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https://www.beatthegmat.com/if-x-neq-5-and-x-is-a-non-zero-integer-what-is-the-value-of-dfrac-x-5-5-x-t332839.html?sid=cb52609d9e8dfb01d7f634f0cde019e7 | 1,702,110,381,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100873.6/warc/CC-MAIN-20231209071722-20231209101722-00583.warc.gz | 713,362,216 | 295,573 | ## If $$x \neq 5$$ and $$x$$ is a non-zero integer, what is the value of $$\dfrac{|x|+5}{5-x}?$$
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### If $$x \neq 5$$ and $$x$$ is a non-zero integer, what is the value of $$\dfrac{|x|+5}{5-x}?$$
by BTGmoderatorLU » Thu Sep 21, 2023 11:48 am
00:00
A
B
C
D
E
## Global Stats
If $$x \neq 5$$ and $$x$$ is a non-zero integer, what is the value of $$\dfrac{|x|+5}{5-x}?$$
1. $$|y+5| < 1-x$$
2. $$\dfrac{\sqrt{x^2}}{\sqrt[3]{x^3}}=-1$$
The OA is D
• Page 1 of 1 | 249 | 583 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.03125 | 3 | CC-MAIN-2023-50 | latest | en | 0.610411 |
https://nlinux.org/how-to-find-intersection-of-two-curves-in-excel/ | 1,638,790,205,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363292.82/warc/CC-MAIN-20211206103243-20211206133243-00620.warc.gz | 480,865,987 | 5,267 | I have two plots making use of scatter plot of which one is non straight and also the various other one is direct. And I call for the intersection of these curves. How need to I continue ?
Edit: The plots are done by using a collection of values in excel.
You are watching: How to find intersection of two curves in excel
Curve 1:
x: 0,0.5,1,1.5,2,2.5,3,3.5,4,4.5,5,5.5,6,6.5,7,7.5,8
y:8.43,8.76,8.27,7.87,7.69,7.76,8.46,8.85,8.34,7.92,7.73,7.79,8.42,8.76,8.27,7.87,7.69
Curve 2: y=8.168
Thanks
Edit-2: In the other question Get collaborates of intersecting suggest of 2 trend lines a trfinish line is made and also then the interarea of those are dealt, which is clearly not possible for mine and also not a duplicate for the question you are referring to.
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edited Sep 11 "17 at 9:37
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EDIT: The following approach is applicable only to graphs wright here straight interpolation is proper and where the linear curve is a continuous horizontal line.
Assuming your information is in columns A,B and C as presented below, the x-coordinate of the intersections have the right to be uncovered making use of the formula below. This formula filled down from D3 provides the outcomes in the table listed below.
=IF(OR(AND(\$B2>=\$C3,\$B3=\$C3)),\$A2+(\$A3-\$A2)*(\$B2-\$C3)/(\$B2-\$B3),"")
If you would certainly clearly explain your requirements, you can gain an acceptable solution.
Here"s the graph through a linear fit to the initially curve (red line) and also the second (constant) curve (purple line).
You can approach this in couple of ways:
You have the right to settle the equation of the linear fit for x as soon as y = 8.168. That provides the allude wright here the 2 straight lines cross (4.040,8.168).
See more: Don'T Take It Personal ( Just One Of Those Days That A Girl Goes Through
You deserve to discover the points wbelow the blue curve equals 8.168. Thesimplest method to do this is by direct interpolation, which assumesthat the line segments in between points deserve to be approximated by adirectly line. For the initially interarea (between points 3 and also 4)8.168 is this fraction of the means between the two points:
(8.27-8.168)/(8.27-7.87) = 0.255
And the x-coordinate is the exact same fraction of the method in between 1 and also 1.5, giving (1.128, 8.168).
The 3rd crossing is coincidentally near the intersection through the direct fit, so let"s view what it is, too: | 743 | 2,550 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.4375 | 3 | CC-MAIN-2021-49 | longest | en | 0.915254 |
https://plainmath.net/other/30963-calculators-rational-functions-produce-vertical-asymptotes-graphing | 1,679,971,628,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296948756.99/warc/CC-MAIN-20230328011555-20230328041555-00062.warc.gz | 518,391,405 | 18,296 | pedzenekO
2021-09-27
On many calculators, graphs of rational functions produce lines at vertical asymptotes. For example, graphing $y=\frac{x-1}{x+1}$ on the window [-4.9, 4.9] by [-4.9, 4.9] produces such a line at x=-1 on the TI-84 Plus and TI-89. But with the window [-4.7, 4.7] by [-4.7, 4.7] on a TI-84 Plus, or [-7.9, 7.9] by [-7.9, 7.9] on a TI-89, the spurious line does not appear. Experiment with this function on your calculator, trying different windows, and try to figure out an explanation for this phenomenon.
hesgidiauE | 185 | 538 | {"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} | 2.71875 | 3 | CC-MAIN-2023-14 | latest | en | 0.877675 |
https://www.thewinningbets.com/predictions-brazil/serie-b/prediction-of-vitoria-ba-criciuma-sc-20-july-2019-3060917 | 1,563,578,421,000,000,000 | text/html | crawl-data/CC-MAIN-2019-30/segments/1563195526386.37/warc/CC-MAIN-20190719223744-20190720005744-00385.warc.gz | 892,001,126 | 14,516 | The championship of Brazil - Serie B has arrived to the day 10 and puts us in front of the match between Vitória - BA and Criciúma - SC, of which we obviously collected the statistics and created the prediction.
Comparing the ranking (subscribe if you cannot see it), we understand that Vitória - BA, compared to Criciúma - SC, has 4 points more. The statistics of the championship indicate that Vitória - BA has proved victorious at home in 1 challenges on 4, with a goal score of 3 facts and 6 suffered. on the other hand, Criciúma - SC dominates his opponent away from home 1 times on 4 games, scoring 3 times and conceding 5 goals.
## Analysis of the State of Form
Vitória - BA FT 1T Pos Criciúma - SC FT 1T Pos
## Team and Championship statistics
Below the statistics on the league and teams of Over and Under and Average of the Goals. A valuable set of data that compares the teams Vitória - BA and Criciúma - SC with their championship of membership of Brazil - Serie B all to refine your bets on Total Goals.
We have calculated an average Total scores, in the matches played by Criciúma - SC, of Goals, while the average calculated on the matches of Vitória - BA is Goals made and suffered.
Let's merge these two values into a single one, and compare it with the championship average.We will note that the merged average, of Gol, is lesser than with the average total goals on all championship, of 2.3 Goals. We base these two values calculated in the first half in a single, and compare it with the championship average.We will note that the melted average, of Gol, is lesser than with the average goals totals across the league, by 0.9 Goals. Let's take these two averages calculated in the second half and let's unite them in one: we will notice that if we compare it with the championship this value, of Gol, is lesser than average of the latter, which instead is 1.4 Gol.
The tables below effectively summarize all that we have written previously, you can compare the average scores values of the teams Vitória - BA and Criciúma - SC with the championship, and we have data also available for the first and second time.
Vitória - BA Criciúma - SC
Average Gol / Match
Under 0.5 Goals
Over 0.5 Goals
Under 1.5 Goals
Over 1.5 Goals
Under 2.5 Goals
Over 2.5 Goals
Under 3.5 Goals
Over 3.5 Goals
Under 4.5 Goals
Over 4.5 Goals
Goal
No Goal
#### Team and Championship statistics Only 1st Half
Vitória - BA Criciúma - SC
Average of the Goals 1T
Under 0.5 Goals 1T
Over 0.5 Goals 1T
Under 1.5 Goals 1T
Over 1.5 Goals 1T
Goal 1T
No Goal 1T
#### Team and Championship statistics Only 2nd Half
Vitória - BA Criciúma - SC
Average of the Goals 2T
Under 0.5 Goals 2T
Over 0.5 Goals 2T
Under 1.5 Goals 2T
Over 1.5 Goals 2T
Goal 2T
No Goal 2T
## Complete Match
Subsequently we show the result of our studies relative to the single results of the meeting. The tables show who will win the game, how many goals will be scored, who will score and how much, everything divided into first and second time, and full time.
The next tables will show the percentages as regards the exact result and the final result
#### Winner of the Match
Percentage
Vitória - BA (Home Win)
Criciúma - SC (Away Win)
Draw
The most probable outcome for this match is attributable to the victory of the team Criciúma - SC.
This table will be useful to visualize, at a glance, the percentage of total scores related to all the possible events of this match.
#### Prediction: Over / Under
0.5 1.5 2.5 3.5 4.5
Over
Under
#### Prediction: Goal / No Goal
Yes No
Both Teams Score
Observing the previous table it is easy to observe that the probability that both teams score is low (41.79%) .
The two teams do not score much and the trend should be repeated The results of the most interesting game are as follows:
#### Prediction: Correct Score
home team Correct Score away team
Vitória - BA Criciúma - SC
H - A 0 1 2 3 4
0
1
2
3
4
## Vitória - BA - Criciúma - SC: First Half
We still have many percentages to show you! Here are the ones related to the first game time of the match.
The two teams do not score much during the first half and the trend should be repeated.
Percentage
0.5 1.5
Yes No
#### Prediction: Correct Score 1T
home team Correct Score away team
H - A 0 1 2
The algorithm indicates the presence low of the probabilistic conditions for which both teams could score a goal in the first half. The percentage is 22.34%.
## Vitória - BA - Criciúma - SC: Second Half
We are able to show you predictions on the second half crossing the historical data in our possession related only to the final phases of the game.
The two teams do not score much during the second half and the trend should be repeated.
Percentage
0.5 1.5
Yes No
#### Prediction: Correct Score 2T
home team Correct Score away team
H - A 0 1 2
As you can see from the probability percentage of 10.25%, in this match there is an eventuality low that both teams score in the second half.
## Vitória - BA - Criciúma - SC: Summary of Predictions
In short, the summary of the prediction of this football match
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### 3 Most Probable Results 2T
?
We wrote everything! If you like the site, and it is useful, we invite you to subscribe to our Social pages and post a positive review. | 1,409 | 5,306 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.078125 | 3 | CC-MAIN-2019-30 | latest | en | 0.952525 |
http://www.wolfram.com/mathematica/new-in-8/graph-and-network-modeling/illustrate-transformations.pt-br.html | 1,531,932,329,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676590295.61/warc/CC-MAIN-20180718154631-20180718174631-00177.warc.gz | 557,961,448 | 2,348 | # Ilustre transformações
Destaque elementos em grafos para ilustrar operações no Mathematica 8.
In[1]:= Xcolor = ColorData[7, "ColorList"][[4 ;; 9]]; color = ColorData["Rainbow"] /@ (Range[6]/6);
In[2]:= Xstyle = {VertexStyle -> Directive[Hue[.6, .3, 1], EdgeForm[Hue[.6, .8, .7]]], VertexSize -> Small, EdgeStyle -> Hue[.6, .8, .5], ImageSize -> {200, 100}};
In[3]:= Xg = Graph[{1 \[UndirectedEdge] 2, 2 \[UndirectedEdge] 3, 3 \[UndirectedEdge] 4, 4 \[UndirectedEdge] 1, 5 \[UndirectedEdge] 1, 5 \[UndirectedEdge] 2}, style];
In[4]:= Xname = {"Line Graph", "Graph Complement", "Graph Power"};
In[5]:= Xinfo = {{HighlightGraph[g, MapThread[Style[#1, #2] &, {EdgeList[g], color}], EdgeStyle -> Directive[Thickness[0.015], Opacity[1]], VertexSize -> Tiny], \!\(\* GraphicsBox[ {Thickness[0.05263157894736842], FaceForm[{RGBColor[0.7, 0.7, 0.7], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, { 0, 1, 0}, {0, 1, 0}}}, {{{9.9999, 18.}, {9.9999, 13.5}, {-0.00009999999999976694, 13.5}, {-0.00009999999999976694, 4.5}, {9.9999, 4.5}, { 9.9999, 0.}, {18.9999, 9.}}}]}, AspectRatio->Automatic, ImageSize->{19., 18.}, PlotRange->{{0., 19.}, {0., 18.}}]\), LineGraph[g, VertexStyle -> Thread[{1, 4, 3, 2, 6, 5} -> color], VertexSize -> 0.3, style]}, {g = CycleGraph[5, style], \!\(\* GraphicsBox[ {Thickness[0.05263157894736842], FaceForm[{RGBColor[0.7, 0.7, 0.7], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, { 0, 1, 0}, {0, 1, 0}}}, {{{9.9999, 18.}, {9.9999, 13.5}, {-0.00009999999999976694, 13.5}, {-0.00009999999999976694, 4.5}, {9.9999, 4.5}, { 9.9999, 0.}, {18.9999, 9.}}}]}, AspectRatio->Automatic, ImageSize->{19., 18.}, PlotRange->{{0., 19.}, {0., 18.}}]\), HighlightGraph[CompleteGraph[5, style], GraphComplement[g]]}, {g = CycleGraph[5, style], \!\(\* GraphicsBox[ {Thickness[0.05263157894736842], FaceForm[{RGBColor[0.7, 0.7, 0.7], Opacity[1.]}], FilledCurveBox[{{{0, 2, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0}, { 0, 1, 0}, {0, 1, 0}}}, {{{9.9999, 18.}, {9.9999, 13.5}, {-0.00009999999999976694, 13.5}, {-0.00009999999999976694, 4.5}, {9.9999, 4.5}, { 9.9999, 0.}, {18.9999, 9.}}}]}, AspectRatio->Automatic, ImageSize->{19., 18.}, PlotRange->{{0., 19.}, {0., 18.}}]\), HighlightGraph[GraphPower[g, 2], g, style]}};
In[6]:= XShowPlotSamples[name_, info_] := Framed[Column[{Style[name, 14, Bold, FontFamily -> "Verdana"], Grid[{info}]}, Alignment -> Center], RoundingRadius -> 9, FrameStyle -> None, Background -> GrayLevel@0.90]; | 1,112 | 2,493 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2018-30 | latest | en | 0.350746 |
https://supportforums.cisco.com/t5/getting-started-with-lans/way-to-aggregation-subnet/td-p/919960 | 1,516,229,476,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084886979.9/warc/CC-MAIN-20180117212700-20180117232700-00458.warc.gz | 820,656,494 | 30,796 | cancel
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## Way to Aggregation Subnet
I have this question.
Say I have the these range of subnets:
172.18.4.128/25
172.18.4.0/25
172.18.5.0/24
172.18.6.0/24
172.18.7.0/24
And the closest super subnet is 172.18.4.0/22
1) I heard from my friend that the lowest number will usually be the super subnet number. Is it true?
2) What is the most fastest way to calculate the super subnet IP?
2 REPLIES
Hall of Fame Super Blue
## Re: Way to Aggregation Subnet
Hi
1) The supernet will be 172.18.4.0/22
2) The way i would work it out is
256 - 252 = 4.
So your networks increase by 4 and will take you the next multiple of 4 ie.
172.18.0.0/22
172.18.4.0/22
172.18.8.0/22
etc..
Hope this makes sense
Jon
Silver
## Re: Way to Aggregation Subnet
Hi Kian
Just to add to what Jon has posted.
Don't get sucked into the "lowest number will be the supernet number" idea.
When you are given IP ranges like your example where they start and finish on a supernet boundry, then yes, the lowest subnet will be the supernet number.
However suppose you were given
172.18.5.0/24
172.18.6.0/24
172.18.7.0/25
172.18.7.128/25
172.18.8.0/23
172.18.10.0/24
172.18.11.0/24
172.18.12.0/24
In this instance the lowest subnet number does not fall cleanly on a supernet boundry and therefore would NOT be the supernet number.
Personally speaking I would summarise these subnets as follows
172.18.5.0/24
172.18.6.0/23
172.18.8.0/22
172.18.12.0/24
There by reducing the number of routes in a routing table from 8 down to 4.
HTH
Best Regards,
Michael
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0 | 536 | 1,629 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2018-05 | latest | en | 0.838938 |
http://forums.wolfram.com/mathgroup/archive/2011/Mar/msg00077.html | 1,721,697,941,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763517931.85/warc/CC-MAIN-20240723011453-20240723041453-00296.warc.gz | 15,708,672 | 7,894 | Re: Vector Runge-Kutta ODE solver with compilation?
• To: mathgroup at smc.vnet.net
• Subject: [mg116870] Re: Vector Runge-Kutta ODE solver with compilation?
• From: DmitryG <einschlag at gmail.com>
• Date: Thu, 3 Mar 2011 05:56:09 -0500 (EST)
• References: <ikihnp\$7sm\$1@smc.vnet.net> <ikl30v\$sra\$1@smc.vnet.net>
```On 2 Mrz., 04:35, DmitryG <einsch... at gmail.com> wrote:
> Continuing efforts to vectorize NDSolve. Now the vectorized code
>
> *********************************************************************
> NN = 1000; tMax = 50;
> x0 = Table[RandomReal[{0, 1}], {i, 1, NN}];
> Equations = x'[t] == - x[t]/(1 + 300 Total[x[t]]^2/NN^2);
>
> Timing[Solution = NDSolve[{Equations, x[0] == x0}, x, {t, 0, tMax},
> MaxSteps -> 1000000]]
>
> xt[t_] := x[t] /. Solution[[1]];
>
> Plot[{xt[t][[1]], xt[t][[2]], xt[t][[3]]}, {t, 0, 50}, PlotStyle ->
> {{Thick, Red}, {Thick, Green}, {Thick, Blue}}, PlotRange -> {0, 1}]
>
> *************************************************************************** **
> computes something, the solution x[t] is avector, but the resulting
> plot differs from the plot generated by the non-vectorized version
>
> *************************************************************************** ***
> NN = 1000; tMax = 50;
> x0 = Table[RandomReal[{0, 1}], {i, 1, NN}];
> IniConds = Table[x[i][0] == x0[[i]], {i, 1, NN}];
> Vars = Table[x[i], {i, 1, NN}];
> Timing[Equations = Table[x[i]'[t] == -x[i][t]/(1 + 300 Sum[x[j][t],
> {j, 1, NN}]^2/NN^2), {i, 1,NN}];]
>
> Timing[Solution = NDSolve[Join[Equations, IniConds], Vars, {t, 0,
> tMax}, MaxSteps -> 1000000)];
>
> x1t[t_] := x[1][t] /. Solution[[1]];
> x2t[t_] := x[2][t] /. Solution[[1]];
> x3t[t_] := x[3][t] /. Solution[[1]];
> Plot[{x1t[t], x2t[t], x3t[t]}, {t, 0, tMax}, PlotStyle -> {{Thick,
> Red}, {Thick, Green}, {Thick, Blue}}]
>
> **************************************************************************
>
> and the RK4 routine above. Although the vectorized program is much
> faster, its results are wrong. Where is the error??
>
> Best,
>
> Dmitry
I see that in my vectorized equation Total[x[t]] is misinterpreted,
possibly because Mathematica does not know x[t] is a vector.
Unfortunately I do not find in Help how to tell Mathematica x[t] is a
vector.
Dmitry
```
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Solving Systems of Equations by the Substitution Method
Indicate whether each system is independent, inconsistent or dependent
x+2y=1
8x+6y=4
x-5y=4
4x+8y = -5
Solution Preview
Indicate whether each system is independent, inconsistent or dependent
x+2y=1 --(1)
8x+6y=4 --(2)
From(1), x = 1-2y
Substitute this in ...
Solution Summary
Solving Systems of Equations by the Substitution Method is investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.
\$2.19 | 164 | 585 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2017-04 | latest | en | 0.854718 |
https://www.sololearn.com/ru/Discuss/2563352/phone-number-validator | 1,725,757,889,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700650926.21/warc/CC-MAIN-20240907225010-20240908015010-00649.warc.gz | 963,152,363 | 188,750 | + 2
# Phone number validator
I was able to pass the challenge with this code, but when I test it the code is flawed. I can start with any number and get valid. https://code.sololearn.com/cX5xUa0DxB2r/?ref=app
26th Oct 2020, 5:37 PM
Nate Toon
6 ответов
+ 8
Can you share a specific test case that isn't behaving correctly? "1234abcd" starts with a number and is invalid and the code correctly indicates "Invalid". "12345678" prints "Valid" which is probably what you want. This is the code I saw when I clicked your link in case you update it: import re num = input() pattern = r"^{^1|8|9}|{^1-9}" if re.search(pattern, num) and len(num) == 8: print("Valid") else: print("Invalid")
27th Oct 2020, 8:53 PM
Josh Greig
+ 3
import re number = input() pattern = r"[189][0-9]{7}\Z" matches = re.match(pattern, number) if matches: print("Valid") else: print('Invalid') This works Well
23rd Jun 2021, 3:43 PM
Avid_Coder
+ 2
I’ve now fixed the code pattern = r"^[1|8|9][0-9]{7}
quot;
28th Oct 2020, 1:51 PM
Nate Toon
0
This one got me through. import re number = input() pattern = r"[189]\d{7}
quot; if re.match(pattern, number): print("Valid") else: print("Invalid")
11th Feb 2022, 3:27 PM
جاراللّه خان
0
import re num = input() pattern = r"^[1|8|9].{7}" if re.match(pattern, num) and len(num) == 8: print("Valid") else: print("Invalid") this one was nice and compact
29th Feb 2024, 2:07 AM
Jamie Enge
- 2
68963279 this prints valid
28th Oct 2020, 1:32 PM
Nate Toon | 499 | 1,461 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2024-38 | latest | en | 0.708111 |
https://courses.lumenlearning.com/intermediatealgebra/chapter/outcome-problem-solving-2/ | 1,709,258,457,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474893.90/warc/CC-MAIN-20240229234355-20240301024355-00032.warc.gz | 183,110,494 | 16,492 | What You Will Learn to Do: Develop and Use a Method for Solving Mathematical Problems
Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus an additional charge if you exceed your data plan; a car rental company charges a daily fee plus an amount per mile driven; a restaurant offers a base price for a burrito plus additional charges for extra toppings, like guacamole and sour cream. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems. | 129 | 649 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2024-10 | latest | en | 0.91055 |
http://fratellicalafuria.com/multiplication-multi-digit | 1,571,396,828,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986682037.37/warc/CC-MAIN-20191018104351-20191018131851-00330.warc.gz | 78,490,466 | 7,353 | Multiplication Multi Digit.ZAP!! Multi Digit Multiplication Game!! Product From . Multi Digit Multiplication On Graph Paper For Neatness TpT. Multiplication Bump Games - Games 4 Gains - Resume Design Ideas
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Or maybe you google math problems ten times a day because you've forgotten how to do any math beyond your basic multiplication to square a two digit number round it first say you need to square | 465 | 2,557 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.578125 | 4 | CC-MAIN-2019-43 | latest | en | 0.928887 |
https://www.safaribooksonline.com/library/view/excel-2007-introduction/00010CWORKS/ | 1,534,577,421,000,000,000 | text/html | crawl-data/CC-MAIN-2018-34/segments/1534221213405.46/warc/CC-MAIN-20180818060150-20180818080150-00324.warc.gz | 967,333,940 | 8,327 | ## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.
No credit card required
## Video Description
This class teaches the basic functions and features of Excel 2007. After an introduction to spreadsheet terminology and Excel's window components, students will learn how to use the Help system and navigate worksheets and workbooks. Then they will enter and edit text, values, formulas, and pictures, and they will save workbooks in various formats. Students will also move and copy data, learn about absolute and relative references, and work with ranges, rows, and columns. This course also covers simple functions, basic formatting techniques, and printing. Finally, students will create and modify charts, and learn how to manage large workbooks.
1. Introduction
1. Introduction 00:01:10
2. Getting Started
1. Understanding the Three Primary Functions of Excel 00:01:45
3. Launching Excel 00:01:20
4. Exploring the Excel Interface 00:03:57
5. Opening a Workbook 00:01:42
6. Examining Formulas 00:01:08
7. Understanding Worksheets 00:01:35
8. Using Zoom 00:01:27
9. Changing Views 00:01:40
3. Basic Worksheet Skills
1. Creating a New Workbook 00:02:18
2. Entering Data 00:05:26
3. Selecting Parts of a Workbook 00:01:21
4. Using Multiple Worksheets 00:02:20
5. Using AutoFill 00:09:27
6. Deleting Ranges 00:03:53
7. Using Undo/Redo 00:01:25
8. Deleting a Worksheet 00:00:58
9. Saving a Workbook 00:04:36
4. Creating Formulas
1. Understanding Order of Operations 00:02:22
2. Entering Simple Formulas 00:02:22
3. Using SUM, MAX, and MIN 00:08:38
4. Using AutoSum 00:03:10
5. Using AVERAGE 00:03:07
6. Creating a Formula 00:03:53
7. Working with Date Arithmetic 00:02:46
8. Using AutoCalculate 00:02:11
9. Relative vs. Absolute Cell Adressing 00:04:45
5. Formatting the Worksheet
1. Adjusting Column Width and Row Height 00:05:57
2. Inserting and Deleting Rows and Columns 00:02:15
3. Hiding Rows and Columns 00:01:58
4. Changing Number Formatting 00:04:03
5. Changing Character Formatting 00:03:03
6. Aligning Data 00:05:59
8. Using Format Painter 00:05:31
9. Using Cell Styles 00:03:19
10. Using Themes 00:01:36
6. Working with Large Worksheets
1. Using Zoom 00:02:17
2. Freezing and Unfreezing Panes 00:03:41
3. Splitting a Window 00:03:13
4. Tiling Multiple Workbooks 00:03:56
5. Using Synchronous Scrolling 00:03:22
7. Using Protection and Comment Boxes
2. Locking and Unlocking Cell Ranges 00:08:34
8. Using Page Setup and Printing
1. Using Spell Check 00:03:55
2. Inserting Page Breaks 00:02:37
3. Using Page Break Preview 00:01:40
4. Using Print Preview 00:02:23
5. Changing the Orientation and Margins 00:02:45
6. Working With Print Areas 00:02:19
7. Setting Print Titles 00:01:58
9. Printing Gridlines and Comments 00:00:58
10. Using the Print Dialog Box 00:01:28
9. Conclusion
1. Conclusion 00:01:19 | 895 | 2,907 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2018-34 | latest | en | 0.755627 |
https://gsddata.com/qa/how-much-do-you-need-to-make-to-afford-a-150k-house.html | 1,627,455,506,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046153531.10/warc/CC-MAIN-20210728060744-20210728090744-00462.warc.gz | 298,761,019 | 9,019 | # How Much Do You Need To Make To Afford A 150k House?
## What house can I afford on 70k a year?
According to Brown, you should spend between 28% to 36% of your take-home income on your housing payment.
If you make \$70,000 a year, your monthly take-home pay, including tax deductions, will be approximately \$4,328..
## What house can I afford on 80k a year?
So, if you make \$80,000 a year, you should be looking at homes priced between \$240,000 to \$320,000. You can further limit this range by figuring out a comfortable monthly mortgage payment. To do this, take your monthly after-tax income, subtract all current debt payments and then multiply that number by 25%.
## How much income is needed for a 300k mortgage?
What income is needed for a 300k mortgage? A \$300k mortgage with a 4.5% interest rate over 30 years and a \$10k down-payment will require an annual income of \$74,581 to qualify for the loan.
## How much do I need to make to afford a 350k house?
Income to Afford a \$350,000 HouseDown Payment2.50%3.50%\$52,500\$50,378\$57,253\$70,000\$47,415\$53,885\$87,500\$44,451\$50,518\$105,000\$41,488\$47,1507 more rows
## How much do I need to make to buy a 170k house?
How much do you need to make to be able to afford a house that costs \$170,000? To afford a house that costs \$170,000 with a down payment of \$34,000, you’d need to earn \$25,366 per year before tax. The monthly mortgage payment would be \$592. Salary needed for 170,000 dollar mortgage.
## How much do I need to make to afford a 160k house?
How much do you need to make to be able to afford a house that costs \$160,000? To afford a house that costs \$160,000 with a down payment of \$32,000, you’d need to earn \$23,874 per year before tax. The monthly mortgage payment would be \$557. Salary needed for 160,000 dollar mortgage.
## Can I buy a house if I only make 20 000 a year?
How Much Mortgage Do I Qualify for If I Make \$20,000 a Year? As discussed above, a home loan lender does not want your monthly mortgage to surpass 28% of your monthly income, which means if you make \$20,000 a year or \$1,676 a month, your monthly mortgage payment should not exceed \$469.
## How much income do I need for a 150k mortgage?
Example Required Income Levels at Various Home Loan AmountsHome PriceDown PaymentAnnual Income\$100,000\$20,000\$30,905.31\$150,000\$30,000\$40,107.97\$200,000\$40,000\$49,310.63\$250,000\$50,000\$58,513.2815 more rows
## How much do I need to make to buy a \$200 K House?
To afford a house that costs \$200,000 with a down payment of \$40,000, you’d need to earn \$29,843 per year before tax. The monthly mortgage payment would be \$696.
## How much do I need to make to afford a 250k house?
How much do you need to make to be able to afford a house that costs \$250,000? To afford a house that costs \$250,000 with a down payment of \$50,000, you’d need to earn \$37,303 per year before tax. The monthly mortgage payment would be \$870. Salary needed for 250,000 dollar mortgage.
## How much income do I need for a 400k mortgage?
To afford a \$400,000 house, for example, you need about \$55,600 in cash if you put 10% down. With a 4.25% 30-year mortgage, your monthly income should be at least \$8178 and (if your income is \$8178) your monthly payments on existing debt should not exceed \$981.
## Is 40k a good starting salary?
A \$40,000 a year salary is slightly less than the median personal income in the U.S., but that doesn’t mean it’s bad. Depending on where you live, your circumstances, and your lifestyle, \$40k a year might be more than enough to live comfortably and put money toward your goals.
## How much do I need to earn for a 150000 mortgage?
So, to borrow £150,000, at most lenders the combined salary of everyone who is going on the mortgage would need to be £37,500. Some lenders will accept £30,000, and a minority of them will offer you a loan of this amount if you earn £25,000.
## Can I buy a house making 40k a year?
Yes, you can! Your mortgage payment including taxes and insurance will be around \$1,178.78. 81 (4.625% rate due to low fico score and low downpayment). Based on the information you provided, your Debt-to-income ratio is around 40% which makes you a qualified buyer.
## How much do you have to make to afford a \$300000 house?
To afford a house that costs \$300,000 with a down payment of \$60,000, you’d need to earn \$52,116 per year before tax. The monthly mortgage payment would be \$1,216. Salary needed for 300,000 dollar mortgage.
## What house can I afford on 50k a year?
A person who makes \$50,000 a year might afford a house worth anywhere from \$180,000 to nearly \$300,000. That’s because salary isn’t the only thing that determines your home buying budget. You also have to factor in credit score, current debts, mortgage rates, and many other factors.
## How much do you have to make a year to afford a \$600000 house?
How much do you need to make to be able to afford a house that costs \$600,000? To afford a house that costs \$600,000 with a down payment of \$120,000, you’d need to earn \$89,528 per year before tax. The monthly mortgage payment would be \$2,089. Salary needed for 600,000 dollar mortgage.
## How much house can I afford making 120k a year?
Multiply Your Annual Income By 2.5 or 3 Simply take your gross income and multiply it by 2.5 or 3, to get the maximum value of the home you can afford. For somebody making \$100,000 a year, the maximum purchase price on a new home should be somewhere between \$250,000 and \$300,000. | 1,471 | 5,555 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2021-31 | latest | en | 0.945563 |
https://mywikibiz.com/index.php?title=Ordinal_number&oldid=6060 | 1,674,971,254,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499700.67/warc/CC-MAIN-20230129044527-20230129074527-00257.warc.gz | 436,020,476 | 20,272 | # Ordinal number
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc., whereas a cardinal number says "how many there are": one, two, three, four, etc. (See How to name numbers.)
Here, we describe the mathematical meaning of transfinite ordinal numbers. They were introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. Ordinals are an extension of the natural numbers different from integers and from cardinals.
Well-ordering is total ordering with transfinite induction, where transfinite induction extends mathematical induction beyond the finite. Ordinals represent equivalence classes of well orderings with order-isomorphism being the equivalence relationship. Each ordinal is taken to be the set of smaller ordinals. Ordinals may be categorized as: zero, successor ordinals, and limit ordinals (of various cofinalities). Given a class of ordinals, one can identify the α-th member of that class, i.e. one can index (count) them. A class is closed and unbounded if its indexing function is continuous and never stops. One can define addition, multiplication, and exponentiation on ordinals, but not subtraction or division. The Cantor normal form is a standardized way of writing down ordinals. There is a many to one association from ordinals to cardinals. Larger and larger ordinals can be defined, but they become more and more difficult to describe. Ordinals have a natural topology.
## Ordinals extend the natural numbers
A natural number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. While in the finite world these two concepts coincide, when dealing with infinite sets one has to distinguish between the two. The notion of size leads to cardinal numbers, which were also discovered by Cantor, while the position is generalized by the ordinal numbers described here.
Whereas the notion of cardinal number is associated to a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets which are called well-ordered (so intimately linked, in fact, that some mathematicians make no distinction between the two concepts). To define things briefly, a well-ordered set is a totally ordered set (given any two elements one defines a smaller and a larger one in a coherent way) in which there is no infinite decreasing sequence (however, there may be infinite increasing sequences). Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labeled 0, the one after that 1, the next one 2, "and so on") and to measure the "length" of the whole set by the least ordinal which is not a label for an element of the set. This "length" is called the order type of the set.
Any ordinal is defined by the set of ordinals that precede it: in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the order type of the ordinals less than it, i.e., the ordinals from 0 (the smallest of all ordinals) to 41 (the immediate predecessor of 42), and it is generally identified as the set {0,1,2,…,41}. Conversely, any set of ordinals which is downward-closed—meaning that any ordinal less than an ordinal in the set is also in the set—is (or can be identified with) an ordinal.
So far we have mentioned only finite ordinals, which are the natural numbers. But there are infinite ones as well: the smallest infinite ordinal is ω, which is the order type of the natural numbers (finite ordinals) and which can even be identified with the set of natural numbers (indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed it can be identified with the ordinal associated to it, which is exactly how we define ω).
File:Omega squared.png
A graphical “matchstick” representation of the ordinal ω². Each stick correspond to an ordinal of the form ω·m+n where m and n are natural numbers.
Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals which we form in this way (the ω·m+n, where m and n are natural numbers) must itself have an ordinal associated to it: and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωω², and much later on ε0 (just to give a few examples of the very smallest—countable—ordinals). We can go on in this way indefinitely far ("indefinitely far" is exactly what ordinals are good at: basically every time one says "and so on" when enumerating ordinals, it defines a larger ordinal).
## Definitions
### Define well-ordered set
A well-ordered set is an ordered set in which every non-empty subset has a least element: this is equivalent (at least in the presence of the axiom of dependent choices) to just saying that the set is totally ordered and there is no infinite decreasing sequence, something which is perhaps easier to visualize. In practice, the importance of well-ordering is justified by the possibility of applying transfinite induction, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation (computer program or game) can be well-ordered in such a way that each step is followed by a "lower" step, then you can be sure that the computation will terminate.
