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https://grindskills.com/is-it-correct-to-use-ln-instead-of-ln-for-natural-logarithm/ | # Is it correct to use ‘Ln’ instead of ‘ln’ for natural logarithm?
In some research papers, authors use ‘Ln’ for natural logarithm instead of ‘ln’. Is it correct?
If there is possibility of confusion, it is best to specify the base, and do so explicitly: $\log_{10}$ or $\log_\mathrm{e}$ or $\ln$ (base e logarithm) or something similar. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9110748767852783, "perplexity": 920.345079349205}, "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-2022-40/segments/1664030335303.67/warc/CC-MAIN-20220929003121-20220929033121-00477.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-11-rational-expressions-and-functions-11-5-solving-rational-equations-practice-and-problem-solving-exercises-page-683/12 | ## Algebra 1
$y=-\frac{1}{3}$
Given: $10+3y=9$ $\frac{10}{y}+\frac{3y}{y}=\frac{9}{y}$ $\implies \frac{10}{y}+3=\frac{9}{y}$ $\implies 3=\frac{9}{y}-\frac{10}{y}$ or, $3y=-1$ Solutions are: $y=-\frac{1}{3}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9831244945526123, "perplexity": 1702.0648567158753}, "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/1585371833063.93/warc/CC-MAIN-20200409091317-20200409121817-00279.warc.gz"} |
http://eccc.hpi-web.de/author/239/ | REPORTS > AUTHORS > YIJIA CHEN:
All reports by Author Yijia Chen:
TR13-065 | 21st April 2013
Yijia Chen, Joerg Flum
#### On Limitations of the Ehrenfeucht-Fraisse-method in Descriptive Complexity
Ehrenfeucht-Fraisse games and their generalizations have been quite successful in finite model theory and yield various inexpressibility results. However, for key problems such as P $\ne$ NP or NP $\ne$ coNP no progress has been achieved using the games. We show that for these problems it is already hard to ... more >>>
TR11-085 | 14th May 2011
Yijia Chen, Joerg Flum, Moritz Müller
#### Hard instances of algorithms and proof systems
Assuming that the class TAUT of tautologies of propositional logic has no almost optimal algorithm, we show that every algorithm $\mathbb A$ deciding TAUT has a polynomial time computable sequence witnessing that $\mathbb A$ is not almost optimal. The result extends to every $\Pi_t^p$-complete problem with $t\ge 1$; however, we ... more >>>
TR11-020 | 20th December 2010
Yijia Chen, Joerg Flum
#### Listings and logics
There are standard logics DTC, TC, and LFP capturing the complexity classes L, NL, and P on ordered structures, respectively. In [Chen and Flum, 2010] we have shown that ${\rm LFP}_{\rm inv}$, the order-invariant least fixed-point logic LFP,'' captures P (on all finite structures) if and only if there is ... more >>>
TR10-008 | 13th January 2010
Yijia Chen, Joerg Flum
#### On optimal proof systems and logics for PTIME
Revisions: 1
We prove that TAUT has a $p$-optimal proof system if and only if $L_\le$, a logic introduced in [Gurevich, 88], is a P-bounded logic for P. Furthermore, using the method developed in [Chen and Flum, 10], we show that TAUT has no \emph{effective} $p$-optimal proof system under some reasonable complexity-theoretic ... more >>>
TR08-083 | 10th June 2008
Yijia Chen, Jörg Flum
#### A logic for PTIME and a parameterized halting problem
In [Blass, Gurevich, and Shelah, 99] a logic L_Y has been introduced as a possible candidate for a logic capturing the PTIME properties of structures (even in the absence of an ordering of their universe). A reformulation of this problem in terms of a parameterized halting problem p-Acc for nondeterministic ... more >>>
TR07-137 | 6th November 2007
Yijia Chen, Jörg Flum, Moritz Müller
#### Lower Bounds for Kernelizations
Among others, refining the methods of [Fortnow and Santhanam, ECCC Report TR07-096] we improve a result of this paper and show for any parameterized problem with a linear weak OR'' and with NP-hard underlying classical problem that there is no polynomial reduction from the problem to itself that assigns to ... more >>>
TR07-106 | 10th September 2007
Yijia Chen, Martin Grohe, Magdalena Grüber
#### On Parameterized Approximability
Combining classical approximability questions with parameterized complexity, we introduce a theory of parameterized approximability.
The main intention of this theory is to deal with the efficient approximation of small cost solutions for optimisation problems.
more >>>
TR06-011 | 2nd January 2006
Yijia Chen, Martin Grohe
#### An Isomorphism between Subexponential and Parameterized Complexity Theory
We establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories.
more >>>
ISSN 1433-8092 | Imprint | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9050574898719788, "perplexity": 3709.894713900622}, "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-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00154-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/concentration-of-water-vs-equilibrium.679911/ | # Concentration of water vs. equilibrium
1. Mar 21, 2013
### christian0710
Hi,
I understand from calculation that the molar concentration of Pure water is 55.5 moles/Liter
Then how come in equilibrium reactions when calculation the dissociation constant, we say that the concentration of water is 1 Molar? This seems like a huge difference to me? What is it I need to understand here?
2. Mar 21, 2013
### DrDu
This is not correct.
What enters the equilibrium constant is the ratio of the concentration of a substance c relative to some standard concentration $c_0$, i.e. $c/c_0$. For diluted substances this standard concentration is $c_0=$1 mol/l (molarity) or 1 mol/ kg (molality) or the like (or more precisely the behaviour at infinite dilution extrapolated to a concentration of 1 mol/l). For solvents etc. we use as a standard state the pure substance, i.e. $c_0=55,5$ mol/l for water. In a dilute solution, the concentration $c$ of water is to an excellent extent equal to $c_0$ so that we can set the ratio equal to 1, at least for calculations with chemical precision.
3. Mar 21, 2013
### christian0710
Okay, so I might understand what you are saying in 2 possible ways, which of the following two ways is the correct?
So if we are measuring the equilibrium H2O + CO2 ⇔ H(+) + HCO3(-) which is a reaction taking place in the blood of a person, then the concentration of water is only 1Molar because a) we have a diluted solution, where water is the solvent so [H2O] =1M ?
or b) the ratio between the initial concentration of water and final concentration ( at products) is almost the same so the ratio between the two concentrations is 1?
4. Mar 21, 2013
### DrDu
The equilibrium constant is a dimensionless quantity as it only depends on the ratios of concentrations to their respective standard concentrations.
Specifically for your reaction
$K=(c_\mathrm{H^+}/1\mathrm{ mol/l})\cdot (c_\mathrm{HCO_3^-}/1\mathrm{ mol/l})/[(c_\mathrm{H_2O}/ 55,5\mathrm{ mol/l})\cdot (c_\mathrm{CO_2}/1\mathrm{ mol/l})]\approx (c_\mathrm{H^+}/1\mathrm{ mol/l})\cdot (c_\mathrm{HCO_3^-}/1\mathrm{ mol/l})/(c_\mathrm{CO_2}/1\mathrm{ mol/l})$
5. Mar 21, 2013
### DrDu
6. Mar 21, 2013
### christian0710
That's exactly the part I don't understand: so you have 55.5M H2O in the denominator in your equation (which makes the fraction 55 times as small) and then you can say it's approximately the same as "≈" removing the 55.5M H2O? How come we can just do that?
7. Mar 21, 2013
### DrDu
The point is that c is to a very good approximation equal to c_0. So their ratio is 1.
E.g. the concentration of water in a solution of CO2 containing 1 mol/l is still about c=(55,5-1) mol/l=54,5 mol/l, hence it differs very little from c_0
8. Mar 21, 2013
### christian0710
PErfect, so the ratio between [H2O]_Start and [H2O]end is 1 :)
Thank you.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9746384024620056, "perplexity": 1529.5703585991687}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2018-30/segments/1531676589892.87/warc/CC-MAIN-20180717183929-20180717203929-00544.warc.gz"} |
https://eprint.iacr.org/2013/498/20130815:072621 | ## Cryptology ePrint Archive: Report 2013/498
Non-Malleable Codes from Two-Source Extractors
Stefan Dziembowski and Tomasz Kazana and Maciej Obremski
Abstract: We construct an efficient information-theoretically non-mall\-eable code in the split-state model for one-bit messages. Non-malleable codes were introduced recently by Dziembowski, Pietrzak and Wichs (ICS 2010), as a general tool for storing messages securely on hardware that can be subject to tampering attacks. Informally, a code $(Enc : \cal M \rightarrow \cal L \times \cal R, Dec : \cal L \times \cal R \rightarrow \cal M)$ is {\em non-malleable in the split-state model} if any adversary, by manipulating {\em independently} $L$ and $R$ (where $(L,R)$ is an encoding of some message $M$), cannot obtain an encoding of a message $M'$ that is not equal to $M$ but is related'' $M$ in some way. Until now it was unknown how to construct an information-theoretically secure code with such a property, even for $\cal M = \{0,1\}$. Our construction solves this problem. Additionally, it is leakage-resilient, and the amount of leakage that we can tolerate can be an arbitrary fraction $\xi < {1}/{4}$ of the length of the codeword. Our code is based on the inner-product two-source extractor, but in general it can be instantiated by any two-source extractor that has large output and has the property of being {\em flexible}, which is a new notion that we define. We also show that the non-malleable codes for one-bit messages have an equivalent, perhaps simpler characterization, namely such codes can be defined as follows: if $M$ is chosen uniformly from $\{0,1\}$ then the probability (in the experiment described above) that the output message $M'$ is not equal to $M$ can be at most $1/2 + \epsilon$.
Category / Keywords:
Original Publication (with major differences): IACR-CRYPTO-2013 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9314429759979248, "perplexity": 1064.649644004496}, "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-2019-51/segments/1575540532624.3/warc/CC-MAIN-20191211184309-20191211212309-00521.warc.gz"} |
https://cs.stackexchange.com/questions/11976/confusion-related-to-time-complexity-of-dynamic-programming-algorithm-for-knapsa | # Confusion related to time complexity of dynamic programming algorithm for knapsack problem
I have this confusion related to the time complexity of the algorithm solving the knapsack problem using dynamic programming
I didn't get how the time complexity of the algorithm came out to be $O(nV^*)$
Since computing each cell in the table is $O(1)$, the running time is just the size of the table. The first coordinate ranges from $1$ to $n$, and the second one from $0$ to the maximal value ever encountered (so it's really a dynamic table), which is $V^*$.
• I have a confusion. We run the dynamic programming with two loops lets say i from 1 to n and another loop v from 0 to nvmax. As mentioned in the picture, they have $V* <= nv_{max}$. So the limit is already decided isn't it? May 13, 2013 at 2:46
• If you do it this way, the complexity is $O(n^2v_{\max})$. May 13, 2013 at 4:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8812634348869324, "perplexity": 183.93375289948253}, "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-2023-14/segments/1679296949355.52/warc/CC-MAIN-20230330163823-20230330193823-00730.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/71654-linearity-matrix-functions-print.html | # Linearity / Matrix functions?
• February 3rd 2009, 06:06 PM
Unenlightened
Linearity / Matrix functions?
I shouldn't have left it so late to ask this, but the answers to any of these would be most helpful...
Is the function mapping a matrix A to the determinant of A linear?
I'm thinking it's non-linear, but simply because I can't think of a transformation matrix that could map [[a,b],[c,d]] <--- (A 2x2 matrix a, b, c, d reading left to right... apologies for absence of Latex!) to ad-bc...
Is the operation mapping f to f''+3f' linear?
(Where f is over the collection of all infinitely differentiable functions)
I know differentiation is linear, but does it hold for two separate derivatives added together?
Fine, A(cx + dy) = c(Ax)+d(Ay) if and only if the function is linear, but how does one actually go about picking these x and y s, or creating a matrix A to test one's theory?
Is a function mapping f to its second derivative linear?
Aye, I'm presuming this is linear, since differentiation is linear (although I'm not sure how to explain that...)
Is the function mapping a matrix A to its trace linear?
This one is linear, right? Because you're just adding two entries together. Again, I'm not sure how to construct the function as a matrix...
Is the function mapping x to 3x + 2 linear?
This is a straight out 'no', right? 3x is fine, but you can't just add a constant like that, right?
Let C = [[1,2],[3,4]]. Is the function mapping A to AC-CA linear?
Again, I'm guessing it's not, for similar reasons to the previous...
Any help on any of these much appreciated.
• February 3rd 2009, 06:32 PM
Isomorphism
Quote:
Originally Posted by Unenlightened
I shouldn't have left it so late to ask this, but the answers to any of these would be most helpful...
Is the function mapping a matrix A to the determinant of A linear?
Idea: What is
$\det( 3A )$?
Quote:
Is the operation mapping f to f''+3f' linear?
(Where f is over the collection of all infinitely differentiable functions)
I know differentiation is linear, but does it hold for two separate derivatives added together?
Fine, A(cx + dy) = c(Ax)+d(Ay) if and only if the function is linear, but how does one actually go about picking these x and y s, or creating a matrix A to test one's theory?
Quote:
Is a function mapping f to its second derivative linear?
Aye, I'm presuming this is linear, since differentiation is linear (although I'm not sure how to explain that...)
For both these questions, you dont need matrices... Check if f and g are mapped as above, is f+g mapped like above....
Quote:
Is the function mapping a matrix A to its trace linear?
This one is linear, right? Because you're just adding two entries together. Again, I'm not sure how to construct the function as a matrix...
Yes it is. Rigorously you say: $\text{Tr }(\alpha A + \beta B) = \alpha \text{Tr } (A) + \beta \text{Tr }(B)$
Quote:
Is the function mapping x to 3x + 2 linear?
This is a straight out 'no', right? 3x is fine, but you can't just add a constant like that, right?
Yes. Show that 0 does not map to 0, thus it is not linear
Quote:
Let C = [[1,2],[3,4]]. Is the function mapping A to AC-CA linear?
Again, I'm guessing it's not, for similar reasons to the previous...
This question is not clear :(
• February 3rd 2009, 07:10 PM
Unenlightened
Thankee koindly :)
Sorry about the last one - it's supposed to be the matrix
(1 2)
(3 4)
And the function is to map A onto A*C - C*A...
Ooh also
How about the function mapping A to A transpose?
Non-linear also?
• February 4th 2009, 10:14 PM
Isomorphism
Quote:
Originally Posted by Unenlightened
Thankee koindly :)
Sorry about the last one - it's supposed to be the matrix
(1 2)
(3 4)
And the function is to map A onto A*C - C*A...
Ooh also
How about the function mapping A to A transpose?
Non-linear also?
They all are linear. Just apply the definition of linearity to get the answer :) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "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": 0.871472954750061, "perplexity": 977.9939448134903}, "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-2014-41/segments/1410657134514.93/warc/CC-MAIN-20140914011214-00299-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/79184/at-most-finitely-many-hamel-coordinate-functionals-are-continuous-different | At most finitely many (Hamel) coordinate functionals are continuous - different proof
If $X$ is a vector space over $\mathbb R$ and $B=\{x_i; i\in I\}$ is a Hamel basis for $X$, then for each $i\in I$ we have a linear functional $a_i(x)$ which assigns to $x$ the $i$-th coordinate, i.e., the functions $a_i$ are uniquely determined by the conditions that $$x=\sum_{i\in I} a_i(x)x_i,$$ where only finitely many summands are non-zero.
If $X$ is a Banach space, then at most finitely many of them can be continuous.
I have learned the following argument from comments in this question.
Suppose that $\{b_i; i\in\mathbb N\}$ is an infinite subset of $B$ such that each $f_{b_i}$ is continuous. W.l.o.g. we may assume that $\lVert{b_i}\rVert=1$.
Let $$y:=\sum_{i=1}^\infty \frac1{2^i}b_i.$$ (Since $X$ is complete, the above sum converges.)
We also denote $y_n:=\sum_{i=1}^n \frac1{2^i}b_i$. Since $y_n$ converges to $y$, we have $f_{b_k}(y)=\lim\limits_{n\to\infty} f_{b_k}(y_n)=\frac1{2^k}$ for each $k\in\mathbb N$. Thus the point $x$ has infinitely many non-zero coordinates, which contradicts the definition of Hamel basis.
I have stumbled upon Exercise 4.3 in the book Christopher Heil: A Basis Theory Primer. Springer, New York, 2011. In this exercise we are working in an infinite-dimensional space $X$. Basically the same notation as I mentioned above is introduced, $a_i$'s are called coefficient functionals and then it goes as follows:
(a) Show by example that it is possible for some particular functional $a_i$ to be continuous.
(b) Show that $a_i(x_j) = \delta_{ij}$ for $i,j\in I$.
(c) Let $J = \{i\in I : a_i \text{ is continuous}\}$. Show that $\sup \{j\in J; \lVert a_j \rVert<+\infty\}$
(d) Show that at most finitely many functionals $a_i$ can be continuous, i.e., $J$ is finite.
(e) Give an example of an infinite-dimensional normed linear space that has a Hamel basis $\{x_i; i\in I\}$ such that each of the associated coefficient functionals $a_i$ for $i\in I$ is continuous.
The part (c) can be shown easily using Banach-Steinhaus theorem (a.k.a. Uniform boundedness principle). But I guess that the author of the book has in mind a different proof for part (d) from what I sketched above, since (c) is an easy consequence of (d) -- so he would probably not be put the exercises in this order. (But maybe I was just trying to read to much between the lines.)
Question: I was not able to find a proof od (d) which uses (c). Do you have some idea how to do this?
NOTE: My question is not about the parts (a), (b), (e). I've included them just for the sake of the completeness, in order to include sufficient context for the question.
-
How about this: If you have a Hamel basis $\{x_i\}$, and replace each $x_i$ by a nonzero scalar multiple of itself, then the result is still a Hamel basis. The corresponding functionals $a_i$ are of course replaced by nonzero scalar multiples of themselves (the multiplier for $a_i$ is the reciprocal of the multiplier for $x_i$). The new functional is continuous iff the original was continuous. If $J$ is infinite, then you can carry out this "replacement" by scalar multiples in such a way that the $\sup$ in (c) is infinite. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9823581576347351, "perplexity": 95.74916535539796}, "config": {"markdown_headings": false, "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-2015-48/segments/1448398456975.30/warc/CC-MAIN-20151124205416-00347-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=2014_AIME_I_Problems/Problem_6&direction=next&oldid=93159 | # 2014 AIME I Problems/Problem 6
## Problem 6
The graphs and have y-intercepts of and , respectively, and each graph has two positive integer x-intercepts. Find .
## Solution 1
Begin by setting to 0, then set both equations to and , respectively. Notice that because the two parabolas have to have positive x-intercepts, .
We see that , so we now need to find a positive integer which has positive integer x-intercepts for both equations.
Notice that if is -2 times a square number, then you have found a value of for which the second equation has positive x-intercepts. We guess and check to obtain .
Following this, we check to make sure the first equation also has positive x-intercepts (which it does), so we can conclude the answer is .
## Solution 2
Let and for the first equation, resulting in . Substituting back in to the original equation, we get .
Now we set equal to zero, since there are two distinct positive integer roots. Rearranging, we get , which simplifies to . Applying difference of squares, we get .
Now, we know that and are both integers, so we can use the fact that , and set and (note that letting gets the same result). Therefore, .
Note that we did not use the second equation since we took advantage of the fact that AIME answers must be integers. However, one can enter into the second equation to verify the validity of the answer.
Note on the previous note: we still must use the second equation since we could also use , yielding This answer however does not check out with the second equation which is why it is invalid.
## Solution 3
Similar to the first two solutions, we deduce that and are of the form and , respectively, because the roots are integers and so is the -intercept of both equations. So the -intercepts should be integers also.
The first parabola gives And the second parabola gives
We know that and that . It is just a fitting coincidence that the average of and is the same as the average of and . That is .
To check, we have Those are the only two prime factors of and , respectively. So we don't need any new factorizations for those numbers.
Thus the common integer value for is .
## See also
2014 AIME I (Problems • Answer Key • Resources) Preceded byProblem 5 Followed byProblem 7 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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https://or.stackexchange.com/questions/4921/optimization-of-strongly-convex-functions-with-approximate-evaluations-of-gradie | # Optimization of strongly convex functions with approximate evaluations of gradient and Hessian
Suppose I want to find the minimum of a differentiable, strongly convex function $$f:\mathbb{R}^n\to\mathbb{R}$$ with constant $$\mu>0$$. That is, for all $$x,y\in\mathbb{R}^n$$, I have that: $$f(y) \geq f(x)+\nabla f(x)^{T}(y-x)+\frac{\mu}{2}\|y-x\|^{2}.$$ If I can efficiently evaluate $$f$$ and its gradient, then it is clear how to find the minimum of $$f$$ with first-order methods. But now assume that I only have access to approximate entries of $$\nabla f(x)$$, say up to an additive error $$\epsilon$$. What is the state of the art result on how small $$\epsilon$$ has to be compared to $$\mu$$ so that I can still ensure convergence of some optimization method to an approximate minimum? Thanks in advance! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9740105867385864, "perplexity": 103.660055405815}, "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-2021-04/segments/1610703519784.35/warc/CC-MAIN-20210119201033-20210119231033-00492.warc.gz"} |
http://www.meritnation.com/cbse/class10/board-papers/math/math/cbse-class-10-math-board-paper-2008-abroad-set-1-solutions/starttest/2OL9gzkyYqUDvlvGQNPsZQ!! | 011-40705070 or
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# Class X: Math, Board Paper 2008, Set-1
General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 30 questions divided into four sections A, B, C and D. Section A comprises of 10 questions of one mark each, Section B comprises of 5 questions of two marks each, Section C comprises of 10 questions of three marks each, and Section D comprises of 5 questions of six marks each.
(iii) All questions in section A are to be answered in one word, one sentence or as per the exact requirements of the question.
(iv) Use of calculators is not permitted.
Question 1
• Q1
Complete the missing entries in the following factor tree:
VIEW SOLUTION
• Q2
If (x + a) is a factor of 2x2 + 2ax + 5x + 10, then find a.
VIEW SOLUTION
• Q3
Show that x = −3 is a solution of x2 + 6x + 9 = 0.
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• Q4
The first term of an A.P. is p and its common difference is q. Find its 10th term.
VIEW SOLUTION
• Q5
If, then find the value of (sin A + cos A) sec A.
VIEW SOLUTION
• Q6
The lengths of the diagonals of a rhombus are 30 cm and 40 cm. Find the side of the rhombus.
VIEW SOLUTION
• Q7
In the figure, PQ || BC and AP: PB = 1: 2. Find
VIEW SOLUTION
• Q8
The surface area of a sphere is 616 cm2. Find its radius.
VIEW SOLUTION
• Q9
A die is thrown once. Find the probability of getting a number less than 3.
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• Q10
Find the class marks of classes 10 − 25 and 35 − 55.
VIEW SOLUTION
• Q11
Find all the zeroes of the polynomial x4 + x3 − 34x2 − 4x + 120, if two of its zeroes are 2 and −2.
VIEW SOLUTION
• Q12
A pair of dice is thrown once. Find the probability of getting the same number on each die.
VIEW SOLUTION
• Q13
If sec 4A = cosec (A − 20°), where 4A is an acute angle, then find the value of A.
OR
In a ΔABC, right-angled at C, if then find the value of sin A cos B + cos A sin B.
VIEW SOLUTION
• Q14
Find the value of k, if the points (k, 3), (6, −2), and (−3, 4) are collinear.
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• Q15
E is a point on the side AD produced of a ||gm ABCD and BE intersects CD at F. Show that ΔABE ΔCFB.
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• Q16
Use Euclid’s Division Lemma to show that the square of any positive integer is either of the form 3m or (3m + 1) for some integer m.
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• Q17
Represent the following pair of equations graphically and write the coordinates of points where the lines intersect y-axis:
x + 3y = 6
2x − 3y = 12
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• Q18
For what value of n are the nth terms of two A.P.’s 63, 65, 67 … and 3, 10, 17 … equal?
OR
If m times the mth term of an A.P. is equal to n times its nth term, then find the (m + n)th term of the A.P.
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• Q19
In an A.P., the first term is 8, nth term is 33, and sum to first n terms is 123. Find n and d, the common difference.
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• Q20
Prove that:
(1+ cot A + tan A) (sin A − cos A) = sin A tan A − cot A cos A
OR
Without using trigonometric tables, evaluate the following:
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• Q21
If P divides the join of A(−2, −2) and B(2, −4) such that , then find the coordinates of P.
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• Q22
The mid-points of the sides of a triangle are (3, 4), (4, 6), and (5, 7). Find the coordinates of the vertices of the triangle.
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• Q23
Draw a right triangle in which the sides containing the right angle are 5 cm and 4 cm. Construct a similar triangle whose sides are times the sides of the above triangle.
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• Q24
Prove that a parallelogram circumscribing a circle is a rhombus.
OR
In figure, AD BC. Prove that AB2 + CD2 = BD2 + AC2.
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• Q25
In the figure, ABC is a quadrant of a circle of radius 14 cm and a semi-circle is drawn with BC as diameter. Find the area of the shaded region.
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• Q26
A peacock is sitting on the top of a pillar, which is 9 m high. From a point 27 m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake, the peacock pounces on it. If their speeds are equal, then at what distance from the hole is the snake caught?
OR
The difference of two numbers is 4. If the difference of their reciprocals is, then find the two numbers.
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• Q27
The angle of elevation of an aeroplane from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. If the plane is flying at a constant height of m, then find the speed, in km/hour, of the plane.
VIEW SOLUTION
• Q28
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.
Using the above, prove the following:
In figure, AB || DE and BC || EF. Prove that AC || DF.
OR
Prove that the lengths of tangents drawn from an external point to a circle are equal.
Using the above, prove the following:
ABC is an isosceles triangle in which AB = AC, circumscribed about a circle, as shown in figure. Prove that the base is bisected by the point of contact.
VIEW SOLUTION
• Q29
If the radii of the circular ends of a conical bucket, which is 16 cm high, are 20 cm and 8 cm, then find the capacity and total surface area of the bucket.
VIEW SOLUTION
• Q30
Find mean, median, and mode of the following data:
Class Frequency 0 − 20 6 20 − 40 8 40 − 60 10 60 − 80 12 80 − 100 6 100 − 120 5 120 − 140 3
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Board Papers 2014, Board Paper Solutions 2014, Sample Papers for CBSE Board, CBSE Boards Previous Years Question Paper, Board Exam Solutions 2014, Board Exams Solutions Maths, Board Exams Solutions English, Board Exams Solutions Hindi, Board Exams Solutions Physics, Board Exams Solutions Chemistry, Board Exams Solutions Biology, Board Exams Solutions Economics, Board Exams Solutions Business Studies, Maths Board Papers Solutions, Science Board Paper Solutions, Economics Board Paper Solutions, English Board Papers Solutions, Physics Board Paper Solutions, Chemistry Board Paper Solutions, Hindi Board Paper Solutions, Political Science Board Paper Solutions, Answers of Previous Year Board Papers, Delhi Board Paper Solutions, All India Board Papers Solutions, Abroad/Foreign Board Paper Solutions, cbse class 12 board papers, Cbse board papers with solutions, CBSE solved Board Papers, ssc board papers. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8200130462646484, "perplexity": 1746.1935294813397}, "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-2017-17/segments/1492917122629.72/warc/CC-MAIN-20170423031202-00077-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/is-the-speed-of-light-wrong.792338/ | # Is the speed of light wrong?
Tags:
1. Jan 14, 2015
### Steven Wallis
Hi there. Just wondering if the speed of light is wrong for a 3 dimensional universe.
As each photon of light traverses space its velocity is 299 792 458 m/s. And since each photon has a wavelength and amplitude, then the actual distance that each photon travels, depending on its wavelength and amplitude, actually travels a longer distance than the straight line that the speed of light is based on.
Is this assumption correct or do I need to do more research into this.
And if what I have stated is correct then would the wavelength and amplitude of the light wave travel its path because it is encountering something that cannot be seen to give it its wave path. Like an airplane that has a wavelength path if no trimming is done to it as it traverses the gas of the atmosphere.
2. Jan 14, 2015
### davenn
Hi Steven and welcome
You need to do more research ..... Light is not photons travelling in the manner you describe
Others may give more details than I can
Dave
3. Jan 14, 2015
### phinds
Also, you should keep in mind that the speed of light is now a DEFINED value, and a defined value can't be wrong (although we COULD end up discovering that photons don't travel at the defined speed). The "speed of light" is actually short-hand for "the universal speed limit" and THAT is what is the defined value. There is no evidence that light has any mass and so it travels at the universal speed limit. If light were found to have mass (very unlikely) it would travel at less than the defined value for the universal speed limit.
4. Jan 14, 2015
### phinds
And by the way, just as a side comment:
When you come up against something that flies utterly in the face of established science, it is not a good idea to start off reaching different conclusions and stating them as correct (not that you did this, exactly) but rather to start off with the assumption that you have made a mistake somewhere and try to find out where it is. If you have NOT made a mistake you will find the flaw in the established science, but that is very unlikely to happen. If you start off thinking that you have overturned established science you are likely to just end up embarrassed.
5. Jan 14, 2015
### Staff: Mentor
The sinusoidally curving line that people use to depict a light wave is not the path that the light literally follows, that is, the light does not literally "snake" back and forth like that. In the classical picture, it represents (as a graph) the variation in the amplitude (strength) and direction of the electric field at different points along the wave. In the quantum picture, it represents something altogether more abstract.
6. Jan 15, 2015
### UncertaintyAjay
So what does it represent in the quantum picture of things?
7. Jan 15, 2015
### Drakkith
Staff Emeritus
I think that's beyond the scope of this thread. If you'd like to know, feel free to start a thread in the Quantum Physics forum.
8. Jan 15, 2015
### Drakkith
Staff Emeritus
That is not how photons work. I highly recommend forgetting about photons and learning the classical physics view of light first, which is that light is an electromagnetic wave. This wave has a wavelength and frequency and propagates outwards at c. The thing that is 'waving' in the wave is the electric and magnetic field vectors, which represent the direction and strength of the electric and magnetic forces that the wave exerts on charged particles. For example, an EM wave passing over an antenna will cause the charges in the metal to oscillate first in one direction and then the other at the same frequency as the EM wave.
9. Jan 15, 2015
### ZapperZ
Staff Emeritus
Reread what jtbell wrote. You seemed to have missed his reference to the electric field of light.
Furthermore, I think you need to start with the classical picture of light first, because you already have a wrong understanding of that, before jumping into a more complex quantum picture of light.
Zz.
10. Jan 15, 2015
### Staff: Mentor
Note also, this isn't how waves in general work. How fast a wave on the ocean is moving is measured in one dimension only as well. The amplitude is not factored into the measurement -- which is good, otherwise it would be difficult to calculate tsunami warning times!
11. Jan 16, 2015
### UncertaintyAjay
@zapper, I do actually have a good picture of the classical theory if light. I think you mistook me for the chap who started this post.
12. Jan 16, 2015
### ZapperZ
Staff Emeritus
Yup, someone pointed that out to me. I apologize.
Zz.
13. Jan 16, 2015
### UncertaintyAjay
No probs, honest misunderstanding. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8575895428657532, "perplexity": 525.0255731991236}, "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-2018-13/segments/1521257645604.18/warc/CC-MAIN-20180318091225-20180318111225-00654.warc.gz"} |
https://arxiv.org/abs/1505.08141 | hep-ex
(what is this?)
# Title: Measurement of differential $J/ψ$ production cross-sections and forward-backward ratio in p+Pb collisions with the ATLAS detector
Abstract: Measurements of differential cross-sections for $J/\psi$ production in p+Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV at the LHC with the ATLAS detector are presented. The data set used corresponds to an integrated luminosity of 28.1 nb$^{-1}$. The $J/\psi$ mesons are reconstructed in the dimuon decay channel over the transverse momentum range $8<p_{\mathrm{T}}<30$ GeV and over the center-of-mass rapidity range $-2.87<y^{*}<1.94$. Prompt $J/\psi$ are separated from $J/\psi$ resulting from $b$-hadron decays through an analysis of the distance between the $J/\psi$ decay vertex and the event primary vertex. The differential cross-section for production of nonprompt $J/\psi$ is compared to a FONLL calculation that does not include nuclear effects. Forward-backward production ratios are presented and compared to theoretical predictions. These results constrain the kinematic dependence of nuclear modifications of charmonium and $b$-quark production in p+Pb collisions.
Comments: 13 pages plus author list + cover pages (26 pages total), 8 figures, 8 tables, submitted to Phys. Rev. C. All figures including auxiliary figures are available at this http URL Subjects: High Energy Physics - Experiment (hep-ex); Nuclear Experiment (nucl-ex) Journal reference: Phys. Rev. C 92, 034904 (2015) DOI: 10.1103/PhysRevC.92.034904 Report number: CERN-PH-EP-2015-103 Cite as: arXiv:1505.08141 [hep-ex] (or arXiv:1505.08141v1 [hep-ex] for this version)
## Submission history
From: Atlas Publications [view email]
[v1] Fri, 29 May 2015 18:41:50 GMT (648kb,D) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9585196375846863, "perplexity": 4803.457586887651}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2017-30/segments/1500549423512.93/warc/CC-MAIN-20170720222017-20170721002017-00627.warc.gz"} |
http://math.stackexchange.com/questions/133793/for-a-1-form-h-why-does-int-gamma-varphih-int-varphi-circ-gammah | # For a $1$-form $h$, why does $\int_\Gamma \varphi^*h=\int_{\varphi\circ\Gamma}h$?
I'm trying to understand why for a differentiable arc $\Gamma:[a,b]\to\Omega$ and a $1$-form $h=fdx+gdy$, then $$\int_\Gamma\varphi^*h=\int_{\varphi\circ\Gamma}h?$$
For background, $\Omega$ is an open set in $\mathbb{C}$, and $\varphi:\Omega\to\mathbb{C}$ a smooth map. For a function $f$, I have the definition $\varphi^*f=f\circ\phi$, (when this makes sense for $f$ of course).
I also have the definitions $$\varphi^*\,dx=\frac{\partial u}{\partial x}\,dx+\frac{\partial u}{\partial y}\,dy, \qquad \varphi^*dy=\frac{\partial v}{\partial x}\,dx+\frac{\partial v}{\partial y}\,dy,$$ where $u$ is the $x$ component of $\varphi$ and $v$ is the $y$ component. For a $1$-form $h=f\,dx+g\,dy$, $$\varphi^*h=(\varphi^*f)\varphi^*\,dx+(\varphi^*g)\varphi^*\,dy.$$
I calculate \begin{align*} \int_\Gamma \varphi^*h &= \int_\Gamma(\varphi^*f)\varphi^*dx+\int_\Gamma (\varphi^*g)\varphi^*dy\\ &= \int_\Gamma(f\circ\varphi)\frac{\partial u}{\partial x}dx+ \int_\Gamma(f\circ\varphi)\frac{\partial u}{\partial y}dy+ \int_\Gamma(g\circ\varphi)\frac{\partial v}{\partial x}dx+ \int_\Gamma(f\circ\varphi)\frac{\partial v}{\partial y}dy\\ &= \int_\Gamma\left((f\circ\varphi)\frac{\partial u}{\partial x}+(g\circ\varphi)\frac{\partial v}{\partial x}\right)dx+\int_\Gamma\left((f\circ\varphi)\frac{\partial u}{\partial y}+(g\circ\varphi)\frac{\partial v}{\partial y}\right)dy \end{align*}
but I don't see if this fits into the form $\int_{\varphi\circ\Gamma}h=\int_{\varphi\circ\Gamma}fdx+\int_{\varphi\circ\Gamma}gdy$? Can it be made to fit? Thanks.
-
Hint: Think of the way you compute an integral of a form in terms of a parametrization to turn it into an ordinary definite integral. – Matt Apr 19 '12 at 19:03
Things become much clearer when you use different names for the variables: You have a curve $\Gamma:\ t\mapsto \bigl(x(t),y(t)\bigr)$ in the $(x,y)$-plane, a map $\phi:\ (x,y)\mapsto (u,v)$ and a $1$-form $\omega$ in the $(u,v)$-plane given as $$\omega=f(u,v)du + g(u,v)dv\ .$$ Then $$\phi^*f=f\circ\phi,\quad \phi^*g=g\circ\phi,\quad \phi^*du=u_xdx + u_y dy,\quad \phi^*dv=v_x dx+v_y dy\ ,$$ and $\phi^*\omega$ becomes $$\phi^*\omega=\bigl((f\circ\phi)u_x+(g\circ\phi)v_x\bigr)dx + \bigl((f\circ\phi)u_y+(g\circ\phi)v_y\bigr)dy=:p(x,y)dx + q(x,y)dy\ .$$ Therefore $$\int_\Gamma \phi^*\omega=\int_a^b\bigl(p(x(t),y(t))\dot x(t) + q(x(t),y(t))\dot y(t)\bigr)dt=\ldots$$ and writing it all out allows you to interpret the resulting integral as $\int_{\phi(\Gamma)} \omega$.
Regarding your first question, we can note that $\int_{\Gamma} \phi ^\ast h=\int_{a}^{b}\Gamma ^\ast \phi ^\ast h =\int_{a}^{b}h\circ\phi \circ \Gamma =\int_{a}^{b}(\phi\circ\Gamma)^\ast h=\int_{\phi \circ\Gamma}h$. Also note that $\phi^\ast dx=d\phi_{x}$, and likewise for $d\phi_y$, which implies that $\phi \ : (x,y) \rightarrow(u,v)$ i.e. is a change of variables. Hopefully this helps! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9986559748649597, "perplexity": 165.36128672202523}, "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-2015-40/segments/1443737911339.44/warc/CC-MAIN-20151001221831-00209-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/pulley-to-shaft-to-pulley-calculation.586600/ | # Pulley-to-Shaft-to-Pulley Calculation?
1. Mar 13, 2012
### playludesc
Hello all! I'm working with a particular pulley configuration and I realized after a few calculations that I don't have an accurate way to calculate one part of the set up.
Here's a quick MSPaint sketch showing what I need to solve for. If I know the outer diameter of both pulleys, and the RPM of the black pulley, how can I solve for the RPM at the outer diameter of the blue pulley? (If you care, this is regarding the jackshaft for my supercharger!)
Thanks very much for your help!
2. Mar 13, 2012
### playludesc
So I figured that this is probably a simple mechanical advantage equation, as the two pulleys simply work like levers on each other. Here's what I ended up with:
Blue pulley diameter = 1.835"
Black pulley diameter = 2.000"
Black pulley RPM = 22,023.5
1.835/2.000 = 0.9175
22,023.5*0.9175 = RPM for Blue pulley of 20,206.5
Can anyone confirm if I've done that correctly; or, more importantly, if mechanical advantage is the right way to calculate this?
3. Mar 13, 2012
### sophiecentaur
If they're on the same shaft (as in the picture) then they would both rotate at the same rate so the speeds of the two belts (?) will be proportional to the diameters. If they are coupled by a belt then the peripheral speeds will be the same.
4. Mar 13, 2012
### playludesc
Right, would the belts' speed be directly proportional, like in my calculation above? Or should I be using a different formula?
5. Mar 13, 2012
### sophiecentaur
Which bit are you asking is right? In one bit you seem to imply that the RPMs would be different. How could that be if they are on the same shaft?
The belt speed 'out' will be belt speed 'in' times the ratio of diameters - it's that simple.
6. Mar 13, 2012
### DennisN
Hi playludesc! I'm not sure I understand your sketch (what does variable and constant mean in the sketch?). If two wheels are connected via a normal axle, they will have the same RPM (revolutions/minute), regardless of the diameter of the wheels (unless you mean some entirely different setup). You may be confusing rotational speed (revolutions/minute) with tangential speed (e.g. centimeters/second).
The circumferences c of the wheels are (d are diameters):
cblack = pi*dblack
cblue = pi*dblue
so the tangential velocities v at the circumference of the wheels will be
vblack = cblack/t = pi*dblack * RPM
vblue = cblue/t = pi*dblue * RPM
If you measure the diameters in cm, the velocities above will be cm/minute (and t is time in minutes).
If you measure the diameters in inches, the velocities above will be inches/minute.
Last edited: Mar 13, 2012
7. Mar 13, 2012
### playludesc
That's what I thought. Sorry if my phrasing was confusing.
I'm just asking if my method of calculating the "out" speed for the blue pulley is correct in my second post. Your posts seem to agree with my method, so we're good!
8. Mar 13, 2012
### playludesc
Thanks very much for taking the time to answer so thoroughly. I think sophie confirmed that I'm on track for my application. Even so, I'll wrap my English degree head around your post and get back to you in a bit!
9. Mar 13, 2012
### jim hardy
http://en.wikipedia.org/wiki/Jackshaft
Aha that solves the mystery. Sophie and dennis both picked up on it.
Is your blue pulley a variable diameter sheave?
What are you supercharging, just out of curiosity?
10. Mar 13, 2012
### playludesc
Nah, blue pulley will be a timing belt pulley, I'm just trying to figure out exactly what size to make it, which in the calculation/selection phase makes it the variable.
I've got a ported M62 going on the H22a4 in my fifth generation Prelude.
11. Mar 13, 2012
### DennisN
Hi again! I might as well also simplify my two velocity equations further. If the pullies have the same RPM (as I suppose), then the velocity relations can be divided to yield the following relation;
vblue/vblack = dblue/dblack
which means e.g.
dblue = (dblack*vblue)/vblack
(d=diameters, v=tangential speeds)
I don't know if it helps you, I'm not sure about the other stuff in your project . (Btw, the t I used before was the period, i.e. the time for one revolution.)
Last edited: Mar 13, 2012
Similar Discussions: Pulley-to-Shaft-to-Pulley Calculation? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8351935148239136, "perplexity": 1975.1705633688869}, "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-2017-51/segments/1512948588294.67/warc/CC-MAIN-20171216162441-20171216184441-00292.warc.gz"} |
https://theinductivistturkey.wordpress.com/2016/01/21/standard-and-non-standard-existential-predictions/ | # Standard and non-standard existential predictions
Existential predictions are very rare in science. For this reason, when scientists can predict the existence of a new entity—a new planet, or a new particle, for example—we feel that something very exciting is going on! But what is going on is not only exciting—it is also very interesting from a philosophical point of view. These predictions raise various philosophically interesting puzzles, which relate to the epistemological, metaphysical, and methodological roles that mathematics can play in scientific representation. In this post, I want to present two main kinds of existential prediction in science and to sketch some philosophical problems related to them. However, I will not address these problems in this context. The interested reader may refer to Ginammi (2016) for a more detailed analysis and for a solution to these problems.1
Typically, these predictions are made possible by an accurate mathematical representation of the physical system (or phenomenon) at issue. This raises questions like: How can we predict the existence of a new, concrete entity on the basis of mathematical, abstract considerations? How can mathematics, which can be defined as the study of possible structures, say something about elements in real structures? How can mathematics, which is developed by mathematicians for completely different purposes, turn out to be so effective in representing physical reality and fostering new discoveries?
### Standard predictions and the DN model
A classical example of existential prediction is the prediction of the planet Neptune. During the first half of the XIXth century, astronomers noticed several anomalies in the astronomical tables for the planet Uranus (Neptune’s neighbour). Briefly speaking, Uranus was not exactly where astronomers expected to find it. The deviation from the expected position was minimal, but enough to raise concerns. There were two possible explanations for this deviation: the first was that the theory (from which the predictions were derived) is wrong; the second was that not the theory, but the initial conditions were wrong. Now, the “theory”, in this case, is Newton’s gravitation theory, and nobody at that time could seriously suppose that Newton’s gravitation theory was wrong! Therefore, astronomers opted for the second possibility, that they missed something in the initial conditions of the Solar system. What could they have missed? Well, an analysis of the Uranus tables suggested that probably there was an unknown body that was altering Uranus’ orbit. Thus, they started working on this, and they eventually made a prediction: that there is an eighth planet in the Solar system—Neptune, indeed. In order to account for the previous anomalies, this planet had to be such-and-such, with such-and-such mass, at a such-and such position, with a such-and-such speed, and so on… Finally, in 1846, the new planet was actually observed, and its prediction was thus confirmed.
Now, which role did mathematics play in this prediction? An important role, of course, but not so decisive. Mathematics seems to have been used in this case only to predict the characteristics of the new planet: based on accurate calculations, astronomers established that, in order to explain away the anomalies, the planet should have had the predicted characteristics. However, mathematics did not really play any role in the prediction; the prediction was suggested by the anomalies. In other words, mathematics did not suggested the existence of anything, it only permitted to precise the conditions under which the new entity could explain away the anomalies.
This kind of existential prediction can be easily accounted by the so-called deductive-nomological model (DN hereafter) proposed by Hempel (1965).2 According to this model, to explain a scientific fact is to derive this fact from a set of laws of nature plus some initial conditions. Analogously, we can also predict a scientific fact by applying the laws to the proper circumstances (described by the initial conditions). However, when the predictions are not confirmed by experience and we end up with anomalies (i.e., when a physical system—previously thought to obey a certain set of laws—appear to behave as if the laws do not apply to it anymore), scientists will try to explain the reasons of this anomalous behaviour. In other words, they will try to “explain away” the anomalies. According to the DN model, this amounts to show how this anomalous behaviour (assuming that the measurements revealing the anomalies are accurate enough) can be derived from a set of laws and initial conditions. They have to options: either they change the laws, or they change the initial conditions. The first option is highly expensive, thus they usually prefer to opt—at least as a first step—for the second one. This option involves a revision of the initial conditions, and this revision may consists in a supplementation of them, thus predicting a new entity. The criteria guiding this implementation are defined by the DN model: the new initial conditions must be such that, by means of them, we must be able to derive (and hence to explain) the behaviour we previously considered anomalous from our set of laws.
We can call “standard” this kind of predictions.3 However, not all existential predictions are like planet Neptune’s one. Let’s give a look to a completely different kind of existential predictions—a kind of prediction that cannot be accounted by means of DN model.
### Non-standard predictions: the omega minus particle case
An example of non-standard prediction is the prediction of the so-called “omega minus particle”, independently predicted by Gell-Mann and Ne’eman in 1962 and then discovered in 1963. We can get a rough—but still precise—comprehension of this prediction by reading the following passage from Ne’eman and Kirsh (1996):4
In 1961 four baryons of spin $\frac{3}{2}$ were known. These were the four resonances $\varDelta^-$, $\varDelta^0$, $\varDelta^+$, $\varDelta^{++}$ which had been discovered by Fermi in 1952. It was not clear that they could not be fitted into an octet, and the eightfold way predicted that they were part of a decuplet or of a family of 27 particles. A decuplet would form a triangle in the $S - I3$ [strangeness-isospin] plane, while the 27 particles would be arranged in a large hexagon. (According to the formalism of SU(3), supermultiplets of 1, 8, 10 and 27 particles were allowed.) In the same year (1961) the three resonances $\varSigma(1385)$ were discovered, with strangeness $-1$ and probable spin $\frac{3}{2}$, which could fit well either into the decuplet or the 27-member family.
At a conference of particle physics held at CERN, Geneva, in 1962, two new resonances were reported, with strangeness $-2$, and the electric charge $-1$ and $0$ (today known as the $\varXi(1530)$). They fitted well into the third course of both schemes (and could thus be predicted to have spin $\frac{3}{2}$). On the other hand, Gerson and Shoulamit Goldhaber reported a ‘failure’: in collisions of $\mathrm{K}^+$ or $\mathrm{K}^0$ with protons and neutrons, one did not find resonances. Such resonances would indeed be expected if the family had 27 members.
The creators of the eightfold way, who attended the conference, felt that this failure clearly pointed out that the solution lay in the decuplet. They saw the pyramid [see fig. above] being completed before their very eyes. Only the apex was missing, and with the aid of the model they had conceived, it was possible to describe exactly what the properties of the missing particle should be! Before the conclusion of the conference Gell-Mann went up to the blackboard and spelled out the anticipated characteristics of the missing particle, which he called ‘omega minus’ (because of its negative charge and because omega is the last letter of the Greek alphabet). He also advised the experimentalists to look for that particle in their accelerators. Yuval Ne’eman had spoken in a similar vein to the Goldhabers the previous evening and had presented them in a written form with an explanation of the theory and the prediction. (pp. 202-203)
In this case, it does not seem that the prediction has been made in order to explain away an anomaly. What should the anomaly be in this case? The fact that there is an empty place in the decuplet scheme cannot be considered an anomaly, because this empty place does not undermine the natural laws at issue. Consider the following hypothetical case. Imagine the prediction of the existence of a tenth spin-$\frac{3}{2}$ baryon turned out to be wrong. This failure could take two different forms:
1. we did not find any particle—at all;
2. we did find a tenth particle, but this tenth particle had completely different characteristics from the ones predicted by the decuplet scheme.
In the second case, we would have a real anomaly, since the measurements cannot be accounted for by our theory. In case (A), instead, the anomaly seems to consist simply in the fact that the symmetry scheme could turn out to have an empty place. But if this were the case, would it be really an anomaly? My answer is: No, it is not! Supposing that experimentalist physicists had not found any new particle corresponding to the characteristics pointed out, should we drop the SU(3) symmetry scheme? This seems unreasonable, for it can still be regarded as a valuable tool for representing the class of spin-$\frac{3}{2}$ baryons.
Hence, the fact that the formalism seems to commit us to the existence of an entity that does not exist cannot be regarded as wrong. There are many cases in which a formalism seems to commit us to entities that we do not regard as actually existing, but still we continue to use those formalisms without worrying about these “fictional” entities. We can consider the case of the applicability of analytic functions to thermodynamic: we know we can treat the critical temperature of a ferromagnet as an analytic function of the number of its dimensions. But, since we cannot calculate the problem for the 3-dimensional magnet, we calculate it for a 4-dimensional magnet, then we expand the function as a power series in a complex plane around the number 4, and finally we plug in the value 3. Now, the problem is that in this procedure we may end up, at a certain point, with dimensions like 3.5, or even $2+3i$! The analytic function is used here just like a formal trick—and it perfectly works! Now, the point is: Should we accept such weird magnets’ dimensions as physically real just because they appear in the formalism? Of course not! Should we abandon such a calculational tool just because it seems to commit us to weird dimensions? Not even! We just accept it as a weird consequence of the mathematical trick we are exploiting.
Cases like these—and the hypothetical failure (A) for the omega minus prediction falls within this group — point out that there is an important distinction to be made here about the representative role of mathematics in physics. On a first approximation, we can say that a mathematical structure can play a representative role without being fully representative; or, in a slightly different terminology, we can say that a mathematical structure playing a representative role can be either perfectly fitting’ or redundant’ (i.e., not perfectly fitting’). In the first case, every element in the mathematical structure plays a representative role; in the second, this is not the case. Importantly, the fact that a mathematical structure is redundant’ does not necessarily undermine its representative effectiveness.
I must say that not everybody would agree with this analysis. Bangu (2008), for example, thinks that in this case there is an anomaly to be explained away, and that the anomaly is precisely the empty place in the decuplet scheme. However, even if Bangu and I do not agree on this point, we both agree on the fact that this prediction cannot be accounted in the same way as other “standard” existential prediction. The reason is that, even if you think that in this case there is an anomaly to be explained away and that the new particle has been predicted in order to explain away this anomaly, still the way in which this prediction has been made is very peculiar. Indeed, look at how Gell-Mann and Ne’eman predicted the characteristics of this new particle: they did not consider the interaction of the new entity with other particles in the scheme. They simply looked at the scheme and they extracted the relevant information out of it! If in the case of the planet Neptune the characteristics of the new entity are derived by considering the interactions of the alleged new entity, in this case the procedure is completely different!
What is even more interesting in this case, is the fact that in this prediction mathematics seems to play a very peculiar role. Mathematics is used here to represent a certain class of particles, but this representation turns out to have a wonderful heuristic potential! Where does this heuristic potential come from? What is really surprising is that it seems that this heuristic potential was already enclosed in the representative effectiveness of the mathematical structure employed. Indeed, the prediction of this new physical entity seems to be motivated only by the mathematics employed. Just to be clear: this does not amount to saying that no empirical fact played a role in shaping the prediction. What I am stressing here, is that the justification for the prediction seems to be purely mathematical—namely, purely based on the mathematical formalism employed.5
This peculiarity of mathematics seems not to be limited to this case or to existential predictions only. In his (1998) famous book,6 Mark Steiner argues that the role of mathematics in contemporary physics is really unique. According to him, very often contemporary physicists draw important consequences about the physical world by relying on purely formal mathematical considerations, or “analogies”, which seem not to be in any sense rooted in the content of the mathematical representations. In this sense, the applicability of mathematics turns out to be “magic” or—as Wigner (1960) would have put it—“miraculous”. Steiner himself justifies the appropriateness of the word “magic” in this context:
Expecting the forms of our notation to mirror those of (even) the atomic world is like expecting the rules of chess to reflect those of the solar system. I shall argue, though, that some of the greatest discoveries of our century were made by studying the symmetries of notation. Expecting this to be any use is like expecting magic to work. (Steiner 1998, p. 72)
### Conclusion
The philosophical problem I have sketched in this post can be summed up as follows: Where does this heuristic effectiveness of mathematics come from? How can a mathematical structure disclose such a heuristic potential? Under which conditions a mathematical structure can reveal its heuristic effectiveness? And finally: Since not all mathematical structures seem to have such a heuristic effectiveness, how can we distinguish between heuristically fruitful mathematical representations and heuristically fruitless ones?
In my article Avoiding Reification I have analysed the prediction of the omega minus particle, I have addressed all these questions, and I have suggested an answer to them. The interested reader can give a look to this article, as well as to all the articles and books quoted in this post.
[1] Ginammi, Michele (2016), “Avoiding reification: Heuristic effectiveness of mathematics and the prediction of the omega minus particle”, Studies in History and Philosophy of Modern Physics, vol. 53, February, pp. 20-27.
[2] Hempel, Carl G. (1965), Aspects of Scientific Explanations, Free Press, New York.
[3] Another example of this kind of predictions is Pauli’s prediction of the neutrino. Also in this case we have an anomaly; the new entity is postulated just in order to explain away the anomaly; and mathematics is used in order to derive the appropriate characteristics of this new entity (appropriate to deduce—together with the proper laws of nature—the behaviour that was previously puzzling).
[4] Ne’man, Yuval. and Kirsh, Yoram (1996), The Particle Hunters, Cambridge University Press, Cambridge.
[5] Other examples of this kind of predictions are Dirac’s prediction of the so-called positron, or Mendeleev’s prediction of new chemical elements on the base of the periodic table. A more recent example is the prediction of the Higgs boson—the so-called “God’s particle” (I must admit, however, that I do not know much about this particular case, therefore I could be wrong on this point). All these cases share the fact that the prediction seems to be justified by purely mathematical considerations.
[6] Steiner, Mark (1998), The Applicability of Mathematics as a Philosophical Problem, Harvard University Press, Cambridge, Mass. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 19, "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": 0.9016088247299194, "perplexity": 545.6497438879717}, "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-2019-43/segments/1570987824701.89/warc/CC-MAIN-20191022205851-20191022233351-00306.warc.gz"} |
https://www.physicsforums.com/threads/magnetic-fields-induced-current.89763/ | # Homework Help: Magnetic fields & induced current.
1. Sep 19, 2005
### bayan
would increasing the speed at which magnet passes through a coil Increase,Reduce or does not change the ammount of electromagnetic induction?
cheers
personally I would go with increase but wanna see if im right
2. Sep 19, 2005
### Staff: Mentor
3. Sep 19, 2005
### bayan
I though the question was like a DC generator.
I know what it mean and the only effect it would have is the emf value, not the current.
Thanx for askin what I base my answer on, it made me re-read the question on the past exams and now I understand. Maybe I shouldn't do homework at 2:20 :rofl:
4. Sep 20, 2005
### bayan
the answer is Increase but why? if it moves faster it will have greater emf which is Voltage how does it increase the ammount of ElectroMagnetic Induction? isn't ammount of ElectroMagnetic induction Current?
thanx
5. Sep 20, 2005
### cliowa
Well, what does the emf value mean? You already mentioned it's voltage.
Now suppose the wire/the coil has a constant resistance (which is a pretty good approximation if you're not running gigantic currents through it): What will happen? You do know Ohm's Law, right?
Here's the answer in white, in case you won't find it out:
You should be easily able to see (using V=R*I) that a higher voltage also leads to a higher current!
Regards...Cliowa
6. Sep 20, 2005
Thanks mate. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8606358766555786, "perplexity": 1935.4735015801905}, "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-2018-43/segments/1539583513548.72/warc/CC-MAIN-20181021010654-20181021032154-00445.warc.gz"} |
https://www.physicsforums.com/threads/basic-normal-force-problems.190629/ | # Basic normal force problems
1. Oct 11, 2007
### mattystew
starting the problem and then direction from there requested
A 38 kg crate rests on a horizontal floor, and a 63 kg person is standing on the crate.
(a) Determine the magnitude of the normal force that the floor exerts on the crate.
(b) Determine the magnitude of the normal force that the crate exerts on the person.
A woman stands on a scale in a moving elevator. Her mass is 58.0 kg, and the combined mass of the elevator and scale is an additional 815 kg. Starting from rest, the elevator accelerates upward. During the acceleration, the hoisting cable applies a force of 9430 N. What does the scale read during the acceleration?
A 19.5 kg sled is being pulled across a horizontal surface at a constant velocity. The pulling force has a magnitude of 77.0 N and is directed at an angle of 30.0° above the horizontal. Determine the coefficient of kinetic friction.
The drawing shows a large cube (mass = 46 kg) being accelerated across a horizontal frictionless surface by a horizontal force P. A small cube (mass = 3.6 kg) is in contact with the front surface of the large cube and will slide downward unless P is sufficiently large. The coefficient of static friction between the cubes is 0.71. What is the smallest magnitude that P can have in order to keep the small cube from sliding downward?
drawing: http://www.webassign.net/CJ/04_47.gif
Last edited: Oct 11, 2007
2. Oct 11, 2007
### learningphysics
can you show some work? what did you try? where are you getting stuck...
3. Oct 11, 2007
### mattystew
everywhere, our teacher didn't teach us this yet so I have no idea where to even start.
4. Oct 11, 2007
### learningphysics
Do you know what the normal force is? For the first problem... part a) take the system of the crate and the person together... what are the forces acting on this system?
5. Oct 11, 2007
### mattystew
gravity, and the floor pushing back on both
6. Oct 11, 2007
### learningphysics
Yes, so what do these forces add to?
So taking the system of the crate and the person together... what is the gravitational force... what is the normal force?
7. Oct 11, 2007
### mattystew
gravity would be 9.8 and normal would be 91
8. Oct 11, 2007
### mattystew
oh wait, 9.8x91=891.8
9. Oct 11, 2007
### learningphysics
63+38 = 101 not 91...
10. Oct 11, 2007
### mattystew
oh whoops. 989.8 then
it's been a long day :P
11. Oct 11, 2007
### learningphysics
lol. no prob. But I think you know more than you're letting on. ;) you solved part a) of the first problem with no troubles...
12. Oct 11, 2007
### mattystew
for part b, I'm not sure if I should use the mass of the crate, or use the force from part a, or add them together?
13. Oct 11, 2007
### learningphysics
For part a), the trick was to analyze the crate+person together as one system.
In part b), you want to analyze the person alone... what are the forces acting on the person?
14. Oct 14, 2007
### mattystew
I figured out the first two problems, but I'm having trouble with the friction problem.
15. Oct 14, 2007
### pooface
Coefficient of Kinetic friction = Kinetic friction force / Normal force.
Kinetic friction force is the force which opposes the force pulling or pushing an object on a surface.
if Kinetic friction force < acting force = object will be sliding/moving.
if Kinetic friction force > acting force = object will not be sliding/moving.
Change masses to their weights. Then see which formulas will give you what you need.
More than welcome to comeback if you are stuck or have questions. Show us what work you have done on the problem and we will see if we can guide you.
16. Mar 3, 2010
### toakley7
For the part B, the forces acting on the person are the weight of the person and the normal force of the crate. Would you set them equal to one another?
Similar Discussions: Basic normal force problems | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8693934679031372, "perplexity": 1038.724433461007}, "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-2017-13/segments/1490218191986.44/warc/CC-MAIN-20170322212951-00119-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.arxiv-vanity.com/papers/1801.08887/ | arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org.
# Neutrino electromagnetic properties: a window to new physics - II
Department of Theoretical Physics, Moscow State University
119992 Moscow, Russia
Joint Institute for Nuclear Research
Dubna 141980, Moscow Region, Russia
E-mail:
###### Abstract:
There is merely a short note on the selected issues of neutrino electromagnetic properties with focus on effects of new physics. The meaning of “new physics” is twofold: 1) a massive neutrino have nonzero electromagnetic properties that can be considered as manifestation of new physics beyond the Standard Model, and 2) in studies of neutrinos electromagnetic interactions new effects are predicted that can lead to new phenomena accessible for observations.
## 1 Electromagnetic properties of neutrino
1. After the Nobel Prize of 2015 in physics was awarded to Arthur McDonald and Takaaki Kajita for the discovery of neutrino oscillations, there can be no doubt that neutrinos are massive particles. It has been know since quite many years [1] that massive neutrinos should have nonzero magnetic moments. Although up to now there are no indications in favour of nonzero neutrino electromagnetic properties, neither from terrestrial experiments nor from astrophysical observations, the electromagnetic properties is one of popular issues related to neutrinos and this problem has been discussed many times in recent literature (see, for instance, [2]-[9] ). A complete review on neutrino electromagnetic properties and neutrino electormagnetic interactions is given in [10]. In [11, 12] an addendum to previous reviews are provided and the most recent new aspects and prospects related to neutrino electromagnetic interactions that have appeared after publication of the review paper [10] are discussed.
It has been often claimed (see, for instance, [7]) that neutrino electromagnetic properties open a window to new physics. In this short note we would like to justify this statement and to focus a discussion on to what extend and what kind of new physics neutrino electromagnetic properties do communicate with.
## 2 Neutrino magnetic moment
Consider the magnetic moment as the most well theoretically appreciated and experimentally studied (constrained) electromagnetic characteristic of neutrinos. Within the initial formulation of the Standard Model neutrinos are massless particles with zero magnetic moment. Thus, the would be nonzero neutrino magnetic moment regardless of its value should indicate the existence of new physics beyond the Standard Model. Indeed, as it has been shown in [1] a minimal extension of the Standard Model with right-handed neutrinos yields for the diagonal magnetic moment of a Dirac neutrino to be proportional to the neutrino mass ,
μDii=3eGFmi8√2π2≈3.2×10−19(mi1 eV)μB, (1)
where is the Bohr magneton. For Majorana neutrinos the diagonal magnetic moments are zero in the neutrino mass basis and only transition (off-diagonal) magnetic moments () can be nonzero in this case.
The value of neutrino magnetic moment (1) is several orders of magnitude smaller than the present experimental limits if to account for the existed constraints on neutrino masses. Note that in general transition magnetic moments are even smaller due to the GIM cancelation mechanism.
The best laboratory upper limit on neutrino magnetic moment has been obtained by the GEMMA collaboration that investigates the reactor antineutrino-electron scattering at the Kalinin Nuclear Power Plant (Russia) [13]. Within the presently reached electron recoil energy threshold of keV the neutrino magnetic moment is bounded from above by the value
μν<2.9×10−11μB (90% C.L.). (2)
This limit, obtained from unobservant distortions in the recoil electron energy spectra, is valid for both Dirac and Majorana neutrinos and for both diagonal and transition moments. The most recent stringent constraint on the electron effective magnetic moment
μνe≤2.8×10−11μB (3)
has been reported by the Borexino Collaboration [14].
A strict astrophysical bound on the neutrino magnetic moment is provided by the observed properties of globular cluster stars and amounts to [15] (see also [16, 17])
(∑i,j∣∣μij∣∣2)1/2≤(2.2−2.6)×10−11μB. (4)
This stringent astrophysical constraint on neutrino magnetic moments is applicable to both Dirac and Majorana neutrinos.
There is a huge gap of many orders of magnitude between the present experimental limits on neutrino magnetic moments and the prediction of a minimal extension of the Standard Model. Therefore, if any direct experimental confirmation of nonzero neutrino magnetic moment were obtained in a reasonable future, it would open a window to new physics beyond a minimal extension of the Standard Model.
Much larger values for a neutrino magnetic moments are predicted in different other extensions of the Standard Model. However, there is a general problem for a theoretical model of how to get large magnetic moment for a neutrino and simultaneously to avoid an unacceptable large contribution to the neutrino mass (see the corresponding discussion in [10] and references therein). If a contribution to the neutrino magnetic moment of an order is generated by physics beyond a minimal extension of the Standard Model at an energy scale characterized by , then the corresponding contribution to the neutrino mass is . Therefore, a particular fine tuning is needed to get large value for a neutrino magnetic moment while keeping the neutrino mass within experimental bounds. Different possibilities to have large magnetic moment for a neutrino were considered in the literature (see in [10]).
A general and termed model-independent upper bound on the Dirac neutrino magnetic moment, that can be generated by an effective theory beyond a minimal extension of the Standard Model, has been derived in [18]: . Note that the corresponding limit for transition moments of Majorana neutrinos is much weaker [19]. Thus, the value of a neutrino magnetic moment once observed experimentally at the level not less than would provide information on the nature of neutrinos. This can be also considered as a view on the realm of new physics.
## 3 Neutrino electric moment
From the most general form of the neutrino electromagnetic vertex function (see for detailed discussion [10]) there are three other sets (in addition to the magnetic moments ) of electromagnetic characteristics that determine a neutrino coupling with real photons (). They are namely the dipole electric moments , anapole moments and millicharges . In the theoretical framework with violation a neutrino can have nonzero electric moments . In the laboratory neutrino scattering experiments for searching the neutrino magnetic moment (like, for instance, the mentioned above GEMMA experiment) the electric moment contributions interfere with those due to magnetic moments. Thus, these kind of experiments also provide constraints on . The astrophysical bounds (4) are also applicable for constraining [15]- [17].
## 4 Neutrino electric millicharge
There are extensions of the Standard Model that allow for nonzero neutrino electric millicharges. This option can be provided by not excluded experimentally possibilities for hypercharhge dequantization or another new physics related with an additional symmetry peculiar for extended theoretical frameworks. Neutrino millicharges are strongly constrained on the level ( is the value of an electron charge) from neutrality of the hydrogen atom.
A nonzero neutrino millicharge would contribute to the neutrino electron scattering in the terrestrial experiments. Therefore, it is possible to get bounds on in the reactor antineutrino experiments GEMMA. The most stringent constraint using the GEMMA data is [20] (see also [21]).
A neutrino millicharge might have specific phenomenological consequences in astrophysics because of new electromagnetic processes are opened due to a nonzero charge. Following this line, the most stringent astrophysical constraint on neutrino millicharges was obtained in [22]. This bound follows from the impact of the neutrino star turning mechanism () [22] that can be charged as a new physics phenomenon end up with a pulsar rotation frequency shift engendered by the motion of escaping from the star neutrinos on curved trajectories due to millicharge interaction with a constant magnetic field.
Even if a neutrino millicharge is vanishing, the electric form factor can still contain nontrivial information about neutrino electromagnetic properties. The corresponding electromagnetic characteristics is determined by the derivative of over at and is termed neutrino charge radius, . A neutrino charge radius (that is indeed the charges radius squared) contributes to the neutrino scattering cross section on electrons and thus can be constrained by the corresponding laboratory experiments [23]. In all (see, for instance, [10]) but one previous studies it was claimed that the effect of the neutrino charge radius can be included just as a shift of the vector coupling constant in the weak contribution to the cross section. However, as it has been recently illustrated, in [24] within the direct calculations of the cross section accounting for all possible neutrino electromagnetic characteristics and neutrino mixing, this is not the fact. The neutrino charge radius dependence of the cross section indeed is more complicated and there is, in particular, the dependence on the interference terms of the type that can’t be obtained just only by the corresponding shift of the constant .
## 6 Neutrino spin precession in magnetic field
One of an important phenomenon among several processes of neutrino electromagnetic interacts is neutrino spin and spin-flavour precession in magnetic fields. The origin of these effects is the neutrino magnetic moment interaction with a transversal magnetic field determined by . The neutrino spin precession in a transverse magnetic field can result in the neutrino helicity flip that can have important phenomenological consequences because an active neutrino can be converted to a sterile one in invironments with a magnetic field. The precession in the magnetic field of the Sun was first considered in [25], a similar effect in magnetic fields of supernovae and neutron stars came into sight for the first time in [1].
## 7 Neutrino spin precession in transversal matter currents
There is a phenomenon of new physics related to the neutrino spin precession in magnetic fields. For many years, until 2004, it was believed that a neutrino helicity precession and the corresponding spin oscillations can be induced by the neutrino magnetic interactions with the transversal magnetic field. A new and very interesting possibility for neutrino spin (and spin-flavour) oscillations engendered by the neutrino interaction with matter background was proposed and investigated in [26]. It was shown that neutrino spin oscillations can be induced not only by the neutrino interaction with a magnetic field but also by neutrino interactions with matter in the case when there is a transversal matter current (or a transversal matter matter polarization). The is no need for neutrino magnetic moment interaction in this case. The origin of the oscillations in the transversal matter currents is the neutrino weak interactions with moving matter and the corresponding mixing between neutrino states and is determined by . This new effect has been explicitly highlighted in [26, 27], recently the existence of this effect was confirmed in [28]. For historical notes reviewing studies and the detailed derivation of the discussed effect see [11, 12] and [29].
