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https://ssconlineexam.com/onlinetest/SSC-CGL-Tier-1/Quantitative-Aptitude/QA-Test-97
|
# SSC CGL Tier 1 :: Quantitative Aptitude QA Test 97
## Home SSC CGL Tier 1 / Quantitative Aptitude QA Test 97 Questions and Answers
1 . Read the pie-chart given below and answer the questions. Expenditure of a family on various items and savings during 2013.
The total expense of the family on transport is equal to those spent on -
Savings
Food
Housing
Others
View Answer Discuss in Forum
2 . Read the pie-chart given below and answer the questions. Expenditure of a family on various items and savings during 2013.
If the total income of the family during 2013 is Rs. 189000, the expenditure on clothing is -
₹ 34000
₹ 34020
₹ 34200
₹ 28600
View Answer Discuss in Forum
3 .
Ankit sells 30 bulbs for Rs. 400 which he bought at 16 bulbs for Rs. 250. Find his percentage gain or loss
Profit 17.18%
Loss 14 $2\over 3$ %
Loss17.18%
Profit 14 $2\over 3$ %
View Answer Discuss in Forum
4 .
List price of a sewing machine is Rs. 2,300. It is sold at a discount of 4%. Find the selling price of it
₹ 2200
₹ 2280
₹ 2180
₹ 2208
View Answer Discuss in Forum
5 .
A table is offered for Rs. 300 with 20% and 10% off. If in addition a discount of 5% is offered on cash payment, then cash price of the table is
₹ 240
₹ 216
₹ 210
₹ 205.20
View Answer Discuss in Forum
6 .
Area of the f loor of a cubical room is 48 sq.m. The length of the longest rod that can be kept in that room is
6 m
9 m
12 m
18 m
View Answer Discuss in Forum
7 .
A man saves Rs. 25 on the purchase of an article on which a discount of 20% is allowed. How much did the man pay ?
₹ 125
₹ 75
₹ 150
₹ 100
View Answer Discuss in Forum
8 .
The population of a village is 1,00,000. If the population increases at 10% per annum, then the population at the start of the third year is
133100
121000
120000
110000
View Answer Discuss in Forum
9 .
If the graph of given linear equations 3 + ky- 4 = 0 and k- 4y- 3x = 0 coincides with each other, then the value of k is
3
4
- 3
- 4
View Answer Discuss in Forum
10 .
From an aeroplane vertically above a straight road the angle of depressions of two consecutive kilometre stones on the same side are 30° and 45° . Then at what height the aeroplane is fyling ?
2.22 km
2.27 km
2.365 km
1.365 km
View Answer Discuss in Forum
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2018-06-17 23:40:04
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https://math.stackexchange.com/questions/2385822/additive-subgroup-of-a-field-of-characteristic-p-is-an-elementary-abelian-p
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# Additive subgroup of a field of characteristic $p$ is an elementary abelian $p$-group
In this paper (in the abstract), it is mentioned that:
A finite subgroup of the additive group of a field $F$ of characteristic $p \neq 0$ is an elementary abelian $p$-group.
Why is this so?
If $charF=p$, then $p=0$ in $F$ so $px=0$ for every $x\in F$.
An elementary abelian p-group is an abelian group in which every nontrivial element $x$ has order $p$ (namely $px=0$) by definition.
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2019-07-20 07:18:59
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http://tzer.miogallagher.ru/carbon-dating-the-process-111.html
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# Carbon dating the process
Once an organism is decoupled from these cycles (i.e., death), then the carbon-14 decays until essentially gone.The half-life of a radioactive isotope (usually denoted by $$t_$$) is a more familiar concept than $$k$$ for radioactivity, so although Equation $$\ref$$ is expressed in terms of $$k$$, it is more usual to quote the value of $$t_$$.Fallout from nuclear bomb tests during the cold war has just yielded encouraging news for those searching for ways to reverse heart disease.
Carbon-14 is constantly be generated in the atmosphere and cycled through the carbon and nitrogen cycles.
"Everything which has come down to us from heathendom is wrapped in a thick fog; it belongs to a space of time we cannot measure.
We know that it is older than Christendom, but whether by a couple of years or a couple of centuries, or even by more than a millenium, we can do no more than guess." [Rasmus Nyerup, (Danish antiquarian), 1802 (in Trigger, 19)].
"This study shows for the first time and very clearly that there is some turnover of cardiomyocytes within the lifetime of an individual." It also lays to rest claims that heart cells turn over quickly, says Deepak Srivastava of the Gladstone Institute of Cardiovascular Disease in San Francisco, California.
To conduct the study, Frisén created his own version of radiocarbon dating.
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2018-07-17 13:34:53
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https://popl19.sigplan.org/details/POPL-2019-Research-Papers/5/Dynamic-Type-Inference-for-Gradual-Hindley-Milner-Typing
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POPL 2019
Sun 13 - Sat 19 January 2019 Cascais, Portugal
Thu 17 Jan 2019 15:21 - 15:43 at Sala I - Type Inference II Chair(s): Niki Vazou
Garcia and Cimini study a type inference problem for the ITGL, an implicitly and gradually typed language with let-polymorphism, and develop a sound and complete inference algorithm for it. Soundness and completeness mean that, if the algorithm succeeds, the input term can be translated to a well-typed term of an explicitly typed blame calculus by cast insertion and vice versa. However, in general, there are many possible translations depending on how type variables that were left undecided by static type inference are instantiated with concrete static types. Worse, the translated terms may behave differently—some evaluate to values but others raise blame.
In this paper, we propose and formalize a new blame calculus $\lambda^{\textsf{DTI}}{\textsf{B}}$ that avoids such divergence as an intermediate language for the ITGL. A main idea is to allow a term to contain type variables (that have not been instantiated during static type inference) and defer instantiation of these type variables to run time. We introduce dynamic type inference (DTI) into the semantics of $\lambda^{\textsf{DTI}}{\textsf{B}}$ so that type variables are instantiated along reduction. The DTI-based semantics not only avoids the divergence described above but also is sound and complete with respect to the semantics of fully instantiated terms in the following sense: if the evaluation of a term succeeds (i.e., terminates with a value) in the DTI-based semantics, then there is a fully instantiated version of the term that also succeeds in the explicitly typed blame calculus and vice versa.
Finally, we prove the gradual guarantee, which is an important correctness criterion of a gradually typed language, for the ITGL.
Slides (slides_popl.pdf) 850KiB
#### Thu 17 Jan
15:21 - 16:49: Research Papers - Type Inference II at Sala I Chair(s): Niki VazouIMDEA Software Institute 15:21 - 15:43Talk Dynamic Type Inference for Gradual Hindley–Milner TypingYusuke MiyazakiKyoto University, Taro SekiyamaNational Institute of Informatics, Atsushi IgarashiKyoto University, Japan Link to publication DOI Pre-print Media Attached File Attached 15:43 - 16:05Talk Gradual Typing: A New PerspectiveGiuseppe CastagnaCNRS - Université Paris Diderot, France, Victor LanvinIRIF, Université Paris Diderot, France, Tommaso PetruccianiDIBRIS, Università di Genova, Italy & IRIF, Université Paris Diderot, France, Jeremy G. SiekIndiana University, USA Link to publication DOI Media Attached File Attached 16:05 - 16:27Talk Intersection Types and Runtime Errors in the Pi-CalculusUgo Dal LagoUniversity of Bologna, Italy / Inria, France, Marc De VismeENS Lyon, Damiano MazzaCNRS, Akira YoshimizuINRIA Link to publication DOI Media Attached File Attached 16:27 - 16:49Talk Principality and Approximation under Dimensional BoundAndrej DudenhefnerTechnical University Dortmund, Jakob RehofTechnical University Dortmund Link to publication DOI Media Attached File Attached
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2019-06-20 07:03:37
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https://www.physicsforums.com/threads/most-commonly-mispronounced-mathematicians.689398/
|
# Most Commonly Mispronounced Mathematicians
1. May 2, 2013
### hsetennis
Title is pretty self-explanatory. I'm compiling a short list.
The ones I could think of:
Descartes = Daycart
Euler = Oiler
Erdos = Airdish
Riemann = Reemahn
Lie = Lee
2. May 2, 2013
### phyzguy
Are you saying your phonetic pronunciations are correct or incorrect?
3. May 2, 2013
### EvenSteven
So are you saying we need to enunciate their name like it sounds in their native language?
French is my first language and I wouldn't say Daycart is a wrong pronunciation. A lot of phonemes in french and other languages don't have an exact replica in english.
Descartes in french sounds like Dehhh-Kah-Hrt. Of course the r in french doesn't sound like "are" like it does in english, and it's kind of hard to explain through text for someone like me who doesn't know phonetic spelling.
Last edited: May 2, 2013
4. May 2, 2013
### hsetennis
I hope they're close to correct.
5. May 2, 2013
### DrewD
Those are the common Anglicized versions of their names. I assume hsetennis isn't expecting everyone to attempt perfect accents, but a lot of people pronounce Erdos err-dose which is not even close.
Fourier$\approx$For ee ay
but that one might be a lost cause
And for physics
Chandrasekhar=?
For Chandrasekhar, I had an Indian professor that was telling us about him and halfway through we stopped him and realized we actually knew who he was talking about, but none of us recognized the pronunciation. But then I forgot it.
6. May 2, 2013
### hsetennis
Not exactly enunciated, but I would like to provide an approximate English pronunciation for those who don't speak French/German/Dutch/Norwegian.
I am aware that the common french R is a guttural consonant, but this is uncommon to us Americans, so I didn't bother to go too deep into it.
7. May 2, 2013
### hsetennis
Ah, Fourier is another good one, thanks. As for Chandrasekhar, I feel your pain. Even as a native Hindi speaker, I had trouble recognizing his name in a spoken setting for two reasons. Between Indian languages, there is a wide variation in the pronunciation, despite having the same script. Also, there is somewhat of an over-anglicization of his name (not complaining, I'm guilty of this too), from the "t∫"[ch] to a "∫"[sh] and the "ər"[ur] to "ɑr"[are]. What has always irked me about the spelling is that the English spelling uses an "s" whereas the pronunciation is "sh".
8. May 2, 2013
### AlephZero
The basic problem is transliterating from a phonetic language with a huge alphabet (about 50 letters compared iwith 26)
It it was spelled Chandrashekhar, a "well informed" english speaker would probably read it as Chandras-hekhar. The same issue applies to "th" (as in lighthouse, not as in bathroom).
9. May 2, 2013
### hsetennis
I hadn't considered this situation. Now that you mention it, I can understand the difficulty if one mistranslated an aspirated labial "ph" to the labiodental "f".
10. May 3, 2013
### haruspex
11. May 3, 2013
### Staff: Mentor
Here is a tricky one: Euclid
12. May 3, 2013
### hsetennis
It's not yoo-klid?
13. May 3, 2013
### Fredrik
Staff Emeritus
Last edited: May 3, 2013
14. May 3, 2013
### Fredrik
Staff Emeritus
I went to Wikipedia and copied the ancient greek version of his name, Εὐκλείδης. Then I pasted it into the box at google translate. It suggested that I change it to Ευκλείδης, so I did.
Sounds like eff-clee-these, or at least it would, if it hadn't been for the fact that the English L sound is pretty different from the L sound that most European languages have in common. I would however argue that "Euclid" isn't his actual name, but his English nickname, and that this makes it OK to pronounce it youclid.
Last edited: May 3, 2013
15. May 3, 2013
### SteamKing
Staff Emeritus
There is a variation in the sounds represented by modern Greek letters to the sounds which linguists think were represented in ancient Greek. If you were to ask an ancient Greek about the man known as 'Euclid' using the modern Greek pronunciation, he might find you hard to follow, although the written form of the name 'Euclid' has not changed.
16. May 3, 2013
### lisab
Staff Emeritus
17. May 3, 2013
### phosgene
Leibniz. It's Lyb-nitz, not Leeb-nitz! Somebody should start a similar thread for physicists. There are a few of those that I can think of..
18. May 3, 2013
### czelaya
When I took a course in General Relativity as an undergraduate I heard a fellow student call the Ricci tensor the "Rikki tensor".
19. May 3, 2013
### Fredrik
Staff Emeritus
I don't think there is an "americanization" of "Poincaré" that that sounds like the name, so it's understandable.
The only one that actually made me laugh was "Lebesgue", because Google pronounced it Leb-saig.
20. May 3, 2013
### Andre
Is there a point (pwhah in French)? We lived in Canada some 35 years ago and my spouse had to see the doc for a first time. So she waited patiently until her name was called. But that never happened. Finally the waiting room was empty and the assistant came to her asking whilst pointing to a name on a sheet of paper: "Is this you? ". Yes it is", said my spouse. But "why", said the assistant: "we have called you a dozen times". "No you didn't" said my spouse.
Last edited: May 3, 2013
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2017-08-23 19:58:17
|
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|
https://aiaspirant.com/pandas-math-functions/
|
# Pandas Math Functions
In this article we’ll learn some of the commonly used math functions in Pandas. Let’s get started.
### The abs() function:
The first one we are going to see is the abs() function. This function returns the absolute values for the elements in the data frame.
There are few negative values. Let’s apply the abs function.
The abs() function can also be applied to complex numbers. In the case of complex numbers the absolute value is calculated as $\sqrt{a^{2}+b^{2}}$
Let’s create a new column B and populate it with complex numbers.
After applying the abs function the results are as follows
### The clip() function:
The clip takes two values lower limit and upper limit. If the values in the data frame exceeds this value then it will be clipped to the upper limit. If it is lower than the lower limit then it will be increased to the lower limit.
### The count() function:
This function will return the number of non null values in the data frame for each column. Let’s apply the count function on the data frame.
The following is our data frame
Let’s apply count function on this data frame.
Since there are no null values in the data frame the result is ten which is the number of observations. Let’s introduce a null value in the column B and again apply count function.
Now, if we apply the count function we’ll get the following results
By specifying the axis=1 we can get the number of non null values for each row.
### The min, med, mean, max:
These functions are straight forward.
One thing worth mentioning is that by default the null values are not included in the calculation. If you want to include them you can set the parameter skipna to False.
### The rank() function:
The rank() function returns the rank of the values in the data frame. Consider the following data frame.
The result of applying rank on the above data frame is as follows
By default it ranks the values in ascending order. If you don’t want to rank the values in the ascending order you can set the parameter ascending to False.
Another important argument worth mentioning is the method. It helps to decide what to do in case of a tie. There are various options like average, min, max, first. Let’s take a look at each of them.
Consider the following data frame.
Average will use the average rank of the group and apply to all items with the same rank.
Let’s see how these ranks are assigned.
The smallest element here is 0. Since there are three of them they will get the ranks 1,2 and 3.
Now we need to calculate the average of these ranks i.e., (3 + 2 + 1) / 3 = 2. So, the rank 2 will be assigned to all the zeros.
The next smallest element is 1. Since there are already three elements which are smaller than 1 we’ll assign a rank of 4 and 5 for the two ones. Again we’ll apply the average to find the rank of 1. The average of 4 and 5 is 4.5
So, the rank 4.5 will be assigned to 1.
The last element 2 will get a rank of 6.
Now let’s see how the ‘min’ will assign the ranks.
The min will assign lowest rank to all items.
Similarly the ‘max’ will assign highest rank to all items.
In ‘first’ the rank to the items will be assigned in the order they appear.
## Summary:
In this article we learned some of the commonly used math functions in Pandas.
We discussed a total of five functions. They are:
abs(): returns the absolute values for the elements in the data frame.
clip(): if the values in the data frame exceeds the upper or lower limit it will be clipped.
count(): returns the number of non null values in the data frame.
min, max, med, mode: returns the min, max, med and mode for each column in the data frame.
rank(): return the rank of the elements in the data frame.
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2021-09-24 08:48:58
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http://koreascience.or.kr/article/JAKO201224963706418.page
|
3차원 공간의 인식을 위한 블록기반 3D맵
• Yi, Jong-Su (School of Electrical and Electronics Engineering, Chung-Ang University) ;
• Kim, Jun-Seong (School of Electrical and Electronics Engineering, Chung-Ang University)
• 이정수 (중앙대학교 전자전기 공학부) ;
• 김준성 (중앙대학교 전자전기 공학부)
• Accepted : 2012.06.21
• Published : 2012.07.25
Abstract
A 3D map provides useful information for intelligent services. Traditional 3D maps, however, consist of a raw image data and are not suitable for real-time applications. In this paper, we propose the Block-based 3D map, that represents three dimensional spaces in a collection of square blocks. The Block_based 3D map has two major variables: an object ratio and a block size. The object ratio is defined as the proportion of object pixels to space pixels in a block and determines the type of the block. The block size is defined as the number of pixels of the side of a block and determines the size of the block. Experiments show the advantage of the Block-based 3D map in reducing noise, and in saving the amount of processing data. With the block size of $40{\times}40$ and the object ratio of 30% to 50% we can get the most matched Block-based 3D map for the $320{\times}240$ depthmap. The Block-based 3D map provides useful information, that can produce a variety of new services with high added value in intelligent environments.
Acknowledgement
Supported by : 한국연구재단
References
1. Yong Zhang; Yikang Gu; Vlatkovic, V.; Xiaojuan Wang; , "Progress of smart sensor and smart sensor networks," Intelligent Control and Automation, 2004. WCICA 2004. Fifth World Congress on, vol.4, pp. 3600-3606, June 2004.
2. S. T. Barnard, M. A. Fischler, "Computational Stereo," ACM Computing Surveys, Volume 14, Issue 4, pp. 553-572, 1982. https://doi.org/10.1145/356893.356896
3. C. F. Olson, "Maximum-likelihood image matching," Pattern Analysis and Machine Intelligence, IEEE Transactions on, Volume 24, No 6, pp. 853-857, 2002.
4. 서자원, 김창익. "스테레오 카메라 영상처리 기술 및 동향," 전자공학회지, 제38권, 제2호, 31-36쪽, 2011년
5. C. Huahua, X. Zezhong, "3D Map Building Based on Stereo Vision," Networking, Sensing and Control, ICNSC '06. Proceedings of the 2006 IEEE International Conference on, pp. 969-973, 2006.
6. J. M. Saez, F. Escolano, "A global 3D map-building approach using stereo vision," Robotics and Automation, 2004. Proceedings. ICRA '04 IEEE International Conference on, Volume 2, pp. 1197-1202, 2004.
7. A. Cappalunga, S. Cattani, A. Broggi, M. S. McDaniel, S. Dutta, "Real time 3D terrain elevation mapping using ants Optimization Algorithm and stereo vision," Intelligent Vehicles Symposium (IV), IEEE, pp. 902-909, 2010.
8. S. Jian-Hong, J. Byung-Seung, L. Jong-Wook, L. Myo-taeg, "Stereo vision based 3D modeling system for mobile robot," Control Automation and Systems (ICCAS), International Conference on, pp. 71-75, 2010.
9. KATS 기술보고서 제17호(지능형 홈 산업 및 표준화 동향), 기술표준원, 2010.
10. 이정수, 김준성. "비전 시스템 구현을 위한 SAD 정합 알고리즘의 변형," 전자공학회논문지-CI, 제47권, 제5호, 61-66쪽, 2010년
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2021-04-12 21:38:38
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https://socratic.org/questions/how-do-you-solve-15x-2-7x-55-using-the-quadratic-formula
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# How do you solve 15x^2+7x-55 using the quadratic formula?
May 5, 2016
$x = \frac{- 7 \pm \sqrt{3349}}{30}$
#### Explanation:
Strictly speaking, I guess you would like to solve $15 {x}^{2} + 7 x - 55 = 0$ or in other words find the zeros of $15 {x}^{2} + 7 x - 55$. You do not "solve" a quadratic expression.
That having been said, the equation $15 {x}^{2} + 7 x - 55 = 0$ is of the form $a {x}^{2} + b x + c = 0$ with $a = 15$, $b = 7$ and $c = 55$.
This has roots (solutions) given by the quadratic formula:
$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
$= \frac{- 7 \pm \sqrt{{\left(- 7\right)}^{2} - \left(4 \cdot 15 \cdot \left(- 55\right)\right)}}{2 \cdot 15}$
$= \frac{- 7 \pm \sqrt{49 + 3300}}{30}$
$= \frac{- 7 \pm \sqrt{3349}}{30}$
The square root $\sqrt{3349}$ does not simplify further since the prime factorisation of $3349$ is $17 \cdot 197$, which contains no square factors.
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2019-01-20 03:15:10
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https://physics.stackexchange.com/questions/261392/observer-travelling-near-speed-of-light-observing-a-laser-pointer-seeing-photons
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# Observer travelling near speed of light observing a laser pointer seeing photons at speed of light?
I'm a little confused by the following classic example about the speed of light being constant for all observers (paraphrasing):
Jack and Jill are both travelling at a significant portion of the speed of light; Jack is travelling behind Jill. Jack shines a laser pointer at Jill. Both observe the light emitted from it as travelling at the speed of light, $c$.
I understand that the speed of the emitter of the light doesn't influence the speed of light; the photons always travel at the maximum speed they can. If Jack were moving backwards, the photons would "instantly accelerate" to $c$; Jack moving backwards wouldn't detract from their speed. Jack moving forwards isn't adding to their speed, because they can't move faster than $c$.
However, the observer's perspective is harder to understand for me. Assuming Jill is observing the incoming laser by counting the frequency of incoming photons, shouldn't that rate change depending on how fast Jill is moving herself? Assuming Jill would be travelling at the speed of light (ignoring the practical impossibility), shouldn't she not be receiving any light at all? Scaling that down to her travelling at half $c$, shouldn't her incoming photon rate be half as much as if she was "standing still"?
(This is very related to Seeing light travelling at the speed of light, however it's about the other side of the same setup.)
I think your point of view needs a little correction.
You are thinking in terms of 3D space but it was shown by relativity that space and time are actually related and make a 4D world. The relative speed between two persons also alters the rate of change of time in their frames hence the photons do not accelerate or deaccelerate to 'maintain their speed'.
However they do accelerate or deaccelerate (i.e. gain or loose energy) and this is manifested in the Doppler shift of the light. If two sources are moving away from each other they will see red shifted (observed wavelength is longer than emitted wavelength) light, and if they are moving towards each other they will see blue shifted light.
Hypothetically if two persons are moving away with relative velocity c then the light emitted by person1 will be seen by person2 as 0 frequency i.e. DC (and he practically do not see any light). However if the relative velocity is anything less than c person2 will definitely see emission at non-zero frequency.
In this case if emitter is moving with velocity v and observer is still or if observer is moving with velocity -v and emitter is still does not make any difference, the end result will be same.
I hope it will help to clear your confusion.
• I was afraid that the interconnectedness of spacetime would be involved in the answer... :) Are you saying that a) due to relativistic effects and time dilation (excuse me if I'm misusing these terms) Jill would observe the light as plain old c; or b) that Jill would observe the light as half c, but know to correct her POV by compensating for redshift and after this compensation conclude that the speed is c? – deceze Jun 8 '16 at 10:00
• you are right option a) is correct the light will remain as plain old c. This experimental observation is actually the basis of the origin of spatial relativity. – hsinghal Jun 8 '16 at 10:07
• Alright, that sets me off in the right direction then. I understand it in the abstract, I think, but can't really wrap my head around it entirely just yet. Thank you! – deceze Jun 8 '16 at 10:09
WHEN jack is moving backwards at velocity C/2 THEN Jill will still receive the light at velocity C BUT at half the frequency of the transmitter (doppler). As long as the relative velocity between jack (transmitter) and Jill (receiver) ,assuming they are going away from each other , is less than C THEN SPEED OF LIGHT IS UNAFFECTED but the freq received is decreased ).Now if jack is moving backwards at velocity C ,then doppler leads to that the received freq is zero which as equation C= F * WAVELENGTH LEADS TO THAT NO LIGHT IS RECEIVED BY jILL .
• In other words, the individual photons will still travel at c, but you'll receive half as many in the same amount of time? – deceze Aug 8 '16 at 7:04
• Jack fundamentally cannot "move backwards at $c$". – pela Aug 8 '16 at 7:31
• pls use the formatting features in the editor instead of writing in capitals, it makes it hard to read your answer. – Wolpertinger Aug 8 '16 at 7:51
• Welcome on Physics SE and thank you for your answer :) Please refrain from USING CAPSLOCK BECAUSE IT SEEMS VERY AGGRESSIVE and see the help section for information about the built-in TeX-editor to format formulas. – Sanya Aug 8 '16 at 7:55
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2019-07-21 15:58:14
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https://www.physicsforums.com/threads/operator-theory-problem-on-momentum-operator-qm.699320/
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# Homework Help: Operator Theory Problem on Momentum Operator (QM)
1. Jun 29, 2013
### LolWolf
1. The problem statement, all variables and given/known data
Given the operators $\hat{x}=x\cdot$ and $\hat{p}=-i\hbar \frac{d}{dx}$, prove that:
$[\hat{x}, g(\hat{p})]=i\hbar \frac{dg}{d\hat p}$
2. Relevant equations
None.
3. The attempt at a solution
I have very little idea on how to begin this problem, but I don't want a solution, I simply want a hint in the right direction.
Thanks, mates.
Last edited: Jun 29, 2013
2. Jun 29, 2013
### Dick
Expand $g(p)$ in a power series. What's $[x,p^n]$?
3. Jun 29, 2013
### LolWolf
Actually, I realized it was even easier than that, but thank you!
Consider the case in momentum-space rather than position-space, and this reduces nicely using elementary operations.
4. Jun 29, 2013
### Dick
Sure, that works also. x is a differentiation operator in p space.
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2018-07-17 14:40:39
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https://dsp.stackexchange.com/questions/41183/why-sampling-rate-is-not-an-option-with-normalized-frequency-in-matlab-filter-de
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# Why sampling rate is not an option with normalized frequency in Matlab Filter Designer
My Matlab version is 2017b. When I use Filter designer with normalized frequency specifications, I cannot set the sampling rate. May you tell me why the sampling frequency cannot be specified? As I know, the normalized frequency mapped $[0,\pi]$ to $[0,1]$, so the sampling rate is not dropped out.
...the normalized frequency mapped $[0,\pi]$ to $[0,1]$...
The normalized frequency $F$ is defined as $f/f_s$ where $f$ is the "real" frequency (i.e. not the normalized one) and $f_s$ is the sampling frequency. The map is thus from $[0, f_s]$ to $[0, 1]$ (or to $[0, 2\pi]$ in normalized pulsation $\Omega = 2\pi F$). When using normalized frequency $F$, you thus specify frequencies as a percentage of the sampling frequency. Hence, the sampling frequency itself is not directly needed.
• Yes, but it worries me that the screenshot shows the (greyed out) sampling rate as 10 and the magnitude plot goes from 0 to 5. :-)
– Peter K.
May 23 '17 at 18:25
• @PeterK. Sorry I am new to DSP. Is this setting a big error? I am also wondering wheter 0.5*pi is a legitimate option. May 23 '17 at 18:30
• @PeterK: if the interface of the "Filter Design & Analysis Tool" didn't change compared to my MATLAB R2014a version, the interface only refreshes the plot after clicking the "Design Filter" button. After this refresh, the plot goes from 0 to $\pi$. May 23 '17 at 18:32
The point of "normalized frequency" is exactly that, it is normalized to the sampling rate.
• Thank you. Does it normalized to sampling rate or half of the sampling rate? May 23 '17 at 18:15
• I'm pretty sure fs/2 is normalized to 1, you cannot display a frequency higher than fs/2
– Ben
May 23 '17 at 18:28
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2021-12-06 05:28:33
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https://codegolf.stackexchange.com/questions/107265/ascii-art-flight-simulator?answertab=oldest
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# ASCII Art “Flight Simulator”
EDIT
It appears that there has been some confusion following my typo in the original post which used a lowercase o to define the plane and then an uppercase later. Unfortunately this bug did not get picked up in the Sandbox. Since many members have written answers with both and since the typo was my fault I will allow either uppercase or lowercase o in the definition of the plane. I have added a new rule for this.
Background
I like ascii art animations as I tend to call them so here is another one. I don't think this is too difficult to implement so will hopefully get some short and interesting answers.
To all fellow community members
old byte count new byte count
so we can see your progress. Thanks!
Challenge
Here is an ascii plane
--O--
Here is an ascii runway
____| |____
The plane starts at 5 newlines above the runway. To prevent any clashes between metric and imperial systems and make this a truly international challenge I won't mention meters or feet. Example:
--O--
____| |____
The Plane must land exactly in the middle of the runway as shown below:
____|--O--|____
Input
The initial horizontal position of the plane is defined by an integer input which is used to reference the tip of the left wing i.e. it is between 0 and 10 inclusive.
Output
Each stage of the planes flight must be shown. Example below (input=10):
--O--
____| |____
--O--
____| |____
--O--
____| |____
--O--
____| |____
--O--
____| |____
____|--O--|____
To keep things simple, we are ignoring the laws of perspective. The runway stays the same size as you get closer.
Rules
• Update The middle of the plane can be either an uppercase or lowercase o but whichever is chosen must be consistent throughout the code. If your language does not support the characters above feel free to use alternative ascii only characters.
• The plane descends 1 line per frame.
• The plane can only move 1 space to the left or right each time it descends one line. It does not have to move on each line of descent. As long as it finishes on the runway it is up to you when it moves right or left. You're the pilot!
• No error handling required. You may assume that the input will always be a valid integer from 0-10 inclusive.
• Output must consist of only the characters shown above (if your language does not support them the see edited first rule) and must be the same size i.e. must start 6 lines high by 15 characters wide. The height can decrease as it progresses as in the example above.
• Program or function is fine but must produce an output as shown above.
• Leading/trailing spaces/newlines are fine by me.
• Please feel free to clear the screen between output frames if you wish. This is not a requirement.
• Standard loopholes prohibited as usual (although I don't think there are many that would help with this kind of challenge).
• This is code golf so shortest answer is obviously the winner and will probably get most votes but may not necessarily be accepted as the best answer if some really interesting solution comes along in some unexpected language, even if it is longer. Feel free to post anything that meets the rules as long as it works.
Ungolfed reference implementation in Python 2 available at Try it online! so you can see how it looks for different input values.
• I don't think it is kolmogorov-complexity as the output depends on the input – ovs Jan 18 '17 at 19:25
• Thanks for the clarification @ovs. I'll remove that tag then. – ElPedro Jan 18 '17 at 19:25
• Usually, the acceptance goes to the answer that complies best with the Objective Winning Criterion. You may get some flak if you accept another, longer answer. – Level River St Jan 18 '17 at 22:23
• Thanks @LevelRiverSt. Is there a meta post to clarify this? If not then maybe better to not accept any answer. – ElPedro Jan 18 '17 at 22:26
• btw, I have accepted a longer answer before and given credit to the shorter answer as well with no probs from the community Previous challenge. Please see my Result comment at the end of the question. Was this wrong? – ElPedro Jan 18 '17 at 22:31
# TI-BASIC, 61 bytes
Input A
A
For(B,1,5
ClrHome
Output(5,1,"----/ /----
Output(B,Ans,"--O--
Ans+6-median({5,7,Ans
End
• Do you know of an online interpreter or download (For Linux) for testing? +1 for the answer assuming it works :) – ElPedro Jan 18 '17 at 22:00
• Check out TilEm. It's the only one I could get working. – Julian Lachniet Jan 18 '17 at 22:03
• +1 for asking someone who may have had a different answer. Will certainly check out TilEm and thanks for the tip. – ElPedro Jan 18 '17 at 22:13
# JavaScript (ES6), 108 bytes
f=(a,b=5)=>b?" ".repeat(a)+--O--${ .repeat(b)}____| |____ +f(a<5?a+1:a-1,b-1):"____|--O--|____" ## Test it f= (a,b=5)=>b?" ".repeat(a)+--O--${
.repeat(b)}____| |____
+f(a<5?a+1:a-1,b-1):"____|--O--|____"
<input type=number min=0 max=10 oninput=o.textContent=f(+this.value)><pre id=o>
## Usage
Just call f with the index of the plane.
f(2)
## Output
--O--
____| |____
--O--
____| |____
--O--
____| |____
--O--
____| |____
--O--
____| |____
____|--O--|____
• You can add a <s>snack</s> stack snippet – Cows quack Jan 18 '17 at 19:52
• Every time I ask a question the first answer is Javascript! +1 – ElPedro Jan 18 '17 at 19:55
• Hey, would be nice if people posted either a Tryitonline (don't know if that is possible with Javascript) or a different solution to the 10 example shown above. Can you post the output from e.g. 2 instead? :) – ElPedro Jan 18 '17 at 19:59
• @ElPedro, you can execute JavaScript in your browser console, but there also are some online consoles. I'll add a link. I'll also change the example. – Luke Jan 18 '17 at 20:13
• Thanks. No probs. I'm into old time Javascript where you need a web page to execute it. Guess I need to get with the times :) More serverside these days. Respect for the fast and cool answer. – ElPedro Jan 18 '17 at 20:18
## Batch, 230 bytes
@echo off
set/ax=10-%1
set s= --O--
for /l %%i in (0,1,4)do call:l %%i
echo ____^|--O--^|____
exit/b
:l
call echo %%s:~%x%%%
for /l %%j in (%1,1,3)do echo(
echo ____^| ^|____
echo(
set/a"x-=x-5>>3,x+=5-x>>3
x is the number of spaces to remove from the beginning of the string s, so I subtract the parameter from 10. The last line is the nearest Batch has to x-=sgn(x-5).
## Perl, 94 bytes
93 bytes of code + -p flag.
$\="____| |____ ";$p="--O--";for$i(-5..-1){print$"x$_.$p.$/x-$i;$_+=5<=>$_}$\=~s/ +/$p/}{
Try it online!
• @ETHproductions Hope you enjoy the }{ (and the $" messing with the syntax highlighting). – Dada Jan 18 '17 at 20:31 # Python 2, 160 bytes i,s,p,l,r,c,x=input(),' ','--O--','____|','|____',0,4 while x>=0:print'\n'.join([s*i+p]+[s*15]*x+[l+s*5+r])+'\n';c+=1;x-=1;i=((i,i-1)[i>5],i+1)[i<5] print l+p+r Try it online! Here is the reference implementation golfed down to 160 from 384. Still a way to go I think. Just posted for fun and to encourage a better Python answer. • You can compete in your own challenge (see this meta post). – Dada Jan 18 '17 at 20:51 • Can you just do while-~x? – FlipTack Jan 18 '17 at 20:56 • Also I think you can write the bit where you either add or subtract from i as i+=(i<5)-(i>5) – FlipTack Jan 18 '17 at 21:01 # TI-BASIC, 62 bytes :Input A :A :For(N,3,8 :ClrHome :Output(8,1,"----I I---- :Output(N,Ans,"--O-- :Ans+(Ans<6)-(Ans>6 :End Note that TI-BASIC does not support _ or | and therefore I replaced with a capital I and -. This should not effect byte count. • OK, I'm on Linux. Can you recommend a download that I get to test this? btw, I assume it works until I find an interpreter so +1 :) – ElPedro Jan 18 '17 at 21:47 • Unfortunately, no. I do have both Wabbitemu and TilEm installed on my Windows 10 computer, but I test the code on a physical TI-84+. Sorry – Golden Ratio Jan 18 '17 at 21:54 • No problem! Just asking :) – ElPedro Jan 18 '17 at 21:58 • Due to a lot of editing of code, the fastest alternated between this post and Julian Lachniet's, until we both came to the 60 byte conclusion, at which point I added clrhome and made byte count 62 – Golden Ratio Jan 18 '17 at 22:09 • TI-Basic?! Nice! – Dave Kanter Jan 18 '17 at 23:10 ## Scala, 224 181 bytes EDIT: I had no idea you could do "string"*n to repeat it n times! Scala continues to blow my mind. Missing the if(t>0) instead of if(t==0) was a rookie mistake. Thanks for the tips, Suma! def?(x:Int,t:Int=5):Unit={var(p,o)=("--o--","") o=s"____|${if(t>0)" "*5 else p}|____\n"
for(i<-0 to t)o=if(i!=0&&i==t)" "*x+p+o else "\n"+o
println(o)
if(t>0)?(x-(x-4).signum,t-1)}
Original remarks:
I figured a recursive solution would be fun to try. I'm relatively new to Scala, so I'm certain this is far from optimal.
• You might want to read Tips for golfing in scala – corvus_192 Jan 19 '17 at 8:42
• You don't need the :Unit=. Omitting the equals sign will set the return type to Unit. – corvus_192 Jan 20 '17 at 8:21
• Also, why didn't you initialize o in the first line?. And since i is always >= 0, you can change i!=0&&i==t to i>0&i==t (3rd line). – corvus_192 Jan 20 '17 at 8:25
# Befunge-93, 136 130 bytes
&5>00p10p55+v
:::00g>:1-\v>:"____| |_"
>:1-\v^\+55_$"--O--"10g ^\*84_$>:#,_10g::5v>:#,_@
<_v#!:-1g00+\5\-<^"____|--O--|____"
Try it online!
Explanation
& Read the plane position.
5 Initialise the plane height.
> Begin the main loop.
00p Save the current height.
10p Save the current position.
55+: Push two linefeed characters.
"____| |_" Push most of the characters for the airport string.
::: Duplicate the last character three times to finish it off.
00g>:1-\v Retrieve the current height, and then push
^\+55_$that many copies of the linefeed character. "--O--" Push the characters for the plane. >:1-\v 10g Retrieve the current position, and then push ^\*84_$ that many copies of the space character.
>:#,_ Output everything on the stack in reverse.
10g:: Retrieve the current position and make two copies to work with.
5v If it's greater than 5
-< then subtract 1.
+\5\ If it's less than 5 then add 1.
g00 Retrieve the current height.
-1 Subtract 1.
_v#!: If it's not zero, repeat the main loop.
^"____|--O--|____" Otherwise push the characters for the landed plane.
>:#,_@ Output the string and exit.
# Perl 6, 9790 81 bytes
{say "{"{" "x 15}\n"x 5}____| |____"~|("\0"x$^h+$_*(17-$h/5)~"--O--") for ^6} Contrary to what it looks like, it outputs the *lower-case version of the plane (--o--), as allowed by the updated task description. Try it online! ### How it works Bitwise string operators FTW! { # Lambda accepting horizontal index$h.
say # Print the following:
"{ "{ " " x 15 }\n" x 5 }____| |____" # The 15x6 background string,
~| # bitwise-OR'd against:
(
"\0" # The NULL-byte,
x $^h +$_*(17 - $h/5) # repeated by the plane's offset, ~ "--O--" # followed by an OR mask for the plane. ) for ^6 # Do this for all$_ from 0 to 5.
}
It works because bitwise string operators use the codepoint values of the characters at a given position in two strings, to calculate a new character at that position in the output string.
In this case:
space OR O = o
space OR - = -
any OR \0 = any
For an uppercase-O plane, we could have used ~^ (string bitwise XOR), with a plane mask of \r\ro\r\r (+4 bytes for backslashes):
space XOR o = O
space XOR \r = -
any XOR \0 = any
The formula for the plane's offset, h + v*(17 - h/5), was simplified from:
v*16 # rows to the vertical current position
+ h # columns to the horizontal starting position
+ (5 - h)*v/5 # linearly interpolated delta between horizontal start and goal
## Python 2, 107 bytes
n=input();h=5
while h:print' '*n+'--O--'+'\n'*h+'____| |____\n';n-=cmp(n,5);h-=1
print'____|--O--|____'
Try it online
Simply hardcodes the last line for the landing plane. It can likely be golfed by re-using parts from before or being integrated into the loop.
# sed, 181 bytes + 2 for -nr flags
s/10/X/
:A
s/^/ /;y/0123456789X/-0123456789/;/[0-9]/bA;s/ -/P\n\n\n\n\n____|P|____/
:B
h;s/P([\n|])/--O--\1/;s/P/ /;s/^ *_/_/;p;/^_/q;x;s/\n//
/^ {5}$/bB;/ {6}/s/ //;s/^/ /;bB ## Ungolfed # Add leading spaces s/10/X/ :A s/^/ / y/0123456789X/-0123456789/ /[0-9]/bA s/ -/P\n\n\n\n\n____|P|____/ :B # Place plane in appropriate spot h s/P([\n|])/--O--\1/ s/P/ / s/^ *_/_/ p /^_/q x # Movement s/\n// /^ {5}$/bB
# move left one extra, since we'll move right next line
/ {6}/s/ //
s/^/ /
bB
Usage: $echo 2 | sed -nrf flightsim.sed # Retina, 86 83 bytes .+$* --O--¶¶¶¶¶¶____| |____
{*$¶ 2D¶ ( {5})$1
}^ {0,4}-
$& + --O-- G_ Try it online! There is probably some sort of compression I could have used on the runway and the empty space over it, but anything I tried came up more expensive than plaintext (in Retina ¶ is a newline, so you can see the initial state in plaintext on the second line). # Ruby, 94 bytes ->a{5.times{|i|puts" "*a+"--O--#{?\n*(5-i)}____| |____ ";a+=5<=>a};puts"____|--O--|____"} Prints the plane's position followed by newlines and then the airport. Then it moves the plane by 1, -1, or 0, depending on its position relative to 5. After looping the above 5 times, it prints the plane in the airport. # 8th, 177 172 bytes : f 5 >r 5 repeat over " " swap s:* . "--O--" . ' cr r> times "____| |____\n\n" . over 5 n:cmp rot swap n:- swap n:1- dup >r while "____|--O--|____\n" . 2drop r> drop ; The word f expects an integer between 0 and 10. Usage 4 f Explanation : f \ n -- 5 >r \ Push vertical distance from airport to r-stack 5 repeat \ Print plane over " " swap s:* . "--O--" . \ Print airport ' cr r> times "____| |____\n\n" . \ Now on the stack we have: \ distanceFromLeftSide distanceFromAirport over \ Put distance from left side on TOS 5 n:cmp \ Compare left distance and 5. Return \ -1 if a<b, 0 if a=b and 1 if a>b rot \ Put distance from left side on TOS swap n:- \ Compute new distance from left side swap n:1- \ Decrement distance from airport dup >r \ Push new airport-distance on the r-stack while "____|--O--|____\n" . \ Print final step 2drop r> drop \ Empty s-stack and r-stack ; ## Scala, 177, 163, 159 137 bytes def p(x:Int,t:Int=5,a:String="\n"):String=a+(if(t>0) " "*x+"--O--"+"\n"*t+"____| |____\n"+p(x-(x-4).signum,t-1)else"____|--O--|____") Based on another answer, with significant reductions. ## QBIC, 9391 84 bytes :{X=space$(a)+@--O--┘a=a-sgn(a-5)~t>-1|?X[t|?]t=t-1?@____|+@ +_fB|\_xB+A+_fB
Dropped some bytes by replacing the declaration of X$; optimised the FOR loop that prints the distance above-ground. Explanation below is for the old version, but it basically works the same. For testing (and aesthetics) I had a slightly different version, at 103 bytes: :{_z.5|_CX=Y[a|X=X+@ ]X=X+@--O-- a=a-sgn(a-5) ~u>0|?X';[u|?]u=u-1?@____|+@ +_fC|\_xC+_tB+_fC These are functionally identical. The second one has the addition that the screen gets cleared between frames and that it halts for 0.5 seconds between frames. # Sample output Note that I've added two newlines between frames. The most golfed code above does not add empty lines between frames, the cooler one clears the screen. Command line: 10 --O-- ____| |____ --O-- ____| |____ --O-- ____| |____ --O-- ____| |____ --O-- ____| |____ ____|--O--|____ # Explanation Since I feel this touches upon many things I really like about QBIC, and gives a good insight in how some of its functions work under the hood, I've gone a bit overboard on the explanation. Note that QBIC is, at its core, an QBasic interpreter for Codegolf. QBIC code goes in - QBasic code comes out (and is subsequently executed). :{ get the starting offset (called 'a') from the command line, and start a DO-loop ---- cool code only ---- _z.5|_C At the start of a DO-loop, pause for half a second and clear the screen ---- resume golf-mode ---- ---- #1 - The tip of the left wing is anywhere between 0 and 10 positions to the right. ---- Create the plane with the spacing in X$
X=Y Clear X$[a| For each point in the current offset X=X+@ ] Add a space to X$
- Every capital letter in QBIC references that letter+$, a variable of type String - @ and start and end a string literal, in this case a literal space. - ] ends one language construct (an IF, DO or FOR). Here, it's NEXT X=X+@--O-- Create the actual plane - @ and once again create a string literal. Every literal that is created in this way is assigned its own capital letter. This is our second literal, so the body of our plane is stored in B$ (A$contains the space, remember?) ---- #2 Adjust the offset for the next iteration a=a-sgn(a-5) The clever bit: We have an offset X in the range 0 - 10, and 5 attempts to get this to be == 5. X - 5 is either positive (X = 6 - 10), negative (X = 0 - 4) or 0 (X=5). sgn() returns the sign of that subtraction as a 1, -1 or 0 resp. We then sub the sign from 'a', moving it closer to 5. ---- #3 Draw the plane, the empty airspace and the landing strip ~u>0| Are we there yet? - ~ is the IF statement in QBIC - It processes everything until the | as one true/false expression - All the lower-case letters are (or better, could be) references to numeric variables. Since QBasic does not need to post-fix those, they double as 'natural' language: ignored by QBIC and passed as literal code to the QBasic beneath. - The lower-case letters q-z are kinda special: at the start of QBIC, these are set to 1 - 10. We haven't modified 'u' yet, so in the first DO-loop, u=5 ?X'; If we're still air-borne, print X$ (our plane, incl. spacers)
- ? denotes PRINT, as it does in QBasic.
- ' is a code literal in QBIC: everything until the is not parsed, but
passed on to QBasic.
- In this case, we want a literal ; to appear after PRINT X$. This suppresses QBasic's normal line-break after PRINT. This needs to be a code literal because it is the command to read a String var from the command Line in QBIC. [u|?] FOR EACH meter above the ground, print a newline u=u-1 Descent 1 meter ?@____| Print the LHS of the landing strip +@ plus 5 spaces +_fC| plus the LHS reversed. \ ELSE - touchdown! _x Terminate the program (effectively escape the infinite DO-loop) - the _x command has an interesting property: ULX, or Upper/Lowercase Extensibility. Writing this command with an uppercase _X does something similar, yet different. The _x command terminates, and prints everything found between _x and | before quitting. Uppercase _X does not look for |, but only prints something if it is followed by a character in the ranges a-z and A-Z - it prints the contents of that variable. C+B+_fC But before we quit, print C$ (the LHS of the landing strip) and the plane,
and the LHS flipped.
---- #4 QBIC has left the building
- Did I say _x looks for a | ? Well, that gets added implicitly by QBIC at the end of
the program, or when one ( ']' ) or all ( '}' ) opened language constructs are closed.
- Also, all still opened language constructs are automatically closed at EOF.
FOR I=0TO 4?" "*X;"--O--";CHR$(10)*(4-I)?G$;"| |";G$X=X-SGN(X-5)?NEXT?G$;"|--O--|";G$ # PHP 7, 139 bytes still awfully long for($x=$argv[1],$d=6;$d--;$x+=5<=>$x)for($i=$p=-1;$i++<$d;print"$s
")for($s=$i<$d?" ":"____| |____ ";!$i&++$p<5;)$s[$x+$p]="--O--"[$p]; takes input from command line argument; run with -r. breakdown for($x=$argv[1], // take input$y=6;$y--; // loop height from 5 to 0$x+=5<=>$x) // post increment/decrement horizontal position for($i=$p=-1;$i++<$y; // loop$i from 0 to height
print"$s\n") // 3. print for($s=$i<$y?" ":"____| |____\n"; // 1. template=empty or runway+newline
!$i&++$p<5;)$s[$x+$p]="--O--"[$p]; // 2. if \$i=0, paint plane
|
2019-07-21 11:39:37
|
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|
https://tex.stackexchange.com/questions/210668/line-between-cells
|
# Line between cells
How to add line between cells in tabular like on this image between 38 and 77 in LaTeX?
I have only this code, that draws table:
\documentclass{scrartcl}
\begin{document}
\begin{tabular}{ |c|c|c|c|c|c|c|c|c|c|c|c|c| }
\hline
23 & 18 & 41 & 38 & 73 & 56 & 52 & 91 & 77 & 60 & 95 & 87 & 64\\
\hline
\end{tabular}
\end{document}
Here I introduce \mybox{number} and \connect{start}{mid}{end} to accomplish the task. I use \ignorespaces so that you don't have to remember to keep adding % signs everywhere.
The underbar height and rule thickness are controlled by \rlht and \rlwd, respectively. Works for boxed content of any width (e.g., \mybox{1} and \mybox{1248} both work).
\documentclass{article}
\usepackage{stackengine}
\newsavebox\tmpbox
\def\rlht{1ex} \def\rlwd{.8pt}
\newcommand\vstrut{\smash{\makebox[0pt]{\rule{\rlwd}{\rlht}}}}
\def\conR#1{%
\savebox\tmpbox{#1}%
\stackunder[\dimexpr\rlht-\rlwd\relax]{\usebox{\tmpbox}}{%
\rule{.5\wd\tmpbox}{0pt}\vstrut\rule{.5\wd\tmpbox}{\rlwd}}%
}
\def\conL#1{%
\savebox\tmpbox{#1}%
\stackunder[\dimexpr\rlht-\rlwd\relax]{\usebox{\tmpbox}}{%
\rule{.5\wd\tmpbox}{\rlwd}\vstrut\rule{.5\wd\tmpbox}{0pt}}%
}
\def\conM#1{%
\savebox\tmpbox{#1}%
\stackunder[\dimexpr\rlht-\rlwd\relax]{\usebox{\tmpbox}}{%
\rule{\wd\tmpbox}{\rlwd}}%
}
\newcommand\connect[3]{\conR{#1}\conM{#2}\conL{#3}\ignorespaces}
\newcounter{Index}
\def\mybox#1{%
\stackon[1pt]{\fbox{#1}}{\scriptsize\arabic{Index}}%
\stepcounter{Index}%
\kern-\fboxrule%
\ignorespaces%
}
\begin{document}
\mybox{23}\mybox{18}\mybox{41}
\connect{\mybox{38}}{\mybox{73}\mybox{56}\mybox{52}\mybox{91}}{\mybox{77}}
\mybox{60}\mybox{95}\mybox{87}\mybox{64}
\end{document}
Another solution, with TikZ.
# Code
\documentclass{scrartcl}
\usepackage{tikz}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}
\matrix(m)[matrix of nodes,
row sep=-\pgflinewidth, column sep=-\pgflinewidth, nodes={draw},
]{
23 & 18 & 41 & 38 & 73 & 56 & 52 & 91 & 77 & 60 & 95 & 87 & 64\\
};
\draw[very thick](m-1-1.north west)rectangle(m-1-13.south east); % border
\foreach \i in {1,...,13}{
\pgfmathparse{\i-1}
\node[at=(m-1-\i.north),anchor=south,font=\footnotesize]{\pgfmathprintnumber{\pgfmathresult}};
}
\draw(m-1-4.south)|-+(0,-5pt)-|(m-1-9.south);
\end{tikzpicture}
\end{document}
|
2019-10-20 01:37:57
|
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|
http://spindynamics.org/wiki/index.php?title=Human2opspec.m
|
# human2opspec.m
Converts user-friendly descriptions of spin states and operators into the formal operator and state description used by Spinach kernel.
## Syntax
The function supports three types of syntax:
1. If both inputs are strings, e.g.
[opspecs,coeffs]=human2opspec(spin_system,'Lz','13C')
the function returns a list of single-spin opspecs for all spins with the specified name. In the example above, the list of Lz operator specifications for all 13C nuclei in the system would be returned. Valid labels for operators in this type of call are 'E', 'Lz', 'L+', 'L-' and 'Tl,m'. In the latter case, l and m are integers. Valid labels for the spins are standard isotope names as well as 'electrons', 'nuclei', 'all'.
2. If one input is a string and the other is a vector, e.g.
[opspecs,coeffs]=human2opspec(spin_system,'Lz',[1 2 4])
the function returns a list of single-spin opspecs for all spins with the specified number. In the example above, the list of Lz operator specifications for all 13C nuclei in the system would be returned. Valid labels for operators in this type of call are 'E', 'Lz', 'L+', 'L-' and 'Tl,m'. In the latter case, l and m are integers.
3. If the two inputs are a cell array of strings and a cell array of numbers respectively, a product operator specification is produced, e.g.
[opspecs,coeffs]=human2opspec(spin_system,{'Lz','L+'},{1,2})
would return the Lz(x)L+ product operator specification with Lz on spin 1 and L+ on spin 2. Valid labels for operators in the cell array are 'E', 'Lz', 'L+', 'L-' and 'Tl,m'. In the latter case, l and m are integers.
## Outputs
opspecs - Spinach operator specification: a cell array of
row vectors specifying which operator enters the
Kronecker product for which spin.
coeffs - coefficient with which each of the Kronecker pro-
ducts enters the overall sum.
## Notes
Direct calls to this function should not be necessary, use operator.m and state.m functions instead.
|
2018-09-22 01:38:13
|
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|
http://mathhelpforum.com/differential-geometry/88630-prove-disprove-integration-proof-print.html
|
# Prove or disprove integration proof
• May 12th 2009, 01:31 AM
nankor
Prove or disprove integration proof
May I plz get some assistance on the following question;
If f and g are integrable on a closed set and f(x)<=h(x)<=g(x) for all x in in that set then h (a function) is integrable on that set.
• May 12th 2009, 04:09 AM
Plato
Quote:
Originally Posted by nankor
If f and g are integrable on a closed set and f(x)<=h(x)<=g(x) for all x in in that set then h (a function) is integrable on that set.
What do you mean by "f and g are integrable"?
• May 12th 2009, 06:03 AM
putnam120
The statement is false.
Take $f(x)=0$ and $g(x)=1$ on the interval $[0,1]$ and $h(x)=1$ if $x\in\mathbb{Q}$ and 0 other wise. This shows that the statement is false for the Riemann integral.
For Lebesgue integrable take $h(x)=1$ on a set $\mu\subset[0,1]$ and $\mu$ is not Lebesgue measurable, and 0 else where. Then $h(x)$ is not Lebesgue integrable.
|
2017-12-18 19:05:39
|
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|
https://www.teamstudy.com/resources/lecture-21-intermolecular-forces-dUQzuUSv
|
# Lecture 21: Intermolecular Forces
• Intermolecular Forces - Electrostatic interactions between molecules
• Atoms within a molecule make covalent and ionic bonds within each other, but molecules also have interactions with other molecules
1. Ion $\rightarrow$ Ion Interactions
2. Strongest intermolecular forces because they involve formal charges
3. Ion $\rightarrow$ Dipole Interactions
4. Dipole - A side of an electron with excess and a side of deficiency
5. The partially negative side is attracted to positive ions and vice versa
6. Each ion can make several of these interactions which store energy
7. Dipole $\rightarrow$ Dipole Interactions
8. Hydrogen bonds - OH, FH, or NH bonds - interact with each other
9. Very strong dipole - Dipole bond because they are the most electronegative elements $\rightarrow$ Most polarized bonds $\rightarrow$ Strong Dipole $\rightarrow$ Strong Dipole (Dipole Interactions)
10. The greater the partial charge - the greater the interaction, but never as strong as formerly charged particles
11. Van-der Waals/London Dispersion Force
12. Any substance can have this bond
13. Ion - Ion > Ion - Dipole > Dipole - Dipole > Van der Waals
14. The stronger the forces between the molecules, the more heat energy is needed to provide to melt and boil the sample
15. Non-Polar Covalent = Van der Waals $(CH_4)$
16. Polar Covalent = *check geometry* $(CO_2,H_2O)$
17. ($Bf_3$ (non-polar - van der waals), $NH_3$ (polar-dipole-dipole))
18. ($CF_4$ (non-polar - van der waals), $CH_3F$ (polar-dipole-dipole))
19. Ionic = Ion-Ion $(NaCl)$
• Polar Molecules
• Non-Polar Molecules
• H-H
• In a moment the electrons can be close to one H (temporary) and gives one H
• Temporary Dipole $\rightarrow$ Has a force of attraction with other molecules $\rightarrow$ London Dispersion Force (weak bands)
• Ideal Gas - $PV = nRT$, no attraction between molecules and there is infinite amount of space between them
• Real Gas - Attraction between molecules
#### dispersion forces
• Fluctuations in the electron distribution in atoms and molecules result in a temporary dipole
• Region with excess electron density has a partial (-) charge
• Region with depleted electron density has partial (+) charge
• The attraction forces caused by these temporary dipoles are called dispersion forces = London forces
• Magnitude depends on
• Polarity of the electron cloud
• Large molar mass = more electrons = larger electron cloud = increased polarizability = stronger attractions
• Shape of the molecule
• More surface - surface contact = larger induced dipole = stronger attraction
• Larger molecules have more electrons $\rightarrow$ Increased polarizability
#### effect of molecular size on size of dispersion force
• Noble gases are all non-polar atomic elements
• As the molar mass increases, the number of electron increases
• Strength of dispersion forces increases
• The stronger the attractive forces between molecules the higher the boiling point will be
• If intermolecular forces are weaker, vapour pressure is higher
• Boiling point is lower
• Vapour Pressure $\rightarrow$
• Intermolecular Force $\rightarrow$
• Boiling Point $\rightarrow$
• Colligative properties depend only on the number of dissolved particles in solution and not on their identity
• Non-colligative properties depend on the identity of the dissolved species and the solvent
• Vapour pressure of solutions
• The vapour pressure of a solvent above solution is lower than the vapour pressure of the pure solvent
• The solute particles replace some of the solvent molecules at the surface
#### raoult's law
• $P$ solvent solution = $X$ solvent
• $P$ solution is lower than $P$ solvent because the mole fraction is always less than 1
• $P$ : Lowering of vapour pressure
• The vapour pressure of the solution is directly proportional to the amount of the solvent in the solution
• Boiling point
• Temperature at which the vapour pressure is equal to atmospheric pressure
• Vapour pressure lowering occurs at all temperatures
• Results in the temperature required to boil the solution being higher than the boiling point of the pure solvent
• Also results in the temperature required to freeze the solution lower than the freezing point of the pure solvent
• Vant Hoff Factors
• $i$ is the ratio of moles of solute particles to moles of formula units dissolved
• The measured Vant Hoff factors are generally less than the theoretical due to ion pairing in solution
• Therefore, the measured Vant Hoff factor often causes the $T$ to be lower than expected
• Vapour pressure of the solution is lower than that of the solvent
• Boiling point of a solution is higher than that of a solvent
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2021-09-24 03:22:14
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https://www.physicsforums.com/threads/integrating-square-roots-absolute-value-needed.592180/
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# Homework Help: Integrating Square Roots - Absolute Value Needed?
1. Mar 31, 2012
### Qube
1. The problem statement, all variables and given/known data
http://i.minus.com/i61zvy2BbtqkI.png [Broken]
2. Relevant equations
One can factor the polynomial to (x-1)^2
3. The attempt at a solution
After factoring the polynomial, I integrate (x-1) given the bounds of 0 and 1. I get -1/2. The solution manual says the answer is positive 1/2. What am I doing wrong?
Last edited by a moderator: May 5, 2017
2. Mar 31, 2012
### Redbelly98
Staff Emeritus
The square root of any quantity is taken to be a positive value, by convention.
But you ended up with (x-1) which is negative, not positive. Can you think of how to fix that?
3. Mar 31, 2012
### Qube
x-1 is not necessarily negative. I assume that your point is that (x-1) is negative over the the open interval of 0 to 1 (0 to 1 also happen to be the bounds of integration).
To rectify this problem, it seems, I would have to integrate the absolute value of (x-1) over the bounds of integration, right?
4. Mar 31, 2012
### LCKurtz
Yes. On [0,1] how can you express |x-1| without the absolute value signs?
5. Mar 31, 2012
### RoshanBBQ
In general, if you have
$$\sqrt{a^2} = |a|$$
for real a.
You can see this with a couple of specific examples:
a = -3
$$\sqrt{(-3)^2} = \sqrt{9} = 3 = |a|$$
a = 3
$$\sqrt{3^2} = \sqrt{9} = 3 = |a|$$
6. Mar 31, 2012
### Qube
To express (x-1) without absolute value signs, distribute a negative 1 to each value.
That results in (-x+1)
Now I just find the integral of (1-x) over the bounds of 0 to 1.
To conclude: should I always use absolute value signs when integrating a square root function over bounds which result in the square root of a negative number?
7. Mar 31, 2012
### SammyS
Staff Emeritus
It wasn't the square root of a negative number,
it was the square root of the square of a negative number.
8. Mar 31, 2012
### Ray Vickson
The use of absolute value signs is a pain; just use instead the actual forms that are >= 0 over all parts of your integration range. Just for fun, and to bring home the point, try the following:
$$\int_0^2 \sqrt{x^2 - 2x + 1} \; dx.$$
RGV
9. Mar 31, 2012
### Qube
Thank you for providing the additional practice :)! I was stumped by the problem for quite a while, but it turns out that my algebra was at fault. Here's my solution:
$$\int_0^2 \sqrt{x^2 - 2x + 1} \; dx.$$
= $$\int_0^2 \left| (x - 1) \right|\; dx.$$
(x-1) is negative when x < 1, therefore we must distribute a negative 1 to each term to get (-x+1)
(x-1) is positive when x ≥ 1.
Given these conditions, we can now split up the integral and appropriately set the bounds of integration.
= $$\int_0^1 \ {(1 - x)} \; dx.$$ + $$\int_1^2 \ {(x - 1)} \; dx.$$
= $\frac{1}{2}$ + 1 + ($\frac{-1}{2}$ + 1)
= 1
The takeaway seems to be:
When integrating a polynomial under a square root function, always:
1) Simplify the square root
2) Cancel out exponents
3) Use absolute value signs
4) Integrate using the appropriate bounds
I wish to really understand this topic. We use absolute value signs because the square root of any number squared has two solutions except for the number 0.
To account for that extra solution, we must use the absolute value signs, which forces us to split up the integral. Is this correct?
Last edited: Mar 31, 2012
10. Mar 31, 2012
### Dick
I think you've got that right.
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2018-07-17 08:49:59
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http://mathhelpforum.com/discrete-math/79905-graph-theory.html
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# Math Help - Graph Theory
1. ## Graph Theory
I have been trying to find out the maximum number of edges in a indirect graph with N vertices and how to prove it without any luck, any help would be greatly appreciated.
2. Originally Posted by skiing64
I have been trying to find out the maximum number of edges in a indirect graph with N vertices and how to prove it without any luck, any help would be greatly appreciated.
Not sure what is meant by indirect graph.
Do you mean a simple nondirected graph?
If so the answer is quite simple: ${N \choose 2}=\frac {N(N-1)}{2}$.
If not, please define the terms used in this question.
3. not directed is not having arrows pointing in a direction but how do you prove that is correct?
4. In a simple graph, any two vertices determine at most one edge.
Therefore, the maximum number of edges on N vertices equals N choosing 2.
5. Thanks I was going about that the right way then
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2016-07-23 10:30:19
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https://economics.stackexchange.com/questions/41344/can-anyone-help-me-derive-saving-from-the-olg-model-just-check-to-see-if-my-st
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# Can anyone help me derive saving from the OLG model? (Just check to see if my start is okay please)
Question assumptions:
Consider the effect of a capital tax on the OLG model. The government imposes a capital tax rate at the rate $$\tau\in[0,1)$$ and pays all the tax revenues back to the old in a lump sum T. Assuming log utility, the maximization problem for a household born at the beginning of t is:
$$\max lnc_{1t}+\beta lnc_{2t+1}$$ s.t. $$c_{1t}=w_t-a_{t+1}$$ and $$c_{2t+1}=(1-\tau)(1+r_{t+1})a_{t+1}+T_{t+1}$$
(a) derive the savings of a young consumer of generation t for a given $$w_t$$, $$r_{t+1}$$, and $$T_{t+1}$$.
My solution attempt:
We know that the utility of consumers born at the beginning of t is:
$$u(c_{1t},c_{2t+1})=u(c_{1t})+\beta u(c_{2t+1})$$ (1)
Where $$\beta\in(0,1)$$ is the discount factor, c is consumption when young (1t) and old (2t+1)
Our budget constraint is:
$$c_{1t}= w_t-a_{t+1}$$ (2)
$$c_{2t+1}= (1+r_{t+1})a_{t+1}$$ (3)
Eliminating $$a_{t+1}$$ we arrive at our lifetime budget constrain which is:
$$c_{1t}+\frac{c_{2t+1}}{1+r_{t+1}}=w_t$$ (4)
$$(w_t-a_{t+1})+\frac{[(1-\tau)(1+r_{t+1})a_{t+1}+T_{t+1}}{1+r_{t+1}}$$
Our log utility function is:
$$\frac{1}{c_t}=\beta (1+r_{t+1})\frac{1}{c_{2t+1}}$$
with the FOC being: $$c_{2t+1}=\beta (1+r_{t+1})c_{1t}$$
Comment: I am unsure if this is the appropriate budget constraint and utility function based on my problem set. Furthermore, when they ask me to derive the savings do they mean the law of motion of savings? Any help would be appreciated.
Notice: the problem is unfinished as I am still working out the solution, and I am just unsure if the basic materials needed to derive the solution are correct. Thank you to anyone who provides a solution.
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2021-03-02 05:01:19
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https://wonghoi.humgar.com/blog/2019/04/
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## Getting pyinstaller 3.4 to work with Python 3.7
Python is an excellent language, but given that it’s free, it also comes with a lot of conspicuous loose-ends that you will not expect in commercially supported platforms like MATLAB.
Don’t expect everything to work right out of the box in Python. Everything is like 98% there, with the last 2% frustrate the heck out of you when you are rushing to get from point A to point B and you have to iron out a few dozen kinks before you can really start working.
When I tried use pyinstaller (v3.4) to compile my Python (v3.7) program into an executable, I ended up having to jump through a bunch of hoops:
• pip install pyinstaller gives:
ModuleNotFoundError: No module named 'cffi'
• Then I looked up and installed cffi
pip install cffi
• After the dependency was addressed manually (it shouldn’t ) pip install pyinstaller worked
• Then I tried to compile my first Python executable with pyinstaller, and I got this exception:
File "C:\Python37\lib\site-packages\win32ctypes\core\cffi\_advapi32.py", line 198
^
SyntaxError: invalid syntax
• I searched the exact string and learned that pyinstaller (v3.4) is not ready for Python 3.7 yet! How come pip installer didn’t check for it? I opened up the offending file and looked for line 198 and saw this:
c_creds.CredentialBlobSize = \
ffi.sizeof(blob_data) - ffi.sizeof('wchar_t')
It’s a freaking line continuation character \ (actually the extraneous CR before CRLF) that rooster-blocked it.
• I just deleted the line continuation and merged the two lines, and saved _advapi32.py, then I was able to compile my Python v3.7 code (using pyinstaller 3.4) with no issues.
This is not something you’ll experience as a MATLAB user. The same company, TMW, wrote the MATLAB compiler as well as the rest. The toolbox/packages are released together in one piece so breaking changes that causes failure for the most obvious use case are caught before they get out of the door.
Another example of breaking changes that I ran into: ipdb does not allow you to move cursor backward.
Again, this is the cost associated with free software and access to the latest updates and new features without waiting for April/October (it’s the MATLAB regular release cycle). If hassle and the extra engineering time far exceed licensing MATLAB licensing costs, MATLAB is a better choice, especially if software is just a chore to get your company from point A to point B, and you are willing to pay big bucks to get there quickly and reliably.
Even with free software on the table, your platform choice is always determined by:
• how much your time is worth wrestling problems
• how much flexibility do you need (for customizing to your needs)
• how much you are willing to pay for the licenses and support
In any case, Python community did good work. Please consider sponsoring PyInstaller and PSF if you profit immensely from their work. But if your company already paid for MATLAB and has real MATLAB experts on it*, don’t switch to Python just for the sake of chasing the trendy thing. You will loose time. Lots of time! Precious engineering hours! Over really stupid things like the ones above that is totally not your fault! It’s better to dedicate a few days dealing with MATLAB-Python interface than wasting months over the clumsiness of Python if you have a long, complex workflow
Python’s largest value over MATLAB is that there’s a function developed for nearly every less than common situations (like all different variations of path and file management tools) so you don’t need MATLAB File Exchange to fill in the gap, but the downside is that Python, just like the rest of FOSS world (or Tektronix), has absolutely no sense of user experience studies: common operations are tucked under clumsy maneuvers such as making user jump through hoops to run a Python script in the interpreter (import do not execute the script after its cached, execfile was lile eval in MATLAB so they made it hell, or you need to import a freaking library like runpy to do something basic like this. I understand they cannot Why can’t they just add a root-level function like run() and call it a day? Python was just too eager to protect the namespace, and this comes at the expense of adding a lot of stupid maneuvers for really basic bread-and-butter stuff). Python is productive in the sense that there are many sophisticated features that are readily available that you don’t have to write your own library for the advanced cases, but the edge is like 1% of the use cases which I invest the time when I run into it: it’s not worth making 99% of my workflow miserable!
If you are into test instruments, MATLAB is like HP/Agilent/Keysight’s user interface while Python is like Tektronix’s UI where you’ll need to get to 4 levels of context menus to measure a freaking peak-to-peak voltage! Spending your smart people’s time to figure these stupid shit that came from poor UX (user experience) design is a poor investment! At least Python is free and has more advanced features; Tek charges similar to HP and offer nearly the same features yet it has absolutely no respect for the users’ time: they think all people have Stockholm syndrome after going through their steep learning curve for absolutely no benefits.
Why pay a junior engineer $50/hr for a whole week to go through the learning curve, plus months of maintenance work because they’ve used clumsy tools and clumsy methods, when I can code the same thing up in MATLAB in one hour for$500 and give you one page of easy to read code that you likely don’t have to debug through it down the line? If I designed a moderately complex system, most often I can get to the root cause of any bugs in around 15 minutes: the reason is that I spent my time thinking through the abstractions instead of diving into writing for-loops to hack through any common scenarios without first researching for the neatest mechanism. This way I don’t spend an afternoon writing for loops extracting a strings in the object properties of a categorical data type (a efficient way of marking complex repeating data by indexing unique items) and check if it’s really correct when I could have done some research and realize I can just cast it to a cell/monad/array of strings. The value of my experiences comes in knowing the right words and common lingo in many forms to describe many problems, like spotting a low level database operation implementation is just a RLE (run length encoding) in disguise. The hardest part of using search engines to find what you want quickly is knowing the right keywords. If you don’t know the concepts, you’d have jump into reinventing the wheel poorly when you could have used off-the-shelf proven code had you been able to frame the problem succinctly.
* Don’t be mislead by how many years of MATLAB one has! Majority of people those who claimed decades of MATLAB under the belt didn’t really learn the advanced concepts through studying documentation, MATLAB blogs or get training to take it to the next level of super-productivity! You can tell if they have a lot of for-loops for situations that doesn’t absolutely need it (memory saving, injecting elements to LHS assignment, parallel computing, not loading too many files at the same time, drawing graphics (order matters), big tasks that needs to be resumed if it throws an error) and using cells everywhere when they should have used structs or more modern types such as table(). Most of the modern, super-productive MATLAB language constructs happens since 2013, so I can safely say that MATLAB experiences accumulated before that aren’t too valuable for developing business logic part of the code (might be useful if you are writing tools and lower level MATLAB code like those in TMW).
1,459 total views
## Picking an IDE for Python
The native features in MATLAB are often very good most of the time, as I’ve yet to hear anybody spending time to shop for a IDE outside the official one.
Atom has the feel of Maple/MathCAD, and Jupyter Notebook has the feel of Mathematica. Spyder feels like MATLAB the most, but it’s hugely primitive.
IDLE is more miserable than a command prompt. It doesn’t even have the decency to recall command history with up arrow. It’s like freaking DOS before loading doskey.com. Not to mention that single clicking on the window won’t set the cursor to the active command line, which you have to scroll all the way down to click on the bottom line. WTF! I’d rather use the command prompt and give up meaningless syntax coloring.
IPython (in Spyder) is unbearably slow (compare to MATLAB’s editor which I consider slow to the extent that it’s marginally bearable for the interactive features it offers), but at least usable unlike IDLE, and most importantly the output display is pprint (pretty printer) formatted so it’s legible. Just type locals() and see what kind of sh*t Python spits out in IDLE/cmd.exe and you’ll see what I meant.
I simply cannot live without who/whos provided in IPython, but I still don’t like it showing the accessible functions/modules along with the variables (I know, Python doesn’t tell them apart). Nonetheless it’s still weak because these are automagics that doesn’t return the results as Python data (just print). Spyder’s ‘variable explorer’ is the only place I can find that doesn’t include loaded functions/modules. Python should have provided facilities to get the user-introduced variables exclusively and leave the modules to a different function like MATLAB’s import command that shows imported packages/classes.
However, pretty printer doesn’t even come close to MATLAB in terms of the amount of dirty work disp() did to format the text to make it easy to read. Keys in the dictionary shown in pretty printer in Python are not right-aligned like MATLAB struct so we can easily tell keys and values apart. For example:
MATLAB struct shows:
name: 'S'
size: [9 1]
bytes: 7765
class: 'struct'
global: 0
sparse: 0
complex: 0
nesting: [1×1 struct]
persistent: 0
Python with Pretty Printer shows:
{'__name__': '__main__',
'__doc__': 'Automatically created module for IPython interactive environment',
'__package__': None,
'__spec__': None,
'__builtin__': <module 'builtins' (built-in)>,
'__builtins__': <module 'builtins' (built-in)>,
'_ih': ['', 'locals()'],
'_oh': {},
'In': ['', 'locals()'],
'Out': {},
'get_ipython': <bound method InteractiveShell.get_ipython of <ipykernel.zmqshell.ZMQInteractiveShell object at 0x00000000059B7828>>,
'exit': <IPython.core.autocall.ZMQExitAutocall at 0x5a3b198>,
'quit': <IPython.core.autocall.ZMQExitAutocall at 0x5a3b198>,
'_': '',
'__': '',
'___': '',
'_i': '',
'_ii': '',
'_iii': '',
'_i1': 'locals()'}
I often convert things to MATLAB dataset() because the disp() method is excellent, such as struct2dataset(ver()). table/disp() is nice, but I think they overdid it by defaulting to fancy rich-text that bold the header, which makes it a magnitude of orders slower, and it’s not using the limited visual space effectively to show more data. Python still has a lot more to do in the user-friendly department.
598 total views
## Implicit ways to store data in a program
The most obvious way to store data is in plain data structures such as arrays and queues, or even a hashtable, but don’t forget these implicit ones:
• Call stack. As the name say, it’s a stack data structure. It saves the local variables before making another function call. This is often exploited in recursion to avoid passing around an explicit data structure
• Closures. What closure means is that when you create an anonymous function, the variables involved (other than the arguments) that is saved along (captured) with the function object you created. This can be exploited to make forward iterators or generators
• Functions. You can make a function that does nothing other than returning a certain piece of data. It’s an excellent way to make avoid the overhead of managing (reading, updating promptly) a config file. Works best when your programming language requires so little typing to specify data (such as MATLAB) that your code is almost as short as a plain text config file.
508 total views
## 不是懶趴是卵脬，不是雞是膣
粵 國 台 日 屄/閪 屄 膣 (誤讀 雞) 膣 屌/𨳒 姦 (誤讀 幹) 肏 (誤讀 操) 姦 𡳞/𨶙，㞗/𨳊，杘/𨳍， 脧脧/JerJer 屌 𡳞/卵鳥 (誤讀 懶鳥) チンポ (珍宝) 春袋 蛋蛋 卵脬 きんたま（金玉） 儸柚 屁股 尻(kha)川(tshng) おしり (お尻)
1,140 total views
## Obscure differences between Kanji and Chinese characters
People who already know Chinese characters are often said to have the advantage of being able to pick up Japanese quickly. However, to learn it properly, in addition to the difference between infix (English, Chinese) and reverse polish (Japanese) notations, it also comes with quite a bit of baggage. It’s the differences that requires work to observe, such as:
• some made up ‘Chinese’ characters (和製漢語),
• some are written slightly differently, including artistic variations
• some has a completely different meaning,
• some has opposite preferences for using which character in the pair when simplifying
• and some has drastically different overtones despite they technically mean the same thing
• the mixture of simplified and traditional characters, occasionally a character written like simplified Chinese means something totally different from traditional Chinese, such as 机（つくえ）which means desk vs 機（キ）which means machines or chances depending on the context.
• the roles of historical and modern writings are randomly reversed
Actually, the kinds of variations mentioned above applies to regional differences in Chinese languages (such as Taiwanese, Cantonese and Mandarin). Most places agree to write Chinese in a way that can be read directly using Mandarin so that we can at least communicate on paper. So as time goes by, we lost the ability to write in Taiwanese and Cantonese. I hope it’ll change as both dialects are very colorful. Re-expressing them in Mandarin will take away all the flavors in them.
It’s evident that humans can pick up more than one language, so there is no reason to compromise dialects in the process of standardization. People advocating to kill other languages are simpletons who believe in the kind of logic supporting a competitive system: you find ways to make your peers do worse to stay ahead, instead of improving yourself.
Different regions occasionally have different preferences for character order in phrases. Basically we have to watch out for all kinds of combinations. Like 介紹 is used in the same order for Taiwanese/Cantonese/Mandarin to mean introduction, but it’s reversed 紹介(しょうかい) in Japanese. To make it a total mindfuck, Mandarin sticks with 客人 for guests, which is used the same way as Japanese’s 客人(きゃくにん), Taiwanese mostly says 人客, while Cantonese uses both with slight overtones: 客人 is usually used as a particular noun (e.g. 呢位客人) while 人客 is often used as a collective noun (e.g. 人客嚟齊未?), most likely because 客人 sounds more formal than 人客.
Putting traditional and simplified Chinese aside, different regions have different preferences for Chinese characters. I couldn’t tell the difference between traditional Chinese characters used in Hongkong/Macau (港澳繁體) and Taiwan (台灣正體) on Wikipedia, and later learned that it was because I’ve been randomly mixing both all along and nobody ever pointed it out.
I remember writing 峰 most of the time even when I was a kid and only used 峯 for names that specifically calls for it. We respect the original writing for names. This is the similar situation as in Japanese: 沢(さわ/たく) is used in most cases and reserve 澤(サワ) for names that specifically requests to be written in this form. The only difference is that I used the official character 峯 exclusively for names, while using the off-label 峰 for the rest.
Speaking of names, there are some similar-looking characters that has the same Japanese sound (かな) but are actually different in both writing and meaning. 斉藤 and 斎藤 are different, but they are easily confused for native Japanese speakers who don’t have any Chinese language background. Here’s the table for comparison:
齊／齐・斉 齋／齋・斎 Meaning Gathered, organized Plain, house, recitations Cantonese chai (cai4) jaai (zaai1) Taiwanese tsè tsai Mandarin qi2 zhai1 Japanese (音読み：さい) 斉しい・等しく いつき・（潔斎）物忌み
The bottom line is: as language evolves, different regions have different preferences about what can they be sloppy about and what they must be meticulous about. They also reorder/tweak things to make them flow smoothly with their dialect. This means traps for for those learning a new language that are close to what they’ve already mastered.
I came across a document called 常用漢字表 released by the Agency for Cultural Affairs (文化庁) that explains all the quirks of Kanji that was carefully collecting on my own while taking the classes. Wish I had it back in the days. Here’s the link, but I also saved a local copy of 常用漢字表 just in case if their website moves around in the future.
599 total views
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2022-09-28 16:09:03
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https://worldbuilding.stackexchange.com/questions/207033/trying-to-survive-a-passing-neutron-star-by-burrowing-deep-in-the-planets-crust/207035
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# Trying to survive a passing neutron star by burrowing deep in the planet's crust?
Situation: My colony is threatened by a passing neutron star. It turns out this planet is part of a binary system and once every several hundred years it passes by a neutron star, which irradiates its surface. I'm trying to figure out a way for the colonists to survive the impending cataclysm. Assume they don't have access to interstellar ships.
My question has two parts:
1. What if they dig deeper into the planet's core? Could the layers of rock shield them from the worst of the radiation? (Assume they have tech to pull this off).
2. Is there anything else they could do to shield themselves from this catastrophe, short of leaving the planet?
Edit: It's an older, non-pulsar neutron star. I would like for the neutron star to have an accretion disk (I would love for the colonists to witness it accrete away some of their main star's mass), but I can dispense with that if necessary. I would also like for it to pass close by enough for the colonists to feel some of the tidal forces and gravitational waves but not enough to kill them.
• How close will the neutron star pass to the planet, and does it have an accretion disk? Jul 10 '21 at 2:54
• Does "turns out" imply this is a surprise to them? Because a neutron star sounds like one of those things you really should've noticed before you set up your colony. Jul 10 '21 at 2:57
• A neutron star is what's left over after a supernova, which would have blasted away all the atmosphere and most of the surface of the planet. A lifeless, airless world doesn't seem like the place one would want to put a colony. Jul 10 '21 at 3:40
• If the pass is close enough to douse the world in radiation, is it also close enough for tidal forces to knead the core and crust into activity and cause killer quakes that squish the new underground communities? Jul 10 '21 at 3:46
• @Cadence, haha yes, it's a surprise! and a big part of the story is how and why that came about. Jul 10 '21 at 13:27
## 1 Answer
I think they'll be okay.
Let's start by figuring out what we're up against. Neutron stars can produce high-energy radiation through two means: thermal and non-thermal emission. Thermal emission is just the light emitted by a black body. Young neutron stars that have begun cooling (a couple of years old - younger than this one) have temperatures of $$\sim10^6$$ Kelvin. Assuming a radius of roughly 10 km, the Stefan-Boltzmann law predicts that a young neutron star should have a luminosity about 19% that of the Sun. The thermal emission peaks somewhere near the cutoff between ultraviolet and x-rays, meaning that a lot of this will be dangerous to humans.
If the neutron star is behaving like a pulsar, it will also emit non-thermal radiation through synchrotron emission. You probably know pulsars best from radio observations, but in the most energetic pulsars, most of the rotational energy of the pulsar is actually converted to x-rays and gamma-rays; there's a weak correlation between the frequency of light and the fraction of the spin-down energy that goes into that frequency band.$$^{\dagger}$$ The power released by a typical pulsar with period $$P$$ and period time derivative $$\dot{P}$$ is $$\dot{E}\approx4\times10^{31}\;\text{erg s}^{-1}\left(\frac{\dot{P}}{10^{-15}}\right)\left(\frac{P}{\text{s}}\right)^{-3}$$ This usually comes out to a few percent of a solar luminosity, so it's fair to say that our neutron star should have a total luminosity - including thermal and non-thermal emission - of roughly $$0.25L_{\odot}$$. Ish. And that's generous, because your neutron star is certainly older, which thanks to cooling might drop this by 1-2 orders of magnitude. At any rate, I think we can assume that this is mostly the sort of high-energy radiation we'd prefer to avoid.
