, -1, 0, -1006760, -388474260])
gp: E = ellinit([0, -1, 0, -1006760, -388474260])
magma: E := EllipticCurve([0, -1, 0, -1006760, -388474260]);
$$y^2=x^3-x^2-1006760x-388474260$$
## Mordell-Weil group structure
$\Z\times \Z/{2}\Z$
### Infinite order Mordell-Weil generator and height
sage: E.gens()
magma: Generators(E);
$P$ = $$\left(3777, 222882\right)$$ $\hat{h}(P)$ ≈ $5.9814062195409619924476448144$
## Torsion generators
sage: E.torsion_subgroup().gens()
gp: elltors(E)
magma: TorsionSubgroup(E);
$$\left(-579, 0\right)$$
## Integral points
sage: E.integral_points()
magma: IntegralPoints(E);
$$\left(-579, 0\right)$$, $$(3777,\pm 222882)$$
## Invariants
sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); Conductor: $$188760$$ = $2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13$ sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); Discriminant: $707490600960$ = $2^{11} \cdot 3 \cdot 5 \cdot 11^{6} \cdot 13$ sage: E.j_invariant().factor() gp: E.j magma: jInvariant(E); j-invariant: $$\frac{31103978031362}{195}$$ = $2 \cdot 3^{-1} \cdot 5^{-1} \cdot 13^{-1} \cdot 109^{3} \cdot 229^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.8793264922537461014822426488\dots$ Stable Faltings height: $0.044993940341277629152141415147\dots$
## BSD invariants
sage: E.rank() magma: Rank(E); Analytic rank: $1$ sage: E.regulator() magma: Regulator(E); Regulator: $5.9814062195409619924476448144\dots$ sage: E.period_lattice().omega() gp: E.omega[1] magma: RealPeriod(E); Real period: $0.15071900933494728383348771185\dots$ sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] magma: TamagawaNumbers(E); Tamagawa product: $4$ = $1\cdot1\cdot1\cdot2^{2}\cdot1$ sage: E.torsion_order() gp: elltors(E)[1] magma: Order(TorsionSubgroup(E)); Torsion order: $2$ sage: E.sha().an_numerical() magma: MordellWeilShaInformation(E); Analytic order of Ш: $4$ = $2^2$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial() gp: ar = ellanalyticrank(E); gp: ar[2]/factorial(ar[1]) magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L'(E,1)$ ≈ $3.6060464793564239725933992839377481334$
## Modular invariants
Modular form 188760.2.a.bd
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
magma: ModularForm(E);
$$q - q^{3} + q^{5} - 4q^{7} + q^{9} - q^{13} - q^{15} - 6q^{17} + O(q^{20})$$
sage: E.modular_degree() magma: ModularDegree(E); Modular degree: 1966080 $\Gamma_0(N)$-optimal: no Manin constant: 1
## Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $II^{*}$ Additive 1 3 11 0
$3$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$5$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$11$ $4$ $I_0^{*}$ Additive -1 2 6 0
$13$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
## Galois representations
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.6.0.1
## $p$-adic regulators
sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
No Iwasawa invariant data is available for this curve.
## Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 188760.bd consists of 3 curves linked by isogenies of degrees dividing 4.
## Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{390})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-429})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-110})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-110}, \sqrt{390})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial. | 2,063 | 5,408 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.234375 | 3 | CC-MAIN-2021-39 | longest | en | 0.231177 |
https://www.englishpedia.net/sentences/a/linear-approximation-in-a-sentence | 1,679,914,151,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296948620.60/warc/CC-MAIN-20230327092225-20230327122225-00360.warc.gz | 852,326,147 | 4,545 | # linear approximation in a sentence
1) The coefficients of the linear approximation are determined as follows.
approximation collocations
2) However, linear function approximation is not the only choice.
3) If __FORMULA__ this is the same as the linear approximation .
## linear approximation example sentences
4) Are these linear approximations appropriate for all investigations in biology?
5) Duration is a linear approximation to the present-value profile.
6) POLE learns complex (nonlinear) functions by piecewise linear approximation .
7) Clearly, exponential approximation produced the best fit, followed closely by linear approximation .
8) For small changes in yield, the linear approximation will be reasonably good.
9) The following theorem states that a Karhunen-Loeve basis is optimal for linear approximations .
10) Instead of working with full polynomials, we can use a linear approximation .
11) Because the linear approximation is only an approximation, __FORMULA__ is different for different reference temperatures.
12) Examples are methods such as Newton's method, fixed point iteration, and linear approximation .
13) The linear approximation is important for maintaining a mathematically tractable analysis of systems perturbed by noisy inputs.
14) The modified duration is a measure of the price sensitivity to yields and provides a linear approximation .
15) NLERP is generally faster than SLERP, but is a linear approximation of a second order calculation.
### example sentences with linearapproximation
16) Linear approximation , on the other hand, seemed to excellently fit the model of intermittent learning.
17) But for large jumps in yield, the linear approximation will become very poor if convexity is high.
18) The tangent plane is the best linear approximation , or linearization, of a surface at a point.
19) The derivative gives the best possible linear approximation , but this can be very different from the original function.
20) If one were to make the linear approximation of , none of the effects under consideration would be obtained.
21) Thus the __FORMULA__ dimensional principal components provide the best linear approximation of rank __FORMULA__ to the observed data matrix __FORMULA__.
22) In the linear approximation that leads to the above acoustic equation, the time average of this flux is zero.
23) Another way of proving the chain rule is to measure the error in the linear approximation determined by the derivative.
24) This equation represents the best linear approximation of the function __FORMULA__ at all points __FORMULA__ within a neighborhood of __FORMULA__.
25) It can be shown that - to a linear approximation - it is always possible to make the field traceless.
26) For these materials a proportional limit stress is defined, below which the errors associated with the linear approximation are negligible.
27) In " linear function approximation " one starts with a mapping __FORMULA__ that assigns a finite-dimensional vector to each state-action pair.
28) The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value.
#### How to use linearapproximation in a sentence
29) Therefore the linear approximation to natural tetration is the only solution of the equation __FORMULA__ and __FORMULA__ which is convex on __FORMULA__.
30) Singular Value Decomposition is a way of factoring matrices into a series of linear approximations that expose the underlying structure of the matrix.
31) Using the formula for linear interpolation, our linear approximation for rm is rm= 12.58 per cent, the same as the exact solution.
32) The viscous stress tensor is only a linear approximation of the stresses around a point __FORMULA__, and does not account for higher-order terms of its Taylor series.
33) This notion of differential is broadly applicable when a linear approximation to a function is sought, in which the value of the increment Δ"x" is small enough.
These examples have been automatically selected and may contain sensitive content that does not reflect the opinions or policies of our website. Please inform us about the inappropriate sentences:
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Complex Sentences with linear approximation
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in a sentence | 973 | 5,081 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5 | 4 | CC-MAIN-2023-14 | latest | en | 0.904223 |
https://www.numbersaplenty.com/1619674540 | 1,686,040,887,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224652494.25/warc/CC-MAIN-20230606082037-20230606112037-00127.warc.gz | 1,010,473,636 | 3,305 | Search a number
1619674540 = 225112669287
BaseRepresentation
bin110000010001010…
…0100010110101100
311011212200222202221
41200202210112230
511304114041130
6424415125124
755065002131
oct14042442654
94155628687
101619674540
1176129a500
123925137a4
131ca734003
14115176588
15972d887a
hex608a45ac
1619674540 has 36 divisors (see below), whose sum is σ = 3738642768. Its totient is φ = 588971680.
The previous prime is 1619674531. The next prime is 1619674571. The reversal of 1619674540 is 454769161.
It is an unprimeable number.
It is a polite number, since it can be written in 11 ways as a sum of consecutive naturals, for example, 332224 + ... + 337063.
It is an arithmetic number, because the mean of its divisors is an integer number (103851188).
Almost surely, 21619674540 is an apocalyptic number.
1619674540 is a gapful number since it is divisible by the number (10) formed by its first and last digit.
It is an amenable number.
1619674540 is an abundant number, since it is smaller than the sum of its proper divisors (2118968228).
It is a pseudoperfect number, because it is the sum of a subset of its proper divisors.
1619674540 is a wasteful number, since it uses less digits than its factorization.
1619674540 is an evil number, because the sum of its binary digits is even.
The sum of its prime factors is 669318 (or 669305 counting only the distinct ones).
The product of its (nonzero) digits is 181440, while the sum is 43.
The square root of 1619674540 is about 40245.1803325566. The cubic root of 1619674540 is about 1174.3816369089.
The spelling of 1619674540 in words is "one billion, six hundred nineteen million, six hundred seventy-four thousand, five hundred forty". | 499 | 1,704 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2023-23 | latest | en | 0.84173 |
https://socratic.org/questions/how-do-you-multiply-3-x-1-1-x-x-1-2-x | 1,723,388,331,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641002566.67/warc/CC-MAIN-20240811141716-20240811171716-00371.warc.gz | 423,234,511 | 6,387 | # How do you multiply 3 /(x - 1) + 1 / (x(x - 1)) = 2 / x?
Jun 5, 2018
Assuming the question really was "solve for "$x$:
$\textcolor{w h i t e}{\text{XXX}} x = - 3$
#### Explanation:
1 Multiply both sides by the Least Common Denominator [namely $x \left(x - 1\right)$]
$\textcolor{w h i t e}{\text{XXX}} 3 x + 1 = 2 \left(x - 1\right)$
2 Simplify
$\textcolor{w h i t e}{\text{XXX}} 3 x + 1 = 2 x - 2$
3 Subtract $2 x$ from both sides
$\textcolor{w h i t e}{\text{XXX}} 3 x + 1 - 2 x = - 2$
4 Simplify $3 x + 1 - 2 x$ to $x + 1$
$\textcolor{w h i t e}{\text{XXX}} x + 1 = - 2$
5 Subtract $1$ from both sides
$\textcolor{w h i t e}{\text{XXX}} x = - 2 - 1$
6 Simplify
$\textcolor{w h i t e}{\text{XXX}} x = - 3$
Jun 5, 2018
$x = - 3$
#### Explanation:
Multiply the first fraction by $x$
$\frac{3 x + 1}{x \left(x - 1\right)} = \frac{2}{x}$
Now multiply both sides by $x \left(x - 1\right)$ to remove the fractions
$3 x + 1 = 2 \left(x - 1\right)$
$3 x + 1 = 2 x - 2$
subtract $2 x$ from both sides
$x + 1 = - 2$
subtract 1 from both sides | 442 | 1,056 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 21, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.78125 | 5 | CC-MAIN-2024-33 | latest | en | 0.620506 |
http://en.m.wikibooks.org/wiki/GLSL_Programming/Unity/Curved_Glass | 1,397,622,497,000,000,000 | text/html | crawl-data/CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00240-ip-10-147-4-33.ec2.internal.warc.gz | 76,179,344 | 8,313 | # GLSL Programming/Unity/Curved Glass
Crystal balls are examples of curved, transparent surfaces.
This tutorial covers refraction mapping and its implementation with cube maps.
It is a variation of Section “Reflecting Surfaces”, which should be read first.
### Refraction MappingEdit
In Section “Reflecting Surfaces”, we reflected view rays and then performed texture lookups in a cube map in the reflected direction. Here, we refract view rays at a curved, transparent surface and then perform the lookups with the refracted direction. The effect will ignore the second refraction when the ray leaves the transparent object again; however, many people hardly notice the differences since such refractions are usually not part of our daily life.
Instead of the `reflect` function, we are using the `refract` function; thus, the fragment shader could be:
``` #ifdef FRAGMENT
void main()
{
float refractiveIndex = 1.5;
vec3 refractedDirection = refract(normalize(viewDirection),
normalize(normalDirection), 1.0 / refractiveIndex);
gl_FragColor = textureCube(_Cube, refractedDirection);
}
#endif
```
Note that `refract` takes a third argument, which is the refractive index of the outside medium (e.g. 1.0 for air) divided by the refractive index of the object (e.g. 1.5 for some kinds of glass). Also note that the first argument has to be normalized, which isn't necessary for `reflect`.
```Shader "GLSL shader with refraction mapping" {
Properties {
_Cube ("Environment Map", Cube) = "" {}
}
Pass {
GLSLPROGRAM
// User-specified uniforms
uniform samplerCube _Cube;
// The following built-in uniforms
// are also defined in "UnityCG.glslinc",
// i.e. one could #include "UnityCG.glslinc"
uniform vec3 _WorldSpaceCameraPos;
// camera position in world space
uniform mat4 _Object2World; // model matrix
uniform mat4 _World2Object; // inverse model matrix
// Varyings
varying vec3 normalDirection;
varying vec3 viewDirection;
#ifdef VERTEX
void main()
{
mat4 modelMatrix = _Object2World;
mat4 modelMatrixInverse = _World2Object; // unity_Scale.w
// is unnecessary because we normalize vectors
normalDirection = normalize(vec3(
vec4(gl_Normal, 0.0) * modelMatrixInverse));
viewDirection = vec3(modelMatrix * gl_Vertex
- vec4(_WorldSpaceCameraPos, 1.0));
gl_Position = gl_ModelViewProjectionMatrix * gl_Vertex;
}
#endif
#ifdef FRAGMENT
void main()
{
float refractiveIndex = 1.5;
vec3 refractedDirection = refract(normalize(viewDirection),
normalize(normalDirection), 1.0 / refractiveIndex);
gl_FragColor = textureCube(_Cube, refractedDirection);
}
#endif
ENDGLSL
}
}
}
```
### SummaryEdit
Congratulations. This is the end of another tutorial. We have seen:
• How to adapt reflection mapping to refraction mapping using the `refract` instruction. | 666 | 2,773 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2014-15 | longest | en | 0.772093 |
https://www.r-bloggers.com/pricing-options-on-multiple-assets-part-1-with-trees/ | 1,591,510,312,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590348523564.99/warc/CC-MAIN-20200607044626-20200607074626-00191.warc.gz | 842,781,787 | 63,428 | # Pricing options on multiple assets (part 1) with trees
June 19, 2012
By
[This article was first published on Freakonometrics - Tag - R-english, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
I am a big fan of trees. It is a very nice way to see how financial pricing works, for derivatives. An with a matrix-based language (R for instance), it is extremely simple to compute almost everything. Even multiple assets options. Let us see how it works. But first, I have to assume that everyone knows about trees (or at least is familiar), and about risk neutral probabilities, and with standard financial derivatives. Just in case, I can upload some old slides of the first course on asset pricing we gave a few years ago at École Polytechnique.
Let us get back on the code to price a (European) call option with trees.The idea is simple. We have to fix the number of periods. Let us start with only one (as described in the slides above). The stock has price and can go either up, and then have price or go down, and have price . And the fundamental theorem of asset pricing says that we do not really care about probabilities of going up, or down. Assuming that we can buy or sell that stock, and that a risk free asset is available on the market, it is possible to price any contingent financial product, like a financial option. Since we know the final value of the option when the stock goes either up, or down, it is possible to replicate the payoff of that option using the stock and the risk free asset. And we can prove that the price of the option is simply
where the probability is the so-called risk neutral probability
So, we’ve done it with only one single period, but it is possible to extend it to multiperiods. The idea is to keep that multiplicative representation of possible values of the stock, and to get a recombinant tree. At step 2, the stock can take only three different values: when up twice, when down twice, or went up and down (or the reverse, but we don’t care, this is the point of recombining). If we write things down, then we can prove that
for some probability. But we do not really care about those closed formula, the goal is to write an algorithm which compute the tree, and return the price of a call option (say). But before starting, we have to make a connection between that model with up and down prices, and the parameter of the Black-Scholes diffusion, for the stock price. The idea is to identify the first and the second moment, i.e.
(where, under the risk neutral probability, the trend is the risk free rate) and
The code might look like that
```n=5; T=1; r=0.05; sigma=.4;S=50;K=50
price=function(n){
u.n=exp(sigma*sqrt(T/n));
d.n=1/u.n
p.n=(exp(r*T/n)-d.n)/(u.n-d.n)
SJ=matrix(0,n+1,n+1)
SJ[1,1]=S
for(i in(2:(n+1)))
{for(j in(1:i)){SJ[i,j]=S*u.n^(i-j)*d.n^(j-1)}}
OPT=matrix(0,n+1,n+1)
OPT[n+1,]=(SJ[n+1,]-K)*(SJ[n+1,]>K)
for(i in(n:1))
{for(j in(1:i)){OPT[i,j]=exp(-r*T/n)*(OPT[i+1,j]*p.n+
(1-p.n)*OPT[i+1,j+1])}}
return(OPT[1,1])
}```
We can plot the evolution of the price, as a function of the number of time periods.
```N=10:400
V=Vectorize(price)(N)
plot(N,V,type="l")```
Note that we can compare with the Black-Scholes price of this call option, given by
where
and
```d1=1/(sigma*sqrt(T))*(log(S/K)+(r+sigma^2/2)*T)
d2=d1-sigma*sqrt(T)
BS=S*pnorm(d1)-K*exp(-r*T)*pnorm(d2)
abline(h=BS,lty=2,col="red")```
The code is clearly not optimal, but at least, we see what’s going on. For instance, we do not need a matrix when we calculate using backward recursions the price of the option. We can just keep a single vector. But this matrix is nice, because we can use in to price American options.
But the great thing with trees, is that we can extend them to model comovments of two underlying prices. For instance, with the code below, we compare the price of an American put option, and the price of European put option.
```price.american=function(n,opt="put"){
u.n=exp(sigma*sqrt(T/n)); d.n=1/u.n
p.n=(exp(r*T/n)-d.n)/(u.n-d.n)
SJ=matrix(0,n+1,n+1)
SJ[1,1]=S
for(i in(2:(n+1)))
{for(j in(1:i)) {SJ[i,j]=S*u.n^(i-j)*d.n^(j-1)}}
OPTe=matrix(0,n+1,n+1)
OPTa=matrix(0,n+1,n+1)
if(opt=="call"){
OPTa[n+1,]=(SJ[n+1,]-K)*(SJ[n+1,]>K)
OPTe[n+1,]=(SJ[n+1,]-K)*(SJ[n+1,]>K)
}
if(opt=="put"){
OPTa[n+1,]=(K-SJ[n+1,])*(SJ[n+1,]<K)
OPTe[n+1,]=(K-SJ[n+1,])*(SJ[n+1,]<K)
}
for(i in(n:1))
{
for(j in(1:i))
{if(opt=="call"){
OPTa[i,j]=max((SJ[i,j]-K)*(SJ[i,j]>K),
exp(-r*T/n)*(OPTa[i+1,j]*p.n+
(1-p.n)*OPTa[i+1,j+1]))}
if(opt=="put"){
OPTa[i,j]=max((K-SJ[i,j])*(K>SJ[i,j]),
exp(-r*T/n)*(OPTa[i+1,j]*p.n+
(1-p.n)*OPTa[i+1,j+1]))}
OPTe[i,j]=exp(-r*T/n)*(OPTe[i+1,j]*p.n+
(1-p.n)*OPTe[i+1,j+1])}}
priceop=c(OPTe[1,1],OPTa[1,1])
names(priceop)=c("E","A")
return(priceop)}```
It is possible to compare those price, obtained on trees, with prices given by closed (approximated) formulas.
```> d1=1/(sigma*sqrt(T))*(log(S/K)+(r+sigma^2/2)*T)
> d2=d1-sigma*sqrt(T)
> (BS=-S*pnorm(-d1)+K*exp(-r*T)*pnorm(-d2) )
[1] 6.572947
> N=10:200
> M=Vectorize(price.american)(N)
> plot(N,M[1,],type='l',col='blue',ylim=range(M))
> lines(N,M[2,],type='l',col='red')
> abline(h=BS,lty=2,col='blue')
> library(fOptions)
> (am=BAWAmericanApproxOption(TypeFlag =
+ "p", S = S,X = K, Time = T, r = r,
+ b = r, sigma =sigma)@price)
[1] 6.840335
> abline(h=am,lty=2,col='red')```
Another great thing with trees, is that it becomes possible to plot to region where it is optimal to exercise our right to sell the stock.
Let us move now to a model with two assets, as suggested by Rubinstein (1994). First, observe that a discretization of two independent Brownian motions will be based on two independent random walk, taking values
i.e. both went up (NW), both went dowm (SE), and one went up while the other went down (either NE or SW).
With independent and symmetric random walks, the probabilities will
be respectively 1/4. Here, the idea is to consider possible correlation,
i.e.
And again, the idea is then to identify the first two moments. This
gives us the following system of equations for the four respective (risk
neutral) probabilities
For those willing to do the maths, please do. The answer should be
and for the last one
The code here looks like that
```price.spead=function(n){
T=1; r=0.05; K=0
S1=105
S2=100
sigma1=0.4
sigma2=0.3
rho=0.5
u1.n=exp(sigma1*sqrt(T/n)); d1.n=1/u1.n
u2.n=exp(sigma2*sqrt(T/n)); d2.n=1/u2.n
v1=r-sigma1^2/2; v2=r-sigma2^2/2
puu.n=(1+rho+sqrt(T/n)*(v1/sigma1+v2/sigma2))/4
pud.n=(1-rho+sqrt(T/n)*(v1/sigma1-v2/sigma2))/4
pdu.n=(1-rho+sqrt(T/n)*(-v1/sigma1+v2/sigma2))/4
pdd.n=(1+rho+sqrt(T/n)*(-v1/sigma1-v2/sigma2))/4
k=0:n
un=matrix(1,n+1,1)
SJ= (S1 * d1.n^k * u1.n^(n-k-1)) %*% t(un) -
un %*%t(S2 * d2.n^k * u2.n^(n-k-1))
OPT=(SJ)*(SJ>K)
for(k in(n:1))
{
OPT0=matrix(0,k,k)
for(i in(1:k))
{
for(j in(1:k))
{OPT0[i,j]=(OPT[i,j]*puu.n+OPT[i+1,j]*pdu.n+
OPT[i,j+1]*pud.n+OPT[i+1,j+1]*pdd.n)*exp(-r*T/n)}}
OPT=OPT0}
return(OPT[1,1])}```
If we look at the details, consider two periods, like on the figure above. The are nine values for the spread,
```> n=2
> SJ
[,1] [,2] [,3]
[1,] 32.02217 84.86869 119.443578
[2,] -47.84652 5.00000 39.574891
[3,] -93.20959 -40.36308 -5.788184```
and the payoff of the option is here
```> OPT
[,1] [,2] [,3]
[1,] 32.02217 84.86869 119.44358
[2,] 0.00000 5.00000 39.57489
[3,] 0.00000 0.00000 0.00000```
So if we go backward of one step, we have the following square of values
```> k=n
> OPT0<-matrix(0,k,k)
> for(i in(1:k))
+ {
+ for(j in(1:k))
+ {
+ OPT0[i,j]=(OPT[i,j]*puu.n+OPT[i+1,j]*pdu.n+
+ OPT[i,j+1]*pud.n+OPT[i+1,j+1]*pdd.n)*exp(-r*T/n)
+ }
+ }
> OPT0
[,1] [,2]
[1,] 22.2741190 58.421275
[2,] 0.5305465 5.977683```
The idea is then to move backward once more,
```> OPT=OPT0
> OPT0<-matrix(0,k,k)
> for(i in(1:k))
+ {
+ for(j in(1:k))
+ {
+ OPT0[i,j]=(OPT[i,j]*puu.n+OPT[i+1,j]*pdu.n+
+ OPT[i,j+1]*pud.n+OPT[i+1,j+1]*pdd.n)*exp(-r*T/n)
+ }
+ }
> OPT0
[,1]
[1,] 16.44106```
Here calculations are much longer,
```> price.spead(250)
[1] 15.66496```
and again, it is possible to use standard approximations to compare that price with a more standard one,
```> (sp=SpreadApproxOption(TypeFlag =
+ "c", S1 = 105, S2 = 100, X = 0,
+ Time = 1, r = .05, sigma1 = .4,
+ sigma2 = .3, rho = .5)@price)
[1] 15.65077 ```
To leave a comment for the author, please follow the link and comment on their blog: Freakonometrics - Tag - R-english.
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Tags: , , , , , , , , , , | 3,082 | 8,997 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.515625 | 4 | CC-MAIN-2020-24 | latest | en | 0.949306 |
http://mathhelpforum.com/calculus/215077-integration-parts-e-print.html | 1,527,075,602,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794865595.47/warc/CC-MAIN-20180523102355-20180523122355-00445.warc.gz | 186,048,285 | 2,950 | # Integration by parts with e
• Mar 19th 2013, 09:16 AM
Integration by parts with e
Hello,
I was just in the middle of trying to solve this Integral when I got very confused. (Angry)
This is the integral that I am trying to solve: http://i1057.photobucket.com/albums/...ps3feb2808.png
I will post the steps and then note when I get confused.
Step 2 (Integrate by Parts): http://i1057.photobucket.com/albums/...pse7c3a335.png
Step 3 (Integrate by Parts again): http://i1057.photobucket.com/albums/...psee06c15a.png
Step 4 (Where I get lost): http://i1057.photobucket.com/albums/...psbe2c1f36.png
I have no problem with integrating by parts, but I start to get lost when the integral left over on the right of the = sign is equal to the integral on the left of the = sign. How does the 5 come into play? What is that process that this problem goes through?
Your help is hugely appreciated! (Nod)
• Mar 19th 2013, 09:21 AM
Plato
Re: Integration by parts with e
Quote:
Originally Posted by Shadow236
Hello,
I have no problem with integrating by parts, but I start to get lost when the integral left over on the right of the = sign is equal to the integral on the left of the = sign. How does the 5 come into play? What is that process that this problem goes through?
If \$\displaystyle y=x-4y\$ then \$\displaystyle 5y=x\$.
• Mar 19th 2013, 09:24 AM | 394 | 1,355 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2018-22 | latest | en | 0.925398 |
https://oeis.org/A286753 | 1,628,039,830,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046154486.47/warc/CC-MAIN-20210803222541-20210804012541-00493.warc.gz | 438,378,321 | 3,943 | The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
A286753 Positions of 0 in A286752; complement of A286753. 3
2, 4, 5, 7, 9, 12, 14, 15, 17, 19, 22, 24, 26, 27, 29, 32, 34, 36, 37, 39, 42, 44, 46, 47, 49, 51, 54, 56, 57, 59, 61, 64, 66, 68, 69, 71, 74, 76, 78, 79, 81, 84, 86, 88, 89, 91, 93, 96, 98, 99, 101, 103, 106, 108, 109, 111, 113, 116, 118, 120, 121, 123, 126 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,1 COMMENTS 2n - a(n) is in {0,1} for n>=1. LINKS Clark Kimberling, Table of n, a(n) for n = 1..10000 FORMULA a(n) = 1 - A286754(n) for n >= 1. EXAMPLE As a word, A286752 = 10100101011010010101101001001..., in which 0 is in positions 2,4,5,7,9,12,... MATHEMATICA s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 12]; (* A003849 *) w = StringJoin[Map[ToString, s]]; w1 = StringReplace[w, {"010010" -> ""}]; st = ToCharacterCode[w1] - 48; (* A286752 *) Flatten[Position[st, 0]]; (* A286753 *) Flatten[Position[st, 1]]; (* A286754 *) CROSSREFS Cf. A003849, A286752, A286754. Sequence in context: A158618 A000788 A053039 * A325543 A214051 A027861 Adjacent sequences: A286750 A286751 A286752 * A286754 A286755 A286756 KEYWORD nonn,easy AUTHOR Clark Kimberling, May 14 2017 STATUS approved
Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
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Last modified August 3 21:16 EDT 2021. Contains 346441 sequences. (Running on oeis4.) | 659 | 1,697 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.25 | 3 | CC-MAIN-2021-31 | latest | en | 0.654538 |
https://cupdf.com/document/application-with-percents.html | 1,716,005,615,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971057260.44/warc/CC-MAIN-20240518023912-20240518053912-00697.warc.gz | 165,301,245 | 23,708 | Standard 3
14
# Application with Percents
Jan 01, 2016
## Documents
Application with Percents. Standard 3. Percent of the Total. Sixty-eight percent of Valley Creek School students attend this year’s homecoming dance. There are 675 students. How many attend the dance? 459 students - PowerPoint PPT Presentation
#### regular price
Welcome message from author
Transcript
Standard 3
1. Sixty-eight percent of Valley Creek School students attend this year’s homecoming dance. There are 675 students. How many attend the dance?
459 students2. Out of the 4,500 people who attend the
rock concert, forty-six percent purchase a T-shirt. How many people buy T-shirts?
2070 people
3. Nina sells ninety-five percent of her 500 cookies at the bake sale. How many cookies does she sell?
475 cookies4. Twelve percent of yesterday’s customers
24 customers
5. The Candy Shack sells 138 pounds of candy on Tuesday. Fifty-two percent of the candy is jelly beans. How many pounds of jelly beans are sold Tuesday?
71.76 pounds6. A fund-raiser at the school raises \$617.50.
Ninety-four percent goes to local charities. How much money goes to charities?
\$580.45
A tip is money given to someone doing a service for you such as a server, hair stylist, cab driver, grocery bagger, ect.
1. The total bill at Jake’s Catfish Place comes to \$35.80. Palo wants to leave a 15% tip. How much money will he have to leave for the tip?
PRICE OF MEAL 35.80 x RATE OF TIP x 0.15 TIP 17900 35800
5.3700
2. Christina makes \$2,550 per month. Her boss promises her a 7% raise. How much more will she make per month?
\$178.503. Ronnie makes \$2,400 per month and puts
6% in a savings account. How much does he save per month?
\$144
In many businesses, sales people are paid on commission – a percent of the total sales they make.
TOTAL COSTx RATE OF COMMISSION
COMMISSION
1. Ramon makes a 4% commission on an \$8,000 pickup truck he sold. How much is his commission?
\$8,000 x 4 % Change % to decimal
\$8,000 x 0.04 Multiply \$ 320 Commission
earned
2. Mia makes a 12% commission on all her sales. This month she sells \$9,000 worth of merchandise. What is her commission?
\$1080 3. Marta sells Sue Anne Cosmetics and gets a
20% commission on all her sales. Last month, she sold \$560.00 worth of cosmetics. How much was her commission?
\$112
Sale prices are sometimes marked 30% off, or better yet 50% off. A 30% discount means you will pay 30% less than the original price. How much money you will save is also known as the amount of the discount.
ORIGINAL PRICE X % DISCOUNT
SAVINGS
1. Tubby Telephones is offering a 25% discount on phones purchased on Tuesday. How much can you save if you buy a phone on Tuesday regularly priced at \$225.00 any other day of the week?
\$225 x 25% Change % to decimal
\$225 x 0.25\$ 56.25 Money saved
2. The regular price for a garden rake is \$10.97 at Sly’s Super Store. This week, Sly is offering a 30% discount. How much is the discount on the rake?
\$10.97 x 30%\$10.97 x 0.3
3.291This means that you will save \$3.29.
3. Christine buys a sweater regularly priced at \$26.80 with a coupon for 20% off any sweater. How much does she save?
\$26.80 x 20%\$26.80 x 0.2
\$5.36 | 864 | 3,225 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5 | 4 | CC-MAIN-2024-22 | latest | en | 0.929608 |
https://www.lybrary.com/an-impromptu-mental-card-routine-p-587011.html | 1,539,730,967,000,000,000 | text/html | crawl-data/CC-MAIN-2018-43/segments/1539583510893.26/warc/CC-MAIN-20181016221847-20181017003347-00390.warc.gz | 995,165,442 | 9,108 | Search All Title Author
# An Impromptu Mental Card Routine
\$15.00
| £11.55 | €13.20 | Ca\$19.91 | Au\$21.48 | ¥1718.54
by Nick Conticello
Tarbell Course contributor Nick Conticello follows his debut The Shadow Placement with three more sensational new effects requiring only an ordinary deck of 52 cards and a clear head to perform. One sleight is required; we include The Widdershins Cut, an effective and simple false cut.
Contents include:
1. THOUGHT CAUGHT: An unprepared deck is shuffled by the performer and cut by the spectator. Two piles of cards are dealt out. The spectator thinks of a card in one of the piles and buries the selection in the deck himself. The performer scans the cards quickly, and removes one card, which turns out to be the selection. No sleights, no outs, no guesswork. The spectator cuts the pack before and after the selection process! This effect introduces two new concepts: the Radar Principle and the Littlecount.
2. AP DIVINATION: The pack is shuffled by the spectator who removes a number of cards at random and counts to a mental selection. The performer finds the card without looking at any of the faces. This one features a new twist on a well-known idea.
3. THE POWER OF FAITH: The finale is perhaps the strongest Smith Myth type effect ever devised. The spectator freely thinks of two related cards with the pack in his own hands. He also shuffles the selection packets. (No counting is involved in the selection process.) The pack is cut in half so one chosen card is in each half. The thought-of cards somehow end up in the same position from the tops of their respective piles. This can be done with nothing more difficult than a false cut, but a handling with a false riffle shuffle will also be disclosed.
Whether or not you like so-called "self-working" card tricks, you will not want to pass up a book which may herald a revolution in the technique of mathematical magic.
1st edition 2014, 19 pages.
word count: 6262 which is equivalent to 25 standard pages of text
SHARE Write your own review Click to leave a voice review or call in the US toll free 1-888-866-9150 product ID: 587011
Reviewed by Tony Bianco
Rating: [5 of 5 Stars!] Date Added: Saturday 08 November, 2014
The first trick alone in the book is worth the asking price. It is finding a selected card that will seem impossible. I have never seen the method in print before. The second trick I have seen be used in other effects but basically you are performing this one to set up the deck for the third effect. The last one is very good also as two selected cards are found located in the same position in two separate halves.
My favorite as you can tell was the first effect but overall definitely worth the purchase.
### This product is listed in the following categories:
Magic > Mentalism & Spiritism
Magic > Cards > Self-Working & Sleightless | 657 | 2,881 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2018-43 | longest | en | 0.944101 |
https://wbb.rail-nation.com/index.php?user-post-list/95456-no1-engineer/ | 1,642,954,205,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320304287.0/warc/CC-MAIN-20220123141754-20220123171754-00605.warc.gz | 647,699,677 | 12,867 | # Posts by No1 Engineer
• ## Chapter 1
yes he will.
Train a will travel 80 km in 30 mins
Train B will travel 60 km in 30 mins thereby the distance of 140km is covered . Assuming the switch is at this point on the track, the supervisor running at 14000 m/ h will cover 6300 meters in 0.45 hrs, which is less than the 0.5 hrs to the train collision.
regards No1 Engineer. | 106 | 374 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2022-05 | latest | en | 0.916663 |
http://g2s3.com/labs/notebooks/SubsurfaceBayesian.html | 1,726,120,532,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651422.16/warc/CC-MAIN-20240912043139-20240912073139-00174.warc.gz | 12,023,058 | 16,063 | $\def\data{ {\bf d}_\rm{obs}} \def\vec{\bf} \def\m{ {\bf m}} \def\map{m_{\nu}} \def\postcov{ \mathcal{C}_{\text{post}} } \def\prcov{ \mathcal{C}_{\text{prior}} } \def\matrix{\bf} \def\Hmisfit{ \mathcal{H}_{\text{misfit}} } \def\diag{\operatorname{diag}} \def\Vr{ {\matrix V}_r } \def\Wr{ {\matrix W}_r } \def\Ir{ {\matrix I}_r } \def\Dr{ {\matrix D}_r } \def\H{ {\matrix H} }$
Bayesian quantification of parameter uncertainty:
Estimating the Gaussian approximation of posterior pdf of the coefficient parameter field in an elliptic PDE
In this example we tackle the problem of quantifying the uncertainty in the solution of an inverse problem governed by an elliptic PDE via the Bayesian inference framework. Hence, we state the inverse problem as a problem of statistical inference over the space of uncertain parameters, which are to be inferred from data and a physical model. The resulting solution to the statistical inverse problem is a posterior distribution that assigns to any candidate set of parameter fields our belief (expressed as a probability) that a member of this candidate set is the true’’ parameter field that gave rise to the observed data.
Bayes’s Theorem
The posterior probability distribution combines the prior pdf $\mu_{\text{prior}}(m)$ over the parameter space, which encodes any knowledge or assumptions about the parameter space that we may wish to impose before the data are considered, with a likelihood pdf $\pi_{\text{like}}(\data \; | \; m)$, which explicitly represents the probability that a given parameter $m$ might give rise to the observed data $\data \in \mathbb{R}^{n_t}$, namely:
Note that infinite-dimensional analog of Bayes’s formula requires the use Radon-Nikodym derivatives instead of probability density functions.
Gaussian prior and noise
The prior
We consider a Gaussian prior with mean ${m}_{\rm prior}$ and covariance $\prcov$, $\mu_{\rm prior} \sim \mathcal{N}({m}_{\rm prior}, \prcov)$. The covariance is given by the discretization of the inverse of differential operator $\mathcal{A}^{-2} = (-\gamma \Delta + \delta I)^{-2}$, where $\gamma$, $\delta > 0$ control the correlation length and the variance of the prior operator. This choice of prior ensures that it is a trace-class operator, guaranteeing bounded pointwise variance and a well-posed infinite-dimensional Bayesian inverse problem.
The likelihood
Here ${\bf f}$ is the parameter-to-observable map that takes a parameter $m$ and maps it to the space observation vector $\data$.
In this application, ${\bf f}$ consists in the composition of a PDE solve (to compute the state $u$) and a pointwise observation of the state $u$ to extract the observation vector $\data$.
The Laplace approximation to the posterior: $\nu \sim \mathcal{N}({\map},\bf \postcov)$
The mean of the Laplace approximation posterior distribution, ${\map}$, is the parameter maximizing the posterior, and is known as the maximum a posteriori (MAP) point. It can be found by minimizing the negative log of the posterior, which amounts to solving a deterministic inverse problem) with appropriately weighted norms,
The posterior covariance matrix is then given by the inverse of the Hessian matrix of $\mathcal{J}$ at $\map$, namely
provided that $\Hmisfit(\map)$ is positive definite.
The generalized eigenvalue problem
In what follows we denote with $\matHmis, \Gpost, \Gprior \in \mathbb{R}^{n\times n}$ the matrices stemming from the discretization of the operators $\Hmisfit(\map)$, $\postcov$, $\prcov$ with respect to the unweighted Euclidean inner product. Then we considered the symmetric generalized eigenvalue problem
where ${\matrix \Lambda} = \diag(\lambda_i) \in \mathbb{R}^{n\times n}$ contains the generalized eigenvalues and the columns of ${\matrix V}\in \mathbb R^{n\times n}$ the generalized eigenvectors such that ${\matrix V}^T \Gprior^{-1} {\matrix V} = {\matrix I}$.
Randomized eigensolvers to construct the approximate spectral decomposition
When the generalized eigenvalues $\{\lambda_i\}$ decay rapidly, we can extract a low-rank approximation of $\matHmis$ by retaining only the $r$ largest eigenvalues and corresponding eigenvectors,
Here, $\Vr \in \mathbb{R}^{n\times r}$ contains only the $r$ generalized eigenvectors of $\matHmis$ that correspond to the $r$ largest eigenvalues, which are assembled into the diagonal matrix ${\matrix{\Lambda}}_r = \diag (\lambda_i) \in \mathbb{R}^{r \times r}$.
The approximate posterior covariance
Using the Sherman–Morrison–Woodbury formula, we write
where ${\matrix{D}}_r :=\diag(\lambda_i/(\lambda_i+1)) \in \mathbb{R}^{r\times r}$. The last term in this expression captures the error due to truncation in terms of the discarded eigenvalues; this provides a criterion for truncating the spectrum, namely that $r$ is chosen such that $\lambda_r$ is small relative to 1.
Therefore we can approximate the posterior covariance as
Drawing samples from a Gaussian distribution with covariance $\Gpost$
Let ${\bf x}$ be a sample for the prior distribution, i.e. ${\bf x} \sim \mathcal{N}({\bf 0}, \Gprior)$, then, using the low rank approximation of the posterior covariance, we compute a sample ${\bf v} \sim \mathcal{N}({\bf 0}, \Gpost)$ as
This tutorial shows:
• Description of the inverse problem (the forward problem, the prior, and the misfit functional)
• Convergence of the inexact Newton-CG algorithm
• Low-rank-based approximation of the posterior covariance (built on a low-rank approximation of the Hessian of the data misfit)
• How to construct the low-rank approximation of the Hessian of the data misfit
• How to apply the inverse and square-root inverse Hessian to a vector efficiently
• Samples from the Gaussian approximation of the posterior
Goals:
By the end of this notebook, you should be able to:
• Understand the Bayesian inverse framework
• Visualise and understand the results
• Modify the problem and code
Mathematical tools used:
• Finite element method
• inexact Newton-CG
• Armijo line search
• Bayes’ formula
• randomized eigensolvers
List of software used:
• FEniCS, a parallel finite element element library for the discretization of partial differential equations
• PETSc, for scalable and efficient linear algebra operations and solvers
• Matplotlib, A great python package that I used for plotting many of the results
• Numpy, A python package for linear algebra. While extensive, this is mostly used to compute means and sums in this notebook.
from __future__ import absolute_import, division, print_function
import dolfin as dl
import math
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from hippylib import *
import logging
logging.getLogger('FFC').setLevel(logging.WARNING)
logging.getLogger('UFL').setLevel(logging.WARNING)
dl.set_log_active(False)
np.random.seed(seed=1)
2. Generate the true parameter
This function generates a random field with a prescribed anysotropic covariance function.
def true_model(prior):
noise = dl.Vector()
prior.init_vector(noise,"noise")
parRandom.normal(1., noise)
mtrue = dl.Vector()
prior.init_vector(mtrue, 0)
prior.sample(noise,mtrue)
return mtrue
3. Set up the mesh and finite element spaces
We compute a two dimensional mesh of a unit square with nx by ny elements. We define a P2 finite element space for the state and adjoint variable and P1 for the parameter.
ndim = 2
nx = 32
ny = 32
mesh = dl.UnitSquareMesh(nx, ny)
Vh2 = dl.FunctionSpace(mesh, 'Lagrange', 2)
Vh1 = dl.FunctionSpace(mesh, 'Lagrange', 1)
Vh = [Vh2, Vh1, Vh2]
print( "Number of dofs: STATE={0}, PARAMETER={1}, ADJOINT={2}".format(
Number of dofs: STATE=4225, PARAMETER=1089, ADJOINT=4225
4. Set up the forward problem
Let $\Omega$ be the unit square in $\mathbb{R}^2$, and $\Gamma_D$, $\Gamma_N$ be the Dirichlet and Neumann portitions of the boundary $\partial \Omega$ (that is $\Gamma_D \cup \Gamma_N = \partial \Omega$, $\Gamma_D \cap \Gamma_N = \emptyset$). The forward problem reads
where $u \in \mathcal{V}$ is the state variable, and $m \in \mathcal{M}$ is the uncertain parameter. Here $\Gamma_D$ corresponds to the top and bottom sides of the unit square, and $\Gamma_N$ corresponds to the left and right sides. We also let $f = 0$, and $u_D = 1$ on the top boundary and $u_D = 0$ on the bottom boundary.
To set up the forward problem we use the PDEVariationalProblem class, which requires the following inputs
• the finite element spaces for the state, parameter, and adjoint variables Vh
• the pde in weak form pde_varf
• the boundary conditions bc for the forward problem and bc0 for the adjoint and incremental problems.
The PDEVariationalProblem class offer the following functionality:
• solving the forward/adjoint and incremental problems
• evaluate first and second partial derivative of the forward problem with respect to the state, parameter, and adojnt variables.
def u_boundary(x, on_boundary):
return on_boundary and ( x[1] < dl.DOLFIN_EPS or x[1] > 1.0 - dl.DOLFIN_EPS)
u_bdr = dl.Expression("x[1]", degree=1)
u_bdr0 = dl.Constant(0.0)
bc = dl.DirichletBC(Vh[STATE], u_bdr, u_boundary)
bc0 = dl.DirichletBC(Vh[STATE], u_bdr0, u_boundary)
f = dl.Constant(0.0)
def pde_varf(u,m,p):
pde = PDEVariationalProblem(Vh, pde_varf, bc, bc0, is_fwd_linear=True)
4. Set up the prior
To obtain the synthetic true paramter $m_{\rm true}$ we generate a realization from the prior distribution.
Here we assume a Gaussian prior, $\mu_{\rm prior} \sim \mathcal{N}(0, \prcov)$ with zero mean and covariance matrix $\prcov = \mathcal{A}^{-2}$, where $\mathcal{A}$ is a differential operator of the form
equipped with Robin boundary conditions $\nabla m \cdot \boldsymbol{n} + \beta m = 0$, where $\beta \propto \sqrt{\gamma\delta}$.
Here $\Theta$ is a s.p.d. anisotropic tensor of the form
gamma = .1
delta = .5
anis_diff = dl.Expression(code_AnisTensor2D, degree=1)
anis_diff.theta0 = 2.
anis_diff.theta1 = .5
anis_diff.alpha = math.pi/4
prior = BiLaplacianPrior(Vh[PARAMETER], gamma, delta, anis_diff, robin_bc=True)
print("Prior regularization: (delta_x - gamma*Laplacian)^order: delta={0}, gamma={1}, order={2}".format(delta, gamma,2))
mtrue = true_model(prior)
objs = [dl.Function(Vh[PARAMETER],mtrue), dl.Function(Vh[PARAMETER],prior.mean)]
mytitles = ["True Parameter", "Prior mean"]
nb.multi1_plot(objs, mytitles)
plt.show()
model = Model(pde,prior, misfit)
Prior regularization: (delta_x - gamma*Laplacian)^order: delta=0.5, gamma=0.1, order=2
5. Set up the misfit functional and generate synthetic observations
To setup the observation operator $\mathcal{B}: \mathcal{V} \mapsto \mathbb{R}^{n_t}$, we generate $n_t$ (ntargets in the code below) random locations where to evaluate the value of the state.
Under the assumption of Gaussian additive noise, the likelihood function $\pi_{\rm like}$ has the form
where $u(m)$ denotes the solution of the forward model at a given parameter $m$.
The class PointwiseStateObservation implements the evaluation of the log-likelihood function and of its partial derivatives w.r.t. the state $u$ and parameter $m$.