Now we don't want to distinguish between two well-ordered sets if they only differ in the "labeling of their elements", or more formally: if we can pair off the elements of the first set with the elements of the second set such that if one element is smaller than another in the first set, then the partner of the first element is smaller than the partner of the second element in the second set, and vice versa. Such a one-to-one correspondence is called an order isomorphism and the two well-ordered sets are said to be order-isomorphic, or similar (obviously this is an equivalence relation). Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the sets as essentially identical, and to seek a "canonical" representative of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set.
So we essentially wish to define an ordinal as an isomorphism class of well-ordered sets: that is, as an equivalence class for the equivalence relation of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set theory. But this is not a serious difficulty. We will say that the ordinal is the order type of any set in the class.
### Definition of an ordinal as an equivalence class
The original definition of ordinal number, found for example in Principia Mathematica, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in ZF and related systems of axiomatic set theory because these equivalence classes are too large to form a set. However, this definition still can be used in type theory and in Quine's set theory New Foundations and related systems (where it affords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).
### Von Neumann definition of ordinals
Rather than defining an ordinal as an equivalence class of well-ordered sets, we can try to define it as some particular well-ordered set which (canonically) represents the class. Thus, we want to construct ordinal numbers as special well-ordered sets in such a way that every well-ordered set is order-isomorphic to one and only one ordinal number.
The ingenious definition suggested by John von Neumann, and which is now taken as standard, is this: define each ordinal as a special well-ordered set, namely that of all ordinals before it. Formally:
A set S is an ordinal if and only if S is totally ordered with respect to set containment and every element of S is also a subset of S.
(Here, "set containment" is another name for the subset relationship.) Such a set S is automatically well-ordered with respect to set containment. This relies on the axiom of well foundation: every nonempty set B has an element b which is disjoint from B.
Note that the natural numbers are ordinals by this definition. For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2 is equal to {0, 1} and so it is a subset of {0, 1, 2, 3}.
It can be shown by transfinite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals.
Furthermore, the elements of every ordinal are ordinals themselves. Whenever you have two ordinals S and T, S is an element of T if and only if S is a proper subset of T, and moreover, either S is an element of T, or T is an element of S, or they are equal. So every set of ordinals is totally ordered. And in fact, much more is true: Every set of ordinals is well-ordered. This important result generalizes the fact that every set of natural numbers is well-ordered and it allows us to use transfinite induction liberally with ordinals.
Another consequence is that every ordinal S is a set having as elements precisely the ordinals smaller than S. This statement completely determines the set-theoretic structure of every ordinal in terms of other ordinals. It is used to prove many other useful results about ordinals. One example of these is an important characterization of the order relation between ordinals: every set of ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set. Another example is the fact that the collection of all ordinals is not a set. Indeed, since every ordinal contains only other ordinals, it follows that every member of the collection of all ordinals is also its subset. Thus, if that collection were a set, it would have to be an ordinal itself by definition; then it would be its own member, which contradicts the axiom of regularity. (See also the Burali-Forti paradox). The class of all ordinals is variously called "Ord", "ON", or "∞".
An ordinal is finite if and only if the opposite order is also well-ordered, which is the case if and only if each of its subsets has a greatest element.
### Other definitions
There are other modern formulations of the definition of ordinal. Each of these is essentially equivalent to the definition given above. One of these definitions is the following. A class S is called transitive if each element x of S is a subset of S, i.e. $$y \in x \in S \Longrightarrow y \in S$$. An ordinal is then defined to be a transitive set whose members are also transitive. It follows from this that the members are themselves ordinals. Note that the axiom of regularity (foundation) is used in showing that these ordinals are well ordered by containment (subset).
## Transfinite induction
### What is transfinite induction?
Transfinite induction holds in any well-ordered set, but it is so important in relation to ordinals that it is worth restating here.
Any property which passes from the set of ordinals smaller than a given ordinal α to α itself, is true of all ordinals.
That is, if P(α) is true whenever P(β) is true for all β<α, then P(α) is true for all α. Or, more practically: in order to prove a property P for all ordinals α, one can assume that it is already known for all smaller β<α.
### Transfinite recursion
Transfinite induction can be used not only to prove things, but also to define them (such a definition is normally said to follow by transfinite recursion - we use transfinite induction to prove that the result is well-defined): the formal statement is tedious to write, but the bottom line is, in order to define a (class) function on the ordinals α, one can assume that it is already defined for all smaller β<α. One proves by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α.
Here is an example of definition by transfinite induction on the ordinals (more will be given later): define a function F by letting F(α) be the smallest ordinal not in the set of F(β) for all β<α. Note how we assume the F(β) known in the very process of defining F: this apparent paradox is exactly what definition by transfinite induction permits. Now in fact F(0) makes sense since there is no β<0, so the set of all F(β) for β<0 is empty, so F(0) must be 0 (the smallest ordinal of all), and now that we know F(0), then F(1) makes sense (and it is the smallest ordinal not equal to F(0)=0), and so on (the and so on is exactly transfinite induction). Well, it turns out that this example is not very interesting since F(α)=α for all ordinals α: but this can be shown, precisely, by transfinite induction.
### Successor and limit ordinals
Any nonzero ordinal has a smallest element, to wit zero. It may or may not have a largest element. For example, 42 has a largest element 41 and ω+6 has a largest element ω+5. On the other hand, ω does not have a largest element since there is no largest natural number. If an ordinal has a largest element α, then it is the next ordinal after α, and it is called a successor ordinal, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is $$\alpha\cup\{\alpha\}$$ since its elements are those of α and α itself.
A nonzero ordinal which is not a successor is called a limit ordinal. One justification for this term is that a limit ordinal is indeed the limit in a topological sense of all smaller ordinals (for the order topology).
When $$\langle \alpha_{\iota} | \iota < \gamma \rangle$$ is a sequence of ordinals indexed by a limit γ and the sequence is increasing, i.e. $$\alpha_{\iota} < \alpha_{\rho}\!$$ whenever $$\iota < \rho,\!$$ we define its limit to be the least upper bound of the set $$\{ \alpha_{\iota} | \iota < \gamma \},\!$$ that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself).
Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if:
There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξ such that ζ<ξ<α.
So in the following sequence
0, 1, 2, ... , ω, ω+1
ω is a limit ordinal because for any ordinal (in this example, a natural number) we can find another ordinal (natural number) larger than it, but still less than ω.
Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite induction rely upon it. Very often, when defining a function F by transfinite induction on all ordinals, one defines F(0), and F(α+1) assuming F(α) is defined, and then, for limit ordinals δ one defines F(δ) as the limit of the F(β) for all β<δ (either in the sense of ordinal limits, as we have just explained, or for some other notion of limit if F does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially for F nondecreasing and taking ordinal values) are called continuous. We will see that ordinal addition, multiplication and exponentiation are continuous as functions of their second argument.
### Indexing classes of ordinals
We have mentioned that any well-ordered set is similar (order-isomorphic) to a unique ordinal number $$\alpha$$, or, in other words, that its elements can be indexed in increasing fashion by the ordinals less than $$\alpha$$. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some $$\alpha$$. The same holds, with a slight modification, for classes of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded, this puts it in class-bijection with the class of all ordinals). So we can freely speak of the $$\gamma$$-th element in the class (with the convention that the “0-th” is the smallest, the “1-th” is the next smallest, and so on). Formally, the definition is by transfinite induction: the $$\gamma$$-th element of the class is defined (provided it has already been defined for all $$\beta<\gamma$$), as the smallest element greater than the $$\beta$$-th element for all $$\beta<\gamma$$.
We can apply this, for example, to the class of limit ordinals: the $$\gamma$$-th ordinal which is either a limit or zero is $$\omega\cdot\gamma$$ (see ordinal arithmetic for the definition of multiplication of ordinals). Similarly, we can consider ordinals which are additively indecomposable (meaning that it is a nonzero ordinal which is not the sum of two strictly smaller ordinals): the $$\gamma$$-th additively indecomposable ordinal is indexed as $$\omega^\gamma$$. The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the $$\gamma$$-th ordinal $$\alpha$$ such that $$\omega^\alpha=\alpha$$ is written $$\varepsilon_\gamma$$. These are called the "epsilon numbers".
### Closed unbounded sets and classes
A class of ordinals is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it (then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function $$F$$ is continuous in the sense that, for $$\delta$$ a limit ordinal, $$F(\delta)$$ (the $$\delta$$-th ordinal in the class) is the limit of all $$F(\gamma)$$ for $$\gamma<\delta$$; this is also the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent).
Of particular importance are those classes of ordinals which are closed and unbounded, sometimes called clubs. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of $$\varepsilon_\cdot$$ ordinals, or the class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded.
A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary and stationary classes are unbounded, but there are stationary classes which are not closed and there are stationary classes which have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality.
Rather than formulating these definitions for (proper) classes of ordinals, we can formulate them for sets of ordinals below a given ordinal $$\alpha$$: A subset of a limit ordinal $$\alpha$$ is said to be unbounded (or cofinal) under $$\alpha$$ provided any ordinal less than $$\alpha$$ is less than some ordinal in the set. More generally, we can call a subset of any ordinal $$\alpha$$ cofinal in $$\alpha$$ provided every ordinal less than $$\alpha$$ is less than or equal to some ordinal in the set. The subset is said to be closed under $$\alpha$$ provided it is closed for the order topology in $$\alpha$$, i.e. a limit of ordinals in the set is either in the set or equal to $$\alpha$$ itself.
## Arithmetic of ordinals
For more details on this topic, see ordinal arithmetic.
There are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the operation or by using transfinite recursion. The Cantor normal form provides a standardized way of writing ordinals. The so-called "natural" arithmetical operations retain commutativity at the expense of continuity.
## Ordinals and cardinals
### Initial ordinal of a cardinal
Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its order-type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal is a cardinal.
The α-th infinite initial ordinal is written $$\omega_\alpha$$. Its cardinality is written $$\aleph_\alpha$$. For example, the cardinality of ω0 = ω is $$\aleph_0$$, which is also the cardinality of ω² or ε0 (all are countable ordinals). So (assuming the axiom of choice) we identify ω with $$\aleph_0$$, except that the notation $$\aleph_0$$ is used when writing cardinals, and ω when writing ordinals (this is important since $$\aleph_0^2=\aleph_0$$ whereas $$\omega^2>\omega$$). Also, $$\omega_1$$ is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and $$\omega_1$$ is the order type of that set), $$\omega_2$$ is the smallest ordinal whose cardinality is greater than $$\aleph_1$$, and so on, and $$\omega_\omega$$ is the limit of the $$\omega_n$$ for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the $$\omega_n$$).
### Cofinality
The cofinality of an ordinal $$\alpha$$ is the smallest ordinal $$\delta$$ which is the order type of a cofinal subset of $$\alpha$$. Notice that a number of authors define confinality or use it only for limit ordinals. The cofinality of a set of ordinals or any other well ordered set is the cofinality of the order type of that set.
Thus for a limit ordinal, there exists a $$\delta$$-indexed strictly increasing sequence with limit $$\alpha$$. For example, the cofinality of ω² is ω, because the sequence ω·m (where m ranges over the natural numbers) tends to ω²; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does $$\omega_\omega$$ or an uncountable cofinality.
The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1. The cofinality of any limit ordinal is at least $$\omega$$.
An ordinal which is equal to its cofinality is called regular and it is always an initial ordinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular which it usually is not. If the Axiom of Choice, then $$\omega_{\alpha+1}$$ is regular for each α. In this case, the ordinals 0, 1, $$\omega$$, $$\omega_1$$, and $$\omega_2$$ are regular, whereas 2, 3, $$\omega_\omega$$, and ωω·2 are initial ordinals which are not regular.
The cofinality of any ordinal α is a regular ordinal, i.e. the cofinality of the cofinality of α is the same as the cofinality of α. So the cofinality operation is idempotent.
## Some “large” countable ordinals
For more details on this topic, see Large countable ordinals.
We have already mentioned (see Cantor normal form) the ordinal ε0, which is the smallest satisfying the equation $$\omega^\alpha = \alpha$$, so it is the limit of the sequence 0, 1, $$\omega$$, $$\omega^\omega$$, $$\omega^{\omega^\omega}$$, etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the $$\iota$$-th ordinal such that $$\omega^\alpha = \alpha$$ is called $$\varepsilon_\iota$$, then we could go on trying to find the $$\iota$$-th ordinal such that $$\varepsilon_\alpha = \alpha$$, “and so on”, but all the subtlety lies in the “and so on”). We can try to do this systematically, but no matter what system is used to define and construct ordinals, there is always an ordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal which limits in this manner a system of construction is the Church-Kleene ordinal, $$\omega_1^{\mathrm{CK}}$$ (despite the $$\omega_1$$ in the name, this ordinal is countable), which is the smallest ordinal which cannot in any way be represented by a computable function (this can be made rigorous, of course). Considerably large ordinals can be defined below $$\omega_1^{\mathrm{CK}}$$, however, which measure the “proof-theoretic strength” of certain formal systems (for example, $$\varepsilon_0$$ measures the strength of Peano arithmetic). Large ordinals can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.
## Topology and ordinals
For more details on this topic, see Order topology.
Any ordinal can be made into a topological space in a natural way by endowing it with the order topology. See the Topology and ordinals section of the "Order topology" article. | 6,234 | 26,849 | {"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} | 4.25 | 4 | CC-MAIN-2023-06 | latest | en | 0.937163 |
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Selecciona nivel educativo | 642 | 2,959 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.640625 | 4 | CC-MAIN-2023-23 | latest | en | 0.518295 |
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## New Brunswick - Application Of Partial Differential Equation In Mechanical Engineering
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https://codereview.stackexchange.com/questions/288312/python-optimization-search-pythagorean-triplets | 1,721,074,732,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514713.74/warc/CC-MAIN-20240715194155-20240715224155-00275.warc.gz | 155,818,500 | 40,016 | # Python optimization search pythagorean triplets
I have an optimization problem. Be it about code, algorithm or maths, that, i don't know. My goal is to find pythagorean triplets satisfying a = b + 1 or a = b - 1. m and n are the triplets generators. The following code produces the desired output, but it's not fast enough from about the 11th or 12th item.
#! /usr/bin/env python3
from time import perf_counter
import numpy as np
from numba import njit, prange
import json, os
@njit
def isPythagoreanTriplet(x:int, y:int) -> bool:
"""
Test if a and b are a valid basis for a pythagorean triplet
"""
return (abs(2*x - y) == 1)
@njit
def findNextPythagoreanTriplet(n:int)->tuple[int]:
"""
Self-explanatory function name
"""
m = 4 * n
while True:
for n in prange(1, m):
m_sq = m * m
n_sq = n * n
a = m_sq - n_sq
b = 2 * m * n
c = m_sq + n_sq
if isPythagoreanTriplet(a, b):
return(2*a, b, c, m, n)
elif isPythagoreanTriplet(b, a):
return(a, 2*b, c, m, n)
m+= 1
def pythagoreanTriplets(compteur:int, n:int=1)->None :
"""
Find compteur pythagorean triplets
"""
while compteur :
a, b, c, m, n = findNextPythagoreanTriplet(n)
store(a, b, c, m, n)
n = m
compteur-= 1
def store(a,b,c, m, n):
"""
Store a pythagorean triplet and its generators in a text file
"""
with open("results.txt", "a") as file_out:
file_out.write(f"{a} {b} {c} {m} {n}\n")
def init_n()->tuple[int]:
"""
Check if result text file exists to start from last stop for n
"""
nb_lignes = 0
n = 1
if os.path.exists("results.txt"):
with open("results.txt", "r") as file_in:
for line in lines:
line = line.strip()
if line:
nb_lignes+= 1
_, _, _, m, n = line.split()
n = m
return (nb_lignes, int(n))
# Driver Code
if __name__ == '__main__' :
nb_items = 12
start = perf_counter()
nb_lignes, n = init_n()
nb_items-= nb_lignes
print(f"Nb items restants : {nb_items}")
pythagoreanTriplets(nb_items, n)
stop = perf_counter()
duree = stop - start
print(f"Durée : {duree:.2f}s")
• Micro-review: PEP-8 recommends snake_case for function names. Commented Dec 8, 2023 at 14:43
I think you're coming at this with the wrong algorithm.
Given a right triangle with sides a, b and (hypotenuse) a+1, we have:
$$a² + b² = (a+1)²$$
So
$$b² = (a+1)² - a²$$ $$= a² + 2a + 1 - a²$$ $$= 2a + 1$$
From this, we can deduce that for any odd length for b (b > 1), we can find a corresponding a and therefore the other two sides.
Using that algorithm, I have this generator (lacking docstring and type annotations):
def generate_triplets():
b = 1
while True:
b += 2
a = b * b // 2
yield (a, b, a+1)
Simple demo:
import itertools
for triangle in itertools.islice(generate_triplets(), 50):
print(triangle) | 877 | 2,653 | {"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": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.296875 | 3 | CC-MAIN-2024-30 | latest | en | 0.621795 |
https://www.thestudentroom.co.uk/showthread.php?t=6979963 | 1,623,976,992,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487634576.73/warc/CC-MAIN-20210617222646-20210618012646-00499.warc.gz | 960,322,600 | 33,592 | # Maths
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#1
I need help with this question:
20) There are only red sweets and yellow sweets in a bag.
There are n red sweets in the bag.
There are 8 yellow sweets in the bag.
Sajid is going to take at random a sweet from the bag and eat it.
He says that the probability that the sweet will be red is
7/10(a) Show why the probability cannot be
7/10
0
2 months ago
#2
(Original post by raidas0001)
I need help with this question:
20) There are only red sweets and yellow sweets in a bag.
There are n red sweets in the bag.
There are 8 yellow sweets in the bag.
Sajid is going to take at random a sweet from the bag and eat it.
He says that the probability that the sweet will be red is
7/10(a) Show why the probability cannot be
7/10
2/10 ,because 8/10 + 2/10 = 10/10 ,since the 10 in the denominator reveals the number of candies in the bag as a whole
0
2 months ago
#3
although this question involves sweets, it should not really be in the Food and Drink forum
2
2 months ago
#4
(Original post by raidas0001)
I need help with this question:
20) There are only red sweets and yellow sweets in a bag.
There are n red sweets in the bag.
There are 8 yellow sweets in the bag.
Sajid is going to take at random a sweet from the bag and eat it.
He says that the probability that the sweet will be red is
7/10(a) Show why the probability cannot be
7/10
0
2 months ago
#5
I thought under 13s weren't allowed on TSR?
0
2 months ago
#6
(Original post by lucydales)
2/10 ,because 8/10 + 2/10 = 10/10 ,since the 10 in the denominator reveals the number of candies in the bag as a whole
No, that's not correct
0
#7
(Original post by Muttley79)
n/(n+8)=7/10
10n=7n+56
3n=56
n=18.66666666666666
if the probability of getting red is 7/10 then there is a decimal number of red sweets and u can't have 0.6666666667 of a sweet.
i am not sure if i'm correct
0
2 months ago
#8
(Original post by raidas0001)
n/(n+8)=7/10
10n=7n+56
3n=56
n=18.66666666666666
if the probability of getting red is 7/10 then there is a decimal number of red sweets and u can't have 0.6666666667 of a sweet.
i am not sure if i'm correct
Yep, looks good.
I'd reword it slightly, since in general all numbers are decimal (it just means base 10).
If the probabilty of getting a red is 7/10, then there must be 18 2/3 sweets. Since the number of sweets must be a whole number it follows you can't get a probability of getting a red being 7/10 .
0
2 months ago
#9
(Original post by Muttley79)
After Sajid has taken the first sweet from the bag and eaten it, he is going to take at random a second sweet from the bag. Given that the probability that both the sweets he takes will be red is 3/5.
Work out the number of red sweets in the bag.
0
2 months ago
#10
(Original post by raidas0001)
I need help with this question:
20) There are only red sweets and yellow sweets in a bag.
There are n red sweets in the bag.
There are 8 yellow sweets in the bag.
Sajid is going to take at random a sweet from the bag and eat it.
He says that the probability that the sweet will be red is
7/10(a) Show why the probability cannot be
7/10
The reason why it can't be 7/10 is that then the number of sweets wouldn't be whole and that does not make sense to have something like half a candy.
If the probability of a red sweet is 7/10 then the probability of a yellow sweet is 3/10 meaning that the number of total sweets will be 8*10/3 which is 26 3/4. This doesn't make sense as mentioned before, non-whole numbers don't make sense.
0
2 months ago
#11
(Original post by raidas0001)
I need help with this question:
20) There are only red sweets and yellow sweets in a bag.
There are n red sweets in the bag.
There are 8 yellow sweets in the bag.
Sajid is going to take at random a sweet from the bag and eat it.
He says that the probability that the sweet will be red is
7/10(a) Show why the probability cannot be
7/10
After Sajid has taken the first sweet from the bag and eaten it, he is going to take at random a second sweet from the bag. Given that the probability that both the sweets he takes will be red is 3/5.
Work out the number of red sweets in the bag.
0
2 months ago
#12
Nope, you need an algebraic explanation for that create a simple expression from teh info given to you and you will realise you just do not get a whol enumber
Last edited by Fiaraziqbal; 2 months ago
0
1 month ago
#13
what exam paper is this question from?
0
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72.49% | 1,349 | 4,745 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2021-25 | latest | en | 0.931184 |
https://math.stackexchange.com/questions/3776639/definition-of-unipotency-in-the-context-of-affine-algebraic-groups | 1,596,826,846,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439737206.16/warc/CC-MAIN-20200807172851-20200807202851-00379.warc.gz | 398,484,049 | 34,231 | # Definition of unipotency in the context of affine algebraic groups
I am reading course notes by Tamás Szamuely called "lectures on linear algebraic groups". The definition he gives for an element of the general linear group to be unipotent is not entirely clear to me. I will present my understanding of it, any remarks whether my understanding is correct or wrong is appreciated.
Szamuely writes: "An endomorphism $$h \in \text{End}(V)$$ is unipotent if $$h-\text{id}_V$$ is nilpotent. (...) Let $$V$$ be a finite-dimensional vector space, $$g\in \text{GL}(V)$$. There exist uniquely determined elements $$g_s,g_u \in \text{GL}(V)$$ with $$g_s$$ semisimple, $$g_u$$ unipotent (...).".
I understand the definition of being unipotent in rings, such as the ring of endomorphisms $$\text{End}(V)$$. However, in the context of groups, such as the general linear group $$\text{GL}(V)$$, it confuses me to make use of the operation of addition and the neutral element $$0$$ with respect to this addition. Both of these do not exist in the general linear group. Do we, in some sense, embed the general linear group into the ring of endomorphisms in order to be able to talk about addition and zero? And if so, am I correct in saying that it is a crucial fact that any affine algebraic group is isomorphic to a Zariski-closed subgroup of the general linear group, because this allows us to define unipotency of elements in any affine algebraic group via the above mentioned embedding, which is not possible for every abstract group?
• At the end of your question you ask about abstract groups. This is absolutely a concept specific to algebraic and Lie groups, and not abstract groups. My answer below explains how to extend the concept more generally to group schemes, but I don't think there's an analogous concept, for example, in finitely generated groups. – Jackson 2 hours ago
You're correct initially—there needs to be some notion of (linear) endomorphism ring to define unipotency. It is straightforward to view $$\mathrm{GL}(V)$$ as sitting inside $$\mathrm{End}(V)$$ to make this definition. And it is true that every affine algebraic group over a field is a closed subgroup of some $$\mathbf{GL}_n$$.
Let $$G$$ be a group consisting of unipotent matrices. Then in some basis all elements of $$G$$ are strictly upper triangular.
We can define unipotency more broadly for an affine group scheme $$G$$ in terms of its linear representations. $$G$$ is a unipotent group scheme if for every nonzero linear representation $$V$$ of $$G$$ there is a nonzero fixed vector $$v$$ (i.e., a nonzero subspace $$kv \subset V$$ on which $$G$$ acts trivially). Then an element $$g \in G(R)$$ is unipotent if the closed subgroup it topologically generates is a unipotent group scheme. | 679 | 2,776 | {"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": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2020-34 | latest | en | 0.920544 |
http://mathhelpforum.com/discrete-math/163284-recurrance-relation.html | 1,529,407,834,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267862248.4/warc/CC-MAIN-20180619095641-20180619115641-00041.warc.gz | 208,966,943 | 9,917 | 1. ## Recurrance Relation
I have asked to solve the linear non-homogeneous equation.
$\displaystyle A(n) = 8A(n-2) -16A(n-4) +F(n) , where F(n) = N^2*4^n$
All these n are subscript of A.
I am not sure how to resolve that F(n).
Can anyone help me.
2. Using a well known theorem and taking into account that $\displaystyle 4$ is not a root of the charactheristic equation, a particular
solution for the complete equation has the form $\displaystyle X(x)=(an^2+bn+c)4^n$.
3. Can you please name the theorem used.
4. If $\displaystyle F(n)=a^nP_m(n)$ where $\displaystyle P_m(n)$ is a polynomial of degree $\displaystyle m$, and $\displaystyle a$ is not a root of the characteristic equation, then a particular solution for the complete is $\displaystyle x(n)=a^nQ_m(n)$, where $\displaystyle Q_m(n)$ is a polynomial of degree $\displaystyle m$. If $\displaystyle a$ is a root of the characteristic equation with multiplicity $\displaystyle s$ then, $\displaystyle x(n)=a^mn^sQ_m(n)$.
Regards. | 280 | 998 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.328125 | 3 | CC-MAIN-2018-26 | latest | en | 0.841219 |
https://math.stackexchange.com/questions/2502144/a-consequence-of-a-linear-transformations-mathcall-x-y-between-two-real | 1,560,914,362,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998882.88/warc/CC-MAIN-20190619023613-20190619045613-00377.warc.gz | 509,105,851 | 39,882 | # a consequence of a linear transformations $\mathcal{L} (X, Y)$ between two real Banach spaces $X$ and $Y$
I read a following statement in an academic paper from Journal of Mathematical Analysis and Applications. Please refer to Lemma 3 in https://ac.els-cdn.com/S0022247X05001897/1-s2.0-S0022247X05001897-main.pdf?_tid=75c416d8-c0c8-11e7-9e29-00000aacb362&acdnat=1509735466_ce1f132e2c3285a21b65e184e2630ecd.
Let $E$ be a real Banach space endowed with complete norm $\| \cdot \|$ and $P$ be a total cone of $E$.
Suppose $B \colon P \to P$ is a bounded linear operator.
Therefore this operator $B$ can be uniquely extended to a bounded linear operator on $\overline{B} \colon \overline{P-P} = E \to E$ such that $\| \overline{B} \| = \| B\|$.
Since there is no proof or any comments regarding this statement in that paper, I did not get why it is true. I was thinking that this statement might be a consequence of the Hahn Banach theorem for linear transformations $\mathcal{L} (X, Y)$ between two real Banach spaces $X$ and $Y$.
In fact, the precondition for such consequences may require the space $Y$ having the extensible property, please refer to Section 10 in this note http://www-personal.umich.edu/~romanv/teaching/2009-10/602/short-history-of-analysis.pdf .
However, regarding the statement I wrote here, they only assumed that $E$ is a real Banach space with a total positive cone $P$. I did not get why is that.
Any idea or suggestion would be much appreciated! Thanks in advance!
• What is the definition of a total cone? – fredgoodman Nov 2 '17 at 23:31
• Thanks @fredgoodman. A cone $P$ is called reproducing if and only if $E = \text{span} (P)$, that is, any element $w$ in $P$ can be expressed as the form of $w = u - v$ where $u, \, v \in P$ (i.e., $E = P-P = \{ u-v \colon u,\, v \in P \}$). If $\text{span} (P)$ is dense in $E$ (i.e., $E = \overline{P-P}$), then we say that $P$ is total. – Paradiesvogel Nov 2 '17 at 23:32
• What do linearity and boundedness mean if the domain of $B$ is not a vector space? If $B$ is (extended to) a linear and continuous operator on the subspace $P-P$ then you get the unique continuous extension from uniform continuity and this extension is again linear. – Jochen Nov 3 '17 at 8:28
• Thanks @Jochen. Let's treat the domain of $B$ as the positive cone $P$ of a real Banach space $E$. Actually, I'm wondering two things. The first one is that how could we extend this bounded linear operator $B \colon P \to P$ to a bounded linear operator $\overline{B} \colon P-P \to P-P$? Would you mind to explain it in detail please? Thanks a lot:) – Paradiesvogel Nov 3 '17 at 8:59
• Whenever you have a dense subspace $L$ of a Banach space $E$ and a bounded linear operator $B:L\to L$ the unique continuous extension $\overline B: E\to E$ has the same norm: By definition $\overline B(x)=\lim B(x_n)$ for every sequence $L\ni x_n\to x$ and hence $\|\overline B(x)\|= \lim \|B(x_n)\| \le \lim\sup \|B\| \|x_n\| = \|B\| \|x\|$ for every $x\in E$. – Jochen Nov 3 '17 at 9:37
The statement is false. One reason it is false is because it turns out continuity of a linear map on a normed cone is not equivalent to Lipschitz-continuity of this map, in strong contrast to the situation of a linear map acting on a normed vector space. Thus the linear extension of a continuous linear map on $P$ to $\mathrm{span}(P)$ will fail to be continuous if it was not Lipschitz. If it was Lipschitz then continuity will still remain satisfied.
Let $e_1,e_2$ be the standard basis of $\Bbb R^2$ and let $e^*_1, e_2^*$ be the dual basis. Let $\Bbb R^2$ have euclidean norm. Denote with $$P_\theta = \{ \alpha\, (\cos\theta\, e_1+\sin\theta \, e_2)+\beta\,e_1\mid \alpha,\beta\in\Bbb R_{≥0}\}$$ the "cone of angle $\theta$". The idea is that for small $\theta$ you will need to subtract really big elements of $P_\theta$ in order to reach $e_2$, as the cone starts collapsing.
For any $p\in P_\theta$ we have with $p=\alpha(\cos\theta\, e_1+\sin\theta \, e_2)+\beta\,e_1$ that $$\|p\|=\sqrt{\alpha^2+\beta^2+2\alpha\beta\cos\theta}≥\alpha, \qquad \left|\frac{e_2^*(p)}{\sin\theta}\right|=\alpha.$$
We will look at a sequence of these cones where the angle $\theta$ collapses to $0$. The functional $\frac{e_2^*}{\sin\theta}$ will remain continuous, because while $\sin\theta\to0$ the cone is also getting thinner with the right speed to guarantee continuity. However the extension to $\Bbb R^2$ will grow unboundedly in norm.
To formalise that look at $P=\sum^+_{k\in\Bbb N}P_{1/k}$ as a cone of $\bigoplus_{k\in\Bbb N}\Bbb R_k^2$. Here $\sum^+$ is supposed to indicate (finite) convex linear combinations, so every element $p\in P$ is the form $$\sum_{k=1}^N p_k\qquad N\in\Bbb N,\ p_k\in P_{1/k}$$ we give the vector space $\bigoplus_{k\in \Bbb N} \Bbb R^2_k$ the norm $$\left\|\sum_k x_k\right\|=\sum_{k}\|x_k\|_2$$ where $\|\cdot\|_2$ is the euclidean norm on $\Bbb R^2$, $E$ should be the completion of this space with this norm. Now the counter-example is that the linear functional $$F:\bigoplus_{k\in\Bbb N}\Bbb R^2_k\to\Bbb R, \qquad \sum_k x_k\mapsto \sum_k \frac{e_2^*(x_k)}{\sin 1/k}$$ is continuous on $P$, but it is not continuous on $\bigoplus_k \Bbb R^2_k=\mathrm{span}(P)$. First we show continuity on $P$:
Let $\sum_k p{k,n}\to \sum_k p_k$ in $P$. It follows that $$\left\|\sum_k (p_{k,n}-p_k)\right\|=\sum_k\|p_{k,n}-p_k\|\to0$$ now only finitely many of the $p_k$ are allowed to be non-zero, so the expression is of the form $$\sum_{k>N}\|p_{k,n}\|+\sum_{k=1}^N\|p_{k,n}-p_k\|≥\sum_{k>N}\alpha_{k,n} +\sum_{k=1}^N\|p_{k,n}-p_k\|$$ it follows $\sum_{k>N}\frac{e_2^*(p_{k,n})}{\sin1/k}=\sum_{k>N}\alpha_{k,n}\to0$. On the other hand $\sum_{k=1}^N\frac{e_2^*(p_{k,n}-p_{k,n})}{\sin1/k}$ must converge to zero, since $\sum_{k=1}^N p_{k,n}\to \sum_{k=1}^N p_k$ in $\bigoplus_{k=1}^N\Bbb R_k^2$, which is a finite dimensional vector space and all linear maps on finite dimensional spaces are continuous. Thus $$F(\sum_k p_{k,n})-F(\sum_k p_k) = \sum_k F(p_{k,n}-p_k)=\sum_{k>N}\alpha_{k,n}+\sum_{k=1}^N \frac{e_2^*(p_{k,n}-p_k)}{\sin1/k}\to0.$$ So continuity of $F$ on $P$ has been verified. $F$ is not continuous $\bigoplus_k \Bbb R^2_k$: Look at the sequence $\sum_k x_{k,n}$ with $x_{k,n}=\sqrt{\sin1/k}\,\delta_{k,n}\cdot e_2$. Now the norm of this sequence is $\|x_{n,n}\|=\sqrt{\sin1/n}$, which converges to $0$. But applying $F$ to this sequence retrieves $\frac1{\sqrt{\sin1/n}}$, which diverges.
I wrote something for a special case before, but I was being a little dense.
Let $P$ be a cone in $E$ such that $E_0 = P - P$ is dense in $E$. Let $B: P \to P$ be as linear as it can be, namely $B(s u + t v) = s B(u) + t B(v)$ for $u, v \in P$ and $s, t \ge 0$. Try to define a linear extension of $B$ to $E_0$ by $\bar B(u-v) = B(u) - B(v)$, for $u, v \in P$. Is this well defined? If $u_1 - v_1 = u_2 - v_2$, then $u_1 + v_2 = u_2 + v_1 \in P$. Thus $B(u_1) + B(v_2) = B(u_1 + v_2) = B(u_2 + v_1) = B(u_2) + B(v_1)$. Therefore, $B(u_1) - B(v_1) = B(u_2) - B(v_2)$. Thus $\bar B$ is a well defined extension to $E_0$. Linearity is easy to check.
Now what about boundedness? If the linear extension $B$ is bounded, it extends uniquely to $E$ by continuity.
It is here that you might need some extra hypotheses on $E$ and $P$. The minimum that you need is the existence of a decomposition $f = f_+ - f_{-}$ with $||f_{\pm}|| \le k ||f||$. For then \begin{aligned} ||\bar B(f)|| &= ||\bar B(f_+ - f_{-})|| \\ &= ||B(f_+) - B(f_{-})|| \le ||B(f_+)|| + ||B(f_{-})|| \\ &\le K(||f_+|| + ||f_{-} ||) \le 2 K k ||f||. \end{aligned}
You would expect better estimates in special cases.