## 8 Conclusions and future prospects
The foreseen progress in constraining neutrino electromagnetic characteristics is related, first of all, with the expected new results from the GEMMA experiment measurements of the reactor antineutrino cross section on electrons at Kalinin Power Plant. The new set of data is expected to arrive next year. The electron energy threshold will be as low as ( or even lower, ). This will provide possibility to test the neutrino magnetic moment on the level of and also to test the millicharge on the level of [20]. For the next future, presently it seems unclear whether further progress in constraining the neutrino electromagnetic characteristics would be achievable with this type of the reactor antineutrino experiment. In this concern, a rather promising claim was made in [30, 31]. It was shown that even much smaller values of the Majorana neutrino transition moments would probably be tested in future high-precision experiments with the astrophysical neutrinos. In particular, observations of supernova fluxes in the JUNO experiment (see [32]- [34]) may reveal the effect of collective spin-flavour oscillations due to the Majorana neutrino transition moment .
To conclude, the existing current constraints on the flavour neutrino charge radius from the scattering experiments differ only by 1 to 2 orders of magnitude from the values calculated within the minimally extended Standard Model with right-handed neutrinos [23]. This indicates that the minimally extended Standard Model neutrino charge radii could be experimentally tested in the near future. Note that there is a need to re-estimate experimental constraints on from the scattering experiments following new derivation of the cross section [24] that properly accounts for the interference of the weak and charge radius electromagnetic interactions and also for the neutrino mixing.
## 9 Acknowledgements
This work was supported by the Russian Foundation for Basic Research under grants No. 16-02-01023 A and No. 17-52-53133 GFEN_a.
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• [34] J.S. Lu, Y.-F. Li and S. Zhou, Getting the most from the detection of Galactic supernova neutrinos in future large liquid-scintillator detectors, Phys. Rev. D 94 (2016) 023006. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9851676225662231, "perplexity": 1399.1022393145279}, "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-2020-29/segments/1593655901509.58/warc/CC-MAIN-20200709193741-20200709223741-00199.warc.gz"} |
https://brilliant.org/practice/coulombs-law-2/ | Electricity and Magnetism
The experiments in the last few explorations show that matter can become electrically charged. That's because matter comprises "positive" protons and "negative" electrons. After losing or gaining electrons or protons, neutral objects gain a net charge, which subjects them to electric forces, but we haven't explored how strong these electric forces are.
We're in a similar position to the "natural philosophers" who pondered the mysteries of electricity in the $1700$s. They knew how to separate charge and move it around but didn't yet have a model that predicts the forces their charges would feel.
This changed in 1782 when Charles-Augustin de Coulomb carefully measured the forces between charges and found they could be described with a simple rule, known today as Coulomb's law.
In this quiz, we'll follow in his footsteps... with a slight update to his equipment.
Coulomb's Law
In our experiments with tape strips, we saw that the electric force is spooky — it acts between objects that aren't touching. Coulomb realized that he could use a common piece of laboratory equipment to measure the strength of the force: a sensitive balance.
We can re-create his experiment with a modern digital scale. First, we'll place a positively charged metal ball on an insulating stand and place that on the scale.
Next, we'll take a negatively charged ball and clamp it to a support, directly above the first ball. What "weight" will the scale show?
Coulomb's Law
The scale gives us an indirect way to measure the amount of electric force, $F_e,$ produced by the electric charge on the two balls. In his experiment, Coulomb hoped to discover a rule that would predict $F_e,$ starting from the amount of charge on each ball.
To measure the electric force directly, we should "zero" the scale when the balls are uncharged — this way, the scale reading will only report the contribution from the electric force.
We can place some positive charge on the ball that's on the scale, and adjust the (positive) electric charge $Q_1$ on the ball above the scale to see what effect it has on the force.
Try playing with the charge in this simulated experiment. When the charge on one of the balls doubles, what happens to the strength of the electric force?
Coulomb's Law
This shows that the electric force $F_\text{e}$ is directly proportional to the charge $Q_1$ on the top ball.
But there's nothing special about the top ball: if we adjust the charge $Q_2$ on the bottom ball instead, we'll observe the same relationship.
This means that if we increased the charge on either ball by a factor of, say, $100,$ while keeping the other constant, the force on both balls also increases by the same factor of $100.$
So, the electric force $F_e$ must be proportional to the product of the two charges. We can write this as $F_\text{e} \propto Q_1 \times Q_2.$
Coulomb's Law
Unlike Newton's law for gravity, the electric force is sometimes attractive, and sometimes repulsive. It turns out that the relationship we just found, $F_e \propto Q_1 \times Q_2,$ also gives us a convenient mathematical trick to predict whether we'll see an attractive or repulsive force.
First, we need to assign negative charges a negative sign, and positive charges a positive sign. For example, if a proton has charge $+q,$ then an electron would have charge $-q.$
How does the sign of the product $Q_1\times Q_2$ relate to the direction of an electric forces?
Hint: to get started, consider case when $Q_1$ is a negative charge and $Q_2$ is also a negative charge. Is the electric force attractive or repulsive?
Coulomb's Law
Select one or more
In the last chapter, we saw that charged objects feel stronger electric forces when they're closer to each other, so this dependence on charge is not the end of the story.
Coulomb used his torsion balance to measure the electric force between metal balls, varying the distance $d$ between their centers.
We can do the same with our version of the experiment, by moving the clamped ball up and down above the ball on the scale.
Try adjusting the distance between the two balls. If you double the distance, what happens to the force?
Coulomb's Law
By moving the balls to different positions, it's also possible to work out the direction of the force.
Bringing together the results of these experiments, we find the following:
The electric force between two stationary point charges $Q_1$ and $Q_2$ that are separated by a distance of $r$ is $F=k_\text{e} \frac{ Q_1 Q_2}{r^2},$ and it's directed along the line connecting the point charges.
If $F$ is negative, then the force points towards the other charge; otherwise, it points away from the other charge.
This relationship is called Coulomb's law, and the constant $k_\text{e}$ is called the Coulomb constant.
Coulomb's Law
We've defined the Coulomb constant $k_\text{e}$, but what is its value?
The number depends on the units we use to measure electric charge. In the SI systems of units, charge is measured in Coulombs, with the symbol $\si{\coulomb}$.
This unit represents a lot of charge — a typical lightning strike delivers $\approx\SI{15}{\coulomb}$ from cloud to ground.
When electric force is measured in Newtons $(\si{\newton}),$ Coulomb's constant has the value $k_\text{e}=\SI[per-mode=symbol]{9e9}{\newton\metre\squared\per\coulomb\squared},$ and in our experiments with the tape, we were probably dealing with excess charges around $\SI{E-7}{\coulomb}.$
Using Coulomb's law, roughly how strong were the forces between the two pieces of tape when we hold the pieces of tape $\SI{10}{\centi\metre}$ apart?
Coulomb's Law
So, net charges of just over $1/\num{10000000}$ of a Coulomb on either tape strip give rise to a Newton, a force big enough to hold up an apple (no joke intended).
Turning back to the particle picture, net charge comes from the imbalance of protons and electrons. Indeed, the charge on an object is simply proportional to the number of protons minus the number of electrons: $Q_\text{net} = +e\times\left(n_p - n_e\right),$ where the charge on a single proton (or minus the charge of an electron) is $+e = \SI{1.6e-19}{\coulomb}.$
Approximately what is the magnitude of $\left(n_p-n_e\right)$ for the tape strips in our experiment?
Coulomb's Law
Though it looks simple, Coulomb's law can already tell us something important about the world — that the protons in the nucleus are bound by immense forces!
All atoms are made of positive protons, negative electrons, and neutral neutrons, but these particles aren't spread evenly through the space inside. The protons and neutrons are packed into a tiny nucleus that's only a few femtometers $\big(\SI[per-mode=symbol]{E-15}{\metre}\big)$ across, which was revealed by atomic physics experiments in the late $19^\text{th}$ century.
The structure of a helium atom, containing two protons, two neutrons, and two electrons
The charge on a proton is $\SI[per-mode=pedagogical]{+1.60E-19}{\coulomb}$. To hold together two protons at a distance of $\SI[per-mode=pedagogical]{2E-15}{\metre},$ how strong is the repulsive electric force between them?
Coulomb's Law
On its own, the electric force between two protons would force them apart and blow the nucleus apart — so there must be "nuclear forces" holding them together that are stronger than the electric forces.
Modern physics incorporates four fundamental forces. On the smallest scales, the strong and weak nuclear forces act — but their strength decreases very quickly with distance. Outside the nucleus, these forces disappear, and electric forces are most important.
On much larger scales than atoms, most matter is electrically neutral, containing an equal balance of protons and electrons (counterintuitively, this is a consequence of the strength of electric forces). This cancels out the electric forces, leaving the weakest force, gravity, to bring the neutral atoms together into planets, stars, and galaxies.
Coulomb's law works very well for two point-like charges such as metal spheres. But our world is full of extremely complicated situations containing many charged particles, spread out over complex shapes. To simplify these situations, we need more tools — the first of which is the electric field.
Let's get started.
Coulomb's Law
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https://ejournals.ph/article.php?id=14878 | ## Home⇛Asia Pacific Journal of Island Sustainability⇛vol. 29 no. 2 (2017)
### Patterns of linear, quadratic and exponential function
Rene V. Torres
#### Abstract:
Generally, when independent variable of a function is used as an exponent, the function is exponential. Hence, the following can be examples of exponential functions: $f\left(x\right)=a{b}^{x}+c$ , $f\left(x\right)=a{e}^{bx}+c$ , or $f\left(x\right)={e}^{a{x}^{2}+bx+c}$ .Deriving functions of these types given the set of ordered pairs is difficult. This study was conducted to derive formulas for the arbitrary constants ${}^{b}$ , and ${}^{c}$ of the exponential function $f\left(x\right)=a{b}^{x}+c$ . It applied the inductive method by using definitions of functions to derive the arbitrary constants from the patterns produced. The findings of the study were: a) For linear, given the table of ordered pairs, equal differences in x produce equal first differences in y; b) for quadratic, given the table of ordered pairs, equal differences in x produce equal second differences in y; and c) for an exponential function, given a table of ordered pairs, equal differences in x produce a common ratio in the first differences in y. The study obtained the following forms: $b=\sqrt[d]{r}$ , $a=\frac{q}{{b}^{n}\left({b}^{d}-1\right)}$ , $c=p-a{b}^{n}$ , Since most models developed used the concept of linear and multiple regressions, it is recommended that pattern analysis be used specifically when data are expressed in terms of ordered pairs. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "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": 0.9552863240242004, "perplexity": 517.1106557192619}, "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-2022-40/segments/1664030335365.63/warc/CC-MAIN-20220929194230-20220929224230-00365.warc.gz"} |
https://www.educator.com/physics/ap-physics-1-2/fullerton/vectors-+-scalars.php?ss=0 | Dan Fullerton
Vectors & Scalars
Slide Duration:
Section 1: Introduction
What is Physics?
7m 38s
Intro
0:00
Objectives
0:12
What is Physics?
0:31
What is Matter, Energy, and How to They Interact
0:55
Why?
0:58
1:05
Matter
1:23
Matter
1:29
Mass
1:33
Inertial Mass
1:53
Gravitational Mass
2:12
A Spacecraft's Mass
2:58
Energy
3:37
Energy: The Ability or Capacity to Do Work
3:39
Work: The Process of Moving an Object
3:45
The Ability or Capacity to Move an Object
3:54
Mass-Energy Equivalence
4:51
Relationship Between Mass and Energy E=mc2
5:01
The Mass of An Object is Really a Measure of Its Energy
5:05
The Study of Everything
5:42
Introductory Course
6:19
Next Steps
7:15
Math Review
24m 12s
Intro
0:00
Outline
0:10
Objectives
0:28
Why Do We Need Units?
0:52
Need to Set Specific Standards for Our Measurements
1:01
Physicists Have Agreed to Use the Systeme International
1:24
The Systeme International
1:50
Based on Powers of 10
1:52
7 Fundamental Units: Meter, Kilogram, Second, Ampere, Candela, Kelvin, Mole
2:02
The Meter
2:18
Meter is a Measure of Length
2:20
Measurements Smaller than a Meter, Use: Centimeter, Millimeter, Micrometer, Nanometer
2:25
Measurements Larger Than a Meter, Use Kilometer
2:38
The Kilogram
2:46
Roughly Equivalent to 2.2 English Pounds
2:49
Grams, Milligrams
2:53
Megagram
2:59
Seconds
3:10
Base Unit of Time
3:12
Minute, Hour, Day
3:20
Milliseconds, Microseconds
3:33
Derived Units
3:41
Velocity
3:45
Acceleration
3:57
Force
4:04
Prefixes for Powers of 10
4:21
Converting Fundamental Units, Example 1
4:53
Converting Fundamental Units, Example 2
7:18
Two-Step Conversions, Example 1
8:24
Two-Step Conversions, Example 2
10:06
Derived Unit Conversions
11:29
Multi-Step Conversions
13:25
Metric Estimations
15:04
What are Significant Figures?
16:01
Represent a Manner of Showing Which Digits In a Number Are Known to Some Level of Certainty
16:03
Example
16:09
Measuring with Sig Figs
16:36
Rule 1
16:40
Rule 2
16:44
Rule 3
16:52
16:57
All Non-Zero Digits Are Significant
17:04
All Digits Between Non-Zero Digits Are Significant
17:07
Zeros to the Left of the Significant Digits
17:11
Zeros to the Right of the Significant Digits
17:16
Non-Zero Digits
17:21
Digits Between Non-Zeros Are Significant
17:45
Zeroes to the Right of the Sig Figs Are Significant
18:17
Why Scientific Notation?
18:36
Physical Measurements Vary Tremendously in Magnitude
18:38
Example
18:47
Scientific Notation in Practice
19:23
Example 1
19:28
Example 2
19:44
Using Scientific Notation
20:02
Show Your Value Using Correct Number of Significant Figures
20:05
Move the Decimal Point
20:09
Show Your Number Being Multiplied by 10 Raised to the Appropriate Power
20:14
Accuracy and Precision
20:23
Accuracy
20:36
Precision
20:41
Example 1: Scientific Notation w/ Sig Figs
21:48
Example 2: Scientific Notation - Compress
22:25
Example 3: Scientific Notation - Compress
23:07
Example 4: Scientific Notation - Expand
23:31
Vectors & Scalars
25m 5s
Intro
0:00
Objectives
0:05
Scalars
0:29
Definition of Scalar
0:39
Temperature, Mass, Time
0:45
Vectors
1:12
Vectors are Quantities That Have Magnitude and Direction
1:13
Represented by Arrows
1:31
Vector Representations
1:47
2:42
Graphical Vector Subtraction
4:58
Vector Components
6:08
Angle of a Vector
8:22
Vector Notation
9:52
Example 1: Vector Components
14:30
Example 2: Vector Components
16:05
Example 3: Vector Magnitude
17:26
19:38
Example 5: Angle of a Vector
24:06
Section 2: Mechanics
Defining & Graphing Motion
30m 11s
Intro
0:00
Objectives
0:07
Position
0:40
An Object's Position Cab Be Assigned to a Variable on a Number Scale
0:43
Symbol for Position
1:07
Distance
1:13
When Position Changes, An Object Has Traveled Some Distance
1:14
Distance is Scalar and Measured in Meters
1:21
Example 1: Distance
1:34
Displacement
2:17
Displacement is a Vector Which Describes the Straight Line From Start to End Point
2:18
Measured in Meters
2:27
Example 2: Displacement
2:39
Average Speed
3:32
The Distance Traveled Divided by the Time Interval
3:33
Speed is a Scalar
3:47
Example 3: Average Speed
3:57
Average Velocity
4:37
The Displacement Divided by the Time Interval
4:38
Velocity is a Vector
4:53
Example 4: Average Velocity
5:06
Example 5: Chuck the Hungry Squirrel
5:55
Acceleration
8:02
Rate At Which Velocity Changes
8:13
Acceleration is a Vector
8:26
Example 6: Acceleration Problem
8:52
Average vs. Instantaneous
9:44
Average Values Take Into Account an Entire Time Interval
9:50
Instantaneous Value Tells the Rate of Change of a Quantity at a Specific Instant in Time
9:54
Example 7: Average Velocity
10:06
Particle Diagrams
11:57
Similar to the Effect of Oil Leak from a Car on the Pavement
11:59
Accelerating
13:03
Position-Time Graphs
14:17
Shows Position as a Function of Time
14:24
Slope of x-t Graph
15:08
Slope Gives You the Velocity
15:09
Negative Indicates Direction
16:27
Velocity-Time Graphs
16:45
Shows Velocity as a Function of Time
16:49
Area Under v-t Graphs
17:47
Area Under the V-T Graph Gives You Change in Displacement
17:48
Example 8: Slope of a v-t Graph
19:45
Acceleration-Time Graphs
21:44
Slope of the v-t Graph Gives You Acceleration
21:45
Area Under the a-t Graph Gives You an Object's Change in Velocity
22:24
Example 10: Motion Graphing
24:03
Example 11: v-t Graph
27:14
Example 12: Displacement From v-t Graph
28:14
Kinematic Equations
36m 13s
Intro
0:00
Objectives
0:07
Problem-Solving Toolbox
0:42
Graphs Are Not Always the Most Effective
0:47
Kinematic Equations Helps us Solve for Five Key Variables
0:56
Deriving the Kinematic Equations
1:29
Kinematic Equations
7:40
Problem Solving Steps
8:13
Label Your Horizontal or Vertical Motion
8:20
Choose a Direction as Positive
8:24
Create a Motion Analysis Table
8:33
8:42
Solve for Unknowns
8:45
Example 1: Horizontal Kinematics
8:51
Example 2: Vertical Kinematics
11:13
Example 3: 2 Step Problem
13:25
Example 4: Acceleration Problem
16:44
Example 5: Particle Diagrams
17:56
20:13
Free Fall
24:24
When the Only Force Acting on an Object is the Force of Gravity, the Motion is Free Fall
24:27
Air Resistance
24:51
Drop a Ball
24:56
Remove the Air from the Room
25:02
Analyze the Motion of Objects by Neglecting Air Resistance
25:06
Acceleration Due to Gravity
25:22
g = 9.8 m/s2
25:25
Approximate g as 10 m/s2 on the AP Exam
25:37
G is Referred to as the Gravitational Field Strength
25:48
Objects Falling From Rest
26:15
Objects Starting from Rest Have an Initial velocity of 0
26:19
Acceleration is +g
26:34
Example 7: Falling Objects
26:47
Objects Launched Upward
27:59
Acceleration is -g
28:04
At Highest Point, the Object has a Velocity of 0
28:19
Symmetry of Motion
28:27
Example 8: Ball Thrown Upward
28:47
Example 9: Height of a Jump
29:23
Example 10: Ball Thrown Downward
33:08
Example 11: Maximum Height
34:16
Projectiles
20m 32s
Intro
0:00
Objectives
0:06
What is a Projectile?
0:26
An Object That is Acted Upon Only By Gravity
0:29
Typically Launched at an Angle
0:43
Path of a Projectile
1:03
Projectiles Launched at an Angle Move in Parabolic Arcs
1:06
Symmetric and Parabolic
1:32
Horizontal Range and Max Height
1:49
Independence of Motion
2:17
Vertical
2:49
Horizontal
2:52
Example 1: Horizontal Launch
3:49
Example 2: Parabolic Path
7:41
Angled Projectiles
8:30
Must First Break Up the Object's Initial Velocity Into x- and y- Components of Initial Velocity
8:32
An Object Will Travel the Maximum Horizontal Distance with a Launch Angle of 45 Degrees
8:43
Example 3: Human Cannonball
8:55
Example 4: Motion Graphs
12:55
Example 5: Launch From a Height
15:33
Example 6: Acceleration of a Projectile
19:56
Relative Motion
10m 52s
Intro
0:00
Objectives
0:06
Reference Frames
0:18
Motion of an Observer
0:21
No Way to Distinguish Between Motion at Rest and Motion at a Constant Velocity
0:44
Motion is Relative
1:35
Example 1
1:39
Example 2
2:09
Calculating Relative Velocities
2:31
Example 1
2:43
Example 2
2:48
Example 3
2:52
Example 1
4:58
Example 2: Airspeed
6:19
Example 3: 2-D Relative Motion
7:39
Example 4: Relative Velocity with Direction
9:40
Newton's 1st Law of Motion
10m 16s
Intro
0:00
Objective
0:05
Newton's 1st Law of Motion
0:16
An Object At Rest Will Remain At Rest
0:21
An Object In Motion Will Remain in Motion
0:26
Net Force
0:39
Also Known As the Law of Inertia
0:46
Force
1:02
Push or Pull
1:04
Newtons
1:08
Contact and Field Forces
1:31
Contact Forces
1:50
Field Forces
2:11
What is a Net Force?
2:30
Vector Sum of All the Forces Acting on an Object
2:33
Translational Equilibrium
2:37
Unbalanced Force Is a Net Force
2:46
What Does It Mean?
3:49
An Object Will Continue in Its Current State of Motion Unless an Unbalanced Force Acts Upon It
3:50
Example of Newton's First Law
4:20
Objects in Motion
5:05
Will Remain in Motion At Constant Velocity
5:06
Hard to Find a Frictionless Environment on Earth
5:10
Static Equilibrium
5:40
Net Force on an Object is 0
5:44
Inertia
6:21
Tendency of an Object to Resist a Change in Velocity
6:23
Inertial Mass
6:35
Gravitational Mass
6:40
Example 1: Inertia
7:10
Example 2: Inertia
7:37
Example 3: Translational Equilibrium
8:03
Example 4: Net Force
8:40
Newton's 2nd Law of Motion
34m 55s
Intro
0:00
Objective
0:07
Free Body Diagrams
0:37
Tools Used to Analyze Physical Situations
0:40
Show All the Forces Acting on a Single Object
0:45
Drawing FBDs
0:58
Draw Object of Interest as a Dot
1:00
Sketch a Coordinate System
1:10
Example 1: Falling Elephant
1:18
Example 2: Falling Elephant with Air Resistance
2:07
Example 3: Soda on Table
3:00
Example 4: Box in Equilibrium
4:25
Example 5: Block on a Ramp
5:01
Pseudo-FBDs
5:53
Draw When Forces Don't Line Up with Axes
5:56
Break Forces That Don’t Line Up with Axes into Components That Do
6:00
Example 6: Objects on a Ramp
6:32
Example 7: Car on a Banked Turn
10:23
Newton's 2nd Law of Motion
12:56
The Acceleration of an Object is in the Direction of the Directly Proportional to the Net Force Applied
13:06
Newton's 1st Two Laws Compared
13:45
Newton's 1st Law
13:51
Newton's 2nd Law
14:10
Applying Newton's 2nd Law
14:50
Example 8: Applying Newton's 2nd Law
15:23
Example 9: Stopping a Baseball
16:52
Example 10: Block on a Surface
19:51
Example 11: Concurrent Forces
21:16
Mass vs. Weight
22:28
Mass
22:29
Weight
22:47
Example 12: Mass vs. Weight
23:16
Translational Equilibrium
24:47
Occurs When There Is No Net Force on an Object
24:49
Equilibrant
24:57
Example 13: Translational Equilibrium
25:29
Example 14: Translational Equilibrium
26:56
Example 15: Determining Acceleration
28:05
Example 16: Suspended Mass
31:03
Newton's 3rd Law of Motion
5m 58s
Intro
0:00
Objectives
0:06
Newton's 3rd Law of Motion
0:20
All Forces Come in Pairs
0:24
Examples
1:22
Action-Reaction Pairs
2:07
Girl Kicking Soccer Ball
2:11
Rocket Ship in Space
2:29
Gravity on You
2:53
Example 1: Force of Gravity
3:34
Example 2: Sailboat
4:00
Example 3: Hammer and Nail
4:49
Example 4: Net Force
5:06
Friction
17m 49s
Intro
0:00
Objectives
0:06
Examples
0:23
Friction Opposes Motion
0:24
Kinetic Friction
0:27
Static Friction
0:36
Magnitude of Frictional Force Is Determined By Two Things
0:41
Coefficient Friction
2:27
Ratio of the Frictional Force and the Normal Force
2:28
Chart of Different Values of Friction
2:48
Kinetic or Static?
3:31
Example 1: Car Sliding
4:18
Example 2: Block on Incline
5:03
Calculating the Force of Friction
5:48
Depends Only Upon the Nature of the Surfaces in Contact and the Magnitude of the Force
5:50
Terminal Velocity
6:14
Air Resistance
6:18
Terminal Velocity of the Falling Object
6:33
Example 3: Finding the Frictional Force
7:36
Example 4: Box on Wood Surface
9:13
Example 5: Static vs. Kinetic Friction
11:49
Example 6: Drag Force on Airplane
12:15
Example 7: Pulling a Sled
13:21
Dynamics Applications
35m 27s
Intro
0:00
Objectives
0:08
Free Body Diagrams
0:49
Drawing FBDs
1:09
Draw Object of Interest as a Dot
1:12
Sketch a Coordinate System
1:18
Example 1: FBD of Block on Ramp
1:39
Pseudo-FBDs
1:59
Draw Object of Interest as a Dot
2:00
Break Up the Forces
2:07
Box on a Ramp
2:12
Example 2: Box at Rest
4:28
Example 3: Box Held by Force
5:00
What is an Atwood Machine?
6:46
Two Objects are Connected by a Light String Over a Mass-less Pulley
6:49
Properties of Atwood Machines
7:13
Ideal Pulleys are Frictionless and Mass-less
7:16
Tension is Constant in a Light String Passing Over an Ideal Pulley
7:23
Solving Atwood Machine Problems
8:02
Alternate Solution
12:07
Analyze the System as a Whole
12:12
Elevators
14:24
Scales Read the Force They Exert on an Object Placed Upon Them
14:42
Can be Used to Analyze Using Newton's 2nd Law and Free body Diagrams
15:23
Example 4: Elevator Accelerates Upward
15:36
Example 5: Truck on a Hill
18:30
Example 6: Force Up a Ramp
19:28
Example 7: Acceleration Down a Ramp
21:56
Example 8: Basic Atwood Machine
24:05
Example 9: Masses and Pulley on a Table
26:47
Example 10: Mass and Pulley on a Ramp
29:15
Example 11: Elevator Accelerating Downward
33:00
Impulse & Momentum
26m 6s
Intro
0:00
Objectives
0:06
Momentum
0:31
Example
0:35
Momentum measures How Hard It Is to Stop a Moving Object
0:47
Vector Quantity
0:58
Example 1: Comparing Momenta
1:48
Example 2: Calculating Momentum
3:08
Example 3: Changing Momentum
3:50
Impulse
5:02
Change In Momentum
5:05
Example 4: Impulse
5:26
Example 5: Impulse-Momentum
6:41
Deriving the Impulse-Momentum Theorem
9:04
Impulse-Momentum Theorem
12:02
Example 6: Impulse-Momentum Theorem
12:15
Non-Constant Forces
13:55
Impulse or Change in Momentum
13:56
Determine the Impulse by Calculating the Area of the Triangle Under the Curve
14:07
Center of Mass
14:56
Real Objects Are More Complex Than Theoretical Particles
14:59
Treat Entire Object as if Its Entire Mass Were Contained at the Object's Center of Mass
15:09
To Calculate the Center of Mass
15:17
Example 7: Force on a Moving Object
15:49
Example 8: Motorcycle Accident
17:49
Example 9: Auto Collision
19:32
Example 10: Center of Mass (1D)
21:29
Example 11: Center of Mass (2D)
23:28
Collisions
21m 59s
Intro
0:00
Objectives
0:09
Conservation of Momentum
0:18
Linear Momentum is Conserved in an Isolated System
0:21
Useful for Analyzing Collisions and Explosions
0:27
Momentum Tables
0:58
Identify Objects in the System
1:05
Determine the Momenta of the Objects Before and After the Event
1:10
Add All the Momenta From Before the Event and Set Them Equal to Momenta After the Event
1:15
Solve Your Resulting Equation for Unknowns
1:20
Types of Collisions
1:31
Elastic Collision
1:36
Inelastic Collision
1:56
Example 1: Conservation of Momentum (1D)
2:02
Example 2: Inelastic Collision
5:12
Example 3: Recoil Velocity
7:16
Example 4: Conservation of Momentum (2D)
9:29
Example 5: Atomic Collision
16:02
Describing Circular Motion
7m 18s
Intro
0:00
Objectives
0:07
Uniform Circular Motion
0:20
Circumference
0:32
Average Speed Formula Still Applies
0:46
Frequency
1:03
Number of Revolutions or Cycles Which Occur Each Second
1:04
Hertz
1:24
Formula for Frequency
1:28
Period
1:36
Time It Takes for One Complete Revolution or Cycle
1:37
Frequency and Period
1:54
Example 1: Car on a Track
2:08
Example 2: Race Car
3:55
Example 3: Toy Train
4:45
Example 4: Round-A-Bout
5:39
Centripetal Acceleration & Force
26m 37s
Intro
0:00
Objectives
0:08
Uniform Circular Motion
0:38
Direction of ac
1:41
Magnitude of ac
3:50
Centripetal Force
4:08
For an Object to Accelerate, There Must Be a Net Force
4:18
Centripetal Force
4:26
Calculating Centripetal Force
6:14
Example 1: Acceleration
7:31
Example 2: Direction of ac
8:53
Example 3: Loss of Centripetal Force
9:19
Example 4: Velocity and Centripetal Force
10:08
Example 5: Demon Drop
10:55
Example 6: Centripetal Acceleration vs. Speed
14:11
Example 7: Calculating ac
15:03
Example 8: Running Back
15:45
Example 9: Car at an Intersection
17:15
Example 10: Bucket in Horizontal Circle
18:40
Example 11: Bucket in Vertical Circle
19:20
Example 12: Frictionless Banked Curve
21:55
Gravitation
32m 56s
Intro
0:00
Objectives
0:08
Universal Gravitation
0:29
The Bigger the Mass the Closer the Attraction
0:48
Formula for Gravitational Force
1:16
Calculating g
2:43
Mass of Earth
2:51
2:55
Inverse Square Relationship
4:32
Problem Solving Hints
7:21
Substitute Values in For Variables at the End of the Problem Only
7:26
Estimate the Order of Magnitude of the Answer Before Using Your Calculator
7:38
7:55
Example 1: Asteroids
8:20
Example 2: Meteor and the Earth
10:17
Example 3: Satellite
13:13
Gravitational Fields
13:50
Gravity is a Non-Contact Force
13:54
Closer Objects
14:14
Denser Force Vectors
14:19
Gravitational Field Strength
15:09
Example 4: Astronaut
16:19
Gravitational Potential Energy
18:07
Two Masses Separated by Distance Exhibit an Attractive Force
18:11
Formula for Gravitational Field
19:21
How Do Orbits Work?
19:36
Example5: Gravitational Field Strength for Space Shuttle in Orbit
21:35
Example 6: Earth's Orbit
25:13
Example 7: Bowling Balls
27:25
Example 8: Freely Falling Object
28:07
Example 9: Finding g
28:40
Example 10: Space Vehicle on Mars
29:10
Example 11: Fg vs. Mass Graph
30:24
Example 12: Mass on Mars
31:14
Example 13: Two Satellites
31:51
Rotational Kinematics
15m 33s
Intro
0:00
Objectives
0:07
0:26
In Degrees, Once Around a Circle is 360 Degrees
0:29
In Radians, Once Around a Circle is 2π
0:34
0:57
1:31
Linear vs. Angular Displacement
2:00
Linear Position
2:05
Angular Position
2:10
Linear vs. Angular Velocity
2:35
Linear Speed
2:39
Angular Speed
2:42
Direction of Angular Velocity
3:05
Converting Linear to Angular Velocity
4:22
Example 3: Angular Velocity Example
4:41
Linear vs. Angular Acceleration
5:36
Example 4: Angular Acceleration
6:15
Kinematic Variable Parallels
7:47
Displacement
7:52
Velocity
8:10
Acceleration
8:16
Time
8:22
Kinematic Variable Translations
8:30
Displacement
8:34
Velocity
8:42
Acceleration
8:50
Time
8:58
Kinematic Equation Parallels
9:09
Kinematic Equations
9:12
Delta
9:33
Final Velocity Squared and Angular Velocity Squared
9:54
Example 5: Medieval Flail
10:24
Example 6: CD Player
10:57
Example 7: Carousel
12:13
Example 8: Circular Saw
13:35
Torque
11m 21s
Intro
0:00
Objectives
0:05
Torque
0:18
Force That Causes an Object to Turn
0:22
Must be Perpendicular to the Displacement to Cause a Rotation
0:27
Lever Arm: The Stronger the Force, The More Torque
0:45
Direction of the Torque Vector
1:53
Perpendicular to the Position Vector and the Force Vector
1:54
Right-Hand Rule
2:08
Newton's 2nd Law: Translational vs. Rotational
2:46
Equilibrium
3:58
Static Equilibrium
4:01
Dynamic Equilibrium
4:09
Rotational Equilibrium
4:22
Example 1: Pirate Captain
4:32
Example 2: Auto Mechanic
5:25
Example 3: Sign Post
6:44
Example 4: See-Saw
9:01
Rotational Dynamics
36m 6s
Intro
0:00
Objectives
0:08
Types of Inertia
0:39
Inertial Mass (Translational Inertia)
0:42
Moment of Inertia (Rotational Inertia)
0:53
Moment of Inertia for Common Objects
1:48
Example 1: Calculating Moment of Inertia
2:53
Newton's 2nd Law - Revisited
5:09
Acceleration of an Object
5:15
Angular Acceleration of an Object
5:24
Example 2: Rotating Top
5:47
Example 3: Spinning Disc
7:54
Angular Momentum
9:41
Linear Momentum
9:43
Angular Momentum
10:00
Calculating Angular Momentum
10:51
Direction of the Angular Momentum Vector
11:26
Total Angular Momentum
12:29
Example 4: Angular Momentum of Particles
14:15
Example 5: Rotating Pedestal
16:51
Example 6: Rotating Discs
18:39
Angular Momentum and Heavenly Bodies
20:13
Types of Kinetic Energy
23:41
Objects Traveling with a Translational Velocity
23:45
Objects Traveling with Angular Velocity
24:00
Translational vs. Rotational Variables
24:33
Example 7: Kinetic Energy of a Basketball
25:45
Example 8: Playground Round-A-Bout
28:17
Example 9: The Ice Skater
30:54
Example 10: The Bowler
33:15
Work & Power
31m 20s
Intro
0:00
Objectives
0:09
What Is Work?
0:31
Power Output
0:35
Transfer Energy
0:39
Work is the Process of Moving an Object by Applying a Force
0:46
Examples of Work
0:56
Calculating Work
2:16
Only the Force in the Direction of the Displacement Counts
2:33
Formula for Work
2:48
Example 1: Moving a Refrigerator
3:16
Example 2: Liberating a Car
3:59
Example 3: Crate on a Ramp
5:20
Example 4: Lifting a Box
7:11
Example 5: Pulling a Wagon
8:38
Force vs. Displacement Graphs
9:33
The Area Under a Force vs. Displacement Graph is the Work Done by the Force
9:37
Find the Work Done
9:49
Example 6: Work From a Varying Force
11:00
Hooke's Law
12:42
The More You Stretch or Compress a Spring, The Greater the Force of the Spring
12:46
The Spring's Force is Opposite the Direction of Its Displacement from Equilibrium
13:00
Determining the Spring Constant
14:21
Work Done in Compressing the Spring
15:27
Example 7: Finding Spring Constant
16:21
Example 8: Calculating Spring Constant
17:58
Power
18:43
Work
18:46
Power
18:50
Example 9: Moving a Sofa
19:26
Calculating Power
20:41
Example 10: Motors Delivering Power
21:27
Example 11: Force on a Cyclist
22:40
Example 12: Work on a Spinning Mass
23:52
Example 13: Work Done by Friction
25:05
Example 14: Units of Power
28:38
Example 15: Frictional Force on a Sled
29:43
Energy
20m 15s
Intro
0:00
Objectives
0:07
What is Energy?
0:24
The Ability or Capacity to do Work
0:26
The Ability or Capacity to Move an Object
0:34
Types of Energy
0:39
Energy Transformations
2:07
Transfer Energy by Doing Work
2:12
Work-Energy Theorem
2:20
Units of Energy
2:51
Kinetic Energy
3:08
Energy of Motion
3:13
Ability or Capacity of a Moving Object to Move Another Object
3:17
A Single Object Can Only Have Kinetic Energy
3:46
Example 1: Kinetic Energy of a Motorcycle
5:08
Potential Energy
5:59
Energy An Object Possesses
6:10
Gravitational Potential Energy
7:21
Elastic Potential Energy
9:58
Internal Energy
10:16
Includes the Kinetic Energy of the Objects That Make Up the System and the Potential Energy of the Configuration
10:20
Calculating Gravitational Potential Energy in a Constant Gravitational Field
10:57
Sources of Energy on Earth
12:41
Example 2: Potential Energy
13:41
Example 3: Energy of a System
14:40
Example 4: Kinetic and Potential Energy
15:36
Example 5: Pendulum
16:55
Conservation of Energy
23m 20s
Intro
0:00
Objectives
0:08
Law of Conservation of Energy
0:22
Energy Cannot Be Created or Destroyed.. It Can Only Be Changed
0:27
Mechanical Energy
0:34
Conservation Laws
0:40
Examples
0:49
Kinematics vs. Energy
4:34
Energy Approach
4:56
Kinematics Approach
6:04
The Pendulum
8:07
Example 1: Cart Compressing a Spring
13:09
Example 2
14:23
Example 3: Car Skidding to a Stop
16:15
Example 4: Accelerating an Object
17:27
Example 5: Block on Ramp
18:06
Example 6: Energy Transfers
19:21
Simple Harmonic Motion
58m 30s
Intro
0:00
Objectives
0:08
What Is Simple Harmonic Motion?
0:57
Nature's Typical Reaction to a Disturbance
1:00
A Displacement Which Results in a Linear Restoring Force Results in SHM
1:25
Review of Springs
1:43
When a Force is Applied to a Spring, the Spring Applies a Restoring Force
1:46
When the Spring is in Equilibrium, It Is 'Unstrained'
1:54
Factors Affecting the Force of A Spring
2:00
Oscillations
3:42
Repeated Motions
3:45
Cycle 1
3:52
Period
3:58
Frequency
4:07
Spring-Block Oscillator
4:47
Mass of the Block
4:59
Spring Constant
5:05
Example 1: Spring-Block Oscillator
6:30
Diagrams
8:07
Displacement
8:42
Velocity
8:57
Force
9:36
Acceleration
10:09
U
10:24
K
10:47
Example 2: Harmonic Oscillator Analysis
16:22
Circular Motion vs. SHM
23:26
Graphing SHM
25:52
Example 3: Position of an Oscillator
28:31
Vertical Spring-Block Oscillator
31:13
Example 4: Vertical Spring-Block Oscillator
34:26
Example 5: Bungee
36:39
The Pendulum
43:55
Mass Is Attached to a Light String That Swings Without Friction About the Vertical Equilibrium
44:04
Energy and the Simple Pendulum
44:58
Frequency and Period of a Pendulum
48:25
Period of an Ideal Pendulum
48:31
Assume Theta is Small
48:54
Example 6: The Pendulum
50:15
Example 7: Pendulum Clock
53:38
Example 8: Pendulum on the Moon
55:14
Example 9: Mass on a Spring
56:01
Section 3: Fluids
Density & Buoyancy
19m 48s
Intro
0:00
Objectives
0:09
Fluids
0:27
Fluid is Matter That Flows Under Pressure
0:31
Fluid Mechanics is the Study of Fluids
0:44
Density
0:57
Density is the Ratio of an Object's Mass to the Volume It Occupies
0:58
Less Dense Fluids
1:06
Less Dense Solids
1:09
Example 1: Density of Water
1:27
Example 2: Volume of Gold
2:19
Example 3: Floating
3:06
Buoyancy
3:54
Force Exerted by a Fluid on an Object, Opposing the Object's Weight
3:56
Buoyant Force Determined Using Archimedes Principle
4:03
Example 4: Buoyant Force
5:12
Example 5: Shark Tank
5:56
Example 6: Concrete Boat
7:47
Example 7: Apparent Mass
10:08
Example 8: Volume of a Submerged Cube
13:21
Example 9: Determining Density
15:37
Pressure & Pascal's Principle
18m 7s
Intro
0:00
Objectives
0:09
Pressure
0:25
Pressure is the Effect of a Force Acting Upon a Surface
0:27
Formula for Pressure
0:41
Force is Always Perpendicular to the Surface
0:50
Exerting Pressure
1:03
Fluids Exert Outward Pressure in All Directions on the Sides of Any Container Holding the Fluid
1:36
Earth's Atmosphere Exerts Pressure
1:42
Example 1: Pressure on Keyboard
2:17
Example 2: Sleepy Fisherman
3:03
Example 3: Scale on Planet Physica
4:12
Example 4: Ranking Pressures
5:00
Pressure on a Submerged Object
6:45
Pressure a Fluid Exerts on an Object Submerged in That Fluid
6:46
If There Is Atmosphere Above the Fluid
7:03
Example 5: Gauge Pressure Scuba Diving
7:27
Example 6: Absolute Pressure Scuba Diving
8:13
Pascal's Principle
8:51
Force Multiplication Using Pascal's Principle
9:24
Example 7: Barber's Chair
11:38
Example 8: Hydraulic Auto Lift
13:26
Example 9: Pressure on a Penny
14:41
Example 10: Depth in Fresh Water
16:39
Example 11: Absolute vs. Gauge Pressure
17:23
Continuity Equation for Fluids
7m
Intro
0:00
Objectives
0:08
Conservation of Mass for Fluid Flow
0:18
Law of Conservation of Mass for Fluids
0:21
Volume Flow Rate Remains Constant Throughout the Pipe
0:35
Volume Flow Rate
0:59
Quantified In Terms Of Volume Flow Rate
1:01
Area of Pipe x Velocity of Fluid
1:05
Must Be Constant Throughout Pipe
1:10
Example 1: Tapered Pipe
1:44
Example 2: Garden Hose
2:37
Example 3: Oil Pipeline
4:49
Example 4: Roots of Continuity Equation
6:16
Bernoulli's Principle
20m
Intro
0:00
Objectives
0:08
Bernoulli's Principle
0:21
Airplane Wings
0:35
Venturi Pump
1:56
Bernoulli's Equation
3:32
Example 1: Torricelli's Theorem
4:38
Example 2: Gauge Pressure
7:26
Example 3: Shower Pressure
8:16
Example 4: Water Fountain
12:29
Example 5: Elevated Cistern
15:26
Section 4: Thermal Physics
Temperature, Heat, & Thermal Expansion
24m 17s
Intro
0:00
Objectives
0:12
Thermal Physics
0:42
Explores the Internal Energy of Objects Due to the Motion of the Atoms and Molecules Comprising the Objects
0:46
Explores the Transfer of This Energy From Object to Object
0:53
Temperature
1:00
Thermal Energy Is Related to the Kinetic Energy of All the Particles Comprising the Object
1:03
The More Kinetic Energy of the Constituent Particles Have, The Greater the Object's Thermal Energy
1:12
Temperature and Phases of Matter
1:44
Solids
1:48
Liquids
1:56
Gases
2:02
Average Kinetic Energy and Temperature
2:16
Average Kinetic Energy
2:24
Boltzmann's Constant
2:29
Temperature Scales
3:06
Converting Temperatures
4:37
Heat
5:03
Transfer of Thermal Energy
5:06
Accomplished Through Collisions Which is Conduction
5:13
Methods of Heat Transfer
5:52
Conduction
5:59
Convection
6:19
6:31
Quantifying Heat Transfer in Conduction
6:37
Rate of Heat Transfer is Measured in Watts
6:42
Thermal Conductivity
7:12
Example 1: Average Kinetic Energy
7:35
Example 2: Body Temperature
8:22
Example 3: Temperature of Space
9:30
Example 4: Temperature of the Sun
10:44
Example 5: Heat Transfer Through Window
11:38
Example 6: Heat Transfer Across a Rod
12:40
Thermal Expansion
14:18
When Objects Are Heated, They Tend to Expand
14:19
At Higher Temperatures, Objects Have Higher Average Kinetic Energies
14:24
At Higher Levels of Vibration, The Particles Are Not Bound As Tightly to Each Other
14:30
Linear Expansion
15:11
Amount a Material Expands is Characterized by the Material's Coefficient of Expansion
15:14
One-Dimensional Expansion -> Linear Coefficient of Expansion
15:20
Volumetric Expansion
15:38
Three-Dimensional Expansion -> Volumetric Coefficient of Expansion
15:45
Volumetric Coefficient of Expansion is Roughly Three Times the Linear Coefficient of Expansion
16:03
Coefficients of Thermal Expansion
16:24
16:59
Example 8: Expansion of an Aluminum Rod
18:37
Example 9: Water Spilling Out of a Glass
20:18
Example 10: Average Kinetic Energy vs. Temperature
22:18
Example 11: Expansion of a Ring
23:07
Ideal Gases
24m 15s
Intro
0:00
Objectives
0:10
Ideal Gases
0:25
Gas Is Comprised of Many Particles Moving Randomly in a Container
0:34
Particles Are Far Apart From One Another
0:46
Particles Do Not Exert Forces Upon One Another Unless They Come In Contact in an Elastic Collision
0:53
Ideal Gas Law
1:18
Atoms, Molecules, and Moles
2:56
Protons
2:59
Neutrons
3:15
Electrons
3:18
Examples
3:25
Example 1: Counting Moles
4:58
Example 2: Moles of CO2 in a Bottle
6:00
Example 3: Pressurized CO2
6:54
Example 4: Helium Balloon
8:53
Internal Energy of an Ideal Gas
10:17
The Average Kinetic Energy of the Particles of an Ideal Gas
10:21
Total Internal Energy of the Ideal Gas Can Be Found by Multiplying the Average Kinetic Energy of the Gas's Particles by the Numbers of Particles in the Gas
10:32
Example 5: Internal Energy of Oxygen
12:00
Example 6: Temperature of Argon
12:41
Root-Mean-Square Velocity
13:40
This is the Square Root of the Average Velocity Squared For All the Molecules in the System
13:43
Derived from the Maxwell-Boltzmann Distribution Function
13:56
Calculating vrms
14:56
Example 7: Average Velocity of a Gas
18:32
Example 8: Average Velocity of a Gas
19:44
Example 9: vrms of Molecules in Equilibrium
20:59
Example 10: Moles to Molecules
22:25
Example 11: Relating Temperature and Internal Energy
23:22
Thermodynamics
22m 29s
Intro
0:00
Objectives
0:06
Zeroth Law of Thermodynamics
0:26
First Law of Thermodynamics
1:00
The Change in the Internal Energy of a Closed System is Equal to the Heat Added to the System Plus the Work Done on the System
1:04
It is a Restatement of the Law of Conservation of Energy
1:19
Sign Conventions Are Important
1:25
Work Done on a Gas
1:44
Example 1: Adding Heat to a System
3:25
Example 2: Expanding a Gas
4:07
P-V Diagrams
5:11
Pressure-Volume Diagrams are Useful Tools for Visualizing Thermodynamic Processes of Gases
5:13
Use Ideal Gas Law to Determine Temperature of Gas
5:25
P-V Diagrams II
5:55
Volume Increases, Pressure Decreases
6:00
As Volume Expands, Gas Does Work
6:19
Temperature Rises as You Travel Up and Right on a PV Diagram
6:29
Example 3: PV Diagram Analysis
6:40
Types of PV Processes
7:52
8:03
Isobaric
8:19
Isochoric
8:28
Isothermal
8:35
8:47
Heat Is not Transferred Into or Out of The System
8:50
Heat = 0
8:55
Isobaric Processes
9:19
Pressure Remains Constant
9:21
PV Diagram Shows a Horizontal Line
9:27
Isochoric Processes
9:51
Volume Remains Constant
9:52
PV Diagram Shows a Vertical Line
9:58
Work Done on the Gas is Zero
10:01
Isothermal Processes
10:27
Temperature Remains Constant
10:29
Lines on a PV Diagram Are Isotherms
10:31
PV Remains Constant
10:38
Internal Energy of Gas Remains Constant
10:40
10:46
Example 5: Removing Heat
11:25
Example 6: Ranking Processes
13:08
Second Law of Thermodynamics
13:59
Heat Flows Naturally From a Warmer Object to a Colder Object
14:02
Heat Energy Cannot be Completely Transformed Into Mechanical Work
14:11
All Natural Systems Tend Toward a Higher Level of Disorder
14:19
Heat Engines
14:52
Heat Engines Convert Heat Into Mechanical Work
14:56
Efficiency of a Heat Engine is the Ratio of the Engine You Get Out to the Energy You Put In
14:59
Power in Heat Engines
16:09
Heat Engines and PV Diagrams
17:38
Carnot Engine
17:54
It Is a Theoretical Heat Engine That Operates at Maximum Possible Efficiency
18:02
It Uses Only Isothermal and Adiabatic Processes
18:08
Carnot's Theorem
18:11
Example 7: Carnot Engine
18:49
Example 8: Maximum Efficiency
21:02
Example 9: PV Processes
21:51
Section 5: Electricity & Magnetism
Electric Fields & Forces
38m 24s
Intro
0:00
Objectives
0:10
Electric Charges
0:34
Matter is Made Up of Atoms
0:37
Protons Have a Charge of +1
0:45
Electrons Have a Charge of -1
1:00
Most Atoms Are Neutral
1:04
Ions
1:15
Fundamental Unit of Charge is the Coulomb
1:29
Like Charges Repel, While Opposites Attract
1:50
Example 1: Charge on an Object
2:22
Example 2: Charge of an Alpha Particle
3:36
Conductors and Insulators
4:27
Conductors Allow Electric Charges to Move Freely
4:30
Insulators Do Not Allow Electric Charges to Move Freely
4:39
Resistivity is a Material Property
4:45
Charging by Conduction
5:05
Materials May Be Charged by Contact, Known as Conduction
5:07
Conductors May Be Charged by Contact
5:24
Example 3: Charging by Conduction
5:38
The Electroscope
6:44
Charging by Induction
8:00
Example 4: Electrostatic Attraction
9:23
Coulomb's Law
11:46
Charged Objects Apply a Force Upon Each Other = Coulombic Force
11:52
Force of Attraction or Repulsion is Determined by the Amount of Charge and the Distance Between the Charges
12:04
Example 5: Determine Electrostatic Force
13:09
Example 6: Deflecting an Electron Beam
15:35
Electric Fields
16:28
The Property of Space That Allows a Charged Object to Feel a Force
16:44
Electric Field Strength Vector is the Amount of Electrostatic Force Observed by a Charge Per Unit of Charge
17:01
The Direction of the Electric Field Vector is the Direction a Positive Charge Would Feel a Force
17:24
Example 7: Field Between Metal Plates
17:58
Visualizing the Electric Field
19:27
Electric Field Lines Point Away from Positive Charges and Toward Negative Charges
19:40
Electric Field Lines Intersect Conductors at Right Angles to the Surface
19:50
Field Strength and Line Density Decreases as You Move Away From the Charges
19:58
Electric Field Lines
20:09
E Field Due to a Point Charge
22:32
Electric Fields Are Caused by Charges
22:35
Electric Field Due to a Point Charge Can Be Derived From the Definition of the Electric Field and Coulomb's Law
22:38
To Find the Electric Field Due to Multiple Charges
23:09
Comparing Electricity to Gravity
23:56
Force
24:02
Field Strength
24:16
Constant
24:37
Charge/ Mass Units
25:01
Example 8: E Field From 3 Point Charges
25:07
Example 9: Where is the E Field Zero?
31:43
Example 10: Gravity and Electricity
36:38
Example 11: Field Due to Point Charge
37:34
Electric Potential Difference
35m 58s
Intro
0:00
Objectives
0:09
Electric Potential Energy
0:32
When an Object Was Lifted Against Gravity By Applying a Force for Some Distance, Work Was Done
0:35
When a Charged Object is Moved Against an Electric Field by Applying a Force for Some Distance, Work is Done
0:43
Electric Potential Difference
1:30
Example 1: Charge From Work
2:06
Example 2: Electric Energy
3:09
The Electron-Volt
4:02
Electronvolt (eV)
4:15
1eV is the Amount of Work Done in Moving an Elementary Charge Through a Potential Difference of 1 Volt
4:28
Example 3: Energy in eV
5:33
Equipotential Lines
6:32
Topographic Maps Show Lines of Equal Altitude, or Equal Gravitational Potential
6:36
Lines Connecting Points of Equal Electrical Potential are Known as Equipotential Lines
6:57
Drawing Equipotential Lines
8:15
Potential Due to a Point Charge
10:46
Calculate the Electric Field Vector Due to a Point Charge
10:52
Calculate the Potential Difference Due to a Point Charge
11:05
To Find the Potential Difference Due to Multiple Point Charges
11:16
Example 4: Potential Due to a Point Charge
11:52
Example 5: Potential Due to Point Charges
13:04
Parallel Plates
16:34
Configurations in Which Parallel Plates of Opposite Charge are Situated a Fixed Distance From Each Other
16:37
These Can Create a Capacitor
16:45
E Field Due to Parallel Plates
17:14
Electric Field Away From the Edges of Two Oppositely Charged Parallel Plates is Constant
17:15
Magnitude of the Electric Field Strength is Give By the Potential Difference Between the Plates Divided by the Plate Separation
17:47
Capacitors
18:09
Electric Device Used to Store Charge
18:11
Once the Plates Are Charged, They Are Disconnected
18:30
Device's Capacitance
18:46
Capacitors Store Energy
19:28
Charges Located on the Opposite Plates of a Capacitor Exert Forces on Each Other
19:31
Example 6: Capacitance
20:28
Example 7: Charge on a Capacitor
22:03
Designing Capacitors
24:00
Area of the Plates
24:05
Separation of the Plates
24:09
Insulating Material
24:13
Example 8: Designing a Capacitor
25:35
Example 9: Calculating Capacitance
27:39
Example 10: Electron in Space
29:47
Example 11: Proton Energy Transfer
30:35
Example 12: Two Conducting Spheres
32:50
Example 13: Equipotential Lines for a Capacitor
34:48
Current & Resistance
21m 14s
Intro
0:00
Objectives
0:06
Electric Current
0:19
Path Through Current Flows
0:21
Current is the Amount of Charge Passing a Point Per Unit Time
0:25
Conventional Current is the Direction of Positive Charge Flow
0:43
Example 1: Current Through a Resistor
1:19
Example 2: Current Due to Elementary Charges
1:47
Example 3: Charge in a Light Bulb
2:35
Example 4: Flashlights
3:03
Conductivity and Resistivity
4:41
Conductivity is a Material's Ability to Conduct Electric Charge
4:53
Resistivity is a Material's Ability to Resist the Movement of Electric Charge
5:11
Resistance vs. Resistivity vs. Resistors
5:35
Resistivity Is a Material Property
5:40
Resistance Is a Functional Property of an Element in an Electric Circuit
5:57
A Resistor is a Circuit Element
7:23
Resistors
7:45
Example 5: Calculating Resistance
8:17
Example 6: Resistance Dependencies
10:09
Configuration of Resistors
10:50
When Placed in a Circuit, Resistors Can be Organized in Both Serial and Parallel Arrangements
10:53
May Be Useful to Determine an Equivalent Resistance Which Could Be Used to Replace a System or Resistors with a Single Equivalent Resistor
10:58
Resistors in Series
11:15
Resistors in Parallel
12:35
Example 7: Finding Equivalent Resistance
15:01
Example 8: Length and Resistance
17:43
Example 9: Comparing Resistors
18:21
Example 10: Comparing Wires
19:12
Ohm's Law & Power
10m 35s
Intro
0:00
Objectives
0:06
Ohm's Law
0:21
Relates Resistance, Potential Difference, and Current Flow
0:23
Example 1: Resistance of a Wire
1:22
Example 2: Circuit Current
1:58
Example 3: Variable Resistor
2:30
Ohm's 'Law'?