To generate the synthetic observation, we first solve the forward problem using the true parameter $m_{\rm true}$. Synthetic observations are obtained by perturbing the state variable at the observation points with a random Gaussian noise. rel_noise is the signal to noise ratio.
ntargets = 300
rel_noise = 0.005
targets = np.random.uniform(0.05,0.95, [ntargets, ndim] )
print( "Number of observation points: {0}".format(ntargets) )
misfit = PointwiseStateObservation(Vh[STATE], targets)
utrue = pde.generate_state()
x = [utrue, mtrue, None]
pde.solveFwd(x[STATE], x, 1e-9)
misfit.B.mult(x[STATE], misfit.d)
MAX = misfit.d.norm("linf")
noise_std_dev = rel_noise * MAX
parRandom.normal_perturb(noise_std_dev, misfit.d)
misfit.noise_variance = noise_std_dev*noise_std_dev
vmax = max( utrue.max(), misfit.d.max() )
vmin = min( utrue.min(), misfit.d.min() )
plt.figure(figsize=(15,5))
nb.plot(dl.Function(Vh[STATE], utrue), mytitle="True State", subplot_loc=121, vmin=vmin, vmax=vmax, cmap="jet")
nb.plot_pts(targets, misfit.d, mytitle="Observations", subplot_loc=122, vmin=vmin, vmax=vmax, cmap="jet")
plt.show()
Number of observation points: 300
6. Set up the model and test gradient and Hessian
The model is defined by three component:
• the PDEVariationalProblem pde which provides methods for the solution of the forward problem, adjoint problem, and incremental forward and adjoint problems.
• the Prior prior which provides methods to apply the regularization (precision) operator to a vector or to apply the prior covariance operator (i.e. to solve linear system with the regularization operator)
• the Misfit misfit which provides methods to compute the cost functional and its partial derivatives with respect to the state and parameter variables.
To test gradient and the Hessian of the model we use forward finite differences.
model = Model(pde, prior, misfit)
m0 = dl.interpolate(dl.Expression("sin(x[0])", degree=5), Vh[PARAMETER])
_ = modelVerify(model, m0.vector(), 1e-12)
(yy, H xx) - (xx, H yy) = 1.4617272123344048e-13
7. Compute the MAP point
We used the globalized Newtown-CG method to compute the MAP point.
m = prior.mean.copy()
solver = ReducedSpaceNewtonCG(model)
solver.parameters["rel_tolerance"] = 1e-6
solver.parameters["abs_tolerance"] = 1e-12
solver.parameters["max_iter"] = 25
solver.parameters["inner_rel_tolerance"] = 1e-15
solver.parameters["GN_iter"] = 5
solver.parameters["globalization"] = "LS"
solver.parameters["LS"]["c_armijo"] = 1e-4
x = solver.solve([None, m, None])
if solver.converged:
print( "\nConverged in ", solver.it, " iterations.")
else:
print( "\nNot Converged")
print( "Termination reason: ", solver.termination_reasons[solver.reason] )
print( "Final cost: ", solver.final_cost )
plt.figure(figsize=(15,5))
nb.plot(dl.Function(Vh[STATE], x[STATE]), subplot_loc=121,mytitle="State", cmap="jet")
nb.plot(dl.Function(Vh[PARAMETER], x[PARAMETER]), subplot_loc=122,mytitle="Parameter")
plt.show()
It cg_it cost misfit reg (g,dm) ||g||L2 alpha tolcg
1 1 6.940520e+03 6.939682e+03 8.387183e-01 -7.334402e+04 1.723648e+05 1.000000e+00 5.000000e-01
2 3 2.515192e+03 2.513892e+03 1.300150e+00 -8.832248e+03 6.111900e+04 1.000000e+00 5.000000e-01
3 4 7.987318e+02 7.953514e+02 3.380398e+00 -3.386902e+03 2.475725e+04 1.000000e+00 3.789893e-01
4 6 4.051231e+02 3.995806e+02 5.542539e+00 -7.930481e+02 9.015037e+03 1.000000e+00 2.286965e-01
5 9 2.371231e+02 2.282068e+02 8.916319e+00 -3.401724e+02 5.145638e+03 1.000000e+00 1.727808e-01
6 3 2.307468e+02 2.218109e+02 8.935908e+00 -1.280167e+01 3.736335e+03 1.000000e+00 1.472308e-01
7 22 1.459150e+02 1.258067e+02 2.010830e+01 -1.738008e+02 2.603111e+03 1.000000e+00 1.228916e-01
8 9 1.435724e+02 1.232348e+02 2.033766e+01 -4.667523e+00 1.153148e+03 1.000000e+00 8.179341e-02
9 43 1.368186e+02 1.091983e+02 2.762021e+01 -1.360922e+01 8.404321e+02 1.000000e+00 6.982759e-02
10 22 1.367776e+02 1.091339e+02 2.764370e+01 -8.206701e-02 1.099361e+02 1.000000e+00 2.525491e-02
11 55 1.367625e+02 1.090441e+02 2.771838e+01 -3.028923e-02 4.834025e+01 1.000000e+00 1.674674e-02
12 39 1.367625e+02 1.090439e+02 2.771853e+01 -2.325655e-05 1.946731e+00 1.000000e+00 3.360692e-03
13 93 1.367625e+02 1.090438e+02 2.771869e+01 -4.135705e-07 1.900557e-01 1.000000e+00 1.050065e-03
Converged in 13 iterations.
Termination reason: Norm of the gradient less than tolerance
Final cost: 136.76246973530718
8. Compute the low rank Gaussian approximation of the posterior
We used the double pass algorithm to compute a low-rank decomposition of the Hessian Misfit. In particular, we solve
The Figure shows the largest k generalized eigenvectors of the Hessian misfit. The effective rank of the Hessian misfit is the number of eigenvalues above the red line ($y=1$). The effective rank is independent of the mesh size.
model.setPointForHessianEvaluations(x, gauss_newton_approx=False)
Hmisfit = ReducedHessian(model, solver.parameters["inner_rel_tolerance"], misfit_only=True)
k = 100
p = 20
print( "Single/Double Pass Algorithm. Requested eigenvectors: {0}; Oversampling {1}.".format(k,p) )
Omega = MultiVector(x[PARAMETER], k+p)
parRandom.normal(1., Omega)
lmbda, V = doublePassG(Hmisfit, prior.R, prior.Rsolver, Omega, k)
nu = GaussianLRPosterior(prior, lmbda, V)
nu.mean = x[PARAMETER]
plt.plot(range(0,k), lmbda, 'b*', range(0,k+1), np.ones(k+1), '-r')
plt.yscale('log')
plt.xlabel('number')
plt.ylabel('eigenvalue')
nb.plot_eigenvectors(Vh[PARAMETER], V, mytitle="Eigenvector", which=[0,1,2,5,10,15])
Single/Double Pass Algorithm. Requested eigenvectors: 100; Oversampling 20.
9. Prior and LA-posterior pointwise variance fields
compute_trace = True
if compute_trace:
post_tr, prior_tr, corr_tr = nu.trace(method="Randomized", r=200)
print( "LA-Posterior trace {0:5e}; Prior trace {1:5e}; Correction trace {2:5e}".format(post_tr, prior_tr, corr_tr) )
post_pw_variance, pr_pw_variance, corr_pw_variance = nu.pointwise_variance(method="Exact")
objs = [dl.Function(Vh[PARAMETER], pr_pw_variance),
dl.Function(Vh[PARAMETER], post_pw_variance)]
mytitles = ["Prior variance", "LA-Posterior variance"]
nb.multi1_plot(objs, mytitles, logscale=False)
plt.show()
LA-Posterior trace 6.537064e-01; Prior trace 1.793511e+00; Correction trace 1.139805e+00
10. Generate samples from Prior and LA-Posterior
nsamples = 5
noise = dl.Vector()
nu.init_vector(noise,"noise")
s_prior = dl.Function(Vh[PARAMETER], name="sample_prior")
s_post = dl.Function(Vh[PARAMETER], name="sample_post")
pr_max = 2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.max()
pr_min = -2.5*math.sqrt( pr_pw_variance.max() ) + prior.mean.min()
ps_max = 2.5*math.sqrt( post_pw_variance.max() ) + nu.mean.max()
ps_min = -2.5*math.sqrt( post_pw_variance.max() ) + nu.mean.min()
vmax = max(pr_max, ps_max)
vmin = max(pr_min, ps_min)
for i in range(nsamples):
parRandom.normal(1., noise)
nu.sample(noise, s_prior.vector(), s_post.vector())
plt.figure(figsize=(15,5))
nb.plot(s_prior, subplot_loc=121,mytitle="Prior sample", vmin=vmin, vmax=vmax)
nb.plot(s_post, subplot_loc=122,mytitle="LA-Posterior sample", vmin=vmin, vmax=vmax)
plt.show()
11. Define a quantify of interest
As a quantity of interest, we consider the log of the flux through the bottom boundary:
where the state variable $u$ denotes the pressure, and $\mathbf{n}$ is the unit normal vector to $\Gamma_b$ (the bottom boundary of the domain).
class FluxQOI(object):
def __init__(self, Vh, dsGamma):
self.Vh = Vh
self.dsGamma = dsGamma
self.n = dl.Constant((0.,1.))#dl.FacetNormal(Vh[STATE].mesh())
self.u = None
self.m = None
self.L = {}
def form(self, x):
def eval(self, x):
u = vector2Function(x[STATE], self.Vh[STATE])
m = vector2Function(x[PARAMETER], self.Vh[PARAMETER])
return np.log( dl.assemble(self.form([u,m])) )
class GammaBottom(dl.SubDomain):
def inside(self, x, on_boundary):
return ( abs(x[1]) < dl.DOLFIN_EPS )
GC = GammaBottom()
marker = dl.FacetFunction("size_t", mesh)
marker.set_all(0)
GC.mark(marker, 1)
dss = dl.Measure("ds", subdomain_data=marker)
qoi = FluxQOI(Vh,dss(1))
12. Compute posterior expectations using MCMC
We compute the mean of the quantity of interest $q$ using MCMC with preconditioned Crank-Nicolson proposal (pCN) and generalized preconditioned Crank-Nicolson proposal (gpCN).
Preconditioned Crank-Nicolson
The pCN algorithm is perhaps the simplest MCMC method that is well-defined in the infinite dimensional setting, that is that ensures a mixing rates independent of the dimension of the discretized parameter space.
For a given Gaussian prior measure $\mu_{\rm prior} \sim \mathcal{N}(m_{\rm prior}, \mathcal{C}_{\rm prior})$, a negative log likelihood function $\Phi(m, \data) = \frac{1}{2}\| {\bf f}(m) - \data \|^2_{\Gamma_{\rm noise}^{-1}}$, the acceptance ratio of pCN is defined as
The algorithm below summarizes the pCN method.
1. Set $k = 0$ and pick $m^{(0)}$
2. Set $v^{(k)} = m_{\rm prior} + \sqrt{1 - \beta^2}(m^{(k)} - m_{\rm prior}) + \beta \xi^{(k)}, \quad \xi^{(k)} \sim \mathcal{N}( 0, \mathcal{C}_{\rm prior} )$
3. Set $m^{(k+1)} = v^{(k)}$ with probability $a(m^{(k)}, v^{(k)})$
4. Set $m^{(k+1)} = m^{(k)}$ otherwise
5. $k \leftarrow k + 1$ and return to 2
Above the parameter $\beta$ controls the step lenght of the pCN proposals. A small $\beta$ will lead to a high acceptance ratio, but the proposed sample will be very similar to the current one, thus leading to poor mixing. On the other hand, a too large $\beta$ will lead to small acceptance ratio, again leading to poor mixing. Therefore, it is important to find the correct trade-off between a large step-size and a good acceptance ratio.
Generalized Preconditioned Crank-Nicolson
gpCN is a generalized version of the pCN sampler. While the proposals of pCN are drown from the prior Gaussian distribution $\mu_{\rm prior}$, proposals in the generalized pCN are drown from a Gaussian approximation $\nu$ of the posterior distribution. More specifically, for a given Gaussian prior measure $\mu_{\rm prior} \sim \mathcal{N}(m_{\rm prior}, \mathcal{C}_{\rm prior})$, a negative log likelihood function $\Phi(m, \data) = \frac{1}{2}\| {\bf f}(m) - \data \|^2_{\Gamma_{\rm noise}^{-1}}$, and a proposal Gaussian distribution $\nu \sim \mathcal{N}(m_\nu, \mathcal{C}_\nu)$, the acceptance ratio of gpCN is defined as
where
If $\nu$ is a good Gaussian approximation of $\mu_{\rm post}$, one expects $\Delta$ to be smaller that $\Phi$, at least in regions of high posterior probability. This suggests that the generalized pCN will have a better acceptance probability than pCN, leading to more rapid sampling. The algorithm below summarizes the gpCN method.
1. Set $k = 0$ and pick $m^{(0)}$
2. Set $v^{(k)} = m_\nu + \sqrt{1 - \beta^2}(m^{(k)} - m_\nu) + \beta \xi^{(k)}, \quad \xi^{(k)} \sim \mathcal{N}( 0, \mathcal{C}_\nu )$
3. Set $m^{(k+1)} = v^{(k)}$ with probability $a_\nu(m^{(k)}, v^{(k)})$
4. Set $m^{(k+1)} = m^{(k)}$ otherwise
5. $k \leftarrow k + 1$ and return to 2
In the code below we ran the chain for 10,000 samples, this is may not be enough to obtain accurate posterior expectation, however it will still give you a feel on how well the chain is mixing.
def run_chain(kernel):
noise = dl.Vector()
nu.init_vector(noise, "noise")
parRandom.normal(1., noise)
pr_s = model.generate_vector(PARAMETER)
post_s = model.generate_vector(PARAMETER)
# Use a sample from LA-posterior as starting point for the chain
chain = MCMC(kernel)
chain.parameters["burn_in"] = 1000
chain.parameters["number_of_samples"] = 10000
chain.parameters["print_progress"] = 10
tracer = QoiTracer(chain.parameters["number_of_samples"])
n_accept = chain.run(post_s, qoi, tracer)
print( "Number accepted = {0}".format(n_accept) )
print( "E[q] = {0}".format(chain.sum_q/float(chain.parameters["number_of_samples"])) )
q = tracer.data
max_lag = 300
integrated_corr_time, lags, acorrs = integratedAutocorrelationTime(q, max_lag)
print ("Integrated autocorrelation time", integrated_corr_time)
plt.figure(figsize=(15,5))
plt.subplot(131)
plt.plot(q, '*b')
plt.title("Trace plot")
plt.subplot(132)
plt.hist(q, normed=True)
plt.title("Histogram")
plt.subplot(133)
plt.plot(lags, acorrs, '-b')
plt.title("Autocorrelation")
plt.ylim([0., 1.])
plt.show()
return tracer
print("Sampling using pCN proposal")
kernel_pCN = pCNKernel(model)
kernel_pCN.parameters["s"] = 0.01
tracer = run_chain(kernel_pCN)
Sampling using pCN proposal
Burn 1000 samples
10.0 % completed, Acceptance ratio 7.0 %
20.0 % completed, Acceptance ratio 5.5 %
30.0 % completed, Acceptance ratio 6.3 %
40.0 % completed, Acceptance ratio 6.5 %
50.0 % completed, Acceptance ratio 7.0 %
60.0 % completed, Acceptance ratio 7.0 %
70.0 % completed, Acceptance ratio 6.9 %
80.0 % completed, Acceptance ratio 6.5 %
90.0 % completed, Acceptance ratio 6.7 %
100.0 % completed, Acceptance ratio 6.6 %
Generate 10000 samples
10.0 % completed, Acceptance ratio 7.4 %
20.0 % completed, Acceptance ratio 7.5 %
30.0 % completed, Acceptance ratio 8.2 %
40.0 % completed, Acceptance ratio 7.9 %
50.0 % completed, Acceptance ratio 7.8 %
60.0 % completed, Acceptance ratio 7.7 %
70.0 % completed, Acceptance ratio 7.7 %
80.0 % completed, Acceptance ratio 7.6 %
90.0 % completed, Acceptance ratio 7.6 %
100.0 % completed, Acceptance ratio 7.6 %
Number accepted = 763
E[q] = 0.5104385815151372
Integrated autocorrelation time 545.423494026
print("Sampling using gpCN proposal")
kernel_gpCN = gpCNKernel(model, nu)
kernel_gpCN.parameters["s"] = 0.9
tracer = run_chain(kernel_gpCN)
Sampling using gpCN proposal
Burn 1000 samples
10.0 % completed, Acceptance ratio 25.0 %
20.0 % completed, Acceptance ratio 20.5 %
30.0 % completed, Acceptance ratio 17.3 %
40.0 % completed, Acceptance ratio 15.5 %
50.0 % completed, Acceptance ratio 15.2 %
60.0 % completed, Acceptance ratio 16.0 %
70.0 % completed, Acceptance ratio 17.4 %
80.0 % completed, Acceptance ratio 16.8 %
90.0 % completed, Acceptance ratio 17.0 %
100.0 % completed, Acceptance ratio 16.5 %
Generate 10000 samples
10.0 % completed, Acceptance ratio 6.5 %
20.0 % completed, Acceptance ratio 8.1 %
30.0 % completed, Acceptance ratio 9.5 %
40.0 % completed, Acceptance ratio 10.7 %
50.0 % completed, Acceptance ratio 11.0 %
60.0 % completed, Acceptance ratio 10.4 %
70.0 % completed, Acceptance ratio 11.0 %
80.0 % completed, Acceptance ratio 11.0 %
90.0 % completed, Acceptance ratio 11.1 %
100.0 % completed, Acceptance ratio 10.8 %
Number accepted = 1085
E[q] = 0.7026869688895735
Integrated autocorrelation time 52.8846366168
Copyright © 2016-2018, The University of Texas at Austin & University of California, Merced. | 7,719 | 25,978 | {"found_math": true, "script_math_tex": 107, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.71875 | 3 | CC-MAIN-2024-38 | latest | en | 0.694023 |
https://grinebiter.com/Numbers/Cardinal/80-eighty.html | 1,571,854,578,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570987835748.66/warc/CC-MAIN-20191023173708-20191023201208-00509.warc.gz | 527,258,675 | 3,398 | Eighty
Here is information about "eighty" that you may find useful and interesting. Number Systems Eighty is a decimal number and can be written with numbers: 80 Binary is a number system with only 0s and 1s. Eighty in binary form is displayed below: 1010000 A Hexadecimal number has a base of 16 which means it includes the numbers 0 to 9 and A through F. Eighty converted to hexadecimal is: 50 Roman Numerals is another number system. Below is eighty in roman numerals: LXXX Scientific Notation Sometimes calculators and scientists shorten numbers using scientific notation. Here is eighty as a scientific notation: 8E+01 Math Here are some math facts about Eighty: Eighty is a rational number and an integer. Eighty is an even number because it is divisible by two. Eighty is divisible by the following numbers: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80 Eighty is not a square number because no number multiplied by itself will equal eighty. Number Lookup Eighty is not the only number we have information about. Go here to look up other numbers.
Translated Here we have translated eighty into some of the most commonly used languages: Chinese: 八十 French: quatre-vingts German: achtzig Italian: ottanta Spanish: ochenta
Currency Here is eighty written in different currencies: US Dollars: \$80 Canadian Dollars: CA\$80 Australian Dollars: A\$80 British Pounds: £80 Indian Rupee: ₹80 Euros: €80
Ordinal The cardinal number eighty can also be written as an ordinal number: 80th Or if you want to write it with letters only: eightieth.
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https://www.philosophyetc.net/2008/02/is-imprecise-credence-rational.html?showComment=1203302460000 | 1,607,017,940,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141729522.82/warc/CC-MAIN-20201203155433-20201203185433-00017.warc.gz | 715,081,611 | 26,962 | ## Friday, February 15, 2008
### Is Imprecise Credence Rational?
If you haven't the faintest clue whether some proposition p is true or false, what subjective probability (credence) should you give it? 1/2? A common answer these days is that you shouldn't give it a precise credence at all. Instead, your credence should be spread over an interval, such as [0,1]. Greater precision than that ought to be based on real knowledge, e.g. of objective probabilities. Mere ignorance doesn't qualify one to make such claims.
Roger White, in his talk 'Evidential Symmetry and Mushy Credence', offers a neat argument for the old-fashioned answer of 1/2,* which goes roughly as follows:
Coin Game: Suppose you're given a fair coin which has 'p' plastered on one side, and '~p' on the other. Moreover, you know that whichever one is true was plastered over the Heads side. You toss the coin and it happens to land on 'p'.
(1) It's a fair coin, so you should initially give P(heads) = 1/2.
(2) This should not changed upon seeing the coin land on 'p' -- you have no idea whether p is the true one or not, so there is no new evidence for you here. So your updated P+(heads) = initial P(heads) = 1/2.
(3) Since the coin landed on 'p', this will be heads-up iff p is true. Hence P+(p) = P+(heads) = 1/2.
(4) But the coin landing on 'p' doesn't tell you anything new about the proposition's truth value. So your prior credence P(p) should also have been 1/2.
Convinced?
* Correction: the argument merely shows that your credence in p shouldn't be imprecise (for that entails, contradictorily, that it should be precisely 1/2). Maybe it should be some other precise value, though; that will depend on the details of what proposition p is.
1. I think 1/2 is a legitimate assumption - the problem arises in that you might be systematically tricked - or that you might be ignoring other information.
Also I think almost any - possibly every, assessment of probability has a known and an unknown probability component.
2. I'm tired, so my brain isn't quite working right, but I think there's a stronger argument working in the other direction: consider two propositions p and q. If you have no idea whether they're true, and no idea whether consistent sentences of first-order logic constructed from them are true [like (p&q) and (p&~q)] then you can't assign all the relevant propositions a subjective probability of 1/2. I think. I think assigning p&q a probability of 1/2 would require assigning p&~q a probability of 0, but my brain isn't quite in the condition to write out a formal proof of that.
3. It's my impression that 50% is the standard answer these days and that this reasoning is well known. It is taken for granted that any other answer is a logical fallacy in one of the Judgment Under Uncertainty books.
The alternative, multi-term probability theory, doesn't gain you anything on Bayesian in terms of predictive power or scope, though it is probably more psychologically realistic and may gain you some accuracy in the context of other known sources of human irrationality.
4. Color me unconvinced.
The labels p and ~p are merely masking the outcome. It would be the same as if you were to flip the coin but cover it before you saw the result. You'd expect a 1/2 chance of getting heads after the flip, but you can't tell because it's covered. Your updated chance is still 1/2.
There seems no legitimate reason to update when the initial move does nothing to move one closer to useful information.
Or maybe I'm talking out of my hat.
5. Jason - right, that's the argument.
Hallq - good point. But perhaps we could reasonably give (p&q) credence of just 1/4, if our ignorance of the atomic sentences p,q is more fundamental? (I'd be curious to hear what others think of this problem.)
Michael - You may be right, but I think the 'mushy credence' solution is more recent. White attributed the view to Jim Joyce and Elliot Sober, for example, both of whom are big names in the contemporary field. (Certainly, the principle of indifference is no longer widely accepted, thanks to Bertrand's paradox, etc.)
Also, I don't think that mushy credence is in any way opposed to Bayesianism. The way White described it, we were simply to model a mushy believer as having a "committee" of homunculi in their head, each one with precise credences subject to Bayesian updating.
6. The problem with your solution is knowing which sentences should be fundamental when we're dealing with real statements. Consider "the present king of Finland is bald," from the point of view of someone who has no idea whether Finland has a king (yes, I know, there are proposals for how to write that one in symbolic logic, but I suspect it would only be the beginning of our troubles).
7. It's tricky, I agree, but it doesn't seem that outrageous to require that a perfectly rational agent be able to work out which sentences are fundamental.
Then again, it's an interesting question to ask what is rational for limited agents (such as real-life humans) who aren't capable of this. How ought we to make judgments under this meta-uncertainty?
8. I gotta say I don't get this.
How can you argue for how much credence to give to an truly unknown proposition by giving an example where the outcome is obviously a 50/50 chance even though the name of the state is covered by tape. You are assuming a dichotomy and therefore artificially setting up the "uncertainty" by strict limits.
Why not try a simpler example without all the fancy notation?
"Yes or No: is 1781 the last four digits of President Bush's phone number"
You could calculate that the odds of a random number being the correct 4 digit number would be 10 to the 4th, but that assumes the number is random. Let's posit a situation where I may know the correct answer, therefore the odds are <10 to the 4th. You have no idea of the odds. Are you still going to claim that the answer to the "yes no question" is 50/50?
Now, admittedly I may completely mis understand the intricacies of the question. On the other hand, you may have over thought it.
9. Gerard - I'm not sure, but that still sounds like a situation where I have more reason to think the numbers aren't correct. (Maybe. It's hard to know what to say there. But that's the advantage of the argument I discuss in the post: it forces us to one clear conclusion.)
I'm not sure I follow your objection to the earlier argument. Which of the four premises do you think is false? Or do you think the conclusion does not follow from the premises? (It looks pretty water-tight to me, but I may be missing something.)
10. Hi, I think my objection is that there is already a defined probability in the example given, and thus is not at all analogous to the question it proposes to explore, which is "If you haven't the faintest clue whether some proposition p is true or false."[emphasis added] In your example, you do, in fact, have more than the faintest clue. You know the odds are 50/50.
Your conclusions do not follow your premiss. You've only proven that for situations where there are known 50/50 odds, the odds remain 50/50
Since the identity of the outcome is masked, multiple permutations do not add more certainty--nor do they subtract certainty--thus they are irrelevant.
So, in the case of unknown probability, or as you say, where you "haven't the faintest clue whether some proposition p is true or false" you cannot ascribe probability of 50/50 since the probability is unknown. Just because the answer is bimodal does not mean that the probability can be rationally assumed to be 50/50. Unknown certainty == unknown odds, not 50/50 odds.
11. Hold on, the example stipulates an objective probability of 50/50 for the coin flip (landing heads), but it doesn't stipulate anything about p. No objective probability for p is discussed at all, and that our subjective probability (credence) for p ought to be 50% is the conclusion of the argument, not anything assumed in the set-up.
Note that the conclusion is that one ought to have 50% credence in p even before the coin flip. So, even if the coin flip never takes place, one's credence should still be 50%. It doesn't depend on being in the peculiar situation with the coin. The coin game simply serves to demonstrate why this is so: if you started with any other credence -- including no credence at all -- then you would end up with the wrong result in this case. Unless, perhaps, you deny premise (4) and think that the flipping of the coin may change your credence in p from undefined to 0.5?
12. I think we are talking cross-purposes.
Your premise was "If you haven't the faintest clue whether some proposition p is true or false, what subjective probability (credence) should you give it? 1/2? "
Which you then follow up with an example with a stipulative probability of 50/50, artificially giving credence with your proposition. You can't stipulate a specific situation that contradicts the premiss, which is unknown probability. In effect, you've merely made an argument by assertion by stipulating a known prior probability in your analogy without regard for your original premiss.
Why not propose a situation where the stipulative prior probability is unknown, as I tried to do in my earlier post. When you start from such a position then you don't wind up with a conclusion of "50/50" but unknown, where you can't assign a fixed level of credence based on any prior probability because there is none.
The probability of unknown is unknown, regardless of whether the answer is expressed in a bimodal fashion.
13. Yes, we're talking at cross-purposes. One last attempt at clarification, then I think I'll have to call it a day...
You need to distinguish between objective and subjective probabilities, and also between the propositions 'p' and 'heads'. Your objections all seem to stem from a failure to carefully distinguish these. (Whenever you speak of a 'probability', I don't know which of the four possible combinations you are talking about.)
In particular, note that there is no contradiction in the following set of claims, all of which are true in the scenario I discuss:
(A) The objective probability of p is unknown. Also, you have no reason to think either p or not-p more likely true. (This is what I mean by the 'haven't the faintest clue' line.)
(B) The objective probability of heads is 1/2
(C) One's subjective credence in heads (both before and after the flip) should be 1/2
(D) One's subjective credence in p (both before and after the flip) should be 1/2.
In particular, note that the stipulation in (B) is compatible with (A). Writing a sentence on a coin doesn't affect the objective probability that the sentence is true (nor does it give you any new evidence for or against its truth). So I haven't a clue what you're talking about when you accuse me [or Roger White] of "stipulating a known prior probability in your analogy without regard for your original premiss."
14. "In particular, note that there is no contradiction in the following set of claims, all of which are true in the scenario I discuss:"
Yes, I would agree. I think the question you asked and the one you answered in you OP are not the same. The imprecision you accuse me of has its orgins in your original premise: "If you haven't the faintest clue whether some proposition p is true or false, what subjective probability (credence) should you give it? 1/2?"
Perhaps if you re-phrased the question to match the answer you have carefully given your conclusions would follow your premises?
I will stipulate a certain amount of ignorance on the subject and concede that I may need to review my own arguments but I do think the distinctions you make to me are post hoc even though you may have meant them to be part of your original premise.
Visitors: check my comments policy first.
Non-Blogger users: If the comment form isn't working for you, email me your comment and I can post it on your behalf. (If your comment is too long, first try breaking it into two parts.) | 2,750 | 12,063 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.4375 | 3 | CC-MAIN-2020-50 | latest | en | 0.954362 |
http://www.danieldollinger.com/2017/01/ | 1,534,485,850,000,000,000 | text/html | crawl-data/CC-MAIN-2018-34/segments/1534221211719.12/warc/CC-MAIN-20180817045508-20180817065508-00242.warc.gz | 476,436,001 | 12,192 | ## What is Arbitrage?
Definition Arbitrage is taking advantage of price differences between identical assets in multiple markets. Those multiple markets could be different in identity or different temporally. Example 1 You are at a garage sale and you see a lamp that costs … Continue reading
## What is Marginal Value?
Introduction Marginal value is marginal benefit minus marginal cost. It is simply how much benefit you get per each additional unit of measure (marginal benefit) minus how much it costs you per each additional unit of measure (marginal cost). Example … Continue reading
## How to Calculate Future Value and Present Value of an Appreciating Asset
Introduction In an earlier post, I defined what future value and what present value is. In this post, I’m going to go through some of the mathematics of how you can calculate those values. How to Calculate Future Value The … Continue reading | 180 | 912 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2018-34 | longest | en | 0.894728 |
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# Problems
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#### Problem 2268. Given a number between 1-365 (a day number in a non leap year) , find what day of what month it is.
Created by: Hassan Dehghani
2
25
#### Problem 44479. Friday or not
Created by: Noriko HOUNOKI
Tags date
0
42
#### Problem 44663. Datetime basics
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0
48
#### Problem 2397. Leap Year
Created by: Kumar Sandeep
Tags leap year
0
76
#### Problem 44668. Day counter function
Created by: Srishti Saha
Tags basic, easy, matlab
0
34
#### Problem 1877. Friday the 13th
Created by: Kevin Hellemans
Tags date
2
35
#### Problem 45328. leap year
Created by: Asif Newaz
1
38
#### Problem 45378. Lost days
Created by: Asif Newaz
2
27
Created by: Jon
Tags time, date
1
39
#### Problem 42352. Days until next NewYear ball drop
Created by: Varoujan
2
38
#### Problem 45365. Count the days
Created by: Asif Newaz
Tags count, date
2
30
#### Problem 2336. Calendar Matrix
Created by: Ned Gulley
Tags date, calendar
3
73
#### Problem 2025. Find the day for a date
Created by: Vivek
2
31
#### Problem 62. Elapsed Time
Created by: Cody Team
Tags time
13
1933
#### Problem 45380. The End of the World
Created by: Asif Newaz
Tags date, calendar, maya
1
23
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# FLUID POWER CONTROL SYSTEM
Dr. K. Sekar
Assistant Professor
## Mechanical Engineering Department
NATIONAL INSTITUTE OF TECHNOLOGY
Calicut, Kerala 673601
Faces of Fluid Mechanics
## Archimedes Newton Leibniz Bernoulli Euler
(C. 287-212 BC) (1642-1727) (1646-1716) (1667-1748) (1707-1783)
## Navier Stokes Reynolds Prandtl Taylor
(1785-1836) (1819-1903) (1842-1912) (1875-1953) (1886-1975)
2
BASIC LAWS OF FLUIDS
Bernoulli’s equation
Reynolds number
Pascal’s law
Boyleslaw
Charles’ law
Gay – Lussac’s law
General Gas laws
Types of Flow
Laminar flow
Turbulent flow
PASCAL’S LAW
FORCE F1
SMALL AREA
A1
PRESSURE
P = F1
P
A1
F2 = P x A2
LARGE AREA
A2
FORCE F2
10
MULTIPLICATION OF FORCES
POWER TRANSMISSION SYSTEM
Power Transmission Methods
1. Mechanical Power Transmission :
2. Electrical Power Transmission Systems
Electrical Network Systems
3.Fluid Power Systems
THE SOURCE OF HYDRAULIC POWER
Hydraulic reservoir
Pumps
• Converts mechanical energy into hydraulic energy
The mechanical energy is delivered in the pump via a prime mover such as electric
motor due to mechanical action. The pump creates a partial vacuum at its inlet. This
permits atmospheric pressure to force the fluid through the inlet line into the pump,
then pushes the fluid into the hydraulic system.
## Non positive displacement pumps
• Low pressure, high-volume flow applications
• Maximum pressure capacity is 250 - 300 psi
• Using for transporting fluids from one location to another location
## Positive displacement pumps
Positive displacement pump ejects a fixed amount of fluid into the hydraulic system
per revolution of pump shaft rotation
Advantages of positive displacement pumps over non positive displacement pumps
• High pressure capability (upto 10000 psi or higher)
• Small, compact size
• High volumetric efficiency
• Small changes in efficiency
• Great flexibility of performance
CLASSIFICATION OF PRINCIPAL TYPES OF
HYDRO - PUMPS
Pumps
## Gear Vane Screw
Ext. gear
Int. gear
Inline Bent axis Stationery Rotating cylinder
Gerotor
cylinder block block
Fixed Variable
displacement displacement
## Variable plate Inclinable Cam/crankshaft driven
Unbalanced Balanced swash plate piston pump
vane pump vane pump
Non positive displacement pumps
Hydrostatic (or) Positive Displacement Pump
## •Positive displacement pumps must be protected against over pressure
•Pressure relief valve is used to protect the pump against over pressure by diverting
pump flow back to the hydraulic tank
Gear pump
## External gear pump, which develops flow by carrying fluid
between the teeth of two meshing gears. One of the gears is
connected to a drive shaft connected to the prime mover. The
second gear is driven as it meshes with the driver gear. Oil
chambers are formed between the gear teeth, the pump
housing, and the side wear plates. The suction side is were
teeth some out of mesh, and it is here that the volume
expands, bringing about a reduction in pressure to below
atmospheric pressure.
Volumetric Displacement and Theoretical Flow Rate
## Gear Pump Nomenclature
1.
2
3
Positive displacement pump Q versus N and P versus Q curves (a)Flow
versus speed curve (b) Flow versus pressure curve at constant pump
speed
Volumetric Efficiency :
Internal gear pump
Lobe pump operation
Gerotor pump operation
Vane pump operation
The following analysis and nomenclature is applicable for vane pumps
VANE PUMP HELD BY SPRING FORCE
Balanced Vane Pump
Piston Pumps
BENT AXIS PUMP
•In this pump, the
pistons are at an
angle to the drive
shaft and Thrust
Plate. The piston
block shaft is
connected to the
drive shaft by a
universal joint,
not shown. The
drive shaft, thrust
plate, piston block
shaft, and piston
block all revolve.
•The connecting rods are attached to the thrust plate and revolve with it, unlike the swash
plate pump where the piston rods slide past a stationary swash plate.
Bent axis piston pump
•Radial Piston Pumps can produce a very smooth flow under extreme pressure.
Generally they are variable-displacement pumps.
## •In variable models, flow
rate changes when the
shaft holding the rotating
pistons is moved with
relation to the casing (in
different models either
the shaft or the casing
moves.) Output can also
be varied by changing the
rotation speed.
In-line Piston Pump/Swash plate
piston pump
Variation in Pump Displacement
Pump Performance Curves
Pump -SYMBOLS
## Basic Hydraulic demo
HYDRAULIC CYLINDERS
ACTUATORS
ROTARY LINEAR
## SINGLE ACTING DOUBLE ACTING
RAM TYPE SPRING TYPE TELESCOPIC TYPE SINGLE ROD DOUBLE ROD
single acting.swf
## Double-acting hydraulic cylinder construction
(A) External view (B) Cutaway view
double acting.swf
Double Acting Cylinder Design
Typical Mechanical Linkages that can be combined with Hydraulic Cylinders
Cylinder Force, Velocity and Power
Operation of Cylinder Cushions
HYDRAULIC CYLINDERS
1
Hydraulic Motors
There are three basic types of hydraulic motors : Gear, Vane, Piston
## Torque Development by a Gear Motor
Vane-type rotary actuator (A)
Clockwise rotation (B) Counter-
clockwise rotation
Gear motors are normally limited to 2000 psi operating pressures and 2400 rpm operating
speeds. They are available with a maximum load capacity of 150gpm. The main advantage
of gear motor are its simple design and subsequent low cost. Hydraulic motors can also be of
thee internal gear design. This type can operate at high pressure and speeds and also has
greater displacements than the external gear motor.
## External Gear Motor
Operation of a vane motor
Inline Piston Motor Operation
Two configurations of in-line piston motors
Motor displacement varies with Swash plate angle
Bent-axis piston motor
Hydraulic Motor Performance
Motor Efficiencies
Basic Hydraulic demo
VALVES
CONTENTS
## • PRESSURE CONTROL VALVES
The most common valves for controlling pressure include relief, reducing,
## • DIRECTION CONTROL VALVES
a summary of the different types, configurations, and uses
## • FLOW CONTROL VALVES
•Orifices •Flow regulator
•Bypass flow regulator •Demand-compensated flow control
## •Pressure-compensated, variable flow-control valve
•. Pressure- and temperature-compensated, variable flow-control valve
•Priority valve.
•Deceleration valve
Check Valve Pilot-operated Check Valve
Ball valve-type manual shutoff Manual shutoff graphic symbol
Spool positions inside two-way valve Spool positions inside four-way valve
Manually actuated four-way valve
Actual solenoid-actuated
directional control valve Single solenoid-actuated, four-
way, two-position spring-offset
directional control valve
Pressure Relief Valve
Needle Valve
Flow Control Valve
Pressure and temperature-
compensated FCV symbol.
## Balanced pool flow divider
Flow directions through a flow control valve
Non-pressure-compensated flow
control valve
Operation of pressure-
compensated flow
control valve
Bourdon tube pressure gauge
Accumulator graphic symbols
Accumulator with pressure switch
## Accumulator supplementing pump flow
Accumulator with a discharge valve
## Accumulator used with a press
Lines and Connections
or
Working Line (Main)
Crossing Lines
## Drain Line Connecting Lines
Flexible Line
Flow Direction
105
Make hose assemblies long enough
and routed in a manner that Left hand drawing show how hose twists
prevents exceeding the minimum because it is bent in one plane while
bend radius recommendations oscillating motion bends in a second plane.
Rerouting the hose eliminates multi plane
bending
SYMBOLS
Thank you
Design of Hydraulic Circuits
Control of single acting hydraulic cylinder Control of double acting hydraulic cylinder
Two way valves can be used to fill and drain
a vessel
Drilling machine application
Schematic shows simple circuit to control cylinder
extension and retraction using a 4-port, 3-position
spool valve.
Double pump hydraulic system
Double acting Pressure intensifier
Pressure intensifier
Counterbalance circuit (A) Hold Cylinder (B) Extend Cylinder (C) Retract Cylinder
Counterbalance valve application
Multispool, 4 way DCV
Parallel circuit with a pressure-compensated Load-sensing pump
pump
Meter-in flow control, extend stroke Meter-out flow control, extend stroke
Cylinder with meter-in flow control of both Cylinder with meter-out flow control of both
strokes strokes
Cylinder with meter-in control of the extend
stroke and meter-out control of the retract
stroke.
## Cylinders with bleed-off flow control
Cylinder synchronization with a flow Motor synchronization with a flow divider
divider
P1 Ap1 P 2( Ap1 AR1 ) F1
and
P2 Ap 2 P3 ( Ap 2 AR 2 ) F2
P1 Ap1 F1 F2
• protection for sudden
extension of cylinder.
Speed control of hydraulic motor using flow control valve
If you take a look at the following picture , let me tell you ... it is not animated. Your eyes are making it move. To
test this, stare at one spot for a couple seconds and everything will stop moving. Or look at the black center of each
circle and it will stop moving. But move your eyes to the next black center and the previous will move after you take
your eyes away from it.... Thanks
Pneumatics: Air Preparation and Components
Operation of air filter
Palm -button valve Limit valve
## Solenoid -actuated directional control valve
Hand-lever operated four -way valve
Operation of a single-acting cylinder
Operation of a double-acting cylinder
Air pilot control of a double-acting cylinder
Cylinder cycle timing system
Two-step speed control system
Two-handed safety control circuit
Control of an air motor
Deceleration air cushion of a pneumatic cylinder
Pneumatic circuit diagram
Position step diagram
Position-step diagram
## Pneumatic circuit diagram
KARNAUGH-VEITCH MAPPING
METHOD
## Transfer of position-step diagram to K-V map
Signal flow diagram with signal flow path
Looping of K-V map and minimization of logic equation
Pneumatic circuit
Semi -Automatic material handling circuit
Automatic Material Handling Circuit
Sequence operation by cam valve
Hydraulic operation of a planning machine
Hydraulic operation of a vertical milling machine
Hydraulic operation of a grinding machine
Hydraulic operation of a press
Pneumatic system for raising and lowering barriers
Many machines are being designed for automatic
operations to be controlled signals from the computers
## 1. Push button switches
2. Limit switches
3. Pressure switches
4. Solenoids
5. Relays
6. Timers
7. Temperature switches
Push button and Limit switches
## Single-pole,single throw type
Pressure and Temperature switches
Relays
Timers
Solenoids and indicators
It is designed by logic function such as AND,OR and NOT for
controlling the operations of industrial equipment and processes
## It is an electromechanical relay control system.
They are more reliable and faster in operation
Smaller size and can be more readily expanded
Require less electrical power and less expensive for the same
number of control functions .
Just like a brains of the PLC it contains a microprocessor with a fixed memory and
an alterable memory.
RAM(Random access memory, store the program for some operations)
Whenever electrical signal is removed, it will store all operations and some
switches function also stored and continuously operation is running.
Sequence of operation of the fluid power system being controlled. PM
connected With CPU only, only the entering or monitoring the program
Program may be pressing keys on the PM’s keypad.
Keypad allows the operator to run programs continuously or single stepped
mode
Various signals received from(OR) send to the fluid power interfaces devices
such as push-button switch, pressure switch, limit switch, motor relay coils,
solenoid coils etc.
It contains a powerful 16-bit, 20-MHz processor, multitasking of up to 64
programs .
12 input and 8 output
PLC control of a hydraulic cylinder
Input output connection diagram
Fluid Power Educational Hobbs Corporation SMC Pneumatics
Foundation www.hobbs-corp.com www.smcusa.com
www.fpef.org
Bimba Manufacturing Lord Corp Fluid Power Dist. Ass’n
www.bimba.com www.lordmpd.com www.fpda.org
Clippard Instrument Lab. Monnier, Inc. Fluid Power Society
www.clippard.com www.monnier.com www.ifps.org
Dresser Norgren Nat’l Fluid Power Ass’n
dresserinstruments.com www.norgren.com www.nfpa.com
## Festo Parker Fluid Power Journal
www.FluidPowerJournal.com
www.festo.com www.parker.com
Thank you | 2,963 | 12,418 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2020-10 | latest | en | 0.623216 |
https://sustainabilitymethods.org/index.php/K-Means_Algorithm_in_Python | 1,716,150,287,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971057922.86/warc/CC-MAIN-20240519193629-20240519223629-00858.warc.gz | 475,097,984 | 10,557 | K-Means Algorithm in Python
Introduction
K-means is one of the simplest unsupervised learning algorithms that solve the clustering problem, in other words, classify a given dataset on a certain number of clusters (assume k clusters) fixed a priori.
Theoretical aspect
The main idea is to define k centroids, one for each cluster. These centroids should be placed in a cunning way because different location causes different result. Therefore, the better choice is to place them as much as possible far away from each other. The next step is to take each point belonging to a given data set and associate it to the nearest centroid. When no point is pending, the first step is completed and an early group is done. At this point, it is needed to recalculate k new centroids as centers of the clusters resulting from the previous step. After that a new binding has to be done between the same data points and the nearest new centroid. A loop has been generated. As a result of this loop the centroids change their location step by step until no more changes are done, so centroids do not move any more.
For a given dataset X containing n multidimensional data points and the number of categories k to be divided, the Euclidean distance is selected as the similarity index and the clustering aims to minimize the sum of the squares of the various types:
where k - cluster centers, n - number of data points, uk - the kth center, and xi the ith point in the data set.
The algorithm consists of the following steps:
1. Place k points in the space represented by the objects that are being clustered. These points are considered as initial group centroids.
2. Assign each object to the group that has the closest centroid.
3. When all objects have been assigned, recalculate the positions of the k centroids.
4. Repeat Step 2 and 3 until the centroids stop moving.
Python Example
RFM analysis
Recency, frequency and monetary (RFM) analysis is a powerful and recognized technique in database marketing. It is widely used to rank the customers based on their prior purchasing history.