• Thanks so much @fredgoodman. Yes, you are right, the paper mainly focus on the Banach space $E= C(X)$ with the positive cone of $C(X)$ which is solid and hence reproducing (i.e., $E = P-P$). But before stating the lemma 3, they generally denote $E$ be a real Banach space with a total cone $P$. Regarding your answer, I thought it might only work in some Banach spaces like $C(X)$ where $X$ is compact, or $C_b(X)$ continuous bounded functions space, or $L^{\infty} (X, \mu)$ essentially bounded $\mu$-measurable functions space. – Paradiesvogel Nov 4 '17 at 22:40
• However, for such $L^{p} (X, \mu)$ space with a Gaussian distribution $\mu$, we cannot always find such $M \gg 0$. For instance, take $p=1$ and $X = \mathbb{R}$ for simplicity. Let $u(x) := - \exp(x)$, which is in $E= L^{p} (\mathbb{R}, \mu)$, but I didn't see we can find a sufficiently large number $M >0$ such that $- \exp(x) + M \geq 0$ for all $x \in \mathbb{R}$. Am I right? Thanks a lot again:) – Paradiesvogel Nov 4 '17 at 22:43
• Right, my answer will only work in special cases, as you say. I wouldn't necessarily trust that the general case is true without digging into the literature. That something is stated (without proof) in a math paper is only an indication that it might be true. – fredgoodman Nov 4 '17 at 23:18
• Oh, I see. Many thanks again @fredgoodman :) That was a huge help to me. May I ask you one more question please? I thought: since $P$ is a total cone of $E$, which means span$\{P\}=P−P$ is dense in $E$, thus we can do like this: as $P$ is a basis for a vector space $P−P$, then for any linear transformation $B \colon P \to P$, there exists a unique linear map (called it linear extension) $\overline{B} \colon P-P \to P-P$ such that $\overline{B} |_{P} = B$. In this connection, $B$ can be extended linearly, and it's what one usually does in linear algebra. – Paradiesvogel Nov 4 '17 at 23:37
• After that, continuous linear extension theorem may apply and yield the result of the statement. Could you please tell me am I right? If not, which step I go wrong? Thanks:) – Paradiesvogel Nov 4 '17 at 23:38 | 3,295 | 9,750 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.109375 | 3 | CC-MAIN-2019-26 | latest | en | 0.865815 |
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# Online Help
###### All Products Maple MapleSim
Home : Support : Online Help : Connectivity : Mathematica Translator : Mma : MmaTranslator/Mma/FactorInteger
MmaTranslator[Mma]
FactorInteger
return a list of prime factors for an integer with its exponents
Calling Sequence FactorInteger(arguments)
Parameters
arguments - Maple translation of the Mathematica command arguments
Description
• The FactorInteger command returns a list of prime factors for a specified integer with its exponents.
Examples
> $\mathrm{with}\left(\mathrm{MmaTranslator}[\mathrm{Mma}]\right):$
Use the command with the Maple translation.
> $\mathrm{FactorInteger}\left(248951\right)$
$\left[\left[{103}{,}{1}\right]{,}\left[{2417}{,}{1}\right]\right]$ (1)
Alternatively, you can use the FromMma command with the evaluate option specified.
> $\mathrm{with}\left(\mathrm{MmaTranslator}\right):$
> $\mathrm{FromMma}\left(\mathrm{FactorInteger\left[248951\right]},\mathrm{evaluate}\right)$
$\left[\left[{103}{,}{1}\right]{,}\left[{2417}{,}{1}\right]\right]$ (2)
See Also
## Was this information helpful?
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https://magiclen.org/hackerrank-java-lambda-expressions/ | 1,686,173,733,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224654016.91/warc/CC-MAIN-20230607211505-20230608001505-00033.warc.gz | 425,083,637 | 9,739 | 題目描述
checkEvenOdd()：這個Lambda表示式必須要回傳一個數字是否為偶數或是奇數。如果是偶數，回傳1；如果是奇數，回傳0。
checkPrime()：這個Lambda表示式必須要回傳一個數字是否為質數或是複合數。如果是質數；回傳0，如果是複合數，回傳1。
checkPalindrome()：這個Lambda表示式必須回傳一個數字是否為回文。如果是回文，回傳0；如果不是回文，回傳1。
參考答案
```import java.io.*;
import java.util.*;
interface performOperation {
int check(int a);
}
class Math {
public static int checker(performOperation p, int num) {
return p.check(num);
}
private static boolean isPrime(int n) {
if(n == 2){
return true;
}
if (n < 2 || n % 2 == 0) {
return false;
}
int nn = (int) java.lang.Math.sqrt(n);
for (int i = 3; i <= nn; i += 2) {
if (n % i == 0) {
return false;
}
}
return true;
}
private static boolean isPalindrome(int n) {
final char[] c = String.valueOf(n).toCharArray();
final int length = c.length;
final int centerIndex = length / 2;
for (int i = 0; i < centerIndex; ++i) {
if (c[i] != c[length - 1 - i]) {
return false;
}
}
return true;
}
public static performOperation checkEvenOdd() {
return a -> a % 2 == 0 ? 0 : 1;
}
public static performOperation checkPrime() {
return a -> isPrime(a) ? 0 : 1;
}
public static performOperation checkPalindrome() {
return a -> isPalindrome(a) ? 0 : 1;
}
}
public class Solution {
public static void main(String[] args) throws IOException {
Math ob = new Math();
int T = Integer.parseInt(br.readLine());
performOperation op;
int ret = 0;
String ans = null;
while (T-- > 0) {
String s = br.readLine().trim();
StringTokenizer st = new StringTokenizer(s);
int ch = Integer.parseInt(st.nextToken());
int num = Integer.parseInt(st.nextToken());
if (ch == 1) {
op = ob.checkEvenOdd();
ret = ob.checker(op, num);
ans = (ret == 0) ? "EVEN" : "ODD";
} else if (ch == 2) {
op = ob.checkPrime();
ret = ob.checker(op, num);
ans = (ret == 0) ? "PRIME" : "COMPOSITE";
} else if (ch == 3) {
op = ob.checkPalindrome();
ret = ob.checker(op, num);
ans = (ret == 0) ? "PALINDROME" : "NOT PALINDROME";
}
System.out.println(ans);
}
}
}``` | 643 | 1,921 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2023-23 | latest | en | 0.125404 |
https://mathzsolution.com/a-path-to-truly-understanding-probability-and-statistics/ | 1,675,476,517,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500080.82/warc/CC-MAIN-20230204012622-20230204042622-00453.warc.gz | 410,260,586 | 21,753 | # A path to truly understanding probability and statistics
I’m embarrassed to say that I have a PhD and hold an asst professorship, but get tripped up when reading statistics research. I am in a field of Business that is similar to IO Psychology or Social Psych. I spend too much time reading applied stats books, but I find even with all the reading I don’t have a firm grasp of what I’m actually doing. Everything is very ‘seat of the pants.’ (As sad as it seems, I think this is not a unique situation among the faculty in the social sciences…) The biggest problem comes when I need to apply a rarely used stat technique. I can find an article from a mathematical stats journal with the equations that would solve my problem, but I don’t have the math to convert those into code. I am forever relying on other prof’s R packages, and crossing my fingers hoping it will work (I can’t even check to verify if it did or not). It’s been over 15 years since I took Calculus and Algebra in undergrad, and I think I want to start at the beginning and truly understand probability and statistics.
I am starting with Gelfand’s Algebra and Trigonometry books for a quick refresher of the basics — I know it’s hard to believe, but in an applied research field we rarely have use for sin or cos. I’m even trying to finally learn how to correctly do a proof, using the books from Velleman (“How to Prove It“) and Houston (“How to Think Like a Mathematician“) — I’m serious about doing this right and understanding the subject. From there I want to move on to (correctly) learn the Calculus and Linear Algebra I need to tackle probability and statistics. I was thinking of using Strang’s Calculus and Algebra books. But Apostol’s Caculus comes highly recommended as well. After that I am completely at a loss. Further, I don’t know how far to go into Calculus or Linear Algebra before I reach diminishing returns. (Apostle introduces Probability in the second half of Vol. 2 — is it vital that I work through everything preceding it before tackling Probability?)
So my question is: if you had to do it over again with the goal of truly, deeply understanding statistics, where would you start? What books are the modern path to deep understanding? I would like to follow a modern path so that I can understand current research in statistics, including Bayesian approaches. But not in a machine learning context (which seems to be the all the rage at the moment), rather a social science / design and analysis of experiments / multilevel modeling context. Perhaps my goal would be the work of Andrew Gelman; his and Hill’s book showed me how I should be looking at modeling and statistics (simulation, uncertainty estimates everywhere, bayesian inference, and so on). How should I go about relearning this material with that end goal in mind?
Update 1: Possible texts, starting from scratch with a focus on proofs and deep understanding. Not necessarily one after another.
Relearn the basics:
Calculus (which one(s), and how deep?):
Linear Algebra (which one(s) and how deep?):
Probability (which one(s)?):
Core Statistics (which one(s)?):
Other suggestions? Again with the goal of understanding and developing (or at least implementing) new methods in hierarchical modelling (generalized and linear).
As someone who started out their career thinking of statistics as a messy discipline, I’d like to share my epiphany regarding the matter. For me, the insight came from Linear Algebra, so I would urge you to push in that direction.
Specifically, once you realize that the sum of squares, $\sum_i X_i^2$, and sum of products, $\sum_i X_i Y_i$, are both inner products (aka dot products), you realize that nearly all of statistics can be thought of as various operations from linear algebra.
If you sample $n$ values from a population, you have an $n$-dimensional vector. The sample mean is a projection of this vector onto the $n$-dimensional all-ones vector. The standard deviation is projection onto the $(n-1)$-dimensional hyperplane normal to the all-ones vector (finally an intuitive reason for the “$n-1$” in the denominator!). Specifically, for the sample variance $s^2$ for sample $X$, here is the linear algebra:
First, we work with deviations from the mean. The mean in linear algebra terms is
$\bar{X}=\frac{\langle X,\mathbf{1}\rangle}{\langle \mathbf{1},\mathbf{1}\rangle} \mathbf{1}$
where $\langle \cdot, \cdot \rangle$ is the inner product and $\mathbf{1}$ is the $n$-dimensional ones vector. Then the deviation from the mean is
$x = X - \bar{X}$
Note that $x$ is constrained to an $(n-1)$-dimensional subspace. The usual equation for variance is
$s^2 = \dfrac{\sum_i (X_i - \bar{X})^2}{n-1}$
For us, that’s
$s^2 = \dfrac{\langle x, x \rangle}{\langle \mathbf{1}, \mathbf{1} \rangle}$
which, without going into too much detail (too late) is a normalized deviation. The trick there is that the new $\mathbf{1}$ has dimension $n-1$.
The other good example is that correlation between two samples is related to the angle between them in that $n$-dimensional space. To see this, consider that the angle between two vectors $v$ and $w$ is:
$\theta = \arccos \dfrac{\langle v, w \rangle}{\|v\|\|w\|}$
where $\|\cdot\|$ is vector length. Compare this to one of the forms for the Pearson Correlation and you will see that $r = \cos \theta$.
There are many other examples, and these have barely been explained here, but I just hope to give an impression of how you can think in these terms. | 1,303 | 5,509 | {"found_math": true, "script_math_tex": 26, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2023-06 | latest | en | 0.93814 |
https://www.weegy.com/?ConversationId=Y7JCKOML | 1,544,698,077,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376824675.15/warc/CC-MAIN-20181213101934-20181213123434-00404.warc.gz | 1,077,703,635 | 9,480 | 20. If the absolute temperature of a gas is 600 K, the temperature in degrees Celsius is A. 273 C. B. 327 C. C. 873 C. D. 705 C.
If the absolute temperature of a gas is 600 K, the temperature in degrees Celsius is 327ºC.
Question
Updated 11/27/2015 12:12:41 AM
Edited by yumdrea [11/27/2015 12:12:39 AM], Confirmed by yumdrea [11/27/2015 12:12:41 AM]
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Home | Contact | Blog | About | Terms | Privacy | © Purple Inc. | 1,019 | 3,473 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2018-51 | latest | en | 0.912036 |
http://webphysics.davidson.edu/physlet_resources/bu_semester2/c03_fieldlines.html | 1,508,699,438,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187825436.78/warc/CC-MAIN-20171022184824-20171022204824-00613.warc.gz | 365,432,487 | 1,517 | #### Counting Field Lines
If field lines come out from a closed surface treat them as positive. If field lines go in to a closed surface treat them as negative.
You should have concluded that the net number of field lines passing through a closed surface is proportional to the charge enclosed by the surface. This is the essence of Gauss' Law. | 70 | 346 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.828125 | 3 | CC-MAIN-2017-43 | latest | en | 0.980517 |
https://www.scienceforums.net/topic/3122-the-difference-between-speed-and-velocity/ | 1,716,209,138,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058278.91/warc/CC-MAIN-20240520111411-20240520141411-00211.warc.gz | 860,688,590 | 27,840 | # The difference between Speed and Velocity
## Recommended Posts
Although this is a very simple question, I yet can't understand it.
What is the difference between speed and velocity?
I know that velocity has a direction, but what differnce does that make? Till now, I learn it off by heart, can anyone explain why does the velocity change as the direction changes?
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SPEED ==> The rate or a measure of the rate of motion, especially: Distance traveled divided by the time of travel.
VELOCITY ==> A vector quantity whose magnitude is a body's speed and whose direction is the body's direction of motion.
(APP & SES 2003 & 2002)
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for futher infomation.
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I understand it now
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• 4 months later...
hmm,velocity and speed are often similar-they have the same numbers
but AVERAGE velocity and AVERAGE speed ain't-they don't have the same numbers in most cases.
i didn't know the true difference between velocity and speed before that.
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For 100m swimming, after the swimmer had finished their race, they have 0 velocity but a speed with greater than 0 m/s.
Why?
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For 100m swimming, after the swimmer had finished their race, they have 0 velocity but a speed with greater than 0 m/s.
aommaster, if you understood what they said on that page (link) then you understood all of it.
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Velocity=Displacement/time
Displacement=0
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aommaster' date=' if you understood what they said on that page (link) then you understood all of it.[/quote']
ok!
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Velocity=Displacement/time
Displacement=0
I suggest you try to understand that link.
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an n dimensional vector needs n mubers to speficy it. For example a 3-dimensional vector can be expressed in terms of three numbers v_1, v_2 and v_3
$\vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k}$
Where the i, j and k are some (usually orthogonal, but not necessarily) unit vectors
or simply by a matrix:
$\vec{v} = \left(\begin{array}{c}v_1\\v_2\\v_3\end{array} \right)$
So the difference between speed and velcotiy is importnat, because velocity (in three dimensions requires 3 numbers to specify it, whereas speed only rrequires one number.
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Velocity=Displacement/time
Displacement=0
The velocity is the derivative with respect to time of displacement. Wht you mentioned is average velocity not instantaneous velocity.
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Another nice way to appreciate difference between speed and velocity, is to analyse a particle moving in a circle with a constant speed v. Even though speed remains the same, the velocity constantly varies due to direction change.
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The velocity is the derivative with respect to time of displacement. Wht you mentioned is average velocity not instantaneous velocity.
I think what he was trying to say is that velocity is given by:
$\lim_{\Delta t \rightarrow 0} \frac{\Delta r}{\Delta t}$
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Or more simply [MATH] \vec{v}=\frac{d\vec{r}}{dt}[/MATH]
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I think what he was trying to say is that velocity is given by:
Are you saying me?
I was trying to use the case of swimming to explain some "tricky"questions to him.
lol
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No, I think he means pulkit.
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I am wrong again........;p
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• 1 month later...
You guys are all a bunch of science nerds. i mean my god who actually goes onto these sites to ask about homework or science stuff! i mean seriously! you guys have no life, you need to get out more
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And you don't see the irony of taking the time to seek out such a forum, signing up, and then posting that comment?
My god your life must be desolate.
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You guys are all a bunch of science nerds. i mean my god who actually goes onto these sites to ask about homework or science stuff! i mean seriously! you guys have no life, you need to get out more
Gobbles, you are so right. Let's see this year I've watched the sun set over the beach in Perth, Australia while drinking a ice cool beer; I've walked on a glacier in Norway in the company of friends; I've had the worst Indian meal ever in Amsterdam and the best in Aberdeen; I've travelled down the Mekong on hydrofoil, I've visited the Botanic Gardens in Bogor, on the island of Java; I've delivered lectures to small groups of students, totalling over three hundred on four continents; I've contributed to sales efforts for my company that have generated in excess of $20,000,000; and throughout it all I've continued to play the piano very badly. Its just as well I still have November and December left to actually do something in. ##### Link to comment ##### Share on other sites Gobbles, you are so right. Let's see this year I've watched the sun set over the beach in Perth, Australia while drinking a ice cool beer; I've walked on a glacier in Norway in the company of friends; I've had the worst Indian meal ever in Amsterdam and the best in Aberdeen; I've travelled down the Mekong on hydrofoil, I've visited the Botanic Gardens in Bogor, on the island of Java; I've delivered lectures to small groups of students, totalling over three hundred on four continents; I've contributed to sales efforts for my company that have generated in excess of$20,000,000; and throughout it all I've continued to play the piano very badly. Its just as well I still have November and December left to actually do [/b']something in.
Ophiolite, I think you are on the wrong forum
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I hope gobbles dies soon........
(just kidding )
It seems odd that YOU signed up for it though.
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Gobbles, you are so right. Let's see this year I've watched the sun set over the beach in Perth, Australia while drinking a ice cool beer; I've walked on a glacier in Norway in the company of friends; I've had the worst Indian meal ever in Amsterdam and the best in Aberdeen; I've travelled down the Mekong on hydrofoil, I've visited the Botanic Gardens in Bogor, on the island of Java; I've delivered lectures to small groups of students, totalling over three hundred on four continents; I've contributed to sales efforts for my company that have generated in excess of \$20,000,000; and throughout it all I've continued to play the piano very badly. Its just as well I still have November and December left to actually do [/b']something in.
This year I went to ottawa on a class trip. It was fun.
## Create an account
Register a new account | 1,581 | 6,749 | {"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} | 3.96875 | 4 | CC-MAIN-2024-22 | latest | en | 0.951114 |
https://www.jiskha.com/archives/2006/11/03 | 1,590,664,303,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347396089.30/warc/CC-MAIN-20200528104652-20200528134652-00496.warc.gz | 783,864,055 | 9,702 | # Questions Asked onNovember 3, 2006
1. ## math
2x + 3y = 78 How do i solve for x and y? 2x=78 78/2=39 so x=39 3y=78 78/3=26 so y=26 Can't be, because 2(39)+ 3(26) does not equl 78 You need TWO equations when you have TWO variables, such as x and y. It can't be done with one equation. BUT, you say, what
2. ## physics
The length of a simple pendulum is 0.82 m and the mass of the particle (the "bob") at the end of the cable is 0.69 kg. The pendulum is pulled away from its equilibrium position by an angle of 6.7 ° and released from rest. Assume that friction can be
3. ## equation complex number
z^4 + 81 = 0 (solve) change it to polar, then take the root. z^4= 81@180 z= 3@180/4 + n90 where n=0, 1, 2, 3 check: z= 3@180/4 + 3*90=3@315 z^4=81@(4*315)=81@1260= 91@180=-81 you can check it at the other roots also. this is what i did: z^4 = - 81 z (81
4. ## Algebra
Is it possible to find the slope of a line if the equation has two ys? Find slope of the line whose equation is 8y = 7 - 2y Solve for y and you'll have a horizontal line. Combine the y terms. 8y+2y=7 10y=7 y=7/10 The general equation for a straight line is
5. ## MVT
How do I explain this problem? A trucker handed in a ticket at a toll booth showing that in 2 h he had covered 159 mi on a toll road with speed limit 65 mph. The trucked was cited for speeding, why? I know it is mean value theorem, but how? His average
6. ## English
I really need help with this Essay question and I haven't got a clue. It is to do with Fitzgeralds "The Great Gatsby" please answer before Monday 6th November. Thankyou. here is the question: Through analysis of patterning, examine Fitzgerald's use of
7. ## Free Fall
Thanks for the help with my previous problems. The acceleration to gravity on the moon is 1.6m/sec^2. If a rock is dropped into a crevasse, how fast it will be going just before it hits bottom 30 sec later? How far below the point of release is the bottom
8. ## complex numbers!
-2-2i in polar form = 2.8 cis 2.35?? -3+4i polar form = 5 cis 2.2??? 1 - sqaureroot(3i) polar form = 2 cis 1.04?? What is cis? never mind I have the right answers. polar form z = r (cos O + i sin O ) is also written in an abbreviated form z = r cis O cos +
9. ## math
the inverse cosine of negative 0.947 Put it in your calculator... The sequence of keys in my old ten dollar calculator is .947 +- 2nd COS Other calculators have different keys strokes. If you want to be neat, type this into a google search window:
10. ## How to sketch graph of...
a differentiable function y =f(x) that has the given properties. 1. local minimum at (1,1), and local maximum at (3,3) 2. local minima at (1,1) and (3,3) 3. local maxima at (1,1) and (3,3) I don't remember how to do this. Thank you.
11. ## ECONOMICS
this is a table with some questions and i don't know how to solve it -------------table-------------------- (1)real domestic output (GDP=DI) in billions \$200 \$250 \$300 \$350 \$400 \$450 \$500 \$550 (2)aggregate expenditures private closed economy billions \$240
12. ## History
Revenue for transportation was to come from these under Henry Clay's American System Very good detailed answer at this site: http://www.answers.com/topic/american-system-economic-plan
13. ## History
Who was the Democratic leader,"Cross of Gold" speech? William Jennings Bryan
14. ## Algebra
Please check for me. 7p - 7/p divided by 10p-10/5p^2 = 7p^2 6x - 6/x * 6x^2/8x-8 = 9x/2 I'm not sure how to answer this one. x/2 divided by 9/x + 5 I think it's x(x + 5) / 18 Without grouping symbols () it is impossible to understand the problems. The ^
15. ## Intermediate Algebra
Am I going in the right track? 8x -1/x + 5x - 3/2x = 16x^2 - 2x/2x^2 + 5x^2 - 3x/2x^2 16x^2 - 2x + 5x^2 - 3x/2x^2 =21x^2 - 5/2x^2 Find the LCD 6/7z- 28, 8/x^2 -4x = 7x(x-4) Simplify 3x^2 - 2x/15x-10 = x/5 Let's start by looking at your first problem: 8x
16. ## to bobpursley
My classmates friends want me to post their questions as well.Thats why there are multiple names is there a problem. Yes. You are posting duplicate or even triplicate questions. That is silly. If we have answered one, the answers to the others follow.
17. ## Math
Hers is another one: 3 people requires 3 weeks and 2 days to do a job. How long would it take for 4 people to do the same job? (Each week is 6 days) Convert time to days only to avoid confusion. Assuming the relation between people and time is linear,
18. ## English!_+ check please!
Rank the following sentences from most formal to least formal by writing the appropriate number in the blank beside each sentence: 1 = most formal; 3 = least formal. /3 ___2__I was pumped. I mean, we had scored the best seats for the show. Awesome ___1__In
19. ## History Essay
I Need help with this essay The Purpose of this essay is to analyze the relationship between wealthy nations and supporting of the arts. -Think about the early years if the renaissance,and the spanish of the ducth today -include examples of artistic
20. ## English
Can anyone check my essay? I could send the rubric for scoring. Thanks You have to include the essay itself, so we can check it. Respond to this post with the essay. Thanks for asking. Whats your question..?
21. ## Eth
My assingment states to view the comparison of the UNited States-Centric. Which I have a compare and contrast sheet I need to fill in about how an average american perceptions about muslim and arab american and christian american groups. Now I've done the
22. ## economics plz help
this is a table with some questions and i don't know how to solve it -------------table-------------------- (1)real domestic output (GDP=DI) in billions \$200 \$250 \$300 \$350 \$400 \$450 \$500 \$550 (2)aggregate expenditures private closed economy billions \$240
23. ## Research Paper
Hi all. It seems that I've run into a problem while starting to write a research paper for my drugs and behavior class. You see we were given an outline to follow and we had to choose any drug related topic. So I'm doing mine on cocaine and the problem I'm
24. ## congruence
What is the difference between the statements segmentAB is congruent to segmentCD & AB is congruent to CD Thanks No difference. Congruence is congruence. A=B B=A
25. ## Interesting Geometry Probem
I am bulding a sculpture out of rock and need help figuring out how much rock to buy... Here are the figures... 1 ton of 1.5" rock covers 107 sq. ft. My sculpture will be comprised of 24 layers of rock 2" thick. Each layer will be aproximatly 3ft. by 3ft.
26. ## Arithmetic Operations
Find a set of 4 distinct positive integers a,b,c,d such that the smallest positive integer that can not be represented by such expressions involving a,b,c,d (instead of 1,2,3,4) is greater than 22.You can use digits exactly once. You are allowed to reuse
27. ## physics
In a room that is 2.13 m high, a spring (unstrained length = 0.30 m) hangs from the ceiling. A board whose length is 1.58 m is attached to the free end of the spring. The board hangs straight down, so that its 1.58-m length is perpendicular to the floor.
28. ## economics
please please please please help me this is due tomorrow!!! this is a table with some questions and i don't know how to solve it (1)real domestic output (GDP=DI) in billions \$200 \$250 \$300 \$350 \$400 \$450 \$500 \$550 (2)aggregate expenditures private closed
29. ## title making
whatis a good inline skating title for something that is aimed at promoting in line skating and incourage skaters to practice safe skating thanks so much this is due tomorrow and i have dinner soon, plz hurry and i thank you soooo much! Skate Safe and
30. ## Stats
Dee's store monthly sales for April, May, June are three independent random variables with means and variances as follow: April: Mean: \$9500 Variance: \$1250 May: Mean: \$7400 Variance: \$1425 June: Mean: \$8600 Variance: \$1610 a) find the mean and standard
this is a table with some questions and i don't know how to solve it (1)real domestic output (GDP=DI) in billions \$200 \$250 \$300 \$350 \$400 \$450 \$500 \$550 (2)aggregate expenditures private closed economy billions \$240 \$280 \$320 \$360 \$400 \$440 \$480 \$520
32. ## Research Paper
I have to write a research paper but I don't know what to pick as a topic. I want something simple. Last time i chose Napoleon and it was overwhealming. Can anyone suggest a simple topic that is easy to research. Preferable historical. Mozart disco John
33. ## English+ check!
Exercise: Add an appropriate /transition word/ or phrase to connect the following pairs of sentences. 1. Our first destination is Ottawa /there/; we plan to visit Montreal and Quebec City. 2. Standing on the rocky ledge, we had an aerial view of the
34. ## Chemistry
True or False or Sometimes true Removing one electron from an atom results in the formation of a postive ion with a 1+Charfe. Of course it is true. Normally atoms have a balance of electrons and protons.
35. ## history
What banking reform finally gave the nation a flexible money supply? This site will also answer this question. http://www.answers.com/topic/american-system-economic-plan
36. ## math - probability
Imagine a LONG bike with a total of 4 wheels - 2 in the front and 2 in the back. For the bike to work at all, at least 1 front wheel and 1 back wheel must be operational. All 4 tires are the same and each, individually, has probability 'p' of failing. What
37. ## math
I have a math test today.What shoud do?
38. ## More related rates.
Heyy, its me again. So last night I asked a related rate question, and I think I still don't get it. I tried it a few differnt ways and I think I'm just missing something...I don't think a speed of a rocket can be 0.018km/hr:S. So heres what happened: A
39. ## Another graph
How do I make a graph of f'(-1) = f'(1) = 0, f'(x) > 0 on (-1,1), f'(x) < 0 for x < -1, f'(x) > 0 for x >1 Thanks. if the derivative is postive on -1 to 1, you have a curve that shope upward as x increases. If the derivative is zero at -1, and 1, thekn at
40. ## complex numbers
-3 + 4i in polar form in pi the magnitude is 5 from pyth theorm. Now the angle. Using i the positive real as the reference axis, the angle from it is PI/2 + arctan4/3
41. ## allgebra
how is linear,quadratic,rational,or exponental realated to arithmetic series 1/4s=4 | 2,889 | 10,347 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.90625 | 4 | CC-MAIN-2020-24 | latest | en | 0.917706 |
https://math.stackexchange.com/questions/405424/find-all-positive-solutions-for-equation/405427 | 1,568,730,599,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514573080.8/warc/CC-MAIN-20190917141045-20190917163045-00559.warc.gz | 588,922,389 | 31,681 | # Find all positive solutions for equation:
$nx^{(n+1)}-(n+1)x^n+1=0$
There's nothing told about $n$, I guess $n \in N$. I would like any kind explanations, thanks! I appreciate your time.
• Guys, sorry. The first x is at power : n+1 . I can't edit it. Someone help me. THanks – Florin M. May 29 '13 at 5:59
Hint: $x =1$ solves the equation. Factoring gives $$(x-1)(nx^n - x^{n-1}-x^{n-2}- \dots -x -1) = 0$$ and you can see $x=1$ is again a zero of the second factor. Factoring further $$(x-1)^2(nx^{n-1} + (n-1)x^{n-2}+\dots+2x+1) = 0$$ and you can conclude that for $n=1,2,\dots$ the equation has a double zero at $1$.
• Thanks again! And the equation : $nx^{n-1} + (n-1)x^{n-2}+\dots+x+1=0$ has no solutions ? – Florin M. May 29 '13 at 6:39
• I guess that for odd $n$ there is another real solution. For even $n$ there are only solution in complex plane with nonzero imaginary part. – UrošSlovenija May 29 '13 at 7:42
• Or differentiate, getting $n (n + 1) (x^n - x^{n - 1})$, which has the zero 1 in common with the original. – vonbrand May 29 '13 at 12:11
Hint: if you distribute the equation, you get the expression $2 - x^{n} = 0$.
• You probably saw the version that looked like the exponent $n+1$ was a factor. – Ross Millikan Jun 6 '13 at 0:23
Let $f(x)=nx^{(n+1)}-(n+1)x^n+1$ $n\in N,x>0$
$f'(x)=n(n+1)x^n-n(n+1)x^{n-1}=n(n+1)x^{n-1}(x-1)$
So we have $f'(x)>0 \forall x>1$
and we have $f'(x)<0\forall 0<x<0$
This implies that the function is increasing for all $x>1$ and decreasing for $0<x<1$
Now $f(1)=0$ as $f(x)$ is increasing for $x>1$ so we have $f(x)>f(1)=0,\forall x>1$
And as it is decresing for $0<x<1$ so we have $f(x)<f(1)$ for $0<x<1$.
So the only (positive)solution of this equation is $x=1$
• Thanks, dear Abhra! $f'(x)<0\forall 0<x<0$, is from $0<x<1$, right? – Florin M. May 29 '13 at 6:55
• yes you are right – Abhra Abir Kundu May 29 '13 at 6:57 | 732 | 1,892 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.3125 | 4 | CC-MAIN-2019-39 | latest | en | 0.877771 |
https://www.tinkutara.com/2023/06/03/find-grad-log-r/ | 1,726,175,253,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651498.46/warc/CC-MAIN-20240912210501-20240913000501-00688.warc.gz | 936,889,260 | 13,338 | Question Number 2 by user1 last updated on 25/Jan/15
$$\mathrm{Find}\:\mathrm{grad}\:{log}\:\mid\boldsymbol{\mathrm{r}}\mid. \\$$
Answered by user1 last updated on 29/Oct/14
$$\mathrm{We}\:\mathrm{have}\:\:\:{r}=\sqrt{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)} \\$$$$\mathrm{log}\:\mid\boldsymbol{\mathrm{r}}\mid=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{log}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right) \\$$$$\mathrm{Now},\:\frac{\partial}{\partial{x}}\mathrm{log}\mid\boldsymbol{\mathrm{r}}\mid=\frac{\mathrm{1}}{\mathrm{2}}\centerdot\frac{\mathrm{2}{x}}{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)}=\frac{{x}}{{r}^{\mathrm{2}} } \\$$$$\mathrm{Similarly},\frac{\partial}{\partial{y}}\mathrm{log}\mid\boldsymbol{\mathrm{r}}\mid=\frac{{y}}{{r}^{\mathrm{2}} },\:\frac{\partial}{\partial{z}}\mathrm{log}\mid\boldsymbol{\mathrm{r}}\mid=\frac{{z}}{{r}^{\mathrm{2}} } \\$$$$\therefore\:\:\:\mathrm{grad}\:\mathrm{log}\mid\boldsymbol{\mathrm{r}}\mid=\frac{\mathrm{1}}{{r}^{\mathrm{2}} }\left(\boldsymbol{\mathrm{i}}{x}+\boldsymbol{\mathrm{j}}{y}+\boldsymbol{\mathrm{k}}{z}\right)=\frac{{r}}{{r}^{\mathrm{2}} } \\$$
Commented by SWPlaysMC last updated on 16/Dec/21
$$\mathrm{Ahh}\:\mathrm{yes},\:\mathrm{the}\:\mathrm{first}\:\mathrm{question}\:\mathrm{ever}\:\mathrm{asked}\:\mathrm{on}\:\mathrm{this}\:\mathrm{app}… \\$$$$\mathrm{What}\:\mathrm{happened}\:\mathrm{to}\:\mathrm{question}\:\mathrm{ID}\:\mathrm{1}?\:\mathrm{Did}\:\mathrm{it}\:\mathrm{EVER}\:\mathrm{exist}? \\$$$${Edit}:\:{All}\:{hail}\:{user}\mathrm{1}! \\$$
Commented by talminator2856791 last updated on 05/Sep/21
$$\:\mathrm{no}\:\mathrm{clue}. \\$$
Commented by talminator2856791 last updated on 05/Sep/21
$$\:\mathrm{it}\:\mathrm{appears}\:\mathrm{that}\:\mathrm{user1}\:\mathrm{also}\:\mathrm{doesnt} \\$$$$\:\mathrm{exist}\:\mathrm{now}. \\$$
Commented by MathematicalUser2357 last updated on 08/Jan/24
$${BUG} \\$$ | 751 | 1,948 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.9375 | 4 | CC-MAIN-2024-38 | latest | en | 0.203549 |
http://financial-dictionary.thefreedictionary.com/loan-to-value+ratio | 1,537,629,965,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267158450.41/warc/CC-MAIN-20180922142831-20180922163231-00256.warc.gz | 88,796,354 | 12,634 | # loan-to-value ratio
Also found in: Dictionary, Acronyms, Wikipedia.
## Loan-to-value ratio (LTV)
The ratio of money borrowed on a property to the property's fair market value.