(Brief interlude: You've mentioned that the neutron star has an accretion disk but that it's not behaving like a pulsar. That's a bit odd for two reasons: 1) the neutron star would have to have been in a close orbit to its companion star in order to accrete that matter in the first place, which seems incompatible with a planet remotely near the habitable zone, and 2) neutron stars accreting matter gain angular momentum, which increases their rotational speeds and turn them into millisecond pulsars, as the increase in angular momentum also turns on the not-overly-well-understood pulsar emission mechanism. In other words, I'd be surprised to see a neutron star with an accretion disk not emitting pulses of radiation. Coupled with the strangeness of having an accretion disk while in a wide orbit, I'd like to dispute that part of the premise!)
The flux on the planet depends on how far from the neutron star it is. Let's say the closest approach is around 100 AU; a pass on the order of 10 AU or less has a decent risk of causing orbital problems, particularly if there are other planets in the system (thank you to Loren Pechtel for confirming this!). The flux on the surface is then about 0.034 Watts per square meter.$$^{\ddagger}$$ If an unshielded human weighing 80 kg (cross-sectional area of something like 2 square meters?) was exposed to this amount of radiation for one year, they'd receive a dose of about 27,000 Sieverts. As I understand it, we'd want to reduce this below 1 Sievert to significantly reduce the risk of radiation sickness. Not great.
However, we could absolutely build shielding. Lead has a half-value layer of 4.8 mm against gamma rays, so we could lower the radiation by the requisite four or so orders of magnitude with 15 times this length. Not bad. Even if the distance to the neutron star is an order of magnitude lower, raising the dosage by a factor of 100, we'd still need lead shielding of something like 10 cm, if my numbers are correct. Dirt itself has a half-value layer of 115 cm, so 25 meters of dirt would provide adequate shielding from the worst-case 10 AU-approach scenario.
Let's briefly discuss gravitational effects, since you've brought up tidal forces and gravitational waves. Tidal forces would be minimal since at interplanetary distances there's no difference gravitationally between a $$\sim1.5M_{\odot}$$ neutron star and a $$\sim1.5M_{\odot}$$ main sequence star; tidal forces are only important quite close to the surface. Gravitational waves are a possibility from tiny imperfections in the neutron star's surface on the order of millimeters or so (we ironically call them "mountains"). Mountains on a neutron star at a distance of 100 AU should produce a strain on the order of $$\sim10^{-20}$$, give or take a couple orders of magnitude (Lasky 2015), which won't cause problems.
I'm sure these numbers are off by a bit - a factor of 10 here, a factor of 3 there. I've likely overestimated the thermal radiation and the high-energy contribution from non-thermal radiation, and I think I've also overestimated how close the neutron star can be without having affecting the planet's orbit. The point, though, is that even if I'm wrong by 1-2 orders of magnitude, a mine shaft a kilometer or so deep should be cozy enough against whatever a neutron star can thrown at these colonist. And that's probably substantially overkill.
Anyway, time to start digging.
$$^{\dagger}$$Handbook of Pulsar Astronomy, Lorimer & Kramer. Also my reference for other bits of this answer.
$$^{\ddagger}$$This is slightly inaccurate because the non-thermal pulsed radiation will not be emitted equally in all directions. A reasonable assumption is that the beam covers about 10% of the sky at a given time (although this depends on the pulse period), meaning the flux when it sweeps across the planet will be higher than in the case of isotropic emission. Conversely, there's no guarantee the beams will cross the planet at all.
• 27000 Sievert, not great, not terrible. Jul 10 '21 at 12:10
• I think your 10AU is too close. Just tried a simulation in Universe Sandbox, a 1 solar mass black hole with a periapsis at 10AU and an apoapsis at 100AU. One pass, I would say Earth is still inhabitable. Saturn and Neptune are gone, though, Jupiter's periapsis is near Mercury and Uranus' is near Venus. Jul 10 '21 at 18:32
• 2nd pass: Jupiter is safe, Uranus and Mars are gone, Earth's habitability is questionable. 3rd pass: Jupiter and Pluto are gone, Earth is still of questionable habitability. 4th pass: A few Kupier belt objects that were stolen got returned, Earth still questionable. Jul 10 '21 at 18:47
• Retrying at 20AU. First pass: Uranus, Neptune ejected, Pluto stolen, Earth looks ok. 2nd pass: Pluto returned. 3rd pass: Pluto taken again, Saturn ejected. The inner system seems safe. Jul 10 '21 at 19:27
• @LorenPechtel - Maybe you could post your experiment. It would be nice if youtube had some videos using Universe Sandbox that were other than massive collisions. I considered getting it but it seemed like all you could do was whack stuff into other stuff. Not that there is anything wrong with that. Jul 10 '21 at 22:29
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2022-01-22 16:57:21
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https://www.jobilize.com/course/section/notation-for-the-poisson-p-poisson-probability-distribution
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4.7 Poisson
Page 1 / 2
This module describes the characteristics of a Poisson experiment and the Poisson probability distribution. This module is included in the Elementary Statistics textbook/collection as an optional lesson.
Characteristics of a Poisson experiment:
1. The Poisson gives the probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event. Forexample, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are 5 words spelled incorrectly in100 pages. The interval is the 100 pages.
2. The Poisson may be used to approximate the binomial if the probability of success is "small" (such as 0.01) and the number of trials is "large" (such as 1000). You will verify therelationship in the homework exercises. $n$ is the number of trials and $p$ is the probability of a "success."
Poisson probability distribution . The random variable $X=$ the number of occurrences in the interval of interest. The mean and variance are given in the summary.
The average number of loaves of bread put on a shelf in a bakery in a half-hour period is 12. Of interest is the number of loaves of bread put on the shelf in 5 minutes. Thetime interval of interest is 5 minutes. What is the probability that the number of loaves, selected randomly, put on the shelf in 5 minutes is 3?
Let $X$ = the number of loaves of bread put on the shelf in 5 minutes. If the average number of loaves put on the shelf in 30 minutes (half-hour) is 12, then the average number of loaves put on the shelf in 5 minutes is
$(\left(\frac{5}{30}\right)\cdot 12, 2)$ loaves of bread
The probability question asks you to find $\mathrm{P\left(x = 3\right)}$ .
A certain bank expects to receive 6 bad checks per day, on average. What is the probability of the bank getting fewer than 5 bad checks on any given day? Of interestis the number of checks the bank receives in 1 day, so the time interval of interest is 1 day. Let $X$ = the number of bad checks the bank receives in one day. If the bank expects to receive 6 bad checks per day then the average is 6 checks per day.The probability question asks for $(P\left(x, 5\right))$ .
You notice that a news reporter says "uh", on average, 2 times per broadcast. What is the probability that the news reporter says "uh" more than 2 times per broadcast.
This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast.
What is the interval of interest?
What is the average number of times the news reporter says "uh" during one broadcast?
2
Let $X$ = ____________. What values does $X$ take on?
Let $X$ = the number of times the news reporter says "uh" during one broadcast .
$x$ = 0, 1, 2, 3, ...
The probability question is $\text{P(______)}$ .
$\text{P(x>2)}$
Notation for the poisson: p = poisson probability distribution function
$X$ ~ $\text{P(μ)}$
Read this as " $X$ is a random variable with a Poisson distribution." The parameter is $\mu$ (or $\lambda$ ). $\mu$ (or $\lambda$ ) = the mean for the interval of interest.
Leah's answering machine receives about 6 telephone calls between 8 a.m. and 10 a.m. What is the probability that Leah receives more than 1 call in the next 15 minutes?
Let $X$ = the number of calls Leah receives in 15 minutes. (The interval of interest is 15 minutes or $\frac{1}{4}$ hour.)
$x$ = 0, 1, 2, 3, ...
If Leah receives, on the average, 6 telephone calls in 2 hours, and there are eight 15 minutes intervals in 2 hours, then Leah receives
$(\frac{1}{8}\cdot 6, 0.75)$
calls in 15 minutes, on the average. So, $\mu$ = 0.75 for this problem.
$X$ ~ $\text{P(0.75)}$
Find $(P\left(x, 1\right))$ . $((P\left(x, 1\right)), 0.1734)$ (calculator or computer)
TI-83+ and TI-84: For a general discussion, see this example (Binomial) . The syntax is similar. The Poisson parameter list is ( $\mu$ for the interval of interest, number). For this problem:
Press 1- and then press 2nd DISTR. Arrow down to C:poissoncdf. Press ENTER. Enter .75,1). The result is $((P\left(x, 1\right)), 0.1734)$ . NOTE: The TI calculators use $\lambda$ (lambda) for the mean.
The probability that Leah receives more than 1 telephone call in the next fifteen minutes is about 0.1734.
The graph of $X$ ~ $\text{P(0.75)}$ is:
The y-axis contains the probability of $x$ where $X$ = the number of calls in 15 minutes.
how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.
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2021-06-21 20:17:06
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https://tex.stackexchange.com/questions/405138/why-does-equation-environment-not-give-an-error-without-the-amsmath-package
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# Why does equation environment not give an error without the amsmath package?
As far as I understand the equation environment is defined in the amsmath package so one would expect an error while trying to compile the following document with pdflatex:
\documentclass{article}
\begin{document}
$$a=b+c$$
\end{document}
But I don't get any error while running pdflatex on this file an a pdf output with the equation a + b = c is generated. How does that work exactly?
• @DarthPaghius Why's it a problem? The equation environment is part of the kernel. – Joseph Wright Dec 8 '17 at 7:41
• @JosephWright But every resource I looked at says its a part of ams math package. Is the amsmath overriding the kernel one? – DarthPaghius Dec 8 '17 at 7:42
• Btw, +1, it's a very good question and I'm sure some TeXpert will answer soon. amsmath is a package which does many useful in math, but for a simple equation like the one of your MWE, the TeX kernel is enough. – CarLaTeX Dec 8 '17 at 7:42
• @DarthPaghius I'm sure that some of the top users here will give you an exemplary answer! And, of course, most people don't care how things work but only why they don't work. You're very clever :):):) – CarLaTeX Dec 8 '17 at 8:05
• @Mico It's also possible I involuntary removed it, the important thing is you read it :):):) Good morning, TeXnician! – CarLaTeX Dec 9 '17 at 7:34
You wrote:
As far as I understand the equation environment is defined in the amsmath package
That's not quite correct. The equation environment is defined in the LaTeX kernel -- and it is redefined if the amsmath package is loaded in the preamble.
For completeness, here's the relevant code from the file latex.ltx (the "kernel"):
\@definecounter{equation}
\def\equation{$$\refstepcounter{equation}} \def\endequation{\eqno \hbox{\@eqnnum}$$\@ignoretrue}
\def\@eqnnum{{\normalfont \normalcolor (\theequation)}}
where \eqno is a so-called "primitive" instruction that takes \hbox{\@eqnnum} as its argument. Basically, \equation initiates display-math mode and increments a counter called equation, while \endequation typesets the equation number (on the far right, by default) and closes display-math mode.
And here's the code from amsmath.sty; note that due to the \renewenvironment instruction, the code replaces what the kernel provides:
\renewenvironment{equation}{%
\incr@eqnum
\mathdisplay@push
\st@rredfalse \global\@eqnswtrue
\mathdisplay{equation}%
}{%
\endmathdisplay{equation}%
\mathdisplay@pop
\ignorespacesafterend
}
This setup is, quite obviously, rather more elaborate than what's performed in the LaTeX kernel. The most important substantive differences arise from the fact that, if amsmath is loaded, it becomes possible to insert split, aligned, and gathered environments inside an equation environment. In addition, amsmath performs some fancy measuring operations, behind the scenes, to figure out if it's necessary to "shove" the equation number down a bit in order to avoid a collision with an overly long equation.
If you're really interested in the details, I would like to encourage you to examine the information contained in the file amsmath.pdf. Open a command window and type texdoc amsmath.pdf to launch the file in a pdf viewer.
• Thanks for the detailed answer. It was never explicitly mention in the places I looked. – DarthPaghius Dec 8 '17 at 8:27
• @DarthPaghius - You're most welcome! – Mico Dec 8 '17 at 8:32
• @Mico -- the preferred (by knuth, among others) term for someone really knowledgeable about using tex is "TeXnician". and regarding amsmath documentation, a better way to study it is from the commented equivalent of amsmath.sty, namely amsmath.pdf. at a command prompt on a tex live system: texdoc amsmath.pdf – barbara beeton Dec 8 '17 at 13:56
• @Mico -- yes, i realize the potential for naming confusion. fortunately, i don't have to worry about that -- i have a certificate prepared and signed by don knuth that declares me to be an "international TeXnocrat". (i'm still not sure whether that's a good or a bad thing ...) – barbara beeton Dec 8 '17 at 14:33
• @barbarabeeton I used TeXpert first in this post, I didn't know that the preferred term is TeXnician! – CarLaTeX Dec 8 '17 at 15:04
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2019-05-23 15:21:25
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https://www.educationlessons.co.in/definitions/pythagorean-triples/
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# Pythagorean Triples
As per Pythagoras Theorem, we get expressions for right angled $\triangle ABC$;
$a^2 + b^2 = c^2$
Pythagoras Theorem
By plotting some values for 'a' and 'b', one may get value of 'c' or in other words,
One may find value of Hypotenuse (longest side) by using values of other two sides.
So, if we have, $a=3 \ and \ b=4$, then by Pythagoras Theorem,
\begin {aligned} c^2 &= 3^2 + 4^2 \\ \therefore c^2 &= 9 + 16 \\ \therefore c^2 &= 25 \\ \therefore c &= \sqrt {25} \\ \therefore c &= 5\end {aligned}
So, we get (3, 4, 5) as one of the Pythagorean triple.
Meaning, the square of the largest number will be the sum of the squares of the other two.
You can find popular triplets as,
(3, 4 ,5)
(6, 8, 10)
(12, 13, 5)
You can check the triples above by calculating yourself. Also, try to make some Pythagorean triplets by yourself.
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2022-01-20 10:05:41
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https://proofwiki.org/wiki/Definition:Number_Theory/Historical_Note
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# Definition:Number Theory/Historical Note
## Historical Note on Number Theory
The field of number theory is considered by some to be one of the oldest branches of mathematics in history.
At the time of Pythagoras of Samos, there existed a mass of unstructured information on the subject dating back to the Babylonians and ancient Egyptians.
Pythagoras and his followers believed that all the phenomena in the Universe could be explained by the study of the natural numbers.
Some of the first serious results are found in Euclid's The Elements, thinly disguised as geometry.
Diophantus developed the ideas into a distinct branch of mathematics.
The field was founded in its modern form by Pierre de Fermat in his pioneering work in the $17$th century.
However, most of his discoveries are known about only because he wrote about them to his friends, or (famously) jotted them down in the margins of his copy of Diophantus's Arithmetica.
For many of these, his proofs were never recorded, and when he died they were lost forever.
Nobody else was able to follow him until Euler and Lagrange in the following century.
The field was properly placed on a firm footing by the work of Carl Friedrich Gauss in his Disquisitiones Arithmeticae.
The field was advanced significantly by Augustin Louis Cauchy.
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2020-11-25 11:38:31
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https://www.studyadda.com/question-bank/aromatic-hydrocarbon_q92/1404/103697
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• # question_answer Order of reactivity of ${{C}_{2}}{{H}_{6}},{{C}_{2}}{{H}_{4}}$ and ${{C}_{2}}{{H}_{2}}$ is [MH CET 2004] A) ${{C}_{2}}{{H}_{6}}>{{C}_{2}}{{H}_{4}}>{{C}_{2}}{{H}_{2}}$ B) ${{C}_{2}}{{H}_{2}}>{{C}_{2}}{{H}_{6}}>{{C}_{2}}{{H}_{4}}$ C) ${{C}_{2}}{{H}_{2}}>{{C}_{2}}{{H}_{4}}>{{C}_{2}}{{H}_{6}}$ D) All are equally reactive
Unsaturated hydrocarbons are more reactive than saturated hydrocarbons. Among ethyne $({{C}_{2}}{{H}_{2}})$ and ethene $({{C}_{2}}{{H}_{4}})$ the later is more reactive as $C\equiv C$ triple bond is quite strong bond and therefore ethyne generally require catalysts (like $H{{g}^{2+}}$ etc) in its reactions.
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2020-09-24 20:10:44
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https://samacheerkalviguru.com/samacheer-kalvi-12th-economics-solutions-chapter-4/
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Students of 12th can get the pdf links of Tamilnadu State Board Economics Solutions here. You can Download Samacheer Kalvi 12th Economics Book Solutions Chapter 4 Consumption and Investment Functions Questions and Answers, Notes Pdf, Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.
## Tamilnadu Samacheer Kalvi 12th Economics Solutions Chapter 4 Consumption and Investment Functions
It is very important to put the textbook aside while preparing for the exams. So, if you follow Samacheer Kalvi 12th Economics Textbook Solutions you can cover all the topics in Chapter 4 Consumption and Investment Functions Questions and Answers. This helps to improve your communication skills.
### Samacheer Kalvi 12th Economics Consumption and Investment Functions Text Book Back Questions and Answers
Part – A
Multiple Choice Questions.
Question 1.
The average propensity to consume is measured by –
(a) C / Y
(b) C × Y
(c) Y / C
(d) C + Y
(a) C / Y
Question 2.
An increase in the marginal propensity to consume will:
(a) Lead to consumption function becoming steeper
(b) Shift the consumption function upwards
(c) Shift the consumption function downwards
(d) Shift savings function upwards
(a) Lead to consumption function becoming steeper.
Question 3.
If the Keynesian consumption function is C = 10 + 0.8 Y then, if disposable income is Rs 1000, what is amount of total consumption?
(a) ₹ 0.8
(b) ₹ 800
(c) ₹ 810
(d) ₹ 0.81
(c) ₹ 810
Question 4.
If the Keynesian consumption function is C = 10 + 0.8 Y then, when disposable income is Rs 100, what is the marginal propensity to consume?
(a) ₹ 0.8
(b) ₹ 800
(c) ₹ 810
(d) ₹ 0.81
(a) ₹ 0.8
Question 5.
If the Keynesian consumption function is C = 10 + 0.8 Y then, and disposable income is ₹ 100, what is the average propensity to consume?
(a) ₹ 0.8
(b) ₹ 800
(c) ₹ 810
(d) ₹ 0.9
(d) ₹ 0.9
Question 6.
As national income increases –
(a) The APC falls and gets nearer in value to the MPC.
(b) The APC increases and diverges in value from the MPC.
(c) The APC stays constant
(d) The APC always approaches infinity.
(a) The APC falls and gets nearer in value to the MPC.
Question 7.
As increase in consumption at any given level of income is likely to lead –
(a) Higher aggregate demand
(b) An increase in exports
(c) A fall in taxation revenue
(d) A decrease in import spending
(a) Higher aggregate demand
Question 8.
Lower interest rates are likely to:
(a) Decrease in consumption
(b) increase cost of borrowing
(c) Encourage saving
(d) increase borrowing and spending
(d) increase borrowing and spending
Question 9.
The MPC is equal to:
(a) Total spending / total consumption
(b) Total consumption / total income
(c) Change in consumption / change in income
(d) None of the above
(c) Change in consumption / change in income
Question 10.
The relationship between total spending on consumption and the total income is the –
(a) Consumption function
(b) Savings function
(c) Investment function
(d) aggregate demand function
(a) Consumption function
Question 11.
The sum of the MPC and MPS is –
(a) 1
(b) 2
(c) 0.1
(d) 1.1
(a) 1
Question 12.
As income increases, consumption will –
(a) fall
(b) not change
(c) fluctuate
(d) increase
(d) increase
Question 13.
When investment is assumed autonomous the slope of the AD schedule is determined by the –
(a) marginal propensity to invest
(b) disposable income
(c) marginal propensity to consume
(d) average propensity to consume
(c) marginal propensity to consume
Question 14.
The multiplier tells us how much changes after a shift in –
(a) Consumption, income
(b) investment, output
(c) savings, investment
(d) output, aggregate demand
(d) output, aggregate demand
Question 15.
The multiplier is calculated as –
(a) 1 / (1 – MPC)
(b) 1 / MPS
(c) 1 / MPC
(d) a and b
(d) a and b
Question 16.
It the MPC is 0.5, the multiplier is –
(a) 2
(b) 1/2
(c) 0.2
(d) 20
(a) 2
Question 17.
In an open economy import ………………………. the value of the multiplier
(a) Reduces
(b) increase
(c) does not change
(d) changes
(a) Reduces
Question 18.
According to Keynes, investment is a function of the MEC and –
(a) Demand
(b) Supply
(c) Income
(d) Rate of interest
(d) Rate of interest
Question 19.
The term super multiplier was first used by –
(a) J.R.Hicks
(b) R.G.D. Allen
(c) Kahn
(d) Keynes
(a) J.R.Hicks
Question 20.
The term MEC was introduced by –
(b) J.M. Keynes
(c) Ricardo
(d) Malthus
(b) J.M. Keynes
Part – B
Answer The Following Questions In One or Two Sentences
Question 21.
What is consumption function?
Meaning of Consumption Function:
1. The consumption function or propensity to consume refers to income consumption relationship. It is a “functional relationship between two aggregates viz., total consumption and gross national income.”
2. Symbolically, the relationship is represented as C = f (Y)
Where,
C = Consumption; Y = Income; f = Function
3. Thus the consumption function indicates a functional relationship between C and Y, where C is the dependent variable and Y is the independent variable, i.e., C is determined by Y. This relationship is based on the ceteris paribus (other things being same) assumption, as only income consumption relationship is considered and all possible influences on consumption are held constant.
Question 22.
What do you mean by propensity to consume?
1. The consumption function or propensity to consume refers to income consumption relationship. It is a “functional relationship between two aggregates viz., total consumption and gross national income.”
2. Symbolically, the relationship is represented as C = f(Y) Where, C = Consumption; Y = Income; f = Function
3. Thus the consumption function indicates a functional relationship between C and Y, where C is the dependent variable and Y is the independent variable, i.e., C is determined by Y. This relationship is based on the ceteris paribus (other things being same) assumption, as only income consumption relationship is considered and all possible influences on consumption are held constant.
Question 23.
Define average propensity to consume (APC)?
Average Propensity to Consume:
1. The average propensity to consume is the ratio of consumption expenditure to any particular level of income.” Algebraically it may be expressed as under:
Where, C = Consumption; Y = Income
APC = $$\frac{C}{Y}$$
Where, C = Consumption; Y = Income.
Question 24.
Define marginal propensity to consume (MPC)?
Marginal Propensity to Consume:
1. The marginal propensity to consume may be defined as the ratio of the change in the consumption to the change in income. Algebraically it may be expressed as under:
MPC = $$\frac { \Delta C }{ \Delta Y }$$
Where, ∆C = Change in Consumption; ∆Y = Change in Income
MPC is positive but less than unity, 0 < $$\frac { \Delta C }{ \Delta Y }$$ < 1.
Question 25.
What do you mean by propensity to save?
1. Thus the consumption function measures not only the amount spent on consumption but also the amount saved.
2. This is because the propensity to save is merely the propensity not to consume.
3. The 45° line may therefore be regarded as a zero – saving line, and the shape and position of the C curve indicate the division of income between consumption and saving.
Question 26.
Define average propensity to save (APS)?
Average Propensity to Save (APS):
1. The average propensity to save is the ratio of saving to income.
2. APS is the quotient obtained by dividing the total saving by the total income. In other words, it is the ratio of total savings to total income. It can be expressed algebraically in the form of equation as under
3. APS = $$\frac{S}{Y}$$ Where, S = Saving; Y = Income
Question 27.
Define Marginal Propensity to Save (MPS)?
Marginal Propensity to Save (MPS):
1. Marginal Propensity to Save is the ratio of change in saving to a change in income.
2. MPS is obtained by dividing change in savings by change in income. It can be expressed algebraically as MPS = $$\frac { \Delta S }{ \Delta Y }$$
∆S = Change in Saving; ∆Y = Change in Income
Since MPC + MPS = 1
MPS = 1 – MPC and MPC = 1 – MPS.
Question 28.
Define Multiplier?
1. The multiplier is defined as the ratio of the change in national income to change in investment.
2. If AI stands for increase in investment and AY stands for resultant increase in income, the multiplier K =AY/AI.
3. Since AY results from AI, the multiplier is called investment multiplier.
Question 29.
Define Accelerator?
1. “The accelerator coefficient is the ratio between induced investment and an initial change in consumption.”
2. Assuming the expenditure of ₹50 crores on consumption goods, if industries lead to an investment of ₹100 crores in investment goods industries, we can say that the accelerator is 2.
3. Accelerator = $$\frac { 100 }{ \Delta Y }$$ = 2
Part – C
Answer The Following Questions In One Paragraph.
Question 30.
State the propositions of Keynes’s Psychological Law of Consumption?
Propositions of the Law:
This law has three propositions:
1. When income increases, consumption expenditure also increases but by a smaller amount. The reason is that as income increases, we wants are satisfied side by side, so that the need to spend more on consumer goods diminishes. So, the consumption expenditure increases with increase in income but less than proportionately.
2. The increased income will be divided in some proportion between consumption expenditure and saving. This follows from the first proposition because when the whole • of increased income is not spent on consumption, the remaining is saved. In this way, consumption and saving move together.
3. Increase in income always leads to an increase in both consumption and saving. This means that increased income is unlikely to lead to fall in either consumption or saving. Thus with increased income both consumption and saving increase.
Question 31.
Differentiate autonomous and induced investment?
Autonomous Investment:
1. Independent
2. Income inelastic
3. Welfare motive
Induced Investment:
1. Planned
2. Income elastic
3. Profit Motive
Question 32.
Explain any three subjective and objective factors influencing the consumption function?
Subjective Factors:
1. The motive of precaution: To build up a reserve against unforeseen contingencies. e.g. Accidents, sickness. ,
2. The motive of foresight: The desire to provide for anticipated future needs. e.g. Old age.
3. The motive of calculation: The desire to enjoy interest and appreciation. Consumption and Investment Functions.
Objective Factors:
1. Income Distribution:
If there is large disparity between rich and poor, the consumption is low because the rich people have low propensity to consume and high propensity to save.
2. Price level:
Price level plays an important role in determining the consumption function. When the price falls, real income goes up; people will consume more and propensity to save of the
society increases.
3. Wage level:
Wage level plays an important role in determining the consumption function and there is positive relationship between wage and consumption. Consumption expenditure increases with the rise in wages. Similar is the effect with regard to windfall gains.
Question 33.
Mention the differences between accelerator and multiplier effect?
Accelerator Effect Multiplier Effect:
1. Accelerator is the numerical value of the relation between an increase in consumption and the resulting increasing in Investment. Multiplier is the ration of the change in national income to change in Investment.
2. Accelerator (β) = $$\frac { \Delta I }{ \Delta C }$$
ΔI = Change in Investment
ΔC = Change in consumption demand Multiplier (K) = $$\frac { \Delta I }{ \Delta C }$$
ΔI = Increase in Investment ΔY = Increase in Income ΔY results from ΔI
3. Accelerator Effects are –
1. Increase in consumer demand.
2. Films get close to fill capacity.
3. Film invest to meet rising demand. Multiplier Effects are
Multiplier Effect:
1. Multiplier is the ration of the change in national income to change in Investment.
2. Multiplier:
Multiplier (K) = $$\frac { \Delta Y }{ \Delta I }$$
ΔI = Increase in Investment
ΔY = Increase in Income
ΔY results from ΔI
Multiplier Effects are:
1. Positive Multiplier an initial increases is an injection (or a decrease in a leakage) leads to a greater final increase in real GDP.
2. Negative Multiplier an initial increases in an injection (or an increase in a leakage) leads to a greater final decrease in real GDP.
Question 34.
State the concept of super multiplier?
Super Multiplier: (k and β interaction):
1. The super multiplier is greater than simple multiplier which includes only autonomous investment and no induced investment, while super multiplier includes induced investment.
2. In order to measure the total effect of initial investment on income, Hicks has combined the k and β mathematically and given it the name of the Super Multiplier.
3. The super multiplier is worked out by combining both induced consumption and induced investment.
Question 35.
Specify the limitations of the multiplier?
1. There is change in autonomous investment.
2. There is no induced investment
3. The marginal propensity to consume is constant.
4. Consumption is a function of current income.
5. There are no time lags in the multiplier process.
6. Consumer goods are available in response to effective demand for them.
7. There is a closed economy unaffected by foreign influences.
8. There are no changes in prices.
9. There is less than full employment level in the economy.
Part – D
Question 36.
Explain Keynes psychological law of consumption function with diagram?
The three propositions of the law:
Proposition (1):
Income increases by ₹ 60 crores and the increase in consumption is by ₹ 50 crores.
Proposition (2):
The increased income of ₹ 60 crores in each case is divided in some proportion between consumption and saving respectively, (i.e., ₹ 50 crores and ₹ 10 crores).
Proposition (3):
As income increases consumption as well as saving increase. Neither consumption nor saving has fallen. Diagrammatically, the three propositions are explained in figure. Here, income is measured horizontally and consumption and saving are measured on the vertical axis. C is the consumption function curve and 45° line represents income consumption equality.
Proposition (1):
When income increases from 120 to 180 consumption also increases from 120 to 170 but the increase in consumption is less than the increase in income, 10 is saved.
Proposition (2):
When income increases to 180 and 240, it is divided in some proportion between consumption by 170 and 220 and saving by 10 and 20 respectively.
Proposition (3):
Increases in income to 180 and 240 lead to increased consumption 170 and 220 and increased saving 20 and 10 than before. It is clear from the widening area below the C curve and the saving gap between 45° line and C curve.
Question 37.
Briefly explain the subjective and objective factors of consumption function?
Subjective Factors:
1. The motive of precaution: To build up a reserve against unforeseen contingencies. e.g. Accidents, sickness
2. The motive of foresight: The desire to provide for anticipated future needs, e.g. Old age
3. The motive of calculation: The desire to enjoy interest and appreciation.
4. The motive of improvement: The desire to enjoy for improving standard of living.
5. The motive of financial independence.
6. The motive of enterprise (desire to do forward trading).
7. The motive of pride.(desire to bequeath a fortune)
8. The motive of avarice.(purely miserly instinct)
Objective Factors:
1. Income Distribution:
If there is large disparity between rich and poor, the consumption is low because the rich people have low propensity to consume and high propensity to save.
2. Price level:
1. Price level plays an important role in determining the consumption function.
2. When the price falls, real income goes up; people will consume more and propensity to save of the society increases.
3. Wage level:
1. Wage level plays an important role in determining the consumption function and there is positive relationship between wage and consumption.
2. Consumption expenditure increases with the rise in wages.
3. Similar is the effect with regard to windfall gains.
4. Interest rate:
1. Rate of interest plays an important role in determining the consumption function.
2. Higher rate of interest will encourage people to save more money and reduces consumption.
5. Fiscal Policy:
When government reduces the tax the disposable income rises and the propensity to consume of community increases.
6. Consumer credit:
1. The availability of consumer credit at easy installments will encourage households to buy consumer durables like automobiles, fridge, computer.
2. This pushes up consumption.
7. Demographic factors:
1. Ceteris paribus, the larger the size of the family, the grater is the consumption.
2. Besides size of family, stage in family life cycle, place of residence and occupation affect the consumption function.
8. Duesenberry hypothesis:
Duesenberry has made two observations regarding the factors affecting consumption.
1. The consumption expenditure depends not only on his current income but also past income and standard of living.
2. Consumption is influenced by demonstration effect. The consumption standards of low income groups are influenced by the consumption standards of high income groups.
9. Windfall Gains or losses:
Unexpected changes in the stock market leading to gains or losses tend to shift the consumption function upward or downward.
Question 38.
Illustrate the working of Multiplier?
Working of Multiplier:
1. Suppose the Government undertakes investment expenditure equal to ₹ 100 crore on some public works, by way of wages, price of materials etc.
2. Thus income of labourers and suppliers of materials increases by ₹ 100 crore. Suppose the MPC is 0.8 that is 80 %.
3. A sum of ₹ 80 crores is spent on consumption (A sum of ₹ 20 Crores is saved).
4. As a result, suppliers of goods get an income of ₹ 80 crores.
5. They intum spend ₹ 64 crores (80% of ₹ 80 cr).
6. In this manner consumption expenditure and increase in income act in a chain like maimer.
The final result is ∆Y = 100 + 100 × 4/5 + 100 × [4/5]2 + 100 × [4/5]3 or,
∆Y = 100 + 100 × 0.8 + 100 × (0.8)2 + 100 × (0.8)3
= 100 + 80 + 64 + 51.2… = 500 .
that is 100 × 1/1 – 4/5
100 × 1/1/5
100 × 5 = ₹ 500 crores
For instance if C = 100 + 0.8Y, I = 100,
Then Y = 100 + 0.8Y + 100
0.2Y = 200
Y = 200/0.2 = 1000 → Point B
If I is increased to 110, then
0.2Y = 210
Y = 210/0.2 = 1050 → Point D
For ₹ 10 increase in I, Y has increased by ₹ 50.
This is due to multiplier effect.
At point A, Y = C = 500
C = 100 + 0.8 (500) = 500; S = 0
At point B, Y = 1000
C = 100 + 0.8 (1000) = 900; S = 100 = I At point D, Y = 1050
C = 100 + 0.8 (1050) = 940; S = 110 = I
When I is increased by 10, Y increases by 50.
This is multiplier effect (K = 5)
K = $$\frac{1}{0.2}$$ = 5
Question 39.
Explain the operation of the Accelerator?
Operation of the Acceleration Principle:
1. Let us consider a simple example. The operation of the accelerator may be illustrated as follows.
2. Let us suppose that in order to produce 1000 consumer goods, 100 machines are required.
3. Also suppose that working life of a machine is 10 years.
4. This means that every year 10 machines have to be replaced in order to maintain the constant flow of 1000 consumer goods. This might be called replacement demand.
5. Suppose that demand for consumer goods rises by 10 percent (i.e. from 1000 to 1100).
6. This results in increase in demand for 10 more machines.
7. So that total demand for machines is 20. (10 for replacement and 10 for meeting increased demand).
8. It may be noted here a 10 percent increase in demand for consumer goods causes a 100 percent increase in demand for machines (from 10 to 20).
9. So we can conclude even a mild change in demand for consumer goods will lead to wide change in investment.
Diagrammatic illustration:
Operation of Accelerator.
1. SS is the saving curve. II is the investment curve. At point E1 the economy is in equilibrium with OY1 income. Saving and investment are equal at OY1 Now, investment is increased from OI2 to OI4.
2. This increases income from OY1 to OY3, the equilibrium point being E3 If the increase in investment by I2 I4 is purely exogenous, then the increase in income by Y1 Y3 would have been due to the multiplier effect.
3. But in this diagram it is assumed that exogenous investment is only by I, I3 and induced investment is by I3I4.
4. Therefore, increase in income by Y1 Y2 is due to the multiplier effect and the increase in income by Y2 Y3 is due to the accelerator effect.
Question 40.
What are the differences between MEC and MEI?
Marginal Efficiency of Capital (MEC):
1. It is based on a given supply price for capital.
2. It represents the rate of return on all successive units of capital without regard to existing capital.
3. The capital stock is taken on the X axis of diagram.
4. It is a “stock” concept.
5. It determines the optimum capital stock in an economy at each level of interest rate.
Marginal Efficiency of Investment (MEI):
1. It is based on the induced change in the price due to change in the demand for capital.
2. It shows the rate of return on just those units of capital over and above the existing capital stock.
3. The amount of investment is taken on the X – axis of diagram.
4. It is a “flow” concept.
5. It determines the net investment of the economy at each interest rate given the capital stock.
### Samacheer Kalvi 12th Economics Consumption and Investment Functions Additional Questions and Answers
part – A
I. Multiple Choice Questions.
Question 1.
Price level plays an important role in determining the ……………………
(a) Consumption function
(b) Income function
(c) Finance function
(d) Price function
(a) Consumption function
Question 2.
The progressive tax system increases the ……………………….. of the people by altering the income distribution in favour of poor?
(a) price level
(b) wage level
(c) propensity to consume
(d) Fiscal policy
(c) propensity to consume
Question 3.
…………………….. means purchase of stocks and shares, debentures, government bonds and equities?
(a) Consumption
(b) Investment
(c) Finance
(d) Saving
(b) Investment
Question 4.
…………………… is influenced by demonstration effect.
(a) Investment
(b) Interest
(c) Expenditure
(d) Consumption
(d) Consumption
Question 5.
Additional investment that is independent of income is called ……………………
(a) Autonomous Investment
(b) Autonomous Consumption
(c) Average Investment
(d) Marginal Investment
(a) Autonomous Investment
Question 6.
Induced investment is motivated?
(a) Investment
(b) Capital
(c) Saving
(d) Profit
(d) Profit
Question 7.
MEI is the expected rate of return on investment as additional units of ……………………
(a) Saving
(b) Investment
(c) Consumption
(d) Expenditure
(b) Investment
Question 8.
Dynamic multiplier is also known as ……………………
(a) Sequence multiplier
(b) Static multiplier
(c) Double multiplier
(d) Single multiplier
(a) Sequence multiplier
Question 9.
The combined effect of interaction of multiplier and accelerator is called ……………………
(a) Super accelerator
(b) Super multiplier
(c) Accelerator
(d) Multiplier
(b) Super multiplier
Question 10.
The tendency to initiate Superior consumption pattern is called ……………………
(a) Accelerator effect
(b) Multiplier effect
(c) Super Multiplier effect
(d) Demonstration effect
(d) Demonstration effect
Question 11.
The multiplier is the reciprocal of one minus ……………………
(a) MPC
(b) MPS
(c) Multiplier
(d) Accelerator
(a) MPC
Question 12.
The concept of multiplier was first developed by ……………………
(a) J.M. Keynes
(b) David Ricardo
(c) R.F. Khan
(d) J.B. Say
(c) R.F. Khan
Question 13.
…………………… the larger size of the family, the greater is the consumption?
(a) Demographic factors
(b) Income Distribution
(c) Duesenberry hypothesis
(d) Wage level
(b) Income Distribution
Question 14.
MPS is the ratio of change in saving to a change in ……………………
(a) profit
(b) money
(c) finance
(d) income
(d) income
Question 15.
Consumption function is called the relationship between ……………………….. and Income?
(a) Money
(b) Consumption
(c) Finance
(d) Investment
(b) Consumption
Question 16.
Consumer’s surplus is useful to the Finance Minister in formulating ……………………….. policies?
(a) Surplus
(b) Consumption
(c) Taxation
(d) Income
(c) Taxation
Question 17.
Consumer surplus is called potential price – ……………………………. price?
(a) real
(b) actual
(c) normal
(d) high
(b) actual
Question 18.
Dynamic multiplier is also known as ………………………. Multiplier.
(a) Sequence
(b) Static
(c) Timeless
(d) Logical
(a) Sequence
Question 19.
Static Multiplier is otherwise known as …………………………… Multiplier.
(a) Dynamic
(b) Leakage
(c) Simultaneous
(d) Multi effect
(c) Simultaneous
Question 20.
The propensity to consume refers to the portion of Income spent on ……………………….
(a) Income
(b) Profit
(c) Expenditure
(d) Consumption
(d) Consumption
Question 21.
………………………. redefined it as investment multiplier.
(a) R.K. Khan
(b) David Ricardo
(c) J.M. Keynes
(d) Marshall
(c) J.M. Keynes
Question 22.
Accelerator Model was made by ……………………
(a) J.M. Keynes
(b) J.M. Clark
(c) R.F. Khan
(d) Marshall
(b) J.M. Clark
Question 23.
The multiplier tells us …………………………. changes after a shift in ……………………
(a) income
(b) investment
(c) aggregate demand
(d) savings
(c) aggregate demand
Question 24.
The simple accelerated model was made by J.M. Clark in ……………………
(a) 1915
(b) 1916
(c) 1914
(d) 1917
(d) 1917
II. Match the following and choose the correct answer by using codes given below
Question 1.
A. Consumption function – (i) Consmption increased
B. Induced Investment – (ii) Borrowings
C. Income Increases – (iii) Subjective and objective
D. Autonomous consumption – (iv) Profit motive
Codes:
(a) A (iii) B (iv) C (i) D (ii)
(b) A (iv) B (i) C (ii) D (iii)
(c) A (i) B (ii) C (iii) D (iv)
(d) A (ii) B (iii) C (iv) D (i)
(a) A (iii) B (iv) C (i) D (ii)
Question 2.
A. MPS – measured – (i) K = 1/MPS
B. Multiplier developed by – (ii) MEC
C. Investment depends on – (iii) ∆S/∆Y
D. Value of multiplier – (iv) R.F. Khan
Codes:
(a) A (i) B (ii) C (iv) D (iii)
(b) A (ii) B (iii) C (i) D (iv)
(c) A (iii) B (iv) C (ii) D (i)
(d) A (iv) B (i) C (iii) D (ii)
(c) A (iii) B (iv) C (ii) D (i)
Question 3.
A. Reduced Investment – (i) 1930
B. Keynes employment dependes on – (ii) Highest interest rate
C. Fall in investment – (iii) Zero
D. Long fun autonomous consumption will – (iv) Investment
Codes:
(a) A (i) B (iii) C (iv) D (ii)
(b) A (ii) B (iv) C (i) D (iii)
(c) A (iii) B (i) C (ii) D (iv)
(d) A (iv) B (ii) C (iii) D (i)
(b) A (ii) B (iv) C (i) D (iii)
Question 4.
A. MPS – (i) AC/AY
B. MPC – (ii) C/Y
C. APS – (iii) S/Y
D. APC – (iv) AS/AY
Codes:
(a) A (iv) B (i) C (iii) D (ii)
(b) A (i) B (ii) C (iv) D (iii)
(c) A (ii) B (iii) C (i) D (iv)
(d) A (iii) B (iv) C (ii) D (i)
(a) A (iv) B (i) C (iii) D (ii)
Question 5.
A. Investment means – (i) Expenditure on capital formation
B. Uses of multiplier – (ii) Consumption forgone
C. Saving is – (iii) Achieve full employment
D. Autonomous investment – (iv) Stocks and shares
Codes:
(a) A (ii) B (i) C (iv) D (iii)
(b) A (iii) B (ii) C (iii) D (iv)
(c) A (iv) B (iii) C (ii) D (i)
(d) A (i) B (iv) C (i) D (ii)
(c) A (iv) B (iii) C (ii) D (i)
III. State whether the statements are true or false.
Question 1.
(i) Keynes propounded the fundamental psychological law of consumption.
(ii) J.M. Keynes has divided factors influencing the consumption function.
(a) Both (i) and (ii) are true
(b) Both (i) and (ii) are false
(c) (i) is true but (ii) is false
(d) (i) is false but (ii) is true
(a) Both (i) and (ii) are true
Question 2.
(i) The kinds of multiplier is called Tax Multiplier, Employment Multiplier, Foreign trade Multiplier, Investment Multiplier.
(ii) Investment means money collecting.
(a) Both (i) and (ii) are true
(b) Both (i) and (ii) are false
(c) (i) is true but (ii) is false
(d) (i) is false but (ii) is true
(d) (i) is false but (ii) is true
Question 3.
(i) The term investment means purchase of stocks and shares, debentures, government bonds and equities.
(ii) The term Investment means expenditure on capital formation.
(a) Both (i) and (ii) are true
(b) Both (i) and (ii) are false
(c) (i) is true but (ii) is false
(d) (i) is false but (ii) is true
(c) (i) is true but (ii) is false
Question 4.
(i) Leakages of multiplier is payment only.
(ii) Leakages of multiplier limitation is called full employment situation.
(a) Both (i) and (ii) are true
(b) Both (i) and (ii) are false
(c) (i) is true but (ii) is false
(d) (i) is false but (ii) is true
(d) (i) is false but (ii) is true
Question 5.
(i) The types of Investment is called Autonomous Investment, Induced Investment.
(ii) Induced Investment is the expenditure on fixed assets and stocks.
(a) Both (i) and (ii) are true
(b) Both 0) and (if) are false
(c) (i) is true but (ii) is false
(d) (i) is false but (ii) is true
(a) Both (i) and (ii) are true
IV. Which of the following is correctly matched:
Question 1.
(a) J.M. Clark – Ceteris Paribus
(b) J.M. Keynes – Psychological law of consumption
(c) R.F. Khan – Accelerator model
(d) Duesenberry – Laissez – faire
(b) J.M. Keynes – Psychological law of consumption
Question 2.
(a) Induced Investment – Profit motive
(b) MEC – Autonomous Investment
(c) MEI – Technology
(d) MPC – Accelerator
(a) Induced Investment – Profit motive
Question 3.
(a) Dynamic Multiplier – Employment
(b) Static Multiplier – Wealth
(c) Accelerator Model – J.M. Clark
(d) Leakage Multiplier – Investment goods
(c) Accelerator Model – J.M. Clark
Question 4.
(a) Afltalion – 1909
(b) Hawtrey – 1914
(c) Bickerdike – 1915
(d) J.M. Clark – 1916
(a) Afltalion – 1909
Question 5.
(a) Aggregate Income – C
(b) Consumption expenditure – IA
(c) Autonomous Investment – Y
(d) Induced Private Investment – IP
(d) Induced Private Investment – IP
V. Which of the following is not correctly matched
Question 1.
(a) Static multiplier – Simultaneous multiplier
(b) Dynamic multiplier – Sequence multiplier
(c) Leakage multiplier – Timeless multiplier
(d) Kinds of multiplier – Tax multiplier
(c) Leakage multiplier – Timeless multiplier
Question 2.
(a) Ratio of the consumption – APC expenditure to Income
(b) Ratio of change in consumption – MPC to change in Income
(c) Ratio of the saving to Income – APS
(d) Ratio of change in saving to change in Income – PSM change in Income
(d) Ratio of change in saving to change in Income – PSM change in Income
Question 3.
(a) Demonstration Effect – Superior consumption pattern
(b) Subjective factors – Psychological feeling
(c) Objective factors – Real and Measurable
(d) Super multiplier – Investment demand
(d) Super multiplier – Investment demand
Question 4.
(a) Average propensity to consume – C/Y
(b) Marginal propensity to consume – AC/AY
(c) Average propensity to consume – S/Y
(d) Marginal propensity to save – AY/AS
(d) Marginal propensity to save – AY/AS
Question 5.
(a) The motive of precaution – Accidents, Sickness
{b) The motive of foresight – Old age
(c) The motive of improvement – Improve standard of living
(d) The motive of calculation – Money collecting
(d) The motive of calculation – Money collecting
VI. Pick the odd one out.
Question 1.
(a) ∆C – Change in consumption
(b) ∆Y – Change in expenditure
(c) ∆S – Change in saving 4
(d) ∆Y – Change in income
(b) ∆Y – Change in expenditure
Question 2.
(a) APC – Algebraically Propensity to Consume
(b) MPC – Marginal Propensity to Consume
(c) APS – Average Propensity to Consume
(d) MPS – Marginal Propensity to Save
(a) APC – Algebraically Propensity to Consume
Question 3.
Keynes’s Law is based on the Assumptions.
(a) Ceteris paribus
(b) Existence of Normal conditions
(c) Existence of a Laissez – Faire
(d) Existence of a Technical attributes
(d) Existence of a Technical attributes
Question 4.
Investment means
(a) Purchase of stocks and shares
(b) Debentures
(c) Government bonds and equities
(d) Bank amount
(d) Bank amount
Question 5.
MEC – Short Run Factors
(a) Supply for the product
(b) Liquid Assets
(c) Sudden changes in Income
(d) Current rate of Investment
(a) Supply for the product
VII. Assertion and Reason.
1. Assertion (A): Keynes Law of propositions – when Income increases, consumption expenditure also increases but by a smaller amount.
Reason (R): Keynes Law of propositions – Increase in Income always lead to an increase in both consumption and saving.
(a) Both ‘A’ and ‘R’ are true and ‘R’ is the correct explanation to ‘A’
(b) Both ‘A’ and ‘R’ are true but ‘R’ is not the correct explanation to ‘A’
(c) ‘A’ is true but ‘R’ is false
(d) ‘A’ is false but ‘R’ is true
(a) Both ‘A’ and ‘R’ are true and ‘R’ is the correct explanation to ‘A’
Question 2.
Assertion (A): J.M. Keynes has influencing consumption function into subjective factors are the Internal factors related to psychological feelings.
Reason (R): J.M. Keynes has influencing consumption function into objective factors are Internal factors are not measurable.
(a) Both ‘A’ and ‘R’ are true and ‘R’ is the correct explanation to ‘A’
(b) Both ‘A’ and ‘R’ are true but ‘R’ is not the correct explanation to ‘A’
(c) ‘A’ is true but ‘R’ is false
(d) ‘A’ is false but ‘R’ is true
(c) ‘A’ is true but ‘R’ is false
Question 3.
Assertion (A): Autonomous Investment is the expenditure on capital formation.
Reason (R): Autonomous Investment is Independent of the change in Income, rate of Interest or rate of profit.
(a) Both ‘A’ and ‘R’ are true and ‘R’ is the correct explanation to ‘A’
(b) Both ‘A’ and ‘R’ are true but ‘R’ is not the correct explanation to ‘A’
(c) ‘A’ is true but ‘R’ is false
(d) ‘A’ is false but ‘R’ is true
(a) Both ‘A’ and ‘R’ are true and ‘R’ is the correct explanation to ‘A’
Question 4.
Assertion (A): MEC – depends on the Demand yield from a capital asset.
Reason (R): MEC – depends on the Supply price of a capital asset.
(a) Both ‘A’ and ‘R’ are true and ‘R’ is the correct explanation to ‘A’
(b) Both ‘A’ and ‘R’ are true but ‘R’ is not the correct explanation to ‘A’
(c) ‘A’ is true but ‘R’ is false
(d) ‘A’ is false but ‘R’ is true
(d) ‘A’ is false but ‘R’ is true
Question 5.
Assertion (A): Keynes theory of the Multiplier Assumption is change in autonomous Investment.
Reason (R): Keynes theory of the Multiplier Assumption is no Induced Investment.
(а) Both ‘A’ and ‘R’ are true and ‘R’ is the correct explanation to ‘A’
(b) Both ‘A’ and ‘R’ are true but ‘R’ is not the correct explanation to ‘A’
(c) ‘A’ is true but ‘R’ is false
(d) ‘A’ is false but ‘R’ is true
(а) Both ‘A’ and ‘R’ are true and ‘R’ is the correct explanation to ‘A’
Part – B
Answer The Following Questions In One or Two Sentences.
Question 1.
Write “Propensity to consume” Equations?
(i) The Average Propensity to Consume = $$\frac{c}{y}$$
(ii) The Marginal Propensity to Consume = $$\frac{∆c}{∆y}$$
(iii) The Average Propensity to Save = $$\frac{x}{y}$$
(iv) The Marginal Propensity to Save = $$\frac{∆s}{∆y}$$
Question 2.
Define “Ceteris paribus”?
Ceteris paribus (constant extraneous variables):
The other variables such as income distribution, tastes, habits, social customs, price movements, population growth, etc. do not change and consumption depends on income alone.
Question 2.
Define “Laissez-Faire” – Capitalist Economy?
Existence of a Laissez – faire Capitalist Economy:
The law operates in a rich capitalist economy where there is no government intervention. People should be free to spend increased income. In the case of regulation of private enterprise and consumption expenditures by the State, the law breaks down.
Question 3.
What do you mean “Windfall Gains” or “Losses”?
Windfall Gains or losses:
Unexpected changes in the stock market leading to gains or losses tend to shift the consumption function upward or downward.
Question 4.
Define “Autonomous consumption”?
Autonomous Consumption:
Autonomous consumption is the minimum level of consumption or spending that must take place even if a consumer has no disposable income, such as spending for . basic necessities.
Part – C
Answer the Following Questions In One Paragraph.
Question 1.
Explain the Keynes Psychological Law’ of consumption assumptions?
Keynes’s Law is based on the following assumptions:
1. Ceteris paribus (constant extraneous variables):
The other variables such as income distribution, tastes, habits, social customs, price movements, population growth, etc. do not change and consumption depends on income alone.
2. Existence of Normal Conditions:
1. The law holds good under normal conditions.
2. If, however, the economy is faced with abnormal and extraordinary circumstances like war, revolution or hyperinflation, the law will not operate.
3. People may spend the whole of increased income on consumption.
3. Existence of a Laissez – faire Capitalist Economy:
1. The law operates in a rich capitalist economy where there is no government intervention.
2. People should be free to spend increased income.
3. In the case of regulation of private enterprise and consumption expenditures by the State, the law breaks down.
Question 2.
Explain the Marginal Efficiency of capital?
Marginal Efficiency of Capital:
1. MEC was first introduced by J.M Keynes in 1936 as an important determinant of autonomous investment.
2. The MEC is the expected profitability of an additional capital asset.
3. It may be defined as the highest rate of return over cost expected from the additional unit of capital asset.
4. Meaning of Marginal Efficiency of Capital (MEC) is the rate of discount which makes the discounted present value of expected income stream equal to the cost of capital.
MEC depends on two factors:
1. The prospective yield from a capital asset.
2. The supply price of a capital asset.
Factors Affecting MEC:
Question 3.
Explain the uses of multiplier?
Uses of multiplier
1. Multiplier highlights the importance of investment in income and employment theory
2. The process throws light on the different stages of trade cycle.
3. It also helps in bringing the equality between S and I.
4. It helps in formulating Government policies.
5. It helps to reduce unemployment and achieve full employment.
Question 4.
Write the Accelerator Assumptions?
Assumptions:
1. Absence of excess capacity in consumer goods industries.
2. Constant capital – output ratio
3. Increase in demand is assumed to be permanent
4. Supply of funds and other inputs is quite elastic
5. Capital goods are perfectly divisible in any required size.
Question 5.
Write the “Leverage Effect” and Equation Explanation?
Leverage Effect:
The combined effect of the multiplier and the accelerator is also called the leverage effect which may lead the economy to very high or low level of income propagation.
Symbolically
Y = C + IA + IP
Y = Aggregate income
C = Consumption expenditure
T = autonomous investment; IP = induced private investment
Part – D
Answer The Following Questions In One Page.
Question 1.
Briefly explain the Leakages of Multiplier?
Leakages of multiplier:
1. The multiplier assumes that those who earn income are likely to spend a proportion of their additional income on consumption.
2. But in practice, people tend to spend their additional income on other items. Such expenses are known as leakages.
Payment towards past debts:
If a portion of the additional income is used for repayment of old loan, the MPC is reduced and as a result the value of multiplier is cut.
Purchase of existing wealth:
1. If income is used in purchase of existing wealth such as land, building and shares money is circulated among people and never enters into the consumption stream.
2. As a result the value of multiplier is affected.
Import of goods and services:
1. Income spent on imports of goods or services flows out of the country and has little chance to return to income stream in the country.
2. Thus imports reduce the value of multiplier.
Non availability of consumer goods:
1. The multiplier theory assumes instantaneous supply of consumer goods following demand.
2. But there is often a time lag.
3. During this gap (D > S) inflation is likely to rise.
4. This reduces the consumption expenditure and there by multiplier value.
Full employment situation:
1. Under conditions of full employment, resources are almost fully employed.
Question 2.
Explain Marginal Propensity to Consume [MPC] and Multiplier with diagram and Diagrammatic explanation?
Marginal propensity to consume and multiplier.
The propensity to consume refers to the portion of income spent on consumption.
The MPC refers to the relation between change in consumption (C) and change in income (Y).
Symbolically MPC = ∆C/∆Y
The value of multiplier depends on MPC
Multiplier (K) = 1/1 – MPC
The multiplier is the reciprocal of one minus marginal propensity to consume.
Since marginal propensity to save is 1 – MPC. (MPC + MPS = 1).
Multiplier is 1/ MPS.
The multiplier is therefore defined as reciprocal of MPS.
Multiplier is inversely related to MPS and directly with MPC.
Numerically if MPC is 0.75, MPS is 0.25 and k is 4.
Using formula k = 1/1 – MPC
1/1 – 0.75 = 1/0.25 = 4
Taking the following values, we can explain the functioning of multiplier.
C = 100 + 0.8 y; 1 = 100 1 = 10
Y = C + I
Y = 100 + 0.8y = 100 + (1000) = 900;
S = 100 = I
After I is raised by 10, now I = 110
Y = 100 + 0.8y + 110
0.2y = 210
Y = $$\frac{210}{0.2}$$ = 1050
Here C = 100 = 0.8 (1050) = 940; S = 110 = 1
Diagrammatic Explanation.
At 45° line y = C + S
It implies the variables in axis and axis are equal.
The MPC is assumed to be at 0.8 (C = 100 + 0.8y)
The aggregate demand (C + I) curve intersects 45° line at point E.
The original national income is 500.
(C = 100 + 0.8y = 100 + 0.8 (500) = 500)
When I is 100, y = 1000, C = 900;
S = 100 = I
The new aggregate demand curve is C+F = 100 + 0.8y + 100 + 10
Y = $$\frac{210}{0.2}$$ = 1050
C = 940; S = 110 = 1
Question 3.
Explain about Marginal Efficiency of Capital [MEC] short run factors and long run factors?
(a) Short – Run Factors
1. Demand for the product:
1. If the market for a particular good is expected to grow and its costs are likely to fall, the rate of return from investment will be high.
2. If entrepreneurs expect a fall in demand for goods and a rise in cost, the investment will decline.
2. Liquid assets:
1. If the entrepreneurs are holding large volume of working capital, they can take advantage of the investment opportunities that come in their way.
2. The MEC will be high.
3. Sudden changes in income:
1. The MEC is also influenced by sudden changes in income of the entrepreneurs.
2. If the business community gets windfall profits, or tax concession the MEC will be high and hence investment in the country will go up.
3. On the other hand, MEC falls with the decrease in income.
4. Current rate of investment:
1. Another factor which influences MEC is the current rate of investment in a particular industry.
2. If in a particular industry, much investment has already taken place and the rate of investment currently going on in that industry is also very large, then the marginal efficiency of capital will be low.
5. Waves of optimism and pessimism:
1. The marginal efficiency of capital is also affected by waves of optimism and pessimism in the business cycle.
2. If businessmen are optimistic about future, the MEC will be likely to be high.
3. During periods of pessimism the MEC is under estimated and so will be low.
(b) Long – Run Factors
The long run factors which influence the marginal efficiency of capital are as follows:
1. Rate of growth of population:
1. Marginal efficiency of capital is also influenced by the rate of growth of population.
2. If population is growing at a rapid speed, it is usually believed that the demand of various types of goods will increase.
3. So a rapid rise in the growth of population will increase the marginal efficiency of capital and a slowing down in its rate of growth will discourage investment and thus reduce marginal efficiency of capital.
2. Technological progress:
1. If investment and technological development take place in the industry, the prospects of increase in the net yield brightens up.
2. For example, the development of automobiles in the 20th century has greatly stimulated the rubber industry, the steel and oil industry etc.
3. So we can say that inventions and technological improvements encourage investment in various projects and increase marginal efficiency of capital.
3. Monetary and Fiscal policies:
Cheap money policy and liberal tax policy pave the way for greater profit margin and so MEC is likely to be high.
4. Political environment:
Political stability, smooth administration, maintenance of law and order help to improve MEC.
5. Resource availability:
Cheap and abundant supply of natural resources, efficient labour and stock of capital enhance the MEC.
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2021-04-16 01:34:13
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http://math.stackexchange.com/questions/52716/whats-an-example-of-a-group-with-equivalent-uniform-structures-where-multiplica/52734
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What's an example of a group with equivalent uniform structures where multiplication is not uniformly continuous?
Say we have a topological group $G$. It's easy to see that if $\cdot: G \times G \rightarrow G$ is uniformly continuous (with respect to either the right or left uniformity), then $G$ must have equivalent uniform structures. I figure the converse is probably false, on the basis that otherwise I would have seen this mentioned somewhere. But I can't think of an example, because I don't know many examples of topological groups, and all the examples I can think of with equivalent uniformities are built out of compact, abelian, or discrete groups, all of which do have uniformly continuous multiplication! Can anyone give me a counterexample? Or are these indeed equivalent?
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Exercise 1.8.c on p.79 of Arhangel'skii-Tkachenko, Topological groups and related structures asks the reader to prove that uniform continuity of the multiplication mapping is equivalent to the group being balanced (having equivalent left and right uniformities). – t.b. Jul 20 '11 at 19:23
...huh. That's a surprise. You should post that as an answer so I can accept it and consider this closed. – Harry Altman Jul 20 '11 at 19:42
Disclaimer: I haven't done the exercise carefully myself (and I'm not really in the mood to), but on Harry's request I'm posting it as an answer.
According to Exercise 1.8.c on page 79 of Arhangel'skii-Tkachenko, Topological groups and related structures, the following are equivalent for a topological group (uniformly continuous means uniformly continuous with respect to both the left and the right uniform structures):
1. The multiplication map $G^{2} \to G$ is uniformly continuous.
2. The multiplication map $G^{n} \to G$ is uniformly continuous for all $n \geq 2$.
3. The group is balanced in the sense that the left and right uniformities are equivalent.
Since you're interested in $3 \implies 1$ that's good enough (provided that it is true).
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I would say that there is not much more to it than the fact that inversion is uniformly continuous on a balanced group. – t.b. Jul 20 '11 at 20:04
Yeah, I feel silly - this is actually easy, but I miscalculated when trying to prove it and figured it probably wasn't true. – Harry Altman Jul 20 '11 at 20:13
@Harry: Well, this happens to all of us :) Once we're convinced of the wrong track, we often need a nudge from outside... Glad I could help in that respect. – t.b. Jul 20 '11 at 20:19
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2014-07-30 21:24:57
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https://wiki.simons.berkeley.edu/doku.php?id=algebra:start&rev=1536192011
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# Simons Institute Wiki
### Site Tools
algebra:start
This is an old revision of the document!
# Algebraic and Geometric Complexity Theory Reading/Discussion Group
Next meeting: Friday, September 7, 2:00 pm in room 116 (if the room is available). Mrinal Kumar will talk about https://arxiv.org/abs/1804.03303.
The usual meeting time is Tuesdays, 4:00pm-5:30pm.
We will not have a meeting during the Boolean Devices workshop.
## Papers
• Kumar, 2018, On top fan-in vs formal degree for depth-3 arithmetic circuits link
• Efremenko, Garg, Oliveira, Wigderson, 2017, Barriers for Rank Methods in Arithmetic Complexity link
• Efremenko, Landsberg, Schenck, Weyman, 2016, The method of shifted partial derivatives cannot separate the permanent from the determinant link
• Oeding, 2016, Border ranks of monomials link
• Bläser, Ikenmeyer, Jindal, Lysikov, 2018, Generalized Matrix Completion and Algebraic Natural Proofs link
Introductions to the representation theory of the general linear group are for example given here:
## Open problems
• Does the class of p-families of polynomially bounded Waring rank equal the class of p-families of polynomially bounded border Waring rank?
• Is the Euclidean closure of the class VF of p-families of polynomially bounded formula size contained in VNP?
## MathJax
This site also supports MathJax for LaTeX. For instance, type this
$\det X = f(\vec x)$.
to get this: $\det X = f(\vec x)$.
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2022-01-22 06:07:08
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https://cds.ismrm.org/protected/17MProceedings/PDFfiles/0208.html
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### 0208
The effects of intra-voxel contrast agent diffusion on the analysis of DCE-MRI data in realistic tissue domains
Ryan Thomas Woodall1, Stephanie L Barnes1, Anna G Sorace2, David A Hormuth II1, C Chad Quarles3, and Thomas E Yankeelov1
1Biomedical Engineering, The University of Texas at Austin, Austin, TX, United States, 2Dell Medical School, The University of Texas at Austin, TX, United States, 3Barrow Neurological Institute, AZ, United States
### Synopsis
Standard compartmental models for quantitative dynamic contrast enhanced MRI (DCE-MRI) typically assume active delivery of contrast agent that is instantaneously distributed within the extravascular extracellular space within each imaging voxel. The goal of this study is to determine the error accumulated in the estimated pharmacokinetic parameters when these assumptions are not satisfied. Using finite element methods to model contrast agent arrival and diffusion throughout realistic tissue domains (obtained from histological stains of tissue sections from a murine cancer model), it was rigorously determined that parameterization error is highest in regions of low vascularity, and lowest in well-perfused regions.
### PURPOSE
Quantitative evaluation of dynamic contrast enhanced MRI (DCE-MRI) allows for the estimation of parameters related to perfusion, vessel permeability, and tissue volume fractions by fitting dynamic signal intensity curves to appropriate pharmacokinetic models (1). These models are based on compartmental analysis which necessarily assumes that the contrast agent concentration rapidly equilibrates within the extravascular space in each voxel. However, there is increasing evidence that this assumption is often violated by the Gadolinium chelates most commonly used for DCE-MRI (2,3). As the diffusivities for Gadolinium chelates range from 1-3 E-4 mm2/s (4), the contrast agent will not be equilibrated throughout an imaging voxel, thereby resulting in inaccurate parameterization. While a previous study has examined this issue using simulated tissue domains (2), we seek to extend those results by comparing the results of a simulated DCE-MRI experiment initialized and constrained with histological volume fractions from entire tumor cross-sections obtained from murine tumor studies (5).
### METHODS
Nude athymic mice were subcutaneously implanted with BT474 cancer cells which were then allowed to grow into ~300 mm3 tumors for six weeks prior to performing DCE-MRI (6). Following imaging, the tumors were extracted, sectioned, and stained for vascularity (CD31) and cellularity (H&E). Stained central slice sections were digitized at high resolution (0.5 mm), and segmented and meshed in MATLAB (Natick, MA) (Figure 1). A finite element model (FEM) with 2 mm resolution (down-sampled from the 0.5 mm resolution) was developed based on the 2D diffusion equation: $$[1], \frac{dC(x,y,t)}{dt}=\nabla\cdot D\nabla C(x,y,t),$$ where $C(x,y,t)$ is the concentration of contrast agent, and $D$ is the diffusivity (set to 2 E-4 mm2/s). Impermeable boundaries were assigned at the tumor periphery and at cell membranes. Flux of contrast agent across the boundaries at blood vessels was defined according to Eq. [2]: $$[2], \nabla C\cdot\hat{n}=P(C_p(t)-C(t)),$$ where $P$ is equal to $K^{trans}\times\frac{V}{S}$, $S$ is the total vessel surface area within a voxel, $K^{trans}$ is the volume transfer coefficient, $V$ is the volume of tissue perfused, $\hat{n}$ is the normal vector with respect to the vessel boundary, and $C_p(t)$ is a population arterial input function (AIF) (6). All vessels are assumed to contain the AIF concentration for each time step, and flux from a vessel does not affect the vascular concentration. Using the resulting contrast agent distribution, a signal intensity is calculated for each MRI voxel at 438 $\mu$m in plane resolution (down-sampled from the 2 mm resolution) at each AIF time step. The extended Tofts’ model (Eq. [3]) is then fit to the simulated signal intensity of each MRI voxel to provide estimates of $K^{trans}$, $v_e$ (extravascular, extracellular volume fraction), and $v_p$ (plasma volume fraction):$$[3], C_t(t)=K^{trans}\int_{0}^{t}C_p(u)exp(\frac{K^{trans}}{v_e}(t-u))du+v_pC_p(t),$$ Finally, the fit values for $K^{trans}$, $v_e$, and $v_p$ are then compared to the histological ($v_e$ and $v_p$) and assigned ($K^{trans}$) model values used in the forward model, and a percent error is calculated for each simulated MRI voxel.
### RESULTS
Figure 2 depicts an example FEM time course of a voxel with a histological $v_p = 0.0135$. The parameterization error for this voxel was -62% for $K^{trans}$, -24% for $v_e$, and 107% for $v_p$. Figure 3 depicts the parameterization error of $K^{trans}$ within each imaging voxel of the total imaging domain. Error in $K^{trans}$ is maximized in poorly perfused regions, such as the necrotic core, while well-perfused voxels exhibit the least error. Figure 4 depicts the relative error in $K^{trans}$ and $v_e$ with increasing distance from a vessel. Of particular note, as the distance from the vessel increases, the parameterization error increases.
### DISCUSSION
Necrotic voxels exhibit high parameterization error, as they are poorly perfused, or only show enhancement due to nearby voxels. Eq. [3] most accurately fits voxels with high vascular surface area. Fit values consistently underestimate the assigned ($K^{trans}$) and histological values ($v_e$ and $v_p$), indicating that the accessible volume (i.e., $v_e$) is frequently less than biological values. Figure 4 demonstrates that Eq. [3] is most accurate in the tissue region nearest to vessels, indicating that error is caused by limited diffusion.
### CONCLUSION
This work demonstrates a major source of error in DCE-MRI parameterization using the standard pharmacokinetic models employed by the field (ourselves included). Having demonstrated these shortcomings, future efforts are aimed at amending the standard model to account for diffusion of contrast agent, which we hypothesize will improve the accuracy of estimated pharmacokinetic parameters.
### Acknowledgements
No acknowledgement found.
### References
(1) Tofts PS, Kermode AG, et al. Measurements of the blood-brain barrier permeability and leakage space using dynamic MR imaging 1. Fundamental Concepts. Magn Reson Med. 1991:17(2):357-67
(2) Barnes SL, Quarles CC, et al. Modeling the Effect of Intra-Voxel Diffusion of Contrast Agent on the Quantitative Analysis of Dynamic Contrast Enhanced Magnetic Resonance Imaging. PlosOne.2014:9(9)
(3) Fluckiger JU, Loveless ME, et al. A diffusion-compensated model for the analysis of DCE-MRI data: theory, simulations and experimental results. Physics in Medicine and Biology.2013:58:1983-1998
(4) Koh TS, Hartono S, et al. In vivo measurement of gadolinium diffusivity by dynamic contrast-enhanced MRI: A preclinical study of human xenografts. Magnetic Resonance in Medicine.2013:69:269-276
(5) Sorace AG, Quarles CC, et al. Trastuzumab improves tumor perfusion and vascular delivery of cytotoxic therapy in a murine model of HER2+ breast cancer: preliminary results. Breast Cancer Res Treat.2016:155(5):273-83
(6) Loveless ME, Halliday J, et al. A quantitative comparison of the influence versus population-derived vascular input functions on dynamic contrast enhanced-MRI in small animals. Magnetic Resonance in Medicine.2012:67:226-236
### Figures
Figure 1A depicts an entire central histology slice of a murine tumor model, for which the entire FEM was run. This domain is split into .438 mm voxels for simulated DCE-MRI. 1B and 1C depict registered histological CD31 and H&E stains of the same tissue location. These stains are used to generate a segmented tissue domain (1D) at 2 μm resolution. Vasculature is labeled yellow, cells green, and the extravascular extracellular space blue. Extracellular space pixels are then meshed as two triangular elements in 1E. Elements unconnected to a blood vessel are not meshed, as contrast agent cannot reach them.
Figure 2 depicts the same FEM voxel shown in Figure 1B-E, at three time points during the simulation, with a diffusivity of 2 E-4 mm2/s. Initially, the elements nearest to the blood vessel are the first to fill with contrast agent. At 19.2 seconds, well past the AIF peak, the contrast agent is still unevenly distributed throughout the simulated voxel. It is not until 80 seconds that the contrast agent is equilibrated within the domain.
Figure 3 depicts the absolute parameterization error of Ktrans in the tumor domain examined in figure 1A. DCE-MRI time courses are generated at the spatial resolution of a voxel from our experimental studies (438 μm). Black voxels indicate infinite parameterization error (due to division by zero). High error on the edges can be attributed to the small portion of the voxel filled with tissue, while high error within the tumor is associated with necrosis. Low error regions correspond to high vascularity. FEM was performed with free diffusion at the boundaries between voxels to accurately simulate the entire tumor domain.
Figure 4 depicts the parameterization error of Ktrans of the maximally perfused voxel within the tumor domain. To obtain each data point, FEM simulation was run using only elements within the given distance from any vessel (x-axis). As the FEM is permitted to include elements further from vessels, the error in Ktrans increases. Parameterization error is minimized when only including elements nearest to blood vessels in the analysis.
Proc. Intl. Soc. Mag. Reson. Med. 25 (2017)
0208
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2021-06-17 23:51:01
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https://socratic.org/questions/how-do-you-solve-a-1-le-4
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How do you solve |a+1|\le 4?
Jun 8, 2018
$- 5 \setminus \le a \setminus \le 3$
Explanation:
By definition, $| x | \setminus \le k \setminus \iff - k \setminus \le x \setminus \le k$
So, $| a + 1 | \setminus \le 4 \setminus \iff - 4 \setminus \le a + 1 \setminus \le 4$
Subtract $1$ from all sides to get
$- 5 \setminus \le a \setminus \le 3$
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2019-12-08 19:31:07
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https://www.mersenneforum.org/showpost.php?s=2558ef9ae8eaa44c600d6ad0f18fbaa9&p=535103&postcount=3
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View Single Post
2020-01-14, 14:45 #3 Dr Sardonicus Feb 2017 Nowhere 17·263 Posts The usual formulation for x and y being in golden proportion is $\frac{x}{y}\;=\;\frac{x\;+\;y}{x}$; the right-hand side clearly is greater than 1. Taking y = 1 gives $\frac{x}{1}\;=\;\frac{x\;+\;1}{x}\text{, or }x^{2}\;-\;x\;-\;1\;=\;0\text{.}$ An illustration is given by the 72-72-36 degree isosceles triangle. The bisector of one of the 72-degree angles divides the opposite side in golden ratio; calling x the length of the base and y the length of the smaller segment of the side opposite the angle bisector, gives the above proportion.
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2021-04-20 05:03:31
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https://blog.forexcycle.com/385/short-term-analysis-december-21-2006/
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# Short Term Analysis – December 21, 2006
USDCAD
USDCAD topped at 1.1587 on 4 hours chart, and further fall towards 1.1400 level can be expected in a couple of days. Key resistance is at 1.1587, only break above this level may signal the resumption of the up trend.
USDJPY
USDJPY is testing the previous high resistance at 118.58, if gives way, further rise towards 119.00 is possible. Key support is at 117.43, a break below this level may signal the reversal of the up trend.
GBPUSD
GBPUSD is forming a sideways consolidation in a range between 1.9433 and 1.9846. Further rise towards 1.9846 can be expected in a couple of days. Near term support is now at 1.9433, only break below this level may signal the resumption of the down trend.
USDCHF
Key support at 1.2115 is broken below, USDCHF has topped at 1.2268 on 4 hours chart, and further fall towards 1.1983 is possible in a couple of days. Key resistance is at 1.2268, only break above this level may signal the resumption of the up trend.
AUDUSD
AUDUSD bottomed at 0.7779 on 4 hours chart and further rise towards 0.7892 previous high is still possible later today. Key support is at 0.7779, only break below this level may signal the resumption of the down trend.
EURUSD
EURUSD is forming a sideways consolidation in a range between 1.3051 and 1.3364. Further rise towards 1.3364 can be expected in a couple of days. Near term support is now at 1.3051, only break below this level may signal the resumption of the down trend.
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2023-03-26 21:35:08
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https://socratic.org/questions/how-do-you-find-the-derivative-of-sin-x-2x-cos-x-2
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# How do you find the derivative of (sin x + 2x) / (cos x - 2)?
Dec 13, 2015
$f ' \left(x\right) = \frac{2 x \sin x - 3}{\cos x - 2} ^ 2$
#### Explanation:
$f \left(x\right) = \frac{\sin x + 2 x}{\cos x - 2}$
This can be differentiate using quotient rule
$f ' \left(x\right) = \frac{\left(\cos x - 2\right) \cdot \frac{d}{\mathrm{dx}} \left(\sin x + 2 x\right) - \left(\sin x + 2 x\right) \cdot \frac{d}{\mathrm{dx}} \left(\cos x - 2\right)}{\cos x - 2} ^ 2$
$\implies f ' \left(x\right) = \frac{\left(\cos x - 2\right) \left(\cos x + 2\right) - \left(\sin x + 2 x\right) \left(- \sin x\right)}{\cos x - 2} ^ 2$
$\implies f ' \left(x\right) = \frac{{\cos}^{2} x - 4 + {\sin}^{2} x + 2 x \sin x}{\cos x - 2} ^ 2$
$\implies f ' \left(x\right) = \frac{\textcolor{red}{{\cos}^{2} x + {\sin}^{2} x} - 4 + 2 x \sin x}{\cos x - 2} ^ 2$
$\implies f ' \left(x\right) = \frac{\textcolor{red}{1} - 4 + 2 x \sin x}{\cos x - 2} ^ 2$
$f ' \left(x\right) = \frac{2 x \sin x - 3}{\cos x - 2} ^ 2$
*${\sin}^{2} x + {\cos}^{2} x = 1$
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2022-07-06 00:53:41
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https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=18&t=39086&p=132212
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## Test 2 Question 4b
$\lambda=\frac{h}{p}$
904901860
Posts: 24
Joined: Mon Jan 08, 2018 8:16 am
### Test 2 Question 4b
"Can any wavelength of light be emitted from the atoms in the lamp? Explain."
I got this question wrong and was wondering if anyone could explain to me what the correct answer is.
Thank you!
aisteles1G
Posts: 117
Joined: Fri Sep 28, 2018 12:15 am
### Re: Test 2 Question 4b
I got this right and I put: "the electron gains energy when it is excited, because it goes up an energy level. To be excited it must absorb the energy of a photon that exactly matches the delta E between energy levels."
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2020-02-25 19:28:13
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https://space.stackexchange.com/questions/35794/how-to-generate-chebyshev-coefficients-to-produce-an-ephemeris-file/37835
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# How to generate Chebyshev coefficients to produce an ephemeris file?
I am currently trying to perform the same interpolation that NASA uses to generate their Development Ephemeris files (Chebyshev coefficients). I have a table with position and velocity of the major planets in equal-spaced intervals. And I need to get this interpolation done, and get the Nth degree Chebyshev coefficients.
This process is well described at Newhall 1988 Numerical Representation of Planetary Ephemerides Celestial Mechanics, vol 45, 1–3, March 1988, pp 305–310 (viewable here as well). But my question is, do I need to implement this (as it will take lots of time) or someone has programmed it before? I mean, is there some MATLAB (could be any other program actually but I'm currently using MATLAB) routine which could make this for me? Maybe some SPICE subroutine?
I'm finding very hard to get this done due to the lack of information available online. If someone knows some explanation/example different from the Newhall one and can share it would be very appreciated.
• Chebyshev interpolation uses unequal intervals. You need samples at specific locations (densely near endpoints, sparsely in the middle) to minimize interpolation error. It’s just like regular interpolation: you have a basis set of polynomials (chebyshev polynomials) with unknown coefficients and you use the values at the unevenly sampled intervals to solve for the coefficients. Have a look at the MATLAB function called Chebfun.m.
– Paul
Apr 27 '19 at 19:34
• @Paul no, in this case the points will be evenly spaced. Have a look at the linked paper in the question (I added a viewable source).
– uhoh
Apr 28 '19 at 0:28
• Great question! In an answer somewhere here (I'll look for it) I think it was explained that the attractiveness of the Chebyshev polynomials was that they provide a clear handle on how large interpolation errors are, and not that they are necessarily best for fitting to orbits. You could use other interpolators for personal work. I'm guessing you want to make Chebyshev polynomial-based ephemerides in order to use the same interpolators that work for the Development Ephemerides, is that right? If so, it seems you'd like to get a hold of a copy of subroutine PVCH as described in Newhall 1988.
– uhoh
Apr 28 '19 at 0:28
• Answers to Is there a way to extract the Chebyshev coefficients for a body from a SPICE kernel? give some more insight into the construction of the kernels, but not on their generation. This is linked from "roll your own" in this answer but alas doesn't fit the bill. I was wondering if Project Pluto would have something to generate kernels but I don't see it. Section IV in adsabs.harvard.edu/full/1983A%26A...125..150N is interesting, but no help
– uhoh
Apr 28 '19 at 0:48
• @uhoh - As an aside, "Doctor Mohawk" and I briefly worked for the same employer. He asked the referenced question when both of us worked for that employer. I didn't know who "Doctor Mohawk" was at the time I answered his question. He knew who I was; my ID is my real name. That I happened to answer in the middle of the night was what prompted his remark "Dave, do ever sleep?" Apparently not. Apr 28 '19 at 7:54
The standard (non-custom) tools for generating Chebyshev polynomial approximations involve evaluating the function to be approximated at the non-uniformly spaced Chebyshev nodes and computing the Chebyshev inner product.
What JPL did in 1988 was highly custom. It was not anything close to the standard Chebyshev fit. They used uniformly spaced data rather than the Chebyshev nodes, and they used an ad hoc least squares approach with exact matches at the end points rather the Chebyshev inner product, and with velocity (in units of au/day) arbitrarily weighted as 2/5 as important as position (in units of au). You will not find an off-the-shelf function that does this.
Moreover, this is what JPL did in 1988. It's been 30+ years since that paper was published. In the interim, vehicles have been sent to Mercury and Pluto (and beyond), and places in between. I highly doubt that the algorithm JPL used in 1988 is what they use now.
It's possible that additional work has been done on this topic since the Newhall paper you linked, but if so, it does not cite that paper.
There is a non-peer-reviewed paper that also allows fitting acceleration data, as well, but acceleration is not used in the JPL ephemerides.
There are probably pieces of software in various places for producing Chebyshev fits of functions. For example, Numerical Recipes includes an algorithm which works when the original function is available. However, for fitting Chebyshev polynomials to coordinates and velocities at discrete times, the Newhall algorithm is my recommendation for several reasons:
• If you only fit positions, but not velocities, the Chebyshev representation will not be faithful to actual spacecraft or planetary trajectories; it will only be correct at the times used for the fit and only for position.
• Even at the points used for the fit, your error will be high.
• Your terminal points (first and last time) will not be sufficiently constrained, so if you represent a trajectory using multiple Chebyshev polynomials, there will be discontinuities.
The Newhall algorithm addresses these problems. I don't see anything better out there, and I've looked around a bit.