3:22
Very Useful Empirical Relationship
3:31
Test if a Material is 'Ohmic'
3:40
Example 4: Ohmic Material
3:58
Electrical Power
4:24
Current Flowing Through a Circuit Causes a Transfer of Energy Into Different Types
4:26
Example: Light Bulb
4:36
Example: Television
4:58
Calculating Power
5:09
Electrical Energy
5:14
Charge Per Unit Time Is Current
5:29
Expand Using Ohm's Law
5:48
Example 5: Toaster
7:43
Example 6: Electric Iron
8:19
Example 7: Power of a Resistor
9:19
Example 8: Information Required to Determine Power in a Resistor
9:55
Circuits & Electrical Meters
8m 44s
Intro
0:00
Objectives
0:08
Electrical Circuits
0:21
A Closed-Loop Path Through Which Current Can Flow
0:22
Can Be Made Up of Most Any Materials, But Typically Comprised of Electrical Devices
0:27
Circuit Schematics
1:09
Symbols Represent Circuit Elements
1:30
Lines Represent Wires
1:33
Sources for Potential Difference: Voltaic Cells, Batteries, Power Supplies
1:36
Complete Conducting Paths
2:43
Voltmeters
3:20
Measure the Potential Difference Between Two Points in a Circuit
3:21
Connected in Parallel with the Element to be Measured
3:25
Have Very High Resistance
3:59
Ammeters
4:19
Measure the Current Flowing Through an Element of a Circuit
4:20
Connected in Series with the Circuit
4:25
Have Very Low Resistance
4:45
Example 1: Ammeter and Voltmeter Placement
4:56
Example 2: Analyzing R
6:27
Example 3: Voltmeter Placement
7:12
Example 4: Behavior or Electrical Meters
7:31
Circuit Analysis
48m 58s
Intro
0:00
Objectives
0:07
Series Circuits
0:27
Series Circuits Have Only a Single Current Path
0:29
Removal of any Circuit Element Causes an Open Circuit
0:31
Kirchhoff's Laws
1:36
Tools Utilized in Analyzing Circuits
1:42
Kirchhoff's Current Law States
1:47
Junction Rule
2:00
Kirchhoff's Voltage Law States
2:05
Loop Rule
2:18
Example 1: Voltage Across a Resistor
2:23
Example 2: Current at a Node
3:45
Basic Series Circuit Analysis
4:53
Example 3: Current in a Series Circuit
9:21
Example 4: Energy Expenditure in a Series Circuit
10:14
Example 5: Analysis of a Series Circuit
12:07
Example 6: Voltmeter In a Series Circuit
14:57
Parallel Circuits
17:11
Parallel Circuits Have Multiple Current Paths
17:13
Removal of a Circuit Element May Allow Other Branches of the Circuit to Continue Operating
17:15
Basic Parallel Circuit Analysis
18:19
Example 7: Parallel Circuit Analysis
21:05
Example 8: Equivalent Resistance
22:39
Example 9: Four Parallel Resistors
23:16
Example 10: Ammeter in a Parallel Circuit
26:27
Combination Series-Parallel Circuits
28:50
Look For Portions of the Circuit With Parallel Elements
28:56
Work Back to Original Circuit
29:09
Analysis of a Combination Circuit
29:20
Internal Resistance
34:11
In Reality, Voltage Sources Have Some Amount of 'Internal Resistance'
34:16
Terminal Voltage of the Voltage Source is Reduced Slightly
34:25
Example 11: Two Voltage Sources
35:16
Example 12: Internal Resistance
42:46
Example 13: Complex Circuit with Meters
45:22
Example 14: Parallel Equivalent Resistance
48:24
RC Circuits
24m 47s
Intro
0:00
Objectives
0:08
Capacitors in Parallel
0:34
Capacitors Store Charge on Their Plates
0:37
Capacitors In Parallel Can Be Replaced with an Equivalent Capacitor
0:46
Capacitors in Series
2:42
Charge on Capacitors Must Be the Same
2:44
Capacitor In Series Can Be Replaced With an Equivalent Capacitor
2:47
RC Circuits
5:40
Comprised of a Source of Potential Difference, a Resistor Network, and One or More Capacitors
5:42
Uncharged Capacitors Act Like Wires
6:04
Charged Capacitors Act Like Opens
6:12
Charging an RC Circuit
6:23
Discharging an RC Circuit
11:36
Example 1: RC Analysis
14:50
Example 2: More RC Analysis
18:26
Example 3: Equivalent Capacitance
21:19
Example 4: More Equivalent Capacitance
22:48
Magnetic Fields & Properties
19m 48s
Intro
0:00
Objectives
0:07
Magnetism
0:32
A Force Caused by Moving Charges
0:34
Magnetic Domains Are Clusters of Atoms with Electrons Spinning in the Same Direction
0:51
Example 1: Types of Fields
1:23
Magnetic Field Lines
2:25
Make Closed Loops and Run From North to South Outside the Magnet
2:26
Magnetic Flux
2:42
Show the Direction the North Pole of a Magnet Would Tend to Point If Placed in the Field
2:54
Example 2: Lines of Magnetic Force
3:49
Example 3: Forces Between Bar Magnets
4:39
The Compass
5:28
The Earth is a Giant Magnet
5:31
The Earth's Magnetic North pole is Located Near the Geographic South Pole, and Vice Versa
5:33
A Compass Lines Up with the Net Magnetic Field
6:07
Example 3: Compass in Magnetic Field
6:41
Example 4: Compass Near a Bar Magnet
7:14
Magnetic Permeability
7:59
The Ratio of the Magnetic Field Strength Induced in a Material to the Magnetic Field Strength of the Inducing Field
8:02
Free Space
8:13
Highly Magnetic Materials Have Higher Values of Magnetic Permeability
8:34
Magnetic Dipole Moment
8:41
The Force That a Magnet Can Exert on Moving Charges
8:46
Relative Strength of a Magnet
8:54
Forces on Moving Charges
9:10
Moving Charges Create Magnetic Fields
9:11
Magnetic Fields Exert Forces on Moving Charges
9:17
Direction of the Magnetic Force
9:57
Direction is Given by the Right-Hand Rule
10:05
Right-Hand Rule
10:09
Mass Spectrometer
10:52
Magnetic Fields Accelerate Moving Charges So That They Travel in a Circle
10:58
Used to Determine the Mass of an Unknown Particle
11:04
Velocity Selector
12:44
Mass Spectrometer with an Electric Field Added
12:47
Example 5: Force on an Electron
14:13
Example 6: Velocity of a Charged Particle
15:25
Example 7: Direction of the Magnetic Force
16:52
Example 8: Direction of Magnetic Force on Moving Charges
17:43
Example 9: Electron Released From Rest in Magnetic Field
18:53
Current-Carrying Wires
21m 29s
Intro
0:00
Objectives
0:09
Force on a Current-Carrying Wire
0:30
A Current-Carrying Wire in a Magnetic Field May Experience a Magnetic Force
0:33
Direction Given by the Right-Hand Rule
1:11
Example 1: Force on a Current-Carrying Wire
1:38
Example 2: Equilibrium on a Submerged Wire
2:33
Example 3: Torque on a Loop of Wire
5:55
Magnetic Field Due to a Current-Carrying Wire
8:49
Moving Charges Create Magnetic Fields
8:53
Wires Carry Moving Charges
8:56
Direction Given by the Right-Hand Rule
9:21
Example 4: Magnetic Field Due to a Wire
10:56
Magnetic Field Due to a Solenoid
12:12
Solenoid is a Coil of Wire
12:19
Direction Given by the Right-Hand Rule
12:47
Forces on 2 Parallel Wires
13:34
Current Flowing in the Same Direction
14:52
Current Flowing in Opposite Directions
14:57
Example 5: Magnetic Field Due to Wires
15:19
Example 6: Strength of an Electromagnet
18:35
Example 7: Force on a Wire
19:30
Example 8: Force Between Parallel Wires
20:47
Intro to Electromagnetic Induction
17m 26s
Intro
0:00
Objectives
0:09
Induced EMF
0:42
Charges Flowing Through a Wire Create Magnetic Fields
0:45
Changing Magnetic Fields Cause Charges to Flow or 'Induce' a Current in a Process Known As Electromagnetic Induction
0:49
Electro-Motive Force is the Potential Difference Created by a Changing Magnetic Field
0:57
Magnetic Flux is the Amount of Magnetic Fields Passing Through an Area
1:17
Finding the Magnetic Flux
1:36
Magnetic Field Strength
1:39
Angle Between the Magnetic Field Strength and the Normal to the Area
1:51
Calculating Induced EMF
3:01
The Magnitude of the Induced EMF is Equal to the Rate of Change of the Magnetic Flux
3:04
Induced EMF in a Rectangular Loop of Wire
4:03
Lenz's Law
5:17
Electric Generators and Motors
9:28
Generate an Induced EMF By Turning a Coil of Wire in a magnetic Field
9:31
Generators Use Mechanical Energy to Turn the Coil of Wire
9:39
Electric Motor Operates Using Same Principle
10:30
Example 1: Finding Magnetic Flux
10:43
Example 2: Finding Induced EMF
11:54
Example 3: Changing Magnetic Field
13:52
Example 4: Current Induced in a Rectangular Loop of Wire
15:23
Section 6: Waves & Optics
Wave Characteristics
26m 41s
Intro
0:00
Objectives
0:09
Waves
0:32
Pulse
1:00
A Pulse is a Single Disturbance Which Carries Energy Through a Medium or Space
1:05
A Wave is a Series of Pulses
1:18
When a Pulse Reaches a Hard Boundary
1:37
When a Pulse Reaches a Soft or Flexible Boundary
2:04
Types of Waves
2:44
Mechanical Waves
2:56
Electromagnetic Waves
3:14
Types of Wave Motion
3:38
Longitudinal Waves
3:39
Transverse Waves
4:18
Anatomy of a Transverse Wave
5:18
Example 1: Waves Requiring a Medium
6:59
Example 2: Direction of Displacement
7:36
Example 3: Bell in a Vacuum Jar
8:47
Anatomy of a Longitudinal Wave
9:22
Example 4: Tuning Fork
9:57
Example 5: Amplitude of a Sound Wave
10:24
Frequency and Period
10:47
Example 6: Period of an EM Wave
11:23
Example 7: Frequency and Period
12:01
The Wave Equation
12:32
Velocity of a Wave is a Function of the Type of Wave and the Medium It Travels Through
12:36
Speed of a Wave is Related to Its Frequency and Wavelength
12:41
Example 8: Wavelength Using the Wave Equation
13:54
Example 9: Period of an EM Wave
14:35
Example 10: Blue Whale Waves
16:03
Sound Waves
17:29
Sound is a Mechanical Wave Observed by Detecting Vibrations in the Inner Ear
17:33
Particles of Sound Wave Vibrate Parallel With the Direction of the Wave's Velocity
17:56
Example 11: Distance from Speakers
18:24
Resonance
19:45
An Object with the Same 'Natural Frequency' May Begin to Vibrate at This Frequency
19:55
Classic Example
20:01
Example 12: Vibrating Car
20:32
Example 13: Sonar Signal
21:28
Example 14: Waves Across Media
24:06
Example 15: Wavelength of Middle C
25:24
Wave Interference
20m 45s
Intro
0:00
Objectives
0:09
Superposition
0:30
When More Than One Wave Travels Through the Same Location in the Same Medium
0:32
The Total Displacement is the Sum of All the Individual Displacements of the Waves
0:46
Example 1: Superposition of Pulses
1:01
Types of Interference
2:02
Constructive Interference
2:05
Destructive Interference
2:18
Example 2: Interference
2:47
Example 3: Shallow Water Waves
3:27
Standing Waves
4:23
When Waves of the Same Frequency and Amplitude Traveling in Opposite Directions Meet in the Same Medium
4:26
A Wave in Which Nodes Appear to be Standing Still and Antinodes Vibrate with Maximum Amplitude Above and Below the Axis
4:35
Standing Waves in String Instruments
5:36
Standing Waves in Open Tubes
8:49
Standing Waves in Closed Tubes
9:57
Interference From Multiple Sources
11:43
Constructive
11:55
Destructive
12:14
Beats
12:49
Two Sound Waves with Almost the Same Frequency Interfere to Create a Beat Pattern
12:52
A Frequency Difference of 1 to 4 Hz is Best for Human Detection of Beat Phenomena
13:05
Example 4
14:13
Example 5
18:03
Example 6
19:14
Example 7: Superposition
20:08
Wave Phenomena
19m 2s
Intro
0:00
Objective
0:08
Doppler Effect
0:36
The Shift In A Wave's Observed Frequency Due to Relative Motion Between the Source of the Wave and Observer
0:39
When Source and/or Observer Move Toward Each Other
0:45
When Source and/or Observer Move Away From Each Other
0:52
Practical Doppler Effect
1:01
Vehicle Traveling Past You
1:05
1:56
Doppler Effect - Astronomy
2:43
Observed Frequencies Are Slightly Lower Than Scientists Would Predict
2:50
More Distant Celestial Objects Are Moving Away from the Earth Faster Than Nearer Objects
3:22
Example 1: Car Horn
3:36
Example 2: Moving Speaker
4:13
Diffraction
5:35
The Bending of Waves Around Obstacles
5:37
Most Apparent When Wavelength Is Same Order of Magnitude as the Obstacle/ Opening
6:10
Single-Slit Diffraction
6:16
Double-Slit Diffraction
8:13
Diffraction Grating
11:07
Sharper and Brighter Maxima
11:46
Useful for Determining Wavelengths Accurately
12:07
Example 3: Double Slit Pattern
12:30
Example 4: Determining Wavelength
16:05
18:04
Example 6: Red Shift
18:29
Light As a Wave
11m 35s
Intro
0:00
Objectives
0:14
Electromagnetic (EM) Waves
0:31
Light is an EM Wave
0:43
EM Waves Are Transverse Due to the Modulation of the Electric and Magnetic Fields Perpendicular to the Wave Velocity
1:00
Electromagnetic Wave Characteristics
1:37
The Product of an EM Wave's Frequency and Wavelength Must be Constant in a Vacuum
1:43
Polarization
3:36
Unpoloarized EM Waves Exhibit Modulation in All Directions
3:47
Polarized Light Consists of Light Vibrating in a Single Direction
4:07
Polarizers
4:29
Materials Which Act Like Filters to Only Allow Specific Polarizations of Light to Pass
4:33
Polarizers Typically Are Sheets of Material in Which Long Molecules Are Lined Up Like a Picket Fence
5:10
Polarizing Sunglasses
5:22
Reduce Reflections
5:26
Polarizing Sunglasses Have Vertical Polarizing Filters
5:48
Liquid Crystal Displays
6:08
LCDs Use Liquid Crystals in a Suspension That Align Themselves in a Specific Orientation When a Voltage is Applied
6:13
Cross-Orienting a Polarizer and a Matrix of Liquid Crystals so Light Can Be Modulated Pixel-by-Pixel
6:26
Example 1: Color of Light
7:30
Example 2: Analyzing an EM Wave
8:49
Example 3: Remote Control
9:45
Example 4: Comparing EM Waves
10:32
Reflection & Mirrors
24m 32s
Intro
0:00
Objectives
0:10
Waves at Boundaries
0:37
Reflected
0:43
Transmitted
0:45
Absorbed
0:48
Law of Reflection
0:58
The Angle of Incidence is Equal to the Angle of Reflection
1:00
They Are Both Measured From a Line Perpendicular, or Normal, to the Reflecting Surface
1:22
Types of Reflection
1:54
Diffuse Reflection
1:57
Specular Reflection
2:08
Example 1: Specular Reflection
2:24
Mirrors
3:20
Light Rays From the Object Reach the Plane Mirror and Are Reflected to the Observer
3:27
Virtual Image
3:33
Magnitude of Image Distance
4:05
Plane Mirror Ray Tracing
4:15
Object Distance
4:26
Image Distance
4:43
Magnification of Image
7:03
Example 2: Plane Mirror Images
7:28
Example 3: Image in a Plane Mirror
7:51
Spherical Mirrors
8:10
Inner Surface of a Spherical Mirror
8:19
Outer Surface of a Spherical Mirror
8:30
Focal Point of a Spherical Mirror
8:40
Converging
8:51
Diverging
9:00
Concave (Converging) Spherical Mirrors
9:09
Light Rays Coming Into a Mirror Parallel to the Principal Axis
9:14
Light Rays Passing Through the Center of Curvature
10:17
Light Rays From the Object Passing Directly Through the Focal Point
10:52
Mirror Equation (Lens Equation)
12:06
Object and Image Distances Are Positive on the Reflecting Side of the Mirror
12:13
Formula
12:19
Concave Mirror with Object Inside f
12:39
Example 4: Concave Spherical Mirror
14:21
Example 5: Image From a Concave Mirror
14:51
Convex (Diverging) Spherical Mirrors
16:29
Light Rays Coming Into a Mirror Parallel to the Principal Axis
16:37
Light Rays Striking the Center of the Mirror
16:50
Light Rays Never Converge on the Reflective Side of a Convex Mirror
16:54
Convex Mirror Ray Tracing
17:07
Example 6: Diverging Rays
19:12
Example 7: Focal Length
19:28
Example 8: Reflected Sonar Wave
19:53
Example 9: Plane Mirror Image Distance
20:20
Example 10: Image From a Concave Mirror
21:23
Example 11: Converging Mirror Image Distance
23:09
Refraction & Lenses
39m 42s
Intro
0:00
Objectives
0:09
Refraction
0:42
When a Wave Reaches a Boundary Between Media, Part of the Wave is Reflected and Part of the Wave Enters the New Medium
0:43
Wavelength Must Change If the Wave's Speed Changes
0:57
Refraction is When This Causes The Wave to Bend as It Enters the New Medium
1:12
Marching Band Analogy
1:22
Index of Refraction
2:37
Measure of How Much Light Slows Down in a Material
2:40
Ratio of the Speed of an EM Wave in a Vacuum to the Speed of an EM Wave in Another Material is Known as Index of Refraction
3:03
Indices of Refraction
3:21
Dispersion
4:01
White Light is Refracted Twice in Prism
4:23
Index of Refraction of the Prism Material Varies Slightly with Respect to Frequency
4:41
Example 1: Determining n
5:14
Example 2: Light in Diamond and Crown Glass
5:55
Snell's Law
6:24
The Amount of a Light Wave Bends As It Enters a New Medium is Given by the Law of Refraction
6:32
Light Bends Toward the Normal as it Enters a Material With a Higher n
7:08
Light Bends Toward the Normal as it Enters a Material With a Lower n
7:14
Example 3: Angle of Refraction
7:42
Example 4: Changes with Refraction
9:31
Total Internal Reflection
10:10
When the Angle of Refraction Reaches 90 Degrees
10:23
Critical Angle
10:34
Total Internal Reflection
10:51
Applications of TIR
12:13
Example 5: Critical Angle of Water
13:17
Thin Lenses
14:15
Convex Lenses
14:22
Concave Lenses
14:31
Convex Lenses
15:24
Rays Parallel to the Principal Axis are Refracted Through the Far Focal Point of the Lens
15:28
A Ray Drawn From the Object Through the Center of the Lens Passes Through the Center of the Lens Unbent
15:53
Example 6: Converging Lens Image
16:46
Example 7: Image Distance of Convex Lens
17:18
Concave Lenses
18:21
Rays From the Object Parallel to the Principal Axis Are Refracted Away from the Principal Axis on a Line from the Near Focal Point Through the Point Where the Ray Intercepts the Center of the Lens
18:25
Concave Lenses Produce Upright, Virtual, Reduced Images
20:30
Example 8: Light Ray Thought a Lens
20:36
Systems of Optical Elements
21:05
Find the Image of the First Optical Elements and Utilize It as the Object of the Second Optical Element
21:16
Example 9: Lens and Mirrors
21:35
Thin Film Interference
27:22
When Light is Incident Upon a Thin Film, Some Light is Reflected and Some is Transmitted Into the Film
27:25
If the Transmitted Light is Again Reflected, It Travels Back Out of the Film and Can Interfere
27:31
Phase Change for Every Reflection from Low-Index to High-Index
28:09
Example 10: Thin Film Interference
28:41
Example 11: Wavelength in Diamond
32:07
Example 12: Light Incident on Crown Glass
33:57
Example 13: Real Image from Convex Lens
34:44
Example 14: Diverging Lens
35:45
Example 15: Creating Enlarged, Real Images
36:22
Example 16: Image from a Converging Lens
36:48
Example 17: Converging Lens System
37:50
Wave-Particle Duality
23m 47s
Intro
0:00
Objectives
0:11
Duality of Light
0:37
Photons
0:47
Dual Nature
0:53
Wave Evidence
1:00
Particle Evidence
1:10
Blackbody Radiation & the UV Catastrophe
1:20
Very Hot Objects Emitted Radiation in a Specific Spectrum of Frequencies and Intensities
1:25
Color Objects Emitted More Intensity at Higher Wavelengths
1:45
1:56
Photoelectric Effect
2:38
EM Radiation Striking a Piece of Metal May Emit Electrons
2:41
Not All EM Radiation Created Photoelectrons
2:49
Photons of Light
3:23
Photon Has Zero Mass, Zero Charge
3:32
Energy of a Photon is Quantized
3:36
Energy of a Photon is Related to its Frequency
3:41
Creation of Photoelectrons
4:17
Electrons in Metals Were Held in 'Energy Walls'
4:20
Work Function
4:32
Cutoff Frequency
4:54
Kinetic Energy of Photoelectrons
5:14
Electron in a Metal Absorbs a Photon with Energy Greater Than the Metal's Work Function
5:16
Electron is Emitted as a Photoelectron
5:24
Any Absorbed Energy Beyond That Required to Free the Electron is the KE of the Photoelectron
5:28
Photoelectric Effect in a Circuit
6:37
Compton Effect
8:28
Less of Energy and Momentum
8:49
Lost by X-Ray Equals Energy and Gained by Photoelectron
8:52
Compton Wavelength
9:09
Major Conclusions
9:36
De Broglie Wavelength
10:44
Smaller the Particle, the More Apparent the Wave Properties
11:03
Wavelength of a Moving Particle is Known as Its de Broglie Wavelength
11:07
Davisson-Germer Experiment
11:29
Verifies Wave Nature of Moving Particles
11:30
Shoot Electrons at Double Slit
11:34
Example 1
11:46
Example 2
13:07
Example 3
13:48
Example 4A
15:33
Example 4B
18:47
Example 5: Wave Nature of Light
19:54
Example 6: Moving Electrons
20:43
Example 7: Wavelength of an Electron
21:11
Example 8: Wrecking Ball
22:50
Section 7: Modern Physics
Atomic Energy Levels
14m 21s
Intro
0:00
Objectives
0:09
Rutherford's Gold Foil Experiment
0:35
Most of the Particles Go Through Undeflected
1:12
Some Alpha Particles Are Deflected Large Amounts
1:15
Atoms Have a Small, Massive, Positive Nucleus
1:20
Electrons Orbit the Nucleus
1:23
Most of the Atom is Empty Space
1:26
Problems with Rutherford's Model
1:31
Charges Moving in a Circle Accelerate, Therefore Classical Physics Predicts They Should Release Photons
1:39
Lose Energy When They Release Photons
1:46
Orbits Should Decay and They Should Be Unstable
1:50
Bohr Model of the Atom
2:09
Electrons Don't Lose Energy as They Accelerate
2:20
Each Atom Allows Only a Limited Number of Specific Orbits at Each Energy Level
2:35
Electrons Must Absorb or Emit a Photon of Energy to Change Energy Levels
2:40
Energy Level Diagrams
3:29
n=1 is the Lowest Energy State
3:34
Negative Energy Levels Indicate Electron is Bound to Nucleus of the Atom
4:03
When Electron Reaches 0 eV It Is No Longer Bound
4:20
Electron Cloud Model (Probability Model)
4:46
Electron Only Has A Probability of Being Located in Certain Regions Surrounding the Nucleus
4:53
Electron Orbitals Are Probability Regions
4:58
Atomic Spectra
5:16
Atoms Can Only Emit Certain Frequencies of Photons
5:19
Electrons Can Only Absorb Photons With Energy Equal to the Difference in Energy Levels
5:34
This Leads to Unique Atomic Spectra of Emitted and Absorbed Radiation for Each Element
5:37
Incandescence Emits a Continuous Energy
5:43
If All Colors of Light Are Incident Upon a Cold Gas, The Gas Only Absorbs Frequencies Corresponding to Photon Energies Equal to the Difference Between the Gas's Atomic Energy Levels
6:16
Continuous Spectrum
6:42
Absorption Spectrum
6:50
Emission Spectrum
7:08
X-Rays
7:36
The Photoelectric Effect in Reverse
7:38
Electrons Are Accelerated Through a Large Potential Difference and Collide with a Molybdenum or Platinum Plate
7:53
Example 1: Electron in Hydrogen Atom
8:24
Example 2: EM Emission in Hydrogen
10:05
Example 3: Photon Frequencies
11:30
Example 4: Bright-Line Spectrum
12:24
Example 5: Gas Analysis
13:08
Nuclear Physics
15m 47s
Intro
0:00
Objectives
0:08
The Nucleus
0:33
Protons Have a Charge or +1 e
0:39
Neutrons Are Neutral (0 Charge)
0:42
Held Together by the Strong Nuclear Force
0:43
Example 1: Deconstructing an Atom
1:20
Mass-Energy Equivalence
2:06
Mass is a Measure of How Much Energy an Object Contains
2:16
Universal Conservation of Laws
2:31
Nuclear Binding Energy
2:53
A Strong Nuclear Force Holds Nucleons Together
3:04
Mass of the Individual Constituents is Greater Than the Mass of the Combined Nucleus
3:19
Binding Energy of the Nucleus
3:32
Mass Defect
3:37
Nuclear Decay
4:30
Alpha Decay
4:42
Beta Decay
5:09
Gamma Decay
5:46
Fission
6:40
The Splitting of a Nucleus Into Two or More Nuclei
6:42
For Larger Nuclei, the Mass of Original Nucleus is Greater Than the Sum of the Mass of the Products When Split
6:47
Fusion
8:14
The Process of Combining Two Or More Smaller Nuclei Into a Larger Nucleus
8:15
This Fuels Our Sun and Stars
8:28
Basis of Hydrogen Bomb
8:31
Forces in the Universe
9:00
Strong Nuclear Force
9:06
Electromagnetic Force
9:13
Weak Nuclear Force
9:22
Gravitational Force
9:27
Example 2: Deuterium Nucleus
9:39
Example 3: Particle Accelerator
10:24
Example 4: Tritium Formation
12:03
Example 5: Beta Decay
13:02
Example 6: Gamma Decay
14:15
Example 7: Annihilation
14:39
Section 8: Sample AP Exams
AP Practice Exam: Multiple Choice, Part 1
38m 1s
Intro
0:00
Problem 1
1:33
Problem 2
1:57
Problem 3
2:50
Problem 4
3:46
Problem 5
4:13
Problem 6
4:41
Problem 7
6:12
Problem 8
6:49
Problem 9
7:49
Problem 10
9:31
Problem 11
10:08
Problem 12
11:03
Problem 13
11:30
Problem 14
12:28
Problem 15
14:04
Problem 16
15:05
Problem 17
15:55
Problem 18
17:06
Problem 19
18:43
Problem 20
19:58
Problem 21
22:03
Problem 22
22:49
Problem 23
23:28
Problem 24
24:04
Problem 25
25:07
Problem 26
26:46
Problem 27
28:03
Problem 28
28:49
Problem 29
30:20
Problem 30
31:10
Problem 31
33:03
Problem 32
33:46
Problem 33
34:47
Problem 34
36:07
Problem 35
36:44
AP Practice Exam: Multiple Choice, Part 2
37m 49s
Intro
0:00
Problem 36
0:18
Problem 37
0:42
Problem 38
2:13
Problem 39
4:10
Problem 40
4:47
Problem 41
5:52
Problem 42
7:22
Problem 43
8:16
Problem 44
9:11
Problem 45
9:42
Problem 46
10:56
Problem 47
12:03
Problem 48
13:58
Problem 49
14:49
Problem 50
15:36
Problem 51
15:51
Problem 52
17:18
Problem 53
17:59
Problem 54
19:10
Problem 55
21:27
Problem 56
22:40
Problem 57
23:19
Problem 58
23:50
Problem 59
25:35
Problem 60
26:45
Problem 61
27:57
Problem 62
28:32
Problem 63
29:52
Problem 64
30:27
Problem 65
31:27
Problem 66
32:22
Problem 67
33:18
Problem 68
35:21
Problem 69
36:27
Problem 70
36:46
AP Practice Exam: Free Response, Part 1
16m 53s
Intro
0:00
Question 1
0:23
Question 2
8:55
AP Practice Exam: Free Response, Part 2
9m 20s
Intro
0:00
Question 3
0:14
Question 4
4:34
AP Practice Exam: Free Response, Part 3
18m 12s
Intro
0:00
Question 5
0:15
Question 6
3:29
Question 7
6:18
Question 8
12:53
Metric Estimation
3m 53s
Intro
0:00
Question 1
0:38
Question 2
0:51
Question 3
1:09
Question 4
1:24
Question 5
1:49
Question 6
2:11
Question 7
2:27
Question 8
2:49
Question 9
3:03
Question 10
3:23
Defining Motion
7m 6s
Intro
0:00
Question 1
0:13
Question 2
0:50
Question 3
1:56
Question 4
2:24
Question 5
3:32
Question 6
4:01
Question 7
5:36
Question 8
6:36
Motion Graphs
6m 48s
Intro
0:00
Question 1
0:13
Question 2
2:01
Question 3
3:06
Question 4
3:41
Question 5
4:30
Question 6
5:52
Horizontal Kinematics
8m 16s
Intro
0:00
Question 1
0:19
Question 2
2:19
Question 3
3:16
Question 4
4:36
Question 5
6:43
Free Fall
7m 56s
Intro
0:00
Question 1-4
0:12
Question 5
2:36
Question 6
3:11
Question 7
4:44
Question 8
6:16
Projectile Motion
4m 17s
Intro
0:00
Question 1
0:13
Question 2
0:45
Question 3
1:25
Question 4
2:00
Question 5
2:32
Question 6
3:38
Newton's 1st Law
4m 34s
Intro
0:00
Question 1
0:15
Question 2
1:02
Question 3
1:50
Question 4
2:04
Question 5
2:26
Question 6
2:54
Question 7
3:11
Question 8
3:29
Question 9
3:47
Question 10
4:02
Newton's 2nd Law
5m 40s
Intro
0:00
Question 1
0:16
Question 2
0:55
Question 3
1:50
Question 4
2:40
Question 5
3:33
Question 6
3:56
Question 7
4:29
Newton's 3rd Law
3m 44s
Intro
0:00
Question 1
0:17
Question 2
0:44
Question 3
1:14
Question 4
1:51
Question 5
2:11
Question 6
2:29
Question 7
2:53
Friction
6m 37s
Intro
0:00
Question 1
0:13
Question 2
0:47
Question 3
1:25
Question 4
2:26
Question 5
3:43
Question 6
4:41
Question 7
5:13
Question 8
5:50
Ramps and Inclines
6m 13s
Intro
0:00
Question 1
0:18
Question 2
1:01
Question 3
2:50
Question 4
3:11
Question 5
5:08
Circular Motion
5m 17s
Intro
0:00
Question 1
0:21
Question 2
1:01
Question 3
1:50
Question 4
2:33
Question 5
3:10
Question 6
3:31
Question 7
3:56
Question 8
4:33
Gravity
6m 33s
Intro
0:00
Question 1
0:19
Question 2
1:05
Question 3
2:09
Question 4
2:53
Question 5
3:17
Question 6
4:00
Question 7
4:41
Question 8
5:20
Momentum & Impulse
9m 29s
Intro
0:00
Question 1
0:19
Question 2
2:17
Question 3
3:25
Question 4
3:56
Question 5
4:28
Question 6
5:04
Question 7
6:18
Question 8
6:57
Question 9
7:47
Conservation of Momentum
9m 33s
Intro
0:00
Question 1
0:15
Question 2
2:08
Question 3
4:03
Question 4
4:10
Question 5
6:08
Question 6
6:55
Question 7
8:26
Work & Power
6m 2s
Intro
0:00
Question 1
0:13
Question 2
0:29
Question 3
0:55
Question 4
1:36
Question 5
2:18
Question 6
3:22
Question 7
4:01
Question 8
4:18
Question 9
4:49
Springs
7m 59s
Intro
0:00
Question 1
0:13
Question 4
2:26
Question 5
3:37
Question 6
4:39
Question 7
5:28
Question 8
5:51
Energy & Energy Conservation
8m 47s
Intro
0:00
Question 1
0:18
Question 2
1:27
Question 3
1:44
Question 4
2:33
Question 5
2:44
Question 6
3:33
Question 7
4:41
Question 8
5:19
Question 9
5:37
Question 10
7:12
Question 11
7:40
Electric Charge
7m 6s
Intro
0:00
Question 1
0:10
Question 2
1:03
Question 3
1:32
Question 4
2:12
Question 5
3:01
Question 6
3:49
Question 7
4:24
Question 8
4:50
Question 9
5:32
Question 10
5:55
Question 11
6:26
Coulomb's Law
4m 13s
Intro
0:00
Question 1
0:14
Question 2
0:47
Question 3
1:25
Question 4
2:25
Question 5
3:01
Electric Fields & Forces
4m 11s
Intro
0:00
Question 1
0:19
Question 2
0:51
Question 3
1:30
Question 4
2:19
Question 5
3:12
Electric Potential
5m 12s
Intro
0:00
Question 1
0:14
Question 2
0:42
Question 3
1:08
Question 4
1:43
Question 5
2:22
Question 6
2:49
Question 7
3:14
Question 8
4:02
Electrical Current
6m 54s
Intro
0:00
Question 1
0:13
Question 2
0:42
Question 3
2:01
Question 4
3:02
Question 5
3:52
Question 6
4:15
Question 7
4:37
Question 8
4:59
Question 9
5:50
Resistance
5m 15s
Intro
0:00
Question 1
0:12
Question 2
0:53
Question 3
1:44
Question 4
2:31
Question 5
3:21
Question 6
4:06
Ohm's Law
4m 27s
Intro
0:00
Question 1
0:12
Question 2
0:33
Question 3
0:59
Question 4
1:32
Question 5
1:56
Question 6
2:50
Question 7
3:19
Question 8
3:50
Circuit Analysis
6m 36s
Intro
0:00
Question 1
0:12
Question 2
2:16
Question 3
2:33
Question 4
2:42
Question 5
3:18
Question 6
5:51
Question 7
6:00
Magnetism
3m 43s
Intro
0:00
Question 1
0:16
Question 2
0:31
Question 3
0:56
Question 4
1:19
Question 5
1:35
Question 6
2:36
Question 7
3:03
Wave Basics
4m 21s
Intro
0:00
Question 1
0:13
Question 2
0:36
Question 3
0:47
Question 4
1:13
Question 5
1:27
Question 6
1:39
Question 7
1:54
Question 8
2:22
Question 9
2:51
Question 10
3:32
Wave Characteristics
5m 33s
Intro
0:00
Question 1
0:23
Question 2
1:04
Question 3
2:01
Question 4
2:50
Question 5
3:12
Question 6
3:57
Question 7
4:16
Question 8
4:42
Question 9
4:56
Wave Behaviors
3m 52s
Intro
0:00
Question 1
0:13
Question 2
0:40
Question 3
1:04
Question 4
1:17
Question 5
1:39
Question 6
2:07
Question 7
2:41
Question 8
3:09
Reflection
3m 48s
Intro
0:00
Question 1
0:12
Question 2
0:50
Question 3
1:29
Question 4
1:46
Question 5
3:08
Refraction
2m 49s
Intro
0:00
Question 1
0:29
Question 5
1:03
Question 6
1:24
Question 7
2:01
Diffraction
2m 34s
Intro
0:00
Question 1
0:16
Question 2
0:31
Question 3
0:50
Question 4
1:05
Question 5
1:37
Question 6
2:04
Electromagnetic Spectrum
7m 6s
Intro
0:00
Question 1
0:24
Question 2
0:39
Question 3
1:05
Question 4
1:51
Question 5
2:03
Question 6
2:58
Question 7
3:14
Question 8
3:52
Question 9
4:30
Question 10
5:04
Question 11
6:01
Question 12
6:16
Wave-Particle Duality
5m 30s
Intro
0:00
Question 1
0:15
Question 2
0:34
Question 3
0:53
Question 4
1:54
Question 5
2:16
Question 6
2:27
Question 7
2:42
Question 8
2:59
Question 9
3:45
Question 10
4:13
Question 11
4:33
Energy Levels
8m 13s
Intro
0:00
Question 1
0:25
Question 2
1:18
Question 3
1:43
Question 4
2:08
Question 5
3:17
Question 6
3:54
Question 7
4:40
Question 8
5:15
Question 9
5:54
Question 10
6:41
Question 11
7:14
Mass-Energy Equivalence
8m 15s
Intro
0:00
Question 1
0:19
Question 2
1:02
Question 3
1:37
Question 4
2:17
Question 5
2:55
Question 6
3:32
Question 7
4:13
Question 8
5:04
Question 9
5:29
Question 10
5:58
Question 11
6:48
Question 12
7:39
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• ## Related Books
### Vectors & Scalars
• Scalars are physical quantities with a magnitude (size) only.
• Vectors are quantities that have magnitude and direction.
• A vector field gives the value of a physical quantity described by that vector.
• Vectors may be added graphically by lining them up tip-to-tail, and drawing a line from the starting point of the first to the ending point of the last.
• Vectors may be broken up into components aligned to coordinate axes for ease of manipulation.
• Vectors may be represented in a variety of forms. Converting between forms may simplify calculations and manipulation.
• The resultant vector (R), or resultant, is the sum of a number of vectors.
### Vectors & Scalars
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
• Intro 0:00
• Objectives 0:05
• Scalars 0:29
• Definition of Scalar
• Temperature, Mass, Time
• Vectors 1:12
• Vectors are Quantities That Have Magnitude and Direction
• Represented by Arrows
• Vector Representations 1:47
• Graphical Vector Subtraction 4:58
• Vector Components 6:08
• Angle of a Vector 8:22
• Vector Notation 9:52
• Example 1: Vector Components 14:30
• Example 2: Vector Components 16:05
• Example 3: Vector Magnitude 17:26
• Example 4: Vector Addition 19:38
• Example 5: Angle of a Vector 24:06
### Transcription: Vectors & Scalars
Hi everyone, and welcome back to educator.com. This lesson is on vectors and scalars.0000
Our objectives are going to be to differentiate between scalar and vector quantities, to use scaled diagrams to represent and manipulate vectors, being able to break up a vector into x and y components, finding the angle of a vector when we are given it's components...0006
...and finally, performing basic mathematical operations on vectors such as addition and subtraction.0021
When we talk about vectors, what we are really talking about are different types of measurements in physics, different quantities, and there are really two types. Scalars and vectors.0028
Scalars are physical quantities that have a magnitude or a size only. They do not need a direction. Things like temperature, mass, and time. I know what you are thinking. Time has a direction, right? forward or backward. Well, not really.0039
When we are talking about direction, we are talking about things like north, south, east, west, up, down, left, right, over yonder, over yander.0055
That sort of direction. Forward or backward when we are talking about just a positive or negative value is not what we are talking about here.0061
On the other hand, vectors are quantities that have a magnitude and a direction. They need a direction to describe them fully, things like a velocity. You have a velocity of 10 m/s in a direction.0071
A force is applied in that direction. Or a momentum, you have a momentum in a specific direction.0083
Vectors, we typically represent by arrows. The direction of the arrow tells you the direction of the vector, obviously, and the length of the arrow represents the magnitude or size. The longer the arrow, the bigger the vector.0091
Let's take a look at a couple of vector representations. Let's call this nice happy blue one A, and this red one down here B.0105
Notice they both have the same direction but A is much smaller than B. A has a smaller magnitude than B. B is the longer arrow, with larger magnitude.0118
Now the other thing that is nice about vectors is as long as you keep their magnitude, their size and their direction the same, you can slide them around anywhere you want. You can move a vector as long as you do not change it's direction or its magnitude.0128
So if we want to, we could take vector A and instead of having it there, we can slide it somewhere over here, for example, give it the same direction and magnitude, make this one go away, and now there is A.0141
With the same value, same direction, same magnitude, we are allowed to move them like that.0152
Let's talk about how we would add up two vectors. A vector such as A and B. The little line over that means it's a vector. If we want to try and put together, A and add it to vector B, to get sum vector, C. The sum of those two vectors.0161
Well, graphically, here is the trick. Take any vectors you want to add, however many there are and if we slide them around so they are lined up tip to tail, we can then find the resultant, the sum of the vectors.0184
So here we have A and B but they are not lined up tip to tail. So, what I am going to do is I am going to redraw these so I am going to put A over here and B, I am going to line up so that it is now tip to tail with A. Hopefully something roughly like that.0199
So now we have A and B lined up so that they are tip to tail. To find the sum of the two vectors, all we have to do is draw a line from the starting point of the first to the ending point of our last vector, that must be the sum of the vectors, C.0217
Alright, does it make a difference what order you add things? Well if you think back to math, B plus A should be the same thing, and it is.0240
Lets prove it. We are going to redraw this now but we're going to do B first.0256
So what I am going to do is I am going to draw B down here, there's roughly B and now I am going to put A on it but I am going to line them up tip to tail, in this direction this time so B comes first and then A.0262
Once again when I go to draw the resultant, I go from the starting point of the first to the ending point of the last. Notice that I have the same thing. Same magnitude, same direction, same vector, same result.0277
Alright, how about graphical vector subtraction? Here we have A again and B. Put the line over them to indicate they are vectors. What do we do for A minus B?0297
The trick here is realizing that A minus B is the same as A plus the opposite of B.0313
What is the opposite of B? Well it is as simple as you might guess.0319
If we have B pointing in this direction with this magnitude, all I have to do is switch it's direction and there is negative b.0324
So if I want A plus negative B, let's just redraw them again, tip to tail. We will start A down here. There is A. Now negative B goes something like this.0333
So A plus negative B, or A - B, we go from the starting point of our first again to the ending point of our last. A plus negative B equals C. Basic vector manipulation.0349
Now when we have these vectors and they are lined up at angles, often times we can simplify our lives from a math perspective if we break them up into component vectors or pieces that add up to the sum.0368
If we pick those pieces carefully, so they line up with an axis, the math gets a whole lot easier and I am a huge fan of easier math.0381
Let's assume that we have some vector, A right here at some angle Θ from the horizontal. We could replace this with a vector along the X axis and a vector along the Y axis.0389
Notice that the blue vector plus the green vector, if we add them together, gives us that A vector, the vector we started with.0405
So we are going to take this A vector and we are going to replace it with this blue one and this green one. Two vectors that are a little simpler to deal with mathematically.0412
Let's call this the X component of A and let's call this the Y component of A. How do we figure out what those are?0420
If you notice, here, we have made a right triangle. Here is our hypotenuse, this is the opposite side because it is the opposite the angle and AX must be the adjacent side, it's beside the angle.0434
Now I can use trig to figure out AY is. AY, since it's the opposite side is going to be equal to A, the hypotenuse times the sine of that angle.0449
On the same note, AX is going to be A times the cosine of Θ because this is the adjacent side. remember SOHCAHTOA?0461
Sine of Θ equals the opposite side divided by the hypotenuse, cosine of Θ equals the adjacent side divided by the hypotenuse and tangent of Θ equals the opposite side divided by the adjacent.0474
All we are doing is we are finding out what this opposite and this adjacent side happens to be. So we can break up this vector A into components AX and AY that are going to be much easier to deal with mathematically.0488
We could also go back to finding the angle of the vector. If we know two of the three sides of these triangles, if we know both of the components, we can find the angle, if we know the hypotenuse and the opposite, we can find the angle.0501
How do we do that though? Well we have to go back to our trig functions.0511
Tangent of Θ equals the opposite over the adjacent side. Therefore Θ must be the inverse tangent of the opposite side divided by the adjacent side.0518
But what if we do not know opposite or adjacent? Well sine Θ is equal to the opposite over the hypotenuse. So if we know opposite over hypotenuse, we can find Θ by taking the inverse sine of the opposite side divided by the hypotenuse.0533
So if you know any two sides of this right triangle you are making with components, you can find the angle using basic trigonometry.0582
Let's talk for a few minutes about vector notation. You can express vectors in many different ways.0589
You can just draw it on a sheet of paper, you can express it mathematically, but want to do this as efficiently as possible. so I am going to show you some examples in 3 dimensions but you can always scale those back to just two dimensions0597
Let's start off by making an axis. We've got YX and lets have a ZX coming out towards us. If we have some vector A, we could express it as having an X component, a Y component, and a Z component.0609
On the other hand though, we could also look at in terms of what are known as unit vectors.0628
If we take a vector of length 1 along the X axis, magnitude of 1 along the X axis, we are going to call that specific vector ihap, length one along the X axis.0636
In the Y axis, we will do the same thing. A vector of unit length, of length 1 in the Y direction, we will call jhap.0647
In the Z direction, same idea. A vector of length 1 in the Z direction we'll call khap. Specific vector constants. So we could write A now, as some value, X value times ihap plus its Y value times jhap plus its Z value times khap.0658
So whatever the X value is, you multiply it by a vector unit length 1 in the X direction.0687
Y value times the unit vector of length one in the Y direction and the Z value times the unit vector of length one in the Z direction.0693
Another way to express vectors. That can be very useful when we get to the point of doing vector addition. Let's assume we have our axis here again.0703
Y, X and Z. And here let's put a vector that is 4 units in the X, 3 in the Y and out toward us, 1.0713
So let's call this point P, 4,3,1, which is defined by some vector P which is 4 units in the X, 3 in the Y and 1 in the Z.0724
Let's also define another vector Q. Let's go 2 units in the X, we won't go any in the Y, zero in the Y and let's come out toward us in the Z direction 1,2,3,4.0741
Let's call that point Q which is 2, 0, 4 and we'll label the vector from that origin to that point vector Q.0752
How do we add these vectors in multiple directions? Well, what we could say is that vector R is going to be equal to Vector P plus vector Q.0762
Therefore, let's write P as equal to 4,3,1 in this bracket notation for vectors and vector Q is equal to 2,0,4 in vector bracket notation.0778
Well if the left hand side is equal to the right here and the left hand side is equal to the right here, then we can add the left hand sides and add the right hand sides, they should still be equal.0799
What we can say then, if we add those two, and add those two. Therefore P plus Q which is equal to our R must be equal to, well, in vector bracket notation, we add up the X components 4 plus 2 is 6.0809
We add up the Y components, 3 plus 0 is 3, and we add up the z components, 1 plus 4 is 5.0838
So the resultant, R, would be 6 units in the X, 3 in the Y and then 5 towards us. Something like that in 3 dimensions.0845
Adding up vectors using that vector notation can make things a lot simpler especially when you don't want to go drawing all of the time.0861
Let's take a look at a vector component problem. A soccer player kicks a ball with the velocity of 10m/s with an angle of 30 degrees above the horizontal. Find the magnitude of the horizontal component, and vertical component of the ball's velocity.0869
I am going to start off with a diagram here. A Y axis, and an X axis and realize that the soccer player is kicking the ball with an initial velocity of 10m/s, so there is our vector, 10m/s at an angle of 30 degrees above the horizontal.0886
We want to know the horizontal component and the vertical component. As you recall, if we want the vertical component, if this is our initial velocity, P, then the Y component of that velocity is going to be V 10m/s times the sine of 30 degrees.0906
10m/s sine 30 should be 5m/s.0934
In similar fashion, the X component of velocity V is going to be V cosine Θ again or 10m/s times the cosine of 30 degrees.0937
Cosine 30 is 0.866 so 10 times that is going to be 8.66m/s. We have broken up V into it's X and Y components.0951
Alright, another one. An airplane flies with the velocity of 750 kmph 30 degrees south of east. What is the magnitude of the plane's eastward velocity?0962
Well let's draw a picture again. North, south, east and west. The airplane flies with a velocity of 750kmph 30 degrees south of east. That means start at east and go 30 degrees south.0977
So I am going to draw it's velocity as roughly that. 750kmph at an angle of 30 degrees south of east. If we want it's eastward velocity, the eastward component, that means we want the X component here.0996
X component of it's velocity, the X is going to be V cosine Θ or 750kmph times the cosine of 30 degrees 0.866 should give us something right around 650kmph.1020
Let's take a look at another one where we have to deal with vector magnitudes. A dog walks a lady 8 meters due north then 6 meters due east, I'm sure you've all seen that before. A big dog, a little person trying to walk it but really the dog is in charge?1040
Determine the magnitude of the dog's total displacement. Well if the dog walks the lady 8m due north, we'll have a vector 8m north and then 6m east. Determine the magnitude of the dog's total displacement.1057
Well if we start it down here and line these up tip to tail so that the total displacement is a straight line from where you start to where you finish, is going to go from here right to there. That's the displacement.1075
How do we find the magnitude of that? Well if we look, that's a right triangle. We can use the pythagorean theorem. A2 plus B2 equals C2 where A is our 8m, B is our 6m, C is going to be our hypotenuse or the displacement.1092
Therefore, this is going to have a magnitude of the square root of 8m2 plus 62 or the square root of 64 plus 36, square root of 100 is going to be 10 meters.1110
Say we wanted to know what this angle is. If we wanted to know that, we could take a look and say, you know, the X component of that green vector is going to be 6m, the Y component must be 8m.1126
Therefore if we wanted that angle, Θ is going to be the inverse tangent of the opposite side over the adjacent which is the inverse tangent of 8m over 6m which comes out to be about 53.1 degrees.1144
That would be our angle, Θ there too.1162
If it had asked us for the angle, it only asked us for the magnitude of the dog's total displacement which we found to be 10 meters.1165
Let's take a look at some more vector addition. A frog hops 4m at an angle 30 degrees north of east.1174
He then hops 6m at an angle of 60 degrees north of west. What is the frog's total displacement from his starting position?1184
This just screams for us to draw a picture here first. So let's draw our axis here. I have a Y axis, and an X and as we look at this, There's our X, here is our Y, The frog starts out 4 meters at an angle of 30 degrees. There is 4 meters at an angle of 30 degrees.1191
Then he is going to go and hop 6m at an angle of 60 degrees north of west. So 6m at an angle of 60 degrees north of west is probably something kind of like that.1202
That angle is 60 degrees north from west and that is 6m long, thats 4m long. What is the frog's total displacement from the starting position.1240
Well, I could find that out graphically, by drawing a line from the starting point of the first to the ending point of the last.1252
Or, if I wanted to do this analytically, or a little bit more exactly, I could take a look if our blue vector is A, A is equal to it's X component is going to be 4m cosine 30 degrees, and it's Y component is going to be 4m sine 30 degrees.1264
Our B vector, there in red, is going to be, well we have got 6m cosine 60 degrees for it's X component, but it is to the left, so let's make sure that's negative and it's Y component is 6m sine 60 degrees.1280
So if I wanted to find the resultant, the sum, vector C. C is just going to be equal to A plus B, so that's going to be 4m cosine 30 degrees. The X component of A, plus the X component B, negative 6m cosine 60 degrees. So that will give us the X component of C.1299
For the Y component, we add their Y components together. 4m sine 30 degrees from A plus 6m sine 60 degrees from B.1324
When I do the math here, I find out that C equals 4 cosine 30 plus negative 6 cosine 60, that is going to be about 0.46m and the Y component 4m sine 30 degrees plus 6m sine 60 degrees comes out to be 7.2 meters.1339
So there is our C vector. 0.46, so not much in the X, 7.2 in the Y. While we are here, let's find out it's magnitude and angle.1360
The magnitude of C, I take the C vector and take it's absolute value, I can find out by using the pythagorean theorem again since I know it's components.1371
That is going to be the square root of 0.462 plus 7.22 it comes out to be about 7.21m.1380
If we wanted it's angle as well, I am expecting a big angle here just by looking at the picture. Θ is going to be equal to the inverse tangent of the opposite side over the adjacent side.1392
The opposite side is the Y, 7.2 over the adjacent 0.46 for an angle of 86.3 degrees which is over here 86.3 degrees north of east.1410
So we could express the vector with magnitude and a direction or we could express it just by leaving it in the vector bracket notation. If we wanted to we could have even written it as 4 6m ihap plus 7.2m jhap. They are all equivalent.1421
Let's take a look at one more sample problem, the angle of a vector. Find the angle Θ depicted by the blue vector below given the X and Y components.1442
Since I am given the opposite side, opposite the angle Θ and the adjacent side, the side next to the angle, but not the hypotenuse, I am going to use the tangent function since tangent of Θ equals opposite over adjacent.1455
Therefore Θ is going to be the inverse tangent of the opposite side over the adjacent side. Or Θ equals the inverse tangent of the opposite side 10 divided by the adjacent, 5.77 or 60 degrees.1468
Hopefully this gets you a good start on vectors and scalars. We will be using them throughout the entire course. They are very important.1494
Thanks for watching educator.com. Make it a great day.1501 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8031191229820251, "perplexity": 3778.153219496149}, "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-2021-39/segments/1631780057366.40/warc/CC-MAIN-20210922132653-20210922162653-00107.warc.gz"} |
http://www.ms.u-tokyo.ac.jp/seminar/2017/sem17-187.html | ## 講演会
### 2017年10月11日(水)
11:00-12:00 数理科学研究科棟(駒場) 128号室
Ahmed Abbes 氏 (CNRS/IHES)
On Faltings' main comparison theorem in p-adic Hodge theory : the relative case (ENGLISH)
[ 講演概要 ]
In the appendix of his 2002 Asterisque article, Faltings roughly sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. I will briefly review the absolute case and then explain some of the key new inputs for the proof of the relative case (joint work with Michel Gros). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9270261526107788, "perplexity": 2282.2778861862066}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2018-39/segments/1537267157216.50/warc/CC-MAIN-20180921151328-20180921171728-00005.warc.gz"} |
https://edurev.in/course/quiz/attempt/-1_Test-Critical-Thickness-of-Insulation-Heat-Transfe/420d375c-5d5c-4dd5-a671-6303202e81b7 | Courses
# Test: Critical Thickness of Insulation & Heat Transfer From Extended Surfaces (Fins) - 2
## 30 Questions MCQ Test RRB JE for Mechanical Engineering | Test: Critical Thickness of Insulation & Heat Transfer From Extended Surfaces (Fins) - 2
Description
This mock test of Test: Critical Thickness of Insulation & Heat Transfer From Extended Surfaces (Fins) - 2 for Mechanical Engineering helps you for every Mechanical Engineering entrance exam. This contains 30 Multiple Choice Questions for Mechanical Engineering Test: Critical Thickness of Insulation & Heat Transfer From Extended Surfaces (Fins) - 2 (mcq) to study with solutions a complete question bank. The solved questions answers in this Test: Critical Thickness of Insulation & Heat Transfer From Extended Surfaces (Fins) - 2 quiz give you a good mix of easy questions and tough questions. Mechanical Engineering students definitely take this Test: Critical Thickness of Insulation & Heat Transfer From Extended Surfaces (Fins) - 2 exercise for a better result in the exam. You can find other Test: Critical Thickness of Insulation & Heat Transfer From Extended Surfaces (Fins) - 2 extra questions, long questions & short questions for Mechanical Engineering on EduRev as well by searching above.
QUESTION: 1
### It is proposed to coat a 1 mm diameter wire with enamel paint (k = 0.1W/mK) to increase heat transfer with air. If the air side heat transfer coefficient is 100 W/m2K, then optimum thickness of enamel paint should be:
Solution:
Ans. (b) Critical radius of insulation (rc) =
= 1mm
∴ Critical thickness of enamel point = = 0.5 mm
QUESTION: 2
Solution:
Ans. (c)
QUESTION: 3
### Upto the critical radius of insulation
Solution:
Ans. (c) The thickness upto which heat flow increases and after which heat flow decreases is termed as Critical thickness. In case of cylinders and spheres it is called 'Critical radius'.
QUESTION: 4
A hollow pipe of 1 cm outer diameter is to be insulated by thick cylindrical insulation having thermal conductivity 1 W/mK. The surfaceheat transfer coefficient on the insulation surface is 5 W/m2K. What isthe minimum effective thickness of insulation for causing the reduction in heat leakage from the insulated pipe?
Solution:
Ans. (c) Critical radius of insulation = 0.2m = 20cm
∴ Critical thickness of insulation ( Δr)C = rc − r1 = 20 - 0.5 = 19.5cm
QUESTION: 5
A copper wire of radius 0.5 mm is insulated with a sheathing ofthickness 1 mm having a thermal conductivity of 0.5 W/m – K. Theoutside surface convective heat transfer coefficient is 10 W/m2 – K. Ifthe thickness of insulation sheathing is raised by 10 mm, then theelectrical current-carrying capacity of the wire will:
Solution:
Ans. (a)
QUESTION: 6
A copper pipe carrying refrigerant at – 200 C is covered by cylindrical insulation of thermal conductivity 0.5 W/m K. The surface heat transfer coefficient over the insulation is 50 W/m2 K. The critical thickness ofthe insulation would be:
Solution:
Ans. (a) Critical radius of insulation ( rc) =
= 0.5m = 0.01m
QUESTION: 7
The temperature variation in a largeplate, as shown in the given figure,would correspond to which of thefollowing condition (s)?
2. Steady-state with variation of k
Select the correct answer using the codes given below:
Codes:
Solution:
Ans. (a)
QUESTION: 8
Water jacketed copper rod “D” m in diameter is used to carry the current. The water, which flows continuously maintains the rod temperature at Toi C during normal operation at “I” amps. The electrical resistance of the rod is known to be “R” Ω /m. If the coolant water ceased to be available and the heat removal diminished greatly, the rod would eventually melt. What is the time required for melting to occur if the melting point of the rod material is Tmp?
[Cp = specific heat, ρ = density of the rod material and L is the length of the rod]
Solution:
Ans. (a)
QUESTION: 9
Consider the following statements:
1. Under certain conditions, an increase in thickness of insulation may increase the heat loss from a heated pipe.
2. The heat loss from an insulated pipe reaches a maximum when theoutside radius of insulation is equal to the ratio of thermalconductivity to the surface coefficient.
3. Small diameter tubes are invariably insulated.
4. Economic insulation is based on minimum heat loss from pipe.
Of these statements
Solution:
Ans. (c)
QUESTION: 10
Consider the following statements:
1. Under certain conditions, an increase in thickness of insulation mayincrease the heat loss from a heated pipe.
2. The heat loss from an insulated pipe reaches a maximum when theoutside radius of insulation is equal to the ratio of thermalconductivity to the surface coefficient.
3. Small diameter tubes are invariably insulated.
4. Economic insulation is based on minimum heat loss from pipe.Of these statements
Solution:
Ans. (c)
QUESTION: 11
An electric cable of aluminium conductor (k = 240 W/mK) is to be insulated with rubber (k = 0.15 W/mK). The cable is to be located in air (h = 6W/m2). The critical thickness of insulation will be:
Solution:
Ans. (c)
QUESTION: 12
Match List-I (Parameter) with List-II (Definition) and select the correct answer using the codes given below the lists:
Nomenclature: h: Film heat transfer coefficient, ksolid: Thermal conductivity of solid, kfluid: Thermal conductivity of fluid, ρ: Density,
c: Specific heat, V: Volume, l: Length.
Solution:
Ans. (a)
QUESTION: 13
What is the critical radius of insulation for a sphere equal to?k = thermal conductivity in W/m-K
h = heat transfer coefficient in W/m2K
Solution:
Ans. (b) Critical radius of insulation for sphere in 2k/h and for cylinder is k/h
QUESTION: 14
In current carrying conductors, if the radius of the conductor is lessthan the critical radius, then addition of electrical insulation isdesirable, as
Solution:
Ans. (b)
QUESTION: 15
Provision of fins on a given heat transfer surface will be more it thereare:
Solution:
Ans. (c)
QUESTION: 16
In current carrying conductors, if the radius of the conductor is lessthan the critical radius, then addition of electrical insulation is desirable, as
Solution:
Ans. (b)
QUESTION: 17
Which one of the following is correct?
The effectiveness of a fin will be maximum in an environment with
Solution:
Ans. (a) The effectiveness of a fin can also be characterized as
It is a ratio of the thermal resistance due to convection to the thermal resistance
of a fin. In order to enhance heat transfer, the fin's resistance should be lower
than that of the resistance due only to convection.
QUESTION: 18
On heat transfer surface, fins are provided
Solution:
Ans. (c) By the use of a fin, surface area is increased due to which heat flow rate
increases. Increase in surface area decreases the surface convection resistance,
whereas the conduction resistance increases. The decrease in convection
resistance must be greater than the increase in conduction resistance in order to
increase the rate of heat transfer from the surface. In practical applications of
fins the surface resistance must be the controlling factor (the addition of fins
might decrease the heat transfer rate under some situations).
QUESTION: 19
In order to achieve maximum heat dissipation, the fin should be designed in such a way that:
Solution:
Ans. (a)
QUESTION: 20
Consider the following statements pertaining to large heat transferrate using fins:
1. Fins should be used on the side where heat transfer coefficient is small
2. Long and thick fins should be used
3. Short and thin fins should be used
4. Thermal conductivity of fin material should be large
Which of the above statements are correct?
Solution:
Ans. (d)
QUESTION: 21
Assertion (A): Nusselt number is always greater than unity.
Reason (R): Nusselt number is the ratio of two thermal resistances, onethe thermal resistance which would be offered by the fluid, if it wasstationary and the other, the thermal resistance associated with convective heat transfer coefficient at the surface.
Solution:
Ans. (a)
QUESTION: 22
Assertion (A): In a liquid-to-gas heat exchanger fins are provided in thegas side.
Reason (R): The gas offers less thermal resistance than liquid
Solution:
Ans. (c)
QUESTION: 23
Extended surfaces are used to increase the rate of heat transfer. When the convective heat transfer coefficient h = mk, the addition ofextended surface will:
Solution:
Ans. (c)
QUESTION: 24
Consider the following statements pertaining to heat transfer throughfins:
1. Fins are equally effective irrespective of whether they are on thehot side or cold side of the fluid.
2. The temperature along the fin is variable and hence the rate of heattransfer varies along the elements of the fin.
3. The fins may be made of materials that have a higher thermalconductivity than the material of the wall.
4. Fins must be arranged at right angles to the direction of flow of theworking fluid.Of these statements:
Solution:
Ans. (d)
QUESTION: 25
Addition of fin to the surface increases the heat transfer if is:
Solution:
Ans. (c) Addition of fin to the surface increases the heat transfer if << 1
QUESTION: 26
A fin of length 'l' protrudes from a surface held at temperature togreater than the ambient temperature ta. The heat dissipation from thefree end' of the fin is assumed to be negligible. The temperaturegradient at the fin tip is:
Solution:
Ans. (a)
QUESTION: 27
The insulated tip temperature of a rectangular longitudinal fin having an excess (over ambient) root temperature of θo is:
Solution:
Ans. (d)
QUESTION: 28
A metallic rod of uniform diameter and length L connects two heat sources each at 500°C. The atmospheric temperature is 30°C. The temperature gradient at the centre of the bar will be:
Solution:
Ans. (d)
QUESTION: 29
The efficiency of a pin fin with insulated tip is:
Solution:
Ans. (b)
QUESTION: 30
A fin of length l protrudes from a surface held at temperature To; it being higher than the ambient temperature Ta. The heat dissipationfrom the free end of the fin is stated to be negligibly small, What is the temperature gradienat the tip of the fin?
Solution:
Ans. (a)
= Negligibly small. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8111669421195984, "perplexity": 2756.3385095015974}, "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-2021-39/segments/1631780055775.1/warc/CC-MAIN-20210917181500-20210917211500-00128.warc.gz"} |
https://mech.subwiki.org/wiki/Special:MobileDiff/432 | # Changes
## Pulley system on a double inclined plane
, 00:07, 13 August 2011
no edit summary
This article is about the following scenario. A fixed triangular wedge has two inclines $I_1$ and $I_2$ making angles $\alpha_1$ and $\alpha_2$ with the horizontal, thus making it a [[involves::double inclined plane]]. A [[involves::pulley]] is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses $m_1$ and $m_2$, resting on the two inclines $I_1$ and $I_2$ respectively. The string is inextensible. The coefficients of static and kinetic friction between $m_1$ and $I_1$ are $\mu_{s1}$ and $\mu_{k1}$ respectively. The coefficients of static and kinetic friction between $m_2$ and $I_2$ are $\mu_{s2}$ and $\mu_{k2}$ respectively. Assume that $\mu_{k1} \le \mu_{s1}$ and $\mu_{k2} \le \mu_{s2}$.
We assume the pulley to be massless so that its moment of inertia can be ignored for the information below.
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https://www.gerad.ca/fr/papers/G-2021-26 | Retour
# Maintenance of a collection of machines under partial observability: Indexability and computation of Whittle index
référence BibTeX
We consider the problem of scheduling maintenance for a collection of machines under partial observations when the state of each machine deteriorates stochastically in a Markovian manner. We consider two observational models: first, the state of each machine is not observable at all, and second, the state of each machine is observable only if a service-person visits them. The agent takes a maintenance action, e.g., machine replacement, if he is chosen for the task. We model both problems as restless multi-armed bandit problem and propose the Whittle index policy for scheduling the visits. We show that both models are indexable. For the first model, we derive a closed-form expression for the Whittle index. For the second model, we propose an efficient algorithm to compute the Whittle index by exploiting the qualitative properties of the optimal policy. We present detailed numerical experiments which show that for multiple instances of the model, the Whittle index policy outperforms myopic policy and can be close-to-optimal in different setups.
, 29 pages
### Document
G2126.pdf (540 Ko) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8380492925643921, "perplexity": 604.1417051918577}, "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-2022-21/segments/1652662529658.48/warc/CC-MAIN-20220519172853-20220519202853-00086.warc.gz"} |
https://web2.0calc.com/questions/algebra_59041 | +0
# Algebra
0
93
1
The integers G and H are chosen such that
$\frac{G}{x+5}+\frac{H}{x^2-4x}=\frac{x^2-3x+5}{x^3+x^2-20x}$
for all real values of $x$ except -5, 0, and 4. Find $H/G$.
Jun 22, 2022
#1
+2448
0
Simplify the left-hand side with a common denominator: $$\large{{(x^2-4x)g + (x+5)h \over x^3 + x^2 - 20x } = {x^2 - 3x + 5 \over x^3 + x^2 - 20x}}$$
Now, recall that the denominators of the fractions are equal, we can ignore them, giving us: $$(x^2 - 4x)g + (x+5)h = x^2 - 3x + 5$$
The only way to get an $$x^2$$ term on the left-hand side is if it comes from the term $$(x^2 - 4x)g$$, meaning $$gx^2= x^2$$
Thus, we know that $$g = 1$$
Likewise, doing the same thing with the $$(x+5)h$$ term, we notice that the coefficient of 5 can only come form the term 5h, meaning $$h = 1$$
Thus, the ratio $${ h \over g} = \color{brown}\boxed{1}$$
Jun 22, 2022 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9877015948295593, "perplexity": 411.42589370194054}, "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-2022-49/segments/1669446710916.70/warc/CC-MAIN-20221202215443-20221203005443-00089.warc.gz"} |
https://www.lessonplanet.com/teachers/discovering-the-magical-pi | # Discovering the Magical Pi
Students calculate pi using data on the circumference and diameter of various objects. They define key vocabulary terms, measure the circumference and diameter of various circular objects, organize the data on a table or chart, and complete a Mystery Ratio worksheet.
Concepts
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https://zims-en.kiwix.campusafrica.gos.orange.com/wikipedia_en_all_nopic/A/Representation_ring | # Representation ring
In mathematics, especially in the area of algebra known as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the) finite-dimensional linear representations of the group. For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of algebraically closed fields of characteristic p where the Sylow p-subgroups are cyclic is also theoretically approachable.
## Formal definition
Given a group G and a field F, the elements of its representation ring RF(G) are the formal differences of isomorphism classes of finite dimensional linear F-representations of G. For the ring structure, addition is given by the direct sum of representations, and multiplication by their tensor product over F. When F is omitted from the notation, as in R(G), then F is implicitly taken to be the field of complex numbers.
Succinctly, the representation ring of G is the Grothendieck ring of the category of finite-dimensional representations of G.
## Examples
• For the complex representations of the cyclic group of order n, the representation ring RC(Cn) is isomorphic to Z[X]/(Xn 1), where X corresponds to the complex representation sending a generator of the group to a primitive nth root of unity.
• More generally, the complex representation ring of a finite abelian group may be identified with the group ring of the character group.
• For the rational representations of the cyclic group of order 3, the representation ring RQ(C3) is isomorphic to Z[X]/(X2 X 2), where X corresponds to the irreducible rational representation of dimension 2.
• For the modular representations of the cyclic group of order 3 over a field F of characteristic 3, the representation ring RF(C3) is isomorphic to Z[X,Y]/(X2 Y 1, XY 2Y,Y2 3Y).
• The continuous representation ring R(S1) for the circle group is isomorphic to Z[X, X 1]. The ring of real representations is the subring of R(G) of elements fixed by the involution on R(G) given by X X 1.
• The ring RC(S3) for the symmetric group on three points is isomorphic to Z[X,Y]/(XY Y,X2 1,Y2 X Y 1), where X is the 1-dimensional alternating representation and Y the 2-dimensional irreducible representation of S3.
## Characters
Any representation defines a character χ:GC. Such a function is constant on conjugacy classes of G, a so-called class function; denote the ring of class functions by C(G). If G is finite, the homomorphism R(G) → C(G) is injective, so that R(G) can be identified with a subring of C(G). For fields F whose characteristic divides the order of the group G, the homomorphism from RF(G) → C(G) defined by Brauer characters is no longer injective.
For a compact connected group R(G) is isomorphic to the subring of R(T) (where T is a maximal torus) consisting of those class functions that are invariant under the action of the Weyl group (Atiyah and Hirzebruch, 1961). For the general compact Lie group, see Segal (1968).
Given a representation of G and a natural number n, we can form the n-th exterior power of the representation, which is again a representation of G. This induces an operation λn : R(G) R(G). With these operations, R(G) becomes a λ-ring.
The Adams operations on the representation ring R(G) are maps Ψk characterised by their effect on characters χ:
${\displaystyle \Psi ^{k}\chi (g)=\chi (g^{k})\ .}$
The operations Ψk are ring homomorphisms of R(G) to itself, and on representations ρ of dimension d
${\displaystyle \Psi ^{k}(\rho )=N_{k}(\Lambda ^{1}\rho ,\Lambda ^{2}\rho ,\ldots ,\Lambda ^{d}\rho )\ }$
where the Λiρ are the exterior powers of ρ and Nk is the k-th power sum expressed as a function of the d elementary symmetric functions of d variables.
## References
• Atiyah, Michael F.; Hirzebruch, Friedrich (1961), "Vector bundles and homogeneous spaces", Proc. Sympos. Pure Math., American Mathematical Society, III: 7–38, MR 0139181, Zbl 0108.17705.
• Bröcker, Theodor; tom Dieck, Tammo (1985), Representations of Compact Lie Groups, Graduate Texts in Mathematics, 98, New York, Berlin, Heidelberg, Tokyo: Springer-Verlag, ISBN 0-387-13678-9, MR 1410059, OCLC 11210736, Zbl 0581.22009
• Segal, Graeme (1968), "The representation ring of a compact Lie group", Publ. Math. IHES, 34: 113–128, MR 0248277, Zbl 0209.06203.
• Snaith, V. P. (1994), Explicit Brauer Induction: With Applications to Algebra and Number Theory, Cambridge Studies in Advanced Mathematics, 40, Cambridge University Press, ISBN 0-521-46015-8, Zbl 0991.20005 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "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": 0.904704749584198, "perplexity": 498.9965139199707}, "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-2021-21/segments/1620243991759.1/warc/CC-MAIN-20210510174005-20210510204005-00027.warc.gz"} |
http://andysresearch.blogspot.com/2006_02_01_archive.html | # Andy's Math/CS page
## Thursday, February 16, 2006
### Fast but Useless
My thesis is about hard functions that are useless. But there's a natural structural condition on functions that makes them useful (and hard): they grow faster than Busy Beaver. In such cases they can act as a timer to solve the halting problem. Is this the end of the story?
A while ago Scott Aaronson mentioned the following question to me while discussing questions around my thesis and a post of his: is there a function asymptotically dominating all recursive functions that *doesn't* allow you to solve the halting problem? He said he'd asked Carl Jokusch, who gave a 'high-powered' proof. I found an easy proof that proved something stronger:
Theorem: Let RAND be the Kolmogorov-random strings; then there's a function f dominating all recursive functions such that RAND is 'immune' (one can't compute an infinite subset of it) even given oracle access to f.
(HALT is Turing reducible to RAND--even truth-table reducible, which is bizarre--so it follows that HALT is not Turing reducible to f. Actually, the theorem remains true if you replace RAND with any immune set.)
Proof: Let B[1](x), B[2](x), ... be a (nonconstructive) enumeration of all total recursive bounds. At each stage we set finitely many values of f to screw up the ith oracle machine M[i]'s attempt to decide an infinite RAND-subset. At stage i we restrict ourselves to setting new values f(x) >B[j](x) for j <= i; this gets the growth condition.
Case I: There's a finite partial completion of f, subject to the constraints/commitments so far, causing M[i] to accept some non-random string x.
Then of course, fix these f-values, along with a big value f(i) to ensure convergence of our partial f to a total function.
Case II: For no such partial completion can M[i] be so induced.
Then M[i] is not a problem: no matter how we set the function hereafter, M[i] can only accept a finite subset of RAND. Otherwise, we could code in the f-values committed so far along with programs for B[1](x), B[2](x), ... B[i](x) into oracle-free machine M', which on input x dovetails over all admissible partial completions of the oracle looking for one that makes M[i] accept x. Then M' accepts an infinite RAND subset, impossible.
So in Case II just fix a big value for f(i) to help
convergence. Continue over all i to get the desired f.
I've also explored a dual question (are there non-r.e.-hard infinite subsets of RAND?), with only partial success:
i) There is no algorithm that computes HALT when given oracle access to any infinite subset of RAND;
ii) There are infinite subsets of the 1/2-random strings (not compressible by a factor of two) that aren't r.e.-hard.
Happy to get questions or comments. Lance Fortnow and others have interesting results about the computational power of 'dense enough' subsets of 1/2-RAND.
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http://math.stackexchange.com/questions/194959/gradient-of-moreau-yosida-regularization | Let $f(x):\Re^n\rightarrow \Re$ be a proper and closed convex function. Its Moreau-Yosida regularization is defined as
$F(x)=\min_yf(y)+\frac{1}{2}\|y-x\|_2^2$
$Prox_f(x)=\arg\min_y f(y)+\frac{1}{2}\|y-x\|_2^2$
Lots of literature say $F(x)$ is Lipschtiz continuous and give explicitly the expression of $\nabla F(x)$ involving $Prox_f(x)$. But I have no idea how to calculate $\nabla F(x)$. Can anyone provide a straightforward method? I know Rockafellar's book gives a proof. But it assumes too much prior knowledge. I am wondering if there is a more elementary method to prove the Lipschtiz continuity and calculate its gradient.
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$\mathbb R$ is a much more common notation for reals than $\mathfrak R$. – user31373 Sep 13 '12 at 15:57
You mentioned the formula $\nabla f_{\mu}(x) = \frac{1}{\mu} ( x - \text{prox}_{\mu f}(x)$. In a lot of important applications, the prox operator of $f$ can be evaluated analytically. In other cases, you might have to solve a convex optimization problem in order to evaluate the prox operator of $f$. Vandenberghe's 236c notes (especially ch. 15 "multiplier methods") are a good resource for this topic. – littleO Oct 9 '13 at 2:34
One relevant fact is that if a function $g$ is strongly convex with parameter $\mu$, then its conjugate $g^*$ is differentiable and $\nabla g^*$ is Lipschitz continuous with parameter $\frac{1}{\mu}$. (See ch. 13 "dual decomposition" in Vandenberghe's 236c notes.)
Another fact is that the conjugate of $f_{(\mu)}$ is given by $f_{(\mu)}^*(y) = f^*(y) + \frac{\mu}{2} \|y\|_2^2$. So $f_{(\mu)}^*$ is strongly convex with parameter $\mu$. (See ch. 15 "multiplier methods" in the 236c notes).
It follows that $f_{(\mu)}^{**} = f_{(\mu)}$ is differentiable and its gradient is Lipschitz continuous with parameter $\frac{1}{\mu}$.
To compute the gradient of $f_{(\mu)}$, if the prox operator of $f$ can't be evaluated analytically, another option is to evaluate the prox operator of $f$ by solving a convex optimization problem.
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You will find the proof of the Lipschitz continuity of $F$ here.
You will not find $\nabla F$ in a straightforward way, i.e., without solving some nonlinear equations. Consider $\nabla f$ as a map from $\mathbb R^n$ to $\mathbb R^n$. Let $\psi$ be the inverse map to $\nabla f$. Then $\nabla F$ is the inverse of $\psi+\mathrm{id}$. Indeed, $\psi$ is the gradient of the Legendre transforms of $f$, which we can call $g$. The Legendre transform converts the infimal convolution of $f$ and $\frac12\|x\|^2$ into the sum of $g$ and $\frac12\|y\|^2$. Then we must take the transform again to return to the $x$-variable. In terms of the gradients this means taking inverses.
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Thanks for the answer. – mining Sep 19 '12 at 21:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9898964166641235, "perplexity": 70.90635481928766}, "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-2016-07/segments/1454701153585.76/warc/CC-MAIN-20160205193913-00293-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.nature.com/articles/s41562-022-01354-2?utm_medium=affiliate&utm_source=commission_junction&utm_campaign=CONR_PF018_ECOM_GL_PHSS_ALWYS_DEEPLINK&utm_content=textlink&utm_term=PID100085446&CJEVENT=73c482bbbc6011ed801c013c0a1cb825&error=cookies_not_supported&code=cfa61656-2146-4413-ad6a-f836d2c3df0e | ## Main
Third-party punishment is a disposition of individuals to punish transgressors or norm violators who have not harmed them directly, and it seems to be universal across cultures1. The dominant explanation is that this disposition is a mechanism for maintaining cooperation2,3,4,5,6. Third-party punishment is unique to humans7,8 and has been well documented in adults1,2,3. However, debates about its evolved propensity9 and motivations10 are ongoing, and its point of emergence in ontogeny remains unknown.
Previous research asserts that even 19-month-old toddlers are willing to punish antisocial individuals in third-party contexts by taking treats away from them11. Young children are willing to incur a cost to avoid interacting with wrongdoers12, intervene against or tattle on moral transgressions13, and seem to expect antisocial actions to be punished14. Moreover, children not only punish wrongdoers but also prioritize helping the victim15. For example, they return a resource to the victim rather than remove a resource from a thief when they have options to punish or help. By age six, children engage in costly third-party punishment; they sacrifice their own resources to punish a transgressor who has acted unfairly16,17 and punish moral transgressors to satisfy both consequentialist and retributive motives18. However, to our knowledge, little to no work has investigated third-party punishment in preverbal infants, and thus its point of emergence in ontogeny remains unknown.
We focused on physical aggression, which is assumed to be salient to preverbal infants. It may therefore function as an intuitive form of punishment and be the most basic form of aggression that infants prefer to intervene against. We specifically focused on the hitting action19 and hitting interactions between agents20,21,22. Infants can discriminate between caressing (positive) and hitting (negative) interactions involving two agents20, and the latter interactions are assumed to be negative from the infants’ viewpoint23. Moreover, not only do infants infer dominance hierarchies (the strong and the weak) from body size24, social interactions25 and relative height26, but they can also discriminate the aggressor from the victim in hitting interactions21. More importantly, infants show aversiveness to the aggressor21, affirm the agents who disturbed (doing negative action to) the aggressor and assume that the aggressor should be hit by other agents22. On the basis of current evidence, these types of actions might be functional as punitive behaviour, and the interaction might be worth interfering for infants.
This study aimed to reveal the developmental origins of third-party punishment in early infancy and determine whether and how preverbal infants punish antisocial agents who have not harmed them directly. We developed a participatory cognitive paradigm by adopting gaze-contingency techniques27,28,29, in which infants can use their gaze to affect agents displayed on a monitor. In this paradigm, fixation on an agent triggers the event of a stone crushing the agent. Prior research that used the same hitting interaction employed in the current study has demonstrated that infants over six months old regard this interaction as negative20,21,22,23 and that eight-month-olds can act on objects on a monitor by their gaze28,29. We therefore chose eight-month-olds as participants in this study. We familiarized infants with a gaze-contingent association between looking at one of two objects or agents and a subsequent punitive event (for example, stones falling and crushing one of the objects or agents; Fig. 1a, Experiment 1). We then compared their tendency to look at each agent before and after the aggressive interaction between agents (Fig. 1b and Supplementary Video 1). If, as a third party, infants are disposed to punish a transgressor, they will increase their selective gaze at the aggressor after watching an aggressive interaction.
## Results
In Experiment 1, 24 eight-month-old infants were familiarized with gaze-contingent events. When the infants fixated on a single object (for example, a red sphere or a blue sphere) or either of two objects presented side by side (for example, red and blue spheres), the contingent event (for example, a square stone falling and crushing the object) occurred in the practical phase. Subsequently, the infants experienced ten identical gaze-contingent events, except that the target objects were two geometric agents with eyes (for example, green and orange geometric shapes) (pretest; Fig. 1b). After watching an aggressive interaction between the geometric agents (one was the aggressor, and the other was the victim), the infants again experienced ten gaze-contingent events identical to the pretest (posttest; Fig. 1b). If the infants sought to punish the transgressor, it is likely that they would increase their selective looks at the aggressor in the posttest phase.