## Loan to Value Ratio
1. In mortgages, the ratio of the amount of a potential mortgage to the value of the property it is intended to finance, expressed as a percentage. It is used as a way to assess the risk of making a particular mortgage loan. A lower loan-to-value ratio is seen as a lower risk to the lender. Most mortgage lenders require a maximum loan-to-value ratio of 75%. That is, a borrower is usually expected to pay for 25% of the value of a property out-of-pocket.
2. More broadly, a ratio of the amount of a potential loan to the asset it is intended to finance. In addition to gauging the risk involved in making the loan, it tells the borrower whether or not the loan can be repaid if he/she sells the asset. This can be important if the borrower becomes unable make payments.
## loan-to-value (LTV) ratio
The relationship between the principal amount of a loan and the appraised value of the property serving as security. A loan of \$80,000 on a property appraised at \$100,000 is an 80 percent LTV.Residential mortgages with an LTV of 80 percent or less qualify for FHA insurance; if the ratio is higher, then borrowers may be required to obtain private mortgage insurance.Generally speaking, the higher the LTV, the higher the interest rate will be because the lender has assumed more risk.Those risks are as follows:(1) When there is little equity in the property, it has a low hostage value; the borrower is more likely to default and walk away from the property because the borrower has little to lose. (2) At foreclosure, the property may not bring a price sufficient to pay off the principal balance of the loan, much less the accrued interest and costs of foreclosure.
## Loan-to-Value Ratio (LTV)
The loan amount divided by the lesser of the selling price or the appraised value.
The LTV and down payment are different ways of expressing the same facts. See Down Payment/Down Payment and LTV.
References in periodicals archive ?
Thus, estimates of the long-term credit risk of a VA mortgage must provide for this risk-sharing; we estimated that 80 percent of the losses are borne by the VA and 20 percent by the mortgage originator regardless of loan-to-value ratio.
Institutions' expected dollar losses are determined primarily by the distribution of loan-to-value ratios within their mortgage portfolios: Higher ratios are associated with higher mortgage default probabilities and loss severity rates.
To some extent, PMI companies compete directly with the FHA and the VA to insure mortgages that have high loan-to-value ratios.
Even while Fannie Mae and Freddie Mac are encouraged to promote lending to lower-income households, their charters may also create barriers to such lending by limiting the risk they may bear: The mortgages they purchase, unless they carry private mortgage insurance or some other form of credit enhancement (for example, recourse to the lender), must have loan-to-value ratios of 80 percent or less.
As with the FHA, the VA's portfolio included a high proportion of loans with high loan-to-value ratios, and these loans had higher default rates than conventional mortgages with similar LTVs, resulting in a relatively large market share.
For most institutions, the distribution of loan-to-value ratios for their mortgage portfolios was not publicly available.
Our estimates, together with institutional knowledge, suggest that there are essentially five different distributions of loan-to-value ratios across mortgage holders and insurers (table B.
Generally, government-insured loans have very high concentrations of mortgages having loan-to-value ratios of 90 percent or higher; we estimated that 93 percent of the loans insured by the VA in 1995 were in this category.
Joint programs among the PMI companies and the secondary market agencies, such as Fannie Mae's Community Home Buyers program and Freddie Mac's Affordable Gold program, allow the borrower to use gifts and other nonborrower sources of funds for part of the down payment on a mortgage with a 95 percent loan-to-value ratio; such sources of funds are not allowed in regular 95 percent loan-to-value ratio programs.
One possible explanation is that PMI companies, when insuring the risk of mortgages with high loan-to-value ratios, provide the best value for individuals with higher-incomes (but little wealth).
And if the Boston experience is representative of that nationally, black and Hispanic applicants would fare worse than white applicants because they have higher obligation and loan-to-value ratios and weaker credit histories.
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Could someone explain why this sentence makes sense: A monopole antenna transfers energy from electrical domain to the electromagnetic domain, hence must contain (equivalently) a resistor, hence ... | 1,626 | 7,263 | {"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": 1, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2016-30 | latest | en | 0.966582 |
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A place to discuss the science of computers and programs, from algorithms to computability.
Formal proofs preferred.
Moderators: phlip, Moderators General, Prelates
### Orcle TM eh?
I know that { (M,x) | M, with an oracle for the Halting Problem, halts on x} is a undecidable..
but is this not recognizable? and if I complement this language will i get not recognizable?
Please, don't abuse the editing option to delete your posts. We don't allow that here. I have restored your post thanks to Google Cache. - phlip
Last edited by bellykellyyang on Fri Mar 16, 2012 9:28 pm UTC, edited 1 time in total.
bellykellyyang
Posts: 3
Joined: Wed Mar 14, 2012 11:26 am UTC
### Re: Orcle TM eh?
Recognisable, as in, you could make a program (presumably, a TM program, not a TM+Oracle program), that succesfully halts for "yes" answers (ie pairs of TM+Oracle programs and inputs such that the program halts on that input), and either unsuccessfully halts or never halts for "no" answers? The normal TM halting question is trivially recognisable in this way.
I'm pretty sure that your question is not recognisable. Consider the program that uses the halting oracle on its input and then halts iff the passed-in program does not halt. If we could make a TM that could recognise even just this TM+Oracle program, then we could pair it with the TM that recognises the TM halting problem, and make a TM that solves the TM halting problem. Which is impossible.
While no one overhear you quickly tell me not cow cow.
but how about watch phone?
phlip
Restorer of Worlds
Posts: 6958
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Location: Australia
### Re: Orcle TM eh?
So are you saying the language is not-recognizable?
But I thought it's recognizable because if HALT-OTM(oracle turing machine) gets input of OTM as turing machine and x as an input, and ask if it will halt, if oracle says "yes it'll halt", then it will go into infinite loop and if it says "no it won't halt" then it will halt. Doesn't this mean it's recognizable since it loops? or it's unrecognizable because it doesn't accept on input? I'm confused..
bellykellyyang
Posts: 3
Joined: Wed Mar 14, 2012 11:26 am UTC
### Re: Orcle TM eh?
I'm pretty sure that your question is not recognisable. Consider the program that uses the halting oracle on its input and then halts iff the passed-in program does not halt. If we could make a TM that could recognise even just this TM+Oracle program, then we could pair it with the TM that recognises the TM halting problem, and make a TM that solves the TM halting problem. Which is impossible.
can you explain this again?
bellykellyyang
Posts: 3
Joined: Wed Mar 14, 2012 11:26 am UTC
### Re: Orcle TM eh?
It seems to me you have a very flimsy grasp of what your question actually means.
To make sure we're on the same page, can you please tell me your understanding of what each of the following terms means: "TM", "OTM", "Halt", "Recognise", "Recognisable", "Decide", "Decidable", and the Halting Problem. Also, while you're at it, proof by contradiction, since that's what a proof of the answer to your question will probably use.
While no one overhear you quickly tell me not cow cow.
but how about watch phone?
phlip
Restorer of Worlds
Posts: 6958
Joined: Sat Sep 23, 2006 3:56 am UTC
Location: Australia
### Re: Orcle TM eh?
Hail phlip, restorer of worlds!
Good answer. Elegant reduction.
philoctetes
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# What units of volume we use in these following circumstances? Volume of eye drops to be taken Capacity of water tank Capacity of syringe Volume of water used for gardening Volume of milk to be bought from the supermarket.
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## For medicines and eye drops, we normally use millilitres (ml). For the capacity of water tanks, we can use litres (L) or kilolitres (kl). For the capacity of syringe, we use ml. For volume of water used for gardening, we use litre (L). For volume of milk to be bought from the supermarket, we can use both ml and litre, depending on the requirement of the household.
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Minimum/Maximum of a Multivariate Function
Description Compute the minimum and maximum values of a multivariate function.
Define a function.
>
${f}{:=}\left({x}{,}{y}\right){→}{3}{+}{\mathrm{sin}}{}\left({x}\right){+}{2}{}{{\mathrm{cos}}{}\left({y}\right)}^{{2}}$ (1)
Compute the minimum of the function.
> $\mathrm{minimize}\left(f\left(x,y\right)\right)$
${2}$ (2)
Compute the maximum of the function.
> $\mathrm{maximize}\left(f\left(x,y\right)\right)$
${6}$ (3)
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posted by jonathan
A 5.0g KBr sample, at 25 degrees celsius, dissolved in 25 ml of water, also at 25 degrees celsius. the final equilibrium temperature of the resulting solution is 18.1 degrees celsius, What is the enthalpy change in J/g and KJ/mole of KBr.
1. DrBob222
q = [mass H2O x specific heat H2O x (Tfinal-Tinitial)]
Substitute and solve for q = delta H for the solution.
q/5 = delta H/g in joules.
delta (H/g) x molar mass KBr = J/mol.
Convert to kJ/mol.
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http://codeforces.com/blog/entry/46162 | 1,571,775,236,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570987823061.83/warc/CC-MAIN-20191022182744-20191022210244-00279.warc.gz | 39,273,396 | 22,994 | ### rumman_sust's blog
By rumman_sust, history, 3 years ago, ,
UPD: I just solved Robert Hood from Kattis online judge. After spending almost a week I've understood the process of rotating calipers. It feels really great to learn a new problem solving technique. I recommend everyone to read this book and visit this site (provided by SuprDewd) for umderstanding rotating calipers technique. Here you can find a basic implementation of rotating calipers for finding diameter of a convex polygon. Also don't forget to look at SuprDewd's comment. Thank you all for helping and encouraging me all the time :)
• +43
» 3 years ago, # | +7 I've found this but I think it's not enough. Moreover, it's in Russian language.
» 3 years ago, # | +3 Rotating Calipers is a very useful technique. So it would be really great If we could discuss about the Topic and related problems here. Following this important thread :)
» 3 years ago, # | +12 Could you please write a detailed blog on it, if you have learned it well.
• » » 3 years ago, # ^ | 0 I'll try thanks.
» 3 years ago, # | +56 I studied the technique some months ago. Unfortunately I don't have all the resources I used while learning, but I remember that the following three links were useful: There are many applications of rotating calipers, but I think the most basic one is to find the diameter of a convex polygon*. Conceptually it's very simple. We put the polygon between the two jaws of a caliper, and tighten. We then rotate the calipers a full circle around the polygon, while keeping the caliper tight at all times. The answer is then the maximum distance between the two jaws at any point during the procedure.The rotating calipers technique is basically just an "angle-sweep-line" version of this. We have two parallel lines that are on either side of the polygon, representing the jaws of the caliper. Initially we could, for example, let one of the lines touch the leftmost point of the polygon and the other line touch the rightmost point of the polygon, and say they are completely vertical (i.e. at an angle pi/2). We then simulate this procedure by rotating the two lines at the same speed, keeping track of which two points the two lines are touching, and then only stopping at the moments these points change.This can then be easily generalized to many applications: Minimum rectangle enclosing a convex polygon: Now we have two calipers that are orthogonal to each other, and again we rotate them around the polygon. This corresponds to having four lines in the simulation, i.e. two pairs of parallel lines, which are then orthogonal to each other so that the interior forms a rectangle. Minimum distance between two convex polygons: Now one of the lines touches one of the polygons, and the other line touches the other polygon. Furthermore, we will initially let one of the lines touch the leftmost point of the corresponding polygon, while the other line touches the rightmost point of the corresponding polygon, and again the lines start out vertical. See the links above for further applications and more detailed explanations.For some example code you can look at my implementation. I have a class that represents a single jaw of the caliper (i.e. a single line), and then below (commented out) is how I use that class to find the diameter of an arbitrary polygon. Note that the code uses doubles and angles (as that was more intuitive for me at the time), but a more robust implementation would only use integers.Finally, here are some practice problems: Would love to hear if others have more problems.*) Finding the diameter of an arbitrary polygon can be reduced to this problem by first taking the convex hull. The same is true for many other applications.
• » » 3 years ago, # ^ | ← Rev. 3 → +8 Thank you so much SuprDewd. It's a great help. I've found the book named "COMPUTATIONAL GEOMETRY WITH THE ROTATING CALIPERS" written by Pirzadeh, Hormoz just before your reply. That's why I didn't check my blog post to see if anyone place a comment or not. Sorry for the late response. I'm reading the book written by Pirzadeh, Hormoz now. Hopefully I'll understand the technique soon and will be able to solve some problems related to it. Thanks again.PS: If you have enough time you can add those resources at here. It will be easy to find for the future problem solvers.
• » » 3 years ago, # ^ | +2 The link SuprDewd gave for COMPUTATIONAL GEOMETRY WITH THE ROTATING CALIPERS is not working anymore. Can someone please provide the pdf ( I assume it was a link to a pdf) of the link ? And thanks SuprDewd, I learnt a lot of things from the links you provided.
• » » » 3 years ago, # ^ | 0 Googling the title gives this page. Can you open the PDF from there?
• » » » » 3 years ago, # ^ | +2 Yes, I can. Thanks.
» 3 years ago, # | +5 Auto comment: topic has been updated by bekar13 (previous revision, new revision, compare).
» 16 months ago, # | ← Rev. 3 → 0 Can we use the intuitive two pointers technique for finding antipodal pairs in convex polygon? Like: j<-0 for i in [0,n): while distance(p[j+1], p[i]) >= distance(p[j+1], p[i]) and j < i: if distance(p[j+1], p[I]) == distance(p[j+1], p[I]): store (p[I],p[j+1]) in cache else: clean cache j += 1 yield points in cache (obviously all incrementation done modulo n)It worked on Robin Hood, but it fells like it has too simple tests. Any pitfalls? Thank you!UPD: Both implementations for robin hood are given here
• » » 10 months ago, # ^ | +3 There is a negative example from this Chinese blogThe O point won't find it's antipodal point A because the of the B point. It may stuck on the C point.
• » » » 7 months ago, # ^ | +5 Isn't that true, that the diameter of the figure on the set equals to the distance between point A and that unnamed point between O and C? If yes (and it seems to be like that), the "wrong" algorithm would actually find the correct answer, and it don't need the length of segment "OA" at all.
• » » » » 7 months ago, # ^ | 0 Yes, I think you're right. But I thought the figure show that you can't find the antipodal point using the binary search approach. For the global diameter, we should find another negative example :) | 1,501 | 6,229 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.78125 | 4 | CC-MAIN-2019-43 | latest | en | 0.938788 |
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# Problem 1805. Cofactor
Solution 304804
Submitted on 15 Aug 2013 by James
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
### Test Suite
Test Status Code Input and Output
1 Pass
%% A = [1 2 3;4 5 6;7 8 9]; i=1; j=1; y_correct = 6; assert(isequal(cofactor(A, i, j),y_correct))
ans = 6
2 Pass
%% A = [1 2 3;4 5 6;7 8 9]; i=2; j=1; y_correct = 6; assert(isequal(cofactor(A, i, j),y_correct))
ans = 6
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1. CJM 2017 (vol 70 pp. 53)
Dantas, Sheldon; García, Domingo; Maestre, Manuel; Martín, Miguel
The Bishop-Phelps-Bollobás property for compact operators We study the Bishop-Phelps-Bollobás property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators, then so do $(C_0(L),Y)$ for every locally compact Hausdorff topological space $L$ and $(X,Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikodým property and $(\ell_1(X),Y)$ has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$ for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$ has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any positive measure $\nu$ and $1\lt p\lt \infty$. For $1\leq p \lt \infty$, if $(X,\ell_p(Y))$ has the BPBp for compact operators, then so does $(X,L_p(\mu,Y))$ for every positive measure $\mu$ such that $L_1(\mu)$ is infinite-dimensional. If $(X,Y)$ has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$ for every $\sigma$-finite positive measure $\mu$ and $(X,C(K,Y))$ for every compact Hausdorff topological space $K$. Keywords:Bishop-Phelps theorem, Bishop-Phelps-Bollobás property, norm attaining operator, compact operatorCategories:46B04, 46B20, 46B28, 46B25, 46E40
2. CJM 2010 (vol 62 pp. 827)
Ouyang, Caiheng; Xu, Quanhua
BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm. Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spacesCategories:46E40, 42B25, 46B20
3. CJM 2003 (vol 55 pp. 969)
Glöckner, Helge
Lie Groups of Measurable Mappings We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $(X,\Sigma,\mu)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group $L^\infty (X,G)$, which is a Fr\'echet-Lie group if $G$ is so. We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum $\bigoplus_{i\in I} L(G_i)$. Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25
4. CJM 2002 (vol 54 pp. 1165)
Blasco, Oscar; Arregui, José Luis
Multipliers on Vector Valued Bergman Spaces Let $X$ be a complex Banach space and let $B_p(X)$ denote the vector-valued Bergman space on the unit disc for $1\le p<\infty$. A sequence $(T_n)_n$ of bounded operators between two Banach spaces $X$ and $Y$ defines a multiplier between $B_p(X)$ and $B_q(Y)$ (resp.\ $B_p(X)$ and $\ell_q(Y)$) if for any function $f(z) = \sum_{n=0}^\infty x_n z^n$ in $B_p(X)$ we have that $g(z) = \sum_{n=0}^\infty T_n (x_n) z^n$ belongs to $B_q(Y)$ (resp.\ $\bigl( T_n (x_n) \bigr)_n \in \ell_q(Y)$). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces $X$ and $Y$. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in $B_p(X)$ are introduced. Categories:42A45, 46E40
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1.6012
VD" = 9.44 X 1.601 = 15.2 ft/s X in V = 13;6„ X 3.54 = 2.88 m/s
### 4.0892
VD = 2.88 X 0.04089 = 0.118 m/s X m (VD" = 0.118 X 129.2 = 15.2 ft/s X in) From Figure 33, Re = 1.5 X 104, and from Figure 31, f = 0.030. From Eq. 16
300 X 12 9.442 in USCS units hfd = 0.030 X 2 Xg4.17 = 9.29 ft
91.5 2.882
in SI units hfd = °.°3° 004089 X 2X9807 = 28.39 m
The valve and fitting losses from Tables 5 and Eq. 20 are 2-in (51-mm) bellmouth, K = 0.05:
In USCS units In SI units
In SI units
2-in (51-mm) gate valve, In USCS units
5.732
hf1 = 0.05 2X321T
1.752
u 5.732
0.026 ft 0.0078 m
In SI units
1g-in (38-mm) gate valve, In USCS units
In SI units
1^-in (38-mm) swing check valve, In USCS units
In SI units
1 752
9.442
The total pipe, valve, and fitting losses are
In USCS units nhf = hfs + hfd + hfi + hf 2 + hf 3 + hf4 + hf5
= 1.84 + 92.9 + 0.026 + 0.48 + 0.082 + 0.263 + 3.46 = 99.05 ft Total variation = ± (0.14 + 0.021 + 0.066 + 1.0) = ± 1.23 ft nhf = hfs + hfd + hf1 + hf 2 + hf 3 + hf4 + hf
= 0.56 + 28.39 + 0.0078 + 0.044 + 0.0250 + 0.0803 + 1.06 = 30.17 m Total variation = ± (0.044 + 0.0062 + 0.02 + 0.32) = ± 0.39 m example 13 Solve Example 12 using resistance coefficients from Tables 6.
Suction pipe:
In SI units
In USCS units In SI units
Discharge pipe,
In USCS units In SI units
(same as in Example 12) (same as in Example 12)
(same as in Example 12) (same as in Example 12)
Valve and fitting losses from Tables 6 and Eq. 20: 2-in (51-mm) bellmouth, K = 0.04
5.732
In USCS units In SI units
2-in (51-mm) SR 90° elbow, In USCS units h1 = 0.04 2X3217 = 0.020 ft
1.752
hf1 = 0.04 ^XSTSÔT = 0.0062 m K = 30 fT fT = 0.019
(from Table 6A)
fs fs
In SI units
In USCS units In SI units
In USCS units
In SI units h
## Renewable Energy 101
Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.
Get My Free Ebook | 921 | 2,433 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.375 | 3 | CC-MAIN-2019-09 | latest | en | 0.82896 |
https://www.dgnpropertysolutions.com/g12ml5su/phase-diagram-of-water-vs-other-substances-30aeb0 | 1,679,806,195,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00104.warc.gz | 819,560,240 | 21,784 | # phase diagram of water vs other substances
(2) the "critical point". Molecules with stronger forces of attraction between them belong to However for other substances, notably water, the line slopes to the left as the diagram for water shows. Molecules of carbon dioxide do not have hydrogen bonds. 28 terms. There are two special points on the curve: BIO - CH. temperature. separate phases. Can you tell me which fuse i replace for drovers side low beam headlight. sublimes directly. low pressures, up to 1000 atmospheres or so. to pack efficiently together. different The critical point is the Strictly, this observation only applies to reasonably compared with 218 atmosphere. This region is known as the supercritical fluid region. States of matter include solid, liquid or gas phases. Phase diagrams are useful to show what will happen when the pressure or temperature moves from one point to another. The definition in temperature is average kinetic energy. forces of attraction between the molecules. Some phase diagrams contain additional information. To extreme pressures above 10000 atm do other solid phases of carbon dioxide force of attraction between them is a rather weak force known as a dispersion The liquid phase appears between the two regions. The liquid phase appears between the two regions. The Carbon dioxide In this phase diagram for water indicate the direction that the solid liquid and liquid gas coexistence lines will move after the addition of solute. He holds bachelor's degrees in both physics and mathematics. In this phase diagram for water indicate the direction that the solid liquid and liquid gas coexistence lines will move after the addition of solute. oxygen atom from another. Figure \(\PageIndex{2}\) shows the phase diagram of water and illustrates that the triple point of water occurs at 0.01°C and 0.00604 atm (4.59 mmHg). lots of empty space in it. Each boundary crossing has its own name depending on the direction the boundary is crossed.When moving from the solid phase to the liquid phase across the solid/liquid boundary, the material is melting.When moving in the opposite direction, liquid phase to solid phase, the material is freezing.When moving between solid to gas phases, the material undergoes sublimation. Mesophases are of particular interest for liquid crystal technology.While phase diagrams look simple at first glance, they contain a wealth of information concerning the material for those who learn to read them. Whether a substance is in a condensed phase -- solid or liquid -- or a gas 51 terms. (1) the "triple point" -- the single set of conditions under which What Is a Volatile Substance in Chemistry? On a phase diagram this causes a crossing of both the solid liquid boundary and the liquid gas boundary. List of Phase Changes Between States of Matter, The Difference Between a Phase and State of Matter, Sublimation Definition (Phase Transition in Chemistry). A water molecule, on the other hand has a shape which has been described (3) Although the triple point of carbon dioxide occurs at such a high This point is called the triple point.The other point of interest is when the pressure and temperature are high enough to be unable to tell the difference between the gas and liquid phases. This is connected In the opposite direction, gas to solid phases, the material undergoes deposition.Changing from liquid phase to gas phase is called vaporization. This video contains plenty of examples and practice. Refer to the phase diagram for carbon dioxide in problem set 60. In this diagram, Point A is in the solid region. When the material is at this pressure and temperature, it can exist in all three phases. At a point on a phase boundary, the substance can be in either one or the other phases that appear at either side of the boundary. very efficiently. These lines are known as phase boundaries. associated with melting and freezing. structure of ordinary ice is a very expanded hexagonal cage structure, with This is an example of a two dimensional phase diagram showing phase boundaries and colored coded phase regions. 6. Point D is the point where all three phases meet. If the bel... My manual for the 2014 jetta doesnt have a fuse box diagram. another. The green line marks the freezing point or transition from liquid to solid the blue line marks the boiling point or transition from liquid to gas and the red line shows the conditions under which a solid can be converted directly to a gas and vice versa. Phase Diagram of Water vs Other Substances: Differences Phase (matter) - Wikipedia Example of the Triple Point | The Flood What is the meaning of triple point of water? S... Not sure if i have the hoses in the right plugs. carbon dioxide cannot occur at ordinary pressures of 1 atm. These phases exist in equilibrium with one another.There are two points of interest on a phase diagram. between red and blue is the boundary between solid and liquid, and is I have a 2003 mini cooper s with a standard cd player i also have ... 34 2012 Ford Fusion Serpentine Belt Diagram, 30 Universal Motorcycle Speedometer Wiring Diagram, 28 Mini Cooper Harman Kardon Amplifier Wiring Diagram. balloon that has been blown up). Molecules of water have quite strong forces of attraction (known as hydrogen 2007 volkswagen jetta fuse box diagram inside and outside needed. This video shows how to replace the deck belt on an lt2000 tractor but this also applies to the lt1000 and many other models. together Substances in this region can take on properties and behaviors of both gas and liquid. Point b is in the liquid phase and point c is in the gas phase. At low pressure and high temperature the substance is in the gas phase. Answer bank liquid pressure atm solid gas temperature c. In this phase diagram for water indicate the direction that the solid liquid and liquid gas coexistence lines will move after the addition of solute. MaddieWalters3. (2) For water, the triple point occurs at a pressure of 4.5 torr, or about The lines on a phase diagram correspond to the dividing lines between two phases. It is also why ice does not form Whether a substance prefers to be in a solid or liquid form depends on how The curve associated with its triple point -- about 5.1 atmosphere at -56 deg C. Liquid Point B is in the liquid phase and Point C is in the gas phase. The only what makes water's phase diagram different?-the steep DOWNWARDS slope on the s-l equilibrium line-this is the evidence that solid water is less dense than liquid water (why ice floats) ... OTHER SETS BY THIS CREATOR. At a point on a phase boundary, the substance can be in either one or the other phases that appear at either side of the boundary.
; | 1,375 | 6,717 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2023-14 | longest | en | 0.924449 |
https://www.khipufieldguide.com/notebook/fieldmarks/catalog/AS183.html | 1,723,560,912,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641079807.82/warc/CC-MAIN-20240813141635-20240813171635-00784.warc.gz | 662,516,204 | 5,895 | Fieldmark Fieldmark XRay
AS183
Pendant Pendant Sum
Group Group Sum
Notes:
Ascher Databook Notes:
1. This is a small piece of cord wound around the pendant for 0.3 cm. between the 2 long knots.
2. AS182-AS186 are associated in that they are tied together. For a comparison of them, see AS182.
3. By spacing, the khipu contains 2 pairs of pendants. Each pendant in the second pair is W with a DB subsidiary.
4. The sum of the pendants in the first pair equals the sum of the pendants in the second pair. Each pair sums to 100.
5. Multiplication by 2 is suggested by the first 4 values. P1 is a W cord with value 85+5; P2 is DB with value 6+4; P3 is W with value 45; and P3s1 is DB with value 3+2. Note that 2(3+2) = 6+4. If you double 45 and keep the tens position and units position separated, you get 2 (4 tens & 5 ones) = 8 tens + 2(5 ones) = 85+5. | 265 | 849 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2024-33 | latest | en | 0.870907 |
https://fmforums.com/topic/77069-selecting-records-at-random/ | 1,675,653,487,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500303.56/warc/CC-MAIN-20230206015710-20230206045710-00689.warc.gz | 283,207,352 | 19,952 | # Selecting Records At Random
This topic is 4348 days old. Please don't post here. Open a new topic instead.
## Recommended Posts
I have a table that contains 397 assessment questions and answers. Is there a way of pulling out a predetermined number of records at random? Would the random function be appropriate here?
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I would define an unstored calculation field (result is Number) = Random.
To get N random records, show all records, sort by the calculation field, go to record number N+1 and omit multiple records [Get (FoundCount) - N].
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I would define an unstored calculation field (result is Number) = Random.
Thanks for the reply comment. So the first step for 40 records as an example would be (40)= Random? I suspect not because this returns a value of zero across all records.
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No, the first step would be to define a calculation field. Enter =
`Random`
as the formula, set the result type to Number, and storage options to 'Do not store…'
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go to record number N+1 and
Sorted the random calculation. If N = 40, what's the best way to go to the 41st record now that they are in random order?
omit multiple records [Get (FoundCount) - N].
Do I use the Get(FoundCount) - N in the specify box of Omit All Records script step?
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what's the best way to go to the 41st record
Use the Go to Record/Request/Page[] script step.
Do I use the Get(FoundCount) - N in the specify box of Omit All Records script step?
Yes.
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Works a treat, thank you very much!
I was expecting it to pull the same 40 records out every time I run the script but I'm delighted that it doesn't. That can only mean that the random value must be changing in each record, at what point does it do that?
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Unstored calculation fields are evaluated "as needed". When you call for records to be sorted by the field, that's when it's needed.
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This topic is 4348 days old. Please don't post here. Open a new topic instead.
## Create an account
Register a new account | 507 | 2,171 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.828125 | 3 | CC-MAIN-2023-06 | latest | en | 0.888762 |
https://math.stackexchange.com/questions/2852962/a-non-constant-harmonic-function-must-have-a-zero | 1,582,979,961,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875149238.97/warc/CC-MAIN-20200229114448-20200229144448-00545.warc.gz | 458,190,082 | 32,183 | A non-constant harmonic function must have a zero
I have the question:
If $u$ is a non-constant harmonic function, it must have a zero. Where is $u$ defined from the complex to the real set?
My approach has been as follows:
Since $u$ is non-constant harmonic, it must have a harmonic conjugate $v$ such that $f = u + iv$ is analytic. For a $z$ in a circle, we can use the Cauchy Integral formula to define $f(z)$ and then separate it into real and imaginary parts.
Doing this, we get the formula from the mean value property for harmonic functions. But I am very confused about the mean value property and how it ensures a zero. Since $u$ is non-constant, does it take both negative and positive values and hence by IVP, it must take zero as well? Any pointers on how to proceed?
• You seem to be asking, when you write "Where is $u$ defined from the complex to the real set?", what is the preimage of the reals. That is, what is $u^{-1}(\mathbb{R})$, which is the same as asking: what is the set $\{z \in \mathbb{C} : u(z) \in \mathbb{R}\}$. Is this what you want to know? I ask because the work you've shown seems to be trying to find a zero, not finding the preimage of the reals. – Eric Towers Jul 16 '18 at 1:53
2 Answers
Suppose $u:\mathbb C\to \mathbb R$ is a nonconstant harmonic function, and that $u$ is never $0.$ Because $u$ is continuous, $u(\mathbb C)$ is a connected subset of $\mathbb R.$ That implies $u(\mathbb C)$ is an interval that doesn't contain $0.$ Thus either $u(\mathbb C)\subset (0,\infty)$ or $u(\mathbb C)\subset (-\infty,0).$ Suppose it's the second case. Let $v$ be a harmonic conjugate of $u$ on $\mathbb C.$ Then $f(z) = e^{u+iv}$ is an entire function such that $|f(z)| = e^{u(z)} \le e^0=1$ everywhere. This implies $f$ is a bounded entire function. By Liouville, $f$ is constant. That implies $u$ is constant, contradiction.
• Nice! Very beautiful! – Hugocito Jul 16 '18 at 2:42
• thank you for the elegant solution! – deadcode Jul 16 '18 at 3:14
The result is true because of the Little Picard Theorem.
Since $u: \mathbb C \to \mathbb R$ is harmonic and non-constant, there is a non-constant holomorphic function $f:\mathbb C \to \mathbb C$ such that $\text{Re}f = u$. The Little Picard Theorem says that $f(\mathbb C) = \mathbb C$ or $f(\mathbb C) = \mathbb C\setminus \{p\}$ for some $p\in \mathbb C$. So, you can take some point of the form $0+i\eta$ on $f(\mathbb C)$. Therefore, there is $z=x+iy$ such that $f(z) = 0+i\eta$ and from that we conclude $u(z) = 0$. | 765 | 2,515 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.546875 | 4 | CC-MAIN-2020-10 | latest | en | 0.901507 |
http://perplexus.info/show.php?pid=7771&cid=50304 | 1,544,846,427,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376826715.45/warc/CC-MAIN-20181215035757-20181215061757-00623.warc.gz | 227,013,737 | 4,586 | All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
perplexus dot info
Killer and Victim (Posted on 2012-08-26)
The sisters of Aaron Green were Betty and Clara; the name of Aaron's girlfriend was Flora Brown and, Duane and Edwin were Flora’s brothers. Their occupations were as follows:
• Aaron was a doctor.
• Betty was a doctor.
• Clara was a lawyer.
• Duane was a doctor.
• Edwin was a lawyer.
• Flora was a lawyer.
One of the six killed one of the other five. Here are some clues to the murder.
1. If the killer and the victim were related, the killer was a man.
2. If the killer and the victim were not related, the killer was a doctor.
3. If the killer and the victim had the same occupation, the victim was a man.
4. If the killer and the victim had different occupations, the victim was a woman.
5. If the killer and the victim were of the same sex, the killer was a lawyer.
6. If the killer and victim were different sexes, the victim was a doctor.
Who was the killer? Who was the victim?
See The Solution Submitted by K Sengupta Rating: 4.6667 (3 votes)
Comments: ( Back to comment list | You must be logged in to post comments.)
Solution (using Excel) Comment 6 of 6 |
I created a series of 6x6 matrices numbered 1 thru 6 for each of the six rules, grouping them into three pairs. Each cell was filled with either T or F according to the stipulations for each rule. From the first two rules, it is clear that neither Clara nor Flora could be the killer since neither was a man or a doctor - the results of rules 1 and 2 respectively.
I then wrote logic comparing the cells in each pair, looking for a 'T' in either corresponding cell for each pair. All three pairs must have at least one 'T' for each cell position.
The only combo that returned a 'T' for all three pairs was Betty as the killer and Duane as the victim.
Posted by hoodat on 2012-12-19 00:33:59
Search: Search body:
Forums (0) | 488 | 1,948 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.71875 | 4 | CC-MAIN-2018-51 | latest | en | 0.977142 |
http://mathhelpforum.com/advanced-statistics/110661-sse-simple-linear-regression.html | 1,481,298,333,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698542712.49/warc/CC-MAIN-20161202170902-00205-ip-10-31-129-80.ec2.internal.warc.gz | 179,257,989 | 10,191 | # Thread: SSE simple linear regression
1. ## SSE simple linear regression
I am given SSE= $\sum_{i=1}^n(y_i-\hat{y}_i)^2$ and am asked to show that it $=\sum_{i=1}^n(y_i-\bar{y})^2-\hat{\beta}_1\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})$
where $\hat{y}_i=\hat{\beta}_0+\hat{\beta}_1x_i$ and $\hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{x}$
So here's what I got:
$\sum_{i=1}^n(y_i-\hat{y}_i)^2$
$=\sum_{i=1}^n(y_i-\hat{\beta}_0-\hat{\beta}_1x_i)^2$
$=\sum_{i=1}^n(y_i-\bar{y}+\hat{\beta}_1\bar{x}-\hat{\beta}_1x_i)^2$
$=\sum_{i=1}^n(y_i-\bar{y}-\hat{\beta}_1(x_i-\bar{x}))^2$
$=\sum_{i=1}^n[(y_i-\bar{y})^2-2\hat{\beta}_1(y_i-\bar{y})(x_i-\bar{x})+\hat{\beta}_1^2(x_i-\bar{x})^2]$
$=\sum_{i=1}^n(y_i-\bar{y})^2-2\hat{\beta}_1\sum_{i=1}^n(y_i-\bar{y})(x_i-\bar{x})+\hat{\beta}_1^2\sum_{i=1}^n(x_i-\bar{x})^2$
and i've got nothing from here.... anyone? thanks
2. You are just one step short. For simplicity convert everything into sums of squares notation, with
$S_{xx} = \sum (x_i - \bar{x})^2, S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y})$ and note that $\hat{\beta_1} = \frac{S_{xy}}{S_{xx}}$ (which I'm assuming you can take as given).