1. Recency: When was the last time the customer made a purchase?
2. Frequency: How many times did the customer purchase?
3. Monetary: How much money did the customer spend?
```import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import datetime as dt
from sklearn.cluster import KMeans
from sklearn.neighbors import NearestNeighbors
from sklearn import metrics
df```
Column1 CustomerID OrderID Date Revenue date_format YEAR month
0 111382 13930366 43771330 20191023 5834 2019-10-23 2019 10
1 111385 11545838 43723657 20191023 5834 2019-10-23 2019 10
Recency
```# last date available in our dataset
df["date_format"].max() # Out: Timestamp('2020-04-19 00:00:00')
# setting max time as now to calculate time differences
now = dt.date(2020,4,19)```
```# group by customer by last date they purchased
df_recency = df.groupby(["CustomerID"] , as_index = False)["date_format"].max()
df_recency.columns = ["CustomerID" , "LastPurchaseDate"]
df_recency["LastPurchaseDate"] = pd.DatetimeIndex(df_recency.LastPurchaseDate).date
CustomerID LastPurchaseDate
0 465132 2019-11-30
1 465164 2020-02-28
2 465198 2020-04-13
3 465204 2019-11-03
4 465211 2020-01-22
```# calculate how often the customers are buying in the last few days
df_recency['Recency'] = df_recency.LastPurchaseDate.apply(lambda x : (now - x).days)
# dropping LastPurchase Date
df_recency.drop(columns=['LastPurchaseDate'],inplace=True)
# checking recencyFrequency
CustomerID Recency
0 465132 141
1 465164 51
2 465198 6
3 465204 168
4 465211 88
Frequency
```frequency_df = df.groupby("CustomerID",as_index = False)["OrderID"].count()
frequency_df = frequency_df.sort_values(by = "OrderID" , ascending=False)
frequency_df = frequency_df.rename(columns={"OrderID" :"Frequency"})
CustomerID Frequency
1525 502561 231
66329 5556237 180
136996 11826511 159
137116 11836775 153
164161 14188909 143
Monetary
```# check summed up spends of customers
monetary_df = df.groupby('CustomerID',as_index=False)["Revenue"].sum()
monetary_df.columns = ['CustomerID','Monetary']
CustomerID Monetary
0 465132 7302
1 465164 578
2 465198 5688
3 465204 1464
4 465211 4760
```# merging these three features
rf = df_recency.merge(frequency_df,left_on='CustomerID',right_on='CustomerID')
# combining with monetary values
rfm = rf.merge(monetary_df,left_on='CustomerID',right_on='CustomerID')
rfm.set_index('CustomerID',inplace=True)
# saving to file
rfm.to_csv('rfm.csv', index=False)
#rf_new = rfm.sort_values(by = "Frequency" , ascending = False)
# checking the dataframe
CustomerID Recency Frequency Monetary
465132 141 1 7302
465164 51 1 578
465198 6 1 5688
465204 168 1 1464
465211 88 1 4760
Segmentation and Evaluation
In the literature several approaches have been proposed to determine the number of clusters for k-mean clustering algorithm, for instance: i) Rule of thumb; ii) Elbow method; iii) Information Criterion Approach; iv) Information Theoretic Approach; v) Silhouette method and vi) Cross-validation.
In this case we will use Elbow method to determine the number of clusters. The basic idea of the elbow rule is to use a squareed distance between the sample points in each cluster and the centroid of the cluster to give a series of K values. The sum of squared errors (SSE) is used as a performance indicator. The method iterates over the K-value and calculates the SSE.
```# creating a copy of the dataframe
rfm_segmentation = rfm.copy()
# find the proper number of clusters
fig,ax = plt.subplots(figsize=(10,8))
wcss = []
for i in range(1,11):
kmeans = KMeans(n_clusters = i, init = 'k-means++', max_iter = 300, n_init = 10, random_state = 0)
kmeans.fit(rfm_segmentation)
wcss.append(kmeans.inertia_)
plt.plot(range(1,11), wcss, 'o')
plt.plot(range(1 , 11) , wcss , '-' , alpha = 0.5)
plt.title('Elbow Method')
plt.xlabel('Number of Clusters')
plt.ylabel('WCSS')
plt.savefig('Elbow_Method.png')
plt.show()```
The graph shows, that the curve changes its behaviour from steep to moderate slope at the point of 4 clusters, therefore, we will choose 3 as the proper number of clusters for k-means segmentation. This will be included in the model as a parameter.
```# instantiating the model
kmeans = KMeans(n_clusters = 3, init = 'k-means++', max_iter = 300, n_init = 10, random_state = 0)
y_kmeans = kmeans.fit_predict(rfm_segmentation)
# creating a column for the cluster
rfm_segmentation['Cluster'] = kmeans.labels_
# checking the column
CustomerID Recency Frequency Monetary Cluster
465132 141 1 7302 0
465164 51 1 578 0
465198 6 1 5688 0
465204 168 1 1464 0
465211 88 1 4760 0
As we applied clustering, we are now able to analyze Recency, Frequency and Monetary values for each group.
Recency rate analysis
```fig, ax = plt.subplots(figsize=(8,8))
plt.title('Recency for each Cluster')
sns.boxplot(rfm_segmentation.Cluster,rfm_segmentation.Recency)
plt.savefig("recency_clusters.png")```
Cluster 0 has high recency rate, respectively, which means it has been the longest for any cluster when it comes to Last Purchase Date.
Cluster 1 and 2 have low recency rate, which is good. They can be our Gold and Silver Customers.
Frequency rate analysis
```fig, ax = plt.subplots(figsize=(8,8))
plt.title('Frequency for each Cluster')
sns.boxplot(rfm_segmentation.Cluster,rfm_segmentation.Frequency);
plt.savefig('frequency_cluster.png')```
Cluster 0 has a low frequency rate, which means consumers in this cluster are not very frequent.
Clusters 1 and 2 have high frequency rates, which puts them even further in the race for Gold and Silver.
Monetary rate analysis
```fig, ax = plt.subplots(figsize=(8,8))
plt.title('Monetary for each Cluster')
sns.boxplot(rfm_segmentation.Cluster,rfm_segmentation.Monetary);
plt.savefig('monetary_cluster.png')```
Cluster 0 has a small monetary rate, which could be referred to Bronze Customers.
Cluster 2 has a medium level monetary rate, which makes this Cluster to be Silver Customers.
Cluster 1 has the highest monetary rate, suggesting this Cluster to be Gold Customers.
Result
We may conclude the following results, based on analyses:
• Gold Customers are in Cluster 1.
• Silver Customers are in Cluster 2.
• Bronze Customers are in Cluster 0.
Visualization
```# getting the values
X = rfm_segmentation.iloc[:, [0,1,2]].values
# 2d plot
fig, ax = plt.subplots(figsize=(10,10))
plt.scatter(X[:, 0], X[:, 1], c=y_kmeans, s=100, cmap='OrRd')
centers = kmeans.cluster_centers_
plt.scatter(centers[:, 0], centers[:, 1], c='black', s=200, alpha=0.5)```
```# checking number of clients per cluster
rfm_segmentation.Cluster.value_counts()```
0 179083 2 581 1 16
Strenghts and Drawbacks of k-means Algorithm
Strenghts:
1. Simple.
2. Fast for low dimensional data.
3. It can find pure sub clusters, if large number of clusters is specified.
4. With a large number of variables, k-Means may be viewed as computationally faster than hierarchical clustering.
5. K-Means is expected to produce tighter clusters than hierarchical clustering.
Drawbacks:
1. Sensitive to the selection of initial cluster center.
2. There is no rule for the decision of value of k and sensitive to initial value , for different initial value there will be different result.
3. This algorithm is easy to be effected by abnormal points.
4. It may contain dead unit problem.
References
The author of this entry is Faezeh Moradi. Edited by Evgeniya Zakharova. | 2,454 | 9,487 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.578125 | 4 | CC-MAIN-2024-22 | latest | en | 0.928206 |
https://www.jiskha.com/questions/23049/i-have-some-problems-that-ive-answered-and-i-need-someone-to-check-to-see-if-im-right | 1,627,691,262,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046154032.75/warc/CC-MAIN-20210730220317-20210731010317-00554.warc.gz | 829,934,621 | 5,227 | # Algebra 2 (check)
I have some problems that I've answered and I need someone to check to see if I'm right. Im just posting the questons and my answer,if its wrong I'll post my work,so here it is bear with me please and thank you.
1)name which sets of numbers to which -28 belongs. my answer = integers,rationals,reals
2)the property illustrated by 7x(9+1)=(9+1)x7 is the. my answer = commutative property of multiplication
3)solve the equation 2/5y=3/14. my answer = 28/15
4)solve the equation 3(absolute value bars x-5)=12. my answer = null set
5)solve the equation (absolute bars y-8)+6=15. my answer = 17 I'll stop here for right now.
3. If y is in the denominator, you are right.
4. No, x=9 works, as does x=1
5. y-8=± 9 so y=8±9
1. 👍
2. 👎
3. 👁
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This is really confusing. I know how to do the equilibrium which also has to do with reaction rates but they didb't give it to me as a choice. Please Please help me. I post other questions that i answered already but no one | 826 | 3,131 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.21875 | 3 | CC-MAIN-2021-31 | latest | en | 0.902572 |
https://stat.ethz.ch/pipermail/r-help/2010-April/234929.html | 1,717,010,150,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059384.83/warc/CC-MAIN-20240529165728-20240529195728-00171.warc.gz | 463,431,251 | 3,296 | # [R] minimization function
Gabor Grothendieck ggrothendieck at gmail.com
Sun Apr 11 18:54:25 CEST 2010
```It works for me:
> x <- 2*c(1,0.5,0.8,0.5,1,0.9, 0.8,0.9,1)
> Dmat <- matrix(x, byrow=T, nrow=3, ncol=3)
> dvec <- numeric(3)
> Amat <- matrix(0,3,4)
> Amat[,1 ] <- c(1,1,1)
> Amat[,2:4 ]<- t(diag(3))
> bvec <- c(3,0,0,0)
>
> solve.QP(Dmat,dvec,Amat,bvec=bvec, meq=1)
\$solution
[1] 1.5 1.5 0.0
\$value
[1] 6.75
\$unconstrained.solution
[1] 0 0 0
\$iterations
[1] 3 0
\$iact
[1] 1 4
>
> R.version.string
[1] "R version 2.10.1 (2009-12-14)"
> win.version()
[1] "Windows Vista (build 6002) Service Pack 2"
[1] "1.4-12"
On Sun, Apr 11, 2010 at 11:31 AM, li li <hannah.hlx at gmail.com> wrote:
> Hi,
> thanks!
>
> I added meq=1 and it did not seem to work. The result is the same as before.
>
>> x <- 2*c(1,0.5,0.8,0.5,1,0.9, 0.8,0.9,1)
>> Dmat <- matrix(x, byrow=T, nrow=3, ncol=3)
>> dvec <- numeric(3)
>> Amat <- matrix(0,3,4)
>> Amat[,1 ] <- c(1,1,1)
>> Amat[,2:4 ]<- t(diag(3))
>> bvec <- c(3,0,0,0)
>>
>> solve.QP(Dmat,dvec,Amat,bvec=bvec, meq=1)
> \$solution
> [1] 1.500000e+00 1.500000e+00 -8.881784e-16
> \$value
> [1] 6.75
> \$unconstrained.solution
> [1] 0 0 0
> \$iterations
> [1] 3 0
> \$Lagrangian
> [1] 4.5 0.0 0.0 0.6
> \$iact
> [1] 1 4
>>
>
>
> 2010/4/11 Gabor Grothendieck <ggrothendieck at gmail.com>
>>
>> Add meq=1 to the arguments.
>>
>> On Sun, Apr 11, 2010 at 9:50 AM, li li <hannah.hlx at gmail.com> wrote:
>> > Hi, thank you very much for the reply!
>> >
>> > Consider minimize quadratic form w'Aw with A be the following matrix.
>> >> Dmat/2
>> > [,1] [,2] [,3]
>> > [1,] 1.0 0.5 0.8
>> > [2,] 0.5 1.0 0.9
>> > [3,] 0.8 0.9 1.0
>> > I need to find w=(w1,w2,w3), a 3 by 1 vector, such that sum(w)=3, and
>> > wi>=0
>> > for all i.
>> >
>> > Below is the code I wrote, using the function solve.QP , however, the
>> > solution for w still have a
>> > negtive component. Can some one give me some suggestions?
>> >
>> > Thank you very much!
>> >
>> >> x <- 2*c(1,0.5,0.8,0.5,1,0.9, 0.8,0.9,1)
>> >> Dmat <- matrix(x, byrow=T, nrow=3, ncol=3)
>> >> dvec <- numeric(3)
>> >> Amat <- matrix(0,3,4)
>> >> Amat[,1 ] <- c(1,1,1)
>> >> Amat[,2:4 ]<- t(diag(3))
>> >> bvec <- c(3,0,0,0)
>> >>
>> >> solve.QP(Dmat,dvec,Amat,bvec=bvec)
>> > \$solution
>> > [1] 1.500000e+00 1.500000e+00 -8.881784e-16
>> > \$value
>> > [1] 6.75
>> > \$unconstrained.solution
>> > [1] 0 0 0
>> > \$iterations
>> > [1] 3 0
>> > \$Lagrangian
>> > [1] 4.5 0.0 0.0 0.6
>> > \$iact
>> > [1] 1 4
>> >
>> >
>> >
>> >
>> >
>> >
>> >
>> >
>> >
>> >
>> >
>> > 2010/4/10 Gabor Grothendieck <ggrothendieck at gmail.com>
>> >>
>> >> Check out the quadprog package.
>> >>
>> >> On Sat, Apr 10, 2010 at 5:36 PM, li li <hannah.hlx at gmail.com> wrote:
>> >> > Hi, thanks for the reply.
>> >> > A will be a given matrix satisfying condition 1. I want to find the
>> >> > vector w that minimizes the
>> >> > quadratic form. w satisfies condition 2.
>> >> >
>> >> >
>> >> > 2010/4/10 Paul Smith <phhs80 at gmail.com>
>> >> >
>> >> >> On Sat, Apr 10, 2010 at 5:13 PM, Paul Smith <phhs80 at gmail.com>
>> >> >> wrote:
>> >> >> >> I am trying to minimize the quardratic form w'Aw, with certain
>> >> >> >> constraints.
>> >> >> >> In particular,
>> >> >> >> (1) A=(a_{ij}) is n by n matrix and it is symmetric positive
>> >> >> definite,
>> >> >> >> a_{ii}=1 for all i;
>> >> >> >> and 0<a_{ij}<1 for i not equal j.
>> >> >> >> (2) w'1=n;
>> >> >> >> (3) w_{i}>=0
>> >> >> >>
>> >> >> >> Analytically, for n=2, it is easy to come up with a result. For
>> >> >> >> larger
>> >> >> n, it
>> >> >> >> seems
>> >> >> >> difficult to obtain the result.
>> >> >> >>
>> >> >> >> Does any one know whether it is possible to use R to numerically
>> >> >> >> compute
>> >> >> it?
>> >> >> >
>> >> >> > And your decision variables are? Both w[i] and a[i,j] ?
>> >> >>
>> >> >> In addition, what do you mean by "larger n"? n = 20 is already large
>> >> >> (in your sense)?
>> >> >>
>> >> >> Paul
>> >> >>
>> >> >> ______________________________________________
>> >> >> R-help at r-project.org mailing list
>> >> >> https://stat.ethz.ch/mailman/listinfo/r-help
>> >> >>
>> >> >>
>> >> >> http://www.R-project.org/posting-guide.html<http://www.r-project.org/posting-guide.html>
>> >> >> and provide commented, minimal, self-contained, reproducible code.
>> >> >>
>> >> >
>> >> > [[alternative HTML version deleted]]
>> >> >
>> >> > ______________________________________________
>> >> > R-help at r-project.org mailing list
>> >> > https://stat.ethz.ch/mailman/listinfo/r-help | 1,873 | 4,602 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2024-22 | latest | en | 0.672359 |
https://www.math-only-math.com/hcf-and-lcm-of-decimals.html | 1,723,319,208,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640822309.61/warc/CC-MAIN-20240810190707-20240810220707-00021.warc.gz | 701,259,112 | 13,430 | # H.C.F. and L.C.M. of Decimals
Steps to solve H.C.F. and L.C.M. of decimals:
Step I: Convert each of the decimals to like decimals.
Step II: Remove the decimal point and find the highest common factor and least common multiple as usual.
Step III: In the answer (highest common factor /least common multiple), put the decimal point as there are a number of decimal places in the like decimals.
Now we will follow the step-by-step explanation on how to calculate the highest common factor and the least common multiple of decimals.
Worked-out examples on H.C.F. and L.C.M. of decimals:
1. Find the H.C.F. and the L.C.M. of 1.20 and 22.5
Solution:
Given, 1.20 and 22.5
Converting each of the following decimals into like decimals we get;
1.20 and 22.50
Now, expressing each of the numbers without the decimals as the product of primes we get
120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5
2250 = 2 × 3 × 3 × 5 × 5 × 5 = 2 × 32 × 53
Now, H.C.F. of 120 and 2250 = 2 × 3 × 5 = 30
Therefore, the H.C.F. of 1.20 and 22.5 = 0.30 (taking 2 decimal places)
L.C.M. of 120 and 2250 = 23 × 32 × 53 = 9000
Therefore, L.C.M. of 1.20 and 22.5 = 90.00 (taking 2 decimal places)
2. Find the H.C.F. and the L.C.M. of 0.48, 0.72 and 0.108
Solution:
Given, 0.48, 0.72 and 0.108
Converting each of the following decimals into like decimals we get;
0.480, 0.720 and 0.108
Now, expressing each of the numbers without the decimals as the product of primes we get
480 = 2 × 2 × 2 × 2 × 2 × 3 × 5 = 25 × 3 × 5
720 = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 24 × 32 × 5
108 = 2 × 2 × 3 × 3 × 3 = 22 × 33
Now, H.C.F. of 480, 720 and 108 = 22 × 3 = 12
Therefore, the H.C.F. of 0.48, 0.72 and 0.108 = 0.012 (taking 3 decimal places)
L.C.M. of 480, 720 and 108 = 25 × 33 × 5 = 4320
Therefore, L.C.M. of 0.48, 0.72, 0.108 = 4.32 (taking 3 decimal places)
3. Find the H.C.F. and the L.C.M. of 0.6, 1.5, 0.18 and 3.6
Solution:
Given, 0.6, 1.5, 0.18 and 3.6
Converting each of the following decimals into like decimals we get;
0.60, 1.50, 0.18 and 3.60
Now, expressing each of the numbers without the decimals as the product of primes we get
60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
150 = 2 × 3 × 5 × 5 = 2 × 3 × 52
18 = 2 × 3 × 3 = 2 × 32
360 = 2 × 2 × 2 × 3 × 3 × 5 = 23 × 32 × 5
Now, H.C.F. of 60, 150, 18 and 360 = 2 × 3 = 6
Therefore, the H.C.F. of 0.6, 1.5, 0.18 and 3.6 = 0.06 (taking 2 decimal places)
L.C.M. of 60, 150, 18 and 360 = 23 × 32 × 52 = 1800
Therefore, L.C.M. of 0.6, 1.5, 0.18 and 3.6 = 18.00 (taking 2 decimal places)
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We have arrived to the most complicated thing we are going to discuss in this class. Unfortunately, we have to warn you that you might find this next stuff a bit complicated. You might not, and that would be great! We will try our best to present the issues in a few different ways, so you have a few different tools to help you understand the issue.
What’s this so very complicated issue? Well, the first part it isn’t that complicated. For example, up until now we have been talking about experiments. Most every experiment has had two important bits, the independent variable (the manipulation), and the dependent variable (what we measure). In most cases, our independent variable has had two levels, or three or four; but, there has only been one independent variable.
What if you wanted to manipulate more than one independent variable? If you did that you would at least two independent variables, each with their own levels. The rest of the book is about designs with more than one independent variable, and the statistical tests we use to analyze those designs.
Let’s go through some examples of designs so can see what we are talking about. We will be imagining experiments that are trying to improve students grades. So, the dependent variable will always be grade on a test.
1. 1 IV (two levels)
We would use a t-test for these designs, because they only have two levels.
1. Time of day (Morning versus Afternoon): Do students do better on tests when they take them in the morning versus the afternoon? There is one IV (time of day), with two levels (Morning vs. Afternoon)
2. Caffeine (some caffeine vs no caffeine): Do students do better on tests when they drink caffeine versus not drinking caffeine? There is one IV (caffeine), with two levels (some caffeine vs no caffeine)
1. 1 IV (three levels):
We would use an ANOVA for these designs because they have more than two levels
1. Time of day (Morning, Afternoon, Night): Do students do better on tests when they take them in the morning, the afternoon, or at night? There is one IV (time of day), with three levels (Morning, Afternoon, and Night)
2. Caffeine (1 coffee, 2 coffees, 3 coffees): Do students do better on tests when they drink 1 coffee, 2 coffees, or three coffees? There is one IV (caffeine), with three levels (1 coffee, 2 coffees, and 3 coffees)
1. 2 IVs, IV1 (two levels), IV2 (two levels)
We haven’t talked about what kind of test to run for this design (hint it is called a factorial ANOVA)
1. IV1 (Time of Day: Morning vs. Afternoon); IV2 (Caffeine: some caffeine vs. no caffeine): How does time of day and caffeine consumption influence student grades? We had students take tests in the morning or in the afternoon, with or without caffeine. There are two IVs (time of day & caffeine). IV1 (Time of day) has two levels (morning vs afternoon). IV2 (caffeine) has two levels (some caffeine vs. no caffeine)
OK, let’s stop here for the moment. The first two designs both had one IV. The third design shows an example of a design with 2 IVs (time of day and caffeine), each with two levels. This is called a 2x2 Factorial Design. It is called a factorial design, because the levels of each independent variable are fully crossed. This means that first each level of one IV, the levels of the other IV are also manipulated. “HOLD ON STOP PLEASE!” Yes, it seems as if we are starting to talk in the foreign language of statistics and research designs. We apologize for that. We’ll keep mixing it up with some plain language, and some pictures.
## 9.1 Factorial basics
### 9.1.1 2x2 Designs
We’ve just started talking about a 2x2 Factorial design. We said this means the IVs are crossed. To illustrate this, take a look at the following tables. We show an abstract version and a concrete version using time of day and caffeine as the two IVs, each with two levels in the design:
Let’s talk about this crossing business. Here’s what it means for the design. For the first level of Time of Day (morning), we measure test performance when some people drank caffeine and some did not. So, in the morning we manipulate whether or not caffeine is taken. Also, in the second level of the Time of Day (afternoon), we also manipulate caffeine. Some people drink or don’t drink caffeine in the afternoon as well, and we collect measures of test performance in both conditions.
We could say the same thing, but talk from the point of view of the second IV. For example, when people drink caffeine, we test those people in the morning, and in the afternoon. So, time of day is manipulated for the people who drank caffeine. Also, when people do not drink caffeine, we test those people in the morning, and in the afternoon, So, time of day is manipulated for the people who did not drink caffeine.
Finally, each of the four squares representing a DV, is called a condition. So, we have 2 IVs, each with 2 levels, for a total of 4 conditions. This is why we call it a 2x2 design. 2x2 = 4. The notation tells us how to calculate the total number of conditions.
### 9.1.2 Factorial Notation
Anytime all of the levels of each IV in a design are fully crossed, so that they all occur for each level of every other IV, we can say the design is a fully factorial design.
We use a notation system to refer to these designs. The rules for notation are as follows. Each IV get’s it’s own number. The number of levels in the IV is the number we use for the IV. Let’s look at some examples:
2x2 = There are two IVS, the first IV has two levels, the second IV has 2 levels. There are a total of 4 conditions, 2x2 = 4.
2x3 = There are two IVs, the first IV has two levels, the second IV has three levels. There are a total of 6 conditions, 2x3 = 6
3x2 = There are two IVs, the first IV has three levels, the second IV has two levels. There are a total of 6 conditions, 3x2=6.
4x4 = There are two IVs, the first IV has 4 levels, the second IV has 4 levels. There are a total of 16 condition, 4x4=16
2x3x2 = There are a total of three IVs. The first IV has 2 levels. The second IV has 3 levels. The third IV has 2 levels. There are a total of 12 condition. 2x3x2 = 12.
### 9.1.3 2 x 3 designs
Just for fun, let’s illustrate a 2x3 design using the same kinds of tables we looked at before for the 2x2 design.
All we did was add another row for the second IV. It’s a 2x3 design, so it should have 6 conditions. As you can see there are now 6 cells to measure the DV.
## 9.2 Purpose of Factorial Designs
Factorial designs let researchers manipulate more than one thing at once. This immediately makes things more complicated, because as you will see, there are many more details to keep track of. Why would researchers want to make things more complicated? Why would they want to manipulate more than one IV at a time.
Before we go on, let’s clarify what we mean by manipulating more than one thing at once. When you have one IV in your design, by definition, you are manipulating only one thing. This might seem confusing at first, because the IV has more than one level, so it seems to have more than one manipulation. Consider manipulating the number of coffees that people drink before they do a test. We could have one IV (coffee), with three levels (1, 2, or 3 coffees). You might want to say we have three manipulations here, drinking 1, 2, or 3 coffees. But, the way we define manipulation is terms of the IV. There is only one coffee IV. It does have three levels. Nevertheless, we say you are only doing one coffee manipulation. The only thing you are manipulating is the amount of coffee. That’s just one thing, so it’s called one manipulation. To do another, second manipulation, you need to additionally manipulate something that is not coffee (like time of day in our previous example).
Returning to our question: why would researchers want to manipulate more than one thing in their experiment. The answer might be kind of obvious. They want to know if more than one thing causes change in the thing they are measuring! For example, if you are measuring people’s happiness, you might assume that more than one thing causes happiness to change. If you wanted to track down how two things caused changes in happiness, then you might want to have two manipulations of two different IVs. This is not a wrong way to think about the reasons why researchers use factorial designs. They are often interested in questions like this. However, we think this is an unhelpful way to first learn about factorial designs.
We present a slightly different way of thinking about the usefulness of factorial designs, and we think it is so important, it get’s its own section.
### 9.2.1 Factorials manipulate an effect of interest
Here is how researchers often use factorial designs to understand the causal influences behind the effects they are interested in measuring. Notice we didn’t say the dependent variables they are measuring, we are now talking about something called effects. Effects are the change in a measure caused by a manipulation. You get an effect, any time one IV causes a change in a DV.
Here is an example. We will stick with this one example for a while, so pay attention… In fact, the example is about paying attention. Let’s say you wanted to measure something like paying attention. You could something like this:
1. Pick a task for people to do that you can measure. For example, you can measure how well they perform the task. That will be the dependent measure
2. Pick a manipulation that you think will cause differences in paying attention. For example, we know that people can get distracted easily when there are distracting things around. You could have two levels for your manipulation: No distraction versus distraction.
3. Measure performance in the task under the two conditions
4. If your distraction manipulation changes how people perform the task, you may have successfully manipulated how well people can pay attention in your task.
### 9.2.2 Spot the difference
Let’s elaborate this with another fake example. First, we pick a task. It’s called spot the difference. You may have played this game before. You look at two pictures side-by-side, and then you locate as many differences as you can find. here is an example:
How many differences can you spot? When you look for the differences, it feels like you are doing something we would call “paying attention”. If you pay attention to the clock tower, you will see that the hands on the clock are different. Ya! One difference spotted.
We could give people 30 seconds to find as many differences as they can. Then we give them another set of pictures and do it again. Every time we will measure how many differences they can spot. So, our measure of performance, our dependent variable, could be the mean number of differences spotted.
### 9.2.3 Distraction manipulation
Now, let’s think about a manipulation that might cause differences in how people pay attention. If people need to pay attention to spot differences, then presumably if we made it difficult to pay attention, people would spot less differences. What is a good way to distract people? I’m sure there are lots of ways to do this. How about we do the following:
1. No distraction condition: Here people do the task with no added distractions. They sit in front of a computer, in a quiet, distraction-free room, and find as many differences as they can for each pair of pictures
2. Distraction condition: Here we blast super loud ambulance sounds and fire alarms and heavy metal music while people attempt to spot differences. We also randomly turn the sounds on and off, and make them super-duper annoying and distracting. We make sure that the sounds aren’t loud enough to do any physical damage to anybody’s ear-drums. But, we want to make them loud enough to be super distracting. If you don’t like this, we could also tickle people with a feather, or whisper silly things into their ears, or surround them by clowns, or whatever we want, it just has to be super distracting.
### 9.2.4 Distraction effect
If our distraction manipulation is super-distracting, then what should we expect to find when we compare spot-the-difference performance between the no-distraction and distraction conditions? We should find a difference!
If our manipulation works, then we should find that people find more differences when they are not distracted, and less differences when they are distracted. For example, the data might look something like this:
The figure shows a big difference in the mean number of difference spotted. People found 5 differences on average when they were distracted, and 10 differences when they were not distracted. We labelled the figure, “The distraction effect”, because it shows a big effect of distraction. The effect of distraction is a mean of 5 spot the differences. It’s the difference between performance in the Distraction and No-Distraction conditions. In general, it is very common to use the word effect to refer to the differences caused by the manipulation. We manipulated distraction, it caused a difference, so we call this the “distraction effect”.
### 9.2.5 Manipulating the Distraction effect
This is where factorial designs come in to play. We have done the hard work of finding an effect of interest, in this case the distraction effect. We think this distraction effect actually measures something about your ability to pay attention. For example, if you were the kind of person who had a small distraction effect (maybe you find 10 differences when you are not distracted, and 9 differences when you are distracted), that could mean you are very good at ignoring distracting things while you are paying attention. On the other hand, you could be the kind of person who had a big distraction effect (maybe you found 10 differences under no distraction, and only 1 difference when you were distracted); this could mean you are not very good at ignoring distracting things while you are paying attention.
Overall now, we are thinking of our distraction effect (the difference in performance between the two conditions) as the important thing we want to measure. We then might want to know how to make people better at ignoring distracting things. Or, we might want to know what makes people worse at ignoring things. In other words we want to find out what manipulations control the size of the distraction effect (make it bigger or smaller, or even flip around!).
Maybe there is a special drug that helps you ignore distracting things. People taking this drug should be less distracted, and if they took this drug while completing our task, they should have a smaller distraction effect compared to people not taking the drug.
Maybe rewarding people with money can help you pay attention and ignore distracting things better. People receiving 5 dollars every time they spot a difference might be able to focus more because of the reward, and they would show a smaller distraction effect in our task, compared to people who got no money for finding differences. Let’s see what this would look like.
We are going to add a second IV to our task. The second IV will manipulate reward. In one condition, people will get 5 dollars for every difference they find (so they could leave the study with lots of money if they find lots of differences). In the other condition, people will get no money, but they will still have find differences. Remember, this will be a factorial design, so everybody will have to find differences when they are distracted and when they are not distracted.
The question we are now asking is: Will manipulating reward cause a change in the size of the distraction effect. We could predict that people receiving rewards will have a smaller distraction effect than people not receiving rewards. If that happened, the data would look something like this:
I’ve just shown you a new kind of graph. I apologize right now for showing this to you first. It’s more unhelpful than the next graph. What I did was keep the x-axis the same as before (to be consistent). So, we have distraction vs. no distraction on the x-axis. In the distraction condition, there are means for spot-the-difference performance in the no-reward (red), and reward (aqua) conditions. The same goes for the no-distraction condition, a red and an aqua bar for the no-reward and reward conditions. We can try to interpret this graph, but the next graph plots the same data in a different way, which makes it easier to see what we are talking about.
All we did was change the x-axis. Now the left side of the x-axis is for the no-reward condition, and the right side is for the reward condition. The red bar is for the distraction condition, and the aqua bar is for the no distraction condition. It is easier to see the distraction effect in this graph. The distraction effect is the difference in size between the red and aqua bars. For each reward condition, the red and aqua bars are right beside each other, so can see if there is a difference between them more easily, compared to the first graph.
No-Reward condition: In the no-reward condition people played spot the difference when they were distracted and when they were not distracted. This is a replication of our first fake study. We should expect to find the same pattern of results, and that’s what the graph shows. There was a difference of 5. People found 5 differences when they were distracted and 10 when they were not distracted. So, there was a distraction effect of 5, same as we had last time.
Reward condition: In the reward condition people played spot the difference when they were distracted and when they were not distracted. Except, they got 5 dollars every time they spotted a difference. We predicted this would cause people to pay more attention and do a better job of ignoring distracting things. The graph shows this is what happened. People found 9 differences when they were distracted and 11 when they were not distracted. So, there was a distraction effect of 2.
If we had conducted this study, we might have concluded that reward can manipulate the distraction effect. When there was no reward, the size of the distraction effect was 5. When there was reward, the size of the distraction effect was 2. So, the reward manipulation changed the size of the distraction effect by 3 (5-2 =3).
This is our description of why factorial designs are so useful. They allow researchers to find out what kinds of manipulations can cause changes in the effects they measure. We measured the distraction effect, then we found that reward causes changes in the distraction effect. If we were trying to understand how paying attention works, we would then need to explain how it is that reward levels could causally change how people pay attention. We would have some evidence that reward does cause change in paying attention, and we would have to come up with some explanations, and then run more experiments to test whether those explanations hold water.
## 9.3 Graphing the means
In our example above we showed you two bar graphs of the very same means for our 2x2 design. Even though the graphs plot identical means, they look different, so they are more or less easy to interpret by looking at them. Results from 2x2 designs are also often plotted with line graphs. Those look different too. Here are four different graphs, using bars and lines to plot the very same means from before. We are showing you this so that you realize how you graph your data matters, and it makes it more or less easy for people to understand the results. Also, how the data is plotted matters for what you need to look at to interpret the results.
## 9.4 Knowing what you want to find out
When you conduct a design with more than one IV, you get more means to look at. As a result, there are more kinds of questions that you can ask of the data. Sometimes it turns out that the questions that you can ask, are not the ones that you want to ask, or have an interest in asking. Because you ran the design with more than one IV, you have the opportunity to ask these kinds of extra questions.
What kinds of new things are we talking about? Let’s keep going with our distraction effect experiment. We have the first IV where we manipulated distraction. So, we could find the overall means in spot-the difference for the distraction vs. no-distraction conditions (that’s two means). The second IV was reward. We could find the overall means in spot-the-difference performance for the reward vs. no-reward conditions (that’s two more means). We could do what we already did, and look at the means for each combination, that is the mean for distraction/reward, distraction/no-reward, no-distraction/reward, and no-distraction/no-reward (that’s four more means, if you’re counting). There’s even more. We could look at the mean distraction effect (the difference between distraction and no-distraction) for the reward condition, and the mean distraction effect for the no-reward condition (that’s two more). I hope you see here that there are a lot of means to look. And they are all different means. Let’s look at all of them together in one graph with four panels.
The purpose of showing all of these means is to orient you to your problem. If you conduct a 2x2 design (and this is the most simple factorial that you can conduct), you will get all of these means. You need to know what you want to know from the means. That is, you need to be able to connect the research question to the specific means you are interested in analyzing.
For example, in our example, the research question was whether reward would change the size of the distraction effect. The top left panel gives us some info about this question. We can see all of the condition means, and we can visually see that the distraction effect was larger in the No-reward compared to the reward condition. But, to “see” this, we need to do some visual subtraction. You need to look at the difference between the red and aqua bars for each of the reward and no-reward conditions.
Does the top right panel tell us about whether reward changed the size of the distraction effect? NO, it just shows that there was an overall distraction effect (this is called the main effect of distraction). Main effects are any differences between the levels of one independent variable.
Does the bottom left panel tell us about whether reward changed the size of the distraction effect? NO! it just shows that there was an overall reward effect, called the main effect of reward. People who were rewarded spotted a few more differences than the people who weren’t, but this doesn’t tell us if they were any less distracted.
Finally, how about the bottom left panel. Does this tell us about whether the reward changed the size of the distraction effect? YES! Notice, the y-axis is different for this panel. The y-axis here is labelled “Distraction Effect”. You are looking at two difference scores. The distraction effect in the no-reward condition (10-5 = 5), and the distraction effect in the Reward condition (11-9 = 2). These two bars are different as a function of reward. So, it looks like reward did produce a difference between the distraction effects! This was the whole point of the fake study. It is these means that were most important for answering the question of the study. As a very last point, this panel contains what we call an interaction. We explain this in the next section.
Pro tip: Make sure you know what you want to know from your means before you run the study, otherwise you will just have way too many means, and you won’t know what they mean.
## 9.5 Simple analysis of 2x2 repeated measures design
Normally in a chapter about factorial designs we would introduce you to Factorial ANOVAs, which are totally a thing. We will introduce you to them soon. But, before we do that, we are going to show you how to analyze a 2x2 repeated measures ANOVA design with paired-samples t-tests. This is probably something you won’t do very often. However, it turns out the answers you get from this method are the same ones you would get from an ANOVA.
Admittedly, if you found the explanation of ANOVA complicated, it will just appear even more complicated for factorial designs. So, our purpose here is to delay the complication, and show you with t-tests what it is that the Factorial ANOVA is doing. More important, when you do the analysis with t-tests, you have to be very careful to make all of the comparisons in the right way. As a result, you will get some experience learning how to know what it is you want to know from factorial designs. Once you know what you want to know, you can use the ANOVA to find out the answers, and then you will also know what answers to look for after you run the ANOVA. Isn’t new knowledge fun!
The first thing we need to do is define main effects and interactions. Whenever you conduct a Factorial design, you will also have the opportunity to analyze main effects and interactions. However, the number of main effects and interactions you get to analyse depends on the number of IVs in the design.
### 9.5.1 Main effects
Formally, main effects are the mean differences for a single Independent variable. There is always one main effect for each IV. A 2x2 design has 2 IVs, so there are two main effects. In our example, there is one main effect for distraction, and one main effect for reward. We will often ask if the main effect of some IV is significant. This refers to a statistical question: Were the differences between the means for that IV likely or unlikely to be caused by chance (sampling error).
If you had a 2x2x2 design, you would measure three main effects, one for each IV. If you had a 3x3x3 design, you would still only have 3 IVs, so you would have three main effects.
### 9.5.2 Interaction
We find that the interaction concept is one of the most confusing concepts for factorial designs. Formally, we might say an interaction occurs whenever the effect of one IV has an influence on the size of the effect for another IV. That’s probably not very helpful. In more concrete terms, using our example, we found that the reward IV had an effect on the size of the distraction effect. The distraction effect was larger when there was no-reward, and it was smaller when there was a reward. So, there was an interaction.
We might also say an interaction occurs when the difference between the differences are different! Yikes. Let’s explain. There was a difference in spot-the-difference performance between the distraction and no-distraction condition, this is called the distraction effect (it is a difference measure). The reward manipulation changed the size of the distraction effect, that means there was difference in the size of the distraction effect. The distraction effect is itself a measure of differences. So, we did find that the difference (in the distraction effect) between the differences (the two measures of the distraction effect between the reward conditions) were different. When you start to write down explanations of what interactions are, you find out why they come across as complicated. We’ll leave our definition of interaction like this for now. Don’t worry, we’ll go through lots of examples to help firm up this concept for you.
The number of interactions in the design also depend on the number of IVs. For a 2x2 design there is only 1 interaction. The interaction between IV1 and IV2. This occurs when the effect of say IV2 (whether there is a difference between the levels of IV2) changes across the levels of IV1. We could write this in reverse, and ask if the effect of IV1 (whether there is a difference between the levels of IV1) changes across the levels of IV2. However, just because we can write this two ways, does not mean there are two interactions. We’ll see in a bit, that no matter how do the calculation to see if the difference scores–measure of effect for one IV– change across the levels of the other IV, we always get the same answer. That is why there is only one interaction for a 2x2. Similarly, there is only one interaction for a 3x3, because there again we only have two IVs (each with three levels). Only when we get up to designs with more than 2 IVs, do we find more possible interactions. A design with three IVS, has four interactions. If the IVs are labelled A, B, and C, then we have three 2-way interactions (AB, AC, and BC), and one three-way interaction (ABC). We hold off on this stuff for much later
### 9.5.3 Looking at the data
It is most helpful to see some data in order to understand how we will analyze it. Let’s imagine we ran our fake attention study. We will have five people in the study, and they will participate in all conditions, so it will be a fully repeated-measures design. The data could look like this:
No Reward
Reward
No Distraction
Distraction
No Distraction
Distraction
subject A B C D
1 10 5 12 9
2 8 4 13 8
3 11 3 14 10
4 9 4 11 11
5 10 2 13 12
Note:
Number of differences spotted for each subject in each condition.
### 9.5.4 Main effect of Distraction
The main effect of distraction compares the overall means for all scores in the no-distraction and distraction conditions, collapsing over the reward conditions.
The yellow columns show the no-distraction scores for each subject. The blue columns show the distraction scores for each subject.
The overall means for for each subject, for the two distraction conditions are shown to the right. For example, subject 1 had a 10 and 12 in the no-distraction condition, so their mean is 11.
We are interested in the main effect of distraction. This is the difference between the AC column (average of subject scores in the no-distraction condition) and the BD column (average of the subject scores in the distraction condition). These differences for each subjecct are shown in the last green column. The overall means, averaging over subjects are in the bottom green row.
All Conditions
No Reward
Reward
Distraction Means
Distraction Effect
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
Difference
subject A B C D AC BD AC.minus.BD
1 10 5 12 9 11 7 4
2 8 4 13 8 10.5 6 4.5
3 11 3 14 10 12.5 6.5 6
4 9 4 11 11 10 7.5 2.5
5 10 2 13 12 11.5 7 4.5
Means 11.1 6.8 4.3
Just looking at the means, we can see there was a main effect of Distraction, the mean for the no-distraction condition was 11.1, and the mean for the distraction condition was 6.8. The size of the main effect was 4.3 (the difference between 11.1 and 6.8).
Now, what if we wanted to know if this main effect of distraction (the difference of 4.3) could have been caused by chance, or sampling error. You could do two things. You could run a paired samples $$t$$-test between the mean no-distraction scores for each subject (column AC) and the mean distraction scores for each subject (column BD). Or, you could run a one-sample $$t$$-test on the difference scores column, testing against a mean difference of 0. Either way you will get the same answer.
Here’s the paired samples version:
##
## Paired t-test
##
## data: AC and BD
## t = 7.6615, df = 4, p-value = 0.00156
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 2.741724 5.858276
## sample estimates:
## mean of the differences
## 4.3
Here’s the one sample version:
##
## One Sample t-test
##
## data: AC - BD
## t = 7.6615, df = 4, p-value = 0.00156
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## 2.741724 5.858276
## sample estimates:
## mean of x
## 4.3
If we were to write-up our results for the main effect of distraction we could say something like this:
The main effect of distraction was significant, $$t$$(4) = 7.66, $$p$$ = 0.001. The mean number of differences spotted was higher in the no-distraction condition (M = 11.1) than the distraction condition (M = 6.8).
### 9.5.5 Main effect of Reward
The main effect of reward compares the overall means for all scores in the no-reward and reward conditions, collapsing over the reward conditions.
The yellow columns show the no-reward scores for each subject. The blue columns show the reward scores for each subject.
The overall means for for each subject, for the two reward conditions are shown to the right. For example, subject 1 had a 10 and 5 in the no-reward condition, so their mean is 7.5.
We are interested in the main effect of reward. This is the difference between the AB column (average of subject scores in the no-reward condition) and the CD column (average of the subject scores in the reward condition). These differences for each subjecct are shown in the last green column. The overall means, averaging over subjects are in the bottom green row.
All Conditions
No Reward
Reward
Reward Means
Reward Effect
No Distraction
Distraction
No Distraction
Distraction
No Reward
Reward
Difference
subject A B C D AB CD CD.minus.AB
1 10 5 12 9 7.5 10.5 3
2 8 4 13 8 6 10.5 4.5
3 11 3 14 10 7 12 5
4 9 4 11 11 6.5 11 4.5
5 10 2 13 12 6 12.5 6.5
Means 6.6 11.3 4.7
Just looking at the means, we can see there was a main effect of reward. The mean number of differences spotted was 11.3 in the reward condition, and 6.6 in the no-reward condition. So, the size of the main effectd of reward was 4.7.
Is a difference of this size likely o unlikey due to chance? We could conduct a paired-samples $$t$$-test on the AB vs. CD means, or a one-sample $$t$$-test on the difference scores. They both give the same answer:
Here’s the paired samples version:
##
## Paired t-test
##
## data: CD and AB
## t = 8.3742, df = 4, p-value = 0.001112
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## 3.141724 6.258276
## sample estimates:
## mean of the differences
## 4.7
Here’s the one sample version:
##
## One Sample t-test
##
## data: CD - AB
## t = 8.3742, df = 4, p-value = 0.001112
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## 3.141724 6.258276
## sample estimates:
## mean of x
## 4.7
If we were to write-up our results for the main effect of reward we could say something like this:
The main effect of reward was significant, t(4) = 8.37, p = 0.001. The mean number of differences spotted was higher in the reward condition (M = 11.3) than the no-reward condition (M = 6.6).
### 9.5.6 Interaction between Distraction and Reward
Now we are ready to look at the interaction. Remember, the whole point of this fake study was what? Can you remember?
Here’s a reminder. We wanted to know if giving rewards versus not would change the size of the distraction effect.
Notice, neither the main effect of distraction, or the main effect of reward, which we just went through the process of computing, answers this question.
In order to answer the question we need to do two things. First, compute distraction effect for each subject when they were in the no-reward condition. Second, compute the distraction effect for each subject when they were in the reward condition.
Then, we can compare the two distraction effects and see if they are different. The comparison between the two distraction effects is what we call the interaction effect. Remember, this is a difference between two difference scores. We first get the difference scores for the distraction effects in the no-reward and reward conditions. Then we find the difference scores between the two distraction effects. This difference of differences is the interaction effect (green column in the table)
All Conditions
No Reward
Reward
Distraction Effects
Interaction Effect
No Distraction
Distraction
No Distraction
Distraction
No Reward
Reward
Difference
subject A B C D A-B C-D (A-B)-(C-D)
1 10 5 12 9 5 3 2
2 8 4 13 8 4 5 -1
3 11 3 14 10 8 4 4
4 9 4 11 11 5 0 5
5 10 2 13 12 8 1 7
Means 6 2.6 3.4
The mean distraction effects in the no-reward (6) and reward (2.6) conditions were different. This difference is the interaction effect. The size of the interaction effect was 3.4.
How can we test whether the interaction effect was likely or unlikely due to chance? We could run another paired-sample $$t$$-test between the two distraction effect measures for each subject, or a one sample $$t$$-test on the green column (representing the difference between the differences). Both of these $$t$$-tests will give the same results:
Here’s the paired samples version:
##
## Paired t-test
##
## data: A_B and C_D
## t = 2.493, df = 4, p-value = 0.06727
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.3865663 7.1865663
## sample estimates:
## mean of the differences
## 3.4
Here’s the one sample version:
##
## One Sample t-test
##
## data: A_B - C_D
## t = 2.493, df = 4, p-value = 0.06727
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## -0.3865663 7.1865663
## sample estimates:
## mean of x
## 3.4
Oh look, the interaction was not significant. At least, if we had set our alpha criterion to 0.05, it would not have met that criteria. We could write up the results like this. The two-way interaction between between distraction and reward was not significant, $$t$$(4) = 2.493, $$p$$ = 0.067.
Often times when a result is “not significant” according to the alpha criteria, the pattern among the means is not described further. One reason for this practice is that the researcher is treating the means as if they are not different (because there was an above alpha probability that the observed idfferences were due to chance). If they are not different, then there is no pattern to report.