We conducted a generalized linear mixed model (GLMM) analysis with a binomial error structure and a logit link function to assess whether watching the aggressive interaction influenced selective looks in the posttest phase. The response variable was infant selective looks at the aggressor (= 1) or the victim (= 0) in the pretest or posttest. The explanatory variables included test type (pretest or posttest) and trial number. We compared models on the basis of the Bayes factor (BF). The model candidates were (1) the null model, (2) a model with the main effect of test type, (3) a model with the main effect of trial number, (4) a model with the main effects of test type and trial number, and (5) a model with the main effect of test type, the main effect of trial number and the interaction between test type and trial number. All models were compared with the null model, and we computed the BF (BF10)—namely, the relative evidence in favour of each model over the null model. We assumed that the prior model probability was uniform, and we evaluated the degree to which the data had changed the prior model odds for each model. We also computed the inclusion BF (ref. 30) (BFincl) for each effect to evaluate how probable the data were under models that included the effect compared with models that excluded the effect. To report BF10 and BFincl, we set the Cauchy distribution with location 0 and scale 1/√2 as a prior distribution for a coefficient parameter31. BFs are sensitive to the prior distribution for model parameters. It is therefore important to check whether the inferences from the data are robust to different prior specifications. We conducted a sensitivity analysis for BFincl, following recommendations made by previous studies regarding Bayesian analysis32,33.
According to Lee and Wagenmakers34, a BF of 1–3 is ‘anecdotal evidence’ or ‘can be considered’, 3–10 is ‘moderate evidence’, 10–30 is ‘strong evidence’ and 30–100 is ‘very strong evidence’ for the alternative hypothesis or model. In contrast, a BF of 1/3–1 is ‘anecdotal evidence’, 1/10–1/3 is ‘moderate evidence’, 1/30–1/10 is ‘strong evidence’ and 1/100–1/30 is ‘very strong evidence’ for the null hypothesis or model. A BF of 1 is ‘no evidence’ in favour of either the alternative hypothesis (model) or the null hypothesis (model).
The model comparison results demonstrated that the data were best represented by the model with the main effect of test type (Table 1). The posterior model probability of the model with the main effect of test type was the largest in the candidate models (P(M|data) = 0.590). The BF10 was 2.473, which indicated anecdotal evidence in favour of this model compared with the null model. Table 2 shows the inclusion probability and BFincl for each effect. On average, the data anecdotally supported the model including the main effect of test type (BFincl = 1.748) and moderately supported the model excluding the main effect of trial (BFincl = 0.139) and the interaction term (BFincl = 0.161). The results of the sensitivity analysis (Fig. 2) robustly supported the model including the main effect of test type against reasonable change in the Cauchy prior width for the effect size, although the evidence was anecdotal. However, the model excluding the main effect of trial and the interaction term was more likely to be supported as the prior width became large. Note that when the Cauchy prior width is zero, the BF equals 1—irrespective of the data. Infants’ selective looks at the aggressor increased in the posttest phase compared with the pretest for the best model. The effect of test type relative to the pretest had a 0.988 probability of being positive (test: posterior median, 0.742; 95% credible interval (CI), (0.102, 1.431); odds ratio (OR), 2.101; Supplementary Table 1). In summary, we found that eight-month-olds were more likely to look selectively towards the aggressor in the posttest than in the pretest; however, this result was inconclusive, as the evidence was anecdotal (Fig. 3a).
We subsequently considered three alternative parsimonious interpretations of selective looks at the aggressor before concluding that looking behaviours involved decision-making regarding punishment. First, the increase in infant selective looks at the aggressor could be due to mere visual preference for said aggressor (for example, preference for a causer of action). To exclude this possibility, in Experiment 2, we tested another group of infants (N = 24) who experienced aggressive interactions identical to those in Experiment 1 but with less negative gaze-contingent events in the pretest and posttest phases. Specifically, materials fell onto an object or agent more softly than in Experiment 1 (Fig. 1a, Experiment 2). If selective looks were driven by preference for the aggressor after watching aggressive interactions, infants would more likely selectively look at the aggressor at posttest even though the gaze-contingent event is less negative. However, if selective looks at the aggressor involved a sense of punishment, then infants would not selectively look at the aggressor at posttest because they have no means to punish the agent. In support of this latter prediction, the model comparison demonstrated that the data were best represented by the null model (Table 1). The posterior model probability of the null model was the largest in the candidate models (P(M|data) = 0.651). The BF10 was 1.000 since the null model was compared with itself. On average, the data moderately supported the model excluding the main effects of test type (BFincl = 0.190) and trial type (BFincl = 0.136), and very strongly supported the model excluding the interaction term (BFincl = 0.032) (Table 2). The results of the sensitivity analysis (Fig. 2) robustly supported the model excluding the two main effects and the interaction term against reasonable change in the Cauchy prior width for the effect size. The model excluding each effect was more likely to be supported as the prior width became large. In the null model, the proportion of an infant’s selective looks at the aggressor was not different from that at the chance level (intercept: posterior median, 0.067; 95% CI, (−0.242, 0.381); OR = 1.070; Supplementary Table 2). In summary, the data moderately supported the idea that eight-month-olds did not change the proportion of selective looks towards the aggressor between the pretest and the posttest (Fig. 3b). We therefore excluded the alternative parsimonious explanation that the increase in infant selective looks at the aggressor was due to a mere visual preference for the aggressor rather than a selective choice for punishment.
A second possible explanation for the increase in infant selective looks at the aggressor in Experiment 1 is a mere expectation that the aggressor would be punished35 as opposed to a sense that punitive action is the consequence of the infants’ intentions. To understand this, in Experiment 3, we decreased the strength of the gaze-contingent association. Specifically, we changed the reinforcement probability between looking at a specific agent and a subsequent punitive event from 100% (Experiment 1) to 50% (chance level) (Fig. 1a, Experiment 3). If infants looked at the aggressive agent because of a mere expectation that the agent would be punished, they would selectively look at the agent at posttest even without a sense of self-agency. However, if infants looked at the aggressive agent due to a sense that the punitive action is a consequence of their intentions (in other words, an understanding of their own causal efficacy), they would not selectively look at the aggressive agent at posttest when they lacked a sense of self-agency. Consistent with this latter prediction, the model comparison demonstrated that the data were best represented by the null model (Table 1). The posterior model probability of the null model was the largest in the candidate models (P(M|data) = 0.705). The BF10 was 1.000 since the null model was being compared with itself. On average, the data moderately supported the model excluding the main effect of test type (BFincl = 0.167), strongly supported the model excluding the main effect of trial type (BFincl = 0.095) and very strongly supported the model excluding the interaction term (BFincl = 0.022) (Table 2). The results of the sensitivity analysis (Fig. 2) robustly supported the model excluding the two main effects and the interaction term against reasonable change in the Cauchy prior width for the effect size. The model excluding each effect was more likely to be supported as the prior width became large. In the null model, the proportion of an infant’s selective looks at the aggressor was not different from that at the chance level (intercept: posterior median, 0.020; 95% CI, (−0.201, 0.238); OR = 1.020; Supplementary Table 3). In summary, the data moderately supported the idea that eight-month-olds did not change the proportion of selective looks towards the aggressor between the pretest and the posttest (Fig. 3c). We therefore excluded the alternative parsimonious explanation that the increase in selective looks at the aggressor was due to a mere expectation that the agent would be punished.
A previous study proposed that infants may consider collisions between geometric figures to be merely negative physical events rather than social interactions23. If this was the case in the present study, infants may have regarded geometric agents as the cause of a negative physical event rather than as aggressors. In Experiment 4, we tested this possibility by recruiting additional eight-month-old infants (N = 24) who were familiar with the same gaze-contingency events but modified the aggressive interactions used in Experiment 1. We tested infants using geometric figures with perceivable ‘animacy or agency’ removed by eliminating their eyes, ability to self-propel and distortion upon contact (Fig. 1a, Experiment 4). If selective looks were driven by infant perception of a geometric figure causing an unpleasant physical event in Experiment 1, then infants would probably selectively look at the causer of physical collisions at posttest. However, if selective looks were driven by infant perception of an aggressive interaction (that is, infants want to punish the agents in Experiment 1), they would not selectively look at the causer of physical collisions at posttest. Consistent with this latter prediction, model comparison demonstrated that the data were best represented by the null model (Table 1). The posterior model probability of the null model was the largest in the candidate models (P(M|data) = 0.544). The BF10 was 1.000 as the null model was being compared with itself. On average, the data anecdotally supported the model excluding the main effect of test type (BFincl = 0.404), moderately supported the model excluding the main effect of trial type (BFincl = 0.109) and strongly supported the model excluding the interaction term (BFincl = 0.062) (Table 2). The sensitivity analysis results (Fig. 2) robustly supported the model excluding the two main effects and the interaction term against reasonable change in the Cauchy prior width for the effect size. The exclusion of each effect was more likely to be supported as the prior width became large; however, the strength of the evidence for excluding the main effect of test type was anecdotal when the prior width was relatively small. In the null model, the proportion of an infant’s selective looks at the causer was not different from that at the chance level (intercept: posterior median, −0.012; 95% CI, (−0.447, 0.427); OR = 0.988; Supplementary Table 4). In summary, the data anecdotally supported the idea that eight-month-olds did not change the proportion of selective looks towards the causer between the pretest and the posttest (Fig. 3d). We thus excluded the non-social explanation that the Experiment 1 results were due to perceiving geometric figures as causing a negative physical event rather than as aggressors.
Finally, we performed Experiment 5 to replicate Experiment 1 for the following reasons. First, the evidence in Experiment 1 was too weak to be conclusive, as we used a new experimental paradigm. Second, there is increasing concern over the lack of replication in psychology research36. We therefore tested another infant group (N = 24) with identical procedures and the same sample size used in Experiment 1. The model comparison results demonstrated that the data were best represented by the model with the main effect of test type (Table 1). The posterior model probability of the model with the main effect of test type was the largest in the candidate models (P(M|data) = 0.795). The BF10 was 24.362, indicating strong evidence in favour of this model compared with the null model. On average, the data strongly supported the model including the main effect of test type (BFincl = 16.179) and moderately supported the model excluding the main effect of trial (BFincl = 0.139) and the interaction term (BFincl = 0.188). The sensitivity analysis results (Fig. 2) robustly supported the model including the main effect of test type in a wide range of the Cauchy prior on the effect size. However, the model excluding the main effect of trial and the interaction term was more likely to be supported as the prior width became large. In the best model, infants’ selective looks at the aggressor increased during the posttest phase compared with the pretest. The effect of test type relative to the pretest had a 0.999 probability of being positive (test: posterior median, 0.870; 95% CI, (0.362, 1.424); OR = 2.387; Supplementary Table 5). In summary, the data strongly supported the idea that eight-month-olds increased the proportion of selective looks towards the aggressor in the posttest compared with the pretest (Fig. 3e). This result indicates the potential of the findings to reflect robust psychological phenomena in early infancy.
The analyses reported above demonstrate that compared with the pretest phase, infants increased selective looking at an aggressor at the posttest phase in Experiment 1 and Experiment 5, but not in the other experiments. However, employing a contrast between the pretest and posttest phases for each experiment did not necessarily elucidate the differences in effect sizes of test type between the experiments37. Therefore, to compare the effect size of the test type for each experiment, we combined all experiment data and estimated the interaction effects between test type (pretest or posttest) and experiment (Experiment 1, Experiment 2, Experiment 3, Experiment 4 or Experiment 5) by using GLMM.
We conducted comparisons of the effect size of test type for each experiment (Supplementary Fig. 1). We calculated the effect size difference of test type between experiments from estimates of the interaction between the test type and the experiment. We checked whether the 95% CIs of the effect size difference excluded zero. The effect size of test type in Experiment 1 was larger than in Experiment 3, and the 95% CI of the effect size difference excluded zero (Exp.1 − Exp.3: posterior median, 0.756; 95% CI, (0.030, 1.498)). However, the 95% CIs of the effect size difference included zero when we compared the test type effect in Experiment 1 with that in Experiments 2, 4 and 5 (Exp.1 − Exp.2: posterior median, 0.581; 95% CI, (−0.157, 1.307); Exp.1 − Exp.4: posterior median, 0.285; 95% CI, (−0.453, 1.023); Exp.1 − Exp.5: posterior median, −0.165; 95% CI, (−0.915, 0.568); see also Supplementary Table 6). The effect size of test type in Experiment 5 was larger than that in Experiments 2 and 3, and the 95% CIs of the effect size difference did not include zero (Exp.5 − Exp.2: posterior median, 0.744; 95% CI, (0.010, 1.488); Exp.5 − Exp.3: posterior median, 0.920; 95% CI, (0.196, 1.656)). However, the 95% CI of the effect size difference included zero when we compared the test type effect in Experiment 5 with that in Experiment 4 (Exp.5 − Exp.4: posterior median, 0.449; 95% CI, (−0.294, 1.198); see also Supplementary Table 6). The 95% CIs of the effect size difference included zero for the pairs in Experiments 2, 3 and 4 (Exp.4 − Exp.2: posterior median, 0.293; 95% CI, (−0.436, 1.037); Exp.4 − Exp.3: posterior median, 0.468; 95% CI, (−0.253, 1.211); Exp.3 − Exp.2: posterior median, −0.175; 95% CI, (−0.904, 0.549); see also Supplementary Table 6). In the above model, the increase in selective looks at an aggressor after the movie phase was larger in Experiments 1 and 5 than in Experiments 2 and 3. However, the increase in selective looks after the movie phase in Experiment 4 was not different from the increase in the main experiments (Experiments 1 and 5) or the other control experiments (Experiments 2 and 3).
## Discussion
We investigated a disposition for third-party punishment of antisocial others in early infancy. After watching an aggressive interaction, infants as young as eight months old selectively looked at the aggressor more often with the apparent intent to punish (Experiment 1). Three control experiments excluded alternative parsimonious interpretations of these increases in selective looks at the aggressor: mere preferential looking at agents (Experiment 2), mere expectation that the agent would be punished (Experiment 3) and perceiving collisions as a negative physical event rather than aggression (Experiment 4). Finally, we replicated Experiment 1 to confirm that our findings indicated robust psychological phenomena (Experiment 5). Importantly, between-experiment differences were not attributable to variation in attention in the movie phase, as the Bayesian one-way analysis of variance results moderately supported the idea that there was no difference in looking time during the movie phase between the experiments (BF10 = 0.23; Supplementary Table 7). In addition, we found that in the main experiments (Experiment 1 and Experiment 5), selective looks at the aggressor after the movie phase tended to increase compared with the control experiments except for Experiment 4. Overall, infants as young as eight months old seem to punish antisocial others in third-party contexts by using their gaze, indicating that third-party punishment emerges much earlier than previously thought11,14,15,16,17,18,19.
Although many developmental studies have revealed that infants can evaluate the moral actions of others11,21,22,38, preverbal infants’ moral behaviour towards others has not been previously investigated. Our findings draw a connection between moral evaluation and moral behaviour among preverbal infants, bringing us closer to elucidating morality in early ontogeny. Furthermore, our findings imply that the primary motivations of punishment are probably intrinsic, rather than extrinsic results of cultural learning9 or higher-order desires to attain benefits for the self (for example, enhancing one’s reputation)10. This outcome might provide crucial evidence for ongoing debates regarding the motivations and evolved propensity underlying third-party punishment. The tendency towards third-party punishment may be engrained in preverbal infants’ minds and may have evolved only in humans.
One might doubt that the selective looks of infants reflect decision-making regarding punishment. Gaze-contingent techniques have been broadly used to investigate decision-making in patients with impaired limbs, such as those with amyotrophic lateral sclerosis39. However, similarity in the underlying mechanism of gaze control between infants and these patients is not evident. Nonetheless, previous research using gaze-contingency techniques demonstrated that infants of the same age showed gaze behaviours for intentional control on the monitor27,29. Furthermore, the three control experiments implied that selective looking behaviour involves punishment-related decisions; infants increased their selective looks at the specific agent (that is, aggressor) only when their gaze was associated with a negative event (that is, punishment; Experiment 2) that consistently occurred (that is, 100% reinforcement; Experiment 3) and when the event provided social information about the agents (that is, who was the aggressor or victim; Experiment 4). In other words, infants changed their behaviour to accomplish their goal only when they perceived the means to punish, had a sense of self-agency for punitive behaviour and were in a situation that called for punishment. They did not change their behaviour if any of these three elements were lacking. Consequently, infant looking behaviours were probably decisions made with the intention to punish.
A point to note is whether the gaze–action association learned during the pretest phase is preserved until the posttest phase even if the movie phase is inserted between the tests. During the movie phase, when infants gazed at the agent, the infants had no contingent events. It is thus possible that the gaze–action association is not preserved until the posttest phase. However, there are differences in the increase of selective looks after the movie phase between the experiments in which infants can learn the association (Experiments 1 and 5) and those where they cannot learn the same (Experiment 3). In addition, if infants were motivated to punish the aggressor, and if the association learning could not be maintained in the beginning of the posttest phase, the punishment rate would be at the chance level in the beginning of the posttest phase and would increase as the trials of the posttest phase elapsed. However, the observed data moderately supported the model excluding the interaction between test type and trial as well as the main effect of trial in Experiments 1 and 5, suggesting that the punishment rate for the aggressor in the posttest phase remained unchanged. We can therefore assume that the association between gaze and contingent event can be kept until the posttest phase.
There are some limitations worth noting. First, infants might think that the victim received a squeeze and thus the other actor should be squeezed as well; previous studies have indicated that infants expect equal treatment of others40,41. However, previous studies demonstrated that infants showed aversiveness to an agent who hit another agent21, affirmed an agent who disturbed the aggressor, and assumed that the aggressor should be hit by other agents22. It therefore seems plausible that infants regarded an agent who hit another agent as negative, thus expecting the aggressor to be punished, and consequently punishing the aggressor with their gaze. However, it may be slightly theory-laden to assert the psychological process of this punitive behaviour. Future studies are needed to identify associated underlying mechanisms. For example, because the aggressive interactions in this study involved multiple behaviours (for example, following the agent around and bumping), explorations on what exactly infants pick up as the critical cue or whether they need to see multiple cues to view interactions between agents as truly aggressive would be valuable.
Second, although our data supported the idea that infants did not change their selective looks between the pretest and posttest in Experiment 4, the evidence for this was weak. This is consistent with the results comparing the effect size of test type between Experiment 4 and the main experiments. These results might be due to the individual differences in animacy perception for objects in Experiment 4. Although we removed the aspect of perceivable ‘animacy or agency’ in Experiment 4 on the basis of a previous study22, the objects seemed to move autonomously to some extent, and thus some infants might perceive the objects to be animates or agents. Finally, although infants showed intentional use of gaze for their decision-making in our study, we do not conclusively know whether the infants were aware that they punished the agent by their gaze. In other words, it is unclear whether the infants looked at the agent with a consciousness of punishment. A previous study proposed a multi-level framework that self-agency is based on complex mechanisms on several levels, ranging from implicit to explicit42. It is interesting to observe the levels of self-agency involved in the punishment behaviours in the current study.
The presented paradigm in which infants can exhibit decision-making in a social context on a monitor might enable new infant cognitive research. Largely owing to limited methodologies as well as immature motor and verbal abilities in infants, most previous studies on infant cognition examined their perception and understanding of events from the viewpoint of a third party—that is, passive responses to physical43 and social24,25,26 events. In contrast, recent research using the gaze-contingent technique has revealed active infant responses to contingent events27,28,29. We incorporated such techniques to investigate behaviour accompanying decision-making regarding others and determined that we can measure infants’ moral behaviour towards others. The application of this paradigm could reveal undiscovered cognitive abilities in preverbal infants.
## Methods
This study was approved by Otsuma Women’s University’s Life Sciences Research Ethics Committee (no. 28-015) and the Behavioral Research Ethics Committee of the Osaka University School of Human Sciences (no. HB020-032).
### Experiment 1
#### Participants
The participants were 24 full-term eight-month-old infants (12 boys and 12 girls; mean age, 8 months 13 days; range, 7 months 13 days to 9 months 27 days). The sample size was determined on the basis of prior infant morality studies11,21,22,38. Eleven additional infants were tested but excluded owing to distress or fussiness (N = 4), or side-looking bias (N = 7, left = 7, right = 0; see the details of the criteria below). The parents provided written informed consent before the experiment and were financially compensated for participation.
#### Apparatus and stimuli
Infant gaze movements were measured using a Tobii TX300 near-infrared eye tracker (Tobii Technology), integrated with a 23-inch computer display (1,280 × 720 pixels). The sampling rate was 120 Hz. Task programming was completed in Visual Basic 2015 Express (Microsoft Corp.) and Tobii SDK (Tobii Technology). In all tasks, when an eye gaze was detected at a point on the display, a translucent red circle with a radius of 25 pixels appeared (Fig. 1a) to facilitate gaze control29. However, during the occurrence of contingent events, the red circle disappeared to allow for focus on said contingent events. The display background was aqua in colour.
The participants’ faces were monitored and recorded with a video camera (Panasonic HC-WX990M). Images on the PC screen (presented to the participants) and images of the participants were synthesized (Picture in Picture) using a video mixer device (Roland, V-1600HD) and recorded on a laptop PC (HP, Elite Book 8570w/CT) with a monitor-capturing device (Avermedia, AVT-C875).
In the practical phase, the first six trials subjected the infants to gaze-contingent events in which fixation on a single object (a red or blue circle positioned alternately on the left or right) for 500 ms resulted in a stone falling and crushing the object. This phase was set to reduce side-looking bias. In four subsequent trials, the infants were presented with two objects side by side (a red circle and a blue circle) instead of a single object. When the infants fixated on either of the two objects for 500 ms, a stone fell and crushed it. The presented position of each object or pair of objects was fixed among the participants.
In the following pretest, the infants experienced gaze-contingent events identical to those in the practical phase except that the targets were two geometric agents with eyes (for example, green and orange squares; pretest in Fig. 1a). The presented position of the geometric agents (left or right) was counterbalanced across participants but consistent between the pretest and posttest within participants.
In the movie phase, the infants were presented with an aggressive interaction animation (20 s in duration) depicting one geometric figure hitting and crashing into another geometric figure20,21,22 (Fig. 1b and Supplementary Video 2). The roles of the geometric figures (aggressor or victim) were counterbalanced between participants. Following the movie phase, the infants completed the posttest phase with gaze-contingent events identical to those of the pretest.
#### Procedure
The infants were fastened in a baby carrier to prevent them from standing up and were placed on their mothers’ laps approximately 60 cm from the monitor. Nine-point calibration was used. The parents were instructed not to watch the monitor and not to talk or interact with their children during the experiment.
The infants experienced ten gaze-contingent events in the practical phase. Then, the infants experienced ten gaze-contingent events in the pretest. In the movie phase, the infants were presented with animated movies of aggressive interactions three times. Finally, the infants experienced ten gaze-contingent events in the posttest. Attractive animated clips (a rotating oval checkerboard) with sound were inserted between trials if infants did not pay attention to the monitor.
#### Data analysis
We excluded data from further analysis if infants showed a side-looking bias, which was defined as looking to one side in more than 12 of the 14 gaze-contingent events (the last four trials of the practical phase and the ten trials of the pretest) (Bayesian binomial test, two-tailed, BF10 = 8.11, moderate evidence in favour of the alternative hypothesis; traditionally, the binomial test gives a P value below 0.05). To compare the proportion of infant selective looks at agents between pretest and posttest, we used GLMMs with a binomial error structure and a logit link function. The response variable was infant selective looks at the aggressor (= 1) or the victim (= 0) in the pretest or posttest. The explanatory variables (fixed effects) were test type (pretest or posttest) and trial number. We set participant identity as a random intercept. To keep the random effects structure “maximal”44, we also included all possible random slopes within participants and correlations.
We compared models on the basis of the BF. The model candidates were (1) the null model, (2) a model with the main effect of test type, (3) a model with the main effect of trial number, (4) a model with the main effects of test type and trial number, and (5) a model with the main effect of test type, the main effect of trial number and the interaction between test type and trial number. All models were compared with the null model, and we computed the BF (BF10), with the relative evidence in favour of each model over the null model (Table 1). We assumed that the prior model probability was uniform and evaluated the degree to which the data had changed the prior model odds for each model. We also computed BFincl (ref. 30) for each effect to evaluate the level of likelihood that the data were under models that included the effect compared with models that excluded the effect (Table 2). BFincl was computed on the basis of inclusion probabilities (that is, the sum of the model probabilities for the models that included the effect) across all models. For reporting BF10 and BFincl, we set the Cauchy distribution with location 0 and scale 1/√2 as a prior distribution for a coefficient parameter31. We also set the default prior (a t distribution with degrees of freedom 3 and scale 2.5) of brms as the prior distribution of an intercept and the standard deviation of random effects. To check whether the main conclusions from the data were robust to different priors, we conducted a sensitivity analysis for BFincl (Fig. 2). We computed BFincl for each effect and set the scale parameter of the Cauchy prior for the effect size from 0.05 to 1.5 in increments of 0.05.
We estimated the posterior distributions of the model parameters and checked the posterior predictive distribution for an infant’s selective looks towards the aggressor for the best model in the model comparison results (Supplementary Fig. 2a). We set the improper prior distribution for a coefficient parameter. Additionally, we set the default prior (a t distribution with degrees of freedom 3 and scale 2.5) of brms as a prior distribution of an intercept and the standard deviation of random effects. The posterior median and a 95% CI were calculated for each parameter.
The computation of BFs and parameter estimation were implemented using the brms package45,46 in R v.4.0.3 (ref. 47). The parameters were estimated with the Markov chain Monte Carlo (MCMC) method, and brms was used as an interface to Stan v.2.21.0 (ref. 48). As a general setting for MCMC sampling, iterations were set to 10,000, burn-in samples were set to 1,000 and the number of chains was set to four. The values of $${\hat{R}}$$ for all parameters were below 1.1, indicating convergence across the four chains; the parameter estimates are shown in Supplementary Table 1 (the best model) and Supplementary Table 8 (the full model). The graphical results of the full model are shown in Fig. 3a. The best model’s posterior predictive distribution for an infant’s selective looks towards the aggressor is shown in Supplementary Fig. 2a. All observed data were inside the 95% prediction interval. The mean times spent looking at the aggressive-interaction animations during the movie phase are shown in Supplementary Table 7.
### Experiment 2
#### Participants
The participants were an additional healthy 24 full-term eight-month-old infants (12 boys and 12 girls; mean age, 8 months 7 days; range, 7 months 17 days to 9 months 3 days). Eighteen additional infants were tested but excluded owing to distress or fussiness (N = 4), experimental error (N = 2) or side-looking bias (N = 12, left = 11, right = 1). All other details were the same as in Experiment 1.
#### Apparatus and stimuli
The movie phase of Experiment 2 used identical apparatus and animations to those in Experiment 1. The gaze-contingent events in Experiment 2 were also identical to those in Experiment 1, but with contact between objects and stones or between geometric figures and stones appearing less negative: materials falling softly hit objects or agents with less force than in Experiment 1 (Fig. 1a, Experiment 2).
#### Procedure
This was identical to Experiment 1.
#### Data analysis
The criteria and analyses of side-looking bias were the same as in Experiment 1, as was the analytic plan. The results of the model comparison and analysis of the effect are shown in Tables 1 and 2, respectively. The sensitivity analysis results for BFincl are shown in Fig. 2. The parameter estimates are shown in Supplementary Table 2 (the best model) and Supplementary Table 9 (the full model). The graphical results of the full model in the model comparison are shown in Fig. 3b. The best model’s posterior predictive distribution for an infant’s selective looks towards the aggressor is shown in Supplementary Fig. 2b. All observed data were inside the 95% prediction interval. The mean times spent looking at the aggressive-interaction animations during the movie phase are shown in Supplementary Table 7.
### Experiment 3
#### Participants
The participants were an additional 24 full-term eight-month-old infants (12 boys and 12 girls; mean age, 8 months 19 days; range, 8 months 0 days to 9 months 22 days). Seven additional infants were tested but excluded owing to distress or fussiness (N = 2), machine trouble (N = 3) or side-looking bias (N = 2, left = 2, right = 0). All other details were the same as in Experiment 1.
#### Apparatus and stimuli
The movie phase of Experiment 3 used identical apparatus and animations as Experiment 1. The gaze-contingent events in Experiment 3 were also identical to those in Experiment 1 except that during the practical phase, the infants were presented with two objects side by side (a red circle and a blue circle) in all ten trials. This modification was to implement a 50% reinforcement probability. In the practical phase, pretest and posttest, when the infants fixated on one of two objects, half of the gaze-contingent events involved the object (or agent) that they looked at, while the other half involved the object (or agent) that they did not look at. The reinforcement order was randomized among infants; however, a given gaze-contingent event was repeated no more than three times (Fig. 1a, Experiment 3).
#### Procedure
This was identical to Experiment 1.
#### Data analysis
The criteria and analyses for side-looking bias were the same as in Experiment 1, as was the analytic plan. The results of the model comparison and analysis of the effect are shown in Tables 1 and 2, respectively. The sensitivity analysis results for BFincl are shown in Fig. 2. The parameter estimates are shown in Supplementary Table 3 (the best model) and Supplementary Table 10 (the full model). The graphical results of the full model in the model comparison are shown in Fig. 3c. The best model’s posterior predictive distribution for an infant’s selective looks towards the aggressor is shown in Supplementary Fig. 2c. All observed data were inside the 95% prediction interval. The mean times spent looking at the aggressive-interaction animations during the movie phase are shown in Supplementary Table 7.
### Experiment 4
#### Participants
The participants were an additional 24 healthy full-term eight-month-old infants (12 boys and 12 girls; mean age, 8 months 13 days; range, 7 months 23 days to 9 months 13 days). Seventeen additional infants were tested but excluded owing to distress or fussiness (N = 7), machine trouble (N = 1), parental intervention (N = 1) or side-looking bias (N = 8, left = 5, right = 3). All other details were the same as in Experiment 1.
#### Apparatus and stimuli
Experiment 4 used the same apparatus as Experiment 1. The gaze-contingent events in the pretest and posttest, as well as the animations in the movie phase, were also identical to those in Experiment 1 with the following exceptions: we divided the eyes of both geometric features into white parts and black parts, with the aim of eliminating perceivable ‘animacy or agency’; we also removed the objects’ ability to self-propel and any distortion upon contact (Fig. 1a,b, Experiment 4; see also Supplementary Video 3).
#### Procedure
See Experiment 1.
#### Data analysis
The criteria and analyses for side-looking bias as well as the analytic plan were the same as in Experiment 1. The results of the model comparison and analysis of the effect are shown in Tables 1 and 2, respectively. The sensitivity analysis results for BFincl are shown in Fig. 2. The parameter estimates are shown in Supplementary Table 4 (the best model) and Supplementary Table 11 (the full model). The graphical results of the full model in the model comparison are shown in Fig. 3d. The best model’s posterior predictive distribution for an infant’s selective looks towards the aggressor is shown in Supplementary Fig. 2d. All observed data were inside the 95% prediction interval. The mean times spent looking at the physical-collision animations during the movie phase are shown in Supplementary Table 7.
### Experiment 5
#### Participants
The participants were an additional 24 full-term eight-month-old infants (11 boys and 13 girls; mean age, 8 months 15 days; range, 7 months 18 days to 9 months 15 days). Eleven additional infants were tested but excluded owing to distress or fussiness (N = 5), machine trouble (N = 2) or side-looking bias (N = 4, left = 4, right = 0). All other details were the same as in Experiment 1.
#### Apparatus, stimuli and procedure
See Experiment 1.
#### Data analysis
The criteria and analyses for side-looking bias and the analytic plan followed those in Experiment 1. The results of the model comparison and analysis of the effect are shown in Tables 1 and 2, respectively. The sensitivity analysis results for BFincl are shown in Fig. 2. The parameter estimates are shown in Supplementary Table 5 (the best model) and Supplementary Table 12 (the full model). The graphical results of the full model in the model comparison are shown in Fig. 3e. The best model’s posterior predictive distribution for infant’s selective looks to the aggressor is shown in Supplementary Fig. 2e. All observed data were inside the 95% prediction interval. The mean times spent looking at the aggressive-interaction animations during the movie phase are shown in Supplementary Table 7.
### Comparison of the effect sizes of test type for each experiment
The results indicating that infants selectively looked at the aggressor in the posttest rather than the pretest only in Experiments 1 and 5 are not sufficient to demonstrate that there were clear differences between the experiments in terms of changes in infants’ looking behaviour between the pretest and posttest phases37. To compare the effect size of the test type for each experiment, we combined all experiment data and estimated the interaction effects between test type and experiment. We used GLMM with a binomial error distribution and a logit link function. The response variable was infant selective looks in the pretest or posttest phase; looking at an aggressor or a causer was treated as 1, and otherwise as 0. The explanatory variables (fixed effects) were test type (pretest or posttest), experiment (Experiment 1, 2, 3, 4 or 5), trial number and the interaction between test type and experiment. Participant identity was set as a random intercept. We also included all possible random slopes within participants and correlations.
The model parameters were estimated with the MCMC method. We used brms45,46 and performed MCMC sampling in the same setting as in the analysis of each experiment. The values of $${\hat{R}}$$ for all parameters were below 1.1, indicating convergence across the four chains. Using MCMC samples of the interaction effects between test type and experiment, we calculated the effect size difference between experiments. Comparisons of the effect of test type for each experiment are shown in Supplementary Fig. 1. The parameter estimates for the model assessing the interaction effects between test type and experiment are shown in Supplementary Table 13. The parameter estimates for the differences in the test type effects between experiments are shown in Supplementary Table 6.
### Post-hoc confirmation of the validity of the sampling design
To assess whether our sampling design of each experiment had sufficient power to detect the effect of test type, we computed simulation-based power, given the actual sample size and the theoretically expected effect size. We simulated new datasets, estimated parameters of the full model with the new data and calculated the 95% CI of the parameter for the effect of test type to set our sampling design, with 24 participants and ten observations per test phase. We set the effect size of test type for the simulation on the basis of a previous meta-analysis study, which investigated infants’ preferences between a prosocial and an antisocial agent49. We randomly generated 100 samples on the basis of the effect size while setting various values for the magnitude of individual difference. Thereafter, we treated the proportion of samples in which the 95% CI of the parameter for the effect of test type did not include zero as a simulated power, given the theoretically expected effect size. Unfortunately, we found that this sampling design was not sufficiently powerful for the range of individual differences estimated from our actual data and the theoretically expected effect sizes. If our sample had been generated from a theoretically expected effect size, our sampling design would have had sufficient power only when the individual difference in the test type effect was small. Although it was not possible to know the magnitude of individual difference in the test type effect a priori in this study, it is advisable to select a larger sample size to conduct a similar paradigm in the future (see the Supplementary Information and Supplementary Fig. 3 for additional information).
### Reporting summary
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https://rocs.hu-berlin.de/courses/complex-systems-2021/script/bifurcation-analysis/ | # Introduction to Complex Systems
Prof. Dirk Brockmann, Winter Term 2021
# Bifurcation Analysis of One-Dimensional Dynamical Systems
Last time we learned how to compute fixpoints and their stability of one-dimensional dynamical systems to investigate their asymptotic behavior.
Now let's look at this in more detail. The examples in the lecture and discussed in the notes were all systems that had parameters. Recall the SIS system
[ \dot x=\alpha\left(x(1-x)-\frac{1}{R_{0}}x\right) ]
that exhibited different types of asymptotic behavior depending on the value of the basic reproduction number $R_0$.
Let's take this perspective now. Instead of asking how does $x(t)$ depend on time we ask:
How does the asymptotic behavior depend on the parameters?
Let's assume we have a dynamical system
[ \dot{x}=f(x;\mu) ] that only depends on one parameter $\mu$. Since one-dimensional dynamical systems are determined by the set of fixpoints, we can use them to see how the behavior changes as $\mu$ is varied (i.e. how the set of fixpoints and their stability changes). Consider the example in the panel below.
This is the dynamical system [ \dot{x}=f(x)=\sin(3x)+x^{2}/5+\mu. ] As we vary the parameter $\mu$ we see that a sequence of events happens to the set of fixpoint. As we decrease the parameter $f(x)$ crosses the $x-$axis and pairs of stable and unstable fixpoints are created or merge and annihilate. These events are called bifurcations. In the above example, they are all defined by the creation or annihilation of fixpoints of different types. Because quite often even a complex function $f(x)$ is just a curve with minima and maxima and if the control parameter $\mu$ is just a constant added to $f(x)$ it pushes the function up and down and generically maxima and minima cross the $x$-axis yielding these bifurcations. They are called saddle-node bifurcations for reasons that will become clear later.
Let's study this in a simple model that has exactly one such saddle node bifurcation: [ \dot{x}=f(x)=x^{2}+\mu ] If we plot $f(x)$ for different values of $\mu$ we see that three different scenarios emerge (Panel 2):
When $\mu<0$ two fixpoints exist, one is stable one is not. The fixpoints are [ x^{\star}=\pm\sqrt{\mu} ] As $\mu$ is increased these fixpoints approach each other and when $\mu=\mu_{c}=0$ the system has one fixpoint that attracts everything from the left but repels everything on the right. It's marginally stable. When $\mu$ is increased further, the fixpoint disappears. In a sense, the stable and unstable fixpoints annihilate by "collision".
Now instead of drawing $f(x)$ for different values of $\mu$ we can draw a Bifucation diagram. This is a plot, that depicts the stationary states (on the y-axis) vs. the control parameter $\mu$ (on the x-axis). Why? Because we are really interested in treating the control parameter as an independent control variable and the asymptotic behavior as the dependent variable. This is then what we get:
This is a bifurcation diagram it illustrates the asymptotic behavior of a dynamical system as a function of its parameter.
Bifurcartion diagrams show comprehensively how a dynamical systems behaves for all possible parameter choices. It therefore provides a global view of the system.
This is often very important because in many applications we would like to know what potential behaviors a system can exhibit, not necessarily what behavior it exhibits at a given parameter choice.
### Transcritical bifurcation
In turns out that in addition to the saddle node bifurcation we can have other types of bifurcations. For example the transcritical bifurcation (Panel 3).
This is when a stable and unstable fixpoint collide and interchange their stability properties. Here's the generic example:
[ \dot{x}=f(x;\mu)=\mu x-x^{2}=x(\mu-x). ] This guy always has a fixpoint at $x=0$ and one at $x=\mu$. However, for $\mu<0$ this second fixpoint is on the left of the origin and for $\mu>0$ it's on the right, see Panel 3.
If we look at the bifurcation diagram, we see that the fixpoints collide and exchange their stability properties:
### Pitchfork bifurcation
There's another type of bifurcation frequently encountered. Let's look at [ \dot{x}=f(x;\mu)=\mu x-x^{3}=x(\mu-x^{2}). ] Here we also have always a fixpoint $x^{\star}=0$. However, for $\mu>0$ we have two additional fixpoints [ x^{\star}=\pm\sqrt{\mu}. ] You can check for yourself in Panel 4.
What happens here is this: Let's start with $\mu<0$. The system has a single stable fixpoint at the origin. As we increase $\mu$ this stable fixpoint gives birth to two new stable fixpoints at $\mu=\mu_{c}=0$ and loses its own stability. The bifurcation diagram looks like a pitchfork which is why this type of bifurcation is called pitchfork bifurcation.
This is what the bifurcation diagram looks like:
It can also happen that an unstable fixpoint gives birth to two new unstable fixpoints and becomes a stable fixed point.
Saddle node, transcritical and pitchfork bifurcations are the most important bifurcations in 1d systems. There are more types of bifurcation but they are not as common.
#### Application: Cell differentiation
The pitchfork bifurcation has some interesting properties that we can use to think about systems that possess a single stable attractor and undergo a slow parameter change and when a critical parameter value is crossed the original stable state becomes unstable and two competing, new stable attractors emerge.
For example we can think of the original singular stable state as a stem cell and a slow parameter change as some changes in the environment that push the system into a parameter regime in which two stable states emerge that reflect different differentiation states that a stem cell can develop into.
So as a qualitative model we could say that the system is governed by the dynamical system
[ \dot{x}=f(x;\mu)=x(\mu-x^{2}). ]
in which $x(t)$ represents the state of a cell and we imagine that the parameter $\mu$ is controlled by some environmental factors.
In addition we say that the state is not fully governed by the above ODE but we also have some noise that can change the state
[ \dot{x}=f(x;\mu)=x(\mu-x^{2})+\text{Noise}. ]
There's a way to model this noise and we can be more specific about it. It's a topic that we will cover later. For now imagine that the noise will add little random changes to the state so that instead of moving along smooth curve there's a bit of a wiggle force acting on the state variable. What we have in mind here becomes clearer if we look at the difference equation again: [ x(t+\Delta t)\approx x(t)+\Delta tf(x(t))+\Delta W(t) ] at every point in time we add a random number $\Delta W(t)$ that is drawn from a normal distribution with a small variance and zero mean. If this is too vague at the moment, don't worry. We will be more specific when we cover stochastic differential equations. Right now, let's just think that as time goes one there's always a little random change in $x(t)$.
Let's now investigate what happens in the above system if we start with a control parameter $\mu<0$ and slowly increasing $\mu$. You can try this with the slider in Panel 5.
We see that in the initial regime the state wiggles around its only stable solution. As we slowly increase the control parameter, even into the region $\mu>0$ the state will remain there for a bit and then either approach the top branch or the bottom branch, so with a 50/50 chance one of the stable attractors. In our analogy of cell differentiation the cell fate is going to one of two possible states.
If we decrease the control parameter again, we will see that the system goes back to the original state.
## Hysteresis
So far, when we changed the control parameter of a system in some equilibrium state $x^{\star}$ the value of this equilibirum changes slowly, even as we pass a critical point. The above example showed that we can go from a regime with one equilibrium to one with two of them and back. But as we change the control parameter the asymptotic state changes continuously from the stable state onto one of the other stable branches and back.
However, the existence and interplay between bifurcations can yield some interesting behavior when many bifurcations exist. Let's look at the following example. Let's say that we have a dynamical system defined by [ \dot{x}=-x(x-1)(x+1)+\mu+\text{Noise}. ] Just like above, we imaging that the system is also under the influence of noise which makes it wiggle around its stable asymptotic state.
The bifurcation diagram of this system is depicted in Panel 6.
The key behavior here is that as the control parameter is changed slowly the system will remain on the stable branch and will not "sense" the emergence of an alternative stable state. However when the bottom stable branch gets annihilated the state has no choice but to jump to the upper branch. This implies a small parameter change can have a massive effect on the state. Also, if one decides to decrease $\mu$ again, the state will not return to the original lower branch, but rather stick to the upper branch until that one disappears when $\mu$ gets too small.
This behavior is known as hysteresis and can play a role in a number of dynamical systems. Because only saddle-node bifurcations are involved, we can expect this behavior to occur generically in many systems that exhibit multi-stability. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8759006857872009, "perplexity": 380.30617797265893}, "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-2022-40/segments/1664030334514.38/warc/CC-MAIN-20220925035541-20220925065541-00468.warc.gz"} |
https://www.softscients.web.id/2019/05/kernel-function-to-machine-learning.html | ## Saturday, May 18, 2019
### buku belajar machine learning dengan matlab - Jenis Kernel pada Machine Learning
Bila anda bekerja dengan kasus machine learning, tentu pemilihan jenis kernel sangatlah penting, ada banyak jenis kernel, karena penulis sedang mempelajari Support Vector Regression yang membutuhkan kernel trick,
clc; clear all;close all;
x = load('data.csv'); %terdiri dari 2 kolom
kernel = @(x) exp(-1*sum(x.^2,2)); %cara kedua
z = kernel(x);
figure()
scatter(x(target==0,1),x(target==0,2))
hold on
scatter(x(target==1,1),x(target==1,2))
xlabel('x'),ylabel('y')
title('Plot Data sebelum hyperplane')
hold off;
figure()
scatter3(x(target==0,1),x(target==0,2),z(target==0))
hold on
scatter3(x(target==1,1),x(target==1,2),z(target==1))
xlabel('x'),ylabel('y'),zlabel('z');
hold off
title('Plot Data setelah hyperplane')
langsung saja disajikan 25 jenis kernel berikut:
1. Linear Kernel
2. Polynomial Kernel
3. Gaussian Kernel
4. Exponential Kernel
5. Laplacian Kernel
6. ANOVA Kernel
7. Hyperbolic Tangent (Sigmoid) Kernel
11. Circular Kernel
12. Spherical Kernel
13. Wave Kernel
14. Power Kernel
15. Log Kernel
16. Spline Kernel
17. B-Spline Kernel
18. Bessel Kernel
19. Cauchy Kernel
20. Chi-Square Kernel
21. Histogram Intersection Kernel
22. Generalized Histogram Intersection Kernel
23. Generalized T-Student Kernel
24. Bayesian Kernel
25. Wavelet Kernel
Penulis langsung copy paste saja
## Kernel Functions
Below is a list of some kernel functions available from the existing literature. As was the case with previous articles, every LaTeX notation for the formulas below are readily available from their alternate text html tag. I can not guarantee all of them are perfectly correct, thus use them at your own risk. Most of them have links to articles where they have been originally used or proposed.
### 1. Linear Kernel
The Linear kernel is the simplest kernel function. It is given by the inner product <x,y> plus an optional constant c. Kernel algorithms using a linear kernel are often equivalent to their non-kernel counterparts, i.e. KPCA with linear kernel is the same as standard PCA.
### 2. Polynomial Kernel
The Polynomial kernel is a non-stationary kernel. Polynomial kernels are well suited for problems where all the training data is normalized.
Adjustable parameters are the slope alpha, the constant term c and the polynomial degree d.
### 3. Gaussian Kernel
The Gaussian kernel is an example of radial basis function kernel.
Alternatively, it could also be implemented using
The adjustable parameter sigma plays a major role in the performance of the kernel, and should be carefully tuned to the problem at hand. If overestimated, the exponential will behave almost linearly and the higher-dimensional projection will start to lose its non-linear power. In the other hand, if underestimated, the function will lack regularization and the decision boundary will be highly sensitive to noise in training data.
### 4. Exponential Kernel
The exponential kernel is closely related to the Gaussian kernel, with only the square of the norm left out. It is also a radial basis function kernel.
### 5. Laplacian Kernel
The Laplace Kernel is completely equivalent to the exponential kernel, except for being less sensitive for changes in the sigma parameter. Being equivalent, it is also a radial basis function kernel.
It is important to note that the observations made about the sigma parameter for the Gaussian kernel also apply to the Exponential and Laplacian kernels.
### 6. ANOVA Kernel
The ANOVA kernel is also a radial basis function kernel, just as the Gaussian and Laplacian kernels. It is said to perform well in multidimensional regression problems (Hofmann, 2008).
### 7. Hyperbolic Tangent (Sigmoid) Kernel
The Hyperbolic Tangent Kernel is also known as the Sigmoid Kernel and as the Multilayer Perceptron (MLP) kernel. The Sigmoid Kernel comes from the Neural Networks field, where the bipolar sigmoid function is often used as an activation function for artificial neurons.
It is interesting to note that a SVM model using a sigmoid kernel function is equivalent to a two-layer, perceptron neural network. This kernel was quite popular for support vector machines due to its origin from neural network theory. Also, despite being only conditionally positive definite, it has been found to perform well in practice.
There are two adjustable parameters in the sigmoid kernel, the slope alpha and the intercept constant c. A common value for alpha is 1/N, where N is the data dimension. A more detailed study on sigmoid kernels can be found in the works by Hsuan-Tien and Chih-Jen.
The Rational Quadratic kernel is less computationally intensive than the Gaussian kernel and can be used as an alternative when using the Gaussian becomes too expensive.
The Multiquadric kernel can be used in the same situations as the Rational Quadratic kernel. As is the case with the Sigmoid kernel, it is also an example of an non-positive definite kernel.
The Inverse Multi Quadric kernel. As with the Gaussian kernel, it results in a kernel matrix with full rank (Micchelli, 1986) and thus forms a infinite dimension feature space.
$k(x,+y)+=+\frac{1}{\sqrt{\lVert+x-y+\rVert^2+++c^2}}$
### 11. Circular Kernel
The circular kernel is used in geostatic applications. It is an example of an isotropic stationary kernel and is positive definite in R2.
### 12. Spherical Kernel
The spherical kernel is similar to the circular kernel, but is positive definite in R3.
### 13. Wave Kernel
The Wave kernel is also symmetric positive semi-definite (Huang, 2008).
$k(x,+y)+=+\frac{\theta}{\lVert+x-y+\rVert+\right}+\sin+\frac{\lVert+x-y+\rVert+}{\theta}$
### 14. Power Kernel
The Power kernel is also known as the (unrectified) triangular kernel. It is an example of scale-invariant kernel (Sahbi and Fleuret, 2004) and is also only conditionally positive definite.
### 15. Log Kernel
The Log kernel seems to be particularly interesting for images, but is only conditionally positive definite.
### 16. Spline Kernel
The Spline kernel is given as a piece-wise cubic polynomial, as derived in the works by Gunn (1998).
However, what it actually mean is:
With
### 17. B-Spline (Radial Basis Function) Kernel
The B-Spline kernel is defined on the interval [−1, 1]. It is given by the recursive formula:
In the work by Bart Hamers it is given by:
Alternatively, Bn can be computed using the explicit expression (Fomel, 2000):
Where x+ is defined as the truncated power function:
### 18. Bessel Kernel
The Bessel kernel is well known in the theory of function spaces of fractional smoothness. It is given by:
where J is the Bessel function of first kind. However, in the Kernlab for R documentation, the Bessel kernel is said to be:
### 19. Cauchy Kernel
The Cauchy kernel comes from the Cauchy distribution (Basak, 2008). It is a long-tailed kernel and can be used to give long-range influence and sensitivity over the high dimension space.
### 20. Chi-Square Kernel
The Chi-Square kernel comes from the Chi-Square distribution:
However, as noted by commenter Alexis Mignon, this version of the kernel is only conditionally positive-definite (CPD). A positive-definite version of this kernel is given in (Vedaldi and Zisserman, 2011) as
and is suitable to be used by methods other than support vector machines.
### 21. Histogram Intersection Kernel
The Histogram Intersection Kernel is also known as the Min Kernel and has been proven useful in image classification.
### 22. Generalized Histogram Intersection
The Generalized Histogram Intersection kernel is built based on the Histogram Intersection Kernel for image classification but applies in a much larger variety of contexts (Boughorbel, 2005). It is given by:
### 23. Generalized T-Student Kernel
The Generalized T-Student Kernel has been proven to be a Mercel Kernel, thus having a positive semi-definite Kernel matrix (Boughorbel, 2004). It is given by:
### 24. Bayesian Kernel
The Bayesian kernel could be given as:
where
However, it really depends on the problem being modeled. For more information, please see the work by Alashwal, Deris and Othman, in which they used a SVM with Bayesian kernels in the prediction of protein-protein interactions.
### 25. Wavelet Kernel
The Wavelet kernel (Zhang et al, 2004) comes from Wavelet theory and is given as:
Where a and c are the wavelet dilation and translation coefficients, respectively (the form presented above is a simplification, please see the original paper for details). A translation-invariant version of this kernel can be given as:
Where in both h(x) denotes a mother wavelet function. In the paper by Li Zhang, Weida Zhou, and Licheng Jiao, the authors suggests a possible h(x) as:
Which they also prove as an admissible kernel function.
Referensi:
http://crsouza.com/2010/03/17/kernel-functions-for-machine-learning-applications | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 2, "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": 0.8327267169952393, "perplexity": 2935.8259661185007}, "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/1585371656216.67/warc/CC-MAIN-20200406164846-20200406195346-00181.warc.gz"} |
https://proofwiki.org/wiki/Prime_iff_Equal_to_Product | # Prime iff Equal to Product
## Theorem
Let $p \in \Z$ be an integer such that $p \ne 0$ and $p \ne \pm 1$.
Then $p$ is prime if and only if:
$\forall a, b \in \Z: p = ab \implies p = \pm a \lor p = \pm b$
## Proof
### Necessary Condition
Let $p$ be a prime number.
Then by definition, the only divisors of $p$ are $\pm 1$ and $\pm p$.
Thus, if $p = a b$ then either $a = \pm 1$ and $b = \pm p$ or $a = \pm p$ and $b = \pm 1$.
$\Box$
### Sufficient Condition
Suppose that:
$\forall a, b \in \Z: p = a b \implies p = \pm a \lor p = \pm b$
This means that the only divisors of $p$ are $\pm 1$ and $\pm p$.
That is, that $p$ is a prime number.
$\blacksquare$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9281812906265259, "perplexity": 123.15336021288987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2020-24/segments/1590347390758.21/warc/CC-MAIN-20200526112939-20200526142939-00006.warc.gz"} |
https://arxiv.org/abs/1904.03858?context=math | math
# Title:The Kikuchi Hierarchy and Tensor PCA
Abstract: For the tensor PCA (principal component analysis) problem, we propose a new hierarchy of algorithms that are increasingly powerful yet require increasing runtime. Our hierarchy is analogous to the sum-of-squares (SOS) hierarchy but is instead inspired by statistical physics and related algorithms such as belief propagation and AMP (approximate message passing). Our level-$\ell$ algorithm can be thought of as a (linearized) message-passing algorithm that keeps track of $\ell$-wise dependencies among the hidden variables. Specifically, our algorithms are spectral methods based on the Kikuchi Hessian matrix, which generalizes the well-studied Bethe Hessian matrix to the higher-order Kikuchi free energies.
It is known that AMP, the flagship algorithm of statistical physics, has substantially worse performance than SOS for tensor PCA. In this work we `redeem' the statistical physics approach by showing that our hierarchy gives a polynomial-time algorithm matching the performance of SOS. Our hierarchy also yields a continuum of subexponential-time algorithms, and we prove that these achieve the same (conjecturally optimal) tradeoff between runtime and statistical power as SOS. Our results hold for even-order tensors, and we conjecture that they also hold for odd-order tensors.
Our methods suggest a new avenue for systematically obtaining optimal algorithms for Bayesian inference problems, and our results constitute a step toward unifying the statistical physics and sum-of-squares approaches to algorithm design.
Comments: 34 pages Subjects: Data Structures and Algorithms (cs.DS); Statistical Mechanics (cond-mat.stat-mech); Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML) MSC classes: 68Q87 ACM classes: F.2.2 Cite as: arXiv:1904.03858 [cs.DS] (or arXiv:1904.03858v1 [cs.DS] for this version)
## Submission history
From: Alexander Wein [view email]
[v1] Mon, 8 Apr 2019 06:26:35 UTC (34 KB) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8424774408340454, "perplexity": 2303.491379446544}, "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-2019-22/segments/1558232256571.66/warc/CC-MAIN-20190521202736-20190521224736-00477.warc.gz"} |
https://www.gradesaver.com/textbooks/science/chemistry/chemistry-and-chemical-reactivity-9th-edition/chapter-13-solutions-and-their-behavior-study-questions-page-505a/25 | Chemistry and Chemical Reactivity (9th Edition)
$34.99\ mmHg$
Number of moles of ethylene glycol: $35.0\ g\div62.07\ g/mol=0.564\ mol$ Number of moles of water: $500\ g\div 18.015\ g/mol=27.754\ mol$ Mole fraction of solvent: $27.754/(27.754+0.564)=0.98$ Vapor pressure of water: $P=XP^0=0.98\times35.7\ mmHg=34.99\ mmHg$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8898718357086182, "perplexity": 4350.491861912631}, "config": {"markdown_headings": false, "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-2022-21/segments/1652663021405.92/warc/CC-MAIN-20220528220030-20220529010030-00612.warc.gz"} |
https://cadabra.science/qa/1398/unexpected-behavior-of-combine | # Unexpected behavior of combine()
Hi,
When I apply the command combine() on
$( \Gamma ^ { a } ) _ { \beta } { } ^ { \gamma } ( \Gamma _ { c d } ) _ { \gamma }{ }^{ \alpha }$,
I get
$( \Gamma _ {c d} \Gamma ^ {a} ) ^ { \alpha } { } _ { \beta }$
while expecting
$( \Gamma ^ {a} \Gamma _ { c d } ) _ { \beta }{ } ^ { \alpha }$.
Am i doing something wrong?
(In case not rendering tex, i added image )
In addition, Is there any way to tell such structures to Cadabra;
$A _ {a} B ^ {a} = - A ^ {a} B _ {a}$.
Since our metric is antisymmetric, the shape of the contraction matters. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8679729104042053, "perplexity": 4728.084949259421}, "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-2020-34/segments/1596439740838.3/warc/CC-MAIN-20200815094903-20200815124903-00234.warc.gz"} |
https://turbomachinery.asmedigitalcollection.asme.org/offshoremechanics/article-abstract/109/2/126/430748/Wave-Loads-and-Motions-of-Long-Structures-in?searchresult=1 | The present paper deals with the effects of wave directionality on the loads and motions of long structures. A numerical procedure based on Green’s theorem is developed to compute the exciting forces and hydrodynamic coefficients due to the interaction of a regular oblique wave train with an infinitely long, semi-immersed floating cylinder of arbitrary shape. The linear transfer function approach is used to determine the wave loads and motions of a structure of finite length in short-crested seas. The effect of wave directionality is expressed as a frequency-dependent, directionally averaged reduction factor for the wave loads and a response ratio for the body motions. Numerical results are presented for the force reduction factor and response ratio of a long floating box subject to a directional wave spectrum with a cosine-power-type energy spreading function.
This content is only available via PDF. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9182840585708618, "perplexity": 804.8829690669453}, "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-2022-21/segments/1652662577757.82/warc/CC-MAIN-20220524233716-20220525023716-00123.warc.gz"} |
https://face2ai.com/math-probability-5-4-the-poisson-distribution/ | # 泊松分布
## 泊松分布的定义和性质 Definition and Properties of the Poisson Distributions
$$f(x|n=3600,p=0.00125)= \begin{cases} \begin{pmatrix} 3600\\x \end{pmatrix}p^x(1-p)^{3600-x}&\text{for }0\leq x\leq 3600\\ 0&\text{otherwise} \end{cases}$$
\begin{aligned} \frac{f(x+1)}{f(x)}&= \frac {\begin{pmatrix}n\\x+1\end{pmatrix}p^{x+1}(1-p)^{n-x-1}} {\begin{pmatrix}n\\x\end{pmatrix}p^{x+1}(1-p)^{n-x-1}}\\ &=\frac{(n-x)p}{(x+1)(1-p)}\\ &\approx\frac{np}{x+1} \end{aligned}
$$f(1)=f(0)\lambda\\ f(2)=f(1)\frac{\lambda}{2}=f(0)\frac{\lambda^2}{2}\\ f(3)=f(2)\frac{\lambda}{3}=f(0)\frac{\lambda^3}{6}\\ \vdots\\ f(n)=f(n-1)\frac{\lambda}{n}==f(0)\frac{\lambda^n}{n!}\\$$
$$\sum^{\infty}_{x=0}f(x)=1$$
$$\sum^{\infty}_{x=0}f(0)\frac{\lambda^n}{n!}=1\\ f(0)\sum^{\infty}_{x=0}\frac{\lambda^n}{n!}=1\\ \text{for :}\sum^{\infty}_{x=0}\frac{\lambda^n}{n!}=e^{\lambda}\\ \text{so :}f(0)=e^{-\lambda}$$
$$f(x|\lambda)= \begin{cases} \frac{e^{-\lambda}\lambda^x}{x!}&\text{for }x=1,2,3,\dots\\ 0&\text{otherwise} \end{cases}$$
Definition Poisson Distribution.Let $\lambda > 0$ .A random variable X has the Poisson Distribution with mean $\lambda$ if the p.f. of $X$ is as follow:
$$f(x|\lambda)= \begin{cases} \frac{e^{-\lambda}\lambda^x}{x!}&\text{for }x=1,2,3,\dots\\ 0&\text{otherwise} \end{cases}$$
### 泊松分布的均值 Mean
Theorem Mean. The mean of Poisson Distribution with p.f. equal to upside is $\lambda$ .
$$E(X)=\sum^{\infty}_{x=0}xf(x|\lambda)$$
\begin{aligned} E(X)&=\sum^{\infty}_{x=0}x\frac{e^{-\lambda}\lambda^x}{x!}\\ &=\sum^{\infty}_{x=1}\frac{e^{-\lambda}\lambda^x}{(x-1)!}\\ &=\lambda\sum^{\infty}_{x=1}\frac{e^{-\lambda}\lambda^{x-1}}{(x-1)!}\\ \text{if we set } y=x-1\\ &=\lambda\sum^{\infty}_{y=0}\frac{e^{-\lambda}\lambda^{y}}{y!} \end{aligned}
### 泊松分布的方差 Varaince
Theorem Variance.The variance of Poisson distribution with mean $\lambda$ is also $\lambda$
\begin{aligned} E[X(X-1)]&=\sum^{\infty}_{x=0}x(x-1)f(x|\lambda)\\ &=\sum^{\infty}_{x=2}x(x-1)f(x|\lambda)\\ &=\sum^{\infty}_{x=2}x(x-1)\frac{e^{-\lambda}\lambda^x}{x!}\\ &=\lambda^2\sum^{\infty}_{x=2}\frac{e^{-\lambda}\lambda^{x-2}}{x-2!}\\ \text{We set }y=x-2\\ E[X(X-1)]&=\lambda^2\sum^{\infty}_{y=0}\frac{e^{-\lambda}\lambda^y}{y!}\\ &=\lambda^2 \end{aligned}
$$Var(X)=E[X^2]-E^2[x]=\lambda^2+\lambda-\lambda^2=\lambda$$
### 泊松分布的距生成函数 m.g.f.
Theorem Moment Generating Function.The m.g.f. of the Poisson distribution with mean $\lambda$ is
$$\psi(t)=e^{\lambda(e^t-1)}$$
for all real $t$
$$\psi(t)=E(e^{tX})=\sum^{\infty}_{x=0}\frac{e^{tx}e^{-\lambda}\lambda^x}{x!}=e^{-\lambda}\sum^{\infty}_{x=0}\frac{(\lambda e^t)^x}{x!}$$
$$\sum^{\infty}_{x=0}\frac{(\lambda e^t)^x}{x!}=e^{\lambda e^t}$$
$$\psi(t)=e^{-\lambda}e^{\lambda e^t}=e^{\lambda(e^t-1)}$$
### 泊松分布随机变量相加
Theorem If the random variable $X_1,\dots,X_k$ are independent and if $X_i$ has Poisson distribution with mean $\lambda_i(i=1,\dots,k)$ ,then the sum $X_1+\dots+X_k$ has the Poisson distribution with mean $\lambda_1+\dots+\lambda_k$
$$\psi(t)=\Pi^k_{i=1}\psi_i(t)=\Pi^k_{i=1}e^{\lambda_i(e^t-1)}=e^{(\lambda_1+\dots+\lambda_k)(e^t-1)}$$
## 二项分布的泊松近似 The Poisson Approximation to Binomial Distributions
Theorem Closeness of Binomial and Pisson Distribution.For each integer n and each $0 < p < 1$ ,let $f(x|n,p)$ denote the p.f. of the binomial distribtuion with parameters $n$ and $p$ .Let $f(x|\lambda)$ denote the p.f. of the Poisson distribution with mean $\lambda$ .Let ${{P_n}}^{\infty}{n=1}$ be a sequence of numbers between 0 and 1 such that $lim{n\to \infty}np_n=\lambda$ . Then
$$lim_{n\to \infty}f(x|n,p_n)=f(x|\lambda)$$
for all $x=0,1\dots$
$$f(x|n,p_n)=\frac{n(n-1)\dots(n-x+1)}{x!}p_n^x(1-p_n)^{n-x}$$
$$f(x|n,p_n)=\frac{\lambda_n^x}{x!}\frac{n}{n}\cdot\frac{n-1}{n}\dots \frac{n-x+1}{n}(1-\frac{\lambda_n}{n})^n(1-\frac{\lambda_n}{n})^{-x}$$
$$lim_{n\to \infty}\frac{n}{n}\cdot\frac{n-1}{n}\dots \frac{n-x+1}{n}(1-\frac{\lambda_n}{n})^{-x}=1$$
$$lim_{n\to \infty}(1-\frac{\lambda_n}{n})^{n}=e^{-\lambda}$$
$$lim_{n\to \infty}f(x|n,p_n)=\frac{e^{-\lambda}\lambda^x}{x!}=f(x|\lambda)$$
Theorem Closeness of Hypergeometric and Poisson Distribution.Let $\lambda>0$ .Let $Y$ have the Poisson distribution with mean $\lambda$ .For each postive integer $T$ ,let $A_T,B_T$ ,and $n_T$ be integers such that $lim_{T\to \infty}n_TA_T/(A_T+B_T)=\lambda$ .Let $X_T$ have the hypergeometric distribution with parameters $A_T,B_T$ and $n_T$ .Tor each fixed $x=0,1,\dots$ ，
$$lim_{T\to \infty}\frac{Pr(Y=x)}{Pr(X_t=x)}=1$$
## 泊松过程 Poisson Processes
Definition Poisson Process.A Poisson process with rate $\lambda$ per unit time is a process that satisfies the following two properties:
i: The number of arrivals in every fixed interval of time of length $t$ has the Poisson distribution with mean $\lambda t$
ii: The numbers of arrivals in every collection of disjoint time intervals are independent
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https://terrytao.wordpress.com/tag/goldbach-conjecture/ | You are currently browsing the tag archive for the ‘Goldbach conjecture’ tag.
Two of the most famous open problems in additive prime number theory are the twin prime conjecture and the binary Goldbach conjecture. They have quite similar forms:
• Twin prime conjecture The equation ${p_1 - p_2 = 2}$ has infinitely many solutions with ${p_1,p_2}$ prime.
• Binary Goldbach conjecture The equation ${p_1 + p_2 = N}$ has at least one solution with ${p_1,p_2}$ prime for any given even ${N \geq 4}$.
In view of this similarity, it is not surprising that the partial progress on these two conjectures have tracked each other fairly closely; the twin prime conjecture is generally considered slightly easier than the binary Goldbach conjecture, but broadly speaking any progress made on one of the conjectures has also led to a comparable amount of progress on the other. (For instance, Chen’s theorem has a version for the twin prime conjecture, and a version for the binary Goldbach conjecture.) Also, the notorious parity obstruction is present in both problems, preventing a solution to either conjecture by almost all known methods (see this previous blog post for more discussion).
In this post, I would like to note a divergence from this general principle, with regards to bounded error versions of these two conjectures:
• Twin prime with bounded error The inequalities ${0 < p_1 - p_2 < H}$ has infinitely many solutions with ${p_1,p_2}$ prime for some absolute constant ${H}$.
• Binary Goldbach with bounded error The inequalities ${N \leq p_1+p_2 \leq N+H}$ has at least one solution with ${p_1,p_2}$ prime for any sufficiently large ${N}$ and some absolute constant ${H}$.
The first of these statements is now a well-known theorem of Zhang, and the Polymath8b project hosted on this blog has managed to lower ${H}$ to ${H=246}$ unconditionally, and to ${H=6}$ assuming the generalised Elliott-Halberstam conjecture. However, the second statement remains open; the best result that the Polymath8b project could manage in this direction is that (assuming GEH) at least one of the binary Goldbach conjecture with bounded error, or the twin prime conjecture with no error, had to be true.
All the known proofs of Zhang’s theorem proceed through sieve-theoretic means. Basically, they take as input equidistribution results that control the size of discrepancies such as
$\displaystyle \Delta(f; a\ (q)) := \sum_{x \leq n \leq 2x; n=a\ (q)} f(n) - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x} f(n) \ \ \ \ \ (1)$
for various congruence classes ${a\ (q)}$ and various arithmetic functions ${f}$, e.g. ${f(n) = \Lambda(n+h_i)}$ (or more generaly ${f(n) = \alpha * \beta(n+h_i)}$ for various ${\alpha,\beta}$). After taking some carefully chosen linear combinations of these discrepancies, and using the trivial positivity lower bound
$\displaystyle a_n \geq 0 \hbox{ for all } n \implies \sum_n a_n \geq 0 \ \ \ \ \ (2)$
one eventually obtains (for suitable ${H}$) a non-trivial lower bound of the form
$\displaystyle \sum_{x \leq n \leq 2x} \nu(n) 1_A(n) > 0$
where ${\nu}$ is some weight function, and ${A}$ is the set of ${n}$ such that there are at least two primes in the interval ${[n,n+H]}$. This implies at least one solution to the inequalities ${0 < p_1 - p_2 < H}$ with ${p_1,p_2 \sim x}$, and Zhang’s theorem follows.
In a similar vein, one could hope to use bounds on discrepancies such as (1) (for ${x}$ comparable to ${N}$), together with the trivial lower bound (2), to obtain (for sufficiently large ${N}$, and suitable ${H}$) a non-trivial lower bound of the form
$\displaystyle \sum_{n \leq N} \nu(n) 1_B(n) > 0 \ \ \ \ \ (3)$
for some weight function ${\nu}$, where ${B}$ is the set of ${n}$ such that there is at least one prime in each of the intervals ${[n,n+H]}$ and ${[N-n-H,n]}$. This would imply the binary Goldbach conjecture with bounded error.