3. Originally Posted by theodds
You are just one step short. For simplicity convert everything into sums of squares notation, with
$S_{xx} = \sum (x_i - \bar{x})^2, S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y})$ and note that $\hat{\beta_1} = \frac{S_{xy}}{S_{xx}}$ (which I'm assuming you can take as given).
$=\sum_{i=1}^n(y_i-\bar{y})^2-2\hat{\beta}_1\sum_{i=1}^n(y_i-\bar{y})(x_i-\bar{x})+\hat{\beta}_1^2\sum_{i=1}^n(x_i-\bar{x})^2$
$=S_{yy}-2\hat{\beta}_1S_{xy}+\hat{\beta}_1\frac{S_{xy}}{S_ {xx}}\cdot S_{xx}$
$=S_{yy}-\hat{\beta}_1S_{xy}$
Good call | 756 | 1,701 | {"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": 17, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5 | 4 | CC-MAIN-2016-50 | longest | en | 0.583067 |
https://www.veyespe.com/what-is-kinematics-physics/ | 1,716,025,354,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971057379.11/warc/CC-MAIN-20240518085641-20240518115641-00751.warc.gz | 964,605,911 | 16,439 | # The concept of what’s kinematics might be confusing for a number of people.
A number of the issues that they are likely to confuse are issues like how does the centripetal force go to get essay writer around?
What may be the force that is certainly not centrifugal around the top rated? And what’s the force that is http://resourcemanagement.wustl.edu/ certainly not centripetal inside the bottom? That is simply to talk about in regards to the most common parts of human physique.
What is it that is slowing down the center of gravity? It’s named the centripetal force. So if you’re inside a automobile along with the bottom is spinning, this really is a place where the centripetal force will be going to.
The center of gravity is generally located inside the center from the ground. But this force can never ever be made unless the force is applied from under.
So the key point that you just ought to realize is that whenever you learn this kind of motion is going to be applied, you are going to be capable of predict it completely. Nonetheless, we are going to attempt to recognize much more about this basic idea.
The classic samedayessay.com reviews way of explanation about what exactly is kinematics is really based on Newton’s physics. But this strategy is rather complicated as well as gives us the errors.
You can not clarify the motion of an object by using the classical mechanics. You should use something else.
We can truly predict the motion of an object applying the basic concepts of what is kinematics. Among these concepts is the center of gravity. Then you canuse the force in the center with the frame.
However, this center is also topic to a degree of acceleration also. That is also on the list of basic concepts of what’s kinematics physics.
However, whenever you discover the center of gravity is spinning around the prime, then you definitely can use it to the bottom from the surface. That is exactly where you could apply the centripetal force.
As quickly as you may apply this force inside the center of gravity, then you can apply it towards the distance on the center of gravity. This really is by far the most critical notion in physics.
However, the moment exactly where the point of influence is located is not a vital factor. When you usually do not know the circumstance, then you definitely should really not make this mistake. If you make a error, then you may have no opportunity to recover in case you are nevertheless alive. | 510 | 2,481 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.265625 | 3 | CC-MAIN-2024-22 | latest | en | 0.959092 |
https://owenduffy.net/blog/?paged=89 | 1,718,494,759,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861618.0/warc/CC-MAIN-20240615221637-20240616011637-00724.warc.gz | 400,113,990 | 16,244 | ## Measuring ambient noise level using a spectrum analyser #2
The article Measuring ambient noise level using a spectrum analyser was a walk through of measuring ambient noise using a spectrum analyser.
This article details a method that uses an online calculator to conveniently perform the calcs that permit more accurate answers by factoring the internal noise of the spectrum analyser into the calcs.
## Step 1: measure instrument noise figure
Measure the noise floor of the instrument with 50Ω input termination using an average power (RMS) detector.
Now calculate the Noise Figure (Field Strength Noise Figure on output report). Continue reading Measuring ambient noise level using a spectrum analyser #2
## Online calculator of ferrite material permeability interpolations – additional materials
I have progressively added some further materials to Ferrite permeability interpolations calculator.
Of interest to European designers is inclusion of three common materials used for HF applications, Ferroxcube’s 4A11, 4B2, and 4C65.
## AE7PD’s transmitting loop measurements
AE7PD documented his measurements of a 3.16m perimeter circular transmitting loop, 1.8m centre height above ground, that he made using 16mm copper tube and a split stator tuning capacitor:
AE7PD gives the radiation efficiency on 20m as 30.5% or -5.2dB.
I present here an alternative analysis of the antenna as measured on 20m.
Assuming the measurements were made with the antenna clear of disturbing conductors etc, and that 5/8″ tube means 16mm OD.
The key measurements were:
• centre frequency 14.165MHz, VSWRmin=1.0;
• VSWR=2.62 bandwidth 22kHz.
A NEC-4.2 model of the antenna at 14MHz was built and calibrated to the measured half power bandwidth (22kHz). Model assumptions include:
• ‘average’ ground (σ=0.005, εr=13);
• Q of the tuning capacitor = 2000;
• conductivity of the loop conductor adjusted to calibrate the model half power bandwidth to measurement.
Note that the model may depart from the actual test scenario in other ways.
Above is the VSWR scan of the calibrated model, the load is matched at centre frequency and half power bandwidth is taken as the range between ReturnLoss=6.99dB points. Continue reading AE7PD’s transmitting loop measurements
## Findling & Siwiak 2012 measurements of an Alexloop – discussion
I mentioned in Findling & Siwiak 2012 measurements of an Alexloop issues with their efficiency calculation.
Above is an extract from (Findling & Siwiak 2012).
(Siwiak & Quick 2018) give an equivalent circuit of lossless loop structure in free space.
When tuned to resonance, the response is simply that of a series RLC circuit where R=Rr (the radiation resistance) which is dependent on frequency, but varies very slowly with frequency compared to the net reactance X.
Above is a NEC simulation of such a loop. Continue reading Findling & Siwiak 2012 measurements of an Alexloop – discussion
## G3CWI 2018 measurements of an Alexloop Walkham
Richard, G3CWI, measured the impedance and bandwidth of a Alexloop Walkham, a popular small transmitting loop (STL). The antenna was situated in the clear at 1.65m centre height above natural ground.
The key measurements were:
• centre frequency 7.014MHz, |Z|=51Ω, VSWR=1.1;
• VSWR=3 bandwidth 16.2kHz.
The step size of the analyser prevented measurement exactly at resonance, but R changes very closely with frequency near resonance so we can estimate it quite well. The above figures can be used to find R close to resonance.
Within the limits of measurement error, we can say that R at resonance should be very close to 51Ω, and VSWRmin close to 1.02. Continue reading G3CWI 2018 measurements of an Alexloop Walkham
## Findling & Siwiak 2012 measurements of an Alexloop
(Findling, A & Siwiak 2012) documented measurements they made of a popular small transmitting loop (STL), an Alexloop Walkham.
Now Alexloops seem to have undergone some evolution, and there does not seem to be a clear list of model names or numbers with features or specifications, so to some extent the antenna is a little non descript.
The article did not document the environment of the test antenna, but Findling explained in correspondence that it was relatively clear of conducting structures and about 1.2m above natural ground.
A NEC-4.2 model of the antenna at 7MHz was built and calibrated to their measured half power bandwidth (19kHz). Model assumptions include:
• ‘average’ ground (σ=0.005, εr=13);
• Q of the tuning capacitor = 1000;
• conductivity of the loop conductor adjusted to calibrate the model half power bandwidth to measurement.
Note that the model may depart from the actual test scenario in other ways, it is challenging to glean all the data that one would like from the article.
Above is an extract from (Findling, A & Siwiak 2012).
Above is the VSWR scan of the calibrated model, the load is matched at centre frequency and half power bandwidth is taken as the range between ReturnLoss=6.99dB points. Continue reading Findling & Siwiak 2012 measurements of an Alexloop
## Single turn coaxial loop resonator analysis
Recent discussion online of a purported commercial HF small transmitting loop (STL) was challenged in analysing the structure, questioning whether such a connection was ‘correct’.
The STL used a main loop resonator and a separate small auxiliary loop for the 50Ω feed, a very common arrangement.
The main loop is a coaxial cable with, in this case, a tuning capacitor inserted between the inner conductors at each end. Above is a diagram of the main loop. Continue reading Single turn coaxial loop resonator analysis
## WW1WW’s matching transformer for an EFHW
At PD7MAA’s BN43-202 matching transformer for an EFHW I gave an estimate of the core loss in PD7MAA’s transformer.
An online expert questioned the analysis and later measurements, and proposed his own transformer design as evidence.
Notably, his transformer uses #61 material and a larger binocular core, a Fair-rite 2861006802 with 2t for a nominal 50Ω primary, giving loss measurements at 7MHz of 0.08dB. Note that the confidence limits of that loss measurement because of the way in which it was obtained (eg a 1% error in the 1120Ω load resistor contributes 0.043dB error to the result), but the measurements do suggest that the loss is probably very low.
Though the loss is low and Return Loss is high at 7MHz, the limits for ReturnLoss>14dB (VSWR<1.5) is 5-18MHz. With compensation, that range may be changed.
## Prediction
Lets apply the method laid out at PD7MAA’s BN43-202 matching transformer for an EFHW.
The best Fair-rite data I can find quickly is a chart of the impedance of a one turn winding.
Scaling from this graph, Xs is close of 35Ω at 7MHz, so lets used that to derive some basic parameters for the core. Continue reading WW1WW’s matching transformer for an EFHW
## AIM 914 produces internally inconsistent results
A new release, AIM914 appeared recently.
In the common theme of one step forward, two steps backwards, this version has defects that were not present in AIM910B.
Let’s review the internal consistency of this part of the display screen.
Most of the values given above are calculated from a single measurement value, and should be internally consistent. That measurement value is translated to different quantities, many based on the stated Zref (75Ω in this case). Continue reading AIM 914 produces internally inconsistent results
## Matched Line Loss of generic RG6/U Quad Shield CCS
This article documents a measurement of Matched Line Loss (MLL) of a 35m test section of generic RG6/U Quad Shield CCS.
It has become impossible in recent years to buy low cost RG6/U with solid centre conductor locally, and the imported product with solid copper conductor is prohibitively expensive (~\$6/m as against \$0.35/m for the CCS).
The CCS cable does have near copper like performance at UHF and above, but what is its behavior at HF?
Above is calculated MLL from a S11 scan of the test section with S/C and O/C termination. There is a little ripple on the response due to measurement error. The graph also has a curve fit, MLL=0.0285f^0.1506 (F in MHz). Continue reading Matched Line Loss of generic RG6/U Quad Shield CCS | 1,900 | 8,251 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2024-26 | latest | en | 0.860815 |
https://www.studypool.com/discuss/480109/suppose-1600-j-of-heat-are-added-to-4-1-mol-of-argon-gas-at-a-constant-pressure-2?free | 1,508,280,394,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187822513.17/warc/CC-MAIN-20171017215800-20171017235800-00308.warc.gz | 983,590,231 | 13,973 | ##### Suppose 1600 J of heat are added to 4.1 mol of argon gas at a constant pressure
label Physics
account_circle Unassigned
schedule 1 Day
account_balance_wallet \$5
Apr 17th, 2015
the heat cacpacity of 4.1 mole Argon is
C = 3/2 n R = 3/2*4.1*8.31 = 51.10J/K
b) 960/51.10 = 18.78 K
c)
PV = nRT
P deltaV = nR delta T
deltaV = nR/P *deltaT = 4.1*8.31/120000*18.78
=0.00533 m^3
Apr 17th, 2015
...
Apr 17th, 2015
...
Apr 17th, 2015
Oct 17th, 2017
check_circle | 204 | 469 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2017-43 | latest | en | 0.871754 |
https://mathematica.stackexchange.com/questions/284725/how-to-riffle-lists-without-cyclic-repetitions | 1,716,614,688,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058773.28/warc/CC-MAIN-20240525035213-20240525065213-00889.warc.gz | 322,612,289 | 40,982 | # How to Riffle lists without cyclic repetitions
To the follow to my previous question How to combine a few lists?.
There are two lists: a={1,2,4,5,7,6,9,2}, b={1,3,4} I would like to get the next lists:
1. ab1={1,1,2,3,4,4,5,7,6,9,2} (the first element of b is placed between the first and second elements of a, the second element of b is placed between the second and third elements of a, etc)
2. ab2={1,2,4,5,7,1,6,3,9,4,2} (similar to 1., but put the elements of the list b from the end of the list a)
3. ab3={1,2,1,4,3,5,4,7,6,9,2} (in this case elements of b can be placed in a specific place of the list a, in particular here the first element of b is placed between the second and third elements of the list a, etc )
• As Flatten may be used to Transpose a 'ragged' array, (Flatten[{a,b},{{2}}]//Flatten)==ab1 May 5, 2023 at 9:07
Using PadRight:
First, the explanation:
PadRight[{a, b}, {2, Max[{Length@a, Length@b}]}, \[Wolf]] // Grid
Keep reading from top to bottom and across. At the end, delete the symbol that was introduced. This prevents the cyclic riffle effect.
(PadRight[{a, b}, {2, Max[{Length@a, Length@b}]}, \[Wolf]] //
Transpose // Flatten) /. \[Wolf] -> Nothing
{1, 1, 2, 3, 4, 4, 5, 7, 6, 9, 2}
Second case is just the reversal of the idea. Don't forget to reverse the generated list at the end.
(PadRight[{Reverse@a, Reverse@b}, {2,
Max[{Length@a, Length@b}]}, \[Wolf]] // Transpose //
Flatten) /. \[Wolf] -> Nothing // Reverse
{1, 2, 4, 5, 7, 1, 6, 3, 9, 4, 2}
For the third case, PadRight provides an additional parameter that can be used to pad elements at the left. Do so with the second list only so as to shift it right by one.
({PadRight[a, Max[{Length@a, Length@b}], \[Wolf]]
, PadRight[b, Max[{Length@a, Length@b}], \[Wolf], 1]
} // Transpose // Flatten) /. \[Wolf] -> Nothing
{1, 2, 1, 4, 3, 5, 4, 7, 6, 9, 2}
Now I have to invest time in learning about Riffle for which two excellent answers already exist.
• Thanks for the interesting explanation! May 5, 2023 at 8:33
You may use "Riffle" like:
1. First element of b at place 2:
a = {1, 2, 4, 5, 7, 6, 9, 2}; b = {1, 3, 4};
Riffle[a, b, {2, 2 Length[b], 2}]
2. Last element of b at second to last place:
Riffle[a, b, {Length[a] - Length[b] + 1, Length[a] + Length[b], 2}]
{1, 2, 4, 5, 7, 1, 6, 3, 9, 4, 2}
1. First element of b at position 3;
Riffle[a, b, {3, (Length[a] + Length[b]) - 3, 2}]
{1, 2, 1, 4, 3, 5, 4, 7, 6, 9, 2}
• Thank you very much! May 5, 2023 at 7:47
riffle[a_, b_, start_, step_] := Riffle[a, b,
Append[step] @
If[Positive @ start, Identity, Reverse] @
(start + {0, Sign[start] step (Length @ b - 1)})]
Examples:
a = {1,2,4,5,7,6,9,2};
b = {1,3,4};
riffle[a, b, 2, 2] == ab1
True
riffle[a, b, -2, 2] == ab2
True
riffle[a, b, 3, 2] == ab3
True
bb = {b1, b2, b3};
riffle[a, bb, 2, 2]
{1, b1, 2, b2, 4, b3, 5, 7, 6, 9, 2}
riffle[a, bb, 3, 2]
{1, 2, b1, 4, b2, 5, b3, 7, 6, 9, 2}
riffle[a, bb, 2, 3]
{1, b1, 2, 4, b2, 5, 7, b3, 6, 9, 2}
riffle[a, bb, -2, 3]
{1, 2, 4, b1, 5, 7, b2, 6, 9, b3, 2}
• Thank you very much! May 5, 2023 at 7:47 | 1,290 | 3,094 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2024-22 | latest | en | 0.812483 |
https://www.brainkart.com/article/Questions-Answers_43386/ | 1,702,083,573,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100781.60/warc/CC-MAIN-20231209004202-20231209034202-00461.warc.gz | 734,880,734 | 14,065 | Home | | Science 8th Std | Questions Answers
# Questions Answers
8th Science : Chapter 2 : Force and Pressure : Text Book Back Questions Answers, Solution
TEXTBOOK EXERCISES
I. Choose the best answer.
1. If we apply force against the direction of motion of the body, then the body will
a. stop moving
b. move with an increased speed
c. move with a decreased speed
d. move in a different direction
[Answer: (a) stop moving]
2. Pressure exerted by a liquid is increased by
a. the density of the liquid
b the height of the liquid column
c. Both a and b
d. None of the above
[Answer: (c) Both (a) & (b)]
3. Unit of pressure is
a. Pascal
b. Nm–2
c. Poise
d. Both a and b
[Answer: (d) Both (a) & (b)]
4. The value of the atmospheric pressure at sea level is
a. 76 cm of mercury column
b. 760 cm of mercury column
c. 176 cm of mercury column
d. 7.6 cm of mercury column
[Answer: (a) 76 cm of mercury column]
5. Pascal’s law is used in
a. hydraulic lift
b. brake system
c. pressing heavy bundles
d. All the above
[Answer: (d) All the above]
6. Which of the following liquids has more viscosity?
a. Grease
b. Water
c. Coconut oil
d. Ghee
[Answer: (a) Grease]
7. The unit of viscosity is
a. Nm2
b. poise
c. kgms–1
d. No unit
[Answer: (b) poise]
II. Fill in the blanks.
1. The pressure of a liquid column increases with the depth of the column.
2. Hydraulic lift works under the principle of Pascal’s Law.
3. The property of surface tension of a liquid surface enables the water droplets to move upward in plants.
4. A simple barometer was first constructed by Torricelli.
III. State true or false. If false, correct the statement.
1. Force acting on a given area is called pressure. [Answer: True]
2. A moving body comes to rest due to friction alone. [Answer: True]
3. A body will sink if the weight of the body is greater than the buoyant force. [Answer: True]
4. One atmosphere is equivalent to 1,00,000 newton force acting on one square metre. [Answer: True]
5. Rolling friction is slightly greater than the sliding friction. [ Answer: False]
6. Friction is the only reason for the loss of energy. [Answer: True]
Correct statement: Rolling friction is slightly lesser than the sliding friction.
7. Liquid pressure decreases with the decrease of depth. [Answer: True]
8. Viscosity depends on the pressure of a liquid. [Answer: True]
IV. Match the following.
a.
Static friction - Viscosity
Kinetic friction - Least friction
Rolling friction - Objects are in motion
Friction between the liquid layers - Objects are sliding
Sliding friction - Objects are at rest
[Answer: i-e, ii-c, iii-b, iv-a, v-d]
i. Static friction (e) objects are at rest
ii. Kinetic friction (c) objects are in motion
iii. Rolling friction (b) least friction
iv. Friction between the liquid layers (a) viscosity
V. Sliding friction (d) objects are sliding
b.
Barometer - reduce friction
Increasing area of contact - Atmospheric pressure
Decreasing area of contact - cause of friction
Lubricants - increases friction
Irregular surface - decreases friction
[Ans : i - b, ii - d, iii - e, iv - a, v - c]
i. Barometer (b) atmospheric pressure
ii. Increase friction (d) increasing area of contact
iii. Decrease friction (e) decreasing area of contact
iv. Lubricants (a) reduce friction
v. Irregular surface (c) cause of friction
V. Complete the analogy.
1. Knot in a thread : Static friction :: Ball bearing : rolling friction
2. Downward force : Weight :: Upward force offered by liquid : Buoyant force
VI. Numerical Problem.
1. A stone weighs 500 N. Calculate the pressure exerted by it, if it makes contact with a surface of area 25 cm2.
Answer:
Given : Weight of a stone F = 500 N
Area A=25 cm2 = 25 × 10-4 m2
To find : Pressure P =?
Formula : Pressure P = F/A
= 500 / [25×10-4]
Solution:
Pressure P = 20 × 104 Nm-2 (or) 20 × 104 Pa
2. In a hydraulic lift, the surface area of the input piston is 10 cm2. The surface area of the output piston is 3000 cm2. A 100 N force applied to the input piston raises the output piston. Calculate the force required to raise the output piston.
Answer:
Solution :
Pressure input on piston,
P = F/A
= 100/ [10×10-4] = 105 N
According to Pascal’s law
P= F/A
105 = F / [3000×10-4 ] = [ F×104 ] / 3000
104 × F = 105 × 3000
F = 3000 × 101
F = 3 × 104N
VII. Consider the statements given below and choose the correct option.
a. Both assertion and reason are true and reason is the correct explanation of assertion.
b. Both assertion and reason are true, but reason is not the correct explanation of assertion.
c. Assertion is true, but reason is false.
d. Both assertion and reason are false.
1. Assertion: Sharp knives are used to cut the vegetables.
Reason: Sharp edges exert more pressure.
[Answer: (a) Both assertion and reason are true and the reason is the correct explanation of the assertion]
2. Assertion: Broad straps are used in bags.
Reason: Broad straps last for long.
[Answer: (b) Both assertion and reason are true, but the reason is not the correct explanation of the assertion]
Correct explanation: The weight of the bags falls on larger area of shoulder. So lesser pressure is produced.
3. Assertion: Water strider slides easily on the surface of water.
Reason: Water strider experiences less buoyant force.
[Answer: (b) If both assertion and reason are true, but reason is not the correct explanation of assertion]
Correct explanation: It is due to the surface tension of water.
VIII. Answer very briefly.
1. Give two examples to verify that a force changes the shape of a body.
Answer:
Force can change the static condition of a body.
(i) If you squeeze a sponge, its shape changes.
(ii) If you pull a rubber band, it becomes longer.
2. Give two examples to verify that a force tends to change the static condition of a body.
Answer: Force can change the static condition of a body.
(i) A rest rubber ball begins to move, when a force applied on it.
(ii) Player applies a force on the stationary football while taking a penalty kick in football match. The force applied by player makes the football move towards the goal.
3. How do you feel when you touch a nail immediately after it is hammered into a wooden plank? Why?
Answer: The nail becomes hot due to friction. Friction changes kinetic energy to heat.
4. How does the friction arise between the surfaces of two bodies in relative motion?
Answer: The force of friction is arised by the interlocking of the irregularities of the two surfaces.
5. Name two instruments which help to measure the pressure of a fluid.
Answer:
(i) Manometer
(ii) Pressure gauge.
6. Define one atmosphere.
Answer: The pressure exerted by the mercury column is considered as the pressure of magnitude ‘one atmosphere’ (1 atm).
7. Why are heavy bags provided with broad straps?
Answer: Broader straps are provided on a back-pack for giving less pressure on the shoulders by providing a larger area of contact with the shoulder.
8. How does surface tension help a plant?
Answer: Water molecules rise up due to surface tension. Xylem tissues are very narrow vessels present in plants. Water molecules are absorbed by the roots and these vessels help the water to rise upward due to “capillarity action” which is caused by the surface tension of water.
9. Which has greater viscosity, oil or honey? Why?
Answer: Honey has greater viscosity.
Reason: Thicker liquids are more viscous than thinner liquids. As honey has greater viscosity, more frictional force will be acting on it.
IX. Answer briefly.
1. Define friction. Give two examples of the utility of friction in day to day life.
Answer:
Friction : Friction is a force that slows down moving objects or prevents stationary objects from moving.
Examples of the utility of friction in day to day life.
(i) Cars and buses are able to move safely on the road because of friction between the treaded tyres and the surface of the road.
(ii) We are able to write on paper only with the help of friction between the pencil or pen and paper.
2. Mention any three ways of minimising friction.
Answer:
(i) By using lubricants : These are applied to surfaces to reduce the friction between the surfaces.
Ex : Oil, wax, grease and castor oil.
(ii) With the help of polishing the surface :
We sprinkle fine powder on the carrom board and then we polish its surface to make smooth so that the striker slides easily on the surface.
(iii) By using ball bearing :
We use leadshots in bearing of a cycle hub because rolling friction is smaller than sliding friction.
3. State Pascal’s law and mention its applications.
Answer:
(i) In an automobile service station, the vehicles are lifted upward using the hydraulic lift, which works as per Pascal’s law.
(ii) The automobile brake system works according to Pascal’s law.
(iii) The hydraulic press is used to make the compressed bundles of cotton or cloth so as to occupy less space.
4. Why is a ball bearing used in a cycle hub?
Answer: The rolling friction is smaller than sliding friction, sliding is replaced by rolling with the usage of ball bearings. So lead shots are used in the bearing of a cycle hub to reduce the friction.
X. Answer in detail.
1. Friction is a necessary evil - Explain.
Answer:
Friction is a necessity in most of our day to day activities. It is desirable in most situations of our daily life.
(i) We can hold any object in our hand due to friction.
(ii) We can walk on the road because of friction. The footwear and the ground help us to .walk without slipping.
(iii) Writing easily with a pen on paper is due to friction.
(iv) Automobiles can move safely due to friction between the tyres and the road. Brakes can be applied due to frictional resistance on brake shoes.
(v) We are able to light a matchstick, sew clothes, tie a knot or fix a nail in the wall because of friction.
Though it is giving a negative effect, in most of our day to day life friction helps us to make our life easy. So, it is called as “necessary evil”.
Disadvantages of friction :
(i) Friction wears out the surfaces rubbing with each other, like screws and gears in machines or soles of shoes.
(ii) To overcome the friction an excess amount of effort has to be given to operate a machine. This leads to wastage of energy.
2. Give the different types of friction and explain each with an example.
Answer:
Friction can be classified into two basic types:
(i) Static friction
(ii) Kinetic friction.
(i) Static friction: The friction experienced by the bodies, which are at rest is called static friction. (E.g : All the objects rigidly placed to be at rest on the Earth, a knot in a thread.)
(ii) Kinetic friction: Friction existing during the motion of bodies is called kinetic friction.
Further, kinetic friction can be classified into two:
(i) Sliding friction
(ii) Rolling friction.
(i) Sliding friction: When a body slides over the surface of another body, the friction acting between the surfaces in contact is called sliding friction.
(ii) Rolling friction: When a body rolls over another surface, the friction acting between the surfaces in contact is called rolling friction.
Rolling friction is less than sliding friction. That is why wheels are provided in vehicles, trolleys, suitcases etc.
3. Describe an experiment to prove that friction depends on the nature of a surface.
Answer:
To understand about the frictional force between the layers of liquid in motion.
Materials required: Different kinds of liquid (coconut oil, honey, water, ghee), glass plates - 4 nos.
Procedure :
(i) Take a small quantity of different kinds of liquid like coconut oil, honey, water and ghee etc., in a cup.
(ii) Place one drop of each liquid on a separate glass plate.
(iii) Next, gently raise one end of the glass plate, one by one, so as to allow the liquid to slide down the smooth surface of the plate.
(iv) Observe the speed of each liquid.
Observation: Each liquid moves with a different speed. Water flows faster than other liquids. Coconut oil flows with a moderate speed. Ghee flows very slowly.
Inference: Between the layers of each liquid, in motion, there is a frictional force parallel to the layers of the liquid. This frictional force opposes the motion of the liquid layers while they are in motion.
4. Explain how friction can be minimised.
Answer:
(i) Using lubricants :
(1) A substance, which reduces the frictional force, is called a lubricant.
E.g : Grease, coconut oil, graphite, castor oil, etc.
(2) The lubricants fill up the gaps in the irregular surfaces between the bodies in contact. This provides a smooth layer thus preventing a direct contact between their rough surfaces.
(ii) Using ball bearing :
Since, the rolling friction is smaller than sliding friction, sliding is replaced by rolling with the usage of ball bearings. We can see lead shots in the bearing of a cycle hub.
5. Describe an experiment to prove that the pressure in a liquid increases with depth.
Answer: Take a plastic bottle. Punch three holes on its side in the same direction, but at different heights. Now pour some water into it and let it flow through the holes. Observe the flow of water.
Inference : The water comes out from all the holes with a different force and falls on the table at points that are at variable distances from the bottle. Water from the lowest hole comes out with the greatest force and falls at a point that is at the maximum distance from the bottle. Water from the topmost hole comes out with the least force and falls at the point that is at the minimum distance from the bottle.
Reason: This activity confirms that the pressure in a liquid varies with the depth of the point of observation in it.
XI. Higher Order Thinking Questions.
1. Why is it not advisable to use a fountain pen while travelling in an aeroplane?
Answer: Fountain pens are built in such a way that the pressure inside them balances the atmospheric pressure at sea level. Since atmospheric pressure decreases with an increase in height above sea level, the pressure inside the pen turns out to be much greater than the air pressure in an aeroplane and the pen stalls leaking.
2. Is there any possibility of making a special device to measure the magnitude of friction directly?
Answer: Yes. Tribometer is a special device to measure the magnitude of friction directly.
3. Vidhya feels that mercury is costly. So, instead of mercury she wants to use water as a barometric liquid. Explain the difficulty of constructing a water barometer.
Answer:
(i) Mercury is commonly used in barometers because of its high density means the height of the column can be a reasonable size to measure atmospheric pressure.
(ii) A barometer using water, for instance, would need to be 13.6 times taller than a mercury barometer to obtain the same pressure difference.
(iii) This is because mercury is 13.6 times more dense than water.
XII. Project Work.
Observe the devices, gadgets or things around you. List out the types of friction involved in each device. How would you minimise the friction? Record your observations and discuss your results with your classmates.
Tags : Force and Pressure | Chapter 2 | 8th Science , 8th Science : Chapter 2 : Force and Pressure
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8th Science : Chapter 2 : Force and Pressure : Questions Answers | Force and Pressure | Chapter 2 | 8th Science
Copyright © 2018-2024 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai. | 3,631 | 15,623 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.09375 | 3 | CC-MAIN-2023-50 | latest | en | 0.760414 |
https://encyclopedia2.thefreedictionary.com/measures+of+central+tendency | 1,624,358,578,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623488517048.78/warc/CC-MAIN-20210622093910-20210622123910-00047.warc.gz | 227,497,846 | 12,533 | # measures of central tendency
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Related to measures of central tendency: standard deviation, measures of dispersion
## measures of central tendency
the different ways of conceptualizing the central or middle position of a group of observations, numbers, etc. There are three measures of central tendency: the mode, the median and the mean. The mode is the value which occurs most often. The median is the value which occupies the central position, having as many values below as above it. The mean (more commonly called the average) is found by adding together each individual value and dividing by the number of cases, or observations.
Sometimes a set of observations will yield a bimodal distribution (where two different values occur most often). Also, if there is an even number of observations there is no central value to represent the median. In this case the median may be taken to lie midway between the two centrally placed values.
Where there are many values in the distribution the approximate value of the median can be calculated by interpolation. The data is first grouped into a set number of bands and the median is taken as lying within the middle group, its value being calculated mathematically by estimating its position from the percentage of cases lying in the lower and higher bands.
The choice of which measure of central tendency to employ is determined by two factors: the level of measurement (see CRITERIA AND LEVELS OF MEASUREMENT) being employed and the amount of dispersion in the set of observations. Where a nominal-level measure is being employed, only the mode should be calculated. For example, if numerical values have been assigned to different types of accommodation, then the mode will show which is the most popular type of accommodation, but both the mean and the median would be meaningless. The median is best used with ordinal-level measures where the relative distances between categories is unknown (although it should be said that many social scientists do use the mean when dealing with ordinal-level variables because of the large number of statistical tests which can then be undertaken). Finally, the mean is generally the best statistic to use with interval level measures, except in those instances in which there are a number of extreme values which skew the distribution. For example, the mean incomes of a group of respondents may be skewed because of the inclusion in the sample of a few high-income earners. In such instances the median is often a better statistic to employ. Another instance where the median might be calculated is where data has been grouped and the ‘highest’ category is open-ended. For example, income might have been grouped in such a way that all earning over £100,000 per annum are grouped together and there is no upper limit to the amount which people in the category earn. In such a case the mean cannot be calculated, but the value of the median can be estimated by the process of interpolation mentioned above. see also MEASURES OF DISPERSION.
Collins Dictionary of Sociology, 3rd ed. © HarperCollins Publishers 2000
References in periodicals archive ?
A fundamental knowledge of the simple measures of central tendency --the mean, the median, and the mode--is essential to understanding basic statistical inference.
Word frequency, repetition, and lexicality effects in word recognition tasks: Beyond measures of central tendency. Journal of Experimental Psychology: General, 128, 32-55.
Cohen takes an accessible, conversation approach when introducing the conceptual foundations and basic of statistic procedures, a practice he continues with material on frequency tables, graphs, distributions, measures of central tendency and variability, standardized scores and the normal distribution, hypothesis testing with one or two samples, internal estimation and the t distribution, the t test for two independent sample means, statistical power and effect size, linear correlation and regression, the matched t test, one-way independent ANOVA and two-way ANOVA, multiple regression and its connection to ANOVA, nonparametric statistics, chi-square tests and statistical tests for ordinal data.
The article gave examples of tasks used on the National Assessment of Educational Progress (NAEP) to assess students' understanding of measures of central tendency. It also provided details about students' responses to the tasks.
According to the research results, most managers are highly aware about statistical methods, but they often use only the basic methods, i.e., graphs, measures of central tendency, measures of dispersion and time series indexes.
* calculate various measures of central tendency and dispersion.
In a more innovative textbook or classroom, students might be using spreadsheets to represent data graphically and compute the various measures of central tendency for a set of data.
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Open / Close | 944 | 4,994 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.640625 | 4 | CC-MAIN-2021-25 | latest | en | 0.947214 |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=133&t=41620 | 1,611,816,408,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610704835901.90/warc/CC-MAIN-20210128040619-20210128070619-00777.warc.gz | 415,222,408 | 11,063 | ## When C, Heat Capacity is not given?
Volume: $\Delta S = nR\ln \frac{V_{2}}{V_{1}}$
Temperature: $\Delta S = nC\ln \frac{T_{2}}{T_{1}}$
VivianaHF2L
Posts: 29
Joined: Fri Sep 28, 2018 12:20 am
### When C, Heat Capacity is not given?
For problem 9.3 the formula for change in temperature is needed but how are we suppose to find Heat Capacity when the number of moles are not given? Without the moles the formula for solving C is not possible.
Andrea Zheng 1H
Posts: 61
Joined: Fri Sep 28, 2018 12:26 am
### Re: When C, Heat Capacity is not given?
If you are referencing problem 9.3 in 6th Edition textbook, for part (a), you would use the equation deltaS=deltaQ/T. You would get deltaQ = +65J and T = 25 + 273 = 298 K. This gives you a deltaS of 65/298 = 0.22J/K
You would use the same process for part (b), except with 373 K instead of 298.