There are differences in opinion among reasonable and expert statisticians on what should or should not be reported. Let’s say we wanted to report the observed mean differences, we would write something like this:
The two-way interaction between between distraction and reward was not significant, t(4) = 2.493, p = 0.067. The mean distraction effect in the no-reward condition was 6 and the mean distraction effect in the reward condition was 2.6.
### 9.5.7 Writing it all up
We have completed an analysis of a 2x2 repeated measures design using paired-samples $$t$$-tests. Here is what a full write-up of the results could look like.
The main effect of distraction was significant, $$t$$(4) = 7.66, $$p$$ = 0.001. The mean number of differences spotted was higher in the no-distraction condition (M = 11.1) than the distraction condition (M = 6.8).
The main effect of reward was significant, $$t$$(4) = 8.37, $$p$$ = 0.001. The mean number of differences spotted was higher in the reward condition (M = 11.3) than the no-reward condition (M = 6.6).
The two-way interaction between between distraction and reward was not significant, $$t$$(4) = 2.493, $$p$$ = 0.067. The mean distraction effect in the no-reward condition was 6 and the mean distraction effect in the reward condition was 2.6.
Interim Summary. We went through this exercise to show you how to break up the data into individual comparisons of interest. Generally speaking, a 2x2 repeated measures design would not be anlayzed with three paired-samples $$t$$-test. This is because it is more convenient to use the repeated measures ANOVA for this task. We will do this in a moment to show you that they give the same results. And, by the same results, what we will show is that the $$p$$-values for each main effect, and the interaction, are the same. The ANOVA will give us $$F$$-values rather than $$t$$ values. It turns out that in this situation, the $$F$$-values are related to the $$t$$ values. In fact, $$t^2 = F$$.
### 9.5.8 2x2 Repeated Measures ANOVA
We just showed how a 2x2 repeated measures design can be analyzed using paired-sampled $$t$$-tests. We broke up the analysis into three parts. The main effect for distraction, the main effect for reward, and the 2-way interaction between distraction and reward. We claimed the results of the paired-samples $$t$$-test analysis would mirror what we would find if we conducted the analysis using an ANOVA. Let’s show that the results are the same. Here are the results from the 2x2 repeated-measures ANOVA, using the aov function in R.
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 4 3.70 0.925 NA NA
Distraction 1 92.45 92.450 58.698413 0.0015600
Residuals 4 6.30 1.575 NA NA
Reward 1 110.45 110.450 70.126984 0.0011122
Residuals1 4 6.30 1.575 NA NA
Distraction:Reward 1 14.45 14.450 6.215054 0.0672681
Residuals 4 9.30 2.325 NA NA
Let’s compare these results with the paired-samples $$t$$-tests.
Main effect of Distraction: Using the paired samples $$t$$-test, we found $$t$$(4) =7.6615, $$p$$=0.00156. Using the ANOVA we found, $$F$$(1,4) = 58.69, $$p$$=0.00156. See, the $$p$$-values are the same, and $$t^2 = 7.6615^2 = 58.69 = F$$.
Main effect of Reward: Using the paired samples $$t$$-test, we found $$t$$(4) =8.3742, $$p$$=0.001112. Using the ANOVA we found, $$F$$(1,4) = 70.126, $$p$$=0.001112. See, the $$p$$-values are the same, and $$t^2 = 8.3742^2 = 70.12 = F$$.
Interaction effect: Using the paired samples $$t$$-test, we found $$t$$(4) =2.493, $$p$$=0.06727. Using the ANOVA we found, $$F$$(1,4) = 6.215, $$p$$=0.06727. See, the $$p$$-values are the same, and $$t^2 = 2.493^2 = 6.215 = F$$.
There you have it. The results from a 2x2 repeated measures ANOVA are the same as you would get if you used paired-samples $$t$$-tests for the main effects and interactions.
## 9.6 2x2 Between-subjects ANOVA
You must be wondering how to calculate a 2x2 ANOVA. We haven’t discussed this yet. We’ve only shown you that you don’t have to do it when the design is a 2x2 repeated measures design (note this is a special case).
We are now going to work through some examples of calculating the ANOVA table for 2x2 designs. We will start with the between-subjects ANOVA for 2x2 designs. We do essentially the same thing that we did before (in the other ANOVAs), and the only new thing is to show how to compute the interaction effect.
Remember the logic of the ANOVA is to partition the variance into different parts. The SS formula for the between-subjects 2x2 ANOVA looks like this:
$$SS_\text{Total} = SS_\text{Effect IV1} + SS_\text{Effect IV2} + SS_\text{Effect IV1xIV2} + SS_\text{Error}$$
In the following sections we use tables to show the calculation of each SS. We use the same example as before with the exception that we are turning this into a between-subjects design. There are now 5 different subjects in each condition, for a total of 20 subjects. As a result, we remove the subjects column.
### 9.6.1 SS Total
We calculate the grand mean (mean of all of the score). Then, we calculate the differences between each score and the grand mean. We square the difference scores, and sum them up. That is $$SS_\text{Total}$$, reported in the bottom yellow row.
All Conditions
Difference from Grand Mean
Squared Differences
No Reward
Reward
No Reward
Reward
No Reward
Reward
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
A B C D A-GrandM B-GrandM C-GrandM D-GrandM (A-GrandM)^2 (B-GrandM)^2 (C-GrandM)^2 (D-GrandM)^2
10 5 12 9 1.05 -3.95 3.05 0.05 1.1025 15.6025 9.3025 0.0025
8 4 13 8 -0.95 -4.95 4.05 -0.95 0.9025 24.5025 16.4025 0.9025
11 3 14 10 2.05 -5.95 5.05 1.05 4.2025 35.4025 25.5025 1.1025
9 4 11 11 0.05 -4.95 2.05 2.05 0.0025 24.5025 4.2025 4.2025
10 2 13 12 1.05 -6.95 4.05 3.05 1.1025 48.3025 16.4025 9.3025
Means 9.6 3.6 12.6 10
Grand Mean 8.95
sums Sums 7.3125 148.3125 71.8125 15.5125
SS Total SS Total 242.95
### 9.6.2 SS Distraction
We need to compute the SS for the main effect for distraction. We calculate the grand mean (mean of all of the scores). Then, we calculate the means for the two distraction conditions. Then we treat each score as if it was the mean for it’s respective distraction condition. We find the differences between each distraction condition mean and the grand mean. Then we square the differences and sum them up. That is $$SS_\text{Distraction}$$, reported in the bottom yellow row.
All Conditions
Distraction Mean - GM
Squared Differences
No Reward
Reward
No Reward
Reward
No Reward
Reward
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
A B C D NDM-GM A DM-GM B NDM-GM C DM-GM D (NDM-GM )^2 A (DM-GM)^2 B (NDM-GM)^2 C (DM-GM)^2 D
10 5 12 9 2.15 -2.15 2.15 -2.15 4.6225 4.6225 4.6225 4.6225
8 4 13 8 2.15 -2.15 2.15 -2.15 4.6225 4.6225 4.6225 4.6225
11 3 14 10 2.15 -2.15 2.15 -2.15 4.6225 4.6225 4.6225 4.6225
9 4 11 11 2.15 -2.15 2.15 -2.15 4.6225 4.6225 4.6225 4.6225
10 2 13 12 2.15 -2.15 2.15 -2.15 4.6225 4.6225 4.6225 4.6225
Means 9.6 3.6 12.6 10
Grand Mean 8.95 No Distraction 11.1 Distraction 6.8
sums Sums 23.1125 23.1125 23.1125 23.1125
SS Distraction SS Distraction 92.45
These tables are a lot to look at! Notice here, that we first found the grand mean (8.95). Then we found the mean for all the scores in the no-distraction condition (columns A and C), that was 11.1. All of the difference scores for the no-distraction condition are 11.1-8.95 = 2.15. We also found the mean for the scores in the distraction condition (columns B and D), that was 6.8. So, all of the difference scores are 6.8-8.95 = -2.15. Remember, means are the balancing point in the data, this is why the difference scores are +2.15 and -2.15. The grand mean 8.95 is in between the two condition means (11.1 and 6.8), by a difference of 2.15.
### 9.6.3 SS Reward
We need to compute the SS for the main effect for reward. We calculate the grand mean (mean of all of the scores). Then, we calculate the means for the two reward conditions. Then we treat each score as if it was the mean for it’s respective reward condition. We find the differences between each reward condition mean and the grand mean. Then we square the differences and sum them up. That is $$SS_\text{Reward}$$, reported in the bottom yellow row.
All Conditions
Reward Mean - GM
Squared Differences
No Reward
Reward
No Reward
Reward
No Reward
Reward
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
A B C D NRM-GM A NRM-GM B RM-GM C RM-GM D (NRM-GM )^2 A (NRM-GM)^2 B (RM-GM)^2 C (RM-GM)^2 D
10 5 12 9 -2.35 -2.35 2.35 2.35 5.5225 5.5225 5.5225 5.5225
8 4 13 8 -2.35 -2.35 2.35 2.35 5.5225 5.5225 5.5225 5.5225
11 3 14 10 -2.35 -2.35 2.35 2.35 5.5225 5.5225 5.5225 5.5225
9 4 11 11 -2.35 -2.35 2.35 2.35 5.5225 5.5225 5.5225 5.5225
10 2 13 12 -2.35 -2.35 2.35 2.35 5.5225 5.5225 5.5225 5.5225
Means 9.6 3.6 12.6 10
Grand Mean 8.95 No Reward 6.6 Reward 11.3
sums Sums 27.6125 27.6125 27.6125 27.6125
SS Reward SS Reward 110.45
Now we treat each no-reward score as the mean for the no-reward condition (6.6), and subtract it from the grand mean (8.95), to get -2.35. Then, we treat each reward score as the mean for the reward condition (11.3), and subtract it from the grand mean (8.95), to get +2.35. Then we square the differences and sum them up.
### 9.6.4 SS Distraction by Reward
We need to compute the SS for the interaction effect between distraction and reward. This is the new thing that we do in an ANOVA with more than one IV. How do we calculate the variation explained by the interaction?
The heart of the question is something like this. Do the individual means for each of the four conditions do something a little bit different than the group means for both of the independent variables.
For example, consider the overall mean for all of the scores in the no reward group, we found that to be 6.6 Now, was the mean for each no-reward group in the whole design a 6.6? For example, in the no-distraction group, was the mean for column A (the no-reward condition in that group) also 6.6? The answer is no, it was 9.6. How about the distraction group? Was the mean for the reward condition in the distraction group (column B) 6.6? No, it was 3.6. The mean of 9.6 and 3.6 is 6.6. If there was no hint of an interaction, we would expect that the means for the reward condition in both levels of the distraction group would be the same, they would both be 6.6. However, when there is an interaction, the means for the reward group will depend on the levels of the group from another IV. In this case, it looks like there is an interaction because the means are different from 6.6, they are 9.6 and 3.6 for the no-distraction and distraction conditions. This is extra-variance that is not explained by the mean for the reward condition. We want to capture this extra variance and sum it up. Then we will have measure of the portion of the variance that is due to the interaction between the reward and distraction conditions.
What we will do is this. We will find the four condition means. Then we will see how much additional variation they explain beyond the group means for reward and distraction. To do this we treat each score as the condition mean for that score. Then we subtract the mean for the distraction group, and the mean for the reward group, and then we add the grand mean. This gives us the unique variation that is due to the interaction. We could also say that we are subtracting each condition mean from the grand mean, and then adding back in the distraction mean and the reward mean, that would amount to the same thing, and perhaps make more sense.
Here is a formula to describe the process for each score:
$$\bar{X}_\text{condition} -\bar{X}_\text{IV1} - \bar{X}_\text{IV2} + \bar{X}_\text{Grand Mean}$$
Or we could write it this way:
$$\bar{X}_\text{condition} - \bar{X}_\text{Grand Mean} + \bar{X}_\text{IV1} + \bar{X}_\text{IV2}$$
When you look at the following table, we apply this formula to the calculation of each of the differences scores. We then square the difference scores, and sum them up to get $$SS_\text{Interaction}$$, which is reported in the bottom yellow row.
All Conditions
Interaction Differences
Squared Differences
No Reward
Reward
No Reward
Reward
No Reward
Reward
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
No Distraction
Distraction
A B C D A-ND-NR+GM B-D-NR+GM C-ND-R+GM D-D-R+GM (A-ND-NR+GM)^2 A (B-D-NR+GM)^2 B (C-ND-R+GM)^2 C (D-D-R+GM)^2 D
10 5 12 9 0.85 -0.85 -0.85 0.85 0.7225 0.7225 0.7225 0.7225
8 4 13 8 0.85 -0.85 -0.85 0.85 0.7225 0.7225 0.7225 0.7225
11 3 14 10 0.85 -0.85 -0.85 0.85 0.7225 0.7225 0.7225 0.7225
9 4 11 11 0.85 -0.85 -0.85 0.85 0.7225 0.7225 0.7225 0.7225
10 2 13 12 0.85 -0.85 -0.85 0.85 0.7225 0.7225 0.7225 0.7225
Means 9.6 3.6 12.6 10
Grand Mean 8.95
sums Sums 3.6125 3.6125 3.6125 3.6125
SS Interaction SS Interaction 14.45
### 9.6.5 SS Error
The last thing we need to find is the SS Error. We can solve for that because we found everything else in this formula:
$$SS_\text{Total} = SS_\text{Effect IV1} + SS_\text{Effect IV2} + SS_\text{Effect IV1xIV2} + SS_\text{Error}$$
Even though this textbook meant to explain things in a step by step way, we guess you are tired from watching us work out the 2x2 ANOVA by hand. You and me both, making these tables was a lot of work. We have already shown you how to compute the SS for error before, so we will not do the full example here. Instead, we solve for SS Error using the numbers we have already obtained.
$SS_ = SS_- SS_ - SS_ - SS_$
$SS_ = 242.95 - 92.45 - 110.45 - 14.45 = 25.6$
We are going to skip the part where we divide the SSes by their dfs to find the MSEs so that we can compute the three $$F$$-values. Instead, if we have done the calculations of the $$SS$$es correctly, they should be same as what we would get if we used R to calculate the $$SS$$es. Let’s make R do the work, and then compare to check our work.
Df Sum Sq Mean Sq F value Pr(>F)
Distraction 1 92.45 92.45 57.78125 0.0000011
Reward 1 110.45 110.45 69.03125 0.0000003
Distraction:Reward 1 14.45 14.45 9.03125 0.0083879
Residuals 16 25.60 1.60 NA NA
A quick look through the column Sum Sq shows that we did our work by hand correctly. Congratulations to us! Note, this is not the same results as we had before with the repeated measures ANOVA. We conducted a between-subjects design, so we did not get to further partition the SS error into a part due to subject variation and a left-over part. We also gained degrees of freedom in the error term. It turns out with this specific set of data, we find p-values of less than 0.05 for all effects (main effects and the interaction, which was not less than 0.05 using the same data, but treating it as a repeated-measures design)
## 9.7 Fireside chat
Sometimes it’s good to get together around a fire and have a chat. Let’s pretend we’re sitting around a fire.
It’s been a long day. A long couple of weeks and months since we started this course on statistics. We just went through the most complicated things we have done so far. This is a long chapter. What should we do next?
Here’s a couple of options. We could work through, by hand, more and more ANOVAs. Do you want to do that? I don’t, making these tables isn’t too bad, but it takes a lot of time. It’s really good to see everything that we do laid bare in the table form a few times. We’ve done that already. It’s really good for you to attempt to calculate an ANOVA by hand at least once in your life. It builds character. It helps you know that you know what you are doing, and what the ANOVA is doing. We can’t make you do this, we can only make the suggestion. If we keep doing these by hand, it is not good for us, and it is not you doing them by hand. So, what are the other options.
The other options are to work at a slightly higher level. We will discuss some research designs, and the ANOVAs that are appropriate for their analysis. We will conduct the ANOVAs using R, and print out the ANOVA tables. This is what you do in the lab, and what most researchers do. They use software most of the time to make the computer do the work. Because of this, it is most important that you know what the software is doing. You can make mistakes when telling software what to do, so you need to be able to check the software’s work so you know when the software is giving you wrong answers. All of these skills are built up over time through the process of analyzing different data sets. So, for the remainder of our discussion on ANOVAs we stick to that higher level. No more monster tables of SSes. You are welcome.
## 9.8 Real Data
Let’s go through the process of looking at a 2x2 factorial design in the wild. This will be the very same data that you will analyze in the lab for factorial designs.
### 9.8.1 Stand at attention
Do you pay more attention when you are sitting or standing? This was the kind of research question the researchers were asking in the study we will look at. In fact, the general question and design is very similar to our fake study idea that we used to explain factorial designs in this chapter.
The paper we look at is called “Stand by your Stroop: Standing up enhances selective attention and cognitive control” (Rosenbaum, Mama, and Algom 2017). This paper asked whether sitting versus standing would influence a measure of selective attention, the ability to ignore distracting information.
They used a classic test of selective attention, called the Stroop effect. You may already know what the Stroop effect is. In a typical Stroop experiment, subjects name the color of words as fast as they can. The trick is that sometimes the color of the word is the same as the name of the word, and sometimes it is not. Here are some examples:
Congruent trials occur when the color and word match. So, the correct answers for each of the congruent stimuli shown would be to say, red, green, blue and yellow. Incongruent trials occur when the color and word mismatch. The correct answers for each of the incongruent stimuli would be: blue, yellow, red, green.
The Stroop effect is an example of a well-known phenomena. What happens is that people are faster to name the color of the congruent items compared to the color of the incongruent items. This difference (incongruent reaction time - congruent reaction time) is called the Stroop effect.
Many researchers argue that the Stroop effect measures something about selective attention, the ability to ignore distracting information. In this case, the target information that you need to pay attention to is the color, not the word. For each item, the word is potentially distracting, it is not information that you are supposed to respond to. However, it seems that most people can’t help but notice the word, and their performance in the color-naming task is subsequently influenced by the presence of the distracting word.
People who are good at ignoring the distracting words should have small Stroop effects. They will ignore the word, and it won’t influence them very much for either congruent or incongruent trials. As a result, the difference in performance (the Stroop effect) should be fairly small (if you have “good” selective attention in this task). People who are bad at ignoring the distracting words should have big Stroop effects. They will not ignore the words, causing them to be relatively fast when the word helps, and relatively slow when the word mismatches. As a result, they will show a difference in performance between the incongruent and congruent conditions.
If we take the size of the Stroop effect as a measure of selective attention, we can then start wondering what sorts of things improve selective attention (e.g., that make the Stroop effect smaller), and what kinds of things impair selective attention (e.g., make the Stroop effect bigger).
The research question of this study was to ask whether standing up improves selective attention compared to sitting down. They predicted smaller Stroop effects when people were standing up and doing the task, compared to when they were sitting down and doing the task.
The design of the study was a 2x2 repeated-measures design. The first IV was congruency (congruent vs incongruent). The second IV was posture (sitting vs. standing). The DV was reaction time to name the word.
### 9.8.2 Plot the data
They had subjects perform many individual trials responding to single Stroop stimuli, both congruent and incongruent. And they had subjects stand up sometimes and do it, and sit-down sometimes and do it. Here is a graph of what they found:
The figure shows the means. We can see that Stroop effects were observed in both the sitting position and the standing position. In the sitting position, mean congruent RTs were shorter than mean incongruent RTs (the red bar is lower than the aqua bar). The same general pattern is observed for the standing position. However, it does look as if the Stroop effect is slightly smaller in the stand condition: the difference between the red and aqua bars is slightly smaller compared to the difference when people were sitting.
### 9.8.3 Conduct the ANOVA
Let’s conduct a 2x2 repeated measures ANOVA on the data to evaluate whether the differences in the means are likely or unlikely to be due to chance. The ANOVA will give us main effects for congruency and posture (the two IVs), as well as one interaction effect to evaluate (congruency X posture). Remember, the interaction effect tells us whether the congruency effect changes across the levels of the posture manipulation.
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 49 2250738.636 45933.4416 NA NA
Congruency 1 576821.635 576821.6349 342.452244 0.0000000
Residuals 49 82534.895 1684.3856 NA NA
Posture 1 32303.453 32303.4534 7.329876 0.0093104
Residuals1 49 215947.614 4407.0942 NA NA
Congruency:Posture 1 6560.339 6560.3389 8.964444 0.0043060
Residuals 49 35859.069 731.8177 NA NA
### 9.8.4 Main effect of Congruency
Let’s talk about each aspect of the ANOVA table, one step at a time. First, we see that there was a significant main effect of congruency, $$F$$(1, 49) = 342.45, $$p$$ < 0.001. The $$F$$ value is extremely large, and the $$p$$-value is so small it reads as a zero. This $$F$$-value basically never happens by sampling error. We can be very confident that the overall mean difference between congruent and incongruent RTs was not caused by sampling error.
What were the overall mean differences between mean RTs in the congruent and incongruent conditions? We would have to look at thos means to find out. Here’s a table:
Congruency mean_rt sd SEM
Congruent 814.9415 111.3193 11.13193
Incongruent 922.3493 118.7960 11.87960
The table shows the mean RTs, standard deviation (sd), and standard error of the mean for each condition. These means show that there was a Stroop effect. Mean incongruent RTs were slower (larger number in milliseconds) than mean congruent RTs. The main effect of congruency is important for establishing that the researchers were able to measure the Stroop effect. However, the main effect of congruency does not say whether the size of the Stroop effect changed between the levels of the posture variable. So, this main effect was not particularly important for answering the specific question posed by the study.
### 9.8.5 Main effect of Posture
There was also a main effect of posture, $$F$$(1,49) = 7.329, $$p$$ =0.009.
Let’s look at the overall means for the sitting and standing conditions and see what this is all about:
Posture mean_rt sd SEM
Sit 881.3544 135.3842 13.53842
Stand 855.9365 116.9436 11.69436
Remember, the posture main effect collapses over the means in the congruency condition. We are not measuring a Stroop effect here. We are measuring a general effect of sitting vs standing on overall reaction time. The table shows that people were a little faster overall when they were standing, compared to when they were sitting.
Again, the main effect of posture was not the primary effect of interest. The authors weren’t interested if people are in general faster when they stand. They wanted to know if their selective attention would improve when they stand vs when they sit. They were most interested in whether the size of the Stroop effect (difference between incongruent and congruent performance) would be smaller when people stand, compared to when they sit. To answer this question, we need to look at the interaction effect.
### 9.8.6 Congruency X Posture Interaction
Last, there was a significant congruency X posture interaction, $$F$$(1,49) = 8.96, $$p$$ = 0.004.
With this information, and by looking at the figure, we can get a pretty good idea of what this means. We know the size of the Stroop effect must have been different between the standing and sitting conditions, otherwise we would have gotten a smaller $$F$$-value and a much larger $$p$$-value.
We can see from the figure the direction of this difference, but let’s look at the table to see the numbers more clearly.
Posture Congruency mean_rt sd SEM
Sit Congruent 821.9232 117.4069 16.60384
Sit Incongruent 940.7855 126.6457 17.91041
Stand Congruent 807.9599 105.6079 14.93521
Stand Incongruent 903.9131 108.5366 15.34939
In the sitting condition the Stroop effect was roughly 941-822 = 119 ms.
In the standing condition the Stroop effect was roughly 904-808 = 96 ms.
So, the Stroop effect was 119-96 = 23 ms smaller when people were standing. This is a pretty small effect in terms of the amount of time reduced, but even though it is small, a difference even this big was not very likely to be due to chance.
### 9.8.7 What does it all mean?
Based on this research there appears to be some support for the following logic chain. First, the researchers can say that standing up reduces the size of a person’s Stroop effect. Fine, what could that mean? Well, if the Stroop effect is an index of selective attention, then it could mean that standing up is one way to improve your ability to selectively focus and ignore distracting information. The actual size of the benefit is fairly small, so the real-world implications are not that clear. Nevertheless, maybe the next time you lose your keys, you should stand up and look for them, rather than sitting down and not look for them.
## 9.9 Factorial summary
We have introduced you to factorial designs, which are simply designs with more than one IV. The special property of factorial designs is that all of the levels of each IV need to be crossed with the other IVs.
We showed you how to analyse a repeated measures 2x2 design with paired samples-tests, and what an ANOVA table would look like if you did this in R. We also went through, by hand, the task of calculating an ANOVA table for a 2x2 between subjects design.
The main point we want you take away is that factorial designs are extremely useful for determining things that cause effects to change. Generally a researcher measures an effect of interest (their IV 1). Then, they want to know what makes that effect get bigger or smaller. They want to exert experimental control over their effect. For example, they might have a theory that says doing X should make the effect bigger, but doing Y should make it smaller. They can test these theories using factorial designs, and manipulating X or Y as a second independent variable.
In a factorial design each IV will have it’s own main effect. Sometimes the main effect themselves are what the researcher is interested in measures. But more often, it is the interaction effect that is most relevant. The interaction can test whether the effect of IV1 changes between the levels of IV2. When it does, researchers can infer that their second manipulation (IV2) causes change in their effet of interest. These changes are then documented and used to test underlying causal theories about the effects of interest.
### References
Rosenbaum, David, Yaniv Mama, and Daniel Algom. 2017. “Stand by Your Stroop: Standing up Enhances Selective Attention and Cognitive Control.” Psychological Science 28 (12): 1864–7. | 16,869 | 66,342 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.34375 | 4 | CC-MAIN-2021-25 | longest | en | 0.962279 |
https://www.java-forums.org/new-java/56717-heap-sort-question-print.html | 1,490,524,951,000,000,000 | text/html | crawl-data/CC-MAIN-2017-13/segments/1490218189198.71/warc/CC-MAIN-20170322212949-00572-ip-10-233-31-227.ec2.internal.warc.gz | 935,071,217 | 4,317 | heap sort question
• 03-15-2012, 01:35 AM
stuckonjava
heap sort question
Can someone explain how the heap-sort algorithm would work on the sequence (16,4,8,10,12).
with explanations of when the insertion and removal to the heaps are required and how they work.
I cannot find anything useful online regarding this.
• 03-15-2012, 02:42 AM
stuckonjava
Re: heap sort question
anyone?
• 03-15-2012, 02:16 PM
stuckonjava
Re: heap sort question
noone
• 03-15-2012, 02:25 PM
KevinWorkman
Re: heap sort question
That is absolutely not how this works. This isn't google or a homework service. How do you think heap sort works? What have you tried? Where are you stuck?
• 03-15-2012, 02:37 PM
stuckonjava
Re: heap sort question
yes I've tried. I don't understand how to begin it. Do I begin it in the tree or can I put them one by one using the insertion sort?
• 03-15-2012, 02:43 PM
KevinWorkman
Re: heap sort question
Do you understand how heap sort works? If so, list the steps it takes and then apply those steps to your input. If not, do a google search of heap sort. I'm sure your class book is another valuable resource for an explanation.
• 03-15-2012, 02:51 PM
JosAH
Re: heap sort question
I wrote a blog article on heap sort once (check the button near the top of this page). Heap sort works with two passes (or phases); the first phase constructs a heap out of a sequence of items and the second phase removes the largest element from the heap and corrects it again until the heap is empty. A big advantage is that it can do this all in place (no auxiliary memory is needed) and it can do it stable big-Oh wise.
kind regards,
Jos
• 03-15-2012, 02:54 PM
stuckonjava
Re: heap sort question
the way i was taught is that in the second phase it removes the smallest numbers first
• 03-15-2012, 02:58 PM
KevinWorkman
Re: heap sort question
Quote:
Originally Posted by stuckonjava
the way i was taught is that in the second phase it removes the smallest numbers first
Either one works, because a heap can be a min-heap or a max-heap. I think most descriptions use a max-heap. The idea is the same though. But these little discrepancies are why I asked you to define heap sort's behavior. If you know how you've been taught, where exactly are you stuck?
• 03-15-2012, 03:09 PM
stuckonjava
Re: heap sort question
I understand the heap sort perfectly, the problem I have is showing the first phase in terms of the list. For eg the list (16,4,8,10,12). How would I show the first phase of this step by step. I can do it using a binary tree but whats the best way to present this without one?
• 03-15-2012, 03:15 PM
KevinWorkman
Re: heap sort question
If you understand heap sort perfectly, I don't know what you're confused about. What is step one in heap sort? What is the first thing that would be done with that input?
• 03-15-2012, 03:25 PM
stuckonjava
Re: heap sort question
ok the first thing would be to put 16 as the root, then the 4 as its left child, but as 4 is smaller than 16 they swap. The 8 would now become the right child of 4, the 10 would become the left child of 16 and swap places with it. The 12 would then become the right child of 10.
• 03-15-2012, 03:32 PM
KevinWorkman
Re: heap sort question
If that's the way you were taught, okay. Like I said, most descriptions reverse that.
So what comes next?
• 03-15-2012, 03:41 PM
stuckonjava
Re: heap sort question
the 4 would be put into the new sorted list(forgot what it is called), and 12 becomes the new root , the 12 and 8 now swap places making 8 the new root. The 8 is nwow put into the new list along wth the 4. 16 now becomes the root, and swaps places with 10 which is then sent into the list with 4 and 8. 12 becomes the new root and is sent into the list with 4,8,10. Now only 16 is left and as there is nothing to compare it with it is sent into the sorted list whose contents now consist of 4,8,10,12,16.
• 03-15-2012, 03:47 PM
KevinWorkman
Re: heap sort question
Actually, I think the extraction is a little more involved than 12 magically becoming the new root. In the descriptions I'm familiar with, the last child is swapped with the root and then the old root is removed, and the new root is "trickled down" to maintain the heap.
• 03-15-2012, 03:55 PM
stuckonjava
Re: heap sort question
Oh, my description was that once the smallest key is the root it is remove into the new list.
• 03-15-2012, 03:58 PM
KevinWorkman
Re: heap sort question
If that's your description, okay, but I'm working from the description I've been familiar with, which is also on wikipedia: Heapsort - Wikipedia, the free encyclopedia
A possible minimum is temporarily set as the root, but only to extract the actual root, which is the max. It could be that your teacher is taking a slightly different approach with heap sort, or it could be that you're misunderstanding a step to mean that the heap is a min-heap. I can't be sure which one is the case.
• 03-15-2012, 04:06 PM
stuckonjava
Re: heap sort question
I though one of the heap rules was that node(v) >= node(v).parent
• 03-15-2012, 04:18 PM
KevinWorkman
Re: heap sort question
Like I said, it really depends on the details of your implementation. There is a chance you're being given a slightly modified heap sort that uses the same idea in a different way. But the "traditional" approach to heap sort is to use a max-heap, where a parent is greater than any of its children (so the opposite of what you're saying). I'm not in your class though, so I really can't say for certain whether that's the case or you're simply confusing something.
Either way though, you're going to want to more fully understand what happens when you take the root out of your heap. | 1,563 | 5,680 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.46875 | 3 | CC-MAIN-2017-13 | longest | en | 0.920267 |
http://math.stackexchange.com/questions/41296/what-is-mathbbcaut-mathbbc-mathbbq?answertab=active | 1,462,003,066,000,000,000 | text/html | crawl-data/CC-MAIN-2016-18/segments/1461860111620.85/warc/CC-MAIN-20160428161511-00027-ip-10-239-7-51.ec2.internal.warc.gz | 178,875,296 | 18,387 | # What is $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$?
Let $Aut(\mathbb{C}/\mathbb{Q})$ be the field automorphisms of $\mathbb{C}$, and $\mathbb{C}^{Aut(\mathbb{C}/\mathbb{Q})}$ the subfield of $\mathbb{C}$ fixed by this group. I supsect that it is equal to $\mathbb{Q}$ but I have difficulty proving it.
Here is what I do : let $x \notin \mathbb{Q}$. We have to find an automorphism not fixing $x$. It is easy to find an embeding $\mathbb{Q}(x) \rightarrow \mathbb{C}$ not sending $x$ on $x$, and use Zorn's lemma to extend it to a maximal subfield (of $\mathbb{C}$) $K \supset \mathbb{Q}(x)$. But it is not true that $K=\mathbb{C}$. So how to handle it ?
-
What's stopping you from extending until you get all the way to $\mathbb{C}$? – Qiaochu Yuan May 25 '11 at 15:34
You might want to take a look in here: mathdl.maa.org/images/upload_library/22/Ford/PaulBYale.pdf – user9413 May 25 '11 at 15:37
You are asserting that $K$ is never equal to $\mathbb{C}$? On what grounds? – Arturo Magidin May 25 '11 at 18:17
The problem is that if I apply Zorn's lemma like I do, then the map $K \rightarrow \C$ can be an isomophism without having $K=C$. – user10676 May 25 '11 at 21:17
Instead of looking for embeddings of $K$ into ${\mathbb C}$, can't you simply apply Zorn Lemma for automorphisms of $K$? The hard part is the first step: for $x$ irrational, find $x \in K$ and an automorphism of $K$ not sending $x$ to $x$, and the Zorn it. – N. S. May 26 '11 at 18:46
In this work in progress I state the following Fundamental Theorem of Galois Theory:
Let $K/F$ be any field extension. The following are equivalent:
(i) For all subextensions $L$ of $K/F$, $K^{\operatorname{Aut}(K/L)} = L$.
(ii) At least one of the following holds:
(a) $K/F$ is algebraic, normal and separable (i.e., a Galois extension in the usual sense), or
(b) $K$ is algebraically closed of characteristic zero.
That's the good news. The bad news is that this "Theorem", which I wrote down several years ago, is actually only a conjecture (it has remained as an open question on MathOverflow for more than a year, which is some indication of its nontriviality, at least). Okay, but there is more good news: the implication (ii) $\implies$ (i) is proven, and is a fairly routine application of basic field theory. Your question is a special case of (ii) $\implies$ (i), so there you go.
Added: Mea culpa, the proof of (ii) $\implies$ (i) in the linked to notes is "$\ldots$". (When you can't prove the big theorem you announce on the first page, you lose some motivation to fill in the other details, it seems.) Instead, please see $\S 10.1$ of my field theory notes in which there is a complete proof of (even a result slightly more general than) (ii) $\implies$ (i). Really, I promise.
-
It is always wonderful when open research questions reach this site, thanks for sharing yours :-) – Asaf Karagila May 25 '11 at 21:14
I think that the article Chandru posted indicates that you can interchange any two irrational algebraic numbers with the same minimal polynomial or any two transcendental numbers with an automorphism of C. Thus the fixed field of Aut(C/Q) is exactly Q. The trick here is not to consider the poset of extension maps of Q(x) into C, but rather to start with an automorphism of the field and then look at the poset of automorphisms of some field extension. The punchline is towards the end of the article.
- | 991 | 3,401 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2016-18 | latest | en | 0.845718 |
https://es.mathworks.com/matlabcentral/answers/1613925-grid-search-and-grid-plot-with-colours | 1,713,268,954,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817081.52/warc/CC-MAIN-20240416093441-20240416123441-00432.warc.gz | 215,702,198 | 33,511 | Grid search and grid plot with colours
4 visualizaciones (últimos 30 días)
Anitha Limann el 18 de Dic. de 2021
Comentada: Anitha Limann el 20 de Dic. de 2021
Hello,
I would like to do a grid search by assining zeros to each grid point first. And then I would like to calculate required values for each grid point and finally I am required to plot the grid with relevant values. here my grid represent error values for each grid point. So I want to plot this grid using some colour scheme method.
Attached here is the type of graph I am looking for.
Below is the code I wrote for this. However I am not sure how much of assigning zeros, for loop and then plotting is correct.
I tried to use plot and quiver fuctions but different length x and y axis appeared to be an error in that case. For now i used surf function, though i am not sure if that is correct. But it gives me a z axis too, which is not what i want.
Please let me know if someone could help me with this.
% Obserevd Z location km/Myr ## 176.7 W, 19.2 S, 64 mm/yr (km/Myr) velocity;
Zlon=-176.324;
Zlat=-18.395;
Zobsw=64;
R=6371; %km
% evaluate error at each grid point
AClon =-177.5:0.01:-176.7;
AClat =-20.5:0.01:-19;
[X,Y]=meshgrid(AClon,AClat);
AC=zeros(151,81);
for i=[1:151]
for j=[1:81]
arclen = distance(Zlon,Zlat,AClon(j),AClat(i)); % arc distant between 1.Z location and 2.each grid point.
w =(Zobsw/(R*sin(theta*pi/180)))*180/pi; % deg/Myr
vl = plate_vel(Zlon,Zlat,AClon(j),AClat(i),w); % i found this plate-vel funtion online. please use any set of data as an example here.
misfit=(sum((Zobsw-vl).^2)./(numel(vl)));
AC(i,j)=misfit;
end
end
surf(X,Y,AC)
18 comentariosMostrar 16 comentarios más antiguosOcultar 16 comentarios más antiguos
Voss el 20 de Dic. de 2021
No problem! Don't hesitate to ask another question if you suspect your calculations are off.
Anitha Limann el 20 de Dic. de 2021
Sure will do. Thank you!
Iniciar sesión para comentar.
Voss el 19 de Dic. de 2021
surf(X,Y,zeros(size(X)),AC,'EdgeColor','none','FaceColor','interp');
set(gca(),'View',[0 90]);
set(gcf(),'Colormap',flip(gray(),1));
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Translated by | 722 | 2,430 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.8125 | 3 | CC-MAIN-2024-18 | latest | en | 0.679248 |
https://socratic.org/questions/how-do-you-differentiate-f-x-x-2sinx-x-6-using-the-quotient-rule | 1,611,735,475,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610704821253.82/warc/CC-MAIN-20210127055122-20210127085122-00502.warc.gz | 556,993,625 | 5,825 | # How do you differentiate f(x)= ( x +2sinx )/ ( x - 6) using the quotient rule?
Jul 14, 2018
$f ' \left(x\right) = - \frac{10 \cos x + 2 \sin x + 6}{x - 6} ^ 2$
#### Explanation:
Here ,
$f \left(x\right) = \frac{x + 2 \sin x}{x - 6}$
Using Quotient Rule ,we get
$f ' \left(x\right) = \frac{\left(x - 6\right) \frac{d}{\mathrm{dx}} \left(x + 2 \sin x\right) - \left(x + 2 \sin x\right) \frac{d}{\mathrm{dx}} \left(x - 6\right)}{x - 6} ^ 2$
$\implies f ' \left(x\right) = \frac{\left(x - 6\right) \left(1 + 2 \cos x\right) - \left(x + 2 \sin x\right) \left(1\right)}{x - 6} ^ 2$
$\implies f ' \left(x\right) = \frac{x + 2 x \cos x - 6 - 12 \cos x - x - 2 \sin x}{x - 6} ^ 2$
$\implies f ' \left(x\right) = \frac{- 10 \cos x - 2 \sin x - 6}{x - 6} ^ 2$
$\implies f ' \left(x\right) = - \frac{10 \cos x + 2 \sin x + 6}{x - 6} ^ 2$ | 387 | 838 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 7, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.4375 | 4 | CC-MAIN-2021-04 | latest | en | 0.289267 |
https://datascience.stackexchange.com/questions/39107/how-to-interpret-sum-of-squared-error | 1,627,308,801,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046152129.33/warc/CC-MAIN-20210726120442-20210726150442-00319.warc.gz | 200,898,385 | 37,566 | How to interpret Sum of squared error
I am working on ANN. I have 2497 training examples and each of them is a vector of 128. So the input size is 128. Number of neurons in hidden layer is 64 and number of output neurons is 6 (since classes are six). My Target vector looks something like this : [0 1 0 0 0 0]. This means that the example belongs to class 2. I have used sigmoid as an activation at all layers and sum of squared error is loss. SSE is computed over one epoch. Total epochs are 10k. my loss starts from around 700 and reduces to 450. Should I say that loss is 18% per example since 450 is the loss for all the 2497 examples. How do I interpret this. Is my model good enough? I know that I should test it on unseen data to be sure of its accuracy but still does this tell anything about the performance at all or not.
ps: I am implementing it in C
• In the last layer, the one with many neurons as classes, you have to use softmax activation – Francesco Pegoraro Oct 3 '18 at 13:00
• How exactly are you computing SSE in a classification task? Can you post the formula? – shadowtalker Oct 3 '18 at 16:19
$$SSE = \sum_n \sum_k (\hat y_{nk} - y_{nk})^2 \\ Brier = \frac{SSE}{N}$$
For observations indexed by $$n$$ and classes indexed by $$k$$.
$$Brier = Reliability - Resolution + Uncertainty$$ | 354 | 1,310 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.71875 | 3 | CC-MAIN-2021-31 | latest | en | 0.921751 |
https://aprove.informatik.rwth-aachen.de/eval/JAR06/JAR_TERM/TRS/TRCSR/Ex7_BLR02_Z.trs.Thm17:POLO_FILTER:NO.html.lzma | 1,718,469,865,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861605.77/warc/CC-MAIN-20240615155712-20240615185712-00313.warc.gz | 94,816,197 | 2,469 | Term Rewriting System R:
[X, XS, N, X1, X2]
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
Termination of R to be shown.
` R`
` ↳Dependency Pair Analysis`
R contains the following Dependency Pairs:
2ND(cons(X, XS)) -> ACTIVATE(XS)
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(ntake(X1, X2)) -> TAKE(X1, X2)
Furthermore, R contains two SCCs.
` R`
` ↳DPs`
` →DP Problem 1`
` ↳Argument Filtering and Ordering`
` →DP Problem 2`
` ↳AFS`
Dependency Pairs:
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(ntake(X1, X2)) -> TAKE(X1, X2)
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
The following dependency pair can be strictly oriented:
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
POL(cons(x1, x2)) = x1 + x2 POL(n__take(x1, x2)) = x1 + x2 POL(TAKE(x1, x2)) = x1 + x2 POL(s(x1)) = 1 + x1 POL(ACTIVATE(x1)) = x1
resulting in one new DP problem.
Used Argument Filtering System:
TAKE(x1, x2) -> TAKE(x1, x2)
ACTIVATE(x1) -> ACTIVATE(x1)
s(x1) -> s(x1)
cons(x1, x2) -> cons(x1, x2)
ntake(x1, x2) -> ntake(x1, x2)
` R`
` ↳DPs`
` →DP Problem 1`
` ↳AFS`
` →DP Problem 3`
` ↳Dependency Graph`
` →DP Problem 2`
` ↳AFS`
Dependency Pair:
ACTIVATE(ntake(X1, X2)) -> TAKE(X1, X2)
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
Using the Dependency Graph resulted in no new DP problems.
` R`
` ↳DPs`
` →DP Problem 1`
` ↳AFS`
` →DP Problem 2`
` ↳Argument Filtering and Ordering`
Dependency Pair:
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
The following dependency pair can be strictly oriented:
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
The following usable rules using the Ce-refinement can be oriented:
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
Used ordering: Polynomial ordering with Polynomial interpretation:
POL(n__from) = 0 POL(from) = 0 POL(activate(x1)) = x1 POL(0) = 0 POL(SEL(x1, x2)) = 1 + x1 + x2 POL(n__take(x1, x2)) = x1 + x2 POL(take(x1, x2)) = x1 + x2 POL(nil) = 0 POL(s(x1)) = 1 + x1
resulting in one new DP problem.
Used Argument Filtering System:
SEL(x1, x2) -> SEL(x1, x2)
s(x1) -> s(x1)
cons(x1, x2) -> x2
activate(x1) -> activate(x1)
nfrom(x1) -> nfrom
from(x1) -> from
ntake(x1, x2) -> ntake(x1, x2)
take(x1, x2) -> take(x1, x2)
` R`
` ↳DPs`
` →DP Problem 1`
` ↳AFS`
` →DP Problem 2`
` ↳AFS`
` →DP Problem 4`
` ↳Dependency Graph`
Dependency Pair:
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, ntake(N, activate(XS)))
take(X1, X2) -> ntake(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfrom(X)) -> from(X)
activate(ntake(X1, X2)) -> take(X1, X2)
activate(X) -> X
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:01 minutes | 1,671 | 4,564 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2024-26 | latest | en | 0.520316 |
https://minuteshours.com/8-18-minutes-in-hours-and-minutes | 1,623,648,318,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487611445.13/warc/CC-MAIN-20210614043833-20210614073833-00102.warc.gz | 364,694,452 | 5,045 | # 8.18 minutes in hours and minutes
## Result
8.18 minutes equals 0 hours and 8.18 minutes
You can also convert 8.18 minutes to hours.
## Converter
Eight point one eight minutes is equal to zero hours and eight point one eight minutes. | 62 | 240 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2021-25 | latest | en | 0.875169 |
https://17centurymaths.com/contents/Euler'smaxmin.htm | 1,718,391,986,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861568.20/warc/CC-MAIN-20240614173313-20240614203313-00392.warc.gz | 61,175,195 | 8,050 | EULER'S
# Methodus Inveniendi Lineas Curvas Maximi Minimive Gaudentes………
## Introduction.
This is another project involving one of Euler's favorite topics : the finding of curves satisfying maximum or minimum properties under various circumstances ; in the first chapter, a general method of finding such curves is set out and numerous examples of its use presented. In the second and third chapters, the work examines increasingly complicated applications of what was later called the Calculus of Variations. At first determined functions are treated, in which the values of the derivatives of the curve are known at any point; later more abstract constructions are introduced in chapters 2 & 3 where Euler deals with functionals related by a general integral formula; for example, the theory developed can be used to find the shape of the curve a body can fall along in the minimum time subject to various forms of resistance, etc. Chapter 4, section 7 sets out the precepts for the 5 kinds of extrema considered ; this is a very useful summary.
It is of course a fundamental work in the establishing the Calculus of Variations by Euler and Lagrange a little later, being a sort of hybrid of ordinary integration and the solving of differential equations satisfying certain conditions ; the functions being integrated are usually not algebraic or transcending, but involve differential formulas that may be expanded out in terms of the usual x, y , p, q, r , etc. , and so special methods are needed to solve such problems. A good place to look initially for a modern appraisal of the work is the article by Craig G. Fraser, 'The Origins of Euler's Variational Calculus' in the Archives of the Exact Sciences. Vol. 47, No. 2 (June 1994), pp. 103-141. Pub. by Springer. However, Chapter 2 of Herman H. Goldstine's classic work ' A History of the Calculus of Variations, from the 17th through the 19th Century', pub. by Springer in 1980, must take pride of place, and should be consulted regularly with this translation, although of course in one chapter only a glimpse of the contents of Euler's book can be set out. Other places to look include of course Wikipedia ; the book by Routh on Rigid Body Dynamics includes a chapter on the Vis Viva which is useful for Addition II ; and of course Salmon' s classic work on columns is useful for Addition I.