However, the parity obstruction blocks such a strategy from working (for much the same reason that it blocks any bound of the form ${H \leq 4}$ in Zhang’s theorem, as discussed in the Polymath8b paper.) The reason is as follows. The sieve-theoretic arguments are linear with respect to the ${n}$ summation, and as such, any such sieve-theoretic argument would automatically also work in a weighted setting in which the ${n}$ summation is weighted by some non-negative weight ${\omega(n) \geq 0}$. More precisely, if one could control the weighted discrepancies
$\displaystyle \Delta(f\omega; a\ (q)) = \sum_{x \leq n \leq 2x; n=a\ (q)} f(n) \omega(n) - \frac{1}{\phi(q)} \sum_{x \leq n \leq 2x} f(n) \omega(n)$
to essentially the same accuracy as the unweighted discrepancies (1), then thanks to the trivial weighted version
$\displaystyle a_n \geq 0 \hbox{ for all } n \implies \sum_n a_n \omega(n) \geq 0$
of (2), any sieve-theoretic argument that was capable of proving (3) would also be capable of proving the weighted estimate
$\displaystyle \sum_{n \leq N} \nu(n) 1_B(n) \omega(n) > 0. \ \ \ \ \ (4)$
However, (4) may be defeated by a suitable choice of weight ${\omega}$, namely
$\displaystyle \omega(n) := \prod_{i=1}^H (1 + \lambda(n) \lambda(n+i)) \times \prod_{j=0}^H (1 - \lambda(n) \lambda(N-n-j))$
where ${n \mapsto \lambda(n)}$ is the Liouville function, which counts the parity of the number of prime factors of a given number ${n}$. Since ${\lambda(n)^2 = 1}$, one can expand out ${\omega(n)}$ as the sum of ${1}$ and a finite number of other terms, each of which consists of the product of two or more translates (or reflections) of ${\lambda}$. But from the Möbius randomness principle (or its analogue for the Liouville function), such products of ${\lambda}$ are widely expected to be essentially orthogonal to any arithmetic function ${f(n)}$ that is arising from a single multiplicative function such as ${\Lambda}$, even on very short arithmetic progressions. As such, replacing ${1}$ by ${\omega(n)}$ in (1) should have a negligible effect on the discrepancy. On the other hand, in order for ${\omega(n)}$ to be non-zero, ${\lambda(n+i)}$ has to have the same sign as ${\lambda(n)}$ and hence the opposite sign to ${\lambda(N-n-j)}$ cannot simultaneously be prime for any ${0 \leq i,j \leq H}$, and so ${1_B(n) \omega(n)}$ vanishes identically, contradicting (4). This indirectly rules out any modification of the Goldston-Pintz-Yildirim/Zhang method for establishing the binary Goldbach conjecture with bounded error.
The above argument is not watertight, and one could envisage some ways around this problem. One of them is that the Möbius randomness principle could simply be false, in which case the parity obstruction vanishes. A good example of this is the result of Heath-Brown that shows that if there are infinitely many Siegel zeroes (which is a strong violation of the Möbius randomness principle), then the twin prime conjecture holds. Another way around the obstruction is to start controlling the discrepancy (1) for functions ${f}$ that are combinations of more than one multiplicative function, e.g. ${f(n) = \Lambda(n) \Lambda(n+2)}$. However, controlling such functions looks to be at least as difficult as the twin prime conjecture (which is morally equivalent to obtaining non-trivial lower-bounds for ${\sum_{x \leq n \leq 2x} \Lambda(n) \Lambda(n+2)}$). A third option is not to use a sieve-theoretic argument, but to try a different method (e.g. the circle method). However, most other known methods also exhibit linearity in the “${n}$” variable and I would suspect they would be vulnerable to a similar obstruction. (In any case, the circle method specifically has some other difficulties in tackling binary problems, as discussed in this previous post.)
One of the most basic methods in additive number theory is the Hardy-Littlewood circle method. This method is based on expressing a quantity of interest to additive number theory, such as the number of representations ${f_3(x)}$ of an integer ${x}$ as the sum of three primes ${x = p_1+p_2+p_3}$, as a Fourier-analytic integral over the unit circle ${{\bf R}/{\bf Z}}$ involving exponential sums such as
$\displaystyle S(x,\alpha) := \sum_{p \leq x} e( \alpha p) \ \ \ \ \ (1)$
where the sum here ranges over all primes up to ${x}$, and ${e(x) := e^{2\pi i x}}$. For instance, the expression ${f(x)}$ mentioned earlier can be written as
$\displaystyle f_3(x) = \int_{{\bf R}/{\bf Z}} S(x,\alpha)^3 e(-x\alpha)\ d\alpha. \ \ \ \ \ (2)$
The strategy is then to obtain sufficiently accurate bounds on exponential sums such as ${S(x,\alpha)}$ in order to obtain non-trivial bounds on quantities such as ${f_3(x)}$. For instance, if one can show that ${f_3(x)>0}$ for all odd integers ${x}$ greater than some given threshold ${x_0}$, this implies that all odd integers greater than ${x_0}$ are expressible as the sum of three primes, thus establishing all but finitely many instances of the odd Goldbach conjecture.
Remark 1 In practice, it can be more efficient to work with smoother sums than the partial sum (1), for instance by replacing the cutoff ${p \leq x}$ with a smoother cutoff ${\chi(p/x)}$ for a suitable choice of cutoff function ${\chi}$, or by replacing the restriction of the summation to primes by a more analytically tractable weight, such as the von Mangoldt function ${\Lambda(n)}$. However, these improvements to the circle method are primarily technical in nature and do not have much impact on the heuristic discussion in this post, so we will not emphasise them here. One can also certainly use the circle method to study additive combinations of numbers from other sets than the set of primes, but we will restrict attention to additive combinations of primes for sake of discussion, as it is historically one of the most studied sets in additive number theory.
In many cases, it turns out that one can get fairly precise evaluations on sums such as ${S(x,\alpha)}$ in the major arc case, when ${\alpha}$ is close to a rational number ${a/q}$ with small denominator ${q}$, by using tools such as the prime number theorem in arithmetic progressions. For instance, the prime number theorem itself tells us that
$\displaystyle S(x,0) \approx \frac{x}{\log x}$
and the prime number theorem in residue classes modulo ${q}$ suggests more generally that
$\displaystyle S(x,\frac{a}{q}) \approx \frac{\mu(q)}{\phi(q)} \frac{x}{\log x}$
when ${q}$ is small and ${a}$ is close to ${q}$, basically thanks to the elementary calculation that the phase ${e(an/q)}$ has an average value of ${\mu(q)/\phi(q)}$ when ${n}$ is uniformly distributed amongst the residue classes modulo ${q}$ that are coprime to ${q}$. Quantifying the precise error in these approximations can be quite challenging, though, unless one assumes powerful hypotheses such as the Generalised Riemann Hypothesis.
In the minor arc case when ${\alpha}$ is not close to a rational ${a/q}$ with small denominator, one no longer expects to have such precise control on the value of ${S(x,\alpha)}$, due to the “pseudorandom” fluctuations of the quantity ${e(\alpha p)}$. Using the standard probabilistic heuristic (supported by results such as the central limit theorem or Chernoff’s inequality) that the sum of ${k}$ “pseudorandom” phases should fluctuate randomly and be of typical magnitude ${\sim \sqrt{k}}$, one expects upper bounds of the shape
$\displaystyle |S(x,\alpha)| \lessapprox \sqrt{\frac{x}{\log x}} \ \ \ \ \ (3)$
for “typical” minor arc ${\alpha}$. Indeed, a simple application of the Plancherel identity, followed by the prime number theorem, reveals that
$\displaystyle \int_{{\bf R}/{\bf Z}} |S(x,\alpha)|^2\ d\alpha \sim \frac{x}{\log x} \ \ \ \ \ (4)$
which is consistent with (though weaker than) the above heuristic. In practice, though, we are unable to rigorously establish bounds anywhere near as strong as (3); upper bounds such as ${x^{4/5+o(1)}}$ are far more typical.
Because one only expects to have upper bounds on ${|S(x,\alpha)|}$, rather than asymptotics, in the minor arc case, one cannot realistically hope to make much use of phases such as ${e(-x\alpha)}$ for the minor arc contribution to integrals such as (2) (at least if one is working with a single, deterministic, value of ${x}$, so that averaging in ${x}$ is unavailable). In particular, from upper bound information alone, it is difficult to avoid the “conspiracy” that the magnitude ${|S(x,\alpha)|^3}$ oscillates in sympathetic resonance with the phase ${e(-x\alpha)}$, thus essentially eliminating almost all of the possible gain in the bounds that could arise from exploiting cancellation from that phase. Thus, one basically has little option except to use the triangle inequality to control the portion of the integral on the minor arc region ${\Omega_{minor}}$:
$\displaystyle |\int_{\Omega_{minor}} S(x,\alpha)^3 e(-x\alpha)\ d\alpha| \leq \int_{\Omega_{minor}} |S(x,\alpha)|^3\ d\alpha.$
Despite this handicap, though, it is still possible to get enough bounds on both the major and minor arc contributions of integrals such as (2) to obtain non-trivial lower bounds on quantities such as ${f(x)}$, at least when ${x}$ is large. In particular, this sort of method can be developed to give a proof of Vinogradov’s famous theorem that every sufficiently large odd integer ${x}$ is the sum of three primes; my own result that all odd numbers greater than ${1}$ can be expressed as the sum of at most five primes is also proven by essentially the same method (modulo a number of minor refinements, and taking advantage of some numerical work on both the Goldbach problems and on the Riemann hypothesis ). It is certainly conceivable that some further variant of the circle method (again combined with a suitable amount of numerical work, such as that of numerically establishing zero-free regions for the Generalised Riemann Hypothesis) can be used to settle the full odd Goldbach conjecture; indeed, under the assumption of the Generalised Riemann Hypothesis, this was already achieved by Deshouillers, Effinger, te Riele, and Zinoviev back in 1997. I am optimistic that an unconditional version of this result will be possible within a few years or so, though I should say that there are still significant technical challenges to doing so, and some clever new ideas will probably be needed to get either the Vinogradov-style argument or numerical verification to work unconditionally for the three-primes problem at medium-sized ranges of ${x}$, such as ${x \sim 10^{50}}$. (But the intermediate problem of representing all even natural numbers as the sum of at most four primes looks somewhat closer to being feasible, though even this would require some substantially new and non-trivial ideas beyond what is in my five-primes paper.)
However, I (and many other analytic number theorists) are considerably more skeptical that the circle method can be applied to the even Goldbach problem of representing a large even number ${x}$ as the sum ${x = p_1 + p_2}$ of two primes, or the similar (and marginally simpler) twin prime conjecture of finding infinitely many pairs of twin primes, i.e. finding infinitely many representations ${2 = p_1 - p_2}$ of ${2}$ as the difference of two primes. At first glance, the situation looks tantalisingly similar to that of the Vinogradov theorem: to settle the even Goldbach problem for large ${x}$, one has to find a non-trivial lower bound for the quantity
$\displaystyle f_2(x) = \int_{{\bf R}/{\bf Z}} S(x,\alpha)^2 e(-x\alpha)\ d\alpha \ \ \ \ \ (5)$
for sufficiently large ${x}$, as this quantity ${f_2(x)}$ is also the number of ways to represent ${x}$ as the sum ${x=p_1+p_2}$ of two primes ${p_1,p_2}$. Similarly, to settle the twin prime problem, it would suffice to obtain a lower bound for the quantity
$\displaystyle \tilde f_2(x) = \int_{{\bf R}/{\bf Z}} |S(x,\alpha)|^2 e(-2\alpha)\ d\alpha \ \ \ \ \ (6)$
that goes to infinity as ${x \rightarrow \infty}$, as this quantity ${\tilde f_2(x)}$ is also the number of ways to represent ${2}$ as the difference ${2 = p_1-p_2}$ of two primes less than or equal to ${x}$.
In principle, one can achieve either of these two objectives by a sufficiently fine level of control on the exponential sums ${S(x,\alpha)}$. Indeed, there is a trivial (and uninteresting) way to take any (hypothetical) solution of either the asymptotic even Goldbach problem or the twin prime problem and (artificially) convert it to a proof that “uses the circle method”; one simply begins with the quantity ${f_2(x)}$ or ${\tilde f_2(x)}$, expresses it in terms of ${S(x,\alpha)}$ using (5) or (6), and then uses (5) or (6) again to convert these integrals back into a the combinatorial expression of counting solutions to ${x=p_1+p_2}$ or ${2=p_1-p_2}$, and then uses the hypothetical solution to the given problem to obtain the required lower bounds on ${f_2(x)}$ or ${\tilde f_2(x)}$.
Of course, this would not qualify as a genuine application of the circle method by any reasonable measure. One can then ask the more refined question of whether one could hope to get non-trivial lower bounds on ${f_2(x)}$ or ${\tilde f_2(x)}$ (or similar quantities) purely from the upper and lower bounds on ${S(x,\alpha)}$ or similar quantities (and of various ${L^p}$ type norms on such quantities, such as the ${L^2}$ bound (4)). Of course, we do not yet know what the strongest possible upper and lower bounds in ${S(x,\alpha)}$ are yet (otherwise we would already have made progress on major conjectures such as the Riemann hypothesis); but we can make plausible heuristic conjectures on such bounds. And this is enough to make the following heuristic conclusions:
• (i) For “binary” problems such as computing (5), (6), the contribution of the minor arcs potentially dominates that of the major arcs (if all one is given about the minor arc sums is magnitude information), in contrast to “ternary” problems such as computing (2), in which it is the major arc contribution which is absolutely dominant.
• (ii) Upper and lower bounds on the magnitude of ${S(x,\alpha)}$ are not sufficient, by themselves, to obtain non-trivial bounds on (5), (6) unless these bounds are extremely tight (within a relative error of ${O(1/\log x)}$ or better); but
• (iii) obtaining such tight bounds is a problem of comparable difficulty to the original binary problems.
I will provide some justification for these conclusions below the fold; they are reasonably well known “folklore” to many researchers in the field, but it seems that they are rarely made explicit in the literature (in part because these arguments are, by their nature, heuristic instead of rigorous) and I have been asked about them from time to time, so I decided to try to write them down here.
In view of the above conclusions, it seems that the best one can hope to do by using the circle method for the twin prime or even Goldbach problems is to reformulate such problems into a statement of roughly comparable difficulty to the original problem, even if one assumes powerful conjectures such as the Generalised Riemann Hypothesis (which lets one make very precise control on major arc exponential sums, but not on minor arc ones). These are not rigorous conclusions – after all, we have already seen that one can always artifically insert the circle method into any viable approach on these problems – but they do strongly suggest that one needs a method other than the circle method in order to fully solve either of these two problems. I do not know what such a method would be, though I can give some heuristic objections to some of the other popular methods used in additive number theory (such as sieve methods, or more recently the use of inverse theorems); this will be done at the end of this post.
I’ve just uploaded to the arXiv my paper “Every odd number greater than 1 is the sum of at most five primes“, submitted to Mathematics of Computation. The main result of the paper is as stated in the title, and is in the spirit of (though significantly weaker than) the even Goldbach conjecture (every even natural number is the sum of at most two primes) and odd Goldbach conjecture (every odd natural number greater than 1 is the sum of at most three primes). It also improves on a result of Ramaré that every even natural number is the sum of at most six primes. This result had previously also been established by Kaniecki under the additional assumption of the Riemann hypothesis, so one can view the main result here as an unconditional version of Kaniecki’s result.
The method used is the Hardy-Littlewood circle method, which was for instance also used to prove Vinogradov’s theorem that every sufficiently large odd number is the sum of three primes. Let’s quickly recall how this argument works. It is convenient to use a proxy for the primes, such as the von Mangoldt function ${\Lambda}$, which is mostly supported on the primes. To represent a large number ${x}$ as the sum of three primes, it suffices to obtain a good lower bound for the sum
$\displaystyle \sum_{n_1,n_2,n_3: n_1+n_2+n_3=x} \Lambda(n_1) \Lambda(n_2) \Lambda(n_3).$
By Fourier analysis, one can rewrite this sum as an integral
$\displaystyle \int_{{\bf R}/{\bf Z}} S(x,\alpha)^3 e(-x\alpha)\ d\alpha$
where
$\displaystyle S(x,\alpha) := \sum_{n \leq x} \Lambda(n) e(n\alpha)$
and ${e(\theta) :=e^{2\pi i \theta}}$. To control this integral, one then needs good bounds on ${S(x,\alpha)}$ for various values of ${\alpha}$. To do this, one first approximates ${\alpha}$ by a rational ${a/q}$ with controlled denominator (using a tool such as the Dirichlet approximation theorem) ${q}$. The analysis then broadly bifurcates into the major arc case when ${q}$ is small, and the minor arc case when ${q}$ is large. In the major arc case, the problem more or less boils down to understanding sums such as
$\displaystyle \sum_{n\leq x} \Lambda(n) e(an/q),$
which in turn is almost equivalent to understanding the prime number theorem in arithmetic progressions modulo ${q}$. In the minor arc case, the prime number theorem is not strong enough to give good bounds (unless one is using some extremely strong hypotheses, such as the generalised Riemann hypothesis), so instead one uses a rather different method, using truncated versions of divisor sum identities such as ${\Lambda(n) =\sum_{d|n} \mu(d) \log\frac{n}{d}}$ to split ${S(x,\alpha)}$ into a collection of linear and bilinear sums that are more tractable to bound, typical examples of which (after using a particularly simple truncated divisor sum identity known as Vaughan’s identity) include the “Type I sum”
$\displaystyle \sum_{d \leq U} \mu(d) \sum_{n \leq x/d} \log(n) e(\alpha dn)$
and the “Type II sum”
$\displaystyle \sum_{d > U} \sum_{w > V} \mu(d) (\sum_{b|w: b > V} \Lambda(b)) e(\alpha dw) 1_{dw \leq x}.$
After using tools such as the triangle inequality or Cauchy-Schwarz inequality to eliminate arithmetic functions such as ${\mu(d)}$ or ${\sum_{b|w: b>V}\Lambda(b)}$, one ends up controlling plain exponential sums such as ${\sum_{V < w < x/d} e(\alpha dw)}$, which can be efficiently controlled in the minor arc case.
This argument works well when ${x}$ is extremely large, but starts running into problems for moderate sized ${x}$, e.g. ${x \sim 10^{30}}$. The first issue is that of logarithmic losses in the minor arc estimates. A typical minor arc estimate takes the shape
$\displaystyle |S(x,\alpha)| \ll (\frac{x}{\sqrt{q}}+\frac{x}{\sqrt{x/q}} + x^{4/5}) \log^3 x \ \ \ \ \ (1)$
when ${\alpha}$ is close to ${a/q}$ for some ${1\leq q\leq x}$. This only improves upon the trivial estimate ${|S(x,\alpha)| \ll x}$ from the prime number theorem when ${\log^6 x \ll q \ll x/\log^6 x}$. As a consequence, it becomes necessary to obtain an accurate prime number theorem in arithmetic progressions with modulus as large as ${\log^6 x}$. However, with current technology, the error term in such theorems are quite poor (terms such as ${O(\exp(-c\sqrt{\log x}) x)}$ for some small ${c>0}$ are typical, and there is also a notorious “Siegel zero” problem), and as a consequence, the method is generally only applicable for very large ${x}$. For instance, the best explicit result of Vinogradov type known currently is due to Liu and Wang, who established that all odd numbers larger than ${10^{1340}}$ are the sum of three odd primes. (However, on the assumption of the GRH, the full odd Goldbach conjecture is known to be true; this is a result of Deshouillers, Effinger, te Riele, and Zinoviev.)
In this paper, we make a number of refinements to the general scheme, each one of which is individually rather modest and not all that novel, but which when added together turn out to be enough to resolve the five primes problem (though many more ideas would still be needed to tackle the three primes problem, and as is well known the circle method is very unlikely to be the route to make progress on the two primes problem). The first refinement, which is only available in the five primes case, is to take advantage of the numerical verification of the even Goldbach conjecture up to some large ${N_0}$ (we take ${N_0=4\times 10^{14}}$, using a verification of Richstein, although there are now much larger values of ${N_0}$as high as ${2.6 \times 10^{18}}$ – for which the conjecture has been verified). As such, instead of trying to represent an odd number ${x}$ as the sum of five primes, we can represent it as the sum of three odd primes and a natural number between ${2}$ and ${N_0}$. This effectively brings us back to the three primes problem, but with the significant additional boost that one can essentially restrict the frequency variable ${\alpha}$ to be of size ${O(1/N_0)}$. In practice, this eliminates all of the major arcs except for the principal arc around ${0}$. This is a significant simplification, in particular avoiding the need to deal with the prime number theorem in arithmetic progressions (and all the attendant theory of L-functions, Siegel zeroes, etc.).
In a similar spirit, by taking advantage of the numerical verification of the Riemann hypothesis up to some height ${T_0}$, and using the explicit formula relating the von Mangoldt function with the zeroes of the zeta function, one can safely deal with the principal major arc ${\{ \alpha = O( T_0 / x ) \}}$. For our specific application, we use the value ${T_0= 3.29 \times 10^9}$, arising from the verification of the Riemann hypothesis of the first ${10^{10}}$ zeroes by van de Lune (unpublished) and Wedeniswki. (Such verifications have since been extended further, the latest being that the first ${10^{13}}$ zeroes lie on the line.)
To make the contribution of the major arc as efficient as possible, we borrow an idea from a paper of Bourgain, and restrict one of the three primes in the three-primes problem to a somewhat shorter range than the other two (of size ${O(x/K)}$ instead of ${O(x)}$, where we take ${K}$ to be something like ${10^3}$), as this largely eliminates the “Archimedean” losses coming from trying to use Fourier methods to control convolutions on ${{\bf R}}$. In our paper, we set the scale parameter ${K}$ to be ${10^3}$ (basically, anything that is much larger than ${1}$ but much less than ${T_0}$ will work), but we found that an additional gain (which we ended up not using) could be obtained by averaging ${K}$ over a range of scales, say between ${10^3}$ and ${10^6}$. This sort of averaging could be a useful trick in future work on Goldbach-type problems.
It remains to treat the contribution of the “minor arc” ${T_0/x \ll |\alpha| \ll 1/N_0}$. To do this, one needs good ${L^2}$ and ${L^\infty}$ type estimates on the exponential sum ${S(x,\alpha)}$. Plancherel’s theorem gives an ${L^2}$ estimate which loses a logarithmic factor, but it turns out that on this particular minor arc one can use tools from the theory of the large sieve (such as Montgomery’s uncertainty principle) to eliminate this logarithmic loss almost completely; it turns out that the most efficient way to do this is use an effective upper bound of Siebert on the number of prime pairs ${(p,p+h)}$ less than ${x}$ to obtain an ${L^2}$ bound that only loses a factor of ${8}$ (or of ${7}$, once one cuts out the major arc).
For ${L^\infty}$ estimates, it turns out that existing effective versions of (1) (in particular, the bound given by Chen and Wang) are insufficient, due to the three logarithmic factors of ${\log x}$ in the bound. By using a smoothed out version ${S_\eta(x,\alpha) :=\sum_{n}\Lambda(n) e(n\alpha) \eta(n/x)}$ of the sum ${S(\alpha,x)}$, for some suitable cutoff function ${\eta}$, one can save one factor of a logarithm, obtaining a bound of the form
$\displaystyle |S_\eta(x,\alpha)| \ll (\frac{x}{\sqrt{q}}+\frac{x}{\sqrt{x/q}} + x^{4/5}) \log^2 x$
with effective constants. One can improve the constants further by restricting all summations to odd integers (which barely affects ${S_\eta(x,\alpha)}$, since ${\Lambda}$ was mostly supported on odd numbers anyway), which in practice reduces the effective constants by a factor of two or so. One can also make further improvements in the constants by using the very sharp large sieve inequality to control the “Type II” sums that arise from Vaughan’s identity, and by using integration by parts to improve the bounds on the “Type I” sums. A final gain can then be extracted by optimising the cutoff parameters ${U, V}$ appearing in Vaughan’s identity to minimise the contribution of the Type II sums (which, in practice, are the dominant term). Combining all these improvements, one ends up with bounds of the shape
$\displaystyle |S_\eta(x,\alpha)| \ll \frac{x}{q} \log^2 x + \frac{x}{\sqrt{q}} \log^2 q$
when ${q}$ is small (say ${1 < q < x^{1/3}}$) and
$\displaystyle |S_\eta(x,\alpha)| \ll \frac{x}{(x/q)^2} \log^2 x + \frac{x}{\sqrt{x/q}} \log^2(x/q)$
when ${q}$ is large (say ${x^{2/3} < q < x}$). (See the paper for more explicit versions of these estimates.) The point here is that the ${\log x}$ factors have been partially replaced by smaller logarithmic factors such as ${\log q}$ or ${\log x/q}$. Putting together all of these improvements, one can finally obtain a satisfactory bound on the minor arc. (There are still some terms with a ${\log x}$ factor in them, but we use the effective Vinogradov theorem of Liu and Wang to upper bound ${\log x}$ by ${3100}$, which ends up making the remaining terms involving ${\log x}$ manageable.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 256, "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": 0.9492769241333008, "perplexity": 192.6532769391502}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2015-32/segments/1438042988924.75/warc/CC-MAIN-20150728002308-00286-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://math.stackexchange.com/users/5580/boumol?tab=activity&sort=comments | # boumol
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Oct8 comment Best Mathematical Logic Books the Style of Which is Like a Mathematics Publication rather than a Logic Publication? @Comeseeconquer: I am afraid I have the same troubles than Carl to follow your question. How do you think it is written "for every $x$, if $x$ is a real number then $x^2 \geq 0$" in a mathematical book? Oct8 comment Colimits in the 2-category of partial functions (which is locally posetal) @Zhen: Why is this true? Given a plain bicomplete category $\mathcal{C}$, there are many ways to look at it as a locally posetal one. Thus, I can imagine it might be the case there is some strange partial order where bicolimits do not exist (while colimits, as a plain category, exist). Is this not possible? Sep7 comment Any Computational Number Theory Book, include software programs for key steps of the proofs of major theorem? Something close (except for what Conifold has said in his comment) can be found in the book "Elementary Number Theory: Primes, Congruences, and Secrets" by William Stein. The book and the software are freely availabe: visit wstein.org/ent and sagemath.org Sep2 comment Colimits in the category of “sets with partial mappings” Thanks for the clarification. I guess you mean "colimits" (since limits are not preserved). May30 comment How did Kurt Gödel's Incompleteness Theorem affect the mathematical world? To start take a look at the papers published in this volume of the Notices AMS ams.org/notices/200604 (in particular I point out the paper by Solomon Feferman). Another related paper worth looking at it is one by Macintyre (take a look at books.google.es/books?id=Tg0WXU5_8EgC&pg=PA3 ) Apr11 comment The idea of “generators” for arbitrary categories I am not able to understand what you mean with "$\{ x \circ f \mid x \in X \}$"; can you tell me what is $f$? Apr11 comment The idea of “generators” for arbitrary categories @espen180: Be careful because the upset generated is not the same thing than the filter generated. The notion of filter is isually used in the context of lattices (a particular kind of partial orders), and filters are always closed under finite meets, while upsets might not be closed under finite meets. Apr11 comment The idea of “generators” for arbitrary categories @Mariano: I am afraid that this wikipedia notion does not coincide with the one I have written above for partial orders (although the same word "generator" is used), so it is not adequate for what I really want to generalize. Mar28 comment nonisomorphic graph drawings You can try to use some software for drawing them, this is always helpful (see for instance the "sage" code in the question ask.sagemath.org/question/3473/… ) Mar28 comment How to explain the power of PA to non-logicians This essentially means that all $\Sigma_1$-setences that are true in the standard model are provable in PA. This result can be find in most of logic books dealing with PA (and Robinson's Q). In particular, you can take a look at the notes by Peter Smith logicmatters.net/igt/godel-without-tears (in the current version is Theorem 17 in Episode 5, page 40) Mar28 comment How to explain the power of PA to non-logicians Do you consider the fact that "PA (indeed Robinson Q is enough) is $\Sigma_1$-complete" strong enough? Mar27 comment Mathematical intro to Turing machines One example is the following "classic" written by Martin Davis amazon.com/Computability-Unsolvability-Prof-Martin-Davis/dp/… Mar19 comment Show that a recursively inseparable pair of recursively enumerable sets exists Yes, there is a standard proof (for two sets whose definiton only involves Turing machines). One place where you can find this is in Theorem 3.3.5 of the wonderful notes "Syllabus Computability Theory" written by Sebastiaan A. Terwijn math.ru.nl/~terwijn/teaching.html After you know these two sets are inseparable you can easily prove the inseparability of sets with a "logic" flavour (like in Biderman answer) Mar18 comment Books (and supporting material) that are useful in deconstructing one's intuition? @Sabysachi: torus solution? Mar15 comment Book on the first-order modal logic @user132181: The standard translation only works for propositional modal logic (and it translates these formulas into first-order, non-modal, formulas). Thus, right now I am afraid that your question does not make sense: could you clarify your question? Mar13 comment When is a Decidable Set Decidable? I suspect Russell is confusing two different uses of the term "undecidable". We have "undecidability in a formal theory" which applies to a sentence and refers to its unprovability. And then we also have "undecidability in a computability setting" which applies to a family (set) of finite objects and refers to the fact that this set cannot be computed by a Turing machine. For this second use there are some people which have advocated for using the word "uncomputable" instead of "undecidable", in order to avoid these misunderstandings. Feb15 comment Complexity of Recursively Inseparable Sets @Easterly: What do you mean here with the word "Complexity"? Jan10 comment Why are mathematical proofs that rely on computers controversial? @DumpsterDoofus (and the upvoters): What is the statement you talk about? Jan5 comment Determining if a theory in first-order logic is decidable @Sid: Use the "informal proof" given by André. In the proof it is crucial that you know that your theory is complete. Indeed, for complete theories (but not in general) it is known that "decidable" coincides with "recursively axiomatizable". Dec22 comment Good textbook on geometries And the recent "trilogy" by Borceux. I think up to now only the first two volumes have been published: amazon.com/An-Axiomatic-Approach-Geometry-Geometric/dp/… amazon.com/An-Algebraic-Approach-Geometry-Geometric/dp/… | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9073200821876526, "perplexity": 850.9573413078532}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "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-2014-42/segments/1413507449615.41/warc/CC-MAIN-20141017005729-00257-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/other-math/thinking-mathematically-6th-edition/chapter-11-counting-methods-and-probability-theory-11-1-the-fundamental-counting-principle-exercise-set-11-1-page-693/16 | ## Thinking Mathematically (6th Edition)
$6561$
The number of ways in which a series of successive things can occur is found by multiplying the number of ways in which each thing can occur. --------- The number of ways to choose the answer to the ... 1st question is 3, 2nd question is 3, 3rd question is 3, ... 7th question is 3, 8th question is 3. By the Fundamental Counting Principle, the total ways of doing this is $3\times 3\times...\times 3=3^{8}=6561$ ways | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9541404843330383, "perplexity": 258.88837085552626}, "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-2019-47/segments/1573496669795.59/warc/CC-MAIN-20191118131311-20191118155311-00148.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_12A_Problems/Problem_24 | # 2006 AMC 12A Problems/Problem 24
## Problem
The expression
is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
## Solution 1
By the Multinomial Theorem, the summands can be written as
and
respectively. Since the coefficients of like terms are the same in each expression, each like term either cancel one another out or the coefficient doubles. In each expansion there are:
terms without cancellation. For any term in the second expansion to be negative, the parity of the exponents of and must be opposite. Now we find a pattern:
if the exponent of is , the exponent of can be all even integers up to , so there are terms.
if the exponent of is , the exponent of can go up to , so there are terms.
if the exponent of is , then can only be 0, so there is term.
If we add them up, we get terms. However, we can switch the exponents of and and these terms will still have a negative sign. So there are a total of negative terms.
By subtracting this number from 2015028, we obtain or as our answer.
## Solution 2
Alternatively, we can use a generating function to solve this problem. The goal is to find the generating function for the number of unique terms in the simplified expression (in terms of ). In other words, we want to find where the coefficient of equals the number of unique terms in .
First, we note that all unique terms in the expression have the form, , where and is some constant. Therefore, the generating function for the MAXIMUM number of unique terms possible in the simplified expression of is
Secondly, we note that a certain number of terms of the form, , do not appear in the simplified version of our expression because those terms cancel. Specifically, we observe that terms cancel when because every unique term is of the form: for all possible .
Since the generating function for the maximum number of unique terms is already known, it is logical that we want to find the generating function for the number of terms that cancel, also in terms of . With some thought, we see that this desired generating function is the following:
Now, we want to subtract the latter from the former in order to get the generating function for the number of unique terms in , our initial goal: which equals
The coefficient of of the above expression equals
Evaluating the expression, we get , as expected.
## Solution 3
Define such that . Then the expression in the problem becomes: .
Expanding this using binomial theorem gives , where (we may omit the coefficients, as we are seeking for the number of terms, not the terms themselves).
Simplifying gives: . Note that two terms that come out of different powers of cannot combine and simplify, as their exponent of will differ. Therefore, we simply add the number of terms produced from each addend. By the Binomial Theorem, will have terms, so the answer is .
## Solution 4
Using stars and bars we know that has or different terms. Now we need to find out how many of these terms are canceled out by . We know that for any term(let's say ) where of the expansion of is only going to cancel out with the corresponding term if only is odd and is even or is even and is odd. Now let's do some casework to see how many terms fit this criteria:
Case 1: is even
Now we know that is even and . Thus is also even or both and are odd or both and are even. This case clearly fails the above criteria and there are 0 possible solutions.
Case 2: is odd
Now we know that is odd and . Thus is odd and is odd and is even or is even and is odd. All terms that have being odd work.
We now need to figure out how many of the terms have as a odd number. We know that can be equal to any number between 0 and 2006. There are 1003 odd numbers between this range and 2007 total numbers. Thus of the terms will cancel out and of the terms will work. Thus there are terms. This number comes out to be (Author: David Camacho) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9183183312416077, "perplexity": 267.3920596225315}, "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-2022-27/segments/1656103619185.32/warc/CC-MAIN-20220628233925-20220629023925-00600.warc.gz"} |
https://stats.stackexchange.com/questions/232465/how-to-compare-models-on-the-basis-of-aic/232494 | # How to compare models on the basis of AIC?
We have two models that use the same method to calculate log likelihood and the AIC for one is lower than the other. However, the one with the lower AIC is far more difficult to interpret.
We are having trouble deciding if it is worth introducing the difficulty and we judged this using a percentage difference in AIC. We found that the difference between the the two AICs was only 0.7%, with the more complicated model having a 0.7% lower AIC.
1. Is the low percentage difference between the two a good reason to avoid using the model with the lower AIC?
2. Does the percentage of difference explain that 0.7% more information is lost in the less complicated model?
3. Can the two models have very different results?
• Possible duplicate of What breaks the comparibility of models with respect to the AIC? – Arun Jose Aug 30 '16 at 11:23
• @ArunJose, it does not seem to be a duplicate. The questions here are quite different. – Richard Hardy Aug 30 '16 at 11:33
• No. This question is not about comparability of models. We already know the models are comparable. This question pertains to what counts as a significant difference in AIC and the trade off between complexity vs. model fit. – Ali Turab Lotia Aug 30 '16 at 11:33
One does not compare the absolute values of two AICs (which can be like $\sim 100$ but also $\sim 1000000$), but considers their difference: $$\Delta_i=AIC_i-AIC_{\rm min},$$ where $AIC_i$ is the AIC of the $i$-th model, and $AIC_{\rm min}$ is the lowest AIC one obtains among the set of models examined (i.e., the prefered model). The rule of thumb, outlined e.g. in Burnham & Anderson 2004, is:
1. if $\Delta_i<2$, then there is substantial support for the $i$-th model (or the evidence against it is worth only a bare mention), and the proposition that it is a proper description is highly probable;
2. if $2<\Delta_i<4$, then there is strong support for the $i$-th model;
3. if $4<\Delta_i<7$, then there is considerably less support for the $i$-th model;
4. models with $\Delta_i>10$ have essentially no support.
Now, regarding the 0.7% mentioned in the question, consider two situations:
1. $AIC_1=AIC_{\rm min}=100$ and $AIC_2$ is bigger by 0.7%: $AIC_2=100.7$. Then $\Delta_2=0.7<2$ so there is no substantial difference between the models.
2. $AIC_1=AIC_{\rm min}=100000$ and $AIC_2$ is bigger by 0.7%: $AIC_2=100700$. Then $\Delta_2=700\gg 10$ so there is no support for the 2-nd model.
Hence, saying that the difference between AICs is 0.7% does not provide any information.
The AIC value contains scaling constants coming from the log-likelihood $\mathcal{L}$, and so $\Delta_i$ are free of such constants. One might consider $\Delta_i = AIC_i − AIC_{\rm min}$ a rescaling transformation that forces the best model to have $AIC_{\rm min} := 0$.
The formulation of AIC penalizes the use of an excessive number of parameters, hence discourages overfitting. It prefers models with fewer parameters, as long as the others do not provide a substantially better fit. AIC tries to select a model (among the examined ones) that most adequately describes reality (in the form of the data under examination). This means that in fact the model being a real description of the data is never considered. Note that AIC gives you the information which model describes the data better, it does not give any interpretation.
Personally, I would say that if you have a simple model and a complicated one that has a much lower AIC, then the simple model is not good enough. If the more complex model is really much more complicated but the $\Delta_i$ is not huge (maybe $\Delta_i<2$, maybe $\Delta_i<5$ - depends on the particular situation) I would stick to the simpler model if it's really easier to work with.
Further, you can ascribe a probability to the $i$-th model via
$$p_i=\exp\left(\frac{-\Delta_i}{2}\right),$$
which provides a relative (compared to $AIC_{\rm min}$) probability that the $i$-th models minimizes the AIC. For example, $\Delta_i=1.5$ corresponds to $p_i=0.47$ (quite high), and $\Delta_i=15$ corresponds to $p_i=0.0005$ (quite low). The first case means that there is 47% probability that the $i$-th model might in fact be a better description than the model that yielded $AIC_{\rm min}$, and in the second case this probability is only 0.05%.
Finally, regarding the formula for AIC:
$$AIC=2k-2\mathcal{L},$$
it is important to note that when two models with similar $\mathcal{L}$ are considered, the $\Delta_i$ depends solely on the number of parameters due to the $2k$ term. Hence, when $\frac{\Delta_i}{2\Delta k} < 1$, the relative improvement is due to actual improvement of the fit, not to increasing the number of parameters only.
TL;DR
1. It's a bad reason; use the difference between the absolute values of the AICs.
2. The percentage says nothing.
3. Not possible to answer this question due to no information on the models, data, and what does different results mean.
• This is the clearest explanation I've ever seen of this mysterious matter. I looked up the article you referenced (pp. 270-272) and your explanation here is a simple and clear but very accurate representation of what the article explains. – Tripartio May 25 '18 at 14:03
• Could you perhaps help with this follow-up question? stats.stackexchange.com/questions/349883/… – Tripartio Jun 5 '18 at 12:13
• @corey979: thank you for the answer. What if several models have the same AIC? What if Δ_i < 2 rule covers several models? The less number of terms the better? – starkid 2 days ago | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.912640392780304, "perplexity": 485.4198903006746}, "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-2021-10/segments/1614178363809.24/warc/CC-MAIN-20210302095427-20210302125427-00048.warc.gz"} |
https://mathoverflow.net/questions/304381/infinite-connected-graphs-isomorphic-to-their-line-graph | Infinite connected graphs isomorphic to their line graph
For any simple, undirected graph $G$, let $L(G)$ denote its line graph.
$G=(\mathbb{Z}, E)$ with $E = \{\{k, k+1\}:k\in \mathbb{Z}\}$ has the property that $G\cong L(G)$.
Is there a connected infinite graph $G$ such that every vertex has more than $2$ neighbors, and $G\cong L(G)$?
There is an embedding of the complete graph on countably many vertices $K_{\aleph_0}$ into its line graph $LK_{\aleph_0}$ as the clique associated to one of the vertices. This induces embeddings $L^n K_{\aleph_0} \to L^{n+1}K_{\aleph_0}$ for all $n$. The iterated union of $L^n K_{\aleph_0}$ under all these embeddings has the desired property. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8287966847419739, "perplexity": 78.09219209271451}, "config": {"markdown_headings": false, "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-2021-04/segments/1610704839214.97/warc/CC-MAIN-20210128071759-20210128101759-00647.warc.gz"} |
https://rupress.org/jem/article/209/13/2515/40992/IL-33-reduces-the-development-of-atherosclerosis | Vol. 205, No. 2, February 11, 2008. Pages 339–346.
The authors regret that Fig. 1 B in this paper was incorrect, as two bands were inadvertently mixed up during figure compilation. None of the conclusions presented in the paper were affected by this error. The corrected Figure 1 is shown below, and the html and pdf versions of the article have been updated. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8817629814147949, "perplexity": 604.17449120532}, "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-2022-27/segments/1656103573995.30/warc/CC-MAIN-20220628173131-20220628203131-00652.warc.gz"} |
https://astarmathsandphysics.com/ib-physics-notes/astrophysics/1202-the-history-of-the-universe.html?tmpl=component&print=1&page= | ## The History of the Universe
Age of Universe Major Process Summary Unknown The Laws of Physics break down at earlier times than this. Unification of Forces There is only one force – all the forces of nature are united as a 'super force'. Inflation Space expands very rapidly, becoming nearly flat. Quark – Lepton era Matter and antimatter are interacting all the time. There is slightly more matter than antimatter. Hadron era Protons and neutrons start to form Helium synthesis Some protons and neutrons combine to form helium nuclei. Matter and antimatter have reacted to leave a small surplus of matter. Radiation era The universe is opaque. Radiation and matter are in equilibrium, with all matter in a plasma. As a soon as an atom forms it is ionized. Formation of atoms The Universe is transparent to radiation. Atoms can form. The Universe is 75% hydrtogen and 25% helium with trace amounts of heavier elements. Formation of stars and galaxies Gravity forces clumping of matter to form stars and galaxies and larger structures. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9075638055801392, "perplexity": 795.4107289425697}, "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-2018-34/segments/1534221210408.15/warc/CC-MAIN-20180816015316-20180816035316-00569.warc.gz"} |
https://nalinkpithwa.com/2021/05/ | # Generalized associative law, gen comm law etc.
Reference : Algebra by Hungerford, Springer Verlag, GTM.
Let G be a semigroup. Given $a_{1}, a_{2}, a_{3}, \ldots, a_{n} \in G$, with $n \geq 3, it is intuitively plausible that there are many ways of inserting parentheses in the expression$latex a_{1}a_{2}\ldots a_{n}\$ so as to yield a “meaningful” product in G of these n elements in this order. Furthermore, it is plausible that any two such products can be proved equal by repeated use of the associative law. A necessary prerequisite for further study of groups and rings is a precise statement and proof of these conjectures and related ones.
Given any sequence of elements of a semigroup G, $(a_{1}a_{2}\ldots a_{n})$ define inductively a meaningful product of $a_{1}, a_{2}, \ldots, a_{n}$ (in this order) as follows: If $n=1$, the only meaningful product is $a_{1}$. If $n>1$, then a meaningful product is defined to be any product of the form $(a_{1}\ldots a_{m})(a_{m+1}\ldots a_{n})$ where $m and $(a_{1}\ldots a_{m})$ and $(a_{m+1} \ldots a_{n})$ are meaningful products of m and $n-m$ elements respectively. (to show that this statement is well-defined requires a version of the Recursion Theorem). Note that for each $n \geq 3$ there may be meaningful products of $a_{1}, a_{2}, \ldots, a_{n}$. For each $n \in \mathcal{N}^{*}$ we single out a particular meaningful product by defining inductively the standard n product $\prod_{i=1}^{n}a_{i}$ of $a_{1}, \ldots, a_{n}$ as follows:
$\prod_{i=1}^{1}a_{i}=a_{1}$, and for $n>1$, $\prod_{i=1}^{n}a_{i}=(\prod_{i=1}^{n-1})a_{n}$
The fact that this definition defines for each $n \in \mathcal{N}^{*}$ a unique element of G (which is clearly a meaningful product) is a consequence of the Recursion Theorem.
Theorem: Generalized Associative Law:
If G is a semigroup and $a_{1}, a_{2}, \ldots, a_{n} \in G$, then any two meaningful products in $a_{1}a_{2}\ldots a_{n}$ in this order are equal.
Proof:
We use induction to show that for every n any meaningful product $a_{1}a_{2} \ldots a_{n}$ is equal to the standard n product $\prod_{i=1}^{n}a_{i}$. This is certainly true for $n=1, 2$. For $n>2$, by definition, $a_{1}a_{2} \ldots a_{n} = (a_{1}a_{2}\ldots a_{m})(a_{m+1} \ldots a_{n})$ for some $m. Therefore by induction and associativity:
$(a_{1}a_{2} \ldots a_{n}) = (a_{1}a_{2} \ldots a_{m})(a_{m} \ldots a_{n}) = (\prod_{i=1}^{m}a_{i})(\prod_{i=1}^{n-m})a_{m+i}$
$= (\prod_{i=1}^{m}a_{i}) ((\prod_{i=1}^{n-m-1}a_{m+i})a_{n} ) = ((\prod_{i=1}^{m})(\prod_{i=1}^{n-m-1}a_{m+i}))a_{n} = (\prod_{i=1}^{n-1}a_{i})a_{n} = \prod_{i=1}^{n}a_{i}$
QED.
Corollary: Generalized Commutative Law:
If G is a commutative semigroup and $a_{1}, \ldots, a_{n}$, then for any permutation $i_{1}, \ldots, i_{n}$ of 1, 2, …,n $a_{1}a_{2}\ldots a_{n} = a_{i_{1}}a_{i_{2}}\ldots a_{i_{n}}$
Proof: Homework.
Definition:
Let G be a semigroup with $a \in G$ and $n \in \mathcal{N}^{*}$. The element $a^{n} \in G$ is defined to be the standard n product $\prod_{i=1}^{n}a_{i}$ with $a_{i}=a$ for $1 \leq i \leq n$. If G is a monoid, $a^{0}$ is defined to be the identity element e. If G is a group, then for each $n \in \mathcal{N}^{*}$, $a^{-n}$ is defined to be $(a^{-1})^{n} \in G$.
It can be shown that this exponentiation is well-defined. By definition, then $a^{1}=a$, $a^{2}=aa$, $a^{3}=(aa)a=aaa, \ldots, a^{n}=a^{n-1}a$…and so on. Note that it is possible that even if $n \neq m$, we may have $a^{n} = a^{m}$.
Regards.
Nalin Pithwa
# A non trivial example of a monoid
Reference : Algebra 3rd Edition, Serge Lang. AWL International Student Edition.
We assume that the reader is familiar with the terminology of elementary topology. Let M be the set of homeomorphism classes of compact (connected) surfaces. We shall define an addition in M. Let $S, S^{'}$ be compact surfaces. Let D be a small disc in S, and $D^{'}$ in $S^{'}$. Let $C, C^{'}$ be the circles which form the boundaries of D and $D^{'}$ respectively. Let $D_{0}, D_{0}^{'}$ be the interiors of D and $D^{'}$ respectively, and glue $S-D_{0}$ to $S^{'}-D_{0}^{'}$ by identifying C with $C^{'}$. It can be shown that the resulting surface is “independent” up to homeomorphism, of the various choices made in preceding construction. If $\sigma, \sigma_{'}$ denote the homeomorphism classes of S and $S^{'}$ respectively, we define $\sigma + \sigma_{'}$ to be the class of the surface obtained by the preceding gluing process. It can be shown that this addition defines a monoid structure on M, whose unit element is the class of the ordinary 2-sphere. Furthermore, if $\tau$ denotes the class of torus, and $\pi$ denotes the class of the projective plane, then every element $\sigma$ of M has a unique expression of the form
$\sigma = n \tau + m\pi$
where n is an integer greater than or equal to 0 and m is zero, one or two. We have $3\pi=\tau+n$.
This shows that there are interesting examples of monoids and that monoids exist in nature.
Hope you enjoyed !
Regards,
Nalin Pithwa
# Algebra is symbolic manipulation though painstaking or conscientious :-)
Of course, I have oversimplified the meaning of algebra. 🙂
Here is an example. Let me know what you think. (Reference: Algebra 3rd Edition by Serge Lang).
Let G be a commutative monoid, and $x_{1}, x_{2}, \ldots, x_{n}$ be elements of G. Let $\psi$ be a bijection of the set of integers $(1,2, \ldots, n)$ onto itself. Then,
$\prod_{v=1}^{n}x_{\psi(v)} = \prod_{v=1}^{n}x_{v}$
Proof by mathematical induction:
PS: if one gets scared by the above notation, one can expand it and see its meaning. Try that.
It is clearly true for $n=1$. We assume it for $n=1$. Let k be an integer such that $\psi(k)=n$. Then,
$\prod_{i}^{n}x_{\psi(v)} = \prod_{1}^{k-1}x_{\psi(v)}.x_{\psi(k)}.\prod_{1}^{n-k}x_{\psi(k+v)}$
$= \prod_{1}^{k-1}x_{\psi(v)}. \psi_{1}^{n-k}x_{\psi(k+v)}.x_{\psi(k)}$
Define a map $\phi$ of $(1,2, \ldots, n-1)$ into itself by the rule:
$\phi(v)=\psi(v)$ if $v
$\phi(v) = \psi(v+1)$ if $v \geq k$
Then,
$\prod_{1}^{n} x_{\psi(v)} = \prod_{1}^{k-1}x_{\phi(v)}. \prod_{1}^{n-k}x_{\phi(k-1+v)} = \prod_{1}^{n-1}x_{\phi(v)}.x_{n}$
which by induction is equal to $x_{1}\ldots x_{n}$ as desired.
Some remarks: As a student, I used to think many a times that this proof is obvious. But it would be difficult to write it. I think this cute little proof is a good illustration of “how to prove obvious things in algebra.” 🙂
Regards,
Nalin Pithwa
# Wisdom of Hermann Weyl w.r.t. Algebra
Important though the general concepts and propositions may be with which the modern and industrious passion for axiomatizating and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity consitute the stock and core of mathematics, and that to master their difficulties requires on the whole hard labour.
—- Prof. Hermann Weyl. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 84, "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": 0.9856687188148499, "perplexity": 384.64283298356344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2021-39/segments/1631780055684.76/warc/CC-MAIN-20210917151054-20210917181054-00592.warc.gz"} |
https://www.physicsforums.com/threads/the-spectrum-of-reflected-light-from-a-planet-question.216954/ | The spectrum of reflected light from a planet question.
1. Feb 21, 2008
BERGXK
Is the spectrum of reflected light of a planet the same as the spectrum of the star near it but just less in magnitude? Like when you do B(Lambda;T) for it you just multiply that answer by the albedo?
2. Feb 21, 2008
mgb_phys
No it depends on the surface of the planet and the atmosphere (if any)
eg. Mars and Jupiter appear a lot redder than the sun.
A rocky object like the moon is 'grey' to a first approximation so you can just use the albedo.
3. Feb 21, 2008
BERGXK
Thanks for the quick reply. I still don't quite get it though. Plancks Spectrum gives an intensity of light at a certain wavelength and temperature right? For a star you can chose the wavelength and calculate the temperature if you assume the star is a blackbody. But how would you do this for a planet? The planet has both emitted and reflected light contributing to its planck spectrum.
I have a problem on my HW that asks me to compute the brightness of the planet relative to the star at certain wavelenghs 450nm 700nm and 2.2um. It asks me to consider both the starlight reflected off the planet and the backbody light emitted from the planet. When i first started this problem I just did B(lambda;Temp of planet)/B(lambda;Temp of star) for all the wavelengths and compared to see what was higher in each case. But this doesnt take into account the reflected light of the planet right? The reflected light and the emitted differ by a factor of albedo and 1-albedo. So my question is can i just do
((B(lambda;Temp of planet)*(1-a)) * (B(lambda;Temp of star)*a)) / B(lambda;Temp of star) to compare total brightness of planet to that of the star?
4. Feb 21, 2008
mgb_phys
The blackbody emission from a planet at 300K with a radius of 3000km is pretty insignificant compared to a star at 6000K with a radius of 700,000Km. You can use the Stefan–Boltzmann law to work it out. At the short wavelengths you are given the planet emits hardly anything.
To a simple approximation you can then work out from the solid angle, the fraction of the solar radiation hitting the planet and assume some albedo. You can also approximate the planet to a 2d disc facing the sun.
I was trying to explain why some planets are coloured due to the rock on their surface or the gases in their atmosphere. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8677576780319214, "perplexity": 584.3565705067167}, "config": {"markdown_headings": false, "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-2016-50/segments/1480698541696.67/warc/CC-MAIN-20161202170901-00058-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/119905/insert-multiple-figures-in-latex/119907 | # Insert multiple figures in Latex
I need to insert 10 figures (in two columns- side by side) in LaTeX that will have one global caption, but also I need to name each figure (1a, 1b, 1c, ... ect.). So they will look like:
1a 1b
1c 1d
1e 1f
1g 1h
1i 1j
Figure 1: plots of....
I would really appreciate if you guys can provide any help.
Below is how to insert two figures. Pls adapt this as per your needs. You need subcaption package.
\documentclass{article}
\usepackage[demo]{graphicx}
\usepackage{subcaption}
\begin{document}
\begin{figure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.8\linewidth]{image1}
\caption{1a}
\label{fig:sfig1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.8\linewidth]{image2}
\caption{1b}
\label{fig:sfig2}
\end{subfigure}
\caption{plots of....}
\label{fig:fig}
\end{figure}
\end{document}
Refer this for information about another method http://texblog.org/2011/05/24/placing-figures-side-by-side-subfig/
This is also similar to what you are looking for - how to put subfigures in several rows
• LaTeX Error: Environment subfigure undefined. What to do now? – user1603548 Mar 1 '14 at 16:58
• @user1603548 You need the subcaption package. I modified the example to reflect this. – Torbjørn T. Mar 17 '14 at 19:22
• This answer does not explain how to change line and there are some subtleties, e.g., put % after \end{subfigure} etc. – Jimmy R. May 3 at 15:34
Instead of you using two environments-subfig and figure, you can just use figure and subfloat
\documentclass{article}
\usepackage{float}
\usepackage[caption = false]{subfig}
\usepackage[demo]{graphicx}
\begin{document}
\begin{figure}
\subfloat[fig 1]{\includegraphics[width = 3in]{something}}
\subfloat[fig 2]{\includegraphics[width = 3in]{something}}\\
\subfloat[fig 3]{\includegraphics[width = 3in]{something}}
\subfloat[fig 4]{\includegraphics[width = 3in]{something}}
\label{some example}
\end{figure}
\end{document}
After ever two figures add \\ or you can adjust the width so that only two figures will fit side by side.
==========================
\usepackage[demo]{graphicx}
\usepackage[final]{graphicx}
• \usepackage[demo]{graphicx} in preamble will do. Note figure-filenames don't need extensions something is fine depending on LaTeX engines executed it will detect which file extension is needed. Hope you might be aware of mwe package that specifically caters to MWE needs. – texenthusiast Jun 19 '13 at 4:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9348520636558533, "perplexity": 3821.1439280339605}, "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-2020-34/segments/1596439738982.70/warc/CC-MAIN-20200813103121-20200813133121-00413.warc.gz"} |
https://math.stackexchange.com/questions/1788756/check-equivalence-of-quadratic-forms-over-finite-fields | # Check equivalence of quadratic forms over finite fields
How to check whether the two quadratic forms $$x_1^2 + x_2^2 \quad \text{(I)}$$ and $$2x_1x_2 \quad \text{(II)}$$
are equivalent on each of the spaces $\mathbb{F}_3^2\, \text{and}\,\mathbb{F}_5^2$? I know that these forms correspond to the two matrices \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} in case (I) and \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} in case (II). But what do I do with those and what difference do the different fields make?
The characteristic of the field makes a big difference in the computations. I would start by comparing the discriminates of the two matrices.
Over both fields, the discriminants are $\pm 1$. The question is: do $\pm1$ differ by a square factor in both fields or not?
Over the field of five elements, $1\cdot 3^2\equiv -1\pmod 5$, but over the field of three elements, $a^2\equiv 1\not\equiv -1\pmod 3$ for all nonzero $a$.
After some work, you can find that the two matrices are cogredient this way: $$\begin{bmatrix}3&1\\1&3\end{bmatrix}\begin{bmatrix}1&0\\0&1\end{bmatrix}\begin{bmatrix}3&1\\1&3\end{bmatrix}=\begin{bmatrix}0&1\\1&0\end{bmatrix}$$
Here's an elementary brute force way to figure this out using just algebra. From $$\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}1&0\\0&1\end{bmatrix}\begin{bmatrix}a&c\\b&d\end{bmatrix}=\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a&c\\b&d\end{bmatrix}=\begin{bmatrix}a^2+b^2&ac+bd\\ac+bd&c^2+d^2\end{bmatrix}=\begin{bmatrix}0&1\\1&0\end{bmatrix}$$
we have already learned that $a^2=-b^2$, $c^2=-d^2$ and $ac+bd=1$. Earlier I noted that $1$ differs from $-1$ by a factor of $3^2$, so I suppose my change of basis matrix has determinant $3$, so $ad-bc=3$.
Now
$$a= a^2c+adb\\ =-b^2c+adb\\ = b(ad-bc)\\ = 3b$$
Similarly $$c= ac^2+dcb\\ =-d^2a+dcb\\ = -d(ad-bc)\\ = -3d = 2d$$
We've consumed the two equations involving squares, and now we combine the other two.
We have that
$$1=ac+bd=6bd+bd=7bd=2bd$$
Hence, $bd=3$ or in other words $d=3/b$. At this point we have $a,c,d$ all in terms of $b$ ($a=3b$, $d=3/b$, $c=2d=1/b$).
Choosing $b=1$, we arrive at $a=d=3$, $b=c=1$.
• Thanks, that helped a lot. But how did you come up with the matrix $P = \begin{pmatrix} 1 & 3 \\ 3 & 1 \end{pmatrix}$ over $\mathbb{F}_5$? – Taufi May 17 '16 at 12:57
• @Taufi Hi: I was just in the process of explaining. It's in there now. Thanks for waiting. – rschwieb May 17 '16 at 13:09
• No, thank you for taking the time to explain the solution so thoroughly. +10 if I could – Taufi May 17 '16 at 13:13
You've wrtten the other way round. For the quadratic form $x_1^2 + x_2^2$, the corresponding matrix is the identity while that for $2x_1x_2$ is $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. The forms are equivalent iff the matrices are congruent. (over $\mathbb{F}_3$ and $\mathbb{F}_5$)
So, suppose $A = P^TIP = P^TP$ over $\mathbb{F}_3$ for some invertible $P$. Then, taking determinants gives $-1 = ( \det P)^2$, which is a contradiction as $-1$ is not a quadratic residue modulo $3$.
Over $\mathbb{F}_5$, you can check that they indeed are equivalent by the matrix $P = \begin{pmatrix} 1 & 3 \\ 3 & 1 \end{pmatrix}$ as we have, $(3x_1+x_2)^2 + (x_1+3x_2)^2 = 2x_1x_2$ in $\mathbb{F}_5$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9848364591598511, "perplexity": 196.17494511587878}, "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-2019-35/segments/1566027314732.59/warc/CC-MAIN-20190819114330-20190819140330-00475.warc.gz"} |
https://theculture.sg/2015/07/what-does-unbiased-in-math-mean/ | # What does unbiased in Math mean?
Some of your school notes will mention and explain the definition of unbiased and consistency. But let us just discuss what unbiased means as we do see it quite often in A-level.
What does its mean to say that is an unbiased estimate of ?
We know that an estimate means its an approximation, meaning that . So unbiased will mean that the . And if we recall expectation is a really intuitive meaning, simply what do you expect? So unbiased indicates that we expect to be but do note that latex s^2\ne \sigma^2\$ still!
Want to learn more? Arrange a math session with KS to get started! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.964445948600769, "perplexity": 1241.5695312640676}, "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-2020-40/segments/1600400211096.40/warc/CC-MAIN-20200923144247-20200923174247-00753.warc.gz"} |
https://nips.cc/Conferences/2015/ScheduleMultitrack?event=5637 | Timezone: »
Poster
Multi-class SVMs: From Tighter Data-Dependent Generalization Bounds to Novel Algorithms
Yunwen Lei · Urun Dogan · Alexander Binder · Marius Kloft
Thu Dec 10 08:00 AM -- 12:00 PM (PST) @ 210 C #77 #None
This paper studies the generalization performance of multi-class classification algorithms, for which we obtain, for the first time, a data-dependent generalization error bound with a logarithmic dependence on the class size, substantially improving the state-of-the-art linear dependence in the existing data-dependent generalization analysis. The theoretical analysis motivates us to introduce a new multi-class classification machine based on lp-norm regularization, where the parameter p controls the complexity of the corresponding bounds. We derive an efficient optimization algorithm based on Fenchel duality theory. Benchmarks on several real-world datasets show that the proposed algorithm can achieve significant accuracy gains over the state of the art. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9228380918502808, "perplexity": 1347.7091387938217}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "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-2020-40/segments/1600400214347.17/warc/CC-MAIN-20200924065412-20200924095412-00312.warc.gz"} |
https://www.physicsforums.com/threads/hanging-sign-torque-and-pivot-points.198256/ | # Hanging sign, torque and pivot points
1. Nov 14, 2007
### Sean77771
1. The problem statement, all variables and given/known data
There is a sign of mass M that is attached to a rigid bar perpendicular to the ground. There is also a rope attached to the sign at the same point, then pulled out at an angle 45 degrees below the horizontal up to the ceiling.
a) Consider the rigid bar to be at equilibrium while the sign is hanging. What are the net force (in Newtons) and the net torque (in N m) acting on the bar?
b) Identify all forces that act on the bar. Write out an equation for the net force on the bar.
c) Choose the point where the bar meets the wall as a pivot point. Write out an equation for the net torque on the bar about that point.
d) Determine the magnitude of the tension in the rope. Take the mass of the sign to be 25kg, the mass of the bar is 5kg, and the length of the bar is 2m.
e) We could choose any point as the pivot and still get the same net torque. Why was the choice in part c a good choice for the pivot?
2. Relevant equations
T_net = Ia
when in equilibrium,
-- F_net = 0
-- T_net = 0
3. The attempt at a solution
I got that, for (a), these are both equal to zero, as the bar is in equilibrium...don't know where to go from here.
2. Nov 15, 2007
### Shooting Star
If you give a rough diagram, I'm sure you'll get some responses. I am unable to visualize the whole picture.
Anyway, in all these kinds of statics problems, the way to proceed is to equate the sum of the horizontal and vertical forces individually to zero; and to equate the moment of all the forces about a conveniently chosen point to zero. Then solve the resulting equations.
Similar Discussions: Hanging sign, torque and pivot points | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9112211465835571, "perplexity": 410.583776358511}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2017-13/segments/1490218189734.17/warc/CC-MAIN-20170322212949-00230-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://twinery.org/wiki/harlowe:subarray | # Twine Wiki
### Special passage tags
harlowe:subarray
This macro is deprecated - it may be removed in a future version of Harlowe.
(subarray: Array, Number, Number) → Array
When given an array, this returns a new array containing only the elements whose positions are between the two number, inclusively.
#### Example usage:
`(subarray: \$a, 3, 4)` is the same as `\$a's (a:3,4)`
#### Rationale:
You can obtain subarrays of arrays without this macro, by using the `'s` or `of` syntax along with an array of positions. For instance, `\$a's (range:4,12)` obtains a subarray of \$a containing its 4th through 12th values. But, for compatibility with previous Harlowe versions which did not feature this syntax, this macro also exists.
#### Details:
If you provide negative numbers, they will be treated as being offset from the end of the array - `-2` will specify the `2ndlast` item, just as 2 will specify the `2nd` item.
If the last number given is larger than the first (for instance, in `(subarray: (a:1,2,3,4), 4, 2)`) then the macro will still work - in that case returning (a:2,3,4) as if the numbers were in the correct order. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8170374035835266, "perplexity": 1721.9497163955334}, "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-2019-43/segments/1570986660231.30/warc/CC-MAIN-20191015182235-20191015205735-00242.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/metal-sphere-radius-carries-charge-q-surrounded-concentric-metal-shell-inner-radius-b-oute-q955047 | A metal sphere of radius a, carries a charge of +q, and is surrounded by a concentric metal shell of inner radius b and outer radius c. The outer shell carries a charge of -q.
a) Within the sphere (r<a)
b) Between the sphere and shell (a<r<b)
c)inside the shell (b<r<c)
d) Outside the shell (r>c)
e) How is the charge distributed on the inner and outer surfaces of the shell?
I know inside the inner sphere E must be zero(a), and I know inside the conducting metal sphere E must also be zero in order for this to be in electrostatic equilibrium(c), but I am unsure how to use Gaussian surfaces to calculate (b). I know but I am unsure how to take into account how the outside of the inner sphere and the inside of the outer sphere are where equal but opposite charges reside. As for d, I know the sphere system will act like a point charge, but once again I am unsure of how to combine the charges and use Gauss' Law. For e) I know that the +q charge is on the outside of the smaller sphere and the -q is on the inside of the larger shell. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8849344849586487, "perplexity": 196.52573252524647}, "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-2015-22/segments/1432207928865.24/warc/CC-MAIN-20150521113208-00111-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.quantumstudy.com/physics/kinematics-5/ | # Motion Under Gravity
Vertical motion under gravity:
Consider a body P projected vertically upwards from the surface of the earth with an initial velocity, u . The body rises to a maximum height at B and then returns (motion BC).
Even through we have shown the motion of the body as ABC, the actual path is always along the same line AB. The picture has been slightly modified for clarity.
Let the displacement of the body (at P) at time t, measured from its initial position A, be denoted by h.
We can now apply the equations . The acceleration, a = – g
(Note the positive direction in the figure, any vector in the opposite direction is negative);
x = h and the initial velocity is u . After time t ,
$\displaystyle (i) v = u – g t$
$\displaystyle (ii) h = ut – \frac{1}{2}g t^2$
$\displaystyle (iii) v^2 = u^2 – 2 g s$
### Other interesting questions that may be posed are :
(a) What’s the maximum height to which it rises ?
(b) What’s the time of flight ?
Let us note that at the point of maximum elevation , B , vB = 0 (it’s got to be zero, if it were not the body would have risen further)
vB = u – g tAB = 0 ;
tAB = time taken for motion AB.
$\displaystyle t_{AB} = \frac{u}{g}$
If the maximum elevation is H (at the point B of course)
vB2 = u2 − 2gH
0 = u2 − 2 g H
$\displaystyle H = \frac{u^2}{2 g}$
When the body reaches the ground again (at the point C), we can write,
$\displaystyle h = u t_{AC} – \frac{1}{2}g {t_{AC}}^2$
Where tAC = time taken for motion AC
$\displaystyle 0 = u t_{AC} – \frac{1}{2}g {t_{AC}}^2$
$\displaystyle t_{AC} = \frac{2 u}{g} = 2 t_{AB}$
The velocity at the point C,
vC = u − gtAC
$\displaystyle = u – g \times \frac{2 u}{g}$
= − u ;
which is equal in magnitude to u (the velocity of projection) but opposite in direction.
In both the examples considered above, the acceleration is constant (or uniform). This may not always be true.
### Particle projected from the top of a cliff
For the motion of a particle projected vertically up or down from some height (h) we have assign the directions of displacement vector, velocity vector and acceleration vector with reference to a system of coordinates.
Let us consider the following cases :
(i) When the particle is projected with an initial velocity v0 in the upward direction and its vertical distance from the point of projection O at a time t be h, then
$\displaystyle h = v_0 t – \frac{1}{2}g t^2$ ; (if it is above the point O)
$\displaystyle – h = v_0 t – \frac{1}{2}g t^2$ ; (if it is below the point O)
We obtain the value of time by putting the given values of h and v0.
If the origin is shifted to the ground level, which is at a depth H below the point of projection,
then h and – h mentioned above are replaced by (H + h) and (H – h) respectively.
Solved Example : A stone is dropped from a balloon ascending with v0 = 2 m/sec, from a height h = 4.8 m. Find the time of flight of the stone (g = 10 m/sec).
Solution: $\large h = – v_0 t + \frac{1}{2} g t^2$
$\large t^2 – \frac{2 v_0}{g} t – \frac{2 h}{g} = 0$
$\large t = \frac{v_0}{g} (1 + \sqrt{\frac{2 g h}{v_0^2} + 1 })$
Putting the values of v0 and h , we obtain,
$\large t = \frac{2}{10} (1 + \sqrt{\frac{2 \times 10 \times 4.8}{2^2} + 1 })$
t = 1.2 sec
Solved Example: Referring to the previous Problem , if the stone reaches the ground after t = 2 second from same initial height of release, find the
(a) speed of the balloon at the time of releasing the stone,
(b) total distance covered by the stone till it reaches the ground level,
(c) the average speed and
(d) average velocity of the stone for the total time of its flight, one from the bottom and the other from the top.