As Dave Hammen mentions, the Newhall paper does include a velocity weighting of 0.4, which they say they arrived at experimentally, though they provide no data.
It is also beneficial, but not necessary, to use points which are arbitrarily spaced in specific circumstances. That is, if we fit a $$n$$-degree polynomial at the $$n+1$$ roots of the $$n+1$$ degree Chebyshev, we minimize the upper bound on the error; see Theorem 16.10 at the link. The roots of a degree $$n+1$$ Chebyshev polynomial are given by
$$t_i = \cos\left(\frac{2i+1}{2n+2}\pi\right)\,\text{ for } i = 0,\dots,n\,\text{.}$$
(Of course, $$t$$ needs to be scaled from $$[-1,1]$$ to your full ephemeris time interval $$[a,b]$$.)
The Newhall paper seems to use points which are equally spaced. According to the equation above, equally spaced points are not optimal, however. Indeed, if you're using the Newhall method, you should fit to the roots (the equation above) and, I think, to the position extrema as well (I have not proved this to my satisfaction yet). Including the extrema ought to also give you the terminal points.
The good news is it ought to only be two hundred or so lines of Python code (based on my own implementation). Might take you about a day to implement.
• @uhoh Fixed it :) Aug 1 '19 at 21:10
• looks great, thank you!
– uhoh
Aug 1 '19 at 22:24
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http://kakaos.com.br/archive/hcl-lone-pairs-06004d
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# hcl lone pairs
atoms around the central atom. One way to identify a lone pair is to draw a Lewis structure.The number of lone pair electrons added to the number of bonding electrons equals the number of valence electrons of an atom. In NI3, each I will have 3 lone pairs (total of 9) and the N will also have 1 lone pair, for a grand total of 10 lone pairs. Indicate Lewis acids and bases. The first arrow originates at one of the lone pairs on the hydroxide oxygen and points to the ‘H’ symbol in the … CCl4 and HCl. Which of the following is an ionic compound? Because lone pairs occupy more space around the central atom than bonding pairs, ... (HCl, CH 2 O, NH 3, and CHCl 3), indicated in blue, whereas others do not because the bond dipole moments cancel (BCl 3, CCl 4, PF 5, and SF 6). To identify lone pairs in a molecule, figure out the number of valence electrons of the atom and subtract the number of electrons that have participated in the bonding. What is the half-life of Ra-226 and how many have passed? Borazine, also known as borazole, is a polar inorganic compound with the chemical formula B 3 H 6 N 3.In this cyclic compound, the three BH units and three NH units alternate.The compound is isoelectronic and isostructural with benzene.For this reason borazine is sometimes referred to as “inorganic benzene”. 6, 8, 9, 10, 13. Chemists usually indicate a bonding pair by a single line, as shown (below). Because the numbers are equal, each water molecule in the liquid could in principle form four hydrogen bonds, two using the δ+ hydrogens and two using the lone pairs. You have 15 grams of radium-226 after 4770 years. A lone pair refers to a pair of valence electrons that are not shared with another atom and is sometimes called a non-bonding pair. In this video we will look at the equation for HCl + H2O and write the products. 0 0. reb1240. Two of the valence electrons in the HCl molecule are shared, and the other six are located on the Cl atom as lone pairs of electrons. This creates the methylammonium ion, CH 3-NH 3 +, which reacts with the chloride (Cl-) ion to produce the salt … So on the left, right, the lone pair on the left of the oxygen didn't do anything. b.!HCl is a weaker acid than HI. One oxygen is bonded to one hydrogen. Note: Geometry refers to the Cl and I, Al and K, Cl and Mg, C and S, Al and Mg. Cl and Mg. SHELL ELECTRON PAIR REPULSION (VSEPR) MODEL. This is simple as they are the Roman numerals above the group or column you’re in. Now for BeCl2 Usually, we only show the bond and lone pair of the central atom but if you want, just for your information, each chlorine atom in the reaction has got 3 lone pairs (the red dots.) HCl Draw mechanism for (1) rearranged and (2) non-rearranged products no Adda Make Proton gone racemic ... all lone pairs, all formal charges, and all the products for each step of the mechanism. molecule. We'd also form water here, so H2O, let me draw that in and show our lone pairs of electrons. ... (HCl), the lone pair on the nitrogen atom forms a dative covalent bond with the hydrogen atom from the HCl. (central atom(s) and outer atoms? ) "A diatomic molecule, a molecule composed of only 2 atoms, must always be linear in shape as the centres of the 2 atoms will always be in a straight line." What is the name of this biochemistry structure? ? a. OH+ OH O H2O + b.CH 3 O H +HCl CH3 O H H +Cl O O +KNH2 O O c. +NH3+K+ a.OHH O N O O OH O N O O H + base acid b. HO O HCN HO O HCN + base acid. A set of common electron-pushing … โมเลกุลที่มี bonding pairs และ lone pairs -ถ้าพิจารณาโมเลกุล H 2 O ตาม Lewis structure จะประกอบด้วย คู่พันธะ O-H จ้านวน 2 … Lone Pair Explained: An easy way to identify the lone pair electrons is by examining the lewis dot structure of the atom. Due to their negative charge they are strongly attracted to positive charge, so anything with a lone pair can act as a nucleophile - it can attack centres of positive charge. ... HCl, H2O, SiH4. Relevance. Lone pair electrons occupy more space than bonding electrons. Drawing the Lewis Structure for HCl (Hydrochloric Acid) Another straight forward Lewis structure. Bond angles will deviate from their ideal values according to the rule that lone pairs repel other electrons more strongly than bonding pairs. Methane (CH 4) - Methane consists of carbon bonded to 4 hydrogen atoms and 0 lone pairs.Steric number = 4. 9 years ago. As it approaches it, the electrons in the hydrogen-chlorine bond are repelled still further towards the chlorine. Top. HCl + NH3 NH + Cl- 1 A) HCl Because It Has The Most Lone Pairs Ob) HCl Because The Chlorine Keeps A Lone Pair O C) Ammonia Because Its Lone Pair Becomes A Bond D) Ammonia Because It Needs A Lone Pair. The presence of lone pair electrons will distort predicted bond angles. HCl Draw mechanism for (1) rearranged and (2) non-rearranged products no Adda Make Proton gone racemic ... all lone pairs, all formal charges, and all the products for each step of the mechanism. Borane requires 2 more electrons on boron to fill the valence shell. Many simple molecules contain two lone pairs around the central atom. repulsive forces on other electron pairs. Get your answers by asking now. Water (H 2 O) - Water has two hydrogen atoms bonded to oxygen and also 2 lone pairs, so its steric number is 4.; Ammonia (NH 3) - Ammonia also has a steric number of 4 because it has 3 hydrogen atoms bonded to nitrogen and 1 lone electron pair. HCl - none. So one lone pair. You started with 240 grams. Join Yahoo Answers and get 100 points today. pairs of electrons about the central atom. 0 0. See the answer. help_outline. Lone pairs continued • The lone pair on the Water is attacking the HCl, pulling the weakly held Hydrogen from (deprotonating) the Chlorine • Note the conservation of charge in the final products compared to the initial reactants • We’ll come back to this reaction later… 1 Answer. Nitrogen has 5 electrons in its outer shell. A double bond is a chemical bond between two chemical elements involving four bonding electrons instead of the usual two. 7 years ago. Draw out the reaction between HCL and NH 3, using full Lewis structures (show all bonds and lone pairs). 1. Still have questions? single pair of electrons. Steric Number Calculation Examples . Remember that this molecule is OH-. Like benzene, … Raffensperger announces new Ga. voting investigation, George Clooney recalls asking wife Amal to marry him, How the 2020 pandemic has permanently changed retail, NFL blindly rolls through an embarrassing weekend, Merriam-Webster's top word of 2020 not a shocker, Movie star's family farm burns down in 'horrible fire', Teaching in the pandemic: 'This is not sustainable', These massive Cyber Monday deals just launched, McDonald's bringing back popular item â nationwide, Economist on stimulus: 'What we've lost is willingness', Missing Fla. boater found alive clinging to capsized boat. H2O, 109.5o The chief was seen coughing and not wearing a mask. By knowing the structure of the compound you can easily identify the bond pair and lone pairs in a compound. A lone pair is an electron pair in the outermost shell of an atom that is not shared or bonded to another atom. VALENCE Lone electron pairs are those electron pairs in the valence shell of an atom in a molecule, ... (HCl). ... What is the electron geometry (or electron arrangement) around an atom in a molecule or ion which is surrounded by zero lone pairs of electrons and four singel bonds. Which of the following contains ionic bonding? Should I call the police on then? lone pair synonyms, lone pair pronunciation, lone pair translation, English dictionary definition of lone pair. In 5-coordinated molecules containing lone pairs, these non-bonding orbitals (which are closer to the central atom and thus more likely to be repelled by other orbitals) will preferentially reside in the equatorial plane. Ammonia can donate its lone pair to boron, making a dative bond which then allows boron to fill its valence shell using the lone pair from the ammonia. 37. One way to identify a lone pair is to draw a Lewis structure.The number of lone pair electrons added to the number of bonding electrons equals the number of valence electrons … Thus there is no direct relationship between them. 4. This will place them at 90° angles with respect to no more than two axially-oriented bonding orbitals. Now , the hydrogen gives its single electron into the covalent bond. A lone pair refers to a pair of valence electrons that are not shared with another atom and is sometimes called a non-bonding pair. Not one, but three! Ammonia has a non-bonding lone pair on nitrogen. The lone pair is used to make a dative bond to a H + ion. 9 years ago. Lone pair electrons have the maximum repulsion, and bond pair electrons the minimum. Multiple bonds are accounted as single electron pairs, and bonded electron pairs as a single pair. The valence or outermost electron shell is assumed to be spherical. configuration that minimize the electron pair repulsions in the valence shell. other. But that does not mean each lone pair leads to a non-bonding orbital. CH4 - none. A double bond is a chemical bond between two chemical elements involving four … Explain why HCl is named hydrochloric acid while HClO3 is named chloric acid. The lone pairs on chlorine make no difference as there is only one bond on it. Ask Question + 100. atom. two (a pair of) valence electrons that are not used to form a covalent bond. 1 0. The lone pair on the right of the oxygen picked up a proton, formed a bond, and so we get this with a plus one formal charge on the oxygen. 1 lone pair. how many lone pair electrons exist in the following compounds HCl, H2C, CH4? This is how bonding works. electrons will distort predicted bond angles. In SO 2, we have one BP–BP interaction and two LP–BP interactions. Water (H 2 O) - Water has two hydrogen atoms bonded to oxygen and also 2 lone pairs, so its steric number is 4.; Ammonia (NH 3) - Ammonia also has a steric number of 4 because it has 3 hydrogen atoms bonded to nitrogen and 1 lone … Lone Pairs (around central atom) 1: Lone Pairs + Single or multiple bonds (around the central atom) 4: Electron Pair Geometry: tetrahedral: Molecular Geometry: trigonal pyramid Electrostatic potential scale 0.25 to 0.7 e … This problem has been solved! Anonymous. Show transcribed image text. The simplest example is when ammonia picks up a proton to form the NH*4* + (ammonium) ion. It has no lone pairs because, again, it only has 1 valence electron - there is no room for any more electrons to surround hydrogen. The total number of lone pairs in NCl3 is. With two bonding pairs and two lone pairs, the oxygen atom has now completed its octet. That means that there will be 9 electrons in the outer shell, expansion of the octet, which is not possible with nitrogen as far as I know. SO2 ) How many lone pairs (non-bounding electron pairs) does the compound possess on atoms? A single lone pair can be found with atoms in the nitrogen group such as nitrogen in ammonia, two lone pairs can be found with atoms in the chalcogen group such as oxygen in water and the halogens can carry three lone pairs such as in hydrogen chloride.. Remember that Hydrogen only needs two electrons to have a full outer shell. Get your answers by asking now. why does decreasing the temperature of a rection decreasing the rate at which the reaction occurs. 1. School of Chemistry > Bristol ChemLabS > Outreach > Resources > VSEPR > Lone Pairs > Examples > HCl HCl. In water, there are two δ+ hydrogens on each molecule and two lone pairs. 43. Herpderp. 2 0. 9 years ago. Arrange each of the following sets of compounds in order of increasing boiling point temperature: F2, Cl2, Br2. the molecule is determined by the number of bonded atoms plus the number of lone Back. The other lone pairs are essentially wasted. 10. lone pair. Lone pairs have a great significance in the chemistry of many compounds. Shape Moreover, by sharing a bonding pair with oxygen, each hydrogen atom now has a full valence shell of two electrons. A lone pair is an electron pair in the outermost shell of an atom that is not shared or bonded to another atom. The equation for the reaction is the following: $H_2O + HCl \rightarrow H_3O^+ + Cl^-$ However, remember that the lone pairs are pairs and therefore if you ever find just one free electron that does not participate, it would mean that the compound has … The proton bonds to the water molecule, using one of the lone pairs, producing H 3 O + . diagram showing lone pairs and bonding pairs of electrons in a molecule or an ion. (a) (b) (c) (d) (e) 41. 0 0. the ability of an atom in a molecule to attract elextrons. Include all lone pairs and formal charge in the products. Geometry is determined by the e.g. There is one lone pair … As shown in the above image, ammonia has one lone pair, water molecule has 2 lone pairs and HCl has 3 lone pairs. ok, but you can have a bond angle in molecules which only have two points eg H2, so Hcl is no differnt right? occupy more space than bonding electrons. Two electrons are shared in a single bond; four electrons are shared in a double bond; and six … These 2 are a non-bonding pair - a lone pair. I'm bloody confused! affect the gross stereochemistry of the molecule. 2. Each bond includes a sharing of electrons between atoms. 109.5 o 107 o 104.5 o . Like benzene, borazine is a colourless liquid. Unscene kid. The extra pairs of electrons on the central atom are called 'lone-pairs'. H2C - 1. Electron pairs arrange themselves to minimize the repulsion between them. Now, as HCL France steps into the next decade, we want to thank every one of our ideapreneurs in France, and around the world, without whom this journey would not have been possible. no lone You only need to draw one stereoisomer of a Methane (CH 4) - Methane consists of carbon bonded to 4 hydrogen atoms and 0 lone pairs.Steric number = 4. Two lone pairs on each O. (central atom(s) and outer atoms? I went to a Thanksgiving dinner with over 100 guests. It is also called a non-bonding pair. Those are some simple examples of why lone pairs exist and what they can do. A. one single bond and 3 lone electron pairs B. three single bonds and 1 lone electron pair C. four single bonds & no lone pairs D. two single bonds and 2 lone electron pairs Methane, CH4 Water , H2O Hydrogen Chloride… HCl is linear for both cases! A non-bonding orbital is usually just an atomic orbital, which you may thus ascribe to a lone pair. If an atom has empty orbitals, the lone pairs can be split into unpaired electrons through hybridization of orbitals and can participate in bonding. Other articles where Lone pair is discussed: chemical bonding: Lewis formulation of a covalent bond: …the chlorine atom are called lone pairs and play no direct role in holding the two atoms together. Then you have 4 hydrogens, each of which share 1 electron with each nitrogen electron. How many lone pairs are in neon? In VSEPR theory the electron pairs on the oxygen atom in water form the vertices of a tetrahedron with the lone pairs … Borazine, also known as borazole, is a polar inorganic compound with the chemical formula B 3 H 6 N 3.In this cyclic compound, the three BH units and three NH units alternate.The compound is isoelectronic and isostructural with benzene.For this reason borazine is sometimes referred to as “inorganic benzene”. Expert Answer . Although lone pairs are clearly smaller than atoms, they need to be closer to the nucleus of an atom than a bonding pair. The geometry of Page 2 4.!Use curved arrows to show the mechanism for each acid-base reaction. How many lone pair and bonding pair in phosphine? Here's a follow up to my video on using Chem Draw including lone pairs. The extra pairs of electrons on the central atom are called 'lone-pairs'. Resources > VSEPR > lone pairs around the central atom angles about a central atom s. The four electron pairs in NCl3 is interaction and two lone apirs extra of. Count as a single pair of ) valence electrons that comprise both the bonding electrons instead of the pairs! Seen coughing and not wearing a mask join Yahoo Answers and get points. Radium-226 after 4770 years so H2O, let me draw that in and show our lone pairs clearly! The four electron pairs ) does the compound you can easily identify the lone pairs of electrons on! Electrons of chlorine covalently bonded F2, Cl2, Br2 of atoms as it approaches it the! The outermost electron shell is assumed to be spherical the atoms in a,! You may thus ascribe to a pair of electrons in the valence shell electron pair in the aber! Δ+ hydrogens on each molecule and two lone apirs other electrostatically in the electron. Water here, so H2O, 109.5o 107o 104.5o, using one the! Themselves hcl lone pairs minimize the repulsion between them and write the products for each step follow... Sharing a bonding pair hcl lone pairs a single pair of valence electrons available fill! Their ideal values according to the bond hcl lone pairs about a central atom molecule an! H 2 O acid or base the compound possess on hcl lone pairs atoms? 109.5o 107o 104.5o diagram lone! Covalently bonded into the covalent bond are not shared or bonded to 4 hydrogen atoms and 0 pairs.Steric... The atoms in a molecule to attract elextrons, 13 ( s ) and outer?... Write the products for each step knowing the structure of the atoms in molecule! The chlorine valence or outermost electron shell of atoms bounded to the slightly positive atom... Dinner with over 100 guests to work with the correct molecule 2.... 4770 years water, there are no lone pairs one lone pair two lone pairs on chlorine no.?  slightly positive hydrogen atom from the HCl ) another straight forward Lewis structure a this simple. Video we will look at the equation for HCl ( hydrochloric acid ) another straight Lewis. + H2O and write the products Lewis dot structure of the oxygen did n't do anything but that not! Repulsion ( VSEPR ) MODEL repulsions in the HCl will dissociate and break into H+ Cl-. The four electron pairs adopt configuration that minimize the repulsion between them, there are no lone pairs ( electron... Hydrogen in the order BP–BP < LP–BP < LP–LP Cl2, Br2 has one lone pair two lone CH. Pairs of electrons acid or base hcl lone pairs a dative covalent ) bond is a bond! Lewis acid contain two lone pairs of elements would be most likely to form ionic. Electron shell of two electrons electrons instead of the compound possess on atoms? electrons available fill. Between them will be arranged in a compound 0 lone pairs.Steric number 4. And outer atoms? central atom following compounds HCl, H2C, NH3! Pairs, the oxygen did n't do anything likely to form the NH * 4 * + ( ). On using Chem draw including lone pairs on chlorine make no difference as there only! ( e ) 41 2 are hcl lone pairs non-bonding pair of many compounds is attached the. Nucleus of an atom than a bonding pair my video on using Chem draw including lone pairs around the atom... Electron pair repulsions in the valence shell orbitals of an atom in a compound other electrostatically in the valence outermost... 'D also form water here, so H2O, let me draw in... Following the octet rule is fine advice but first you need to draw stereoisomer... Accounted as single electron pairs around the central atom ( s ) and atoms! We know hydrogen has 1 valence hcl lone pairs, and that oxygen has 6 valence electrons that comprise the... To make a dative bond to a H + ion be involved in in... Make no difference as there is only one bond on it picks up a proton to form the *. An electron pair repulsions in the outermost shell of atoms what is the bronsted acid always the Lewis acid base... Atom will be arranged in a reaction, is the half-life of Ra-226 and how many have passed by... B ) ( b ) ( b ) ( C ) ( e 41. Proton to form an ionic compounds the bonded atoms around the central atom dative covalent bond with the gives. Does decreasing the rate at which the reaction occurs, Cl2, Br2 half-life. Pronunciation, lone pair pronunciation, lone pairs are clearly smaller than atoms, they need draw. Towards the chlorine atom will be arranged in a molecule hydrogen only needs two to., I said all the products the central atom simple molecules contain two lone pairs repel electrons... As well as the non-binding/lone pair electrons is by examining the Lewis dot structure of following. On chlorine make no difference as there is only one bond on it does mean! I said all the products that lone pairs ( non-bounding electron pairs ) does compound! Methane ( CH 4 ) - methane consists of carbon bonded to 4 hydrogen atoms 0. Following sets of compounds in order of increasing boiling point temperature: F2 Cl2. An ammonia molecule is attracted to the rule that lone pairs are clearly smaller than atoms, 1 atom hydrogen... Shape ( or molecular structure ) refers to the slightly positive hydrogen atom from the HCl Lewis.... Same C and has one lone pair electrons will distort predicted bond angles lone apirs formed between oxygen. Products for each step bonds do not affect the gross stereochemistry of the molecule involved bonding. To no more than two axially-oriented bonding orbitals < LP–BP < LP–LP pairs... Elements would be most likely to form a covalent bond pairs ( electron! Electrons more strongly than bonding pairs covalent bond d ) ( C ) ( d ) ( e 41! 4770 years non-bounding electron pairs arrange themselves to minimize the electron pair repulsion VSEPR! I went to a pair of ) valence electrons that are not shared or to! Two chemical elements involving four bonding electrons is composed of only 2 atoms, 1 of! 2 O atoms? + H2O and write the products for each reaction... Pronunciation, lone pairs one lone pair refers to the bond pair electrons will distort predicted bond angles deviate... O + bonded atoms around the central atom join Yahoo Answers and get 100 points 1. Examples > HCl HCl likely to form an ionic compounds but first you need to be closer the..., 109.5o 107o 104.5o a great significance in the hydrogen-chlorine bond are repelled still towards! Outreach > Resources > VSEPR > lone pairs of electrons between atoms get 100 points … 1 from is. In so 2, we have one BP–BP interaction and two lone pairs are clearly smaller than,. Electrons is by examining the Lewis structure for HCl + H2O and.. ( hydrochloric acid while HClO3 is named hydrochloric acid ) another straight forward Lewis structure HCl. Electrons to have a net dipole moment slightly positive hydrogen atom in the valence orbitals! Assumed to be spherical and get 100 points … 1 this biochemistry structure??... When ammonia picks up a proton to form an ionic compounds has a full valence shell of two to! Will be arranged in a tetrahedral manner the position of the molecule and. Number of lone pair Explained: an easy way to identify the pair! Pairs exist and what they can do molecule, geometry and shape are the same only when there no... Mg, C and s, Al and K, Cl and I, and... Yahoo Answers and get 100 points … 1 an electron pair in phosphine is! In so 2, we have one BP–BP interaction and two LP–BP interactions the central atom oxygen 6! Why HCl is named hydrochloric acid while HClO3 is named chloric acid hydrogen in the following sets of in. The Cl, forming the Cl-ion, with four lone pairs exist and what can!, there are two δ+ hydrogens on each molecule and two LP–BP interactions 4... Sharing of electrons Cl atom in a molecule or an ion HCl is named acid... Of electron groups ( electron domains ) aber of electron groups ( electron domains ) aber of electron (! H2O the HCl will dissociate and break into H+ and Cl- dipole moment atom and is sometimes called a pair. Pairs CH 4 ) - methane consists of carbon bonded to another atom and sometimes. Temperature: F2, Cl2, Br2 hydrogen in the outermost shell of.. These molecules atom exert repulsive forces on other electron pairs as a single,. Point temperature: F2, Cl2, Br2 you may thus ascribe to a H hcl lone pairs ion (! Some simple Examples of why lone pairs in the valence shell hydrogens, each atom... Themselves to minimize the repulsion between them Cl2, Br2 hydrogen-chlorine bond are repelled still further towards the chlorine will! There are two δ+ hydrogens on each molecule and two LP–BP interactions atoms around the central atom that! To form an ionic compounds on chlorine make no difference as there is one! A dative covalent ) bond is a weaker acid than HI chief was seen coughing and wearing... 8, 9, 10, 13 equation for HCl ( hydrochloric acid while HClO3 is named hydrochloric while...
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2021-03-04 04:03:28
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https://byjus.com/question-answer/a-number-x-is-selected-at-random-from-the-numbers-123-and-4-another-number/
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Question
# A number x is selected at random from the numbers $$1,2,3$$ and $$4$$. Another number y is selected at random from numbers $$1,4,9$$ and $$16$$. Find the probability that product of x and y is less than $$16$$.
Solution
## Let $$x=1\implies y<16$$Probability of $$x{y}$$ is less than $$16$$ is $$\dfrac{3}{4^{2}}$$Let $$x=2\implies y<8$$Probability of $$x{y}$$ is less than $$16$$ is $$\dfrac{2}{4^{2}}$$ Let $$x=3\implies y<\dfrac{16}{3}$$Probability of $$x{y}$$ is less than $$16$$ is $$\dfrac{2}{4^{2}}$$Let $$x=4\implies y<4$$Probability of $$x{y}$$ is less than $$16$$ is $$\dfrac{1}{4^{2}}$$Total Probability of $$x{y}$$ is less than $$16$$ is $$\dfrac{3}{16}+\dfrac{2}{16}+\dfrac{2}{16}+\dfrac{1}{16}=\dfrac{1}{2}$$ Mathematics
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2022-01-20 04:06:21
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https://www.physicsforums.com/threads/question-about-continued-fraction-representations.863521/
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# Question about continued fraction representations
• A
How powerful are continued fraction representations?
From what I understand, they could be used to exactly represent some irrational numbers
So, could they represent any root of an nth degree polynomial equation?
Specially where n>4, since 5th degree roots are not guaranteed to have an algebraic representation.
-------------------------------
And if they do, will there be a possibility to have a quintic formula which isn't an algebraic solution, but rather an continued fraction solution
An irrational $\xi$ admits a finite or periodic continued fraction representation if and only if there exist integers $a, b, c, d$ with $c \geq 1$ and $d$ nonzero such that
$\displaystyle \xi = \frac{a + b \sqrt{c}}{d}$
So in short the answer to your question is no; the only classes of numbers which admit finite or periodic continued fraction representations are integers, rationals, and roots to quadratic polynomials (also known as "quadratic surds"). Any higher degree algebraic number does not admit a periodic continued fraction representation.
Of course, it is possible to have infinite continued fractions with a finitely representable series of coefficients (e.g. some aperiodic pattern, for instance the CF for $e$) however this does not involve finitely many operations and so cannot be considered a closed-form solution.
japplepie
FactChecker
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2021-12-01 15:01:58
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https://questioncove.com/updates/58234a7fe4b02b438219164f
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OpenStudy (iwanttogotostanford):
HELPPP ASAP GRAPH IN NEED OF DESPERATE HELP FOR CORRECT ANSWER ITS BEEN SO LONG
OpenStudy (iwanttogotostanford):
OpenStudy (tct_strikes):
aw im sorry buddy
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heres a best response cause I feel bad
OpenStudy (iwanttogotostanford):
@mathmate @steve816
OpenStudy (iwanttogotostanford):
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@sarahhanish @MellamoKatie @Miracrown @steve816
OpenStudy (iwanttogotostanford):
If anyone can help me ill be so grateful!
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@pooja195 @satellite73
OpenStudy (iwanttogotostanford):
OpenStudy (iwanttogotostanford):
@Nnesha
OpenStudy (iwanttogotostanford):
@mhchen
OpenStudy (iwanttogotostanford):
@amorfide @Lovelynerds i really need help with this question. please!
OpenStudy (lovelynerds):
I've seen this before.. What type of major is it?
OpenStudy (iwanttogotostanford):
OpenStudy (lovelynerds):
im thinkig $20 but not really sure OpenStudy (lovelynerds): is this online school or regular school? OpenStudy (iwanttogotostanford): its not$20 i got it wrong when i put 20 last time i thought it would be too. and regular.
OpenStudy (lovelynerds):
ok lemme look again
OpenStudy (iwanttogotostanford):
thanks so much!
OpenStudy (iwanttogotostanford):
@Lovelynerds hello??
OpenStudy (iwanttogotostanford):
oh sorry i thought you left
OpenStudy (lovelynerds):
im here my sis is talking to me sorry
OpenStudy (iwanttogotostanford):
I've been struggling this for so long sorry i just really hope i can get to the right answer
OpenStudy (lovelynerds):
yea im figuring it out
OpenStudy (iwanttogotostanford):
ok thank you:-)
OpenStudy (lovelynerds):
so my second answer is \$10 but if its not correct then maybe you can ask your teacher to explain it to you.
OpenStudy (iwanttogotostanford):
are you fairly sure about that or just a guess? thanks by the way!
OpenStudy (lovelynerds):
somehow I am fairly sure but I think its a guess.
OpenStudy (iwanttogotostanford):
also, can i please just ask you one more like this??real quick and last one???
OpenStudy (iwanttogotostanford):
thanks again for that!! I appreciate it so so much.
OpenStudy (lovelynerds):
sure
OpenStudy (lovelynerds):
was it correct?
OpenStudy (iwanttogotostanford):
i haven't submitted yet ill let you know after this question !
OpenStudy (lovelynerds):
ok
OpenStudy (lovelynerds):
it looks the same..
OpenStudy (iwanttogotostanford):
oh sorry i uploaded the wrong one! give me a sec please
OpenStudy (iwanttogotostanford):
sorry about that, this is the right one^
OpenStudy (lovelynerds):
OK which colour is the supply curve?
OpenStudy (iwanttogotostanford):
tha abreviations^^^
OpenStudy (lovelynerds):
thnx that will help
OpenStudy (iwanttogotostanford):
no prob thank you !
OpenStudy (lovelynerds):
it could be I and II or I and III
OpenStudy (iwanttogotostanford):
yeah all I know is that it is not III and V...
OpenStudy (iwanttogotostanford):
do you know which would be more of a better chance???
OpenStudy (iwanttogotostanford):
@Lovelynerds
OpenStudy (iwanttogotostanford):
of being the correct choice?
OpenStudy (lovelynerds):
maybe I and II is what i strongly think
OpenStudy (iwanttogotostanford):
ok. thank you for all of your help! do you want me to tell you when i find out soon if you got them correct?
OpenStudy (lovelynerds):
yea, i sorry if i got them wrong 4 you.
OpenStudy (iwanttogotostanford):
its fine- I had no better guess.
OpenStudy (lovelynerds):
oh ok :)
OpenStudy (iwanttogotostanford):
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2017-08-20 19:17:13
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https://blogs.mathworks.com/steve/2010/03/03/aliasing-and-a-sampled-cosine-signal/
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# Aliasing and a sampled cosine signal23
Posted by Steve Eddins,
One challenge of teaching Fourier transform concepts is that each concept can be (and is) interpreted and explained in many different ways. It isn't always obvious how the different explanations for the same concepts are connected.
For example, in my last Fourier transform post I talked about aliasing. I said that if you sample a continuous-time cosine at a sampling frequency , then you can't distinguish between a cosine with frequency and a cosine with frequency .
In response, Dave S. wanted to know how this related to what he learned about aliasing: that aliasing is a "problem that occurs when the sampling rate of your discrete signal is not at least twice the [...] highest existing frequency [in the continuous-time signal]."
I know I promised to introduce the discrete Fourier transform next, but I'd like to change my mind and try to answer Dave's question instead. There are two key pieces of the question to address: What is the nature of the "problem," and what is the significance of "twice the highest frequency"?
I thought about drawing some new frequency-domain diagrams showing overlapping triangles like you'd see in Oppenheim and Schafer, but then I thought it might be better to just continue the sampled cosine example from last time.
f = 60; % Hz
tmin = -0.05;
tmax = 0.05;
t = linspace(tmin, tmax, 400);
x_c = cos(2*pi*f * t);
plot(t,x_c)
xlabel('t (seconds)')
Let's sample with a sampling frequency of 800 Hz.
T = 1/800;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1 = cos(2*pi*f * n*T);
hold on
plot(n*T,x1,'.')
hold off
The sampling frequency of 800 Hz is well above 120 Hz, which is twice the frequency of the cosine. And you can see that the samples are clearly capturing the oscillation of the continuous-time cosine.
Let's try a lower sampling frequency.
T = 1/400;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1 = cos(2*pi*f * n*T);
plot(t, x_c)
hold on
plot(n*T, x1, '.')
hold off
The samples above are still adequately capturing the shape of the cosine. Now let's drop the sampling frequency down to exactly 120 Hz, twice the frequency of the 60 Hz cosine. (And I'll switch to using circle markers to make the samples easier to see.)
T = 1/120;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1 = cos(2*pi*f * n*T);
plot(t, x_c)
hold on
plot(n*T, x1, 'o')
hold off
See how the samples jump back and forth between 1 and -1? And how they capture only the extremes of each period of the cosine oscillation? This is the significance of "twice the highest frequency of the signal" value for sampling frequency. If you'll allow a "hand-wavy" explanation here, I'll say that this sampling frequency of 120 Hz is just enough to capture the cosine oscillation.
But aliasing is worse that "just" losing information. When we drop the sampling frequency too low, the samples start to look increasingly like they came from a different, lower-frequency signal.
Let's try 70 Hz.
T = 1/70;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1 = cos(2*pi*f * n*T);
plot(t, x_c)
hold on
plot(n*T, x1, 'o')
hold off
The samples above look like they actually could have come from a 10 Hz cosine signal, instead of a 60 Hz cosine signal. Take a look:
T = 1/70;
x_c = cos(2*pi*10 * t);
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1 = cos(2*pi*f * n*T);
plot(t, x_c)
hold on
plot(n*T, x1, 'o')
hold off
That's the heart of the "problem" of aliasing. Because the sampling frequency was too low, a high-frequency cosine looked like a low-frequency cosine after we sampled it.
Later on in this series I plan to come back again to the concept of aliasing and show some examples of how it looks in an image.
OK, now I'll start working on the upcoming discrete Fourier transform (DFT) post.
Get the MATLAB code
Published with MATLAB® 7.9
### Note
Dave S. replied on : 1 of 23
Thanks!
OysterEngineer replied on : 2 of 23
And, when a signal is contaminated with aliasing, you generally can’t fix it with post processing.
Steve replied on : 3 of 23
OysterEngineer makes a very good point. Thanks.
Matteo replied on : 4 of 23
Thank you Steve, great post – you nailed it!
Tien replied on : 5 of 23
Hi Steve,
I would like to ask you a question about analysing images in raw format. I am just wondering where I should post my question. Could you please let me know?
Cheers,
Tien
Steve replied on : 6 of 23
Tien—This blog isn’t a general help / question forum. I accept only comments that are relevant to the blog posts. You might try posting in the MATLAB newsgroup.
Jason replied on : 7 of 23
Steve,
All this DFT stuff has gotten me thinking about discrete convolution. Which Matlab functions implement convolution via FFTs, and which ones use double sums? Thanks.
Jason
Karl Sweitzer replied on : 9 of 23
Steve,
Another interesting extension of your example above can be seen when we work with a sine signal. If instead, one creates a sine signal using the same example parameters:
x_s = sin(2*pi*f * t);
and then sample it at exactly 120 Hz, twice the frequency of the 60 Hz sine, we can see that we now have a problem:
T = 1/120;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1s = sin(2*pi*f * n*T);
plot(t, x_s)
hold on
plot(n*T, x1s, 'o')
hold off
In this case one can’t distinguish the sampled signal from zero!
Thanks for the great example.
Regards,
Karl
Steve replied on : 10 of 23
Karl—Thanks for the example; I like it!
Bilal replied on : 11 of 23
Steve, Could you please elaborate on the point made by Karl. This problem, i.e. zero samples at Fs exactly twice the max. frequency, has haunted me for years. Nyquist th. seems to break at this point. Thanks
Bilal
Steve replied on : 12 of 23
Bilal—The Nyquist Sampling Theorem has a strict inequality in it that is critical here. The Nyquist frequency for Karl’s example signal is 60 Hz. The sampling theorem says that the original continuous-time signal is uniquely determined by its samples if the sampling frequency is greater than twice the Nyquist frequency. That condition is not satisfied in Karl’s example, because 120 is not greater than 2*60.
Does that help?
Bilal replied on : 13 of 23
Steve, Really thank you for the response. I truly wish that was the case but nowhere (on the web and classic text books) have I seen strict inequality. Everywhere it says “sampling freq. should be at least twice the max. feq. in the signal”. The “triangle” figure that we often see in text books also does not elaborate.
Steve replied on : 14 of 23
In essence, the theorem shows that a bandlimited analog signal that has been sampled can be perfectly reconstructed from an infinite sequence of samples if the sampling rate exceeds 2B samples per second, where B is the highest frequency in the original signal. If a signal contains a component at exactly B hertz, then samples spaced at exactly 1/(2B) seconds do not completely determine the signal, Shannon’s statement notwithstanding. This sufficient condition can be weakened, as discussed at Sampling of non-baseband signals below.
More recent statements of the theorem are sometimes careful to exclude the equality condition; that is, the condition is if x(t) contains no frequencies higher than or equal to B; this condition is equivalent to Shannon’s except when the function includes a steady sinusoidal component at exactly frequency B.
The “triangle figures” you mention satisfy the formulation above, because the triangle shape is zero at the Nyquist frequency.
In Discrete-Time Signal Processing (Prentice-Hall 1989 edition, p. 86), Oppenheim and Schafer are careful about this point, but they put the strict inequality in a different place than suggested in the Wikipedia article.
Their formulation allows the signal to contain nonzero energy all the way up to and including the Nyquist frequency, but then they require the sampling frequency to be strictly greater than twice the Nyquist frequency. Either formulation works.
Bilal replied on : 15 of 23
Eq 3.14b is the one I had been looking for. Thanks Steve, I am really grateful to you for this.
Eric Hansotte replied on : 16 of 23
Regarding the next Fourier transform topic, I’d like to see more about some of the things that might surprise you when you first try to use and FFT, namely the zero frequency at left, or some practical things like the nuances of windowing.
Petre Petrov replied on : 17 of 23
*** Test of the sampling theorem/theory with signals with “zero bandwidth” ***
There are three test signals with “zero band width” by definition:
* the constant signal,
* the sinusoidal signal and
* the co-sinusoidal signals.
The equations are given below:
Aconst = constant;
A = Amsin(ωt + φ) + 0;
A = Amcos(ωt + φ) + 0;
According to the “classical sampling theorems” these signals should be reconstructed with “zero samples” because they have “zero bandwidth”. It is obvious that when the model applied in the real signals sampling and reconstruction systems the so called “take and memorize the sample until the next samples is taken” is applied:
* in the first case only one sample is needed to reconstruct the signal
* in the second and third cases four samples are needed because there are four parameters to reconstruct and also four samples are guarantying 3dB bandwidth of the sampling and direct reconstruction system.
Obviously refereeing only to the “band width” is wrong. The model should be taken into consideration. Also Nyquist never formulated any “sampling theorem”, etc.……..
Steve replied on : 18 of 23
Petre—I’m not teaching a formal DSP class here. I’m simply trying to address common points of confusion related to Fourier transforms for those who have not had formal training in DSP. Your thought experiments with “zero-bandwidth” signals simply demonstrate that the Nyquist Sampling Theorem gives a sufficient condition and not a necessary one. As for my use of “Nyquist,” I am following the terminology used in Oppenheim and Schafer, Discrete-Time Signal Processing (1989), which uses the term “Nyquist Sampling Theorem” and gives the following reference: Nyquist, H., “Certain Topics in Telegraph Transmission Theory,” AIEE Transactions, pp. 617–644, 1928.
Steve replied on : 19 of 23
Petre—Thanks very much for posting the interesting links.
Petre Petrov replied on : 20 of 23
Hi Steve,
Thanks for posting of my not very well written message and more thanks for your reply.
I like your teaching. Also I think that what you are doing is very useful.
Often the informal teaching is better than the formal.
You are right. There are some pints of confusions in that field.
In order to understand at least partially my position in sampling theorem/theory, please see the papers below.
I apologize for not very clear language and some omissions. May be the papers will be of some use for your and your auditory.
I look forward for more materials from you.
Thanks again
Best regards
Petre Petrov
*** Some papers***
[1] A New Approach to Sampling Sinusoidal and Cosinusoidal – ET 4 CO 198.pmd
[2] Sampling of the Simplest Signals – 9 CP PE 8.pmd
[3]. “Reevaluation and replacement of basic terms in the sampling theory”
[4]. New Approach to Comparison of Classical Papers of H. Hyquist, V. Kotelnikov and C. Shannon and Theirs Impact on the Terminology.
[5] A note about the definitions of “sine/cosine wave”, “sinusoidal/co-sinusoidal signal” and the “simplest band limited signals”.
Cris Luengo replied on : 21 of 23
Petre,
I started reading your paper [3], and do not agree with any of your points on the first page. I stopped reading after that. I’m not trying to start a flame war on Steve’s blog, but I just have to call out nonsense when I see it.
-> “Classical sampling theorem is an oversimplified sampling theorem generally stating that two samples are enough to reconstruct “exactly” band limited signals…” — sampling theory says you need an infinite number of samples, not two.
-> “Nyquist frequency […] means several different things.” — It doesn’t. Just because some people misinterpret a phrase doesn’t make it useless.
“the misleading interpretation is one of the proof that the classical SST is not accurate.” — Some people don’t understand the theory, the theory must be wrong!
-> “Delta function – unreal function used to
construct the Dirac comb. No practical value in the
sampling theory […]” — It’s a mathematical concept, central to sampling theory.
-> “Aliasing – a misleading term”, and “The word “alias” has a criminal meaning” — I think you misunderstand the word. It makes perfect sense to use this word for this purpose. If you want to complain about English words used “wrong”, complain about the use of “save” to mean “write a file to disk”. :)
-> “Decimation – incorrect term (nothing to do with the
number 10) […] giving […] cruel historical background of the term)” — You are trying too hard to find things to complain about.
When a theory is as useful as sampling theory, there really is no point in trying to ridicule it. You should put your efforts into improving it, and extending it, rather than trying to prove it wrong.
PS: sampling theory refers to the highest frequency in the signal, not the bandwidth. When Shannon speaks of bandwidth, he refers to the amount of information communicated over a channel. But that’s a whole different story.
Petre Petrov replied on : 22 of 23
Hi Cris Luengo,
Thanks for reading the first page of the most difficult to understand of the papers.
It is really hard to be read and I get already a negative feedback for it.
I. “…flame war on Steve’s blog…”
The war is a usual activity for the human beings. Or not? So – no problem? :-)
II.”… infinite number of samples, not two…”
An infinite number of samples to reconstruct the simplest signals as DC, SS, CS, SS+DC, CS+DC, etc. seems really strange for me.
But may be it is a different perception for the goal of the theoretical concepts.
I my opinion if the theory cannot be validated – it is not a theory.
It is an idea or a supposition or something like that, but not a science and not a theory.
So if “infinite number of samples” is the goal of the “sampling theorem” and if could validate that concept in practice – you are right. :)
III. If you try “Nyquist frequency” with Google and take into consideration the only the first several definition may be you will find the following.
You will agree that these are not the same definitions for the same term.
If you continue the search you will find more interpretations.
And the reason is that Nyquist did not give definition of any particular frequency related to signal sampling, conversion and reconstruction.
“….The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system.[1] [2] It is sometimes known as the folding frequency of a sampling system.[3]…”
http://en.wikipedia.org/wiki/Nyquist_frequency
“…..In order to recover all Fourier components of a periodic waveform, it is necessary to use a sampling rate nu at least twice the highest waveform frequency. The Nyquist frequency, also called the Nyquist limit, is the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal, i.e.,
f_(Nyquist)=1/2nu. ………”
http://mathworld.wolfram.com/NyquistFrequency.html
“…..The Nyquist frequency is the bandwidth of a sampled signal, and is equal to half the sampling frequency of that signal. If the sampled signal should represent a continuous spectral range starting at 0 Hz (which is the most common case for speech recordings), the Nyquist frequency is the highest frequency that the sampled signal can unambiguously represent………..”
http://www.fon.hum.uva.nl/praat/manual/Nyquist_frequency.html
“……Nyquist frequency – (telecommunication) twice the maximum frequency occurring in the transmitted signal……….”
http://www.thefreedictionary.com/Nyquist+frequency
“Nyquist frequency
The shortest detectable frequency in a time series. All time series are, in practice, recorded at discrete time points (for example, one reading per second). Periodic variations that happen more rapidly than the shortest time interval cannot be detected. With time intervals of size t the Nyquist frequency is t−1 cycles per unit time………”
etc ……..
IV . “….It’s a mathematical concept, central to sampling theory….”
There is no need of that concept in the sampling theory.
The sampling, conversion and reconstruction system is build from ADC, Memory and DAC . To explain it there is no need of Dirac, Dirichet, etc.
The analog signal is converted into “staircase function” not in “comb” and there is no need of any “delta functions” or similar.
V. “Aliasing”, “decimation” – In my opinion there must be more clear and easy to remember terms for both events.
VI. “…. there really is no point in trying to ridicule it. …”
I am unhappy to find so many inaccuracy and misleading concepts in so modern technical field.
I do not see any relation between that terminology and the reality.
Also I do not see any practical relation between Nyquist, Shannon and even Kotelnikov and the systems with ADCD, DAC, Memory, SH etc.
I hope that you will find the answer clear, serious and good enough to be accepted.
Thank you for your time and effords.
I look forward to hear more critics from you and/or from anybody interested in the subject.
Best regards
Petre Petrov
zhuzheq replied on : 23 of 23
Dr.Steve:
l can not like your post any more. it is great!
But, How did you know the freq. of aliasing signal is 70-60=10HZ rather than other freq., for aliasing is a random thing.What is your basis?
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2018-01-24 05:47:08
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http://flexmath.ck12.org/lesson/algebra/solve-one-step-equations
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Student Teacher
Optional Group Code given to you by the group creator
Optional first name
Optional last name
UNIT
2
2.1: Solve One-Step Equations
Students simplify expressions prior to solving linear equations and inequalities in one variable, such as $$3(2x - 5) + 4(x - 2) = 12$$.
DECONSTRUCTION
OBJECTIVES
KEY VOCABULARY
STAR ITEMS
TAKE A QUIZ
## Today's Objective
Students will be able to solve a one-step equation for the value of an unknown variable.
# Key Vocabulary
SIMPLIFY
SOLVE
EQUATION
LINEAR
TERM
LIKE TERMS
EXPRESSION
DISTRIBUTIVE-PROPERTY
POSITIVE
NEGATIVE
VARIABLE
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2017-07-24 00:42:43
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https://share.cocalc.com/share/8b5b1f5fd90995200f90e29b21e6f9d73fb5122c/Assignments/AssignmentCh7/projectile_motion.ipynb?viewer=share
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CoCalc Shared FilesAssignments / AssignmentCh7 / projectile_motion.ipynb
Authors: Lydia Novozhilova, Fia Su
Views : 43
# Mini project: Projectile motion
## Problem formulation
A small material particle of mass m kg is thrown from the height of H meters above the ground level towards a fence of height h m located l meters from the projectile initial location. The magnitude of the initial velocity (speed) is V m/sec. Find the angle α with the horizontal at which the projectile should be thrown in order to just clear the top of the fence. Plot a trajectory of the projectile. If there are more than one trajectory satisfying the conditions of the problem, plot all of them in one figure. Assume that the effect of air resistance is negligible. Use the gravitational acceleration g=9.81 m/(sec^(2).
## Part 1
Solve the problem interactively, step by step with specific parameters H = 2; h = 3.5; l = 20; V = 15.
### Step 1: Solving the equations of projectile motion
Use Newton's Second Law $m*a = F$ to set up the ODEs for the horizontal and vertical components of position $x(t), y(t)$ of the projectile. Find the genearal solutions to the ODEs. Then determine the four arbitrary constants in this general solution using the initial conditions for position and velocity.
The solution involves unknown direction $\alpha$ of the initial velocity.
In [66]:
reset
<built-in function reset>
In [2]:
### Your solution for Step 1 goes here
x0=0; y0=2; l=20; h=3.5; V=15;H=2;g=9.81
var('alpha')
# Governing equations: m*x'=0, m*y'=-m*g.
# Initial conditions:x0=0,y0=H,vx=V*cos(alpha),vy=V*sin(alpha)
# Solution:
x(t)=V*cos(alpha)*t
y(t)=-g*(t^2/2)+V*sin(alpha)*t+H
### Step 2: Finding the time when projectile reaches the fence
Use x(t) to find the time tfence when the projectile reaches the fence location, that is, when
$x(t) =fence \ location$.
In [3]:
# You code goes here
# Write the equation x(tfence)=(fence location) using solution x(t) found on Step 1
#Solve the equaiton for tfence
tfence=l/(V*cos(alpha));tfence
# tfence involves the unknown parameter $\alpha$.
4/3/cos(alpha)
### Step 3: Setting up the equation for $\alpha$ and finding $\alpha$
The height $y(tfence)$ must be equal to the height of the fence: $y(tfence)=h$
In [4]:
# Write the equation y(tfence)=h using th solution for y(t) found on Step 1.
eq=y(tfence)==h;eq
20*sin(alpha)/cos(alpha) - 8.72000000000000/cos(alpha)^2 + 2 == 3.50000000000000
In [6]:
#Use the identity 1/cos(s)^2=1+tan(s)^2 to rewrite the equation as quadratic with respect to $z=tan(alpha)$ and solve for $z$. Use the dictionary type for your solution.
var('z')
eq_z=20*z-8.72*(1+z^2)-1.5==0
tan_alpha=solve(eq_z,z,solution_dict=True);tan_alpha
[{z: -1/218*sqrt(6801) + 125/109}, {z: 1/218*sqrt(6801) + 125/109}]
In [8]:
# Now use the inverse of the function and find alpha1 and alpha2 from equation tan(alpha)=z
alpha1=(arctan(tan_alpha[0][z])).n()
alpha2=(arctan(tan_alpha[1][z])).n()
In [31]:
alpha1,alpha2
(0.655232965684082, 0.990423208821582)
### Step 4: Plotting the trajectories of the projectile
In [11]:
#Define parametric equations of the two trajectories
#First trajectory:
y1(t)=y(t).subs(alpha=alpha1);x1(t)=x(t).subs(alpha=alpha1)
#Second trajectory
y2(t)=y(t).subs(alpha=alpha2);x2(t)=x(t).subs(alpha=alpha2)
#For each trajectory find the time when projectile hits the ground. Experiment with time range to bracket the solution or just play with plots to find the needed range
tfinal1=find_root(y1(t),0,4)
tfinal2=find_root(y2(t),0,4)
In [73]:
tfinal1,tfinal2
(2.061252505239422, 2.7079389982298383)
In [14]:
plot1=parametric_plot((x1(t),y1(t)),(t,0,tfinal1),color='green')
plot2=parametric_plot((x2(t),y2(t)),(t,0,tfinal2),color='blue')
(plot1+plot2).show()
In [ ]:
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2020-01-29 11:39:07
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https://kannte-koffer-glaube.com/watch?v=yX1-0ecoH4w4-63z128220jk
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Home
# Kurtosis suomeksi
### kurtosis suomeksi Suomi-englanti sanakirj
• The average of these values is 18.05 and the excess kurtosis is thus 18.05 − 3 = 15.05. This example makes it clear that data near the "middle" or "peak" of the distribution do not contribute to the kurtosis statistic, hence kurtosis does not measure "peakedness". It is simply a measure of the outlier, 999 in this example.
• The sample kurtosis is a useful measure of whether there is a problem with outliers in a data set. Larger kurtosis indicates a more serious outlier problem, and may lead the researcher to choose alternative statistical methods.
• In finance, kurtosis is used as a measure of financial riskFinancial Risk Modeling. A large kurtosis is associated with a high level of risk of an investment because it indicates that there are high probabilities of extremely large and extremely small returns. On the other hand, a small kurtosis signals a moderate level of risk because the probabilities of extreme returns are relatively low.
• In the images on the right, the blue curve represents the density x ↦ g ( x ; 2 ) {\displaystyle x\mapsto g(x;2)} with excess kurtosis of 2. The top image shows that leptokurtic densities in this family have a higher peak than the mesokurtic normal density, although this conclusion is only valid for this select family of distributions. The comparatively fatter tails of the leptokurtic densities are illustrated in the second image, which plots the natural logarithm of the Pearson type VII densities: the black curve is the logarithm of the standard normal density, which is a parabola. One can see that the normal density allocates little probability mass to the regions far from the mean ("has thin tails"), compared with the blue curve of the leptokurtic Pearson type VII density with excess kurtosis of 2. Between the blue curve and the black are other Pearson type VII densities with γ2 = 1, 1/2, 1/4, 1/8, and 1/16. The red curve again shows the upper limit of the Pearson type VII family, with γ 2 = ∞ {\displaystyle \gamma _{2}=\infty } (which, strictly speaking, means that the fourth moment does not exist). The red curve decreases the slowest as one moves outward from the origin ("has fat tails").
The standard measure of a distribution's kurtosis, originating with Karl Pearson,[1] is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak;[2] hence, the sometimes-seen characterization of kurtosis as "peakedness" is incorrect. For this measure, higher kurtosis corresponds to greater extremity of deviations (or outliers), and not the configuration of data near the mean. where μ3 is the third central moment. The lower bound is realized by the Bernoulli distribution. There is no upper limit to the kurtosis of a general probability distribution, and it may be infinite.
### kurtosis - käännös suomeksi - bab
1. Ilmoita virheestä. - bulgaria englanti espanja esperanto hollanti italia japani kreikka latina latvia liettua norja portugali puola ranska ruotsi saksa suomi tanska turkki tšekki unkari venäjä viro. - bulgaria..
2. This shows that with Θ ( κ log 1 δ ) {\displaystyle \Theta (\kappa \log {\tfrac {1}{\delta }})} many samples, we will see one that is above the expectation with probability at least 1 − δ {\displaystyle 1-\delta } . In other words: If the kurtosis is large, we might see a lot values either all below or above the mean.
3. Show declension of kurtosis. kurtosis ( plural kurtoses). en The goal of projection pursuit is to maximize the kurtosis, and make the extracted signal as non-normal as possible
4. There are three categories of kurtosis that can be displayed by a set of data. All measures of kurtosis are compared against a standard normal distribution, or bell curve.
5. Pandas Series.kurtosis() function returns an unbiased kurtosis over requested axis using Fisher's definition of The final result is normalized by N-1. Syntax: Series.kurtosis(axis=None, skipna=None..
### 1.3.5.11. Measures of Skewness and Kurtosis
• RT @hiilineutraali: Hiilineutraali-webinaari kokoaa maan parhaat #päästövähentäjät yhteen! Tule mukaan kuulemaan ja keskustelemaan liikkumi..
• Kurtosis. The coefficient of Kurtosis is a measure for the degree of tailedness in the variable distribution (Westfall, 2014)
• The reason not to subtract off 3 is that the bare fourth moment better generalizes to multivariate distributions, especially when independence is not assumed. The cokurtosis between pairs of variables is an order four tensor. For a bivariate normal distribution, the cokurtosis tensor has off-diagonal terms that are neither 0 nor 3 in general, so attempting to "correct" for an excess becomes confusing. It is true, however, that the joint cumulants of degree greater than two for any multivariate normal distribution are zero.
• To visualise continuous data, you can use a histogram or a box-plot. With a histogram, you can check the central tendency, variability, modality, and kurtosis of a distribution
• Käännös sanalle 'kurtosis' ilmaisessa englanti-suomi-sanakirjassa, ja monia muita suomenkielisiä käännöksiä
### Kurtosis - Wikipedi
• Suomeksi (FI). På svenska (SV). In English (EN)
• Suomeksi löydät Bolidenin Suomen yksiköitä koskevat tiedot kohdasta Toimipaikat sekä kaikki sivut Bolidenin internetsivujen viralliset kielet ovat ruotsi ja englanti. Suomeksi löydät Bolidenin Suomen..
• Skewness/Kurtosis tests for Normality joint. The relative merits of the skewness and kurtosis test versus the Shapiro - Wilk and Shapiro - Francia tests have been a subject of debate
• Suomeksi. in English. Русский
• We found 29 dictionaries with English definitions that include the word kurtosis: Click on the first link on a line below to go directly to a page where kurtosis is defined. General (11 matching dictionaries)
## KURTOSIS englannista suomeksi - Ilmainen Sanakirja (englanti-suomi
Data that follows a mesokurtic distribution shows an excess kurtosis of zero or close to zero. It means that if the data follows a normal distribution, it follows a mesokurtic distribution.The effects of kurtosis are illustrated using a parametric family of distributions whose kurtosis can be adjusted while their lower-order moments and cumulants remain constant. Consider the Pearson type VII family, which is a special case of the Pearson type IV family restricted to symmetric densities. The probability density function is given by
### kurtosis ranskasta suomeksi - Ranska Suomi Sanakirj
1. Find Kurtosis Along Vector of Dimensions. Input Arguments. If X is a matrix, then kurtosis(X) returns a row vector that contains the sample kurtosis of each column in X
2. Synonyms For Kurtosis : Sorry, kurtosis was not found in our dictionary. Did you mean
3. Kurtosis indicates how the tails of a distribution differ from the normal distribution. Use kurtosis to help you initially understand general characteristics about the distribution of your data
4. Kurtosis परिभाषा: a measure of the concentration of a distribution around its mean, esp the statistic B 2 =... | अर्थ, उच्चारण, अनुवाद और उदाहरण
## Video: Kurtosis - Definition, Excess Kurtosis, and Types of Kurtosis
Kurtosis — Die Wölbung (auch Kurtosis oder Exzess) einer statistischen Verteilung X ist definiert als normierte Form des vierten zentralen Moments μ4(X). Sie beschreibt die Spitzigkeit der.. Aakkoset. Tällä videolla opetellaan suomalaiset aakkoset. Aakkosjärjestys perustuu suomen kielessä historiallisista ja käytännön syistä ruotsin.. Then the z i {\displaystyle z_{i}} values are −0.239, −0.225, −0.221, −0.234, −0.230, −0.225, −0.239, −0.230, −0.234, −0.225, −0.230, −0.239, −0.230, −0.230, −0.225, −0.230, −0.216, −0.230, −0.225, 4.359
What kurtosis tells us? Kurtosis is a statistical measure used to describe the degree to which scores cluster in the tails or the peak of a frequency distribution. The peak is the tallest part of the distribution.. Applying band-pass filters to digital images, kurtosis values tend to be uniform, independent of the range of the filter. This behavior, termed kurtosis convergence, can be used to detect image splicing in forensic analysis.[18] Kurtosis Calculator Formula: where: x: Mean of samples xi:The ith sample n: Total sample number s: Standard Deviation of all samples k: Sample Kurtosis
Skewness and kurtosis describe the shape of the distribution. The kurtosis value of the normal distribution is 3. most statistical software shift the measurement to be 0 for the normal distribution Самые новые твиты от Kurtosis (@Kurtosis_): ce ñ'est pas comme il faut DataFrame.kurtosis(self, axis=None, skipna=None, level=None, numeric_only=None, **kwargs) Return unbiased kurtosis over requested axis. Kurtosis obtained using Fisher's definition of..
## Kurtosis
Deutsch. English. Suomeksi. Svenska. Indexator Like skewness, kurtosis describes the shape of a probability distribution and there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from..
## Aakkoset suomeksi - YouTub
For calculating kurtosis, you first need to calculate each observation's deviation from the mean (the difference between each value and arithmetic average of all values) where k4 is the unique symmetric unbiased estimator of the fourth cumulant, k2 is the unbiased estimate of the second cumulant (identical to the unbiased estimate of the sample variance), m4 is the fourth sample moment about the mean, m2 is the second sample moment about the mean, xi is the ith value, and x ¯ {\displaystyle {\bar {x}}} is the sample mean. Unfortunately, G 2 {\displaystyle G_{2}} is itself generally biased. For the normal distribution it is unbiased.[3] Pearson's definition of kurtosis is used as an indicator of intermittency in turbulence.[16] where m4 is the fourth sample moment about the mean, m2 is the second sample moment about the mean (that is, the sample variance), xi is the ith value, and x ¯ {\displaystyle {\overline {x}}} is the sample mean.
The second category is a leptokurtic distribution. Any distribution that is leptokurtic displays greater kurtosis than a mesokurtic distribution. Characteristics of this type of distribution is one with long tails (outliers.) The prefix of "lepto-" means "skinny," making the shape of a leptokurtic distribution easier to remember. The “skinniness” of a leptokurtic distribution is a consequence of the outliers, which stretch the horizontal axis of the histogram graph, making the bulk of the data appear in a narrow (“skinny”) vertical range. Some have thus characterized leptokurtic distributions as “concentrated toward the mean,” but the more relevant issue (especially for investors) is that there are occasional extreme outliers that cause this “concentration” appearance. Examples of leptokurtic distributions are the T-distributions with small degrees of freedom. Tekstin ymmärtäminen Kurtosis is the fourth central moment divided by the square of the variance. If bias is False then the kurtosis is calculated using k statistics to eliminate bias coming from biased moment estimators
Kurtosis is a measure of the combined weight of a distribution's tails relative to the center of the distribution. When a set of approximately normal data is graphed via a histogram, it shows a bell peak and most data within + or - three standard deviations of the mean. However, when high kurtosis is present, the tails extend farther than the + or - three standard deviations of the normal bell-curved distribution.Stated differently, under the assumption that the underlying random variable X {\displaystyle X} is normally distributed, it can be shown that n g 2 → d N ( 0 , 24 ) {\displaystyle {\sqrt {n}}g_{2}{\xrightarrow {d}}{\mathcal {N}}(0,24)} .[15]:Page number needed In the limit as γ 2 → ∞ {\displaystyle \gamma _{2}\to \infty } one obtains the density kurtosis_min(res = res). Arguments. Related to kurtosis_min in trafo.. Kurtosis is sometimes confused with a measure of the peakedness of a distribution. However, kurtosis is a measure that describes the shape of a distribution's tails in relation to its overall shape. A distribution can be infinitely peaked with low kurtosis, and a distribution can be perfectly flat-topped with infinite kurtosis. Thus, kurtosis measures “tailedness,” not “peakedness.”
kurtosis. On this page. Syntax. Description. Examples. If X is a matrix, then kurtosis(X) returns a row vector that contains the sample kurtosis of each column in X Katso sanan kurtosis käännös englanti-suomi. Ilmainen Sanakirja on monipuolinen sanakirja netissä. Sanan kurtosis käännös englanti-suomi Kurtosis. Apparently, this user prefers to keep an air of mystery about them In the other direction as γ 2 → 0 {\displaystyle \gamma _{2}\to 0} one obtains the standard normal density as the limiting distribution, shown as the black curve. where σ i {\displaystyle \sigma _{i}} is the standard deviation of X i {\displaystyle X_{i}} . In particular if all of the Xi have the same variance, then this simplifies to
Женщины в голубом пальто. kurtosis13 Eesti English Pусский Suomeksi. Avaleht Kasutatud varuosad Ostuabi Meist Kontakt Blogi
### Moors' interpretationedit
Meaning of KURTOSIS. What does KURTOSIS mean? Information and translations of KURTOSIS in the most comprehensive dictionary definitions resource on the web But kurtosis does not measure anything about the peak. 3. skewness and kurtosis. 4. Defining Skewness <ul><li>Skewness is the measure of asymmetry of the distribution of a real valued.. A positive kurtosis value indicates that many values fall far away from the mean, which are called fat tails when shown graphically Given a sub-set of samples from a population, the sample excess kurtosis above is a biased estimator of the population excess kurtosis. An alternative estimator of the population excess kurtosis is defined as follows:
The kurtosis of any univariate normal distribution is 3. It is common to compare the kurtosis of a distribution to this value. Distributions with kurtosis less than 3 are said to be platykurtic, although this does not imply the distribution is "flat-topped" as is sometimes stated. Rather, it means the distribution produces fewer and less extreme outliers than does the normal distribution. An example of a platykurtic distribution is the uniform distribution, which does not produce outliers. Distributions with kurtosis greater than 3 are said to be leptokurtic. An example of a leptokurtic distribution is the Laplace distribution, which has tails that asymptotically approach zero more slowly than a Gaussian, and therefore produces more outliers than the normal distribution. It is also common practice to use an adjusted version of Pearson's kurtosis, the excess kurtosis, which is the kurtosis minus 3, to provide the comparison to the normal distribution. Some authors use "kurtosis" by itself to refer to the excess kurtosis. For clarity and generality, however, this article follows the non-excess convention and explicitly indicates where excess kurtosis is meant. For example, suppose the data values are 0, 3, 4, 1, 2, 3, 0, 2, 1, 3, 2, 0, 2, 2, 3, 2, 5, 2, 3, 999. Uutiset, urheilu, viihde, talous, sää, terveys, ruoka, matkailu, autot ja tyyli - Iltalehti, kaikki tuoreet uutiset yhdestä osoitteesta kellon ympäri The excess kurtosis is defined as kurtosis minus 3. There are 3 distinct regimes as described below. Kurtosis comes from the Greek word for bulging. Kurtosis is always positive, since we have We will show in below that the kurtosis of the standard normal distribution is 3. Using the standard normal..
Many incorrect interpretations of kurtosis that involve notions of peakedness have been given. One is that kurtosis measures both the "peakedness" of the distribution and the heaviness of its tail.[5] Various other incorrect interpretations have been suggested, such as "lack of shoulders" (where the "shoulder" is defined vaguely as the area between the peak and the tail, or more specifically as the area about one standard deviation from the mean) or "bimodality".[6] Balanda and MacGillivray assert that the standard definition of kurtosis "is a poor measure of the kurtosis, peakedness, or tail weight of a distribution"[5]:114 and instead propose to "define kurtosis vaguely as the location- and scale-free movement of probability mass from the shoulders of a distribution into its center and tails".[5] The coefficient of Skewness is a measure for the degree of symmetry in the variable distribution (Sheskin, 2011). Negatively skewed distributionor Skewed to the leftSkewness <0 Normal distributionSymmetricalSkewness = 0 Positively skewed distributionor Skewed to the rightSkewness > 0 KURTOSIS. Kurtosis tells you the height and sharpness of the central peak, relative to that of a standard bell curve Käännös sanalle kurtosis englannista suomeksi. Suomienglantisanakirja.fi on suomen ja englannin kääntämiseen keskittyvä ilmainen sanakirja
where g 1 {\displaystyle g_{1}} is the sample skewness m 3 / m 2 3 / 2 {\displaystyle m_{3}/m_{2}^{3/2}} . For non-normal samples, the variance of the sample variance depends on the kurtosis; for details, please see variance. and the z i 4 {\displaystyle z_{i}^{4}} values are 0.003, 0.003, 0.002, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.003, 0.002, 0.003, 0.003, 360.976. Kurtosis (from the Greek word κυρτός, kyrtos or kurtos, meaning bulging) is a measure of the Positive kurtosis (leptokurtic) indicates a distribution more outlier-prone than predicted by a normal.. For two random variables, X and Y, not necessarily independent, the kurtosis of the sum, X + Y, is
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Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers What does kurtosis mean? kurtosis is defined by the lexicographers at Oxford Dictionaries as The sharpness of the peak of a frequency-distribution curve suomeksi Tulosarkistosta voit hakea vanhojenkin pelien tulokset helposti ja nopeasti. Valitse vain peli ja haluamasi aikaväli. Katso tulokset kurtosis0. Pro. Block or report user. Report or block kurtosis0. Hide content and notifications from this user
Is kurtosis a scrabble word? Kurtosis is worth 12 points in Scrabble, and 13 points in Words with Friends. There are 8 letters in kurtosis: I K O R S S T U Eine Kurtosis mit Wert 0 ist normalgipflig (mesokurtisch) Hinweis: Häufig werden die Begriffe Exzess und Kurtosis synonym verwendet, allerdings bezeichnet der Exzess den Kurtosis-Koeffizienten Several well-known, unimodal and symmetric distributions from different parametric families are compared here. Each has a mean and skewness of zero. The parameters have been chosen to result in a variance equal to 1 in each case. The images on the right show curves for the following seven densities, on a linear scale and logarithmic scale: CFI offers the Financial Modeling & Valuation Analyst (FMVA)™FMVA® CertificationJoin 350,600+ students who work for companies like Amazon, J.P. Morgan, and Ferrari certification program for those looking to take their careers to the next level. To keep learning and advancing your career, the following resources will be helpful:
## Skewness and Kurtosis
One cannot infer that high or low kurtosis distributions have the characteristics indicated by these examples. There exist platykurtic densities with infinite support, kurtosis muilla kielillä. Portugali curtose kurtosis ranska > suomi. kurtosis, coefficient d'aplatissement. huipukkuus, kurtoosi Kurtosis is a measure of the peakedness of a distribution. Like skewness statistics, it is not of much Kurtosis is sometimes used in conjunction with the skewness statistics to determine whether an.. In probability theory and statistics, kurtosis (from Greek: κυρτός, kyrtos or kurtos, meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes the shape of a probability distribution and there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Different measures of kurtosis may have different interpretations. suomeksi. teaching materials
Kurtosis definition is - the peakedness or flatness of the graph of a frequency distribution especially with respect to the concentration of values near the mean as compared with the normal distribution Excess kurtosis. Kurtosis measures the fatness of the tails of a distribution. Positive excess kurtosis means that distribution has fatter tails than a normal distribution
Leptokurtic indicates a positive excess kurtosis. The leptokurtic distribution shows heavy tails on either side, indicating the large outliers. In finance, a leptokurtic distribution shows that the investment returns may be prone to extreme values on either side. Therefore, an investment whose returns follow a leptokurtic distribution is considered to be risky. Kurtosis. 英 [kɜː'təʊsɪs] 美 [kɜː'toʊsɪs]. When we measure the kurtosis of a distribution, we are measuring its peakedness. 度量一条频率分布曲线的尖削度就是度量其顶部的峰态 For investors, high kurtosis of the return distribution implies that the investor will experience occasional extreme returns (either positive or negative), more extreme than the usual + or - three standard deviations from the mean that is predicted by the normal distribution of returns. This phenomenon is known as kurtosis risk.The first category of kurtosis is a mesokurtic distribution. This distribution has kurtosis statistic similar to that of the normal distribution, meaning that the extreme value characteristic of the distribution is similar to that of a normal distribution. In other words, kurtosis identifies whether the tails of a given distribution contain extreme values. Along with skewnessPoisson DistributionThe Poisson Distribution is a tool used in probability theory..
### The Pearson type VII familyedit
Kurtosis is a measure of the tailedness of the probability distribution. An increased kurtosis (>3) can be visualized as a thin bell with a high peak whereas a decreased kurtosis corresponds to a.. flow-kurtosis. Description. Reduce stream factory to calculate the sample excess kurtosis of streamed data values Suomeksi. Vapo Group - Sustainable Everyday Living. We are an international company, with a strategy of satisfying people's basic needs
..0.0, kurtosis=-1.3599999999999999) DescribeResult(nobs=4, minmax=(-1.2828087129930659 skewness=0.48089217736510326, kurtosis=-1.1471008824318165) DescribeResult(nobs=4.. Choose 'Distributional plots and tests' Select 'Skewness and kurtosis normality tests'. On the other hand, Kurtosis represents the height and sharpness of the central peak relative to..
## Kurtosis (K) Vose Softwar
In probability theory and statistics, kurtosis is a measure of the tailedness of the probability distribution of a real-valued random variable normal dağılım için kurtosis 3 olduğundan, -1.5,+1.5 kurtosis değeri arasında kalan alan normal dağılan veriyi temsil eder. bir dağılımın kurtosisi yani basıklığı ne kadar fazlaysa, dağılım normal'e.. In English Suomeksi 3. Mitä evästeitä käytetään? Jotkin evästeet ovat sivustomme teknisen toiminnan ja käytön vuoksi välttämättömiä. Nämä evästeet eivät kerää käyttäjästä tietoa, jota voitaisiin hyödyntää.. Kurtosis is a statistical measure used to describe the distribution of observed data around the mean. It is sometimes referred to as the volatility of volatility
### What is Kurtosis? Simply Psycholog
1. Интересное. Интересное. Интересное. Просмотр. Просмотр. Просмотр. Больше. Поиск
2. In statistics, kurtosis describes the shape of the probability distribution curve and there are 3 main types. More specifically, kurtosis refers to the tails or the 2 ends of the curve
3. Rohkem infot Nõustun. eesti keeles. in english suomeksi по-русски
## Interpretation of Skewness, Kurtosis, CoSkewness - Finance Trai
Kurtosis and Skewness. Kurtosis refers to a measure of the degree to which a given distribution is more or less 'peaked', relative to the normal distribution. The concept of kurtosis is very useful in.. where μ4 is the fourth central moment and σ is the standard deviation. Several letters are used in the literature to denote the kurtosis. A very common choice is κ, which is fine as long as it is clear that it does not refer to a cumulant. Other choices include γ2, to be similar to the notation for skewness, although sometimes this is instead reserved for the excess kurtosis. Kurtosis is a measure of whether the data in a data set are heavy-tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers. Data sets with low kurtosis tend to have light tails, or lack of outliers Statistics - Kurtosis - The degree of tailedness of a distribution is measured by kurtosis. It tells us the extent to which the distribution is more or less outlier-prone (heavier or l
## KURTOSIS POLITICAL STRATEGIST (@ceokurtosis) Instagramiss
Kurtosis is all about the tails of the distribution — not the peakedness or flatness. It is used to describe the extreme values in one versus the other tail. It is actually the measure of outliers present in the.. Kurtosis - online store, Karachi, Pakistan - rated 4 based on 4 reviews Keep it up girls (y) See more of Kurtosis on Facebook The final type of distribution is a platykurtic distribution. These types of distributions have short tails (paucity of outliers.) The prefix of "platy-" means "broad," and it is meant to describe a short and broad-looking peak, but this is an historical error. Uniform distributions are platykurtic and have broad peaks, but the beta (.5,1) distribution is also platykurtic and has an infinitely pointy peak. The reason both these distributions are platykurtic is that their extreme values are less than that of the normal distribution. For investors, platykurtic return distributions are stable and predictable, in the sense that there will rarely (if ever) be extreme (outlier) returns.A different measure of "kurtosis" is provided by using L-moments instead of the ordinary moments.[19][20]
Statistics - Kurtosis - The degree of tailedness of a distribution is measured by kurtosis. It tells us the extent to which the distribution is more or less outlier-prone (heavier or l The variance of the sample kurtosis of a sample of size n from the normal distribution is[14] Deutsch-Englisch-Übersetzung für: kurtosis. kurtosis in anderen Sprachen: Deutsch - Englisch skewness, moments and kurtosis introduction the measures of central tendency and variation discussed in previous chapters do not reveal the entire story about All densities in this family are symmetric. The kth moment exists provided m > (k + 1)/2. For the kurtosis to exist, we require m > 5/2. Then the mean and skewness exist and are both identically zero. Setting a2 = 2m − 3 makes the variance equal to unity. Then the only free parameter is m, which controls the fourth moment (and cumulant) and hence the kurtosis. One can reparameterize with m = 5 / 2 + 3 / γ 2 {\displaystyle m=5/2+3/\gamma _{2}} , where γ 2 {\displaystyle \gamma _{2}} is the excess kurtosis as defined above. This yields a one-parameter leptokurtic family with zero mean, unit variance, zero skewness, and arbitrary non-negative excess kurtosis. The reparameterized density is
## Kurtosis (@Kurtosis_) Твитте
Suomea suomeksi 1. Suomea suomeksi 1 Tuoreimmat uutiset. Näkökulmia yhteiskuntaan, kulttuuriin, hyvinvointiin ja tieteeseen. Laadukkaita timanttiartikkeleja ja koukuttavaa datajournalismia Kurtosis is a bit difficult. And we're not that concerned about. But understand these concepts of skewness and kurtosis, and be slightly circumspect when you interpret that kurtosis rather have.. kurtosis 어떻게 사용되는 지 Cambridge Dictionary Labs에 예문이 있습니다. This guarantees that directional selection will produce skew and positive kurtosis proportional to the peakshift, given the..
Alternative measures of kurtosis are: the L-kurtosis, which is a scaled version of the fourth L-moment; measures based on four population or sample quantiles.[3] These are analogous to the alternative measures of skewness that are not based on ordinary moments.[3] The kurtosis can now be seen as a measure of the dispersion of Z2 around its expectation. Alternatively it can be seen to be a measure of the dispersion of Z around +1 and −1. κ attains its minimal value in a symmetric two-point distribution. In terms of the original variable X, the kurtosis is a measure of the dispersion of X around the two values μ ± σ. D'Agostino's K-squared test is a goodness-of-fit normality test based on a combination of the sample skewness and sample kurtosis, as is the Jarque–Bera test for normality. Is it possible to calculate the skewness and kurtosis from an image just using the functions. scipy.stats.kurtosis scipy.stats.skew. When I applied it showed an array and not a single value
## python - Calculate skewness and Kurtosis - Stack Overflo
Note that in these cases the platykurtic densities have bounded support, whereas the densities with positive or zero excess kurtosis are supported on the whole real line. Oikeusasiamies moittii poliisia. Oikeusasiamiehen mielestä poliisi on antanut sakkoja ilman perustetta. Uudenmaan raja oli huhtikuussa suljettuna koronaepidemian takia. Poliisi valvoi rajaa. Poliisi antoi.. Read stories about Kurtosis on Medium. Discover smart, unique perspectives on Kurtosis and the topics that matter most to you like skewness, data science, statistics, central limit theorem.. Wikipedia - see also. kurtosis. Advertizing ▼. All translations of Kurtosis excess Outotec.com suomeksi. Online services
### Other well-known distributionsedit
Types of Kurtosis • Platykurtic distributions are flat distribution. • Note: Skewness and kurtosis are measures which compare two or more distributions in terms of their degree of departure from normality Many translated example sentences containing kurtosis - Russian-English dictionary and search engine for Russian translations Definition of Excess kurtosis in the Financial Dictionary - by Free online English dictionary and encyclopedia. What does Excess kurtosis mean in finance Suomeksi Distributions with kurtosis greater than 3 (excess kurtosis greater than 0) are called leptokurtic When method=fisher, the coefficient of kurtosis is estimated using the unbiased estimator for the..
### Variance under normalityedit
Stream Tracks and Playlists from Kurtosis on your desktop or mobile device A distribution with positive excess kurtosis is called leptokurtic, or leptokurtotic. "Lepto-" means "slender".[8] In terms of shape, a leptokurtic distribution has fatter tails. Examples of leptokurtic distributions include the Student's t-distribution, Rayleigh distribution, Laplace distribution, exponential distribution, Poisson distribution and the logistic distribution. Such distributions are sometimes termed super-Gaussian.[9] An excess kurtosis is a metric that compares the kurtosis of a distribution against the kurtosis of a normal distribution. The kurtosis of a normal distribution equals 3. Therefore, the excess kurtosis is found using the formula below:Kurtosis is a statistical measure that defines how heavily the tails of a distribution differ from the tails of a normal distribution. In other words, kurtosis identifies whether the tails of a given distribution contain extreme values.Distributions with zero excess kurtosis are called mesokurtic, or mesokurtotic. The most prominent example of a mesokurtic distribution is the normal distribution family, regardless of the values of its parameters. A few other well-known distributions can be mesokurtic, depending on parameter values: for example, the binomial distribution is mesokurtic for p = 1 / 2 ± 1 / 12 {\displaystyle p=1/2\pm {\sqrt {1/12}}} .
### Kurtosis convergenceedit
.. kurtosis (TR). Level. Home. > kurtosis (tr). Overview I would bet that this is true for the estimates of kurtosis and skewness. Someone want to post some research on this? $\endgroup$ - Peter Westfall Nov 11 '17 at 22:53 where the z i {\displaystyle z_{i}} values are the standardized data values using the standard deviation defined using n rather than n − 1 in the denominator.
kurtosis. English. (wikipedia kurtosis). Noun. (kurtoses). (statistics) A measure of peakedness of a probability distribution, defined as the fourth cumulant divided by the square of the variance of the.. kurtosis - WordReference English-Greek Dictionary. kurtosis. [links]. UK:*UK and possibly other pronunciationsUK and possibly other pronunciations/kəˈtəʊsɪs/US:USA pronunciation: respellingUSA.. A reason why some authors favor the excess kurtosis is that cumulants are extensive. Formulas related to the extensive property are more naturally expressed in terms of the excess kurtosis. For example, let X1, ..., Xn be independent random variables for which the fourth moment exists, and let Y be the random variable defined by the sum of the Xi. The excess kurtosis of Y is
### Sanan kurtosis määritelmät
The types of kurtosis are determined by the excess kurtosis of a particular distribution. The excess kurtosis can take positive or negative values, as well as values close to zero. Listen to the best Kurtosis shows
A distribution with negative excess kurtosis is called platykurtic, or platykurtotic. "Platy-" means "broad".[10] In terms of shape, a platykurtic distribution has thinner tails. Examples of platykurtic distributions include the continuous and discrete uniform distributions, and the raised cosine distribution. The most platykurtic distribution of all is the Bernoulli distribution with p = 1/2 (for example the number of times one obtains "heads" when flipping a coin once, a coin toss), for which the excess kurtosis is −2. Such distributions are sometimes termed sub-Gaussian distribution, originally proposed by Jean-Pierre Kahane[11] and further described by Buldygin and Kozachenko.[12] Tietosuoja Käyttöehdot Saavutettavuus. FI. Suomeksi Kurtosis considers the shape of the peaks in the probability distribution of data. The third classification for kurtosis is platykurtic. Platykurtic distributions are those that have slender tails 13 seuraajaa, 0 seurattavaa, 0 julkaisua. Katso käyttäjän KURTOSIS POLITICAL STRATEGIST (@ceokurtosis) Instagram-kuvat ja -videot
KURTOSIS. Greek 'kyrtosis,' meaning convexity. Measure of relative data value concentration in the A normal distribution has a kurtosis of 3. Higher kurtosis distributions show fatter tails with more.. suomeksi. Translative singular form of suomi. suomeksi. In Finnish. museoksi A platykurtic distribution shows a negative excess kurtosis. The kurtosis reveals a distribution with flat tails. The flat tails indicate the small outliers in a distribution. In the finance context, the platykurtic distribution of the investment returnsInternal Rate of Return (IRR)The Internal Rate of Return (IRR) is the discount rate that makes the net present value (NPV) of a project zero. In other words, it is the expected compound annual rate of return that will be earned on a project or investment. is desirable for investors because there is a small probability that the investment would experience extreme returns.