Click here for the 1st Chapter : Concerning the Method of finding the maxima and minima of curved lines generally.
This is an introductory chapter in which most of what lies ahead in the following chapters is surveyed perhaps in a rather abstract view. Thus, Euler distinguishes between the 'ordinary' max. and min. of a given curve, and the present situation, where the curve itself is to be varied in a given domain to produce an extreme value. A great deal of careful argument is gone into, the relevance of which may only become apparent on reading further chapters. The use of an integral Zdx as the formula carrying the formula was revolutionary at the time, as was the use of x, y, p, q, etc. as variables; a most convenient notation, not involving derivatives directly greater than first order. It is a good idea to return to this introductory chapter and to refresh oneself on the ideas presented. See H. Goldstine p.67 onwards for a review of this chapter and the next.
Click here for Chapter 2a : A Method for finding the absolute maxima and minima for curved lines.
This is a long chapter containing much material, which I have split into two parts for computing convenience. Here Euler has been able to set out the conditions for the max. or min. of a curve contained within an integral in a straight forwards manner, and depending on y, p, q, etc.. The basic idea consists of extending some arbitrary y coordinate of the max. or min. curve by the infinitesimal amount nv, calculating the changes arising in dZ at neighbouring points and equating these changes to zero, so allowing Euler to set up a necessary condition for a max. or min. in terms of simple differential equations, depending on the form of dZ , and into an increasing number of terms. Numerous examples are given, the work is continued in part 2b. At present determinate formulas only are considered.
Click here for Chapter 2b : A Method for finding the absolute maxima and minima for curved lines.
Click here for Chapter 3 :On finding max./min. …. with indeterminate magnitudes present. This is a long chapter in which the mathematical procedures needed to resolve various physical problems are set out ; such problems involve bodies sliding down unknown curves according to some law of resistance, with the aim of minimizing the time of falling, etc. Thus a number of unknown functions are present in the integrand to be minimized. This leads Euler to an extension of his original scheme, so that the calculations become more involved, but still follow the same general lines. See H. Goldstine p.73 onwards for a review of this chapter.
Click here for Chapter 4 :Concerning the use of the method now treated in the resolution of various kinds of questions. This is another long chapter ; however, in section 7 Euler has summarized the 5 main methods he has set up in chapter 3 for finding optimal curves under various conditions ; the examples help make the work of chapter 3 much easier to understand; one wonders why he took so long to introduce his examples….. See H. Goldstine p.84 onwards for a review of this chapter.
Click here for Chapter 5 : A method of finding that curve among all the curves with a given property, which may be endowed with the property of being a max. or min.
This is another long chapter ; here Euler considers the introduction of an extra condition to be satisfied by the max. or min. curve, and satisfied by all the curves : i.e. an isoperimetric condition ; this involves the use of extending two neighbouring applied lines by independent incremental amounts in defining the common property, as well as the above single applied line, or y-coordinate. See H. Goldstine p.93 onwards for a review of this chapter. Many examples are given in a most thorough investigation.
Click here for Chapter 6 : A method for determining that curve among the curves endowed with several common properties, which may be provided with the max. or min. property.
In this final chapter, before moving on to appendices, Euler examines the problem of variation when some number of common conditions are present, and the max. min. value of a curve is to be found satisfying these conditions. The third proposition lands Euler in trouble when he tries to generalize and to produce the same result extended by his new line of reasoning, that he found before from the general method introduced. Nevertheless, the old method works, and is used to solve three more examples.
Click here for Appendix 1A : The curves associated with elastic laminas.
In this appendix, later added to the main work, Euler sets out to show that his method of finding maxima or minima curves associated with generalized functions in the form of integrals, can be applied to finding the shape of loaded laminas or ribbons, as had then recently been established in a straightforward method from mechanics by Daniel Bernoulli, following on the earlier work of his uncle, James Bernoulli. Most of the first part of this appendix, so subdivided for convenience, is given over to finding the nine classes or kinds of shapes adopted by a flexed lamina under different end conditions. An English translation exists already in Isis (1933) by Oldfather et al., of which I have just become aware, and have not referred to here.
Click here for Appendix 1B : The curves associated with elastic laminas cont'd.
This completes the deflections and vibrations of beams as presented by Euler at this time; he was to correct an error spotted by Daniel Bernoulli, which we will deal with here soon.
Click here for Appendix II : The motion of projectiles in a non-resisting medium, determined by the method of maxima and minima .
This completes Euler's work at this time; in this last section is the germ of the calculus of variations; Euler is able to derive the equations of motion of projectiles and orbits according to the vis viva principle ; this is very close to the correct theory, though he knows his effort is not complete, due to the state of physics at the time.
Click here for E296 : The elements of the calculus of variations.
Soon after Lagrange sent Euler a letter and his method for shortening Euler's calculations in this treatise, Euler produced this paper in which he sets out Lagrange's contributions in a clear manner.
Click here for E297 : The analytical explanation of the method of maxima and minima.
In this second paper, Euler revisits his formulas and rewrites his main results in term of variations; this is probably the best place to look if you do not have the desire or the time to read the above book. The most satisfactory thing of course is to do both.
Ian Bruce. Aug. 8th , 2013 latest revision. Copyright : I reserve the right to publish this translated work in book form. You are not given permission to sell all or part of this translation as an e-book. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. See note on the index page.Please feel free to contact me if you wish by clicking on my name here, especially if you have any relevant comments or concerns. | 2,017 | 9,538 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2024-26 | latest | en | 0.927876 |
http://zebra0.com/actionscript/class/fishcode.php | 1,508,738,730,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187825700.38/warc/CC-MAIN-20171023054654-20171023074654-00243.warc.gz | 591,972,758 | 2,378 | # OOPs: Creating a Bobber Class
I will just explain the code for the bobbing fish.
This is the code for the fish movie:
`1234567891011121314151617181920212223242526272829303132333435` ```//variables var angle:Number=0; var distance:Number=20; //how high up and down the y value goes var origY:Number=fish.y; //where the fish started var rate:Number=0.5; //how fast it bobs up and down, if 0 there is no bobbing var speed:Number=5; //x distance fish moves var turnAround:Boolean=false; //fish can leave and enter on other side or turnaround //initialize this.addEventListener(Event.ENTER_FRAME,frames); //functions; function frames(e:Event):void{ fish.y = origY + Math.sin(angle) * distance; angle = angle + rate; fish.x += speed; if (fish.x > stage.stageWidth + fish.width) {//went off on right if (turnAround) { fish.scaleX *= -1;//turn around or flip over horizontally speed=speed*-1; }//turnAround else { fish.x = 0 - fish.width;//reenter on left } } if (fish.x < 0 - fish.width) {//went off on left if (turnAround) { fish.scaleX *= -1;//turn around speed=speed*-1; } else { fish.x = stage.stageWidth + fish.width;//reenter on right } } }//frames```
CODE | 315 | 1,161 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2017-43 | longest | en | 0.523593 |
https://www.cnblogs.com/alexanderkun/p/10010871.html | 1,719,333,342,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198866143.18/warc/CC-MAIN-20240625135622-20240625165622-00784.warc.gz | 630,817,743 | 9,641 | ## 前言
I是图片信息矩阵也就是[224,224,3]，通过前面的cnn也就是所谓的sequence-sequence模型中的encoder，我用的是vgg19，得到a，这里的a其实是[14*14,512]=[196,512]，很形象吧，代表的是图片被分成了这么多个区域，后面就看我们单词注意在哪个区域了，大家可以先这么泛泛理解。通过了本文要讲的Attention之后得到z。这个z是一个区域概率，也就是当前的单词在哪个图像区域的概率最大。然后z组合单词的embedding去训练。
## attention的内部结构是什么？
def _get_initial_lstm(self, features):
with tf.variable_scope('initial_lstm'):
features_mean = tf.reduce_mean(features, 1)
w_h = tf.get_variable('w_h', [self.D, self.H], initializer=self.weight_initializer)
b_h = tf.get_variable('b_h', [self.H], initializer=self.const_initializer)
h = tf.nn.tanh(tf.matmul(features_mean, w_h) + b_h)
w_c = tf.get_variable('w_c', [self.D, self.H], initializer=self.weight_initializer)
b_c = tf.get_variable('b_c', [self.H], initializer=self.const_initializer)
c = tf.nn.tanh(tf.matmul(features_mean, w_c) + b_c)
return c, h
y向量代表的就是feature。
c和y在输入到tanh之前要做个全连接，代码如下。
w = tf.get_variable('w', [self.H, self.D], initializer=self.weight_initializer)
b = tf.get_variable('b', [self.D], initializer=self.const_initializer)
w_att = tf.get_variable('w_att', [self.D, 1], initializer=self.weight_initializer)
h_att = tf.nn.relu(features_proj + tf.expand_dims(tf.matmul(h, w), 1) + b) # (N, L, D)
def _project_features(self, features):
with tf.variable_scope('project_features'):
w = tf.get_variable('w', [self.D, self.D], initializer=self.weight_initializer)
features_flat = tf.reshape(features, [-1, self.D])
features_proj = tf.matmul(features_flat, w)
features_proj = tf.reshape(features_proj, [-1, self.L, self.D])
return features_proj
out_att = tf.reshape(tf.matmul(tf.reshape(h_att, [-1, self.D]), w_att), [-1, self.L]) # (N, L)
alpha = tf.nn.softmax(out_att)
context = tf.reduce_sum(features * tf.expand_dims(alpha, 2), 1, name='context') #(N, D)
def _attention_layer(self, features, features_proj, h, reuse=False):
with tf.variable_scope('attention_layer', reuse=reuse):
w = tf.get_variable('w', [self.H, self.D], initializer=self.weight_initializer)
b = tf.get_variable('b', [self.D], initializer=self.const_initializer)
w_att = tf.get_variable('w_att', [self.D, 1], initializer=self.weight_initializer)
h_att = tf.nn.relu(features_proj + tf.expand_dims(tf.matmul(h, w), 1) + b) # (N, L, D)
out_att = tf.reshape(tf.matmul(tf.reshape(h_att, [-1, self.D]), w_att), [-1, self.L]) # (N, L)
alpha = tf.nn.softmax(out_att)
context = tf.reduce_sum(features * tf.expand_dims(alpha, 2), 1, name='context') #(N, D)
return context, alpha
ok，回到我们的image_caption中，看下图
https://segmentfault.com/a/1190000011744246
posted on 2018-11-24 08:56 alexanderkun 阅读(4456) 评论(0编辑 收藏 举报 | 836 | 2,611 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2024-26 | latest | en | 0.239299 |
http://www.financialwisdomforum.org/gummy-stuff/sampling-Frontier.htm | 1,508,541,670,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187824471.6/warc/CC-MAIN-20171020230225-20171021010225-00855.warc.gz | 454,921,895 | 4,907 | reSampled Frontier ... and portfolio allocation
motivated by e-mail from Levi F.
an Old Frontier
Once upon a time (1952) Markowitz introduced Modern Portfolio Theory and the so-called Efficient Frontier.
(He shared a Nobel prize for this work.)
>Efficient frontier? Huh?
That Efficient Frontier stuff goes like this:
Our portfolio contains N assets, with Mean Returns r1, r2, ...
We allocate fractions of our portolio to these N assets and we want to select these fractions, namely x1, x2, ... so that:
1. x1 + x2 + ... + xN = 1
The fractions add to "1" since 100% of our portfolio is devoted to these N components
2. The Volatility (or Standard Deviation) of your portfolio, SD is prescribed
The Volatility is sometimes called "Risk" (for no logical reason) ... so you're picking your Risk level.
This Volatility (or Standard Deviation) will depend upon the returns of the N assets as well as your allocations x1, x2 etc.
3. Determine the allocations so that your Average (or "expected") Return, namely R = x1r1 + x2r2 + ... + xNrN, is a maximum.
>And how do you do all that?
You pick a set of fractions devoted to the N assets ...
>You pick x1, x2 etc.?
Yes, then you see what R-value that gives.
Then you pick another set of x's ...
>Which add up to 1, right?
Yes. That's #1, above. For example, you might have x1=0.60 and x2=0.25 and x3=0.15 meaning (for three assets) 60% + 25% + 15%.
Anyway, you see what R you get. Then you pick another ...
>Again and again until you get the largest R? That's a lot of work, eh?
A computer can do all the work, but I don't want to talk about the Efficient Frontier. Instead, I want to ... >Wait! What's that frontier look like? The typical picture assocated with the Frontier is shown here along with a selected Standard Deviation and the maximum R-value. You can see the relationship to the Sharpe Ratio. >You have something against Sharpe? Who me? Actually, the big problem is: We look at the historical data for our assets, extract the Means and Variances and Covariances (and any other stuff we may need), determine our allocations and assume that all this will conveniently stay fixed for the future.
>Then we can relax, eh?
Yes ... and that ain't good!
Note that you could also prescribe the Return you'd like (that's R) and vary the allocations until you get the minimum SD. (That'd be the minimum volatility and, if you believe that volatility = "Risk", then it's the minimum risk portfolio which provides that prescribed return.) Or, you could just find the point which gives the minimum ... uh, "Risk". See the chart (using a sample 3-asset portfolio) (Not surprisingly, it's got lots of bonds.) One problem with this stuff (besides the fact that we rely heavily upon historical data) is that the Frontier moves about when we make small changes to the data. >And the "optimal" allocations? Yes, they move about too.
>So what do you intend to do?
Talk about a sampled Frontier ... or, better, a reSampled Frontier
>Beg pardon?
It's an idea promoted by Michaud (1989) and Jorian (1992).
a New Frontier
Suppose we have a bunch of historical returns for our assets. Then we can calculate things like their Mean, Variance (which is Volatility2) and Covariance. That'd give us a set of fixed numbers we can use to construct our Efficient Frontier. But these numbers are actually estimates of future Means and Variances ... and future is not that fixed, eh?
Here's a way to introduce some randomness into future values:
Take random samples of the set of historical returns. For example, if we have a set of returns, say 10.2%, 8.5%, 6.7% and 9.4% we could select a random sample of these four returns as 8.5%, 8.5%, 9.4% and 10.2% where, after making our selection, we put that return back into the set ... else the selection of "random" returns would be the same as the original set! Use this random sample to calculate an Efficient Frontier. Repeat steps 1 and 2 a jillion times ... each time generating a Frontier. Stare in awe (and trepidation) at the variation in those jillion Frontiers >And the variation in associated allocations, eh? Exactly.
Taking a set of historical returns and sampling from that set (again and again) is a neat way to get randomness from a fixed set of returns.
You never have to worry about assuming some kind of distribution (normal? lognormal?) or "fat tails".
I used it myself, in sensible withdrawals
To see the effect of changing the returns (for four assets with prescribed Means and Volatilities),
you can try this neat spreadsheet by Michael Kishinevsky.
2. Select four (monthly) Means and Standard Deviations and an (annualized) Risk-free Rate.
3. Press F9 to get a bunch of random returns for each asset (selected from four normal distributions with the prescribed Means and SDs).
4. Look affectionately at the Efficient Frontier.
5. Repeat step 3 (and 4).
We noted above that one can prescribe the Return and get the minimum Volatility (for that Return) ... hence an Efficient Frontier as in Figure 1A. However, we can also plot the Volatility against the return, like Figure 1B (which is easier to interpret ) Let's consider a 3-asset Portfolio: The Portfolio has a certain allocation of resources. (Example: 50% + 30% + 20% ) That gives a particular Mean Return and Volatility. We modify the allocations and reduce the Volatility to a minimum while maintaining the Return. (Example: 35% + 37% +28% gives Min. Volatility) (It's got the same return with a smaller volatility.) Clearly we prefer the blue portfolio to our original red portfolio. We repeat this ritual for a bunch of Returns (each time minimizing the Volatility). This gives the Efficient Frontier curve. >Did you say that one can prescribe the Return? Can I ask for 100%? Can I ask for ...? Within reason ... say between the minimum and maximum asset returns. Figure 1A Figure 1B
For just three assets you can try this spreadsheet:
>How do I get a feel for what the appropriate allocations might be?
Or how that allocation might change as the returns are reSampled?
Or if I start with different assets and different returns ... or?
Yes, well ... you might want to play the FRONTIER GAME
>A game? How useful is that?
On a scale from 1 to 10, it's a 3, alongside Solitaire
... but it does illustrate the effect of reSampling the same set of returns and how the "optimal" allocation can change quite dramatically.
Some References: | 1,542 | 6,408 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.265625 | 3 | CC-MAIN-2017-43 | latest | en | 0.920007 |
https://math.stackexchange.com/questions/1156751/do-there-exist-any-cycles-for-these-number-sequences | 1,556,014,263,000,000,000 | text/html | crawl-data/CC-MAIN-2019-18/segments/1555578596571.63/warc/CC-MAIN-20190423094921-20190423120921-00528.warc.gz | 493,827,334 | 33,410 | Do there exist any cycles for these number sequences?
We define, for $k\in\mathbb{N}$, the sequence $\left(S_{k,n}\right)_{n\in\mathbb{N}}$: $$S_{k,1}=k,\;\;\; S_{k,n+1}=p_1q_1\cdots p_mq_m \text{ (written out in decimal)}$$ Where $p_1^{q_1}*\cdots *p_m^{q_m}$, with $p_1<\cdots<p_m$ is the ordered prime factorisation of $S_{n}$.
For example: $$S_{28,1}=28$$ Now since $28=2^2*7^1$: $$S_{28,2}=2271$$ Since $2271=3^1*757^1$: $$S_{28,3}=317571$$ Etcetera.
We see that, at least in this particular case, the sequence grows rapidly. This turns out to be a general phenomenon. However, sometimes the sequence can decrease: $$S_{512,1}=512=2^9\;\;\;\;\; S_{512,2}=29$$ Motivated by this, I am wondering wether or not there exists a $k\in\mathbb{N}$ such that the sequence $\left(S_{k,n}\right)_{n\in\mathbb{N}}$ has cycles (i.e. is periodic).
By doing computer research, I have found that there exist no cycles which have highest element $\leq 10^7$. My conjecture is that no cycles exist for any $k$, however I am not sure where to start a proof.
For further investigation we might also consider the problem in other number systems (with a different base). Is there any base for which there are cycles?
EDIT: The answer to the last question is yes. Thanks to Joffan we have found bases in which there even exist fixed points.
Some of my further attempt at fixed points:
Note that in order to find fixed points in base $x$, we will need to find an $n\in\mathbb{N}$ such that there exist prime numbers $p_1<\cdots<p_n$ and positive integers $q_1,\cdots,q_n$ satisfying:
$$p_1^{q_1}*\cdots*p_n^{q_n}=p_1x^{2n-1}+q_1x^{2n-2}+\cdots+p_nx+q_n$$ We are particularly interested in the case $x=10$.
Maybe it is easier to first solve the equation for $x$. For example, for $n=1$ the equation is: $$p^q=px+q$$ Which gives us $x=p^{q-1}-\frac{q}{p}$. So we would like to find a prime number $p$ and a positive integer $q$ such that $p^{q-1}-\frac{q}{p}=10$ (probably no luck here).
For $n=2$ we get: $$p_1^{q_1}p_2^{q_2}=p_1x^3+q_1x^2+p_2x+q_2$$ Which gives us some nastier solutions for $x$, but will maybe give us more chance to end up with the magical $10$ since we have more $p_i$s and $q_i$s to choose from.
Maybe someone can even find a proof that there exist no fixed points for base $10$, which of course will also be a big step. Please do not hesitate to post any suggestions that come to mind.
• The base 10 representation makes your iterated representation somewhat "meaningless" in terms of the preceding prime factorization. The fact that you are appending the powers after the primes makes it even more somewhat "meaningless". I think this is probably a very, very hard question to attack with theoretical math, and so probably the only hope is that if there is a cycle that doesn't start with too large of a number, or have too long a cycle length, then a computer program will find it. – user2566092 Feb 19 '15 at 22:23
• Thank you for your feedback. I agree that this sequence might not be that "meaningful" from a theoretical point of view, especially because of the dependence of the number system used. However, I don't feel like that should stop us. If you don't like base $10$, you can always try to work in another base. Maybe someone with better programming skills than me will find a way to easily check for sequences in other number systems, and it would be pretty cool if we actually found one! I don't care about the meaning, I'm just interested in some fun maths. Any ideas will help! Many thanks in advance. – Uncountable Feb 19 '15 at 22:39
• Are you familiar with the $3n + 1$ conjecture? That deals with stuff that is seemingly even more mathematically "meaningful" than your sequences but Paul Erdos famously said that he believed the problem was perhaps beyond current mathematics. – user2566092 Feb 19 '15 at 22:44
• Fairly trivially, perhaps, we can make $k^k = kk$ in base $n=k^{k-1}-1$ - for example, $3^3 = 33$ base $8$. – Joffan Feb 20 '15 at 0:26
• \$Joffan Very nice! I hadn't thought of that yet. Well done, sir. – Uncountable Feb 20 '15 at 0:29 | 1,216 | 4,087 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.421875 | 3 | CC-MAIN-2019-18 | latest | en | 0.850157 |
https://oeis.org/A141088 | 1,597,509,760,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439740929.65/warc/CC-MAIN-20200815154632-20200815184632-00394.warc.gz | 430,770,617 | 3,770 | The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
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A141088 a(n) = prime(2^n) - 2*n. 0
1, 3, 13, 45, 121, 299, 705, 1603, 3653, 8141, 17841, 38849, 83991, 180475, 386063, 821609, 1742503, 3681095, 7754039, 16290007, 34135987, 71378525, 148948093, 310248193, 645155147, 1339484145, 2777105075, 5750078991, 11891268343, 24563311249, 50685770105 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,2 LINKS FORMULA a(n) = A033844(n) - A005843(n). EXAMPLE a(1) = prime(2^1) - 2*1 = prime(2) - 2 = 3 - 2 = 1; a(4) = prime(2^4) - 2*4 = prime(16) - 8 = 53 - 8 = 45. MAPLE seq(ithprime(2^n)-2*n, n=1..24); # Emeric Deutsch, Aug 27 2008 MATHEMATICA Table[Prime[2^n]-2n, {n, 30}] (* Harvey P. Dale, Oct 11 2014 *) PROG (PARI) a(n) = prime(2^n) - 2*n; \\ Michel Marcus, Mar 25 2019 CROSSREFS Cf. A033844, A005843. Sequence in context: A136520 A212416 A058934 * A187915 A115128 A140420 Adjacent sequences: A141085 A141086 A141087 * A141089 A141090 A141091 KEYWORD nonn AUTHOR Juri-Stepan Gerasimov Jul 31 2008 EXTENSIONS Corrected and extended by Emeric Deutsch, Aug 27 2008 More terms from Harvey P. Dale, Oct 11 2014 STATUS approved
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Last modified August 15 12:33 EDT 2020. Contains 336499 sequences. (Running on oeis4.) | 601 | 1,606 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2020-34 | latest | en | 0.543892 |
http://www.perlmonks.org/?node_id=774042 | 1,508,703,462,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187825436.78/warc/CC-MAIN-20171022184824-20171022204824-00390.warc.gz | 544,413,650 | 6,338 | No such thing as a small change PerlMonks
### Re^2: why the array index has to start at 0??
by Zen (Deacon)
on Jun 23, 2009 at 14:47 UTC ( #774042=note: print w/replies, xml ) Need Help??
This argument by Dijkstra is silly. Yes, I said it. Doesn't matter who says it; if the argument is aesthetic, that's not a reason. Consider:
N1..Nn (N sub 1 to N sub n)
My notation is "nicer," therefore it's better? No! The offset argument is better for the 0 discussion, but in the end it comes down to the generally accepted culture of using 0. If you work in a vacuum, by all means set it to be whatever your favorite number is.
• Comment on Re^2: why the array index has to start at 0??
Replies are listed 'Best First'.
Re^3: why the array index has to start at 0??
by roboticus (Chancellor) on Dec 21, 2012 at 16:32 UTC
Zen:
I don't think it's silly, rather it just moves the question a bit. As you indicate, the question is merely moved to "which is the better way of expressing a range". It seems that Djikstra prefers "0 <= i < N" to "1<= i < N+1". However, he doesn't say why that's any better. I agree that it looks better, but there's another formulation "0 < i <= N" that looks just as good. Why isn't that just as good? He leaves the question open.
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Notices? | 522 | 1,833 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2017-43 | longest | en | 0.953973 |
https://getrevising.co.uk/revision-cards/c3-differentiation-2 | 1,526,888,058,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794863967.46/warc/CC-MAIN-20180521063331-20180521083331-00097.warc.gz | 585,712,724 | 13,284 | # C3 Differentiation
HideShow resource information
## CHAIN RULE
The Chain Rule
Example:
1- Bring the power to the front:
2- Take 1 away from remaining power:
3- Differentiate between the brackets:
4- Multiply this answer the the number in front of the brackets:
1 of 4
## Special Case
Special Case
This is when it is X= instead of Y.
Example:
1 - Differentiate:
2 - Put 1 over:
3 - Simplify further if neccersary:
2 of 4
## Product Rule
Product Rule
Example:
1 - Choose your U and V values:
2 - Differentiate these both:
3 - Substitute these values into the formula:
4 - Pull out the common factors:
5 - Simplify further if neccersary:
3 of 4
## Quotient Rule
Quotient Rule
Example:
1 - Write down your u and v values:
2 - Differentiate these values:
3 - Substitute into the following formula:
4 - Pull out common factors:
5 - Simplify further if neccersary: | 237 | 896 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.484375 | 3 | CC-MAIN-2018-22 | latest | en | 0.743229 |
https://math.stackexchange.com/questions/2272672/conjugate-gradient-with-inaccurate-gradient-information | 1,561,075,791,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627999291.1/warc/CC-MAIN-20190620230326-20190621012326-00226.warc.gz | 522,757,478 | 34,907 | I am trying to minimize an unconstrained nonlinear optimization problem using conjugate gradient. I don't have exact gradient ($\nabla f(x)$) information, but I do have an approximate gradient ($\nabla f_a(x)$) such that $\| \nabla f(x) - \nabla f_a(x)\| \leq \delta$ for some constant $\delta >0$ for all $x \in \mathbb{R}^n$. Given the fact that $-\nabla f_a$ is a descent direction, will the conjugate gradient with this approx. gradient converge to some local minima ? If not true in general, can we say something for strictly convex quadratic functions ?
• No such luck; even for strictly convex functions, all you can say is that the algorithm will eventually find itself in a region where $\|\nabla f(x)\|\le\delta$. It may then converge to a point that may or may not be a local minimum, or it may wander endlessly in that region without converging. – Rahul May 9 '17 at 6:51
• Can you please elaborate your answer a bit more. What I follow from your reply is that $\nabla f_a$ is driven to 0 using the simple conjugate gradient. Does every algorithm which uses descent direction at each iteration always converge to local min. ? If this fact is true, then steepest descent should work with approx gradient. If we somehow make sure that conjugate gradient creates descent direction at each iteration, can we comment on convergence. If that is true, then we can ask about how to make sure conjugate directions are descent. – user402940 May 9 '17 at 15:13 | 364 | 1,461 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2019-26 | latest | en | 0.840083 |
https://answers.sap.com/questions/9867808/need-to-have-correct-unit-price-in-commercial-invo.html | 1,600,686,315,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400201601.26/warc/CC-MAIN-20200921081428-20200921111428-00076.warc.gz | 280,282,277 | 24,213 | # Need to have correct unit price in Commercial Invoice
Hi Friends,
Generally we have a Parent item and two child item in Feeder system ,For example if Parent item have Net Value \$100 child item have value 0 ,once its transferred to GTS the value of Parent item becomes 0 and child item value is proportionately divided between two child item.and the Commercial invoice ( output ) shows total quantity and the value of Parent item .Unit price =Total Value of Parent/ delivered quantity.
Now in some instance the child item is delivered partially.The quantity of Parent is 20 unit child item is delivered as 4,12,4 qty ,both are delivered separately..
Issue is appearing that since there is separate Customs Declaration document for Parent item=20 Qty and Child item 4 qty,12 qty and 3 qty .
In Commercial Invoice of Child item where there is 4 qty the Net value is appearing as \$100 the unit price is \$100/4 = \$25 ,whereas for 12 qty the net value appearing is \$100 unit price is \$100/12 = \$8.33 , as per this logic there is discrepancy in unit price of same material but with different quantity.
We want to restructure the unit price such that it should be \$100 /20 qty=\$5
In commercial Invoice 1 where 4 quantity is delivered value should be 4qtyX\$5=\$20
In commercial Invoice 2 where 12 quantity is delivered value should be 12qtyX\$5=\$60
commercial Invoice 3 where 3 quantity is delivered value should be 4qtyX\$5=\$20
Can the experts have there comments how to restructure this.
Thanks & Regards
Amit
10|10000 characters needed characters exceeded
### Related questions
• Posted on Mar 04, 2013 at 03:44 PM
Hi Amit,
I find it impossible to follow your description of the problem. For example, you don't say from which system the Commercial Invoice is created, and what the "Child Items" are - are they batch-splits, or something else? And what Billing Relevance value is being used in the Item Category? Is it 'A' or 'M', or something different?
Perhaps if you could simply refer to the quantities and values that you see in the ERP feeder system and in the GTS Export Declaration, it would be easier.
Let's suppose that your ordered quantity of 20 has so far been partially delivered as quantity 4, and then (later) quantity 12. What quantities and values do you see in the Billing Document and Declaration?
Regards,
Dave | 542 | 2,358 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2020-40 | latest | en | 0.907716 |
http://mathwire.blogspot.com/2011/12/last-snowman-standing-game.html | 1,369,297,764,000,000,000 | text/html | crawl-data/CC-MAIN-2013-20/segments/1368703035278/warc/CC-MAIN-20130516111715-00066-ip-10-60-113-184.ec2.internal.warc.gz | 176,614,680 | 19,695 | ## Thursday, December 15, 2011
### Last Snowman Standing Game
The snowmen face off in this game of addition facts. But beware! A toss of the die may mean the sun melts a snowman. Students practice addition facts as they try to be the last snowman standing because in this game the first person to remove all of his/her snowmen loses the game!
Download the Last Snowman Standing Game for directions and game mats for three different versions of the game:
• Sum of Two Dice Version to practice addition facts
• Difference of Two Dice Version to practice subtraction facts
• One Die Toss for a simplified version to analyze the probability of a die toss
Data Collection: The directions for each version also include directions for data collection and analysis of the outcomes of the games. Be sure to incorporate these activities, if at all possible, as games offer a highly motivational study in probability. Students love to "play games" to collect data. They're also eager to analyze games so that they learn how the game works and what strategies they can use to improve their odds of winning.
Differentiation: This game offers many opportunities to differentiate the activity. First of all, teachers are able to select from three different versions. Secondly, each teacher should differentiate the game analysis to meet the instructional level of his/her students. Most students can handle the questions with teacher guidance. Older students and talented primary students may be challenged to analyze the game and answer the questions in small groups. | 307 | 1,558 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2013-20 | longest | en | 0.928354 |
https://number.academy/52866 | 1,721,433,364,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514972.64/warc/CC-MAIN-20240719231533-20240720021533-00211.warc.gz | 376,679,605 | 11,595 | # Number 52866 facts
The even number 52,866 is spelled 🔊, and written in words: fifty-two thousand, eight hundred and sixty-six. The ordinal number 52866th is said 🔊 and written as: fifty-two thousand, eight hundred and sixty-sixth. The meaning of the number 52866 in Maths: Is it Prime? Factorization and prime factors tree. The square root and cube root of 52866. What is 52866 in computer science, numerology, codes and images, writing and naming in other languages. Other interesting facts related to 52866.
## Interesting facts about the number 52866
### Asteroids
• (52866) 1998 ST23 is asteroid number 52866. It was discovered by LONEOS from Anderson Mesa on 9/17/1998.
## What is 52,866 in other units
The decimal (Arabic) number 52866 converted to a Roman number is (L)MMDCCCLXVI. Roman and decimal number conversions.
#### Length conversion
52866 kilometers (km) equals to 32850 miles (mi).
52866 miles (mi) equals to 85080 kilometers (km).
52866 meters (m) equals to 173443 feet (ft).
52866 feet (ft) equals 16114 meters (m).
#### Time conversion
(hours, minutes, seconds, days, weeks)
52866 seconds equals to 14 hours, 41 minutes, 6 seconds
52866 minutes equals to 1 month, 1 week, 1 day, 17 hours, 6 minutes
### Codes and images of the number 52866
Number 52866 morse code: ..... ..--- ---.. -.... -....
Sign language for number 52866:
Number 52866 in braille:
Images of the number
Image (1) of the numberImage (2) of the number
More images, other sizes, codes and colors ...
## Share in social networks
### Advanced math operations
#### Is Prime?
The number 52866 is not a prime number. The closest prime numbers are 52861, 52879.
#### Factorization and factors (dividers)
The prime factors of 52866 are 2 * 3 * 3 * 3 * 11 * 89
The factors of 52866 are
1, 2, 3, 6, 9, 11, 18, 22, 27, 33, 54, 66, 89, 99, 178, 198, 267, 297, 534, 594, 801, 979, 1602, 1958, 2403, 2937, 4806, 5874, 8811, 52866. show more factors ...
Total factors 32.
Sum of factors 129600 (76734).
#### Powers
The second power of 528662 is 2.794.813.956.
The third power of 528663 is 147.750.634.597.896.
#### Roots
The square root √52866 is 229,926075.
The cube root of 352866 is 37,531174.
#### Logarithms
The natural logarithm of No. ln 52866 = loge 52866 = 10,875516.
The logarithm to base 10 of No. log10 52866 = 4,723176.
The Napierian logarithm of No. log1/e 52866 = -10,875516.
### Trigonometric functions
The cosine of 52866 is 0,751031.
The sine of 52866 is -0,660267.
The tangent of 52866 is -0,879148.
## Number 52866 in Computer Science
Code typeCode value
52866 Number of bytes51.6KB
Unix timeUnix time 52866 is equal to Thursday Jan. 1, 1970, 2:41:06 p.m. GMT
IPv4, IPv6Number 52866 internet address in dotted format v4 0.0.206.130, v6 ::ce82
52866 Decimal = 1100111010000010 Binary
52866 Decimal = 2200112000 Ternary
52866 Decimal = 147202 Octal
52866 Decimal = CE82 Hexadecimal (0xce82 hex)
52866 BASE64NTI4NjY=
52866 MD59bc445a5089c2b9babbb9629f0af909e
52866 SHA174dd478fdcb687a7d91a096c3c802f0a1e7b0ffd
52866 SHA224097968b75c999e27d83de2b68cd5ed59a148bda43fa73bf597011166
52866 SHA25611c15f0e44a68d4a321bbd78a7df43bb89496fd2633524e87a3cbab13b28cb3e
52866 SHA3848d8baa9f52b06c262c7e95eafb3b3523db52bfb158fa7fb83b0b4b6d189ed46f1e81c0a7912fb24c924e64c2f8da4f45
More SHA codes related to the number 52866 ...
If you know something interesting about the 52866 number that you did not find on this page, do not hesitate to write us here.
## Numerology 52866
### Character frequency in the number 52866
Character (importance) frequency for numerology.
Character: Frequency: 5 1 2 1 8 1 6 2
### Classical numerology
According to classical numerology, to know what each number means, you have to reduce it to a single figure, with the number 52866, the numbers 5+2+8+6+6 = 2+7 = 9 are added and the meaning of the number 9 is sought.
## № 52,866 in other languages
How to say or write the number fifty-two thousand, eight hundred and sixty-six in Spanish, German, French and other languages. The character used as the thousands separator.
Spanish: 🔊 (número 52.866) cincuenta y dos mil ochocientos sesenta y seis German: 🔊 (Nummer 52.866) zweiundfünfzigtausendachthundertsechsundsechzig French: 🔊 (nombre 52 866) cinquante-deux mille huit cent soixante-six Portuguese: 🔊 (número 52 866) cinquenta e dois mil, oitocentos e sessenta e seis Hindi: 🔊 (संख्या 52 866) बावन हज़ार, आठ सौ, छयासठ Chinese: 🔊 (数 52 866) 五万二千八百六十六 Arabian: 🔊 (عدد 52,866) اثنان و خمسون ألفاً و ثمانمائة و ستة و ستون Czech: 🔊 (číslo 52 866) padesát dva tisíce osmset šedesát šest Korean: 🔊 (번호 52,866) 오만 이천팔백육십육 Danish: 🔊 (nummer 52 866) tooghalvtredstusinde og ottehundrede og seksogtreds Hebrew: (מספר 52,866) חמישים ושניים אלף שמונה מאות שישים ושש Dutch: 🔊 (nummer 52 866) tweeënvijftigduizendachthonderdzesenzestig Japanese: 🔊 (数 52,866) 五万二千八百六十六 Indonesian: 🔊 (jumlah 52.866) lima puluh dua ribu delapan ratus enam puluh enam Italian: 🔊 (numero 52 866) cinquantaduemilaottocentosessantasei Norwegian: 🔊 (nummer 52 866) femtito tusen åtte hundre og sekstiseks Polish: 🔊 (liczba 52 866) pięćdziesiąt dwa tysiące osiemset sześćdziesiąt sześć Russian: 🔊 (номер 52 866) пятьдесят две тысячи восемьсот шестьдесят шесть Turkish: 🔊 (numara 52,866) elliikibinsekizyüzaltmışaltı Thai: 🔊 (จำนวน 52 866) ห้าหมื่นสองพันแปดร้อยหกสิบหก Ukrainian: 🔊 (номер 52 866) п'ятдесят дві тисячі вісімсот шістдесят шість Vietnamese: 🔊 (con số 52.866) năm mươi hai nghìn tám trăm sáu mươi sáu Other languages ...
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The content of the comments is the opinion of the users and not of number.academy. It is not allowed to pour comments contrary to the laws, insulting, illegal or harmful to third parties. Number.academy reserves the right to remove or not publish any inappropriate comment. It also reserves the right to publish a comment on another topic. Privacy Policy. | 2,101 | 6,148 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2024-30 | latest | en | 0.810085 |
https://metacpan.org/dist/Math-Random-Normal-Leva/source/lib/Math/Random/Normal/Leva.pm | 1,642,823,031,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320303729.69/warc/CC-MAIN-20220122012907-20220122042907-00601.warc.gz | 418,013,825 | 5,576 | ``````package Math::Random::Normal::Leva;
use strict;
use warnings;
our \$VERSION = "0.04";
use Exporter qw(import export_to_level);
our @EXPORT_OK = qw(gbm_sample random_normal);
use Math::Random::Secure qw(rand);
use Carp qw(confess);
Math::Random::Normal::Leva - generate normally distributed PRN using Leva method
This document describes Math::Random::Normal::Leva version 0.02
use Math::Random::Normal::Leva;
my @normal = map { random_normal() } 1..1000;
Generates normally distributed pseudorandom numbers using algorithm described
in the paper "A Fast Normal Random Number Generator", Joseph L. Leva, 1992
(L<http://saluc.engr.uconn.edu/refs/crypto/rng/leva92afast.pdf>)
=cut
Returns a random number sampled from the normal distribution.
=over 4
=item I<\$rand>
is the value of the stock initially
=cut
# This algorithm comes from the paper
# "A Fast Normal Random Number Generator" (Leva, 1992)
sub random_normal {
my \$rand = shift || \&rand;
my (\$s, \$t) = (0.449871, -0.386595); # Center point
my (\$a, \$b) = (0.19600, 0.25472);
my \$nv;
while (not defined \$nv) {
my (\$u, \$v) = (\$rand->(), 1.7156 * (\$rand->() - 0.5));
my \$x = \$u - \$s;
my \$y = abs(\$v) - \$t;
my \$Q = \$x**2 + \$y * (\$a * \$y - \$b * \$x);
if (\$Q >= 0.27597) {
next if (\$Q > 0.27846 || \$v**2 > -4 * \$u**2 * log(\$u));
}
\$nv = \$v / \$u;
}
return \$nv;
}
=back
=head2 gbm_sample(\$price, \$vol, \$t, \$r, \$q, \$rand)
Generates a random sample price of a stock following Geometric Brownian Motion after t years.
=over 4
=item I<\$price>
is the value of the stock initially
=item I<\$vol>
is the annual volatility of the stock
=item I<\$t>
is the time elapsed in years
=item I<\$r>
is the annualized drift rate
=item I<\$q>
is the annualized dividend rate
=item I<\$rand>
custom rand generated if not passed will use Math::Random::Secure::rand
=back
note: all rates are taken as decimals (.06 for 6%)
=cut
sub gbm_sample {
my (\$price, \$vol, \$time, \$r, \$q, \$rand) = @_;
confess('All parameters are required to be set: generate_gbm(\$price, \$annualized_vol, \$time_in_years, \$r_rate, \$q_rate)')
if grep { not defined \$_ } (\$price, \$vol, \$time, \$r, \$q);
return \$price * exp((\$r - \$q - \$vol * \$vol / 2) * \$time + \$vol * sqrt(\$time) * random_normal(\$rand));
}
1;
__END__
Please report any bugs or feature requests via GitHub bug tracker at
L<http://github.com/binary-com/perl-Math-Random-Normal-Leva/issues>.
Binary.com C<< <binary at cpan.org> >> | 783 | 2,508 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.0625 | 3 | CC-MAIN-2022-05 | latest | en | 0.600863 |
https://instrumentauto.com/1993/Feb_38424.html | 1,632,273,918,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057303.94/warc/CC-MAIN-20210922011746-20210922041746-00011.warc.gz | 377,893,421 | 6,658 | • Home
• How To Calculate Grinding Media Weight
# How To Calculate Grinding Media Weight
• ### How To Calculate Raw Mill Grinding Media
how to calculate raw mill grinding media. Sand mill can be used glass ball, zirconia balls, steel ball for grinding media, each medium is best to use a uniform particle size into the cylinder before the media should be cleaned in advance, screeningGrinding with a certain medium to be based on the fineness of the material to be polished and viscosity to choose
• ### How To Calculate Grinding Rate Of Ball Mill
how to calculate grinding media in a ball mill in cement industry. Re how to estimate the wear rate for Ball mill - International .The ball charge mill consists of grinding media in various sizes to ensure .. Max. power is calculated at Ch1 30 of total power consumption.
• ### 200 Ton Capacity Grinding Mill Media Charge Calculation
The modeling of dry grinding of quartz in tumbling media The grinding of quartz sand to produce high purity silica flour was studied using ceramic balls, ceramic cylinders or flint pebbles in a laboratory mill and three full-scale closed circuit mills of 2.2, 2.3 and 2.8 m internal diameter.
• ### Grinding To Nanosizes Effect Of Media Size And Slurry
Apr 01, 2015 In order to calculate the maximum volume of grinding zones in the mill, we assume that each media particle has twelve nearest neighbours upper limit for hexagonal close-packing HCP a lower number will apply in practice and calculate the maximum total volume of grinding zones in the mill as a fraction of the void volume between the media.
• ### How To Calculate Grp For Digital Campaigns By Mikhail
Dec 20, 2017 A Few Thoughts on Media Planning. In my opinion, no one ad or campaign is going to make or break a company usually. A healthy mix of distributing quality content via organic and paid media
• ### Calculation For Grinding Media In Ball Mill
how to calculate the media in ball mill - Farmine Machinery. grinding media charge calculation for bead broyeur 192. grinding media charge calculation for bead broyeur 192 calculate and select ball mill ball size for optimum grinding. in grinding, selecting calculate the correct or optimum ball size that allows for the best and optimumideal or target grind size to be achieved by your ball mill ...
• ### Calculation Of Grinding Media Of Cement Mill
5 Calculation of the Top Size Grinding Media 6 Ball Charges analysis calculator Weight and Surface of the Grinding Charges 7 Ball Charge Makeup Calculator 8 Ball Mill Simulation 9 Grinding Media Wear Rate Monochamber Mill 10 Grinding Media Wear Rate 2 chambers Mill 11 Marked Ball Test Calculator Section 2 Volume Load Power Kit 12
• ### Formula For Putting Grinding Media In Mill Crusher Mills
The effect of grinding media performance on milling and . formula properties were achieved solid content 50 weight- and specific slurry density 1.5 kgl. mass of the grinding media in the mill generates much more.
• ### Calculation Rotating Grinding Stone
cement ball mill grinding media calculation. Re How Can I calculate new ball size and weight desing for ball . Our ball mill has got two compartments. Tonnage of grinding media u can calculate
• ### Amit 135 Lesson 6 Grinding Circuit Mining Mill Operator
Grinding takes place in more open space which makes the retention time longer and adjustable compared to crushers. Theoretical size reduction and power ranges for different grinding mills image 135-6-1 AGSAG Mills Autogenous Grinding AG Mill. Wet or dry Primary, coarse grinding up to 400 mm feed size Grinding media is grinding feed
• ### Grinding Wheel Speed Calculator Norton Abrasives
Grinding Wheel Speed Calculator All Norton grinding wheels are marked with a maximum operating speed in RPM. Most machines, and especially CNC machines, use Surface Feet Per Minute SFPM as an input, which requires operators to do the conversion.
• ### Metal Weight Calculator Priyank Steels
Priyank Steels introduces metal weight calculator to calculate weights of metals, metal weights in lbs, metal weights in kgs including weight of steel, stainless steel, stainless steel 200 series, stainless steel 300 series, nickel, aluminum, brass, copper, zinc, titanium,
• ### Identify And Calculate The Mean Median And Mode
Mar 24, 2020 Remember, the mean is calculated by adding the scores together and then dividing by the number of scores you added. In this case, the mean would be 2 4 add the two middle numbers, which equals 6. Then, you take 6 and divide it by 2 the total number of scores you added together, which equals 3. So, for this example, the median is 3.
• ### Grinding Media Milling Balls Ceramic Grinding Media
Grinding Media Grinding media are the means used to crush or grind material in a mill. It comes in different forms such as alumina oxide balls, ceramic cylinders, or soda lime glass. At Norstone Inc., we offer all types of medias used for grinding, deagglomeration, polishing, deburring, fillers, proppants, spacers, refractory beds and shot peening.
• ### A Discussion On The Measurement Of Grinding Media Wear
Jul 01, 2016 3. Conclusion. This article aimed to review and discuss the available literature on the wear of grinding media. The consumption of the grinding media represents an expressive part of the grinding operational costs, reaching up to 50 of it. During grinding, a combination of abrasion, corrosion and impact results in wear of grinding media.