### Particle Projected from Top & Bottom
We see that s1 is downward ,
$\displaystyle s_1 = v_1 t + \frac{1}{2}g t^2$ ……(i)
$\displaystyle s_2 = v_2 t – \frac{1}{2}g t^2$ ……..(ii)
(i) + (ii) gives
⇒ s1 + s2 = (v1 + v2)t
$\displaystyle t = \frac{h}{v_1 + v_2}$ …….(iii)
v1 + v2 = Relative velocity and
h = initial relative distance of separation.
Solved Example : A stone is released form the top of a cliff. Another particle is simultaneously projected with v = 20 m/sec. from the bottom of the cliff. If they meet after 2 seconds, find the height of the cliff.
Solution: Let after a time t, they meet s1 downward
$\displaystyle s_1 = \frac{1}{2}g t^2$ …..(1)
Since , s2 is upward
$\displaystyle s_2 = v_2 t – \frac{1}{2}g t^2$
Putting v2 = v ,we obtain ,
$\displaystyle s_2 = v t – \frac{1}{2}g t^2$ …(2)
(1) + (2)
⇒ s1 + s2 = v t
⇒ h = v t.
Putting v = 20 m/sec and t = 2 sec.
We obtain h = (20) (2) = 40 m.
Exercise : A body is released from a height and falls freely towards the earth. Exactly 1 sec. later another body is released. What is the distance between the bodies 2 sec. after the released of the second body if , g = 9.8 m/s2 ?
Exercise : Two particles are simultaneously projected upwards from the top and bottom of a cliff of height h = 20 m. If the speed of one is double that of the other and they meet after a time t = 2 second, find their speed of projection. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8914072513580322, "perplexity": 719.8344479207071}, "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-2022-49/segments/1669446710890.97/warc/CC-MAIN-20221202014312-20221202044312-00250.warc.gz"} |
http://eprint.iacr.org/2012/525 | ## Cryptology ePrint Archive: Report 2012/525
Computing endomorphism rings of abelian varieties of dimension two
Gaetan Bisson
Abstract: Generalizing a method of Sutherland and the author for elliptic curves, we design a subexponential algorithm for computing the endomorphism ring structure of ordinary abelian varieties of dimension two over finite fields. Although its correctness and complexity bound rely on several assumptions, we report on practical computations showing that it performs very well and can easily handle previously intractable cases.
Category / Keywords: foundations / hyperelliptic curves, complex multiplication, isogenies | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9224725961685181, "perplexity": 1271.683978504095}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2017-04/segments/1484560280730.27/warc/CC-MAIN-20170116095120-00513-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://philarchive.org/rec/GUASDA | # Semantic Dispositionalism and Non-Inferential Knowledge
Philosophia 42 (3):749-759 (2014)
# Abstract
The paper discusses Saul Kripke's Normativity Argument against semantic dispositionalism: it criticizes the orthodox interpretation of the argument, defends an alternative reading and argues that, contrary to what Kripke himself seems to have been thinking, the real point of the Normativity Argument is not that meaning is normative. According to the orthodox interpretation, the argument can be summarized as follows: (1) it is constitutive of the concept of meaning that its instances imply an ought, but (2) it is not constitutive of the concept of a disposition that dispositions imply an ought, hence (3) no dispositional analysis of meaning can work. According to my alternative reading, the point of the argument is another one, namely that while (1) dispositionalism is committed to the thesis that speakers have non-inferential knowledge of their unmanifested linguistic dispositions, (2) speakers, as a matter of fact, do not have such a knowledge. A point that is in principle independent from the issue of the normativity of meaning.
# Author's Profile
Andrea Guardo
Università degli Studi di Milano | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9256842732429504, "perplexity": 2248.54342556166}, "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-2022-49/segments/1669446711344.13/warc/CC-MAIN-20221208150643-20221208180643-00126.warc.gz"} |
https://physics.stackexchange.com/questions/378409/wick-theorem-performing-contractions-in-the-right-order | # Wick Theorem: Performing contractions in the right order
The first line is one of four terms that one gets after applying Wick theorem to the time-ordered product of these field operators and as far as i understand it is just a short-hand notation for which operators have to be contracted in this case.
I dont unterstand how to get the right order of the contractions as shown in the second line, given the first line. First guess was naively performing the contractions shown in the first line as they appear from left to right, starting with $a_{\textbf{p´}}^-$ contracted with $A^{\nu}(y)$ and then $b_{\textbf{q´}}^-$ contracted with $\overline{\psi}(y)$ and so on... which is obviously not the right way to do it.
Of course only from physical reasoning the ordering of the contractions in the second line makes perfect sense (for example: the Fermion propagator should stand between the gamma matrices). But i am looking for rigorous rules how to do the ordering in a strict way.
Has anyone an idea how to explain them in an easy way?
• This site supports MathJax (LaTeX) notation for mathematical formulae and it's generally preferable to use that instead of embedded images. However, I'm not fully sure how to typeset Wick contractions, so that might require some additional help. – Emilio Pisanty Jan 6 '18 at 23:29
• @EmilioPisanty: how would you do the connecting lines above and below the expression? – flippiefanus Jan 7 '18 at 3:05
• @flippiefanus As I said, I don't know how to typeset the Wick contractions in the image; if I did, I would have edited it directly. In standard LaTeX there appears to be a simplewick package that can do this, but I suspect it won't be available on MathJax. This needs attention from someone who actually works on QFT to be typeset correctly. – Emilio Pisanty Jan 7 '18 at 3:15
• On the one hand i know that a contraction of two fields should just be a C number but on the other hand that would mean i can simply pick a any order for writing down the contractions. But that can obviously not be true otherwise i could for example write the contraction with $\overline{\psi}(x)$ and $\psi (y)$ at the beginning and thus have, after conctracting all the other contractions afterwards, the fermion propagator at the beginning of the expression for the amplitude and not between expressions for external lines as it should be the case by looking at the corresponding feynman diagram – Johnny90 Jan 8 '18 at 23:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9019018411636353, "perplexity": 262.754463399489}, "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-2019-35/segments/1566027314752.21/warc/CC-MAIN-20190819134354-20190819160354-00229.warc.gz"} |
https://www.physicsforums.com/threads/physical-pendulum.205200/ | # Physical pendulum
1. Dec 18, 2007
### JolleJ
[SOLVED] Physical pendulum
1. The problem statement, all variables and given/known data
A physical pendulum constists of a rod (length = 0,30 m og mass = 0,80 kg). The rod rotates around its center. On this rod there is a moveable mass (mass=0,80kg).
Now the period of the pendulum is T = 0,83 s.
The problem to solve is: How far below the center of the rod is the mass placed?
2. Relevant equations
$$T=2\pi*\sqrt{\frac{I}{m*g*a}}$$
From what I have read a is the distance between the center of the rod and the center of gravity.
Also: For a rod: I = 1/12 * m * l^2, where l is the length of the rod.
And for the mass shifted from the center I think it is I = m*x^2 - where r is the shifted distance. (From Steiners equation where I0 = 0).
3. The attempt at a solution
Moment of inertia I:
I = 1/12*0,8*0,3^2+0,8*x^2
Center of gravity a:
a = 0,80*x/1,6 = 0,5*x
Then this equation:
$$T=2\pi*\sqrt{\frac{I}{m*g*a}}$$
is solved with respect to for T = 0,83 s. However this gives L = Ø. That is, a non existing result.
I really hope that someone can help me, and tell me what I've done wrong.
2. Dec 18, 2007
### Staff: Mentor
How did you conclude this?
3. Dec 18, 2007
### JolleJ
Just by entering:
solve(0.83=2*%pi*sqrt((1/12*0.8*0.3^2+0.8*x^2)/(1.6*9.82*0.5*x)),x);
on a calculator. It says "false".
4. Dec 18, 2007
### Staff: Mentor
Crank it out by hand and check for yourself.
5. Dec 18, 2007
### JolleJ
Okay, now I tried. I also get "false"...
First I did something wrong apparently and got 0,023 m / 0,31 m... Not sure what it was though.
However; still L = Ø. :s Something is wrong with the equation that I've made.
6. Dec 18, 2007
### Staff: Mentor
OK, you're right: Given the values that you posted, the resulting quadratic has no real solutions. I don't see anything wrong with the method used, so I suspect that there's a typo (or error) in one or more of the values.
7. Dec 18, 2007
### JolleJ
Very strange, given that it's an examination task.
However, thank you very much (once again, again!). :D
I will write my teacher and ask him, if he's made a mistake. :)
8. Dec 19, 2007
### rl.bhat
When the rod rotates around its center, the period is infinity. When a mass is attached to the rod the period is affected by only due to this mass, and its distance from the center. So T = 2*pi*sqrt(m*a^2/m*g*a) = 2*pi*sqrt(a/g)
9. Dec 19, 2007
### Staff: Mentor
This is true.
No, you can't just ignore the mass distribution of the rod. You must consider the rotational inertia of the entire system.
No. That's the period of a simple pendulum; this must be treated as a physical pendulum.
10. Dec 19, 2007
### rl.bhat
Moment of inertia I:
I = 1/12*0,8*0,3^2+0,8*x^2
This I is about the center of the rod. In the formula we want MI about CM.
So I = Icm + M(x/2)^2 or Icm = I - M(x/2)^2 where M is the total mass.
Just by entering:
solve(0.83=2*%pi*sqrt((1/12*0.8*0.3^2+0.8*x^2)/(1.6*9.82*0.5*x)),x);
on a calculator. It says "false".
Rewright it as
0.83 = 2*pi*sqrt[(1/12*0.8*0.3^2 + 0.4*x^2)/(1.6*9.82*0.5x)] and solve.
11. Dec 20, 2007
### Shooting Star
We all tried to, but b^2-4ac of the resulting quadratic is negative. Also, the thing, at least to me, is neater if you use CGS. The eqn I got is x^2 -(34.24)x + 600 = 0.
12. Dec 20, 2007
### rl.bhat
The eqn I got is x^2 -(34.24)x + 600 = 0.
It should be x^2 -(34.24)x + 150 = 0.
13. Dec 20, 2007
### Staff: Mentor
No, we need the moment of inertia about the point of suspension, which is the middle of the rod.
Not sure what that's supposed to be. If you mean that to be the MI about the center of mass of the system, that's not quite right. Since the CM is at a point x/2 from the middle of the rod, the MI about that point would be: 1/12ML^2 + M(x/2)^2 + M(x/2)^2 = 1/12ML^2 + 1/2Mx^2. (Of course, this MI is irrelevant, as the CM is not the point of suspension.)
14. Dec 20, 2007
### JolleJ
Thank you very much, all of you, however I just got respons from my teacher, who said there was an error in the question. It should be T = 0,835 s. And now everything works.
Thanks again !
15. Dec 20, 2007
### Shooting Star
I have not repeated the calculation, but I have a strong doubt that changing the time period by such a small amount would resolve the problem. I ask the OP to do the calculation once more, even if he's fed up by this msg from me.
16. Dec 20, 2007
### JolleJ
It works now, so there's no problem. :) Thanks though. :D
17. Dec 20, 2007
### Staff: Mentor
Yep, it works. Just by a hair.
18. Dec 20, 2007
### Shooting Star
Well, it was a close shave then.
19. Dec 20, 2007
Haha. :) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9322846531867981, "perplexity": 1573.5123465173242}, "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-2016-50/segments/1480698541692.55/warc/CC-MAIN-20161202170901-00326-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://trickledown.wordpress.com/2008/06/08/learning-more-about-derivatives/ | ## Learning More About Derivatives
Today we’ll look at some tips on how to figure out derivatives in calculus. Disclaimer: I’m no math pro, so this might not all be accurate.
*Update*
A really nice basic demonstration of how the to use the difference quotient and how it is related to some of the basic rules of differentiation is located at http://www.intmath.com/Differentiation/3_Derivative-first-principles.php.
One important thing is to pin down what the independent variable(s) is/are in the equations you will be differentiating–generally denoted in letters at the end of the alphabet, like x, y, or z in many situations, such if you’re given the functions f(x), g(y), or h(z)–and figuring out what variables you’ll be differentiating with respect to. This is important, because for any constant in the function (generally denoted by letters at the beginning of the alphabet such as a, b, c, d, etc in many situations) the derivative will be zero (unless the constant is multiplied by the variable you’re differentiating with respect to)–examine the constant rule:
• Constant rule: if f is the constant function f(x) = c, for any number c, then for all x $f'(x)=0$.
and the constant factor/constant multiple rules of differentiation.
When looking at functions with more than one variable, the variables you’re not differentiating with respect to are also treated like constants, like when taking partial derivatives.
Here are some examples of where pinning down the constants, and realizing that their derivative will be zero, will be useful, in combination with a handful of differentiation rules.
For example, in this explanation of the proof of the second mean value theorem in the book Mathematics for Economists, using Rolle’s theorem, part of the proof involves taking the second derivative of the equation $R(x)=f(x)-f(a)-(x-a)f'(a)-K(x-a)^2$, where a is a constant.
Now, taking the derivative of this equation involves the sum rule, also called the combination rule, the product rule, the power rule, the constant rule, and the constant factor/constant multiple rules of differentiation, as well as the basic distributive property of multiplication.
So, to get R'(x), first of all, you can separately take the derivatives of all the terms separated by plus (+) signs, according to the sum rule/the combination rule. So, taking the derivatives of each of the terms:
First, d/dx(f(x))=f'(x). Next, d/dx(f(a))=0 since a is a constant, by the constant rule–since a is a constant, f(a) also is a constant, since for each input of a function there is a unique output, and the constant rule says that the derivative of a constant is 0. Next, we have (x-a)f'(a). Using the distributive property of multiplication, this factors into xf'(a) + -af'(a). Next, to find d/dx(xf'(a)), we use the constant factor/constant multiple rule. Since a is a constant, on its own f'(a) would be zero, but since it’s multiplied by x, we multiply it by the derivative of x which is 1, so d/dx(xf'(a))=f'(a). Next, we have d/dx(-af'(a)). Here both a and f'(a) are constants, so both d/dx(a) and d/dx(f'(a)) are 0, and d/dx(-af'(a))=0. Next we have $-K(x-a)^2$. First we can multiply out $K(x-a)^2$, from which we get $-K(x^2-2ax+a^2)$, so we get $-Kx^2 +2Kax -Ka^2$ (Note: we could also have used the composite function/chain rule here instead of multiplying out $K(x-a)^2$, see the end of the post for more details). Now taking the derivative of this, d/dx ( $-Kx^2$)=-2Kx by the power rule and the constant factor/constant multiple rule, and d/dx(2Kax)=2ka by the same rules, and d/dx($-Ka^2$)=0 since a is a constant, by the constant rule.
This leaves us with R'(x)=f'(x) + 0 + -f'(a) + 0 + -2Kx + 2Ka. Now, to take the derivative of R'(x), which will be the second derivative of R(x), which will be R”(x).
First, d/dx(f'(x))=f”(x). Next, d/dx(-f'(a))=0 since a is a constant. Next, d/dx(-2Kx)=-2K by the power rule and the constant factor/constant multiple rule. Next, d/dx(2Ka)=0 since 2, K, and a are constants, and the derivative of a constant equals 0.
This leaves us with R”(x)=f”(x) + -2K. Further on in the proof, using Rolle’s theorem, you find that there is a number d such that R”(d)=0. Thus, R”(d)=f”(d) + -2K, which equals 0=f”(d) +-2K, which leaves us with f”(d)=2K. The rest of the proof is in the book.
Hopefully this is all correct (it might not be, I’m no math pro), and demonstrates how all sorts of different differentiation rules are used in calculus problems!
• Note 1: we could have used the composite function/chain rule instead of multiplying out $-K(x-a)^2$ (see the section on composition of functions also). By the composite function rule, for the derivative of $(x-a)^2$ we would set q(x)=u=(x-a) , and p(u)=y=$u^2$. We would set f(x)=p(q(x))=y. Then dy/dx=du/dx * dy/du. Now, du/dx, where u=(x-a), equals 1 by the sum rule/the combination rule, the power rule, and the constant rule. Next, dy/du where y=$u^2$ is 2u, by the power rule. Then we want dy/dx=du/dx * dy/du. So that’s 2u * 1. Substituting in u=(x-a) we get 2(x-a) or 2x-2a. So to find the derivative of $-K(x-a)^2$, we use the constant factor/constant multiple rule for the -K, and the derivative of $-K(x-a)^2$, is -2K(x-a) or -2Kx+2Ka like we figured out above.
• Note 2: We didn’t even get to the product rule/multiplication rule!!! The product rule can be used in equations where you have more than one term containing the independent variable that you are differentiating with respect to–you can use the product rule when those terms are multiplied together. You can read more about the product rule in the Mathematics for Economists book here. For example, if you had $f(x)=x * x^2$, you could easily multiply the terms together to get $f(x)=x^3$ then use the power rule to get $f'(x)=3x^2$, but for the sake of example let’s use the product rule. If we set x=u and $x^2=v$, and f(x)=uv, using product rule, f'(x)=v * du/dx + u * dv/dx. Here, du/dx=1 by the power rule, and dv/dx=2x by the power rule. Then $f'(x) = v * 1 + u *2x= x^2 * 1 + x * 2x = x^2 + 2x^2 = 3x^2$, which shows you that that’s what we would have gotten if we had just multiplied the terms together and taken the derivative using the power rule. The product rule can be helpful in cases where it would be simpler to product rule than it would be to multiply the terms together first then take the derivative.
The Rushmore math picture is from this Math at the Movies webpage.
Also see: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 20, "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": 0.9578456878662109, "perplexity": 490.77494668414425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2017-13/segments/1490218191353.5/warc/CC-MAIN-20170322212951-00044-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://planetmath.org/MultiplicativeFilter | # multiplicative filter
For any ring $A$, any set $S\subset A$ and any element $x\in A$, we use the notation
$(S:x):=\{a\in A\ ax\in S\}$
Let $A$ be a commutative ring with unity, and let $\mathcal{I}(A)$ be the set of all ideals of $A$.
• A Multiplicative Filter of $A$ is a filter $\mathcal{F}$ on $\mathcal{I}(A)$ such that $I,J\in\mathcal{F}\Rightarrow IJ\in\mathcal{F}$.
• A Gabriel Filter of $A$ is a filter $\mathcal{F}$ on $\mathcal{I}(A)$ such that
$[I\in\mathcal{F},J\in\mathcal{I}(A)\textrm{ and }\forall x\in I,(J:x)\in% \mathcal{F}]\Rightarrow J\in\mathcal{F}$
Note that Gabriel Filters are also Multiplicative Filters.
Title multiplicative filter MultiplicativeFilter 2013-03-22 16:48:22 2013-03-22 16:48:22 jocaps (12118) jocaps (12118) 6 jocaps (12118) Example msc 03E99 msc 54A99 Gabriel Filter Multiplicative Filter | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 15, "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": 0.9944595098495483, "perplexity": 854.8826784069121}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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/1585371612531.68/warc/CC-MAIN-20200406004220-20200406034720-00023.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=51&t=39648&p=134464 | ## 5.33
Michelle Song 1G
Posts: 30
Joined: Wed Nov 14, 2018 12:23 am
### 5.33
According to the solutions manual, the answer is a because the reaction is endothermic. The solutions manual says that the value of the equilibrium constant is larger in Flask 2 than in Flask 1, which is consistent with an increase in temperature of an endothermic reaction. I do not understand why an increase in equilibrium constant shows that a reaction is endothermic.
Last edited by Michelle Song 1G on Thu Jan 10, 2019 10:46 pm, edited 1 time in total.
Michelle Song 1G
Posts: 30
Joined: Wed Nov 14, 2018 12:23 am
### Re: 5.33
Also, why does decreasing the volume favor the formation of X2? I understand that decreasing the volume increases the pressure but I do not understand why this would lead to an increase in reactants.
Alondra Olmos 4C
Posts: 32
Joined: Fri Sep 28, 2018 12:21 am
### Re: 5.33
An increase in the equilibrium constant suggests the reaction is favoring the products. In order to produce more product, the reaction requires heat which makes it endothermic
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https://artofproblemsolving.com/wiki/index.php?title=2005_PMWC_Problems/Problem_T8&oldid=113374 | # 2005 PMWC Problems/Problem T8
## Problem
An isosceles right triangle is removed from each corner of a square piece of paper so that a rectangle of unequal sides remains. If the sum of the areas of the cut-off pieces is and the lengths of the legs of the triangles cut off are integers, find the area of the rectangle.
## Solution
Since the figure in the middle is a rectangle, the isosceles triangles on opposite vertices are congruent. Let be a leg of the first two, and the other two. The sum of the areas of the triangle is then . (Remember that the sides are of unequal lengths, so we exclude ). Since squares , we can reduce our search to even integers, and a short bit of trial and error yield works.
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# Unidirectional reflectionless light propagation at exceptional points
Yin Huang
• Department of Optoelectrics Information Science and Engineering, School of Physics and Electronics, Central South University, Changsha, Hunan 410012, China
• Key Laboratory of Optoelectronic Devices and Systems of the Ministry of Education and Guangdong Province, Shenzhen University, Shenzhen 518060, China
• Other articles by this author:
/ Yuecheng Shen
• Department of Biomedical Engineering, Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130, USA
• Other articles by this author:
/ Changjun Min
• Key Laboratory of Optoelectronic Devices and Systems of the Ministry of Education and Guangdong Province, Shenzhen University, Shenzhen 518060, China
• Other articles by this author:
/ Shanhui Fan
/ Georgios Veronis
• Corresponding author
• School of Electrical Engineering and Computer Science, Louisiana State University, Baton Rouge, LA 70803, USA
• Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA
• Email
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Published Online: 2017-05-20 | DOI: https://doi.org/10.1515/nanoph-2017-0019
## Abstract
In this paper, we provide a comprehensive review of unidirectional reflectionless light propagation in photonic devices at exceptional points (EPs). EPs, which are branch point singularities of the spectrum, associated with the coalescence of both eigenvalues and corresponding eigenstates, lead to interesting phenomena, such as level repulsion and crossing, bifurcation, chaos, and phase transitions in open quantum systems described by non-Hermitian Hamiltonians. Recently, it was shown that judiciously designed photonic synthetic matters could mimic the complex non-Hermitian Hamiltonians in quantum mechanics and realize unidirectional reflection at optical EPs. Unidirectional reflectionlessness is of great interest for optical invisibility. Achieving unidirectional reflectionless light propagation could also be potentially important for developing optical devices, such as optical network analyzers. Here, we discuss unidirectional reflectionlessness at EPs in both parity-time (PT)-symmetric and non-PT-symmetric optical systems. We also provide an outlook on possible future directions in this field.
## 1 Introduction
Exceptional points (EPs), which are branch point singularities of the spectrum, associated with the coalescence of both eigenvalues and corresponding eigenstates, lead to interesting phenomena, such as level repulsion and crossing, bifurcation, chaos, and phase transitions in open quantum systems described by non-Hermitian Hamiltonians [1], [2], [3], [4]. EPs have been studied in lasers [5], coupled dissipative dynamical systems [6], mechanics [7], electronic circuits [8], gyrokinetics of plasmas [9], and atomic as well as molecular systems [10]. In the past few years, EPs in non-Hermitian parity-time (PT)-symmetric systems have attracted considerable attention [11]. In quantum mechanics, non-Hermitian Hamiltonians may still possess entirely real energy spectra as long as they respect PT-symmetry. In general, the Hamiltonian H=p2/2+V(r), associated with a complex potential V(r), is PT-symmetric as long as the complex potential satisfies the condition V(r)=V*(−r) [11]. However, once a non-Hermiticity parameter exceeds a certain threshold, the eigenvalues of the Hamiltonian cease to be all real. This threshold is associated with the appearance of the EP, where the eigenvalue branches merge and PT-symmetry breaks down. Based on the close analogy between the Schrödinger equation in quantum mechanics and the wave equation in optics, PT-symmetry in optics requires that n(r)=n*(−r), which implies that the real part of the refractive index n(r) should be an even function of position, whereas the imaginary part must be an odd function [12], [13].
The constructed PT-symmetric optical structures with balanced gain and loss can lead to a range of extraordinary phenomena, including novel beam refraction [14], [15], power oscillation [16], [17], loss-induced transparency [18], nonreciprocal nonlinear light transmission [17], [19], [20], perfect absorption [21], [22], [23], teleportation [24], optical switching [25], optical tunneling [26], mode conversion [27], super scattering [28], and various other novel nonlinear effects [29], [30], [31]. In addition, there has been significant progress in using PT-symmetric periodic optical structures to attain unidirectional light reflectionlessness [12], [13]. In such structures, the reflection is zero when measured from one end of the structure at optical EPs and nonzero when measured from the other end.
Unidirectional reflectionlessness at EPs can be understood by considering a general two-port optical scattering system, as shown in Figure 1. The optical properties of this system can be described by the scattering matrix S defined by [32]
Figure 1:
Scattering matrix S of a two-port optical system.
${H}_{L}^{+}$ and ${H}_{R}^{+}$ are the complex magnetic field amplitudes of the incoming modes at the left and right ports, respectively. Similarly, ${H}_{L}^{-}$ and ${H}_{R}^{-}$ are the complex magnetic field amplitudes of the outgoing modes from the left and right ports, respectively.
$(HR−HL−)=S(HL+HR+)=(trbrft)(HL+HR+),$(1)
where HL+ and HR+ are the complex magnetic field amplitudes of the incoming modes at the left and right ports, respectively. Similarly, HL and HR are the complex magnetic field amplitudes of the outgoing modes from the left and right ports, respectively. In addition, t is the complex transmission coefficient, whereas rf and rb are the complex reflection coefficients for light incidence from the left (forward direction) and from the right (backward direction), respectively. We note that, because PT-symmetric structures are in general reciprocal, the transmission coefficients in both directions must be the same [33]. However, the reflection coefficients from the left and right directions are not constrained by reciprocity and can therefore be different. We also note that the scattering matrix S as defined here [Equation (1)], which is in general nonsymmetrical, is different from the typical convention used in electromagnetics [22]. In general, the matrix S is non-Hermitian, and its corresponding complex eigenvalues are ${\lambda }_{\text{s}}^{±}=t±\sqrt{{r}_{f}{r}_{b}}.$ Its eigenstates, which are ${\Psi }_{±}=\left(1,±\sqrt{\frac{{r}_{f}}{{r}_{b}}}\right)$ for rb≠0, are not orthogonal. The two eigenvalues and the corresponding eigenstates can be coalesced and form EPs. Such non-Hermitian degeneracies represent scattering states with unidirectional reflectionless propagation in the forward (rf=0, rb≠0) or backward (rb=0, rf≠0) direction.
In the case of unidirectional reflectionless propagation in the forward direction (rf=0, rb≠0), the scattering matrix S eigenvalues ${\lambda }_{\text{s}}^{±}$ coalesce into λc=t and the eigenstates Ψ± coalesce into the only eigenstate Ψc=(1,0)T. Because the scattering matrix eigenstates coalesce into the only eigenstate Ψc=(1,0)T, they no longer form a complete basis. We note that the eigenstate Ψc corresponds, through Equation (1), to a well-defined physical scattering state with (HL+, HR+)=Ψc=(1,0)T and (HL, HR)=λcΨc=(t,0)T. In other words, the eigenstate corresponds to a state with unidirectional reflectionless propagation for light incident from the left. A similar discussion holds for the case of unidirectional reflectionless propagation in the backward direction (rb=0, rf≠0).
The two complex eigenvalue solutions of the scattering matrix S construct a multivalued Riemann surface [1], [3]. They are on two branches of the Riemann surface. The two branches, ${\lambda }_{\text{s}}^{±}=t±\sqrt{{r}_{f}{r}_{b}},$ represent two superposition states between forward and backward light scatterings, including both transmission and reflection. The solutions of two branches coalesce and lead to EPs, corresponding to singularity points in the complex Riemann surface [1]. The reflection in both directions is due to the superposition of such two solutions [34]. At the EPs, light scattering is a destructive interference of two branch solutions. Therefore, the corresponding modal interference between two branch solutions at the EPs suppresses the reflection in one direction but not the other [34]. EPs exist in a larger family of non-Hermitian Hamiltonians [34]. Unidirectional reflectionless propagation can also be observed in non-PT-symmetric optical systems with unbalanced gain and loss [34].
In this paper, we review the fundamental physics and the latest developments on unidirectional reflectionless light propagation at EPs. In Section 2, we first review the unidirectional reflectionless propagation at EPs in PT-symmetric optical systems. Section 3 is devoted to reviewing unidirectional reflectionless propagation at EPs in non-PT-symmetric optical systems. Finally, in Section 4, we conclude this review and offer our perspectives on possible future directions in this field.
## 2 Unidirectional reflectionless propagation in PT-symmetric systems
To realize a PT-symmetric optical structure with unidirectional invisibility, Lin et al. used coupled mode theory (CMT) to theoretically investigate a 1D optical structure consisting of a periodic grating with refractive index distribution along the propagation direction n(z)=n0+n1cos (2βz)+in2sin(2βz) for |z|<L/2 (Figure 2A) [12]. This grating is embedded in a homogeneous medium having a uniform refractive index n0 for |z|>L/2 (Figure 2A). Here, β is the grating wave number. The terms n1cos(2βz) and n2sin(2βz) correspond to the periodic distribution of the real and imaginary parts (gain and loss) of the complex refractive index in the grating, respectively. For PT-symmetric optical systems, the eigenvalues of the S matrix (Figure 1) must be either a unimodular $\left(|{\lambda }_{\text{s}}^{±}|=1\right),$ or a nonunimodular inverse conjugate pair $\left[{\lambda }_{\text{s}}^{±}=\frac{1}{{\left({\lambda }_{\text{s}}^{±}\right)}^{\text{*}}}\right]$ [22]. In the former case, the system is in the exact PT phase, whereas, in the latter one, it is in the broken symmetry phase [21], [22], [35]. For the complex periodic structure considered in Figure 2A, the transition from one phase to another takes place when n1=n2. At n1=n2, the eigenvalues of the S matrix and their corresponding eigenvectors coalesce and form an EP, indicating that the phase transition takes place exactly at this EP. In fact, in quantum mechanics, the phase transition due to a non-Hermitian degeneracy is another phenomenon associated with EPs [36], [37].
Figure 2:
Unidirectional invisibility of a PT-symmetric Bragg scatterer.
(A) The wave entering from the left does not recognize the existence of the periodic structure and goes through the sample entirely unaffected. On the contrary, a wave entering the same grating from the right experiences enhanced reflection. (B) Exact numerical evaluation [from Equation (1)] of transmission T and reflection R coefficients for a Bragg grating for n0=1, n1=10−3, L=12.5 π, and β=100. In case of a PT grating, the system is at the EP when n1=n2. In this case, RL is diminished (up to n1,22~10−6; see inset) for a broad frequency band, whereas RR is enhanced [12].
Figure 2B shows the transmission and reflection coefficients as a function of the detuning δ=βk, where k=ωn0/c. At n1=n2 (EP), the reflection in the left direction RL=|rf|2 is zero, whereas the reflection in the right direction RR=|rb|2 is nonzero. Meanwhile, the transmission (T=|t|2) is unity, which implies that the Bragg scatterer is invisible when the light is incident from left (light propagating as if the Bragg scatterer is absent). Note that the transmitted wave cannot be detected from interference measurements, as the phase ϕt of the transmission coefficient t is zero at n1=n2 (EP). In addition, the transmission delay time τt=dϕt/dk is zero at n1=n2, which indicates that the time of flight of the light transmitted through a certain distance is the same whether there is a Bragg scatterer or not. Therefore, reflectionlessness in the proposed PT-symmetric Bragg scatterer results in invisibility. It is also clear that the reflection coefficients are symmetric, that is, RL=RR, when no gain or loss is included in the system (n2=0). In addition, the generalized power representation $T+\sqrt{{R}_{L}{R}_{R}},$ relates all the elements in the scattering matrix S and is essentially the power summation of the superposition of the two eigenstates [Equation (1)] [38]. Note that the reflection of an incident wave from the left side of the structure is subunitary, whereas the reflection from the right side is superunitary. Therefore, T+RR in this system (Figure 2A) can be larger than one (Figure 2B) [38]. When the reflection in the forward and backward directions are equal, we obtain $T+\sqrt{{R}_{L}{R}_{R}}=T+R=1,$ which is the power conservation relation for an optical system without gain or loss (Figure 2B).
Later on, Longhi reconsidered the scattering properties of this sinusoidal PT-symmetric Bragg scatterer (Figure 2A) using modified Bessel functions of the first kind [39]. His analytical results show that the unidirectional reflectionlessness only occurs for Bragg scatterers with short length L at EPs and breaks down for extremely long length scatterers. Note that the derivation of coupled-mode equations in Ref. [12] was based on multiscale asymptotic techniques and the rotating wave approximation [40]. Subsequently, Sarisaman demonstrated unidirectional reflectionlessness and invisibility in the TE and TM modes of a PT-symmetric slab system consisting of a separated pair of balanced gain and loss layers with a gap [41]. Kalish et al. reported that one-way invisibility could be obtained in randomly layered optical media with PT-symmetric refractive index [42]. Midya designed unidirectionally invisible complex optical crystals with balanced gain and loss based on nonrelativistic supersymmetry transformations [43]. Fu et al. demonstrated the existence of two EPs in a waveguide system consisting of zero-index materials with PT symmetry, which could induce unidirectional transparency [44]. Rivolta and Maes observed unidirectional visibility in a structure consisting of a waveguide and a finite chain of side-coupled resonators [45]. Fleury et al. constructed a unidirectional reflectionless optical system using two metasurfaces characterized by a PT-symmetric impedance distribution [46].
Regensburger et al. performed scattering experiments on a periodic PT-symmetric temporal Bragg scatterer by imposing a periodic phase modulation only within a finite time window [13]. Outside this time window, the periodic potential does not have an effect on light traveling, and the Bragg scatterer reflects light coming from both sides of the scatterer. When gain and loss are added to the phase modulation in a PT-symmetric fashion within the time window, unidirectional invisibility occurs at EPs. Hahn et al. first observed totally asymmetric diffraction in a PT-symmetric photonic lattice at EPs using vector-field holographic interference of two elliptically polarized pump beams on azobenzene-doped polymer thin films [47].
PT-symmetry in surface plasmon polaritons (SPPs) has been demonstrated by Yang and Mei [48]. SPPs, which are surface waves propagating along the interface between a metal and a dielectric [49], can concentrate electromagnetic energy at volumes of subwavelength scale and enable the manipulation of light beyond the diffraction limit [50], [51], [52], [53], [54], [55], [56], [57], [58], [59]. Yang and Mei introduced a periodic PT-symmetric modulation on the effective refractive index of a metal-dielectric waveguide in the cylindrical coordinate system. After a coordinate transformation, they realized a 3D cylindrical unidirectional PT-cloak at EPs for SPPs. Zhu et al. have also demonstrated a one-way invisible cloak based on a transformed PT-symmetric optical potential at EPs [60].
Unidirectional reflectionlessness can also be attained in optical coupled-resonator systems at EPs [61], [62]. Jin et al. studied the scattering of rhombic ring form coupled resonators with enclosed synthetic magnetic flux [62]. The scattering center is a two-arm Aharonov-Bohm interferometer. The magnetic flux induces nonreciprocal tunneling phase. In the presence of balanced gain and loss, the rhombic ring structure is under reflection PT-symmetry, which induces asymmetric reflection and reciprocal transmission. The optical gain and loss in the coupled-resonator optical systems can be experimentally realized using InGaAsP quantum wells and chrome or graphene layers, respectively, on top of the resonator [63], [64], [65], [66], [67].
Feng et al. have demonstrated an approach that leads to unidirectional reflectionless light propagation in a microscale silicon-on-insulator (SOI) waveguide platform without using PT-symmetric structures with balanced gain and loss [68]. They used only purely passive materials without gain, which relaxes the requirements of the fabrication, as optical gain is difficult to achieve using conventional complementary metal-oxide-semiconductor silicon technology. As shown in Figure 3A, they introduced a periodic passive PT-symmetric modulation of the dielectric permittivity along the direction of propagation Δε=cos(qz)−iδsin(qz) [instead of ε=cos(qz)−iδsin(qz)] into the SOI waveguides. Here, δ controls the relative strength and phase between the sinusoidal and cosinusoidal modulations, and q is the wave vector of the fundamental mode at the wavelength of 1550 nm.
Figure 3:
Optical properties of a designed passive unidirectional reflectionless PT metamaterial on an SOI platform.
(A) Schematic of the passive PT metamaterial on an SOI platform. Periodically arranged PT optical potentials with modulated dielectric permittivity of Δε=cos(qz)−iδsin(qz) (4nπ/q+π/qz≤4nπ/q+2π/q) are introduced into an 800-nm-wide and 220-nm-thick Si waveguide embedded in SiO2. The period is 4π/q=575.5 nm. (B and C) Dependences of reflectance of the passive PT metamaterial on the modulation length with different values of δ at the wavelength of 1550 nm in forward (B) and backward (C) directions, respectively. In contrast to the monotonic decrease in the forward direction, reflection in the backward direction first reaches a minimum value of Rb=0 at δ=1 and then increases as δ becomes larger. (D) Periodically arranged 760-nm-wide sinusoidal-shaped combo structures are applied on top of an 800-nm-wide Si waveguide embedded inside SiO2 to mimic PT optical potentials, in which imaginary part modulation is implemented with 14 nm Ge/24 nm Cr structures, and 51 nm Si layers are for real part modulation. The designed PT metamaterial consists of 25 sets of top-modulated combo structures with a period of 575.5 nm and a width of 143.9 nm for each sinusoidal-shaped combo. (E) SEM picture of the device, where the boxed area indicates a unit cell. (F) Measured reflection spectra of the device through the waveguide coupler for both directions over a broad band of telecom wavelengths from 1520 to 1580 nm. Red and blue curves are Gaussian fits of raw data in forward (black) and backward (green) directions, respectively [68].
The eigenvalues of the S matrix of the structure in Figure 3A are both unimodular but with an additional attenuation term included in the system for 0<δ<1. For δ>1, the eigenvalues are a nonunimodular inverse conjugate pair with the same attenuation term. Therefore, δ=1 is the phase transition point corresponding to the EP, where both eigenvalues and eigenvectors become degenerate. Figure 3B and C shows that, at δ=1, the reflection in the backward direction (Rb) is zero, whereas the reflection in the forward direction (Rf) approaches ~40% as the modulation length L increases.
For practical implementation, periodically arranged sinusoidal-shaped combo structures are applied on top of a Si waveguide embedded inside SiO2 to mimic the passive PT modulation on a macroscopic scale, in which imaginary part modulation is implemented with 14 nm Ge/24 nm Cr structures and 51 nm Si layers are used for real part modulation (Figure 3D). Figure 3E shows the fabricated passive PT-symmetric waveguide. The sinusoidal-shaped combo structures are first patterned in polymethyl methacrylate (PMMA) by electron beam lithography followed by evaporation and lift-off of Si and Ge/Cr in two steps. After these steps, the Si waveguide is formed by electron beam lithography and dry etching. The measured reflection spectra of the fabricated passive waveguide are in agreement with the theoretical predictions (Figure 3F). Asymmetric light propagation in this passive system is associated with the fact that EPs exist in a larger family of non-Hermitian Hamiltonians [34]. Using a similar method, Feng et al. proposed a passive PT-symmetric metawaveguide on an SOI platform for asymmetric interferometric light-light switching with a weak control beam, which enables coherent perfect absorption of a strong signal beam, based on the asymmetric reflection of the metawaveguide near the EP [69]. The high ratio between the strong signal beam and the weak control beam is in contrast to previous approaches of controlling light in all-optical active devices [22], [23], [70], [71], [72], [73], [74] and is promising for low-power next-generation optical networks.
In addition to the approach described above to achieve passive PT-symmetric SOI waveguides on a microscale platform, Jia et al. introduced a simple route to implement passive PT symmetry in the modal effective index of large-area (~cm2) organic thin-film waveguides [75]. They used interference lithography to write a sinusoidal grating profile into a thin film of photoresist and deposit high extinction coefficient blue pigment copper phthalocyanine (CuPc) to coat the “windward” grating facets (Figure 4A). This arrangement results in a modal effective index with a passive PT-symmetric profile of the form $\Delta {\stackrel{˜}{n}}_{\text{eff\hspace{0.17em}}}$ (z)≈(Δneff/2)[1+cos(qz)]+i(Δkeff/2)[1−sin(qz)] for the fundamental guided mode. Figure 4B shows scanning electron micrographs (SEMs) of a fabricated composite waveguide before (top panel) and after (bottom panel) planarization of a top photoresist layer.
Figure 4:
Unidirectional reflectionlessness of passive PT-symmetric organic thin-film waveguides.
(A) Waveguide composition calculated to support the fundamental leaky TE mode together with its corresponding position-dependent complex effective index neff+ikeff displayed in the top panel. (B) Cross-sectional SEM images of a typical CuPc-coated photoresist grating before (top) and after (bottom) planarization with the top photoresist layer. Faint contrast between the upper and lower photoresist layers remains visible in the bottom panel, confirming that the internal grating structure is maintained. (C) Diagram showing the optical measurement setup, where a laser beam is Kretschmann coupled into the leaky TE0 mode supported by the composite waveguide layer. The intensity of the 0 order reflected and −1 order diffracted beam is measured as a function of incidence angle for forward and backward illumination corresponding to light incident on the left and right sides of the prism, respectively. (D) Dispersion diagram illustrating the process of unidirectional scattering within the complex index modulated waveguide that occurs at its EP. (E) Reflectivity and −1 order diffraction efficiencies measured as a function of incidence angle (in the glass substrate) for a bare waveguide with no CuPc. The diffracted intensities for forward and backward incidence peaks near the Littrow condition (θinc~56°), which is designed to coincide with resonant in-coupling to the waveguide mode marked by the reflectivity dip. (F) The same measurement carried out for a waveguide with CuPc (dPC=6 nm, θdep=55°) targeted to achieve δ≈1 [75].
The optical property of the passive PT-symmetric waveguide is investigated using Kretschmann-coupled diffraction in Littrow [76]: first, the incident laser beam is evanescently coupled into the fundamental TE mode of the composite waveguide; then, the reverse-going waveguide mode is generated by grating reflection and subsequently evanescently coupled out to air (Figure 4C). Thus, measuring the diffraction efficiency, defined as |Edif/Einc|2, for left and right incident light provides a proxy probe of the forward- and backward-going modal reflectivity within the waveguide. Here, Edif and Einc are the incident plane wave and diffractive wave amplitudes, respectively. At EPs, light in-coupled into the forward-going (left to right) mode can scatter into the backward-going (right to left) mode, giving rise to Littrow diffraction but not vice versa (Figure 4D).
As shown in Figure 4E, when δkeffneff=0 (neither gain nor loss is included in the waveguide layer), the measured diffraction efficiency as a function of incidence angle for a TE-polarized 640 nm probe beam shows that the backward and forward Littrow diffraction efficiencies are similar. In contrast, at the EP, where δ=1, the backward Littrow diffraction efficiency is almost suppressed (Figure 4F). The reflectivity in the backward and forward directions is equal even in the presence of gain or loss. This is similar to the transmission in the structure of Figure 2B, which is the same in both directions regardless of the presence of gain or loss (Figure 2B). In addition, Yan and Giebink exploited vapor-deposited organic small molecules as a platform to realize passive PT-symmetry breaking and demonstrated unidirectional reflectionlessness at EPs in a composite organic thin film via complex refractive index modulation [77]. Zhu et al. designed a passive PT-symmetric grating with asymmetric diffraction arising from the spontaneous PT-symmetry breaking at EPs in a wide range of incidence angles [78].
The use of gain media in optical waveguides can lead to active optical devices and to material loss compensation [79], [80], [81], [82], [83]. Hahn et al. investigated the unidirectional reflectionlessness at EPs in PT-symmetric gratings based on active dielectric-loaded long-range SPP (DL-LRSPP) waveguides [84]. To obtain a PT-symmetric profile, they designed a step-in-width grating structure consisting of a thin Ag stripe on an active polymer bottom cladding with an active polymer ridge (Figure 5A). The polymer used is PMMA doped with IR140 dye.
Figure 5:
Optical properties of a designed passive unidirectional reflectionless DL-LRSPP PT-symmetric grating structure.
(A) 3D view of a DL-LRSPP PT-symmetric grating structure. (B) Real and imaginary parts of effective index near the nR crossing region of the ssb0 (fundamental) and sab3 (asymmetric high-order) modes. Vertical red dashed lines indicate four widths selected for a PT-symmetric grating. (C) Schematic of the real and imaginary index distribution within one period. (D) Reflectance from the left side Rl and the right side Rr of a passive PT-symmetric grating operating at the EP [84].
Normally, in a typical waveguide, it is impossible to achieve a PT-symmetric profile because both the real (nR) and the imaginary (nI) parts of the effective index change monotonically with the geometric parameters. Thanks to the coupling between the fundamental mode and an asymmetric high-order mode of the waveguide, when the ridge width is within specific ranges, the effective index of the DL-LRSPP waveguide has the potential to achieve a PT-symmetric profile. Figure 5B shows the effective index of the DL-LRSPP waveguide without gain as a function of the ridge width for a 35-nm-thick Ag stripe at a wavelength of 880 nm. One observes a large dip in the imaginary part of the effect index nI and a peak then a dip in the real part of the effect index nR. By arranging four widths within one step-in-width grating period in the sequence w3, w2, w4, w1, a passive PT-symmetric profile is obtained (Figure 5A and C). The thickness of the Ag stripe tag=35 nm was chosen to have ΔnR=ΔnI, which corresponds to an EP. Figure 5D shows the unidirectional reflection light propagation in such a PT-symmetric grating at its EP. When the gain of IR140-doped PMMA is included to make the average value of nI zero or positive, the maximum reflection in the left direction (Rl) is greatly enhanced, and the grating can even become an active PT-symmetric grating. Moreover, by placing a gain waveguide within a cavity formed by concatenating two gratings with opposite nonreflective ends, a high-performance laser can be created at EPs [85].
In all cases presented above, to achieve PT symmetry, the dielectric permittivity profile is chosen so that ε(z)=ε(−z), whereas the magnetic permeability is equal to the one of free space (μ=μ0). If, however, magnetic materials (μμ0) are considered, a similar requirement must be imposed on the magnetic permeability, that is, μ(z)=μ(−z), to satisfy the PT symmetry condition $n\left(z\right)=\sqrt{\epsilon \left(z\right)\mu \left(z\right)}=\sqrt{{\epsilon }^{\ast }\left(-z\right){\mu }^{\ast }\left(-z\right)}={n}^{\ast }\left(-z\right)\text{\hspace{0.17em}}$ [86]. Interestingly, Li et al. explored the possibility of cross-matching ε(z) to μ(−z), namely, ε(z)=μ(−z), to establish PT symmetry [87]. More specifically, they investigated a 1D photonic crystal (PhC) consisting of alternating purely dielectric slabs with ε1=2.25+, μ1=1 and purely magnetic slabs with μ2=2.25−, ε2=1 (Figure 6A). These two types of media have matched gain and loss coefficients. Note that ${n}_{1}=\sqrt{{\epsilon }_{1}{\mu }_{1}}=\sqrt{{\epsilon }_{2}^{\ast }{\mu }_{2}^{\ast }}={n}_{2}^{\ast },$ so that the spatial PT-symmetry still holds.
Figure 6:
Unidirectional reflectionlessness of a PT-symmetric 1D PhC with cross-matching ε(z) to μ*(−z).
(A) Permittivity (ε) and permeability (μ) profiles of a 1D PhC stacking alternating purely dielectric slabs with ε=2.25+ and purely magnetic slabs with μ=2.25− with equal filling ratio and varying γ. (B) Forward and (C) backward reflectance (in natural log scale) for the PhC in (A) with a finite thickness of four unit cells. (D) Splitting of eigenvalues of the effective constitutive matrix C−1. The scattered symbols (solid lines) are the numerical results (analytical effective medium results) [87].
Figure 6B and C shows the forward and backward reflection spectra, respectively, as a function of γ. The dips (red color in Figure 6B), reaching a value of zero, correspond to unidirectional reflection. To provide further insight into the unidirectional character in reflection, the eigenvalue behavior of constitutive matrix C−1 against γ is shown in Figure 6D. The optical property of a single unit cell can be described by the constitutive matrix C−1
$(Hy2−Hy1Ex2−Ex1)=ik0a2C−1(Ex2+Ex1Hy2+Hy1),$(2)
where Ex1, Hy1 are the fields on the left side of the cell, Ex2, Hy2 are the fields on the right side, and a is the lattice constant. The constitutive matrix C−1 can be written in terms of the S matrix [Equation (1)] of a single unit cell through a bilinear transformation
$C−1=2ik0aBS−IS+IB−1, with I=(1001 ) and B=(111−1 ).$(3)
Because unidirectional reflectionlessness occurs at an EP of the scattering matrix S, we can equivalently study the eigenvalue behavior of the matrix C−1. The eigenvalues of matrix C−1 are pairs of two purely real numbers for γ<0.107, coincide at an EP for γ=0.107, and then split into a complex conjugate pair (Figure 6D). This eigenvalue behavior is equivalent to the transition of PT-symmetric Hamiltonians in quantum mechanics from the PT-symmetric phase to the PT-broken phase [11], [13], [88], [89]. The EP of the constitutive matrix C−1 is actually the EP of the S matrix, which leads to the unidirectional reflectionless behavior shown in Figure 6B and C.
When more material potentials, such as the magnetic permeability, are considered, additional degrees of freedom exist in establishing PT symmetry. An antisymmetric PT-photonic structure under combined PT operations, i.e. n(−z)=−n(z), with balanced positive- and negative-index materials, has been reported, where the magnetic permeability was required to satisfy μ(−z)=−μ(z) [90]. Studies showed that light propagation in optical lattices of driven cold atoms with PT-antisymmetric susceptibilities, i.e. χ(z)=−χ(−z), exhibited EPs (also known as non-Hermitian degeneracies), at which complete unidirectional reflectionless light propagation was observed [91], [92].
In addition, a few recent studies of multidimensional PT-symmetric systems have demonstrated intriguing properties not found in 1D PT-symmetric systems, such as conical diffraction, third-order EPs, continuous rings of EPs, and rotating input by asymmetric coupling between wave vectors [14], [93], [94], [95], [96], [97], [98]. Most recently, Fan et al. considered 2D PT-symmetric PhCs, whose non-Hermitian primitive cell is an integer multiple of the primitive cell of the underlying Hermitian system [15]. Similar to many other low-dimensional PT systems, PT-symmetric PhCs can also exhibit unidirectional reflection behavior when light comes from the left and right sides of PhCs.
## 3 Unidirectional reflectionless propagation in non-PT-symmetric systems
As we have mentioned in the previous sections, there is a large family of non-Hermitian Hamiltonians that can produce merged branches of eigensolutions through accidental degeneracy [34]. Therefore, can a non-PT-symmetric system, in which the relation n(−z)=n(z) for the refractive index profile does not hold, lead to asymmetric reflection? It has been reported that unidirectional reflectionlessness can be realized in a two-layer non-PT-symmetric slab structure by tuning both the real and imaginary parts of the refractive index of each layer at EPs [99].
Moreover, Horsley et al. completely suppressed scattering in one direction in a planar inhomogeneous dielectric structure, in which the spatial distributions of the real and imaginary parts of the dielectric permittivity are related by Kramers-Kronig relations [100]. Specifically, they considered a monochromatic electromagnetic wave propagating in the xy plane within a medium with a positive background contribution εb plus a spatially varying part α in the dielectric permittivity ε: ε(x)=εb(x) (Figure 7A).
Figure 7:
Unidirectional reflectionlessness of a planar inhomogeneous dielectric structure in which the spatial distributions of the real and imaginary parts of the dielectric permittivity are related by Kramers-Kronig relations.
(A) A wave propagating in the xy plane in an inhomogeneous medium with permittivity ε(x) (indicated by blue shading). The TE polarization has an electric field pointing only along z and the TM polarization has a magnetic field pointing only along z. When the real (blue) and imaginary (red) parts of the permittivity are related to one another by the Kramers-Kronig relations (as in B), then the reflection vanishes for all angles of incidence. (B) Permittivity profile ε(x) given by Equation (7) for parameters A=2 and ξ=0.1λ with λ=2π/k0. (C and D) Absolute values of the electric field of a line source placed at positions x=−5λ (C) and x=+5λ (D) in the permittivity profile. The region between the vertical dashed lines in (C) indicates the region plotted in (B). (E and F) As in (C and D), with identical parameters, but taking only the real (E) and imaginary (F) parts of the permittivity. The absence of any oscillations in (C) shows that the reflection is completely suppressed for incidence from the left for all incident angles [100].
If we expand the scattered field induced by the variation α(x) as a series, ${e}_{s}=\sum _{n}{e}_{\text{s}}^{\left(n\right)},$ and solve the Helmholtz equation for the TE polarization, the first and nth terms in this series can be expressed as
$es(1)(x)=−E0k02∫dk2πG(k)α˜(k−K)eikx,$(4)
$es(n)(x)=−k02∫dk2π∫dk′2πG(k)α˜(k−k′) e˜s(n−1)(k′)eikx,$(5)
where $\stackrel{˜}{\alpha }$ is the spatial Fourier transform of α(x), G(k) is the retarded Green function, $K={\left({\epsilon }_{b}{k}_{0}^{2}-{k}_{y}^{2}\right)}^{1/2}$ and ky determines the angle of incidence. If $\stackrel{˜}{\alpha }$(k<0)=0, then the first item ${e}_{\text{s}}^{\left(1\right)}\left(x\right)$ is composed of only right-going waves for any value of K or, in other words, for any angle of incidence. Based on Equation (5), every successive term also contains only right-going waves if ${e}_{\text{s}}^{\left(1\right)}$ is composed of right-going waves and $\stackrel{˜}{\alpha }$(k<0)=0. Interestingly, the requirement $\stackrel{˜}{\alpha }$(k<0)=0 can be satisfied when the permittivity profile α(x) is holomorphic (analytic) in the upper half complex plane, i.e. Im(x)≥0, and is of Kramers-Kronig type [101]:
$Re[α(x)]=1πP∫−∞∞Im[α(s)]s−xds,$(6)
where P is the principal part of the integral. We consider a specific example with $\alpha \left(x\right)=A\frac{\text{i}-x/\xi }{1+{\left(x/\xi \right)}^{2}}.$ The overall permittivity profile of the medium is then
$ε(x)=εb+Ai−x/ξ1+(x/ξ)2,$(7)
which is plotted in Figure 7B. Here, ξ and A set the spatial scale and amplitude of the profile, respectively.
The one-way scattering behavior is confirmed in Figure 7C and D, which show electric field profiles for a point source placed at either side of x=0. The absence of reflection is clearly observed when light is incident from the left. Figure 7E and F shows the electric field when only the real (Figure 7E) or the imaginary (Figure 7F) parts of the permittivity profile [Equation (7)] are considered. In fact, if the imaginary part of α(x) is set to be symmetric about x=0, then the real part calculated based on Equation (6) is antisymmetric and vice versa. Thus, the spatial Kramers-Kronig relations can generate permittivity profiles that exhibit PT-symmetry [α(−x)=α*(x)]. In other words, compared to the specific class of PT-symmetric complex profiles, the permittivity profiles associated with Kramers-Kronig relations are more general nonreflecting profiles, which can achieve unidirectional reflectionlessness at EPs.
The refractive index modulation profiles that we have considered so far for synthesizing PT-symmetric or non-PT-symmetric optical structures require careful tuning of both the real and imaginary parts of the refractive index of the constituent materials, which increases the difficulty in practical realization. Therefore, the question arises as to whether it is possible to achieve asymmetric reflection in a non-PT-symmetric system by tuning only the real or only the imaginary part of the refractive index of the material. Feng et al. recently demonstrated unidirectional reflectionless light transport at EPs in a conventional large-sized nonperiodic multilayer structure, which was fabricated by alternating thin film depositions of lossy amorphous silicon and lossless silica layers on a cleaned glass wafer using plasma-enhanced chemical vapor deposition (43 nm silica/9 nm silicon/26 nm silica/23 nm silicon; Figure 8A) [102]. The refractive index of silica at the wavelength of interest is 1.46, and the complex refractive index of amorphous silicon is 4.86+iγ, so that the proposed structure is clearly non-PT-symmetric. Using the transfer matrix theory [103], one can show that, by modulating only the imaginary part of the refractive index of amorphous silicon, the eigenvalues of the S matrix of this multilayer system coalesce and form an EP when γ=−0.65. The reflection spectra of the structure for both forward and backward directions were numerically calculated and experimentally measured, as shown in Figure 8B and C, respectively. Unidirectional reflectionless light transport is clearly observed around the wavelength of 520 nm, where the reflection in the forward direction is significantly suppressed due to the existence of the EP.
Figure 8:
Unidirectional reflectionlessness of a non-PT-symmetric large-sized nonperiodic multilayer structure consisting of lossy amorphous silicon and lossless silica layers.
(A) SEM pictures of the cross-section of the fabricated EP structure. The scale bar corresponds to 50 nm. (B and C) Numerically calculated and experimentally measured reflection spectra of the EP structure from 450 to 600 nm, respectively, for both forward (red) and backward (black) incidence. (D and E) Photographs of the wafer-scale EP structure in forward and backward directions, respectively, with a 10-nm band-pass filter centered at the wavelength of 520 nm [102].
In the presence of loss, this optical system is analogous to open quantum systems that are subjected to dissipation and characterized by complex non-Hermitian Hamiltonians. Thus, the existence of EPs in this purely lossy non-PT-symmetric optical system also provides an opportunity to control the unidirectional reflection of light. Because of the associated unidirectional reflectionless light propagation at the EP, this large-sized non-PT-symmetric multilayer structure can form imaging in reflection under sunlight illumination only from the backward direction. An image formed in the backward direction is shown in Figure 8E, whereas an image cannot be formed in the forward direction due to the nonreflecting behavior of the structure (Figure 8D).
A couple of subsequent studies showed theoretically that unidirectional reflectionless light propagation can be realized in similar two-layer non-PT-symmetric lossy dielectric slab structures with modulation of the imaginary part of the refractive index of each slab [104], [105]. Ge and Feng further pointed out that an optical reciprocity-induced symmetry, which is related to the amplitude ratio of the incident waves in the scattering eigenstates, can lead to an EP. Because optical reciprocity holds in general and does not rely on PT symmetry, the unidirectional reflectionlessness at EPs can be obtained in optical systems with unbalanced gain and loss and even in the absence of gain [105]. In addition, Kang et al. designed a non-PT-symmetric ultrathin metamaterial to exhibit one-way zero reflection at EPs [106].
Forming an EP of the S matrix through tuning the geometric parameters of a structure rather than the refractive index profile was proposed in a non-PT-symmetric plasmonic waveguide-cavity system consisting of two metal-dielectric-metal (MDM) stub resonators side coupled to an MDM waveguide (Figure 9A) [107]. Tuning the geometry rather than the refractive index can reduce the difficulty in the experimental realization of such structures. Among different plasmonic waveguide structures, MDM plasmonic waveguides are of particular interest [108], [109], [110], [111], [112], [113], [114], [115], [116], because they support modes with deep subwavelength scale over a very wide range of frequencies extending from DC to visible [117] and are relatively easy to fabricate [118], [119]. The waveguide widths w, w1, and w2 are set to be 50, 20, and 100 nm, respectively (Figure 9A). The metal is silver and the dielectric is air. To obtain unidirectional reflectionless propagation, the MDM stub lengths L1, L2, as well as the distance between the stubs L, are optimized using the scattering matrix theory [120], [121] to minimize the amplitude of the reflection coefficient in the forward direction |rf| at the optical communication wavelength of λ0=1.55 μm. Note that, if one tunes the refractive index of a material to form an EP, the optimized refractive index will be complex. However, if one tunes the geometric parameters to obtain an EP, these parameters are restricted to be purely real.
Figure 9:
Unidirectional reflectionlessness of a non-PT-symmetric system consisting of a MDM plasmonic waveguide side coupled to two MDM stub resonators.
(A) Schematic of an MDM plasmonic waveguide side coupled to two MDM stub resonators. (B) Reflection spectra for the structure of (A) calculated for both forward and backward directions using the finite-difference frequency-domain (FDFD) method (solid lines) and the scattering matrix theory (circles). Results are shown for w=50 nm, w1=20 nm, w2=100 nm, L1=175 nm, L2=365 nm, and L=561 nm. Also shown are the reflection spectra calculated using FDFD for lossless metal (blue solid line). (C and D) Real and imaginary parts of the eigenvalues of the scattering matrix S as a function of the distance L between the two MDM stub resonators. The black and red lines correspond to eigenvalues ${\lambda }_{\text{s}}^{+}=t+\sqrt{{r}_{f}{r}_{b}},$ and ${\lambda }_{\text{s}}^{-}=t-\sqrt{{r}_{f}{r}_{b}},$ respectively. All other parameters are as in (B). (E) Spectra of the generalized power $T+\sqrt{{R}_{f}{R}_{b}}$ (black) and of the differential generalized power (red), defined as the derivative of the generalized power with respect to frequency $d\left[T+\sqrt{{R}_{f}{R}_{b}}\right]/df,$ calculated using FDFD. All parameters are as in (B). (F) Phase spectra of the reflection coefficients in the forward (rf, black) and backward (rb, red) directions. All parameters are as in (B) [107].
Figure 9B shows the reflection spectra for the structure of Figure 9A calculated for both forward and backward directions for L1=175 nm, L2=365 nm, and L=561 nm. The results verify that the optimized structure of Figure 9A is unidirectional reflectionless at f=193.4 THz (λ0=1.55 μm). Figure 9C and D shows the real and imaginary parts, respectively, of the eigenvalues ${\lambda }_{\text{s}}^{±}$ of the scattering matrix S [Equation (1)] as a function of the distance L between the two MDM stub resonators. We observe that the real and imaginary parts of the two eigenvalues indeed collapse for L=561 nm. As discussed above, this is the optimal distance between the two stubs, which minimizes the reflection in the forward direction. In Figure 9C and D, we observe the level repulsion in the real parts of the eigenvalues, as well as the level crossing in their imaginary parts, which resembles a system also described by a non-Hermitian Hamiltonian matrix consisting of two coupled damped oscillators [4]. In open quantum systems, a repulsion (crossing) for the real part of the energy and a crossing (repulsion) for the imaginary part of the energy in the 2D complex energy plane are required around EPs [3], [4]. Unlike the Hermitian case, the levels approach each other in the form of a cusp rather than a smooth approach because of the plain square-root behavior of the singularity (Figure 9C) [1], [4].
Similar to other classical optical systems that have EPs [102], we observe that a generalized power decreasing phase and a generalized power increasing phase are divided by the EP at f=193.4 THz (Figure 9E). In addition, an abrupt phase change in the differential generalized power spectrum is observed at the EP as well (Figure 9E). These results are essentially due to the fact that the reflection coefficient in the forward direction rf approaches zero at the EP. Figure 9F shows that the phase of the reflection coefficient in the forward direction undergoes an abrupt π jump, when the frequency is crossing over the EP, which actually resembles the phase transition from the PT-symmetric phase to the PT broken phase in optical PT-symmetric systems [12], [13], [17], [38], [102], [104]. Such an abrupt π-phase jump in the reflection coefficient in the forward direction confirms the existence of the EP in this plasmonic system and that the unidirectional reflectionlessness in the system is directly associated with this EP. In contrast, the phase of the reflection coefficient in the backward direction does not undergo an abrupt jump and varies smoothly with frequency. In addition, the reflection is not unidirectional if the system is lossless (Figure 9B). This is different from other classical optical analogues of quantum systems, such as the plasmonic analogue of electromagnetically induced transparency [122], [123], which can be realized in both lossless and lossy optical systems. In addition, the formation of an EP and the resulting unidirectional reflectionlessness can also be implemented using plasmonic waveguide-cavity systems based on other plasmonic two-conductor waveguides, such as 3D plasmonic coaxial waveguides [124], [125].
Unlike PT-symmetric systems, most non-PT-symmetric optical systems with modulation of the real or the imaginary part of the refractive index of the material typically exhibit unidirectional reflectionless propagation only within a very narrow wavelength range around the EP (see, for example, Figures 8C and 9B) [102], [104], [105], [106], [107]. This is especially true for nonperiodic non-PT-symmetric systems, which are relatively easy to fabricate and more compact. Although Yang et al. demonstrated broadband unidirectional reflectionless light transport at an EP in a periodic ternary-layered structure consisting of lossy and lossless dielectrics [126], large-sized periodic structures are not easy to implement in densely integrated optical chips.
A compact non-PT-symmetric plasmonic waveguide-cavity system consisting of two MDM stub resonators with unbalanced gain and loss side coupled to an MDM waveguide was recently theoretically investigated (Figure 10A) [127]. This non-PT-symmetric structure can exhibit unidirectional reflectionlessness in the forward direction at λ0=1.55 μm. Moreover, light reflection in the forward direction in this system is close to zero in a broad wavelength range.
Figure 10:
Unidirectional reflectionlessness of a non-PT-symmetric system consisting of a MDM plasmonic waveguide side coupled to two MDM stub resonators with unbalanced gain and loss.
(A) Schematic of a perfect absorber unit cell consisting of an MDM plasmonic waveguide side coupled to two MDM stub resonators. (B) Reflection and transmission spectra for the structure of A calculated for light incident from both the forward and backward directions using FDFD (solid lines) and CMT (circles). Results are shown for w=50 nm, w1=10 nm, w2=25 nm, h1=67.5 nm, and h2=53 nm. The metal is silver and the dielectric is air. The left and right stubs are filled with SiO2 doped with CdSe quantum dots (εA=4.0804−j0.6) and InGaAsP (εB=11.38+j0.41), respectively. Also shown are the absorption spectra in the forward direction calculated using FDFD (green solid line). (C and D) Magnetic field amplitude profiles for the structure of (A) at f=193.4 THz (λ0=1.55 μm), when the fundamental TM mode of the MDM waveguide is incident from the left and right, respectively. All parameters are as in (B). (E and F) Magnetic field amplitude in the middle of the MDM waveguide, normalized with respect to the field amplitude of the incident fundamental TM waveguide mode in the middle of the waveguide, when the mode is incident from the left and right, respectively. The two vertical dashed lines indicate the left boundary of the left stub and the right boundary of the right stub. All parameters are as in (B) [127].
Using the temporal CMT [128], [129], it can be shown that the unidirectional reflectionlessness condition coincides with the broadband near-zero reflection condition if
$1τ2=1τ1=1τ, 1τ01=1τ(e2αL+1), 1τ02=0, ω01=ω02=ω0, cos(2βL)=−1,$(8)
where ω01 and ω02 are the resonance frequencies of the two resonators, 1/τi, i=1, 2, are the decay rates of the resonator mode amplitudes due to the power escape through the waveguide, 1/τ0i, i=1, 2, are the decay (growth) rates due to the internal loss (gain) in the resonators, and L is the distance between the two resonators. α and β are the real and imaginary parts, respectively, of the complex propagation constant of the fundamental mode in the waveguide at λ0=1.55 μm. The left and right stubs are filled with SiO2 doped with CdSe quantum dots (εA=4.0804−j0.6) and InGaAsP (εB=11.38+j0.41), respectively. The real parts of εA and εB are fixed, whereas the imaginary parts are tuned to satisfy Equation (8). In Figure 10B, we observe that not only the system exhibits unidirectional reflectionless light propagation at the resonance frequency of f0=193.4 THz (λ0=1.55 μm) but also the off-resonance reflection in the forward direction beyond f0=193.4 THz is significantly suppressed in a broad frequency range.
More interestingly, as shown in Figure 10B, the on-resonance reflection in the backward direction is unity. This is due to the fact that the right stub behaves as a lossless stub [$\frac{1}{{\tau }_{02}}=0,$ Equation (8)]; therefore, it acts as a perfect reflector at the resonant frequency (f0=193.4 THz). The gain of the material filling the right stub compensates the material loss in the metal. Thus, the on-resonance transmission for light incident from both the forward and backward directions is zero, and for light incident from the left, the on-resonance absorption is unity. We note here that the existence of an EP-induced unidirectional reflectionless mode with unity transmission in PT-symmetric systems with balanced gain and loss (Figure 2) is commonly referred to as unidirectional invisibility, whereas the existence of an EP-induced unidirectional reflectionless mode with zero transmission corresponds to unidirectional perfect absorption. Ramezani et al. also obtained unidirectional perfect absorption in passive Fano disk resonators at EPs based on a similar idea [130].
The unidirectional perfect absorption in the forward direction can be observed in the magnetic field distributions. When the waveguide mode is incident from the right (backward direction), there is no transmission, and the incident and reflected fields form a strong interference pattern (Figure 10D and F). On the contrary, when the waveguide mode is incident from the left (forward direction), there is hardly any reflection or transmission (Figure 10C and E). In addition, by cascading multiple non-PT-symmetric optical structures with different resonant absorption frequencies, and taking advantage of their broadband near-zero reflection property, an ultra-broadband near-total absorber can be realized. It should be noted that the proposed non-PT-symmetric waveguide-cavity system can also be realized using other nanophotonic structures such as microring and PhC cavities [131], [132]. In such lossless structures, gain media are not required to implement the proposed waveguide-cavity systems, as there is no internal loss in the resonators.