Kurtosis is defined as the measure of thickness or heaviness of the given distribution for the random • The distribution with kurtosis equal to 3 is known as mesokurtic. A random variable which follows.. kurtosis. synonyms - similar meaning - 42 Kurtosis It indicates the extent to which the values of the variable fall above or below the mean and Within Kurtosis, a distribution could be platykurtic, leptokurtic, or mesokurtic, as shown belo Now by definition of the kurtosis κ {\displaystyle \kappa } , and by the well-known identity E [ V 2 ] = var [ V ] + [ E [ V ] ] 2 , {\displaystyle E[V^{2}]=\operatorname {var} [V]+[E[V]]^{2},} The exact interpretation of the Pearson measure of kurtosis (or excess kurtosis) used to be disputed, but is now settled. As Westfall notes in 2014[2], "...its only unambiguous interpretation is in terms of tail extremity; i.e., either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution)." The logic is simple: Kurtosis is the average (or expected value) of the standardized data raised to the fourth power. Any standardized values that are less than 1 (i.e., data within one standard deviation of the mean, where the "peak" would be), contribute virtually nothing to kurtosis, since raising a number that is less than 1 to the fourth power makes it closer to zero. The only data values (observed or observable) that contribute to kurtosis in any meaningful way are those outside the region of the peak; i.e., the outliers. Therefore, kurtosis measures outliers only; it measures nothing about the "peak".
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https://love2d.org/forums/viewtopic.php?p=231860
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## [SOLVED][STI]crash when map:resize
Questions about the LÖVE API, installing LÖVE and other support related questions go here.
Forum rules
gotokiichi
Prole
Posts: 6
Joined: Wed Jan 29, 2020 7:41 pm
### [SOLVED][STI]crash when map:resize
I'm using karai17's STI library for my little project. Thanks karai17, it realy helps!
But when I try to implement zoom in or out the tiled map, the program just crashes every runtime without any error infomation to me. I have located the error to map:resize. If I delete this line, everything's ok and stable. I really need this function for zoom in and out the stage map, because the function map:draw with scale will also scale the drawsize of it, so that I need map:resize to change the drawarea back to original. Pls help and suggestion, thanks in advance!
I attach the love file and copy my urgly code as below:
mouse drag to pan the map
mouse wheel up/down to scale the map
Code: Select all
sti = require('sti')
map = sti("assets/map01.lua")
main_canvas = love.graphics.newCanvas()
Game = {}
Game.tileWidth = 48
Game.tileHeight = 48
Game.tileRowNum = 30
Game.tileColNum = 30
Game.xScale = 1
Game.yScale = 1
Game.windowWidth = 48*15
Game.windowHeight = 48*10
local cam, map
local flag_XYRecorded
local dx, dy
local x_start, y_start
local scale --scale for sti
local main_canvas
local floor = math.floor
function clamp(value, min, max)
if max then
if value > max then return max end
end
if min then
if value < min then return min end
end
return value
end
function initCamera()
flag_XYRecorded = false
dx, dy = 0, 0
x_start, y_start = 0, 0
scale = 1
end
function drawWorld(dx, dy)
love.graphics.setCanvas(main_canvas)
love.graphics.clear()
width_change = floor(Game.tileRowNum*Game.tileWidth/scale)
height_change = floor(Game.tileColNum*Game.tileHeight/scale)
map:resize(width_change,height_change)
map:draw(dx,dy,scale, scale)
love.graphics.setCanvas()
love.graphics.draw(main_canvas, 0, 0, 0, 1)
end
function love.wheelmoved(x, y)
if y > 0 then
--Mouse wheel moved up
scale = scale + 0.1
elseif y < 0 then
--Mouse wheel moved down
scale = scale - 0.1
end
scale = clamp(scale, 0.5, 1.0)
end
----------------
map = sti("assets/map01.lua")
initCamera()
main_canvas = love.graphics.newCanvas(Game.tileRowNum*Game.tileWidth, Game.tileColNum*Game.tileHeight)
end
function love:update(dt)
-- print('update')
if love.mouse.isDown(1) then -- mouse key1 pressed
if not flag_XYRecorded then
local x, y = love.mouse.getPosition()
x_start = x - dx
y_start = y - dy
flag_XYRecorded = true
else
local x, y = love.mouse.getPosition()
dx = x - x_start
dy = y - y_start
end
else -- mouse key1 release
if flag_XYRecorded then
local x, y = love.mouse.getPosition()
dx = x - x_start
dy = y - y_start
dx = clamp(dx, floor(Game.windowWidth/Game.xScale-Game.tileWidth*Game.tileRowNum), 0)
dy = clamp(dy, floor(Game.windowHeight/Game.yScale-Game.tileHeight*Game.tileColNum), 0)
flag_XYRecorded = false
end
end
map:update(dt)
end
function love:draw()
drawWorld(dx,dy)
--debug info
local x, y = love.mouse.getPosition()
love.graphics.setColor(1, 0, 0)
love.graphics.print('scr: '..x..' '..y, 10, 10)
love.graphics.print('map: '..(x-dx*Game.xScale)..' '..(y-dy*Game.yScale), 10, 40)
love.graphics.print('init: '..dx..' '..dy, 10, 70)
love.graphics.print('start: '..x_start..' '..y_start, 10, 100)
love.graphics.print('scale: '..scale, 10, 130)
love.graphics.setColor(1, 1, 1)
end
Attachments
test_sti.love
Last edited by gotokiichi on Thu Jan 30, 2020 2:26 pm, edited 2 times in total.
JJSax
Prole
Posts: 38
Joined: Fri Apr 04, 2014 3:59 am
### Re: [STI]crash when map:resize
I'm not seeing anywhere in there where it calls map:resize(). Perhaps you copied over the code that isn't crashing by mistake. It would surely help to see how you're calling it as well.
Edit: also let us know what the error you're getting is.
gotokiichi
Prole
Posts: 6
Joined: Wed Jan 29, 2020 7:41 pm
### Re: [STI]crash when map:resize
JJSax wrote:
Thu Jan 30, 2020 1:20 am
I'm not seeing anywhere in there where it calls map:resize(). Perhaps you copied over the code that isn't crashing by mistake. It would surely help to see how you're calling it as well.
Edit: also let us know what the error you're getting is.
Pls forgive a sleepy dog:P
I have changed the code and attached the runable love file.
JJSax
Prole
Posts: 38
Joined: Fri Apr 04, 2014 3:59 am
### Re: [STI]crash when map:resize
gotokiichi wrote:
Thu Jan 30, 2020 2:36 am
Pls forgive a sleepy dog:P
I have changed the code and attached the runable love file.
You are definitely forgiven.
I'll preface this by saying I'm not an expert. But I think I fixed it and I have a theory of why it works. It didn't crash right away and just eventually had the error. I dug into the init for sti and found that Map:resize(w,h) sets a new canvas every time it's called. The wiki says that love.graphics.newCanvas can be slow when ran repeatedly, like in a draw loop like you had it. This not only would cause your FPS to drop, but I think it was causing some issues with running too many canvases. Here is what I did. (just the draw loop and the wheel moved functions)
Code: Select all
function drawWorld(dx, dy)
love.graphics.setCanvas(main_canvas)
love.graphics.clear()
-- removed the resize function from here
map:draw(dx,dy,scale, scale)
love.graphics.setCanvas()
love.graphics.draw(main_canvas, 0, 0, 0, 1)
end
function love.wheelmoved(x, y)
if y > 0 then
--Mouse wheel moved up
scale = scale + 0.1
elseif y < 0 then
--Mouse wheel moved down
scale = scale - 0.1
end
-- put it here so it's not called all the time, only when it's needed.
width_change = floor(Game.tileRowNum*Game.tileWidth/scale)
height_change = floor(Game.tileColNum*Game.tileHeight/scale)
map:resize(width_change,height_change)
scale = clamp(scale, 0.5, 1.0)
end
Side note, is there a reason you set main_canvas at the top, then on a separate line make it local? It doesn't pose an issue as far as I know and you're not the only one who I've seen do it, but just curious why you don't set it local to begin with.
gotokiichi
Prole
Posts: 6
Joined: Wed Jan 29, 2020 7:41 pm
### Re: [STI]crash when map:resize
JJSax wrote:
Thu Jan 30, 2020 3:06 am
gotokiichi wrote:
Thu Jan 30, 2020 2:36 am
Pls forgive a sleepy dog:P
I have changed the code and attached the runable love file.
You are definitely forgiven.
I'll preface this by saying I'm not an expert. But I think I fixed it and I have a theory of why it works. It didn't crash right away and just eventually had the error. I dug into the init for sti and found that Map:resize(w,h) sets a new canvas every time it's called. The wiki says that love.graphics.newCanvas can be slow when ran repeatedly, like in a draw loop like you had it. This not only would cause your FPS to drop, but I think it was causing some issues with running too many canvases. Here is what I did. (just the draw loop and the wheel moved functions)
Code: Select all
function drawWorld(dx, dy)
love.graphics.setCanvas(main_canvas)
love.graphics.clear()
-- removed the resize function from here
map:draw(dx,dy,scale, scale)
love.graphics.setCanvas()
love.graphics.draw(main_canvas, 0, 0, 0, 1)
end
function love.wheelmoved(x, y)
if y > 0 then
--Mouse wheel moved up
scale = scale + 0.1
elseif y < 0 then
--Mouse wheel moved down
scale = scale - 0.1
end
-- put it here so it's not called all the time, only when it's needed.
width_change = floor(Game.tileRowNum*Game.tileWidth/scale)
height_change = floor(Game.tileColNum*Game.tileHeight/scale)
map:resize(width_change,height_change)
scale = clamp(scale, 0.5, 1.0)
end
Side note, is there a reason you set main_canvas at the top, then on a separate line make it local? It doesn't pose an issue as far as I know and you're not the only one who I've seen do it, but just curious why you don't set it local to begin with.
Thanks for you such prompt help! You are right, I made a test with counting map:resize time and every time the program just crashed on the 653th resizing. Moving map:resize to wheelmoved callback funtion as your suggestion would delay that crash but it still came on the 653th resizing. No body could escape the end:(
I think only I can do now is freezing the map scale functino to future.
And for that local canvas, there's no reason for that, I just follow BYTEPATH tutorial:D
You can find it here: https://github.com/adnzzzzZ/blog/issues/15
pgimeno
Party member
Posts: 2164
Joined: Sun Oct 18, 2015 2:58 pm
### Re: [STI]crash when map:resize
Yes, JJSax has correctly identified the source of the problem. To prevent it from happening, you can help the garbage collector a bit, e.g. by adding a collectgarbage() call after the resize.
For me it didn't crash, it just became more and more sluggish over time, but the problem disappeared after I added the collectgarbage() call.
The cause of the problem seems to be that LuaJIT does not have information about how many resources a certain object takes, and therefore it does not prioritize: it doesn't collect garbage until many objects (perhaps thousands of them) have accumulated. If those objects happen to be big canvases, the graphics card can't cope with so many of them.
gotokiichi
Prole
Posts: 6
Joined: Wed Jan 29, 2020 7:41 pm
### Re: [STI]crash when map:resize
pgimeno wrote:
Thu Jan 30, 2020 12:13 pm
Yes, JJSax has correctly identified the source of the problem. To prevent it from happening, you can help the garbage collector a bit, e.g. by adding a collectgarbage() call after the resize.
For me it didn't crash, it just became more and more sluggish over time, but the problem disappeared after I added the collectgarbage() call.
The cause of the problem seems to be that LuaJIT does not have information about how many resources a certain object takes, and therefore it does not prioritize: it doesn't collect garbage until many objects (perhaps thousands of them) have accumulated. If those objects happen to be big canvases, the graphics card can't cope with so many of them.
Thanks pgimeno, I will take a try of collectgarbage()!
Yes, it works!
Last edited by gotokiichi on Thu Jan 30, 2020 2:25 pm, edited 1 time in total.
grump
Party member
Posts: 612
Joined: Sat Jul 22, 2017 7:43 pm
### Re: [STI]crash when map:resize
Calling :release() on the old canvas should help as well and is a more localized solution for the problem at hand.
pgimeno
Party member
Posts: 2164
Joined: Sun Oct 18, 2015 2:58 pm
### Re: [SOLVED][STI]crash when map:resize
Eh, good point, thanks grump!
Only problem is that the canvas "belongs" to the lib, so you need to dig into the lib's internals. Maybe the STI maintainers can add code to the library to release the old canvas when creating a new one.
@ gotokiichi That's a guarantee against crashing, but as JJSax said, it's best if you minimize the map:resize() calls, limiting them to the instants where the zoom level changes. An alternative, or addendum, to what he did is to keep the last zoom value in a variable, and only call map:resize() when the zoom actually changes, i.e. when the last zoom differs from the newly calculated zoom.
### Who is online
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2020-07-03 22:49:35
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https://www.hackmath.net/en/math-problem/4038
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# Wall diagonal
Calculate the length of wall diagonal of the cube whose surface is 384 cm square.
Result
u = 11.314 cm
#### Solution:
$S=384 \ \text{cm}^2 \ \\ \ \\ S=6a^2 \ \\ \ \\ a=\sqrt{ S/6 }=\sqrt{ 384/6 }=8 \ \text{cm} \ \\ \ \\ u=\sqrt{ 2 } \cdot \ a=\sqrt{ 2 } \cdot \ 8 \doteq 8 \ \sqrt{ 2 } \doteq 11.3137 \doteq 11.314 \ \text{cm}$
Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!
Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):
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2020-04-09 19:09:31
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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, "math_score": 0.46796518564224243, "perplexity": 1623.2842606343663}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371876625.96/warc/CC-MAIN-20200409185507-20200409220007-00016.warc.gz"}
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https://www.esaral.com/q/the-given-distribution-shows-the-number-of-runs-scored-by-some-top-batsmen-of-the-world-in-one-day-international-cricket-matches-80864
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# The given distribution shows the number of runs scored by some top batsmen of the world in one day international cricket matches :
Question:
The given distribution shows the number of runs scored by some top batsmen of the world in one day international cricket matches :
Find the mode of the data.
Solution:
Modal class = 4000 – 5000
Mode $=\ell+\left\{\frac{\mathbf{f}-\mathbf{f}}{\boldsymbol{2} \mathbf{f}-\mathbf{f}_{\mathbf{0}}-\mathbf{f}_{2}}\right\} \times \mathrm{h}$
$=4000+\left\{\frac{18-4}{2 \times 18-4-9}\right\} \times 1000$
$=4000+\left\{\frac{14}{23}\right\} \times 1000$
= 4608.69
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2023-03-22 19:43:25
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http://mathematica.stackexchange.com/questions/32079/import-audio-in-ulaw-format
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# Import audio in uLaw format?
How do I import a uLaw-format audio file into Mathematica? Here's an example:
For reference, here's the same file in wav format:
As I understand it, the uLaw format contains raw µ-law audio data. In this respect, it should be very similar (or identical) to AU, which Mathematica supports, except there is no header.
I have tried:
Import["hello-world.ulaw", "AU"]
but Mathematica returns:
Import::fmterr: Cannot import data as AU/SND format.
From the documentation, it looks like there is probably some combination of parameters and options that will work, but the Import/Export documentation doesn't go into much detail on how to actually use the various available options. (Or at least, not in a way that makes sense to me!)
-
You can import the data in the file with BinaryReadList["hello-world.uLaw"]. This gives a long list of values (between 2 and 255). Normally one would turn these into sound with SampledSoundList but it does not lead to an intelligible sound in this case, so the data must be encoded in some way. – bill s Sep 11 '13 at 4:11
@SWB You can run sox from within Mathematica, and even read through a pipe. Import["!sox ...", "WAV"] will read whatever sox writes to stdout. This is going to give you a seamless and practical, but maybe not "beautiful" solution. If you need to see how Mma reads AU, your can do spelunking, starting with the SystemFiles/Formats folder, figure out how it works, and re-use the bits to read your files. I expect it to be doable with a few hours of work, but personally I don't think it's worth the effort when there's already a convenient solution. – Szabolcs Sep 13 '13 at 4:24
@SWB To give some more guidance, start with importing some AU file to trigger loading all relevant functions. Then take a look ?SystemConvertAudioDump*. It looks like there are some useful pieces there (mentions of μlaw). These spelunking tools will come handy. – Szabolcs Sep 13 '13 at 4:31
Thanks, @Szabolcs! It hadn't occurred to me to use sox as an importer for Mathematica, and I didn't know until now how to efficiently study Mathematica's internal implementation. After a few minutes with Spelunk, I've found that importing an AU file results in a call to SystemConvertAudioDumpImportAudioFile, which finally calls \$InstallationDirectory\SystemFiles\Converters\Binaries\Windows-x86-64\Audio.exe over a MathLink connection to perform the import. Therefore, it looks like the only solution will be to use an external helper like sox. – SWB Sep 13 '13 at 15:20
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2015-08-02 16:24:29
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http://mathhelpforum.com/advanced-algebra/63721-prove-matrix-not-invertible-if-eignvalue-0-a.html
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# Math Help - Prove that a matrix in NOT invertible if an eignvalue is 0
1. ## Prove that a matrix in NOT invertible if an eignvalue is 0
^^
Thank you very much!
2. Originally Posted by ishan
^^
Thank you very much!
In the basis that diagonalizes the matrix, the matrix can be written :
$\begin{pmatrix}
\lambda_1 & 0 & 0 &\ldots & 0 \\
0 & \lambda_2 & 0 & \ldots & 0 \\
0 & 0 & \lambda_3 & \ldots & 0 \\
\vdots & \vdots & \vdots & \ddots & 0 \\
0 & 0 & 0 & \ldots & \lambda_n \end{pmatrix}$
Where $\lambda_1, \dots, \lambda_n$ are the eigenvalues of the matrix.
The determinant in this base is obviously the product of the $\lambda_i$
So if one of the eigenvalues is 0, the determinant is 0.
And since the determinant doesn't change with the basis, we have proved that det A = 0, and hence it is not invertible.
Edit : waaaah 39th post
3. The determinant of a matrix is equal to the product of the eigenvalues. Hence the determinant is 0 and the matrix is not invertible.
4. If $A$ has $0$ as an eigenvalue then it means $A\bold{x} = 0\bold{x}$ for $\bold{x} \not = \bold{0}$.
Therefore, $A\bold{x} = \bold{0}$ for $\bold{x}\not = \bold{0}$.
This means $A$ is not invertible.
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2014-03-13 17:08:22
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http://www.journaltocs.ac.uk/index.php?action=browse&subAction=subjects&publisherID=8&journalID=801&pageb=1&userQueryID=&sort=&local_page=&sorType=&sorCol=
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for Journals by Title or ISSN for Articles by Keywords help
Subjects -> MATHEMATICS (Total: 886 journals) - APPLIED MATHEMATICS (72 journals) - GEOMETRY AND TOPOLOGY (20 journals) - MATHEMATICS (656 journals) - MATHEMATICS (GENERAL) (42 journals) - NUMERICAL ANALYSIS (19 journals) - PROBABILITIES AND MATH STATISTICS (77 journals) MATHEMATICS (656 journals) 1 2 3 4 | Last
1 2 3 4 | Last
Annals of the Institute of Statistical Mathematics [SJR: 0.931] [H-I: 31] [1 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 1572-9052 - ISSN (Online) 0020-3157 Published by Springer-Verlag [2352 journals]
• Estimation of the tail exponent of multivariate regular variation
• Authors: Moosup Kim; Sangyeol Lee
Pages: 945 - 968
Abstract: Abstract In this study, we consider the problem of estimating the tail exponent of multivariate regular variation. Since any convex combination of a random vector with a multivariate regularly varying tail has a univariate regularly varying tail with the same exponent under certain conditions, to estimate the tail exponent of the multivariate regular variation of a given random vector, we employ a weighted average of Hill’s estimators obtained for all of its convex combinations, designed to reduce the variability of estimation. We investigate the asymptotic properties and evaluate the finite sample performance of the weighted average of Hill’s estimators. A simulation study and real data analysis are provided for illustration.
PubDate: 2017-10-01
DOI: 10.1007/s10463-016-0574-9
Issue No: Vol. 69, No. 5 (2017)
• The limit distribution of weighted $$L^2$$ L 2 -goodness-of-fit statistics
under fixed alternatives, with applications
• Authors: L. Baringhaus; B. Ebner; N. Henze
Pages: 969 - 995
Abstract: Abstract We present a general result on the limit distribution of weighted one- and two-sample $$L^2$$ -goodness-of-fit test statistics of some hypothesis $$H_0$$ under fixed alternatives. Applications include an approximation of the power function of such tests, asymptotic confidence intervals of the distance of an underlying distribution with respect to the distributions under $$H_0$$ , and an asymptotic equivalence test that is able to validate certain neighborhoods of $$H_0$$ .
PubDate: 2017-10-01
DOI: 10.1007/s10463-016-0567-8
Issue No: Vol. 69, No. 5 (2017)
• A doubly sparse approach for group variable selection
• Authors: Sunghoon Kwon; Jeongyoun Ahn; Woncheol Jang; Sangin Lee; Yongdai Kim
Pages: 997 - 1025
Abstract: Abstract We propose a new penalty called the doubly sparse (DS) penalty for variable selection in high-dimensional linear regression models when the covariates are naturally grouped. An advantage of the DS penalty over other penalties is that it provides a clear way of controlling sparsity between and within groups, separately. We prove that there exists a unique global minimizer of the DS penalized sum of squares of residuals and show how the DS penalty selects groups and variables within selected groups, even when the number of groups exceeds the sample size. An efficient optimization algorithm is introduced also. Results from simulation studies and real data analysis show that the DS penalty outperforms other existing penalties with finite samples.
PubDate: 2017-10-01
DOI: 10.1007/s10463-016-0571-z
Issue No: Vol. 69, No. 5 (2017)
• Statistical inference with empty strata in judgment post stratified
samples
• Authors: Omer Ozturk
Pages: 1029 - 1057
Abstract: Abstract This article develops estimators for certain population characteristics using a judgment post stratified (JPS) sample. The paper first constructs a conditional JPS sample with a reduced set size K by conditioning on the ranks of the measured observations of the original JPS sample of set size $$H \ge K$$ . The paper shows that the estimators of the population mean, median and distribution function based on this conditional JPS sample are consistent and have limiting normal distributions. It is shown that the proposed estimators, unlike the ratio and regression estimators, where they require a strong linearity assumption, only need a monotonic relationship between the response and auxiliary variable. For moderate sample sizes, the paper provides a bootstrap distribution to draw statistical inference. A small-scale simulation study shows that the proposed estimators based on a reduced set JPS sample perform better than the corresponding estimators based on a regular JPS sample.
PubDate: 2017-10-01
DOI: 10.1007/s10463-016-0572-y
Issue No: Vol. 69, No. 5 (2017)
• Smoothed jackknife empirical likelihood for the difference of two
quantiles
• Authors: Hanfang Yang; Yichuan Zhao
Pages: 1059 - 1073
Abstract: Abstract In this paper, we propose a smoothed estimating equation for the difference of quantiles with two samples. Using the jackknife pseudo-sample technique for the estimating equation, we propose the jackknife empirical likelihood (JEL) ratio and establish the Wilk’s theorem. Due to avoiding estimating link variables, the simulation studies demonstrate that JEL method has computational efficiency compared with traditional normal approximation method. We carry out a simulation study in terms of coverage probability and average length of the proposed confidence intervals. A real data set is used to illustrate the JEL procedure.
PubDate: 2017-10-01
DOI: 10.1007/s10463-016-0576-7
Issue No: Vol. 69, No. 5 (2017)
• Additional aspects of the generalized linear-fractional branching process
• Authors: Nicolas Grosjean; Thierry Huillet
Pages: 1075 - 1097
Abstract: Abstract We derive some additional results on the Bienyamé–Galton–Watson-branching process with $$\theta$$ -linear fractional branching mechanism, as studied by Sagitov and Lindo (Branching Processes and Their Applications. Lecture Notes in Statistics—Proceedings, 2016). This includes the explicit expression of the limit laws in both the subcritical cases and the supercritical cases with finite mean, and the long-run behavior of the population size in the critical case, limits laws in the supercritical cases with infinite mean when the $$\theta$$ process is either regular or explosive, and results regarding the time to absorption, an expression of the probability law of the $$\theta$$ -branching mechanism involving Bell polynomials, and the explicit computation of the stochastic transition matrix of the $$\theta$$ process, together with its powers.
PubDate: 2017-10-01
DOI: 10.1007/s10463-016-0573-x
Issue No: Vol. 69, No. 5 (2017)
• Efficient estimation of quasi-likelihood models using B -splines
• Authors: Minggen Lu
Pages: 1099 - 1127
Abstract: Abstract We consider a simple yet flexible spline estimation method for quasi-likelihood models. We approximate the unknown function by B-splines and apply the Fisher scoring algorithm to compute the estimates. The spline estimate of the nonparametric component achieves the optimal rate of convergence under the smooth condition, and the estimate of the parametric part is shown to be asymptotically normal even if the variance function is misspecified. The semiparametric efficiency of the model can be established if the variance function is correctly specified. A direct and consistent variance estimation method based on the least-squares estimation is proposed. A simulation study is performed to evaluate the numerical performance of the spline estimate. The methodology is illustrated on a crab study.
PubDate: 2017-10-01
DOI: 10.1007/s10463-016-0575-8
Issue No: Vol. 69, No. 5 (2017)
• On coupon collector’s and Dixie cup problems under fixed and random
sample size sampling schemes
• Authors: James C. Fu; Wan-Chen Lee
Pages: 1129 - 1139
Abstract: Abstract Suppose an urn contains m distinct coupons, labeled from 1 to m. A random sample of k coupons is drawn without replacement from the urn, numbers are recorded and the coupons are then returned to the urn. This procedure is done repeatedly and the sample sizes are independently identically distributed. Let W be the total number of random samples needed to see all coupons at least l times $$(l \ge 1)$$ . Recently, for $$l=1$$ , the approximation for the first moment of the random variable W has been studied under random sample size sampling scheme by Sellke (Ann Appl Probab, 5:294–309, 1995). In this manuscript, we focus on studying the exact distributions of waiting times W for both fixed and random sample size sampling schemes given $$l \ge 1$$ . The results are further extended to a combination of fixed and random sample size sampling procedures.
PubDate: 2017-10-01
DOI: 10.1007/s10463-016-0578-5
Issue No: Vol. 69, No. 5 (2017)
• Moment convergence of regularized least-squares estimator for linear
regression model
• Authors: Yusuke Shimizu
Pages: 1141 - 1154
Abstract: Abstract In this paper, we study the uniform tail-probability estimates of a regularized least-squares estimator for the linear regression model. We make use of the polynomial type large deviation inequality for the associated statistical random fields, which may not be locally asymptotically quadratic. Our results enable us to verify various arguments requiring convergence of moments of estimator-dependent statistics, such as the mean squared prediction error and the bias correction for AIC-type information criterion.
PubDate: 2017-10-01
DOI: 10.1007/s10463-016-0577-6
Issue No: Vol. 69, No. 5 (2017)
• Collapsibility of some association measures and survival models
• Authors: P. Vellaisamy
Pages: 1155 - 1176
Abstract: Abstract Collapsibility deals with the conditions under which a conditional (on a covariate W) measure of association between two random variables Y and X equals the marginal measure of association. In this paper, we discuss the average collapsibility of certain well-known measures of association, and also with respect to a new measure of association. The concept of average collapsibility is more general than collapsibility, and requires that the conditional average of an association measure equals the corresponding marginal measure. Sufficient conditions for the average collapsibility of the association measures are obtained. Some interesting counterexamples are constructed and applications to linear, Poisson, logistic and negative binomial regression models are discussed. An extension to the case of multivariate covariate W is also analyzed. Finally, we discuss the collapsibility conditions of some dependence measures for survival models and illustrate them for the case of linear transformation models.
PubDate: 2017-10-01
DOI: 10.1007/s10463-016-0580-y
Issue No: Vol. 69, No. 5 (2017)
• Semiparametric efficient estimators in heteroscedastic error models
• Authors: Mijeong Kim; Yanyuan Ma
Abstract: Abstract In the mean regression context, this study considers several frequently encountered heteroscedastic error models where the regression mean and variance functions are specified up to certain parameters. An important point we note through a series of analyses is that different assumptions on standardized regression errors yield quite different efficiency bounds for the corresponding estimators. Consequently, all aspects of the assumptions need to be specifically taken into account in constructing their corresponding efficient estimators. This study clarifies the relation between the regression error assumptions and their, respectively, efficiency bounds under the general regression framework with heteroscedastic errors. Our simulation results support our findings; we carry out a real data analysis using the proposed methods where the Cobb–Douglas cost model is the regression mean.
PubDate: 2017-10-13
DOI: 10.1007/s10463-017-0622-0
• Purely sequential bounded-risk point estimation of the negative binomial
mean under various loss functions: one-sample problem
• Authors: Nitis Mukhopadhyay; Sudeep R. Bapat
Abstract: Abstract A negative binomial (NB) distribution is useful to model over-dispersed count data arising from agriculture, health, and pest control. We design purely sequential bounded-risk methodologies to estimate an unknown NB mean $$\mu (>0)$$ under different forms of loss functions including customary and modified Linex loss as well as squared error loss. We handle situations when the thatch parameter $$\tau (>0)$$ may be assumed known or unknown. Our proposed methodologies are shown to satisfy properties including first-order asymptotic efficiency and first-order asymptotic risk efficiency. Summaries are provided from extensive sets of simulations showing encouraging performances of the proposed methodologies for small and moderate sample sizes. We follow with illustrations obtained by implementing estimation strategies using real data from statistical ecology: (1) weed count data of different species from a field in Netherlands and (2) count data of migrating woodlarks at the Hanko bird sanctuary in Finland.
PubDate: 2017-10-13
DOI: 10.1007/s10463-017-0620-2
• A generalized partially linear framework for variance functions
• Authors: Yixin Fang; Heng Lian; Hua Liang
Abstract: Abstract When model the heteroscedasticity in a broad class of partially linear models, we allow the variance function to be a partial linear model as well and the parameters in the variance function to be different from those in the mean function. We develop a two-step estimation procedure, where in the first step some initial estimates of the parameters in both the mean and variance functions are obtained and then in the second step the estimates are updated using the weights calculated based on the initial estimates. The resulting weighted estimators of the linear coefficients in both the mean and variance functions are shown to be asymptotically normal, more efficient than the initial un-weighted estimators, and most efficient in the sense of semiparametric efficiency for some special cases. Simulation experiments are conducted to examine the numerical performance of the proposed procedure, which is also applied to data from an air pollution study in Mexico City.
PubDate: 2017-10-04
DOI: 10.1007/s10463-017-0619-8
• General rank-based estimation for regression single index models
• Authors: Huybrechts F. Bindele; Ash Abebe; Karlene N. Meyer
Abstract: Abstract This study considers rank estimation of the regression coefficients of the single index regression model. Conditions needed for the consistency and asymptotic normality of the proposed estimator are established. Monte Carlo simulation experiments demonstrate the robustness and efficiency of the proposed estimator compared to the semiparametric least squares estimator. A real-life example illustrates that the rank regression procedure effectively corrects model nonlinearity even in the presence of outliers in the response space.
PubDate: 2017-09-20
DOI: 10.1007/s10463-017-0618-9
• Hazard rate estimation for left truncated and right censored data
• Authors: Sam Efromovich; Jufen Chu
Abstract: Abstract Left truncation and right censoring (LTRC) presents a unique challenge for nonparametric estimation of the hazard rate of a continuous lifetime because consistent estimation over the support of the lifetime is impossible. To understand the problem and make practical recommendations, the paper explores how the LTRC affects a minimal (called sharp) constant of a minimax MISE convergence over a fixed interval. The corresponding theory of sharp minimax estimation of the hazard rate is presented, and it shows how right censoring, left truncation and interval of estimation affect the MISE. Obtained results are also new for classical cases of censoring or truncation and some even for the case of direct observations of the lifetime of interest. The theory allows us to propose a relatively simple data-driven estimator for small samples as well as the methodology of choosing an interval of estimation. The estimation methodology is tested numerically and on real data.
PubDate: 2017-09-20
DOI: 10.1007/s10463-017-0617-x
• A constructive hypothesis test for the single-index models with two groups
• Authors: Jun Zhang; Zhenghui Feng; Xiaoguang Wang
Abstract: Abstract Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramér–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramér–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures.
PubDate: 2017-09-13
DOI: 10.1007/s10463-017-0616-y
• Hybrid schemes for exact conditional inference in discrete exponential
families
• Authors: David Kahle; Ruriko Yoshida; Luis Garcia-Puente
Abstract: Abstract Exact conditional goodness-of-fit tests for discrete exponential family models can be conducted via Monte Carlo estimation of p values by sampling from the conditional distribution of multiway contingency tables. The two most popular methods for such sampling are Markov chain Monte Carlo (MCMC) and sequential importance sampling (SIS). In this work we consider various ways to hybridize the two schemes and propose one standout strategy as a good general purpose method for conducting inference. The proposed method runs many parallel chains initialized at SIS samples across the fiber. When a Markov basis is unavailable, the proposed scheme uses a lattice basis with intermittent SIS proposals to guarantee irreducibility and asymptotic unbiasedness. The scheme alleviates many of the challenges faced by the MCMC and SIS schemes individually while largely retaining their strengths. It also provides diagnostics that guide and lend credibility to the procedure. Simulations demonstrate the viability of the approach.
PubDate: 2017-09-04
DOI: 10.1007/s10463-017-0615-z
• Smoothed nonparametric tests and approximations of p -values
• Authors: Yoshihiko Maesono; Taku Moriyama; Mengxin Lu
Abstract: Abstract We propose new smoothed sign and Wilcoxon’s signed rank tests that are based on kernel estimators of the underlying distribution function of the data. We discuss the approximations of the p-values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon’s tests in the sense of Pitman’s asymptotic relative efficiency, and the differences between the ordinary and new tests converge to zero in probability. Under the null hypothesis, the main terms of the asymptotic expectations and variances of the tests do not depend on the underlying distribution. Although the smoothed tests are not distribution-free, making use of the specific kernel enables us to obtain the Edgeworth expansions, being free of the underlying distribution.
PubDate: 2017-08-23
DOI: 10.1007/s10463-017-0614-0
• Testing equality between several populations covariance operators
• Authors: Graciela Boente; Daniela Rodriguez; Mariela Sued
Abstract: Abstract In many situations, when dealing with several populations, equality of the covariance operators is assumed. An important issue is to study whether this assumption holds before making other inferences. In this paper, we develop a test for comparing covariance operators of several functional data samples. The proposed test is based on the Hilbert–Schmidt norm of the difference between estimated covariance operators. In particular, when dealing with two populations, the test statistic is just the squared norm of the difference between the two covariance operators estimators. The asymptotic behaviour of the test statistic under both the null hypothesis and local alternatives is obtained. The computation of the quantiles of the null asymptotic distribution is not feasible in practice. To overcome this problem, a bootstrap procedure is considered. The performance of the test statistic for small sample sizes is illustrated through a Monte Carlo study and on a real data set.
PubDate: 2017-08-20
DOI: 10.1007/s10463-017-0613-1
• Erratum to: A doubly sparse approach for group variable selection
• Authors: Sunghoon Kwon; Jeongyoun Ahn; Woncheol Jang; Sangin Lee; Yongdai Kim
PubDate: 2017-07-25
DOI: 10.1007/s10463-017-0612-2
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2017-10-23 00:37:42
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https://brilliant.org/problems/trevor-rides-jake-piggyback-i/
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Trevor rides Jake piggyback I
Calculus Level 4
It's a beautiful Saturday morning. The wind is light yet refreshing, the sun warm and mellow, and the Trevor is riding Jake piggyback upon the verdant turf of the first golf course.
Starting at $$(0,0)$$, Trevor tells his serf to advance north by $$1$$ meter and turn $$90^{\circ}$$ clockwise. Jake, with his incredibly exact eyesight, does just that. At each nth step, he walks $$\frac{1}{n}$$ meter and turns $$90^{\circ}$$ clockwise.
What is the displacement in meters Trevor and Jake have travelled from the origin? If it can be expressed in the form $$\frac{1}{2} \sqrt{\ln^{2} a + \frac{\pi^{2}}{b}}$$, where $$a$$ and $$b$$ are positive integers, what is $$a+b$$?
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2019-01-23 23:17:15
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https://traumstyling.com/78muoslm/a-line-intersecting-a-plane-in-real-life-f43862
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5. the intersection of a plane and a line not in the plane is ___ line. Opposite Rays! rail road tracks off in the distance. a taut piece of thread. *Why is it important that these lines intersect?-In my picture, it is important that the lines intersect because they are train tracks and by them intersecting, the trains that travel on the tracks are able to go many different ways, like they are able to turn or keep staright just by these lines intersecting. Let’s consider what the UK did in 2004, when it relocated passenger checkpoints from arrival stations inside the British geographical border to the stations that travellers depart from in France and Belgium. As naïve as this question may be, attempts to draw lines all over the globe has provided fodder for much of the content in this volume. So, it's possible for the intersection of a line to be the null set, a point or a line. A plane is a two-dimensional surface and like a line, it extends up to infinity. Toufiq Elahi ID : 153002030 Department : Computer Science and Engineering (CSE) 2. Geometry is the branch of mathematics which deals with the measurement, properties and relationships of points, lines, angles, surfaces and solids. 7. Line 4. (f) If two lines intersect, then exactly one plane contains both lines (Theorem 3). Intersecting lines meet at one point ONLY. Find the point of intersection for the infinite ray with direction (0, -1, -1) passing through position (0, 0, 10) with the infinite plane with a normal vector of (0, 0, 1) and which passes through [0, 0, 5]. Best real-world examples of line/plane intersections Hi! Thanks So the point of intersection can be determined by plugging this … Send to friends and colleagues. SOLUTION Step 1 Draw a vertical plane. Session 16: Intersection of a Line and a Plane Course Home Syllabus ... Use OCW to guide your own life-long learning, or to teach others. What is the difference between a postulate and a theorem? Make sure you label your points, lines, and planes clearly, and refer to them by name when writing explanations. Not only is it tidy numerically, but it is also the only outcome that can be solved for in a single cartesian plane. Referring to (2), intersecting lines lie in a plane. Postulate! Use the diagram. 3. Slope! In two dimensions (i.e., the Euclidean plane), two lines which do not intersect are called parallel. A line contains at least two points (Postulate 1). 16. Well taking that same plane a, b, c, d if I took one edge let's say a, b so I'm going to say line segment a, b line segment a, b intersects this plane a, b, c, d it also intersects this plane a, b, e, f. Which means it could be parallel to this bottom face, so the bottom face is c, d, h and g. Download files for later. Postulate 1-5: The intersection of two planes is a line. 1. stars, galaxies, any 3 things that are related on position. Just two planes are parallel, and the 3rd plane cuts each in a line : The intersection of the three planes is a line : The intersection of the three planes is a point : Each plane cuts the other two in a line ... therefore the three planes intersect in a line. Names given to the pairs of angles in parallel lines are Step 3 Draw the line of intersection. Step 3 Draw the line of intersection. Since we found a single value of t from this process, we know that the line should intersect the plane in a single point, here where t = − 3. The novice mathematician in me would argue that the simplest outcome is when the line is contained within the plane. I have a few examples that aren't too unique or wild, but I am wondering what YOUR best real-world examples of objects or situations are for the following intersection instances as I introduce the concept of plane and line intersections to my students next Monday. Daniel Rossi’s illustrations are bound to two dimensions by their very nature as drawings and their simple use of vectors reference mathematical clarity. − 2t = 6. t = − 3. The term 'Geometry' is derived from the Greek word 'Geometron'. It is a point, if the line is not contained on the plane, and a line, if the line is contained on the plane. Sketch a plane and a line that is in the plane. (g) If a point lies outside a line, then exactly one plane contains both the line and the point (Theorem 2). the crease in a folded sheet of wrapping paper. We asked him to create a series of illustrations for Volume 35 that would do visually what our contributors have done in writing. Parallel Lines : In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be … Consequently, what a geographic border represents in the 21st century is far from ubiquitous. Step 2 Draw a second plane that is horizontal. In real life, while railroad tracks, the edges of sidewalks, and the markings on streets are all parallel, the tracks, sidewalks, and streets go up and down hills and around curves. Perpendicular Lines! What type of geometric object is the intersection of a line and a plane? 26. 4 − 2t = 10. Parallel lines as a purely geometric concept must be completely flat and exist on a single plane. Parallel And Perpendicular With Real Life Examples. Geometry In The Real World..Brianna Ford 3rd Block Ms.Sharpe 2012-2013: Home; Table of Contents! For the best answers, search on this site https://shorturl.im/ZOtlk. Topics Parallel & Perpendicular. Skew Lines! Plane-Line Intersection B 25. A line is said to be perpendicular to another line if the two lines intersect at a right angle. line. Two planes always intersect in a line as long as they are not parallel. The first option is that the line intersects the plane in a single point, the second and third options occur when the line and plane are parallel. Example 7: Draw and label the intersection of line and ray at point . 4 − 2t = 10. The first option is that the line intersects the plane in a single point, the second and third options occur when the line and plane are parallel. The planes : 6x-8y=1 , : x-y-5z=-9 and : -x-2y+2z=2 are: Intersecting … Let the planes be specified in Hessian normal form, then the line of intersection must be perpendicular to both n_1^^ and n_2^^, which means it is parallel to a=n_1^^xn_2^^. Shade the plane. The vertical angles are opposite angles with a common vertex (which is the point of intersection). How do you solve a proportion if one of the fractions has a variable in both the numerator and denominator? Postulate 5: If two points lie in a plane, then the line joining them lies in that plane. Sketch two planes that intersect in a line. (1) To uniquely specify the line, it is necessary to also find a particular point on it. A line and a point not on it 3. What do you think of the answers? Such an intersection of two planes is shown below. Since we found a single value of t from this process, we know that the line should intersect the plane in a single point, here where t = − 3. You can draw an infinite parallel line between the two parallel lines $\overline{PQ}$ and $\overline{RS}$in the given plane. When a line is contained within the ground plane of a region or territory the complexities of history, politics, sociology, economics, stories, and beliefs come into play. Given the algebraic simplicity of this outcome, why then, when we impose a line on a surface outside of Euclidean’s safety net does it cause so many issues? (4) You are right. MMonitoring Progressonitoring Progress Help in English and Spanish at BigIdeasMath.com 4. Parallel And Perpendicular With Real Life Examples. Either, they will not meet at all, or the line will be contained within the plane altogether. Task. Calculator won't calculate sin divided by anything, shows error. Parallel Line Examples in Real Life. ∠a + … Shade this plane a different color. Name the intersection of ⃖PQ""⃗ and line k. 6. If two lines intersect, then exactly one plane contains both lines (Theorem 3). 6 − 3t − 2 − 2t + 3t = 10. Intersecting Lines! While these checkpoints were very much inside the administrative region of another territory, the British government had essentially extended itself into the heart of continental Europe, creating a fragmented border with countries it does not even physically touch. The same concept is of a line-plane intersection. But, in general they need not belong to a plane. Plane 6. Parallel Lines! Use dashed lines to show where one plane is hidden. c. Sketch a plane and a line that intersects the plane at a point. Shade the plane. Provide students with either the Geometric Figures Picture Cards or the Geometric Figures in Real-life Objects Picture Cards. 5. SOLUTION Step 1 Draw a vertical plane. a knot in a piece of thread. Which is to say, reconcile the relationship between a line and the space it is imposed upon. Two or more lines which share exactly one common point are called intersecting lines. Model representing intersecting line segments using two pieces of … You can sign in to give your opinion on the answer. When making geometric drawings, you need to be sure to be clear and label. Sketch a plane and a line that does not intersect the plane. Ray 5. This example shows one of the many ways that we are all faced with borders in our daily routine. Where the plane can be either a point and a normal, or a 4d vector (normal form), In the examples below (code for both is provided).. Also note that this function calculates a value representing where the point is on the line, (called fac in the code below). Shade this plane a different color. Sketch two planes that intersect in a line. Plane! A line is said to be perpendicular to another line if the two lines intersect at a right angle. for parallel the stripes on the American flag. plane. In three-dimensional geometry, there exist an infinite number of lines perpendicular to a given line. Collinear Points! Use the diagram. Toufiq Elahi ID : 153002030 Department : Computer Science and Engineering (CSE) 2. Still have questions? Also, ∠b and ∠d are vertical angles and equal to each other. Modify, remix, and reuse (just remember to cite OCW as the source.) Thank you for signing up to The Site Magazine's newsletter. 6 − 3t − 2 − 2t + 3t = 10. Any two intersecting planes (that are not the same plane) form a line of intersection; this is the three-dimensional analog of a point of intersection for two lines. In this situation, (after substituting the equation of the line into the equation of the plane and solving for the scalar) you would produce an outcome of zero. Previous Points Lines and Planes. Sign up for a biweekly newsletter to receive great, original content straight to your inbox. Intersecting lines I need real life examples of these and I'm really stuck. the intersection of 2 planes is a ___ 2 _ points determine a line. Step 3 Draw the line of intersection. Geometry has a long and rich history. ∠a + … This common point exists on all these lines and is called the point of intersection. SOLUTION Step 1 Draw a vertical plane. SOLUTION a. b. c. Sketching Intersections of Planes Sketch two planes that intersect in a line. In analytic geometry there are three possible outcomes when calculating the relationship between a line and a plane. MMonitoring Progressonitoring Progress Help in English and Spanish at BigIdeasMath.com 4. Our newsletter includes exclusive content. Parallel Lines 4. Name the intersection of plane A and plane B. Help please!!!! Editor Michael Taylor explores here the problems that arise when two dimensions are forced into the infinite. Draw your answer. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. Topics Parallel & Perpendicular. Heres a Python example which finds the intersection of a line and a plane. In analytic geometry there are three possible outcomes when calculating the relationship between a line and a plane. Intersecting Lines 7. For 13-16, use geometric notation to explain each picture in as much detail as possible. 2. a road and dirt on the side of it, maps, pitcher and first base line, 3. city roads, lanes of traffic, book pages, tv screen lines, 4. crosses, addition signs, t-joints, fire-blocking in stud walls. It means that two or more than two lines meet at a point or points, we call those point/points intersection point/points. point. Postulate 6: If two planes intersect, then their intersection is a line. 3 _ non collinear points determine a plane. This property of being perpendicular is the relationship between two lines which meet at a right angle (90 degrees). Parallel Lines 8. Inductive Reasoning! Intersection Postulate Plane-Line Intersection: If a plane and a line intersect, then their intersection is either a point or a line. Either, they will not meet at all, or the line will be contained within the plane altogether. I can think that, for instance, a software which needs to find the intersections of a line, can benefit from always having an intersection point which might lead to simpler code, but is it really used? Consider a line l that intersects a plane at a right angle (in other words, wherever an angle measurement is taken around the line with respect to the plane, it is always 90°). − 2t = 6. t = − 3. Delimiting in the context of the contemporary city is complicated: no longer the finite, territorial boundary of the nation state, borders now are varied and fragmented, physical or artificial, and often have different repercussions for different people. Concept must be completely flat and exist a line intersecting a plane in real life a single plane the projective plane use geometric notation to each! Common intersection point in a line is said to be perpendicular to line... Even if they were all coincident they would still belong to a line you can sign in to your. Our contributors have done in writing, then exactly one point ( Theorem 1 ), any 3 things are. Remember to cite OCW as the source. something close to a line does! The numerator and denominator there any real life application which uses the projective plane a common vertex which! Even if they were all coincident they would still belong to a line reuse just! X-Y-5Z=-9 and: -x-2y+2z=2 are: intersecting … However, in his operative he... Then is a line intersect, then exactly one point ( Theorem 1 ) him... First things i think of when i think parallel and intersecting lines i need real life examples of 4! ⃗ and line k. 6 153002030 Department: Computer Science and Engineering ( CSE ) 2 folded sheet of paper! Perpendicular lines step 3 Draw the line will be contained within the plane altogether line k and plane Q! Computer Science and Engineering ( CSE ) 2 lines step 3 Draw the line said! Means that two or more lines which do not intersect the plane as possible a newsletter! Life examples of the 4 ways to determine a plane in writing the many ways that are. _ points determine a line and chimney line ) are called parallel explain each in! Points determine a line is said to be the null set, a point a... Heres a Python example which finds the intersection of plane a and B are in same.! Pair of vertical angles and are equal imposed upon me would argue that the line will be contained within plane! The intersection of ⃖PQ '' '' ⃗ and line k. 6 they were all coincident would... Progressonitoring Progress Help in English and Spanish at BigIdeasMath.com 4 extends up to the print issue are to! Computer Science and Engineering ( CSE ) 2 content straight to your inbox argue that the simplest outcome is the... The scalar then is a two-dimensional surface and like a line and a point or points, we those... ___ 2 _ points determine a plane at the same point which finds the intersection 2. That can be determined by plugging … two intersecting lines lie in plane! Only is it tidy numerically, but it is also the only a line intersecting a plane in real life that can be solved in. Makes two dimensional drawings look realistic present within contemporary urbanism of illustrations for Volume 35 that do! Done in writing much detail as possible English and Spanish at BigIdeasMath.com 4 uniquely specify the line will be within. Joining them lies in that plane intersection point/points they would still belong to a plane and a line said. As much detail as possible planes toward the endless possibilities present within contemporary urbanism contained within plane. Concept must be completely flat and exist on a single cartesian plane a line intersecting a plane in real life names are given to these angles the! Site Magazine 's newsletter, line segments PQ and RS denote parallel lines in the same point in both numerator. Two intersecting lines i need real life examples of perpendicular lines are denoted by the symbol of ( ⊥.... Reappropriated, and planes toward the endless possibilities present within contemporary urbanism as much as... One point ( Theorem 1 ) then is a qualitative exercise shows one of fractions... 2 Draw a second plane that is horizontal me would argue that the simplest outcome is the. Is a qualitative exercise intersection postulate Plane-Line intersection: if two lines intersect then! Possibility that the line, it is imposed upon is parallel to the Site Magazine 's newsletter of intersection the! Angle of intersection is either a point for Volume 35 that would visually... Endless possibilities present within contemporary urbanism possible for the best answers, search on this Site https //shorturl.im/ZOtlk... Is something close to a plane is hidden examples of these and i 'm really.... Two different lines that intersect in exactly one common point are called parallel general they not. Are perpendicular ( 2 ), intersecting lines form a pair of vertical.! Or certification for using OCW any real life examples of perpendicular lines denoted. 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To explain each Picture in as much detail as possible intersection at all or. -X-2Y+2Z=2 are: intersecting … However, a point to explain each Picture in as much detail as.... Learn about, 1 planes: 6x-8y=1,: x-y-5z=-9 and: -x-2y+2z=2 are: intersecting However...
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2021-04-13 05:35:31
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http://rpg.stackexchange.com/questions/19442/player-characters-as-mounts/19463
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# Player characters as mounts
I am asking about an edge case based on the three paragraphs under "Encounters with PC Mounts" on DMG 46. So far as I can tell these paragraphs are not reproduced anywhere else, and these paragraphs are why I am asking this question. Please bear this in mind when you answer; thank you.
When a PC with Mounted Combat uses another PC as a mount, how do their turns and movement interact?
I've got a player who wants to build a pixie (Tiny) with the Mounted Combat feat, and ride another (Medium) PC as its mount. The players involved are fine with however this works out. The Mounted Combat mechanics are pretty clear, and there are often different rules for an NPC or a PC:
• The mount and rider act on the rider’s initiative count.
• A monster and its mount have separate turns, whereas an adventurer and his or her mount have a single turn.
• An adventurer and his or her mount have a shared set of actions: a standard action, a move action, and a minor action. However, they each have their own free actions.
...But then there are three paragraphs on DMG 46, beneath the heading "Encounters with PC Mounts." I have snipped out the relevant bits:
[...] You can allow the PCs and the creatures they ride to get their own sets of actions, especially if a character rides a powerful, intelligent monster such as a dragon. However, at that point you have effectively added an additional member to the party. [...] You should use this rule if the mount’s level is at the party’s level or higher, or if its level is no more than two below the characters’ level. [...]
This is clearly counter to the PC Mounted Combat rules in many ways, and is more akin to the rules for mounted NPCs. Obviously a PC is an intelligent mount (despite anecdotal evidence to the contrary), and in this case the level of mount and rider will be equal. With that in mind...
1. Should the mounted PC be treated as a normal mount for a PC, losing his turn and becoming subordinate to the rider?
2. If 1 is "no", do they share initiative counts?
3. If 1 is "no", can the rider use his move action to make the mounted PC move, or force the mount to act by spending his own actions in any other way described by the Mounted Combat rules?
4. Can the mounted PC use the rider's base skill check bonuses as described in the Mounted Combat feat (e.g. a human with a pixie on his shoulder uses the pixie's Stealth modifier)?
5. How would feats like Holy Steed, which gives a rider's mount bonuses to defenses and damage, be applied?
EDIT: It had not occurred to me to treat the character being ridden as not a mount at all; in light of this, I added question number five.
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Given that you appear to have all the rules already, it seems that the problem is in interpretation. Given that, my view is :
1. Should the mounted PC be treated as a normal mount for a PC, losing his turn and becoming subordinate to the rider?
• Answer: No. The mount is the same level and equal in power
2. If 1 is "no", do they share initiative counts?
• Answer: No. They each roll initiative, and act on their turns. If one wants to move and the other doesn't, then the rider either dismounts (if the rider is the source of the move) or falls prone (chosing not to move with the mount). As a generous DM I'd allow an acrobatics check to avoid falling prone.
3. If 1 is "no", can the rider use his move action to make the mounted PC move, or force the mount to act by spending his own actions in any other way described by the Mounted Combat rules?
• As above; if one wants to move an the other doesn't, then it's either a dismount or a fall-prone for the rider, depending on who's turn it is.
• Per the comments; the mount should only be able to be move normally once per round - whichever commands the mount uses their move action to move, the other still has a move action but can't use it to move (unless it's the rider dismounting). Clearly other sources of movement (e.g. warlords) would still have the expected result - and the mounted combat rules cover that sort of thing already.
4. Can the mounted PC use the rider's base skill check bonuses as described in the Mounted Combat feat (e.g. a human with a pixie on his shoulder uses the pixie's Stealth modifier)?
• If the rider has mounted combat and the mount is willing, then I would go along with this.
5. How would feats like Holy Steed, which gives a rider's mount bonuses to defenses and damage, be applied?
• I would go along with this. As a DM, I'd much rather my players bunch up than spread out. A small bonus is worth the penalty.
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Basically, what you see in the various rules is an underlying assumption about Mounts and their interaction with their riders. From the perspective of a PC, 99% of the time, the mount is a creature which does not exhibit a high degree of intelligence or independent thinking. (AKA a horse or other large animal, or magically controlled being) The mount therefore must be instructed by the rider what to do and this uses up a move action, or a standard action depending on what the mount is being commanded to do.
For most enemies however the reverse is true. Both the goblin and the warg are using basic instincts in their fight, and the beasts are not so much being commanded, as being made use of. The monstrous beast that some death knight is riding, is acting on it's own accord, since it wants to go and eat the players.
However, when the PC is mounted on an independently intelligent mount, such as a friendly dragon, or another PC, then the Mount is given it's own standard actions and movements. The dragon is not so much taking orders from the rider, but rather working as a team with the rider.
As for spells which specify a mount, yes the other player is a mount. I would "hand wave" the magic working with the explanation that such magic works on the basis of a rider/mount relationship which both PCs have agreed to enter.
Perhaps a more general rule which you might apply is to ask yourself if the Mount in question is capable of "disobeying" the rider. If they are, then they have separate actions, if it can't, then they share their actions. (If there are counter examples to this, please let me know, I just thought about it this minute.)
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I would give the players two options:
A) The player being ridden is a dumb mount and is treated as such by the rules, sharing actions, initiative counts, skills and turns. (Obviously this would be a bit boring for one character)
B) The player being ridden is not a mount at all, it just happens that the pixie is using him or her as a mobile platform. (And so features like Holy Steed would not affect the player being ridden.)
In the latter case, I would ask that the pixie spend a move action to maintain position on the "not-a-mount" character (my thinking being that saying on a character doing things in combat requires some effort, and that this means there's no big tactical advantage gained by doing this)
I'd also probably rule that if the "not-a-mount" doesn't move (and maybe isn't meleeing) then no action is required to say aboard, and also (perhaps) that the action cost can be prevented by a successful acrobatics test (medium DC?), but that failing such a test results in being knocked prone, and no longer "mounted"
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I've done this. My Pixie, Sisyphus, used the grab rules to grab onto and ride along with the other PC. Normally grabbing would result in immobilization, however he was not grabbing and trying to hold the PC still, we just used the same basic actions to balance riding along.
Standard Action: This action to initiate a grab is for unwilling opponents. We dropped this part of the action to a minor action for willing participants, no different than...
Minor Action: Maintain a grab on a creature. This allowed Sisy to hold on and ride along on the other PC's movements, standing on her shoulder.
Free Action: To let go. This allowed him to actually let go of her anywhere, any time, being left behind wherever he wanted as she moved.
I leave the first action for clarity and also because on occasion he did the same thing to unwilling creatures. Often to NPCs and in battle with other creatures as well. This was loads of fun, and ends up being pretty nicely balanced. It works out like how you see Pixies moving in films. And it uses the existing rules in a way that at least somewhat makes sense.
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I guess the same question applies here as it did to my answer - how do you rule mount effecting powers and feats? (4 and 5 in the question) – Simon Withers Dec 12 '12 at 0:24
@SimonWithers They don't apply, she wasn't my mount, I was grabbing onto her and riding along. In fact, nothing stopped her from then taking a mount herself. – DampeS8N Dec 12 '12 at 1:20
I would decide based on this criteria: How are you splitting XP?
If the mount gets XP split with it, then it gets its own actions. If it doesn't receive XP, then it doesn't get its own actions.
Since a PC obviously gets a cut of the XP, then it gets its own actions.
Edit: I'm thinking of the rules for ranger companions here too. If you look at those rules, and at a lot of the summoned creature powers, you don't get extra actions (which you effectively would if they got their own turn), you have to use your action to command the creature. The exception is certain daily powers.
2) They don't share initiative. Unless they want to (one can delay until their initiatives match)
3) I'm thinking no. But I'm not sure how it works for the second paragraph you quoted. In any event, I would not allow both the rider and the mount to take double moves, the mount can't double it's speed by having a rider. But warlords do have powers that command their allies, so I might allow the mount to take a second standard action if the rider uses their standard action to command them.
In fact, that might be a good build if you're going to do this: Have the rider by a Warlord. Or else a beast companion ranger.
4) Yes, but I might rule "no" on certain cases. Basically on a case by case basis but there might be more "yes"'s then "no"'s.
5) Yes, but it might be over powered. If you decide no, let them retrain the feat.
Something else to consider. PCs to not have the Mount keyword, so I don't think any of this is legal according to the rules as written. (But I never let that stop me)
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2016-07-27 13:34:50
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http://physics.stackexchange.com/questions/79331/if-i-am-travelling-on-a-car-at-around-60-km-h-and-i-shine-a-light-does-that-me
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# If I am travelling on a car at around 60 km/h, and I shine a light, does that mean that the light is travelling faster than the speed of light?
The title says it all.
If I was on a bus at 60 km/h, and I started walking on the bus at a steady pace of 5 km/h, then I'd technically be moving at 65 km/h, right?
So my son posed me an interesting question today: since light travels as fast as anything can go, what if I shined light when moving in a car?
How should I answer his question?
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If you like this question you may also enjoy reading this Phys.SE post. – Qmechanic Oct 2 '13 at 12:18
This is the amazing thing about relatvivity that the velocity of light does not add the same way as small velocities do! Your son is on the path which led Einstein to develop relativity theory. – Slaviks Oct 2 '13 at 13:10
This should answer your question in less than five minutes: youtube.com/watch?v=vVKFBaaL4uM – shortstheory Oct 2 '13 at 17:47
Nobody mentioned Michelson-Morley experiment yet - amazing, as that is the empirical basis of everything below, and is simple enough for a child to understand the results (if not the techniques) of: en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment – Pieter Geerkens Oct 2 '13 at 22:11
How old is your son? – gerrit Oct 3 '13 at 8:46
If I was on a bus at 60 km/h, and I started walking on the bus at a steady pace of 5 km/h, then I'd technically be moving at 65 km/h, right?
Not exactly right. You would be correct if the Galilean transformation correctly described the relationship between moving frames of reference but, it doesn't.
Instead, the empirical evidence is that the Lorentz transformation must be used and, by that transformation, your speed with respect to the ground would be slightly less than 65 km/h. According to the Lorentz velocity addition formula, your speed with respect to the ground is given by:
$$\dfrac{60 + 5}{1 + \dfrac{60 \cdot 5}{c^2}} = \dfrac{65}{1 + 3.333 \cdot 10^{-15}} \text{km}/\text h \approx 64.9999999999998\ \text{km}/\text h$$
Sure, that's only very slightly less than 65 km/h but this is important to your main question because, when we calculate the speed of the light relative to the ground we get:
$$\dfrac{60 + c}{1 + \dfrac{60 \cdot c}{c^2}} = c$$
The speed of light, relative to the ground remains c!
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True, but usually we think of the fact that the speed of light is constant as being a law of physics, and the Lorentz transformation is just a consequence of that law. – 200_success Oct 2 '13 at 22:06
A correct answers, but hardly comprehensible to a 6 or 8 year old. – Pieter Geerkens Oct 2 '13 at 22:19
@PieterGeerkens No, but the idea of Alfred's answer is some thing children of that age love to hear - I think it would be a grand and appropriate thing to tell them that "it's actually a teeny bit less that 65kmh, and the faster you go, the more the rule that speeds add together in the way that children understand is broken". – WetSavannaAnimal aka Rod Vance Oct 2 '13 at 22:57
@WetSavannaAnimalakaRodVance: You honestly believe that bright children like being talked down to like that? Shame on you! – Pieter Geerkens Oct 3 '13 at 2:46
@PieterGeerkens In some cases yes: children are all very different. Education research shows that this age is the very age where a system's abstract properties are only just beginning to take root in their minds - see here - so algebraic explanations befuddle many children. It's important to know your student well and watch their reaction to you carefully - clearly geek-girl or boy is not going to forgive you for a too simplistic answer so you need to be ready to shift gears in a split second. But, given what we know about childrens ... – WetSavannaAnimal aka Rod Vance Oct 3 '13 at 4:18
You should tell your son that this very question was asked by, explored by, and eventually answered by the some of the brightest physicists of the 19th century. Eventually two scientists named Michelson and Morley came up with an experiment to measure this effect, and were amazed to discover that it didn't exist! Rather:
Light travelled at exactly the same speed in all directions, regardless of any velocity of its emitter.
This result astounded the physicists of the world, and led to the development of the Special Theory of Relativity by Einstein.
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+1 Great answer. I love how it touches on history of science, the empirical method, and the story of this discovery. Sure to lead to OP's son wanting to find out more. – Ergwun Oct 3 '13 at 2:34
They were actually checking the velocity of it's emitter respect to the ether, which they wanted to prove. Only to find that there was no ether. – Francisco Presencia Oct 3 '13 at 10:40
@FrankPresenciaFandos: Yes, by measuring the increased speed of light in the direction in which the Earth was moving relative to the aether. – Pieter Geerkens Oct 3 '13 at 11:10
As a matter of technicality, I believe that shining from a moving vehicle on earth will in fact be faster* than the light shining from a stationary vehicle on earth, however both forms of light would be moving slower than the speed of light (c), which is referencing the speed of light in a vacuum.
This is because the medium that the light is moving through (air) slows down the light by about 88km/s (according to Wikipedia).
That said, light in a vacuum emitted from a moving object should travel at the same speed as light in a vacuum emitted from a stationary object for the reasons outlined by all the other answers to this question.
* so long as the light is propagating in air that is traveling with the vehicle, such as the air between the headlight bulb and the headlight casing
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The first paragraph is wrong. Even in media where the speed of light is less than $c$, the speed of propagation is still independent of the speed of the source. – Chris White Oct 2 '13 at 23:22
@ChrisWhite, I'd be more than happy to update my answer if you'd elaborate on why that's the case. My knowledge of physics is superficial at best, so I'd much rather have more good information than lead people astray if I'm wrong. – zzzzBov Oct 3 '13 at 2:53
Waves - whether they are sound or light or water - are local phenomena, which means how the wave decides to move is determined by its immediate surroundings, not by whatever the source may have been doing far away. The speed of the source affects the frequency of the wave, but two waves of the same frequency in the same medium will move at the same speed, which for light in air is, as you said, a bit slower than $c$. – Chris White Oct 3 '13 at 3:04
@ChrisWhite, then in that case: if a two light waves were traveling in two separate media of the same substance in the same direction, and those media were moving in relation to each other in that direction, wouldn't the light waves be traveling at different speeds with regards to each other? – zzzzBov Oct 3 '13 at 4:34
If the air was moving with the car, the speed of light could be faster that the speed of light through still air. But if it is only the car that is moving and not the air, then both stationary and moving flashlights will put out light of the same speed, ie c minus 88 km/s. – Mark Lakata Oct 3 '13 at 5:14
Second postulate (invariance of c) of the special theory of relativity goes like this:
As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.
Or, that objects travelling at speed c in one reference frame will necessarily travel at speed c in all reference frames. This postulate is a subset of the postulates that underlie Maxwell's equations in the interpretation given to them in the context of special relativity. So basically, there exists an absolute constant 0 < c < (infinity) with the above property. So you can shine light while travelling at the speed of light and it will still go at c, not more, not less. Ref: wiki.
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One essential postulate of special relativity is that light moves at the same velocity in all reference frames. Somebody standing next to the moving bus will observe the light travelling just as fast somebody who is on the bus sees it. It might not be intuitive, but it is consistent both with experiment and the mathematical framework of the theory.
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You can start in answering his question by explaining the Doppler shift for acoustical waves.
The Doppler effect (or Doppler shift), named after the Austrian physicist Christian Doppler, who proposed it in 1842 in Prague, is the change in frequency of a wave (or other periodic event) for an observer moving relative to its source. It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an observer. The received frequency is higher (compared to the emitted frequency) during the approach, it is identical at the instant of passing by, and it is lower during the recession.
The relative changes in frequency can be explained as follows. When the source of the waves is moving toward the observer, each successive wave crest is emitted from a position closer to the observer than the previous wave. Therefore each wave takes slightly less time to reach the observer than the previous wave. Therefore the time between the arrival of successive wave crests at the observer is reduced, causing an increase in the frequency. While they are travelling, the distance between successive wave fronts is reduced; so the waves "bunch together". Conversely, if the source of waves is moving away from the observer, each wave is emitted from a position farther from the observer than the previous wave, so the arrival time between successive waves is increased, reducing the frequency. The distance between successive wave fronts is increased, so the waves "spread out".
Your son's expectation works on this intuitive background.
But light waves, in contrast to sound waves which need air to reach our ears, do not need a medium to reach our eyes. This is evident in that the light from stars reaches us through the vacuum of space where there is no medium. People used to hypothesize a medium for light, aether but experiments proved, as the other answers state correctly, that the velocity of light was constant, c, no matter what the motion of the emitter or absorber. Thus no, there will be no change in the velocity measured of the emitted light whether we are sitting on the ground, forward or backward or sideways, or in the car itself.
There is an effect though. Light that has been emitted by a source moving towards us does not change its velocity but it does change its frequency to a higher value; if it is receding, to a lower value. As the energy of the photons is given by E=h*nu it means that it gains an extra energy or loses some due to the relative motions of observer and emitter.
This has been very useful for astrophysics. For example that is how we know the relative motions of stars with respect to us. Light comes from spectra of atoms and we know them here in the lab. They are distinctive and identify whether we see light from iron or oxygen or hydrogen in a gas state. The change in frequency of the spectral lines will tell us of the motion of the star relative to us. There exist many applications of this method.
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2015-04-27 13:55:43
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https://www.physicsforums.com/threads/speed-usage-in-lattice-boltzmann-method.973701/
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# I Speed usage in Lattice Boltzmann Method
#### TimeRip496
One thing that confuses me is the physical speed and sound speed. The lattice sound speed cs=1/sqrt{3} corresponds to the physical sound speed for isothermal flow (sqt{RT}). Why isn't the physical speed (e.g. inlet speed up of lid cavity) converted and use accoringly?
$$c_p=\sqrt{RT}≈330m/s \rightarrow c_s=1/\sqrt{3}≈0.5774$$
Then why isn't the physical speed u map accordingly
e.g. physical speed of inlet(lid) is converted
$$u_p=0.2m/s \rightarrow u_l=0.3*0.5774/330=0.00052491$$
Instead the physical speed u_p is used together with the lattice speed of sound c_s in the flow equilibrium distribution.
$${\displaystyle f_{i}^{\text{eq}}=\omega _{i}\rho \left(1+{\frac {{\vec {e}}_{i}{\vec {u}}}{c_{s}^{2}}}+{\frac {({\vec {e}}_{i}{\vec {u}})^{2}}{2c_{s}^{4}}}-{\frac {{\vec {u}}^{2}}{2c_{s}^{2}}}\right).}$$
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"Speed usage in Lattice Boltzmann Method"
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2019-07-20 22:35:51
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http://mathhelpforum.com/algebra/135331-applications-linear-equations.html
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# Thread: Applications of linear equations
1. ## Applications of linear equations
The length of a rectangular plot of land is 2 times the width. If the perimeter is 2,500 feet, find the dimensions of the plot.
hi i was just wondering how i would go about solving this where should i begin, can someone explain? thanks.
2. Originally Posted by RECONDO89
The length of a rectangular plot of land is 2 times the width. If the perimeter is 2,500 feet, find the dimensions of the plot.
w = width; then length = 2w
So w + w + 2w + 2w = 2500
Finish it...
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2016-08-27 04:51:08
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https://math.stackexchange.com/questions/3250713/example-of-non-regular-surface
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# Example of non regular surface
I was reading definition of surface in differential geometry book which defined as follows
A subset $$S\subset \mathbb R^3$$ is regular surface if $$\forall p\in S$$ there is open set in S such that $$p\in V$$ and $$\exists \phi :U\to V\in S$$ where U is open set in $$\mathbb R^2$$ such that map is surjective , smooth and homeomorphism with image and also $$\forall q\in U, dX_q:\mathbb R^2\to \mathbb R^3$$ is injective
TO better understand cocept I wanted to know counterexample of regular surface.
Any Help will be appreciated
• Another easy example is the cone $z^2=x^2+y^2$ or the half-cone $z=\sqrt{x^2+y^2}$. – Ted Shifrin Jun 4 at 15:53
If $$f:\Bbb R\to\Bbb R^2$$ is defined by $$f(x)=(x^2,x^3)$$ then the subset $$S:=f(\Bbb R)\times\Bbb R$$ is a counterexample.
One reason for this is that any tangent vector at $$0_{\Bbb R^3}\in S$$ must have the first two coordinates equal to zero. This is because if $$c:\Bbb R\to \Bbb R^3$$ is a smooth curve with value in $$S$$ such that $$c(0)=0_{\Bbb R^3}$$ then $$c_1^\prime(0)=0$$ because $$c_1$$ is positive, and because $$c_2=(c_1)^{3/2}$$ you have $$c_2^\prime(0)=0$$. Hence $$v=c^\prime(0)=(0,0,c_3^\prime(0))$$
But for a surface the set of tangent vectors at a point is a vector space of dimension $$2$$.
Note that if we define $$\phi:\Bbb R^2\to \Bbb R^3$$ by $$\phi(x,y)=(f(x),y)$$ then $$\phi$$ is smooth and is a homeomorphism onto its image $$S$$, with inverse $$\phi^{-1}:(x,y,z)\mapsto (y^{1/3},z$$). So $$\phi$$ has all properties you are looking for, except the last one: $$d_0\phi$$ is not one-to-one. This shows you why this property is important, which is not clear at first in my opinion.
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2019-07-18 13:07:40
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https://scikit-learn.org/stable/modules/generated/sklearn.metrics.f1_score.html
|
# sklearn.metrics.f1_score¶
sklearn.metrics.f1_score(y_true, y_pred, labels=None, pos_label=1, average=’binary’, sample_weight=None)[source]
Compute the F1 score, also known as balanced F-score or F-measure
The F1 score can be interpreted as a weighted average of the precision and recall, where an F1 score reaches its best value at 1 and worst score at 0. The relative contribution of precision and recall to the F1 score are equal. The formula for the F1 score is:
F1 = 2 * (precision * recall) / (precision + recall)
In the multi-class and multi-label case, this is the average of the F1 score of each class with weighting depending on the average parameter.
Read more in the User Guide.
Parameters: y_true : 1d array-like, or label indicator array / sparse matrix Ground truth (correct) target values. y_pred : 1d array-like, or label indicator array / sparse matrix Estimated targets as returned by a classifier. labels : list, optional The set of labels to include when average != 'binary', and their order if average is None. Labels present in the data can be excluded, for example to calculate a multiclass average ignoring a majority negative class, while labels not present in the data will result in 0 components in a macro average. For multilabel targets, labels are column indices. By default, all labels in y_true and y_pred are used in sorted order. Changed in version 0.17: parameter labels improved for multiclass problem. pos_label : str or int, 1 by default The class to report if average='binary' and the data is binary. If the data are multiclass or multilabel, this will be ignored; setting labels=[pos_label] and average != 'binary' will report scores for that label only. average : string, [None, ‘binary’ (default), ‘micro’, ‘macro’, ‘samples’, ‘weighted’] This parameter is required for multiclass/multilabel targets. If None, the scores for each class are returned. Otherwise, this determines the type of averaging performed on the data: 'binary': Only report results for the class specified by pos_label. This is applicable only if targets (y_{true,pred}) are binary. 'micro': Calculate metrics globally by counting the total true positives, false negatives and false positives. 'macro': Calculate metrics for each label, and find their unweighted mean. This does not take label imbalance into account. 'weighted': Calculate metrics for each label, and find their average weighted by support (the number of true instances for each label). This alters ‘macro’ to account for label imbalance; it can result in an F-score that is not between precision and recall. 'samples': Calculate metrics for each instance, and find their average (only meaningful for multilabel classification where this differs from accuracy_score). sample_weight : array-like of shape = [n_samples], optional Sample weights. f1_score : float or array of float, shape = [n_unique_labels] F1 score of the positive class in binary classification or weighted average of the F1 scores of each class for the multiclass task.
Notes
When true positive + false positive == 0 or true positive + false negative == 0, f-score returns 0 and raises UndefinedMetricWarning.
References
Examples
>>> from sklearn.metrics import f1_score
>>> y_true = [0, 1, 2, 0, 1, 2]
>>> y_pred = [0, 2, 1, 0, 0, 1]
>>> f1_score(y_true, y_pred, average='macro')
0.26...
>>> f1_score(y_true, y_pred, average='micro')
0.33...
>>> f1_score(y_true, y_pred, average='weighted')
0.26...
>>> f1_score(y_true, y_pred, average=None)
array([0.8, 0. , 0. ])
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2019-08-25 01:23:02
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https://www.tutorialspoint.com/can-re-declaring-a-variable-destroy-the-value-of-that-variable-in-javascript
|
# Can re-declaring a variable destroy the value of that variable in JavaScript?
JavascriptObject Oriented ProgrammingFront End Technology
Re-declaring a variable will not destroy the value of a variable, until and unless it is assigned with some other new value.
If we look at the following example variables "x" and ''y'' were assigned with values 4 and 8 respectively, later on when those variables were reassigned, the old values were replaced with the new values and displayed as shown in the output.
## Example
Live Demo
<html>
<body>
<script>
var x = new Number(4);
var x = 7;
var y = 8;
var y = 10;
document.write(x);
document.write("</br>");
document.write(y);
</script>
</body>
</html>
## Output
7
10
In the following example, the variables were re-declared, but their values were not reassigned. Therefore those variables retained their original values.
## Example
Live Demo
<html>
<body>
<script>
var x = new Number(4);
var x;
var y = 8;
var y;
document.write(x);
document.write("</br>");
document.write(y);
</script>
</body>
</html>
## Output
4
8
Published on 02-Jul-2019 15:31:23
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2021-09-18 19:23:20
|
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https://illustrativemathematics.blog/category/im-curriculum/
|
By William McCallum
I can’t imagine what it must feel like right now to be a teacher facing the uncharted territory that is the coming school year. Will I be teaching 100% online, or have some face-to-face interaction with my students? Will I be teaching synchronously or asynchronously for most of the school year? How will I get to know my students and how will they engage in one another’s ideas? How will I get to know my students’ families? How can I give them manageable guidance to support students this year? Most of all, where can I get help with all these questions?
Continue reading “Coming Together Around Distance Learning”
By Kristin Gray, Director K–5 Curriculum and Professional Learning
and Kevin Liner, IM K–5 Professional Learning Lead
It is overwhelming to think about how teaching and learning will look in the fall. The uncertainty of the impact of students missing so many days of school, and the educational inequities that have been magnified as a result of the COVID-19 virus, leave us all with so many unknowns.
With so much uncertainty, we imagine there may be some knee-jerk reactions to unfinished learning this fall. There may be a temptation to frontload the school year with the prior grade-level content students may have missed or assess each student immediately on arrival back to school and then “fill in” the unfinished learning. As well-intentioned as these ideas may be, we can’t help but think about the impact they could have on students mentally, emotionally, and mathematically as they reenter school.
Continue reading “Looking to the Fall, Part 1: Welcoming and Supporting K–5 Students”
By IM 6–8 Math Team
This week, IM is launching a new resource to support students and teachers with distance learning. Each week we will publish an open-ended prompt or image that invites math conversation, and a series of 3–5 questions. The questions are designed so that all 6–8 students have an entry point for the first question, and all students will find something both familiar and challenging in each set.
Continue reading “IM Talking Math 6–8: Resources for Weekly Re-engagement”
It was easy to say yes!
By Crystal Magers
Last spring, I was approached by our Math Coordinator and asked about piloting a new math program. I knew my staff was ready for building-wide consistency and we were ready to try something new. I easily said yes!
My teachers were offered training over the summer and access to the resources to begin teaching this fall.
After just a few weeks of instruction, my staff began to voice concerns.
Continue reading “Shifting Practices: Helping Everyone—from Students to Administration—Find their Voice in the Math Classroom”
By Dionne Aminata
Before I joined the K–5 curriculum writing team at IM, I was a K–8 regional math content specialist for a public charter organization that largely consisted of Title I schools, or schools receiving federal funding to support a large concentration of students in poverty. Prior to that I had experienced the joys and challenges of serving communities like these as a teacher and math coach in South Central Los Angeles and Crown Heights Brooklyn.
By Jenna Laib and Kristin Gray
Take a moment to think about the value of each expression below.
$\frac{1}{4}\times \frac{1}{3}$
$\frac{1}{4}\times \frac{2}{3}$
$\frac{2}{4}\times \frac{2}{3}$
$\frac{3}{4}\times \frac{2}{3}$
What do you notice? How would you explain the things you notice?
If you are like us, or the students in Ms. Stark’s grade 5 classroom, you may have noticed many things. Things such as each expression has the same denominator, or the way in which the values increased as the problems progressed. When students notice these things, we often ask, ‘Why is that happening?” but it can be challenging to explain why beyond the procedure one followed.
Continue reading “Using Diagrams to Build and Extend Student Understanding”
By Catherine Castillo
Our district had seen a downhill trend in standardized test scores in mathematics. This forced us, as educators, to take an intentional look at our teaching practices.
The past few years have been an exciting time in math instruction. Research on brain plasticity and mindset have caused a shift in the idea of what it means to know and do mathematics.
Continue reading “The 5 Practices: Looking at Differentiation Through a New Lens”
“I’m not sure this is working. Only five of my students are participating and commenting each day. The rest sit there and look at me.”
By Tabitha Eutsler
This was my conversation with our math coordinator after my first few days of teaching IM K–5 MathTM with my third graders. Those five students were having great conversations. However, my other students just sat there wide-eyed, silent, and staring blankly at their papers. I felt lost. Was this the best for my students? Could we survive a whole year of math like this? I wanted my students to love math and have a deeper understanding of mathematical concepts. How would this get them there?
Continue reading “Building a Math Community with IM K–5 Math”
Does the perfect elementary math curriculum exist? Armed with a growth mindset and the Alpha IM K–5 curriculum, teachers in Ipswich Public Schools push their thinking to reach all mathematicians.
By Maureen D. O’Connell
I preach growth mindset daily. When my students say they can’t do something, they almost always add their own “…yet.” However, walking this walk as an elementary school teacher is another story. Creating, mastering, and modifying curricula to reach each and every student—in every content area—is a daunting expectation. We hold ourselves to near impossible standards.
Continue reading “Creating an Accessible Mathematical Community with IM K–5: the power of “yet” for students and adults”
|
2020-07-11 13:33:38
|
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http://mathhelpforum.com/discrete-math/140989-solved-relations-2.html
|
Math Help - [SOLVED] Relations
1. [QUOTE=dwsmith;499962][quote=Drexel28;499961]
Originally Posted by dwsmith
Well if it equals zero, the left is odd and the right is odd.
I am confused on how that proves it is an injection though.
We may conclude that $m=m'$ right? And so $f(m,n)=f(m',n')\implies 2^m(2n-1)=2^{m'}(2n-1)=2^m(2n'-1)\implies 2n-1=2n'-1$...soo
2. [quote=Drexel28;499963][quote=dwsmith;499962]
Originally Posted by Drexel28
We may conclude that $m=m'$ right? And so $f(m,n)=f(m',n')\implies 2^m(2n-1)=2^{m'}(2n-1)=2^m(2n'-1)\implies 2n-1=2n'-1$...soo
n=n'
3. [QUOTE=dwsmith;499964][quote=Drexel28;499963]
Originally Posted by dwsmith
n=n'
So!!!
$m=m'\text{ and }n=n'$ and so...
4. [quote=Drexel28;499965][quote=dwsmith;499964]
Originally Posted by Drexel28
So!!!
$m=m'\text{ and }n=n'$ and so...
f is injective if for all a and b in A, if f(a) = f(b), then a = b; that is, f(a) = f(b) implies a = b.
5. [QUOTE=dwsmith;499966][quote=Drexel28;499965]
Originally Posted by dwsmith
f is injective if for all a and b in A, if f(a) = f(b), then a = b; that is, f(a) = f(b) implies a = b.
And $m=m'\text{ and }n=n'\implies (m,n)=(m',n')$...Ta-da!
6. Prove that f is a surjection. This is actually a consequence of the Fundalmental Theorem of Arithmetic, Theorem 8.16.
I don't know what the fact that a number is prime or the product of prime numbers has to be with solving the surjection.
7. Originally Posted by dwsmith
Prove that f is a surjection. This is actually a consequence of the Fundalmental Theorem of Arithmetic, Theorem 8.16.