• ### Statistics Calculator Median
This calculator computes the median from a data set To calculate the median from a set of values, enter the observed values in the box above. Values must be numeric and may be separated by commas, spaces or new-line. You may also copy and paste data into the text box. You do not need to specify whether the data is from a population or a sample ...
• ### White Ceramic Grinding Balls Size 13mm Weight 1kg
White Ceramic Grinding Balls - Size 13mm - Weight 1kg - Milling Media for Ball Mill - by Inoxia 3.4 out of 5 stars 6 ratings. Price 27.37 Specifications for this item. Brand Name Inoxia Chemicals Ean 0029882792698 Grit Material Ceramic Number of Items 1 Part
• ### Ball Mill Grinding Media Calculation Formula
Apr 4, 2018 An online calculator lets you calculate Top Ball Size of Grinding Media for your mill. Use this Equation and Method to properly grind your ore. Chat. Cement mill notebook SlideShare. Jan 7, 2015 4.8 t m3 Specific gravity is 7.8-7.9 t m3 Example Calculate the charge of grinding balls of a three-compartment mill for dry grinding of ...
• ### What Is Formula For Calculating Grinding Media Balls Bulk
Dec 04, 2012 TECHNICAL NOTES 8 GRINDING R. P. King Mineral . complicated in practice and it is not possible to calculate of the balls and m the density of the media. and potential energy of the grinding media. More detailed
• ### The Effect Of Grinding Media J Performance On Milling
Different grinding media materials varying in specific weight, bead size and ... calculate and evaluate specific energy as well as stress intensity ... grinding media filling grade of 80 volume- mill working volume. The stirrer tip speed was set at 6 ms and 10 ms.
• ### Weight And Surface Of Grinding Charges
Weight and surface of grinding charges. The Art Of Sharing and...Imagination. Home. About Us. Services.
• ### The Grinding Balls Bulk Weight Measuring
Sep 22, 2017 We have considered earlier in detail how to calculate the bulk weight of new grinding media and grinding media loaded into the mill. For many years experience of industrial tests conduction, Energosteel Company specialists noted, not all enterprises have the ability for correctly and quickly measure the grinding balls bulk weight.
• ### The Bulk Weight Of Grinding Balls
Mar 14, 2017 Below, we give the main points that reveal the determining process the bulk weigh of grinding media grinding balls. To measure the bulk weigh of grinding media uses container with regular geometric shape cube and volume no less than 1.0 cubic meter. Pay your attention, calculations results will be more accurate if cube has large capacity.
• ### 5 Grinding Considerations For Improving Surface Finish
Jan 26, 2021 It is best to calculate the dressing overlap ratio which takes into account the dresser width when developing a new process or changing the dressing tool type. The overlap ratio is the number of times any one point on the grinding wheel face will contact the dresser face as the dresser moves across the wheel. | 1,816 | 8,516 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.90625 | 3 | CC-MAIN-2021-39 | latest | en | 0.857953 |
https://www.offthekuff.com/wp/?p=86915 | 1,623,808,367,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487621699.22/warc/CC-MAIN-20210616001810-20210616031810-00608.warc.gz | 787,135,798 | 22,459 | # Fundraising: 2018 vs the rest of the decade
When I posted about the Q2 Congressional finance reports, I said I would try to put the totals in some more context at a later time. This is where I do that. Take a look at this table:
```
Dist 2012 2014 2016 Total 2018
=============================================================
CD02 50,168 0 14,217 64,385 843,045
CD03 0 0 0 0 153,559
CD06 145,117 13,027 27,339 185,483 358,960
CD07 76,900 74,005 68,159 219,064 2,321,869
CD08 14,935 0 0 14,935 25,044
CD10 51,855 9,994 6,120 67,969 171,955
CD12 10,785 80,216 525 91,526 106,715
CD14 1,187,774 35,302 21,586 1,244,662 105,067
CD17 0 0 39,642 39,642 67,000
CD21 57,058 0 70,714 127,772 1,594,724
CD22 40,303 0 24,584 64,887 405,169
CD23 1,802,829 2,671,926 2,198,475 6,673,230 2,256,366
CD24 6,252 10,001 21,914 39,167 61,324
CD25 12,235 32,801 55,579 100,615 199,047
CD26 11,273 0 0 11,273 94,235
CD27 399,641 301,255 23,558 724,454 93,570
CD31 0 67,742 28,317 96,059 1,618,359
CD32 79,696 10,215 0 89,911 1,916,601
CD36 2,597 25,213 0 27,810 516,859
Total 3,927,360 3,251,481 2,600,204 9,780,045 12,909,468
```
The first three columns are the total amounts raised by the November candidate in the given district for the given year. Some years there were no candidates, and some years the candidate reported raising no money. The fourth column is the sum of the first three. Note that with the exception of CD23 in 2014, these are all totals raised by challengers to Republican incumbents.
The numbers speak for themselves. With five months still go so, Democratic Congressional challengers have raised more so far this cycle than the challengers in the previous three cycles combined. The combined amount raised this year is three times what was raised in 2012, four times what was raised in 2014, and five times what was raised in 2016. Candidates this year outraised the three-year total in their districts everywhere except CDs 14 (due to Nick Lampson’s candidacy in 2012), 27 (due to two cycles’ worth of decent funding), and 23, the one true swing district where the big money is always raised.
It’s been said many times and I’ll say it again: We’ve never seen anything like this before. The reasons for it are well-explored, and the conditions that have given rise to it are (I fervently hope) singular, but it all happened. Is this a unicorn that we’ll never see again, or will it be the first step towards something different, more like this year even if not quite as much? I’d say that depends to some extent on how successful this year ends up being, and how committed everyone is to making this be more than a one-time thing. It’s a good start, but there is a whole lot more that can still be done. | 989 | 3,207 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2021-25 | latest | en | 0.830064 |
http://math.stackexchange.com/questions/740010/help-me-understand-equivalence-classes-and-relations | 1,469,478,433,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824345.69/warc/CC-MAIN-20160723071024-00072-ip-10-185-27-174.ec2.internal.warc.gz | 150,620,670 | 24,335 | # Help me understand 'equivalence classes' and relations
I'm reading up on binary relations and I understand them to be a mapping from one set into another.
However I'm having problems understand 'equivalence classes'. My book only gives a pretty dry explanation.
If you could give a good tangible explanation of equivalence relations and equivalence classes, that would be great!
-
Can you explain in what sense a binary relation is a mapping from one set into another? – anon Apr 4 '14 at 21:11
@anon In the sense that binary relations can be seen as sort of "loose" functions (may not be defined everywhere and may have more than one image). To be able to define the properties of equivalence relations you need that the "domain" and the "codomain" are one and the same, though. – A.P. Apr 5 '14 at 10:55
Let $X$ be a finite set and $R \subseteq X^2$ be some binary relation. Then $R$ is very similar to a directed graph that has elements of $X$ as vertices, or in other words, $R$ can be drawn as dots and arrows. The following example depicts a graph and corresponding relation. $$\begin{array}{ccc} 1 &\to& \stackrel{\curvearrowright} 2 \\ \downarrow & & \updownarrow\\ 3 & \gets & 4 \end{array}$$ $$\{(1,2),(1,3),(2,2),(2,4),(4,2),(4,3)\}$$
Now, to talk about equivalence classes we need an equivalence relation. How an appropriate graph looks like?
• It has to be reflexive, that is, each vertex $v$ has to have a loop $\stackrel{\curvearrowleft}v$ like $2$ in the previous example.
• It has to be symmetric, i.e. each edge of the graph has to be bidirectional, like $\{2,4\}$ in the above diagram. If all the edges are bidirectional, then such graph is called undirected.
• It has to be transitive. This means that the graphs includes all the possible "shortcuts", i.e. if you can get from vertex $u$ to $v$, then there is an edge $u \to v$, or in case of undirected graphs $u \leftrightarrow v$.
The above conditions together imply that if some two vertices are connected, then they belong to a clique (a graph that has all the possible edges, modulo loops), that is, each connected component forms a clique. These cliques are precisely the equivalence classes. In the following diagram we have two: $\{1,2,3,4\}$ and $\{5,6,7\}$ (loops omitted for clarity).
$$\begin{array}{ccc} 1 &\leftrightarrow & 2 & & 5 & \leftrightarrow & 6\\ \updownarrow & \swarrow\hspace{-10pt}\nearrow\hspace{-10pt}\nwarrow\hspace{-10pt}\searrow & \updownarrow & &\updownarrow & \swarrow\hspace{-10pt}\nearrow\\ 3 & \leftrightarrow & 4 & & 7 \end{array}$$
Observe, that if we would add edge $4 \to 5$ then symmetry would require also $5 \to 4$ and then transitivity would add all the other edges. The two clusters would merge into a single equivalence class, i.e. the graph would become the $K_7$ clique.
This intuition also holds for infinite sets, but it's hard to draw infinite graphs.
I hope this helps $\ddot\smile$
-
I never thought of equivalence relations as groupoids, thanks! – A.P. Apr 4 '14 at 22:22
@A.P. See here. Also, MathWorld suggests an inverse intuition: <<A useful way to think of a groupoid is as a parametrized equivalence relation on $B$, as follows. Given a groupoid over $B$, define an equivalence relation on $B$ by $\alpha(g)∼\beta(g)$ for each $g$ in $G$. This equivalence relation is "parameterized" because there may be more than one element in $G$ which give rise to the same equivalence, that is, $g$ and $g'$ such that $\alpha(g)=\alpha(g')$ and $\beta(g)=\beta(g')$.>> – dtldarek Apr 4 '14 at 22:36
@A.P. On the other hand, I prefer the graph-theory-based intuition. In fact, this was my primary intention and I was quite surprised by your comment. Anyway, I'm glad I could help $\ddot\smile$ – dtldarek Apr 4 '14 at 22:44
Thanks for the pointers. I just happen to be biased towards recognising category-like graphs as such. ;) – A.P. Apr 4 '14 at 22:44
An equivalence relation on a set $X$ is a binary relation $R\subset X\times X$ such that the following three properties hold:
• Reflextivity: $xRx$, or $(x,x)\in R$ for every $x$ in $X$.
• Symmetry: $xRy \leftrightarrow yRx$ or $(x,y)\in R$ if and only if $(y,x)\in R$.
• Transitivity: $xRy \wedge yRz \rightarrow xRz$, or that if $(x,y), (y,z)\in R$, then $(x,z)\in R$.
An equivalence class is a subset $E$ of $X$ such that the following holds: if $x,y\in E$, then $(x,y)\in R$ (i.e. $xRy$), and if $x$ is an element of $E$ and $y$ is an element of $X$ such that $xRy$, then $y$ is also an element of $E$. That is, it is closed under the relation and only contains the elements that are 'equivalent'.
The idea is that an equivalence relation gives us a new way of saying whether or not two objects should be viewed as essentially the same thing; the equivalence classes are then the sets after grouping all the equivalent things together, and they form a way of partitioning the set into these 'sets of equivalent things'.
This mention of partitions is actually an important thing; it ends up that a partition of a set uniquely determines an equivalence relation, and vice-a-versa. It is not hard to show that the set of equivalence classes is a partition of $X$ and given a partition, we can define a relation $R$ where $(x,y)\in R$ if and only if $x$ and $y$ lie in the same part of the partition; it can be checked rather easily that this is an equivalence relation.
-
I will assume that you know the definition of an equivalence relation on a set $A$.
Here is an example which may help: Consider the set of all integers, $\mathbb{Z}$, and define and equivalence relation on $\mathbb{Z}$ by $a$ ~ $b$ if $a = b + 3k$ for some integer $k$. Then there are three equivalence classes, $A_0, A_1, A_2$ where $A_j$ represents all the integers that leave a remainder of $j$, $0 \leq j \leq 2$ when divided by the number $3$. Note that we have decomposed $\mathbb{Z}$ as $A_0 \cup A_1 \cup A_2 = \mathbb{Z}$ and that the sets $A_j$ are disjoint.
-
Actually, binary relations are more like "loose" mappings: they do not need to be defined everywhere on their domain and any point can have more than one image.
Formally, an equivalence relation on a set $X$ is a binary relation $\rho$ defined on $X$, that is a subset $\rho$ of $X\times X$, such that for every $x,y,z\in X$:
• $x\rho x$ (reflexivity)
• $x\rho y$ implies $y\rho x$ (symmetry)
• $x\rho y$ and $y\rho z$ imply $x\rho z$ (transitivity)
where we (customarily) write $x\rho y$ for $(x,y)\in\rho$. Now, from these properties we can see that $\rho$ induces a partition on $X$, namely a collection $P$ of subsets of $X$ such that
1. $\varnothing\notin P$
2. the union of all elements of $P$ is the whole space $X$
3. $S_1\cap S_2=\varnothing$ for any two distinct $S_1,S_2\in P$
Indeed, for every $x\in X$ define $[x]=\{y\in X : x\rho y\}$. Then:
1. by reflexivity $x\in[x]\neq\varnothing$
2. $X\supseteq \bigcup_{x\in X} [x] \supseteq \bigcup_{x\in X} \{x\} = X$
3. $z\in[x]\cap[y]\neq\varnothing$ implies $x\rho y$ by transitivity and symmetry, which then implies that $[x]=[y]$
Therefore the set $P=\{[x]:x\in X\}$ is indeed a partition of $X$. The equivalence classes of $\rho$ are exactly the elements of this partition.
Vice-versa, a partition $Q$ of $X$ induces an equivalence relation $\sigma$ on $X$ simply by defining $x\sigma y$ iff $x,y\in S$ for some $S\subseteq X$ (prove it!).
Finally, it is a nice, not so difficult, exercise to show that those two operations are actually the inverse of each other: the partition $P$ induced by a relation $\rho$ induces again $\rho$ and the relation $\sigma$ induced by a partition $Q$ induces again $Q$.
-
Binary relations and equivalence classes are two basic notions that we all learn at the very beginning of school. By this I am saying that you actually very well know what they are and how they work.
It is only due to the bad influence of Bourbaki, the formalism that was important to address some problem in logic (like consistency) that those dry expositions have permeated areas of mathematics in which they are absolutely not needed, and even become detrimental.
I will use pictures to explain what are these two notions. I will use pictures because it is with pictures we (I) learned (in the first year we entered school in our life) what this concepts are.
Binary relation
In kindergarten they used to gives us pictures of a row of bees and a row of flower pots. We were supposed to draw arrows indicating what flowers each bee likes. There were different requirements about the preferences depending on the exercise and the skill the teacher wanted to teach. But a totally freely chosen set of arrows going from the bees to the flowers is what a relation is.
If we require, for example, that a bee has only one favorite flower then we get a a relation that is what we call a function from bees to flowers.
If we add to this the requirement, for example, that each flower should have no more than one bee that likes it, they we get a relation that is what we call an injective function from the bees to the flowers.
And so on.
Equivalence class
We had this other type of exercise in kindergarten. We were given a figure with many objects: flowers, tools, people, ...
We were given a feature, and we were supposed to enclose in blobs figures that had that feature in common.
For example, we could be told to group figures by color, or by use, or by kingdom, etc.
Something like this.
This breaking into classes is what an equivalence relation is.
-
Your example deals with a very specific equivalence relation, namely one where there are two objects in each equivalence class. You picked a nice approach, but that's a bad example. – dtldarek Apr 4 '14 at 22:03
I suggested an improvement, because I think your post is a bit misleading (notice that I did not downvote). However, nobody forces you to take good advice (I won't edit your answer against your will). On the other hand, I would say that your last comment is dangerously close to name-calling and there are users that might take offence. Perhaps you have a bad day, it happens to all of us. Nevertheless, people in this community have generally good intentions, please consider this the next time you will be writing a reply. – dtldarek Apr 4 '14 at 22:25
Indeed, in elementary school we are given some basics about set theory and binary relations. I would like to point out, though, that you only described binary relations in general (without a word about equivalence relations). Moreover, what you describe as equivalence classes are just subsets. For example suppose we are asked to group the figures in the picture by colour: then the paint bucket would go in the same set as the brush, which would go in the same set as the baseball glove. Thus by transitivity the bucket and the glove are in the same subset, although they don't share any colour. – A.P. Apr 4 '14 at 22:40 | 2,900 | 10,879 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2016-30 | latest | en | 0.930454 |
http://pygments.org/demo/6046116/ | 1,534,838,300,000,000,000 | text/html | crawl-data/CC-MAIN-2018-34/segments/1534221218070.73/warc/CC-MAIN-20180821073241-20180821093241-00656.warc.gz | 341,245,755 | 3,486 | # Demo entry 6046116
Joss TestCode
Submitted by Joss Test on Sep 21, 2016 at 18:06
Language: Python 3. Code size: 2.2 kB.
```#This allows the use of sys.exit to quit the program
import sys
#This allows us to use pi
import math
#the use of \n in a printed string escapes to a new line
print ("Welcome to the Area of a Shape Calculator")
print ("*****************************************\n")
option = input("Calculate the area of a (R)ectangle, (C)ircle ot (T)riangle? Type (E)xit to exit. >> ")
option = option.upper()
if option == "R":
rectangle()
elif option == "C":
circle()
elif option == "T":
triangle()
elif option == "E":
exit()
else:
print ("Hmm not quite sure about that entry - I will return you to the menu\n")
def rectangle():
height = float(input("Please enter the height of the rectangle? "))
width = float(input("Please enter the width of the rectangle? "))
area_rect = height * width
print (area_rect,"\n")
def circle():
area_circle = math.pi * radius ** 2
print (round(area_circle,2),"\n")
def triangle():
print ("Dummy function - where area of a triangle would appear\n\n")
def exit():
print ("Many thanks for using the area calculator - Goodbye")
sys.exit
#Non of the above functions will run until they are called
#The program will start by calling the menu function
#menu1 is a demonstartion only of how """three quotes""" lays text out on the screen
print ("Welcome to the Area of a Shape Calculator")
print ("*****************************************\n\n")
option = input(("""Please choose a shape:\n
Rectangle - Press 1
Circle - Press 2
Triangle - Press 3
>> """))
if option == "1":
rectangle()
elif option == "2":
circle()
elif option == "3":
triangle()
elif option == "4":
exit()
else:
print ("Hmm not quite sure about that entry - I will return you to the menu\n") | 462 | 1,815 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2018-34 | latest | en | 0.707299 |
https://practicaldev-herokuapp-com.global.ssl.fastly.net/antogarand/php-return-true-to-win---writeup-part-1-9bo | 1,611,415,966,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703538082.57/warc/CC-MAIN-20210123125715-20210123155715-00779.warc.gz | 506,321,458 | 40,274 | ## DEV Community is a community of 555,288 amazing developers
We're a place where coders share, stay up-to-date and grow their careers.
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# PHP: Return true to win - WriteUp (Part 1)
Antony Garand ・7 min read
# Introduction
Returntrue.win is a website containing 16 PHP challenges where we must return true using the least amount of characters in the given context.
The challenges demonstrate many interesting quirks from PHP and are of greatly varying difficulty, which is why solving and understanding those can help us gain a better understanding of the language.
This post will cover the first 8 solutions, while an upcoming part 2 will cover the remaining ones.
Now is the time to stop reading and try to solve the challenges yourself before reading on the solutions!
Website: https://returntrue.win
## Level 1
#### Challenge
``````function foo(\$x)
{
return \$x;
}
``````
#### Solution
``````foo(!0);
``````
#### Explanation
The obvious solution would be to enter `true`, but that would be 4 bytes and the shortest answer uses only half of those! We can save two bytes by using `!0` instead.
The reason `!0` works is as the logical not (!) operator will type cast the value to a boolean first, and return the opposite of this result.
As `0` is converted to `false`, we get `!false`, which is `true`.
See: Boolean type casting for a list of values converted to false.
## Level 2
#### Challenge
``````function foo(\$x)
{
return \$x('gehr') === 'true';
}
``````
#### Solution
``````foo(str_rot13)
``````
#### Explanation
Our first guess could be to create an anonymous function which always returns `'true'`, such as the following:
``````foo(function(){return'true';})
``````
But this isn't the shortest solution.
The key to the shortest 9 bytes score is the 'gehr' argument given to the function, which is exactly `true` with each letter shifted by 13 characters.
As it turns out, php has the str_rot13 function which performs this exact operation!
PHP doesn't like crashing, and it does so by trying to be smart with its conversions.
When we're using a constant which is not defined, the following behavior is used:
If you use an undefined constant, PHP assumes that you mean the name of the constant itself, just as if you called it as a string (CONSTANT vs "CONSTANT").
From: PHP Constant syntax
Combine this with Variable Functions
This means that if a variable name has parentheses appended to it, PHP will look for a function with the same name as whatever the variable evaluates to, and will attempt to execute it.
And we've got a working solution!
## Level 3
#### Challenge
``````function foo(\$x)
{
return (\$x >= 1) && (\$x <= 1) && (\$x !== 1);
}
``````
#### Solution
``````foo(!0)
// Or
foo(1.)
``````
#### Explanation
The first requirement here is get a variable which is `<=` and `>=` than 1.
As those are numeric comparisons, PHP will convert both sides of the equation to numbers.
The second requirement is for our variable to not be the integer `1`.
Booleans converted to numbers will manage to pass this condition:
FALSE will yield 0 (zero), and TRUE will yield 1 (one).
From the doc: Integer: Casting from boolean
This gives us our first answer, passing `true` will succeed in all the checks.
The second solution is based on PHP's handling of its different numeric types.
As you may know, PHP has both floats and integers.
As they are different types, `1.0` and `1` returns true to the Not identical operator.
Our solution is therefore the float value of `1`, which can be written `1.`
## Level 4
#### Challenge
``````function foo(\$x)
{
\$a = 0;
switch (\$a) {
case \$x:
return true;
case 0:
return false;
}
return false;
}
``````
#### Solution
``````foo(0);
``````
#### Explanation
As `\$a` is `0` an we need the program flow to enter our `case \$x`, the solution is to set `\$x` to `0`.
Not much of a challenge here!
## Level 5
#### Challenge
``````function foo(\$x)
{
return strlen(\$x) === 4;
}
``````
#### Solution
``````foo(💩);
``````
#### Explanation
The tricky part here is getting the shortest solution, which is only 1 character long yet `strlen` will return 4.
The solution resides in the difference between `strlen` and `mb_strlen`, which handles multibyte strings differently:
strlen() returns the number of bytes rather than the number of characters in a string.
[mb_strlen()] A multi-byte character is counted as 1.
As utf8 supports up to 4 bytes per character, our poop emoji returns `4` on the `strlen` function while being only one character.
References:
strlen documentation
mb_strlen documentation
## Level 6
#### Challenge
``````function foo(\$x)
{
return \$x === \$x();
}
``````
#### Solution
``````// Not working anymore, 22 characters
foo((\$x=session_id)(\$x).\$x);
// Shortest? working solution: 33 characters
foo(\$x=function()use(&\$x){return \$x;})
// Alternative working solution, 42 characters
foo(\$GLOBALS[x]=function(){return\$GLOBALS[x];})
// Runner up, 44 characters
foo(new class{function __invoke(){return\$this;}})
``````
#### Explanation
This one is among the most interesting challenges of the lot, and I've spent a lot of time finding the shortest solution.
Note that at the time of this writing, the shortest solution will NOT work due to additional sandboxing done on the website, but it does work locally if you have sessions enabled.
There are four different solutions which I'd like to explore here.
##### Shortest
``````(\$x=session_id)(\$x).\$x
// Which is equivalent to
\$x = 'session_id';
\$x(\$x);
foo(\$x);
``````
Firstly, the essence of this solution is the session_id function.
This function is used as an getter/setter:
• When called without an argument, it will return the stored value.
• When called with an argument, it will assign its inner value and return an empty string.
The first part of this solution is setting the `\$x` variable to `'session_id'` (As string, because of PHP's handling of undefined constants).
The second part is calling session_id, which works once again thanks to PHP's Variable functions, with `\$x` as argument. As `\$x` is a string, this will set the content of `session_id` to `'session_id'`.
The result of our `session_id(\$x)` call is an empty string, which isn't what we want to send to `foo`.
In order to send `session_id` to `foo`, we concatenate `\$x` to the empty string we previously had, which results in the `'session_id'` string.
Finally, the foo function will compare our entry variable `\$x`, aka. `'session_id'`, to the result of `\$x()`, aka. `session_id()`.
As we did set our `session_id` to `'session_id'`, both strings are the same and we passed the comparison!
##### Self returning function using use
``````\$x=function()use(&\$x){return \$x;}
``````
This solution is pretty straightforward: Create a function which uses itself, and returns itself.
Reference: Anonymous functions, specifically example #3
##### Globals
``````\$GLOBALS[x]=function(){return\$GLOBALS[x];}
// Which is equivalent to
\$GLOBALS[x] = function() {
return \$GLOBALS[x];
};
foo(\$GLOBALS[x]);
``````
This solution is similar to the previous one, but uses the \$GLOBALS array instead of inheriting variables from the parent scope.
##### Callable class returning itself
``````new class {
function __invoke()
{
return \$this;
}
}
``````
Since PHP7, we can create Anonymous Classes.
This lets us create a new class with the __invoke magic method implemented.
This magic method lets us return `\$this` when we call our object as a function to solve the challenge!
## Level 7
#### Challenge
``````function foo(stdClass \$x)
{
\$x = (array) \$x;
return \$x[0];
}
``````
#### Solution
``````foo((object)[!0]);
``````
#### Explanation
This solution resides in how PHP type casts to object.
As we are typecasting an array with `[0 => true]` to an object, we are creating a generic stdClass with the `0` property having the `true` value.
Once this is type casted back to an array with the `\$x = (array) \$x;` line, the property `0` of our object gets back to the new array's index 0.
## Level 8
#### Challenge
``````class Bar {}
function foo(Bar \$x)
{
return get_class(\$x) != 'Bar';
}
``````
#### Solution
``````foo(new class extends Bar{});
``````
#### Explanation
This solution depends on Object Inheritance in PHP.
We can create a new child class which extends `Bar` as an argument. As the given object is a new class, get-class will return the name of the new anonymous class and not 'Bar'.
# Conclusion
Those were lots of original solutions to the challenge!
Most of those solutions did produce warnings and notices, therefore most of the odd behavior should be detected in a proper development environment.
I would like to thank:
If you do know similar oddities or challenge websites, please do send them my way so I can solve and write about them!
# References
## Discussion
Comment deleted
Josh Cheek
You can't really figure out when user code is malicious, and escaping won't work here b/c the input itself must be evaluated. IDK how it's implemented, my guess is that it's running on a sandboxed server (eg has memory/processor/duration thresholds set, which will kill the program if you exceed them, has abusable features like http and system commands disabled). The comment about `session_id` would support this hypothesis. There are other options, though. I've done things like this by shipping them off to eval.in, which does its own sandboxing. You could also compile php to web assembly and run it in the user's browser (guessing this would take quite a bit of work, but it should be possible).
Antony Garand
This all depends on how you validate your user input!
This may not be a simple `eval`, but perhaps a docker container launched specifically for this test and destroyed afterwards.
I know that's how pwnfixrepe.at does evaluate untrusted code safely, and therefore there are most likely similar mechanisms in place here.
robencom
Hehe I like a PHP challenge when I see one, but when one of the answers is a POOP emoji, then you'll know that this challenge has passed the limits of logic :)
Ben Sinclair
I'm glad you explained this because I had no clue what I was supposed to do from the website itself! | 2,486 | 10,268 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2021-04 | latest | en | 0.897626 |
http://slideplayer.com/slide/7898497/ | 1,716,477,171,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058642.50/warc/CC-MAIN-20240523142446-20240523172446-00518.warc.gz | 25,571,939 | 19,227 | # Conventional current: the charges flow from positive to negative electron flow: the charges move from negative to positive the “flow of electrons” Hand.
## Presentation on theme: "Conventional current: the charges flow from positive to negative electron flow: the charges move from negative to positive the “flow of electrons” Hand."— Presentation transcript:
conventional current: the charges flow from positive to negative electron flow: the charges move from negative to positive the “flow of electrons” Hand rules: we will use our right hand when we are using conventional current and we will use our left hand when we use electron flow. We draw a force that goes into the page as an x. We draw a force that comes out of the page as a ○.
First Hand Rule We can find the direction of the magnetic field around a wire for a given current. Imagine holding a length of insulated wire with your right hand. Keep your thumb pointed in the direction of the conventional current. The fingers of your hand circle the wire and point in the direction of the magnetic field.
If we used the left hand rule, our thumb would be directed against the conventional current but our fingers would still point in the direction of the magnetic field.
Second Hand Rule A current in a solenoid creates a magnetic field with the field from each coil adding to the others. The solenoid has a north and a south pole and it is in itself a magnet. We can determine the direction of the magnetic field produced by an electromagnet if we know the direction of the current. Imagine holding an insulated coil with your right hand. If you then curl your fingers around the loops in the direction of the conventional current, your thumb will point toward the north pole of the electromagnet. If we used the left hand rule, our thumb would be directed against the conventional current.
1.If a current carrying wire is bent into a loop, why is the magnetic field inside the loop stronger than the magnetic field outside? 2.Suppose you are lost in the woods but have a compass with you. Unfortunately the red paint marking the north pole of the compass needle has worn off. You have a flashlight with a battery and a length of wire. How could you identify the north pole of the compass?
Third Hand Rule We can figure out the direction of the magnetic force when the current and magnetic field are known. Point the fingers of your right hand in the direction of the magnetic field and point your thumb in the direction of the conventional current in the wire. The palm of your hand will be facing in the direction of the force acting on the wire. If you use your left hand you will point your thumb against the conventional current.
If we have two wires next to each other and their currents are in the same direction they will be attracted to each other. If their currents are in opposite directions they will repel each other.
The force on a current-carrying wire in a magnetic field is equal to the product of the current, the length of the wire, and the magnetic field strength. F=BIL where F is the magnetic force (Newtons), I = current (amps), L = length (m), and B is the magnetic field strength (Tesla) - 1 Tesla = 1 N/A*m - If the wire is not perpendicular the equation becomes F =BIL*sin θ. If the wire becomes parallel to the magnetic field sin θ becomes zero and there is no magnetic force.
Example: A wire that is 0.50 m long and carrying a current of 8.0A is at right angles to a 0.40-Tesla magnetic field. How strong is the force that acts on the wire? Example: How much current will be required to produce a force of 0.38N on a 10.0cm length of wire at right angles to a 0.49-Tesla field?
Fourth Hand Rule We can use the fourth hand rule to find the direction of the forces on the charged in a conductor that is moving in the magnetic field. To generate current, either the conductor can move through a magnetic field or a magnetic field can move past the conductor. It is the relative motion between the wire and the magnetic field that produces the current. This process is called electromagnetic induction.
To find the force on the charged in the wire hold your right hand so that your thumb points in the direction in which the wire is moving and your fingers point in the direction of the magnetic field. The palm of your hand will point in the direction of the conventional current. This is how an electrical generator works. It converts mechanical energy to electrical energy. An electrical generator consists of a number of wire loops placed in a strong magnetic field. The wire is wound around an iron core to increase the strength of the magnetic field. The wire loops rotate through the magnetic field.
Download ppt "Conventional current: the charges flow from positive to negative electron flow: the charges move from negative to positive the “flow of electrons” Hand."
Similar presentations | 1,012 | 4,900 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.828125 | 4 | CC-MAIN-2024-22 | latest | en | 0.919463 |
https://usethinkscript.com/threads/convert-amibroker-make-or-break-mob-indicator.85/ | 1,670,289,466,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446711064.71/warc/CC-MAIN-20221205232822-20221206022822-00384.warc.gz | 627,962,628 | 26,445 | # Convert Amibroker Make or Break (MOB) Indicator
#### Billy101
##### New member
Hi Guys,
Since this is my first post, before asking for some help I´d like to share a very useful toll (for me) to identify waves, on below link you´ll find thinkscript for Elliot Wave Oscillator with breakout bands as found on esignal, helps pretty well to identify wave 3 and wave 5.
Now I´d like to ask for some help from more experienced users, I came across with this code that seems to be MOB indicator (another esignal), I´m a newbie on thinkscrip so if some could give a hand will be much appreciated.
Awsome forum, look forward sharing trading ideas!
Thanks!
Rich (BB code):
``````_SECTION_BEGIN("MOB Simulation");
SetChartOptions(0,chartShowArrows|chartShowDates);
_N(Title = StrFormat("{{NAME}} - {{INTERVAL}} {{DATE}} Open %g, High %g, Low %g, Close %g (%.1f%%) {{VALUES}}", O, H, L, C, SelectedValue( ROC( C, 1 ) ) ));
//Plot Colored Candles :)
PlotOHLC(O ,H ,L ,C ,"Price",IIf(C>O,colorGreen,colorRed),styleCandle);
Offset = 5; //Recommended to use two sheets: one with 5 and another with 7, or maybe other offset value
Avgmov = Offset * MA (abs(ROC(C,1)) ,20);
per = LastValue(Avgmov) ;
numberOfBars = Cum(1);
Range = 0.01;
PS = TroughBars(L, per, 1) == 0;
Title = Title + StrFormat("AVGMOV %g|%g|%g
", Avgmov,per, numberOfBars);
xa = LastValue(ValueWhen (PS,numberOfBars,1)) ;//x from last trough
Ya = LastValue(ValueWhen (PS,L,1)) ;//y (Low) last trough
PR = PeakBars(H,per, 1) == 0;
xb = LastValue(ValueWhen (PR,numberOfBars,1)) ;//x from last peak
Yb = LastValue(ValueWhen (PR,H,1)) ;//y (High) last peak
Title = Title + StrFormat("PS %g|%g|%g|%g|%g|%g", PS,xa,ya,PR,xb,yb);
Trough_ReTest = abs((L/ya)-1) <Range;
Peak_ReTest = abs((H/yb)-1) <Range;
Trough_Cross = Cross(ya,C);
Peak_Cross = Cross(C,yb);
//UP = upSwing DN = downSwing
UP = xb>xa;//upSwing
DN = xa>xb;//DownSwing
RT23_6 = IIf(UP,yb-(yb- ya)*0.236, IIf(DN,ya+ (yb-ya)*0.236,-1e10) );
RT38_2 = IIf(UP,yb-(yb- ya)*0.382, IIf(DN,ya+ (yb-ya)*0.382,-1e10) );
RT50_0 = IIf(UP,yb-(yb- ya)*0.500, IIf(DN,ya+ (yb-ya)*0.500,-1e10) );
RT61_8 = IIf(UP,yb-(yb- ya)*0.618, IIf(DN,ya+ (yb-ya)*0.618,-1e10) );
RT78_6 = IIf(UP,yb-(yb- ya)*0.786, IIf(DN,ya+ (yb-ya)*0.786,-1e10) );
RT12_7 = IIf(UP,yb-(yb- ya)*1.27, IIf(DN,ya+ (yb-ya)*1.27,-1e10) );
RT16_1 = IIf(UP,yb-(yb- ya)*1.61, IIf(DN,ya+ (yb-ya)*1.61,-1e10) );
RT=
IIf(UP,-100* (yb-L)/(yb- ya),
100*(H-ya)/( yb-ya));//Retracement_ Value
InZone = C<yb & C>ya;//use it for filter to receive only signals that are in in the Retracement zone.
Sell = Peak_ReTest OR trough_Cross;
Filter = 1;
//Plot(C,"C",1, 64);
Plot(IIf(numberOfBars>xa, ya,-1e10) ,"Bottom" ,colorBrown, 1+8);
Plot(IIf(numberOfBars>xb, yb,-1e10) ,"Top",colorBrown,1+8);
xab = IIf(xb>xa,xb, xa);
//Retracements
Plot(IIf(numberOfBars>= xab+1,RT23_6,-1e10), "R2 23.6% Retr.",5,styleLine | styleNoTitle | styleDots);
Plot(IIf(numberOfBars>= xab+1,RT38_2,-1e10), "R1 38.2% Retr.",5,styleLine | styleNoTitle | styleDots);
Plot(IIf(numberOfBars>= xab+1,RT50_0,-1e10), "ZR 50.0% Retr.",colorBlue, styleLine | styleNoTitle | styleDots);
Plot(IIf(numberOfBars>= xab+1,RT61_8,-1e10), "S1 61.8% Retr.",colorDarkRed, styleLine | styleNoTitle | styleDots);
Plot(IIf(numberOfBars>= xab+1,RT78_6,-1e10), "S2 78.6% Retr.",colorDarkRed, styleLine | styleNoTitle |styleDots);
// Plot the MOB Cloud
Plot(IIf(numberOfBars>= xab+1,RT12_7,-1e10), "127% ext.",colorBrightGreen, styleNoTitle | styleNoLabel | styleLine);
Plot(IIf(numberOfBars>= xab+1,RT16_1,-1e10), "161% ext.",colorBrightGreen, styleNoTitle | styleNoLabel | styleLine);
CondA=IIf(numberOfBars>= xab+1,RT12_7,-1e10);
CondB=IIf(numberOfBars>= xab+1,RT16_1,-1e10);
PlotOHLC(Condb,Condb,Conda,Conda,"",ColorRGB(30,130,30),styleCloud, styleNoTitle | styleNoLabel);
GraphXSpace = 0.5;
//Plot(Gauss2ord(C,5),"M5",4,1);
//GraphXSpace = 1.5;
//Title = Name()+" per = "+WriteVal(per, 1.0) +" Close = "+WriteVal(C, 1.2)+ " ("+WriteVal( ROC(C,1), 1.2)+"%)" +" Current Correction = "+WriteVal(RT, 1.0)+"%";
//Plot( Volume,"V", ParamColor("Color", colorBlueGrey ), ParamStyle( "Style", stylehidden| styleOwnScale | styleThick, maskHistogram ), 2 );
_SECTION_END( );``````
Last edited by a moderator:
#### Bluephire1914
##### New member
2019 Donor
The script is not working - can you please link the TOS chart setup tos.mx?
Last edited:
##### Well-known member
VIP
The code in the first post is NOT Thinkscript it is from another trading platform altogether and would need to be converted to Thinkscript to be useful... That's why you are getting multiple errors...
Last edited by a moderator:
#### Billy101
##### New member
[email protected] or [email protected]
Hi everyone,
I´d like to ask if what I´m trying to achieve is feasible on TOS and maybe hear if it can be done easily.
Ok, here´s the thing, there´s a tool on @dvanced 6et called MO8, make or [email protected] that allows the user to select a previous pivot and from that point a target zone in the future is drawn on the chart.
After a couple hours I found that on the Y axis (price) most of the time this area is drawn applying a fib retracement taking extensions 1.2 / 1.4 / 1.5 / 1.6 as the middle line and then adding +/- 5% and that identifies the zone.
Then you have the total lenght of the box and two timing marks (I´m working on the math on this).
For the ones interested on looking at 42 examples, here´s a list ";" separated to import in excel.
Currently I´m using this study http://tos.mx/p2FsUAl this is one of the many versions of ZigZag study, and I´d like to know if is it possible to count the swings and manually tell the code by an input variable which swing must be taken to draw a retracement from, and then plotting the zone on the forward swing.
I´m having a hard time shifting the plot
Thanks,
Billy.
Example with GE on GET and TOS
;LongSwing;;ABSL(a-b);Duration Long;ShortSwing;;ABSC(a-b);Duration Short;Target;len MOB;;Mark#1;Mark#2
1;4,7892;9,4143;4,6251;28;9,4143;7,755;1,6593;11;1,2;;;;
2;45,907;31,294;14,613;48;31,294;39,118;7,824;17;1,5;;;;
3;29,957;41,5992;11,6422;38;41,5992;35,3;6,29920000000001;9;1,5;;;;
4;9,959;13,531;3,572;16;13,531;11,501;2,03;12;1,2;;;;
5;14,381;9,2834;5,0976;30;9,2834;11,882;2,5986;37;1,5;;;;
6;8,4462;10,799;2,3528;11;10,799;8,2985;2,5005;18;1,2;;;;
7;13,499;10,76;2,739;9;10,76;12,777;2,017;12;1,2;;;;
8;17,517;14,93;2,587;12;14,93;17,513;2,583;10;1,6;;;;
9;14,431;10,295;4,136;23;10,295;12,335;2,04;7;1,2;;;;
10;36,13;15,56;20,57;131;15,56;22,771;7,211;64;1,6;;;;
11;270,53;225,79;44,74;59;225,79;241,73;15,94;18;1,2;;;;
12;241,73;217,5;24,23;16;217,5;247,48;29,98;29;1,6;;;;
13;114,76;145,12;30,36;23;145,12;115,37;29,75;15;1,2;;;;
14;134,39;162,59;28,2;52;162,59;142,91;19,68;12;1,2;;;;
15;105,47;128,96;23,49;23;128,96;117,5;11,46;24;1,2;;;;
16;100,95;114,02;13,07;24;114,02;105,47;8,55;17;1,3;;;;
17;90,89;110,72;19,83;42;110,72;100,95;9,77;7;1,2;;;;
18;10,853;7,4183;3,4347;95;7,4183;9,0818;1,6635;30;1,6;;;;
19;4,52;7,74;3,22;16;7,74;5,74;2;23;1,2;;;;
20;8,43;13,62;5,19;48;13,62;11,14;2,48;32;1,6;;;;
21;8,67;10,91;2,24;21;10,91;9,754;1,156;23;1,3;;;;
22;9,617;11,802;2,185;34;11,802;10,194;1,608;9;1,4;;;;
23;4,5;10,893;6,393;59;10,893;6,5823;4,3107;21;1,6;;;;
24;29,16;16,724;12,436;32;16,724;24,27;7,546;38;1,4;;;;
25;12,275;26,56;14,285;49;26,56;15,14;11,42;34;1,6;;;;
26;174,09;144,37;29,72;39;144,37;179,1;34,73;25;1,2;98;;61;0
27;37,71;45,56;7,85;28;45,56;41,25;4,31;61;1,2;128;;30;52
28;121,89;157,03;35,14;81;157,03;138,07;18,96;19;1,2;69;;28;47
29;3,9061;5,3617;1,4556;66;5,3617;3,7;1,6617;137;1,2;321;;58;95
30;79,12;63,42;15,7;32;63,42;74,103;10,683;18;1,6;110;;47;77
31;144,97;165,48;20,51;27;165,48;144,84;20,64;32;1,2;111;;92;0
32;31,53;47,75;16,22;130;47,75;38,03;9,72;31;1,2;117;;69;113
33;11,33;6,59;4,74;19;6,59;8,61;2,02;16;1,2;35;;10;18
34;90,34;143,31;52,97;33;143,31;122,98;20,33;16;1,5;50;;27;46
35;486,72;272,41;214,31;56;272,41;490,76;218,35;56;1,2;184;;69;113
36;292,72;397,69;104,97;12;397,69;272,41;125,28;16;1,2;62;;25;42
37;1243,49;886,12;357,37;32;886,12;1208;321,88;9;1,4;28;;14;26
38;13,69;16,124;2,434;28;16,124;14,266;1,858;10;1,5;24;;4;8
39;152,57;134,7;17,87;32;134,7;152;17,3;24;1,6;64;;18;31
40;47348,1;39782,26;7565,84;14;39782,26;68976,48;29194,22;50;1,2;158;;50;82
41;71,999;53,823;18,176;13;53,823;70,279;16,456;17;1,4;72;;27;45
42;52,03;63,1;11,07;46;63,1;51,409;11,691;38;1,2;120;;40;66
Last edited by a moderator:
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http://what-when-how.com/Tutorial/topic-3734bj/JavaScript-Novice-to-Ninja-529.html | 1,685,800,641,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224649293.44/warc/CC-MAIN-20230603133129-20230603163129-00353.warc.gz | 55,169,798 | 3,937 | Java Reference
In-Depth Information
context.lineWidth = 4;
The fillRect() method can draw a filled-in rectangle. The first two parameters are the
coordinates of the top-left corner, the third parameter is the width, and the last parameter
is the height. The following produces a filled-in blue rectangle in the top-left corner of the
canvas at coordinates (10,10) that is 100 pixels wide and 50 pixels high:
context.fillRect(10,10,100,50);
The strokeRect() method works in the same way, but produces a rectangle that is not
filled in. This will draw the outline of a rectangle underneath the last one:
context.strokeRect(10,100,100,50);
Straight lines can be drawn employing the moveTo() and lineTo() methods. These
methods can be used together to produce a path. Nothing will actually be drawn onto the
canvas until the stroke() method is called. The following example will draw a thick red
T shape onto the canvas by moving to the coordinates (150,50), then drawing a horizontal
line 30 pixels long, and finally moving to the middle of that line and drawing a vertical line
40 pixels long:
context.beginPath();
context.moveTo(130, 50);
context.lineTo(180, 50);
context.moveTo(155, 50);
context.lineTo(155, 90);
context.strokeStyle = "#c00";
context.lineWidth = 15;
context.stroke();
The arc() method can be used to draw an arc of a given radius from a particular point.
The first two parameters are the coordinates of the center of the arc; the next parameter is
the radius, followed by the start angle, then the finish angle (note that these are measured in
radians). The last parameter is a Boolean value that says whether the arc should be drawn
counter-clockwise. The following example will draw a yellow circle of radius 30 pixels at
center (200,200), since Math.PI * 2 represents a full turn:
Search WWH ::
Custom Search | 422 | 1,829 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2023-23 | latest | en | 0.75363 |
http://en.wikipedia.org/wiki/Satisfiability_Modulo_Theories | 1,430,156,732,000,000,000 | text/html | crawl-data/CC-MAIN-2015-18/segments/1429246659254.83/warc/CC-MAIN-20150417045739-00042-ip-10-235-10-82.ec2.internal.warc.gz | 93,695,599 | 19,474 | # Satisfiability modulo theories
(Redirected from Satisfiability Modulo Theories)
In computer science and mathematical logic, the satisfiability modulo theories (SMT) problem is a decision problem for logical formulas with respect to combinations of background theories expressed in classical first-order logic with equality. Examples of theories typically used in computer science are the theory of real numbers, the theory of integers, and the theories of various data structures such as lists, arrays, bit vectors and so on. SMT can be thought of as a form of the constraint satisfaction problem and thus a certain formalized approach to constraint programming.