Recently, Yang et al. performed the first direct measurement of asymmetric backscattering (reflection) in a microcavity. Their setup consists of an erbium-doped silica microtoroid whispering-gallery modes (WGM) resonator that allows for in- and out-coupling of light through two single-mode optical fiber waveguides (Figure 11A) [133]. They have also previously reported PT-symmetry breaking in two coupled WGM silica microtoroid resonators with balanced gain and loss [134]. Here, to probe the asymmetric backscattering of the WGMs, they used two silica nanotips as Rayleigh scatterers (Figure 11A). The microcavity is an open system, and the corresponding effective Hamiltonian is
Figure 11:
Experimental observation of scatterer-induced asymmetric backscattering in a WGM resonator side-coupled to two waveguides.
(A) Illustration of a WGM resonator side-coupled to two waveguides, with the two scatterers enabling the dynamical tuning of the modes. cw and ccw are the clockwise and counterclockwise rotating intracavity fields. acw(ccw) and bcw(ccw) are the field amplitudes propagating in the waveguides. β is the relative phase angle between the scatterers. Inset shows the optical microscope image of the microtoroid resonator, the tapered fiber waveguides (horizontal lines), and the two silica nanotips denoting the scatterers (diagonal lines on the left and right sides of the resonator). (B) Varying the size and the relative phase angle of a second scatterer helps to dynamically change the frequency detuning (splitting) and the linewidths of the split modes revealing avoided crossings (top) and an EP (bottom). (C and D) When there is no scattering center in or on the resonator, light coupled into the resonator through the first waveguide in the cw (C-i) [or ccw (D-i)] direction couples out into the second waveguide in the cw (C-i) [or ccw (D-i)] direction: the resonant peak in the transmission and no signal in the reflection. (C-ii and D-ii) When a first scatterer is placed in the mode field, resonant peaks are observed in both the transmission and the reflection regardless of whether the light is input in the cw (C-ii) or ccw (D-ii) direction. (C-iii and D-iii) When a second scatterer is suitably placed in the mode field, for the cw input, there is no signal in the reflection output port (C-iii), whereas, for the ccw input, there is a resonant peak in the reflection, revealing asymmetric backscattering for the two input directions. Inset in D-iii compares the two backscattering peaks in C-iii and D-iii [133].
$H=(ΩcABΩc),$(9)
which is, in general, non-Hermitian. The real parts of the diagonal elements Ωc are the frequencies, and their imaginary parts are the decay rates of the resonant traveling waves. The complex-valued off-diagonal elements A and B are the backscattering coefficients, which describe the scattering from the clockwise (cw) [counterclockwise (ccw)] to the counterclockwise (clockwise) propagating wave (Figure 11A). The complex eigenvalues of H are
$Ω±=Ωc±AB,$(10)
and its eigenvectors are
$Ψ±=(A,±B).$(11)
When the first scatterer is introduced into the WGM volume, frequency splitting (Ω+ and Ω) can be observed in the transmission spectra due to scatterer-induced modal coupling between the cw and ccw propagating modes. Subsequently, the relative position (i.e. relative phase angle β) and the size of the second scatterer are tuned by nanopositioners to bring the system to an EP (Figure 11B), which, as mentioned above, is a non-Hermitian degeneracy identified by the coalescence of the complex frequency eigenvalues (Ω+ and Ω) and the corresponding eigenstates Ψ+. As a result, asymmetric backscattering (unidirectional reflectionlessness) of the WGMs can be achieved (A=0, or B=0).
In the absence of scatterers, a resonance peak appears in the transmission and no signal is observed in the reflection (backscattering), when the light is incident from the cw direction, as a result of wave vector matching (Figure 11C-i). Similarly, there is a resonance peak in the transmission and no signal in the reflection when the light is coming from the ccw direction (Figure 11D-i). In the presence of the first scatterer, two split resonance modes are observed in the transmission and reflection spectra regardless of whether the signal is coming from the cw or the ccw direction (Figure 11C-ii and D-ii). This implies that the field inside the resonator is composed of modes propagating in both cw and ccw directions due to the modal coupling between these two modes. Note that reflection in the cw and ccw directions is symmetric (red line in Figure 11C-ii and blue line in Figure 11D-ii). When the second scatterer is introduced and its position and size are tuned to form an EP, the transmission in the two different directions is still symmetric, whereas the reflection is asymmetric (red line in Figure 11C-iii and blue line in Figure 11D-iii). The reflection in the cw direction vanishes, whereas the reflection in the ccw direction supports a resonance peak. This unidirectional zero backscattering phenomenon can support chiral behavior [133], [135], [136], [137], [138] and enable directional emission of a WGM microlaser at EPs. It is also noteworthy that the EP in this non-PT-symmetric microcavity system is induced by tuning of the geometric parameters, such as the size of scatterers and the relative phase angle between scatterers (Figure 11B), rather than tuning of the refractive index of the materials.
## 4 Conclusions and outlook
Asymmetric light transport is important for several key applications in photonic circuits [19], [64], [138], [139]. In this paper, we reviewed unidirectional reflectionless light propagation at EPs. We first discussed the large body of recent works on unidirectional invisibility using PT-symmetric systems with balanced gain and loss. We then discussed how it is possible to achieve one-way zero reflection at EPs through a periodic passive PT-symmetric modulation of the dielectric permittivity. When additional material potentials, such as the magnetic permeability, are considered, additional degrees of freedom exist in establishing PT symmetry to obtain unidirectional reflectionlessness at EPs. In addition, EPs exist in a larger family of non-Hermitian Hamiltonians. As an example, a medium with a permittivity profile, which is an analytic function in the upper or lower half of the complex position plane, and its real and imaginary parts are related by the spatial Kramers-Kronig relations, shows unidirectional nonreflecting behavior. This finding is more general than previous results associated with PT symmetry. The proposed non-PT-symmetric optical systems also provide a simple way to realize asymmetric reflection by solely tuning the imaginary part of the permittivity of materials. In addition, systems that exhibit unidirectional perfect absorption have also been demonstrated. We finally reviewed unidirectional reflectionless light propagation at EPs in non-PT-symmetric structures through tuning of their geometric parameters rather than the refractive index profiles of the materials. These designs can reduce the difficulty in the experimental realization of such structures.
Unidirectional reflectionless light propagation is an intriguing phenomenon. Based on unidirectional invisibility in acoustics, a noninvasive, shadow-free, fully invisible acoustic sensor with PT symmetry was recently demonstrated [140]. Unidirectional invisibility at EPs in optics may also have remarkable implications for noninvasive sensing. In addition, applying unidirectional perfect absorption designs in optical structures based on electro-optic, absorptive, and nonlinear materials with outstanding properties could advance the development of nanophotonic devices, such as switches, modulators, and devices for imaging. In addition, EPs in plasmons could lead to nanoscale photonic devices with novel functionalities.
## Acknowledgments
The authors want to acknowledge financial support from the Foundation of the Key Laboratory of Optoelectronic Devices and Systems of the Ministry of Education and Guangdong Province (GD201601), the National Science Foundation (NSF) grants 1102301 and 1254934, the National Natural Science Foundation of China (NSFC) grants 61605252 and 61422506, and the Air Force Office of Scientific Research (AFOSR) grants FA9550-12-1-0471 and FA9550-17-1-0002.
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Revised: 2017-03-22
Accepted: 2017-03-31
Published Online: 2017-05-20
Citation Information: Nanophotonics, Volume 6, Issue 5, Pages 977–996, ISSN (Online) 2192-8614,
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Optics Express, 2017, Volume 25, Number 22, Page 27283 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 41, "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": 0.8502192497253418, "perplexity": 4179.507130209947}, "config": {"markdown_headings": true, "markdown_code": false, "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-2019-39/segments/1568514574665.79/warc/CC-MAIN-20190921211246-20190921233246-00219.warc.gz"} |
https://www.meritnation.com/ask-answer/question/roh-na-ro-na-1-2-h2owrite-the-order-of-reactivity-of-alcohol/alcohols-phenols-and-ethers/11732215 | # ROH+Na = RO^- Na^+ +1/2 H2O Write the order of reactivity of alcohols b/w 1Degree,2degree and 3 degree
Dear Student,
Given Reaction is :-
R-OH + Na --> RO- Na+$\frac{1}{2}$ H2
As Alcohols is weaker acids than water. This can be explained by the reaction of water with an alkoxide
In above reaction, the water is a better proton donor that is a stronger acid than alcohol. An alkoxide ion is a better proton acceptor than hydroxide ion (-OH), which shows that alkoxides are the stronger base that is sodium ethoxide is a stronger base than sodium hydroxide. So, Alcohol act as Bronsted bases as well and It is due to the presence of unshared electron pairs on oxygen that makes them proton acceptors.Now The acidic character In alcohols occurs due to the presence of polar nature of O–H and the electron-releasing groups like methyl (– CH3), ethyl (–C2H5 ) increase the electron density on oxygen tend to decrease the polarity of O-H bond. This will decrease the acid strength. For this reason, the acid strength of alcohols decreases in the following order: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.8769824504852295, "perplexity": 3016.415505548547}, "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-2023-06/segments/1674764500028.12/warc/CC-MAIN-20230202133541-20230202163541-00374.warc.gz"} |
https://asmedigitalcollection.asme.org/vibrationacoustics/article-abstract/107/1/112/441315/Proposed-Solution-Methodology-for-the-Dynamically?redirectedFrom=fulltext | The equations which describe the three-dimensional motion of an unbalanced rigid disk in a shaft system are nonlinear and contain dynamic-coupling terms. Traditionally, investigators have used an order analysis to justify ignoring the nonlinear terms in the equations of motion, producing a set of linear equations. This paper will show that, when gears are included in such a rotor system, the nonlinear dynamic-coupling terms are potentially as large as the linear terms. Because of this, one must attempt to solve the nonlinear rotor mechanics equations. A solution methodology is investigated to obtain approximate steady-state solutions to these equations. As an example of the use of the technique, a simpler set of equations is solved and the results compared to numerical simulations. These equations represent the forced, steady-state response of a spring-supported pendulum. These equations were chosen because they contain the type of nonlinear terms found in the dynamically-coupled nonlinear rotor equations. The numerical simulations indicate this method is reasonably accurate even when the nonlinearities are large.
This content is only available via PDF. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8662294745445251, "perplexity": 419.6276765705477}, "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-2019-43/segments/1570986675409.61/warc/CC-MAIN-20191017145741-20191017173241-00016.warc.gz"} |
https://link.springer.com/article/10.1007%2Fs10853-007-2429-5 | Journal of Materials Science
, Volume 43, Issue 9, pp 2971–2977
# The effective oxidation pressure of indium–oxygen system
• E. Ricci
• T. Lanata
• D. Giuranno
• E. Arato
Open Access
Interface Science
## Abstract
A theoretical model on oxygen transport at the surface of liquid metals has been validated by dynamic surface tension measurements performed on liquid Indium as test metal. The oxygen contamination conditions have been obtained at different oxygen partial pressures under both low total pressure (Knudsen regime) and inert atmospheric pressure (Fick regime) conditions, confirming that an oxide removal regime occurs under an oxygen partial pressure much higher than the equilibrium one (the “Effective Oxidation Pressure”). Experimental results are reported which give a further insight on the relative importance of the various processes due to the oxygen mass transport between the liquid metal and the gas phase. The critical aspects involved in surface tension measurements of liquid metals, related to the problem of liquid metal–oxygen interactions, are also underlined.
## Keywords
Surface Tension Liquid Metal Oxygen Partial Pressure In2O Surface Tension Measurement
## Introduction
The interactions between liquid metals and their environments (vapour phase and working atmosphere) can be understood by analysing the surface properties of the metals. The capillary properties, in particular the surface tension, are strongly affected by surface cleanliness, which depends on the surrounding atmosphere and, for liquid metals in particular, on its oxygen content. Oxygen can either dissolve into the liquid phase or form an oxide film on the surface of the liquid metal [1, 2]. In most cases the Gibbs Adsorption Equation [3] or the derived adsorption isotherms [4, 5, 6, 7], which account for the decrease in the interfacial free energies when the oxygen activity increases, are applied to explain the mechanism of oxygen adsorption at the liquid metal–gas interface. The differences in the interpretations arise principally from the difficulties of performing surface properties’ determinations using controlled values of oxygen present in the gas phase. In addition, when dealing with high-temperature measurements of metallic systems, vaporisation phenomena generally occur and have to be taken into account.
Several experiments [8] have clearly shown that a flux of metallic vapours occurs in the direction opposite to the oxygen flow. The effect of oxygen partial pressure on the vapourisation rate of metals, which depends linearly on the oxygen content of the surrounding gaseous atmosphere, has been verified [9]. Moreover, for those molten metals that form volatile oxides there will also be a flow of metal oxides leaving the liquid surface. In this case, the oxygen transfer from the gas phase to the condensed phase must account for the double contribution of molecular oxygen and the oxygen linked as oxide. Many theoretical and experimental studies of these effects are available in literature for solid metals [10]. In fact, purification processes by means of oxide and sub-oxide evaporation at high temperature are extensively used in metal refining. The dependence of the rate of the oxidation on the gas composition at a high temperature was discussed theoretically in [11].
To have a complete approach to the study of oxygen interactions with the liquid metal and of its influence on the material properties, in particular the surface properties, it is clear that a correlation needs to be established between the kinetics of vapour fluxes and thermodynamic equilibrium. The estimation of both the degree of contamination and the mechanism of the gas mass-transfer at the liquid metal–gas interface have been provided by different theoretical models [12, 13, 14]. The application of these models is even more useful when dealing with the measurements of the surface properties of metals of particular technological interest, such as silicon [15, 16], tin [17, 18] or aluminium [19, 20].
The aim of this paper is to obtain a further confirmation of the main result previously demonstrated [18, 19]: i.e. the true operative parameters of the surface oxidation/de-oxidation conditions can be described through the definition of the Effective Oxygen Pressure [1]. In fact, the description presented here concerns the dynamic surface tension measurements of liquid indium, performed to determine the interactions of oxygen with that metal through the validation of the theoretical oxidation/de-oxidation curve under both low total pressure and atmospheric pressure.
## Experimental
The high melting temperature and chemical reactivity of most metals increase the experimental difficulties in performing thermophysical properties’ measurements. The lack of accuracy can be caused by the numerous sources of errors, for example, impurities coming from the base material, from the container material or the environment. However, some uncertainties on measurements could still exist, essentially due to the shortage of the experimental evaluation of the oxygen content.
Among the thermophysical properties, the surface tension is certainly the parameter most sensitive to the interaction of the oxygen and the metal. This is one of the reasons why surface tension measurements of pure metals are so much affected by a great uncertainty [20]. In turn, performing systematic dynamic surface tension measurements, the liquid metal surface evolution can be successfully observed as a function of the surrounding atmosphere.
The large-drop technique is a suitable method for performing systematic dynamic surface tension measurements on liquid metals and alloys [21, 22]. In this case, the solid support is a special circular crucible with sharp edges. The design of the edges of the crucible blocks the triple line at an “apparent” contact angle that is much higher than the real one and the axis-symmetry of the drop can be imposed. A further advantage of this method is the high accuracy due to the large drop used, in fact the size of the drop was chosen so that the error in surface tension did not exceed 1%.
In addition, due to the difficulties in performing reliable measurements at high temperature, some standard criteria have to be follow in the building of the apparatus and in the procedures, as described in [23]. The experimental apparatus used for the dynamic surface tension determinations has already been described in [18]. Recently it has been supplemented with three solid-state electrodes (POAS-Setnag®) with internal metal/oxide reference, each kept at its optimal working temperature, allowing the oxygen partial pressure to be monitored at different sites in the apparatus: (a) inside the chamber; (b) at the inlet of the feed gas; (c) at the outlet of the exhaust gas. In addition, a high-precision flow meter and a micro-leak valve have allowed a precise and controlled amount of the gas flux inside the chamber and an ion gun and gas supply system have allowed the cleaning of the molten sample surface by argon ion sputtering.
Samples of the highest purity (99.9999 Marz grade) Indium of about 2.5 g were mechanically abraded and chemically cleaned with an organic solvent in an ultrasonic bath. The sample was placed in a non-oriented monocrystalline alumina (sapphire) crucible (r = 5.5 mm). The crucible with the sample was laid on an alumina holder sliding in the experimental apparatus. When the apparatus conditions (temperature, oxygen partial pressure) were reached, the sample was introduced into the centre of the furnace by a pushrod magnetic manipulator. The drop profile of the sample in the crucible was acquired by a CCD camera and processed with adhoc acquisition software [24] in a LABView® environment. This acquisition procedure allows real-time surface tension measurements to be made and, at the same time, the values of the surface tension and the other parameters (sample temperature and oxygen partial pressure) to be followed. Due to its high performance in terms of time of acquisition and reliability (up to 10 points per second with an accuracy up to 0.1%), it is particularly suitable for both static and dynamic measurements. The magnification factor of the system was calculated for each shape acquisition with an internal reference. The density, as a function of temperature, was introduced from literature data [25]. When compared to other sources [26, 27], the differences in the density data of the Indium were less than 0.5% over the temperature range investigated.
To obtain the actual experimental curve, indicating the oxygen tensioactive effect on the liquid metal an experimental isobaric procedure was utilised, where the imposed temperature was changed under the constant oxygen partial pressure of the feed gas until the effect of oxygen is found [18]. The tensioactivity is revealed by the change of the temperature coefficient sign. The advantage of the isobaric procedure over the isothermal one was that the variations in temperature can be set automatically.
As saturation conditions strongly depend on temperature, a temperature modulation corresponds to a modulation of the saturation level and it is then equivalent to a modulation of oxygen concentration in the liquid phase [23]. The results of the application of the isobaric procedure to molten indium are reported and discussed below.
## Results and discussion
The experimental tests were performed varying the temperature at a rate = 5 K/min, under both Knudsen regime and inert gas flux, in the temperature ranges 533–833 K and 773–1,073 K, respectively. Surface tension measurements were made at 1 min intervals. In Fig. 1, an example of a test performed under the Knudsen regime (Ptot ≈ 10−3 Pa) is shown, the oxygen partial pressure being constant and equal to 2.6 × 10−4 Pa. The temperature was increased and decreased between 533 and 833 K. The temperature modulation allows some significant inversions in the surface tension behaviour to be noted. The meaning of the ‘inversions’ are clearly explained in [23]. Following the behaviour shown in Fig. 1, if a negative temperature coefficient is assumed characteristic of pure In, the oxygen tension-active effect over the liquid becomes evident between point ‘1’ and ‘2’. When temperature increases, the slope of the surface tension versus temperature is positive showing the presence of an oxygen adsorption [23]. The position of ‘2’ can be approximately symmetrical to that of ‘1’ as a sign of the reversibility oxygen dissolution in the metal drop. The portion of the curve between ‘1’ and ‘2’ reflects the surface adsorption due to the segregation of the oxygen dissolved into the liquid bulk (consequence of the variation of temperature) more than the effect of the external imposed oxygen partial pressure. The experimental conditions of the inversion (‘1’ or ‘2’ coordinates) are linked to tension-active effect, but not necessarily to the formation of stable oxides on the surface [28] and this effect is identified by the sign inversion (actually from negative to positive) of the temperature coefficient of the surface tension. Positive temperature coefficients are not usually reported in literature, probably because they are considered the consequence of heavy contamination. On the other hand, in our experiments, the temperature coefficients for what we consider a ‘clean’ surface are easily repeatable, and the sign inversions, when they occur, are easily detectable. Considering several consecutive decreasing and increasing temperature segments, the behaviour of surface tension with the temperature is almost univocal: its symmetry derives from the temperature dependence of the oxygen solubility product which is responsible for the bulk oxygen segregation towards the surface.
This is particularly evident looking at Fig. 2, where the data of the same test of Fig. 1 (only for the segments characterised by a temperature variation) are shown in the (γ, T) plane.
In the light of these facts, the identification of the sign inversion with the saturation of the interface, which is the incoming formation of an oxide phase, appears to be a reasonable assumption [23]. The same behaviour had already been observed for a tin/oxygen system [18] and aluminium/oxygen system [19]. However, in the case of tin, a slight difference between the theoretical Effective Oxygen Pressure curve and experimentally available ($$P_{{\text{O}}_{\text{2}} }$$, T) values exists: the latter are higher than the thermodynamic value and the theoretical curve, due to the contribution of the linked oxygen in the form of sub-oxides and the oxygen diffusion inside the bulk phase [29]. In the In–O system under Knudsen regime, the oxygen fluxes to and from the oxidised surface originated in the vaporisation of a compact oxide layer (In2O3) during its decomposition into other oxides and a three-phase mono-variant equilibrium was assumed [29]. These conditions are the same as those given by the model presented by Castello et al. [1] but with the calculated results shifted towards higher residual pressure values because the sub-oxide vapour pressures (in particular In2O) were maximised under the three-phase equilibrium conditions [30].
Under atmospheric pressure, the molecular diffusion is assumed to be the main process controlling the exchange of matter between the liquid metal and the surrounding atmosphere. In this case, chemical reactions in the gaseous layer are also taken into account and it is possible to verify that the liquid metal is under steady-state conditions, which are different from those of thermodynamic equilibrium. When an inert gas flux (Ar-N60; Ptot = 105 Pa) was experimentally applied, different behaviours were found, depending on both the initial status of the metal drop and the gas flow rate. In Fig. 3 the results of a test performed under a low gas flow rate (0.05 L/min; $$P_{{\text{O}}_{\text{2}} } {\text{ = 5}} \times {\text{10}}^{ - {\text{3}}} \;{\text{Pa}}$$ in the inlet inert gas), in the temperature range 773–1,073 K are shown. The oxygen partial pressure $$P_{{\text{O}}_{\text{2}} } {\text{ = 4}} \times {\text{10}}^{ - {\text{13}}} {\text{ Pa}}$$ was measured inside the test chamber. Under these conditions the inversion points were not observed during the experiment (timeexp > 16 h) and the surface tension behaviour indicated that the liquid metal drop was always in the de-oxidation regime as foreseen by the Effective Oxygen Pressure curve calculated for the In–O system. In Fig. 4 an example of a test carried out at constant $$P_{{\text{O}}_{\text{2}} } {\text{ = 5}} \times {\text{10}}^{ - {\text{3}}} \;{\text{Pa}}$$ in the inlet inert gas, while varying the temperature is reported. The variation in the temperature over time was set at 5 K/min, the temperature was in the range 773–1,073 K and the oxygen flux was maintained at a constant value of 0.05 L/min. In the first 8 h, it was possible to observe a sequence of inversion points that progressively disappear and an increase of the value of the surface tension corresponding to the lowest temperature. In the meantime, a progressive decrease in the oxygen partial pressure (from $$P_{{\text{O}}_{\text{2}} } {\text{ = 10}}^{ - {\text{4}}}$$ to $$P_{{\text{O}}_{\text{2}} } {\text{ = 10}}^{ - {\text{13}}} \;{\text{Pa}}$$) inside the test chamber over time was monitored by the sensors.
The fundamental point is that the entire test was carried out in surface de-oxidation conditions, for which reason the trend reported in the figure is not correlated to the value of $$P_{{\text{O}}_{\text{2}} }$$ inside the test chamber, but is linked to the initial concentration of the dissolved oxygen in the metal sample. The inversion point marks a concentration threshold of dissolved oxygen in the drop above which oxygen manifests its tensioactive character in relation to Indium. This threshold concentration should correspond to the saturation level of oxygen in the metal, and the point at which it begins to form an oxide on the metal surface. When the temperature is reduced the solubility of oxygen is also reduced [31] and it then reaches its saturation concentration on the surface and in the bulk. In fact, when lowering the temperature, if the drop is “clean”, the surface tension increases at first, according to the behaviour of pure liquid, then as saturation is approached, the oxygen content in the liquid surface increases and the surface tension begins to decrease. The maximum surface tension for a decrease in temperature or the minimum surface tension when temperature increases corresponds to incoming oxidation points.
As the system is in surface de-oxidation conditions, the oxygen level inside the sample falls over time and, in the test considered, after more than 30 h its concentration is such that even at the minimum test temperature (773 K) it is not sufficient to saturate the drop and form an oxide film on the surface. The trend further shows that the liquid metal drop with a relatively high oxygen content does not immediately achieve equilibrium with the surrounding atmosphere: the exchange of material occurring at the interface brings the metal to steady-state conditions with its surroundings only after many hours. This behaviour is confirmed by studying in more detail the dependence of the temperature coefficient on the oxygen concentration dissolved in the metal. For this reason a selection of the experimental data shown in Fig. 4, was drawn in a temperature–surface tension graph and presented in Fig. 5. The series are identified according to their temporal sequence (the trend over time is also presented by the arrow) and differ only with respect to the moment at which the inversion in the sign of the temperature coefficient occurred. In the linear part (T > 973 K), that characterises the trends before the inversion point, all the tests are very similar: independent of their timing of the tests and independent of whether the temperature was increasing or decreasing. The experimental findings are very significant: (a) the surface tension data do not vary by more than 2 mN/m in the “linear section” (T > 973 K); (b) the temperature coefficient of the surface tension of indium is independent in a finite range of dissolved oxygen concentration of values of the same concentration.
A similar behaviour was already observed for the tin–oxygen system and described in detail in [28].
Finally, in Fig. 6, the Effective Oxygen Pressure curve calculated for the In–O system is shown (bold line). This quantity is the true limit value of the oxygen partial pressure, discriminating between the surface oxidation and de-oxidation regimes for the In–O system and is, as expected, several orders of magnitude (up to 30 o.m.) higher than the corresponding value of the thermodynamic equilibrium gas/condensed phase [32] ($$P_{{\text{O}}_{\text{2}} }^{\text{s}}$$-broken line). Liquid mass-transfer phenomena are also considered [33] so that only the liquid metal–gas interface could be assumed stationary, and the oxidation pressure becomes proportional to the solubility of oxygen in the metal [23]. When all the experimental points (e.g. under Knudsen regime and under inert gas flux), which characterised the inversion of the surface tension versus temperature measured during the dynamic surface tension tests are reported in the Effective Oxygen Pressure diagram, a very good agreement is observed (Fig. 6).
## Conclusions
The phenomena of oxygen transport at the molten metal–atmosphere interface and particularly the mechanisms whereby a molten metal or alloy surface is kept clean even in the presence of O2 impurities in the gas, play a very important role in most high-tech processes.
From the results obtained from our models, it is possible to straightforwardly explain the mechanisms which keep a molten metallic system oxygen-free, despite the thermodynamic driving force, due to an oxygen flux in the opposite direction to the oxide bond.
The experimental results based on the In–O system using dynamic surface tension measurements show once again that the experimental oxidation/de-oxidation transitions occur at values very close to those predicted by the models and, therefore, higher than the thermodynamic values, owing to the non-negligible contribution of the linked oxygen in the form of the volatile In2O sub-oxide.
The stationary drop assumption leads to the definition of an Effective Oxygen Pressure curve that well describes the indium experimental data: the data show a very good agreement with the theory when liquid mass-transfer effects are taken into account.
## Notes
### Acknowledgement
The authors wish to thank Paolo Costa for his contributions to this work.
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Arato E, Ricci E, Fiori L, Costa P (2005) J Cryst Growth 282:5 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8916922211647034, "perplexity": 1730.5580690399709}, "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-2019-04/segments/1547584445118.99/warc/CC-MAIN-20190124014810-20190124040810-00181.warc.gz"} |
https://www.jiskha.com/display.cgi?id=1512611835 | # Chemistry
posted by Michele
Lithium 7 decays losing one alpha particle what is left?
7H
1
1. bobpursley
loses 2 protons, 2 neutrons
so atomic number goes from 3 to 1, and atomic mass goes from 7 to 3. Tritium is left. 1H3
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https://stats.stackexchange.com/questions/107844/kalman-filter-transition-matrix | Kalman filter transition matrix
Hi guys I am trying to writ e a code on python to correct forecast data using Kalman Filter. I am following the equations and recommendations in this link : http://www.swarthmore.edu/NatSci/echeeve1/Ref/Kalman/ScalarKalman.html. At some point, I have written a function which satisfies me since I don't know whatsoever how to use the pykalman module to correct estimations. I am using just one parameter (unidimensional and univariate time series), so my matrices form will be a numerical value. My main problem is what is the transition matrix? In many paper they just use a transition matrix of 1 but I think this is not rigorous since it would mean that: forecast(t) = A * forecast(t-1) We do not have a constant value of forecast so what would be my transition matrix?
Hypothesis 1: Can I just try to see a correlation coefficient between two estimated values then take the mean of those coefficients for my first day of forecast.By the way this transition matrix will be just an initialization and will change for each forecast following
The transition matrix relates state t and state t-1.
If we write the temporal coherence equation like this
$$x_t = \Psi x_{t-1} + \epsilon_p$$
This is the temporal model. This model tells you what is the tendency of your system. When no measurement is found, the system will follow this tendency. When it is found, there is a trade-off between where the measurement says the track should go and where the temporal model says it should go.
$\Psi$ is the transition matrix then.
You can have different types of transition matrix, for instance, temporal brownian motion, where $\Psi = I$, meaning that the next state is the last one plus some noise.
Another possibility would be constant velocity.
Imagine an easy example in 1d. We are tracking the position of an object and its velocity. It is just the same equation as above, in this particular case.
$$\begin{bmatrix} x_t\\ vel_t \end{bmatrix} = \begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix} \begin{bmatrix} x_t\\ vel_{t-1} \end{bmatrix} + \epsilon_p$$
Then, if you multiply terms, you get
$$x_t = x_t + vel_t + \epsilon_{p,x}$$ $$vel_t = vel_{t-1} + \epsilon_{p,vel}$$
This example would be, as the second equation tells us, a constant velocity model.
If you still have doubts, there is a nice explanation of Kalman Filter here: http://web4.cs.ucl.ac.uk/staff/s.prince/book/book.pdf Chapter 19.
I think you should consult a clearer description. (I like a lot the one from Matlab (which you can easily port).
Especially your formula is wrong it is:
predicted state space $\tilde X(t) = A *$corrected state space $\hat X(t-1)$
the forecast would be $h(X)$.
Most important even though you just observe one dimension your state space can be much bigger. Your idea with correlation coefficient would imply a two dimensional one.
• Thanks can you help me on one of my post here:stackoverflow.com/questions/24736263/… Jul 14 '14 at 12:26
• Sorry - I'm not a python user. So you will need someone else. Jul 14 '14 at 13:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9427351951599121, "perplexity": 465.5975027701146}, "config": {"markdown_headings": false, "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-2021-39/segments/1631780054023.35/warc/CC-MAIN-20210917024943-20210917054943-00528.warc.gz"} |
https://cds.cern.ch/collection/Theses?as=1 | # CERN Accelerating science
ABOUT THESES
The theses collection aims to cover as well as possible all theses in particle physics and its related fields.
The collection starts with the thesis of Feynman, defended in 1942, and covers now alltogether more than 3000 theses. Most of the documents are held as hard copies, theses from later years are available electronically. (Note also that many theses are not physically held by the CERN Library.)
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Precision physics in Hadronic Tau Decays / Rodriguez-Sanchez, Antonio In the Standard Model of Particle Physics, which gives a very precise description of nature at many different scales, hadronic tau decays occur through the interaction of two weak charged currents mediated by the W boson, so it becomes a very nice test of electroweak interactions [...] 2018. - 214 p.
2021-10-21
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Search for pentaquark state and measurement of CP violation with the LHCb detector / Liu, Xuesong Testing the standard model (SM) of particle physics and searching for new physics beyond it, explaining the asymmetry between matter and antimatter in the universe is the frontier of particle physics researches [...] CERN-THESIS-2019-423 - 165 p.
2021-10-19
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Tuning of parameters sensitive to colour reconnection in the Pythia 8 event generator using top-quark-antiquark pair production events recorded with the ATLAS detector / Caspar, Maximilian In this master thesis, Monte Carlo models describing the reorientation of QCD colour interactions (”colour reconnection”) in an event are fitted to experimental data taken at ATLAS [...] CERN-THESIS-2021-170 - 81 p.
2021-10-18
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Measurement of the Higgs boson coupling to $b$-quarks in the associated production with a vector boson with the ATLAS detector / Gargiulo, Simona This thesis presents the measurement of the Higgs boson coupling to a pair of $b$-quarks [...] CERN-THESIS-2021-169 -
2021-10-18
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Search for heavy resonance decaying into a photon and a Higgs bosonwith the ATLAS detector at the LHC / Chen, Boping A search for heavy resonance decaying into a photon and a Higgs boson is performed, where the Higgs boson continually decaying into a pair of b-quarks [...] CERN-THESIS-2020-369 -
2021-10-18
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Measurement of the mixing parameters of neutral charm mesons and search for indirect \textit{\textbf{CP}} violation with $D^0\to K^0_S\pi^+\pi^-$ decays at LHCb / Hilton, Martha Mixing is the time-dependent phenomenon of a neutral meson (in this case charm meson $D^0$) changing into its anti-particle ($\bar{D}^0$) and vice versa [...] CERN-THESIS-2021-168 - 262 p.
2021-10-17
08:42
Characterisation of a High-Voltage Monolithic Active Pixel Sensor Prototype for Future Collider Detectors / Kroeger, Jens The physics goals and operating conditions at existing and proposed high-energy colliders, such as the Compact Linear Collider (CLIC), pose challenging demands on the performance of their detector systems [...] CERN-THESIS-2021-167 -
2021-10-15
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Search for flavor-changing neutral-current interactions between the top quark and the Higgs boson in dileptonic same-charge final states with the ATLAS detector / Geyik, Marvin Emin The analysis presented in this thesis focuses on the search for flavor-changing neutral-current (FCNC) interactions involving the top quark and the Higgs boson at a center-of-mass energy of $\sqrt{s}=13\,$TeV with the ATLAS detector [...] CERN-THESIS-2021-166 - 107 p.
2021-10-14
17:50
Probing the CP nature of the Higgs coupling to top quarks with the ATLAS experiment at the LHC / Gouveia, Emanuel Since the Higgs boson with a mass of 125 GeV was discovered by the ATLAS and CMS experiments at CERN, the study of the properties of this particle has been a priority of the Large Hadron Collider (LHC) physics programme [...] CERN-THESIS-2021-165 - 267 p.
2021-10-14
17:36
Algorithms for processing of large data sets using distributed architectures and load balancing / Subrt, Ondrej Modern experiments in high energy physics impose great demands on the reliability, the efficiency, and the data rate of Data Acquisition System (DAQ) [...] CERN-THESIS-2020-368 - 214 p. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9746503829956055, "perplexity": 1959.8177304498754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "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-2021-43/segments/1634323588102.27/warc/CC-MAIN-20211027053727-20211027083727-00314.warc.gz"} |
http://math.stackexchange.com/users/93448/lucian?tab=activity | Lucian
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Apr 25 awarded Nice Answer Apr 20 comment Finite series that gives Beta function at integers Can someone help me see how this identity arises ? - Probably in the same way as this one... Apr 9 comment Mid-sections and angles Why do we have $B=2\alpha$ and $C=2\beta$ ? Apr 4 awarded Enlightened Apr 4 awarded Nice Answer Mar 27 revised solving the inequalty Fixed Row Alignment. Mar 26 awarded Nice Question Mar 23 awarded Nice Question Mar 23 asked Elegantly Proving that $~\sqrt[5]{12}~-~\sqrt[12]5~>~\frac12$ Mar 19 comment Evaluating an integral involving Beta function Hint: What substitution transforms $(t,z)$ into $(0,1)$ ? Mar 8 comment Program to create graph with modified bessel function @xsr: In that case, may I recommend Mathematica $($ the offline version of Wolfram Alpha $),$ or Maple. Mar 7 comment Solving an infinite series See also Basel problem. Mar 7 answered Program to create graph with modified bessel function Mar 6 comment Primes of the form $x^3+y^3+z^3 - 3xyz$ Mar 6 revised Simplify expression in Boolean algebra Typo Corrected. Mar 5 comment Integral involving Bessel function $J_0$. The general solution can be found here. Mar 5 comment Nested radical $\sqrt{x+\sqrt{x^2+\cdots\sqrt{x^n+\cdots}}}$ A related question. There is no closed form, though, that's the whole point. Mar 5 comment Calculating $\int_0^{\pi/2} (x \sin(x))^n dx$ Try to find a recurrence relation, if at all possible. Mar 5 comment Sum of a particular Series Differentiate with regard to $\alpha,$ and use Euler's formula to rewrite the new expression in terms of the Jacobi $\theta$ function. If even this fails, then all hope is lost. Mar 5 comment Let $p$ a prime number, then prove that : $\sum \limits_{k=0}^{p} \binom {p}{k} \binom{p+k}{k} \equiv 2^p +1 \pmod{p^2}$ Would rewriting $\displaystyle{p+k\choose k}~=~\displaystyle(-1)^k~{-p-1\choose k}$ help in any way ? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9230751395225525, "perplexity": 814.376919864773}, "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-2016-18/segments/1461860111392.88/warc/CC-MAIN-20160428161511-00016-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.arxiv-vanity.com/papers/1411.5693/ | arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org.
# Quantum versus Classical Annealing of Ising Spin Glasses
Bettina Heim Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland Troels F. Rønnow Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland Sergei V. Isakov Google, Brandschenkestrasse 110, 8002 Zurich, Switzerland Matthias Troyer Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland
###### Abstract
The strongest evidence for superiority of quantum annealing on spin glass problems has come from comparing simulated quantum annealing using quantum Monte Carlo (QMC) methods to simulated classical annealing [G. Santoro et al., Science 295, 2427(2002)]. Motivated by experiments on programmable quantum annealing devices we revisit the question of when quantum speedup may be expected for Ising spin glass problems. We find that even though a better scaling compared to simulated classical annealing can be achieved for QMC simulations, this advantage is due to time discretization and measurements which are not possible on a physical quantum annealing device. QMC simulations in the physically relevant continuous time limit, on the other hand, do not show superiority. Our results imply that care has to be taken when using QMC simulations to assess quantum speedup potential and are consistent with recent arguments that no quantum speedup should be expected for two-dimensional spin glass problems.
With first archeological records dating back more than six thousand years annealing_first, thermal annealing is likely to be the oldest optimization method in human history. By first heating a material and then letting it cool down slowly, it can relieve internal stresses and achieve a lower energy state. Inspired by thermal annealing, Kirkpatrick and co-authors suggested a similar approach to find the ground states of combinatorial optimization problems more than three decades ago kirkpatrick1983. In particular, they studied Ising spin glasses with spins described by the Hamiltonian
Hc=−∑i
where takes the values and represents the orientation of the spin on lattice site . The couplings between spin and are denoted by and are local fields.
Non-convex optimization problems, such as finding the ground state of this Ising spin glass Barahona1982, are important in many areas of science and industry. Other typical problems include job scheduling jobschedule, circuit minimization Knuth, and chain optimization scan_chain_opt which are all non-deterministic polynomially (NP) hard problems Cook-ACM-1971. A consequence of NP-hardness is that there exists a polynomial time mapping from one problem to the other. Thus, any method to efficiently find solutions to the Ising spin glass problem would provide an efficient way of solving other important problems.
Applying the Metropolis algorithm Metropolis, Kirkpatrick et al. demonstrated that using “simulated annealing” (SA) – simulating the process of cooling Ising spin glasses – is an excellent method to minimize . Starting from a high temperature where the system thermalizes quickly, the temperature is slowly decreased towards zero. Thermal excitations allow the system to escape from local minima and relax into a low-energy state with energy equal or close to that of the ground state sa_and_sa_conv_cond. We will refer to the difference between the final energy and as the residual energy .
Quantum annealing (QA) Ray1989; Finnila1994; Kadowaki1998; idea_of_qa; qareview uses a similar idea but employs quantum tunneling instead of thermal excitations to escape from local minima. QA can be advantageous in systems with narrow but tall barriers, which are easier to tunnel through than to thermally climb over. To perform QA of Ising spin glasses, an additional non-commuting kinetic term is added, usually by applying a transverse magnetic field. The time-dependent Hamiltonian of QA is then given by
Hq=−∑i
where and are Pauli - and -operators, respectively. The transverse field is initially much larger then the couplings, , and the spins start out aligned in the -direction. During quantum annealing is slowly reduced to zero such that at the end of the annealing process we recover the Hamiltonian of the initial Ising spin glass problem. On a perfectly coherent quantum device, this algorithm idea_of_qa would find the ground state of the spin glass in question with probability approaching unity, provided that the annealing time is sufficiently long to stay adiabatically in the ground state landau; zener. Quantum annealing can also be performed at non-zero temperature, for example on spin glass material Brooke1999 or in programmable devices by the Canadian company D-Wave systems Johnson2011.
QA can also be implemented as a simulation on a classical computer. While the simulation of unitary time evolution scales exponentially with the system size, QA can be efficiently performed using stochastic dynamics in a path integral quantum Monte Carlo (QMC) simulation santoro1; santoro2. There, the partition function of the Ising model in a transverse field is mapped to that of a classical Ising model in one higher dimension corresponding to the imaginary time direction suzuki_orig. We call this algorithm simulated quantum annealing (SQA). In Fig. 1A the two-dimensional lattice on which we perform SA and in Fig. 1B the three-dimensional lattice on which SQA is performed after the path-integral mapping. Details of the simulation algorithms and annealing schedules used in the Letter are discussed in the Supplementary Material.
The strongest evidence for quantum annealing being superior to classical annealing for Ising spin glass instances comes from a comparison of the performance of SQA and SA santoro2; santoro1; qareview. Upon increasing the annealing time the residual energy was seen to drop faster in SQA than in SA, indicating that quantum tunneling may indeed be advantageous in finding low energy states. However, recent studies of the performance of the D-Wave devices failed to see indications of quantum speedup article_eth, although the device performance was consistent with that of a quantum annealer Boixo:2014ej. Furthermore, in contrast to Refs. santoro2; santoro1 no advantage of SQA over SA was seen.
In order to investigate these seemingly contradictory results we first show, in Fig. 2A, that we can reproduce the results of Ref. santoro1. Additionally, we show the best results of 32 independent SA simulations, which corresponds to roughly the same computational effort as the SQA simulations, as illustrated in Fig. 1C. In either case we see that, as in Ref. santoro1, the scaling of SQA is superior to that of SA.
However, these simulations were all performed with a finite number of time slices and a corresponding non-zero time step , which we refer to as a discrete time SQA (DT-SQA) simulation. Discrete time steps incur time discretization errors of order . To obtain accurate thermal averages for the quantum system one has to either extrapolate DT-SQA results to or perform a continuous time SQA simulation (CT-SQA) that works directly in the limit ctq.
Repeating the same simulations using CT-SQA in Fig. 2B we see that the CT-SQA result has an entirely different behavior than the corresponding DT-SQA curve. While the performance is improved for fewer than Monte Carlo steps (MCS, corresponding to one attempted update per spin), the residual energy saturates for longer annealing times, at a level higher than that reached by SA. While the time discretization error in DT-SQA is of no concern for its use as a classical optimization algorithm 111The time discretization error vanishes at the end of the annealing schedule when the transverse field is switched off and all remaining terms in the Hamiltonian commute, it does not reflect our expectations for a physical quantum device, for which the continuous time limit is relevant. Hence, the circumstances under which SQA outperforms SA depend on whether we use SQA as a quantum inspired classical algorithm, or as simulation of a physical system. Understanding the role of time discretization in SQA is important both to estimate the performance of experimental quantum annealers as well as to tune SQA as a classical algorithm.
## Effects of Time Discretization and Temperature:
To understand the role of time discretization we have gone beyond the single spin glass instance of Ref. santoro1 and studied 1000 random spin glass instances on an square lattice with periodic boundary conditions. We use the the same distribution, choosing the uniformly from the interval , and set all local fields to zero () and obtain the exact ground state energy using the spin glass server. Following the procedure of Ref. santoro2, the initial state in Fig. 2 was prepared by precooling. Comparing the DT-SQA curves with and without precooling (Fig. 2B), we find that precooling only results in constant offset but does not improve the scaling. We thus omit precooling from our simulations.
In Fig. 3A we show the residual energy as a function of annealing time for various Trotter numbers . As expected, for , DT-SQA converges towards the continuous time limit. For the chosen temperature of , convergence is achieved for . We find the same surprising behavior already indicated in Fig. 2: Although the initial scaling is better in the continuous limit, lower residual energies are reached at a finite time step size. Comparing and , a lower residual energy of is found for compared to for , despite the computational effort being four times smaller.
Analyzing the residual energies as a function of temperature with a constant number of time slices, as shown in Fig. 3B, leads to a similar observation. For , the DT-SQA results match well with the CT-SQA results (shown in the Supplementary Material), indicating that 64 time slices are sufficient to converge to the continuous time limit. At lower temperatures deviations from CT-SQA are seen and the larger time step in DT-SQA allows to eventually find states with lower energy than in CT-SQA – consistent with the results of changing . A closer look at Fig. 3B shows that at lower temperatures lower energies can be reached. This fact, which is confirmed in the continuous time limit shown in the Supplementary Material, is encouraging for a potential weak quantum speedup article_eth for SQA over SA in the zero-temperature limit.
For all choices of and we considered, the residual energy saturates at some point, indicating that the simulations consistently get stuck in some local minimum during annealing. We will discuss reasons for this behavior below.
## Quantum Annealing as a Classical Optimization Method:
When discussing SQA as a classical optimization algorithm, we can search the final configuration for the time slice (or time interval in continuous time) with the lowest energy. This improves the results if the spin alignment along the imaginary time axis is incomplete at the end of the annealing. However, we have to take into consideration the increased computational effort of QMC simulations compared to SA. The number of Monte Carlo steps needs to be multiplied by for DT-SQA and by for CT-SQA.
Plotting the residual energy as a function of total computational effort in Fig. 4A we find that – in agreement with Ref. santoro1 – with suitable chosen temperature and number of time steps, DT-SQA outperforms SA. The optimal choice depends on the desired computational effort and the envelope seems to outperform SA, although the asymptotic behavior when we anneal for longer times seems similar.
In order to use SQA as a classical optimization algorithm it is thus advantageous to use a small time step for short annealing times, since the continuous time limit has a more rapid initial decrease of . When annealing for longer times a lower temperature and larger time step are preferred, since that way we reach lower asymptotic residual energies. To reach the lowest energies, rather large time steps of order unity are preferred, where the system consists of few moderately coupled individual replicas instead of a more tightly coupled continuous path of configurations. We note finally that, as we show in the supplementary material, even CT-SQA with suitably chosen temperature can outperform SA when used as a classical optimizer.
## Quantum Annealing of a Physical System:
While the Monte Carlo dynamics in SQA is not the same as the unitary or open systems dynamics of a physical quantum annealer, it is very similar since it captures tunneling and quantum entanglement. In particular, if thermalization (at least within a local minimum) is fast compared to the annealing time, SQA is expected to reliably capture the performance of physical QA, as has been seen in the case of the D-Wave devices Boixo:2014ej.
To use SQA as a tool to estimate the performance of hardware-based QA we have to take the continuous time limit and use either CT-SQA, or DT-SQA with a large enough number of time slices to be converged to the continuous time limit. We may measure only properties that are experimentally accessible and thus instead of picking the time slice with lowest energy, we either have to average the residual energy over all time slices, or measure it just at one randomly chosen imaginary time to mimic the process of measurement in a quantum system.
Figure 4B shows the slightly higher residual energy obtained this way as a function of the number of MCS. We find that increasing the temperature slightly over that when SQA is used as a classical optimizer helps performance. For more details we refer to the Supplementary Material. Lower temperatures are again preferred for longer annealing times. While SQA outperforms SA for short annealing times, the asymptotic scaling of the envelope seems worse for SQA.
We find that for short annealing times, up to MCS, SQA still outperforms SA when choosing an appropriate temperature but the asymptotic scaling is better for SA.
## Discussion and Outlook:
Carefully investigating evidence for quantum annealing outperforming classical annealing for spin glass instances, we found that, surprisingly, the performance advantage previously observed for path-integral QMC annealing compared to classical annealing santoro1; santoro2 is due to the use of large imaginary time steps in the path integral and choosing the lowest energy over all time slices. When taking the physical limit of continuous time and measuring the average energy, the advantage vanishes.
We also found that SQA tends to get stuck in local minima more than SA. This may be understood by the more deterministic dynamics of QA, preferring a subset of low-energy states over others Matsuda:2009ji. Repeating SQA can thus get consistently stuck in similar local minima. SA, on the other hand, starts in a random state at high temperatures and thus explores the configuration space more evenly. The more deterministic nature of SQA can also explain the counterintuitive result that for some choices of parameters (see Fig. 4) the residual energy may increase when annealing more slowly. As pointed out by Ref. Crosson:2014ez, perturbing a quantum annealing schedule, for example by annealing faster, can excite a system out of a local minimum in which QA is stuck and thus help to ultimately find a better solution.
Our results also resolve the discrepancy between the observed superiority of SQA over SA santoro1; santoro2 and recent arguments that two-dimensional spin glasses should not see any quantum speedup in QA Katzgraber:2014cy. It will be interesting to explore if three-dimensional spin glasses or spin glasses with long range couplings exhibit indications of superiority for QA. When investigating the powers of QA for such spin glasses or for problem instances derived from applications, it will be important to compare to both discrete and continuous time SQA. The former being relevant for the assessment of the powers of SQA as a classical optimization algorithm and the latter for evidence of potential quantum speedup on quantum annealing devices.
We thank H.G. Katzgraber, G. Santoro, and I. Zintchenko for useful discussions, G. Santoro for providing the spin glass instance used in Ref. santoro1, and Canadian Maple Syrup for inspiration bet. This work was supported by the Swiss National Science Foundation through the National Competence Center in Research QSIT and by the European Research Council through ERC Advanced Grant SIMCOFE. M.T. acknowledges the hospitality of the Aspen Center for Physics, supported by SNF grant 1066293. The spin glass server sgserver was used to obtain the ground states for our problem instances. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8970726728439331, "perplexity": 815.1087101311284}, "config": {"markdown_headings": true, "markdown_code": false, "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-2020-24/segments/1590347436466.95/warc/CC-MAIN-20200603210112-20200604000112-00240.warc.gz"} |
https://www.preprints.org/manuscript/201612.0057/v1 | Preprint Article Version 1 This version is not peer-reviewed
Using the Outskirts of Galaxy Clusters to Determine their Mass Accretion Rate
Version 1 : Received: 9 December 2016 / Approved: 9 December 2016 / Online: 9 December 2016 (16:38:17 CET)
A peer-reviewed article of this Preprint also exists.
De Boni, C. Using the Outskirts of Galaxy Clusters to Determine Their Mass Accretion Rate. Galaxies 2016, 4, 79. De Boni, C. Using the Outskirts of Galaxy Clusters to Determine Their Mass Accretion Rate. Galaxies 2016, 4, 79.
Journal reference: Galaxies 2016, 4, 79
DOI: 10.3390/galaxies4040079
Abstract
We explore the possibility of using the external regions of galaxy clusters to measure their mass accretion rate (MAR). The main goal is to provide a method to observationally investigate the growth of structures on the nonlinear scales of galaxy clusters. We derive the MAR by using the mass profile beyond the splashback radius, evaluating the mass of a spherical shell and the time it takes to fall in. The infall velocity of the shell is extracted from N-body simulations. The average MAR returned by our prescription in the redshift range z=[0, 2] is within 20-40% of the average MAR derived from the merger trees of dark matter haloes in the reference N-body simulations. Our result suggests that the external regions of galaxy clusters can be used to measure the mean MAR of a sample of clusters.
Subject Areas
galaxies; clusters; general; methods N-body simulations; cosmology; theory | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8221006393432617, "perplexity": 1928.1344225234436}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2018-30/segments/1531676591216.51/warc/CC-MAIN-20180719183926-20180719203926-00560.warc.gz"} |
https://en.wikibooks.org/wiki/Applicable_Mathematics/Odds | # Applicable Mathematics/Odds
"Odds" is a way of expressing the likelihood of an event.
The more usual way of expressing the likelihood of an event is its "probability" (the percentage of future trials which are expected to produce the event: so in tossing a coin believed to be fair, we would assign a probability of 50% (or one half, or 0.5) to the event "heads").
The ODDS of an event, however, is the ratio of the probability of the event happening to the probability of the even not happening (i.e. the ODDS of a fair coin landing heads is 50%:50% = 1:1 = 1). It is the ODDS we are using when we use a phrase like "it is 50/50 whether I get the job" or "The chances of our team winning are 2 to 1".
Or, For example, when rolling a fair die, there is one chance that you will roll a 1 and five chances that you will not. The odds of rolling a 1 are 1:5, or 1 to 5. This can also be expressed as ${\displaystyle 1/5}$ or 0.2 or 20%, but these forms are likely to be misunderstood as normal probabilities rather than odds.
To convert odds to probability, you add the two parts, and this is the denominator of the fraction of your probability. The first part becomes the numerator. Thus, 1:5 becomes 1/(1+5) or 1/6.
To convert probability to odds, you use the numerator as the first number, then subtract the numerator from the denominator and use it as the second number. Thus, 1/6 becomes 1:(6-1) or 1:5. | {"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": 0.9441046118736267, "perplexity": 349.4338758739066}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2018-17/segments/1524125948617.86/warc/CC-MAIN-20180426222608-20180427002608-00114.warc.gz"} |
http://physics.stackexchange.com/questions/62448/voltage-of-open-circuit | Voltage of open circuit
A battery with emf $\varepsilon$ and internal resistance $r$ is connected with a resistor $R$ in the following open circuit. What is the voltage $V_{ab}=V_a-V_b$?
The answer is $- \varepsilon$. "No current. There is no voltage change across R and r.". But I don't really understand why ... I was thinking intuitively it should be $0$? Then thinking of how to get 0, I was thinking ... $V_a = - \varepsilon$ since its on the negative terminal, the $V_b = + \varepsilon$ since its on the positive terminal. But $V_a - V_b = -2 \varepsilon$ ... how do I make sense of this?
-
But Va−Vb=−2ε ... how do I make sense of this?
It's incorrect to write $V_a = \varepsilon$. The voltage $\varepsilon$ is across the battery.
Try this: place a ground symbol on the wire between the battery and the $a$ terminal; this is your zero node or the place you put the black lead of your voltmeter.
Now, if you place the red lead on the terminal of the battery connected to the resistor, you'll measure $\varepsilon$ volts.
If you place the red lead on the other side of the resistor, you'll measure $\varepsilon + V_R$ volts but $V_R = 0V$ so you still measure $\varepsilon$ volts.
But note that now you're measuring the voltage $V_{ba}$ since the read lead is connected to terminal b and the black lead is on terminal a.
So, $V_{ba} = \varepsilon = -V_{ab}$
-
Think of the open as infinite resistance. Your current will be $I = E/(R+\infty) = 0$. Now, voltage dropping across every resistor is proportional to the current and resistance: $V(r) = I * r$. Since infinite resistance is infinitely larger than R, all $E$ will drop across $\infty$ and 0v is left to R. Particluarly, $V_R = 0\mathrm{v} * R = 0\mathrm{v}$. Meantime, $V_\infty = 0\mathrm{v} * \infty = E$. The only way to get a finite value with one of multipliers 0 is to multiply it by $\infty$. Actually, $0 * \infty$ is uncertainty. But here we know that it must be E since $V_\infty$ is the only place where all voltage, generated by supply, must be dropped.
-
or think of it as a capacitor in steady state :) – nonagon Apr 27 '13 at 12:12
Ok , Potential is work done by electrostatic force as you move from A to B per Couloumb. So transfer 1 coulomb from A to B and as you move through the battery, E is directed from +ve plate to -ve plate . So work done by $\vec{E}$ is ($V_-) - (V_+$) , which means $-EMF$.
And yes that is an assumption that charge must reach the positive plate with the same velocity with which it entered the negative plate .
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8807223439216614, "perplexity": 384.3766375956819}, "config": {"markdown_headings": false, "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-2014-35/segments/1408500826679.55/warc/CC-MAIN-20140820021346-00433-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://milkyeggs.com/math/localizations-of-a-ring-at-specific-elements-correspond-to-sections-of-a-sheaf/ | ### Localizations of a ring at specific elements correspond to sections of a sheaf
One fundamental observation in algebraic geometry is that there are a number of powerful correspondences between ring localizations and algebraic sheaves. Although elementary, this post will outline one such correspondence, which is that the localizations of a ring to a specific element can be exactly thought of as the sections of the sheaf of regular functions over the distinguished open set of . (Consider this post to be a self-directed refresher! It has been a long time since I learned this material.)
Suppose that is a finitely generated algebra over an algebraically closed field . One might naturally wonder if there is any geometric interpretation of the localization of this ring at a single one of its elements. Say we take (where ) and consider the local ring , i.e. quotients of the form for and nonnegative integers . If we consider and as polynomials to be evaluated, then these quotients only make sense if is always nonzero. We can in fact show that the local ring corresponds exactly to the sections of the sheaf over the distinguished open set .
Consider the natural homomorphism which maps a formal fraction to the regular function which is given by the actual quotient of polynomials . This mapping is straightforwardly injective: suppose that on ; then on and on the complement of by assumption, so everywhere. Hence the preimage of such a function is exactly 0.
Surjectivity is less trivial. Obviously, every function of the form is regular, but is every function in globally representable in the form ? Yes. Suppose that is a regular function in . I will sketch out the argument in broad strokes.
First, by the definition of a regular function, is representable as a quotient of polynomials, , in the neighborhood of every point . Distinguished open sets are in a sense the “smallest open sets” of the Zariski topology (i.e. they form a basis of the topology), so if we shrink these neighborhoods enough, they become distinguished open sets in their own right, say for every such point . These open neighborhoods together cover and, conversely, we may say that the variety is the intersection of every or, equivalently, the single variety .
We have essentially managed to glue together all our little pieces of information about the regular function into a single piece of information about , but how can we go further? Translating into the algebraic setting, we know of course that , but here we can use the Nullstellensatz to yield a more tractable result; in particular, it tells us that , or that (as an element of ) is a member of the radical of the ideal which is generated by finitely many elements .
Now we know that for finitely many . Let us define , recalling from before that each comes from the local representation of as near . Now taking any arbitrary point with a local representation of , we can check algebraically that and hence . Therefore is a valid global representation of over .
February 10th, 2023 | Posted in Math | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9663792848587036, "perplexity": 215.83783340237753}, "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-2023-14/segments/1679296949533.16/warc/CC-MAIN-20230331020535-20230331050535-00369.warc.gz"} |
https://winnerscience.com/2011/05/05/ | ## dot or scalar product of two unit vectors
Last time I have written about the properties of unit vectors that is:
Dot product of unit vectors :
i.i=1
j.j=1
k.k=1
i.j=j.k=k.i=0
Do you know how the above results come? If your answer is no, then let us discuss it:
I have already explained in my earlier articles that dot product or scalar product between two vectors A and B is given as:
A.B = AB cos θ
where θ is the angle between A and B. A and B are magnitudes of A and B.
As i the unit vector along x axis Continue reading “dot or scalar product of two unit vectors”
## Coordinate systems
Coordinate system is used to represent any point, say P(x, y, z ) in space. There are many methods by which this can be done, but there are three simple methods, which we will discuss in this article.
Types of coordinate systems are :
I. Cartesian or rectangular coordinate system.
II Cylindrical coordinate system
III Spherical coordinate system.
Today I will discuss briefly about the cartesian or rectangular coordinate system: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9717716574668884, "perplexity": 580.5712453466418}, "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-2021-25/segments/1623488286726.71/warc/CC-MAIN-20210621151134-20210621181134-00522.warc.gz"} |
https://www.physicsforums.com/threads/advanced-integral-question.703092/ | 1. Jul 26, 2013
### JBrandonS
1. The problem statement, all variables and given/known data
Find $\int_0^\infty e^{-\beta x^2 - \alpha/x^2}dx$
2. Relevant equations
This is from a mathematical techniques of engineering and physics course by Dr. Feynman. Methods it uses are generalizing the function, differentiation under the integral sign and complexifying.
3. The attempt at a solution
I have tried everything I know over the course of a few days. Taylor series expansion does not work. differentiation wrt alpha or beta do not work. I am unable to find a general term I can introduce to allow me to change the formula into something that is workable. I am completely at a loss here.
2. Jul 26, 2013
### Jufro
To make your life easier you can expand the integral from -∞ to ∞ taking advantage of the fact that it is an even function.
Then I would just expect that you would want to think of the real line as being part of the complex plane and think of a half circle of radius r (r → ∞) centered around the origin.
This should make it easy as zero is a pole and you can use the Cauchy integral theorem.
3. Jul 26, 2013
### JBrandonS
Thanks, I'll look into doing it that way. I have never used the Cauchy integral theorem so its going to take some time. Oh well, time to learn something new. :)
4. Jul 26, 2013
### lurflurf
After differentiating you need to change variables to express the derivative in terms of the original integral. Also you will need to do the integral for one value of alpha.
5. Jul 26, 2013
### JBrandonS
Could you elaborate a bit more? I understand what you are say I am just not seeing how it can be applied to this problem.
6. Jul 26, 2013
### haruspex
Two problems with that. How does the integrand behave as r tends to infinity? Why is there a pole anywhere?
7. Jul 26, 2013
### haruspex
lurflurf is suggesting writing the integrand as a product then integrating by parts. See what you get.
Edit: But I'm not having much luck with that, so maybe I misinterpreted lurflurf's suggestion.
Last edited: Jul 27, 2013
8. Jul 27, 2013
### jackmell
I don't think so. Let me see if I can help without getting in Lurflur's way:
Let's work the problem backwards. Suppose we have the differential equation:
$$\frac{dI}{d\alpha}=kI,\quad I(\alpha_0)=g$$
Then surely, $I(\alpha)=e^{k\alpha}+c$
Now, let's just cheat a little bit in the interest of learning how to do this problem. Hope that's ok. When I blindly plug in that integral into Mathematica, I get for the solution:
$$I=\frac{e^{-2\sqrt{\alpha}\sqrt{-\beta}}\pi}{2\sqrt{-\beta}}$$
Now compare that expression to the differential equation. That looks like we have to make some sort of substitution in the integral on the right of :
$$\frac{dI}{d\alpha}=\int_0^{\infty}\left(-\frac{1}{x^2}\right) e^{\beta x^2} e^{-\alpha/x^2} dx$$
using $u=f(\sqrt{-\beta},\sqrt{\alpha},x)$. Well there you go, that's what math is all about! You need to try things. How about:
$$u=\sqrt{-\beta}\sqrt{\alpha}x$$
Huh? No? Alright, how about $u=\frac{\sqrt{\alpha}}{\sqrt{-\beta}} x$? How about
$u=\frac{\sqrt{\alpha}}{\sqrt{-\beta}} 1/x$
Now, if these don't work, just keep trying other substitutions to see if you can get it into the expression:
$$\frac{dI}{d\alpha}=kI$$
Just worry about doing that much first. I don't have it yet either. I got it close though so I think we're on the right track and I just need to fiddle with it a little more, like you.
Edit: Ok, think I made a slight mistake. The DE we get I believe will be in the form:
$$\frac{dI}{d\alpha}=h(\alpha) I$$
which is still first-order linear that we can easily solve. See what you get.
Last edited: Jul 27, 2013
9. Jul 27, 2013
### lurflurf
jackmell's suggestion is good, but I think it is hard to guess the form without knowing the answer in advance.
The backwards method would be to change variable u=sqrt(a/b)/x then recognize the integral and its derivative are related by multiplication by a constant so e^(k x) is the integral with k=sqrt(a/b)
so first find
$$\left. \int_0^\infty \! e^{-\beta x^2-\alpha /x^2} \,\mathrm{dx} \right] _{\alpha=0}= \int_0^\infty \! e^{-\beta x^2-0 /x^2} \,\mathrm{dx}$$
then show
$$\dfrac{d}{d \alpha}\int_0^\infty \! e^{-\beta x^2-\alpha /x^2} \,\mathrm{dx}=\int_0^\infty \! (-1/x^2) e^{-\beta x^2-\alpha /x^2} \,\mathrm{dx}$$
note it is not always possible to move the derivative inside the integral, but it is in this case as the convergence is rapid.
change variables to see that
$$\dfrac{d}{d \alpha}\int_0^\infty \! e^{-\beta x^2-\alpha /x^2} \,\mathrm{dx}=\int_0^\infty \! (-1/x^2) e^{-\beta x^2-\alpha /x^2} \,\mathrm{dx}=k \int_0^\infty \! e^{-\beta x^2-\alpha /x^2} \,\mathrm{dx}$$
for some k
finally deduce the integral
10. Jul 27, 2013
### vanhees71
The idea was already good. Let's treat the problem as differential equation initial-value problem as a function of $\alpha$. We have from #1 the function
$$I(\alpha)=\int_0^{\infty} \mathrm{d} x \exp(-\beta x^2-\alpha/x^2).$$
Now take the derivative
$$I'(\alpha)=-\int_0^{\infty} \mathrm{d} x \frac{1}{x^2} \exp(-\beta x^2-\alpha/x^2).$$
In this integral substitute
$$x=\sqrt{\frac{\alpha}{\beta}}1/u,$$
which leads you to the equation
$$I'(\alpha)=-\sqrt{\frac{\beta}{\alpha}}I(\alpha).$$
This you can easily solve by separation. Together with the initial condition (Gaussian integral!)
$$I(\alpha=0)=\frac{\sqrt{\pi}}{2 \sqrt{\beta}}$$
you find the unique result
$$I(\alpha)=\frac{\sqrt{\pi} \exp(-2\sqrt{\alpha \beta})}{2 \sqrt{\beta}}.$$
11. Jul 27, 2013
### JBrandonS
You rock, Never thought of that substitution.
Thanks for all the help guys. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "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": 0.9705204367637634, "perplexity": 431.7841190502615}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2018-13/segments/1521257646914.32/warc/CC-MAIN-20180319120712-20180319140712-00734.warc.gz"} |
http://bln.curtisbright.com/2013/03/11/minimal-polynomial-misconception/ | Minimal polynomial misconception
Yesterday I had a discussion with a friend about computing minimal polynomials. For example, say you are given algebraic numbers $\alpha$ and $\beta$ as specified by minimal polynomials which have degrees $n$ and $m$. How do you compute the minimal polynomial of $\alpha+\beta$, for example?
The method I was taught, which seems to be fairly standard (e.g., see Dummit and Foote Chapter 13.2) is the following:
Multiply $\alpha+\beta$ by $\alpha^i\beta^j$ for each $i=0{,}~1{,}~\dotsc{,}~n-1$ and $j=0{,}~1{,}~\dotsc{,}~m-1$, and use the minimal polynomials of $\alpha$ and $\beta$ to reduce the resulting expressions to linear combinations of $\alpha^i\beta^j$ (again with $0\leq i<n$ and $0\leq j<m$). In other words, what you are doing is computing a matrix $M$ which satisfies
$(\alpha+\beta)\left[\alpha^i\beta^j\right]_{i,j} = M\left[\alpha^i\beta^j\right]_{i,j}$
where $[\alpha^i\beta^j]_{i,j}$ is a column vector containing the $\alpha^i\beta^j$.
From this we see that $\alpha+\beta$ is an eigenvalue of $M$ and therefore is a root of the characteristic polynomial $p_M$ of $M$. If $p_M$ is irreducible then it will be the minimal polynomial of $\alpha+\beta$, but in general the minimal polynomial will divide $p_M$, and so it will be necessary to factor $p_M$.
However, once $p_M$ has been factored, how does one tell which factor is the required minimal polynomial? The obvious answer is to evaluate each factor at $\alpha+\beta$ and see which one gives zero. “You could do that numerically”, my friend says, and I respond with “…or symbolically”. But then he asks if I will always be able to determine if the symbolic expression is zero or not. Well, I hadn’t thought of that, and I admitted I wasn’t sure, but claimed “anything you can do numerically I can do symbolically!”
I spent several hours yesterday trying to solve that problem, but eventually had to go to bed. This morning, after looking in Cohen I found the following passage:
…it does not make sense to ask which of the irreducible factors $\alpha+\beta$ is a root of, if we do not specify embeddings in $\mathbb{C}$…
Wait, what? I got excited as I realized I had just uncovered a misconception of mine!
Note that if the conjugates of $\alpha$ are $\alpha_i$ then they are all “symbolically identical”: $\mathbb{Q}(\alpha_i)$ is isomorphic for each conjugate. From that, I had assumed that the values of $\alpha_i+\beta_j$ would also be symbolically identical for all conjugates of $\alpha$ and $\beta$. Not true! As a trivial counterexample, if $\alpha$ is a root of $x^3-2$ and $\beta$ is a root of $x^3+2$ then two possible values for $\alpha+\beta$ are $0$ and $\sqrt[3]{2}\sqrt{3}i$, and these have very different algebraic behaviour.