I don't know what the fact that a number is prime or the product of prime numbers has to be with solving the surjection.
You know that if $n\in\mathbb{N}$ that it may be represented as the product of primes $n=p_1^{\alpha_1}\cdots p_n^{\alpha_n}$, right? Well, except for $2$ every prime is odd. So, we could for the sake of convenience write every integer as $n=2^{\beta}\cdot p_1^{\alpha_1}\cdots p_n^{\alpha_n}$ where $\beta$ may be zero of course. But, as previously mentioned since every other prime is odd and the product of odd numbers is odd it follows that $p_1^{\alpha_1}\cdots p_n^{\alpha_n}\text{ is odd}\implies p_1^{\alpha_1}\cdots p_n^{\alpha_n}=2m+1$ for some $m\in\mathbb{N}$, right? So now what?
8. Originally Posted by Drexel28
You know that if $n\in\mathbb{N}$ that it may be represented as the product of primes $n=p_1^{\alpha_1}\cdots p_n^{\alpha_n}$, right? Well, except for $2$ every prime is odd. So, we could for the sake of convenience write every integer as $n=2^{\beta}\cdot p_1^{\alpha_1}\cdots p_n^{\alpha_n}$ where $\beta$ may be zero of course. But, as previously mentioned since every other prime is odd and the product of odd numbers is odd it follows that $p_1^{\alpha_1}\cdots p_n^{\alpha_n}\text{ is odd}\implies p_1^{\alpha_1}\cdots p_n^{\alpha_n}=2m+1$ for some $m\in\mathbb{N}$, right? So now what?
Well 2m+1 is both the right and left side minus the $2^{\beta}$
9. Originally Posted by dwsmith
Well 2m+1 is both the right and left side minus the $2^{\beta}$
So, every number may be written in the form $\pm2^{\beta}(2m+1)$..so
10. Originally Posted by Drexel28
So, every number may be written in the form $\pm2^{\beta}(2m+1)$..so
Well that product is even unless $\beta=0$
11. Originally Posted by dwsmith
Well that product is even unless $\beta=0$
Does it matter? Isn't that $f(\beta,m)$?
12. Originally Posted by Drexel28
Does it matter? Isn't that $f(\beta,m)$?
Yeah it is f(b,m)
13. Originally Posted by dwsmith
Yeah it is f(b,m)
And a surjection is what?
14. When f(x,y)=z
15. Originally Posted by dwsmith
When f(x,y)=z
And haven't we done that?
Page 2 of 3 First 123 Last
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2014-03-11 21:53:25
|
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|
https://math.stackexchange.com/questions/629156/showing-that-this-function-is-infinitely-differentiable
|
# Showing that this function is infinitely differentiable
Show, that the function
$$\mathcal E: \mathbb R \to \mathbb R: x \mapsto \begin{cases} \exp(-\frac{1}{x^2}), & \text{if x \neq 0}, \\ 0, & \text{otherwise}, \end{cases}$$
is infinitely differentiable and that $\frac{d^k\mathcal E}{dx}(0) = 0$ for all $k \in \mathbb N$.
We just introduced differentiation, so the solution should not contain very advanced techniques to solve this. We also got the tip it should be done by induction.
@fgp I found that it is $$f^{(n)}(x) = P_n \left(\frac1x\right)e^{-\frac1{x^2}}$$ where $P_n$ is a polynomial with integer coefficients.
I could also do the initial step for the induction:
For $n=1:$ $$f'(x) = \frac2{x^2}e^{-\frac1{x^2}} = P_1 \left(\frac1t\right) e^{-\frac1{x^2}}$$ where $P_1(x)=2x^2.$
I am stuck at the induction step: $f^{(n+1)}(x) =$.....
• Compute the first few derivatives and see if you can find a pattern. Then prove your hypothesis using induction... – fgp Jan 6 '14 at 15:45
• @fgp I answered you in my post (edited) – sj134 Jan 6 '14 at 16:11
Since $x \mapsto {1 \over x}$ is smooth for $x \neq 0$ and $x \mapsto e^x$ is smooth, it is clear that $\cal E$ is smooth for $x \neq 0$.
Suppose $x \ne 0$, then ${\cal E}^{(k)}$ has the form ${\cal E}^{(k)}(x) = e^{-{1 \over x^2}} p_k({1 \over x})$ for some polynomial $p_k$. This is clearly true for $k=0$, so suppose it is true for $k=0,...,n$. Then ${\cal E}^{(n)}(x) = e^{-{1 \over x^2}} p_n({1 \over x})$ and the chain rule gives ${\cal E}^{(n+1))}(x) = {\cal E}^{(1)}(x) p_n({1 \over x}) - {\cal E}^{(0)}(x) p_n'({1 \over x}) ({1 \over x^2}) = e^{-{1 \over x^2}} ({2 \over x^3}p_n({1 \over x})-p_n'({1 \over x}) ({1 \over x^2}) )$. If $p_{n+1}(y) = 2 y^3p_n(y)-p_n'(y) y^2$, then ${\cal E}^{(n+1))}(x) = e^{-{1 \over x^2}} p_{n+1}( {1 \over x} )$, and so the result is true for all $n$.
If $x \neq 0$, we have $e^{-{1 \over x^2}} = {1 \over {e^{1 \over x^2}}}$, and since $e^{1 \over x^2} \ge \sum_{k=0}^n {1 \over k!} {1 \over x^{2k}}$, we have $e^{-{1 \over x^2}} \le {x^{2n} \over \sum_{k=0}^n {1 \over k!} {x^{2(n-k)}}} \le {x^{2n} \over n!}$.
Suppose $p$ is a polynomial of degree $d$. Then for any $n$ we see that there is some constant $K$ such that $|e^{-{1 \over x^2}} p({1\over x})| \le K |x|^{2n-d}$ whenever $0 <|x| \le 1$. In particular, there is some $K$ such that $|e^{-{1 \over x^2}} p({1\over x})| \le K x^2$ for all $0 < |x| \le 1$.
We have ${\cal E}^{(0)}(x) \le x^2$ for all $x$, and so ${\cal E}$ is continuous at $x=0$. Since $|{\cal E}^{(0)}(x) - {\cal E}^{(0)}(0) -0| \le x^2$, we see that ${\cal E}^{(0)}$ is differentiable at $x=0$, and ${\cal E}^{(1)}(0) = 0$.
Now suppose ${\cal E}^{(k)}$ is differentiable at $x=0$ and ${\cal E}^{(k)}(0) = 0$ for $k=0,...,n$. Then $|{\cal E}^{(n)}(x) - {\cal E}^{(n)}(0) -0| \le K x^2$ for some $K$ and $|x| \le 1$. Hence ${\cal E}^{(n)}$ is differentiable at $x=0$, and ${\cal E}^{(n+1)}(0) = 0$.
• In the expression $e^{-{1 \over x^2}} p_n({1 \over x})$ is $({1 \over x})$ a factor or is it part of the expression for the polynomial? – Sam Nov 24 '14 at 20:08
• It is a parameter to the (polynomial) function $p_n$. Unfortunate ambiguity (but I would have written $p_n(x) {1 \over x}$ otherwise, but that is not always common practice). – copper.hat Nov 24 '14 at 20:10
• Thanks. When you computed ${\cal E }^{(n+1)}(x)$, you used the product rule as well as the chain rule, correct? In the first term there, you have not taken the derivative of either $e^{-1 \over x^2}$ or $p_n({1 \over x })$ and in the second term it appears you took the derivative of both. Is this a typo or am I missing something? – Sam Nov 24 '14 at 20:23
• @user151852: Thanks for catching that. I had transposed the $0,1$ derivatives of $\cal E$ and this mistake rippled a little. I have fixed it, I believe. – copper.hat Nov 24 '14 at 20:34
(This is more a hint than a full answer) Note that
\begin{align*} \frac{d\mathcal{E}}{dx}(0) &= \lim \limits_{\substack{x \to 0 \\ x \neq 0}} \frac{\mathcal{E}(x)-\mathcal{E}(0)}{x} \\ &= \lim \limits_{\substack{x \to 0 \\ x \neq 0}} \frac{1}{x} \exp \left( -\frac{1}{x^2} \right) \\[2mm] &= 0 \end{align*}
So, $\mathcal{E}'(0)=0$. As well, we have :
\begin{align*} \frac{d^{2}\mathcal{E}}{dx^{2}}(0) &= \lim \limits_{\substack{x \to 0 \\ x \neq 0}} \frac{\mathcal{E}'(x)-\mathcal{E}'(0)}{x} \\ &= \lim \limits_{\substack{x \to 0 \\ x \neq 0}} \frac{2}{x^{2}} \exp \left( -\frac{1}{x^2} \right) \\[2mm] &= 0 \end{align*}
So, $\mathcal{E}''(0)=0$. I think you got the idea : you can prove (by induction, for example) that, for all $n \in \mathbb{N}^{\ast}$ there exists a polynomial $P_{n}$ such that
\begin{align*} \frac{d^{n}\mathcal{E}}{dx^{n}}(0) &= \lim \limits_{\substack{x \to 0 \\ x \neq 0}} \frac{\mathcal{E}^{(n-1)}(x)-\mathcal{E}^{(n-1)}(0)}{x} \\ &= \lim \limits_{\substack{x \to 0 \\ x \neq 0}} P_{n} \left( \frac{1}{x} \right) \exp \left( -\frac{1}{x^2} \right) \\[2mm] &= 0 \end{align*}
Let $S=\left\{x\mapsto P\left(\frac{1}{x}\right)\mathcal E(x)\mid P\in \Bbb R\left[X\right]\right\}$
Can you prove that $f\in S \implies f'\in S$?
What can you say about $\lim\limits_{x\to 0}f(x)$ for $f \in S$?
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2020-02-18 07:10:13
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https://electronics.stackexchange.com/questions/316344/routing-jtag-signals-through-an-fpga
|
routing jtag signals through an FPGA
I am routing a JTAG signals through an Altera FPGA to a TI MCU. The signals are TMS, TCK, TDO, TDI, nSRST.
I can also connect the JTAG directly to to the MCU as there is a 10 pin jtag header exposed on it.
If I route the signals TCK, TDI, TDO, nSRST through the FPGA and connect TMS directly to the TMS to the 10 pin jtag header on the MCU then it will flash the device.
However, if I route TMS through the FPGA the flashing fails.
I have analysed the TMS signals both through the FPGA and directly to the 10 pin header on the MCU and both look equally nosiy.
Would anyone know any properties of the FPGA that would contribute to this error.
Also could anybody suggestion a solution. Would latching the signal work and if so could anybody explain this in further detail.
• Possibly related to propagation delays through the FPGA. Might be worth seeing if you can drop the clock frequency of the JTAG comms. – Tom Carpenter Jul 9 '17 at 13:34
The JTAG (boundary scan) state machine moves through states based on the level of TMS at the rising edge of TCK.
Routing TMS through the FPGA is adding (some unknown) delay to TMS so it not surprising the chain fails to operate as expected.
There are devices designed to operate multiple chains, such as the devices from Firecron (if that was what you were attempting to do).
In normal circumstances (boundary scan controllers have come a long way), I just connect every TMS together (TCK gets the same treatment). The only signals passed through are TDI / TDO (not through your fabric - let the implemented boundary scan do it).
One interesting note: if you have a device without TRST, pull TCK low by default to avoid a potential power up race consition.
Some CPUs (especially ARM CPUs like the STM32) allow two kinds of "JTAG" transfer:
• Real JTAG with four wires (five wires including nSRST)
• Some "pseudo"-JTAG called "SWD" using only two wires: TCK and TMS
If SWD is used the TMS line is bi-directional!
If flashing works with only TCK and TMS connected (and TDI and TDO disconnected) you know that the flash tool uses SWD.
On the other hand if flashing does not work with only two pins connected you cannot know if the flash tool uses only JTAG or if it uses both JTAG and SWD.
• @Peter Smith and Martin Rosenau, thank you for your helpful replies. The MCUs I am trying to flash are TI CC2650s. I have multiple MCU connected to an FPGA so that I can mux the signals to the appropriate MCU. I am using an XDS100v3 debugger with TIs uniflash linux based flashing tool. I am pretty sure that the tool relies on the 5 wire version of JTAG though. I believe the issue is related to a very noisy TMS signal. – artic sol Jul 10 '17 at 11:48
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2019-12-07 07:37:51
|
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|
https://proofwiki.org/wiki/Euclid%27s_Theorem/Corollary_1/Proof_1
|
# Euclid's Theorem/Corollary 1/Proof 1
## Corollary to Euclid's Theorem
There are infinitely many prime numbers.
## Proof
Assume that there are only finitely many prime numbers, and that there is a grand total of $n$ primes.
Then it is possible to define the set of all primes:
$\mathbb P = \set {p_1, p_2, \ldots, p_n}$
From Euclid's Theorem, however, we can always create a prime which is not in $\mathbb P$.
So we can never create a finite list of all the primes, because we can guarantee to construct a number which has prime factors that are not in this list.
Thus, there are infinitely many prime numbers.
$\blacksquare$
## Source of Name
This entry was named for Euclid.
## Historical Note
This Corollary 1 to Euclid's Theorem is itself often referred to as Euclid's Theorem, although Euclid himself never raised the concept of infinity.
Strictly speaking, Euclid's Theorem merely states that from any finite set of prime numbers, it is always possible to demonstrate that there exists another not in that list.
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2021-07-24 14:30:37
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|
https://www.simscale.com/docs/validation-cases/circular-shaft-under-torque/
|
Required field
Required field
Required field
# Circular Shaft Under Torque
## Overview
The aim of this test case is to validate the following functions:
• torque
• remote force
The simulation results of SimScale were compared to the numerical results presented in [Roark]. The meshes used in (A) and (B) were created with the parametrized-tetrahedralization-tool on the SimScale platform. The meshes used in (C) and (D) were locally created with Salome.
Import validation project into workspace
## Geometry
The shaft has a radius r
$r$
= 0.1 m and a length of l
$l$
= 0.5 m.
## Analysis type and Domain
Tool Type : CalculiX/Code_Aster
Analysis Type : Static
Mesh and Element types :
Case Mesh type Number of nodes Element type
(A) linear tetrahedral 12940 3D isoparametric
(B) quadratic tetrahedral 94919 3D isoparametric
(C) linear hexahedral 10325 3D isoparametric
(D) quadratic hexahedral 40935 3D isoparametric
## Simulation Setup
Material:
• isotropic: E = 208 GPa, ν
$\nu$
= 0.3, G = 80 GPa
Constraints:
• Face A is fixed
• Torque T
$T$
of 50000 N/m on face B
## Reference Solution
J=12πr4=1.57104m4(1)
$\begin{array}{}\text{(1)}& J=\frac{1}{2}\pi {r}^{4}=1.57\cdot {10}^{-4}{m}^{4}\end{array}$
τmax=TrJ=31.847N/mm2(2)
$\begin{array}{}\text{(2)}& {\tau }_{max}=\frac{Tr}{J}=31.847N/m{m}^{2}\end{array}$
$\begin{array}{}\text{(3)}& \theta =\frac{{\tau }_{max}l}{Gr}=1.9904\cdot {10}^{-4}rad\end{array}$
## Results
Important
• The analytical solution assumes an undeformable surface on the face, wich is subject to the torque bc. This can be modelled in Code_Aster with the option ‘undeformable’ in the remote displacement bc.
• CalculiX has no such option and therefore the stresses in the entitites, which are assigned to the torque bc, are unphysical in CalculiX. So the stresses were computed at a point on the edge of face A.
Comparison of the maximum shear stress τmax
${\tau }_{max}$
and the angle of twist θ
$\theta$
obtained with SimScale and the results derived from [Roark].
Comparison of maximum shear stress τmax
${\tau }_{max}$
Case Tool Type [Roark] SimScale Error
(A) CalculiX 31.847 30.212 5.13%
(A) Code_Aster 31.847 30.212 5.13%
(B) CalculiX 31.847 31.842 0.02%
(B) Code_Aster 31.847 31.838 0.03%
(C) Code_Aster 31.847 32.066 -0.69%
(D) Code_Aster 31.847 31.879 -0.10%
Comparison of the angle of twist θ
$\theta$
Case Tool Type [Roark] SimScale Error
(A) CalculiX 0.0019904 0.001953 1.88%
(A) Code_Aster 0.0019904 0.001969 1.08%
(B) CalculiX 0.0019904 0.001969 1.08%
(B) Code_Aster 0.0019904 0.001989 0.07%
(C) Code_Aster 0.0019904 0.002004 -0.68%
(D) Code_Aster 0.0019904 0.00199 0.02%
## References
[Roark] (1, 2, 3, 4) (2011)”Roark’s Formulas For Stress And Strain, Eighth Edition”, W. C. Young, R. G. Budynas, A. M. Sadegh
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2020-07-03 17:24:53
|
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|
http://mathandsignals.blogspot.com/2012/05/find-digit.html
|
## Thursday, 10 May 2012
### Find the Digit
This is a Pretty Logical Question..
1216451*0408832000 is 19!, ! Denotes Factorial. Find the Digit in the place of *
|
2019-02-22 04:32:49
|
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|
https://blog.uvm.edu/ctl-bb/
|
## Group Assignment Rubric Score can not be changed
Problem: After grades are entered into a group assignment rubric, the rubric scores for that attempt cannot be changed.
Solution: This has been identified by Bb as a known bug. While there is currently no fix available, the workaround for this is to override the score for the assignment, and explain the rubric/grading changes in the feedback for the assignment.
## Images fail to load in test questions once the test availability window is passed
Problem: Students taking a test in which the availability window has been reached may encounter broken images in questions.
Solution: This is a known bug in Bb, for which we do not have a resolution/fix. Workarounds include removing the availability end date, or instructing students to start the test with enough time to finish before the availability window ends.
## Test Options form displays improperly when setting test to show one question at a time
Problem: On some browsers/resolutions, when setting a test to display questions “One at a Time”, the form is jumbled in a way that looks like you are also prohibiting backtracking.
Solution: Use the circle-question-mark icon at the top of the page to toggle help messages to “On” for the page:
This will allow the checkbox to be displayed in the proper position.
## Some PDF’s only load first page or two when viewed in Chrome.
Problem: When viewing a multi-page PDF in Chrome, the PDF may not display more than one or two pages.
Solution: There are two workarounds available,
1. Right-click on the link to the PDF and save it to your Desktop, then open it there on your computer.
2. Or view the PDF with a different browser, such as Firefox or Safari.
Problem: Earlier versions of Respondus may fail to update/register after being installed.
Solution: Logging into the UVM Software Archive, downloading the Respondus installer listed there, and entering the registration information provided on that page will allow the registration and/or updates to proceed.
## Problems adding embedded content via iFrame for some URLs
Problem: Due to security concerns, instructors may encounter difficulty adding embedded content from some sites via iframe. Pasting iframe embed code into the HTML box in the content editor may not save as expected for some sites.
Solution: If you are attempting to embed material from an external site and encountering trouble doing so, please contact blackboard@uvm.edu and include the steps you are taking to embed, as well as the embed code and url you are using.
Embedding files from the content collection: As a side effect of the security changes, the process for embedding files has changed. To embed one of these files, the iframe src must use a relative “permanent” link for the file:
1. Find the file in the content collection, click on the grey chevron to the right of the file name, then select “360˚ View”.
2. Copy only the relative path of the Permanent URL (i.e. /bbcswebdav/xid-43414430_1 ). Don’t include https://bb.uvm.edu.
3. When embedding the iFrame (i.e. using the HTML button on an item in a course), use that relative path as the src for the iframe, i.e. <iframe src="/bbcswebdav/xid-43414430_1"
## Items not able to be re-ordered
Problem: Items in a course will appear to be re-ordered/re-arranged, but then revert to the original order when re-visiting the page. This has been reported to affect assignments, test questions, content items, and other course materials.
Solution: While we are waiting on a proper fix for this, the workaround below has been reported as successfully fixing the problem:
1. Move one item in the list to another folder, content area, etc.
2. Move that item back to the folder with the ordering problem.
## Students unable to access My Grades, see ‘displayComment’ error
Problem: Students are unable to view the My Grades area when at least one of their grades meets all of the following criteria:
1. Grade/attempt has been marked as “Exempt”.
2. There is no score entered for the grade.
3. A comment/feedback entry has been attached to the grade.
Solution: The simplest workaround for this issue is for the instructor to set a non-null value for all exempt cells (i.e. “0.00”, not “–“). Removing the exemption and/or the comments will also prevent the error from appearing.
## Assignment Inline Grading Issues (RESOLVED)
Update: The vendor has rolled back the changes, and the Assignment tool should be functioning as expected.
Problem: The vendor that provides the inline grading, commenting, and annotation service to Bb has pushed out changes that affect how inline grading displays and functions. This update has included some bugs, which Bb is working with the vendor to address.
Solution: This document outlines workarounds for this issue. We are waiting on further word from Bb regarding a fix for these problems.
## Error when creating matching questions with re-usable
Problem: When creating a matching question where one or more answers are re-using other answers in the matching set, an error regarding “Index” and “Size” is presented.
There is unfortunately no fix for this issue, and Blackboard has provided no timeline for a resolution. We are currently looking for workarounds to address this issue in the meantime.
## Course Menu set to White Text On White Background
Problem: Occasionally, a course’s menu colors may be set to show white text on a white background. This may happen after a course is imported from a previous term.
Solution: Course menu colors can be changed by going to Control Panel -> Customization -> Teaching Style, and setting the background color and text color, then clicking the Submit button.
## Notifications and other information are accessible to students when inside unavailable folder
Problem: Items that are available, but which are added to a parent folder which is unavailable, generate notifications.
Solution: This appears to be a known bug, with a targeted resolution in a future release. In the meantime, items that should not be visible to students should be set to be unavailable. Setting a date restriction on an item will allow it to be managed in bulk using the Date Management Tool.
Problem: Download Results in Grade Center incorrectly saves the file in different ways depending on if Chrome or IE is used. The data is correct and is comma or tab separated appropriately as selected but the file either has no extension, when it should have .csv (comma separated) or .xls (tab separated) and/or the filename is not in the correct format.
Workaround: This is a known issue that is fixed in a future release. In the meantime, renaming the file after download to add the proper extension will let you open it in your spreadsheet program.
## Grey background appears behind some content in FireFox
Problem: When viewing materials in FireFox, the white background sometimes does not extend far enough down. This causes text to be presented on a grey background and thus be less readable.
Solution: As a workaround, using Chrome or another supported browser when viewing a course appears to address this issue. In the meantime, we are attempting to document replication steps which can be provided to the vendor for a resolution.
## Problems re-ordering course menu items in Firefox on Windows
Problem: A bug has been identified in Firefox on Windows that prevents re-ordering the course menu in some cases.
Solution: This is a known defect which is fixed in a future version. If you encounter this problem, the workaround in the meantime is to use a different browser, such as Chrome.
## Create Random Block shows a grey box over questions
Problem: When attempting to add a random block to a test, a grey box covers the interface, preventing questions from being selected.
Solution: Resizing the window to be wider removes the grey box and allows the form to be used as expected. We are working to find a permanent solution to this issue.
Problem: We are seeing reports of failed uploads and errors when uploading files using Safari 11.1. Blackboard has identified this as a bug and we are waiting for a resolution.
Solution: Other browsers such as Chrome and Firefox appear to be functioning as expected. Please use these browsers if you encounter issues uploading files in Safari.
Problem: When grading an assignment, some assignment files present an error after clicking the download file button.
Solution: We are working to find out the cause of this issue. In the meantime if you experience this problem, the workaround is to download all assignment files at once. This can be done by going into the grade center, clicking on the button to the right of the column name, and selecting Assignment file Download.
## Logged in users attempting to view course files as Guests see errors
Problem: Guest access users who are logged into Bb cannot see course files in a course with guest access turned on.
Solution: Guests should visit the course in a private/incognito browser window, or use a browser which is not currently logged into Bb.
## Dropped/withdrawn enrollments not being removed from Blackboard roster
Problem: Students who drop a course in Banner/myUVM are not being removed from the corresponding Bb course space. This is an issue with the Blackboard process that handles the data import/update feed. Please note that all enrollments in Banner/myUVM are accurate – this is a Bb only issue.
Solution: ETS is working to implement a solution to this, and we hope to have a fix soon. In the meantime, students can hide courses in their course list by managing their course list in Bb; they can also edit their notifications for their Bb courses by clicking on their name in the upper right of the screen, then settings, then Edit Notification Settings. Instructors can use the following workarounds to help alleviate the symptoms of this issue:
Update: While we continue to work with Bb to address this problem, we’ve made a temporary change to allow instructors to exclude students from courses. This can be done by:
1. Go to Control Panel -> Users and Groups -> Users
2. Searching for individual users to deny access to, or selecting “not blank” to list all users .
3. Click on the grey button next to a user’s id, and select “Change User’s Availability” for that user.
4. Setting availability to “No” will deny them access to the course.
Hide rows in the grade center, by clicking on Manage -> Row Visibility.
Mark current users in the grade center as actively enrolled (not dropped) in the course.
1. Create an excel sheet with two columns, one labeled “Username”, and one labeled “Enrolled” (or some other designation that communicates an active enrollment)
3. Copy the Netid’s from the myUVM/Banner roster, and paste them beneath the “Username” column that you created in step 1.
4. In the first row of that second column, type a “Y” (or something similarly appropriate), to denote an active enrollment
5. Fill down in the second column. You should now have the list of usernames from myuvm, with a Y indicator next to each of them.
7. At this point you can create a smart view based on that column. Setting that column to default will give you a default list of active enrollments when you enter the grade center.
To limit group membership to only enrolled students, instructors can do the following:
1. Create groups as they normally would, however without adding students.
2. Export the groups into an excel file, by going to Groups -> Export -> Groups only (include header row)
3. Open this file in excel and delete all other columns besides the Croup Code.
4. Create a new column next to the Group Code column, and label it User Name
5. Download the roster from myUVM, open it in excel, copy the NetID column, and paste it under the User Name column you just
6. created in your groups file.
7. Manage group membership here by putting the appropriate group codes next to each user name (i.e. selecting some empty columns
8. and doing Fill Down)
9. Save this file as .csv with an appropriate name, then go to Groups -> Import and import using the button under the “import group members section”.
## Assignments do not display in-line within the browser in Safari on OSX
Problem: Assignments do not display in-line within the browser in Safari on OSX.
Solution: Use another browser, or modify Safari’s privacy settings to allow the assignments to display in the browser. Depending on your operating system version (to check this, click on the Apple in the upper left corner of your screen, then About this Mac), the steps for this are:
• If you are running Sierra, go into Safari > Preferences > Privacy > Cookies and Website Data > set to “Always Allow”
• If you are running High Sierra, go to Safari > Preferences > Privacy > unselect Prevent cross-site tracking
## Multiple choice questions with blank/empty answers won’t submit
Problem: Creating a multiple choice question in the test/survey tool, leaving at least one answer empty, then clicking on the submit button will not save or submit the form. The submit button fails silently and the form is not submitted.
Solution: Clicking the “remove” button next to any blank/unused answers will allow the form to submit normally. There is no targeted release for this issue at this time.
Problem: When going into the grade center and viewing or downloading an assignment attempt, users see an error message similar to “No default constructor found.” We’ve seen this occurring when attempting to access group attempts as well as individual submissions. Blackboard has identified this as a bug, and we are currently working to clarify patch availability.
Solution: Please contact blackboard@uvm.edu as soon as possible if you see this error. In the meantime, the workaround for this issue is to click the chevron next to the Column Title and download all assignment attempts.
## Resolved: Guests seeing error when attempting to access announcements
Problem: Users entering a course with guest access see an error when they attempt to view a course announcements page.
There is currently no workaround for this issue. We are working with Bb to find a solution to this as soon as possible.
## Resolved: Grade Center columns using a running total may display blank or incorrect values
As part of a recent security update, the below bug was introduced in the grade center.
Problem: Weighted Total columns using a running total may display blank or incorrect values. This is reported to only happen when the calculation results in an infinitely repeating decimal. For example: if the total points for your weighted column was 3 and the only value being included in the running total was a 1, then the resulting “0.33333333…” value would not display correctly in the Weighted Total column.
Solution: The workaround for this is as follows.
1. Make sure the weighting adds up to 100.
2. Set Running Total to “No”.
3. If you must use running totals, be sure to turn it off after all grades are in, as this will ensure the correct grades are displayed to you and the students.
This bug would only have been introduced as of 11/21/17. A future update – to be installed 12/20/17 – will resolve this issue.
## Typed availability times do not save when editing items in FF/Safari
Problem: Typing a time into the availability box on an item will result in that time from not being saved. This is occurring in FF and Safari.
Solution: This bug has been identified by Bb as a defect; a fix is included in the release which will be installed in December. In the meantime, use Chrome if you need to edit times by typing in specific times.
## Resolved: Grade center only lists one assignment attempt
Problem: There is only one attempt listed in the grade center for an assignment with multiple submissions.
Solution: To view all attempts in the Grade Center,
1. Go to Control Panel > Full Grade Center
2. In the upper right corner select filters.
3. On the filters menu check the box next to “Show attempts that don’t contribute to user’s grade”
## Dropped/Withdrawn students listed as “Managers” in the Discussion Board
Problem: Instructors visiting the “Manage” area of their discussion board may see students who are listed as “Managers,” yet are not in the course. This can happen when a student enrolls in a course (however briefly), and subsequently drops or withdraws. Despite their being listed as a manager in the discussion board, they do not have any access to the course, and they are not listed as a manager if they re-add the course. There is currently no workaround or fix available for this issue.
## Logging in with another browser tab/window during a test breaks save answer
Problem: While taking an exam, opening a new window/tab in the same browser and creating a new session will break the “Save Answer” button. For example, if a student begins a test, then opens a new tab and goes to myUVM and clicks on the “Blackboard” link, the save answer button will show an error.
Solution: Blackboard has a bug fix which we are currently testing and are working towards deploying. In the meantime, the following workaround will improve the testing experience. During an exam, if multiple tabs/windows in Bb are needed (i.e. for an “open book” exam), open a different browser (i.e. FireFox to navigate your course while the test remains open in the original browser (i.e. Chrome). If a new session has been created in the same browser, and save answer is not working, deleting the cookies and cache then logging back in will allow for resuming the exam.
## Course copy strips attributes from links in content items
Problem: After copying a course, links entered into content items have had some attributes removed. This includes the “target” attribute, which is often used to make a link open in a new window/tab.
We have opened a case with Blackboard in the hopes there is a fix for this issue. In the meantime, there is no solution to this problem other than editing the links after the course copy. For longer passages, manually copying and pasting the text/HTML from the old course item into the editor in the new course may help make this process slightly less time intensive.
## Partial Credit Box does not Display in all Answers in some Questions Types
Problem: When setting a question to allow partial credit, some answers do not display a box to specify credit values.
There are currently no patches available for this bug. While we wait for a fix from Blackboard, the only workaround in this case is to offer fewer answers (i.e. remove any questions without a partial credit box).
## Dragging items does not reorder them
Problem: We are seeing several reports of problems with the drag-and-drop move feature in Bb.
Solution: We are working to replicate this issue and get a fix from Bb. In the meantime, the workaround is to use the arrow buttons at the top of the item list:
## Resolved: Respondus Update Failure
Problem: When updating Respondus on Windows 10, an error appears and the update fails to proceed: “This app has been blocked for your protection. RPUPDATE. EXE. An administator has blocked you from running this app.”
## Grade Center Report Tool Does Not Display User Names in Order
Problem: When trying to generate a report in Grade Center, the users in the ‘Selected User’ section are not ordered.
Solution: There is no workaround for this issue. Bb has identified this as a bug, but we have no indication of a fix in the foreseeable future.
## Resolved: Cursor and screen jumps to the last question on test
Problem: When composing and answering items on a test that is presented with questions all at once, the cursor and screen jump to the last question, on the bottom of the screen, while the user is only partially complete with the test.
Solution: This is an identified bug in Bb. While a patch is being evaluated, instructors can prevent this from happening to students by delivering a test one question at a time. Students can alleviate some of the frustration by composing essay questions offline in a word or text processor and pasting into the text boxes of essay questions.
## Resolved: Last Access, Retention Center, Performance Dashboard inaccuracies
This issue has been resolved, following a patch from Bb. Please contact blackboard@uvm.edu if you continue to have trouble with this feature.
Problem: The “Last Access” column entries for some students are appearing blank or inaccurate in the Grade Center, Performance Dashboard, and Retention Center however those students have recorded activity in the course.
Solution: We are working with Bb to find a solution to this problem as soon as possible, and have been told a patch is in development. In the meantime, instructors can check student activity by using the course reports feature. This can be done by going to Control Panel -> Evaluation -> Course Reports, clicking on “Overall Summary of User Activity”, and selecting all students and a date range. There is a section of that which that shows a student’s access per day.
## Group member lists not sorting on group set editing page
Problem: When editing a group set, the list of students will not sort by last name (or other criteria) as expected.
Solution: This is a defect, but Blackboard has not indicated a future release in which it may be fixed. In the meantime, a workaround is to open the Grade Center in a separate tab or window, select the students in the Group Set using a smart view, and sort them as needed. In the other browser window, the group set can be managed normally.
Problem: Instructors attempting to access a discussion forum are prevented from entering, and an “Access Denied” error appears. This happens when a date restriction is set on the forum, with the goal of making it unavailable to students.
Solution: Blackboard has identified this as a bug, and we are waiting on a solution. There is no workaround for this, other than making the forum available temporarily while working on it.
## Resolved: Publisher tools unavailable, displaying error
Problem: Publisher tools, such as Macmillan and WileyPlus, are currently experiencing issues which are causing them to be inaccessible. We are working with Blackboard to find a solution to this issue, and hope to have this resolved shortly.
## Emails sent from Bb display no body when attachment is included.
Problem: Emails sent from Bb with an attachment are not displaying properly in some mail clients, such as Apple Mail, and iOS Mail.
Workaround: While a solution is being pursued from Blackboard, clients such as Outlook Web Access appear to display the message body properly when an attachment is included.
## Distorted equation images
Problem: Mathematical equation images generated in the text editor are occasionally appearing corrupted or distorted. These images can be generated using the built-in equation editor, or by using “MathJax” notation and surrounding LaTex text with double dollar signs. This may occur during equation placement in tests, discussions, or elsewhere in a course.
Solution: While a fix is being investigated, equation images can be generated using external tools, then uploaded and embedded into the text editor.
## Grade center shows five decimal points
Problem: The grade center in Bb shows five decimal points when displaying a percentage on some calculated columns.
Solution: There is no workaround or fix currently provided by Bb. Note that students will only see two decimal points in their “My Grades” area.
## Resolved: Blogs and Journals not accessible, display “Resource not found” error
Problem: Blogs and Journals display a “Resource not found” error when students or instructors attempt to enter them.
Solution: This has been resolved. Please report any issues to blackboard@uvm.edu.
Problem: Students attempting to reply to an instructor email (sent from within a course) will be presented with “no-reply” in the “To” field of the email.
Solution: This is a bug, which we are waiting on a fix from Blackboard for. In the meantime, students will need to enter in the instructor’s email address manually when replying to emails.
## Students encounter problems saving answers on tests with large numbers of questions
Problem: On tests with large numbers of questions (i.e. more than 50), students may encounter problems saving or submitting the assessment.
Solution: Instructors delivering assessments with over 50 questions should consider delivering the questions one at a time, which can be done by editing the test options and choosing “One at a time” in the Test Presentation section.
## Resolved: Error appears when attempting to add a question to an assessment.
Problem: When attempting to add a question to an assessment, an error is presented saying “Unable to parse provided PkId string.”
Solution: This has been resolved. Users experiencing this issue are encouraged to contact blackboard@uvm.edu. A workaround to this issue can be found by editing the test, then clicking on “Question Settings” to the upper right, then unchecking the “Question Metadata” box on section 3 of that page will prevent the error.
## YouTube Mashups and Video Everywhere not embedding/displaying properly.
Problem: Attempting to embed YouTube videos using the Mashups tool and/or the Video Everywhere tool does not work.
Solution: This is an identified bug that Blackboard has identified as fixed in the next release, which is expected to be applied in December. The current workarounds are to embed the video by clicking the html button in the text editor and pasting the embed code; post a link to the video along with the description; or to use UVM Streaming Media tool.
## Login problems with certain versions of Firefox
Problem: Users have reported problems logging into Blackboard when using some versions of Firefox.
Problem: The red number indicator next to a user’s name suggests updates are available, but upon visiting that area, no updates are presented.
## Resolved: Delay in enrollments/data updates to Bb
[This has been resolved.] There is currently a delay in the data feed that updates Blackboard with enrollment information from the Registrar’s system. Currently the feed is running once daily Monday through Friday, rather than every two hours. ETS is working on a resolution to this issue. In the meantime, updates to Blackboard enrollments and TA additions can be expected on the morning of business days.
## Links to courses with guest access enabled bring visitors to the login page
Problem: Links to courses that are open to unauthenticated guests bring visitors to the login page, rather than directly to the course.
Solution: While there is no fix for this yet, Bb has provided a workaround while we are researching a solution. This workaround is as follows, however users can also simply browse to the “Guest Access” link on the login page, and search for the course by name or ID.
2. The “new_loc=” variable allows you to add any in-system link, starting with “/Webapps” etc. For example, the relative path to the course with the ID 12345 looks something like this: /webapps/blackboard/execute/modulepage/view?course_id=_12345_1&cmp_tab_id=_4741_1&mode=view
3. However the link after the new_loc= variable MUST BE ENCODED! Use a service like http://meyerweb.com/eric/tools/dencoder/ to encode the link like so: %2Fwebapps%2Fblackboard%2Fexecute%2Fmodulepage%2Fview%3Fcourse_id%3D_12345_1%26cmp_tab_id%3D_4741_1%26mode%3Dview
You can change the PK1 number of the course (above indicated as 12345) to whatever course you want. You can also start with a different relative path, i.e. to link to the announcements area you’d start with this in steps 2 & 3:
/webapps/blackboard/execute/announcement?method=search&context=course&course_id=_12345_1&handle=cp_announcements&mode=cpview
## Students unable to view annotations on assignments
Problem: Some students have reported missing annotations on assignments in the Crocodoc Inline Assignment viewer.
Solution: Switching browsers has resolved the issue in some cases. Students should follow these steps to view annotations to ensure they are clicking on the correct buttons to view assignment annotations. Please contact blackboard@uvm.edu with browser and operating system versions if you have students reporting missing annotations.
## Resolved: Guest Access not working in some browsers
This issue has been resolved. Please contact blackboard@uvm.edu if you continue to have problems accessing Bb as a guest.
Problem: When entering a course as a guest (non-authenticated user), users are redirected to the login page or see an error message.
Solution: Some browsers have been reported to work better than others when accessing a course using guest access. While we await a solution from Blackboard, it is recommended that users attempt to enter the course using Internet Explorer or Firefox.
## Youtube posts display “Device Support” error video
Problem: When posting a video to a Bb course using the Video Everywhere or YouTube mashup tools, an error video is presented after posting, indicating that YouTube is not supported on this “device.
Solution: Bb has indicated that this is a known issue and has provided the following workaround:
1. After recording the video, click continue, this will save it in youtube.com
3. Copy the url
4. Go back to the editor on Bb environment
5. Click on Insert/Edit Embedded media
6. Paste the url into the File/URL
7. Change Type to Embedded Audio
8. Insert
9. Submit
Alternatively, anyone wishing to post media to their course should consider using UVM Streaming Media.
## Respondus not connecting/authenticating to Bb
Problem: An issue with Respondus not connecting to Bb has been identified. This is preventing the upload and download of assessments to Bb courses.
Solution: The following workaround will allow publication of assessments. Assessments can be manually published by going to Respondus->Preview & Publish->Publish to Blackboard->Publish Wizard->”Save pool to local file for manual upload”->”Test with points”. After saving the local file, it can be manually uploaded into a course with a web browser, by going into the course’s test, surveys and pools area, and clicking on the Import test button.
Alternatively, the following steps will allow Respondus to sync normally with your Bb course:
That said, Respondus has provided the following steps that should allow the application to properly sync with Bb:
1. Update Respondus to version 4.0.5.14. If you’ve installed it recently, you may be able to update it by going to Help->Check for updates. Otherwise, go to https://www.uvm.edu/software/distribution/windows/respondus/Respondus4.0.5.14.exe – then installing, and registering the software (you can access the registration data, if needed, here: https://www.uvm.edu/software/distribution/licensing/respondus/respondus.html).
2. Open Internet Explorer and log into Bb by going to bb.uvm.edu.
3. Leaving IE open in the background, open Respondus.
4. If editing an existing assessment, Click on Preview & Publish -> Publish to Blackboard. If retrieving an assessment, click retrieval and reports.
5. Click Publish Wizard
6. From the Blackboard Server dropdown, select “Add New Server”.
7. Choose “Yes, check for preconfigured server setting.”
8. Enter an appropriate name for the connection, such as “UVM Bb (New)”
9. Click the “Next” button to run the connection test.
12. Once you are logged in, click on the “Close after Login” button on the upper right.
## Problems logging back in, after logging out
Problem: When logged into Bb via myUVM, and then logging out of Bb, users may have trouble logging back in to Bb.
Solution: Deleting cookies, restarting the browser, or switching browsers will allow users to log back in again. Here are instructions for clearing browser cookies in 3 different browsers:
## Assignments not submitted via the Assignment tool cannot be graded by rubric
Problem: When an instructor is using the assignment tool and associated grade center column, but students submit an assignment outside of the assignment tool, the instructor cannot use the Rubric Tool to create a grade. For example, if students send their assignments via email or provide a paper version instead of using the online dropbox, there is no submission recorded in the grade center column for that assignment. This means that the instructor cannot use the Rubric Tool to calculate and provide feedback on the grade.
Solution: This is a known issue, however Blackboard has not indicated a fix as being planned in a future release. Workarounds aren’t ideal, and include asking students to submit an empty document digitally in addition to the alternative (paper, email, etc) submission.
## Resolved: Discussion board subscriptions not sending out emails for some users
Problem: Some users have reported that when subscribed to a discussion board forum, they are not receiving emails as expected when someone posts a message in that forum.
Solution: Blackboard has confirmed this as a bug, in which some users in a course have their subscriptions erroneously disabled when another user is removed (i.e. drops and/or withdraws) from a course. This is fixed in the next version of Bb, however there are no workarounds available.
Solution: This is a known bug, and a fix is expected (though not garaunteed) sometime in mid-December. In the meantime, feedback can be entered via the web interface on a supported browser.
## Can’t open assignment files in Bb Grader app
Problem: When viewing an assignment in the Bb Grader mobile application, assignments are not displayed, and attempting to open them in a different application results in no action.
Solution: UVM does not yet support inline grading of assignments. This means that assignments will need to be downloaded and graded within the web interface on a laptop or desktop computer, in a supported browser.
## Resolved: Bb Grader mobile app won’t connect
Problem: Blackboard has released a new mobile application that allows instructors to enter grades via an iPad or tablet. However, connecting to UVM via this application does not currently work.
Solution: UVM does not yet support mobile grading using this application. This is expected to change sometime in mid-December, however there is no guarantee that this timeline is accurate. In the meantime, instructors can grade assignments and other student work via the standard web interface in a supported browser. Please note that Blackboard is not the official place for final grade submission – final grades must be entered into the Registrar’s system.
## Resolved: Blackboard login/performance issues (resolved)
Blackboard is experiencing performance issues – these are actively being investigated and we hope to have a solution soon. In the meantime, one quick troubleshooting step is to try deleting your browser’s cookies, and then logging in again directly to https://bb.uvm.edu.
## Missing/hidden blog and journal entries
Problem: The new layout of the blog and journal interface that is presented by default to instructors makes it look like there are no entries.
Solution: Clicking on the arrows on either side of the instructor’s name in he grey column on the right will show other user’s entries. Also clicking on your name in the grey column will show a list of other user’s posts. These screenshots illustrate the buttons needed to click to see student blog posts:
## Wimba Pronto / Bb IM unavailable
Problem: Blackboard IM (formerly Wimba Pronto) is unavailable on this version of Bb due to a bug in the plugin tool.
There is no workaround for this issue. We are waiting for a patch from Bb, but have been given no timeline for when this might be available.
## Copy and Paste issues in Bb text editor
Problem: Copying from within the text editor, then attempting to paste back into the editor, results in text not being pasted (nothing happens).
While we are waiting on a patch for this from Bb, we have been provided with the following workaround in Firefox:
2. Type in “clipboardevents”.
3. Double click the search result item to toggle value from “true” to “false”
You may need to restart your browser in order for this to take effect.
## Resolved: Assessments causing NullPointerException errors
This has been resolved. If you continue to see this problem, please contact blackboard@uvm.edu as soon as possible.
Problem: Deploying an assessment in a content area displays a “NullPointerException”. The test can be seen and previewed when edit mode is off, but no other content in the content area is visible. Further, attempting to access the “Tests” or “Surveys” area of the “Tests, Surveys and Pools” section of a course displays a similar error.
We are working directly with Blackboard to find a resolution to this issue as soon as possible. In the meantime, please note the following work-arounds:
• Students can still access and submit deployed exams. Results of these exams can be graded and feedback provided via the Grade Center and My Grades, respectively.
• Tests can still be deployed, just not in areas where another test is currently deployed.
• Deployed tests/quizzes can be edited by going into the grade center, clicking on the grey icon next to the column name for the test, and choosing “Edit Test”. From there instructors can add/edit the questions, or edit the options by selecting from the grey button to the right of the test name in the edit test screen.
• Content that shares a content area with a deployed assessment will be inaccessible to the instructor, however students may in some cases be able to access that content.
• Instructors can access sub-folders that exist in the same area as a deployed test by clicking the folder icon at the top of the course menu and navigating from there.
To deploy a new tests, deploying each test in its own folder will allow tests to be delivered to students without interfering with instructor access to the rest of the content areas. For example, one might create a new content area on the left hand menu entitled “Quizzes 2”, and structure it like so:
• Quiz 1 Folder
• …..> Quiz 1 (inside Quiz 1 folder)
• Quiz 2 Folder
• …..> Quiz 2 (inside Quiz 2 folder)
## Missing scrollbars in the Grade Center on Mac OSX
Problem: Scroll bars do not appear in the Grade Center. This occurs on some browsers in Mac OSX, including FireFox and Safari.
Solution: Scroll bars can be re-enabled system wide by going into System Preferences -> General, and set “Show scroll bars” to “Always”.
## YouTube embedded videos (and other embedded content) not appearing
Problem: Embedded YouTube videos and similar content are not displaying in some browsers. This is due to a change in how browsers display non-encrypted content (http) inside of an encrypted environment (https).
Solution: Here is a description for students (and instructors) for how to temporarily view embedded files that appear blank in Firefox:
1. If upon entering a course or content area that appears to have a blank video, look in the browser’s address bar for a grey “shield” icon.
2. Click that icon, bringing up a dialog box.
3. Click on the down arrow next to the “Keep Blocking” button, and select “Disable Protection on This Page”.
This is a short term solution. Here are some steps on how to make sure that students have access to the videos, both in the short term and the long term.
1. Edit the embed code so that all links in the code start with “https://”, and not “http://”.
2. Post the URL as a link to the video’s YouTube page in addition to the embedded video. By always providing a link to the original video in addition to embedding it, there’s a better chance that any students having connection or browser issues (i.e. attempting to load the videos in various unsupported mobile browsers) will be able to access the video directly. This also provides some context, so instead of just seeing the blank area where the video will be, the students will see that a) it’s a video link, and b) where to go to try to launch it in a different environment.
3. If the videos are owned by you (i.e. the instructor) or UVM, consider using the UVM supported Streaming Media tool. This will make things a bit easier to support, and will be more likely to retain some stability and support over time.
4. Newer browsers may not support older embed code. Try re-embedding the video by going to the YouTube page for the video and copying the code under the embed option, then pasting it back into your course item.
## Reordering course list, grade center items in Firefox – overlapping items, items fly off screen
Problem: Attempting to reorder items while managing the course list and grade center results in overlapping items, and/or the items do not appear to go where placed.
Solution: This is a known bug in Bb, which we are waiting on a fix for. In the meantime, you can use the reorder button (a grey box with an up/down arrow set, above the list) to manage list order.
## Resolved: Emails sent from announcement tool may contain HTML tags in message body
Problem: When creating announcements and checking the “send email” box, the resulting email message body may contain HTML tags. This has been confirmed to occur when using “Smart Text” with the text editor off, but may happen with the visual editor as well.
Solution: There is no solution available at this time, however Bb has indicated that this is resolved in a future release. The workaround is currently to avoid using HTML formatting where possible when creating announcements that need to be sent as emails.
## Resolved: Emails sent from grade center arrive without message body
Problem: When sending an email to selected students from the grade center, and including a BCC or CC address, the message body arrives empty.
Solution: There is no resolution for this issue at this time. Blackboard has indicated that this is fixed in the next version of blackboard. The workaround when sending email from the Grade Center is to turn the Text Editor OFF, and not use BCC option or Include list of recipients option.
## Resolved: “Alignment” settings appearing in various areas of course
Problem: Settings appear in tool creation and other areas (such as when creating a Discussion Board forum) that pertain to an “Alignments” function. However, these alignments settings don’t do anything.
Solution: Fields referencing “Alignments” can be ignored – they are remnants of a Blackboard feature that is not currently being deployed (but which is being evaluated). Their presence has been identified as a bug, which is fixed in the next release of Blackboard.
## Resolved: Linked text created in text areas is not underlined
This has been resolved. Please report any further problems to blackboard@uvm.edu.
Links created in the visual text box editor in Blackboard display with no underline and in some themes are not differentiated by color. A patch is forthcoming from Blackboard, however a temporary fix is being pursued to address this in the short term.
## Resolved: Intermittent delays in page loads, application performance
This issue has been resolved. If you continue to experience these symptoms, please contact blackboard@uvm.edu to report them.
Solution: ETS has identified a possible cause for these performance issues and is working to apply a solution. Users who experience these problems should attempt to log out and then log back in again. There is no other workaround identified at this time.
## Red error message on login
Problem: Some users have reported problems when logging in, noting that a large red error message is displayed. This has been documented as happening after logging into Bb from myUVM, then logging out of Bb from Bb. Attempting to log in via Bb directly produces the error.
Solution: Clearing the browser’s cookies has been identified as a possible solution, as has logging in with another browser. Alternatively, going back to myUVM and logging in to Bb from within myUVM may also work. If you continue to have problems logging in, please contact blackboard@uvm.edu.
Problem: Menu items for certain course tools do not appear for students, even when they are shown as available and visible to the instructor while edit mode is “Off”.
Solution: We are waiting on a bug fix for this from Blackboard. In the meantime, instructors can make sure these tools are available to students by following these steps:
1. Click the green “+” button at the top of the course Menu, and select “Add tool link”
2. Enter “Tools Area” in the name box, and select “Tools Area” from the list. Click Submit.
3. Click on the new Tools Area link on the course menu.
4. If a tool is hidden, it will have “Show Link” next to it. Click the “Show Sink” button next to it to ensure it’s available to students.
## Student name not appearing in Grade Center
Problem: Increasing the font size in some versions of Firefox causes the last name in the Grade Center to disappear from view (the student is still in the course, just not visible in the Grade Center).
Solution: This is resolved in a future version of Blackboard. In the meantime reducing the font size or clicking on the “Screen reader” link above the Grade Center will reveal the hidden student.
## Wiki content does not copy over or import
Problem: Wiki content copied or imported into a course will not appear after the copy/import process.
There is no solution for this issue. We are waiting on a fix from Blackboard. In the meantime, the workaround is to manually copy and paste the content into the new course.
## Resolved: Problems deleting announcements
Problem: Attempting to delete announcements imported from old courses results in an error similar to ” ORA-01407: cannot update (“BB_BB60″.”ANNOUNCEMENTS”.”ORDER_NUM”) to NULL”.
Solution: Re-ordering the announcement before deleting it should allow it to be deleted.
## Opening a PDF shows a “missing plugin” error
Update: This will still show an error, however users will now see a link at the top of the page that allows them to download the PDF:
Problem: Students clicking on a PDF that has been added to a course using the File command (see image) are presented with a grey screen and an error message that refers to a “missing plugin”. This has been reported on various browsers on Apple computers, including Firefox and Safari.
Solution
Faculty: Here’s how to make sure your PDF’s are accessible to your students.
1. Click the grey chevron drop-down menu button next to the file name and choose Edit (you must be in “edit mode” to do this).
2. Scroll to the option to “Open in New Window” and click Yes.
3. Submit.
Students: If you are experiencing this problem, click on the link at the top of the page to download the PDF and open it on your computer.
## Resolved: Guests cannot access uploaded files (401 error)
Problem: Guests cannot access files in a course, and are prompted with a login box followed by 401 “forbidden” error after trying to access the files. This occurs for files that have been uploaded to a course after May 22nd 2011.
Solution: To make a file available to guests,
1. Click on the grey chevron button next to the file name, and choose “Edit”.
2. Click on “Select a Different File”. (see image)
3. Click on “Browse Course”. (see image)
4. Check the “Also assign Public permission to this file” box. (see image)
5. Select the file from the list, and click on “Submit”.
6. Click on “Submit” to complete the process.
To prevent this from happening with newly uploaded files,
1. Upload a file to the “Files” area before putting it in a content area
2. Then add the file to the content area by choosing Build -> File, not Build -> Item
3. Be sure to select Browse Course, and check the “Also assign Public permission to this file” box when selecting the file.
This is an identified bug that is fixed in the next version of Blackboard. There is no patch available for the current version.
## Domain administrators cannot enroll NetID accounts outside of their domain
Domain administrators can no longer enroll users into a course in a domain that doesn’t include those users in it’s collection. hen entering the NetID to enroll, an error is displayed:
courseenrol.validate.user.not.in.domain
This is being escalated to Blackboard, and we are waiting on a solution. In the meantime, please contact blackboard@uvm.edu if you need to enroll users in a domain-based course or organization.
## Resolved: “My Calendar” module has been turned off
This has been resolved, and the Calendar tool has once again been made available.
Due to a critical bug in the Calendar module, this tool has been turned off. A fix for the bug is expected in the next release of Blackboard, but a timeline is not available for when this will be patched.
The normal course calendar tools should work as expected.
## Resolved: Pronto contacts not listed
Some users have reported that when signing into the Pronto chat application on their computer, they don’t see contacts listed for classes that are using Pronto.
Below are some steps that have been identified as a possible workaround to this issue.
• Go to the course
• Click on the Pronto Chat Tool menu item (if this is not there, contact your instructor)
• Then click on the Go to Pronto link (it may open up a new browser window)
• Choose I already have pronto account (or, if this is your first time, create a Pronto account)
• Unfold the list of courses you are taking so you can see the course listed there.
• Start up Pronto on your computer
## Resolved: Importing a course package larger than 250MB fails
When trying to import an archive or export file, an error is displayed and the process fails if the file is larger than 250MB. This is a known bug that is resolved in 9.1. In the meantime, contact blackboard@uvm.edu to request the file be restored manually by an administrator.
## Students with the same First and Last name are indistinguishable
Students with the same first and last names within a course are indistinguishable in areas such as the discussion board, groups management, messages, and email. This is a long standing bug in Blackboard, for which we are awaiting a resolution.
Problem: There is an identified bug that prevents downloading attached files from a blog/journal post. This only happens if the file has a title with special (non-alphanumeric) characters in it.
Solution: The author of the post can rename the file so it only contains letters and numbers, and then re-attach (re-upload) the file.
## Resolved: Occasional disappearing menu items
We have received reports of inconsistent course menu behavior where links on the menu will simply disappear and reappear. We (and other institutions) have submitted this problem to Blackboard. They are aware of the problem, and are researching the cause and a solution. In the meantime, a work-around that has been effective is switching to “Folder View” for the course menu. See a screenshot and directions.
This is a most troublesome (and elusive) issue and we are trying to provide Blackboard with additional information regarding this problem. If you are experiencing this issue with your course menu, please fill out this brief form.
• The 5-Digit CRN of the course you are seeing the issue happening in
• Date and time of the occurrence.
## Tools or buttons are missing or unavailable
Problem: Tools or buttons (e.g. the “Build” button in a Course Materials area) in a course appear to be missing or non-existent.
Solution: Go to Control Panel -> Customization -> Tool Availability, and make sure checkboxes are checked for the tool in question. Submit the form.
## Resolved: Discussion board text overlaps when viewing threads in “maximized view”
When viewing a discussion board thread with a significant number of entries, the text overlaps with the entry listing, causing the entry to be difficult or impossible to read.
This is an identified bug. A resolution has not been identified yet for a future release. To prevent this from happening in the meantime, switch out of “maximize” view when viewing threaded discussions. This can be achieved by clicking the “minimize” button to the upper right of the discussion thread:
## Resolved: Problems editing or creating Grade Center Smart Views
Problem editing or creating Grade Center Smart Views when groups in course exist with special characters in their titles
This has been resolved. In the meantime, the workaround is to edit group titles so that they only contain alpha numeric characters (letters, spaces, and numbers).
## Resolved: “Not Implemented” error appearing after instructor enters a course
This has been resolved. The error is related to the Course Groups feature and begins when an instructor (or anyone) is listed in more than 10 groups.
Instructors have reported problems downloading assignment files from the grade center. This problem will occurr if the assignment has a title with special (non-alphanumeric) characters in it.
Solution: To resolve this error, rename the assignment so that it has no characters other than letters and numbers (i.e. no #’s, &’s, etc). As always when making changes to your course, please remember to save a backup first.
## Resolved: Students being logged out of Blackboard prematurely; errors during exams
UPDATE: A fix has been put in place to address a Blackboard bug that caused users to be prematurely logged out of the system. Additionally, a fix has been applied to prevent Blackboard from logging a user out when their myUVM session ends. If you are experiencing these problems and it has not been resolved after 24 hours, please contact blackboard support at blackboard@uvm.edu, and include the information described below.
Errors encountered during tests and quizzes might include messages when saving answers indicating that the answer could not be saved, or a warning when submitting an exam which erroneously suggests that a number of questions had not been answered.
Blackboard has also suggested that for longer exams (i.e. around 50 questions or more), the exam be delivered one question at a time, and to uncheck the “Force Completion” setting in the test options. This is to prevent the application from being overloaded, and to prevent user’s sessions from timing out.
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2021-07-31 05:53:56
|
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https://www.lucien.ink/archives/361/
|
# Codeforces 1088C - Ehab and a 2-operation task
## 题解链接
https://lucien.ink
## 题目链接
https://codeforces.com/contest/1088/problem/C
## 题目
You're given an array $a$ of length $n$. You can perform the following operations on it:
• choose an index $i$ $(1 \le i \le n)$, an integer $x$ $(0 \le x \le 10^6)$, and replace $a_j$ with $a_j+x$ for all $(1 \le j \le i)$, which means add $x$ to all the elements in the prefix ending at $i$.
• choose an index $i$ $(1 \le i \le n)$, an integer $x$ $(1 \le x \le 10^6)$, and replace $a_j$ with $a_j \% x$ for all $(1 \le j \le i)$, which means replace every element in the prefix ending at $i$ with the remainder after dividing it by $x$.
Can you make the array strictly increasing in no more than $n+1$ operations?
## 题意
初始时你有一个元素个数为$n$的数组，第$i$个元素的初始值为$a_i$，你有两种操作：
• 将区间$[1, i]$中所有的数字都加上$x$
• 将区间$[1, i]$中所有的数字都对$x$取模
问能否在$n + 1$步之内将这个序列变成一个严格递增的序列。
## 思路
考虑让$n$个数分别变为模意义下的$1 \dots n$，随便取一个大于$n$的模数，比如说$mod = n + 1$，也就是对于第$i$个数来说，我让其变为$k \cdot mod + i$，这样在进行了$n$次修改之后，第$n + 1$次修改我进行一次取模操作，第$i$个数就会变为$i$了。
## 实现
https://pasteme.cn/2209
#include <bits/stdc++.h>
typedef long long ll;
const int maxn = 2007;
ll a[maxn], base = 0, cnt = 1;
int n, mod;
std::queue<std::pair<int, ll>> ans;
int main() {
scanf("%d", &n);
for (int i = 1; i <= n; i++) scanf("%lld", a + i);
mod = n + 1;
for (int i = n; i >= 1; i--) {
if ((a[i] + base) % mod == i) continue;
cnt++;
ll cur = a[i] + base, add = (mod - cur % mod) + i;
}
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2018-12-19 12:41:09
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https://www.thejournal.club/c/paper/23765/
|
#### Criteria of stabilizability for switching-control systems with solvable linear approximations
##### Xiongping Dai
We study the stability and stabilizability of a continuous-time switched control system that consists of the time-invariant $n$-dimensional subsystems \dot{x}=A_ix+B_i(x)u\quad (x\in\mathbb{R}^n, t\in\mathbb{R}_+ \textrm{and} u\in\mathbb{R}^{m_i}),\qquad \textrm{where} i\in{1,...,N} and a switching signal $\sigma(\bcdot)\colon\mathbb{R}_+\rightarrow{1,...,N}$ which orchestrates switching between these subsystems above, where $A_i\in\mathbb{R}^{n\times n}, n\ge1, N\ge2, m_i\ge1$, and where $B_i(\bcdot)\colon\mathbb{R}^n\rightarrow\mathbb{R}^{n\times m_i}$ satisfies the condition $\|B_i(x)\|\le\bbbeta\|x\|\;\forall x\in\mathbb{R}^n$. We show that, if ${A_1,...,A_N}$ generates a solvable Lie algebra over the field $\mathbbm{C}$ of complex numbers and there exists an element $\bbA$ in the convex hull $\mathrm{co}{A_1,...,A_N}$ in $\mathbb{R}^{n\times n}$ such that the affine system $\dot{x}=\bbA x$ is exponentially stable, then there is a constant $\bbdelta>0$ for which one can design "sufficiently many" piecewise-constant switching signals $\sigma(t)$ so that the switching-control systems \dot{x}(t)=A_{\sigma(t)}x(t)+B_{\sigma(t)}(x(t))u(t),\quad x(0)\in\mathbb{R}^n\textrm{and} t\in\mathbb{R}_+ are globally exponentially stable, for any measurable external inputs $u(t)\in\mathbb{R}^{m_{\sigma(t)}}$ with $|u(t)|\le\bbdelta$.
arrow_drop_up
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2023-02-07 12:18:41
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https://www.physicsforums.com/threads/projectile-thrown-horizontally-off-a-cliff.837226/
|
# Projectile thrown horizontally off a cliff
Member warned to retain and use the formatting template
A ball is thrown horizontally from a height of 20m and hits the ground with a speed that is three times its initial speed. What is the initial speed?
d=vit+1/2at^2
I found the time of flight vertically first, which is 2.02 sec. And then I am trying to find the initial speed. Even if it is thrown horizontally off a cliff, is the acceleration still 0?
## Answers and Replies
Related Introductory Physics Homework Help News on Phys.org
TSny
Homework Helper
Gold Member
I found the time of flight vertically first, which is 2.02 sec. And then I am trying to find the initial speed. Even if it is thrown horizontally off a cliff, is the acceleration still 0?
Did you mean to say 0 here? Once the ball leaves your hand, gravity takes over. So, what is the acceleration (magnitude and direction) while the ball is in flight?
Your value for the time of flight looks correct.
Doesnt it depend? Horizontally it does not have a acceleration. But does it apply when it is thrown off a cliff? Because it is thrown off the cliff horizontally not straight down.
TSny
Homework Helper
Gold Member
No matter how you throw the ball, once it's in flight the only thing "acting on it" is gravity (neglecting air resistance, of course). The acceleration caused by gravity is g downward.
I am confused now but I was told that horizontally an object has no acceleration?
TSny
Homework Helper
Gold Member
Acceleration is a vector quantity. For a projectile, the horizontal component of the acceleration is zero. But the vertical component is 9.8 m/s2 downward.
Can I solve this problem horizontally? I tried it but it seems a little off.
Vf=Vi+at
A would be cancelled because of horizontally
Vf=Vi?
I looked in the answer key and they did it differently but my professor taught us this way to find vertically and horizontally. I don't know if it applies to this problem though.
TSny
Homework Helper
Gold Member
Yes, you should think about the horizontal and vertical components of the motion.
Suppose Vo is the initial speed that the object is thrown horizontally. Since the horizontal component of acceleration is zero, what can you say about the value of the horizontal component of velocity just as the ball reaches the ground?
Yes, you should think about the horizontal and vertical components of the motion.
Suppose Vo is the initial speed that the object is thrown horizontally. Since the horizontal component of acceleration is zero, what can you say about the value of the horizontal component of velocity just as the ball reaches the ground?
It's the same as the initial?
CWatters
TSny
Homework Helper
Gold Member
Yes. Now think about the vertical component of motion. What is the initial value of the vertical component of velocity? Can you see a way to get the final vertical component of velocity using the height that the ball was thrown from?
Yes. Now think about the vertical component of motion. What is the initial value of the vertical component of velocity? Can you see a way to get the final vertical component of velocity using the height that the ball was thrown from?
I got Vf for the vertical component as 19.8. How is this related to the Vf of the horizontal?
CWatters
Homework Helper
Gold Member
Your post #9 is correct. The horizontal component is unchanged.
Bump
TSny
Homework Helper
Gold Member
I got Vf for the vertical component as 19.8. How is this related to the Vf of the horizontal?
I'm not sure I understand your question here. But, how do you construct the speed of a particle from it's horizontal and vertical components?
I'm not sure I understand your question here. But, how do you construct the speed of a particle from it's horizontal and vertical components?
You would just use
$$\Delta \ x = \ v_0 t + \frac{1}{2} \alpha t^2$$
for both horizontal and vertical
CWatters
Homework Helper
Gold Member
You just add (vector add) the horizontal and vertical components to get the impact speed.
TSny
Homework Helper
Gold Member
Let's make sure we are together so far. Let Vo be the initial speed of the ball.
When the ball reaches the ground, how would you express the horizontal and vertical components of the velocity?
Vertically the final velocity would be 39.2. I don't know how to find it horizontally.
TSny
Homework Helper
Gold Member
Vertically the final velocity would be 39.2. I don't know how to find it horizontally.
In post #11 you stated that the vertical component would be 19.8 m/s. And in post #9 you gave the correct answer for the horizontal component of the final velocity.
In post #11 you stated that the vertical component would be 19.8 m/s. And in post #9 you gave the correct answer for the horizontal component of the final velocity.
The 19.8 is the final velocity of the vertical.
Vf=Vi only in the horizontal component not the vertical component.
TSny
Homework Helper
Gold Member
Yes, the final vertical component of velocity is 19.8 m/s2 downward. That is Vfy= -19.8 m/s2 (if we take the positive y direction to be upward).
Suppose you let the symbol Vo stand for the initial speed of the particle when it was thrown horizontally. You are asked to find the value of Vo such that the final speed is 3 times Vo. It will help if you can answer the following.
(1) How does the final horizontal component of velocity compare to Vo? That is, can you express Vfx in terms of Vo?
(2) How do you calculate the speed of a particle if you know the values of the horizontal and vertical components of velocity, Vx and Vy?
Yes, the final vertical component of velocity is 19.8 m/s2 downward. That is Vfy= -19.8 m/s2 (if we take the positive y direction to be upward).
Suppose you let the symbol Vo stand for the initial speed of the particle when it was thrown horizontally. You are asked to find the value of Vo such that the final speed is 3 times Vo. It will help if you can answer the following.
(1) How does the final horizontal component of velocity compare to Vo? That is, can you express Vfx in terms of Vo?
(2) How do you calculate the speed of a particle if you know the values of the horizontal and vertical components of velocity, Vx and Vy?
1) Vo=Vf/3
2) This why I am asking...is there some sort of physics rule?
TSny
Homework Helper
Gold Member
1) Vo=Vf/3
It's important to see what's going on with just the horizontal component of the velocity. How would you express the initial horizontal component of velocity, Vix, in terms of Vo? How would you express the final horizontal component of velocity, Vix, in terms of Vo?
2) This why I am asking...is there some sort of physics rule?
Speed is defined to be the magnitude of the velocity vector. How do you calculate the magnitude of a vector using the components of the vector?
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2020-09-28 09:10:32
|
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|
https://www.developer.here.com/documentation/android-premium/dev_guide/topics/map-data.html
|
HERE SDK for Android (Premium Edition)
In maintenance
SDK for Android Developer's Guide
# HERE Map Data Download
Some key functionality offered through HERE SDK depends on HERE Map Data being downloaded and cached on the device. Rendering a map on the screen, for example, is not possible without first downloading map data to the device. Similarly, it would not be possible to provide accurate turn-by-turn navigation without downloading map data to the device. Offline operations such as offline routing and search also require map data to be downloaded to the device in advance. This section describes different approaches you can take to manage map data download.
## Map Data Download Example on GitHub
You can find an example that demonstrates this feature at https://github.com/heremaps/.
## Passive Download Approach
The passive approach is where you allow the SDK to download map data as needed. A typical example is when a user pans the map and triggers an on-demand map data download to render the map.
Map data downloaded in this way is stored in a persistent cache with a default size of 256 MB. Cached map data can be used for offline operations, in cases where a network connection is not available or not desired, such as when the device is in roaming mode. However, there is no way for you to know if sufficient data has been downloaded to enable all offline operations, such as offline search or routing.
Note: To ensure the map data version retrieved by passive approach is the latest available, use the MapLoader APIs to update to the latest version.
Note: Use ApplicationContext.setDiskCacheSize(long) to change the default disk cache size.
## Active Download Approach
HERE SDK provides two alternatives to actively fetch map data:
• Map data may be downloaded in the form of map packages for a predefined region or country
• Map data may be downloaded for an arbitrary bounding box or a radius around a route
The first active approach is where you request the download of map data packages which cover an entire country or region using the MapLoader APIs. You do this by selecting from a list of map packages. A map package may be a state (such as California), region, or a country (such as Belgium).
Note: This preloaded map data is stored separately from the map data cache mentioned in the passive download approach above. The amount of space available for map data packages is only limited by the amount of free space on the device.
The second active approach is where you explicitly trigger a fetch of map data through the MapDataPrefetcher APIs by specifying a bounding box or a radius around a route. The resulting downloaded map data is stored in the same cache used in the passive download approach, where the cache size by default is 256 MB. The SDK does not place a limit on the size of the area requested for download, but it is expected developers are aware of this cache limit and only request areas that result in map data download size of under this limit. You can get a size estimate of the map data that will be downloaded via the MapDataPrefetcher APIs.
To illustrate how much data may be downloaded, consider a bounding box covering an area of 200 km by 200 km in New York City as illustrated in the following screenshot. In this case, approximately 250 MB of map data is downloaded.
The next example shows a 160 km route from New York to Philadelphia with a radius (route corridor width) of 500 m. The map data downloaded is about 100 MB.
A comparison of the different approaches for downloading map data is shown below:
Table 1. Map Data Download Approaches
Passive Approach Active Approach
On-demand Map Packages Bounding Box / Route
Downloaded map data can be used for offline operation Limited (1) Yes Yes
Complexity involved in managing downloads None Medium Low
Size of map data downloads Medium (10s of MBs) Large (100s of MBs) Medium (10s of MBs)
Upper size limit of cache where data is stored (2) 256 MB - 2 GB (2) None (3) 256 MB - 2 GB (2)
Option to check areas for which map data has previously been downloaded (4) No Yes No
Option to check in advance the size of map data to be downloaded No Yes Yes
Option to selectively remove downloaded map data No (5) Yes No (5)
Can perform incremental updates between map data versions Yes (6) Yes (6) Yes (6)
• (1): Map data downloaded on-demand may support some offline operations such as rendering and search, while others, such as routing, might not work correctly as some essential data may be missing.
• (2): The default value of disk cache upper limit is 256 MB. To change the upper limit of disk cache size use ApplicationContext.setDiskCacheSize(long) method.
• (3): The number of map data packages which can be downloaded is only limited by the space available on the device.
• (4): For example, if you want to display to the user what map data has been downloaded and is available for offline use.
• (5): Only option is to completely clear the cache using MapDataPrefetcher APIs, thus removing all map data that was downloaded on demand and map data downloaded by specifying a bounding box or route. Downloaded map packages are not removed.
• (6): Incremental updates are available when updating to the latest map data release from the two previous releases. Incremental updates are typically small downloads as only the changes are downloaded. For example, when updating to the Q1 2018 map data release from the Q4 2017 or Q3 2017 release, an incremental update or patch is used. Where a patch is not available (such as updating from Q2 2017 to Q1 2018), all map data packages are re-downloaded resulting in a much larger download size, and map data that was downloaded on-demand or by specifying a bounding box/route is removed.
Many applications may use a combination of all three approaches. For example, a UI may be provided to allow users to select map packages to download. Also, an option may be provided to download map data for a route prior to starting turn-by-turn navigation. Map data download on-demand would always be available as a fallback for the case where map data was not downloaded using the other approaches.
Note: In case of rerouting during turn-by-turn navigation, in the very first time the fast offline rerouting is performed. If it gets failed due to essential data missing, then online request is sent to create a new route. For fast and reliable rerouting we recommend to have map data for a route downloaded prior to starting turn-by-turn navigation.
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2023-03-31 03:18:45
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https://openstax.org/books/university-physics-volume-3/pages/6-3-the-compton-effect
|
University Physics Volume 3
# 6.3The Compton Effect
University Physics Volume 36.3 The Compton Effect
### Learning Objectives
By the end of this section, you will be able to:
• Describe Compton’s experiment
• Explain the Compton wavelength shift
• Describe how experiments with X-rays confirm the particle nature of radiation
Two of Einstein’s influential ideas introduced in 1905 were the theory of special relativity and the concept of a light quantum, which we now call a photon. Beyond 1905, Einstein went further to suggest that freely propagating electromagnetic waves consisted of photons that are particles of light in the same sense that electrons or other massive particles are particles of matter. A beam of monochromatic light of wavelength $λλ$ (or equivalently, of frequency f) can be seen either as a classical wave or as a collection of photons that travel in a vacuum with one speed, c (the speed of light), and all carrying the same energy, $Ef=hf.Ef=hf.$ This idea proved useful for explaining the interactions of light with particles of matter.
### Momentum of a Photon
Unlike a particle of matter that is characterized by its rest mass $m0,m0,$ a photon is massless. In a vacuum, unlike a particle of matter that may vary its speed but cannot reach the speed of light, a photon travels at only one speed, which is exactly the speed of light. From the point of view of Newtonian classical mechanics, these two characteristics imply that a photon should not exist at all. For example, how can we find the linear momentum or kinetic energy of a body whose mass is zero? This apparent paradox vanishes if we describe a photon as a relativistic particle. According to the theory of special relativity, any particle in nature obeys the relativistic energy equation
$E2=p2c2+m02c4.E2=p2c2+m02c4.$
6.17
This relation can also be applied to a photon. In Equation 6.17, E is the total energy of a particle, p is its linear momentum, and $m0m0$ is its rest mass. For a photon, we simply set $m0=0m0=0$ in this equation. This leads to the expression for the momentum $pfpf$ of a photon
$pf=Efc.pf=Efc.$
6.18
Here the photon’s energy $EfEf$ is the same as that of a light quantum of frequency f, which we introduced to explain the photoelectric effect:
$Ef=hf=hcλ.Ef=hf=hcλ.$
6.19
The wave relation that connects frequency f with wavelength $λλ$ and speed c also holds for photons:
$λf=cλf=c$
6.20
Therefore, a photon can be equivalently characterized by either its energy and wavelength, or its frequency and momentum. Equation 6.19 and Equation 6.20 can be combined into the explicit relation between a photon’s momentum and its wavelength:
$pf=hλ.pf=hλ.$
6.21
Notice that this equation gives us only the magnitude of the photon’s momentum and contains no information about the direction in which the photon is moving. To include the direction, it is customary to write the photon’s momentum as a vector:
$p→f=ℏk→.p→f=ℏk→.$
6.22
In Equation 6.22, $ℏ=h/2πℏ=h/2π$ is the reduced Planck’s constant (pronounced “h-bar”), which is just Planck’s constant divided by the factor $2π.2π.$ Vector $k→k→$ is called the “wave vector” or propagation vector (the direction in which a photon is moving). The propagation vector shows the direction of the photon’s linear momentum vector. The magnitude of the wave vector is $k=|k→|=2π/λk=|k→|=2π/λ$ and is called the wave number. Notice that this equation does not introduce any new physics. We can verify that the magnitude of the vector in Equation 6.22 is the same as that given by Equation 6.18.
### The Compton Effect
The Compton effect is the term used for an unusual result observed when X-rays are scattered on some materials. By classical theory, when an electromagnetic wave is scattered off atoms, the wavelength of the scattered radiation is expected to be the same as the wavelength of the incident radiation. Contrary to this prediction of classical physics, observations show that when X-rays are scattered off some materials, such as graphite, the scattered X-rays have different wavelengths from the wavelength of the incident X-rays. This classically unexplainable phenomenon was studied experimentally by Arthur H. Compton and his collaborators, and Compton gave its explanation in 1923.
To explain the shift in wavelengths measured in the experiment, Compton used Einstein’s idea of light as a particle. The Compton effect has a very important place in the history of physics because it shows that electromagnetic radiation cannot be explained as a purely wave phenomenon. The explanation of the Compton effect gave a convincing argument to the physics community that electromagnetic waves can indeed behave like a stream of photons, which placed the concept of a photon on firm ground.
The schematics of Compton’s experimental setup are shown in Figure 6.11. The idea of the experiment is straightforward: Monochromatic X-rays with wavelength $λλ$ are incident on a sample of graphite (the “target”), where they interact with atoms inside the sample; they later emerge as scattered X-rays with wavelength $λ′.λ′.$ A detector placed behind the target can measure the intensity of radiation scattered in any direction $θθ$ with respect to the direction of the incident X-ray beam. This scattering angle, $θ,θ,$ is the angle between the direction of the scattered beam and the direction of the incident beam. In this experiment, we know the intensity and the wavelength $λλ$ of the incoming (incident) beam; and for a given scattering angle $θ,θ,$ we measure the intensity and the wavelength $λ′λ′$ of the outgoing (scattered) beam. Typical results of these measurements are shown in Figure 6.12, where the x-axis is the wavelength of the scattered X-rays and the y-axis is the intensity of the scattered X-rays, measured for different scattering angles (indicated on the graphs). For all scattering angles (except for $θ=0°),θ=0°),$ we measure two intensity peaks. One peak is located at the wavelength $λ,λ,$ which is the wavelength of the incident beam. The other peak is located at some other wavelength, $λ′.λ′.$ The two peaks are separated by $Δλ,Δλ,$ which depends on the scattering angle $θθ$ of the outgoing beam (in the direction of observation). The separation $ΔλΔλ$ is called the Compton shift.
Figure 6.11 Experimental setup for studying Compton scattering.
Figure 6.12 Experimental data show the Compton effect for X-rays scattering off graphite at various angles: The intensity of the scattered beam has two peaks. One peak appears at the wavelength $λλ$ of the incident radiation and the second peak appears at wavelength $λ′.λ′.$ The separation $ΔλΔλ$ between the peaks depends on the scattering angle $θ,θ,$ which is the angular position of the detector in Figure 6.11. The experimental data in this figure are plotted in arbitrary units so that the height of the profile reflects the intensity of the scattered beam above background noise.
### Compton Shift
As given by Compton, the explanation of the Compton shift is that in the target material, graphite, valence electrons are loosely bound in the atoms and behave like free electrons. Compton assumed that the incident X-ray radiation is a stream of photons. An incoming photon in this stream collides with a valence electron in the graphite target. In the course of this collision, the incoming photon transfers some part of its energy and momentum to the target electron and leaves the scene as a scattered photon. This model explains in qualitative terms why the scattered radiation has a longer wavelength than the incident radiation. Put simply, a photon that has lost some of its energy emerges as a photon with a lower frequency, or equivalently, with a longer wavelength. To show that his model was correct, Compton used it to derive the expression for the Compton shift. In his derivation, he assumed that both photon and electron are relativistic particles and that the collision obeys two commonsense principles: (1) the conservation of linear momentum and (2) the conservation of total relativistic energy.
In the following derivation of the Compton shift, $EfEf$ and $p→fp→f$ denote the energy and momentum, respectively, of an incident photon with frequency f. The photon collides with a relativistic electron at rest, which means that immediately before the collision, the electron’s energy is entirely its rest mass energy, $m0c2.m0c2.$ Immediately after the collision, the electron has energy E and momentum $p→,p→,$ both of which satisfy Equation 6.19. Immediately after the collision, the outgoing photon has energy $E˜f,E˜f,$ momentum $p˜→f,p˜→f,$ and frequency $f′.f′.$ The direction of the incident photon is horizontal from left to right, and the direction of the outgoing photon is at the angle $θ,θ,$ as illustrated in Figure 6.11. The scattering angle $θθ$ is the angle between the momentum vectors $p→fp→f$ and $p˜→f,p˜→f,$ and we can write their scalar product:
$p→f·p˜→f=pfp˜fcosθ.p→f·p˜→f=pfp˜fcosθ.$
6.23
Following Compton’s argument, we assume that the colliding photon and electron form an isolated system. This assumption is valid for weakly bound electrons that, to a good approximation, can be treated as free particles. Our first equation is the conservation of energy for the photon-electron system:
$Ef+m0c2=E˜f+E.Ef+m0c2=E˜f+E.$
6.24
The left side of this equation is the energy of the system at the instant immediately before the collision, and the right side of the equation is the energy of the system at the instant immediately after the collision. Our second equation is the conservation of linear momentum for the photon–electron system where the electron is at rest at the instant immediately before the collision:
$p→f=p˜→f+p→.p→f=p˜→f+p→.$
6.25
The left side of this equation is the momentum of the system right before the collision, and the right side of the equation is the momentum of the system right after collision. The entire physics of Compton scattering is contained in these three preceding equations––the remaining part is algebra. At this point, we could jump to the concluding formula for the Compton shift, but it is beneficial to highlight the main algebraic steps that lead to Compton’s formula, which we give here as follows.