For part (c), the entropy change in part (a) was higher, because temperature is in the denominator, and thus a lower temperature will result in a higher change in entropy. | 312 | 1,023 | {"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} | 3.640625 | 4 | CC-MAIN-2021-04 | latest | en | 0.897206 |
https://zbmath.org/?q=1108.34020 | 1,642,652,396,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320301720.45/warc/CC-MAIN-20220120035934-20220120065934-00617.warc.gz | 1,130,135,892 | 9,439 | zbMATH — the first resource for mathematics
On the existence of minimal and maximal solutions of discontinuous functional Sturm-Liouville boundary value problems. (English) Zbl 1108.34020
The authors study the existence of minimal and maximal solutions of the discontinuous functional Sturm-Liouville boundary value problems
$-\frac{d}{dt}(\mu (t)u^{\prime }(t))=\lambda g(t,u,u(t),u^{\prime }(t))\text{ a.e. in } J=[t_{0},t_{1}],\tag{P}$
$a_{0}u(t_{0})-b_{0}u^{\prime }(t_{0})=c_{0},\quad a_{1}u(t_{1})+b_{1}u^{\prime }(t_{1})=c_{1},$
where $$\lambda ,$$ $$a_{j},$$ $$b_{j}$$ $$\in \mathbb{R}^{+},c_{j}\in \mathbb{R}$$ for $$j=0,1,\mu \in C(J,(0,\infty ))$$ and $$g:$$ $$J\times C(J)\times \mathbb{ R\times R\rightarrow }\mathbb{R}$$ is a given function. First, they give an existence result on $$(P)$$ where the second argument of $$g$$ is a fixed function in $$C(J).$$ Then, they study the dependence of the solution set of $$(P)$$ on the fixed function. By a fixed-point result for multifunctions, they give existence results for minimal and maximal solutions of $$(P).$$
MSC:
34B24 Sturm-Liouville theory 34B15 Nonlinear boundary value problems for ordinary differential equations
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https://www.codeproject.com/Articles/11719/Cyclomatic-Code-Complexity-Analysis-for-Microsoft?fid=218002&df=90&mpp=25&sort=Position&spc=Relaxed&select=1239494&tid=1239494 | 1,503,304,691,000,000,000 | text/html | crawl-data/CC-MAIN-2017-34/segments/1502886107744.5/warc/CC-MAIN-20170821080132-20170821100132-00644.warc.gz | 888,267,016 | 24,322 | 13,091,920 members (61,337 online)
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Posted 21 Sep 2005
# Cyclomatic Code Complexity Analysis for Microsoft .NET Applications
, 21 Sep 2005
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Code Complexity is a measure that provides a single ordinal number which can be compared to the complexity of other programs. It is one of the most widely accepted static software metrics and is intended to be independent of language and language format.
## Introduction
Cyclomatic Code Complexity was first introduced by Thomas McCabe in 1976. In 1976, Thomas McCabe published a paper arguing that code complexity is defined by its control flow. Since that time, others have identified different ways of measuring complexity (e.g. data complexity, module complexity, algorithmic complexity, call-to, call-by, etc.). Although these other methods are effective in the right context, it seems to be generally accepted that control flow is one of the most useful measurements of complexity, and high complexity scores have been shown to be a strong indicator of low reliability and frequent errors.
## Overview
This measure provides a single ordinal number that can be compared to the complexity of other programs. It is one of the most widely accepted static software metrics and is intended to be independent of language and language format.
Code Complexity is a measure of the number of linearly-independent paths through a program module and is calculated by counting the number of decision points found in the code (if, else, do, while, throw, catch, return, break etc.).
### Technical Specification
Cyclomatic Complexity for a software module calculated based on graph theory is based on the following equation:
`CC=E-N+p`
Where
• CC = Cyclomatic Complexity
• E = the number of edges of the graph
• N = the number of nodes of the graph
• p = the number of connected components
Further academic information on the specifics of this can be found here.
From a layman’s perspective the above equation can be pretty daunting to comprehend. Fortunately there is a simpler equation which is easier to understand and implement by following the guidelines shown below:
• Add 1 for each of the following keywords or their equivalent: `if`, `while`, `repeat`, `for`, `and`, `or`.
• Add 1 for each `case` in a `switch` statement.
Let’s look at a few examples to understand how the code complexity is calculated.
#### Example 1
```public void ProcessPages()
{
while(nextPage !=true)
{
if((lineCount<=linesPerPage) && (status != Status.Cancelled) && (morePages == true))
{
//....
}
}
}```
In the code above, we start with 1 for the routine, add 1 for the `while` loop, add 1 for the `if`, and add 1 for each `&&` for a total calculated complexity of 5.
#### Example 2
```public int getValue(int param1)
{
int value = 0;
if (param1 == 0)
{
value = 4;
}
else
{
value = 0;
}
return value;
}```
In the code above, we start with 1 for the routine, add 1 for the `if`, and add 1 for the `else` for a total calculated complexity of 3.
Members that have high code complexity should be reviewed for possible refactoring.
The SEI provides the following basic risk assessment based on the value of code:
Cyclomatic ComplexityRisk Evaluation
1 to 10a simple program, without very much risk
11 to 20a more complex program, moderate risk
21 to 50a complex, high risk program
> 50an un-testable program (very high risk)
## Tools
There are several free tools available which help one analyze the code complexity:
• devMetrics by Anticipating minds have a free community edition available for analyzing metrics for C# projects.
• Reflector Add-In: Code Metrics can be used to analyze .NET assemblies and show design quality metrics. This add-in is to be used in conjunction with Lutz Roeder’s Reflector.
• It is very easy to compute as illustrated in the example.
• Unlike other complex measurements, it can be computed immediately in the development cycle (which makes it agile friendly).
• It provides a good indicator of the ease of code maintenance.
• It can help focus testing efforts.
• It makes it easy to find complex code for formal review.
A list of licenses authors might use can be found here
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Visual Studio .NET 2005 ssaSpot28-Sep-05 8:14 ssaSpot 28-Sep-05 8:14
I believe the 2005 version of Visual Studio has a code analyser and the Cyclomatic Code Complexity is just one messures it provides if you pass a certain threshold.
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Article Copyright 2005 by Saikalyan Akunuri | 1,127 | 4,851 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2017-34 | latest | en | 0.917567 |
http://www.mapleprimes.com/questions/87782-ODE-Wpower-SeriesPLOTTING | 1,501,094,376,000,000,000 | text/html | crawl-data/CC-MAIN-2017-30/segments/1500549426372.41/warc/CC-MAIN-20170726182141-20170726202141-00272.warc.gz | 487,609,282 | 22,661 | # Question:ODE w/power series....PLOTTING
## Question:ODE w/power series....PLOTTING
Maple
I have two IVP ODE's that i can find the approximate solutions of but I cant figure out how to plot them correctly
for the first one i have tried this..
it needs to be plotted on the interval [1,4]
> with(plots);
> plot((-1)-1+x+(1/2)*(-1+x)^2+(-1+x)^3+(2/3)*(-1+x)^4+(37/60)*(-1+x)^5+(127/360)*(-1+x)^6+(31/140)*(-1+x)^7+(185/2016)*(-1+x)^8+(53/1620)*(-1+x)^9-(391/151200)*(-1+x)^10-(1097/90720)*(-1+x)^11-(160903/11975040)*(-1+x)^12-(969289/97297200)*(-1+x)^13-(4366883/681080400)*(-1+x)^14-(5242229/1513512000)*(-1+x)^15+O((-1+x)^16), x = 1 .. 4);
the error message i recieved...
'Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct.'
the second problem asks me to find the exact solution using dsolve and the first 20 terms of the ODE and the recurrence relation, all of which i have found i just need to know how to plot them on the interval [-1,1]
any and all help would be greatly appreciated. im an engineering student and im losing my mind cause its finals week and i need to study not do maple hah
| 384 | 1,212 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2017-30 | latest | en | 0.72318 |
https://physics.stackexchange.com/questions/309104/is-possible-to-control-the-direction-of-light-colliding-to-each-other?noredirect=1 | 1,624,571,920,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623488559139.95/warc/CC-MAIN-20210624202437-20210624232437-00530.warc.gz | 396,945,630 | 39,980 | # Is possible to control the direction of light colliding to each other?
I read this question, and I thought the following thing:
As photons can collide with each other, can we use photons to control the direction of another photon? As far as I know photons have a acceleration, so if we can throw a photon extremely close to the collision point, this photon will have a small acceleration and will just hit the another photon and change the direction of both..
for example:
1. photon 1 is thrown
2. when photon 1 is very close to the collision point photon 2 is thrown
3. Both photons collide to each other and goes to different direction (like pool balls)
• "like pool balls" is the issue. To deal with photons you need to deal with quantum theory and this doesn't work out like classical mechanics with pool balls. – StephenG Feb 1 '17 at 18:26
Photons do not collide with each other. Or at least only very very rare. In this rare process the direction of the scattered photon has an angular dependency. That means you cant really control the scattered photon to go into a specific direction. Can we use photons to control the direction of another photon? No, not with your two photon set up.
Of course you could build some set up where a light beam activates a 'machine' that bends another light beam into a specific direction..
Photon-photon collisions do happen. As Mr. Puh said, it's very uncommon. In technical terms, the scattering cross-section is very low. High-energy photons have a higher cross-section than low-energy photons, so they are much more likely to collide. The probability of a collision goes up with frequency to the 6th power, but unfortunately I can't find a quick comparison of how likely the collision is compared to particles like electrons or protons.
Trying to deflect photons via collision runs into Heisenberg's uncertainty principle. With pool balls, we can know where they are and how fast they're moving to an incredibly high degree of accuracy. We can predict the outgoing momentum very well. With light, we'll need to narrow down the width of our target area to that cross-section that I mentioned before. (I really wish I could find a number for that.) This means the light's momentum will be uncertain, and small changes in the momentum could result in a different direction for the other photon.
One could come up with a probability distribution that describes where the photons are most likely to go afterward, but it would be much less precise than if you did this with larger objects like pool balls or even two atomic nuclei.
One other small item: You say that photons have an acceleration. This is typically not true when they're passing through empty space. They only change speed when they're moving from one medium to another (like from air to water). I'm not sure why you're bringing in acceleration here; it doesn't seem to have much to do with the question.
I hope that helps!
Photon photon scattering belongs to the framework of quantum mechanics, as photons are elementary particles of the standard model.
This means that there is a probability distribution of scattering and the calculations depend on quantum electrodynamics , with the evaluation of Feynman diagrams.
This diagram for frequencies of light lower than twice the mass of an electron calculated will give the probability. The four electromagnetic vertices mean that the probability is small, order of (1/137)^4. So a photon passes a photon most probably undisturbed.
For energies where pair production can take place, an e+e- pair for example, there are only two electromagnetic vertices, and then the probability gets higher, and gamma gamma scattering has been proposed to form a collider. | 757 | 3,727 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2021-25 | latest | en | 0.953301 |
https://mathematica.stackexchange.com/questions/91700/defining-symbolic-expectation-function?noredirect=1 | 1,696,017,879,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510528.86/warc/CC-MAIN-20230929190403-20230929220403-00468.warc.gz | 423,255,574 | 44,697 | # Defining symbolic expectation function
I want to define the standard expectation $E$ operator in Mathematica. In particular, I want it to satisfy,
$E[ c + a \cdot X_1^{n_1} X_2^{n_2} X_3^{n_3} X_4^{n_4} + b \cdot Y_1^{m_1} Y_2^{m_2} Y_3^{m_3} Y_4^{m_4} ] = c + a \cdot E[ X_1^{n_1} X_2^{n_2} X_3^{n_3} X_4^{n_4}] + b \cdot E[ Y_1^{m_1} Y_2^{m_2} Y_3^{m_3} Y_4^{m_4} ]$
$c, a, b$ are deterministic constants, and where $n_i$ and $m_j$ are positive integers and, in particular, can take on the value $0$. I do not want to enforce a particular distribution function on these random variables. (It's even better if the code can incorporate arbitrary number of products and powers of the random variables).
In Mathematica, I will associate the random variables $X_i, Y_j$ as functions, say, of the form randX[i], randY[j], so any value that does not match the randX[i] and randY[j] form will be regarded as deterministic constants. And the resulting moments should look like, in Mathematica,
expect[c + a * randX[1]^n1 * randX[2]^n2 * randX[3]^n3 * randX[4]^n4 +
b * randY[1]^m1 * randY[2]^m2 * randY[3]^m3 * randY[4]^m4]
c + a * expect[randX[1]^n1 * randX[2]^n2 * randX[3]^n3 * randX[4]^n4] +
b * expect[randY[1]^m1 * randY[2]^m2 * randY[3]^m3 * randY[4]^m4]
The difficulty I'm having is that I find it hard to write patterns and rules that take on the zero-value powers of those random variables.
• Does the issue of $n_i = 0$ or $m_i = 0$ actually result in something that you would need to handle differently? Do those instances get changed to 1 such that by the time that the expect function would operate, randX[1]^5 * randX[2]^0 * randX[3]^4 would be changed to randX[1]^5 * randX[3]^4 ?
– JimB
Aug 16, 2015 at 0:20
• @JimBaldwin No, not in particular. Aug 16, 2015 at 0:45
• Not sure if your response is for my first question or the second question. Maybe if you showed what rules you've constructed so far (and even the ones that don't do what you want them to do), that would be helpful.
– JimB
Aug 16, 2015 at 0:57
• @JimBaldwin Thanks for the suggestions! I'm currently now working off of solution of Jens (see below) at the moment. In particular, I'm following the usual rules of algebra that $X_1^5 X_2^0 X_3^4 = X_1^5 X_3^4$. Aug 16, 2015 at 1:02
• Good. At point you'll need to convert things like expect[randX[1]^2] to the associated moments of the random variables that you're considering. This example (assuming that the moments exist) would result in $\sigma^2+\mu^2$ (if $\sigma^2$ is the variance of randX[1] and $\mu$ is the mean of randX[1]. Do you have a convention for naming the moments of X[i] and Y[i] and the expectations of their products?
– JimB
Aug 16, 2015 at 1:07
Here is one way of defining the linearity of your expectation value.
To allow for powers as they arise in the definition of the variance, I just added rules that expand such powers when possible (it is only possible when there is a sum under the power, that's why I restrict the pattern expect[Power[expr_Plus, n_]] to expressions expr with head Plus), and recognize any complete expectation value expect[...] as a constant that can be pulled out from under a surrounding expect. In the treatment of Power, the case n == 1 is also matched (because Power has a definition for a Default second argument set equal to 1).
Clear[expect]
expect[expr_Plus] := Map[expect, expr]
expect[Times[x_, y__]] /; (FreeQ[x, randX[_] | randY[_]]) :=
x expect[Times[y]]
expect[Times[x_expect, y__]] := x expect[Times[y]]
expect[expr_?(FreeQ[#, randX[_] | randY[_]] &)] := expr
expect[Power[expr_Plus, n_]] := expect[Expand[Power[expr, n]]]
expect[Power[x_expect, n_]] := x^n
expect[
c + a*randX[1]^n1*randX[2]^n2*randX[3]^n3*randX[4]^n4 +
b*randY[1]^m1*randY[2]^m2*randY[3]^m3*randY[4]^m4]
There is nothing special that needs to be done for powers of 0. Powers of 0 make the random variable disappear, so that case is covered. In deciding whether to pull out constants, I use a PatternTest of the form expr_?(...) check that the random variables don't appear in expr (unless wrapped by expect, which when raised to any power counts as a constant, too - that's the last definition).
With the rules for Power and Times involving expect itself, we also get the desired simplification of the variance:
expect[(randX[1] - expect[randX[1]])^2]
-expect[randX[1]]^2 + expect[randX[1]^2]
• Thanks for the answer. Is it possible to expand your solution? In particular, there are often cases when one needs to consider, say $Z := c + a X_1^{n_1} X_2^{n_2} X_3^{n_3} X_4^{n_4}$ and consider the computation $E[ (Z - E[Z])^2 ]$ (i.e. the variance of $Z$); and likewise, consider the other central moments of $Z$. In this case, your current code would generate something like expect[ expect[randZ] ] and does not pull out the deterministic expect[randZ] inside. Thanks (in particular, since I'm a novice to Mathematica)! Aug 16, 2015 at 1:00
• What @Jens gave you will do the job if you use the Expand function. z = c + a*randX[1]^n1*randX[2]^n2*randX[3]^n3*randX[4]^n4; Simplify[expect[Expand[z^2]] - Expand[expect[z]^2]] results in a^2 (-expect[randX[1]^n1 randX[2]^n2 randX[3]^n3 randX[4]^n4]^2 + expect[randX[1]^(2 n1) randX[2]^(2 n2) randX[3]^(2 n3) randX[4]^( 2 n4)]).
– JimB
Aug 16, 2015 at 1:24
• Thanks again! This does indeed work --- however, it seems to require the user to know the algebraic identity that $Var(Z) := E[ (Z - E[Z])^2 ] = E[Z^2] - ( E Z )^2$, which is fine for the variance. But I'm not sure how does one think about using Expand when one considers say, the 3rd centralized moment (for skewness) $E[ (Z - E[Z])^3 ]$, as expanding that would still involve a computation of a term like $E [ E[Z] Z ]$ and the code Jens currently has doesn't allow for the computation $E [ E[Z] Z ] = E[Z] E[Z] = ( E[Z] )^2$. Same comment applies for other higher centralized moments. Aug 16, 2015 at 1:32
• Might this work: Expand[expect[Expand[(z - ez)^3]] /. ez -> expect[z]]. And, of course, instead of 3 one could put in any positive integer.
– JimB
Aug 16, 2015 at 1:51
I recently ran into this same question and came up with a slightly more convenient solution based on Jens answer that I think is different enough to be worth sharing. First I define two functions, RandomVariableQ and MakeRandomVariable
(*atoms are not RVs unless explicitly marked*)
RandomVariableQ[expr_?AtomQ] := False
(*recurse down expression tree until we reach an atom or expectation value*)
RandomVariableQ[expr_ /; (!AtomQ[expr]) && (Head[expr] =!= AngleBracket)] := Or @@ Map[RandomVariableQ,List @@ expr]
(*expectation values are not RVs*)
RandomVariableQ[expr__AngleBracket] := False
(*make a symbol into a RV*)
MakeRandomVariable[x__Symbol] := (# /: RandomVariableQ[#] := True;) & /@ List[x]
MakeRandomVariable can be used to mark a particular symbol as being a random variable that cannot be factored out of an expectation. RandomVariableQ returns true if an expression contains a random variable that is not wrapped in an expectation value.
I then define AngleBracket as the expectation value
(*Linear*)
AngleBracket[expr_Plus] := AngleBracket /@ expr
AngleBracket[Times[x_, y__]] /; !RandomVariableQ[x] := x AngleBracket[Times[y]]
(*Expectation values only operate on RVs*)
AngleBracket[expr_ /; !RandomVariableQ[expr]] := expr
(*Expand before taking expectation values*)
AngleBracket[Power[expr_Plus, n_]] := AngleBracket[Expand[Power[expr, n]]]
AngleBracket[Times[expr_Plus, x_?RandomVariableQ]] := AngleBracket[Expand[expr x]]
Note here I am using RandomVariableQ instead of FreeQ. This allows the code to correctly recognize that arbitrary functions of expectation values can be factored out of an expectation value, i.e $\left\langle f(\left\langle x \right\rangle) \right \rangle = f(\left\langle x \right\rangle)$
We can test the code by computing the first 5 central moments of a random variable
MakeRandomVariable[x];
Table[AngleBracket[(x - AngleBracket[x])^n], {n, 2, 5}] // TableForm
which produces
$$\begin{array}{c} \left\langle x^2\right\rangle -\langle x\rangle ^2 \\ \left\langle x^3\right\rangle -3 \left\langle x^2\right\rangle \langle x\rangle +2 \langle x\rangle ^3 \\ \left\langle x^4\right\rangle -4 \left\langle x^3\right\rangle \langle x\rangle +6 \left\langle x^2\right\rangle \langle x\rangle ^2-3 \langle x\rangle ^4 \\ \left\langle x^5\right\rangle -5 \left\langle x^4\right\rangle \langle x\rangle +10 \left\langle x^3\right\rangle \langle x\rangle ^2-10 \left\langle x^2\right\rangle \langle x\rangle ^3+4 \langle x\rangle ^5 \\ \end{array}$$
as desired. This is an improvement on the current solution which cannot simplify expressions such as expect[expect[randX[1]]^2 randX[1]] because it does not recognize that expect[randX[1]]^2 can be pulled out of the expectation value. This also works as expected with multiple random variables. | 2,713 | 8,875 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.21875 | 3 | CC-MAIN-2023-40 | latest | en | 0.841493 |
http://math.stackexchange.com/questions/296793/if-lim-x-to-inftyfx-a-lim-x-to-inftyfx-b-then-lim-x-to-in/296809 | 1,464,739,680,000,000,000 | text/html | crawl-data/CC-MAIN-2016-22/segments/1464053252010.41/warc/CC-MAIN-20160524012732-00214-ip-10-185-217-139.ec2.internal.warc.gz | 181,993,801 | 18,007 | # If $\lim_{x\to+\infty}f(x)=a$, $\lim_{x\to+\infty}f'(x)=b$, then $\lim_{x\to+\infty}f'(x)=0$
I'm trying to solve this question:
Let $f:(c,+\infty)\to \mathbb R$ be differentiable. If the limits $\lim_{x\to+\infty}f(x)=a$ and $\lim_{x\to+\infty}f'(x)=b$ exist with $a\in \mathbb R$, then $b=0$.
There is a hint, it says $f(n+1)-f(n)=f'(x_n)$, where $x_n\to \infty$
I didn't understand this hint.
I need help, any clue is welcome.
Thanks a lot
-
The question title seems to make a stronger claim than the one in the question body (and the stronger claim is wrong.) – Trevor Wilson Feb 7 '13 at 1:46
Counterexample showing that the condition $\lim_{x\to\infty}f'(x) = b$ is necessary: $f(x) = \frac{1}{x}\sin(x^2)$. – Ayman Hourieh Feb 7 '13 at 9:29
Apply the mean value theorem to $[n, n+1]$ to find $x_n \in (n, n+1)$ so that: $$f'(x_n) = \frac{f(n+1) - f(n)}{(n+1) - n} = f(n+1) - f(n)$$
Now evaluate: $$\lim_{n\to\infty} f'(x_n) = \lim_{n\to\infty}\left(f(n+1) - f(n)\right)$$
Since $\lim_{x\to\infty} f'(x)$ exists, what do you conclude?
-
then we have $\lim_{n\to\infty}f'(x_n)=\lim_{n\to\infty}\left(f(n+1) - f(n)\right)= a-a=0$? – user42912 Feb 7 '13 at 1:58
@user42912 Yes! – Ayman Hourieh Feb 7 '13 at 2:14
thank you very much for your answer! – user42912 Feb 7 '13 at 3:13
Nice answer (+1). – Mhenni Benghorbal Feb 7 '13 at 7:05
Just to understand this result intuitively, suppose $b>0$. Then $b>b/2>0$ and so from some point onwards on the $X$ axis the derivative satisfies $f'(x)>b/2$ and so from that point onwards the function $f$ is increasing at a rate of no less then $b/2>0$. Clearly such a function will thus tend to $\infty$ and not to a finite $a$ as given. The case $b<0$ is treated similarly.
The hint suggested a very clear solution as the other answers show. For diversity, and since I like formal solutions that follow an intuitively clear informal argument, here is a solution along the lines in the preceding paragraph.
Assume to the contrary that $b\ne 0$. It will suffice to consider the case $b>0$ as the case $b<0$ will follow by considering the function $g(x)=-f(x)$. Since $\lim _{x\to \infty }f'(x)=b$ there exists $x_0$ such that for all $x>x_0$ holds that $f'(x)>b/2$ (take $\epsilon = b/2$ in the definition of limit). Consider any $y>x_0$ and apply the Mean Value Theorem on $[x_0,y]$ to obtain some $x_0<t<y$ such that $f'(t)=(f(y)-f(x_0))/(y-x_0))$. Rearrange to get $f(y)=f'(t)(y-x_0)+f(x_0)$. Recalling that $f'(t)>b/2$ we obtain $f(y)>y-x_0+f(x_0)$. As $x_0$ is fixed it follows that $\lim _{y\to \infty }f(y)=\infty$, a contradiction.
- | 954 | 2,596 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.3125 | 4 | CC-MAIN-2016-22 | latest | en | 0.799133 |
https://cs.stackexchange.com/questions/9447/algorithm-for-building-a-suffix-array-in-time-on-log2-n/9466 | 1,631,867,075,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780055601.25/warc/CC-MAIN-20210917055515-20210917085515-00114.warc.gz | 250,790,589 | 37,622 | # Algorithm for building a suffix array in time $O(n \log^2 n)$
I've been working with suffix arrays lately, and I can't find an efficient algorithm for building a suffix array which is easy to understand. I have seen in many sites that there is an $O(n \log^2 n)$ algorithm, but I can't understand it, as many important details are omitted. There's an example at Top Coder.
Could someone introduce me an efficient algorithm for suffix array construction, which is easy to comprehend?
You can compute the suffix array in linear time with the DC-3 Algorithm. This is a super-cool fancy algorithm that can be implemented in 50 lines of readable C++ code - one of my all-time favorites. The source code is contained in the original paper. If you can compare two characters in constant time and the alphabet size is $n^{O(1)}$, then the DC3 algorithm runs in $O(n)$ time.
Here's a good explanation for the $O(n\log^2n)$ algorithm: http://www.stanford.edu/class/cs97si/suffix-array.pdf. Actually, by using linear time sorting, the same approach gives $O(n\log n)$ time complexity.
I agree with A. Schultz that DC-3 is super-cool. It is also not very complicated, but the $O(n\log^2 n)$ is still simpler. | 292 | 1,201 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.703125 | 3 | CC-MAIN-2021-39 | latest | en | 0.916969 |
http://openstudy.com/updates/50c4b148e4b0c673d53ddbcd | 1,448,838,654,000,000,000 | text/html | crawl-data/CC-MAIN-2015-48/segments/1448398460263.61/warc/CC-MAIN-20151124205420-00113-ip-10-71-132-137.ec2.internal.warc.gz | 178,795,143 | 11,162 | ## ggrree 2 years ago Thermodynamics question: http://gyazo.com/ae831c4c7ae3325cfce6f280cfa37b88 show that this is true.
• This Question is Open
1. soty2013
easy.. use partial diffrentiation method...
2. Carl_Pham
You need the "Anti-recipe" for working with second derivatives of the fundamental relation, which you'll find in H. B. Callen "Thermodynamics", somewhere around Chapter 2 or 3. My copy of Callen is at work, so if you haven't figured it out by then, I'll see if I can dig up what you need when I get there.
3. Carl_Pham
OK, it doesn't look like Callen's recipe is useful. But we can do it another way. First, A is a natural function of T,V and N. We'll ignore N here, because it's just held constant. So the total differential of A is, by definition:$dA = \left(\frac{\partial A}{\partial T}\right)_V dT + \left(\frac{\partial A}{\partial V}\right)_T dV$Divide through by dT and hold p constant:$\left(\frac{\partial A}{\partial T}\right)_p = \left(\frac{\partial A}{\partial T}\right)_V + \left(\frac{\partial A}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_p$Now there is a general theorem for partial variations of a function X(Y,Z) of several variables, which produces this general result:$\left(\frac{\partial X}{\partial Y}\right)_Z = - \left(\frac{\partial X}{\partial Z}\right)_Y \left(\frac{\partial Z}{\partial Y}\right)_X$It's derived in Callen Appendix A, and probably in some diff eq or calculus books. Using this on the second term above gives you your first equation. Maybe there's a way to get there in one step, but I'm not seeing it.
4. Carl_Pham
Darn, I skipped a step. You have to use the chain rule first on the second term of the second equation:$\left(\frac{\partial A}{\partial T}\right)_p = \left(\frac{\partial A}{\partial T}\right)_V + \left(\frac{\partial A}{\partial p}\right)_T \left(\frac{\partial p}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_p$Now you can use that XYZ rule on the last term to get what you want.
5. Carl_Pham
Now to get to the second equation, remember that the differential of A turns out to be:$dA = -S dT - p dV$You get this by taking the Legendre transform of the canonical differential of internal energy U (dU = TdS - pdV) which you remember from intro thermo, since A is the Legendre transform of U. You can immediately identify:$\left(\frac{\partial A}{\partial T}\right)_V = - S$ and$\left(\frac{\partial A}{\partial V}\right)_T = -p$Using these in the second equation in the first post above:$\left(\frac{\partial A}{\partial T}\right)_p = -S - p \left(\frac{\partial V}{\partial T}\right)_p$Now we use that XYZ theorem the other way, to bring the p inside the derivatives:$\left(\frac{\partial A}{\partial T}\right)_p = -S + p \left(\frac{\partial V}{\partial p}\right)_V \left(\frac{\partial p}{\partial T}\right)_V$Rewrite the product as a fraction, inverting the derivative on the bottom, and you've got your second equation. This is obviously not the way the author did it, since he meant the second equation to follow from the first, but I don't have a quick isight into what he did. | 884 | 3,112 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.59375 | 4 | CC-MAIN-2015-48 | longest | en | 0.864153 |
https://forum.freecadweb.org/search.php?st=0&sk=t&sd=d&sr=posts&author_id=19043 | 1,597,258,411,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439738913.60/warc/CC-MAIN-20200812171125-20200812201125-00174.warc.gz | 304,617,296 | 7,693 | ## Search found 80 matches
Wed Aug 29, 2018 10:27 pm
Forum: Open discussion
Topic: [It already exists, sorry for the unnecessary noise!]Feature request: named measurements on sketch (non-constraining)
Replies: 7
Views: 364
### Re: [It already exists, sorry for the unnecessary noise!]Feature request: named measurements on sketch (non-constraining
saved me from lots of frustration No worries, that details did frustrate me before i found the work around, that's probably a missing feature, maybe I should submit a request. Oh and you seem to imply it in your first mail, but for sake of completion these blue datum measures cannot be use in the s...
Wed Aug 29, 2018 8:29 pm
Forum: Forum français
Topic: modelisation avion rc
Replies: 30
Views: 2256
### Re: modelisation avion rc
GlouGlou wrote:
Wed Aug 29, 2018 5:45 pm
tu dis ça parce que tu es Quebecois?
Ah nan mais il se trouve que je suis aux US depuis qq temps et ces systemes non metriques me sortent par les yeux...
J'aurai du mettre les "sarcastic quotes" comme ils disent ici... "super logique"
hehe
Wed Aug 29, 2018 5:42 pm
Forum: Forum français
Topic: modelisation avion rc
Replies: 30
Views: 2256
### Re: modelisation avion rc
C'est quoi l'unité derrière les pouces, au fait? Les cotes sont en pieds ' - pouces " et fraction de pouces : 1' - 7" 3/4 = 1 feet + 7/12 feet + 3/4*1/12 feet ~ 47.6cm Simple je te dis... Le pire c'est qd tu essais de comprendre les dimensions reelle des planches de bois dans les bricoramas du coin...
Wed Aug 29, 2018 5:30 pm
Forum: Open discussion
Topic: [It already exists, sorry for the unnecessary noise!]Feature request: named measurements on sketch (non-constraining)
Replies: 7
Views: 364
### Re: [It already exists, sorry for the unnecessary noise!]Feature request: named measurements on sketch (non-constraining
Thanks! This is EXACTLY what I wanted. Sorry, I should have found it myself. Now it seems so obvious. Note that if you create a blue/non constraining measure, to name it (for example to use it as a parameter in another sketch/object) you can't double click on it to just change the name: it will swi...
Sun Aug 12, 2018 1:26 am
Forum: Users Showcase
Topic: backpack design challenge
Replies: 21
Views: 4505
### Re: backpack design challenge
One of the coolest design I saw recently is this armadillo shape shell backpack: https://mypangolin.com/collections/renegade Maybe easier to design than some completely soft fabric surface? Each of the segments are flat and stiff and bent in 1 direction, there is an inside liner to fill the gap betw...
Tue Aug 07, 2018 3:47 pm
Forum: Users Showcase
Topic: Voxel Workbench
Replies: 35
Views: 10681
### Re: Voxel Workbench
I have created a workbench to draw in 3D using cubes (those annoying voxels :lol: ): What do you think? Javier. Javier, a bit late to the subject, but if you ever search for ideas of new features for your bench I would recommend you have a look at: https://www.3dslash.net/index.php Their tool set i...
Fri Aug 03, 2018 12:46 pm
Topic: Help with 3D Printing
Replies: 19
Views: 605
### Re: Help with 3D Printing
This middle thingy is higher than the rest so one layer doesn't take much time. There should be some time gap between layers. Ding ding, this is the money comment (as long as the model has no errors and the slicer is properly configured). You will need a very effective ventilation or the clamp side...
Thu Jul 19, 2018 3:32 pm
Topic: Datum entity vs Body
Replies: 33
Views: 1364
### Re: Datum entity vs Body
saso wrote:
Thu Jul 19, 2018 6:30 am
what I have been arguing for in the past months
Thanks Saso, that was a lot more argumented than my half baked example, I guess I missed this thread, good read
Cheers
Wed Jul 18, 2018 10:29 pm
Forum: Assembly
Topic: Another approach to assembly solver (A2plus)
Replies: 1753
Views: 92358
### Re: Another approach to assembly solver (A2plus)
kbwbe, Thanks for your work on updating the assembly bench! I've skimmed though the thread, so sorry if I missed it. Is it possible to simulate gear/chains angle constraints with your assembly solver? I do not expect it's realistic to do a full interference analysis to move the gears depending on th...
Wed Jul 18, 2018 9:58 pm | 1,174 | 4,225 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2020-34 | latest | en | 0.653546 |
https://docs.matrixorigin.cn/1.2.0/MatrixOne/Reference/Operators/operators/logical-operators/not/ | 1,718,838,973,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861853.72/warc/CC-MAIN-20240619220908-20240620010908-00386.warc.gz | 190,121,094 | 20,322 | # NOT,!