So the whole problem of computing the minimal polynomial of $\alpha+\beta$ was not well defined unless you specify which roots $\alpha$, $\beta$ you are talking about—for example, by specifying them numerically. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "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": 0.9204105734825134, "perplexity": 136.85051027997653}, "config": {"markdown_headings": false, "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-2020-05/segments/1579251783000.84/warc/CC-MAIN-20200128184745-20200128214745-00076.warc.gz"} |
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# Oxygen content
## Video transcript
Let's talk about oxygen content. And I'm going to actually spell it out two ways. One is the full word oxygen content, or the full term. And I'm also going to give you the shorthand, way you might see. Sometimes it's written this way, CaO2. And the C is the content. The little a is arterial. And the O2 is oxygen. So what it means exactly and the way we think of it as, how much oxygen is there. How much is there? And we measure it in milliliters, per 100 milliliters of blood. So per 100 ml. and. Sometimes you might see deciliter instead of milliliter. Let me just quickly jot that down. That equals 1 deciliter per 100 milliliters of blood. So this is the definition. Now, let's use this definition right away. Let's see if you can think through this idea. So let's imagine I go down and I decide to get 1 pint of blood taken from my left arm. Let's, instead of bint, let me write pint. And this is my left arm. And let's say I'm in a huge rush this day. So I decide that I also want to get another needle stuck in my right arm. And they also draw blood out of my right arm, at the same moment, the same time. So the same kind of blood, same hemoglobin concentration, and same amount of oxygen in my lungs when I was getting the blood drawn. Except for some reason, maybe this needle, this second one was larger. And they were able to get more blood out-- 2 pints. Now, some smart wise guy walks by and says, hey, which side, your left or your right, were you're able to get a higher oxygen content from. Now, just looking at the picture, you might be tempted to say, well, oxygen content. Sounds like the right side is the winner. But actually, this is kind of a trick question because it's per 100 milliliters. So you got remember, it's a certain volume that we're thinking about. And in this case, since we know that the blood was drawn at the same moment from my two arms. And I have no reason to believe that the left versus right had a higher oxygen saturation. I would say, actually probably the two had the same oxygen content. That would be my guess based on this set up. So that's one important thing to remember that it's per 100 ml. So let's just keep that in mind. And now let me actually just jot down for you the exact equation, kind of the formula. If want to mathematically calculate oxygen content, how would that look? Well, CaO2 is quicker to write. So let me just jot that down. And the units on this are milliliters of oxygen per, I said, 100 milliliters of blood. So these are the units here. And this is going to equal-- to figure this out, I need to know the hemoglobin concentration. And there it's the grams of hemoglobin per 100 milliliters of blood. And then, I have to multiply this by a constant. And the constant is 1.34. And what that number is, is it's telling me the milliliters of oxygen that I can expect to bind for each gram of hemoglobin. So that's actually quite a nice little number to have handy because now you can see that the units are about to cancel. This will cancel with this. And I end up with our correct units. But there's one more thing I have to add in here which is the oxygen saturation. Remember, this O2 saturation. And if I know the O2 saturation, remember, there's this nice little curve. This is O2 saturation. And if I'm looking at just the arterial side, I could write, S little a O2. And I could compare to the partial pressure in the arterial side of oxygen. And remember, we have these little S-shaped curves. These S-shaped curves. And all I want to point out is that, for any increase in my PaO2, in the partial pressure of oxygen, I'm going to have an increase in the O2 saturation. So there's an actual relationship there. And we usually measure this in percentage. Percentage of oxygen that is bound to hemoglobin. And so this is the same thing here, as a certain percentage. So this whole top part of the formula, then, this whole bit in my brackets really is telling me about hemoglobin bound to oxygen. Now remember, that's not the only way that oxygen actually travels in the blood. Let me write out this second way that oxygen likes to get around. And the second way is when it dissolves in the blood. So this is all going to be plus. And the second part of the equation is the partial pressure of oxygen. And this is measured in millimeters of mercury. So that's the unit. And this is times, now this is another constant, 0.003. And then, keep track of the units here because we have to end up with these units. So you know everything has to cancel out to end up with that. So I have milliliters of oxygen on top. And I'm going to want to cancel my millimeters of mercury. So take that times 100 milliliters of blood. So these are the units on the bottom. And they end up the same as we just worked through. We've got this crosses out with that. And my units are going to end up perfect. And this bottom bit, that I'm going to put in purple brackets. This bit tells me about dissolved oxygen. So I have my oxygen bound to hemoglobin. And I have my dissolved oxygen. These are the two parts of my formula. So let me actually just quickly, before I move on, circle in blue, then, the important parts that I want you keep your eyeballs on. There is the total O2 content, hemoglobin, oxygen saturation, and partial pressure of oxygen. And remember, this guy influences this guy. And we saw that on the O2 curve that I just drew. Let me just bring it up again, so I can remind you what I'm talking about. In this graph, you can see how the two are related. There's a very nice relationship between the two. So this is my formula for calculating the total oxygen content. So let's actually use this formula. Let's think through this. And when I think through it, I always go through all of my four variables. Let me just jot them down here. So we keep track of them. Let's do PaO2, SaO2, and then hemoglobin and the total oxygen content. These are my four variables. Now, let's do a little problem together. Let me make a little bit of space. And let's say I have two little containers. And the first container, this first one is full of blood. Here's a B for blood. And here's a second container full of plasma. Remember, plasma is a part of the blood. But it's not all of the blood. Plasma specifically does not have any red blood cells or any hemoglobin. So let me just write that down. No hemoglobin in the plasma side. Just to make sure we don't lose track of that fact. Now, plasma is yellow colored. So let me just make it yellow colored here. Make sure we clearly see that that's plasma. And blood I'm going to keep as a red color. So now, we have our two containers full of plasma and blood. So now, let's say, I decide to increase the partial pressure of oxygen in the air. So it's going to diffuse in here. And it's going to diffuse in here. So I increase the partial pressure oxygen in the air. And it's going to diffuse into those two liquids. It's going to dissolve into those liquids. So my question is, as we go through one by one, each of these four variables, I want you to think through if they go up, if they go down, or if they stay the same. So let's start with the first one, PaO2. Well, if the oxygen is going to diffuse into those liquids, then I would say the partial pressure of oxygen in the liquid would go up. Now, it's a little bit confusing to use the words PaO2 in this case, or even down here, CaO2 or SaO2. Because we're not really talking about arterial blood here. We're just talking about blood. And we're not talking about arterial plasma. We're just talking a plasma because there's no artery connected to these two tanks of fluid. But the concept is the same. So the partial pressure of oxygen is going to go up in the blood. And it's going to go up in the plasma because it just dissolves into those liquids. Now, what about saturation of oxygen? Well, O2 saturation goes up in the blood. Remember, there's a relationship, we said, between PaO2 and oxygen saturation. So it's going to cause the SaO2 to go up here. Whereas on the plasma side there is no hemoglobin. So of course, there's going to be no change here. I would say, not applicable because there is no hemoglobin. So how could you have an oxygen saturation curve for hemoglobin? Now, what about the third variable, hemoglobin concentration? Remember, that was grams per 100 milliliters of blood. Well, I'm not talking about adding or subtracting hemoglobin. So there should be no change here. I'll write, no change. And on the other side of the plasma side, again, there is no hemoglobin. So it's not going to affect that at all. It's not really applicable. Plasma, again, does not have hemoglobin. So in terms of the total oxygen content, or the CaO2, what I expected to go up in the blood, definitely, it's definitely going to go up because the dissolved part of the equation goes up. But even the hemoglobin bound to oxygen part of the equation goes up because we said, the SaO2 went up. That's an interesting point. On the other side on the plasma side, it also increases. But only a little bit because here you only have the contribution from the PaO2. You have no contribution from any of the oxygen bound to hemoglobin because, again, there is no hemoglobin. So this problem illustrates some of the ideas, specifically around trying to tie-in an increase in the partial pressure of oxygen to how that could affect the saturation of oxygen. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8154330849647522, "perplexity": 535.4193551293603}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "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-2022-21/segments/1652662543264.49/warc/CC-MAIN-20220522001016-20220522031016-00264.warc.gz"} |
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# Gaussian q-distribution
In mathematical physics and probability and statistics, the Gaussian q-distribution is a family of probability distributions that includes, as limiting cases, the uniform distribution and the normal (Gaussian) distribution. It was introduced by Diaz and Teruel,[clarification needed] is a q-analogue of the Gaussian or normal distribution.
The distribution is symmetric about zero and is bounded, except for the limiting case of the normal distribution. The limiting uniform distribution is on the range -1 to +1.
## Definition
The Gaussian q-density.
Let q be a real number in the interval [0, 1). The probability density function of the Gaussian q-distribution is given by
${\displaystyle s_{q}(x)={\begin{cases}0&{\text{if }}x<-\nu \\{\frac {1}{c(q)}}E_{q^{2}}^{\frac {-q^{2}x^{2}}{[2]_{q}}}&{\text{if }}-\nu \leq x\leq \nu \\0&{\mbox{if }}x>\nu .\end{cases}}}$
where
${\displaystyle \nu =\nu (q)={\frac {1}{\sqrt {1-q}}},}$
${\displaystyle c(q)=2(1-q)^{1/2}\sum _{m=0}^{\infty }{\frac {(-1)^{m}q^{m(m+1)}}{(1-q^{2m+1})(1-q^{2})_{q^{2}}^{m}}}.}$
The q-analogue [t]q of the real number ${\displaystyle t}$ is given by
${\displaystyle [t]_{q}={\frac {q^{t}-1}{q-1}}.}$
The q-analogue of the exponential function is the q-exponential, Ex
q
, which is given by
${\displaystyle E_{q}^{x}=\sum _{j=0}^{\infty }q^{j(j-1)/2}{\frac {x^{j}}{[j]!}}}$
where the q-analogue of the factorial is the q-factorial, [n]q!, which is in turn given by
${\displaystyle [n]_{q}!=[n]_{q}[n-1]_{q}\cdots [2]_{q}\,}$
for an integer n > 2 and [1]q! = [0]q! = 1.
The Cumulative Gaussian q-distribution.
The cumulative distribution function of the Gaussian q-distribution is given by
${\displaystyle G_{q}(x)={\begin{cases}0&{\text{if }}x<-\nu \\[12pt]\displaystyle {\frac {1}{c(q)}}\int _{-\nu }^{x}E_{q^{2}}^{-q^{2}t^{2}/[2]}\,d_{q}t&{\text{if }}-\nu \leq x\leq \nu \\[12pt]1&{\text{if }}x>\nu \end{cases}}}$
where the integration symbol denotes the Jackson integral.
The function Gq is given explicitly by
${\displaystyle G_{q}(x)={\begin{cases}0&{\text{if }}x<-\nu ,\\\displaystyle {\frac {1}{2}}+{\frac {1-q}{c(q)}}\sum _{n=0}^{\infty }{\frac {q^{n(n+1)}(q-1)^{n}}{(1-q^{2n+1})(1-q^{2})_{q^{2}}^{n}}}x^{2n+1}&{\text{if }}-\nu \leq x\leq \nu \\1&{\text{if}}\ x>\nu \end{cases}}}$
where
${\displaystyle (a+b)_{q}^{n}=\prod _{i=0}^{n-1}(a+q^{i}b).}$
## Moments
The moments of the Gaussian q-distribution are given by
${\displaystyle {\frac {1}{c(q)}}\int _{-\nu }^{\nu }E_{q^{2}}^{-q^{2}x^{2}/[2]}\,x^{2n}\,d_{q}x=[2n-1]!!,}$
${\displaystyle {\frac {1}{c(q)}}\int _{-\nu }^{\nu }E_{q^{2}}^{-q^{2}x^{2}/[2]}\,x^{2n+1}\,d_{q}x=0,}$
where the symbol [2n − 1]!! is the q-analogue of the double factorial given by
${\displaystyle [2n-1][2n-3]\cdots [1]=[2n-1]!!.\,}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 13, "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": 0.9361697435379028, "perplexity": 1882.9839803963523}, "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-2021-04/segments/1610703537796.45/warc/CC-MAIN-20210123094754-20210123124754-00229.warc.gz"} |
http://utsv.net/solid-mechanics/7-rate-form-constitutive-expressions/time-stepping-algorithm | # Time-Stepping Algorithm
In hypoelasticity, we required the use of the so-called “Jaumann rate of Cauchy Stress”, , because is not work conjugate with or , while is. However, we still need , because is the stress tensor that we need. The Cauchy stress (and its rate) have a well-understood physical meaning (whereas and do not have physical meaning) – namely, is defined in the “spatial” coordinate system. Recall once more that this is necessary in order to have a consistent frame of reference for all of the elements in the finite element simulation. Thus, before we can consider the problem “solved” for any particular instance in time, we need to add an additional “step” to our solution procedure that obtains from . We can do this by simply solving for from our previous expression for the Jaumann rate of Cauchy Stress (eq. 3 in Section 7: Hypoelasticity). Thus, during each time step, the Jaumann rate is the stress that is used for the constitutive relationship, but the Cauchy stress is what is needed for the equation of motion (e.x. stress equilibrium).
It turns out that this same multi-step time-stepping “algorithm” will work for linear infinitesimal elasticity as well. We already mentioned that is analogous to the linear infinitesimal stress tensor, and is analogous to the linear infinitesimal strain tensor, . In other words, the work-conjugate pair – defined in material coordinates – that is used in linear infinitesimal elasticity, can be thought of as a simplified version of . and we know to be similarly work-conjugate and invariant to rigid body rotation.
Thus, one possible expression in linear infinitesimal elasticity that is analogous to our hypoelastic expression (eq. 7 in Section 7: Hypoelasticity) could be:
where is a fourth-order tensor that relates linear infinitesimal stress, , to linear infinitesimal strain, , as we will see in the next chapter.
Obtaining the stress in spatial coordinates can be accomplished via eq. 2 in Section 4: Alternative Measures of Stress, namely, . After performing this time derivative, we could then rearrange to find the rate of Cauchy stress, , in terms of . However, it may be desirable to take a different approach in order to avoid having to explicitly calculate the deformation gradient, , or its time-derivative .
Additionally, let’s keep in mind that the linear infinitesimal stress and strain are not exactly equal to and , respectively. Thus, obtaining the Cauchy stress from the expression for (eq. 2 in Section 4: Alternative Measures of Stress) may not be appropriate here because the linear infinitesimal stress is not exactly equal to .
note: We cannot always simply replace the linear infinitesimal stress and strain with and . Although this is what we did, above, it was not derived because it is not mathematically correct. The above expression is shown primarily for illustrative purposes. It turns out that it is possible to relate and using , exactly, but this requires that we use the so-called Saint Venant-Kirchhoff strain energy density function.
Let’s instead consider our previous definitions of the Truesdell and the Jaumann rate, which, for infinitesimal elasticity, state that:
Here, we should note that the term was eliminated since for infinitesimal deformations.
Since and , we can see that:
(1)
From Chapter 3, we know that eq. 1 conveniently reduces (with reasonable approximation) to:
(2)
Or,
(3)
In eq. 2, we recall that is the “Rate of Deformation Tensor.”
In linear infinitesimal elasticity, is often simply referred to as the “strain rate.”
Some authors write the constitutive relationship for linear infinitesimal elasticity as . The physical nature of and was illustrated in the example at the end of Section 4: Alternative Measures of Stress, where we saw that the two strain measures gave different results, even for the infinitesimal deformation case. Ultimately, however, one can argue that we are free to choose either stress-strain pair in linear infinitesimal elasticity. Applying the Jaumann operator to either constitutive expression results in the equation 2. Thus, our “time-stepping algorithm,” which we will see shortly, is independent of our “interpretation” of linear infinitesimal stress and strain.
The Kirchhoff stress, , is defined as . Sometimes is used in place of , in eq. 2 for example, so that the small deformation approximation of does not need to be used.
Obtaining the stress rate in spatial coordinates simply requires that we solve for from eq. 3 in Section 7: Hypoelasticity, namely,
(4)
An alternative proof of eq. 4, for infinitesimal elasticity, is given in Appendix D.1
Thus, our constitutive expressions, in rate-form, for both linear infinitesimal elasticity and hyperelasticity (hypoelasticity), involve the Jaumann rate. For hypoelasticity, we had eq. 7 in Section 7: Hypoelasticity, while for linear infinitesimal elasticity, we have eq. 3. The following time-stepping algorithm, which allows us to obtain Cauchy (“spatial”) stress is identical for both hyperelasticity and linear infinitesimal elasticity.
note that the bases do not always remain orthogonal when there is shear (recall the second figure in Section 6: Phenomenological and Micromechanical Models, for example). The Jaumann rate in eq. 4 assumes orthogonal bases, which is a simplification.
General Framework For Problem Solving
1. Use an appropriate work-conjugate constitutive relationship:
or
2. Obtain a spatial representation of stress by first finding from eq. 4, and then finding from
3. Solve the equation of motion for the next increment in time:
The majority of this text is devoted to “1” and “2” (in a theoretical sense). Good course sequences on FEA spend a great deal of time on the implementation of these steps – in particular on step “3.” Recall that the expression given in step “3” was derived in Chapter 4.
The above “time-stepping algorithm” is just one possibility. The purpose here is merely to show one particular methodology that FEA software use for solving real-life problems. This methodology has only been shown to apply to isotropic elastic materials. Different time-stepping algorithms are surely used for other situations.
Plasticity
Obtaining Cauchy stress from the Jaumann rate is what is most often done by FEA software, even though the Truesdell rate form is better (exact) for elasticity. The Jaumann rate is known to have some problems with shear behavior due to the assumption of orthogonal bases [Valanis]. In addition, for highly compressible materials, the Truesdell rate can give significantly better results [Bazant]. Perhaps, then, the main reason that the Jaumann rate is preferred over the Truesdell rate (and other rates) has to do with plasticity.
In plasticity, many assumptions are made. The Truesdell rate turns out not to be good for plasticity, in general, because the dilatational (volume-changing) terms that relate and the Truesdell rate are ignored in plasticity. So, suffice to say, plasticity is a complex topic that will not be covered in this introductory text, but the Jaumann rate, , is often used in FEA for solid elements, because it is computationally efficient and handles plasticity better than the Truesdell rate.
A general overview of plasticity equations used in FEA can be found in [Belytschko] and in [Simo]. Where finite strain elasticity and plasticity (e.x. rubber elastoplasticity) are considered, good theoretical references include [Lee], [Lubarda], and [Volokh]. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.953247606754303, "perplexity": 787.6621303326589}, "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-2019-09/segments/1550247481122.31/warc/CC-MAIN-20190216210606-20190216232606-00267.warc.gz"} |
https://www.physicsforums.com/threads/potential-energy-per-unit-length-in-a-string-sin-wave.705259/ | # Potential energy per unit length in a string (sin wave)
1. Aug 11, 2013
### PsychonautQQ
1. The problem statement, all variables and given/known data
Given that the stretched length of a string is Δx(1+1/2(dy/dx)^2) show that the potential energy per unit length is equal to
1/2F(dy(x,t)/dx)^2
2. Relevant equations
potential energy = kx^2
cos(kx-wt)
idk really...
3. The attempt at a solution
The fact that the stretched length equals
Δx(1+1/2(dy/dx)^2)
can be derived from the fact that the strings stretched length is equal to (x^2+x(dy/dx)^2)^1/2
and then simplified with binomial expansion.
according to my answer key, the first step to the solution of solving potential energy per unit length is understanding that it is equal to
(FΔx(1+1/2(dy/dx)^2)-Δx) / Δx
The part about this step that I don't understand is why is the -Δx term in the numerator? Wouldn't the work per unit length just equal (Force * the amount it is stretched / Δx)? which would equal
FΔx(1+1/2(dy/dx)^2) / Δx.
yeah.. I don't understand why the -Δx is in the numerator i guess sums up my concerns.
2. Aug 11, 2013
### TSny
Should that be F(Δx(1+1/2(dy/dx)^2)-Δx) / Δx?
Yes, that's right. But "the amount it is stretched" is not the same as "the stretched length". For example, suppose I have a spring that has an unstretched length of 20 cm. (This is the length from one end of the spring to the other end when the spring is not stretched.) Then I stretch it until it has a "stretched length" of 30 cm. (This is the length from one end of the spring to the other when it is stretched.) The "amount it is stretched" would be 10 cm.
3. Aug 11, 2013
### PsychonautQQ
Cool, i'm still a little confused how that is represented in the equation by the term -Δx
4. Aug 11, 2013
### TSny
Δx is the length of a section of string before it was stretched.
Δx(1+y'2/2) is the length of the same section after it has been stretched.
So, how would you write an expression for the amount the section has been stretched?
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1. CMB Online first
Otsubo, Noriyuki
Homology of the Fermat tower and universal measures for Jacobi sums We give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson's adelic beta functions, in a similar manner to Ihara's definition of $\ell$-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve. Keywords:Fermat curves, Ihara-Anderson theory, Jacobi sumsCategories:11S80, 11G15, 11R18
2. CMB 2009 (vol 53 pp. 58)
Dąbrowski, Andrzej; Jędrzejak, Tomasz
Ranks in Families of Jacobian Varieties of Twisted Fermat Curves In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series. Keywords:Fermat curve, Jacobian variety, elliptic curve, canonical heightCategories:11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15
top of page | contact us | privacy | site map | | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8568849563598633, "perplexity": 1671.8343306748582}, "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-2016-18/segments/1461860114285.32/warc/CC-MAIN-20160428161514-00080-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.mathdoubts.com/sin-angle-difference-identity/ | # Sin of Angle difference formula
## Formula
$\sin{(A-B)}$ $\,=\,$ $\sin{A}\cos{B}$ $-$ $\cos{A}\sin{B}$
It is called sine of angle difference identity. It states that the sin of subtraction of two angles is equal to the subtraction of products of sine and cosine of both angles.
### Introduction
Sine functions are often appeared with subtraction of two angles. In order to deal them, sine of difference of two angles identity is derived in trigonometry.
#### Mathematical form
The sine of difference of two angles formula can be written in several ways, for example $\sin{(A-B)}$, $\sin{(x-y)}$, $\sin{(\alpha-\beta)}$, and so on but it is popularly written in the following three mathematical forms.
$(1) \,\,\,\,\,\,$ $\sin{(A-B)}$ $\,=\,$ $\sin{A}\cos{B}$ $-$ $\cos{A}\sin{B}$
$(2) \,\,\,\,\,\,$ $\sin{(x-y)}$ $\,=\,$ $\sin{x}\cos{y}$ $-$ $\cos{x}\sin{y}$
$(3) \,\,\,\,\,\,$ $\sin{(\alpha-\beta)}$ $\,=\,$ $\sin{\alpha}\cos{\beta}$ $-$ $\cos{\alpha}\sin{\beta}$
#### Use
The sine of angle difference property is mainly used in two cases.
1. To expand sin of difference of two angles as the subtraction of products of sine and cosine functions.
2. To simplify the subtraction of products of sine and cosine functions as the sine of subtraction of two angles.
#### Proof
Learn how to derive the angle difference sin identity in mathematical form by geometrical approach in trigonometry.
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https://cs.stackexchange.com/questions/56252/problem-with-update-in-dynamic-bayesian-networks | # Problem with update in Dynamic Bayesian Networks
Consider the following Bayesian network:
I want to impose constraints that state that a node can only be true (1) if at least one of its parents are true (1). So, for node $C$, the constraint takes the form $P(C=1|A=0,B=0) = 0$.
Now I want to take into account time. So it seems like the logical thing to do is to construct a dynamic Bayesian network from the above structure, as follows:
So, the update for a given node, say $C$ to updated value $C'$ obeys the following probability $P(C'=1|A,B,C)$. The leaf nodes, say $A$, obeys the update $P(A'=1|A)$.
Issue: I'm a little bit confused on how to ensure that the constraints I wish to impose are held after the variables are updated. For example, let $P(A'=1|A=1) = 1-\alpha$ and $P(B'=1|B=1) = 1-\beta$, where $\alpha,\beta>0$. This allows for the possibility of the realized values of both $A'$ and $B'$ becoming false (0), where $C'$ could be true, violating the constraint stated earlier.
Question: How do I ensure that the constraints are maintained after the variables are updated? Under the current set up, it does not seem like the dynamic Bayesian network shown above is capable of doing this.
Idea: Does the update need to be performed starting from the leaf nodes? That is, update $A'$ and $B'$ first, then once their values have been realized, update $C'$, and so on, down the DAG? Is this the standard way of doing this?
Update: Would the following representation allow for the constraints to be represented?
Update #2: Does anyone have any ideas? I'm really stumped on this. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8905877470970154, "perplexity": 223.2960161390843}, "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-2019-26/segments/1560627998250.13/warc/CC-MAIN-20190616142725-20190616164540-00027.warc.gz"} |
http://owl.fish.washington.edu/Athaliana/quast_results/results_2018_05_10_15_04_07/report.html | # QUAST
Quality Assessment Tool for Genome Assemblies by CAB
Contigs are ordered from largest (contig #1) to smallest. FRCurve: Y is the total number of aligned bases divided by the reference length, in the contigs having the total number of at most X. Contigs are broken into nonoverlapping 100 bp windows. Plot shows number of windows for each GC percentage. Plot shows number of contigs with GC percentage in a certain range.
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{{ qualities }}
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{ "# contigs" : "is the total number of contigs in the assembly.", "Largest contig" : "is the length of the longest contig in the assembly.", "Total length" : "is the total number of bases in the assembly.", "Reference length" : "is the total number of bases in the reference.", "# contigs (>= 0 bp)" : "is the total number of contigs in the assembly that have size greater than or equal to 0 bp.", "Total length (>= 0 bp)" : "is the total number of bases in the contigs having size greater than or equal to 0 bp.", "N50" : "is the contig length such that using longer or equal length contigs produces half (50%) of the bases of the assembly. Usually there is no value that produces exactly 50%, so the technical definition is the maximum length x such that using contigs of length at least x accounts for at least 50% of the total assembly length.", "NG50" : "is the contig length such that using longer or equal length contigs produces half (50%) of the bases of the reference genome. This metric is computed only if a reference genome is provided.", "N75" : "is the contig length such that using longer or equal length contigs produces 75% of the bases of the assembly. Usually there is no value that produces exactly 75%, so the technical definition is the maximum length x such that using contigs of length at least x accounts for at least 75% of the total assembly length.", "NG75" : "is the contig length such that using longer or equal length contigs produces 75% of the bases of the reference genome. This metric is computed only if a reference genome is provided.", "L50" : "is the minimum number of contigs that produce half (50%) of the bases of the assembly. In other words, it's the number of contigs of length at least N50.", "LG50" : "is the minimum number of contigs that produce half (50%) of the bases of the reference genome. In other words, it's the number of contigs of length at least NG50. This metric is computed only if a reference genome is provided.", "L75" : "is the minimum number of contigs that produce 75% of the bases of the assembly. In other words, it's the number of contigs of length at least N75.", "LG75" : "is the minimum number of contigs that produce 75% of the bases of the reference genome. In other words, it's the number of contigs of length at least NG75. This metric is computed only if a reference genome is provided.", "NA50" : "is N50 where the lengths of aligned blocks are counted instead of contig lengths. I.e., if a contig has a misassembly with respect to the reference, the contig is broken into smaller pieces. This metric is computed only if a reference genome is provided.", "NGA50" : "is NG50 where the lengths of aligned blocks are counted instead of contig lengths. I.e., if a contig has a misassembly with respect to the reference, the contig is broken into smaller pieces. This metric is computed only if a reference genome is provided.", "NA75" : "is N75 where the lengths of aligned blocks are counted instead of contig lengths. I.e., if a contig has a misassembly with respect to the reference, the contig is broken into smaller pieces. This metric is computed only if a reference genome is provided.", "NGA75" : "is NG75 where the lengths of aligned blocks are counted instead of contig lengths. I.e., if a contig has a misassembly with respect to the reference, the contig is broken into smaller pieces. This metric is computed only if a reference genome is provided.", "LA50" : "is L50 where aligned blocks are counted instead of contigs. I.e., if a contig has a misassembly with respect to the reference, the contig is broken into smaller pieces.", "LGA50" : "is LG50 where aligned blocks are counted instead of contigs. I.e., if a contig has a misassembly with respect to the reference, the contig is broken into smaller pieces.", "LA75" : "is L75 where aligned blocks are counted instead of contigs. I.e., if a contig has a misassembly with respect to the reference, the contig is broken into smaller pieces.", "LGA75" : "is LG75 where aligned blocks are counted instead of contigs. I.e., if a contig has a misassembly with respect to the reference, the contig is broken into smaller pieces.", "Average %IDY" : "is the average of alignment identity percent (Nucmer measure of alignment accuracy) among all contigs.", "# misassemblies" : "is the number of positions in the assembled contigs where the left flanking sequence aligns over 1 kbp away from the right flanking sequence on the reference (relocation) or they overlap on more than 1 kbp (relocation) or flanking sequences align on different strands (inversion) or different chromosomes (translocation).", "# misassembled contigs" : "is the number of contigs that contain misassembly events.", "Misassembled contigs length" : "is the number of total bases contained in all contigs that have one or more misassemblies.", "# relocations" : "is the number of relocation events among all misassembly events. Relocation is a misassembly where the left flanking sequence aligns over 1 kbp away from the right flanking sequence on the reference, or they overlap by more than 1 kbp and both flanking sequences align on the same chromosome.", "# translocations" : "is the number of translocation events among all misassembly events. Translocation is a misassembly where the flanking sequences align on different chromosomes.", "# interspecies translocations" : "is the number of interspecies translocation events among all misassembly events. Interspecies translocation is a misassembly where the flanking sequences align on different references (based on alignments to the combined reference).", "# inversions" : "is the number of inversion events among all misassembly events. Inversion is a misassembly where it is not a relocation and the flanking sequences align on opposite strands of the same chromosome.", "# local misassemblies" : "is the number of local misassemblies. We define a local misassembly breakpoint as a breakpoint that satisfies these conditions:
1. Two or more distinct alignments cover the breakpoint.
2. The gap between left and right flanking sequences is less than 1 kbp.
3. The left and right flanking sequences both are on the same strand of the same chromosome of the reference genome.
", "# scaffold gap size misassemblies" : "is the number of scaffold gap size misassemblies. We define scaffold gap size misassembly as a breakpoint where the flanking sequences combined in scaffold on the wrong distance. These misassemblies are not included in the total number of misassemblies. ", "# possibly misassembled contigs": "is the number of contigs that contain large unaligned fragment (default min length is 500 bp) and thus could possibly contain interspecies translocation with unknown reference.", "# possible misassemblies" : "is the number of putative interspecies translocations in possibly misassembled contigs if each large unaligned fragment is supposed to be a fragment of unknown reference.", "# intergenomic misassemblies" : "is the number of all found and putative (possible) interspecies translocations.", "# structural variations" : "is the number of misassemblies matched with structural variations.", "# unaligned mis. contigs" : "is the number of contigs that have the number of unaligned bases more than 50% of contig length and a misassembly event in their aligned fragment. Note that such misassemblies are not counted in # misassemblies and other misassemblies statistics.", "# fully unaligned contigs" : "is the number of contigs that have no alignment to the reference sequence.", "Fully unaligned length" : "is the total number of bases contained in all fully unaligned contigs.", "# partially unaligned contigs" : "is the number of contigs that have at least one alignment to the reference sequence but also have at least one unaligned fragment of length ≥ unaligned-part-size threshold.", "Partially unaligned length" : "is the total number of unaligned bases in all partially unaligned contigs.", "# ambiguous contigs" : "is the number of contigs that have reference alignments of equal quality in multiple locations on the reference.", "Ambiguous contigs length" : "is the total number of bases contained in all ambiguous contigs.", "Genome fraction (%)" : "is the total number of aligned bases in the reference, divided by the genome size. A base in the reference genome is counted as aligned if there is at least one contig with at least one alignment to this base. Contigs from repeat regions may map to multiple places, and thus may be counted multiple times in this quantity.", "GC (%)" : "is the total number of G and C nucleotides in the assembly, divided by the total length of the assembly.", "Reference GC (%)" : "is the total number of G and C nucleotides in the reference, divided by the total length of the reference.", "# mismatches per 100 kbp" : "is the average number of mismatches per 100000 aligned bases.", "# mismatches" : "is the number of mismatches in all aligned bases.", "# indels per 100 kbp" : "is the average number of indels per 100000 aligned bases.", "# indels" : "is the number of indels in all aligned bases", "# indels (<= 5 bp)" : "is the number of indels of length less than or equal to 5 bp", "# indels (> 5 bp)" : "is the number of indels of length greater than 5 bp", "Indels length" : "is the number of total bases contained in all indels", "# genes" : "is the number of genes in the assembly (complete and partial), based on a user-provided annotated list of gene positions in the reference genome. A gene counts as 'partially covered' if the assembly contains at least 100 bp of this gene but not the whole gene.", "# operons" : "is the number of operons in the assembly (complete and partial), based on a user-provided annotated list of operon positions in the reference genome. An operon counts as 'partially covered' if the assembly contains at least 100 bp of this operon but not the whole operon.", "# predicted genes (unique)" : "is the number of unique genes in the assembly found by a gene prediction tool.", "# predicted genes (>= 0 bp)" : "is the number of found genes having length greater than or equal to 0 bp.", "Cumulative length" : "plot shows the growth of assembly contig lengths. On the x-axis, contigs are ordered from largest (contig #1) to smallest. The y-axis gives the size of the x largest contigs in the assembly.", "Nx" : "plot shows the Nx metric value as x varies from 0 to 100. Nx is the minimum contig length y such that using contigs of length at least y accounts for at least x% of the total assembly length.", "NGx" : "plot shows the NGx metric value as x varies from 0 to 100. NGx is the minimum contig length y such that using contigs of length at least y accounts for at least x% of the bases of the reference genome. This metric is computed only if a reference genome is provided.", "NAx" : "plot shows the NAx metric value as x varies from 0 to 100. NAx is computed similarly to Nx, but based on lengths of aligned blocks instead of contig lengths. Contigs are broken into aligned blocks at misassembly breakpoints. NAx is the minimum block length y such that using blocks of length at least y accounts for at least x% of the bases of the assembly. This metric is computed only if a reference genome is provided.", "NGAx" : "plot shows the NGAx metric value as x varies from 0 to 100.NGAx is computed similarly to NGx, but based on lengths of aligned blocks instead of contig lengths. Contigs are broken at misassembly breakpoints. NGAx is the minimum block length y such that using blocks of length at least y accounts for at least x% of the bases of the reference genome. This metric is computed only if a reference genome is provided.", "GC content" : "plot shows the distribution of GC percentage among the contigs, i.e., the total number of bases in contigs with such GC content. Typically, the distribution is approximately Gaussian. However, for some genomes it is not Gaussian. For assembly projects with contaminants, the GC distribution of the contaminants often differs from the reference genome and may give a superposition of multiple curves with different peaks.", "Duplication ratio" : "is the total number of aligned bases in the assembly (i.e. Total length - Fully unaligned length - Partially unaligned length), divided by the total number of aligned bases in the reference (see the Genome fraction (%) metric). If the assembly contains many contigs that cover the same regions of the reference, its Duplication ratio may be much larger than 1. This may occur due to overestimating repeat multiplicities and due to small overlaps between contigs, among other reasons.", "Largest alignment" : "is the length of the largest continuous alignment in the assembly. This metric is always equal to the Largest contig metric but it can be smaller if the largest contig of the assembly contains a misassembly event.", "Total aligned length" : "is the total number of aligned bases in the assembly.", "Avg contig read support" : "is the average coverage of contigs that have large unique alignments to the reference. Read coverage is extracted from contig names (SPAdes/Velvet naming scheme only).", "# N's" : "is the total number of uncalled bases (N's) in the assembly.", "# N's per 100 kbp" : "is the average number of uncalled bases (N's) per 100000 assembly bases.", "# similar correct contigs" : "is the number of correct contigs similar among > 50% assemblies (see Icarus for visualization).", "# similar misassembled blocks" : "is the number of misassembled blocks similar among > 50% assemblies (see Icarus for visualization)." } | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8387795686721802, "perplexity": 4709.7543917094035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2022-21/segments/1652662588661.65/warc/CC-MAIN-20220525151311-20220525181311-00154.warc.gz"} |
https://voiceofwave.com/what-is-the-difference-between-wave-and-particle-properties-of-light | # What is the difference between wave and particle properties of light?
1
Date created: Fri, Nov 5, 2021 4:33 PM
Date updated: Thu, Jan 20, 2022 5:05 PM
Content
FAQ
Those who are looking for an answer to the question «What is the difference between wave and particle properties of light?» often ask the following questions:
### 👋 Does light have more wave or particle properties?
Both. E = mc2(squared) wave is an energy form while m is mass (particle) einstein proved that light acts as wave and particle at the same time.
### 👋 What is the difference between particle and wave theory of light?
Particle theory of light can explain Photoelectric Effect,Compton effect,Pair production.... wave theory of light can explain interference,refraction...
### 👋 What is difference between particle and wave?
The difference between the particle and waves are: The particle is defined as the small quantity of matter under the consideration… The wave is defined as the propagating dynamic distrubance. The energy of the wave is calculated based on the wavelength and velocity.
I think the difference between these two is you can see light and you can't see sound waves. But Light travels faster then sound. EXAMPLE: on a rainy day when lightening strike the sound comes after.
We've handpicked 25 related questions for you, similar to «What is the difference between wave and particle properties of light?» so you can surely find the answer!
Can a particle have properties like a wave?
• Means light (wave) can have properties like a particle. Later in 1924 De- Broglie also postulated that a particle of velocity (v) and mass (m) i.e momentum (mv)=p , can behave like a wave of wavelength of Lamda. So, in summary any pcarticle with velocity and mass can have wavelength. This is also called Matter Waves.
Can electrons act as wave and particle properties?
Yes, it is possible.
What is the wave particle of light?
photon
What is wave particle duality of light?
• Wave-Particle Duality of Light Quantum theory describes that matter, and light consists of minute particles that have properties of waves that are associated with them. Light consists of particles known as photons and matter are made up of particles known as protons, electrons, and neutrons.
Which type of experiment demonstrates that electron has properties of wave and that light has properties of particle?
photoelectric effect
What are the wave properties of light?
bfghdsbfjhfeuofgyewhrlofjkygsdfwejklfhdugewhjfgfhdgfsjdfdfge
Is it true that light has properties of both a particle and a wave?
Photons are the fundamental particles of light, they exhibit wave-particle duality, which means they show properties of both waves and particles.
How is light particle and wave?
Einstein believed light is a particle (photon) and the flow of photons is a wave… He maintained that photons have energy equal to "Planck's constant times oscillation frequency," and this photon energy is the height of the oscillation frequency while the intensity of light is the quantity of photons.
Is light a particle or wave?
• In an approximate way, light is both a particle and a wave. But in an exact representation, light is neither a particle nor a wave, but is something more complex. As a metaphor, consider a cylindrical can of beans.
Is light a wave or particle?
Light Is Also a Particle!
Now that the dual nature of light as "both a particle and a wave" has been proved, its essential theory was further evolved from electromagnetics into quantum mechanics. Einstein believed light is a particle (photon) and the flow of photons is a wave.
Is light particle or wave theory?
• Photoelectric Effect and Wave Theory of Light It is generally argued that the photoelectric effect provides evidence for the particle nature of light as the photoelectrons are almost released instantaneously when the light hits the detector.
Is light wave or a particle?
Einstein believed light is a particle (photon) and the flow of photons is a wave. The main point of Einstein's light quantum theory is that light's energy is related to its oscillation frequency.
Light is a wave or particle?
wavelike particles... a mishmash of both
What is the difference between light ray and wave?
• Light ray is defined as the path along which light travels. Hence, a ray is geometrical concept. Now , precise definition of wave: A wave is a disturbance which varies from time to time and from point to point.
What two properties of light confirm that light is wave?
Polarization and Interference confirm light is a wave.
How does matter exhibit both wave and particle properties?
• Just like light, it seemed that matter exhibited both wave and particle properties under the right circumstances. Obviously, massive objects exhibit very small wavelengths, so small in fact that it's rather pointless to think of them in a wave fashion.
Why do electrons have both wave and particle properties?
Electron and atom diffraction
Experiments proved atomic particles act just like waves… The energy of the electron is deposited at a point, just as if it was a particle. So while the electron propagates through space like a wave, it interacts at a point like a particle. This is known as wave-particle duality.
What is the wave particle duality of light?
That phrase is used to refer to the fact that light, and all other electromagnetic radiation, behaves like both a series of waves and a stream of little bullets (particles). If you build an experiment that looks for characteristics of waves, light provides them. And if the experiment is designed to detect characteristics of particles, light provides those too. So if someone asks "is light really a wave or a particle ?", the best, most honest answer is: "Yes".
What makes light a particle and a wave?
Nothing. It's something else that we don't quite understand. But sometimes it acts as a particle (quanta) and some times as a wave.
What makes light a wave or a particle?
• 1. Light as a wave: Light can be described (modeled) as an electromagnetic wave. In this model, a changing electric field creates a changing magnetic field. This changing magnetic field then creates a changing electric field and BOOM - you have light. Unlike many other waves (sound, water waves,...
What experiment showed that light had wave properties?
Interference.
Which properties of light defines light as a wave?
It can be reflected, refracted and polarised. It also shows the phenomenon of interference. (Young's double slit experiment) The above are the properties of waves. Light shows these properties and thus, this defines light as a wave.
What is the main difference between radio wave and a light wave?
The difference is their wavelengths. That means that their frequencies are different, and also the amount of energy carried by each photon.
Is a light a particle or wave?
Light Is Also a Particle!
Now that the dual nature of light as "both a particle and a wave" has been proved, its essential theory was further evolved from electromagnetics into quantum mechanics. Einstein believed light is a particle (photon) and the flow of photons is a wave.
Is light a particle and a wave?
#### Is light a particle or a wave?
• "Light" is both a particle and a wave (the photon, like all other elementary particles, can be understood as a localized vibration in its quantum field that can be measured as particle-like or wave-like). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9101040363311768, "perplexity": 726.8157345001109}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2022-05/segments/1642320303385.49/warc/CC-MAIN-20220121131830-20220121161830-00072.warc.gz"} |
http://umj-old.imath.kiev.ua/authors/name/?lang=en&author_id=2146 | 2019
Том 71
№ 11
# Antoniouk A. Vict.
Articles: 9
Anniversaries (Ukrainian)
### Anatolii Mykhailovych Samoilenko (on his 80th birthday)
Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 3-6
Article (English)
### Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point
Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1299–1317
The Dirichlet problem for the heat equation in a bounded domain $G ⊂ ℝ^{n+1}$ is characteristic because there are boundary points at which the boundary touches a characteristic hyperplane $t = c$, where c is a constant. For the first time, necessary and sufficient conditions on the boundary guaranteeing that the solution is continuous up to the characteristic point were established by Petrovskii (1934) under the assumption that the Dirichlet data are continuous. The appearance of Petrovskii’s paper was stimulated by the existing interest to the investigation of general boundary-value problems for parabolic equations in bounded domains. We contribute to the study of this problem by finding a formal solution of the Dirichlet problem for the heat equation in a neighborhood of a cuspidal characteristic boundary point and analyzing its asymptotic behavior.
Article (English)
### Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds
Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1299–1316
We study the dependence on initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may not be everywhere twice differentiable, we show that, under certain monotonicity conditions on the coefficients and curvature of the manifold, there are estimates exponential in time for the continuity of a diffusion process with respect to initial data. These estimates are combined with methods of the theory of absolutely continuous functions to achieve the first-order regularity of solutions with respect to initial data. The suggested approach neither appeals to the local stopping time arguments, nor applies the exponential mappings on the tangent space, nor uses imbeddings of a manifold to linear spaces of higher dimensions.
Article (English)
### Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds
Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1454–1472
We find sufficient conditions on coefficients of diffusion equation on noncompact manifold, that guarantee non-explosion of solutions in a finite time. This property leads to the existence and uniqueness of solutions for corresponding stochastic differential equation with globally non-Lipschitz coefficients.
Proposed approach is based on the estimates on diffusion generator, that weakly acts on the metric function of manifold. Such estimates enable us to single out a manifold analogue of monotonicity condition on the joint behaviour of the curvature of manifold and coefficients of equation.
Article (English)
### Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?
Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1011–1034
It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations on manifolds and semigroups, we study $C^{∞}$ regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional separation of curvature.
Article (English)
### Nonlinear-estimate approach to the regularity of infinite-dimensional parabolic problems
Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 579–596
We show how the use of nonlinear symmetries of higher-order derivatives allows one to study the regularity of solutions of nonlinear differential equations in the case where the classical Cauchy-Liouville-Picard scheme is not applicable. In particular, we obtain nonlinear estimates for the boundedness and continuity of variations with respect to initial data and discuss their applications to the dynamics of unbounded lattice Gibbs models.
Article (Ukrainian)
### Higher-Order Relations for Derivatives of Nonlinear Diffusion Semigroups
Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 117-122
We show that a special choice of the Cameron–Martin direction in the characterization of the Wiener measure via the formula of integration by parts leads to a set of natural representations for derivatives of nonlinear diffusion semigroups. In particular, we obtain a final solution of the non-Lipschitz singularities in the Malliavin calculus.
Article (Ukrainian)
### Noncommutative central limit theorem for Gibbs temperature states
Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 299–306
For Gibbs temperature states, the scheme of the proof of the noncommutative central limit theorem is given by using the commutative central limit theorem for corresponding Euclidean measures. Applications are constructed for the model of a temperature-anharmonic crystal and the generalized Ising model with compact continuous configuration space.
Article (Ukrainian)
### Essential self-adjointness of Dirichlet operators of Gibbs measures on infinite products of manifolds
Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 4–11
We obtain the conditions of essential self-adjointness of Dirichlet operators of Gibbs measures with essential domains consisting of smooth cylindrical functions. It is proved that certain spaces of smooth functions are invariant under the action of the semigroup of the Dirichlet operator. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8935868740081787, "perplexity": 395.8319489700055}, "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-2020-45/segments/1603107875980.5/warc/CC-MAIN-20201021035155-20201021065155-00467.warc.gz"} |
https://www.physicsforums.com/threads/calculate-the-spectral-interval-lmax-lmin-and-dl-in-which-all-the-lines-of-the-ly.636788/ | # Calculate the spectral interval (λmax, λmin, and Δλ) in which all the lines of the Ly
1. Sep 18, 2012
### b0094
1. The problem statement, all variables and given/known data
Calculate the spectral interval (λmax, λmin, and Δλ) in which all the lines of the Lyman, Balmer, and Pashcen series (excluding their continuous parts) of the H-atom are found.
2. Relevant equations
Here is where I am confused.. do I use the Balmer's formula for H?
frequency = [1/4 - 1/n^2] * 3.29*10^(15)s^-1
3. The attempt at a solution
The problem is.. I don't know what energy levels these series correspond to? Is this something I am just supposed to memorize??
2. Sep 18, 2012
### rl.bhat
Re: Calculate the spectral interval (λmax, λmin, and Δλ) in which all the lines of th
For Lyman series, transition from n = 2 to n = 1 for λmax and n= ∞ to n = 1 for λmin.
Similarly for the other series.
3. Sep 18, 2012
### b0094
Re: Calculate the spectral interval (λmax, λmin, and Δλ) in which all the lines of th
Is the equation I provided correct in this case??
Similar Discussions: Calculate the spectral interval (λmax, λmin, and Δλ) in which all the lines of the Ly | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.909635603427887, "perplexity": 2939.947092285028}, "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-2017-43/segments/1508187826114.69/warc/CC-MAIN-20171023145244-20171023165244-00471.warc.gz"} |
https://physics.stackexchange.com/questions/391652/adjoint-of-creation-and-anihilation-operators-and-problem-with-expected-values-o | # Adjoint of creation and anihilation operators and problem with expected values of $\hat a$ and $\hat a^{\dagger}$ for coherent state
The definition of the adjoint operator of an operator $\hat A$ is
$$(\vec x|\hat A \vec y) = (\hat A \vec x| \vec y) \quad \forall x, y \in \mathcal{H}$$
where $(\cdot|\cdot)$ is the inner product of a Hilbert space.
So before this definition came into play, I innocently tried to do the expected values of $\hat a$ and $\hat a^\dagger$ for the coherent state $\left|\alpha\right>$, getting:
$$\left<\alpha\right|\hat a \left|\alpha\right> = \alpha \left<\alpha|\alpha\right> = \alpha$$
and
$$\left<\alpha\right|\hat a^\dagger \left|\alpha\right> = \alpha^* \left<\alpha|\alpha\right> = \alpha^*$$
which I think are correct.
But then, I thought that the expected value of $\hat a$ and $\hat a^\dagger$ should be the same, since one is the transposed complex-conjugated operator of the other, and that usually means that one is the adjoint of the other, but with the definition of adjoint operator and getting different values for the brakets I did before, it's clear that they're not their adjoint.
So my question is:
• Did I suppose or do anything wrong? Is this definition I gave for adjoint operator not applicable to this case?
• If it's correct, do $\hat a$ and $\hat a^\dagger$ have any operator corresponding to its adjoint, or if the transposed complex-conjugated operator of $\hat A$ is not the adjoint of $\hat A$ then $$\hat A doesn't have an adjoint operator? • If they do have an adjoint, which are they? ## 2 Answers There’s nothing wrong with what you did except with your incorrect notion that the expectation values and its adjoint should be the same. Indeed you have just shown that, for \hat a and \hat a^\dagger, one is the complex conjugate of the other. One intuitive way to see this is to remember that, up to constants$$ \hat a\sim \hat x+i\hat p\, , \qquad \hat a^\dagger \sim \hat x -i \hat p \tag{1} $$with \hat x and \hat p self-adjoint, so their expectations values are real. Using this together with Eq.(1) clearly shows how one can anticipate that the expectation values of \hat a and \hat a^\dagger will be complex conjugate in pairs. The creation and annihilation operators are true adjoint operators, yes. In the case of the coherent state, we have$$(\alpha|a^{\dagger}\alpha)=(a\alpha|\alpha)=a^{*}(\alpha|\alpha).
I think you may be forgetting that the complex inner product satisfies $(c\vec{v}|\vec{w})=c^{*}(\vec{v}|\vec{w})$ and not $(c\vec{v}|\vec{w})=c(\vec{v}|\vec{w})$.
If this isn't where your confusion lies, comment below or edit your question to more clearly explain why you think that $a$ and $a^{\dagger}$ should have the same expectation value.
I hope this helps!
• Ok, then after seeing this answer, I think that maybe my problem is with the inner product. When doing $(a\alpha|\alpha)$ you're treating the first $\alpha$ as a ket? In my definition of adjoint, I thought that $(\hat A \vec x|\vec y)=<\vec x|\hat A|\vec y>$, but in your answer it doesn't seem like it... I don't quite get it – Mr. Nobody Mar 11 '18 at 22:59
• @Mr.Nobody - $({\hat A} \vec{x} | \vec{y} ) = \langle\vec{x} | {\hat A}^\dagger | \vec{y} \rangle$. – Prahar Mitra Mar 11 '18 at 23:10
• So my mistake was trying to translate the definition so I could understand it into $\left<x\right|A\left|y\right>=\left<y\right|A^\dagger\left|x\right>$ which seems that isn't always true am I right? – Mr. Nobody Mar 12 '18 at 0:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.9523477554321289, "perplexity": 282.36230560061455}, "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-2021-25/segments/1623487608856.6/warc/CC-MAIN-20210613131257-20210613161257-00524.warc.gz"} |
https://chemistry.stackexchange.com/questions/137182/reason-for-same-frequency-of-spectral-line-in-infrared-and-visible-region | # Reason for same frequency of spectral line in infrared and visible region
I encountered this question while trying to understand the basics of spectroscopy and atomic structure.
Which electronic transition in Balmer series of hydrogen atom has same frequency as that of n=6 to n=4 transition in Helium positive ion?
I am aware that lines of Balmer series correspond to visible region of electromagnetic spectrum and that to Brackett series correspond to infrared region.
How is it that the frequency of a spectral line belonging to infrared region be same as that of line belonging to visible region knowing that frequency is characteristic property unique to an electromagnetic radiation?
• The Balmer series is said to be visible in case of Z=1 $\mathrm{n_1 = 2}$, The spectral lines are not in different regions. – Safdar Faisal Jul 29 '20 at 11:13
• What do you mean by 'the spectral lines are not in different regions'? Spectral lines have different wavelengths and therefore should belong to different regions. – anushka verma Jul 29 '20 at 11:40
• But that is if we consider them to be emitted from the same atom. Here you are asked to compare between two different atoms. – Safdar Faisal Jul 29 '20 at 11:47
According to Bohr's theory of the structure of the hydrogen atom, the atom is defined as a system with a positively charged nucleus that has electrons revolving around it in discrete orbits of fixed size and energy and stated that the orbits were held stationary using electrostatic forces.
Upon calculation of the various parameters of the model, we get the energy of the $$\mathrm{n}^{\text{th}}$$ orbital in the hydrogen atom to be:
$$E=\frac{\mathrm{-13.6}}{\mathrm{n^2}} \mathrm{eV} \tag{1}$$
If we state that the atom to be considered were to be made mono-electronic and had an atomic number of Z, satisfying the conditions for the Bohr's model to work, we get that the energy of the $$\mathrm{n}^{\text{th}}$$ orbital to be:
$$E= -13.6 \cdot\frac{\mathrm{Z^2}}{\mathrm{n}^2} \mathrm{eV} \tag{2}$$
A spectral line is observed when the electron de-excites itself from a higher energy state to a lower energy state that is more stable. This is done by emitting a photon whose energy is the same as that of the difference of energy between the two levels. Applying $$\frac{\mathrm{hc}}{\lambda} =E$$, and $$\frac{\mathrm{c}}{\lambda} = f$$ we get
$$f = \frac{\mathrm{c}}{\lambda} = \mathrm{c} R_H\big[\mathrm{\frac{1}{n_1^2}-\frac{1}{n_2^2}\big]Z^2} \tag{3}$$
Here $$R_H$$ is the Rydberg constant whose value is $$\approx\pu{109,678 cm^-1}$$
In the question, you have been asked to figure out which line in the Balmer series($$\mathrm{n_1} = 2$$) of hydrogen would have the same frequency as that of the $$3^\text{rd}$$ line in the Brackett series($$\mathrm{n_1} = 4$$). I believe you would know how to solve the question and if not would now be able to via equating the frequencies of the two spectral lines.
Moving on to your doubt which is based on a misconception.
How is it that the frequency of a spectral line belonging to infrared region be same as that of line belonging to visible region knowing that frequency is characteristic property unique to an electromagnetic radiation?
The issue with this statement here is that you are assuming that the Balmer series is always visible for all atoms. This is only true for hydrogen as the the wavelengths provided for the different series are based on measurements taken on the hydrogen atom. That is, the Paschen series is infrared for hydrogen and this may change if the atom changes. As the atomic number increases, the frequency becomes greater as you can see from equation ($$3$$). This implies that at higher atomic numbers, there is a greater shift towards the UV region.
TL;DR: What you say is correct, frequency is a characteristic property unique for a certain electromagnetic radiation. However, the Balmer series is defined to be visible specifically for the hydrogen atom and not for any other mono-electronic atom/ion. As the atomic number becomes greater, a shift towards the UV region takes place. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 13, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8260621428489685, "perplexity": 279.99219732846643}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2021-17/segments/1618039594808.94/warc/CC-MAIN-20210423131042-20210423161042-00536.warc.gz"} |
http://clay6.com/qa/48775/the-coefficient-of-x-in-the-expansion-of-x-4-large-frac-is | # The coefficient of $x^{32}$ in the expansion of $(x^4-\large\frac{1}{x^3})^{15}$ is
$\begin{array}{1 1}(A)\;-15C_3\\(B)\;15C_4\\(C)\;-15C_5\\(D)\;15C_2\end{array}$
Toolbox:
• $T_{r+1}=nC_r a^{n-r} b^r$
$T_{r+1}=15C_r (x^4)^{15-r}(-x^{\Large\frac{1}{3}})^r$
$\Rightarrow 15C_r x^{60-4r-3r}(-1^r)$
$\Rightarrow 15C_r (-1)^r x^{60-7r}$
$\Rightarrow 60-7r=32$
$\Rightarrow r=4$
$\therefore T_{r+1}=T_{4+1}=15C_4 (-1)^4x^{32}$
$\therefore$ Required coefficient =$15C_4$
Hence (B) is the correct answer. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9932730793952942, "perplexity": 515.8015503027199}, "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-2018-17/segments/1524125946199.72/warc/CC-MAIN-20180423203935-20180423223935-00435.warc.gz"} |
https://economics.stackexchange.com/questions/29645/difference-between-shocks-innovations-and-disturbances | # Difference between shocks, innovations and disturbances
In macroeconomics and macroeconomic models, shocks, innovations, and disturbances are very prominent and often mentioned in the literature.
In general, is there a difference between macroeconomic shocks, disturbances, and innovations, or can these terms be used interchangeably?
I don't think there is really a consensus and the definition depend on the field and context. Within applied macroeconomics, I prefer the definitions used by Blanchard and Watson (1986) and Bernanke (1986) and others. Valerie Ramey gives a good discussion of the definition in "Macroeconomic shocks and their propagation" in the Handbook of macroeconomics, vol. 2. You can see a discussion in a YouTube video here.
In short, a shock is an economically meaningful primitive exogenous force. We can think of these as the "empirical counterparts to the shocks we discuss in our theories." An innovation, on the other hand, is the residual from a reduced form VAR or similar model. An identifying assumption may establish a link between the innovation and the shock, but they are distinct concepts.
Within this context, the definition of a disturbance is less clear. Ramey mentions in the Handbook chapter that "shocks are most closely related to the structural disturbances in a simultaneous equation system."
• Ramey, Valerie A. "Macroeconomic shocks and their propagation." In Handbook of macroeconomics, vol. 2, pp. 71-162. Elsevier, 2016.
In Greene "Econometric Analysis" the disturbance of linear regression, $$\epsilon_t = y_t - \mathbf{x_t}\boldsymbol{\beta}$$
is estimated with the residual, $$e_t = y_t - \mathbf{x_t b}$$
In the text, disturbances $$\epsilon_t$$ are also called innovations. Qualitative events in an economy that impact a process or model's variables or error term are shocks.
Here's an example of why your question is difficult and depends on the framework.
Case 1: Take the classical OLS case $$Y = \beta X + \epsilon$$.
In this case, the $$\epsilon$$ can thought of as an error term which represents noise around the response. So, the mean of the response is $$\beta X$$ but on any particular response is not going to be exactly equal to that mean because of $$\epsilon$$ which represents the variation around that mean.
Case 2: Take a simple dynamic model in econometrics such as the ADL(1,0) so that $$Y_t = \rho Y_{t-1} + \beta X_{t} + \epsilon_t$$.
In this case, $$\epsilon_t$$ is not really noise because it's going to stay in $$Y_{t}$$ after that period is finished so it's actually part of the model. It's almost like an exogenous variable rather than noise so, to me, the best term in this case for $$\epsilon$$ would probably be innovation. It has what economists refer to as a "permanent" effect on the response.
So, my point is that each case can be different. This paper by Qin explains all of this in much more detail and more clearly. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8748782277107239, "perplexity": 808.8751324063563}, "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-2019-43/segments/1570987779528.82/warc/CC-MAIN-20191021143945-20191021171445-00147.warc.gz"} |
https://collegephysicsanswers.com/openstax-solutions/wavelength-4653textrm-mutextrmm-observed-hydrogen-spectrum-transition-ends-nf-5 | Question
A wavelength of $4.653\textrm{ }\mu\textrm{m}$ is observed in a hydrogen spectrum for a transition that ends in the $n_f = 5$ level. What was $n_i$ for the initial level of the electron?
7
Solution Video | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.998015820980072, "perplexity": 494.73433287720894}, "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-2020-05/segments/1579250595787.7/warc/CC-MAIN-20200119234426-20200120022426-00433.warc.gz"} |
https://tex.stackexchange.com/questions/319717/theorem-environment-interaction-with-enumerate-itemize | # Theorem environment : interaction with enumerate/itemize
My question is related to this one : Is it possible to skip the first line in a theorem environment? Actually, I am looking for a way to implement the solution given there in a somewhat systematic manner.
More precisely : whenever one starts a theorem with a list, the first item of the list appears on the same line as the theorem header ("Theorem"), even if one specifies \newline in the theorem style. I would like to force Latex to skip a line, even in this case.
One of the simplest solution suggested in the question above is to add a empty item at the beginning of the list :
\begin{thm}
\begin{enumerate}
\item[]
\item First actual item
\item Second actual item
\end{enumerate}
\end{thm}
My question is : is there a systematic way to add this blank item at the beginning of every theorem, by adding it into the theoremstyle command ? (Or at the beginning of every enumerate environment ?)
The idea would be that when one types :
\begin{thm}
Body of the theorem
\end{thm}
Latex would understand :
\begin{thm}
\begin{enumerate}
\item[]
\end{enumerate}
Body of the theorem
\end{thm}
My code and my theorem style are the following :
\documentclass{article}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\newtheoremstyle{exercisestyle}% % Name
{}% % Space above
{}% % Space below
{}% % Body font
{}% % Indent amount
{}% % Punctuation after theorem head
{\newline}% % Space after theorem head, ' ', or \newline
{\thmname{#1}\thmnumber{ #2} --- \thmnote{#3}}% % Theorem head spec (can be left empty, meaning normal')
\theoremstyle{exercisestyle}
\newtheorem{exercice}{Exercice}
\begin{document}
\begin{exercice}[Title]
\begin{enumerate}
\item First item
\item Second item
\end{enumerate}
\end{exercice}
\end{document}
• What is it supposed to do if theorem body has no list? – Bernard Jul 16 '16 at 21:48
• @Bernard The idea was to add a list with one empty item at the beginning of each theorem, regardless of whether the theorem starts with a list or not. But as ilFuria pointed out, this may not be very good for theorems starting at the end of a page. – user75362 Jul 17 '16 at 8:29
• What I meant with my question was this: even if the theorem doesn't begin with a list (or has no list), will it begin on a different line from the label? – Bernard Jul 17 '16 at 9:43
• @Bernard Yes, but that is taken care of automatically by the \newline command in \theoremstyle. – user75362 Jul 17 '16 at 13:47
The amsthm package manual has it covered in §2.1. You can obtain it via texdoc amsthm on your system, or on CTAN
I loosely quote that section:
The best way to avoid these problems is to allow the list to start on a new line. One way to accomplish this is to follow the theorem head (and \label, if present) by the command \leavevmode. (For more information, see section 4.3.1.) However, if the theorem comes near the bottom of a page, the list might move to the top of the next page, leaving an orphaned heading. If this happens, it must be addressed as an exception, to be taken care of after the text is final; at that point, the recommended fix is to call on the needspace [NDS] package, or insert an explicit page break.
Also:
An alternative method of starting a new line after the heading is to provide a \newtheoremstyle{break}; the definition is given below on page 9. Like the \leavevmode approach, a break theorem environment is not perfect; known limitations accompany the definition.
The aforementioned definition on "page 9" is:
This style will break after the theorem heading and start a new line.
\newtheoremstyle{break}%
{}{}%
{\itshape}{}%
{\bfseries}{}% % Note that final punctuation is omitted.
{\newline}{}
This style can be used for a theorem beginning with a list. When used with enumerate and an AMS document class, all items are properly labeled and will link. However, the vertical spacing needs help; a conflict between definitions prevents the first item from starting on a new line—it looks almost the same as a default theorem beginning with an enumerated list. To repair this problem, begin the theorem like this:
\begin{breakthm}[...]
\leavevmode \vspace{-\baselineskip}
\begin{enumerate}
\leavevmode alone will leave a full blank line after the theorem head. One more problem may arise: if the theorem starts close to the end of a page, the list could be split to a new page, leaving an orphaned heading. Make a note to address this when the text is final; then call on the needspace [NDS] package, or insert an explicit page break.
In order to reach your goal automatically, you could try to do something like this:
\documentclass{article}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\newtheoremstyle{exercisestyle}% % Name
{}% % Space above
{}% % Space below
{}% % Body font
{}% % Indent amount
{}% % Punctuation after theorem head
{\newline}% % Space after theorem head, ' ', or \newline
{\thmname{#1}\thmnumber{ #2} --- \thmnote{#3}}% % Theorem head spec (can be left empty, meaning normal')
\theoremstyle{exercisestyle}
\newtheorem{exercice}{Exercice}
\newenvironment{breakingex}[1][]{%
\begin{exercice}[#1]\leavevmode\vspace{-\baselineskip}%
}{\end{exercice}}
\begin{document}
\begin{breakingex}[Title]
\begin{enumerate}
\item First item
\item Second item
\end{enumerate}
\end{breakingex}
\end{document}
Which yields the following result:
Please do mind that it's possible you will have to create some more environments like this, or to use the default setting with some overrides, in order to overcome page breaks. Refer to the amsthm and your judgement manual to achieve the best solution.
• Thank you very much for your answer, @ilFuria ! I was aware of the \leavevmode command, but not of the elegant way to suppress the vertical space afterwards. However, the solution you propose (to add \leavevmode \vspace{-\baselineskip} at the beginning of each theorem starting with a list) is still not automatic, is it ? Is there a way to do it automatically ? to unify the cases where the theorem starts with a list and where it does not ? – user75362 Jul 17 '16 at 8:36
• @user75362 what you can do, and I tried to do it, is create a new environment which wraps the theorem: this way you can embed the \leavevmode \vspace{-\baselineskip} in the starting definition. You still can't automate the page break management – Moriambar Jul 17 '16 at 9:05
• Thank you, it does create several types of environments, but this is better than having to add a line at the beginning of each (breaking) theorem ! – user75362 Jul 17 '16 at 9:43
• @JPi I think for the sake of clarity: what you are doing is simply exit the vertical mode. I honestly don't know whether there are any other reasons. I created my example using the suggestions in the amsthm manual – Moriambar Jul 17 '16 at 17:51
• @JPi -- i discovered yesterday that if a ~ is the first thing after \begin{thm} (for any theorem object), the vertical space above the theorem object is suppressed. this is undesirable (and undocumented); and how to fix it is not obvious. that is one reason to use \leavevmode – barbara beeton Feb 16 '18 at 13:44
I can propose a simple solution with ntheorem: its break style does not seem to have the same problem with page breaks, so I patched it to do what you want:
\documentclass{article}
\usepackage{amsmath}
\usepackage[thmmarks, amsmath, thref, hyperref]{ntheorem}
\usepackage{amssymb}
\usepackage{enumitem}
\usepackage{lipsum}
\makeatletter
\newtheoremstyle{exobreak}%
##1\ ##2\theorem@separator}\hbox{\strut}}}]}%
##1\ ##2\,\textemdash\,##3 \theorem@separator}\smallskip\hbox{\strut}}}]}
\makeatother
\theorempostwork{\vskip-0.5\topsep}
\theoremstyle{exobreak}
\theorembodyfont{\normalfont}
\theoremseparator{}
\newtheorem{exercice}{Exercice}
\begin{document}
\vspace*{33\baselineskip}
\lipsum[3]
\begin{exercice}[Title]
\begin{enumerate}[itemsep=0pt, font =\bfseries, wide=0pt]
\item First item
\item Second item
\item Third item
\end{enumerate}
\end{exercice}
\lipsum[4]
\begin{exercice}[Another title]
\begin{enumerate}[itemsep=0pt, font =\bfseries, wide=0pt]
\item First item
\item Second item
\item Third item
\end{enumerate}
\end{exercice}
\lipsum[4]
\end{document} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8882349729537964, "perplexity": 1266.9889655611983}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2019-22/segments/1558232256381.7/warc/CC-MAIN-20190521122503-20190521144503-00490.warc.gz"} |
https://quant.stackexchange.com/questions/18010/using-fx-atm-rr-bf-volatility-to-estimate-smile | # Using FX ATM/RR/BF Volatility to Estimate Smile
Suppose $S$ is some FX rate, EUR/USD say, and $\sigma_{S}(K,T)$ is the implied volatility for some option written on $S$, sourced from the surface $\sigma_{S}(\cdot,\cdot)$ (alternatively, consider the implied volatility surface $(\Delta,T)\mapsto\sigma_{S}(\Delta,T)$, commonly used for FX implied volatility data).
Suppose we have the following term-structure data (maturity tenors $T$ for 1D, 1W, 2W, 3W, 1M, ..., 1Y, ..., 10Y):
1. Forward rates $F(\cdot)$
2. Risk-free zero rates $r_{d}(\cdot)$ and $r_{f}(\cdot)$ for both currencies
3. ATM volatilities $\sigma_{S}(K^{*},\cdot)$ or $\sigma_{S}(\Delta^{*},\cdot)$ (the ATM strike or delta, denoted with the asterisk, is not known)
4. $25\Delta$ "Risk Reversal" $RR(\cdot)$
5. $25\Delta$ "Butterfly" $BF(\cdot)$
Is there anyway to use (1)-(5) to approximate $\sigma_{S}(K,T)$ given the option's $(K,T)$?
(The motivation for asking this question is that I have access to (1)-(5) through an automated data import tool, but not the entire surface, and would like to price some options using the data that can be imported automatically. The valuation is not being done for trading purposes, so an absolutely precise valuation is not needed, only an estimate.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9766451120376587, "perplexity": 2607.095677434393}, "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-2020-29/segments/1593655882051.19/warc/CC-MAIN-20200703122347-20200703152347-00385.warc.gz"} |
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