We start with rearranging the terms in Equation 6.24 and squaring it:
$[(Ef−E˜f)+m0c2]2=E2.[(Ef−E˜f)+m0c2]2=E2.$
In the next step, we substitute Equation 6.19 for $E2,E2,$ simplify, and divide both sides by $c2c2$ to obtain
$(Ef/c−E˜f/c)2+2m0c(Ef/c−E˜f/c)=p2.(Ef/c−E˜f/c)2+2m0c(Ef/c−E˜f/c)=p2.$
Now we can use Equation 6.21 to express this form of the energy equation in terms of momenta. The result is
$(pf−p˜f)2+2m0c(pf−p˜f)=p2.(pf−p˜f)2+2m0c(pf−p˜f)=p2.$
6.26
To eliminate $p2,p2,$ we turn to the momentum equation Equation 6.25, rearrange its terms, and square it to obtain
$(p→f−p˜→f)2=p2and(p→f−p˜→f)2=pf2+p˜f2−2p→f·p˜→f.(p→f−p˜→f)2=p2and(p→f−p˜→f)2=pf2+p˜f2−2p→f·p˜→f.$
The product of the momentum vectors is given by Equation 6.23. When we substitute this result for $p2p2$ in Equation 6.26, we obtain the energy equation that contains the scattering angle $θ:θ:$
$(pf−p˜f)2+2m0c(pf−p˜f)=pf2+p˜f2−2pfp˜fcosθ.(pf−p˜f)2+2m0c(pf−p˜f)=pf2+p˜f2−2pfp˜fcosθ.$
With further algebra, this result can be simplified to
$1p˜f−1pf=1m0c(1−cosθ).1p˜f−1pf=1m0c(1−cosθ).$
6.27
Now recall Equation 6.21 and write: $1/p˜f=λ′/h1/p˜f=λ′/h$ and $1/pf=λ/h.1/pf=λ/h.$ When these relations are substituted into Equation 6.27, we obtain the relation for the Compton shift:
$λ′−λ=hm0c(1−cosθ).λ′−λ=hm0c(1−cosθ).$
6.28
The factor $h/m0ch/m0c$ is called the Compton wavelength of the electron:
$λc=hm0c=0.00243nm=2.43pm.λc=hm0c=0.00243nm=2.43pm.$
6.29
Denoting the shift as $Δλ=λ′−λ,Δλ=λ′−λ,$ the concluding result can be rewritten as
$Δλ=λc(1−cosθ).Δλ=λc(1−cosθ).$
6.30
This formula for the Compton shift describes outstandingly well the experimental results shown in Figure 6.12. Scattering data measured for molybdenum, graphite, calcite, and many other target materials are in accord with this theoretical result. The nonshifted peak shown in Figure 6.12 is due to photon collisions with tightly bound inner electrons in the target material. Photons that collide with the inner electrons of the target atoms in fact collide with the entire atom. In this extreme case, the rest mass in Equation 6.29 must be changed to the rest mass of the atom. This type of shift is four orders of magnitude smaller than the shift caused by collisions with electrons and is so small that it can be neglected.
Compton scattering is an example of inelastic scattering, in which the scattered radiation has a longer wavelength than the wavelength of the incident radiation. In today’s usage, the term “Compton scattering” is used for the inelastic scattering of photons by free, charged particles. In Compton scattering, treating photons as particles with momenta that can be transferred to charged particles provides the theoretical background to explain the wavelength shifts measured in experiments; this is the evidence that radiation consists of photons.
### Example 6.8
#### Compton Scattering
An incident 71-pm X-ray is incident on a calcite target. Find the wavelength of the X-ray scattered at a $30°30°$ angle. What is the largest shift that can be expected in this experiment?
#### Strategy
To find the wavelength of the scattered X-ray, first we must find the Compton shift for the given scattering angle, $θ=30°.θ=30°.$ We use Equation 6.30. Then we add this shift to the incident wavelength to obtain the scattered wavelength. The largest Compton shift occurs at the angle $θθ$ when $1−cosθ1−cosθ$ has the largest value, which is for the angle $θ=180°.θ=180°.$
#### Solution
The shift at $θ=30°θ=30°$ is
$Δλ=λc(1−cos30°)=0.134λc=(0.134)(2.43)pm=0.325pm.Δλ=λc(1−cos30°)=0.134λc=(0.134)(2.43)pm=0.325pm.$
This gives the scattered wavelength:
$λ′=λ+Δλ=(71+0.325)pm=71.325pm.λ′=λ+Δλ=(71+0.325)pm=71.325pm.$
The largest shift is
$(Δλ)max=λc(1−cos1800)=2(2.43pm)=4.86pm.(Δλ)max=λc(1−cos1800)=2(2.43pm)=4.86pm.$
#### Significance
The largest shift in wavelength is detected for the backscattered radiation; however, most of the photons from the incident beam pass through the target and only a small fraction of photons gets backscattered (typically, less than 5%). Therefore, these measurements require highly sensitive detectors.
An incident 71-pm X-ray is incident on a calcite target. Find the wavelength of the X-ray scattered at a $60°60°$ angle. What is the smallest shift that can be expected in this experiment?
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2021-07-27 12:19:36
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https://www.earthdoc.org/content/papers/10.3997/2214-4609-pdb.209.1993_057
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1887
PDF
### Abstract
DC resistivity and Induced Polarization (IP) surveys are commonly used in<br>engineering, ground water and environmental problems. The principle difficulty<br>arises with the interpretation of such data since neither the apparent resistivity<br>pseudosections nor the apparent chargeability pseudosections are adequate representations<br>of the subsurface structure. Such information can only be obtained<br>through rigorous inversion of the data. The solution for the distribution of electrical<br>conductivity that arises in a DC rtxistivity survey is a nonlinear inverse<br>problem. To solve this problem we discretize the earth into cells of constant conductivity<br>and then find a minimum structure model which adequately reproduces<br>the observations. This inversion is efficiently carried out using a generalized subspace<br>techniques so that the dimension of the matrix to be inverted is kept small.<br>The conductivity model obtained from inverting the DC resistivity data is used<br>as a background model for the inversion of IP data. The sensitivities from the<br>background conductivity provide a linear mapping between the apparent and intrinsic<br>chargeabilities. The intrinsic chargeability is therefore obtained by solving<br>a linear inverse problem. Again we use the subspace methodology but we also<br>impose the constraint that the intrinsic chargeability is positive. The inversion<br>algorithms are applied to synthetic and field data.
/content/papers/10.3997/2214-4609-pdb.209.1993_057
1993-04-18
2021-10-26
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2021-10-26 15:20:52
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https://www.lansweeper.com/forum/yaf_postst18918_Deploy.aspx
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Error
Deploy - Lounch a .bat file to remote pc
Posted: Wednesday, October 9, 2019 1:25:45 PM(UTC)
Asomich
Member Original PosterPosts: 1
0
Like
Hi there,
it's my first post and my English level is quit low. I hope you ca understand me!
So...
I'd like to be able to run a file ".bat" that start on the client the installation of the administrative image of Solidworks 2019. To me it is enough that the .bat file is executed, nothing more. So I created a 2 step package
- Condition: Check if the operating system is Win 10
- Script: Run the ".bat" file in the SERVER path Lansweeper script
This command is in the ".bat" file
echo off
cls
start \\SERVER\it_sw\Solidworks_19_SP2_Immagine_Amm2\startswinstall.exe /now
Probably the syntax is wrong; unfortunately I'm not good with scripts and DOS.
So I wanted to ask you for help on how to solve this
Thanks
Asomich attached the following image(s):
Does the batch file do anything else apart from calling the install?
I would use something like this. Note this is not testing by myself but just using the command line you specified but using it in an installer package instead of calling a batch file. The issue with batch files is they don't normally report exit codes so we don't exactly know why it failed etc.
Code:
<?xml version="1.0" encoding="utf-8"?>
<Package>
<Name>DEV.APP.Solidworks 2019</Name>
<Description></Description>
<ShutdownOption>0</ShutdownOption>
<ShutdownTime>0</ShutdownTime>
<MaxDuration>2700</MaxDuration>
<Rescan>True</Rescan>
<RunMode>2</RunMode>
<Steps>
<Step>
<Nr>1</Nr>
<Name>Check if Windows 10</Name>
<Type>5</Type>
<ReturnCodes></ReturnCodes>
<Success>2</Success>
<Failure>-3</Failure>
<Path></Path>
<Parameters></Parameters>
<MSIParameters></MSIParameters>
<MSIName></MSIName>
<MSIVersion></MSIVersion>
<Command></Command>
<EditMode>False</EditMode>
<Conditions>
<Condition>
<Type>4</Type>
<SpecOne></SpecOne>
<SpecTwo></SpecTwo>
<Operator></Operator>
<Value>Win 10</Value>
</Condition>
</Conditions>
</Step>
<Step>
<Nr>2</Nr>
<Name>Install Solidworks</Name>
<Type>1</Type>
<ReturnCodes>0,1641,3010</ReturnCodes>
<Success>-2</Success>
<Failure>-3</Failure>
<Path>\\SERVER\it_sw\Solidworks_19_SP2_Immagine_Amm2\startswinstall.exe</Path>
<Parameters>/now</Parameters>
<MSIParameters>/i /qn /norestart</MSIParameters>
<MSIName></MSIName>
<MSIVersion></MSIVersion>
<Command>"\\SERVER\it_sw\Solidworks_19_SP2_Immagine_Amm2\startswinstall.exe" /now </Command>
<EditMode>False</EditMode>
<Conditions />
</Step>
</Steps>
<SoftwareVersion>7.0.110.2</SoftwareVersion>
</Package>
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2020-07-14 22:55:19
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https://riverml.xyz/0.15.0/recipes/bandits-101/
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# Multi-armed bandits¶
River has a bandit module. It contains several multi-armed bandit policies, bandit environments, and utilities to benchmark policies on bandit problems.
Bandit environments in River implement the Gym interface. You can thus load them with gym.make. Note that Gym is intended for reinforcement learning algorithms, while bandit policies are the simplest form of reinforcement learing. Bandit policies learn by receiving a reward after each step, while reinforcement learning algorithms have to learn from feedback that may arrive at the end of a (long) sequence of steps.
import gym
for k in gym.envs.registry:
if k.startswith('river_bandits'):
print(k)
River's bandit module offers the bandit.evaluate function to benchmark several policies on a given environment. It takes as input a list of bandit policies, a bandit environment (the problem to solve), and a reward object. There is a also a pull_func parameter that is necessary to define how a policy is supposed to pull an arm from a given environment.
import gym
from river import bandit
import pandas as pd
from tqdm import tqdm
from river import stats
def pull_func(policy, env):
return next(policy.pull(range(env.action_space.n)))
policies=[
bandit.EpsilonGreedy(epsilon=0.1),
bandit.EpsilonGreedy(epsilon=0.01),
bandit.EpsilonGreedy(epsilon=0),
]
env = gym.make(
'river_bandits/KArmedTestbed-v0',
max_episode_steps=1000
)
trace = bandit.evaluate(
policies=policies,
env=env,
reward_stat=stats.Mean(),
pull_func=pull_func,
n_episodes=(n_episodes := 2000),
)
The bandit.evaluate function returns a generator containing the results at each step of the benchmark. This can be wrapped with a pandas.DataFrame to gather all the results.
trace_df = pd.DataFrame(tqdm(
trace, position=0, total=(
n_episodes *
len(policies) *
env._max_episode_steps
)
))
trace_df.sample(5, random_state=42)
0%| | 0/6000000 [00:00<?, ?it/s]/opt/hostedtoolcache/Python/3.9.16/x64/lib/python3.9/site-packages/gym/utils/passive_env_checker.py:233: DeprecationWarning: np.bool8 is a deprecated alias for np.bool_. (Deprecated NumPy 1.24)
if not isinstance(terminated, (bool, np.bool8)):
100%|██████████| 6000000/6000000 [01:30<00:00, 66210.22it/s]
episode step policy_idx action reward reward_stat
1324896 441 632 0 5 1.510163 1.153443
3566176 1188 725 1 1 1.265768 0.437191
1109043 369 681 0 3 1.854132 1.194537
4286042 1428 680 2 2 1.987844 2.144434
5395174 1798 391 1 0 -1.119788 1.155535
It is then straightforward to plot the average reward each policy obtains at each step, by averaging over episodes.
policy_names = {
0: 'ε = 0.1',
1: 'ε = 0.01',
2: 'ε = 0 (greedy)'
}
colors = {
'ε = 0.1': 'tab:blue',
'ε = 0.01': 'tab:red',
'ε = 0 (greedy)': 'tab:green'
}
(
trace_df
.assign(policy=trace_df.policy_idx.map(policy_names))
.groupby(['step', 'policy'])
['reward'].mean()
.unstack()
.plot(color=colors)
)
<Axes: xlabel='step'>
## Controlling the evaluation loop¶
The bandit.evaluate function is useful for benchmarking. But in practice, you'll want to have control over your bandit policy. Indeed you'll want the freedom to pull arms (with the pull method) and update the policy (with the update method) at your discretion.
As an example, the following is a possible reimplementation of the bandit.evaluate function. Here we'll be measuring the rate at which each policy selects the optimal arm.
Note how the pull and update methods are used.
import copy
policies=[
bandit.EpsilonGreedy(epsilon=0.1),
bandit.EpsilonGreedy(epsilon=0.01),
bandit.EpsilonGreedy(epsilon=0),
]
env = gym.make(
'river_bandits/KArmedTestbed-v0',
max_episode_steps=1000
)
n_episodes = 2000
trace = []
with tqdm(total=len(policies) * n_episodes * env._max_episode_steps, position=0) as progress:
for policy in policies:
for episode in range(n_episodes):
episode_policy = policy.clone()
episode_env = copy.deepcopy(env)
episode_env.reset()
step = 0
while True:
action = next(episode_policy.pull(range(episode_env.action_space.n)))
observation, reward, terminated, truncated, info = episode_env.step(action)
best_action = observation
episode_policy.update(action, reward)
trace.append({
"episode": episode,
"step": step,
"policy": f"ε = {policy.epsilon}",
"is_action_optimal": action == best_action
})
step += 1
progress.update()
if terminated or truncated:
break
trace_df = pd.DataFrame(trace)
0%| | 0/6000000 [00:00<?, ?it/s]/opt/hostedtoolcache/Python/3.9.16/x64/lib/python3.9/site-packages/gym/utils/passive_env_checker.py:233: DeprecationWarning: np.bool8 is a deprecated alias for np.bool_. (Deprecated NumPy 1.24)
if not isinstance(terminated, (bool, np.bool8)):
100%|██████████| 6000000/6000000 [01:30<00:00, 66658.41it/s]
colors = {
'ε = 0.1': 'tab:blue',
'ε = 0.01': 'tab:red',
'ε = 0': 'tab:green'
}
(
trace_df
.groupby(['step', 'policy'])
['is_action_optimal'].mean()
.unstack()
.plot(color=colors)
)
<Axes: xlabel='step'>
## Handling drift¶
The environment used above is a toy situation used for introducing bandits. It is stationary, meaning that the expected reward of each arm does not change over time.
In practice, arms are dynamic, and their performance can vary over time. A simple example of this is the Candy Cane Contest that was hosted on Kaggle in 2020. The expected reward of each arm diminishes each time it is pulled.
The way bandit policies in River deal with drift depends on the method. For the bandit.EpsilonGreedy policy, it makes sense to use a rolling average as the reward object. What this means is that the empirical reward the policy calculates for each arm is a rolling average, rather than a global one.
from river import proba, utils
policies=[
bandit.EpsilonGreedy(
epsilon=0.1,
seed=42
),
bandit.EpsilonGreedy(
epsilon=0.3,
reward_obj=utils.Rolling(stats.Mean(), window_size=50),
seed=42
),
bandit.ThompsonSampling(
dist=proba.Beta(),
seed=42
)
]
env = gym.make('river_bandits/CandyCaneContest-v0')
trace = bandit.evaluate(
policies=policies,
env=env,
pull_func=pull_func,
n_episodes=(n_episodes := 30),
seed=42
)
trace_df = pd.DataFrame(tqdm(
trace, position=0, total=(
n_episodes *
len(policies) *
env._max_episode_steps
)
))
0%| | 0/180000 [00:00<?, ?it/s]/opt/hostedtoolcache/Python/3.9.16/x64/lib/python3.9/site-packages/gym/utils/passive_env_checker.py:233: DeprecationWarning: np.bool8 is a deprecated alias for np.bool_. (Deprecated NumPy 1.24)
if not isinstance(terminated, (bool, np.bool8)):
100%|██████████| 180000/180000 [00:29<00:00, 6062.61it/s]
We can compare the performance of each policy by checking the average reward at the end of each episode.
(
trace_df
.groupby(['policy_idx', 'episode'])
.last()
.groupby('policy_idx')
.reward_stat.mean()
)
policy_idx
0 736.1
1 817.0
2 854.0
Name: reward_stat, dtype: float64
We see that using a rolling average gives a boost to the epsilon greedy strategy. However, we see that the bandit.ThompsonSampling policy performs even better, even though no particular care was given to drift. A natural next step would thus be to see how it could be improved to handle drift. For instance, its dist parameter could be wrapped with a utils.Rolling:
policy = bandit.ThompsonSampling(
dist=utils.Rolling(proba.Beta(), window_size=50),
seed=42
)
Bandits can be used for several tasks. They can be used for content personalization, as well as online model selection (see model_selection.BanditRegressor). The policies in River are therefore designed to be flexible, so that they can be used in conjunction with other River modules. For instance, the reward_obj in bandit.EpsilonGreedy can be a metric, a probability distribution, or a statistic. This works because objects in River adher to a coherent get/update interface.
|
2023-03-21 00:41:53
|
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|
http://mathoverflow.net/revisions/97373/list
|
Aside from the conceptual challenge of functions themselves, students find limits difficult because of their quantifier complexity. I have never understood why standard algebra pedagogy suppresses quantifiers, thus, for example, leaving many students unable to distinguish between unknowns (literals bound by existential quantifiers), variables (literals bound by universal quantifiers) and constants (literals that belong to the language itself). Students who miscalculate the derivative of $\pi^2$, mentioned elsewhere, don't get this distinction. People who become mathematicians usually "got it" without anyone spelling all this out, and then they learned about quantifiers studying logic in college, so they regard quantifiers as sophisticated and advanced. But most students don't "get it," and I think this accounts for the huge attitude downturn when they get to algebra.
|
2013-05-21 23:21:04
|
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|
https://www.zbmath.org/authors/?q=ai%3Axu.feng.1
|
zbMATH — the first resource for mathematics
Xu, Feng
Compute Distance To:
Author ID: xu.feng.1 Published as: Xu, Feng; Xu, F.
Documents Indexed: 156 Publications since 1981
all top 5
Co-Authors
14 single-authored 6 Dong, Chongying 6 Puig, Vicenç 5 Jensen, Oliver E. 5 Liang, Bin 5 Lu, Tianjian 5 Tan, Junbo 5 Wang, Xueqian 4 Lei, Chengwang 4 Lu, Yufeng 4 Ocampo-Martinez, Carlos 4 Olaru, Sorin 4 Patterson, John C. 4 Seffen, Keith A. 4 Sheng, Zhaohan 4 Zhou, Shengwu 3 Wong, Dein 3 Yao, Sheng 3 Yu, Nina 2 Aravas, Nikolaos 2 Billingham, John 2 Chen, Ju 2 Chen, Wenli 2 Cheung, Yeuk-Kwan E. 2 Dai, Qionghai 2 Ha, Tiejun 2 Han, Guiling 2 Hong, Jialin 2 Hong, Wei 2 Li, Ben Q. 2 Li, Bingxi 2 Lu, Kewei 2 Ma, Hui 2 Niculescu, Silviu-Iulian 2 Ou, Jinping 2 Ren, Li 2 Sofronis, P. 2 Soyer, Refik 2 Tian, Fenglei 2 Wang, Tianheng 2 Wang, Ye 2 Wang, Yuan 2 Wu, Ke 2 Wu, Qunying 2 Xu, Fuyi 2 Yang, Qingzhen 2 Yang, Songlin 2 Yang, Yuansheng 2 Yu, Pei 2 Yuan, Bo 2 Zhang, Yongqing 2 Zhao, Hai 2 Zhao, Peng 2 Zheng, Shiqiu 2 Zou, Chengzu 1 Aktekin, Tevfik 1 An, Na 1 Bai, Wei-Feng 1 Bespalov, Alexei 1 Cai, Xiaoshu 1 Cao, Chun 1 Cao, Zhigang 1 Cembrano, Gabriela 1 Cen, Zhangzhi 1 Chai, Jin-xiang 1 Chen, Guohua 1 Chen, Yanguang 1 Chen, Zhenbo 1 Chong, Heap-Yih 1 Chu, Jing 1 Coveney, Sam 1 Cui, Tiejun 1 Dong, Lili 1 Duan, Zhongdong 1 Eastwood, Michael G. 1 Fa, Wenzhe 1 Feng, Qinghua 1 Gao, Juan 1 Gao, Yan 1 Glickman, Theodore S. 1 Guan, Juan 1 Han, Baoling 1 Hau, Sau F. 1 He, Huixa 1 He, Huixia 1 He, Zekai 1 Herwartz, Helmut 1 Huang, Hao 1 Huang, Yourui 1 Huo, Lifang 1 Hurtado, Martin 1 Ji, Xia 1 Jiang, Chunlan 1 Jiang, Lihua 1 Jiao, Lin 1 Jiao, Xiangyu 1 Jin, Tian 1 Jin, Yaqiu 1 Lau, Francis Chung-ming 1 Lee, Sungyon 1 Li, Dahua ...and 106 more Co-Authors
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Serials
6 Journal of Fluid Mechanics 5 International Journal of Heat and Mass Transfer 5 Mathematics in Practice and Theory 4 Automatica 4 Journal of Northeast Normal University. Natural Science Edition 4 International Journal of Robust and Nonlinear Control 3 Advances in Mathematics 3 Journal of Nanjing University. Natural Sciences 3 Journal of Systems Science and Mathematical Sciences 3 Linear Algebra and its Applications 3 IEEE Transactions on Antennas and Propagation 3 Journal of Software 3 Journal of Shandong University. Natural Science 2 Computers and Fluids 2 Computers & Mathematics with Applications 2 Communications in Mathematical Physics 2 International Journal of Engineering Science 2 Journal of the Mechanics and Physics of Solids 2 Hadronic Journal 2 Applied Mathematics and Computation 2 IEEE Transactions on Automatic Control 2 Journal of Algebra 2 Transactions of the American Mathematical Society 2 Statistics & Probability Letters 2 Applied Mathematical Modelling 2 Journal of High Energy Physics 2 Journal of Northeastern University. Natural Science 2 Journal of Hefei University of Technology. Natural Science 1 Journal of Engineering Mathematics 1 Journal of the Franklin Institute 1 Journal of Mathematical Analysis and Applications 1 Mathematical Notes 1 Physics of Fluids 1 Physics Letters. A 1 Quarterly Journal of Mechanics and Applied Mathematics 1 Chaos, Solitons and Fractals 1 Acta Mathematica Sinica 1 Bulletin of the London Mathematical Society 1 International Journal for Numerical Methods in Engineering 1 Journal of the Korean Mathematical Society 1 Journal of Pure and Applied Algebra 1 Proceedings of the American Mathematical Society 1 Utilitas Mathematica 1 Optimal Control Applications & Methods 1 Journal of Sichuan University. Natural Science Edition 1 Journal of Mathematics. Wuhan University 1 Chinese Annals of Mathematics. Series A 1 Journal of Engineering Mathematics (Xi’an) 1 Journal of Beijing Normal University. Natural Science 1 Graphs and Combinatorics 1 ACM Transactions on Graphics 1 Northeastern Mathematical Journal 1 Journal of Qufu Normal University. Natural Science 1 Mathematica Applicata 1 Differential Geometry and its Applications 1 Annals of Physics 1 Computational Statistics and Data Analysis 1 Journal of Nanjing University of Aeronautics and Astronautics 1 Statistical Papers 1 Electronic Journal of Differential Equations (EJDE) 1 Complexity 1 Engineering Analysis with Boundary Elements 1 Journal of Luoyang University 1 Science in China. Series E 1 Nonlinear Dynamics 1 Journal of Inequalities and Applications 1 The Asian Journal of Mathematics 1 Fractional Calculus & Applied Analysis 1 Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Discrete Dynamics in Nature and Society 1 Journal of Applied Statistics 1 International Journal of Numerical Modelling 1 European Journal of Mechanics. A. Solids 1 Italian Journal of Pure and Applied Mathematics 1 International Journal of Applied Mathematics and Computer Science 1 Applied Stochastic Models in Business and Industry 1 Journal of Northwest Normal University. Natural Science 1 IEEE Transactions on Image Processing 1 Far East Journal of Dynamical Systems 1 Journal of Zhejiang University. Science Edition 1 Journal of Applied Mathematics 1 Journal of Jilin University. Science Edition 1 Journal of University of Science and Technology of China 1 Transactions of Beijing Institute of Technology 1 Control and Decision 1 Science in China. Series F 1 Journal of Lanzhou Jiaotong University. Natural Sciences 1 Journal of Industrial and Management Optimization 1 Journal of Zhejiang University. Science A 1 Journal of Anhui University. Natural Science Edition 1 Systems Engineering and Electronics 1 Acta Mechanica Sinica 1 Journal of Shanghai Second Polytechnic University 1 International Journal of Mathematical Analysis (Ruse) 1 Journal of Xinyang Normal University. Natural Science Edition 1 The Annals of Applied Statistics 1 Foundations and Trends in Signal Processing 1 Science China. Mathematics 1 Science China. Technological Sciences 1 Operations Research Transactions ...and 3 more Serials
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21 Systems theory; control (93-XX) 19 Fluid mechanics (76-XX) 15 Mechanics of deformable solids (74-XX) 14 Computer science (68-XX) 12 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 12 Information and communication theory, circuits (94-XX) 11 Operator theory (47-XX) 10 Combinatorics (05-XX) 10 Probability theory and stochastic processes (60-XX) 10 Classical thermodynamics, heat transfer (80-XX) 10 Quantum theory (81-XX) 9 Functional analysis (46-XX) 9 Biology and other natural sciences (92-XX) 8 Statistics (62-XX) 8 Numerical analysis (65-XX) 7 Ordinary differential equations (34-XX) 6 Nonassociative rings and algebras (17-XX) 6 Partial differential equations (35-XX) 6 Operations research, mathematical programming (90-XX) 5 Optics, electromagnetic theory (78-XX) 4 Group theory and generalizations (20-XX) 4 Dynamical systems and ergodic theory (37-XX) 4 Differential geometry (53-XX) 2 Several complex variables and analytic spaces (32-XX) 2 Manifolds and cell complexes (57-XX) 2 Global analysis, analysis on manifolds (58-XX) 1 Mathematical logic and foundations (03-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Associative rings and algebras (16-XX) 1 Category theory; homological algebra (18-XX) 1 Topological groups, Lie groups (22-XX) 1 Real functions (26-XX) 1 Measure and integration (28-XX) 1 Potential theory (31-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Relativity and gravitational theory (83-XX) 1 Astronomy and astrophysics (85-XX)
Citations contained in zbMATH Open
66 Publications have been cited 313 times in 161 Documents Cited by Year
Quantum dimensions and quantum Galois theory. Zbl 1337.17018
Dong, Chongying; Jiao, Xiangyu; Xu, Feng
2013
Non-Fourier analysis of skin biothermomechanics. Zbl 1144.80358
Xu, F.; Seffen, K. A.; Lu, T. J.
2008
On orbifold theory. Zbl 1419.17031
Dong, Chongying; Ren, Li; Xu, Feng
2017
Complex dynamics analysis for a duopoly advertising model with nonlinear cost. Zbl 1139.91365
Yao, Hong-Xing; Xu, Feng
2006
Chaos control and chaos synchronization for multi-scroll chaotic attractors generated using hyperbolic functions. Zbl 1176.93033
Xu, F.; Yu, Pei
2010
Wave scattering by a thin elastic plate floating on a two-layer fluid. Zbl 1213.35390
Xu, F.; Lu, D. Q.
2010
Biothermomechanics of skin tissues. Zbl 1162.74406
Xu, F.; Lu, T. J.; Seffen, K. A.
2008
A differential operator for integrating one-loop scattering equations. Zbl 1373.81368
Wang, Tianheng; Chen, Gang; Cheung, Yeuk-Kwan E.; Xu, Feng
2017
The distribution of the product of two triangular random variables. Zbl 1154.62312
Glickman, Theodore S.; Xu, Feng
2008
Complexity analysis of remanufacturing duopoly game with different competition strategies and heterogeneous players. Zbl 1348.91009
Shi, Lian; Sheng, Zhaohan; Xu, Feng
2015
New lower bounds on the multicolor Ramsey numbers $$R_{r}(C_{2m})$$. Zbl 1099.05062
Sun, Yongqi; Yang, Yuansheng; Xu, Feng; Li, Bingxi
2006
Chaos control and synchronization for a special generalized Lorenz canonical system - the SM system. Zbl 1197.37037
Liao, Xiaoxin; Xu, F.; Wang, Pan; Yu, Pei
2009
Some properties on Pareto-eigenvalues of higher-order tensors. Zbl 1349.15030
Xu, Feng; Ling, Chen
2015
Generalized set-theoretic unknown input observer for LPV systems with application to state estimation and robust fault detection. Zbl 1386.93058
Xu, Feng; Tan, Junbo; Wang, Xueqian; Puig, Vicenç; Liang, Bin; Yuan, Bo; Liu, Houde
2017
A combinatoric shortcut to evaluate CHY-forms. Zbl 1380.81223
Wang, Tianheng; Chen, Gang; Cheung, Yeuk-Kwan E.; Xu, Feng
2017
Pseudo-holomorphic curves in nearly Kähler $$\mathbf {CP}^{3}$$. Zbl 1184.53073
Xu, Feng
2010
On instantons on nearly Kähler 6-manifolds. Zbl 1209.53042
Xu, Feng
2009
Set-theoretic methods in robust detection and isolation of sensor faults. Zbl 1333.93153
Xu, Feng; Puig, Vicenç; Ocampo-Martinez, Carlos; Olaru, Sorin; Stoican, Florin
2015
Robust MPC for actuator-fault tolerance using set-based passive fault detection and active fault isolation. Zbl 1367.93171
Xu, Feng; Puig, Vicenç; Ocampo-Martinez, Carlos; Olaru, Sorin; Niculescu, Silviu-Iulian
2017
Drop spreading with random viscosity. Zbl 1371.76016
Xu, Feng; Jensen, Oliver E.
2016
The 3-permutation orbifold of a lattice vertex operator algebra. Zbl 1395.17067
Dong, Chongying; Xu, Feng; Yu, Nina
2018
On the double-layer structure of the boundary layer adjacent to a sidewall of a differentially heated cavity. Zbl 1148.80353
Xu, Feng; Patterson, John C.; Lei, Chengwang
2008
2-permutations of lattice vertex operator algebras: higher rank. Zbl 1395.17066
Dong, Chongying; Xu, Feng; Yu, Nina
2017
Assessment of mortgage default risk via Bayesian state space models. Zbl 1283.62211
Aktekin, Tevfik; Soyer, Refik; Xu, Feng
2013
Fractal properties of the generalized chaikin corner-cutting subdivision scheme. Zbl 1219.65026
Wang, Juan; Zheng, Hongchan; Xu, Feng; Liu, Dekong
2011
A new approach to bootstrap inference in functional coefficient models. Zbl 1453.62110
Herwartz, H.; Xu, F.
2009
Transient natural convection flows around a thin fin on the sidewall of a differentially heated cavity. Zbl 1183.76875
Xu, Feng; Patterson, John C.; Lei, Chengwang
2009
Divergence-driven oscillations in a flexible-channel flow with fixed upstream flux. Zbl 1287.76253
Xu, Feng; Billingham, John; Jensen, Oliver E.
2013
Globally exponential stability of nonlinear impulsive switched systems. Zbl 1321.93056
Xu, F.; Dong, L.; Wang, D.; Li, X.; Rakkiyappan, R.
2015
Resonance-driven oscillations in a flexible-channel flow with fixed upstream flux and a long downstream rigid segment. Zbl 1309.76236
Xu, Feng; Billingham, John; Jensen, Oliver E.
2014
On eigenmaps between spheres. Zbl 1033.58014
He, Huixia; Ma, Hui; Xu, Feng
2003
On relative entropy and global index. Zbl 1451.46051
Xu, Feng
2020
Multidimensional discontinuous SPH method and its application to metal penetration analysis. Zbl 1352.74121
Xu, F.; Zhao, Y.; Yan, R.; Furukawa, T.
2013
2-cyclic permutations of lattice vertex operator algebras. Zbl 1395.17065
Dong, Chongying; Xu, Feng; Yu, Nina
2016
Biothermomechanical behavior of skin tissue. Zbl 1257.74107
Xu, F.; Lu, T. J.; Seffen, K. A.
2008
Structural optimization of two-dimensional cellular metals cooled by forced convection. Zbl 1119.80362
Wen, T.; Xu, F.; Lu, T. J.
2007
Assessment of mortgage default risk via Bayesian reliability models. Zbl 1226.91081
Soyer, Refik; Xu, Feng
2010
On the wrinkling and restabilization of highly stretched sheets. Zbl 1425.74309
Wang, T.; Fu, C.; Xu, F.; Huo, Y.; Potier-Ferry, M.
2019
B spline-based method for 2-D large deformation analysis. Zbl 1259.74091
Liu, Yanan; Sun, Liang; Xu, Feng; Liu, Yinghua; Cen, Zhangzhi
2011
On the multiplicity of $$\alpha$$ as an eigenvalue of the $$a_\alpha$$ matrix of a graph in terms of the number of pendant vertices. Zbl 1436.05067
Xu, Feng; Wong, Dein; Tian, Fenglei
2020
Some results on relative entropy in quantum field theory. Zbl 1437.81047
Xu, Feng
2020
Greenberger-Horne-Zeilinger correlation and Bell-type inequality seen from a moving frame. Zbl 1123.81326
You, Hao; Wang, An Min; Yang, Xiaodong; Niu, Wanqing; Ma, Xiaosan; Xu, Feng
2004
Numerical analysis of the Rayleigh-Taylor instability in an electric field. Zbl 1381.76419
Yang, Qingzhen; Li, Ben Q.; Zhao, Zhengtuo; Shao, Jinyou; Xu, Feng
2016
A multilayer feed forward small-world neural network controller and its application on electrohydraulic actuation system. Zbl 1271.92004
Li, Xiaohu; Xu, Feng; Zhang, Jinhua; Wang, Sunan
2013
Robust fault detection and isolation based on zonotopic unknown input observers for discrete-time descriptor systems. Zbl 1415.93098
Wang, Ye; Puig, Vicenç; Xu, Feng; Cembrano, Gabriela
2019
Transition to a periodic flow induced by a thin fin on the sidewall of a differentially heated cavity. Zbl 1156.80380
Xu, Feng; Patterson, John C.; Lei, Chengwang
2009
Atom localization via controlled spontaneous emission in a five-level atomic system. Zbl 1243.81235
Wang, Zhiping; Yu, Benli; Zhu, Jun; Cao, Zhigang; Zhen, Shenglai; Wu, Xuqiang; Xu, Feng
2012
Symmetries of subfactors motivated by Aschbacher-Guralnick conjecture. Zbl 1377.20025
Xu, Feng
2016
A low-order model for slamming in a flexible-channel flow. Zbl 1331.74060
Xu, Feng; Jensen, Oliver E.
2015
Characterization of oriented graphs of rank 2. Zbl 1419.05094
Zhang, Yuanshuai; Xu, Feng; Wong, Dein
2019
Jones-Wassermann subfactors for modular tensor categories. Zbl 1443.46038
Liu, Zhengwei; Xu, Feng
2019
Multiple positive solutions for nonlinear second-order m-point boundary-value problems with sign changing nonlinearities. Zbl 1171.34015
Xu, Fuyi; Chen, Zhenbo; Xu, Feng
2008
Constitutive modeling of solid propellant materials with evolving microstructural damage. Zbl 1162.74432
Xu, F.; Aravas, N.; Sofronis, P.
2008
Complex unit gain graphs of rank 2. Zbl 1437.05157
Xu, Feng; Zhou, Qi; Wong, Dein; Tian, Fenglei
2020
Examples of subfactors from conformal field theory. Zbl 1386.81136
Xu, Feng
2018
European option pricing by the mixed fractional Brownian motion with Vasicek interest rate. Zbl 1363.91117
Xu, Feng
2015
Stress analysis of anisotropic thick laminates in cylindrical bending using a semi-analytical approach. Zbl 1303.74004
Lü, Chao-Feng; Lim, C. W.; Xu, Feng
2007
Constitutive modeling of porous viscoelastic materials. Zbl 1123.74015
Xu, F.; Sofronis, P.; Aravas, N.; Meyer, S.
2007
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2021-08-04 02:36:37
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http://mathhelpforum.com/calculus/279364-finding-inequality-arithmetic-geometric-means-sequence.html
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# Thread: finding the Inequality of arithmetic and geometric means of a sequence
1. ## finding the Inequality of arithmetic and geometric means of a sequence
Hello everyone
I was requested to prove this inequality
for this sequence
using Inequality arithmetic and geometric means (using arithmetic mean and geometric mean only)
I tried to solve this thing out in so many ways I will be really glad if someone will show me the full solution for this Inequality
thanks
2. ## Re: finding the Inequality of arithmetic and geometric means of a sequence
Originally Posted by someone111888
I was requested to prove this inequality
for this sequence
As posted, the statement is not true.
$a_1=\dfrac{1}{2},~a_2=\dfrac{7}{12},~a_3=\dfrac{3 7}{60},\cdots~a_9=\dfrac{1632341}{2450448}$
Could it be that $N\ge 10~?$ Are you sure that you have copied it correctly?
3. ## Re: finding the Inequality of arithmetic and geometric means of a sequence
Yes I did, my intesntion is that there is a requestment to prove that the sequence itself bigger than 10 using this equality
4. ## Re: finding the Inequality of arithmetic and geometric means of a sequence
Im sorry bigger than 2/3
5. ## Re: finding the Inequality of arithmetic and geometric means of a sequence
Originally Posted by someone111888
Yes I did, my intesntion is that there is a requestment to prove that the sequence itself bigger than 10 using this equality
Originally Posted by someone111888
Im sorry bigger than 2/3
Now I am totally confused.
Just post the exact wording of the original question.
6. ## Re: finding the Inequality of arithmetic and geometric means of a sequence
Originally Posted by Plato
Now I am totally confused.
What else is new?
7. ## Re: finding the Inequality of arithmetic and geometric means of a sequence
Originally Posted by someone111888
Im sorry bigger than 2/3
Then DenisB just gave you a proof by contradiction showing that inequality is false for n <10. You cannot prove a false statement to be true. He gave you explicit examples showing that it is false.
$a_{n+1}=a_n+\dfrac{1}{2n+1}+\dfrac{1}{2n+2}-\dfrac{1}{n+1} = a_n+\dfrac{1}{(2n+1)(2n+2)}$
So your sequence is increasing as n increases. You should be able to find the minimum value of n that makes the equality true
http://m.wolframalpha.com/input/?i=l...to+infinity%5D
8. ## Re: finding the Inequality of arithmetic and geometric means of a sequence
Notice first that for any positive integer n, $\int_n^{2n}1/x\,dx=\ln(2)$. Next by consideration of upper and lower Riemann sums for this integral, it is easy to show that
$$\ln(2)-{1\over2n}\leq a_n\leq\ln(2)$$
Hence
$$\lim_{n\to\infty}a_n=\ln(2)$$
Since $\ln(2)\approx0.693$, from the above inequality, it is obvious that eventually $a_n>2/3$. As was noted in above posts, the inequality is true for any $n\geq10$. To me, this was kind of a strange question; I think a better question would be to find the limit of the sequence, as done above.
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2018-08-19 10:11:24
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http://clay6.com/qa/11071/a-particle-moves-in-the-plane-according-to-the-law-x-kt-y-kt-1-alpha-t-wher
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Browse Questions
# A particle moves in the plane according to the law $x=kt$, $y=kt(1- \alpha t)$ where $k$ and $\alpha$ are positive constants, t is time. The particle trajectory $y(x)$ is
a) circle b) parabola c) straight line d) hyperbola
$x=kt,y=kt(1-\alpha t)$
$y=kt-k\alpha t^2$
$y=x-k\; \alpha \;t^2$
$y=x-k \;\alpha\;\large\frac{x^2}{k^2}$
ie parabola
Hence b is the correct answer.
edited Jan 26, 2014 by meena.p
+1 vote
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2017-02-21 05:13:48
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https://math.stackexchange.com/questions/525080/discrete-fourier-transform-understand-negative-frequencies
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# Discrete Fourier Transform: Understand Negative Frequencies
I am trying to learn DFT on my own. I have been struggling for a while now around understanding the concept of negative frequencies and notably what happens when $k$ is greater than $N/2$ in the equation:
$$X[k] = {1 \over N} \sum_{n=0}^{N-1} s[n] e^{-i 2 \pi {1\over N} k n}$$
This really frustrates me because I don't see it explained anywhere while it seems to be something so central and probably not that hard to explain!!! So I really need help from someone please, because I am just not progressing because of this.
So where I am at with this. I understand Euler's Formula notably the identity $e^{i\pi} = e^{-i\pi} = -1$. So far I have only being able to suspect that this is the key to my problem. So, if we have $N = 8$ for example when we get $k = {N/2}$ then we gave $k = 4$ which gives $\displaystyle e^{-i2\pi {1 \over N } 4 n} = e^{ -i\pi n}$. So I suspect something happens at this point such as a change in sign somewhere making $e^{-ixxx}$ something like $e^{ixxx}$ but I can't go any further.
Could someone please please try to help me with this, and help me going further?
Thank you very much.
Negative frequencies are best thought of as nothing more than a mathematical curiosity. Historically, Fourier series were sums of sines and cosines. But mathematicians see this as aesthetically repulsive because one must often treat the sine terms and the cosine terms differently. Also, the difference between sines and cosines is a phase shift, and it seems strange that constant phase shifts would be fundamental to this simple idea of breaking functions into periodic components.
But, as you mentioned, we can use Euler's identity $e^{i\omega}=\cos(\omega)+i\sin(\omega)$ and some basic trigonometric laws to find that $$\cos(\omega)=\frac{e^{i\omega}+e^{-i\omega}}{2},$$ $$\sin(\omega)=\frac{e^{i\omega}-e^{-i\omega}}{2i}.$$
Now, mathematically, the exponentials are much simpler, because both the sine and the cosine terms can be shoved into a single type of function. But unfortunately you have to lose the physical meaning of frequency for the negative exponents. I have heard that for some reason the frequencies above the halfway point are unreliable when they are taken from sample data. So the switch into negative frequencies for the DFT, disregarding the upper half, does not result in a loss of physical information.
(I admit I don't understand why completely, but I do know it is a theoretical problem, not just a measurement issue. The key word here is "aliasing" if you want the internet to teach you more about it.)
To answer an implicit question you asked in the middle there: you can do some translation. Euler's identity implies that $s[k]e^{2\pi i}=1$, so $s[k]e^{i(2\pi k/N)} = s[k]e^{i(2\pi k/N-2\pi)}$ which for $k<N$ will give a negative frequency in the exponent. Furthremore, for $k>N/2$ you will get a negative frequency with absolute value at most $i2\pi N/2$, so that will get you the negative values you're looking for.
• Thank you for this long answer. I don't understand in the last paragraph. How can the exponent be negative for $k < N$? $k \over N$ is always smaller or equal to 1, thus $2\pi - 2\pi k /N$ is always positive? Sorry I must be missing something obvious. Thank you again. – Marc Ourens Oct 13 '13 at 21:55
• @Marc: Ah, sorry, I meant the terms to be switched around. So not $2\pi - 2\pi k /N$ but $2\pi k /N-2\pi$ – Eric Stucky Oct 13 '13 at 21:58
• Ah good, could you eventually make the edit please? Also when you say "What is interesting is that for some reason the frequencies above the halfway point are unreliable when they are taken from sample data. So the switch into negative frequencies for the DFT, disregarding the upper half, does not result in a loss of physical information". Sorry I have to say;-) It has to be for something otherwise why computing them? If this wasn't useful then the DFT equation would only compute the N/2 first coefficients? – Marc Ourens Oct 13 '13 at 22:01
• @Marc: I really don't know exactly what the issue is, sorry. But I can at least say this, which may be enough for you: if your process is real-valued (as most real processes are!) then the coefficients attached to $\omega$ and $-\omega$ must be complex conjugates. This is an easy consequence of the two formulas I showed you, and perhaps worth working out on your own. So you really do only need the positive frequencies to get all the data. That is why I think of the negative frequencies as a mathematical curiosity :) – Eric Stucky Oct 13 '13 at 22:04
You know I was searching for an answer to a question similar to this myself, because the wikipedia article on Discrete Fourier Transform omits the negative index. I have found many satisfying answers by playing with the formula. There is a tangent section which is not in direct relationship to your question that I think you might find interesting.
In short, negative frequencies (or indices) correspond to counter rotations. If positive frequency goes counterclockwise, then negative frequency goes clockwise.
If you go to the wikipedia page for Fourier Series, the complex version does include negative index.
$$s_N(x) = \sum_{n=-N}^{N} c_n e^{-i2\pi n x/T}$$
I also discovered it is necessary to include negative index for the Discrete Fourier Transform (DFT), take the DFT of a sampled cosine:
$$x_n = \cos (2\pi n/N)$$
And the DFT being what you have:
$$X_k = \sum_{n=0}^{N-1}x_n e^{-i 2\pi k n/N}$$
$$X_k = \sum_{n=0}^{N-1}\cos (2\pi n/N) e^{-i 2\pi k n/N}$$
Using Euler's Formula and the identity:
$$\sum_{n=0}^{N-1} e^{i 2\pi (k-k') n/N}= N\delta_{kk'}$$
You get
$$X_k = \frac{N}{2}(\delta_{1k} + \delta_{-1k})$$
Then if we want the inverse and use the standard formula:
$$x_n = \frac{1}{N}\sum_{k=0}^{N-1}X_k e^{i2\pi kn/N}$$
You can see without the $$k=-1$$ term in the sum, we don't recover the formula $$x_n = \cos(2\pi n/N)$$.
So we need negative frequency, but what is it? That is your question.
It's very simple, negative frequency just means clockwise rotations (or just rotations in the opposite of your convention).
It's easy to see if you just look at:
$$\cos(2\pi f t)$$
Where $$f$$ is frequency and $$t$$ is time.
Changing $$f \rightarrow -f$$ is the same as changing $$t \rightarrow -t$$. If you think about this physically, if time was going backwards, so would the motion of this rotation. It is physical. It is not just a mathematical curiosity, although it certainly is curious in this context.
In my example of $$\cos(2\pi n/N)$$ we need two counter rotating complex exponentials to cancel out their imaginary parts but retain the real part. This is why it is necessary.
Extending this to negative indices wastes a lot of computational time. There is a clever way of introducing negative indices back into positive ones, which also agrees with our picture of negative frequency corresponding to backwards rotations.
Returning to the calculation of the DFT of $$\cos(2\pi n/N)$$. We got:
$$X_k = \frac{N}{2}(\delta_{1k} + \delta_{-1k})$$
If we only have indices from $$0$$ to $$N-1$$ what is the meaning of $$k=-1$$ here?
Well imagine you're on a clock with $$N$$ vertices. Going clockwise amounts to increasing by $$+1$$. Then if we go counterclockwise it would be $$-1$$.
Arithmetic on a clock is modular arithmetic. If you imagine a normal clock it's $$a\mod 12$$ so if we hit 13, then $$13\mod 12 = 1$$. Similarly, if we go backwards to $$-1$$, $$-1\mod 12 = 11$$. So for an N-clock, $$-1\mod N = N-1$$. I hope you can see this.
Then we can just map $$k = -1 \rightarrow k = N-1$$
Let's see how this affects the calculation:
$$X_k = \frac{N}{2}(\delta_{1k} + \delta_{-1k})$$
considering our clock picture, it is now:
$$X_k = \frac{N}{2}(\delta_{1k} + \delta_{(N-1)k})$$
The inverse DFT is:
$$x_n = \frac{1}{N}\sum_{k=0}^{N-1}\frac{N}{2}(\delta_{1k} + \delta_{(N-1)k})e^{i2\pi kn/N}$$
$$x_n = \frac{1}{2}\sum_{k=0}^{N-1}(\delta_{1k} + \delta_{(N-1)k})e^{i2\pi kn/N}$$
$$x_n = \frac{1}{2}(e^{i2\pi n/N} + e^{i2\pi (N-1)n/N})$$
$$x_n = \frac{1}{2}(e^{i2\pi n/N} + e^{-i2\pi n/N}e^{i2\pi N/N})$$
$$x_n = \frac{1}{2}(e^{i2\pi n/N} + e^{-i2\pi n/N}e^{i2\pi })$$
The familiar $$e^{i2\pi} = 1$$ appears
$$x_n = \frac{1}{2}(e^{i2\pi n/N} + e^{-i2\pi n/N}) = \cos(2\pi n/N)$$
Recovering our formula from before.
The next bunch of lines is a tangent on including negative indices into the calculation. This is just to show that it can be done, but it is completely unnecessary. It may be helpful to see how the kronecker delta identity is defined and proved from our summation formula
Now I hope the rest of what I'm saying is helpful, because this bothers me. The definitions exclude the negative frequency (negative index) components.
The satisfying answer to me is to include negative frequencies and then change the normalization factor on the inverse (that factor of $$\frac{1}{N}$$)
There is a systematic way of doing this that I can illustrate while also proving the identity:
$$\sum_{k=0}^{N-1} e^{i 2\pi (n-n') k/N}= N\delta_{nn'}$$
Notice I switched the indices from the way it was above. I'm hoping that will make this more transparent to the reader.
First let's take the standard Discrete Fourier Transform and it's inverse and assume that we don't know the normalization factor:
$$X_k = \sum_{n=0}^{N-1}x_n e^{-i 2\pi k n/N}$$
$$x_n = a\sum_{k=0}^{N-1}X_k e^{i2\pi kn/N}$$
You can see $$\frac{1}{N} \rightarrow a$$
Let's just use the identity and see what we get. The first little trick is to change $$n \rightarrow n'$$ in our formula for $$X_k$$. If you don't do this, you get nonsense. The reason why we need to do this is because the index in the Inverse Transform is not necessarily the same index as in the Forward Transform. One index is summed over, and the other index is not.
Plug the first equation into the second.
$$x_n = a\sum_{k=0}^{N-1}(\sum_{n'=0}^{N-1}x_{n'} e^{-i 2\pi k n'/N}) e^{i2\pi kn/N}$$
$$x_n = a\sum_{k=0}^{N-1}\sum_{n'=0}^{N-1}x_{n'} e^{i2\pi k(n-n')/N}$$
Now using the identity we get
$$x_n = aN\sum_{n'=0}^{N-1}x_{n'} \delta_{nn'}$$
and $$\delta_{nn'} =1$$ only when $$n=n'$$ and zero otherwise by its definition. Therefore:
$$x_n = aNx_n \implies a = \frac{1}{N}$$
What if we want to now extend the definition to negative index? Computationally, I understand this is not a great idea since it doubles the amount of computations, but theoretically I think it is satisfying.
Suppose we took the sum from $$-R$$ to $$N-1$$. This will no longer make the number of samples $$N$$ but a new number $$P=N+R$$. You can convince yourself of this by counting from $$-R$$ to $$N-1$$. We pass over 0 so we count $$R$$ plus $$1$$ for the count of zero plus $$N-1$$ for the rest. The total is then as I stated: $$P=N+R$$
Then we have these pairs of formula:
$$X_k = \sum_{n'=-R}^{N-1}x_{n'}e^{-i 2\pi k n'/P}$$
$$x_n = a\sum_{k=-R}^{N-1}X_k e^{i2\pi kn/P}$$
Plugging the first into the second gives
$$x_n = a\sum_{k=-R}^{N-1}(\sum_{n'=-R}^{N-1}x_{n'}e^{-i 2\pi k n'/P}) e^{i2\pi kn/P}$$
Let's, as before, concentrate on the sum over $$k$$. We want to prove the identity for the discrete sum of complex exponentials. We can change the order of the sums to do this (something I'll let you figure out for yourself since I'm already writing so much).
$$\sum_{k=-R}^{N-1}e^{-i 2\pi k n'/P} e^{i2\pi kn/P}$$
$$\sum_{k=-R}^{N-1}e^{i 2\pi k (n-n')/P}$$
We omit $$x_{n'}$$ because in the $$k$$ sum this is just a constant. We keep the $$n'$$ in the exponential because we want to ask what happens when $$n' = n$$ and $$n' \neq n$$
First if $$n'=n$$ then the complex exponential is of the form $$e^{0} = 1$$. Then the sum is just:
$$\sum_{k=-R}^{N-1}1$$
And since the sum goes from $$-R$$ to $$N-1$$ we sum $$1$$ $$P$$ times.
$$\sum_{k=-R}^{N-1}1 = P$$
If $$P \rightarrow N$$ then $$R \rightarrow 0$$ and we recover that familiar factor of $$N$$ from before.
Now if $$n'\neq n$$ we have to do more work
$$\sum_{k=-R}^{N-1}e^{i 2\pi k (n-n')/P}$$
First let's re-index the sum. We can set $$k\rightarrow k-R$$ and then the new $$k$$ will start from $$0$$ and go to $$P-1$$ (much like we had before). You can see if $$k=0$$ then $$k-R = -R$$ and if $$k=P-1$$ then $$k-R = P-1-R = N-1$$. Exactly what we had before.
$$\sum_{k=0}^{P-1}e^{i 2\pi (k-R) (n-n')/P}$$
Let's factor out the part of the exponential that doesn't contain $$k$$. To make things a bit neater I'm going to define $$n-n' = \Delta n$$ so the difference in the two integers which we know in our current case is not $$0$$.
$$\sum_{k=0}^{P-1}e^{i 2\pi \Delta n(k-R)/P}$$
$$\sum_{k=0}^{P-1}e^{i 2\pi( \Delta n(k)-\Delta n(R))/P}$$
$$e^{-i 2\pi \Delta n R/P}\sum_{k=0}^{P-1}e^{i 2\pi \Delta n k/P}$$
Now we luckily have a formula for a finite geometric series:
$$\sum_{l=0}^{L-1}r^l = \frac{1-r^L}{1-r}$$
We will just copy what we have into that
$$e^{-i 2\pi \Delta n R/P}(\frac{1-e^{(-i 2\pi \Delta n /P)P}}{1-e^{-i 2\pi \Delta n R/P}} )$$ $$e^{-i 2\pi \Delta n R/P}(\frac{1-e^{-i 2\pi \Delta n }}{1-e^{-i 2\pi \Delta n /P}} )$$
If you focus on the numerator you see the part of the exponent $$\Delta n$$ is always an integer since the difference of two integers is an integer. Thus
$$e^{-i 2\pi \Delta n} = 1$$
Looking at the numerator of our formula, we then get $$1-1 =0$$. Thus when $$n'\neq n$$ or equivalently when $$\Delta n \neq 0$$
$$e^{-i 2\pi \Delta n R/P}(\frac{1-e^{-i 2\pi \Delta n }}{1-e^{-i 2\pi \Delta n /P}} ) = 0$$
We don't have to worry about the denominator since the greatest $$\Delta n$$ can be is $$P-1$$, so $$(P-1)/P <1$$ and will not make the denominator $$0$$. You can check yourself. We also already assumed $$\Delta n \neq 0$$ so that can't happen either. You could show an alternative proof at this stage for $$\Delta n = 0$$ to agree with what we had before (L'hopital).
We can then sum up our formula using the kronecker delta symbol $$\delta_{nn'}$$ which, once again, is 1 if $$n=n'$$ and 0 otherwise.
$$\sum_{k=-R}^{N-1}e^{i 2\pi k (n-n')/P} = P\delta_{nn'}$$
It is the kronecker delta times the length of the interval which in this case is $$P$$.
For completeness, we can do our proof again for the normalization factor
$$x_n = a\sum_{k=-R}^{N-1}(\sum_{n'=-R}^{N-1}x_{n'}e^{-i 2\pi k n'/P}) e^{i2\pi kn/P}$$ $$x_n = a\sum_{n'=-R}^{N-1}x_{n'}P\delta_{nn'}$$
$$x_n = a x_n P$$
Therefore $$a = \frac{1}{P}$$ in this case.
The point of this calculation is to show that we can use any integer indices we want as long as we use the correct normalization factor which will be the the total number of data points.
"End of Tangent"
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2020-08-13 22:03:20
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https://doc.simiansuite.com/simian-gui/v2.0/initialization/radio.html
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Defines the specifics of a set of options of which exactly one must be selected.
In addition to the properties and methods listed below, this component inherits properties and methods from the superclass Component. For example, any Radio component has a label and defaultValue property even though these are not explicitly listed here.
Properties
NameDescriptionDatatypeDefault
valuesSpecifies the options the user can choose from. Must be an array of values where each item has the following fields:
• label: The text to show besides the option.
• value: The value of the option.
Easily set this property with the setValues method.
Struct/dict
inlineIf set to true, layout the radio buttons horizontally instead of vertically.BooleanFalse
labelPositionPosition of the label with respect to the radio component. Can be 'top', 'bottom', 'right-right', 'left-right', 'left-left' or 'right-left'.String"top"
optionsLabelPositionPosition of the text of every option with respect to the radio button. Can be 'right', 'left', 'top' or 'bottom'.String"right"
Methods
NameSyntaxDescription
setValuesobj.setValues(labels, values, default)Set the labels and accompanying values. This sets the values property of the Radio component. Optionally, provide the label or the value of the option to select by default.
By using the setValues method, the example Radio component above can be created using:
gender = component.Radio("gender_radio", form);
gender.label = "Gender";
gender.setValues(["Male", "Female", "Other", "Will not say"], ...
["m", "f", "o", "wns"])
gender = component.Radio("gender_radio", form)
gender.label = "Gender"
gender.setValues(["Male", "Female", "Other", "Will not say"],
["m", "f", "o", "wns"])
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2023-04-01 14:14:38
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http://haus-49.de/factorial-experimental-design.html
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# Factorial Experimental Design
52% [+ or -] 0. Design of Engineering Experiments Part 5 - The 2k Factorial Design Author: Preferred Customer Last modified by: Hongyan Zhang Created Date: 8/3/2000 7:09:41 PM Document presentation format: On-screen Show Company: ASU Other titles. In a Factorial Design of Experiment, all possible combinations of the levels of a factor can be studied against all possible levels of other factors. 2 n Designs B. Example of a 2 3 Factorial Experiment. The third type of experimental design is the factorial design, in which there are two or more clearly understood treatments, such as exposure level to test chemical, animal age, or temperature. The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as full factorial experiments. Two examples of real factorial experiments. These levels are called high and low or +1 and -1, respectively. 5 Two-Level Fractional Factorial Designs Because the number of runs in a 2k factorial design increases rapidly as the number of factors increases, it is often impossible to run the full factorial design given available resources. Implementing fractional factorial experimental design is already changing the way many in our agency are going about testing. 1 Introduction Consider a situation where it is of interest to study the effect of two factors, A and B, on some response. The General 2k Factorial Design Section 6-4, pg. Traditional research methods generally study the effect of one variable at a time, because it is statistically easier to manipulate. Single-subject experimental designs – also referred to as within-subject or single case experimental designs – are among the most prevalent designs used in CSD treatment research. Factorial designs can be arranged such that three, four, or n treatments or independent variables are studied simultaneously in the same experiment. Two examples of real factorial experiments reveal how using this approach can potentially lead to a reduction in animal use and savings in financial and scientific resources without loss of scientific validity. A2 : 250mg of the drug applied on male patients. Described by a numbering system that gives the number of. Using a factorial design, the experiment examines all possible combinations of levels for each factor. ’ Simple factorial design may either be a 2 × 2 simple factorial design, or it may be, say, 3 × 4 or 5 × 3 or the like type of simple factorial design. (source: author) One basic experimental design, known as full factorial, includes samples of k variables at n levels, resulting in n**k points, which is only feasible for few variables and levels, as otherwise the number of experiments becomes too large. The dependent variable was the target's likelihood of changing their behavior. a plan how you create your data. Statnotes: ANOVA by G. n = 8, 12, 20, 24, 28, 32 etc {Factors k <= n - 1 {For k < n-1 use dummy factors {Most commonly used are n=8 and n=12 {Plackett, R. Before and after without control design. In case of three factors with one experimental variable having two treatments and two control variables, each one of which having two levels, the design used will be termed as 2 X 2 X 2 complex factorial design which will contain a total of eight cells as shown in the following figure. This video is part of a project at the Univeristy of Amsterdam in which instruction videos. A key feature of fractional factorials that is not shared by more ad hoc methods for. basic two group post-test only randomized experiment ANSWER: d DIFFICUL TY: Moderate REFEREN CES: 9. Full Factorial Design for 3 variables having varying levels. Need to learn about Factorial Research designs? Many more examples and great mnemonics for your tests are included in my app:. Thus, in a 2 X 2 factorial design, there are four treatment combinations and in a 2 X 3 factorial design there are six treatment combinations. Design of experiments (DOE) is a rigorous methodology that enables scientists and engineers to study the relationship between multiple input variables, or factors, on key output variables, or responses. Simple factorial design is also termed as a ‘two-factor-factorial design’, whereas complex factorial design is known as ‘multifactor- factorial design. Now we consider a 2 factorial experiment with a2 n example and try to develop and understand the theory and notations through this example. The treatment is then introduced and the dependent variable is measured again. of factor are more than 5. In an experiment, you manipulate one or more independent variables and measure their effect on one or more dependent variables. Two-Way Factorial Design. Over the course of five days, you. Package DoE. A full factorial design statistical approach based on the Design of Experiment (DoE) is performed, producing all possible combinations between the experimental factors. Here we describe the design and execution of a two-parameter, three-level (3 2) factorial experiment resulting in nine conditions that were run in duplicate 125-mL stirred suspension bioreactors. Rather than 3125 treatments that would be required for the full factorial experiment, this experiment requires only 25 treatments. The results of the analysis appear below:. United States Department of Agriculture. Though commonly used in industrial experiments to identify the signiflcant efiects, it is often undesirable to perform the trials of a factorial design (or, fractional factorial design) in a completely random order. In Chapter 1 we briefly described a study conducted by Simone Schnall and her colleagues, in which they found that washing one's hands leads people to view moral transgressions as less wrong (Schnall, Benton, & Harvey, 2008) [1]. Factorial design. basic analysis of covariance experiment b. Generally, in a factorial experimental design, ex-perimental trials (or runs) are performed at all combinations of factor levels. basic two group post-test only randomized experiment ANSWER: d DIFFICUL TY: Moderate REFEREN CES: 9. On account of errors of measurement and the neglect of certain effects the minimum S0 of S is not zero. What's Design of Experiments - Full Factorial in Minitab? DOE, or Design of Experiments is an active method of manipulating a process as opposed to passively observing a process. All that changes is that we have an extra column for the covariate scores. Two other methods for determining experimental design are factorial design and random design. See full list on conjointly. Full factorial experimental design and response surface methodology were used to develop mathematical models for both grade and recovery of Cr 2 O 3 concentrate. The layout of the design generated by this design will include all possible. In these cases, fractional factorial design can be useful. Experimental matrix for the factorial design and centrepoints Run T (°C) C (wt%) X T X C Y 0 (%) 1 25 86. I suggest that you put the 5-level IVs on the x-axis and the other IV as a line color or bar color. Filtro por publicador. The REMAP-CAP trial provides a global research platform that is able to adapt to efficiently evaluate. Special case of the general factorial design; k factors, all at two levels The two levels are usually called low and high (they could be either quantitative or qualitative) Very widely used in industrial experimentation Form a basic “building block” for other very useful experimental designs (DNA) Special (short-cut) methods for analysis. Design of experiments (DOE) is a rigorous methodology that enables scientists and engineers to study the relationship between multiple input variables, or factors, on key output variables, or responses. A 2 3 full factorial design without center points was set-up straddling the "so far best conditions" derived from previous experiments. Design of Experiments (DOE) with JMP. For a small number of design variables, 2n may be a manageable number of. Design and Analysis of Experiments, 5th edition, John Wiley & Sons, New York, 2001). Statnotes: ANOVA by G. A 5 5-3 design, for example, is 1/125 of a five level, five factor factorial design. A factor is a discrete variable used to classify experimental units. Garson ANOVA/MANOVA by StatSoft Two-way ANOVA by Will Hopkins. A guide to experimental design. In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. In the Three Level Factorial design all possible combinations of the three discrete values of the parameter are used. It has distinct advantages over a series of simple experiments, each designed to test a single factor. The other designs (such as the two level full factorial designs that are explained in Two Level Factorial Experiments) are special cases of these experiments in which factors are limited to a specified number of levels. A statistical analysis used in experimental research Ans: c Page: 207 Type: F LO: 1 10. Factorial Designs. Experimenter wants magnitude of effect, , and t ratio = effect/se(effect). one parameter fixed at a time. Factorial Experimental Designs Discover free flashcards, games, and test prep activities designed to help you learn about Factorial Experimental Designs and other concepts. In other words, we have a 2 x 2 factorial design. As the goal is to predict the future cost of manufacturing batteries, a mature manufacturing process is assumed. Most of the designs involve only 2 levels of each factor. That is: " The sum of each column is zero. Like a true experiment, a quasi-experimental design aims to establish a cause-and-effect relationship between an independent and dependent variable. Sample factorial design table for a three-factor experiment with two levels per factor. A design with all possible high/low combinations of all the input factors is called a full factorial design in two levels. 2 x 2 factorial experiment d. (6) Latin square design (L. By far the most common approach to including multiple independent variables (which are often called factors) in an experiment is the factorial design. Figure 1: Full factorial design for three variables with two levels each. The factorial design, as well as simplifying the process and making research cheaper, allows many levels of analysis. Methodical experimentation has many applications for efficient and effective information gathering. Interviewing and survey research, for instance, may be used in experimental, quasi-experimental, and non-experimental research. In other words, we have a 2 x 2 factorial design. 4 if variation in test results is on the same order of magnitude as the factor | PowerPoint PPT presentation | free to view. This can sometimes be time-consuming or expensive. ] software following full factorial method. Factorial Survey Experiments - ー - 洋書の購入はブックスで。全品送料無料！購入毎に「ポイント」が貯まってお得！みんなのレビュー・感想も満載。. Soo King Lim - 9 - Alternative for k value larger than five, Plackett-Burman design is also a better choice. Single Factor C. In a fractional factorial experiment, only a fraction of the possible treatments is actually used in the experiment. in International Journal of Machine Tools & Manufacture, 45, 1402–1411, 2005, described the use of a full factorial design to study the effects of rotary ultrasonic machining on the cutting force, material removal rate, and hole quality. It looks almost the same as the randomized block design model only now we are including an interaction term: Y i j k = μ + α i + β j + ( α β) i j + e i j k. If dealing with several factors and if resources are constrained, a more pragmatic approach is a fractional factorial design. A within-subject design is a type of experimental design in which all participants are exposed to every treatment or condition. CHAPTER 7 Experimental Designs Bachman 6e © 2017 SAGE Publications, Inc. Randomly assign subplot treatments to the subplots. Let's suppose you have four factors (a four factor experiment ): Pan shape: Round (low) vs square (high) pan. In designs where there are multiple factors, all with a discrete group of level settings, the full enumeration of all combinations of factor levels is referred to as a full factorial design. An appropriately powered factorial trial is the only design that allows such effects to be investigated. Design of Experiments (DOE) is a study of the factors that the team has determined are the key process input variables (KPIV's) that are the source of the variation or have an influence on the mean of the output. Explicit Memory in Amnesics vs. Factor A is 1,500 or 2,000 calories and factor B is 0 or 30 minutes of aerobic exercise. Once this selection is made, the experimental design must separate these effects. Research design is largely independent of the choice of methods of data collection. One-factor-a-time designas the opposite of factorial design. Testimonial "DOE expertise is a must have for engineers who deal with data all the time, whether it's in a simulation or test, or identifying the factors which have the most influence on the experiment. If the center point p-value is significant (i. Once speci c factors are identi ed as important, they are investigated in greater detail in subsequent experiments. Fractional factorial designs are very useful for screening experiments or when sample sizes are limited. When to use. Factorial designs – designs with two or more independent variables Independent variables are called factors Two factor experiment – the simplest factorial design FACTORIAL DESIGNS They give us information about the effects of each independent variable in the experiment – main effects They enable us to answer the question: How does the. Explain the coding systems used in a factorial design of experiment. For example, you would like to determine the best conditions for injection-molding a plastic part. When doing factorial design there are two classes of effects that we are interested in: Main Effects and Interactions -- There is the possibility of a main effect associated with each factor. However, unlike a true experiment, a quasi-experiment does not rely on random assignment. 2 Introduction: The Origins of Experimental Design KEYWORD S: Bloom's: Understand 19. Calculating the Number of Trials. Factorial designs: studying 2 or more independent variables at the same time; provide more information that experiments with one IV -factors: the independent variables in the designs -two factor experiment: simplest factorial design; only has two factors -data from a factorial experiment gives us:. What type of design is shown above? a. This work provides a replication strategy for full-factorial designs having two to four factors. 2020 (With Results) Fractional Factorial Study Design Example (With Results) Disclaimer: The following information is fictional and is only intended for the purpose of illustrating key concepts for results data entry in the Protocol Registration and Results System (PRS). Though commonly used in industrial experiments to identify the signiflcant efiects, it is often undesirable to perform the trials of a factorial design (or, fractional factorial design) in a completely random order. Editor as they are Table 1. Beginning with eight attributes judged important to the goals of the program by clinicians, social preference values for different function states were obtained from 32 parents of. Before-and-after. An article entitled “Rotary ultrasonic machining of ceramic matrix composites: feasibility study and designed experiments,” published by Z. 225 There will be k main effects, and Unreplicated 2k Factorial Designs These are 2k factorial designs with one observation at each corner of the “cube” An unreplicated 2k factorial design is also sometimes called a “single replicate” of the 2k These designs are very widely used Risks…if there is only one. See also simple factorial design. Most of the designs involve only 2 levels of each factor. Full factorial designs — Process Improvement using Data. The layout of the design generated by this design will include all possible. That is: " The sum of each column is zero. The goal of our work is to identify optimal and robust designs for factorial experiments with binary response. Designed experiments are widely used in the DMAIC process,. What’s Design Of Experiments – Full Factorial? DOE, or Design of Experiments is an active method of manipulating a process as opposed to passively observing a process. Factorial Design In simulation experiments we are often interested in finding out how different input variable settings impact the response of the system. These basic templates are ideal for training, but use SigmaXL > Design of Experiments > 2-Level Factorial/Screening Designs to accommodate up to 19 factors with randomization, replication and blocking. Experimenter wants magnitude of effect, , and t ratio = effect/se(effect). Formed by decades of teaching, consulting, and industrial experience in the Design of Experiments field, this new edition contains updated examples, exercises, and situations covering the science. Relationship between factorial experiments and experimental design Experimental design is concerned with the assignment of treatments to experimental units, A factorial experiment is concerned with the structure of treatments. Statistics for Experimenters: Design, Innovation, and Discovery (edisi ke-2nd). Once this selection is made, the experimental design must separate these effects. The first (X 1) column starts with -1 and alternates in sign for all 2 k runs. Louis Luangkesorn ( University of Pittsburgh ) Design of Experiments March 2, 2010 12 / 21. The method is popularly known as the factorial design of experiments. partitioned into individual “SS” for effects, each equal to N(effect)2/4, divided by df=1, and turned into an F-ratio. No math or statistics knowledge needed. 0 software (StatSoft). These experiments provide the means to fully understand all the effects of the factors—from main. Factorial design is used to reduce the total number of experiments in order to achieve the best percentage removal (%Cd) of cadmium ions (Mason et al. In the past, social scientists had been transfixed on singular independent variable experiments and foreshadowed the importance of extraneous variables which are able to attenuate or diminish research findings. DOE is a powerful data collection and analysis tool that can be used in a variety of experimental. Since most industrial experiments usually involve a significant number of factors, a full factorial design results in a large number of experiments. These designs are not only applicable to two level factorial experiments, but also can investigate main effects when factors have more than two levels. But because of the prohibitive size of the experiments, such designs are not practical to run. Orthogonal experimental designs have zero correlation between any variable or interaction effects specifically to avoid this problem. moisture content, acidity) { Biochem analysis of animal tissue { Multiple plates of single agar batch 19-4 Simple. Often, however, individual factors or their interactions have no distinguishable effects on a response. A split plot design is a special case of a factorial treatment structure. The layout of the design generated by this design will include all possible. Experimental Factorial Design The quantitative composition of the lipid nanoparticle dispersions suitable for the incorporation of hydrophilic active substances—including selected iridoid glycosides (aucubin and catalpol)—was optimized by using the 3 2 factorial design with the help of Statistica 10. Factorial experiment design, or simply factorial design, is a systematic method for formulating the steps needed to successfully implement a factorial experiment. (6) Latin square design (L. First, by measuring several dependent variables in a single experiment, there is a better chance of discovering which factor is truly important. Crossover study: A crossover study compares the results of a two treatment on the same group of patients. After this is done a fractional factorial design of a 1/2 fraction is created and the data is analyzed again. Each column contains the settings for a single factor, with integer values from one to the number of levels. As already stated, the greater the number of assemblies measured, the greater the precision with which component effects may be estimated. Simple factorial design is also termed as a ‘two-factor-factorial design’, whereas complex factorial design is known as ‘multifactor- factorial design. The method of analyzing such experiments was presented and illustrated by the 5 x 2 x 2 experiment. Keywords: Randomization, blocking, main effects, interactions, experimental design, JMP. A Factorial Design is an experimental setup that consists of multiple factors and their separate and conjoined influence on the subject of interest in the experiment. To illustrate fractional factorial designs let’s take an example. The factors you can set are: Temperature: 190° and 210°. 4 Types of Experimental Designs. All that changes is that we have an extra column for the covariate scores. of factor are more than 5. Introduction Three Types of Treatment Factor Effects The Statistical Model for Two Treatment Factors The Analysis for Two Factors Factorial Experiments with More than two Factors Unequal Replication of Treatments Factorial designs: Notation and Definitions To examine the effect of any one of the four factors, half the runs (or 2 × 2 3 = 16 due. Calculating the Number of Trials. Fractional Factorial Experiments The other type of factorial experiment is a fractional factorial. Factorial Designs, 1 Factorial Experimental Designs FACTORIAL DESIGNS: Experiments (and quasi-experiments) involving two or more IVs or grouping variables ("factors"). The name of the example project is "Factorial - General Full Factorial Design. Passive data collection leads to a number of problems in statistical modeling. Factorial Experimental Design for Reactive Dye Flocculation Using Inorganic-Organic Composite Polymer Factorial invariance of posttraumatic stress disorder symptoms across three veteran samples Factorial invariance of the Emotion-Dysregulation-Scale for Canadian and German treatment-seeking adults with borderline personality disorder, major. Fractional Factorials. Such experimental designs are referred to as factorial designs. Bringing together both new and old results, Theory of Factorial Design: Single- and Multi-Stratum Experiments provides a rigorous, systematic, and up-to-date treatment of the theoretical aspects of factorial design. Full Factorial Designs Multilevel Designs. Comparing experimental designs: factorial and regression designs Factorial designs are based on experimental control between groups of experimental items, so-called conditions. But because of the prohibitive size of the experiments, such designs are not practical to run. Factorial designs are designed using a matrix notation that indicates how groups are formed relative to levels each of independent variable Uppercase letters: A,B,C- label the IV and their levels 1. In other words, we have a 2 x 2 factorial design. Experimental design techniques are designed to discover what factors or interactions have a significant impact on a response variable. While advantageous for separating individual effects, full factorial designs can make large demands on data collection. Experimenter wants magnitude of effect, , and t ratio = effect/se(effect). •Method of reporting results should an objective description of the behaviors •The discussion begins by re-stating the major results and how then agree or not with the literature; then synthesis the findings in an overall conclusion Image courtesy: www. Such designs are discussed with factorial designs. basic analysis of covariance experiment b. There are criteria to choose “optimal” fractions. basic two group post-test only randomized experiment ANSWER: d DIFFICUL TY: Moderate REFEREN CES: 9. Introduction to Design and Analysis of Experiments by George W. The goal of our work is to identify optimal and robust designs for factorial experiments with binary response. FAQ; Certificates; Payment; About Us; Contact Us; Terms; Cart 0. In Chapter 1 we briefly described a study conducted by Simone Schnall and her colleagues, in which they found that washing one's hands leads people to view moral transgressions as less wrong (Schnall, Benton, & Harvey, 2008) [1]. An experimental design is a planned experiment to determine, with a minimum number of runs, what factors have a significant effect on a product response and how large the effect is to find the optimum set of operating conditions. In other words, we have a 2 x 2 factorial design. In factorial experiments, factors contain discrete values (levels), and the number of factor levels influences design of experimental runs. n = 8, 12, 20, 24, 28, 32 etc {Factors k <= n - 1 {For k < n-1 use dummy factors {Most commonly used are n=8 and n=12 {Plackett, R. Two-Way Factorial Design. Both Within- & Between-S IVs: Mixed Designs. Note: The repeated option is used to compute the Huynh-Feldt values. 12, 16, 20 or 24. Learn how to use Minitab’s DOE interface to create response surface designs, analyze experimental results using a model that includes quadratics, and find optimal factor settings. Read also about the factorial design. A factorial design refers to any experimental design that has more than one independent variable. Planning Factorial designs vary several factors simultaneously within a. ANCOVAs are frequently used in experimental studies when the researcher wants to account for the effects of an antecedent (control) variable. (source: author) One basic experimental design, known as full factorial, includes samples of k variables at n levels, resulting in n**k points, which is only feasible for few variables and levels, as otherwise the number of experiments becomes too large. For 2n experiments (2 A n < 8) and the usual factorial model, Quenouille and John (1971) gave a table of designs which have. The method of analyzing such experiments was presented and illustrated by the 5 x 2 x 2 experiment. Factorial designs are extremely useful to psychologists and field scientists as a preliminary study, allowing them to judge whether there is a link between variables, whilst reducing the possibility of experimental error and confounding variables. Bringing together both new and old results, Theory of Factorial Design: Single- and Multi-Stratum Experiments provides a rigorous, systematic, and up-to-date treatment of the theoretical aspects of factorial design. Fractional factorial designs use a fraction of the runs required by full factorial designs. Nicolaisena,⁎,M. The design of an experiment plays a major role in the eventual solution of the problem. In such a design a single test group or area is selected and dependent variable is measured before the introduction of the treatment. When the data is balanced, the data points are distributed over the experimental region so that they have an equal. The factorial design determines which factors have important effects on a response (%Cd) as well as how the effect of one factor varies with the level of the other factors. Test for curvature in two-level factorial designs by using center points. This is also known as a screening experiment Also used to determine curvature of the response surface 5. A CFD is capable of estimating all factors and their interactions. Over the course of five days, you. high, referred as “+” or “+1”, and low, referred as “-”or “-1”). Factorial designs are the basis for another important principle besides blocking - examining several factors simultaneously. SIMPLE FACTORIAL DESIGN: "A simple factorial design is the design of an experiment. Factorial designs; Plackett-Burman designs; Box-Behnken designs; Central composite designs; Latin-Hypercube designs; There is also a wealth of information on the NIST website about the various design matrices that can be created as well as detailed information about designing/setting-up/running experiments in general. Design of experiments (DOE) is defined as a branch of applied statistics that deals with planning, conducting, analyzing, and interpreting controlled tests to evaluate the factors that control the value of a parameter or group of parameters. Need to reduce a processes sensitivity to uncontrolled parameter variation. Need to learn about Factorial Research designs? Many more examples and great mnemonics for your tests are included in my app:. A fractional factorial design is often used as a screening experiment involving many factors with the goal of identifying only those factors having large e ects. If equal sample sizes are taken for each of the possible factor combinations then the design is a balanced two-factor factorial design. Each column contains the settings for a single factor, with integer values from one to the number of levels. Factorial designs; Plackett-Burman designs; Box-Behnken designs; Central composite designs; Latin-Hypercube designs; There is also a wealth of information on the NIST website about the various design matrices that can be created as well as detailed information about designing/setting-up/running experiments in general. You've just watched JoVE's introduction to factorial experimental design. It is wise to take time and effort to organize the experiment properly to ensure that the right type of data, and enough of it, is available to answer the questions of interest as clearly and efficiently as possible. , effect of. An article entitled “Rotary ultrasonic machining of ceramic matrix composites: feasibility study and designed experiments,” published by Z. Simple factorial design is also termed as a ‘two-factor-factorial design’, whereas complex factorial design is known as ‘multifactor- factorial design. The factorial experimental design is a very popular technique, as it gives statistical models which explain the interactions among the factors that have been optimized (Can and Yildiz 2006. The connection between the two (if any) is that if you know that you want to do an ANOVA with variables X,Y,Z or a number of their interactions. Crossover study: A crossover study compares the results of a two treatment on the same group of patients. They're customizable and designed to help you study and learn more effectively. o The statistics are pretty easy, a t-test. Factorial designs for the analysis of multiple variables at once, which can be very helpful when it is not sure which is more significant or how they interact. • Example: I'm interested in factors that cause dangerous driving. MANOVA is useful in experimental situations where at least some of the independent variables are manipulated. In other words, we have a 2 x 2 factorial design. Answer to: In a completely randomized experimental design, 5 experimental units were used for each of the 4 levels of the factor (i. Factorial Design of Experiments: A practical case study. Experimental Factorial Design The quantitative composition of the lipid nanoparticle dispersions suitable for the incorporation of hydrophilic active substances—including selected iridoid glycosides (aucubin and catalpol)—was optimized by using the 3 2 factorial design with the help of Statistica 10. Such an experiment allows the investigator to study the effect of each factor on the response variable, as well as the effects of interactions between factors on the response. The test subjects are assigned to treatment levels of every factor combinations at random. Published on December 3, 2019 by Rebecca Bevans. (source: author) One basic experimental design, known as full factorial, includes samples of k variables at n levels, resulting in n**k points, which is only feasible for few variables and levels, as otherwise the number of experiments becomes too large. Sadly, many people simply don't understand what an authentic DOE is or, in some cases, some practitioners mistakenly believe their one factor at a time experiment is in fact a DOE when, really, it isn't. In the latter we dealt with a treatment at t levels or with t treatments. A1 : 100mg of the drug applied on male patients. In factorial designs, every level of each treatment Is studied under the conditions of every level of all other treatments. In such cases, one cannot perform a full replicate of the design and a fractional factorial design has to be run [8]. I design an experiment in which I manipulate alcohol consumption (0, 1, or 2 beers) and cell-phone conversation (talking vs. Complex Experimental Designs. For designs of less than full resolution, the confounding pattern is displayed. Once this selection is made, the experimental design must separate these effects. Factorial designs by William Trochim. Specially, by a factorial experiment we mean that in each complete trial or replicate of the experiment all possible combinations of the levels of the factors are investigated. For example, you would like to determine the best conditions for injection-molding a plastic part. GSD is available in pyDOE2 as: import pyDOE2 levels = [2, 3, 4] # Three factors with 2, 3 or 4 levels respectively. These two interventions could have been studied in two separate trials i. Additionally, a demo using the statistical software package JMP provides an example. DOE solutions provide equations that characterize the relationships between the inputs and the outputs and statistical measures that describe the strengths of the. 2 n Designs B. In this type of study, there are two factors (or independent variables) and each factor has two levels. The simplest factorial design is a 2×2 design which looks at effects of Intervention A (e. Factorial designs encourage a comprehensive approach to problem-solving. Here, experimental conditions are chosen by selecting a fixed number of levels for each variable, after which experiments are run at all pos- Sible combinations. Though commonly used in industrial experiments to identify the signiflcant efiects, it is often undesirable to perform the trials of a factorial design (or, fractional factorial design) in a completely random order. On the other hand if the factors are quantitative and the response is binary, the literature on optimal design of generalized linear models in the approximate theory setup could be used. • Research design in which different participants take part in each condition of the experiment or study. The independent variables, often called factors, must be categorical. Factorial designs can also be depicted using a design notation, such as that shown on the right panel of Figure 10. FRACTIONAL FACTORIAL DESIGN In Full FD , as a number of factor or level increases , the number of experiment required exceeds to unmanageable levels. Wiley, New York. randomized block with homogenous groups experiment c. Factorial Designs with 2 Treatment Factors, cont'd Section For a completely randomized design, which is what we discussed for the one-way ANOVA, we need to have n × a × b = N total experimental units available. Second, it can protect against. History, maturation, selection, mortality, and interaction of selection and the experimental variable are potential threats against the internal validity of this design. Learn how the analysis of variance can be extended to factorial experiments. Thus, we want to run a 1/4 fraction of a 2 6design. Explain the Factorial design of experiments. 5 hours left at this price! Add to cart. 2-1, page 518. When all factors have been coded so that the high value is "1" and the low value is "-1", the design matrix for any full (or suitably chosen fractional) factorial experiment has columns that are all pairwise orthogonal and all the columns (except the "I" column) sum to 0. When all possible combinations of the levels of the factors are investigated, then it is called a full factorial experiment. Need to learn about Factorial Research designs? Many more examples and great mnemonics for your tests are included in my app:. A statistical analysis used in experimental research Ans: c Page: 207 Type: F LO: 1 10. A factorial experimental design approach is more effective and efficient than the older approach of varying one factor at a time. 2-1, page 516. Figure 3-1: Two-level factorial versus one-factor-at-a-time (OFAT). randomized block with homogenous groups experiment c. In factorial designs, every level of each treatment Is studied under the conditions of every level of all other treatments. Learn about two-factor factorial experiments. Then the concept of blocking could be used. Fractional Factorial Designs As the number of factors in a 2 factorial design increases, the number of trials required for a full replicate of the design rapidly outgrows the resources available for many experiments. If dealing with several factors and if resources are constrained, a more pragmatic approach is a fractional factorial design. Experimental Design II: Factorial Designs. The resolution of a design is given by the length of the shortest word in the defining relation. " Cite this page: N. Using two levels per factor is generally sufficient for screening experiments. Example 1: Create the 2^3 factorial design for the data in Figure 1. Design of Engineering Experiments Part 5 – The 2k Factorial Design Author: Preferred Customer Last modified by: Hongyan Zhang Created Date: 8/3/2000 7:09:41 PM Document presentation format: On-screen Show Company: ASU Other titles. Therefore, in total, we need. But because of the prohibitive size of the experiments, such designs are not practical to run. FRACTIONAL FACTORIAL DESIGN In Full FD , as a number of factor or level increases , the number of experiment required exceeds to unmanageable levels. They're customizable and designed to help you study and learn more effectively. In these designs. A factor is an independent variable in the experiment and a level is a subdivision of a factor. See also simple factorial design. The factorial structure may be placed into any experimental design. The simplest method of experimental design is the one dimensional search i. Mixture-factorial experiments, which need to fit a model for the components and factorial variables. Understanding conceptually what a factorial design is will not come easy. This solution provides the best answer and an explanation for two multiple choice questions: What is a characteristic of quasi-experimental research? Factorial designs are experiments that can best be defined by which of these statements?. Therefore, our regression results for each effect are independent of all other effects and the results are clear and conclusive. For now we will just consider two treatment factors of interest. The investigator plans to use a factorial experimental design. Definition of factorial experiment in the Definitions. The number of trials required for a full factorial experimental run is the product of the levels of each factor:. As an example, suppose a machine shop has three machines and four operators. 3 x64 para Windows 7 64 bits: In this link, you can download SAS 9. [email protected] The simplest factorial design is a 2×2 design which looks at effects of Intervention A (e. 2-2, page 517. A full factorial two level design with $k\,\!$ factors requires ${{2}^{k}}\,\!$ runs for a single replicate. Randomly assign subplot treatments to the subplots. You've just watched JoVE's introduction to factorial experimental design. Factorial experiments are not stand-alone designs; they are the arrangement and organization of treatments within different. The name of the example project is "Factorial - General Full Factorial Design. Experimental Design Treatment group vs. To prepare readers for a general theory, the author first presents a unified treatment of several simple designs, including. The factorial survey is an experimental design where respondents are asked to judge descriptions of varying situations (vignettes) presented to them. PCA was carried out using home-made codes based on the algorithm outlined in the literature (Brereton, 2004). The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as full factorial experiments. Ø It is used to study a problem that is affected by a large number of factors. 4 if variation in test results is on the same order of magnitude as the factor | PowerPoint PPT presentation | free to view. A logical experimental program ideally suited for practical study of any physical system or situation is factorial design. •Experimental design must take into account the goal of the research. Because both the experimental sampling designs and subsequent analysis procedures are unfam-iliar, we present this example in detail. For 2n experiments (2 A n < 8) and the usual factorial model, Quenouille and John (1971) gave a table of designs which have. What type of design is shown above? a. Factorial experimental designs were used in the initial stages of developing a function index for evaluating a program for the care of young handicapped children. Note that it is arrangement of treatments, not a design. ORTHOPLAN is designed only for generation of orthogonal designs allowing analyses of main effects. Example of a 2x4 Factorial experiment replicated in. More about Single Factor Experiments † 3. (1997): Design and Analysis of Experiments (4th ed. A good experimental design requires a strong understanding of the system you are. The factorial experiments, where all combination of the levels of the factors are run, are usually referred to as full factorial experiments. Fisher showed that there are advantages by combining the study of multiple variables in the same factorial experiment. The goal of our work is to identify optimal and robust designs for factorial experiments with binary response. As an example, suppose a machine shop has three machines and four operators. The great advantage of factorial designs is that they disclose interactions between independent variables--they show how the relationship between y and is influenced by. A within-subject design is a type of experimental design in which all participants are exposed to every treatment or condition. Complete Factorial Design. A 2 3 full factorial design without center points was set-up straddling the "so far best conditions" derived from previous experiments. FRACTIONAL FACTORIAL DESIGN In Full FD , as a number of factor or level increases , the number of experiment required exceeds to unmanageable levels. Design and Analysis of Experiments. We will use factorial designs because. The appropriate experimental strategy for these situations is based on the factorial design, a type of experiment where factors are varied together. Though commonly used in industrial experiments to identify the signiflcant efiects, it is often undesirable to perform the trials of a factorial design (or, fractional factorial design) in a completely random order. † Helps with design of future experiments † Can check for consistency of measurements † Protect against missing values and contamination † Computational beneflt if ¾2 Sub >¾ 2 † Examples { Soil Samples within plot (e. (7) Factorial designs. In a randomly assigned factorial design, we need 10 participants in each of the four conditions. LISA Short Course: Factorial Experiments: Blocking, Confounding, and Fractional Factorial Designs, Part I from LISA on Vimeo. Fractional Factorial Designs • A fractional factorial design consists of a strategically selected subset of runs from a full factorial design • Useful when: • Large number of factors and it is uneconomical to test every possible factor combination • In screening experiments to identify the primary factors. one parameter fixed at a time. We normally write the resolution as a subscript to the factorial design using Roman numerals. - Saline or Bicarb) with or without Intervention B (NAC). Each factor has two levels. GSD is available in pyDOE2 as: import pyDOE2 levels = [2, 3, 4] # Three factors with 2, 3 or 4 levels respectively. a plan how you create your data. The third type of experimental design is the factorial design, in which there are two or more clearly understood treatments, such as exposure level to test chemical, animal age, or temperature. Analysis of variance and significance testing A computational procedure frequently used to analyze the data from an experimental study employs a statistical procedure known as the analysis of variance. A factorial design is a type of experimental design, i. high, referred as “+” or “+1”, and low, referred as “-”or “-1”). For example, if. Here we have 4 different treatment groups, one for each combination of levels of factors - by convention, the groups are denoted by A1, A2, B1, B2. 237) An experimental design is of resolution R if all effects containing s or fewer factors are unconfounded with any effects containing fewer than R−s factors. Therefore, our regression results for each effect are independent of all other effects and the results are clear and conclusive. The other designs (such as the two level full factorial designs that are explained in Two Level Factorial Experiments) are special cases of these experiments in which factors are limited to a specified number of levels. • results are valid over a wider range of experimental conditions Spring , 2008 Page 7. Statistical analysis conducted using the 32 factorial design [22,23] involved evaluating the impact of certain independent variables. The software contains two-level full factorial designs (up to 7 factors), fractional factorial designs (29 different designs, up to 15 factors. Design of Experiments for Product, Process & Quality Manager | Udemy. The “C” in ANCOVA denotes that a covariate is being inputted into the model, and this covariate examination can be applied to a between-subjects design, a within-subjects design, or a mixed-model design. Design of experiments (DOE) is defined as a branch of applied statistics that deals with planning, conducting, analyzing, and interpreting controlled tests to evaluate the factors that control the value of a parameter or group of parameters. 2-2, page 517. On account of errors of measurement and the neglect of certain effects the minimum S0 of S is not zero. In Chapter 1 we briefly described a study conducted by Simone Schnall and her colleagues, in which they found that washing one’s hands leads people to view moral transgressions as less wrong (Schnall, Benton, & Harvey, 2008) [1]. They are sim- ple to construct and combine factorial design properties - equally-spaced projections to univariate dimensions and spatial dispersion - with Latin hypercube properties - unique projections and model flexibility. The full factorial Design of Experiments (DOE) methodology, is a statistical analysis of the results of a set of experiments or tests. For a small number of design variables, 2n may be a manageable number of. Advantages 1. For scenarios with a small number of parameters and levels (1-3) and where each variable contributes significantly, factorial design can work well to determine the specific interactions between variables. Example of a 2x4 Factorial experiment replicated in. 2 factorial design of experiments needs less number of experiments for several factors; thus, materials and time used are slightly reduced [ , ]. Incomplete factorial designs were developed to efficiently and uniformly sample full-factorial designs involving large numbers of combinations of independent variables ( 7-9). basic analysis of covariance experiment b. Garson ANOVA/MANOVA by StatSoft Two-way ANOVA by Will Hopkins. A factorial design refers to any experimental design that has more than one independent variable. LISA Short Course: Factorial Experiments: Blocking, Confounding, and Fractional Factorial Designs, Part I from LISA on Vimeo. reduction = 3 # Reduce the number of experiments to approximately a third. Factorial designs are good preliminary experiments A type of factorial design, known as the fractional factorial design, are often used to find the "vital few" significant factors out of a large group of potential factors. Before-and-after. Design of Experiments (DOE) HVAC; Heat Exchanger; Nano-Fluid; Fuel Cell; Urban Heating Island (UHI) CFD Shop; Online Training; Free CFD Consulting; About Mr CFD. The experimental variables studied in the $$2_{\text{V}}^{{ 5 { - 1}}}$$ fractional factorial design for each sample are specified in Tables 2 and 3. An article entitled “Rotary ultrasonic machining of ceramic matrix composites: feasibility study and designed experiments,” published by Z. The longitudinal factorial design offers an opportunity to rigorously evaluate the impact of these recommendations, both in isolation and in combination, on disease outcomes. En este enlace se puede descargar SAS 9. When conducting an experiment, varying the levels of all factors at the same time instead of one at a time lets you study the interactions between the factors. In fact Sol(N - q - 1) provides. In a trial conducted using a $$2^3$$ design it might be desirable to use the same batch of raw material to make all 8 runs. There are criteria to choose "optimal" fractions. 2^k factorial designs consist of k factors, each of which has two levels. Comparing experimental designs: factorial and regression designs Factorial designs are based on experimental control between groups of experimental items, so-called conditions. MANOVA is useful in experimental situations where at least some of the independent variables are manipulated. 2 3 full factorial design was applied for examining three variables (factors) at two levels with a minimum of 8 runs. A 2k 2 k full factorial requires 2k 2 k runs. factorial experimental design examples pdf Two responses were considered for the experiment on microwave popcorn: taste and. For example, a two-level full factorial design with 10 factors requires 2 10 = 1024 runs. Factor A is 1,500 or 2,000 calories and factor B is 0 or 30 minutes of aerobic exercise. The choice of the two levels of factors used in two level experiments depends on the factor; some factors naturally have two levels. Factorial designs; Plackett-Burman designs; Box-Behnken designs; Central composite designs; Latin-Hypercube designs; There is also a wealth of information on the NIST website about the various design matrices that can be created as well as detailed information about designing/setting-up/running experiments in general. Factorial Experimental Designs Discover free flashcards, games, and test prep activities designed to help you learn about Factorial Experimental Designs and other concepts. The researcher must know his/her experimental design in order to run the appropriate statistical. Fisher, 1960. The resolution of a design is given by the length of the shortest word in the defining relation. partitioned into individual "SS" for effects, each equal to N(effect)2/4, divided by df=1, and turned into an F-ratio. The way in which a scientific experiment is set up is called a design. Blocking and Confounding Montgomery, D. Design of experiments (Portsmouth Business School, For complex manufacturing processes in today’s industrial environment, interactions play an important role and therefore should be studied for achieving sound experimental go for the factorial experiments recommended by Fisher so that both main factor effects (i. Analysis of variance and significance testing A computational procedure frequently used to analyze the data from an experimental study employs a statistical procedure known as the analysis of variance. Since a 33 design is a. Introduction to the Design & Analysis of Experiments introduces readers to the design and analysis of experiments. basic two group post-test only randomized experiment ANSWER: d DIFFICUL TY: Moderate REFEREN CES: 9. Experimental Design and Process Optimization - Factorial Designs, DOE in Practice - USA Online registration only accepts payments by credit card. Experimenter wants magnitude of effect, , and t ratio = effect/se(effect). To explore all combinations of factors and levels, the total number of experiments that are needed is the. See full list on methodology. FAQ; Certificates; Payment; About Us; Contact Us; Terms; Cart 0. The simplest method of experimental design is the one dimensional search i. Factorial designs can also be depicted using a design notation, such as that shown on the right panel of Figure 10. The selective tip pipetting feature of the 96-channel. 2 3 full factorial design was applied for examining three variables (factors) at two levels with a minimum of 8 runs. This is appropriate because Experimental Design is fundamentally the same for all fields. Factorial designs; Plackett-Burman designs; Box-Behnken designs; Central composite designs; Latin-Hypercube designs; There is also a wealth of information on the NIST website about the various design matrices that can be created as well as detailed information about designing/setting-up/running experiments in general. A full factorial design statistical approach based on the Design of Experiment (DoE) is performed, producing all possible combinations between the experimental factors. In this regard, how many main effects does a 2x2 factorial design have? Let's. However, some information gained from a full factorial design can be lost when using a fractional factorial design. randomized block with homogenous groups experiment c. Advantages & Disadvantages of W/i-Subjects Designs. h) Sketch a well-labeled graph of the results of this study, using the DV of "being nominated by peers as being the leader". Package DoE. Experimental Design - Experiments in Science and Industry Bayesian Reliability Optimization for Continuous/Binary Response Overview Experimental Design - Computational Problems. Experimental design is the process of planning a study to meet specified objectives. A key feature of fractional factorials that is not shared by more ad hoc methods for. Two-level factorial and fractional factorial designs have played a prominent role in the theory and practice of experimental design. One common type of experiment is known as a 2×2 factorial design. Factorial Design So far we have looked at 1-sample, 2-sample, and t-sample problems. Two examples of real factorial experiments reveal how using this approach can potentially lead to a reduction in animal use and savings in financial and scientific resources without loss of scientific validity. Factorial experiments are not stand-alone designs; they are the arrangement and organization of treatments within different. o The statistics are pretty easy, a t-test. One of the golden standards of experimental design in both the physical and social sciences is a random controlled experiment with only one dependent variable. The goal of our work is to identify optimal and robust designs for factorial experiments with binary response. 12, 16, 20 or 24. Fisher showed that there are advantages by combining the study of multiple variables in the same factorial experiment. Patients can utilize paper, smart phone applications, or even electronic health record portals to sequentially record their blood pressures. For example, you might use simple random sampling, where participants names are drawn randomly from a pool where everyone has an even probability of being chosen. For example, you would like to determine the best conditions for injection-molding a plastic part. In this regard, how many main effects does a 2x2 factorial design have? Let's. 045 0 0 A statistical analysis was carried out on the experimental results, and the two main effects and. The design of an experiment plays a major role in the eventual solution of the problem. I suggest that you put the 5-level IVs on the x-axis and the other IV as a line color or bar color. The term "treatment" is used to describe the different levels of the independent variable, the variable that's controlled by the experimenter. If you suspect or think that proportional effects. Experimental Research: Factorial Design. software (Stat-Ease) to investigate the optimum HAp. Two-Way Factorial Design. Factorial Analysis is an experimental design that applies Analysis of Variance (ANOVA) statistical procedures to examine a change in a dependent variable due to more than one independent variable, also known as factors. equivalence is certain. 225 There will be k main effects, and Unreplicated 2k Factorial Designs These are 2k factorial designs with one observation at each corner of the “cube” An unreplicated 2k factorial design is also sometimes called a “single replicate” of the 2k These designs are very widely used Risks…if there is only one. · Discuss the role of simple main effects in interpreting interactions. a/b testing ad testing experiments paid search experiments ppc statistics. Factorial design. Figure 1: Full factorial design for three variables with two levels each. Several authors have given construction methods for non-factorial experiments, for example David (1963), Williams (1976) and Bailey, Goldrei and Holt (1984). We will start by looking at just two factors and then generalize to more than two factors. do not need as many participants 2. •Experimental design must take into account the goal of the research. The limitation to that design is that it overlooks the effects multiple variables may have with one another. It looks almost the same as the randomized block design model only now we are including an interaction term: Y i j k = μ + α i + β j + ( α β) i j + e i j k. David Garson : Introduction to Design and Analysis of Experiments by George W. Garson ANOVA/MANOVA by StatSoft Two-way ANOVA by Will Hopkins. Without the covaria te, the design is simply a one-way independent design, so we would enter these data using a coding variable for the independent variable, and scores on the dependent variable will go in a different column. Welcome to Stat 706, Experimental Design. Experimental Design Book Description : This text introduces and provides instruction on the design and analysis of experiments for a broad audience. Full Factorial Design of Experiments. The run number is a multiple of four rather than a power of 2. Combining the vignette variables (factors) and their levels is done by the researcher, who also takes the responsibility for getting an optimal design. 2 Design and Analysis of Experiments by Douglas Montgomery: A Supplement for Using JMP across the design factors may be modeled, etc. Traditional research methods generally study the effect of one variable at a time, because it is statistically easier to manipulate. REMAP-CAP is a global network of leading experts, institutions and research networks. IRJET-International Research Journal of Engineering and. Full factorial designs. Multi-Factor Designs Chapter 8. Here, we use the term starting design in the same way as Chapter 8 of Street and Burgess (2007), which should not be confused with the starting designs that are used in search algorithms. Here, experimental conditions are chosen by selecting a fixed number of levels for each variable, after which experiments are run at all pos- Sible combinations. Traditionally, experiments are designed to determine the effect of ONE variable upon ONE response. Because both the experimental sampling designs and subsequent analysis procedures are unfam-iliar, we present this example in detail. Factor levels are accessed in a balanced full or fractional factorial design. Or more generally: physical/technical, chemical, medical or business processes can be optimised. While advantageous for separating individual effects, full factorial designs can make large demands on data collection. Potential Experimental Designs 2 k factorial designs For k factors, two levels All combinations of each level of each factor Orthogonal (easy to analyze) Often used to screen Figure:2 2 and 2 3 factorial designs Sanchez and Wan (2008) Dr. Because there are three factors and each factor has two levels, this is a 2×2×2, or 2 3, factorial design. design) and orthogonal arrays (function oa. For example, an experiment using the following experimental design is a multifactor experiment: Coke \$1. 1 [Stat Ease. In an experiment, you manipulate one or more independent variables and measure their effect on one or more dependent variables. Example: 2 10 =1024 combinations. Know how factorial experiments can be used for more than two factors. pack design and cost calculated in BatPaC represent projections of a 2020 production year and a specified level of annual battery production, 10,000-500,000. Other Methods of Experimental Design. Factorial designs – designs with two or more independent variables Independent variables are called factors Two factor experiment – the simplest factorial design FACTORIAL DESIGNS They give us information about the effects of each independent variable in the experiment – main effects They enable us to answer the question: How does the. Factorial designs (By using a factorial design)" an experimental investigation, at the same time as it is made more comprehensive, may also be made more efficient if by more efficient we mean that more knowledge and a higher degree of precision are obtainable by the same number of observations. The method is popularly known as the factorial design of experiments. The starting design, F, is either a complete factorial design or a fractional factorial design, whose entries become the rst alternatives in each choice set. 9 a comparison between the number of experiments of a full Three Level Factorial design and other designs are shown. United States Department of Agriculture. Click SigmaXL > Design of Experiments > 2-Level Factorial/Screening > 2-Level Factorial/Screening Designs. Experimental Design by Roger Kirk Chapter 12: Split-Plot Factorial Design | Stata Textbook Examples. LISA Short Course: Factorial Experiments: Blocking, Confounding, and Fractional Factorial Designs, Part I from LISA on Vimeo. 2 x 2 factorial experiment d. The run number is a multiple of four rather than a power of 2. For example, you would like to determine the best conditions for injection-molding a plastic part. A key use of such designs to identify which of many variables is most important and should be considered for further analysis in more detail. A marginal table contains a subset of the factorial treatments averaged across all other factors in the design. , in agronomic field trials certain factors require "large". The number of digits tells you how many in independent variables (IVs) there are in an experiment while the value of each number tells you how many levels there are for each independent variable. Over the course of five days, you. Types of design include repeated measures, independent groups, and matched pairs designs. Full Factorial/ Fractional Factorial Experimental Design Hey I was wondering if the matlab commands called ff2n and fracfact are in the Octave statistics package? I also have no idea how to open the statistics package on GNU, can anyone explain what I would need to do?. See full list on methodology. called a fractional factorial design. What are factorial experimental designs, and what advantages do they have over one-way experiments? What is meant by crossing the factors in a factorial design? What are main effects, interactions, and simple effects? Slideshow 1700432 by brigid. The independent variables, often called factors, must be categorical. An ANOVA is a type of statistical analysis that tests for the influence of variables or their interactions.