## 运算符说明
`NOT,!` 逻辑运算符用作于逻辑非运算。如果操作数为零，则返回结果为 `true`；如果操作数非零，则返回结果为 `false`；如果操作数为 `NOT NUL` 则返回 `NULL`
## 语法结构
``````> NOT|! value
``````
## 示例
``````mysql> select not 0;
+-------+
| not 0 |
+-------+
| true |
+-------+
1 row in set (0.02 sec)
mysql> select ! null;
+-------+
| !null |
+-------+
| NULL |
+-------+
1 row in set (0.00 sec)
mysql> select ! 1;
+-------+
| !1 |
+-------+
| false |
+-------+
1 row in set (0.01 sec)
``````
``````create table t1 (a boolean,b bool);
insert into t1 values (0,1),(true,false),(true,1),(0,false),(NULL,NULL);
mysql> SELECT * FROM T1;
+-------+-------+
| a | b |
+-------+-------+
| false | true |
| true | false |
| true | true |
| false | false |
| NULL | NULL |
+-------+-------+
5 rows in set (0.01 sec)
mysql> select not a and not b from t1;
+-----------------+
| not a and not b |
+-----------------+
| false |
| false |
| false |
| true |
| NULL |
+-----------------+
5 rows in set (0.00 sec)
mysql> select * from t1 where !(a=false);
+------+-------+
| a | b |
+------+-------+
| true | false |
| true | true |
+------+-------+
2 rows in set (0.00 sec)
`````` | 425 | 1,206 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2024-26 | latest | en | 0.393914 |
http://blog.csdn.net/nymph_h/article/details/77141849 | 1,503,080,646,000,000,000 | text/html | crawl-data/CC-MAIN-2017-34/segments/1502886105086.81/warc/CC-MAIN-20170818175604-20170818195604-00028.warc.gz | 52,444,379 | 14,860 | # 【AC自动机+矩阵快速幂】poj2778 DNA Sequence
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1. 若ch[I][j] ==0 ,某个节点没有某一字符的儿子，考虑将其指向其失配节点的儿子：ch[I][j] = ch[f[I]][j];Tire数就变成Tire图了。
2.若val[f[I]]==1,某节点失配节点是危险节点，则将此节点标为危险节点。
#define _CRT_SECURE_NO_WARNINGS
#include<iostream>
#include<cstdio>
#include<cstring>
#include<queue>
using namespace std;
typedef long long ll;
const int maxn = 101;
const int mod = 100000;
int n, l;
struct Matrix{
ll mat[101][101];
int r, c;
Matrix(int x){ memset(mat, 0, sizeof(mat)); r = x; c = x; }
Matrix operator*(const Matrix &b) const{
Matrix res(b.r);
for (int i = 0; i <= b.r; i++){
for (int j = 0; j <= b.r; j++){
for (int k = 0; k <= b.r; k++){
res.mat[i][j] = (res.mat[i][j] + mat[i][k] * b.mat[k][j]) % mod;
}
}
}
return res;
}
};
Matrix Qpow(Matrix x, ll n){
Matrix res(x.c), tmp = x;
for (int i = 0; i <= x.c; i++) res.mat[i][i] = 1;
while (n){
if (n & 1) res = res *tmp;
n >>= 1;
tmp = tmp*tmp;
}
return res;
}
struct Trie{
int ch[maxn][26];
int val[maxn];
int last[maxn];
int f[maxn];
int sz;
int idx(char x){
if (x == 'A') return 0;
if (x == 'T') return 1;
if (x == 'C') return 2;
if (x == 'G') return 3;
}
void insert(char *s){
int u = 0, n = strlen(s);
for (int i = 0; i < n; i++){
int c = idx(s[i]);
if (!ch[u][c]){
ch[u][c] = sz++;
}
u = ch[u][c];
}
val[u] = 1;
}
void init(){
memset(ch, 0, sizeof(ch));
memset(val, 0, sizeof(val));
sz = 1;
for (int i = 0; i < n; i++){
char str[55]; scanf("%s", str);
insert(str);
}
}
void build(){
queue<int> q;
memset(f, 0, sizeof(f));
memset(last, 0, sizeof(last));
for (int i = 0; i < 26; i++){
int u = ch[0][i];
if (u){ q.push(u); }
}
while (!q.empty()){
int r = q.front(); q.pop();
for (int i = 0; i < 26; i++){
int u = ch[r][i];
if (!u) {
ch[r][i] = ch[f[r]][i];
continue;
};
q.push(u);
int v = f[r];
while (v && !ch[v][i]) v = f[v];
f[u] = ch[v][i];
last[u] = val[f[u]] ? f[u] : last[f[u]];
if (val[f[u]] == 1) val[u] = 1;
}
}
}
Matrix getM(){
Matrix mp(sz);
for (int i = 0; i < sz; i++){
for (int j = 0; j < 4; j++){
if (!val[i] && !val[ch[i][j]])
mp.mat[i][ch[i][j]]++;
}
}
return mp;
}
};
Trie ac;
int main(){
while (~scanf("%d%d", &n, &l)){
ac.init();
ac.build();
Matrix m = ac.getM();
m = Qpow(m, l);
ll ans = 0;
for (int i = 0; i < ac.sz; i++){
ans = (ans + m.mat[0][i]) % mod;
}
cout << ans << endl;
}
}
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文章分类
文章存档 | 1,015 | 2,390 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2017-34 | longest | en | 0.140741 |
https://www.infocomm.ky/what-is-a-decision-tree-glossary-of-business-terms/ | 1,718,779,229,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861806.64/warc/CC-MAIN-20240619060341-20240619090341-00358.warc.gz | 717,617,752 | 12,314 | # What is a Decision Tree?
A decision tree, as its name indicates, is a scheme in which the possible consequences derived from the achievement of actions are configured according to an order. This resource helps to decide, to make decisions. Not only is this an inexpensive concept, but it can be really useful for day-to-day tasks. It will help us to know which activities to prioritize and the weight that each one of them receives.
## Selection criteria for a decision tree
In order to select one or the other, we must base ourselves on a series of criteria and make certain decisions as they arise. Obviously the statistics and probability come into play when it comes to making decisions.
The selection criteria when choosing one decision or another in the tree will have to be chosen by us, since the decisions and actions that we take into account depend on us. The probability will do its thing so that the actions that are taken are one or the other.
In this way, we can avoid falling into doing certain actions that do not suit us because they harm us, as long as we are clear about the structure of our decision tree.
## The structure of a decision tree
It is about the way in which the decisions we want to carry out are materialized. It is about giving an order to the decisions that we are going to make, as well as some probabilities that we think they would have. We can take the following decision tree as an example:
Source: support.office.com.
Finally, comment that the probabilities of a phenomenon occurring or not within the tree depends on how well or badly we know each of the cases. If we know that an action or decision has a greater danger, it will be assigned more or less probability.
Categories D | 349 | 1,736 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2024-26 | latest | en | 0.974378 |
http://www.reddit.com/r/askscience/comments/16rixu/will_the_universe_ever_reach_a_point_at_which/ | 1,369,143,797,000,000,000 | text/html | crawl-data/CC-MAIN-2013-20/segments/1368700074077/warc/CC-MAIN-20130516102754-00063-ip-10-60-113-184.ec2.internal.warc.gz | 672,670,282 | 14,940 | [–] 2 points3 points ago
Follow-up question: if entropy is no longer increasing, would time essentially cease as well? If there was no longer any change in the system, whether towards higher or lower entropy, then there would be no way to measure the passage of time.
[–] 1 point2 points ago
If no enthropy can be expended to watch entrophy not changing then no one can watch it not changing.
[–] 1 point2 points ago
Then assume the role of an outside observer. One who can see into this Universe without actually entering it.
[–] -1 points0 points ago
I'd say, for the universe to be a truly closed system, outside observing would be impossible.
[–] 1 point2 points ago
this answer always bugged me, because if you ever were talking about entropy in a subsystem, then it doesn't apply at all. So lets say we have an isolated box with maximum entropy inside it. Time clearly does not stop for it.
[–] 0 points1 point ago
But does anything meaningfully occur inside the box? The box might exist in time, since it is in an area that, external to it, has non-maximum entropy. But inside there is maximum entropy. Ergo, nothing in there is changing, it has changed as much as it can and the capability for further change is exhausted without an influx of more energy.
This might not mesh well with how we conceive of time, but what effectively marks the passage of time in the box?
[–] 0 points1 point ago
When a material is at thermal equilibrium it has reached a local maximum in entropy. When two gasses mix, and finally become "completely mixed" it is not like all meaning disappears. The gas is exerting a constant pressure on the box, it is at some constant temperature, all the particles are whizzing around with some kinetic energy defined by that temperature (or vice versa).
Just because a system has reached a local maximum of entropy, does not mean all interest is lost.
To reach a total possible maximum --> Quantum fun begins. That goes beyond my field of understanding. Possibly beyond our current collective knowledge.
[–] 4 points5 points ago
[–]Nuclear Engineering | Powerplant Technology 2 points3 points ago
It's not necessary that total entropy in the universe increases, only that it doesn't decrease. Notice I say total, because it is possible for entropy to decrease locally. But as far as the universe(the system that is defined as the system that contains all other systems) is concerned, entropy cannot decrease. It is possible that entropy in the universe may increase to a maximum, where the universe would reach thermal equilibrium and have no energy available for work. In this case, the change in total entropy of the universe would go to zero. This is known as the Heat Death of the Universe.
[–][deleted] ago
[deleted]
[–] 1 point2 points ago
Not supposed to do that here according to forum rules, but great read | 633 | 2,865 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2013-20 | latest | en | 0.962572 |
https://estudyassistant.com/mathematics/question12060978 | 1,627,494,687,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046153739.28/warc/CC-MAIN-20210728154442-20210728184442-00210.warc.gz | 257,335,584 | 17,468 | , 21.06.2019 16:00 Destinationz
# Svetlana's hair is 4 4 centimeters ( cm cm) long. her hair grows 1 . 5 cm 1.5cm per month. svetlana wants her hair to be less than 1 6 cm 16cm long. write an inequality to determine the number of months, m m, svetlana can allow her hair to grow so that it is less than 1 6 cm 16cm long.
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Svetlana's hair is 4 4 centimeters ( cm cm) long. her hair grows 1 . 5 cm 1.5cm per month. svetlana...
Questions
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Questions on the website: 13559995 | 416 | 1,156 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.453125 | 3 | CC-MAIN-2021-31 | latest | en | 0.870343 |
scifitheory.blogspot.com | 1,526,962,794,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794864624.7/warc/CC-MAIN-20180522034402-20180522054402-00188.warc.gz | 260,501,782 | 16,709 | ## Saturday, January 5, 2013
### LIGHT
Isn't it amazing that light travels with a speed
which is equal to speed of light.
let me try to explain the difference between speed of light and the speed with which light travels.
While doing special relativity there appears a fundamental constant
say x whose value is equal to 3 * 10^8 and this constant has a dimension of LT^-1 ( m/ s )
Next we will use the famous Einstein's eqn.. m= M/ sqrt(1-v2/c2)
Now from this eqn since the rest mass of photons is 0, so it must travel with the speed equals to c, Now these photons are nothing but light, so we now know that light has to travel with the velocity . Of x which in turn is equal to 3 * 10^8 m/s, , Now u know that in some medium light can travel with speed much less than 3* 10^8 m/s
so let me sum up things - there is a fundamental constant whose value is equal to 3*10^8 m/s, Since light is nothing but mass less photon so light travels with speed equal to 3*10^8 m/s. | 261 | 972 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.359375 | 3 | CC-MAIN-2018-22 | longest | en | 0.935832 |
https://rdrr.io/github/CharlotteJana/momcalc/man/symbolicMoments.html | 1,618,769,116,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618038507477.62/warc/CC-MAIN-20210418163541-20210418193541-00516.warc.gz | 589,440,712 | 8,864 | # symbolicMoments: Symbolic calculation of moments In CharlotteJana/momcalc: Symbolic Moment Calculation
## Description
Compute the moments of a multivariate variable X = X[1], ..., X[n] and return the formula as quoted expression.
## Usage
```1 2``` ```symbolicMoments(distribution, missingOrders, mean = NA, cov = NA, var = NA, simplify = TRUE) ```
## Arguments
`distribution` string specifying the (multivariate) distribution of X. The following values are possible: `"zero"` sets all moments to 0, `"normal"` calculates the moments of a centralized multivariate normal distribution, `"lognormal"` calculates the raw moments of a multivariate lognormal distribution, `"gamma"` calculates the raw moments of a multivariate gamma distribution, `"NA"` sets all moments to NA. `missingOrders` numeric vector or matrix. Each row gives the order of a moment that shall be calculated. `mean` vector or list with expected values. Entry `mean[[i]]` should contain a value for E(X[i]). `cov` matrix or nested list. Entry `cov[i][j]` (or `cov[[i]][[j]]` respectivly) should contain a value for the covariance Cov(X[i], X[j]). `var` optional. If `n = 1`, `cov` would only contain one entry - the variance Var(X). This value can be passed to parameter `var` instead of `cov`. `simplify` bool indiciating if the resulting expressions should be simplified. Function `Simplify` from package Deriv is used for simplification.
## Value
A list where each element is a quoted expression. The i-th element of this list gives a formula for the moment whose order is given in the i-th row of `missingOrders`. If `simplify = TRUE`, the returned value may as well be a vector or number.
## Note
The calculation of the central moments of a multivariate normal distribution is based on function `callmultmoments` of package symmoments. If the calculation for a multivariate gamma distribution leads to NaNs, then the values of cov and mean do not fit to a gamma distribution. All entries of cov should be positive and the diagonals should be large with respect to the other entries. More specifically, the inequations
mean[i] > sum_(k != i) mean[k]*cov[i,k]/cov[i,i]
should be satisfied for all i in 1:n.
## Examples
``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26``` ```# raw moments of a one dimensional gamma distribution symbolicMoments(distribution = "gamma", missingOrders = as.matrix(1:3, ncol = 1), mean = "μ", var = "σ") # raw moments of a one dimensional lognormal distribution symbolicMoments(distribution = "lognormal", missingOrders = as.matrix(1:2, ncol = 1), mean = 2, var = 1, simplify = FALSE) # evaluate the result symbolicMoments(distribution = "lognormal", missingOrders = as.matrix(1:2, ncol = 1), mean = 2, var = 1, simplify = TRUE) #### central moments of a four dimensional normal distribution #### missingOrders <- matrix(c(4, 0, 0, 0, 3, 1, 0, 0, 2, 2, 0, 0, 2, 1, 1, 0, 1, 1, 1, 1), ncol = 4, byrow = TRUE) cov <- matrix(c("σ11", "σ12", "σ13", "σ14", "σ12", "σ22", "σ23", "σ24", "σ13", "σ23", "σ33", "σ34", "σ14", "σ24", "σ34", "σ44"), ncol = 4, byrow = TRUE) symbolicMoments("normal", missingOrders, mean = "μ", cov = cov) ```
CharlotteJana/momcalc documentation built on Oct. 17, 2019, 7:21 a.m. | 927 | 3,243 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2021-17 | latest | en | 0.732963 |
https://hundred-worries.com/en/advices/2709 | 1,624,279,914,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623488273983.63/warc/CC-MAIN-20210621120456-20210621150456-00028.warc.gz | 286,588,024 | 11,814 | # Chimney for the boiler room: how to do the calculation of sizes
The main function that the chimney should perform for the boiler room is to exhaust the flue gases from the boilers into the atmosphere and dissipate them in this space. She also has an additional function: they should create natural cravings resulting from the difference between the temperature in the firebox and the outside.
We will introduce you to the varieties of smoke channels, which are classified on the basis of design features and pipe material. Here you will learn how to calculate the geometric parameters for a specific example. Our tips will help determine the type and size of the chimney.
The content of the article:
• Types of chimneys
• Self-supporting boiler tubes
• Column smoke structures
• Truss pipes
• Mast chimney pipes
• Materials for the construction of pipes boiler
• Calculation of pipe parameters
• Determination of the height of the pipe during natural thrust
• Pipe diameter calculation
• Conclusions and useful video on the topic
## Types of chimneys
In large boilers, natural draft cannot ensure proper combustion; here it is forcedly created using smoke pumps. The combustion process and the discharge of its products into the atmosphere should bring as little harm to the surrounding the environment and not cause emergency situations as a result of pressure in the furnaces exceeding the norm.
Constructional pipes for boiler rooms very different from each other and in the form of the supporting structure, and the material of manufacture. On the first sign, there are several types of pipes.
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A photo of
Chimney is a significant constructive component of a boiler house of an enterprise and a private house. It depends on the efficiency of the unit
The height and dimensions of the smoke channel must be calculated with extreme precision, since the size of the channel depends on the combustion process
For the formation of impeccable thrust required to take into account the wind rose in the region, barometric pressure and the average data on air temperature
The design of the chimney should provide full flue gas without causing harm to the owners and users of closely located buildings and the environment
Means and devices for maintenance of the chimney should be thought out, envisaged and designed.
When designing a chimney, it is mandatory to take into account the type of the boiler and the type of fuel. In the case of using a combined unit, the pipe is selected, focusing on the safe removal of gases of maximum temperature
The flow capacity of the chimney is determined taking into account the volume of combustion products at maximum load of the unit
The boiler room pipe is a rather high structure exposed to natural phenomena. The design is designed to ensure stability in strong winds and the ability to extinguish lightning strikes.
An important component of boiler room equipment
The need for accurate calculations of pipe dimensions
Significance of design before installation
Chimney construction requirements
Providing serviceability
Matching chimney to boiler type
Bandwidth Rules
Stability and lightning protection
### Self-supporting boiler tubes
Such vertical constructions are single- or multi-barreled. They divert combustion products from boilers and boilers.
They are used regardless of the type of fuel, but subject to certain requirements:
1. The temperature of the flue gases passing through the self-supporting pipes should not exceed 350 degrees C.
2. Combustion products must not be chemically aggressive.
3. The optimal snow load for self-supporting structures is 250 kg per kW. cm, wind - 30 kg per kV. cm in the conditions of the II wind region.
Install a self-supporting pipe on the roof, and fix inside the building. Its design features provide the possibility of transportation and installation on site, because it consists of separate sections representing 3-ply sandwich pipe. The foundation is attached to the foundation using anchors.
Inside the pipe there is a layer made of strong steel that is not amenable to the effects of substances released during combustion. The outer layer protects against weathering.
Chimneys for large boiler houses are often self-supporting. This building, built on an individual project and having its own infrastructure
The parameters of smoke structures must comply with the requirements set forth in regulatory documents. Their calculation is based on such factors as the number of boilers, power, type of fuel. Be sure to take into account standards for emissions into the atmosphere. In some cases, chimneys are equipped with a platform, a ladder, an inspection hatch, light fencing.
### Column smoke structures
The pipe of this type consists of an outer shell of high carbon steel and inserted into it internal trunks of different diameters of stainless steel for the removal of gases. The design is fixed in an anchor basket embedded in the foundation. They can be either 1 or several. That inside did not settle a condensate, use heat insulation.
Part of the column tube. Here in the section you can see that several trunks of various diameters of stainless steel are placed inside
The advantage of this design solution is a long operational period, the prospect of connecting several boilers. The thickness of the steel and the brand is chosen based on the temperature and aggressiveness of the combustion products.
The diameter of each barrel can reach one and a half meters, and if you plan to use a common gas duct for several boilers, then a diameter of about 3 m is needed. To avoid condensation, barrels are covered thermal insulation.
Install near-front chimneys for boiler rooms attached to the house or built-in ones. Attach them to the wall of the building with the use of brackets. Components of the chimney are trunks and frame or anchor fasteners.
The barrel has 3 layers: inside is stainless steel, then heat insulation and galvanized steel. Pipes are intended for boiler rooms where boilers work on gas or liquid fuel.
The most common facade pipes are located along the outer wall of the building. When choosing a steel grade and pipe wall thickness, take into account the chemical composition of the exhaust gases and their temperature
Near-facade and facade pipes transfer the weight load through an additional lower foundation and wind one through vibration-proof fastenings. This type of chimneys, in terms of material costs, is the most economical due to the lack of supporting structures and a solid foundation.
The modular system used to create the exhaust barrels allows for easy replacement of damaged parts.
### Truss pipes
Such a metal structure consists of pipes mounted on a durable self-supporting truss-type column. The farm, in turn, is fixed in an anchor basket, poured into the foundation. Farm type flues are suitable for use in regions with dangerous seismological conditions.
Farm type construction includes from 1 to 6 stems. The column is made of rolled steel. The profile pipe can have a square or a triangle in cross section. It depends on the number of trunks.
To prevent corrosion, gas vents are coated with a primer, then painted.
The barrel for removal of gases is composed of modules consisting of 3 layers:
• internal, in contact with the combustion products directly and made of special-grade stainless steel;
• 5-6 cm thick, playing the role of thermal insulation;
• external, protecting the insulating layer from the negative effects of the environment.
For anti-corrosion coatings use paints containing a large percentage of zinc. In some structures inside the column there may be stairs and platforms that facilitate maintenance. Structural elements of pipes of this type are relatively lightweight and this facilitates both their transportation and installation work.
### Mast chimney pipes
The central element of the mast pipe is a supporting tower - a three- or four-masted tower, to which chimneys are attached. All components of the construction are assembled on the basis in the form of a concrete pad, starting from the bottom and gradually moving upwards. Used when assembling a rivet connection or use screws.
The supporting structure of the mast pipe is assembled from steel profiles interconnected by braces and corners. With its base, the column rests on the foundation, and is fixed by its anchoring.
Usually, individual elements are transported to the installation site and assembled as a designer. It takes this process quite a bit of time - a few hours. The height of the chimney can reach a maximum of 28.5 m. Stability of the chimney provide stiffeners - steel delays with a cross section of 1.6 to 2 cm. They compensate for the action of transverse forces.
## Materials for the construction of pipes boiler
Smoke extraction systems are made of different materials - bricks, steel, ceramics, and polymer. Brick chimneybuilt over brick stoves and fireplaces, it has good mechanical strength, excellent heat capacity, a sufficiently high degree of fire safety.
There are also a lot of flaws in these structures, so in modern construction fully brick chimneys are becoming increasingly rare. Regulatory documents limit the height of brick pipes 30-70 m, and a diameter of 0.6-8 m.
On the walls of a brick pipe with a lot of protrusions and recesses inside, there is always a lot of condensate, soot containing sulfur oxides. The latter, reacting with water, forms acids that actively destroy the brick.
Surface irregularities, narrowing of the passage as a result of the gradual increase of the soot layer, cause a decrease in the speed of passage of smoke and tipping over in the flue duct.
More resistant to condensation and external factors ceramic flues, they have high refractoriness. But this system is heavy, because inside there are metal rods, giving it extra strength. Hence the requirements for the mandatory installation of a separate foundation, supports, which increases the complexity and cost of installation.
Polymer flue pipes are appropriate in boiler rooms with a maximum temperature of 250 degrees C, during installation gas water heaters. They are lightweight, flexible and durable, but relevant only for gas equipment.
A device for the exhaust of stainless steel - an assembly consisting of individual elements of the chimney, interconnected by means of fittings: tees, nozzles, deflectors, tees, taps. Steel chimneys equip mainly gas boilers.
Installation of such a chimney can be performed after the construction of the building in a short time. There is a wide range of fittings, so the pipe can be given any configuration.
The modular chimney can be easily dismantled and moved to another location. The advantage of the design is its low weight, which allows to do without the foundation, resistance to moisture, slight soot deposition on the inner walls, high flue passage rate gases.
Sanitary standards allow the use of steel pipes for the construction of chimneys with a height of more than 30 m, an exception is possible only if less than 5 tons of multi-ash is used per day. fuel. The reason is that the service life of such facilities is 10 years, and if high sulfur fuel is used, it is significantly reduced.
The varieties, the body of which is made of steel alloy, are coaxial chimneys, with design specifics and features of operation which we recommend to get acquainted.
## Calculation of pipe parameters
To determine the height and diameter of the chimney for the boiler room, it is necessary to perform aerodynamic design calculations. The diameter depends on the capacity of individual boilers or the whole boiler room.
The combustion of fuel and the effective removal of smoke is greatly influenced by thrust, which requires constant air supply to the firebox to create. This is provided both naturally and artificially.
If a smoke pump is built into the system, then the height of the pipe is not critical. This parameter is important mainly to account for harmful emissions into the atmosphere. To determine samotyagu, you need a mandatory calculation of the height and section of the pipe.
### Determination of the height of the pipe during natural thrust
To create a normal natural thrust, it is necessary to observe the condition of equality of the thrust force and the total resistance that occurs during the movement of flue gases through the gas channels of the boiler and the path chimney. To provide such a thrust is possible under the condition of a small gas resistance, when the height of the pipe does not exceed 60 m.
This scheme will simplify the process of calculating the main parameters of the pipe for removal of the combustion products of any fuel in the furnaces of the boiler room
Regulatory documents regulating the location and calculation of chimney height, are SNiP41-01-2003, SP 7.13130.2009.
It is also necessary to take into account the recommendations set forth in the instructions for the boiler, in particular, the following requirements:
1. From the grate to the top point of the pipe should not be less than 5 m.
2. Above a flat roof without a high fence, the pipe should rise not less than 0.5 m.
3. In relation to the height of the fence and the ridge of the roof, the pipe should exceed their level by 0.5 m if it is within a m and a half from these structures.
4. When the chimney is removed from the parapet and the ridge at a distance of 1.5 to 3 m, its upper point should coincide with their height level.
With an incorrectly calculated chimney height, many problems can arise and the main one is air turbulence or a wind backpressure zone. The fire in the furnace can extinguish strong gusts of wind.
When constructing a chimney, it is necessary to take into account the design of the roof, the thickness of the roofing pie, the distance to the enclosing elements and the ridge, the rules of fire safety (+)
The implementation of fire safety rules is also a prerequisite for the design of the boiler tube. It is necessary to isolate the structures adjacent to the pipe.
So that sparks from the ventilation holes on the pipe do not fall on the roof when it is made of combustible material, the height of the construction increase by 0.5 m. The boiler room pipe should be removed from tall buildings and trees at least 2 m.
The height of the pipe is determined depending on the roof construction. If the roof is multi-level, they take into account height differences, but the base in all cases is the same - ridge height (+)
Since the optimal thrust arises due to the difference between the total density of the gases leaving the chimney and the outside air column equal in height, the calculation is performed according to the formula:
The height of the smoke channel is independently calculated by this formula. All values can be taken from the documentation attached to the heating equipment.
The calculation is quite complicated, it is better if it is performed by experts. Parameters affecting pipe height:
1. Coefficient A characterizes the meteorological situation of the region.
2. Mi is the mass of flue gases that pass through the pipe per unit of time.
3. F - the rate at which the particles formed during combustion settle.
4. Spdki and Sfi - indicators of the concentration of various substances in the flue gas.
5. V is the volume of gas.
6. T is the difference between the temperatures of the air entering and leaving the pipe.
If a boiler room located in the annexe to the house, the latter becomes a hindrance. It is necessary that in this case the tip of the pipe should be located above the zone of the wind backwater. Otherwise, the heating equipment will not function normally.
In order to determine by what amount a pipe should be grown, the highest point on the house is found, a direct angle of 45 degrees is made through it with the surface of the earth. The space below this line is a zone of wind pressure, and the chimney should be located above it.
### Pipe diameter calculation
To calculate the diameter of the pipe there is a formula:
S = m / (ρr x w),
Here m is the fuel consumption in 1 hour, w is the speed of movement of the flue gases, ρr is the air density in the working conditions, it is determined by the formula: Where tо is the outside air temperature, pBn is the air density in normal conditions = 1.2932 kg / m3.
The table will help determine the value of air density ρg in the working conditions without performing complex calculations. The value of the density of flue gases to simplify the calculations is assumed to be equal to the density of air (+)
Let 50 kg of solid fuel burn in a boiler in an hour, then in a second it will be 50: 3600 = 0.013888 kg. The speed of movement of flue gases - 2 m per second. At an air temperature of -4 degrees C, the air density is 0.6881 kg per cubic meter. m. Then S = 0.013888: (0.6881 x 2) = 0.01092 square meters. m = 92 square meters. cm. For a round section d = √4 x 92: 3.14 = 10.83 cm.
The diameter of a cylindrical chimney can be calculated using a different formula: d = 1000 / 1.163 x (r x Q√H), where r is a coefficient depending on the type of fuel used. For coal it is 0.03, for firewood 0.045, for gas 0.016, liquid fuel - 0.024.
## Conclusions and useful video on the topic
Video with a visual demonstration of the process of calculating the height of the smoke channel for arranging the boiler room:
Here, the author of the video shared his own experience in calculating and installing a chimney for a solid fuel boiler:
Another video to help the amateur designer:
It is not so important on what kind of fuel the boilers in the boiler room work. In any case, do not do without the flue gas removal system. The main requirements that chimney pipes must meet are good traction and throughput, sustained environmental standards.
Do you want to ask a question on a controversial or ambiguous point that you met when reading the information? There is useful information on the topic of the article, which you would like to share with site visitors? Please write comments in the box below.
#### Manifold heating own hands: circuit assembly nuancesBoiler House Arrangement
Defining task in designing the independent heating system is a uniform distribution of the heat carrier. This task is in the heating system performs control and adjustment unit - manifold.By proper... | 3,771 | 18,425 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2021-25 | latest | en | 0.915935 |
http://forum.arduino.cc/index.php?topic=246914.msg1765182 | 1,481,054,862,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698541995.74/warc/CC-MAIN-20161202170901-00399-ip-10-31-129-80.ec2.internal.warc.gz | 94,804,583 | 9,853 | Go Down
### Topic: Random stepper motor movement using randomSeed() or random() help appreciated (Read 478 times)previous topic - next topic
#### timlong
Hello
How can I apply a random rotation to and fro for a stepper motor please? I have adapted the sketch below, and could add a series of turns up and down but would prefer it to be random. I am planning to get the motor to play a guitar chaotically...
I am using this motor, PF35T - 49L4:
http://forum.arduino.cc/index.php?topic=181956.0
Quote
// Include the Stepper Library
#include <Stepper.h>
// Map our pins to constants to make things easier to keep track of
const int pwmA = 3;
const int pwmB = 11;
const int brakeA = 9;
const int brakeB = 8;
const int dirA = 12;
const int dirB = 13;
// The amount of steps for a full revolution of your motor.
// 360 / stepAngle
const int STEPS = 48;
// Initialize the Stepper class
Stepper myStepper(STEPS, dirA, dirB);
void setup() {
// Set the RPM of the motor
myStepper.setSpeed(30);
// Turn on pulse width modulation
pinMode(pwmA, OUTPUT);
digitalWrite(pwmA, HIGH);
pinMode(pwmB, OUTPUT);
digitalWrite(pwmB, HIGH);
// Turn off the brakes
pinMode(brakeA, OUTPUT);
digitalWrite(brakeA, LOW);
pinMode(brakeB, OUTPUT);
digitalWrite(brakeB, LOW);
// Log some shit
Serial.begin(9600);
}
void loop() {
// Move the motor X amount of steps
myStepper.step(STEPS);
Serial.println(STEPS);
// Pause
delay(2000);
// Move the motor X amount of steps the other way
myStepper.step(-STEPS);
Serial.println(-STEPS);
// Pause
delay(2000);
}
#### MarkT
#1
##### Jun 14, 2014, 11:44 amLast Edit: Jun 14, 2014, 01:14 pm by MarkT Reason: 1
You probably need to compute a random absolute position within a fixed range, then
move to that - otherwise you have no bounds on the movement (which is currently a
random walk)
[ BTW the AccelStepper library has an absolute moveTo() method which might
be more suitable ]
[ I will NOT respond to personal messages, I WILL delete them, use the forum please ]
Go Up
Please enter a valid email to subscribe | 564 | 2,041 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2016-50 | longest | en | 0.693355 |
http://learn.arcade.academy/en/latest/chapters/13_lists/lists.html | 1,579,472,197,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250595282.35/warc/CC-MAIN-20200119205448-20200119233448-00426.warc.gz | 93,600,470 | 12,397 | # 13. Introduction to Lists¶
## 13.1. Data Types¶
So far this book has shown four types of data:
• String (a string is short for “string of characters,” which normal people think of as text.)
• Integer
• Floating point
• Boolean
Python can display what type of data a value is with the type function. For example, what type of data is a 3?
# Print the type of data a 3 is:
print(type(3))
<class 'int'>
This type function isn’t useful for other programming in this book, but it is good to demonstrate the types of data introduced so far.
Here we set x to the four types of data we’ve used so far, and call the type command to see how Python classifies that data:
x = 3
print("x =", x, "and is of type:", type(x))
x = 3.145
print("x =", x, "and is of type:", type(x))
x = "Hi there"
print("x =", x, "and is of type:", type(x))
x = True
print("x =", x, "and is of type:", type(x))
The output:
x = 3 and is of type: <class 'int'>
x = 3.145 and is of type: <class 'float'>
x = Hi there and is of type: <class 'str'>
x = True and is of type: <class 'bool'>
Note
More than one coin to collect? Use a list!
The two new types of data introduced in this chapter are Lists and Tuples. Lists are similar to another data structure called an array. A list can be resized, but an array can not. A course in data structures will teach you the details, but it that is beyond the scope of this book. Try running the following commands in the interactive Python shell and see what is displayed:
x = (2, 3, 4, 5)
print("x =", x, "and is of type:", type(x))
x = [2, 3, 4, 5]
print("x =", x, "and is of type:", type(x))
The output:
x = (2, 3, 4, 5) and is of type: <class 'tuple'>
x = [2, 3, 4, 5] and is of type: <class 'list'>
## 13.2. Working With Lists¶
You’ve created grocery lists, to-do lists, bucket lists, but how do you create a list on the computer?
To create a list and print it out, try the following:
x = [10, 20]
print(x)
The output:
[10, 20]
To print an individual element in a list:
print(x[0])
The output:
10
This number with the item’s location is called the index. Note that list locations start at zero. So a list or array with 10 elements does not have an element in spot [10]. Just spots [0] through [9]. It can be very confusing to create an list of 10 items and then not have an item 10, but most computer languages start counting at 0 rather than 1.
Think of a list as an ice cube tray that holds numbers, as shown in Figure 7.2. The values are stored inside each tray spot, and written on the side of the tray are numbers starting at zero that identify the location of each spot.
Attention
Don’t mix the index and the value!
Remember, there are two sets of numbers to consider when working with a list of numbers: the position and the value. The position, also known as index, refers to where a value is. The value is the actual number stored at that location. When working with a list or array, make sure to think if you need the location or the value.
It is easy to get the value given the location, but it is harder to get the location given the value. Chapter 15 is dedicated to answering how to find the location of a particular value.
You can also access elements from the back-side of an array using negative numbers. (Not all languages support this.) For example:
x = [10, 20, 30]
print(x[-1])
The output:
30
A program can assign new values to an individual element in a list. In the case below, the first spot at location zero (not one) is assigned the number 22.
x = [1, 2]
print(x)
x[0] = 22
print(x)
[1, 2]
[22, 2]
Also, a program can create a “tuple.” This data type works just like a list, but with two differences. First, it is created with parentheses rather than square brackets. Second, it is not possible to change the tuple once created. See below:
x = (1, 2)
print(x)
x[0] = 22
print(x)
[1, 2]
Traceback (most recent call last):
File "<pyshell#18>", line 4, in <module>
x[0] = 22
TypeError: 'tuple' object does not support item assignment
As can be seen from the output of the code above, we can’t assign an item in the tuple a new value. Why would we want this limitation? First, the computer can run faster if it knows the value won’t change. Second, some lists we don’t want to change, such as a list of RGB colors for red. The color red doesn’t change, therefore an immutable tuple is a better choice.