## Basic terminology
Formally speaking, an SMT instance is a formula in first-order logic, where some function and predicate symbols have additional interpretations, and SMT is the problem of determining whether such a formula is satisfiable. In other words, imagine an instance of the Boolean satisfiability problem (SAT) in which some of the binary variables are replaced by predicates over a suitable set of non-binary variables. A predicate is basically a binary-valued function of non-binary variables. Example predicates include linear inequalities (e.g., $3x+ 2y - z \geq 4$) or equalities involving uninterpreted terms and function symbols (e.g., $f(f(u, v), v) = f(u, v)$ where $f$ is some unspecified function of two unspecified arguments.) These predicates are classified according to each respective theory assigned. For instance, linear inequalities over real variables are evaluated using the rules of the theory of linear real arithmetic, whereas predicates involving uninterpreted terms and function symbols are evaluated using the rules of the theory of uninterpreted functions with equality (sometimes referred to as the empty theory). Other theories include the theories of arrays and list structures (useful for modeling and verifying software programs), and the theory of bit vectors (useful in modeling and verifying hardware designs). Subtheories are also possible: for example, difference logic is a sub-theory of linear arithmetic in which each inequality is restricted to have the form $x - y > c$ for variables $x$ and $y$ and constant $c$.
Most SMT solvers support only quantifier free fragments of their logics.
## Expressive power of SMT
An SMT instance is a generalization of a Boolean SAT instance in which various sets of variables are replaced by predicates from a variety of underlying theories. Obviously, SMT formulas provide a much richer modeling language than is possible with Boolean SAT formulas. For example, an SMT formula allows us to model the datapath operations of a microprocessor at the word rather than the bit level.
By comparison, answer set programming is also based on predicates (more precisely, on atomic sentences created from atomic formula). Unlike SMT, answer-set programs do not have quantifiers, and cannot easily express constraints such as linear arithmetic or difference logic—ASP is at best suitable for boolean problems that reduce to the free theory of uninterpreted functions. Implementing 32-bit integers as bitvectors in ASP suffers from most of the same problems that early SMT solvers faced: "obvious" identities such as x+y=y+x are difficult to deduce.
Constraint logic programming does provide support for linear arithmetic constraints, but within a completely different theoretical framework.
## SMT solver approaches
Early attempts for solving SMT instances involved translating them to Boolean SAT instances (e.g., a 32-bit integer variable would be encoded by 32 bit variables with appropriate weights and word-level operations such as 'plus' would be replaced by lower-level logic operations on the bits) and passing this formula to a Boolean SAT solver. This approach, which is referred to as the eager approach, has its merits: by pre-processing the SMT formula into an equivalent Boolean SAT formula we can use existing Boolean SAT solvers "as-is" and leverage their performance and capacity improvements over time. On the other hand, the loss of the high-level semantics of the underlying theories means that the Boolean SAT solver has to work a lot harder than necessary to discover "obvious" facts (such as $x + y = y + x$ for integer addition.) This observation led to the development of a number of SMT solvers that tightly integrate the Boolean reasoning of a DPLL-style search with theory-specific solvers (T-solvers) that handle conjunctions (ANDs) of predicates from a given theory. This approach is referred to as the lazy approach.
Dubbed DPLL(T),[1] this architecture gives the responsibility of Boolean reasoning to the DPLL-based SAT solver which, in turn, interacts with a solver for theory T through a well-defined interface. The theory solver need only worry about checking the feasibility of conjunctions of theory predicates passed on to it from the SAT solver as it explores the Boolean search space of the formula. For this integration to work well, however, the theory solver must be able to participate in propagation and conflict analysis, i.e., it must be able to infer new facts from already established facts, as well as to supply succinct explanations of infeasibility when theory conflicts arise. In other words, the theory solver must be incremental and backtrackable.
## SMT for undecidable theories
Most of the common SMT approaches support decidable theories. However, many real-world systems can only be modelled by means of non-linear arithmetic over the real numbers involving transcendental functions, e.g. an aircraft and its behavior. This fact motivates an extension of the SMT problem to non-linear theories, e.g. determine whether
$\begin{array}{lr} & (\sin(x)^3 = \cos(\log(y)\cdot x) \vee b \vee -x^2 \geq 2.3y) \\ & \wedge \left(\neg b \vee y < -34.4 \vee \exp(x) > {y \over x}\right) \end{array}$
where
$b \in {\mathbb B}, x,y \in {\mathbb R}$
is satisfiable. Then, such problems become undecidable in general. (It is important to note, however, that the theory of real closed fields, and thus the full first order theory of the real numbers, are decidable using quantifier elimination. This is due to Alfred Tarski.) The first order theory of the natural numbers with addition (but not multiplication), called Presburger arithmetic, is also decidable. Since multiplication by constants can be implemented as nested additions, the arithmetic in many computer programs can be expressed using Presburger arithmetic, resulting in decidable formulas.
Examples of SMT solvers addressing Boolean combinations of theory atoms from undecidable arithmetic theories over the reals are ABsolver,[2] which employs a classical DPLL(T) architecture with a non-linear optimization packet as (necessarily incomplete) subordinate theory solver, and iSAT [1], building on a unification of DPLL SAT-solving and interval constraint propagation called the iSAT algorithm.[3]
## SMT solvers
The table below summarizes some of the features of the many available SMT solvers. The column "SMT-LIB" indicates compatibility with the SMT-LIB language; many systems marked 'yes' may support only older versions of SMT-LIB, or offer only partial support for the language. The column "CVC" indicates support for the CVC language. The column "DIMACS" indicates support for the DIMACS format.
Projects differ not only in features and performance, but also in the viability of the surrounding community, its ongoing interest in a project, and its ability to contribute documentation, fixes, tests and enhancements.
Platform Features Notes
Name OS License SMT-LIB CVC DIMACS Built-in theories API SMT-COMP [2]
ABsolver Linux CPL v1.2 No Yes linear arithmetic, non-linear arithmetic C++ no DPLL-based
Alt-Ergo Linux, Mac OS, Windows CeCILL-C (roughly equivalent to LGPL) partial v1.2 and v2.0 No No empty theory, linear integer and rational arithmetic, non-linear arithmetic, polymorphic arrays, enumerated datatypes, AC symbols, bitvectors, record datatypes, quantifiers OCaml 2008 Polymorphic first-order input language à la ML, SAT-solver based, combines Shostak-like and Nelson-Oppen like approaches for reasoning modulo theories
Barcelogic Linux Proprietary v1.2 empty theory, difference logic C++ 2009 DPLL-based, congruence closure
Beaver Linux, Windows BSD v1.2 No No bitvectors OCaml 2009 SAT-solver based
Boolector Linux GPLv3 v1.2 No No bitvectors, arrays C 2009 SAT-solver based
CVC3 Linux BSD v1.2 Yes empty theory, linear arithmetic, arrays, tuples, types, records, bitvectors, quantifiers C/C++ 2010 proof output to HOL
CVC4 Linux, Mac OS, Windows BSD Yes Yes rational and integer linear arithmetic, arrays, tuples, records, inductive data types, bit-vectors, strings, and equality over uninterpreted function symbols C++ 2010 version 1.4 released July 2014
Decision Procedure Toolkit (DPT) Linux Apache No OCaml no DPLL-based
iSAT Linux Proprietary No non-linear arithmetic no DPLL-based
MathSAT Linux Proprietary Yes Yes empty theory, linear arithmetic, bitvectors, arrays C/C++, Python, Java 2010 DPLL-based
MiniSmt Linux LGPL partial v2.0 non-linear arithmetic 2010 SAT-solver based, Yices-based
OpenCog Linux AGPL No No No probabilistic logic, arithmetic. relational models C++, Scheme, Python no subgraph isomorphism
OpenSMT Linux, Mac OS, Windows GPLv3 partial v2.0 Yes empty theory, differences, linear arithmetic, bitvectors C++ 2011 lazy SMT Solver
SatEEn ? Proprietary v1.2 linear arithmetic, difference logic none 2009
SMTInterpol Linux, Mac OS, Windows LGPLv3 v2.0 uninterpreted functions, linear real arithmetic, and linear integer arithmetic Java 2012 Focuses on generating high quality, compact interpolants.
SMCHR Linux, Mac OS, Windows GPLv3 No No No linear arithmetic, nonlinear arithmetic, heaps C no Can implement new theories using Constraint Handling Rules.
SMT-RAT Linux, Mac OS GPLv3 v2.0 No No linear arithmetic, nonlinear arithmetic C++ no Toolbox offering theory solver modules for the development of SMT solvers for nonlinear real arithmetic (NRA). Example embedding in OpenSMT available.
SONOLAR Linux, Windows Proprietary partial v2.0 bitvectors C 2010 SAT-solver based
Spear Linux, Mac OS, Windows Proprietary v1.2 bitvectors 2008
STP Linux, OpenBSD, Windows, Mac OS MIT partial v2.0 Yes No bitvectors, arrays C, C++, Python, OCaml, Java 2011 SAT-solver based
SWORD Linux Proprietary v1.2 bitvectors 2009
UCLID Linux BSD No No No empty theory, linear arithmetic, bitvectors, and constrained lambda (arrays, memories, cache, etc.) no SAT-solver based, written in Moscow ML. Input language is SMV model checker. Well-documented!
veriT Linux, OS X BSD partial v2.0 empty theory, rational and integer linear arithmetics, quantifiers, and equality over uninterpreted function symbols C/C++ 2010 SAT-solver based
Yices Linux, Mac OS, Windows Proprietary v2.0 No Yes rational and integer linear arithmetic, bit-vectors, arrays, and equality over uninterpreted function symbols C 2014 Source code is available online
Z3 Linux, Mac OS, Windows, FreeBSD MIT v2.0 Yes empty theory, linear arithmetic, nonlinear arithmetic, bitvectors, arrays, datatypes, quantifiers C/C++, .NET, OCaml, Python, Java 2011
## Applications
SMT solvers are useful both for verification, proving the correctness of programs, software testing based on symbolic execution, and for synthesis, generating program fragments by searching over the space of possible programs.
### Verification
Computer-aided verification of software programs often uses SMT solvers. A common technique is to translate preconditions, postconditions, loop conditions, and assertions into SMT formulas in order to determine if all properties can hold.
There are many verifiers built on top of the Z3 SMT solver. Boogie is an intermediate verification language that uses Z3 to automatically check simple imperative programs. The [3] verifier for concurrent C uses Boogie, as well as Dafny for imperative object-based programs, Chalice for concurrent programs, and Spec# for C#. F* is a dependently typed language that uses Z3 to find proofs; the compiler carries these proofs through to produce proof-carrying bytecode.
There are also many verifiers built on top of the Alt-Ergo SMT solver. Here is a list of mature applications:
• Why3, a platform for deductive program verification, uses Alt-Ergo as its main prover;
• CAVEAT, a C-verifier developed by CEA and used by Airbus; Alt-Ergo was included in the qualification DO-178C of one of its recent aircraft;
• Frama-C, a framework to analyse C-code, uses Alt-Ergo in the Jessie and WP plugins (dedicated to "deductive program verification");
• SPARK, uses Alt-Ergo (behind GNATprove) to automate the verification of some assertions in Spark 2014;
• Atelier-B can use Alt-Ergo instead of its main prover (increasing success from 84% to 98% on the ANR Bware project benchmarks);
• Rodin, a B-method framework developed by Systerel, can use Alt-Ergo as a back-end;
• Cubicle, an open source model checker for verifying safety properties of array-based transtion systems.
• EasyCrypt, a toolset for reasoning about relational properties of probabilistic computations with adversarial code.
Many SMT solvers implement a common interface format called SMTLIB2. The LiquidHaskell tool implements a refinement type based verifier for Haskell that can use any SMTLIB2 compliant solver, e.g. CVC4, MathSat, or Z3.
### Symbolic-execution based analysis and testing
An important class of applications for SMT solvers is symbolic-execution based analysis and testing of programs (e.g., concolic testing), aimed particularly at finding security vulnerabilities. Important actively-maintained tools in this category include SAGE from Microsoft Research, KLEE, and S2E. SMT solvers that are particularly useful for symbolic-execution applications include Z3, STP, Z3str2, and Boolector.
## Notes
1. ^ Nieuwenhuis, R.; Oliveras, A.; Tinelli, C. (2006), "Solving SAT and SAT Modulo Theories: From an Abstract Davis-Putnam-Logemann-Loveland Procedure to DPLL(T)", Journal of the ACM (PDF) 53 (6), pp. 937–977.
2. ^ Bauer, A.; Pister, M.; Tautschnig, M. (2007), "Tool-support for the analysis of hybrid systems and models", Proceedings of the 2007 Conference on Design, Automation and Test in Europe (DATE'07), IEEE Computer Society, p. 1, doi:10.1109/DATE.2007.364411
3. ^ Fränzle, M.; Herde, C.; Ratschan, S.; Schubert, T.; Teige, T. (2007), "Efficient Solving of Large Non-linear Arithmetic Constraint Systems with Complex Boolean Structure", JSAT Special Issue on SAT/CP Integration (PDF) 1, pp. 209–236 | 3,340 | 14,621 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.3125 | 3 | CC-MAIN-2015-18 | longest | en | 0.903771 |
https://www.gamedev.net/forums/topic/487932-index-buffer-in-opengl/ | 1,513,300,745,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948551501.53/warc/CC-MAIN-20171215001700-20171215021700-00379.warc.gz | 707,518,370 | 35,733 | OpenGL Index buffer in OpenGL
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Hello, How is an index buffer in OpenGL organized? I understand, how vertex arrays are passed to OpenGL, but I don't understand how to explain OpenGL, which vertex corresponds to which triangle. For example, I need to pass a square. How could this be done, using index buffers? Thank you in advance, Ilya.
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An index buffer is a list of numbers. Each of this numbers is called an index.
If your are drawing triangles the first 3 indices in your IB are for the first triangle,
the indices 4 to 6 are for the second triangle and so on.
So an index buffer which looks like this:
0 1 2 | 1 2 3
would use the 0th, 1st and 2nd vertex from your vertex buffer for the first triangle
and the 1st, 2nd, 3rd vertex for the second triangle.
HtH
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The easiest way to understand indexed rendering is to draw a picture of a square and number the vertices:
3---------2| / || / || / |0---------1
Put the vertices into the vertex buffer in the same order they are numbered in the picture. Then, just use the index buffer to select those vertices and draw them:
0 1 2 3
GL_TRIANGLE_LIST:
0 1 2 0 2 3
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With a single thread, everything work fine. The problem arise when im starting the thread for a second time, nothing get rendered, but the encoding work fine. For example, if i start the thread with 3 files to process, all of them render fine, but if i start the thread again (with the same batch of files or not...), OpenGL fail to render anything.
Im pretty sure it has something to do with the rendering context (or maybe the window DC?). Here a snippet of my OpenGL class:
bool CGLEngine::Initialize(HWND hWnd) { hDC = GetDC(hWnd); if(!SetupPixelFormatDescriptor(hDC)){ ReleaseDC(hWnd, hDC); return false; } hRC = wglCreateContext(hDC); wglMakeCurrent(hDC, hRC); // more code ... return true; } void CGLEngine::Shutdown() { // some code... if(hRC){wglDeleteContext(hRC);} if(hDC){ReleaseDC(hWnd, hDC);} hDC = hRC = NULL; }
The full source code is available here. The most relevant files are:
Thx in advance if anyone can help me.
• This article uses material originally posted on Diligent Graphics web site.
Introduction
Graphics APIs have come a long way from small set of basic commands allowing limited control of configurable stages of early 3D accelerators to very low-level programming interfaces exposing almost every aspect of the underlying graphics hardware. Next-generation APIs, Direct3D12 by Microsoft and Vulkan by Khronos are relatively new and have only started getting widespread adoption and support from hardware vendors, while Direct3D11 and OpenGL are still considered industry standard. New APIs can provide substantial performance and functional improvements, but may not be supported by older hardware. An application targeting wide range of platforms needs to support Direct3D11 and OpenGL. New APIs will not give any advantage when used with old paradigms. It is totally possible to add Direct3D12 support to an existing renderer by implementing Direct3D11 interface through Direct3D12, but this will give zero benefits. Instead, new approaches and rendering architectures that leverage flexibility provided by the next-generation APIs are expected to be developed.
There are at least four APIs (Direct3D11, Direct3D12, OpenGL/GLES, Vulkan, plus Apple's Metal for iOS and osX platforms) that a cross-platform 3D application may need to support. Writing separate code paths for all APIs is clearly not an option for any real-world application and the need for a cross-platform graphics abstraction layer is evident. The following is the list of requirements that I believe such layer needs to satisfy:
Lightweight abstractions: the API should be as close to the underlying native APIs as possible to allow an application leverage all available low-level functionality. In many cases this requirement is difficult to achieve because specific features exposed by different APIs may vary considerably. Low performance overhead: the abstraction layer needs to be efficient from performance point of view. If it introduces considerable amount of overhead, there is no point in using it. Convenience: the API needs to be convenient to use. It needs to assist developers in achieving their goals not limiting their control of the graphics hardware. Multithreading: ability to efficiently parallelize work is in the core of Direct3D12 and Vulkan and one of the main selling points of the new APIs. Support for multithreading in a cross-platform layer is a must. Extensibility: no matter how well the API is designed, it still introduces some level of abstraction. In some cases the most efficient way to implement certain functionality is to directly use native API. The abstraction layer needs to provide seamless interoperability with the underlying native APIs to provide a way for the app to add features that may be missing. Diligent Engine is designed to solve these problems. Its main goal is to take advantages of the next-generation APIs such as Direct3D12 and Vulkan, but at the same time provide support for older platforms via Direct3D11, OpenGL and OpenGLES. Diligent Engine exposes common C++ front-end for all supported platforms and provides interoperability with underlying native APIs. It also supports integration with Unity and is designed to be used as graphics subsystem in a standalone game engine, Unity native plugin or any other 3D application. Full source code is available for download at GitHub and is free to use.
Overview
Diligent Engine API takes some features from Direct3D11 and Direct3D12 as well as introduces new concepts to hide certain platform-specific details and make the system easy to use. It contains the following main components:
Render device (IRenderDevice interface) is responsible for creating all other objects (textures, buffers, shaders, pipeline states, etc.).
Device context (IDeviceContext interface) is the main interface for recording rendering commands. Similar to Direct3D11, there are immediate context and deferred contexts (which in Direct3D11 implementation map directly to the corresponding context types). Immediate context combines command queue and command list recording functionality. It records commands and submits the command list for execution when it contains sufficient number of commands. Deferred contexts are designed to only record command lists that can be submitted for execution through the immediate context.
An alternative way to design the API would be to expose command queue and command lists directly. This approach however does not map well to Direct3D11 and OpenGL. Besides, some functionality (such as dynamic descriptor allocation) can be much more efficiently implemented when it is known that a command list is recorded by a certain deferred context from some thread.
The approach taken in the engine does not limit scalability as the application is expected to create one deferred context per thread, and internally every deferred context records a command list in lock-free fashion. At the same time this approach maps well to older APIs.
In current implementation, only one immediate context that uses default graphics command queue is created. To support multiple GPUs or multiple command queue types (compute, copy, etc.), it is natural to have one immediate contexts per queue. Cross-context synchronization utilities will be necessary.
Swap Chain (ISwapChain interface). Swap chain interface represents a chain of back buffers and is responsible for showing the final rendered image on the screen.
Render device, device contexts and swap chain are created during the engine initialization.
Resources (ITexture and IBuffer interfaces). There are two types of resources - textures and buffers. There are many different texture types (2D textures, 3D textures, texture array, cubmepas, etc.) that can all be represented by ITexture interface.
Resources Views (ITextureView and IBufferView interfaces). While textures and buffers are mere data containers, texture views and buffer views describe how the data should be interpreted. For instance, a 2D texture can be used as a render target for rendering commands or as a shader resource.
Pipeline State (IPipelineState interface). GPU pipeline contains many configurable stages (depth-stencil, rasterizer and blend states, different shader stage, etc.). Direct3D11 uses coarse-grain objects to set all stage parameters at once (for instance, a rasterizer object encompasses all rasterizer attributes), while OpenGL contains myriad functions to fine-grain control every individual attribute of every stage. Both methods do not map very well to modern graphics hardware that combines all states into one monolithic state under the hood. Direct3D12 directly exposes pipeline state object in the API, and Diligent Engine uses the same approach.
Shader Resource Binding (IShaderResourceBinding interface). Shaders are programs that run on the GPU. Shaders may access various resources (textures and buffers), and setting correspondence between shader variables and actual resources is called resource binding. Resource binding implementation varies considerably between different API. Diligent Engine introduces a new object called shader resource binding that encompasses all resources needed by all shaders in a certain pipeline state.
API Basics
Creating Resources
Device resources are created by the render device. The two main resource types are buffers, which represent linear memory, and textures, which use memory layouts optimized for fast filtering. Graphics APIs usually have a native object that represents linear buffer. Diligent Engine uses IBuffer interface as an abstraction for a native buffer. To create a buffer, one needs to populate BufferDesc structure and call IRenderDevice::CreateBuffer() method as in the following example:
BufferDesc BuffDesc; BufferDesc.Name = "Uniform buffer"; BuffDesc.BindFlags = BIND_UNIFORM_BUFFER; BuffDesc.Usage = USAGE_DYNAMIC; BuffDesc.uiSizeInBytes = sizeof(ShaderConstants); BuffDesc.CPUAccessFlags = CPU_ACCESS_WRITE; m_pDevice->CreateBuffer( BuffDesc, BufferData(), &m_pConstantBuffer ); While there is usually just one buffer object, different APIs use very different approaches to represent textures. For instance, in Direct3D11, there are ID3D11Texture1D, ID3D11Texture2D, and ID3D11Texture3D objects. In OpenGL, there is individual object for every texture dimension (1D, 2D, 3D, Cube), which may be a texture array, which may also be multisampled (i.e. GL_TEXTURE_2D_MULTISAMPLE_ARRAY). As a result there are nine different GL texture types that Diligent Engine may create under the hood. In Direct3D12, there is only one resource interface. Diligent Engine hides all these details in ITexture interface. There is only one IRenderDevice::CreateTexture() method that is capable of creating all texture types. Dimension, format, array size and all other parameters are specified by the members of the TextureDesc structure:
TextureDesc TexDesc; TexDesc.Name = "My texture 2D"; TexDesc.Type = TEXTURE_TYPE_2D; TexDesc.Width = 1024; TexDesc.Height = 1024; TexDesc.Format = TEX_FORMAT_RGBA8_UNORM; TexDesc.Usage = USAGE_DEFAULT; TexDesc.BindFlags = BIND_SHADER_RESOURCE | BIND_RENDER_TARGET | BIND_UNORDERED_ACCESS; TexDesc.Name = "Sample 2D Texture"; m_pRenderDevice->CreateTexture( TexDesc, TextureData(), &m_pTestTex ); If native API supports multithreaded resource creation, textures and buffers can be created by multiple threads simultaneously.
Interoperability with native API provides access to the native buffer/texture objects and also allows creating Diligent Engine objects from native handles. It allows applications seamlessly integrate native API-specific code with Diligent Engine.
Next-generation APIs allow fine level-control over how resources are allocated. Diligent Engine does not currently expose this functionality, but it can be added by implementing IResourceAllocator interface that encapsulates specifics of resource allocation and providing this interface to CreateBuffer() or CreateTexture() methods. If null is provided, default allocator should be used.
Initializing the Pipeline State
As it was mentioned earlier, Diligent Engine follows next-gen APIs to configure the graphics/compute pipeline. One big Pipelines State Object (PSO) encompasses all required states (all shader stages, input layout description, depth stencil, rasterizer and blend state descriptions etc.). This approach maps directly to Direct3D12/Vulkan, but is also beneficial for older APIs as it eliminates pipeline misconfiguration errors. With many individual calls tweaking various GPU pipeline settings it is very easy to forget to set one of the states or assume the stage is already properly configured when in fact it is not. Using pipeline state object helps avoid these problems as all stages are configured at once.
While in earlier APIs shaders were bound separately, in the next-generation APIs as well as in Diligent Engine shaders are part of the pipeline state object. The biggest challenge when authoring shaders is that Direct3D and OpenGL/Vulkan use different shader languages (while Apple uses yet another language in their Metal API). Maintaining two versions of every shader is not an option for real applications and Diligent Engine implements shader source code converter that allows shaders authored in HLSL to be translated to GLSL. To create a shader, one needs to populate ShaderCreationAttribs structure. SourceLanguage member of this structure tells the system which language the shader is authored in:
When sampling a texture in a shader, the texture sampler was traditionally specified as separate object that was bound to the pipeline at run time or set as part of the texture object itself. However, in most cases it is known beforehand what kind of sampler will be used in the shader. Next-generation APIs expose new type of sampler called static sampler that can be initialized directly in the pipeline state. Diligent Engine exposes this functionality: when creating a shader, textures can be assigned static samplers. If static sampler is assigned, it will always be used instead of the one initialized in the texture shader resource view. To initialize static samplers, prepare an array of StaticSamplerDesc structures and initialize StaticSamplers and NumStaticSamplers members. Static samplers are more efficient and it is highly recommended to use them whenever possible. On older APIs, static samplers are emulated via generic sampler objects.
The following is an example of shader initialization:
Creating the Pipeline State Object
After all required shaders are created, the rest of the fields of the PipelineStateDesc structure provide depth-stencil, rasterizer, and blend state descriptions, the number and format of render targets, input layout format, etc. For instance, rasterizer state can be described as follows:
PipelineStateDesc PSODesc; RasterizerStateDesc &RasterizerDesc = PSODesc.GraphicsPipeline.RasterizerDesc; RasterizerDesc.FillMode = FILL_MODE_SOLID; RasterizerDesc.CullMode = CULL_MODE_NONE; RasterizerDesc.FrontCounterClockwise = True; RasterizerDesc.ScissorEnable = True; RasterizerDesc.AntialiasedLineEnable = False; Depth-stencil and blend states are defined in a similar fashion.
Another important thing that pipeline state object encompasses is the input layout description that defines how inputs to the vertex shader, which is the very first shader stage, should be read from the memory. Input layout may define several vertex streams that contain values of different formats and sizes:
// Define input layout InputLayoutDesc &Layout = PSODesc.GraphicsPipeline.InputLayout; LayoutElement TextLayoutElems[] = { LayoutElement( 0, 0, 3, VT_FLOAT32, False ), LayoutElement( 1, 0, 4, VT_UINT8, True ), LayoutElement( 2, 0, 2, VT_FLOAT32, False ), }; Layout.LayoutElements = TextLayoutElems; Layout.NumElements = _countof( TextLayoutElems ); Finally, pipeline state defines primitive topology type. When all required members are initialized, a pipeline state object can be created by IRenderDevice::CreatePipelineState() method:
// Define shader and primitive topology PSODesc.GraphicsPipeline.PrimitiveTopologyType = PRIMITIVE_TOPOLOGY_TYPE_TRIANGLE; PSODesc.GraphicsPipeline.pVS = pVertexShader; PSODesc.GraphicsPipeline.pPS = pPixelShader; PSODesc.Name = "My pipeline state"; m_pDev->CreatePipelineState(PSODesc, &m_pPSO); When PSO object is bound to the pipeline, the engine invokes all API-specific commands to set all states specified by the object. In case of Direct3D12 this maps directly to setting the D3D12 PSO object. In case of Direct3D11, this involves setting individual state objects (such as rasterizer and blend states), shaders, input layout etc. In case of OpenGL, this requires a number of fine-grain state tweaking calls. Diligent Engine keeps track of currently bound states and only calls functions to update these states that have actually changed.
Direct3D11 and OpenGL utilize fine-grain resource binding models, where an application binds individual buffers and textures to certain shader or program resource binding slots. Direct3D12 uses a very different approach, where resource descriptors are grouped into tables, and an application can bind all resources in the table at once by setting the table in the command list. Resource binding model in Diligent Engine is designed to leverage this new method. It introduces a new object called shader resource binding that encapsulates all resource bindings required for all shaders in a certain pipeline state. It also introduces the classification of shader variables based on the frequency of expected change that helps the engine group them into tables under the hood:
Static variables (SHADER_VARIABLE_TYPE_STATIC) are variables that are expected to be set only once. They may not be changed once a resource is bound to the variable. Such variables are intended to hold global constants such as camera attributes or global light attributes constant buffers. Mutable variables (SHADER_VARIABLE_TYPE_MUTABLE) define resources that are expected to change on a per-material frequency. Examples may include diffuse textures, normal maps etc. Dynamic variables (SHADER_VARIABLE_TYPE_DYNAMIC) are expected to change frequently and randomly. Shader variable type must be specified during shader creation by populating an array of ShaderVariableDesc structures and initializing ShaderCreationAttribs::Desc::VariableDesc and ShaderCreationAttribs::Desc::NumVariables members (see example of shader creation above).
Static variables cannot be changed once a resource is bound to the variable. They are bound directly to the shader object. For instance, a shadow map texture is not expected to change after it is created, so it can be bound directly to the shader:
m_pPSO->CreateShaderResourceBinding(&m_pSRB); Note that an SRB is only compatible with the pipeline state it was created from. SRB object inherits all static bindings from shaders in the pipeline, but is not allowed to change them.
Mutable resources can only be set once for every instance of a shader resource binding. Such resources are intended to define specific material properties. For instance, a diffuse texture for a specific material is not expected to change once the material is defined and can be set right after the SRB object has been created:
m_pSRB->GetVariable(SHADER_TYPE_PIXEL, "tex2DDiffuse")->Set(pDiffuseTexSRV); In some cases it is necessary to bind a new resource to a variable every time a draw command is invoked. Such variables should be labeled as dynamic, which will allow setting them multiple times through the same SRB object:
m_pSRB->GetVariable(SHADER_TYPE_VERTEX, "cbRandomAttribs")->Set(pRandomAttrsCB); Under the hood, the engine pre-allocates descriptor tables for static and mutable resources when an SRB objcet is created. Space for dynamic resources is dynamically allocated at run time. Static and mutable resources are thus more efficient and should be used whenever possible.
As you can see, Diligent Engine does not expose low-level details of how resources are bound to shader variables. One reason for this is that these details are very different for various APIs. The other reason is that using low-level binding methods is extremely error-prone: it is very easy to forget to bind some resource, or bind incorrect resource such as bind a buffer to the variable that is in fact a texture, especially during shader development when everything changes fast. Diligent Engine instead relies on shader reflection system to automatically query the list of all shader variables. Grouping variables based on three types mentioned above allows the engine to create optimized layout and take heavy lifting of matching resources to API-specific resource location, register or descriptor in the table.
This post gives more details about the resource binding model in Diligent Engine.
Setting the Pipeline State and Committing Shader Resources
Before any draw or compute command can be invoked, the pipeline state needs to be bound to the context:
m_pContext->SetPipelineState(m_pPSO); Under the hood, the engine sets the internal PSO object in the command list or calls all the required native API functions to properly configure all pipeline stages.
The next step is to bind all required shader resources to the GPU pipeline, which is accomplished by IDeviceContext::CommitShaderResources() method:
m_pContext->CommitShaderResources(m_pSRB, COMMIT_SHADER_RESOURCES_FLAG_TRANSITION_RESOURCES); The method takes a pointer to the shader resource binding object and makes all resources the object holds available for the shaders. In the case of D3D12, this only requires setting appropriate descriptor tables in the command list. For older APIs, this typically requires setting all resources individually.
Next-generation APIs require the application to track the state of every resource and explicitly inform the system about all state transitions. For instance, if a texture was used as render target before, while the next draw command is going to use it as shader resource, a transition barrier needs to be executed. Diligent Engine does the heavy lifting of state tracking. When CommitShaderResources() method is called with COMMIT_SHADER_RESOURCES_FLAG_TRANSITION_RESOURCES flag, the engine commits and transitions resources to correct states at the same time. Note that transitioning resources does introduce some overhead. The engine tracks state of every resource and it will not issue the barrier if the state is already correct. But checking resource state is an overhead that can sometimes be avoided. The engine provides IDeviceContext::TransitionShaderResources() method that only transitions resources:
m_pContext->TransitionShaderResources(m_pPSO, m_pSRB); In some scenarios it is more efficient to transition resources once and then only commit them.
Invoking Draw Command
The final step is to set states that are not part of the PSO, such as render targets, vertex and index buffers. Diligent Engine uses Direct3D11-syle API that is translated to other native API calls under the hood:
ITextureView *pRTVs[] = {m_pRTV}; m_pContext->SetRenderTargets(_countof( pRTVs ), pRTVs, m_pDSV); // Clear render target and depth buffer const float zero[4] = {0, 0, 0, 0}; m_pContext->ClearRenderTarget(nullptr, zero); m_pContext->ClearDepthStencil(nullptr, CLEAR_DEPTH_FLAG, 1.f); // Set vertex and index buffers IBuffer *buffer[] = {m_pVertexBuffer}; Uint32 offsets[] = {0}; Uint32 strides[] = {sizeof(MyVertex)}; m_pContext->SetVertexBuffers(0, 1, buffer, strides, offsets, SET_VERTEX_BUFFERS_FLAG_RESET); m_pContext->SetIndexBuffer(m_pIndexBuffer, 0); Different native APIs use various set of function to execute draw commands depending on command details (if the command is indexed, instanced or both, what offsets in the source buffers are used etc.). For instance, there are 5 draw commands in Direct3D11 and more than 9 commands in OpenGL with something like glDrawElementsInstancedBaseVertexBaseInstance not uncommon. Diligent Engine hides all details with single IDeviceContext::Draw() method that takes takes DrawAttribs structure as an argument. The structure members define all attributes required to perform the command (primitive topology, number of vertices or indices, if draw call is indexed or not, if draw call is instanced or not, if draw call is indirect or not, etc.). For example:
DrawAttribs attrs; attrs.IsIndexed = true; attrs.IndexType = VT_UINT16; attrs.NumIndices = 36; attrs.Topology = PRIMITIVE_TOPOLOGY_TRIANGLE_LIST; pContext->Draw(attrs); For compute commands, there is IDeviceContext::DispatchCompute() method that takes DispatchComputeAttribs structure that defines compute grid dimension.
Source Code
Full engine source code is available on GitHub and is free to use. The repository contains two samples, asteroids performance benchmark and example Unity project that uses Diligent Engine in native plugin.
AntTweakBar sample is Diligent Engine’s “Hello World” example.
Atmospheric scattering sample is a more advanced example. It demonstrates how Diligent Engine can be used to implement various rendering tasks: loading textures from files, using complex shaders, rendering to multiple render targets, using compute shaders and unordered access views, etc.
Asteroids performance benchmark is based on this demo developed by Intel. It renders 50,000 unique textured asteroids and allows comparing performance of Direct3D11 and Direct3D12 implementations. Every asteroid is a combination of one of 1000 unique meshes and one of 10 unique textures.
Finally, there is an example project that shows how Diligent Engine can be integrated with Unity.
Future Work
The engine is under active development. It currently supports Windows desktop, Universal Windows and Android platforms. Direct3D11, Direct3D12, OpenGL/GLES backends are now feature complete. Vulkan backend is coming next, and support for more platforms is planned.
• I've started building a small library, that can render pie menu GUI in legacy opengl, planning to add some traditional elements of course.
It's interface is similar to something you'd see in IMGUI. It's written in C.
Early version of the library
I'd really love to hear anyone's thoughts on this, any suggestions on what features you'd want to see in a library like this? | 5,898 | 28,392 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2017-51 | longest | en | 0.910258 |
http://mymathforum.com/algebra/25067-power-series.html | 1,566,786,576,000,000,000 | text/html | crawl-data/CC-MAIN-2019-35/segments/1566027330962.67/warc/CC-MAIN-20190826022215-20190826044215-00196.warc.gz | 126,229,766 | 8,690 | My Math Forum power series
Algebra Pre-Algebra and Basic Algebra Math Forum
February 21st, 2012, 03:50 AM #1 Newbie Joined: Feb 2012 Posts: 17 Thanks: 0 power series show that e^i?=cos???+i sin?? ?, using the power series for e^x, cos?x and sin?x
February 21st, 2012, 04:07 AM #2 Global Moderator Joined: Dec 2006 Posts: 20,942 Thanks: 2210 I think you intended Euler's formula: $e^{i\theta}\,=\,\cos\theta\,+\,i\sin\theta.$
February 21st, 2012, 07:19 AM #3 Senior Member Joined: Jul 2011 Posts: 227 Thanks: 0 Re: power series We have to prove: $e^{ix}=\cos(x)+i\sin(x)$ Using the Maclaurin series: $\cos(x)=\sum_{k=0}^{+\infty} \frac{(-1)^k x^{2k}}{(2k)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots$ $\sin(x)=\sum_{k=0}^{+\infty} \frac{(-1)^k x^{2k+1}}{(2k+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\ldots$ $\cos(x)+i\sin(x)= \left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots\right)+i\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\ldots\right)$ $=1+ix-\frac{x^2}{2!}-\frac{ix^3}{3!}+\frac{x^4}{4!}+\frac{ix^5}{5!}-\frac{x^6}{6!}-\frac{ix^7}{7!}+\ldots$ $=\sum_{k=0}^{+\infty}\frac{(ix)^k}{k!}=e^{ix}$
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Contact - Home - Forums - Cryptocurrency Forum - Top | 659 | 1,577 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 7, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2019-35 | latest | en | 0.62629 |
https://www.lespressobarmercurio.com/tasty-food/how-long-is-a-standard-size-hotdog.html | 1,669,919,335,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710829.5/warc/CC-MAIN-20221201153700-20221201183700-00528.warc.gz | 923,778,747 | 19,112 | # How Long Is A Standard Size Hotdog?
Not only do the manner in which they are cooked and the additions that are made to them vary, but also their dimensions fluctuate. The length of a regular hot dog is around 6 inches (15 cm), whereas the length of a ‘footlong’ hot dog is approximately 12 inches (30 cm). The following are just a few of the numerous and popular ways that hot dogs may be served around the country:
There is a wide range of sizes for hot dogs, from cocktail wieners, which are approximately 2 inches long, all the way up to the legendary foot-long hot dogs that are popular at sporting events. The normal length of a hot dog, which is six inches, is the most common and is typically supplied in packages of ten.
## What size hot dogs should you serve at a restaurant?
How big should the hot dogs be that you serve? Vendors have discussed this topic for years, and as time has progressed, there has been a small trend away from a ″regular″ hot dog and toward the 1/4-pound hot dog. 8:1 ″Normal″ Hot Dogs When you go to the supermarket near you, you’ll see that the 8:1 portion size is by far the most prevalent.
## How much do 70 hot dogs weigh?
How much do 70 hot dogs weigh? Because there are eight hot dogs in a pack of one pound of hot dogs, the weight of a single hot dog is two ounces. Which indicates that 70 equals 140 ounces. If we divide 140 by the ratio of 16 ounces to a pound, we get a total weight of 8.75 pounds for your hot dogs.
## How many hot dogs are in a pack?
• The packages of standard national brands typically contain eight hot dogs and eight buns in total.
• Although larger packs of dogs (ten to twelve) sold at a discount and smaller packs of premium dogs (four to six) are available, the standard number of dogs sold in a pack is eight.
• How many kilograms do seventy hot dogs have?
See also: How Do I Place A Hotdog Stand In Front Of Grocery Store?
Because there are eight hot dogs in a pack of one pound of hot dogs, the weight of a single hot dog is two ounces.
## What temperature do hot dogs need to be cooked to?
Despite the fact that hot dogs are cooked during the manufacturing process, it is strongly advised that they be re-heated to a temperature of at least 165 degrees Fahrenheit (75 degrees Celsius) before being consumed.
## What is the standard length of a hot dog?
Regarding the first issue, the results of my comprehensive internet study revealed that the length of a standard hot dog is 4.8 inches, but the length of a bun-length dog is 6 inches. It would appear that some individuals prefer shorter hot dogs since this provides more area for ″spill-over″ condiments on the bun.
## What is the length of a hot dog bun?
• Roll the balls into cylinders that are 4 and a half feet in length to make the hot dog buns.
• Reduce the height of the cylinders by making them somewhat flatter; as the dough rises more in the middle, this will result in a top that is softly rounded rather than lofty.
• When making buns with soft sides, set them a half inch apart on a baking sheet that has been seasoned well so that they may come closer to one another as they rise.
## What is the average width of a hot dog?
Detailed scientific measurements have revealed that the diameter of a normal hot dog is between between one and two and a half centimeters. This gives us a radius for the hot dog of around 1.25 centimeters. (This value is 2.726 centimeters in diameter according to precise measurements of hot dogs; nevertheless, the diameter of your real hot dog may be different.)
## How many inches is an Oscar Mayer hot dog?
Kraft Frozen Oscar Mayer Hot and Spicy Hot Dog, 6 inch — 1 each.
## How big is Costco hotdog?
The availability of quick meal alternatives helps justify the expense of an annual membership, especially considering that customers can buy more food for \$1.50 than they ever could before. The traditional pairing of a hot dog with a can of soda consisted of a can of soda with a capacity of 12 ounces and a quarter-pound hot dog. As of right now, it includes a 20 oz.
## How big is a Chicago dog?
The average beef hot dog has a weight of 1/8 of a pound, which is equal to 2 ounces (57 grams). The most traditional variety of hot dog has a natural casing, which produces a distinct ″snap″ as it is bit into. The buns are of a high-gluten kind that was specifically developed to withstand steam warming, as is frequently done in the S.
## Are bun size hot dogs bigger?
It came to my attention that the Bun Size Beef Hot Dogs are longer but really thinner than the Ball Park Beef Hot Dogs, which leads me to believe that the number of each may not change all that significantly from one another.
## Why are hot dogs so small?
Since the average weight of a hot dog is around 1.6 ounces, the number 10 was chosen as the de facto standard to equal one pound. This type of production on a broad scale is the reason why the majority of us purchase hot dogs and hot dog buns in grocery stores nowadays. In addition, the hot dogs are typically offered in packages of ten. There is a direct correlation between the two.
## Why are there 10 hotdogs and 8 buns?
″Sandwich rolls, also known as hot dog buns, often come in packages of eight since the buns are baked in clusters of four in pans that have the capacity to contain eight rolls,″ says one source. ″Despite the fact that baking pans are now available in configurations that allow baking 10 or even 12 at a time, the eight-roll pan is still the most popular.″
## What is a 6 1 hotdog?
Hot dog made from beef that is completely cooked and skinless, measuring 6.0 inches in length and seasoned to provide the classic taste sensation. Cooking times may vary. Unless otherwise specified, all timeframes are calculated based on a fully thawed product. A serving and holding temperature of at least 140 degrees Fahrenheit is recommended.
## What sides go with hot dogs?
1. Here are some of our favorite easy sides to serve with hot dogs, ranging from traditional coleslaw to fantastic corn on the cob that has been grilled. grilled corn on the cob served with a mayonnaise flavored with calamansi
2. Smoky Coleslaw.
3. Pasta salad topped with feta cheese, parsley, and grilled vegetables
5. The famous Gina Mae’s Baked Beans | 1,426 | 6,273 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.171875 | 3 | CC-MAIN-2022-49 | longest | en | 0.95463 |
https://testbook.com/question-answer/the-base-and-height-of-a-triangle-are-in-the-ratio--602a712a6f7c8796fc838b3d | 1,638,241,007,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964358903.73/warc/CC-MAIN-20211130015517-20211130045517-00299.warc.gz | 648,508,784 | 31,901 | # The base and height of a triangle are in the ratio 3 : 4. If the area is 150 cm2, its height in cm is:
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1. 20
2. 5
3. 15
4. 12
Option 1 : 20
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## Detailed Solution
Given:
Ratio of base and height of the triangle is 3 : 4 and area of the triangle is 150 cm2
Concept Used:
If base of a triangle is 'b' and height be 'h' then area of the triangle is (1/2) × b × h
Calculation:
Ratio of base and height of the triangle is 3 : 4
Let, base be 3x and height be 4x
Accordingly,
(1/2) × 3x × 4x = 150
⇒ 6x2 = 150
⇒ x2 = 25
⇒ x = 5
Height of the triangle is 4x = 4 × 5 = 20 cm
∴ The height of the triangle in cm is 20 cm. | 305 | 830 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.28125 | 4 | CC-MAIN-2021-49 | latest | en | 0.828459 |
https://nicefreedatingall.com/what-is-the-meaning-of-slope.php | 1,624,349,432,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623488512243.88/warc/CC-MAIN-20210622063335-20210622093335-00484.warc.gz | 364,868,613 | 10,934 | ## What is the meaning of slope
verb (used without object), sloped, slop·ing. to have or take an inclined or oblique direction or angle considered with reference to a vertical or horizontal plane; slant. to move at an inclination or obliquely: They sloped gradually westward. Keeping this fact in mind, by definition, the slope is the measure of the steepness of a line. In real life, we see slope in any direction. However, in math, slope is defined as you move from left to right. I repeat we always measure slope going from left to right. This is very important! There are four types of slope you can encounter. A slope can be.
In real life, we see slope in any direction. However, in math, slope is defined as you meaing from left to right. An undefined slope: A slope is undefined if you neither move to the right nor to the left. Whzt, you are dealing ghe a vertical line where the possibility is to either slpoe up or dlope. How to find the slope Learn how to compute the slope using thhe rise and the run or slopw points.
Undefined slope A thorough explanation of what it means for a slope to be undefined. Graphing slope Learn how to graph the slope using the slope and a point. Slope intercept form Learn how to find the slope intercept form. Point slope form Learn how to find the point slope form. Slope calculator Given two points, this calculator will calculate the slope and the slope intercept form of whah line.
Midpoint of a line segment Find out how to find the midpoint of a line segment using the midpoint formula. Direct variation. Close Help. Entering your problem is easy to do. Just type! Your problem wyat appear on a Web page exactly the way you enter it here.
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What is slope? Definition and real-life examples. If you have ever walked up or down a hill, then you have already experienced a real life example of slope. As you go uphill, you may feel like you are spending lots of energy to get yourself to move. Homepage Geometry lessons What is slope Meabing to find the slope Learn how to compute the slope using the rise and the run or 2 points. Related topics Slope calculator Given two points, this calculator will calculate the slope and the slope intercept form of a line.
Direct variation A couple of real life examples of direct variation explained. Do you have great problems about slope? Share what is the meaning of slope here with the solutions! Enter Your Title Showcase your problem! Upload Pictures or Graphics optional [? Recent Articles. Check out some of our top basic slops lessons. Close Help Entering your problem is easy to do. Close Help Do you have some pictures or graphics to add?
slope noun [C] (SURFACE) a surface that rises at an angle, esp. a hill or mountain, or the angle at which something rises: Students learn to ski on gentle slopes in a straight line. Snow had . Aug 29, · Definition of Slope The slope of a line is the ratio of the amount that y increases as x increases some amount. Slope tells you how steep a line . Slope is the rise over the run, the change in 'y' over the change in 'x', or the gradient of a line. Check out this tutorial to learn about slope!