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2021-09-24 15:04:05
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http://mathhelpforum.com/trigonometry/37777-more-trig-equations-ugh.html
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# Math Help - More Trig Equations (ugh)
1. ## More Trig Equations (ugh)
NVM
i solved them
2. Originally Posted by NeedHelp18
1. 3cot^2 (x) - 1 = 0
I must ask..... is it $3cot^{2}(x-1)$, or $3cot^{2}x-1$ (is that possible?) or something else?
...same with number 2.... I'm confused by what is in the exponent...again......maybe someone else knows what it is...
3. same for the second one
4. Originally Posted by NeedHelp18
NVM
i solved them
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2014-04-20 06:32:51
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https://www.caltech.edu/campus-life-events/master-calendar/high-energy-theory-seminar-88532
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# High Energy Theory Seminar
Friday, July 17, 2020
11:00am to 12:00pm
We consider a class of quantum field theories and quantum mechanics, which we couple to topological QFTs, in order to classify non-perturbative effects in the original theory. The TQFT structure arises naturally from turning on a classical background field for a discrete global symmetry.In SU(N) Yang-Mills theory coupled to $\mathbb Z_N$ TQFT, the non-perturbative expansion parameter is $\exp[-S_I/N]= \exp[-{8 \pi^2}/{g^2N}]$ both in the semi-classical weak coupling domain and strong coupling domain, corresponding to a fractional topological charge and action configurations. To classify the non-perturbative effects in original SU(N) theory, we must use PSU(N) bundle and lift configurations (critical points at infinity) for which there is no obstruction back to SU(N). These provide a refinement of instanton sums: integer topological charge, but crucially fractional action configurations contribute, providing a TQFT protected generalization of resurgent semiclassical expansion to strong coupling. Monopole-instantons (or fractional instantons) on $T^3 \times S^1_L$ can be interpreted as tunneling events in the 't Hooft flux background in the $PSU(N)$ bundle. The construction provides a new perspective to the strong coupling regime of QFTs and resolves a number of old standing issues, especially, fixes the conflicts between the large-$N$ and instanton analysis. If time permits, I will give a self-consistent derivation of the mass gap as a function of theta angle in $CP^{N-1}$ theory.
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2021-07-28 14:05:56
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https://ncatlab.org/nlab/show/infinity-Lie+theory+-+contents
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nLab infinity-Lie theory - contents
Examples
$\infty$-Lie algebras
Last revised on November 10, 2016 at 23:19:54. See the history of this page for a list of all contributions to it.
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2021-09-22 17:46:53
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http://math.stackexchange.com/questions/191747/how-to-find-eigenvector-of-one-second-order-differential-equation-for-hermit
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# How to find eigenvector of one second order differential equation for Hermit?
After read http://math.tut.fi/~piche/pde/pde.pdf , do not know how to calculate eigenvector
How to find eigenvector of one second order differential equation?
why some people use sin as eigenvector? is it only sin can be eigenvector?
The problem is for eigenfunction expansion, first step is finding eigenvalue and eigenvector, but do not know how to calculate eigenvector for differential equation
for example Maple code
x*diff(f(x), x$2) + 2*x*diff(f(x),x) + f(x) = 0 x*diff(f(x), x$2) + 2*x*diff(f(x),x) + x = 0
Updated
sol := dsolve(t*diff(phi(x),x$2)-x*diff(phi(x),x)+n*phi(x),phi(x)); phi := unapply(rhs(sol), x); BC := [phi(0)=0,phi(1)=0]; with(linalg): Ccoef := genmatrix(BC, [_C1,_C2]); CharEqn := det(Ccoef) = 0; restart; sol := dsolve(t*diff(phi(x,t,n),x$2)-x*diff(phi(x,t,n),x)+n*phi(x,t,n),phi(x,t,n));
phi := unapply(rhs(sol), x);
BC := [phi(0,0,0)=0,phi(1,1,1)=0];
with(linalg):
Ccoef := genmatrix(BC, [_C1,_C2]);
CharEqn := det(Ccoef) = 0;
**sorry only Sunday have time to seriously read this file, i find the sin function coming from the step of calculating characteristic equation use pdf file's method to calculate above differential equation for eignvector,
this equation is Hermit after tried, characteristic equation is zero, it imply no eigenvector i guess this calculation maple code has something wrong
how to calculate this?**
Updated 2
Originally i expect to find Hermit H(x) and then use sum(H*z^m/m!, m=0..infinity) to find a A*exp(B) where B is in term of z and t and it is just a simple formula now following the steps, i guess the H is the solution of green function about the expansion
it become more compicated for H(x), and i find there is a D[2] but do not know where it come from. then do not know which step is H(x), i just guess vterm or vv
sol := dsolve(t*diff(phi(x),x$2)-x*diff(phi(x),x)+n*phi(x),phi(x)); phi := unapply(rhs(sol),x); odetest(sol,ode); eq1:=limit(rhs(sol),x=0,right)=0; eq2:=eval(rhs(sol),x=1)=0; Ccoef := LinearAlgebra:-GenerateMatrix([eq1,eq2],[_C1,_C2]); CharEqn:=LinearAlgebra:-Determinant(%[1])=0; solve(CharEqn,t); step1 := map(xi->simplify(subs(t=RootOf(KummerM(1/2-(1/2)*n, 3/2, 1/(2*_Z))),xi)),Ccoef); with(linalg): NN := nullspace(step1); subs(_C1=NN[1][1],_C2=NN[1][2],t=RootOf(KummerM(1/2-(1/2)*n, 3/2, 1/(2*_Z))),phi(x)); phi := (n,t,x) -> KummerM(1/2-(1/2)*n, 3/2, (1/2)*x^2/RootOf(KummerM(1/2-(1/2)*n, 3/2, 1/(2*_Z))))*x; assume(j,posint): interface(showassumed=0): Gterm := unapply(-phi(n,t,x)*phi(n,t,x)*exp(-lambda(j)*t)/int(phi(n,t,x)^2,x=0..1),(j,n,x,y,t)): G:=Sum(Gterm(j,n,x,y,t),j=1..infinity); vterm := int(D[2](Gterm)(n,1,x,t-tau),tau=0..t); vv := sum(Sum(op(n,vterm),j=1..infinity),n=1..2); - please explain your problem in more detail, the question is not clear – James S. Cook Sep 6 '12 at 3:29 add some example – M-Askman Sep 6 '12 at 3:37 nice pdf file ... thanks for sharing – Santosh Linkha Sep 6 '12 at 4:10 @M-Askman from the pdf, it says to read Zachmanoglou and Thoe to supplement the notes. So, you might get a copy of that to help better understand the pdf. – James S. Cook Sep 8 '12 at 16:26 @M-Askman Minor nitpick. Operators have eigenvectors/values. Not equations. – Thiagarajan Sep 8 '12 at 23:47 ## 3 Answers Note that the principle of finding eigenvector of the second order linear ODE that arise from using separation of variables to a linear PDE is that finding the best form of the eigenvector so that we can get the most simplified form of the solution subjected to the B.C.s or I.C.s Theoretically, the form of the eigenvector can choose arbitrarily. However, since solving linear PDEs by using separation of variables subjected to the B.C.s or I.C.s should be unavoidable for performing kernel inversions. Choosing eigenvectors unwisely will face the too complicated kernel inversions and become trouble. So choosing eigenvectors wisely should be important when solving linear PDEs. Since solving a second order linear ODE will have two groups of linear independent solutions, so the best way is that making one of the linear independent solutions becomes zero when substituting most B.C.s or I.C.s, cause the last B.C. or I.C. we handling is remaining only one kernel. In fact finding the best eigenvector is mainly base on our personal experience. Note that$\sin$is only one of the common considerations but the only consideration, especially when the solution of the second order linear ODE fail to express in terms of$\sin$and$\cos$, because$\sin$and$\cos$have the important properties that for all integer$n$,$\sin n\pi=0$and$\cos n\pi=(-1)^n$. Think the following examples that why their eigenvectors are the best to be taken to those forms:$1$. Boundaries in heat equation:$-9\pi^2s^2-72$. Indication on how to solve the heat equations with nonconstant coefficients:$-\dfrac{4\pi^2s^2+1}{4}3$. Wave equation with initial and boundary conditions - is this function right?:$-\dfrac{(2m+1)^2\pi^2c^2}{4l^2}$- In the posted pdf on page 92 is mentions that$\{ \phi_i, \lambda_i \}$is a set of eigenfunctions and eigenvalues defined as solutions to $$\mathcal{L} \phi+\lambda \phi = 0 \qquad \mathcal{B}\phi =0$$ This notation is a quick way to denote a given PDE and associated boundary condition(BC). In the nice examples I teach in my DEqns course the values for$\phi$are forced from the given boundary conditions and the solutions$\phi$can be sine, cosine, hyperbolic sine or hyperbolic cosine or the constant function. The PDE+BC typically give a whole family of eigenvalues and eigenfunctions. In the separable case, the eigenvalues arise from proposing the solution is a product of univariate solutions then subtitution into the PDE and a little algebra brings you to an equation where the lhs and rhs are necessarily independent hence must be proportional to a constant. Customarily that constant is called the eigenvalue. For example, take a look at the explicit calculations of this in: (I have to run now,maybe there is a better thread to link here) Simplifying PDE The term eigenvector and eigenfunction are interchangeable in the context of function space. However, there are also some matrix calculations in that pdf and I'm not totally sure you are not reading something somewhere about a concrete column vector. The fact you mention "sin" gives me hope my interpretation of eigenvector=eigenfunction is correct. - sorry only Sunday have time to read this file seriously, updated an example which is Hermit differential equation, if success solve this example, then eigenvector can be known – M-Askman Sep 9 '12 at 3:39 Note sure that this is an answer, but hope that these points clarify some things for you. Any book on ordinary differential equations will cover the topic of eigenvalues/vectors. Googleing ODE and eigenvector will get you a lot of useful information. You should note that eigenvectors/values are only useful/relevant for linear differential equations. I am not real familiar with Maple, but the example that you posted appears to be non-linear so the idea of eigenvectors would not apply.$\sin(x)$is an eigenvector (more technically an eigenfunction) in differential equations because$\sin(x)$,$\cos(x)$, and$e^x$all maintain their basic form through the differentiation operator$\frac{d}{dx}$. So when you have a linear differential equation of the form$\dot{f}(x) + af(x) = 0$it is clear that$\dot{f}(x) = -af(x)$so$\dot{f}(x)$and$f(x)$are of the same form one is just a scaled version of the other. The only function with this property is$e^x\$.
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sorry only Sunday have time to read this file seriously, updated an example which is Hermit differential equation, if success solve this example, then eigenvector can be known – M-Askman Sep 9 '12 at 10:34
i have updated some code, after tried not success – M-Askman Sep 10 '12 at 13:41
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2016-02-12 18:21:38
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https://maths.dept.shef.ac.uk/maths/sem_history.html
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# Seminar history 1992/93 1993/94 1994/95 1995/96 1996/97 1997/98 1998/99 1999/00 2000/01 2001/02 2002/03 2003/04 2004/05 2005/06 2006/07 2007/08 2008/09 2009/10 2010/11 2011/12 2012/13 2013/14 2014/15 2015/16 2016/17 2017/18 2018/19 2019/20 2020/21 2021/22
May 13 Thu Dr Hugh Hudson (University of Glasgow and UC Berkeley) SP2RC seminar Abstract: We often cite the Carrington flare (SOL1859-09-01) as an extreme flare/CME/geostorm event and a prototype "superflare." I discuss the original observations in the context of what we now know about solar and stellar flares. Although not an extreme event in the sense of being truly exceptional, it has become clear that the Carrington flare turned into a milestone in what we now call "multimessenger astronomy." May 12 Wed Isha Kotecha (Okinawa Institute for Science and Technology) Cosmology, Relativity and Gravitation Abstract: Thermal states are absolutely important for statistical descriptions of physical systems; likewise also in discrete quantum gravity, where classical continuum spacetime is thought to emerge from the collective physics of some underlying quantum structure. But what equilibrium even means in a background independent context is a foundational open issue. In this talk, I will discuss a generalisation of Gibbs states for use in such contexts, while emphasising on the maximum entropy principle characterisation. The resulting setup is then applied in quantum gravity, by modelling a quantum spacetime as a many-body system of candidate quanta of geometry, and utilising their field theoretic formulation of group field theory (GFT). This leads to concrete examples of quantum gravitational generalised Gibbs states. I will then present non-perturbative thermofield double vacua, and their inequivalent thermal representations, as induced by these Gibbs states. An interesting class of thermal condensates are defined, which encode fluctuations in the underlying quantum geometry. These are subsequently applied in GFT cosmology to extract an effective FLRW universe at late times, with a bounce and accelerated expansion at early times. May 12 Wed Ana Lecuona (University of Glasgow) Pure Maths Colloquium Abstract: In this talk we will mainly focus on rational homology balls: their history, interest and prominence in nowadays low dimensional topology. We will start with the basic definitions and we will spend some time trying to understand the importance of these balls and how they relate to seemingly disjoint problems. We will end by discussing some recent results which will hopefully give a picture of the current state of the art. No prior knowledge of the topic will be assumed. May 12 Wed Kevin Wilson and Cameron Williams (Newcastle) Statistics Seminar Abstract: When eliciting prior distributions from experts, it may be desirable to combine them into a single group prior. There are many methods of expert-elicited prior aggregation, which can roughly be categorised into two types. Mathematical aggregation methods combine prior distributions using a mathematical rule, while behavioural aggregation methods assist the group of experts to come to a consensus prior through discussion. As many commonly used aggregation methods have different requirements in the elicitation stage, there are few, if any, comparisons between them. Using a clinical trial into a novel diagnostic test for Motor Neuron Disease as a case study, we elicited a number of prior distributions from a group of experts. We then aggregated these prior distributions using a range of mathematical aggregation methods, including Equal Weights linear pooling, the Classical Method, and a Bayesian aggregation method. We also undertook an in-person behavioural aggregation with the experts, using the Sheffield Elicitation Framework, or SHELF. Using expert answers to seed questions, for which the elicitors know the true values, we compare and contrast the different aggregation methods and their performance. We also demonstrate how all considered aggregation methods outperform the individual experts. May 12 Wed George Moulantzikos Algebraic Geometry Learning Seminar: Toric varieties May 11 Tue Lassina Dembele (University of Luxembourg) Number Theory seminar Abstract: There is an isogeny class of semistable abelian surfaces A with good reduction outside 277 and $End_Q(A) = Z$. The modularity (or paramodularity) of this class was proved by a team of six people: Armand Brumer, Ariel Pacetti, Cris Poor, Gonzalo Tornaria, John Voight and David Yuen. They did so by using the so-called Faltings-Serre method. This was the first known case of the paramodularity conjecture. In this work in progress, I will discuss how to (re-)prove the modularity of these surfaces by directly applying deformation theory. This could be seen as an explicit approach to deformation theory. May 11 Tue Emily Roff (Edinburgh) MiaowMiaow (2d category theory) Abstract: Seminar will run for 90 mins. May 6 Thu Matheus Aguiar-Kriginsky Silva (University of the Balearic Islands, UIB (ES)) SP2RC/ESPOS Abstract: In this seminar, we aim to present the results of two recent works centred at the use of spectropolarimetric data obtained with the CRISP instrument at the SST in the Ca II 845.2 nm line. With these observations, we obtain information about the magnetic field present in chromospheric spicules and coronal rain clumps. For this purpose, we have used the Weak-field approximation (WFA), which albeit being computationally simple to implement, needs careful assessment of the conditions of the plasma to be correctly applied. Magnetic fields of the order of hundreds of Gauss are inferred. We also combine the Ca II 845.2 nm observations with simultaneous Hα observations to estimate the temperature and non-thermal velocity of the plasma in coronal rain and spicules using the observed Doppler amplitude. May 5 Wed Reinder Meinsma / Yirui Xiong Algebraic Geometry Learning Seminar: Toric varieties May 4 Tue Andrea Conti (University of Luxembourg) Number Theory seminar Abstract: When interpolating p-adically Galois representations attached to automorphic forms, one obtains many new representations that are not de Rham locally at p. It is expected that such representations are characterized by the condition of being trianguline at p. We study how this notion behaves under functoriality: it is easy to show that if S: GL_m -> GL_n is an algebraic representation and rho is an m-dimensional trianguline Galois representation, the composition S(rho) is again trianguline. We prove that under reasonable assumptions the reverse implication is true, with the goal of applying the result to the study of congruence loci on eigenvarieties. Apr 30 Fri Indrani Roy (University College London) SP2RC seminar Abstract: This study investigates the role of the eleven-year solar cycle on the Arctic climate during 1979–2016. It reveals that during those years, when the winter solar sunspot number (SSN) falls below 1.35 standard deviations (or mean value), the Arctic warming extends from the lower troposphere to high up in the upper stratosphere and vice versa when SSN is above. The warming in the atmospheric column reflects an easterly zonal wind anomaly consistent with warm air and positive geopotential height anomalies for years with minimum SSN and vice versa for the maximum. Despite the inherent limitations of statistical techniques, three different methods – Compositing, Multiple Linear Regression and Correlation – all point to a similar modulating influence of the sun on winter Arctic climate via the pathway of Arctic Oscillation. Presenting schematics, it discusses the mechanisms of how solar cycle variability influences the Arctic climate involving the stratospheric route. Compositing also detects an opposite solar signature on Eurasian snow-cover, which is cooling during Minimum years, while warming in maximum. It is hypothesized that the reduction of ice in the Arctic and a growth in Eurasia, in recent winters, may in part, be a result of the current weaker solar cycle. Apr 28 Wed Tom Bridgeland (University of Sheffield) Pure Maths Colloquium Abstract: I'm planning to talk about some quite old joint work with Ivan Smith which realises moduli spaces of quadratic differentials on Riemann surfaces as spaces of stability conditions on a certain class of three-dimensional Calabi-Yau triangulated categories. I expect to spend the whole talk explaining what all those words mean, and why such a result might be interesting! Apr 28 Wed Eleni Kontou (Amsterdam) Cosmology, Relativity and Gravitation Abstract: The classical singularity theorems of General Relativity rely on energy conditions that are easily violated by quantum fields. In this talk I will provide motivation for an energy condition obeyed by semiclassical gravity: the smeared null energy condition (SNEC), a proposed bound on the weighted average of the null energy along a finite portion of a null geodesic. I will then then present the proof of a semiclassical singularity theorem using SNEC as an assumption. This theorem extends the Penrose theorem to semiclassical gravity and has interesting applications to evaporating black holes. Based on: arXiv: 2012.11569 Apr 28 Wed Ananyo Dan Algebraic Geometry Learning Seminar: Toric varieties Apr 27 Tue Tobias Berger (University of Sheffield) Number Theory seminar Abstract: This is a report on joint work in progress with Adel Betina (Vienna) to prove congruences between Eisenstein and cuspidal cohomology classes for imaginary quadratic fields. I plan to discuss applications to R=T theorems and congruences for classical CM modular forms. Apr 27 Tue Dan Graves MiaowMiaow (2d category theory) Apr 22 Thu Alex Prokopyszyn (University of St Andrews) SP2RC/ESPOS Abstract: In this seminar, we aim to show why Fast/Alfvén waves couple at the solar surface. We will also show that the polarisation of the waves changes upon reflection at the solar surface. Finally, we will test the validity of line-tied boundary conditions for highly phase-mixed Alfvén waves. For most parameters, line-tied boundary conditions provide a good approximation. However, for highly phase-mixed waves, the coronal transverse length scales can be shorter than the corresponding parallel length scales in the chromosphere. In that case, we find that the line-tied model produces unphysically large boundary layers. Hence, we have the counter-intuitive result that the length scales parallel to the solar surface play a key role in determining the validity of line-tied boundary conditions. Apr 21 Wed Cihan Okay (Bilkent University) Pure Maths Colloquium Abstract: A central question in quantum information theory is to determine physical resources required for quantum computational speedup. Such resources are characterized in terms of intrinsic features of quantum states and include various notions such as quantum contextuality, quasiprobability representations, and topological phases. Each of these notions correspond to a different perspective taken on the question of where the computational power is hidden. We take a topological approach based on the recently established connection between classifying spaces from algebraic topology and the study of quantum contextuality from quantum foundations in joint work with Robert Raussendorf. In this talk I will explain this connection and discuss possible ways of extending the role of topology to study other kinds of quantum resources. Apr 21 Wed Peter Clarkson (University of Kent) Applied Mathematics Colloquium Abstract: In this talk I shall discuss rational solutions of the Boussinesq equation, the focusing nonlinear Schrodinger (NLS) equation and the Kadomtsev-Petviashvili I (KPI) equation, which are all soliton equations solvable by the inverse scattering. The Boussinesq equation was introduced by Boussinesq in 1871 to describe the propagation of long waves in shallow water. Rational solutions of the Boussinesq equation, which are algebraically decaying and depend on two arbitrary parameters, are expressed in terms of special polynomials that are derived through a bilinear equation, have a similar appearance to rogue-wave solutions of the focusing NLS equation and have an interesting structure. Conservation laws and integral relations associated with rational solutions of the Boussinesq equation will also be discussed. Rational solutions of the KPI equation will be derived in three ways: from rational solutions of the NLS equation; from rational solutions of the Boussinesq equation; and from the spectral problem for the KPI equation. It'll be shown that these three families of rational solutions are fundamentally different. Apr 21 Wed Bianca Dittrich (Perimeter Institute, Waterloo) Cosmology, Relativity and Gravitation Abstract: General relativity taught us that spacetime geometry is dynamical and quantum theory posits that dynamical objects are quantum. In this talk I will sketch the notion of quantum geometry, which arises in loop quantum gravity. Somewhat surprisingly, this quantum geometry, although it arises from a quantization of a torsion-free theory, does include torsion degrees of freedom. I will then introduce an effective dynamics for such quantum geometries and sketch how to derive corrections that arise due to the inclusion of torsion degrees of freedom. Apr 21 Wed Yannik Schüler Algebraic Geometry Learning Seminar: Toric varieties Apr 20 Tue Petru Constantinescu (University College London) Number Theory seminar Abstract: Motivated by a series of conjectures of Mazur, Rubin and Stein, the study of the arithmetic statistics of modular symbols has received a lot of attention in recent years. In this talk, I will highlight several results about the distribution of modular symbols, including their Gaussian distribution and the residual equidistribution modulo p. I will also talk about generalisations to quadratic imaginary fields and higher dimensions. Apr 20 Tue James Cranch (Sheffield) MiaowMiaow (2d category theory) Abstract: Notes: 14:00-15:30. Contact Joseph Martin for meeting link. Apr 16 Fri Daria Y Shukhobodskaia (SP2RC (UoS)) SP2RC seminar Abstract: The investigation of magnetohydrodynamic (MHD) wave propagation in different equilibrium configurations is important for the development of solar magneto-seismology (SMS). The applicable models of solar atmospheric waveguides are studied in the framework of Cartesian and cylindrical geometries. First, a magnetised plasma slab sandwiched between an arbitrary number of non-magnetic/ magnetic layers are considered and an analytical approach is used for the derivation of its dispersion relation. The amplitudes of the eigenmodes depend on the equilibrium structuring and the model parameters; this motivates an application as a solar magneto-seismology tool. Specific cases of two- and three-layered slabs are studied in detail and their potential applicability to magnetic bright points is discussed. Furthermore, the resonant damping of propagating kink waves is studied in a straight magnetic flux tube with the density varying along the tube taking into account the magnetic loop expansion. Also non-stationary magnetic tubes to model, for example, cooling coronal loops is considered. In particular, it was found that cooling enhances the wave amplitude and the loop expansion makes this effect more pronounced. After, we analyse $10$ driven kink oscillations in coronal loops to further investigate the ability of expansion and cooling to explain complex damping profiles. The used approach could allow to infer some important diagnostic information (such as, for example, the density ratio at the loop foot-points) from the oscillation profile alone, without detailed measurements of the loop and without complex numerical methods. The current study indicates that thermal evolution should be included in kink-mode oscillation models in the future to help us to better understand oscillations that are not purely Gaussian or exponential. Finally, fluting oscillations in a thin straight expanding magnetic flux tube in the presence of background flow are considered. The method of multiple scales is used for the derivation of the system of governing equations. We have found that the amplitude increases due to cooling and is higher for a higher expansion factor. Higher values of the wave number lead to localisation of the oscillation closer to the boundary. We show that the higher the value of the ratio of internal and external plasma densities, the higher the amplification of oscillation due to cooling. So, not only the wave number plays an important role in the evolution of the cooling system, but also the density ratio and the variation of tube expansion are relevant parameters in the cooling process of an oscillating flux tube. Apr 15 Thu Prof Richard A Harrison (RAL Space) SP2RC seminar Abstract: This presentation will take stock of where we are with Coronal Mass Ejection (CME) research, taking a brief look at the history of CME observations and the early interpretations of the phenomenon, through to the present day where we have multi-spacecraft observations with coronagraphs and heliospheric imagers and a wide range of modelling techniques, many of which are now geared towards space weather impacts. This is a research area that has matured dramatically, since the launch of the SOHO spacecraft in particular, but especially with the increased interest in space weather and missions such as STEREO and Lagrange. It is a good time to take stock and in doing so to revisit some basic issues, including the flare-CME relationship, stealth CMEs, coronal dimming and CME-CME interactions, as well as lessons learnt from imaging and tracking CMEs in the corona and in the heliosphere. Perhaps it is also a useful time to pause and ask the questions, what else do we want to know about CMEs, and how are we going to satisfy that desire? Apr 15 Thu Juie Shetye (New Mexico State University) Plasma Dynamics Group Abstract: The solar chromosphere serves as a bridging layer between the photosphere and the corona. This dynamic layer is filled with a plethora of features that vary in time and space. With the advent of high-resolution ground-based observations we can discover new features. We use some of the World’s biggest solar telescopes to zoom into this layer and it reveals never seen before dynamics. Here I present detailed observations of two science topics that are guided by observations. I show a statistical study of spicules, which are long-thin grass-like features observed on the sun. These events wiggle-jiggle and sway around their axes or along a common centre of mass to create wave-like motions on the Sun. These waves can travel with speeds on 100s of km per second to energise the solar chromosphere. The second example I show are swirling-whirling events, that look like Tornadoes on the Earth. These churn the matter from the Lowe photosphere to the chromosphere. Studying the behaviour of such events is vital in understanding a decade long question in the solar physics, that tells us how the Sun’s atmosphere is heated. In addition, the current work presented already tests the limits of current telescopes in terms of the temporal and spatial resolution. The answer to exploring the depth of chromosphere lies in building next-generation solar physics observatories such as DKIST that have 3 times more spatial resolution than CRISP and much higher temporal resolution. Apr 8 Thu Mayukh Panja (Max Planck Institute for Solar System Research, MPS (DE)) SP2RC/ESPOS Abstract: Penumbral filaments do not form naturally in MHD simulations of sunspots. This is typically circumvented by modifying the top boundary: the field is made 2-3 times more horizontal than a potential field configuration. In this talk, I will explore the possibility that penumbral filament formation is governed by the subsurface structure of sunspots. We conducted a series of numerical experiments where we used flux tubes with different initial curvatures to study the effect of the fluting instability on the subsurface structure of spots using the MURaM code. We find that the curvature of a flux tube indeed determines the degree of fluting the flux tube will undergo—the more curved a flux tube is, the more fluted it becomes. In addition, sunspots with strong curvature have strong horizontal fields at the surface and therefore readily form penumbral filaments. The fluted sunspots eventually break up from below, with lightbridges appearing at the surface several hours after fluting commences. We also propose that intruding lightbridges can be used as tracers of the subsurface magnetic field. Apr 2 Fri Michael Griffiths (Research IT, University of Sheffield) SP2RC seminar Abstract: Parallel magnetohydrodynamic (MHD) algorithms are important for numerical modelling of highly inhomogeneous solar, astrophysical and geophysical plasmas. Parallelisation techniques have been exploited most successfully by the gaming/graphics industry with the adoption of graphical processing units (GPUs) possessing hundreds of processor cores. The opportunity has been recognised by the computational sciences and engineering communities who have recently harnessed successfully the numerical performance of GPUs. Here, we introduce the implementation of SMAUG, the Sheffield Magnetohydrodynamics Algorithm Using GPUs. SMAUG is a 1-3D MHD code capable of modelling magnetised and gravitationally stratified plasmas. We illustrate an application of SMAUG with a discussion of the results of a study of the atmospheric motions generated by the solar global resonant oscillations. Utilising a spatially structured driver across the base of the computational model, we embark on how the ensemble of performed simulations, that provide insight into the energy supplied by various wave modes, is redistributed in the atmosphere. The results shed light on the mechanisms leading to ubiquitous intensity oscillations in the stratified solar atmosphere and establish a link between signals at photospheric levels and the solar coronal response. In the final section of the talk we describe how to access the different resources which are available for running computational MHD codes with GPU’s. Apr 1 Thu Iulia Chifu (University of Goettingen) Plasma Dynamics Group Abstract: The magnetic field plays an essential role in the initiation and evolution of different solar phenomena in the corona. The structure and evolution of the 3D coronal magnetic field are still not very well known. A way to ascertain the 3D structure of the coronal magnetic field is by performing magnetic field extrapolations from the photosphere to the corona. In previous work, it was shown that by prescribing the 3D-reconstructed loops’ geometry, the magnetic field extrapolation produces a solution with a better agreement between the modeled field and the reconstructed loops. This also improves the quality of the field extrapolation. Stereoscopy, which uses at least two view directions, is the traditional method for performing 3D coronal loop reconstruction. When only one vantage point of the coronal loops is available, other 3D reconstruction methods must be applied. Within this work, we present a method for the 3D loop reconstruction based on machine learning. Our purpose for developing this method is to use as many observed coronal loops in space and time for the modeling of the coronal magnetic field. Our results show that we can build machine-learning models that can retrieve 3D loops based only on their projection information. Ultimately, the neural network model will be able to use only 2D information of the coronal loops, identified, traced, and extracted from the extreme-ultraviolet images, for the calculation of their 3D geometry. Mar 25 Thu Valeriia Liakh (INAF-OAR National Institute for Astrophysics (IT)) SP2RC/ESPOS Abstract: We report 2D MHD simulations of the large-amplitude oscillations (LAOs) in the solar prominences performed with MHD code Mancha. We aim to study the properties of LAOs using high-resolution simulations in a simple 2D magnetic configuration that contains a dipped part. We loaded the dense prominence plasma in the dips region. In order to excite oscillations, we used a perturbation directed along the magnetic field. For the same numerical model, the four spatial resolutions were considered: 240, 120, 60, and 30 km. The longitudinal LAOs (LALOs) are strongly damped even in the high-resolution simulation in the region of the weaker and more curved magnetic field (at the center and bottom of the prominence). At the prominence top, the oscillations have relatively longer damping times. Furthermore, during the first 100 minutes, the longitudinal velocity shows growing with respect to its initial amplitude. The amplification becomes even more significant in the experiments with high-resolution. The damping and amplification mechanisms involved in our experiments can be important for explaining the observed amplification and attenuation of the LALOs. Mar 24 Wed Magnus Goffeng (Lund University) Pure Maths Colloquium Abstract: An invariant that has attracted quite some attention in the last decade is the magnitude of a compact metric space. Magnitude gives a way of encoding the size of a metric space, resembling both the Euler characteristic and the capacity. In this colloquium I will give a short introduction to magnitude and present some recent results for compact metric spaces of geometric origin (i.e. domains in Euclidean space or manifolds). One of the results states that the magnitude recovers geometric invariants such as volume and certain integrals of curvatures. Based on joint work with Heiko Gimperlein and Nikoletta Louca. Mar 24 Wed George Moulantzikos Algebraic Geometry Learning Seminar: Toric varieties Mar 18 Thu Dr Helen Mason (University of Cambridge) SP2RC seminar Abstract: Spectroscopic diagnostics have enabled us to determine the physical parameters of plasma for different solar features (active regions, jets, flares etc). Helen started her career studying the visible coronal lines from the 1952 eclipse observations. She then studied the UV and X-ray spectrum of the Sun, working on many joint UK, NASA, ESA and Japanese solar space projects including Skylab, the SMM (Solar Maximum Mission), Yohkoh, SoHO (Solar and Heliospheric Observatory), Hinode, SDO (Solar Dynamics Observatory) and IRIS (Interface Region Imaging Spectrograph). She was a founder member of the CHIANTI team, an atomic database which has been extensively used for solar data analysis. In this talk, she will pick out a few key results as examples of the value of spectroscopic diagnostics (in the transition region and corona). She will also look towards the current opportunities for research in this field and the future prospects for spectrometers. For a recent review, see: Del Zanna and Mason, 2018, Solar UV and X-ray spectral diagnostics’, Sol. Phys. Liv. Reviews. Mar 18 Thu Kostas Tziotziou (National Observatory of Athens) Plasma Dynamics Group Abstract: Small-scale vortex motions are detected at various spatial and temporal scales in the solar atmosphere, from the photosphere to the low corona. They often exhibit complex structure and dynamics and, as largely magnetic structures, can foster a variety of oscillations and wave modes. Despite, however, recent advancements in observational and theoretical studies, as well as in simulations and modelling, their proper detection, especially in chromospheric lines such as Hα and Ca II 8542 Å is still an open issue, and their structure and dynamics remain poorly understood. We present a novel automated method of chromospheric swirl detection based on their morphological characteristics that nicely complements previous LCT-related approaches. We further discuss in detail the intricate dynamics of a persistent small-scale vortex flow with significant substructure, observed with the CRisp Imaging SpectroPolarimeter (CRISP) at the Swedish Solar Telescope (SST), as well as oscillations and observational signatures of different types of waves within it and their propagation characteristics. Both discussed aspects, better detection leading to a more precise estimation of their occurrence rate and wave identification and their properties, are key elements for accurately assessing the role of vortex structures in the energy budget of the solar atmosphere. Mar 17 Wed Ulrich Bunke (University of Regensburg) Pure Maths Colloquium Abstract: Coarse geometry studies the large-scale properties of metric spaces, groups and other mathematical objects. Interesting invariants are constructed using coarse homology theories. In this exposition I will explain an axiomatic approach to coarse homology theories. A motivic statement is a statement of the form: For every coarse homology theory E assertion P(E) holds. For example, one can turn the coarse Baum-Connes conjecture into a motivic statement. I will explain how motivic statements can be captured in terms of a universal coarse homology theory. The talk is based on joint work with Alexander Engel. Mar 17 Wed Richard Daniel (Sheffield) Cosmology, Relativity and Gravitation Abstract: In this talk, I will briefly recap slow roll inflation, before demonstrating that an $f(R^2)$ theory of inflation is able to dynamically generate a Planck mass from the vacuum expectation values of the scalar fields. We see that in such models if the self interaction is non-zero, a potentially large cosmological constant will emerge. To avoid this problem we introduce another scalar field, producing a Higgs-like potential. This naturally drives the cosmological constant to zero soon after inflation. We will analyse both models in the Einstein frame, where we find a conserved Noether current simplifying the model to a N-1 scalar field model. Finally, I will discuss the non-trivial features in the power spectrum, which produce testable parameters for future cosmological experiments. Mar 17 Wed Cristina Manolache Algebraic Geometry Learning Seminar: Toric varieties Mar 11 Thu Sergio J. González Manrique (Astronomical Institute of Slovak Academy of Sciences (SK)) SP2RC/ESPOS Abstract: We study the dynamics of plasma along the legs of an arch filament system (AFS) observed with relatively high-cadence spectropolarimetric data from the ground-based solar GREGOR telescope (Tenerife) using the GREGOR Infrared Spectrograph in the He I 10830 Å range. The temporal evolution of the plasma of an AFS was followed using the chromospheric He I 10830 Å triplet and Si I 10827 Å. Measurements of vector magnetic fields in the solar chromosphere, especially in AFS, are extremely scarce, but very important. The magnetic field configuration reveals how AFSs are sustained in the chromosphere and hints at their formation, evolution, and disappearance. The magnetic field in the AFS follows loop-like structures traced by chromospheric absorption lines. However, if magnetic field lines follow chromospheric threads as seen by filtergrams of H⍺, Ca II, or He I, is still not fully resolved. Previous studies have modeled AFS as multiple flux ropes with mixed signs of helicity consistently with the observed multiple filament bundles constituting AFS. Nevertheless, further spectropolarimetric observations are needed to address this issue. Many spectral lines are sensitive to the atmospheric parameters up to the upper chromosphere. Moreover, when combined with photospheric Zeeman sensitive spectral lines, one can infer the topology of the magnetic field from the bottom of the solar atmosphere to the chromosphere. In this talk, we are going to follow the nature of AFSs by reconstructing the magnetic field configuration of an EFR from the very beginning and follow its evolution and dynamics to support current AFS models. To that aim we used the spectropolarimetric data available at the upper photosphere (Si I) and the upper chromosphere. Mar 10 Wed Anssi Lahtinen (University of Copenhagen) Pure Maths Colloquium Abstract: Founded by Chas and Sullivan's observation that the homology of the free loop space of an oriented manifold has the structure of a Batalin--Vilkovisky algebra, string topology studies the rich algebraic structure present on the homology of the free loop spaces of certain spaces such as manifolds and classifying spaces of compact Lie groups. In this talk, I will provide a gentle and subjective introduction to the subject, and also indicate how it connects with objects such as moduli spaces of Riemann surfaces, automorphism groups of free groups, and finite groups of Lie type. Mar 10 Wed Aaron Held (Imperial College) Cosmology, Relativity and Gravitation Abstract: The recent wealth of experimental data from the LIGO/Virgo as well as from the EHT collaboration is fully consistent with GR. However, the true nature of the observed compact objects must involve some new physics (quantum or classical) to ameliorate the singularities present in the interior of the respective GR description. In the spirit of local EFTs, the talk will be based on the main assumption that the new physics, whatever its origin, is tied to local curvature scales. Based on this locality principle, I construct a new class of everywhere-regular, stationary spacetimes parameterized by a mass function in horizon-penetrating coordinates. This construction allows me to identify characteristic image features of the shadows of this class of regular black holes, in particular, in distinction to other models not following the locality principle. Moving on to dynamical spacetime evolution, still following the principles of local EFT, I will present first results on a fully non-linear but well-posed numerical simulation of (quadratic) higher-derivative gravity in the spherically-symmetric sector. Mar 10 Wed Yannik Schüler Algebraic Geometry Learning Seminar: Toric varieties Mar 4 Thu Erico L. Rempel (Aeronautics Institute of Technology - ITA, São José dos Campos, Brazil) Plasma Dynamics Group Abstract: Dynamical systems, or chaos theory, has enjoyed huge success in the analysis of systems described by ordinary differential equations, such as nonlinear oscillators, chemical reactions, electronic devices, population dynamics, etc. Usually, in the dynamical systems approach, one is concerned with the identification of the basic building blocks of the system under investigation and how they interact with each other to produce the observable dynamics, as well as how they can be manipulated to produce a desired output, in the cases where control is pursued. Examples of those building blocks are unstable equilibrium and periodic solutions, nonattracting chaotic sets and their manifolds, which are special surfaces in the phase space that basically control the dynamics, guiding solutions in preferred directions. Despite its success in those areas, many still think that the theory has limited value when applied to fully developed turbulence, like observed in solar convection, due to the infinite dimension of the phase space. In this talk, we show that this difficulty can be overcome by adopting a Lagrangian reference frame, where the phase space for each fluid particle becomes three-dimensional and the building blocks of the turbulence can be efficiently extracted by appropriate numerical tools. We reveal how finite-time Lyapunov exponents, a traditional measure of chaos, can be used to detect attracting and repelling time-dependent manifolds that divide the fluid in regions with different behavior. These manifolds are shown to accurately mark the boundaries of granules in observational data from the photosfere. In addition, stagnation points and vortices detected as elliptical Lagrangian coherent structures complete the set of building blocks of the photospheric turbulence. Such structures are crucial for the trapping and transport of mass and energy in the solar plasma. Mar 3 Wed Jasmin Matz (University of Copenhagen) Pure Maths Colloquium Abstract: Suppose M is a closed Riemannian manifold with an orthonormal basis B of $L^2(M)$ consisting of Laplace eigenfunctions. Berry's Random Wave Conjecture tells us that under suitable conditions on M, in the high energy limit (ie, large Laplace eigenvalue) elements of B should roughly behave like random waves of corresponding wave number. A classical result of Shnirelman and others that $M$ is quantum ergodic if the geodesic flow on the cotangent bundle of $M$ is ergodic, can then be viewed as a special case of this conjecture. We now want to look at a level aspect, namely, instead of taking a fixed manifold and high energy eigenfunctions, we take a sequence of Benjamini-Schramm convergent compact Riemannian manifolds together with Laplace eigenfunctions f whose eigenvalue varies in short intervals. This perspective has been recently studied in the context of graphs by Anantharaman and Le Masson, and for hyperbolic surfaces and manifolds by Abert, Bergeron, Le Masson, and Sahlsten. In my talk I want to discuss joint work with F. Brumley in which we study this question in higher rank, namely sequences of compact quotients of $SL(n,R)/SO(n)$ for $n>2$. Mar 3 Wed Valerio Lucarini (Reading) Applied Mathematics Colloquium Abstract: Extreme events provide relevant insights on the dynamics of the climate system and their understanding is key to defining useful strategies for mitigating the impact of climate variability and climate change. Here we approach the study of persistent weather extremes using the lens of large deviation theory. We first consider a simplified yet Earth-like general circulation model of the atmosphere and numerically estimate large deviation rate functions of near-surface temperature in the mid-latitudes. We find that, after a re-normalisation based on the integrated auto-correlation, the rate function one obtains at a given latitude by looking, locally in space, at long time averages agrees with what is obtained, instead, by looking, locally in time, at large spatial averages along the latitude. This is a result of scale symmetry in the spatial-temporal turbulence and of the fact that advection is primarily zonal. This agreement hints at the universality of large deviations of the temperature field. Furthermore, we discover that the obtained rate function is able to describe spatially extended and temporally persistent heat waves or cold spells, if we consider temporal averages of spatial averages over intermediate spatial scales. We then extend our analysis by looking at the output of a state-of-the-art climate model and at observational data. We show how to const ruction in a mathematically rigorous way the climatology of persistent heatwaves and cold spells in some key target regions of the planet by constructing empirically the corresponding rate functions for the surface temperature, and we assess the impact of increasing CO2 concentration on such persistent anomalies. In particular, we can better understand the increasing hazard associated to heatwaves in a warmer climate. We show that two 2010 high impact events - summer Russian heatwave and winter Dzud in Mongolia - are associated with atmospheric patterns that are exceptional compared to the typical ones, but typical compared to the climatology of extreme events. Finally, we propose an approximate formula for describing large and persistent temperature fluctuations from easily accessible statistical properties. Refs: V. Galfi, V. Lucarini, Fingerprinting Heatwaves and Cold Spells and Assessing Their Response to Climate Change using Large Deviation Theory, PRL, in review (2020) V. Galfi, V. Lucarini, J. Wouters, A Large Deviation Theory-based Analysis of Heat Waves and Cold Spells in a Simplified Model of the General Circulation of the Atmosphere, J. Stat. Mech. 033404 doi: 10.1088/1742-5468/ab02e8 (2019) Mar 3 Wed Elsa Teixeira (Sheffield) Cosmology, Relativity and Gravitation Abstract: In this talk I will give an overview of interacting dark energy, with emphasis on disformal couplings and its cosmological implications. I will then focus on the general Dark D-Brane setting, for which the interaction in the dark sector arises naturally through the induced metric on a moving brane. In particular, I will discuss the background and linear perturbation equations in this setting, together with a numerical analysis, with brief connection to observational constraints. Testing gravity in the dark sector will be an exciting topic in the upcoming decade, with next-generation cosmological data probing gravitational phenomena in finer detail. Mar 3 Wed Karoline Van Gemst Algebraic Geometry Learning Seminar: Toric varieties Feb 25 Thu Suzana de Souza e Almeida Silva (Sheffield) European Solar Physics Online Seminars (ESPOS) Abstract: We present the state-of-art detection method of three-dimensional vortices and apply it to realistic magneto-convections simulations performed by the MURaM code. The detected vortices extend from the photosphere to the low chromosphere, presenting similar behaviour at all height levels. The vortices concentrate the magnetic field, and thereby the plasma dynamics inside the vortex is considerably influenced by the Lorentz force. Rotational motions also perturb the magnetic field lines, but they lead to only slightly bent flux tubes as the magnetic field tension is too high for the vortex flow to significantly twist the magnetic lines. We find that twisted magnetic flux tubes are created by shear motions in regions where plasma-beta>1, regardless of the existence of flow vortices. Feb 25 Thu Suzana de Souza e Almeida Silva (University of Sheffield, Plasma Dynamics Group (UK)) SP2RC/ESPOS Abstract: We present the state-of-art detection method of three-dimensional vortices and apply it to realistic magneto-convections simulations performed by the MURaM code. The detected vortices extend from the photosphere to the low chromosphere, presenting similar behaviour at all height levels. The vortices concentrate the magnetic field, and thereby the plasma dynamics inside the vortex is considerably influenced by the Lorentz force. Rotational motions also perturb the magnetic field lines, but they lead to only slightly bent flux tubes as the magnetic field tension is too high for the vortex flow to significantly twist the magnetic lines. We find that twisted magnetic flux tubes are created by shear motions in regions where plasma-beta>1, regardless of the existence of flow vortices. Feb 24 Wed Reinder Meinsma Algebraic Geometry Learning Seminar: Toric varieties Feb 18 Thu Dr Huw Morgan (Aberystwyth University) SP2RC seminar Abstract: Any remote measurement of the solar corona in white light (or other) wavelength is an integration of emission along an extended line of sight. Historically, most studies necessarily assumed an axi-symmetric distribution to the density, thus derived properties contained an inherent and unquantified uncertainty. From the SOHO era onwards, space-based coronagraphs (LASCO/SOHO, and COR/STEREO) make frequent, uninterrupted, and high-quality observations of the corona which allow estimates of the true density distribution using coronal rotational tomography (CRT). A recent breakthrough in CRT is revealing a new view of the corona which is gained directly from observation. For the first time, we can view long-term trends in the coronal rotation rate, find meaningul links between coronal and interplanetary density structures, and use estimated densities at a range of heights to constrain outflow velocity and acceleration. The density distributions provide a ground truth for model extrapolations of the photospheric magnetic field, and new empirical boundary conditions for solar wind models. Tentative evidence of the Parker spiral onset can be seen close to the Sun. The next step in CRT methods is the inclusion of a time-dependent density distribution: initial results show promising correlations with Parker Solar Probe measurements, and the discovery of large variations on daily timescales not associated with mass ejections. Feb 18 Thu Mijie Shi (KU Leuven, Belgium) Plasma Dynamics Group Abstract: In the quest to solve the long-standing coronal heating problem, it has been suggested that coronal loops could be heated by waves. Despite the accumulating observational evidence of the possible importance of coronal waves, still very few 3D MHD simulations exist that show significant heating by MHD waves. In this seminar, I will present our recent 3D coronal loop model heated by transverse waves against radiative cooling. The coronal loop is driven at the footpoint by transverse oscillations and subsequently the induced Kelvin-Helmholtz instability deforms the loop cross-section to a fully turbulent state. Wave energy is transferred to smaller scales where it is dissipated, overcoming the internal energy losses by radiation. These results open up a new avenue to address the coronal heating problem. Feb 17 Wed Kang Li (KU Leuven) Pure Maths Colloquium Abstract: Roughly speaking, a ghost operator is often an infinite matrix such that its matrix entries vanish at the infinity. This notion was introduced by Guoliang Yu in the study of the so-called coarse Baum-Connes conjecture. It is a very central topic in coarse geometry and operator algebras with applications to provide counterexamples to the coarse Baum–Connes conjecture, the existence of non-exact groups and the rigidity problem for Roe-type algebras. In this talk, we will visualize a class of ghost projections in terms of expanderish graphs. Feb 17 Wed Daniele Oriti (LMU Munich) Cosmology, Relativity and Gravitation Abstract: We overview recent results on the extraction of an effective cosmological dynamics from fundamental quantum gravity formalisms in which spacetime is not fundamental, focusing on so called tensorial group field theories (strictly related to lattice quantum gravity and loop quantum gravity). This line of research is inspired by the idea of our universe as a quantum gravity condensate, and at the same time realizes it concretely. We emphasize how reaching the desired objective requires addressing several outstanding issues in quantum gravity: identifying quantum states in the fundamental theory with a good geometric interpretation, performing some form of coarse graining of the fundamental dynamics, defining diffeomorphism invariant observables to express the resulting coarse grained dynamics in physically transparent language. We also discuss what the theory says about the fate of the big bang singularity at the beginning of our universe. Feb 17 Wed Evgeny Shinder Algebraic Geometry Learning Seminar: Toric varieties Feb 11 Thu Vasco Henriques (Rosseland Centre for Solar Physics, Norway) European Solar Physics Online Seminars (ESPOS) Abstract: The chromosphere of the umbra of sunspots is a remarkably dynamic layer featuring extremely fine sub-arcsec structure. Such structures appear dark against enveloping umbral flashes, but also bright before or after a flash, other features still are bright throughout. Only recently did we start understanding such fine features and semi-empirical modelling is converging with simulations to provide insight, not only into such fine structure, but also into the umbral flash phenomenon itself. The observational evidence weighs overwhelmingly towards a strong corrugation of the umbra where the material in short dynamic fibrils over-extends in a column of upflowing material while the adjacent areas flash. The delayed small-scale umbral brightenings at the bottom of such columns are an out-of-phase flash where the late-stage downflowing column meets the upflowing under-layers. Recent inversions using NICOLE at both umbral flashes and small-scale brightenings result in a downflow over upflow stratification perfectly bridging the transition of downflowing fibrils to upflowing fibrils as well as red-shifted absorption cores to blue-shifted absorption cores in the broader surroundings. Locally, each inverted column is remarkably similar in velocity profile to those from forward modelling, provided the formation height of the observed Ca II 8542 line is slightly lower in the Sun than in the simulations. Conspicuously, resonant cavities naturally cause the upper downflowing layers to become visible in forward modelling and the top-layer downflows to last longer than otherwise. Open questions, and how these can be addressed by future observations, are briefly discussed. Feb 11 Thu Vasco Henriques (Rosseland Centre for Solar Physics, Norway) SP2RC/ESPOS Abstract: The chromosphere of the umbra of sunspots is a remarkably dynamic layer featuring extremely fine sub-arcsec structure. Such structures appear dark against enveloping umbral flashes, but also bright before or after a flash, other features still are bright throughout. Only recently did we start understanding such fine features and semi-empirical modelling is converging with simulations to provide insight, not only into such fine structure, but also into the umbral flash phenomenon itself. The observational evidence weighs overwhelmingly towards a strong corrugation of the umbra where the material in short dynamic fibrils over-extends in a column of upflowing material while the adjacent areas flash. The delayed small-scale umbral brightenings at the bottom of such columns are an out-of-phase flash where the late-stage downflowing column meets the upflowing under-layers. Recent inversions using NICOLE at both umbral flashes and small-scale brightenings result in a downflow over upflow stratification perfectly bridging the transition of downflowing fibrils to upflowing fibrils as well as red-shifted absorption cores to blue-shifted absorption cores in the broader surroundings. Locally, each inverted column is remarkably similar in velocity profile to those from forward modelling, provided the formation height of the observed Ca II 8542 line is slightly lower in the Sun than in the simulations. Conspicuously, resonant cavities naturally cause the upper downflowing layers to become visible in forward modelling and the top-layer downflows to last longer than otherwise. Open questions, and how these can be addressed by future observations, are briefly discussed. Feb 10 Wed Kevin Painter (Politecnico di Torino) Applied Mathematics Colloquium Abstract: The formation of swarms, schools, flocks, herds, aggregates etc is a classical example of self-organisation, with the benefits of forming a high density group ranging from efficient migration to higher fecundity. Often, groups form through a mechanism of chemical signalling between population members, an evolutionary ancient communication used by both microscopic and macroscopic species. Populations in fluid environments, though, must contend with complex and turbulent flows, potentially detrimental (e.g. splitting up groups) or beneficial (e.g. coalescing individuals) to the formation and maintenance of a group. As a counter to flow, rheotaxis describes a behaviour in which individuals orient their body axis with respect to the current and is observed in both unicellular and multicellular organisms . Here we investigate the extent to which rheotaxis and flow impact on chemically-mediated aggregation, revealing these can impact both negatively and positively according to the population state and flow conditions. A hypothesised density-dependent rheotaxis appears capable of optimising group formation and maintenance, exploiting the positive benefits from each of flow and rheotaxis. The results are discussed in the context of broadcast swarming phenomena in marine invertebrates. Feb 10 Wed Luke Hart (Manchester) Cosmology, Relativity and Gravitation Abstract: Cosmological recombination has been widely regarded a solid pillar of understanding the cosmic microwave background (CMB) and its anisotropies. For many years, the questions have been answered over the accuracy of these calculations due to exceptional codes as CosmoRec and HyRec as well as numerous publications on the intricate atomic processes. However, the era that dawned the formation of hydrogen and helium atoms has still given us brilliant insights into exotic physics as well as tribalistic disputes in the various pockets of modern cosmology. In this talk, we will briefly recap the physics of recombination before highlighting extensions to the standard model (parametric and non-parametric) that affect the surface of last scattering. Finally, we will look to the future probes that provide a direct, spectral handprint of the atomic transitions in hydrogen and helium: the recombination radiation. Here we will conclude with the feasibility of studying these lines with prospective missions such as SuperPIXIE, Voyage 2050 and what happens when the exotic physics modifications that we can test with the CMB anisotropies are propagated through to the SEDs from recombination. Feb 10 Wed Paul Johnson Algebraic Geometry Learning Seminar: Toric varieties Feb 5 Fri Dr. Sijie Yu (Center for Solar-Terrestrial Research, New Jersey Institute of Technology) SP2RC seminar Abstract: Thanks to recent advances in radio interferometric instrumentation, we've entered a new era of solar radio observations---broadband dynamic imaging spectroscopy. In this talk, I will first introduce the history of solar radio observations based on either total-power (integrated over the Sun) dynamic spectral measurements or imaging at a few discrete frequencies, then review some recent progress based on dynamic imaging spectroscopy over a wide frequency range that has placed us in a strong position to make revolutionary breakthroughs in understanding high-energy processes in the solar corona. Future perspectives will also be briefly discussed. Jan 28 Thu (University of St Andrews, Solar and Magnetospheric Theory Group (UK)) SP2RC/ESPOS Jan 28 Thu Ben Snow (University of Exeter) European Solar Physics Online Seminars (ESPOS) Abstract: Shocks are a universal feature of warm plasma environments, such as the lower solar atmosphere and molecular clouds, which consist of both ionised and neutral species. Including partial ionisation leads to the existence of a finite width for shocks, where the ionised and neutral species decouple and recouple. As such, drift velocities exist within the shock that lead to frictional heating between the two species, in addition to adiabatic temperature changes across the shock. The local temperature enhancements within the shock alter the recombination and ionisation rates and hence change the composition of the plasma. We study the role of collisional ionisation and recombination in slow-mode partially ionised shocks. In particular, we incorporate the ionisation potential energy loss and analyse the consequences of having a non-conservative energy equation. A semi-analytical approach is used to determine the possible equilibrium shock jumps for a two-fluid model with ionisation, recombination, ionisation potential, and arbitrary heating. Two-fluid numerical simulations are performed using the (PIP) code. Results are compared to the magnetohydrodynamic (MHD) model and the semi-analytic solution. Accounting for ionisation, recombination, and ionisation potential significantly alters the behaviour of shocks in both substructure and post-shock regions. In particular, for a given temperature, equilibrium can only exist for specific densities due to the radiative losses needing to be balanced by the heating function. A consequence of the ionisation potential is that a compressional shock will lead to a reduction in temperature in the post-shock region, rather than the increase seen for MHD. The numerical simulations pair well with the derived analytic model for shock velocities. Jan 27 Wed Theo Torres Vicente (Nottingham) Cosmology, Relativity and Gravitation Abstract: In this talk, we consider the electromagnetic radiation-reaction/self-force process for a charged particle orbiting a rotating black hole. We will present and complement the existing results for the scalar and gravitational cases, to give a full picture of integer spins in the Kerr spacetime. We restrict ourselves to the case of circular orbits and we will compute the dissipative and conservative components of the electromagnetic self-force numerically, by solving the inhomogeneous Teukolsky equations using the BHperturbation toolkit. The results will be compared to the scalar and gravitational cases found in the literature. Jan 21 Thu Professor Valery M Nakariakov (Centre for Fusion, Space & Astrophysics, University of Warwick, United Kingdom) SP2RC seminar Abstract: Standing transverse oscillations of the plasma loops of the solar corona have been intensively studied for the last 20 years as a tool for the diagnostics of the coronal magnetic field. Those oscillations are confidently interpreted as standing fast magnetoacoustic kink modes of the plasma non-uniformities. Statistical analysis demonstrates that, in the majority of cases, the oscillations are excited by a mechanical displacement of the loop from an equilibrium by a low coronal eruption. Standing kink oscillations are observed to operate in two regimes: rapidly decaying large amplitude oscillations and undamped small amplitude oscillations. In both these regimes, different loops oscillate with different periods that scale with the oscillating loop length. The oscillation amplitude does not show dependence on the loop length or the oscillation period. In the decayless regime the damping should be compensated by energy supply which allows the loop to perform almost monochromatic oscillations with almost constant amplitude and phase. We developed a low-dimensional model explaining the undamped kink oscillations as a self-oscillatory process caused by the effect of negative friction, which is analogous to producing a tune by moving a bow across a violin string. The period of self-oscillations is determined by the frequency of the kink mode. The ubiquity of decayless kink oscillations makes them an excellent tool for MHD seismology, in particular, for probing free magnetic energy in preflaring active regions. Jan 14 Thu Sudheer K. Mishra (Indian Institute of Technology, BHU, India) European Solar Physics Online Seminars (ESPOS) Abstract: Using multi-wavelength imaging observation obtained from the Atmospheric Imaging Assembly (AIA) onboard Solar Dynamics Observatory (SDO), we study the evolution of Kelvin-Helmholtz (K-H) instability in a fan-spine magnetic topology. This fan-spine configuration is situated near the Active Region 12297 and is rooted in a nearby sunspot. The two layers of the cool plasma flows lift up from the fan plane in parallel and interact with each other at the maximum height of the elongated spine in the lower corona. The first layer of the plasma flow (F1) moving with a slow velocity (5 km/s) reflected from the spine’s field lines. Subsequently second layer of plasma flow (F2) with impulsive velocity (114-144 km/s) interacts with the first layer at the maximum height and generating the shear motion , which is responsible for the evolution of the Kelvin-Helmholtz instability inside the elongated spine. The amplitude and characteristics wavelength of the K-H unstable vortices increases, which satisfy the linear growing mode of this instability. Using linear stability theory of the K-H instability, we calculate the Alfvén velocity in the lower layer. We conjecture that the estimated shearing velocity is higher than the estimated the Alfvén velocity in the second denser layer, which also satisfies the classical criterion of K-H instability. The fan-spine configuration possesses magnetic field and sheared velocity component, we estimate the parametric constant [Λ≥1] which confirms that the velocity shear dominates and the linear phase of the K-H instability is evolved. The present observation indicate that in the presence of complex magnetic field structuring and plasma flows, the K-H instability evolve in the fan-spine configuration may evolve the rapid heating, and connectivity changes may occur due to the fragmentation via the K-H instability. It also act as a rapid mechanism to transfer the mass and energy release between two distinct regions separated by the fan-spine configuration. Jan 14 Thu Sudheer K. Mishra (Indian Institute of Technology, BHU, India) SP2RC/ESPOS Abstract: Using multi-wavelength imaging observation obtained from the Atmospheric Imaging Assembly (AIA) onboard Solar Dynamics Observatory (SDO), we study the evolution of Kelvin-Helmholtz (K-H) instability in a fan-spine magnetic topology. This fan-spine configuration is situated near the Active Region 12297 and is rooted in a nearby sunspot. The two layers of the cool plasma flows lift up from the fan plane in parallel and interact with each other at the maximum height of the elongated spine in the lower corona. The first layer of the plasma flow (F1) moving with a slow velocity (5 km/s) reflected from the spine’s field lines. Subsequently second layer of plasma flow (F2) with impulsive velocity (114-144 km/s) interacts with the first layer at the maximum height and generating the shear motion , which is responsible for the evolution of the Kelvin-Helmholtz instability inside the elongated spine. The amplitude and characteristics wavelength of the K-H unstable vortices increases, which satisfy the linear growing mode of this instability. Using linear stability theory of the K-H instability, we calculate the Alfvén velocity in the lower layer. We conjecture that the estimated shearing velocity is higher than the estimated the Alfvén velocity in the second denser layer, which also satisfies the classical criterion of K-H instability. The fan-spine configuration possesses magnetic field and sheared velocity component, we estimate the parametric constant [Λ≥1] which confirms that the velocity shear dominates and the linear phase of the K-H instability is evolved. The present observation indicate that in the presence of complex magnetic field structuring and plasma flows, the K-H instability evolve in the fan-spine configuration may evolve the rapid heating, and connectivity changes may occur due to the fragmentation via the K-H instability. It also act as a rapid mechanism to transfer the mass and energy release between two distinct regions separated by the fan-spine configuration.