Improve your vocabulary with English Vocabulary in Use from Cambridge. Learn the words you need to communicate with confidence. Breaking the ice and throwing caution to the wind Weather idioms, Part 3.
Definitions Clear explanations of natural written and spoken English. Click on the arrows to change the translation direction. Follow us. Choose a dictionary. Clear explanations of natural written and spoken English. Usage explanations of natural written and spoken English. Word Lists. Choose your language. My word lists. Tell us about this example sentence:. The word in the example sentence does not match the entry word.
The sentence contains offensive content. Cancel Submit. Your feedback will be reviewed. B2 a surface that lies at an angle to the horizontal so that some points on it are higher than others :. B2 part of the side of a hill or mountain :. Snow had settled on some of the higher slopes. There's a very steep slope just before you reach the top of the mountain. Slanting, sloping, leaning, tilting.
Want to learn more? The football pitch sloped at the south end, so one half of the game had to be played uphill. Synonyms incline formal.
Related word sloping. Phrasal verb slope off. Students learn to ski on gentle slopes in a straight line. The path slopes down to the house. Examples of slope. Estimated slope coefficients and standard errors which are not reported here can be obtained from the author. From the Cambridge English Corpus. Second, it limits the researcher to investigating only differences in level across the two cohorts, not differences in shape or slope. Figure 4 illustrates the four different patterns resulting from group differences in slope and intercept.
In addition, early depressive symptoms had a significant negative quadratic effect that indicated that the slope was less steep at higher depressive symptom levels. The slope of the regression line passing through these columns reflects the optimal leastsquares trajectory based on an average of their scores on each variable.
Equations 2. Determining the slope of the sequestration isocline requires an additional assumption. Survival across months showed an overall negative slope for the survival of each species with time of exposure in the field figs 3 and 4.
Clinoform geometries shape, slope , height and lateral extent and clinofacies sedimentary structure, original fabric and components can be considered as unaltered. Note the pervasive matrix recrystallization microspar , and the general increase in grain size towards the upper slope. However, the blocks were found in close association with one another on a small patch of slope. Conversely, the outer ramp and upper slope of the clinoform is marked by abundant autochthonous red algae and large benthic foraminifers.
The rapid accumulation of sediments and steep accretionary foresets also induces slope instability and provides a source for sandy debris flows.
In prolongation of axis, but without connecting to it, there is a tiny axial ridge inconspicuously marked on anterior slope of medial posterior border. See all examples of slope. These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. Collocations with slope. Click on a collocation to see more examples of it.
From the Hansard archive. Example from the Hansard archive. Contains Parliamentary information licensed under the Open Parliament Licence v3. See all collocations with slope. Translations of slope in Chinese Traditional. See more. Need a translator? Translator tool. What is the pronunciation of slope? Browse sloop-of-war BETA. Test your vocabulary with our fun image quizzes. Image credits. Word of the Day telescope.
## 4 thoughts on “What is the meaning of slope”
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2. Vuzilkree:
Right now to my favs. I mean the tutorial
3. Dikasa:
Or pta nhi tumhe kya chigg achi nhi lgi
4. Vudorg:
Dusty Luck ooh yes please, recommendations for metal | 1,961 | 9,474 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.796875 | 4 | CC-MAIN-2021-25 | latest | en | 0.931871 |
https://www.teacherspayteachers.com/Product/Discerning-Digits-A-math-probability-Game-1768107 | 1,485,244,138,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560284352.26/warc/CC-MAIN-20170116095124-00535-ip-10-171-10-70.ec2.internal.warc.gz | 975,720,672 | 50,484 | # Discerning Digits - A math probability Game
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### PRODUCT DESCRIPTION
The Discerning Digits Activity is similar to the Discarding Digits Activity that I created. However, it is more challenging because the students have to ensure that digits are placed correctly relative to the other digits in the same row and same column. Consequently, some of my students prefer this game to the Discarding Digits game.
Materials required:
• Student Activity: Discerning Digits (2 pages), 1 copy per student
• A single random number generated to be used by the instructor
Through the choices that they have to make in this activity, students have the opportunity to experiment with different strategies while they observe the outcomes of this probability game. Since each game takes approximately five minutes to complete, students may revise their strategies as they play additional games throughout the class period.
When this activity is completed, an examination of different strategies, and student rationales for using them, can lead to an interesting class discussion.
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0 ratings | 297 | 1,343 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2017-04 | longest | en | 0.904253 |
https://www.coursehero.com/file/210982/Cpsc-440-Problem-Set-2-Question-1/ | 1,498,303,994,000,000,000 | text/html | crawl-data/CC-MAIN-2017-26/segments/1498128320257.16/warc/CC-MAIN-20170624101204-20170624121204-00542.warc.gz | 877,067,188 | 24,414 | Cpsc 440 Problem Set 2 Question 1
# Cpsc 440 Problem Set 2 Question 1 - hw2p1.h ifndef HW2P1_H...
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Unformatted text preview: hw2p1.h 2/25/2008 # ifndef HW2P1_H # define HW2P1_H typedef struct lagrange_polynomial * lagrange_polynomial ; lagrange_polynomial newLagrangePolynomial ( void ) ; void freeLagrangePolynomial ( lagrange_polynomial lp ) ; double evaluateLagrangePolynomial ( lagrange_polynomial lp , double x ) ; void getEquispacedNodes ( double min , double max , int count , double * nodes ) ; void getChebyshevNodes ( double min , double max , int count , double * nodes ) ; double f1 ( double x ) ; double f2 ( double x ) ; # endif 1 hw2p1.c 4/1/2008 # include < math . h > # include < stdio . h > # include < stdlib . h > # include "hw2p1.h" # define PI 3.141592653589793 # define MIN_VECTOR_SIZE 10 # define N_MIN 2 # define N_MAX 40 # define X_MIN- 1 # define X_MAX 1 struct lagrange_polynomial { int num_data ; int capacity ; double * x_values ; double * y_values ; } ; lagrange_polynomial newLagrangePolynomial ( void ) { lagrange_polynomial lp = malloc ( sizeof ( struct lagrange_polynomial ) ) ; lp- > num_data = ; lp- > capacity = MIN_VECTOR_SIZE ; lp- > x_values = malloc ( lp- > capacity * sizeof ( double ) ) ; lp- > y_values = malloc ( lp- > capacity * sizeof ( double ) ) ; return ( lp ) ; } void freeLagrangePolynomial ( lagrange_polynomial lp ) { free ( lp- > x_values ) ; free ( lp- > y_values ) ; free ( lp ) ; } void addDataPoint ( lagrange_polynomial lp , double x_value , double y_value ) { if ( lp- > num_data = = lp- > capacity ) { 1 hw2p1.c 4/1/2008 lp- > capacity * = 2 ; lp- > x_values = realloc ( lp- > x_values , lp- > capacity * sizeof ( double ) ) ; lp- > y_values = realloc ( lp- > y_values , lp- > capacity * sizeof ( double ) ) ; } lp- > x_values [ lp- > num_data ] = x_value ; lp- > y_values [ lp- > num_data ] = y_value ; lp- > num_data + + ; } double evaluateLagrangePolynomial ( lagrange_polynomial lp , double x ) { double lp_result = ; for ( int j = ; j < lp- > num_data ; j + + ) { double cur_term = 1 ; for ( int i = ; i < lp- > num_data ; i + + ) { if ( i ! = j ) { double xi = lp- > x_values [ i ] ; double xj = lp- > x_values [ j ] ; cur_term * = ( x- xi ) / ( xj- xi ) ; } } lp_result + = lp- > y_values [ j ] * cur_term ; } return ( lp_result ) ; } int main ( int argc , char * argv ) { // Extract the command-line...
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## This note was uploaded on 07/19/2008 for the course CPSC 440 taught by Professor Vladimirrokhlin during the Spring '08 term at Yale.
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Cpsc 440 Problem Set 2 Question 1 - hw2p1.h ifndef HW2P1_H...
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Ask a homework question - tutors are online | 920 | 3,098 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.078125 | 3 | CC-MAIN-2017-26 | longest | en | 0.423036 |
https://www.esaral.com/q/mark-the-tick-against-the-correct-answer-in-the-following-98403 | 1,718,712,950,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861752.43/warc/CC-MAIN-20240618105506-20240618135506-00047.warc.gz | 665,023,687 | 11,974 | # Mark the tick against the correct answer in the following:
Question:
Mark the tick against the correct answer in the following:
Let $S$ be the set of all real numbers and let $R$ be a relation on $S$, defined by $a R b \Leftrightarrow|a-b| \leq 1$. Then, $R$ is
A. reflexive and symmetric but not transitive
B. reflexive and transitive but not symmetric
C. symmetric and transitive but not reflexive
D. an equivalence relation
Solution:
According to the question,
Given set $S=\{\ldots \ldots,-2,-1,0,1,2 \ldots . .\}$
And $R=\{(a, b): a, b \in S$ and $|a-b| \leq 1\}$
Formula
For a relation $\mathrm{R}$ in set $\mathrm{A}$
Reflexive
The relation is reflexive if $(a, a) \in R$ for every $a \in A$
Symmetric
The relation is Symmetric if $(a, b) \in R$, then $(b, a) \in R$
Transitive
Relation is Transitive if $(a, b) \in R \&(b, c) \in R$, then $(a, c) \in R$
Equivalence
If the relation is reflexive, symmetric and transitive, it is an equivalence relation.
Check for reflexive
Consider, $(a, a)$
$\therefore|\mathrm{a}-\mathrm{a}| \leq 1$ and which is always true.
Ex_if $a=2$
$\therefore|2-2| \leq 1 \Rightarrow 0 \leq 1$ which is true.
Therefore , R is reflexive ……. (1)
Check for symmetric
$a R b \Rightarrow|a-b| \leq 1$
$b R a \Rightarrow|b-a| \leq 1$
Both can be true.
Ex $_{-}$If $\mathrm{a}=2$ and $\mathrm{b}=1$
$\therefore|2-1| \leq 1$ is true and $|1-2| \leq 1$ which is also true.
Therefore , R is symmetric ……. (2)
Check for transitive
$a R b \Rightarrow|a-b| \leq 1$
b R c $\Rightarrow|b-c| \leq 1$
$\therefore|\mathrm{a}-\mathrm{c}| \leq 1$ will not always be true
Ex_a $=-5, b=-6$ and $c=-7$
$\therefore|6-5| \leq 1,|7-6| \leq 1$ are true But $|7-5| \leq 1$ is false.
Therefore , R is not transitive ……. (3)
Now, according to the equations (1), (2), (3)
Correct option will be (A) | 651 | 1,845 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.34375 | 4 | CC-MAIN-2024-26 | latest | en | 0.717401 |
https://angrybearblog.com/2010/02/how-to-build-economic-model | 1,722,918,671,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640476479.26/warc/CC-MAIN-20240806032955-20240806062955-00549.warc.gz | 79,257,510 | 19,057 | # How to Build an Economic Model
Let us see if we can translate my previous post on job selection into an economic model. Start with a basic formula:
(1) AcceptOffer = a(1) + a(2)*w + a(3)*b + a(4)*oa + a(5)*t
where a is a constant, w is wages, b is benefits, oa is “opportunity for advancement” and t is treatment received in the workplace.
The first observation we make is that several of these variables are difficult to quantify—and even more difficult to objectify. So let’s start with the easy ones.
w is very identifiable: reported (on a per capita aggregate basis), subject to enforcement penalties (e.g., minimum wage laws), and used in “downstream applications” (e.g., tax filings) and therefore relatively verifiable.
b is (1) known to be non-negative and (2) often variable within, let alone between, organizations. (Vacation time, sick days, insurances offered and costs to the employee all may vary depending on level, time of service, location of office, etc.)
This could present a problem, but here we can use standard economic theory to our advantage. We do not know the amount of b, but we can assume that the employer is rational, and is offering a total compensation to the worker that s/he expects will be less than or equal to the marginal product of that person’s labor. We therefore can reasonably assume that b is related to w. If we then review the available aggregate data we can approximate that benefits offered will be approximately a certain percentage of w—and that workers will assume that assumption (and, in most cases, verify that assumption within a margin of error) before accepting the job.
We then restate the equation as
(2a) AcceptOffer = a(1) + a(2’)*w’ + a(4)*oa + a(5)*t
where w’ is the weighted combination of w and b above, and a(2’) is the restated coefficient.
If we then assume that all parties have full information of the ratio of wages to benefits, then a(2’) = a(2)=a(3), so we simplify to:
(2b) AcceptOffer = a(1) + a(2)*w’ + a(4)*oa + a(5)*t
We now have to consider opportunities for advancement and treatment. Here, we have two problems that are difficult, possibly insurmountable, for modeling.
The first is a lack of measurability. There are no public records for “didn’t get promoted.” Nor, except in extreme cases, is there a way to measure treatment by supervisors. The data that might be available&mdassh;lawsuits, official complaints, even Human Resources files (for which there are significant privacy considerations)—is all negative and, accordingly, skewed (biased). This is because (a) ninety percent or so of all workers and/or bosses will never have a complaint filed against them and (b) the ability to file a complaint may be present because the general work atmosphere is more amenable to filing one than not, so the presence of a complaint is not in itself a good or bad thing for the overall measure.
The second is that tolerances vary by person. To use an absurd example, people who use “Every Breath You Take” for their wedding may be more likely to tolerate attentions that others view as harassment. Similarly, forcing people to clock out for a “smoke break” will be viewed differently depending upon whether one is a smoker or not. General policies are just that—general.
So, if we are building an economic model, we must come up with a reasonable approximation of these last two variables. The most direct way to do this is the standard method: assume each individual has their own Utility Curve, and “prices” accordingly.
Based on their preferences and options, then, we map the compensation required to offset negative consequences from oa and t. While the variables still are not directly observable, we can make a simplifying assumption:
Assume that the compensation required to do the work is a factor of w’.
Have to work in the sewer system? Change w’ to compensate. Need to work the night shift and/or weekends? Same type of adjustment. Boss clearly favors buxom blondes and you’re a petite redhead? Adjust current salary requirements to compensate for lowered opportunity for advancement/promotion. You’re a b.b. who will have trouble getting work done because the boss will harass you? Adjust accordingly.
We assume—due to the constraint: a lack of available data—that we can reduce “a(4)*oa + a(5)*t” to some proportion of w that will compensate the worker for the environment into which they are being placed. If we further assume that the worker has complete information as to hisser preferences, the worker will not accept a job that does not offer that level of compensation.
So we can restate equation 2(b) using the Utility Curve assumption. Assume
(3a) a(6)*w” =a(4)*oa + a(5)*t
such that w” also proportionate to w(and therefore w’ as well) and a(6) is the coefficient selected by the individual that makes the offered wage compensatory to the opportunities for advancement and expected treatment on a Present Value basis.
We can then reduce equation (2b) to
(3b) AcceptOffer = a(1) + a(2)*w’ + a(6)*w”
or, given that (a) w” is proportionate to w and w’ and (b) that the multiplier in most cases is 1, and (c) the constant (e.g., signing bonus) can be assumed without loss of generality to be 0,
(3b) AcceptOffer =
to indicate that the value varies with individuals.
To concretize the example, assume that a redhead and a blonde, as above, are both offered a job. Assume further that the redhead’s compensation requirement—lower-but-still-positive opportunity for advancement—is lower than the blonde’s for will-be-harassed-and-work-will-be-impeded. That is
(r) < (b)
There are four possibilities:
1. The offered wage will be below(r), in which case neither will accept the job
2. The offered wage will be below (b) but above (r), in which case one of the two positions will be filled
3. The offered wage will be above (b), in which case both will accept the offer and the company will have offered a higher wage than was required to fill both positions. (That the offer is what the company believes will be the employees’ s marginal product of labor [MPL] is a collateral issue.), or
4. The company will negotiate with each, offering the redhead (r) and the blonde (b), and everyone will be happy—so long as initial expectations were accurate (or, if you prefer, the new employees both had full information).
Note also that there is a learning process for both the applicant and the employer. Offers and demands will be adjusted based on historic data (if both decline the offer, the next candidates of similar background will be offered more, and perceptions of growth (improvements in experience and/or education by the worker).
If we generalize this, we note that there is a distribution of (due to Individual Preferences). If we further make simplifying assumptions—e.g., a normal distribution of among the population—we come to the conceit of the “reservation wage,” and all the economic literature that is attendant upon it.
So that is how you build an economic model. The question then becomes: how do you use it? A relatively short (though it does incorporate a micro model) discussion of that continues below the fold.
The problem—if it is one (I’m inclined to argue it is; YMMV)—is that, having built a model in which all the proxying assumptions are “simplified” into a single variable, we lose some granularity, having made a trade-off for the sake of measurement.
Accordingly, a change in the “reservation wage” may not in itself tell us whether the real wage has gone up or the work environment has become, on balance, more or less acceptable.
Again, an example, one that will be familiar to students of microeconomics. You are given two choices: (1) you can receive \$100 right here and now or (2) you can travel a known distance (say, five miles)—with a finite chance of death or injury—and receive \$1,000,000 on completion.
Surroundings, at this point, matter. If the five miles traveled is as an American soldier in full uniform walking outside of the Green Zone, you might well choose \$100. Even when you are native to the area, the choice may vary: walking five miles through one gang’s territory is a different option than traveling that distance through different territories. Or even a pure environmental matter may have an effect: Walking five miles through a desert with no canteen, or having to swim five miles from shore in a dangerous waters, is not the same risk as walking five miles down an unpaved dirt road in the middle of the day. Even if you would be required “only” to walk down a heavily-used interstate highway with no shoulder or sidewalks, discretion may be the better part of valor.
Over time, through the “learning process” (op cit. Arrow, 1962, as all good op cits must), the dollars offered will be adjusted so that the payouts balance on a risk-adjusted basis. (Collaterally, there may be other reasons for the greater payouts; signaling by any other name.)
Now suppose the landscape changes. There is a canteen every half-mile in the desert. A gang is run out of its territory, or takes over another’s territory. The Green Zone becomes larger.
The balance has changed; the risk is different. The job is demonstrably different, and therefore requires lower (higher) compensation. But the difference has nothing directly to do with the base salary/benefits requirement and everything to do with the overall attractiveness and/or treatment received.
If we were to forget that, we would conclude that there is more demand for the job itself, and therefore people are willing to take a lower salary. If, on the other hand, we keep in mind that there are more factors to the reservation wage than just the salary itself, we realize that producing a more pleasant work atmosphere is beneficial to our firm, as it enables us both to present a good face to clients and to reduce our cost of labor.
The first type of economist probably should be avoided, as he adds very little value to the discussion of how to use the model. | 2,235 | 10,011 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.40625 | 3 | CC-MAIN-2024-33 | latest | en | 0.949543 |
https://ethereum.stackexchange.com/questions/1527/how-to-delete-an-element-at-a-certain-index-in-an-array/39302 | 1,701,798,419,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100555.27/warc/CC-MAIN-20231205172745-20231205202745-00637.warc.gz | 274,606,576 | 48,573 | How to delete an element at a certain index in an array?
Anyone know how to delete an element in a array? Is there any built in method to do that?
If not, does anyone know of how to implement such a method?
• Do you want to just delete the element, or delete it and move everything down an index? Feb 21, 2016 at 1:20
• I would like everything down an index so if i delete a[2] then a[3] becomes a[2] Feb 21, 2016 at 1:42
Use the `delete` operator to delete the element:
``````delete array[index];
``````
If you don't want to leave a gap, you need to move each element manually:
``````contract test{
uint[] array = [1,2,3,4,5];
function remove(uint index) returns(uint[]) {
if (index >= array.length) return;
for (uint i = index; i<array.length-1; i++){
array[i] = array[i+1];
}
delete array[array.length-1];
array.length--;
return array;
}
}
``````
If you don't care about the order, you can also just copy the last element into the empty spot, then delete the last element.
• Thanks! I just have an issue with the function you gave when trying to make it more " general : `function remove(uint[] array,uint index) returns(uint[]) {` gives me `Error: Expression has to be an lvalue. array.length--;` Also , does that method can be adapted to work on array of all types ( struct , etc ) ? Feb 21, 2016 at 2:35
• I'm not sure about decreasing the length of arrays in memory, as opposed to storage. You can copy it all over to a new array, I guess, or just remove that line and leave the trailing zeros. You won't save any gas by making a memory array shorter, anyway. This method should work on any type, except mappings, since `delete` doesn't really make sense in mappings. Feb 21, 2016 at 2:41
• Decreasing array length will automatically clean up the storage slots occupied by the out-of-bounds elements. So the line `delete array[array.length-1];` is redundant. Moreover it adds 5000 gas to the transaction since gas refund applies only in the case when storage is reset from non-zero to zero value. If it's set from zero to zero (added by compiler) it costs 5000 gas. Feb 11, 2018 at 20:02
• The best way I tested that works is to copy the last element to the deleted spot, call delete on the last index, then decrement the array length. This is a constant amount of work, not linear. If you care about the order, you should have an additional mapping from this element to next, that you maintain during addition and deletion. Sep 21, 2018 at 20:03
• From 0.6.0, pop() is the only way to reduce size of array. "It is no longer possible to resize storage arrays by assigning a new value to their length" - docs.soliditylang.org/en/v0.8.11/… Feb 17, 2022 at 21:31
This constant operation works without preserving order:
``````uint[] internal array;
// Move the last element to the deleted spot.
// Remove the last element.
function _burn(uint index) internal {
require(index < array.length);
array[index] = array[array.length-1];
array.pop();
}
``````
To preserve order on recall without incurring the gas cost of shifting right-of-gap values, you'll need an additional mapping between each element's index to its successor's index that you need to maintain during insertion and deletion: `mapping(uint => uint) private indexAfter;`
• Best answer!!!! Aug 1, 2021 at 4:29
• Definitely the best one Oct 31, 2021 at 9:39
• Since solidity 0.6.0 it should not work ethereum.stackexchange.com/questions/80743/… Aug 23, 2022 at 10:33
• Nice solution. Works on 0.8.0 Sep 6, 2022 at 22:20
``````contract Test {
uint[] array = [1,2,3,4,5];
function remove(uint index) returns(uint[]) {
if (index >= array.length) return;
for (uint i = index; i<array.length-1; i++){
array[i] = array[i+1];
}
array.length--;
return array;
}
}
``````
I removed the line `delete array[array.length-1];` before `array.length--;`. This makes the function cheaper by 5000 gas. The compiler will automatically clean up unoccupied slots when array length is decreased. Double storage resetting adds 5000 gas.
• I found this answer only to work on storage arrays, can you confirm? The array.length--; line will throw an exception. When changing it so that there is no error anymore it will not remove the last element of the array.
– Nico
May 18, 2018 at 8:40
• @Nico, yup, array.length-- only works on storage arrays, for memory arrays, the length is constant after it's assigned. if you want a memory array of different length, you need to declare a new memory array. Jul 31, 2019 at 12:10
• What is the purpose of the return statement at the end of the function? Dec 9, 2019 at 2:40
• array.pop() instead of array.length-- for solidity > 0.6 even for storage arrays. Aug 20, 2020 at 19:16
Most of the previous answers directly modify the array length to reduce its length.
Since Solidity 0.6.0 this is no longer possible
Member-access to length of arrays is now always read-only, even for storage arrays. It is no longer possible to resize storage arrays assigning a new value to their length. Use push(), push(value) or pop() instead, or assign a full array, which will of course overwrite existing content. The reason behind this is to prevent storage collisions by gigantic storage arrays.
https://docs.soliditylang.org/en/v0.6.2/060-breaking-changes.html
You can fix medvedev1088's answer with:
``````contract Test {
uint[] array = [1,2,3,4,5];
function remove(uint index) returns(uint[]) {
if (index >= array.length) return;
for (uint i = index; i<array.length-1; i++){
array[i] = array[i+1];
}
array.pop();
return array;
}
}
``````
Notice: `array.pop();` instead of `array.length--;`
``````pragma solidity ^0.4.11;
contract TestArray {
uint[] public original;
uint[] public newOr;
event Log(uint n, uint a, uint b, uint c);
function TestArray(){
original.push(1);
original.push(2);
original.push(3);
original.push(4);
}
function test(){
newOr = remove(original, 1);
Log(newOr.length, newOr[0], newOr[1], newOr[2]);
}
function remove(uint[] array, uint index) internal returns(uint[] value) {
if (index >= array.length) return;
uint[] memory arrayNew = new uint[](array.length-1);
for (uint i = 0; i<arrayNew.length; i++){
if(i != index && i<index){
arrayNew[i] = array[i];
} else {
arrayNew[i] = array[i+1];
}
}
delete array;
return arrayNew;
}
}
``````
• How Could I delete the complete array? Sep 22, 2017 at 0:49
• You need to delete all elements one by one. The array itself cannot not be deleted as it's a storage variable and lives forever in the contract space. Feb 8, 2018 at 4:39
delete a assigns the initial value for the type to a. I.e. for integers it is equivalent to a = 0, but it can also be used on arrays, where it assigns a dynamic array of length zero or a static array of the same length with all elements reset. For structs, it assigns a struct with all members reset.
> I have implemented it, may be helpful to understand by this simple example
**
And if we remove the element using index it will not leave the gap.
**
``````contract UserRecord {
constructor() public { owner = msg.sender; }
modifier onlyOwner {
require(msg.sender == owner);
_;
}
struct User {
bytes32 userEmail;
uint index;
}
mapping (bytes32 => User) private users;
bytes32[] private usersRecords;
event LogNewUser(bytes32 indexed userEmail, uint index);
function setUseremail(bytes32 _userEmail) public onlyOwner returns(bool success){
users[_userEmail].userEmail = _userEmail;
users[_userEmail].index = usersRecords.push(_userEmail) -1;
emit LogNewUser(
_userEmail,
users[_userEmail].index
);
return true;
}
//this will delete the user at particular index and gap will be not there
function deleteUser(bytes32 _userEmail) public onlyOwner returns(uint index){
require(!isUser(_userEmail));
uint toDelete = users[_userEmail].index;
bytes32 lastIndex = usersRecords[usersRecords.length-1];
usersRecords[toDelete] = lastIndex;
users[lastIndex].index = toDelete;
usersRecords.length--;
}
}
``````
A solution that consumes a bit more gas compared to others.
``````// Preload any custom data through other functions
function removeIndex(uint256 index) external { | 2,095 | 8,079 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.578125 | 3 | CC-MAIN-2023-50 | latest | en | 0.915797 |
krasia-demertzis.gr | 1,604,106,488,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107912593.62/warc/CC-MAIN-20201031002758-20201031032758-00253.warc.gz | 346,578,927 | 17,951 | What is parity in Physics?
This might sound confusing but in reality parity is among the most commonly-used words in the world of Physics, that is why it truly is nonetheless so vital.
In Physics, parity refers for the scenario exactly where precisely the same equations work for each initial situations inside a test method. Considering that each solutions look exactly the same in a model of matter, they are able to be treated as if they may be identical and hence both options will look precisely the same for any test of your laws of physics.
In a predicament where the initial conditions are identical, the forces acting on an object cannot be distinct. An example of this will be a tank made up of the exact same materials but with distinct tanks made up of distinct components. In each cases, they’ve the exact same weight, and neither can move because of a difference in their mass.
In numerous techniques, this concept is additional crucial than the principle point. You will need to understand how your physics principles and problems will work, just before you apply them. One of the methods to accomplish that is by utilizing the above instance. When a tank with distinctive contents are placed within the same environment, the tanks can not move for the reason that they can’t move around and have the exact same mass.
The much more difficult a technique is, the far more physical variables involve, the more complicated the system, the a lot more difficult the equations involved, the far more complicated it truly is to have a result. These challenges are known as physical laws and by asking the above query, the answer to what’s parity in Physics will provide you with an insight into why the equations of motion cannot be manipulated.
So how is parity measured? In case you buy essay possess a pendulum that is produced up of two objects, certainly one of which has to move as well as the other which does not, you may figure out the distinction in time amongst these two pendulums bymeasuring the time it requires for the pendulum to produce a complete cycle in between the two moving objects. If these two pendulums have the identical mass, the pendulum will oscillate in between them in the identical rate. The lower the difference in time, the extra momentum is needed to bring the pendulum back to its original position.
Parity in Physics does not mean that a single object can do anything that a second object can do. It simply signifies that there is certainly only one particular path the pendulum will take, in the very same speed and in the very same path. The opposite of parity in Physics is known as a parabolic equation, this is a mathematical approach that produces distinctive benefits on account of the fact that you will find no ‘numerical variables’ to be regarded.
Most equations in Physical laws are obviously distinctive to each and every individual trouble. There’s no ‘one size fits all’ resolution to an issue that you are operating on. The most beneficial way to establish which can be suitable for you will be to do a trial run initial and test each of the different http://pier.macmillan.yale.edu/ systems until you get something that works properly for you.
Some persons do not know what parity in Physics is, but after they have mastered the notion, they realize that it makes a massive distinction in their life and in their Physics writing. Additionally, it assists to offer them an notion with the complexities that exists in Physics and it could assist them with their Physics calculations.
Of course, lots of individuals will not take the time to find out this mainly because they do not think that their physical technique is any superior, or they https://buyessay.net/ feel that it isn’t time-consuming. But bear in mind, in Physics, every thing is about learning about a physical method, how it works, and what impact it is going to have on our world.
Sometimes, the scenario can be a little bit extra complex and it can be hard to predict which things will make it a lot easier or make it tougher, but ultimately, the only point that genuinely matters is which components with the program will affect which other parts. parts of the program.
In Physics, among the best strategies to understand what’s parity in Physics is to practice it. Never wait for a person else to inform you, just get available and make an effort to find out!
Facebook Honey | 875 | 4,418 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.6875 | 4 | CC-MAIN-2020-45 | latest | en | 0.96141 |
https://stacks.math.columbia.edu/tag/0534 | 1,725,773,661,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700650960.90/warc/CC-MAIN-20240908052321-20240908082321-00389.warc.gz | 526,268,423 | 6,442 | Lemma 15.21.6. Let $R \to S$ be an injective integral ring map. Let $M$ be a finitely presented module over $R[x_1, \ldots , x_ n]$. If $M \otimes _ R S$ is flat over $S$, then $M$ is flat over $R$.
Proof. Choose a presentation
$R[x_1, \ldots , x_ n]^{\oplus t} \to R[x_1, \ldots , x_ n]^{\oplus r} \to M \to 0.$
Let's say that the first map is given by the $r \times t$-matrix $T = (f_{ij})$ with $f_{ij} \in R[x_1, \ldots , x_ n]$. Write $f_{ij} = \sum f_{ij, I} x^ I$ with $f_{ij, I} \in R$ (multi-index notation). Consider diagrams
$\xymatrix{ R \ar[r] & S \\ R_\lambda \ar[u] \ar[r] & S_\lambda \ar[u] }$
where $R_\lambda$ is a finitely generated $\mathbf{Z}$-subalgebra of $R$ containing all $f_{ij, I}$ and $S_\lambda$ is a finite $R_\lambda$-subalgebra of $S$. Let $M_\lambda$ be the finite $R_\lambda [x_1, \ldots , x_ n]$-module defined by a presentation as above, using the same matrix $T$ but now viewed as a matrix over $R_\lambda [x_1, \ldots , x_ n]$. Note that $S$ is the directed colimit of the $S_\lambda$ (details omitted). By Algebra, Lemma 10.168.1 we see that for some $\lambda$ the module $M_\lambda \otimes _{R_\lambda } S_\lambda$ is flat over $S_\lambda$. By Lemma 15.21.5 we conclude that $M_\lambda$ is flat over $R_\lambda$. Since $M = M_\lambda \otimes _{R_\lambda } R$ we win by Algebra, Lemma 10.39.7. $\square$
## Comments (0)
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• 2 comment(s) on Section 15.21: Descent of flatness along integral maps
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https://www.physicsforums.com/threads/speed-of-light-in-any-man-made-vehicle.71682/ | 1,723,780,876,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641333615.45/warc/CC-MAIN-20240816030812-20240816060812-00026.warc.gz | 701,044,759 | 24,020 | # Speed of light in any man made vehicle
In summary, the conversation discusses a theoretical experiment involving two capsules moving at 1/4 the speed of light and facing each other with mirrors. The idea is that by observing the reflected images in the mirrors, it may be possible to observe speeds greater than the speed of light. However, there are several flaws in this theory and the concept of c, the speed of light, is not relative. Ultimately, the experiment is not feasible and the conversation ends with a suggestion to forget the idea and focus on a simpler scenario with a single moving car and a fixed mirror.
i do not know if the theory behind this will work, but seems plausable
now as far as i know, we have not attained the speed of light in any man made vehicle, however, 1/4 the speed of light seems a much more plausable situation. in this experiment, the set up looks somthing like this
the back of each one of those capsules is a flat edge that has a miror face and a small camera on it (if 1/4th the speed of light is not unbearable for humans then it is a small capsule for veiwing instead) now each of hese are spinning on an axis the opposite direction of each other so that for a few brief miliseconds, the two back faces are facing each other, they are each spining at 1/4th the speed of light, am i wrong to asume that say, if you weere on the capsule, s you veiwed the other face/mirror moving away from you, first of all, you have a velocity of -1/4th the speed of light, you are veiwing the other going at a velocity of 1/4th the spead of light, so basically, you see the other face mving away from you at 1/2 the spead of light, now with the mirrors on these cpsules, the whole process is reversed, the light is refracted off the thing going 1/4th velocity witch then finds you at another -1/4th, this adds andother 1/2 the spead of light to it, the total should in theory be the spead of light, it should be at least observable.
i realize there are many many flaws in this experiment, namely that I am not sure that 1/4th spead of light has ever been attained, however, this is the spead of light , would actually veiwing the process be possible as veiwing needs light to work?
at least tell me if this idea is at all interesting.
Edit: apparently i can't get the graphic to load, no matter, it looks just like 2 centrifuges moving in opposite directions right next to each other, the explination in the post should explain it somewhat
Last edited by a moderator:
I have no idea what you are getting at with this apparatus. If the two capsules are each moving away from each other at 1/4 light speed (with respect to the earth), then their speed relative to each other is not 1/2c, but only about 0.47c. In any case, so what?
well, let's say c = 4mph, we have 2 cars, each's rear end is a big mirror with a mirror tainted window so that the passenger looking back can see out, but to anyone behind, it looks like a mirror, these 2 cars are faced back to back, they each start driving at 1mph away from each other, the passenger inside the left moving car is moving at a velocity of -1mph right, his brother in the other car is also moving at 1mph and is therefore going at 1mph right, if there were no mirrors, the first brother (Bob) sees his brother's (Joe) car moving away from him at 2mph. however, with the mirrors, the image of his own car is reflected off of Joe's car going 1mph right and out of this Joe sees himself moving away at 4mph, in other words, c. if i had a vidio camera and mirors and such i could probably explain this better, but i dont, so i cant. the apperatus i first mentioned was simply the more plausable way, i thought of ataining 1/4c, however, in this case, a simple adding to that 1/4c would allow the camera, or in this case, Bob, to observe more than 4mph, more than c, more than the speed of light.
a simple adding to that 1/4c would allow the camera, or in this case, Bob, to observe more than 4mph, more than c, more than the speed of light.
There are an awful lot of problems with that whole scenario, but I'll just point out the simplest one. c is not relative. It's the one constant against which others can be measured. Your receding images would redshift themselves flat and you wouldn't see anything.
well, let's say c = 4mph, we have 2 cars, each's rear end is a big mirror with a mirror tainted window ...
I think I finally see what you are saying. First, to avoid pointless diversion, forget having two cars driving away from each other. (If you insist on that scenario, you must add the velocities relativistically.) You can make your point with a mirror fixed with respect to the Earth and a single car moving away from it at whatever (sub-light) speed you like. What I think you are saying is that the image one sees in the mirror appears to move at twice the speed that the car moves. (After all, if one is a distance D from a plane mirror, then one's image appears to be at a distance of 2D.) An interesting thought!
The first thing to realize is that the image in the mirror is not a "thing", so that even if it appears to move faster than c, so what?
To work out the details, realize that the apparent position of the image seen at any time depends on where the mirror was when the light reflected from it, not where the mirror is now. Also realize that with respect to the observers in the car, the light still moves at speed c.
My calculation says that the apparent speed of the image would be $2vc/(c+v)$ if the car moves away from the mirror. (I hope someone can double check that.) Thus the apparent speed of the receding image ranges from 2v (for low speeds) to a limit of c as the speed of the car approaches that of light.
On the other hand, if the car drives towards the mirror, the image would seem to approach the car at a speed of $2vc/(c-v)$!
You want an image to attain the speed of light? Then I think that this could work.
Imagine that you have your hand in front of a light source at a fairly close distance. Then 1000 light years away, there is a giant shell which has a screen. First switch on the light and wait. Eventually the shadow will fall on the giant screen. Now move your hand rapidly in front of the light source. If you move it fast enough, the shadow on the giant screen should travel faster than c.
But if you want to travel faster than light, why not consider this.
Correct me if i am wrong, but if you pass light through a material with sufficiently high refracitve index or through Bose-Einsten condensate, then you can slow the light down
so much that, humans can move faster than it.
Last edited by a moderator:
siddharth said:
Correct me if i am wrong, but if you pass light through a material with sufficiently high refracitve index or through Bose-Einsten condensate, then you can slow the light down
so much that, humans can move faster than it.
Not just that, but to "slow" it down to 0 km/s! I think an ant can move faster than that.
Zz.
but, and i am probably very wrong, but if we can slow down light, how can we say taht speed of light is a constant?
i soppose there would be really no use for this expriment lol the car would allow it to see itself moving at speed of light, would probably do no good lol,
also as for the car, yes it would see twice it's own speed if one of the mirors was fixed, but, it isnt, there are 2 caqrs moving away from each other at 1/4c anyway, taking this slowing down of light one step furthur, might it be possible to not only slow it down but to stop it? and tehn perhaps, to reverse it?
but, and i am probably very wrong, but if we can slow down light, how can we say taht speed of light is a constant?
i soppose there would be really no use for this expriment lol the car would allow it to see itself moving at speed of light, would probably do no good lol,
also as for the car, yes it would see twice it's own speed if one of the mirors was fixed, but, it isnt, there are 2 caqrs moving away from each other at 1/4c anyway, taking this slowing down of light one step furthur, might it be possible to not only slow it down but to stop it? and tehn perhaps, to reverse it?
This is a perfect example of making sure that one clearly understands the FULL extent on how things are defined in physics.
The speed of light is a constant as defined in VACUUM. The speed of light in a medium changes to lower value due to scattering, absortion-reemission, etc. Furthermore, one needs to clearly know HOW the speed of light is typically measured, especially in a dispersive medium. We define such speed via its GROUP VELOCITY.
When light is stopped in the Lena Hau's experiment, it means that both its energy and phase are "stored" in the medium, which in this case is a very cold gas. So the info about the absorbed light is stored coherently! This is very different than light being absorbed by a "black" paper, for example, because the absorbing material loses any coherent info and energy is absorbed completely.
When we start testing and "theorizing" various exotic properties of light, it is imperative that we are very clear on how exactly its properties are defined and measured. Without that, things will make for a very confusing picture.
Zz.
but, and i am probably very wrong, but if we can slow down light, how can we say taht speed of light is a constant?
No, we can't slow light down. If you are thinking about refraction, refraction involves interaction with atoms, making it appear to slow down. But light goes from atom to atom at the speed of light.
also as for the car, yes it would see twice it's own speed if one of the mirors was fixed, but, it isnt, there are 2 caqrs moving away from each other at 1/4c anyway...
Please reread my post where I describe how images are seen in a moving mirror. All that matters is the relative speed of the mirror and the car. Having two cars just introduces more room for confusion.
## What is the speed of light in any man made vehicle?
The speed of light is approximately 299,792,458 meters per second. This is a universal constant that does not change regardless of the vehicle or method of transportation.
## Why is the speed of light important in man made vehicles?
The speed of light is important in man made vehicles as it is the fastest possible speed that any object can travel. In order to efficiently and effectively transport people and goods, it is important for man made vehicles to be designed to reach speeds as close to the speed of light as possible.
## Is it possible for man made vehicles to travel at the speed of light?
Currently, it is not possible for man made vehicles to travel at the speed of light. The laws of physics prevent any object with mass from reaching the speed of light. However, advancements in technology and research are constantly being made to improve the speed and efficiency of man made vehicles.
## How fast can man made vehicles travel in comparison to the speed of light?
Man made vehicles can travel at incredibly high speeds, but they are still significantly slower than the speed of light. The fastest man made vehicle to date is NASA's Juno spacecraft, which reached a speed of 165,000 miles per hour. This is only a small fraction of the speed of light.
## What are the potential implications of traveling at the speed of light in man made vehicles?
If man made vehicles were able to travel at the speed of light, it would revolutionize transportation and open up new possibilities for space exploration and interstellar travel. It would also significantly decrease travel time and make long distance travel more accessible. However, it would also require significant advancements in technology and safety measures to ensure the safety of passengers and cargo at such high speeds.
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# QA – Placement Quizzes | Progressions | Question 9
For n positive integers, if their product is nn, then what will be their sum?
(A)
Equal to n+(1/n)
(B)
Equal to n
(C)
A negative integer
(D)
Never less than n2
Explanation:
Clearly, since the given integers are positive, their sum can\’t be negative.
Also, since the numbers are all integers their sum can\’t be a fraction.
Let\’s take 1, 3 and 9. The product of these three integers is 27 = 33
This can also be written as nn where n=3.
As we can see, the sum of these 3 integers is not equal to 3.
Therefore, we are left with the fourth option.
Quiz of this Question
Please comment below if you find anything wrong in the above post
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# HW7_Solutions_Problem_Set - AEM 250 Fall 2008 HW#7...
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AEM 250, Fall 2008 HW #7 Solutions 1. (Harris p. 104, #1) The graph below shows the supply and demand curve in the current generation without any consideration for the future. We can solve for the equilibrium price by setting Q d =Q s : 200 – 5P = 5P 200 = 10P 20 = P So, the equilibrium price is \$20 per unit. Substituting this price back into either the supply or demand curve equation solves for a market quantity of 100 units of oil. At any quantity the net benefit is equal to the distance between the demand curve (the willingness to pay) and the supply curve (the marginal costs). We can express this algebraically as the difference between the price on the demand curve and the price on the supply curve: MNB = (40 - 0.2Q) - (0.2Q) MNB = 40 - 0.4Q We can express this graphically in the graph below. Marginal net benefits of consumption are positive until we reach the equilibrium quantity of 100 units.
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2. (Harris p. 104, #2) To calculate the marginal net benefit function in the second generation, we apply a discount factor of four.
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https://solvedlib.com/n/3-c-p-at-bt-7,2576352 | 1,695,897,242,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510387.77/warc/CC-MAIN-20230928095004-20230928125004-00350.warc.gz | 586,433,354 | 17,588 | # 3 C P(AT /BT ) =7
###### Question:
3 C P(AT /BT ) =7
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https://quizgecko.com/q/what-is-the-cube-root-of-64-oor68 | 1,726,357,961,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651601.85/warc/CC-MAIN-20240914225323-20240915015323-00392.warc.gz | 442,280,638 | 33,332 | # What is the cube root of 64?
#### Understand the Problem
The question is asking to find the cube root of the number 64, which involves determining what number, when multiplied by itself three times, equals 64.
4
The final answer is 4
#### Steps to Solve
1. Express the problem mathematically
To find the cube root of 64, we need to determine what number $x$ satisfies the equation:
$$x^3 = 64$$
1. Solve for the cube root
To find the value of $x$, take the cube root of both sides of the equation:
$$x = \sqrt[3]{64}$$
1. Calculate the cube root
Recognize that $64 = 4 imes 4 imes 4$, therefore
$$x = 4$$
The final answer is 4 | 180 | 643 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2024-38 | latest | en | 0.844714 |
https://community.airtable.com/t/can-i-count-how-many-times-something-appear-in-a-range-of-entries/12622 | 1,550,950,756,000,000,000 | text/html | crawl-data/CC-MAIN-2019-09/segments/1550249530087.75/warc/CC-MAIN-20190223183059-20190223205059-00122.warc.gz | 525,953,180 | 5,187 | # Can I count how many times something appear in a range of entries?
#1
I have a sheet called fixtures and it is filtered into seasons, each entry represents one match and has players listed that played in a match. I want to be able to count how many times a player appeared in a each season. Any ideas? The seasons are currently identified by a ‘single-select’, and each fixture is numbered individually regardless of season.
#2
Are the players linked to - and listed in - another table? If that’s the case, that will make it a lot easier.
Greetz,
André
#3
Hi @Andre_Zijlstra , yes all the players link to another table.
#4
I tested 2 tables (Fixtures and Players) of an imaginary u11 team. 7 players are on the Staring VII and 3 are subs. It would look like this:
Fixtures:
Players:
Count fields give you the number a player is selected in Starting VII and Subs.
A formula field adds up both the Count fields. And added to this, at the bottom you see numbers that:
1. you should be able to divide by 7
2. you should be able to divide by 3
3. you should be able to divede by 10
Regards,
André
#5
@Andre_Zijlstra thanks, that’s very similar to what I have. The main difference is I have 13 seasons of fixtures with each season identified by a single select field. I’m looking to count players appearances by season, any ideas?
#6
The easiest way would probably be to create a formula field per season / for the starting players and a formula field per season for the subs. Something like this:
Of course you would need a field to identify the season.
In the table where you find the players, you need Rollup fields (per season, start, sub).
For the starting players:
For the subs:
And that will like like this:
Of course you may choose the option to create a view per season. In that view you can hide the fields that contain data of other seasons.
Hopefully that will work for you!
Greetz,
André
#7
Thanks @Andre_Zijlstra, that looks like the route I should be pursuing. One difference is I have 11 fields for the different starters rather than linking them all in the same field. I’m sure there’s a work around for this. Thanks again for your help.
#8
Are those 11 fields related to the positions on the field? In other words, do you bring a sub into a position or just one player for another?
#9
I just have three additional fields for the subs, and discount any subs who were not brought on.
Regards,
Sam | 579 | 2,441 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2019-09 | latest | en | 0.966269 |
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