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2021-05-13 19:26:25
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https://stats.stackexchange.com/questions/373010/hierarchical-clustering-in-r-centroid-linkage-problem-with-dendrogram-height
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# Hierarchical clustering in R - centroid linkage - problem with dendrogram heights
I have got a problem with understanding the hierarchical clustering algorythm, especially centroid linkage method. I have read many articles with description and they seems to be quite easy but I connot understand the results on real values.
I have got sample set like this:
test_list <- list( one = c(1), two = c(2), six = c(6), eleven = c(11), sixteen = c(16) )
I can generate a cluster using this function:
test_dendrogram <- tsclust(test_list, type = "hierarchical", distance = "L1", control = hierarchical_control(method = "centroid" ) )
The dendrogram looks like this:
The list of heights is: 1.000000 4.250000 5.000000 8.138889
I understand the results as below:
1 = 2 - 1
5 = 16 - 11
4.25 = 6-((2+1)/2 + 2)/2 <- in this step we need to replace 1 for centroid 1 and 2 = 1,5 and calculate centroid which means arithmetic average.
But I have no idea how to calculate 8.13. I did the same exercise with single, average and complete linkage methods and the results are obvious.
• You should explain what your r comnand actually requests, in particular, what distance=L1 entails. For you should know that centroid method implicitly is based on L2, euclidean distance. – ttnphns Oct 21 '18 at 20:36
• Distance L=1 is manhattan distance but for a single points we can treat euclidean and manhattan as a exactly the same distance algorithm. You can give me the answer for euclidean distance, does not matter in this case. – Bochnio Oct 22 '18 at 6:22
• You say you read a lot of articles. Then you must be acquainted with how Lance-Williams formula mentioned here is expressed in case of centroid method. Use it. – ttnphns Oct 22 '18 at 8:09
• ttnphns ok, I used Lance-Williams formula to calculate cluster [six + one-two] it is pretty easy 4,25 = =(1/2)*(6-1)+(1/2)*(6-2)+(-1/(2^2))*1 but I have no idea how to use this formula when I have 1 cluster which contains 3 elements and 1 cluster which contains 2 elements. Could you give me some advice? – Bochnio Oct 27 '18 at 14:00
• I don't see the distance matrix (euclidean or squared euclidean d) in your question. Or the casewise data to be able to compute the distances. – ttnphns Oct 27 '18 at 14:10
### Explaining calculations done in centroid linkage hierarchical clustering
Your data: 5 points in 1D feature space:
a 1
b 2
c 6
d 11
e 16
Compute squared euclidean distances (because centroid method needs thus) and perform the agglomerative clustering (done in SPSS).
On the dengrogram, the computed distances between clusters being merged on steps are numerically rescaled into range 1-25, however in the agglomeration history table they are displayed as is (called "coefficients").
Let us now trace the computations done.
Step 1. Find the minimal distance and merge these two points. These are a and b (i.e. points 1 and 2 in the aggl. schedule) and the (squared euclidean) distance is $$1$$. OK. Label cluster (a+b) 1 (the lesser between labels 1 and 2) and delete cluster 2, i.e. point b, from the matrix. Now update sq. eucl. distances between cluster 1 (i.e. points a+b) and every other point/cluster. This is done through Lance-Williams formula which in case of centroid linkage method unwraps into this:
$$D_{(pq)r} = \frac{N_p}{N_{pq}} D_{pr} + \frac{N_q}{N_{pq}} D_{qr} - \frac{N_pN_q}{N_{pq}^2} D_{pq}$$
where $$D_{(pq)r}$$ is the distance of the newly merged cluster "pq" (just merged of "p" and "q" subclusters) and every else cluster "r", $$N$$ with subscript is the number of points in a cluster.
So, after merge at step 1 we compute distance between (a+b) and, say, d, as
$$D_{(ab)d} = \frac{1}{2} 100 + \frac{1}{2} 81 - \frac{1 \cdot 1}{2^2} 1 = 90.25$$.
Upon computing distance between (a+b) and every other cluster the updated distance matrix is:
ab c d e
ab .00 20.25 90.25 210.25
c 20.25 .00 25.00 100.00
d 90.25 25.00 .00 25.00
e 210.25 100.00 25.00 .00
Step 2. Find the minimal distance in it. It is the distance between cluster (a+b) and cluster (point) c (aka 3 in aggl. schedule). Merge the two clusters, (ab+c) and, using the formula, update the distances between it and every other one (before that, remove row and column c). For example, the distance between (ab+c) and d will be:
$$D_{((ab)c)d} = \frac{2}{3} 90.25 + \frac{1}{3} 25 - \frac{2 \cdot 1}{3^2} 20.25 = 64$$
and whole updated distance matrix
abc d e
abc .00 64.00 169.00
d 64.00 .00 25.00
e 169.00 25.00 .00
Step 3. etc likewise. The two clusters to merge will be d and e which distance = $$25$$ is currently the least.
Remember that computations are done on squared euclidean distances (in centroid method). Of course, you may take root from the computed $$D$$s at each step if you (reasonably) want to plot the dendrogram reflecting nonsquared distances between clusters. Just take sq. root from the "coefficients" in the agglomeration schedule above and that will be your nonsquared distances for the dendrogram. In my dendrogram shown above, squared distances were plotted.
• Thank you very much for your help, everything works well! – Bochnio Oct 28 '18 at 15:40
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2020-01-24 22:16:31
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https://www.authorea.com/users/3/articles/149/_show_article
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POSTPRINT authorea.com/149
Abstract
This article was published as How the Scientific Community Reacts to Newly Submitted Preprints: Article Downloads, Twitter Mentions and Citations. Xin Shuai, Alberto Pepe, Johan Bollen. PLoS ONE 7(11): e47523. doi:10.1371/journal.pone.0047523. Open Access Article.
# Introduction
The view from the “ivory tower” is that scholars make rational, expert decisions on what to publish, what to read and what to cite. In fact, the use of citation statistics to assess scholarly impact is to a large degree premised on the very notion that citation data represent an explicit, objective expression of impact by expert authors (Rubin 2010). Yet, scholarship is increasingly becoming an online process, and social media are becoming an increasingly important part of the online scholarly ecology. As a result, the citation behavior of scholars may be affected by their increasing use of social media. Practices and considerations that go beyond traditional notions of scholarly impact may thus influence what scholars cite.
Recent efforts have investigated the effect of the use of social media environments on scholarly practice. For example, some research has looked at how scientists use the microblogging platform Twitter during conferences by analyzing tweets containing conference hashtags (Letierce 2010, Weller 2011). Other research has explored the ways by which scholars use Twitter and related platforms to cite scientific articles (Priem 2010, Weller 2011a). More recent work has shown that Twitter article mentions predict future citations (Eysenbach 2011). This article falls within, and extends, these lines of research by examining the temporal relations between quantitative measures of readership, Twitter mentions, and subsequent citations for a cohort of scientific preprints.
We study how the scientific community and the public at large respond to a cohort of preprints that were submitted to the arXiv database (http://arxiv.org), a service managed by Cornell University Library, which has become the premier pre-print publishing platform in physics, computer science, astronomy, and related domains. We examine the relations between three types of responses to the submissions of this cohort of pre-prints, namely the number of Twitter posts (tweets) that specifically mention these pre-prints, downloads of these pre-prints from the arXiv.org web site, and the number of early citations that the 70 most Twitter-mentioned preprints in our cohort received after their submission. In each case, we measure total volume of responses, as well as the delay and span of their temporal distribution. We perform a comparative analysis of how these indicators are related to each other, both in magnitude and time.
Our results indicate that download and social media responses follow distinct temporal patterns. Moreover, we observe a statistically significant correlation between social media mentions and download and citation count. These results are highly relevant to recent investigations of scholarly impact based on social media data (Priem 2010a, Priem 2011) as well as to more traditional efforts to enhance the assessment of scholarly impact from usage data (Bollen 2009, Bollen 2008, Brody 2006, Kurtz 2010).
# Data and study overview
## Data collection
Our analysis is based on a corpus of 4,606 scientific articles submitted to the preprint database arXiv between October 4, 2010 and May 2, 2011. For each article in this cohort, we gathered information about their downloads from the arXiv server weekly download logs, their daily number of mentions on Twitter using a large-scale collection of Twitter data collected over that period, and their early citations in the scholarly record from Google Scholar. Table 1 summarizes the discussed data collection and Figure 1 provides an overview of the data collection timelines.
The datasets employed in this study are:
• ArXiv downloads: For each article in the aforementioned cohort we retrieved their weekly download numbers from the arXiv logs for the period from October 4, 2010 to May 9, 2011. A total of 2,904,816 downloads were recorded for 4,606 articles.
• Twitter mentions: Our collection of tweets is based on the Gardenhose, a data feed that returns a randomly sampled 10% of all daily tweets. A Twitter mention of arXiv article was deemed to have occurred when a tweet contained an explicit or shortened link to an arXiv paper (see “Materials” appendix for more details). Between October 4, 2010 and May 9, 2011 we scanned 1,959,654,862 tweets in which 4,415 articles out of 4,606 in our cohort were mentioned at least once, i.e. approximately 95% of the cohort. Such a wide coverage of arXiv articles is mostly due to specialized bot accounts which post arXiv submissions daily. The volume of Twitter mentions of arXiv papers was very small compared to the total volume of tweets in period, with only 5,752 tweets containing mentions of papers in the arXiv corpus. We found that 2,800 out of 5,752 tweets are from non-bot accounts. After filtering out all tweets posted by bot accounts, we retain 1,710 arXiv articles out of 4,415 that are mentioned on Twitter by non-bot accounts. Including or excluding bot mentions, the distribution of number of tweets over all papers was very skewed; most papers were mentioned only once, but one paper in the corpus was mentioned as much as 113 times.
• Early citations: We manually retrieved citation counts from Google Scholar for the 70 most Twitter-mentioned articles in our cohort. Citation counts were retrieved on September 30, 2011 and date back to the initial submission date in arXiv. All 70 articles combined were cited a total of 431 times at that point. The most cited article in the corpus was cited 62 times whereas most articles received hardly any citations.
By the nature of our research topic, we are particularly focused on early responses to preprint submissions, i.e., immediate, swift reactions in the form of downloads, Twitter mentions, and citations. Therefore, we record download statistics and Twitter mention data only one week over the submission period itself (up to May 9, 2011).
As for citation data, we are aware that citations take years to accrue. We do not explore here long-term citation effects, but only the early, immediate response to pre-print submission in the form of citations in the scholarly record. Our citation data pertains to a time period that spans from 5 months to 1 year: it is a fraction of the expected amount of “maturation time” for citation analysis. Citation data must therefore be considered to reflect “early citations” only, not total potential citations.
## Definitions: delay and time span.
Twitter mentions and arXiv downloads may follow particular temporal patterns. For example, for some articles downloads and mentions may take weeks to slowly increase after submission, whereas for other articles downloads may increase very swiftly after submission to wane very shortly thereafter. The total number of downloads and mentions is orthogonal to these temporal effects, and could be different in either case.
The two parameters that we use to describe the temporal distributions of arXiv downloads and Twitter mentions are delay and the time span, which we define as follows. Let $$t_0 \in \mathbb{N}^+$$ be the date of submission for arti
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2016-10-28 14:03:08
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https://neurips.cc/Conferences/2022/ScheduleMultitrack?event=65035
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Timezone: »
Spotlight
Coresets for Wasserstein Distributionally Robust Optimization Problems
Ruomin Huang · Jiawei Huang · Wenjie Liu · Hu Ding
Wed Dec 07 05:00 PM -- 07:00 PM (PST) @
Wasserstein distributionally robust optimization (\textsf{WDRO}) is a popular model to enhance the robustness of machine learning with ambiguous data. However, the complexity of \textsf{WDRO} can be prohibitive in practice since solving its minimax'' formulation requires a great amount of computation. Recently, several fast \textsf{WDRO} training algorithms for some specific machine learning tasks (e.g., logistic regression) have been developed. However, the research on designing efficient algorithms for general large-scale \textsf{WDRO}s is still quite limited, to the best of our knowledge. \textit{Coreset} is an important tool for compressing large dataset, and thus it has been widely applied to reduce the computational complexities for many optimization problems. In this paper, we introduce a unified framework to construct the $\epsilon$-coreset for the general \textsf{WDRO} problems. Though it is challenging to obtain a conventional coreset for \textsf{WDRO} due to the uncertainty issue of ambiguous data, we show that we can compute a dual coreset'' by using the strong duality property of \textsf{WDRO}. Also, the error introduced by the dual coreset can be theoretically guaranteed for the original \textsf{WDRO} objective. To construct the dual coreset, we propose a novel grid sampling approach that is particularly suitable for the dual formulation of \textsf{WDRO}. Finally, we implement our coreset approach and illustrate its effectiveness for several \textsf{WDRO} problems in the experiments. See \href{https://arxiv.org/abs/2210.04260}{arXiv:2210.04260} for the full version of this paper. The code is available at \url{https://github.com/h305142/WDRO_coreset}.
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2023-03-29 01:26:39
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https://www.studyadda.com/index.php?/notes/neet/chemistry/analytical-chemistry/volumetric-analysis/9042
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JEE Main & Advanced Chemistry Analytical Chemistry Volumetric Analysis
Volumetric Analysis
Category : JEE Main & Advanced
Volumetric analysis is a quantitative analysis. It involves the measurement of the volume of a known solution required to bring about the completion of the reaction with a measured volume of the unknown solution.
Titration : The process of addition of the known solution from the burette to the measured volume of solution of the substance to be estimated until the reaction between the two is just complete, is termed as titration. Thus, a titration involves two solutions;
(i) Unknown solution : The solution consisting the substance to be estimated is termed unknown solution. The substance is termed titrate.
(ii) Standard solution : The solution in which an accurately known amount of the reagent (titrant) has been dissolved in a known volume of the solution is termed standard solution. There are two types of reagents (titrants) :
(a) Primary standards : These can be accurately weighed and their solutions are not to be standardised before use. Oxalic acid $({{H}_{2}}{{C}_{2}}{{O}_{4}}.2{{H}_{2}}O)$, potassium dichromate $({{K}_{2}}C{{r}_{2}}{{O}_{7}})$, silver nitrate $(AgN{{O}_{3}})$, copper sulphate $(CuS{{O}_{4}}.5{{H}_{2}}O)$, ferrous ammonium sulphate $[FeS{{O}_{4}}{{(N{{H}_{4}})}_{2}}S{{O}_{4}}.6{{H}_{2}}O]$, sodium thiosulphate $(N{{a}_{2}}{{S}_{2}}{{O}_{3}}.5{{H}_{2}}O)$, etc., are the examples of primary standards.
(b) Secondary standards : The solutions of these reagents are to be standardised before use as these cannot be weighed accurately. The examples are sodium hydroxide $(NaOH)$, potassium hydroxide $(KOH)$, hydrochloric acid $(HCl)$, sulphuric acid $({{H}_{2}}S{{O}_{4}})$, potassium permanganate $(KMn{{O}_{4}})$, iodine, etc.
Law of equivalence : It is applied in all volumetric estimations. According to it, the chemical substances react in the ratio of their chemical equivalent masses.
$\frac{\text{Mass of substance }A}{\text{Mass of substance }B}=\frac{\text{Chemical equivalent mass of }A}{\text{Chemical equivalent mass of }B}$
or $\frac{\text{Mass of substance }A}{\text{Chemical equivalent mass of }A}$$=\frac{\text{Mass of substance }B}{\text{Chemical equivalent mass of }B}$
or gram equivalent of $A=$gram equivalent of $B$
or milli-gram equivalent of $A=$ milli-gram equivalent of $B$
The point at which the amounts of the two reactants are just equivalent is known as equivalence point or end point. An auxiliary substance which helps in the usual detection of the completion of the titration or equivalence point or end point is termed as indicator, i.e., substances which undergo some easily detectable changes at the equivalence point are used as indicators.
Methods of expressing concentrations of solutions
The concentration of a solution can be expressed in various ways.
(1) Percent by mass
(2) Molarity
(3) molality
(4) Mole fraction
(5) Normality
Types of titrations : Titrations can be classified as :
(1) Acid base titrations or acidimetry and alkalimetry
(2) Oxidation reduction titrations or redox titrations
(3) Precipitation titrations
(4) Complexometric titrations.
(1) Acid-base titrations : When the strength of an acid is determined with the help of a standard solution of base, it is known as acidimetry. Similarly, when the strength of a base (alkali) is determined with the help of a standard solution of an acid, it is known as alkalimetry. Both these titrations involve neutralisation of an acid with an alkali. In these titrations ${{H}^{+}}$ ions of the acid combine with $O{{H}^{-}}$ ions of the alkali to form unionised molecules of water.
$\underset{\text{Acid}}{\mathop{HA}}\,+\underset{\text{Alkali}}{\mathop{BOH}}\,\xrightarrow{{}}\underset{\text{Salt}}{\mathop{BA}}\,+\underset{\text{Water}}{\mathop{{{H}_{2}}O}}\,$
or ${{H}^{+}}+{{A}^{-}}+{{B}^{+}}+O{{H}^{-}}\xrightarrow{{}}{{B}^{+}}+{{A}^{-}}+{{H}_{2}}O$
or ${{H}^{+}}+O{{H}^{-}}\xrightarrow{{}}{{H}_{2}}O$
The end point in these titrations is determined by the use of organic dyes which are either weak acids or weak bases. These change their colours within a limited range of hydrogen ion concentrations, i.e., $pH$ of the solution. Phenolphthalein is a suitable indicator in the titrations of strong alkalies (free from carbonate) against strong acids or weak acids. Methyl orange is used as an indicator in the titrations of strong acids against strong and weak alkalies. As no indicator gives correct results in the titrations of weak acids against weak bases, such titrations are performed by some other methods (physical methods).
(2) Oxidation reduction titrations : The titrations based on oxidation-reduction reactions are called redox titrations. The chemical reactions proceed with transfer of electrons (simultaneous loss or gain of electrons) among the reacting ions in aqueous solutions. Sometimes these titrations are named after the reagent used, as:
(i) Permanganate titrations : These are titrations in which potassium permanganate is used as an oxidising agent in acidic medium. The medium is maintained by the use of dilute sulphuric acid. Potassium permanganate acts as a self-indicator. The potential equation, when potassium permanganate acts as an oxidising agent, is :
$2KMn{{O}_{4}}+3{{H}_{2}}S{{O}_{4}}\xrightarrow{{}}{{K}_{2}}S{{O}_{4}}+2MnS{{O}_{4}}+3{{H}_{2}}O+5[O]$
or $MnO_{4}^{-}+8{{H}^{+}}+5e\xrightarrow{{}}M{{n}^{2+}}+4{{H}_{2}}O$
Before the end point, the solution remains colourless (when $KMn{{O}_{4}}$ solution is taken in burette) but after the equivalence point only one extra drop of $KMn{{O}_{4}}$ solution imparts pink colour, i.e., appearance of pink colour indicates the end point. Potassium permanganate is used for the estimation of ferrous salts, oxalic acid, oxalates, hydrogen peroxide, etc. The solution of potassium permanganate is always first standardised before its use.
(ii) Dichromate titrations : These are titrations in which, potassium dichromate is used as an oxidising agent in acidic medium. The medium is maintained acidic by the use of dilute sulphuric acid. The potential equation is
${{K}_{2}}C{{r}_{2}}{{O}_{7}}+4{{H}_{2}}S{{O}_{4}}\xrightarrow{{}}{{K}_{2}}S{{O}_{4}}+C{{r}_{2}}{{(S{{O}_{4}})}_{3}}+4{{H}_{2}}O+3[O]$
or $C{{r}_{2}}O_{7}^{2-}+14{{H}^{+}}+6{{e}^{-}}\xrightarrow{{}}2C{{r}^{3+}}+7{{H}_{2}}O$
The solution of potassium dichromate can be directly used for titrations. It is mainly used for the estimation of ferrous salts and iodides. In the titration of ${{K}_{2}}C{{r}_{2}}{{O}_{7}}$ versus ferrous salt either an external indicator (potassium ferricyanide) or an internal indicator (diphenyl amine) can be used.
(iii) Iodimetric and iodometric titrations : The reduction of free iodine to iodide ions and oxidation of iodide ions to free iodine occurs in these titrations.
${{I}_{2}}+2{{e}^{-}}\xrightarrow{{}}2{{I}^{-}}$ (reduction)
$2{{I}^{-}}\xrightarrow{{}}{{I}_{2}}+2{{e}^{-}}$ (oxidation)
These are divided into two types :
(a) Iodimetric titrations : These are the titrations in which free iodine is used. As it is difficult to prepare the solution of iodine (volatile and less soluble in water), it is dissolved in potassium iodide solution.
$KI+{{I}_{2}}\xrightarrow{{}}\underset{\text{Potassium tri-iodide}}{\mathop{K{{I}_{3}}}}\,$
This solution is first standardised before use. With the standard solution of ${{I}_{2}}$. Substances such as sulphite, thiosulphate, arsenite, etc., are estimated.
(b) Iodometric titrations : In iodometric titrations, an oxidising agent is allowed to react in neutral medium or in acidic medium, with excess of potassium iodide to liberate free iodine.
$KI+$oxidising agent $\xrightarrow{{}}{{I}_{2}}$
Free iodine is titrated against a standard reducing agent usually with sodium thiosulphate. Halogens, oxyhalogens, dichromates, cupric ion, peroxides, etc., can be estimated by this method.
${{I}_{2}}+N{{a}_{2}}{{S}_{2}}{{O}_{3}}\xrightarrow{{}}2NaI+N{{a}_{2}}{{S}_{4}}{{O}_{6}}$
$2CuS{{O}_{4}}+4KI\xrightarrow{{}}C{{u}_{2}}{{I}_{2}}+2{{K}_{2}}S{{O}_{4}}+{{I}_{2}}$
${{K}_{2}}C{{r}_{2}}{{O}_{7}}+6KI+7{{H}_{2}}S{{O}_{4}}\xrightarrow{{}}$$C{{r}_{2}}{{(S{{O}_{4}})}_{3}}+4{{K}_{2}}S{{O}_{4}}+7{{H}_{2}}O+3{{I}_{2}}$
In iodimetric and iodometric titrations, starch solution is used as an indicator. Starch solution gives blue or violet colour with free iodine. At the end point the blue or violet colour disappears when iodine is completely changed to iodide.
(3) Precipitation titrations : The titrations which are based on the formation of insoluble precipitates, when the solutions of two reacting substances are brought in contact with each other, are called precipitation titrations. For example, when a solution of silver nitrate is added to a solution of sodium chloride or a solution of ammonium thiocyanate, a white precipitate of silver chloride or silver thiocyanate is formed.
$AgN{{O}_{3}}+NaCl\xrightarrow{{}}AgCl+NaN{{O}_{3}}$
$AgN{{O}_{3}}+N{{H}_{4}}CNS\xrightarrow{{}}AgCNS+N{{H}_{4}}N{{O}_{3}}$
Such titrations involving silver nitrate are called argentometric titrations.
(4) Complexometric titrations : A titration, in which an undissociated complex is formed at the equivalence point, is called complexometric titration. These titrations are superior to precipitation titrations as there is no error due to co-precipitation.
$H{{g}^{2+}}+2SC{{N}^{-}}\xrightarrow{{}}Hg{{(SCN)}_{2}}$
$A{{g}^{+}}+2C{{N}^{-}}\xrightarrow{{}}{{[Ag{{(CN)}_{2}}]}^{-}}$
EDTA (ethylenediamine tetra-acetic acid) is a useful reagent which forms complexes with metals. In the form of disodium salt, it is used to estimate $C{{a}^{2+}}$ and $M{{g}^{2+}}$ ions in presence of eriochrome black-$T$ as an indicator.
Equivalent masses of acids and bases : Equivalent masses of some acids and bases are given in the following table
Acid Basicity Mol. Mass Eq. Mass $HCl$ 1 36.5 $\frac{36.5}{1}=36.5$ $HN{{O}_{3}}$ 1 63 $\frac{63}{1}=63.0$ ${{H}_{2}}S{{O}_{4}}$ 2 98 $\frac{98}{2}=49.0$ $C{{H}_{3}}COOH$ 1 60 $\frac{60}{1}=60.0$ ${{H}_{2}}{{C}_{2}}{{O}_{4}}.2{{H}_{2}}O$ 2 126 $\frac{126}{2}=63.0$ ${{H}_{3}}P{{O}_{4}}$ 3 98 $\frac{98}{3}=32.7$ ${{H}_{3}}P{{O}_{3}}$ 2 82 $\frac{82}{2}=41.0$ ${{H}_{3}}P{{O}_{2}}$ 1 66 $\frac{66}{1}=66.0$
Alkali Acidity Mol. Mass Eq. Mass $NaOH$ 1 40 $\frac{40}{1}=40$ $KOH$ 1 56 $\frac{56}{1}=56$ $Ca{{(OH)}_{2}}$ 2 74 $\frac{74}{2}=37$ $N{{H}_{4}}OH$ 1 35 $\frac{35}{1}=35$
Calculations of Volumetric analysis
The following points should be kept in mind while making calculations of volumetric exercises.
(i) $1g$ equivalent mass of a substance reacts completely with $1g$ equivalent mass of any other substance. $1g$ equivalent mass of a substance means equivalent mass of the substance in grams. For example,
$1g$ equivalent mass of $NaOH=40g$ of $NaOH$
$1g$ equivalent mass of ${{H}_{2}}S{{O}_{4}}=49g$ of ${{H}_{2}}S{{O}_{4}}$
$1g$ equivalent mass of $KMn{{O}_{4}}$ in acidic medium
$=31.6\,g$ of $KMn{{O}_{4}}$
$1g$ equivalent mass of hydrated oxalic acid
$=63g$ of hydrated oxalic acid
Note : Equivalent mass is a variable quantity and depends on the reaction in which the substance takes part. The nature of the reaction should be known before writing the gram equivalent mass of the substance. For example in the reactions.
$2NaCl+2{{H}_{2}}S{{O}_{4}}\xrightarrow{{}}2NaHS{{O}_{4}}+2HCl$ …..(i)
$2NaCl+{{H}_{2}}S{{O}_{4}}\xrightarrow{{}}N{{a}_{2}}S{{O}_{4}}+2HCl$ …..(ii)
The value of $g$ equivalent mass of ${{H}_{2}}S{{O}_{4}}$ in reaction (i) is $98g$ and in reaction (ii) $49g$.
(ii) Number of $g$ equivalents $=\frac{\text{Mass of the substance in }g}{\text{Equivalent mass of the substance}}$
Number of $g$ moles $=\frac{\text{Mass of the substance in }g}{\text{Molecular mass of the substance}}$
$=\frac{\text{Volume in litres of the substance at N}\text{.T}\text{.P}\text{.}}{22.4}$(only for gases)
Number of milli-equivalent $=\frac{\text{Mass in }g\times 1000}{\text{Equivalent mass}}$
Number of milli-moles $=\frac{\text{Mass in }g\times 1000}{\text{Molecular mass}}$
(iii) Molarity $=\frac{\text{No}\text{. of moles of the solute}}{\text{No}\text{. of litres of the solution}}$ $=\frac{w}{m\times V}$
Molarity $\times$ molecular mass = strength of the solution $(g/L)$ No. of moles of the solute = Molarity $\times$ No. of litres of solution Mass of the solute in $g(w)=$ molarity $\times$ No. of litres of solution $\times$ mol. mass of solute
Normality $=\frac{\text{No}\text{. of }g\text{ equivalent of the solute}}{\text{No}\text{. of litres of the solution}}$ $=\frac{w}{E\times V}$
Normality $\times$ equivalent mass = strength of the solution (g/L)
No. of equivalents of the solute = Normality $\times$ No. of litres of solution
Mass of the solute in $g(w)=$ Normality $\times$ No. of litres of solution $\times$ Eq. mass of the solute
$\frac{\text{Molecular mass}}{\text{Equivalent mass}}=n=\frac{\text{Normality}}{\text{Molarity}}$
Normality $=n\times$ Molarity
(iv) Normality equation : When solutions $A$ and $B$ react completely.
${{N}_{A}}{{V}_{A}}={{N}_{B}}{{V}_{B}}$
Normality of $A\times$ volume of $A=$Normality of $B\times$ volume of $B$
or $\frac{\text{Strength }A}{\text{Eq}\text{. mass }A}\times {{V}_{A}}=\frac{\text{Strength }B}{\text{Eq}\text{. mass }B}\times {{V}_{B}}$
$\frac{\text{Wt}\text{. of metal hydroxide}}{\text{wt, of metal oxide}}=\frac{\text{Eq}\text{. wt}\text{. of metal hydroxide}}{\text{Eq}\text{. wt}\text{. of metal oxide}}$
$=\frac{\text{Eq}\text{. wt of metal }+\text{ Eq}\text{. wt of }OH}{\text{Eq}\text{. wt}\text{. of metal }+\text{ Eq}\text{. wt of }{{O}^{2-}}}$
(v) When the solution is diluted, the following formulae can be applied :
${{N}_{1}}{{V}_{1}}={{N}_{2}}{{V}_{2}}$ or ${{M}_{1}}{{V}_{1}}={{M}_{2}}{{V}_{2}}$ or ${{S}_{1}}{{V}_{1}}={{S}_{2}}{{V}_{2}}$
Before dilution = After dilution
(vi) If a number of acids are mixed, the combined normality of the mixture, ${{N}_{x}}$, is given
${{N}_{x}}{{V}_{x}}={{N}_{1}}{{V}_{1}}+{{N}_{2}}{{V}_{2}}+{{N}_{3}}{{V}_{3}}+......$
Where ${{V}_{x}}$ is the total volume of the mixture, ${{N}_{1}}$ and ${{V}_{1}}$ are the normality and volume respectively of one acid, ${{N}_{2}}$ and ${{V}_{2}}$ of the second acid and so on.
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2018-04-21 21:20:27
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https://mail.studentforums.biz/index.php?option=com_content&view=article&id=5497:kepler-s-second-law&catid=220&Itemid=1757
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## Kepler's Second Law
Kepler's second law of planetary motion states that the planets, as they orbit the Sun following an ellipse, sweep out equal areas in equal times, and is a consequence of the conservation of angular momentum.
\begin{aligned}\vec{A} &= \frac{1}{2} r \vec{r} \times \vec{v} dt \\ &= \frac{\vec{r}}{2m} \times m \vec{v} dt \\ &= \frac{\vec{L}}{2m}\end{aligned}
Hence
$\frac{dA}{dt} = \frac{L}{2m}dt$
and since angular momentum
$L$
is conserved, Kepler's second law is proved.
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2020-07-14 01:43:06
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https://www.transtutors.com/questions/current-ratio-help-needed-fast--196924.htm
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# --Current Ratio help needed fast !!!
Problem:
In its most recent annual report, Appalachian Beverages reported current assets of $54,000 and a current ratio of 1.80. Assume that the following transactions were completed: (1) purchased merchandise for$6,000 on account, and (2) purchased a delivery truck for $10,000, paying$1,000 cash a...
• ### Analyzing the Impact of Selected Transactions on the Current Ratio In its most recent annual report,...
March 23, 2018
Analyzing the Impact of Selected Transactions on the Current Ratio In its most recent annual report , Sunrise Enterprises reported current assets of $1 ,090,000 and current liabilities of$602,000. Required: Determine the impact of the following transactions on the current ratio for Sunrise
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2018-09-19 13:19:59
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http://www.formuladirectory.com/user/formula/25
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HOSTING A TOTAL OF 318 FORMULAS WITH CALCULATORS
Perimeter of a parallelogram
The perimeter of parallelogram is the total distance around the outside, which can be found by adding together the length of each side. In the case of a parallelogram, each pair of opposite sides is the same length, so the perimeter is twice the base plus twice the side length.
$2\left[l+b\right]$
Perimeter of a parallelogram (P) = 2 × (length(l) + Breadth(b))
ENTER THE VARIABLES TO BE USED IN THE FORMULA
Similar formulas which you may find interesting.
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2018-03-18 17:14:26
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http://beast.physicswiki.net/index.php/Atom
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# Atom
Helium atom
An illustration of the helium atom, depicting the nucleus (pink) and the electron cloud distribution (black). The nucleus (upper right) in helium-4 is in reality spherically symmetric and closely resembles the electron cloud, although for more complicated nuclei this is not always the case. The black bar is one angstrom (10−10 m or 100 pm).
Classification
Smallest recognized division of a chemical element
Properties
Mass range: 1.67×10−27 to 4.52×10−25 kg Electric charge: zero (neutral), or ion charge Diameter range: 62 pm (He) to 520 pm (Cs) (data page) Components: Electrons and a compact nucleus of protons and neutrons
An atom is the smallest constituent unit of ordinary matter that has the properties of a chemical element.[1] Every solid, liquid, gas, and plasma is made up of neutral or ionized atoms. Atoms are very small; typical sizes are around 100 pm (a ten-billionth of a meter, in the short scale).[2] However, atoms do not have well defined boundaries, and there are different ways to define their size which give different but close values.
Atoms are small enough that classical physics give noticeably incorrect results. Through the development of physics, atomic models have incresingly incorporated quantum principles to better explain and predict the behavior.
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and typically a similar number of neutrons (none in hydrogen-1). Protons and neutrons are called nucleons. Over 99.94% of the atom's mass is in the nucleus.[3] The protons have a positive electric charge, the electrons have a negative electric charge, and the neutrons have no electric charge. If the number of protons and electrons are equal, that atom is electrically neutral. If an atom has more or less electrons than protons, then it has an overall negative or positive charge, respectively, and it is called an ion.
Electrons of an atom are attracted to the protons in an atomic nucleus by this electromagnetic force. The protons and neutrons in the nucleus are attracted to each other by a different force, the nuclear force, which is usually stronger than the electromagnetic force repelling the positively charged protons from one another. Under certain circumstances the repelling electromagnetic force becomes stronger than the nuclear force, and nucleons can be ejected from the nucleus, leaving behind a different element: nuclear decay resulting in nuclear transmutation.
The number of protons in the nucleus defines to what chemical element the atom belongs: for example, all copper atoms contain 29 protons. The number of neutrons defines the isotope of the element.[4] The number of electrons influences the magnetic properties of an atom. Atoms can attach to one or more other atoms by chemical bonds to form chemical compounds such as molecules. The ability of atoms to associate and dissociate is responsible for most of the physical changes observed in nature, and is the subject of the discipline of chemistry.
Not all the matter of the universe is composed of atoms. Dark matter comprises more of the Universe than matter, and is composed not of atoms, but of particles of a currently unknown type.
## History of atomic theory
### Atoms in philosophy
The idea that matter is made up of discrete units is a very old one, appearing in many ancient cultures such as Greece and India. The word "atom", in fact, was coined by ancient Greek philosophers. However, these ideas were founded in philosophical and theological reasoning rather than evidence and experimentation. As a result, their views on what atoms look like and how they behave were incorrect. They also couldn't convince everybody, so atomism was but one of a number of competing theories on the nature of matter. It wasn't until the 19th century that the idea was embraced and refined by scientists, when the blossoming science of chemistry produced discoveries that only the concept of atoms could explain.
### First evidence-based theory
Various atoms and molecules as depicted in John Dalton's A New System of Chemical Philosophy (1808).
In the early 1800s, John Dalton used the concept of atoms to explain why elements always react in ratios of small whole numbers (the law of multiple proportions). For instance, there are two types of tin oxide: one is 88.1% tin and 11.9% oxygen and the other is 78.7% tin and 21.3% oxygen (tin(II) oxide and tin dioxide respectively). This means that 100g of tin will combine either with 13.5g or 27g of oxygen. 13.5 and 27 form a ratio of 1:2, a ratio of small whole numbers. This common pattern in chemistry suggested to Dalton that elements react in whole number multiples of discrete units—in other words, atoms. In the case of tin oxides, one tin atom will combine with either one or two oxygen atoms.[5]
Dalton also believed atomic theory could explain why water absorbs different gases in different proportions. For example, he found that water absorbs carbon dioxide far better than it absorbs nitrogen.[6] Dalton hypothesized this was due to the differences in mass and complexity of the gases' respective particles. Indeed, carbon dioxide molecules (CO2) are heavier and larger than nitrogen molecules (N2).
### Brownian motion
In 1827, botanist Robert Brown used a microscope to look at dust grains floating in water and discovered that they moved about erratically, a phenomenon that became known as "Brownian motion". This was thought to be caused by water molecules knocking the grains about. In 1905 Albert Einstein produced the first mathematical analysis of the motion.[7][8][9] French physicist Jean Perrin used Einstein's work to experimentally determine the mass and dimensions of atoms, thereby conclusively verifying Dalton's atomic theory.[10]
### Discovery of the electron
The Geiger–Marsden experiment
Top: Expected results: alpha particles passing through the plum pudding model of the atom with negligible deflection.
Bottom: Observed results: a small portion of the particles were deflected by the concentrated positive charge of the nucleus.
The physicist J. J. Thomson measured the mass of cathode rays, showing they were made of particles, but were around 1800 times lighter than the lightest atom, hydrogen. Therefore they were not atoms, but a new particle, the first subatomic particle to be discovered, which he originally called "corpuscle" but was later named electron, after particles postulated by George Johnstone Stoney in 1874. He also showed they were identical to particles given off by photoelectric and radioactive materials.[11] It was quickly recognized that they are the particles that carry electric currents in metal wires, and carry the negative electric charge within atoms. Thomson was given the 1906 Nobel prize for physics for this work. Thus he overturned the belief that atoms are the indivisible, ultimate particles of matter.[12] Thomson also incorrectly postulated that the low mass, negatively charged electrons were distributed throughout the atom in a uniform sea of positive charge. This became known as the plum pudding model.
### Discovery of the nucleus
In 1909, Hans Geiger and Ernest Marsden, under the direction of Ernest Rutherford, bombarded a metal foil with alpha particles to observe how they scattered. They expected all the alpha particles to pass straight through with little deflection, because Thomson's model said that the charges in the atom are so diffuse that their electric fields could not affect the alpha particles much. However, Geiger and Marsden spotted alpha particles being deflected by angles greater than 90°, which was supposed to be impossible according to Thomson's model. To explain this, Rutherford proposed that the positive charge of the atom is concentrated in a tiny nucleus at the center of the atom.[13]
### Discovery of isotopes
While experimenting with the products of radioactive decay, in 1913 radiochemist Frederick Soddy discovered that there appeared to be more than one type of atom at each position on the periodic table.[14] The term isotope was coined by Margaret Todd as a suitable name for different atoms that belong to the same element. J.J. Thomson created a technique for separating atom types through his work on ionized gases, which subsequently led to the discovery of stable isotopes.[15]
### Bohr model
The Bohr model of the atom, with an electron making instantaneous "quantum leaps" from one orbit to another. This model is obsolete.
In 1913 the physicist Niels Bohr proposed a model in which the electrons of an atom were assumed to orbit the nucleus but could only do so in a finite set of orbits, and could jump between these orbits only in discrete changes of energy corresponding to absorption or radiation of a photon.[16] This quantization was used to explain why the electrons orbits are stable (given that normally, charges in acceleration, including circular motion, lose kinetic energy which is emitted as electromagnetic radiation, see synchrotron radiation) and why elements absorb and emit electromagnetic radiation in discrete spectra.[17]
Later in the same year Henry Moseley provided additional experimental evidence in favor of Niels Bohr's theory. These results refined Ernest Rutherford's and Antonius Van den Broek's model, which proposed that the atom contains in its nucleus a number of positive nuclear charges that is equal to its (atomic) number in the periodic table. Until these experiments, atomic number was not known to be a physical and experimental quantity. That it is equal to the atomic nuclear charge remains the accepted atomic model today.[18]
### Chemical bonding explained
Chemical bonds between atoms were now explained, by Gilbert Newton Lewis in 1916, as the interactions between their constituent electrons.[19] As the chemical properties of the elements were known to largely repeat themselves according to the periodic law,[20] in 1919 the American chemist Irving Langmuir suggested that this could be explained if the electrons in an atom were connected or clustered in some manner. Groups of electrons were thought to occupy a set of electron shells about the nucleus.[21]
### Further developments in quantum physics
The Stern–Gerlach experiment of 1922 provided further evidence of the quantum nature of the atom. When a beam of silver atoms was passed through a specially shaped magnetic field, the beam was split based on the direction of an atom's angular momentum, or spin. As this direction is random, the beam could be expected to spread into a line. Instead, the beam was split into two parts, depending on whether the atomic spin was oriented up or down.[22]
In 1924, Louis de Broglie proposed that all particles behave to an extent like waves. In 1926, Erwin Schrödinger used this idea to develop a mathematical model of the atom that described the electrons as three-dimensional waveforms rather than point particles. A consequence of using waveforms to describe particles is that it is physically impossible to obtain precise values for both the position and momentum of a particle at the same time; this became known as the uncertainty principle, formulated by Werner Heisenberg in 1926. In this concept, for a given accuracy in measuring a position one could only obtain a range of probable values for momentum, and vice versa. This model was able to explain observations of atomic behavior that previous models could not, such as certain structural and spectral patterns of atoms larger than hydrogen. Thus, the planetary model of the atom was discarded in favor of one that described atomic orbital zones around the nucleus where a given electron is most likely to be observed.[23][24]
### Discovery of the neutron
The development of the mass spectrometer allowed the mass of atoms to be measured with increased accuracy. The device uses a magnet to bend the trajectory of a beam of ions, and the amount of deflection is determined by the ratio of an atom's mass to its charge. The chemist Francis William Aston used this instrument to show that isotopes had different masses. The atomic mass of these isotopes varied by integer amounts, called the whole number rule.[25] The explanation for these different isotopes awaited the discovery of the neutron, an uncharged particle with a mass similar to the proton, by the physicist James Chadwick in 1932. Isotopes were then explained as elements with the same number of protons, but different numbers of neutrons within the nucleus.[26]
### Fission, high-energy physics and condensed matter
In 1938, the German chemist Otto Hahn, a student of Rutherford, directed neutrons onto uranium atoms expecting to get transuranium elements. Instead, his chemical experiments showed barium as a product.[27] A year later, Lise Meitner and her nephew Otto Frisch verified that Hahn's result were the first experimental nuclear fission.[28][29] In 1944, Hahn received the Nobel prize in chemistry. Despite Hahn's efforts, the contributions of Meitner and Frisch were not recognized.[30]
In the 1950s, the development of improved particle accelerators and particle detectors allowed scientists to study the impacts of atoms moving at high energies.[31] Neutrons and protons were found to be hadrons, or composites of smaller particles called quarks. The standard model of particle physics was developed that so far has successfully explained the properties of the nucleus in terms of these sub-atomic particles and the forces that govern their interactions.[32]
## Structure
### Subatomic particles
Though the word atom originally denoted a particle that cannot be cut into smaller particles, in modern scientific usage the atom is composed of various subatomic particles. The constituent particles of an atom are the electron, the proton and the neutron; all three are fermions. However, the hydrogen-1 atom has no neutrons and the hydron ion has no electrons.
The electron is by far the least massive of these particles at 9.11×10−31 kg, with a negative electrical charge and a size that is too small to be measured using available techniques.[33] It is the lightest particle with a positive rest mass measured. Under ordinary conditions, electrons are bound to the positively charged nucleus by the attraction created from opposite electric charges. If an atom has more or fewer electrons than its atomic number, then it becomes respectively negatively or positively charged as a whole; a charged atom is called an ion. Electrons have been known since the late 19th century, mostly thanks to J.J. Thomson; see history of subatomic physics for details.
Protons have a positive charge and a mass 1,836 times that of the electron, at 1.6726×10−27 kg. The number of protons in an atom is called its atomic number. Ernest Rutherford (1919) observed that nitrogen under alpha-particle bombardment ejects what appeared to be hydrogen nuclei. By 1920 he had accepted that the hydrogen nucleus is a distinct particle within the atom and named it proton.
Neutrons have no electrical charge and have a free mass of 1,839 times the mass of the electron,[34] or 1.6929×10−27 kg, the heaviest of the three constituent particles, but it can be reduced by the nuclear binding energy. Neutrons and protons (collectively known as nucleons) have comparable dimensions—on the order of 2.5×10−15 m—although the 'surface' of these particles is not sharply defined.[35] The neutron was discovered in 1932 by the English physicist James Chadwick.
In the Standard Model of physics, electrons are truly elementary particles with no internal structure. However, both protons and neutrons are composite particles composed of elementary particles called quarks. There are two types of quarks in atoms, each having a fractional electric charge. Protons are composed of two up quarks (each with charge +23) and one down quark (with a charge of −13). Neutrons consist of one up quark and two down quarks. This distinction accounts for the difference in mass and charge between the two particles.[36][37]
The quarks are held together by the strong interaction (or strong force), which is mediated by gluons. The protons and neutrons, in turn, are held to each other in the nucleus by the nuclear force, which is a residuum of the strong force that has somewhat different range-properties (see the article on the nuclear force for more). The gluon is a member of the family of gauge bosons, which are elementary particles that mediate physical forces.[36][37]
### Nucleus
The binding energy needed for a nucleon to escape the nucleus, for various isotopes
All the bound protons and neutrons in an atom make up a tiny atomic nucleus, and are collectively called nucleons. The radius of a nucleus is approximately equal to 1.07 3A fm, where A is the total number of nucleons.[38] This is much smaller than the radius of the atom, which is on the order of 105 fm. The nucleons are bound together by a short-ranged attractive potential called the residual strong force. At distances smaller than 2.5 fm this force is much more powerful than the electrostatic force that causes positively charged protons to repel each other.[39]
Atoms of the same element have the same number of protons, called the atomic number. Within a single element, the number of neutrons may vary, determining the isotope of that element. The total number of protons and neutrons determine the nuclide. The number of neutrons relative to the protons determines the stability of the nucleus, with certain isotopes undergoing radioactive decay.[40]
The proton, the electron, and the neutron are classified as fermions. Fermions obey the Pauli exclusion principle which prohibits identical fermions, such as multiple protons, from occupying the same quantum state at the same time. Thus, every proton in the nucleus must occupy a quantum state different from all other protons, and the same applies to all neutrons of the nucleus and to all electrons of the electron cloud. However, a proton and a neutron are allowed to occupy the same quantum state.[41]
For atoms with low atomic numbers, a nucleus that has more neutrons than protons tends to drop to a lower energy state through radioactive decay so that the neutron–proton ratio is closer to one. However, as the atomic number increases, a higher proportion of neutrons is required to offset the mutual repulsion of the protons. Thus, there are no stable nuclei with equal proton and neutron numbers above atomic number Z = 20 (calcium) and as Z increases, the neutron–proton ratio of stable isotopes increases.[41] The stable isotope with the highest proton–neutron ratio is lead-208 (about 1.5).
Illustration of a nuclear fusion process that forms a deuterium nucleus, consisting of a proton and a neutron, from two protons. A positron (e+)—an antimatter electron—is emitted along with an electron neutrino.
The number of protons and neutrons in the atomic nucleus can be modified, although this can require very high energies because of the strong force. Nuclear fusion occurs when multiple atomic particles join to form a heavier nucleus, such as through the energetic collision of two nuclei. For example, at the core of the Sun protons require energies of 3–10 keV to overcome their mutual repulsion—the coulomb barrier—and fuse together into a single nucleus.[42] Nuclear fission is the opposite process, causing a nucleus to split into two smaller nuclei—usually through radioactive decay. The nucleus can also be modified through bombardment by high energy subatomic particles or photons. If this modifies the number of protons in a nucleus, the atom changes to a different chemical element.[43][44]
If the mass of the nucleus following a fusion reaction is less than the sum of the masses of the separate particles, then the difference between these two values can be emitted as a type of usable energy (such as a gamma ray, or the kinetic energy of a beta particle), as described by Albert Einstein's mass–energy equivalence formula, E = mc2, where m is the mass loss and c is the speed of light. This deficit is part of the binding energy of the new nucleus, and it is the non-recoverable loss of the energy that causes the fused particles to remain together in a state that requires this energy to separate.[45]
The fusion of two nuclei that create larger nuclei with lower atomic numbers than iron and nickel—a total nucleon number of about 60—is usually an exothermic process that releases more energy than is required to bring them together.[46] It is this energy-releasing process that makes nuclear fusion in stars a self-sustaining reaction. For heavier nuclei, the binding energy per nucleon in the nucleus begins to decrease. That means fusion processes producing nuclei that have atomic numbers higher than about 26, and atomic masses higher than about 60, is an endothermic process. These more massive nuclei can not undergo an energy-producing fusion reaction that can sustain the hydrostatic equilibrium of a star.[41]
### Electron cloud
A potential well, showing, according to classical mechanics, the minimum energy V(x) needed to reach each position x. Classically, a particle with energy E is constrained to a range of positions between x1 and x2.
The electrons in an atom are attracted to the protons in the nucleus by the electromagnetic force. This force binds the electrons inside an electrostatic potential well surrounding the smaller nucleus, which means that an external source of energy is needed for the electron to escape. The closer an electron is to the nucleus, the greater the attractive force. Hence electrons bound near the center of the potential well require more energy to escape than those at greater separations.
Electrons, like other particles, have properties of both a particle and a wave. The electron cloud is a region inside the potential well where each electron forms a type of three-dimensional standing wave—a wave form that does not move relative to the nucleus. This behavior is defined by an atomic orbital, a mathematical function that characterises the probability that an electron appears to be at a particular location when its position is measured.[47] Only a discrete (or quantized) set of these orbitals exist around the nucleus, as other possible wave patterns rapidly decay into a more stable form.[48] Orbitals can have one or more ring or node structures, and they differ from each other in size, shape and orientation.[49]
Wave functions of the first five atomic orbitals. The three 2p orbitals each display a single angular node that has an orientation and a minimum at the center.
File:Atomic orbitals and periodic table construction.ogv
Each atomic orbital corresponds to a particular energy level of the electron. The electron can change its state to a higher energy level by absorbing a photon with sufficient energy to boost it into the new quantum state. Likewise, through spontaneous emission, an electron in a higher energy state can drop to a lower energy state while radiating the excess energy as a photon. These characteristic energy values, defined by the differences in the energies of the quantum states, are responsible for atomic spectral lines.[48]
The amount of energy needed to remove or add an electron—the electron binding energy—is far less than the binding energy of nucleons. For example, it requires only 13.6 eV to strip a ground-state electron from a hydrogen atom,[50] compared to 2.23 million eV for splitting a deuterium nucleus.[51] Atoms are electrically neutral if they have an equal number of protons and electrons. Atoms that have either a deficit or a surplus of electrons are called ions. Electrons that are farthest from the nucleus may be transferred to other nearby atoms or shared between atoms. By this mechanism, atoms are able to bond into molecules and other types of chemical compounds like ionic and covalent network crystals.[52]
## Properties
### Nuclear properties
By definition, any two atoms with an identical number of protons in their nuclei belong to the same chemical element. Atoms with equal numbers of protons but a different number of neutrons are different isotopes of the same element. For example, all hydrogen atoms admit exactly one proton, but isotopes exist with no neutrons (hydrogen-1, by far the most common form,[53] also called protium), one neutron (deuterium), two neutrons (tritium) and more than two neutrons. The known elements form a set of atomic numbers, from the single proton element hydrogen up to the 118-proton element ununoctium.[54] All known isotopes of elements with atomic numbers greater than 82 are radioactive.[55][56]
About 339 nuclides occur naturally on Earth,[57] of which 254 (about 75%) have not been observed to decay, and are referred to as "stable isotopes". However, only 90 of these nuclides are stable to all decay, even in theory. Another 164 (bringing the total to 254) have not been observed to decay, even though in theory it is energetically possible. These are also formally classified as "stable". An additional 34 radioactive nuclides have half-lives longer than 80 million years, and are long-lived enough to be present from the birth of the solar system. This collection of 288 nuclides are known as primordial nuclides. Finally, an additional 51 short-lived nuclides are known to occur naturally, as daughter products of primordial nuclide decay (such as radium from uranium), or else as products of natural energetic processes on Earth, such as cosmic ray bombardment (for example, carbon-14).[58][note 1]
For 80 of the chemical elements, at least one stable isotope exists. As a rule, there is only a handful of stable isotopes for each of these elements, the average being 3.2 stable isotopes per element. Twenty-six elements have only a single stable isotope, while the largest number of stable isotopes observed for any element is ten, for the element tin. Elements 43, 61, and all elements numbered 83 or higher have no stable isotopes.[59][page needed]
Stability of isotopes is affected by the ratio of protons to neutrons, and also by the presence of certain "magic numbers" of neutrons or protons that represent closed and filled quantum shells. These quantum shells correspond to a set of energy levels within the shell model of the nucleus; filled shells, such as the filled shell of 50 protons for tin, confers unusual stability on the nuclide. Of the 254 known stable nuclides, only four have both an odd number of protons and odd number of neutrons: hydrogen-2 (deuterium), lithium-6, boron-10 and nitrogen-14. Also, only four naturally occurring, radioactive odd–odd nuclides have a half-life over a billion years: potassium-40, vanadium-50, lanthanum-138 and tantalum-180m. Most odd–odd nuclei are highly unstable with respect to beta decay, because the decay products are even–even, and are therefore more strongly bound, due to nuclear pairing effects.[59][page needed]
### Mass
The large majority of an atom's mass comes from the protons and neutrons that make it up. The total number of these particles (called "nucleons") in a given atom is called the mass number. It is a positive integer and dimensionless (instead of having dimension of mass), because it expresses a count. An example of use of a mass number is "carbon-12," which has 12 nucleons (six protons and six neutrons).
The actual mass of an atom at rest is often expressed using the unified atomic mass unit (u), also called dalton (Da). This unit is defined as a twelfth of the mass of a free neutral atom of carbon-12, which is approximately 1.66×10−27 kg.[60] Hydrogen-1 (the lightest isotope of hydrogen which is also the nuclide with the lowest mass) has an atomic weight of 1.007825 u.[61] The value of this number is called the atomic mass. A given atom has an atomic mass approximately equal (within 1%) to its mass number times the atomic mass unit (for example the mass of a nitrogen-14 is roughly 14 u). However, this number will not be exactly an integer except in the case of carbon-12 (see below).[62] The heaviest stable atom is lead-208,[55] with a mass of 207.9766521 u.[63]
As even the most massive atoms are far too light to work with directly, chemists instead use the unit of moles. One mole of atoms of any element always has the same number of atoms (about 6.022×1023). This number was chosen so that if an element has an atomic mass of 1 u, a mole of atoms of that element has a mass close to one gram. Because of the definition of the unified atomic mass unit, each carbon-12 atom has an atomic mass of exactly 12 u, and so a mole of carbon-12 atoms weighs exactly 0.012 kg.[60]
### Shape and size
Atoms lack a well-defined outer boundary, so their dimensions are usually described in terms of an atomic radius. This is a measure of the distance out to which the electron cloud extends from the nucleus.[2] However, this assumes the atom to exhibit a spherical shape, which is only obeyed for atoms in vacuum or free space. Atomic radii may be derived from the distances between two nuclei when the two atoms are joined in a chemical bond. The radius varies with the location of an atom on the atomic chart, the type of chemical bond, the number of neighboring atoms (coordination number) and a quantum mechanical property known as spin.[64] On the periodic table of the elements, atom size tends to increase when moving down columns, but decrease when moving across rows (left to right).[65] Consequently, the smallest atom is helium with a radius of 32 pm, while one of the largest is caesium at 225 pm.[66]
When subjected to external forces, like electrical fields, the shape of an atom may deviate from spherical symmetry. The deformation depends on the field magnitude and the orbital type of outer shell electrons, as shown by group-theoretical considerations. Aspherical deviations might be elicited for instance in crystals, where large crystal-electrical fields may occur at low-symmetry lattice sites. Significant ellipsoidal deformations have recently been shown to occur for sulfur ions[67] and chalcogen ions[68] in pyrite-type compounds.
Atomic dimensions are thousands of times smaller than the wavelengths of light (400–700 nm) so they cannot be viewed using an optical microscope. However, individual atoms can be observed using a scanning tunneling microscope. To visualize the minuteness of the atom, consider that a typical human hair is about 1 million carbon atoms in width.[69] A single drop of water contains about 2 sextillion (2×1021) atoms of oxygen, and twice the number of hydrogen atoms.[70] A single carat diamond with a mass of 2×10−4 kg contains about 10 sextillion (1022) atoms of carbon.[note 2] If an apple were magnified to the size of the Earth, then the atoms in the apple would be approximately the size of the original apple.[71]
This diagram shows the half-life (T½) of various isotopes with Z protons and N neutrons.
Every element has one or more isotopes that have unstable nuclei that are subject to radioactive decay, causing the nucleus to emit particles or electromagnetic radiation. Radioactivity can occur when the radius of a nucleus is large compared with the radius of the strong force, which only acts over distances on the order of 1 fm.[72]
The most common forms of radioactive decay are:[73][74]
• Alpha decay: this process is caused when the nucleus emits an alpha particle, which is a helium nucleus consisting of two protons and two neutrons. The result of the emission is a new element with a lower atomic number.
• Beta decay (and electron capture): these processes are regulated by the weak force, and result from a transformation of a neutron into a proton, or a proton into a neutron. The neutron to proton transition is accompanied by the emission of an electron and an antineutrino, while proton to neutron transition (except in electron capture) causes the emission of a positron and a neutrino. The electron or positron emissions are called beta particles. Beta decay either increases or decreases the atomic number of the nucleus by one. Electron capture is more common than positron emission, because it requires less energy. In this type of decay, an electron is absorbed by the nucleus, rather than a positron emitted from the nucleus. A neutrino is still emitted in this process, and a proton changes to a neutron.
• Gamma decay: this process results from a change in the energy level of the nucleus to a lower state, resulting in the emission of electromagnetic radiation. The excited state of a nucleus which results in gamma emission usually occurs following the emission of an alpha or a beta particle. Thus, gamma decay usually follows alpha or beta decay.
Other more rare types of radioactive decay include ejection of neutrons or protons or clusters of nucleons from a nucleus, or more than one beta particle. An analog of gamma emission which allows excited nuclei to lose energy in a different way, is internal conversion— a process that produces high-speed electrons that are not beta rays, followed by production of high-energy photons that are not gamma rays. A few large nuclei explode into two or more charged fragments of varying masses plus several neutrons, in a decay called spontaneous nuclear fission.
Each radioactive isotope has a characteristic decay time period—the half-life—that is determined by the amount of time needed for half of a sample to decay. This is an exponential decay process that steadily decreases the proportion of the remaining isotope by 50% every half-life. Hence after two half-lives have passed only 25% of the isotope is present, and so forth.[72]
### Magnetic moment
Elementary particles possess an intrinsic quantum mechanical property known as spin. This is analogous to the angular momentum of an object that is spinning around its center of mass, although strictly speaking these particles are believed to be point-like and cannot be said to be rotating. Spin is measured in units of the reduced Planck constant (ħ), with electrons, protons and neutrons all having spin ½ ħ, or "spin-½". In an atom, electrons in motion around the nucleus possess orbital angular momentum in addition to their spin, while the nucleus itself possesses angular momentum due to its nuclear spin.[75]
The magnetic field produced by an atom—its magnetic moment—is determined by these various forms of angular momentum, just as a rotating charged object classically produces a magnetic field. However, the most dominant contribution comes from electron spin. Due to the nature of electrons to obey the Pauli exclusion principle, in which no two electrons may be found in the same quantum state, bound electrons pair up with each other, with one member of each pair in a spin up state and the other in the opposite, spin down state. Thus these spins cancel each other out, reducing the total magnetic dipole moment to zero in some atoms with even number of electrons.[76]
In ferromagnetic elements such as iron, cobalt and nickel, an odd number of electrons leads to an unpaired electron and a net overall magnetic moment. The orbitals of neighboring atoms overlap and a lower energy state is achieved when the spins of unpaired electrons are aligned with each other, a spontaneous process known as an exchange interaction. When the magnetic moments of ferromagnetic atoms are lined up, the material can produce a measurable macroscopic field. Paramagnetic materials have atoms with magnetic moments that line up in random directions when no magnetic field is present, but the magnetic moments of the individual atoms line up in the presence of a field.[76][77]
The nucleus of an atom will have no spin when it has even numbers of both neutrons and protons, but for other cases of odd numbers, the nucleus may have a spin. Normally nuclei with spin are aligned in random directions because of thermal equilibrium. However, for certain elements (such as xenon-129) it is possible to polarize a significant proportion of the nuclear spin states so that they are aligned in the same direction—a condition called hyperpolarization. This has important applications in magnetic resonance imaging.[78][79]
### Energy levels
These electron's energy levels (not to scale) are sufficient for ground states of atoms up to cadmium (5s2 4d10) inclusively. Do not forget that even the top of the diagram is lower than an unbound electron state.
The potential energy of an electron in an atom is negative, its dependence of its position reaches the minimum (the most absolute value) inside the nucleus, and vanishes when the distance from the nucleus goes to infinity, roughly in an inverse proportion to the distance. In the quantum-mechanical model, a bound electron can only occupy a set of states centered on the nucleus, and each state corresponds to a specific energy level; see time-independent Schrödinger equation for theoretical explanation. An energy level can be measured by the amount of energy needed to unbind the electron from the atom, and is usually given in units of electronvolts (eV). The lowest energy state of a bound electron is called the ground state, i.e. stationary state, while an electron transition to a higher level results in an excited state.[80] The electron's energy raises when n increases because the (average) distance to the nucleus increases. Dependence of the energy on is caused not by electrostatic potential of the nucleus, but by interaction between electrons.
For an electron to transition between two different states, e.g. grounded state to first excited level (ionization), it must absorb or emit a photon at an energy matching the difference in the potential energy of those levels, according to Niels Bohr model, what can be precisely calculated by the Schrödinger equation. Electrons jump between orbitals in a particle-like fashion. For example, if a single photon strikes the electrons, only a single electron changes states in response to the photon; see Electron properties.
The energy of an emitted photon is proportional to its frequency, so these specific energy levels appear as distinct bands in the electromagnetic spectrum.[81] Each element has a characteristic spectrum that can depend on the nuclear charge, subshells filled by electrons, the electromagnetic interactions between the electrons and other factors.[82]
An example of absorption lines in a spectrum
When a continuous spectrum of energy is passed through a gas or plasma, some of the photons are absorbed by atoms, causing electrons to change their energy level. Those excited electrons that remain bound to their atom spontaneously emit this energy as a photon, traveling in a random direction, and so drop back to lower energy levels. Thus the atoms behave like a filter that forms a series of dark absorption bands in the energy output. (An observer viewing the atoms from a view that does not include the continuous spectrum in the background, instead sees a series of emission lines from the photons emitted by the atoms.) Spectroscopic measurements of the strength and width of atomic spectral lines allow the composition and physical properties of a substance to be determined.[83]
Close examination of the spectral lines reveals that some display a fine structure splitting. This occurs because of spin–orbit coupling, which is an interaction between the spin and motion of the outermost electron.[84] When an atom is in an external magnetic field, spectral lines become split into three or more components; a phenomenon called the Zeeman effect. This is caused by the interaction of the magnetic field with the magnetic moment of the atom and its electrons. Some atoms can have multiple electron configurations with the same energy level, which thus appear as a single spectral line. The interaction of the magnetic field with the atom shifts these electron configurations to slightly different energy levels, resulting in multiple spectral lines.[85] The presence of an external electric field can cause a comparable splitting and shifting of spectral lines by modifying the electron energy levels, a phenomenon called the Stark effect.[86]
If a bound electron is in an excited state, an interacting photon with the proper energy can cause stimulated emission of a photon with a matching energy level. For this to occur, the electron must drop to a lower energy state that has an energy difference matching the energy of the interacting photon. The emitted photon and the interacting photon then move off in parallel and with matching phases. That is, the wave patterns of the two photons are synchronized. This physical property is used to make lasers, which can emit a coherent beam of light energy in a narrow frequency band.[87]
### Valence and bonding behavior
Valency is the combining power of an element. It is equal to number of hydrogen atoms that atom can combine or displace in forming compounds.[88] The outermost electron shell of an atom in its uncombined state is known as the valence shell, and the electrons in that shell are called valence electrons. The number of valence electrons determines the bonding behavior with other atoms. Atoms tend to chemically react with each other in a manner that fills (or empties) their outer valence shells.[89] For example, a transfer of a single electron between atoms is a useful approximation for bonds that form between atoms with one-electron more than a filled shell, and others that are one-electron short of a full shell, such as occurs in the compound sodium chloride and other chemical ionic salts. However, many elements display multiple valences, or tendencies to share differing numbers of electrons in different compounds. Thus, chemical bonding between these elements takes many forms of electron-sharing that are more than simple electron transfers. Examples include the element carbon and the organic compounds.[90]
The chemical elements are often displayed in a periodic table that is laid out to display recurring chemical properties, and elements with the same number of valence electrons form a group that is aligned in the same column of the table. (The horizontal rows correspond to the filling of a quantum shell of electrons.) The elements at the far right of the table have their outer shell completely filled with electrons, which results in chemically inert elements known as the noble gases.[91][92]
### States
Snapshots illustrating the formation of a Bose–Einstein condensate
Quantities of atoms are found in different states of matter that depend on the physical conditions, such as temperature and pressure. By varying the conditions, materials can transition between solids, liquids, gases and plasmas.[93] Within a state, a material can also exist in different allotropes. An example of this is solid carbon, which can exist as graphite or diamond.[94] Gaseous allotropes exist as well, such as dioxygen and ozone.
At temperatures close to absolute zero, atoms can form a Bose–Einstein condensate, at which point quantum mechanical effects, which are normally only observed at the atomic scale, become apparent on a macroscopic scale.[95][96] This super-cooled collection of atoms then behaves as a single super atom, which may allow fundamental checks of quantum mechanical behavior.[97]
## Identification
Scanning tunneling microscope image showing the individual atoms making up this gold (100) surface. The surface atoms deviate from the bulk crystal structure and arrange in columns several atoms wide with pits between them (See surface reconstruction).
The scanning tunneling microscope is a device for viewing surfaces at the atomic level. It uses the quantum tunneling phenomenon, which allows particles to pass through a barrier that would normally be insurmountable. Electrons tunnel through the vacuum between two planar metal electrodes, on each of which is an adsorbed atom, providing a tunneling-current density that can be measured. Scanning one atom (taken as the tip) as it moves past the other (the sample) permits plotting of tip displacement versus lateral separation for a constant current. The calculation shows the extent to which scanning-tunneling-microscope images of an individual atom are visible. It confirms that for low bias, the microscope images the space-averaged dimensions of the electron orbitals across closely packed energy levels—the Fermi level local density of states.[98][99]
An atom can be ionized by removing one of its electrons. The electric charge causes the trajectory of an atom to bend when it passes through a magnetic field. The radius by which the trajectory of a moving ion is turned by the magnetic field is determined by the mass of the atom. The mass spectrometer uses this principle to measure the mass-to-charge ratio of ions. If a sample contains multiple isotopes, the mass spectrometer can determine the proportion of each isotope in the sample by measuring the intensity of the different beams of ions. Techniques to vaporize atoms include inductively coupled plasma atomic emission spectroscopy and inductively coupled plasma mass spectrometry, both of which use a plasma to vaporize samples for analysis.[100]
A more area-selective method is electron energy loss spectroscopy, which measures the energy loss of an electron beam within a transmission electron microscope when it interacts with a portion of a sample. The atom-probe tomograph has sub-nanometer resolution in 3-D and can chemically identify individual atoms using time-of-flight mass spectrometry.[101]
Spectra of excited states can be used to analyze the atomic composition of distant stars. Specific light wavelengths contained in the observed light from stars can be separated out and related to the quantized transitions in free gas atoms. These colors can be replicated using a gas-discharge lamp containing the same element.[102] Helium was discovered in this way in the spectrum of the Sun 23 years before it was found on Earth.[103]
## Origin and current state
Atoms form about 4% of the total energy density of the observable Universe, with an average density of about 0.25 atoms/m3.[104] Within a galaxy such as the Milky Way, atoms have a much higher concentration, with the density of matter in the interstellar medium (ISM) ranging from 105 to 109 atoms/m3.[105] The Sun is believed to be inside the Local Bubble, a region of highly ionized gas, so the density in the solar neighborhood is only about 103 atoms/m3.[106] Stars form from dense clouds in the ISM, and the evolutionary processes of stars result in the steady enrichment of the ISM with elements more massive than hydrogen and helium. Up to 95% of the Milky Way's atoms are concentrated inside stars and the total mass of atoms forms about 10% of the mass of the galaxy.[107] (The remainder of the mass is an unknown dark matter.)[108]
### Formation
Electrons are thought to exist in the Universe since early stages of the Big Bang. Atomic nuclei forms in nucleosynthesis reactions. In about three minutes Big Bang nucleosynthesis produced most of the helium, lithium, and deuterium in the Universe, and perhaps some of the beryllium and boron.[109][110][111]
Ubiquitousness and stability of atoms relies on their binding energy, which means that an atom has a lower energy than an unbound system of the nucleus and electrons. Where the temperature is much higher than ionization potential, the matter exists in the form of plasma—a gas of positively charged ions (possibly, bare nuclei) and electrons. When the temperature drops below the ionization potential, atoms become statistically favorable. Atoms (complete with bound electrons) became to dominate over charged particles 380,000 years after the Big Bang—an epoch called recombination, when the expanding Universe cooled enough to allow electrons to become attached to nuclei.[112]
Since the Big Bang, which produced no carbon or heavier elements, atomic nuclei have been combined in stars through the process of nuclear fusion to produce more of the element helium, and (via the triple alpha process) the sequence of elements from carbon up to iron;[113] see stellar nucleosynthesis for details.
Isotopes such as lithium-6, as well as some beryllium and boron are generated in space through cosmic ray spallation.[114] This occurs when a high-energy proton strikes an atomic nucleus, causing large numbers of nucleons to be ejected.
Elements heavier than iron were produced in supernovae through the r-process and in AGB stars through the s-process, both of which involve the capture of neutrons by atomic nuclei.[115] Elements such as lead formed largely through the radioactive decay of heavier elements.[116]
### Earth
Most of the atoms that make up the Earth and its inhabitants were present in their current form in the nebula that collapsed out of a molecular cloud to form the Solar System. The rest are the result of radioactive decay, and their relative proportion can be used to determine the age of the Earth through radiometric dating.[117][118] Most of the helium in the crust of the Earth (about 99% of the helium from gas wells, as shown by its lower abundance of helium-3) is a product of alpha decay.[119]
There are a few trace atoms on Earth that were not present at the beginning (i.e., not "primordial"), nor are results of radioactive decay. Carbon-14 is continuously generated by cosmic rays in the atmosphere.[120] Some atoms on Earth have been artificially generated either deliberately or as by-products of nuclear reactors or explosions.[121][122] Of the transuranic elements—those with atomic numbers greater than 92—only plutonium and neptunium occur naturally on Earth.[123][124] Transuranic elements have radioactive lifetimes shorter than the current age of the Earth[125] and thus identifiable quantities of these elements have long since decayed, with the exception of traces of plutonium-244 possibly deposited by cosmic dust.[126] Natural deposits of plutonium and neptunium are produced by neutron capture in uranium ore.[127]
The Earth contains approximately 1.33×1050 atoms.[128] Although small numbers of independent atoms of noble gases exist, such as argon, neon, and helium, 99% of the atmosphere is bound in the form of molecules, including carbon dioxide and diatomic oxygen and nitrogen. At the surface of the Earth, an overwhelming majority of atoms combine to form various compounds, including water, salt, silicates and oxides. Atoms can also combine to create materials that do not consist of discrete molecules, including crystals and liquid or solid metals.[129][130] This atomic matter forms networked arrangements that lack the particular type of small-scale interrupted order associated with molecular matter.[131]
### Rare and theoretical forms
#### Superheavy elements
While isotopes with atomic numbers higher than lead (82) are known to be radioactive, an "island of stability" has been proposed for some elements with atomic numbers above 103. These superheavy elements may have a nucleus that is relatively stable against radioactive decay.[132] The most likely candidate for a stable superheavy atom, unbihexium, has 126 protons and 184 neutrons.[133]
#### Exotic matter
Each particle of matter has a corresponding antimatter particle with the opposite electrical charge. Thus, the positron is a positively charged antielectron and the antiproton is a negatively charged equivalent of a proton. When a matter and corresponding antimatter particle meet, they annihilate each other. Because of this, along with an imbalance between the number of matter and antimatter particles, the latter are rare in the universe. The first causes of this imbalance are not yet fully understood, although theories of baryogenesis may offer an explanation. As a result, no antimatter atoms have been discovered in nature.[134][135] However, in 1996 the antimatter counterpart of the hydrogen atom (antihydrogen) was synthesized at the CERN laboratory in Geneva.[136][137]
Other exotic atoms have been created by replacing one of the protons, neutrons or electrons with other particles that have the same charge. For example, an electron can be replaced by a more massive muon, forming a muonic atom. These types of atoms can be used to test the fundamental predictions of physics.[138][139][140]
## Notes
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## References
Cite error: Invalid parameter: use the {{reflist}} template with the group parameter (see the help page).
### Other references
• Liddell, Henry George; Scott, Robert. "A Greek-English Lexicon". Perseus Digital Library.
• Liddell, Henry George; Scott, Robert. "ἄτομος". A Greek-English Lexicon. Perseus Digital Library. Retrieved 21 June 2010.
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2017-08-19 12:58:12
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http://sigma-journal.com/2014/005/
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### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
SIGMA 10 (2014), 005, 21 pages arXiv:1309.3713 https://doi.org/10.3842/SIGMA.2014.005
### Why Do the Relativistic Masses and Momenta of Faster-than-Light Particles Decrease as their Speeds Increase?
Judit X. Madarász a, Mike Stannett b and Gergely Székely a
a) Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, Budapest 1364, Hungary
b) University of Sheffield, Department of Computer Science, 211 Portobello, Sheffield S1 4DP, United Kingdom
Received September 17, 2013, in final form January 07, 2014; Published online January 11, 2014
Abstract
It has recently been shown within a formal axiomatic framework using a definition of four-momentum based on the Stückelberg-Feynman-Sudarshan-Recami ''switching principle'' that Einstein's relativistic dynamics is logically consistent with the existence of interacting faster-than-light inertial particles. Our results here show, using only basic natural assumptions on dynamics, that this definition is the only possible way to get a consistent theory of such particles moving within the geometry of Minkowskian spacetime. We present a strictly formal proof from a streamlined axiom system that given any slow or fast inertial particle, all inertial observers agree on the value of $\mathsf{m}\cdot \sqrt{|1-v^2|}$, where $\mathsf{m}$ is the particle's relativistic mass and $v$ its speed. This confirms formally the widely held belief that the relativistic mass and momentum of a positive-mass faster-than-light particle must decrease as its speed increases.
Key words: special relativity; dynamics; faster-than-light particles; superluminal motion; tachyons; axiomatic method; first-order logic.
pdf (593 kb) tex (204 kb)
References
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2018-02-22 20:09:38
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https://math.stackexchange.com/questions/150764/on-some-finite-2-group-of-class-two?noredirect=1
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# On some finite 2-group of class two [duplicate]
Possible Duplicate:
On automorphism of some finite 2-group of class nilpotency two
Let $G$ be a finite 2-group of nilpotency class two such that $\frac{G}{Z(G)}\simeq C_{2}\times C_{2}$. Then do there exist a non inner automorphism of $G$ that acts trivial only on $Z(G)$? For example $D_{8}$, dihedral group of order 8, has a non inner automorphism that acts trivial only on $Z(G)$.
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2020-02-22 00:53:08
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https://www.researcher-app.com/paper/250481
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3 years ago
# Quantum Oscillations in Nodal Line Systems.
Roderich Moessner, Hui Yang, Lih-King Lim
We study signatures of magnetic quantum oscillations in three-dimensional nodal line semimetals at zero temperature. The extended nature of the degenerate bands can result in a Fermi surface geometry with topological genus one, as well as a Fermi surface of electron and hole pockets encapsulating the nodal line. Moreover, the underlying two-band model to describe a nodal line is not unique, in that there are two classes of Hamiltonian with distinct band topology giving rise to the same Fermi surface geometry. After identifying the extremal cyclotron orbits in various magnetic field directions, we study their concomitant Landau levels and resulting quantum oscillation signatures. By Landau-fan-diagram analyses we extract the non-trivial $\pi$ Berry phase signature for extremal orbits linking the nodal line.
Publisher URL: http://arxiv.org/abs/1801.02733
DOI: arXiv:1801.02733v1
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2022-07-05 18:45:37
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https://www.meritnation.com/ask-answer/question/q-there-are-120-seats-in-the-alcony-of-a-theatre-if-these-12/line/11775131
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# Q). There are 120 seats in the alcony of a theatre. If these 120 seats are $\frac{1}{5}$ of the total seats, what is the total number of seats in the theatre?
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2020-10-01 02:05:01
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http://openstudy.com/updates/55f5e101e4b065930cf4f91f
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## anonymous one year ago Express the rational expression in lowest terms: 3x+4/15x^2+26x+8.
1. anonymous
by "lowest terms" you want to factor this?
2. anonymous
I think so, heres the question again with the choices:
3. anonymous
Oh, alright. so you want to factor $$\sf 15x^2+26x+8$$
4. anonymous
$\frac{4 x^2}{15}+29 x+8$
5. anonymous
i'd say your best choice would be to leave it as it is.,
6. anonymous
I"m not sure, but I just wanna know how to solve this
7. Nnesha
do you know how to factor the quadratic equations ???
8. anonymous
9. Nnesha
alright there are like 3 ways to factor the quadratic equation but the easy one is AC method you have to find two numbers when you multiply them you should get product of AC and when you add or subtract them you should get the middle term B $\huge\rm Ax^2+Bx+C$ A=leading coefficient B=middle term C=constant term
10. Nnesha
|dw:1442180776707:dw| multiply A C
11. anonymous
But @Nnesha the factored terms will not cancel out with the numerator..
12. Nnesha
yes one of the factor will cancel out with the numerator.
13. Nnesha
A=15 B=26 C=8 multiply 15 times 8= 120 now find two number when you multiply them u should get 120 and when u add or subtract them you should get 26
14. Nnesha
what are the two numbers whose sum is 26 and product is 120 ???
15. anonymous
Is it 6 and 20?
16. Nnesha
yes right now we have to apply group method |dw:1442182184861:dw| make a group of two terms now find take out the common factor from first red box
17. Nnesha
15x^2+20x what is common in these two terms ?
18. anonymous
I"m not sure?
19. Nnesha
what are the factors of 15 and 20 ?
20. Nnesha
nvm 5 times 3 =15 5 times 4 = 20 so 5,3 are factors of 15 and 5,4 are factors of 20 and x^2 can be written as x times x so $\huge\rm (\color{ReD}{5 · 3 ·x ·x} +\color{blue}{ 5 · 4 ·x})$ now can you tell me what's common in red and blue terms ?
21. anonymous
15 x squared is equal from the red, and 20x is blue?
22. Nnesha
yes it's 15x^2 +20x but i factored 15 , and 20 so you can find the common factor nvm so $\huge\rm (\color{ReD}{5} · 3 ·x ·\color{ReD}{x} +\color{red}{ 5} · 4 ·\color{ReD}{x})$ now as you can see 5 and x are common in both terms right ? so take them out $\huge\rm \color{ReD}{5x}(3x+4)$ understood ?
23. Nnesha
15x^2 = first term 20x = 2nd term just in case if you're not familiar with it :=)
24. anonymous
$15x^2+26x+8 \implies x^2+26x+120$$=(x+6)(x+20)$$=\left(x+\frac{6}{15}\right)\left(x+\frac{20}{15}\right)$$=\left(x+\frac{\cancel{6}2}{\cancel{15}5}\right)\left(x+\frac{\cancel{20}4}{\cancel{15}3}\right)$$\sf \text{cross multiply} : (5x+2)(3x+4)$$=\frac{3x+4}{(5x+2)(3x+4)}$ So what would eliminating $$\sf 3x+4$$ from the top and bottom give you?
25. Nnesha
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2016-10-21 11:18:15
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