## 13.3. Create an Empty List¶
Occasionally we need to create a list that is empty. We’ll use this in a bit when we start with an empty list and build on it. How do I create an empty list? Easy:
# Create an empty list
my_list = []
## 13.4. Iterating (Looping) Through a List¶
If a program needs to iterate through each item in a list, such as to print it out, there are two types of for loops that can do this.
The first method to iterate through each item in a loop is by using a “for-each” loop. This type of loop takes a collection of items, and loops the code once per item. It will take a copy of the item and store it in a variable for processing.
The format of the command:
for item_variable in list_name:
Here are some examples:
my_list = [101, 20, 10, 50, 60]
for item in my_list:
print(item)
101
20
10
50
60
Programs can store strings in lists too:
my_list = ["Spoon", "Fork", "Knife"]
for item in my_list:
print(item)
Spoon
Knife
Fork
Lists can even contain other lists. This iterates through each item in the main list, but not in sublists.
my_list = [[2, 3], [4, 3], [6, 7]]
for item in my_list:
print(item)
[2,3]
[4,3]
[6,7]
The other way to iterate through a list is to use an index variable and directly access the list rather than through a copy of each item. To use an index variable, the program counts from 0 up to the length of the list. If there are ten elements, the loop must go from 0 to 9 for a total of ten elements.
The length of a list may be found by using the len function. Combining that with the range function allows the program to loop through the entire list.
my_list = [101, 20, 10, 50, 60]
for index in range(len(my_list)):
print(my_list[index])
101
20
10
50
60
This method is more complex, but is also more powerful. Because we are working directly with the list elements, rather than a copy, the list can be modified. The for-each loop does not allow modification of the original list.
## 13.5. Looping With Both An Index And Element¶
If you want both the index, like a for i in range gives you, and the element, like a for item in my_list gives you, the proper Python-ic way to use the enumerate function like this:
for index, value in enumerate(my_list):
print(index, value)
## 13.6. Adding to a List¶
New items may be added to a list (but not a tuple) by using the append command. For example:
my_list = [2, 4, 5, 6]
print(my_list)
my_list.append(9)
print(my_list)
[2, 4, 5, 6]
[2, 4, 5, 6, 9]
Side note: If performance while appending is a concern, it is very important to understand how a list is being implemented. For example, if a list is implemented as an array data type, then appending an item to the list is a lot like adding a new egg to a full egg carton. A new egg carton must be built with thirteen spots. Then twelve eggs are moved over. Then the thirteenth egg is added. Finally the old egg carton is recycled. Because this can happen behind the scenes in a function, programmers may forget this and let the computer do all the work. It would be more efficient to simply tell the computer to make an egg carton with enough spots to begin with. Thankfully, Python does not implement a list as an array data type. But it is important to pay attention to your next semester data structures class and learn how all of this works.
To create a list from scratch, it is necessary to create a blank list and then use the append function to build it based upon user input:
Creating a list of numbers from user input
# Create an empty list
my_list = []
for i in range(5):
user_input = input( "Enter an integer: ")
user_input = int(user_input)
my_list.append(user_input)
print(my_list)
Enter an integer: 4
[4]
Enter an integer: 5
[4, 5]
Enter an integer: 3
[4, 5, 3]
Enter an integer: 1
[4, 5, 3, 1]
Enter an integer: 8
[4, 5, 3, 1, 8]
If a program needs to create an array of a specific length, all with the same value, a simple trick is to use the following code:
Create an array with 100 zeros
1 2 # Create an array with 100 zeros. my_list = [0] * 100
## 13.7. Summing or Modifying a List¶
Creating a running total of an array is a common operation. Here’s how it is done:
Summing the values in a list v1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 # Copy of the array to sum my_list = [5, 76, 8, 5, 3, 3, 56, 5, 23] # Initial sum should be zero list_total = 0 # Loop from 0 up to the number of elements # in the array: for index in range(len(my_list)): # Add element 0, next 1, then 2, etc. list_total += my_list[index] # Print the result print(list_total)
The same thing can be done by using a for loop to iterate the array, rather than count through a range:
Summing the values in a list v2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 # Copy of the array to sum my_list = [5, 76, 8, 5, 3, 3, 56, 5, 23] # Initial sum should be zero list_total = 0 # Loop through array, copying each item in the array into # the variable named item. for item in my_list: # Add each item list_total += item # Print the result print(list_total)
Numbers in an array can also be changed by using a for loop:
Doubling all the numbers in a list
1 2 3 4 5 6 7 8 9 10 11 # Copy of the array to modify my_list = [5, 76, 8, 5, 3, 3, 56, 5, 23] # Loop from 0 up to the number of elements # in the array: for index in range(len(my_list)): # Modify the element by doubling it my_list[index] = my_list[index] * 2 # Print the result print(my_list)
However version 2 does not work at doubling the values in an array. Why? Because item is a copy of an element in the array. The code below doubles the copy, not the original array element.
Bad code that doesn’t double all the numbers in a list
1 2 3 4 5 6 7 8 9 10 11 # Copy of the array to modify my_list = [5, 76, 8, 5, 3, 3, 56, 5, 23] # Loop through each element in myArray for item in my_list: # This doubles item, but does not change the array # because item is a copy of a single element. item = item * 2 # Print the result print(my_list)
## 13.8. Slicing Strings¶
Strings are actually lists of characters. They can be treated like lists with each letter a separate item. Run the following code with both versions of x:
Accessing a string as a list
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x = "This is a sample string" #x = "0123456789" print("x=", x) # Accessing the first character ("T") print("x[0]=", x[0]) # Accessing the second character ("h") print("x[1]=", x[1]) # Accessing from the right side ("g") print("x[-1]=", x[-1]) # Access 0-5 ("This ") print("x[:6]=", x[:6]) # Access 6 to the end ("is a sample string") print("x[6:]=", x[6:]) # Access 6-8 print("x[6:9]=", x[6:9])
Strings in Python may be used with some of the mathematical operators. Try the following code and see what Python does:
Adding and multiplying strings
1 2 3 4 5 6 7 8 a = "Hi" b = "There" c = "!" print(a + b) print(a + b + c) print(3 * a) print(a * 3) print((a * 2) + (b * 2))
It is possible to get a length of a string. It is also possible to do this with any type of array.
Getting the length of a string or list
1 2 3 4 5 a = "Hi There" print(len(a)) b = [3, 4, 5, 6, 76, 4, 3, 3] print(len(b))
Since a string is an array, a program can iterate through each character element just like an array:
for character in "This is a test.":
print(character)
Exercise: Starting with the following code:
1 2 months = "JanFebMarAprMayJunJulAugSepOctNovDec" n = int(input("Enter a month number: "))
Print the three month abbreviation for the month number that the user enters. (Calculate the start position in the string, then use the info we just learned to print out the correct substring.)
## 13.9. Secret Codes¶
This code prints out every letter of a string individually:
1 2 3 4 plain_text = "This is a test. ABC abc" for c in plain_text: print(c, end=" ")
Computers do not actually store letters of a string in memory; computers store a series of numbers. Each number represents a letter. The system that computers use to translate numbers to letters is called Unicode. The full name for the encoding is Universal Character Set Transformation Format 8-bit, usually abbreviated UTF-8.
The Unicode chart covers the Western alphabet using the numbers 0-127. Each Western letter is represented by one byte of memory. Other alphabets, like Cyrillic, can take multiple bytes to represent each letter. A partial copy of the Unicode chart is below:
Value Character Value Character Value Character Value Character 40 ( 61 = 82 R 103 g 41 ) 62 > 83 S 104 h 42 63 ? 84 T 105 i 43 64 @ 85 U 106 j 44 , 65 A 86 V 107 k 45 66 B 87 W 108 l 46 . 67 C 88 X 109 m 47 / 68 D 89 Y 110 n 48 0 69 E 90 Z 111 o 49 1 70 F 91 [ 112 p 50 2 71 G 92 113 q 51 3 72 H 93 ] 114 r 52 4 73 I 94 ^ 115 s 53 5 74 J 95 _ 116 t 54 6 75 K 96 117 u 55 7 76 L 97 a 118 v 56 8 77 M 98 b 119 w 57 9 78 N 99 c 120 x 58 : 79 O 100 d 121 y 59 ; 80 P 101 e 122 z 60 < 81 Q 102 f
For more information about ASCII (which has the same values as Unicode for the Western alphabet) see:
http://en.wikipedia.org/wiki/ASCII
For a video that explains the beauty of Unicode, see here:
This next set of code converts each of the letters in the prior example to its ordinal value using UTF-8:
plain_text = "This is a test. ABC abc"
for c in plain_text:
print(ord(c), end=" ")
This next program takes each UTF-8 value and adds one to it. Then it prints the new UTF-8 value, then converts the value back to a letter.
plain_text = "This is a test. ABC abc"
for c in plain_text:
x = ord(c)
x = x + 1
c2 = chr(x)
print(c2, end="")
The next code listing takes each UTF-8 value and adds one to it, then converts the value back to a letter.
simple_encryption.py
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 # Sample Python/Pygame Programs # Simpson College Computer Science # http://programarcadegames.com/ # http://simpson.edu/computer-science/ # Explanation video: http://youtu.be/sxFIxD8Gd3A plain_text = "This is a test. ABC abc" encrypted_text = "" for c in plain_text: x = ord(c) x = x + 1 c2 = chr(x) encrypted_text = encrypted_text + c2 print(encrypted_text)
Finally, the last code takes each UTF-8 value and subtracts one from it, then converts the value back to a letter. By feeding this program the output of the previous program, it serves as a decoder for text encoded by the prior example.
simple_decryption.py
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 # Sample Python/Pygame Programs # Simpson College Computer Science # http://programarcadegames.com/ # http://simpson.edu/computer-science/ # Explanation video: http://youtu.be/sxFIxD8Gd3A encrypted_text = "Uijt!jt!b!uftu/!BCD!bcd" plain_text = "" for c in encrypted_text: x = ord(c) x = x - 1 c2 = chr(x) plain_text = plain_text + c2 print(plain_text)
## 13.10. Associative Arrays¶
Python is not limited to using numbers as an array index. It is also possible to use an associative array. An associative array works like this:
1 2 3 4 5 6 7 8 9 10 11 # Create an empty associative array # (Note the curly braces.) x = {} # Add some stuff to it x["fred"] = 2 x["scooby"] = 8 x["wilma"] = 1 # Fetch and print an item print(x["fred"]) `
You won’t really need associative arrays for this class, but I think it is important to point out that it is possible. | 4,548 | 15,610 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2020-05 | latest | en | 0.876168 |
https://thisisbeep.com/how-much-is-37-bhp/ | 1,701,213,111,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100016.39/warc/CC-MAIN-20231128214805-20231129004805-00328.warc.gz | 660,309,846 | 11,911 | # How much is 37 bhp?
## How much is 37 bhp?
a power conversion table
boiler horse- power horse- power
35 460.33
36 473.48
37 486.64
38 499.79
## What does 95PS mean in cars?
95PS – PS stands for PferdStarke (literally, ‘horse strength’ in German). This is basically metric horsepower as opposed to the imperial or mechanical measure of horsepower denoted by HP or BHP. SE – This particular make and model defines its trim levels by S, SE and FR.So SE is a mid-range trim level.
What is kW in a car?
Kilowatts (kW) is a measure of power equal to 1000W of energy. Although converted into BHP for acceleration and speed figures kW is a more relevant unit for vehicles that don’t measure fuel or MPG instead measuring Kilowatt Hours and Miles per KiloWatt Hours.
What is BHP vs hp?
One source cites horsepower as the output of a mechanical or electrical system, while the brake horsepower is the output of only the engine (the exact opposite of what we state above). In this definition, hp determines the efficiency of an entire system, while bhp is just for the engine.
### What is the best 150hp outboard?
Environmentally friendly. Before going further,it would be prudent to note that all 150hp outboard engines now sold new in Australia meet the very latest non-road spark ignition engines emissions
• Clean,green and easy to use.
• Modern 150hp outboard engines.
• Evinrude E-TEC G2 150hp.
• Honda BF150.
• Mercury 150hp.
• Suzuki DF150A.
• Yamaha F150.
• ### How many amperes is a 150hp motor?
Since 1 HP is equal to 746 W simply multiply the horsepower by 746 for this conversion. Since motors are not 100% efficient, the number of amps needed to achieve the power output is actually higher depending on the efficiency. To find this, divide the horsepower by the efficiency η in decimal form.
How do you convert kW to horsepower?
– kW: Kilowatts or 1000Watts – Hp: Horsepower or horse power – E: Engine efficiency.
How to convert hp to kW?
Formulas to convert,calculate,transform from Hp to kW: E: Motor efficiency.
• Examples of conversions from Hp to kW: Four motors of 1.5 Hp and 80% efficiency are connected to a transformer,how many kW in total have the motors?.
• Common efficiencies for motor: 1) Design NEMA B,individual speed 1200,1800,3600 RPM. | 575 | 2,276 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.203125 | 3 | CC-MAIN-2023-50 | latest | en | 0.886833 |
http://www.javaprogrammingforums.com/%20loops-control-statements/16654-finding-two-thirds-root-number-using-loops-printingthethread.html | 1,508,708,998,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187825464.60/warc/CC-MAIN-20171022203758-20171022223758-00093.warc.gz | 480,460,950 | 4,190 | # Finding the Two-Thirds Root of a Number Using Loops
• July 8th, 2012, 03:30 PM
tryingtolearn
Finding the Two-Thirds Root of a Number Using Loops
Hello everyone!
I am trying to create a program that can display the two thirds root of a number that I provide.
I have yet to find any clear explanation of my problem online.
The format of the answer should be like this:
The two thirds root of xxx = xxx.xxx
A two thirds root of x should be equal to x^3/2, correct?
Well I am trying to have the program solve for that x^3/2 without using Math.pow, and instead using a loop of any kind.
Thank you for any help,
tryingtolearn
• July 8th, 2012, 06:37 PM
pbrockway2
Re: Finding the Two-Thirds Root of a Number Using Loops
In case you aren't aware of it, Java doesn't have an exponentiation operator. People often say "x^2/3" as a shorthand, but using standard Java they mean something like a call to Math.pow(). ^ really means something quite different in Java.
(Also note that the 2/3 root is 2/3, not 3/2)
(A) The 2/3 root of a number is the cube root of the square of that number.
Think about positive numbers as a simplification that might serve as starting point.
(B) The 2/3 root of zero is zero and the 2/3 root grows as the number whose root is being found grows. What I mean is that big numbers have big 2/3 roots and small numbers have small 2/3 roots.
This suggests we might find a 2/3 root by playing 20 questions. (or rather, n questions where n is the limit in the for loop you are supposed to use.)
Code :
```make a bigGuess and a littleGuess LOOP: is (bigGuess+littleGuess)/2 too big? If so make bigGuess=(bigGuess+littleGuess)/2 is (bigGuess+littleGuess)/2 too small? If so make littleGuess=(bigGuess+littleGuess)/2```
The initial guesses could take advantage of the fact that (C) for numbers bigger than 1 the 2/3 root is less than the number, while for numbers less than 1 the 2/3 root is bigger than the number.
-----
Newton and Halley both came up with iterative methods. But the problem of doubling a cube is much older with geometric (Greek) and iterative approximations (Indian) going back to c400BCE.
• July 8th, 2012, 07:54 PM
tryingtolearn
Re: Finding the Two-Thirds Root of a Number Using Loops
Quote:
Originally Posted by pbrockway2
In case you aren't aware of it, Java doesn't have an exponentiation operator. People often say "x^2/3" as a shorthand, but using standard Java they mean something like a call to Math.pow(). ^ really means something quite different in Java.
(Also note that the 2/3 root is 2/3, not 3/2)
(A) The 2/3 root of a number is the cube root of the square of that number.
Think about positive numbers as a simplification that might serve as starting point.
(B) The 2/3 root of zero is zero and the 2/3 root grows as the number whose root is being found grows. What I mean is that big numbers have big 2/3 roots and small numbers have small 2/3 roots.
This suggests we might find a 2/3 root by playing 20 questions. (or rather, n questions where n is the limit in the for loop you are supposed to use.)
Code :
```make a bigGuess and a littleGuess LOOP: is (bigGuess+littleGuess)/2 too big? If so make bigGuess=(bigGuess+littleGuess)/2 is (bigGuess+littleGuess)/2 too small? If so make littleGuess=(bigGuess+littleGuess)/2```
The initial guesses could take advantage of the fact that (C) for numbers bigger than 1 the 2/3 root is less than the number, while for numbers less than 1 the 2/3 root is bigger than the number.
-----
Newton and Halley both came up with iterative methods. But the problem of doubling a cube is much older with geometric (Greek) and iterative approximations (Indian) going back to c400BCE.
Thank you for your reply, but I am still having trouble understanding.
let's say I wanted to find the 2/3 root of 100.
In the post above it says I should take the square of 100, which is 10, and then find the cube root of 10, which is 3. something.
So I guess I just don't understand the bigGuess/littleGuess concept.
Thanks for your patience and help so far!
tryingtolearn
• July 8th, 2012, 08:24 PM
pbrockway2
Re: Finding the Two-Thirds Root of a Number Using Loops
Quote:
let's say I wanted to find the 2/3 root of 100
The 2/3 root of 100 is a number whose cube is 10000 (100 squared). That's (A). In other words we want to find something, where something*something*something is 10000.
That something lies between 1 and 10000. That's (C).
I'll take littleGuess=1 and bigGuess=10000.
LOOP starts:
I check (littleGuess+bigGuess)/2 which is 5000.5
5000.5*5000.5*5000.5 is 125037503750.125 and that's too big (the cube should be 10000). So I revise my guess and make bigGuess=5000.5 and go back to the top of the loop...
-----
I check (littleGuess+bigGuess)/2 which is 2500.75
2500.75*2500.75*2500.75 is 15639066719.171875 and that's still too big. So I make bigGuess=2500.75 and try again...
-----
Check (littleGuess+bigGuess)/2 which is 1250.875. The cube is 1957229434.263671875. bigGuess becomes 1250.875
Check (littleGuess+bigGuess)/2 which is 625.9375. The cube is 245240906.585693359375. bigGuess becomes 625.9375
Check (littleGuess+bigGuess)/2 which is 313.46875. The cube is 30802272.332489013671875. bigGuess becomes 313.46875
Check (littleGuess+bigGuess)/2 which is 157.234375. The cube is 3887250.213802337646484375. bigGuess becomes 157.234375
Check (littleGuess+bigGuess)/2 which is 79.1171875. The cube is 495236.357871532440185546875. bigGuess becomes 79.1171875
Check (littleGuess+bigGuess)/2 which is 40.05859375. The cube is 64281.662188470363616943359375. bigGuess becomes 40.05859375
Check (littleGuess+bigGuess)/2 which is 20.529296875. The cube is 8652.113846175372600555419921875. Too small! bigGuess stays at 40.05859375 and littleGuess becomes 20.529296875.
Check (littleGuess+bigGuess)/2 which is 30.2939453125. The cube is 27801.454087962396442890167236328. (too big) bigGuess becomes 30.2939453125
Check (littleGuess+bigGuess)/2 which is 25.41162109375. The cube is 16409.566686896956525743007659912. bigGuess becomes 25.41162109375
Check (littleGuess+bigGuess)/2 which is 22.970458984375. The cube is 12120.178596764206304214894771576.
-----
The loop constantly evaluates the cube of (littleGuess+bigGuess)/2 and updates one or other of the guesses based on whether the cube is too high or too low (compared with 10000=100^2).
• July 8th, 2012, 08:44 PM
helloworld922
Re: Finding the Two-Thirds Root of a Number Using Loops
Try taking a look at this: optimized pow() approximation
edit: | 1,890 | 6,525 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.671875 | 4 | CC-MAIN-2017-43 | longest | en | 0.943909 |
https://pressbooks.nscc.ca/algebratrigonometryopenstax/chapter/linear-functions/ | 1,718,693,737,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861747.46/warc/CC-MAIN-20240618042809-20240618072809-00705.warc.gz | 431,889,752 | 55,746 | # 25 Linear Functions
### Learning Objectives
In this section you will:
• Represent a linear function.
• Determine whether a linear function is increasing, decreasing, or constant.
• Interpret slope as a rate of change.
• Write and interpret an equation for a linear function.
• Graph linear functions.
• Determine whether lines are parallel or perpendicular.
• Write the equation of a line parallel or perpendicular to a given line.
Just as with the growth of a bamboo plant, there are many situations that involve constant change over time. Consider, for example, the first commercial maglev train in the world, the Shanghai MagLev Train ((Figure)). It carries passengers comfortably for a 30-kilometer trip from the airport to the subway station in only eight minutes[1] .
Suppose a maglev train travels a long distance, and maintains a constant speed of 83 meters per second for a period of time once it is 250 meters from the station. How can we analyze the train’s distance from the station as a function of time? In this section, we will investigate a kind of function that is useful for this purpose, and use it to investigate real-world situations such as the train’s distance from the station at a given point in time.
### Representing Linear Functions
The function describing the train’s motion is a linear function, which is defined as a function with a constant rate of change. This is a polynomial of degree 1. There are several ways to represent a linear function, including word form, function notation, tabular form, and graphical form. We will describe the train’s motion as a function using each method.
#### Representing a Linear Function in Word Form
Let’s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.
• The train’s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at constant speed.
The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by 1 second, the corresponding distance (output) increases by 83 meters. The train began moving at this constant speed at a distance of 250 meters from the station.
#### Representing a Linear Function in Function Notation
Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the slope-intercept form of a line, where$\,x\,$is the input value,$\,m\,$is the rate of change, and$\,b\,$is the initial value of the dependent variable.
$\begin{array}{cc}\text{Equation form}\phantom{\rule{2em}{0ex}}\hfill & y=mx+b\hfill \\ \text{Function notation}\phantom{\rule{2em}{0ex}}\hfill & f\left(x\right)=mx+b\hfill \end{array}$
In the example of the train, we might use the notation$\,D\left(t\right)\,$where the total distance$\,D\,$is a function of the time$\,t.\,$The rate,$\,m,\,$is 83 meters per second. The initial value of the dependent variable$\,b\,$is the original distance from the station, 250 meters. We can write a generalized equation to represent the motion of the train.
$D\left(t\right)=83t+250$
#### Representing a Linear Function in Tabular Form
A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in (Figure). From the table, we can see that the distance changes by 83 meters for every 1 second increase in time.
Can the input in the previous example be any real number?
No. The input represents time so while nonnegative rational and irrational numbers are possible, negative real numbers are not possible for this example. The input consists of non-negative real numbers.
#### Representing a Linear Function in Graphical Form
Another way to represent linear functions is visually, using a graph. We can use the function relationship from above,$\,D\left(t\right)=83t+250,\,$to draw a graph as represented in (Figure). Notice the graph is a line. When we plot a linear function, the graph is always a line.
The rate of change, which is constant, determines the slant, or slope of the line. The point at which the input value is zero is the vertical intercept, or y-intercept, of the line. We can see from the graph that the y-intercept in the train example we just saw is$\,\left(0,250\right)\,$and represents the distance of the train from the station when it began moving at a constant speed.
Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line$\,f\left(x\right)=2x+1.\,$
Ask yourself what numbers can be input to the function. In other words, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product.
### Linear Function
A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line
$f\left(x\right)=mx+b$
where$\,b\,$is the initial or starting value of the function (when input,$\,x=0\,$), and$\,m\,$is the constant rate of change, or slope of the function. The y-intercept is at$\,\left(0,b\right).$
### Using a Linear Function to Find the Pressure on a Diver
The pressure,$\,P,$in pounds per square inch (PSI) on the diver in (Figure) depends upon her depth below the water surface,$\,d,$in feet. This relationship may be modeled by the equation,$\,P\left(d\right)=0.434d+14.696.\,$Restate this function in words.
To restate the function in words, we need to describe each part of the equation. The pressure as a function of depth equals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths.
#### Analysis
The initial value, 14.696, is the pressure in PSI on the diver at a depth of 0 feet, which is the surface of the water. The rate of change, or slope, is 0.434 PSI per foot. This tells us that the pressure on the diver increases 0.434 PSI for each foot her depth increases.
### Determining Whether a Linear Function Is Increasing, Decreasing, or Constant
The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant. For an increasing function, as with the train example, the output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in (Figure)(a). For a decreasing function, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in (Figure)(b). If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in (Figure)(c).
### Increasing and Decreasing Functions
The slope determines if the function is an increasing linear function, a decreasing linear function, or a constant function.
• $f\left(x\right)=mx+b\,$is an increasing function if$\,m>0.$
• $f\left(x\right)=mx+b\,$is a decreasing function if$\,m<0.$
• $f\left(x\right)=mx+b\,$is a constant function if$\,m=0.$
### Deciding Whether a Function Is Increasing, Decreasing, or Constant
Some recent studies suggest that a teenager sends an average of 60 texts per day[2] . For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or constant.
1. The total number of texts a teen sends is considered a function of time in days. The input is the number of days, and output is the total number of texts sent.
2. A teen has a limit of 500 texts per month in his or her data plan. The input is the number of days, and output is the total number of texts remaining for the month.
### Writing an Equation for a Linear Function Given Two Points
If$\,f\,$is a linear function, with$\,f\left(3\right)=-2,$and$\,f\left(8\right)=1,$find an equation for the function in slope-intercept form.
We can write the given points using coordinates.
$\begin{array}{ccc}\hfill f\left(3\right)& =& -2\to \left(3,-2\right)\hfill \\ \hfill f\left(8\right)& =& 1\to \left(8,1\right)\hfill \end{array}$
We can then use the points to calculate the slope.
$\begin{array}{ccc}\hfill m& =& \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\hfill \\ & =& \frac{1-\left(-2\right)}{8-3}\hfill \\ & =& \frac{3}{5}\hfill \end{array}$
Substitute the slope and the coordinates of one of the points into the point-slope form.
$\begin{array}{ccc}\hfill y-{y}_{1}& =& m\left(x-{x}_{1}\right)\hfill \\ \hfill y-\left(-2\right)& =& \frac{3}{5}\left(x-3\right)\hfill \end{array}$
We can use algebra to rewrite the equation in the slope-intercept form.
$\begin{array}{ccc}\hfill y+2& =& \frac{3}{5}\left(x-3\right)\hfill \\ \hfill y+2& =& \frac{3}{5}x-\frac{9}{5}\hfill \\ \hfill y& =& \frac{3}{5}x-\frac{19}{5}\hfill \end{array}$[/hidden-answer]
### Try It
If$\,f\left(x\right)\,$is a linear function, with$\,f\left(2\right)=–11,$and$\,f\left(4\right)=-25,$write an equation for the function in slope-intercept form.
$y=-7x+3$
### Modeling Real-World Problems with Linear Functions
In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.
### How To
Given a linear function$\,f\,$and the initial value and rate of change, evaluate$\,f\left(c\right).$
1. Determine the initial value and the rate of change (slope).
2. Substitute the values into$\,f\left(x\right)=mx+b.$
3. Evaluate the function at$\,x=c.$
### Using a Linear Function to Determine the Number of Songs in a Music Collection
Marcus currently has 200 songs in his music collection. Every month, he adds 15 new songs. Write a formula for the number of songs,$\,N,$in his collection as a function of time,$\,t,$the number of months. How many songs will he own at the end of one year?
The initial value for this function is 200 because he currently owns 200 songs, so$\,N\left(0\right)=200,$which means that$\,b=200.$
The number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. Therefore we know that$\,m=15.\,$We can substitute the initial value and the rate of change into the slope-intercept form of a line.
We can write the formula$\,N\left(t\right)=15t+200.$
With this formula, we can then predict how many songs Marcus will have at the end of one year (12 months). In other words, we can evaluate the function at$\,t=12.$
$\begin{array}{ccc}N\left(12\right)& =& 15\left(12\right)+200\hfill \\ & =& 180+200\hfill \\ & =& 380\hfill \end{array}$
Marcus will have 380 songs in 12 months.
#### Analysis
Notice that N is an increasing linear function. As the input (the number of months) increases, the output (number of songs) increases as well.
### Using a Linear Function to Calculate Salary Based on Commission
Working as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilya’s weekly income$\,I,$depends on the number of new policies,$\,n,$he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies and earned$920. Find an equation for$\,I\left(n\right),$and interpret the meaning of the components of the equation.
The given information gives us two input-output pairs:$\,\left(3,760\right)\,$and$\,\left(5,\text{92}0\right).\,$We start by finding the rate of change.
$\begin{array}{ccc}\hfill m& =& \frac{920-760}{5-3}\hfill \\ & =& \frac{160}{2 \text{policies}}\hfill \\ & =& \text{\}80\,\text{per}\,\text{policy}\hfill \end{array}$
Keeping track of units can help us interpret this quantity. Income increased by $160 when the number of policies increased by 2, so the rate of change is$80 per policy. Therefore, Ilya earns a commission of $80 for each policy sold during the week. We can then solve for the initial value. $\begin{array}{cccc}\hfill I\left(n\right)& =& 80n+b\hfill & \\ \hfill 760& =& 80\left(3\right)+b\hfill & \phantom{\rule{1em}{0ex}}\text{ }\text{when}\,n=3,I\left(3\right)=760\hfill \\ \hfill 760& -& 80\left(3\right)=b\hfill & \\ \hfill 520& =& b\hfill & \end{array}$ The value of$\,b\,$is the starting value for the function and represents Ilya’s income when$\,n=0,\,$or when no new policies are sold. We can interpret this as Ilya’s base salary for the week, which does not depend upon the number of policies sold. We can now write the final equation. $I\left(n\right)=80n+520$ Our final interpretation is that Ilya’s base salary is$520 per week and he earns an additional $80 commission for each policy sold.[/hidden-answer] ### Using Tabular Form to Write an Equation for a Linear Function (Figure) relates the number of rats in a population to time, in weeks. Use the table to write a linear equation. number of weeks, w 0 2 4 6 number of rats, P(w) 1000 1080 1160 1240 [reveal-answer q=”fs-id1450415″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1450415″] We can see from the table that the initial value for the number of rats is 1000, so$\,b=1000.$ Rather than solving for$\,m,$we can tell from looking at the table that the population increases by 80 for every 2 weeks that pass. This means that the rate of change is 80 rats per 2 weeks, which can be simplified to 40 rats per week. $P\left(w\right)=40w+1000$ If we did not notice the rate of change from the table we could still solve for the slope using any two points from the table. For example, using$\,\left(2,1080\right)\,$and$\,\left(6,1240\right)$ $\begin{array}{ccc}\hfill m& =& \frac{1240-1080}{6-2}\hfill \\ & =& \frac{160}{4}\hfill \\ & =& 40\hfill \end{array}$ [/hidden-answer] Is the initial value always provided in a table of values like (Figure)? No. Sometimes the initial value is provided in a table of values, but sometimes it is not. If you see an input of 0, then the initial value would be the corresponding output. If the initial value is not provided because there is no value of input on the table equal to 0, find the slope, substitute one coordinate pair and the slope into$\,f\left(x\right)=mx+b,\,$and solve for$\,b.$ ### Try It A new plant food was introduced to a young tree to test its effect on the height of the tree. (Figure) shows the height of the tree, in feet,$\,x\,$months since the measurements began. Write a linear function,$\,H\left(x\right),$where$\,x\,$is the number of months since the start of the experiment. x 0 2 4 8 12 H(x) 12.5 13.5 14.5 16.5 18.5 [reveal-answer q=”fs-id2238094″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2238094″] $H\left(x\right)=0.5x+12.5$ [/hidden-answer] ### Graphing Linear Functions Now that we’ve seen and interpreted graphs of linear functions, let’s take a look at how to create the graphs. There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by using the y-intercept and slope. And the third method is by using transformations of the identity function$\,f\left(x\right)=x.$ #### Graphing a Function by Plotting Points To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values. The input values and corresponding output values form coordinate pairs. We then plot the coordinate pairs on a grid. In general, we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph. For example, given the function,$\,f\left(x\right)=2x,$we might use the input values 1 and 2. Evaluating the function for an input value of 1 yields an output value of 2, which is represented by the point$\,\left(1,2\right).\,$Evaluating the function for an input value of 2 yields an output value of 4, which is represented by the point$\,\left(2,4\right).\,$Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error. ### How To Given a linear function, graph by plotting points. 1. Choose a minimum of two input values. 2. Evaluate the function at each input value. 3. Use the resulting output values to identify coordinate pairs. 4. Plot the coordinate pairs on a grid. 5. Draw a line through the points. ### Graphing by Plotting Points Graph$\,f\left(x\right)=-\frac{2}{3}x+5\,$by plotting points. [reveal-answer q=”fs-id1420870″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1420870″] Begin by choosing input values. This function includes a fraction with a denominator of 3, so let’s choose multiples of 3 as input values. We will choose 0, 3, and 6. Evaluate the function at each input value, and use the output value to identify coordinate pairs. $\begin{array}{cc}\hfill x=0& \phantom{\rule{2em}{0ex}}f\left(0\right)=-\frac{2}{3}\left(0\right)+5=5⇒\left(0,5\right)\hfill \\ \hfill x=3& \phantom{\rule{2em}{0ex}}f\left(3\right)=-\frac{2}{3}\left(3\right)+5=3⇒\left(3,3\right)\hfill \\ \hfill x=6& \phantom{\rule{2em}{0ex}}f\left(6\right)=-\frac{2}{3}\left(6\right)+5=1⇒\left(6,1\right)\hfill \end{array}$ Plot the coordinate pairs and draw a line through the points. (Figure) represents the graph of the function$\,f\left(x\right)=-\frac{2}{3}x+5.$ [/hidden-answer] #### Analysis The graph of the function is a line as expected for a linear function. In addition, the graph has a downward slant, which indicates a negative slope. This is also expected from the negative, constant rate of change in the equation for the function. ### Try It Graph$\,f\left(x\right)=-\frac{3}{4}x+6\,$by plotting points. [reveal-answer q=”2406831″]Show Solution[/reveal-answer][hidden-answer a=”2406831″][/hidden-answer] #### Graphing a Function Using y-intercept and Slope Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its y-intercept, which is the point at which the input value is zero. To find the y-intercept, we can set$\,x=0\,$in the equation. The other characteristic of the linear function is its slope. Let’s consider the following function. $f\left(x\right)=\frac{1}{2}x+1$ The slope is$\,\frac{1}{2}.\,$Because the slope is positive, we know the graph will slant upward from left to right. The y-intercept is the point on the graph when$\,x=0.\,$The graph crosses the y-axis at$\,\left(0,1\right).\,$Now we know the slope and the y-intercept. We can begin graphing by plotting the point$\,\left(0,1\right).\,$We know that the slope is the change in the y-coordinate over the change in the x-coordinate. This is commonly referred to as rise over run,$\,m=\frac{\text{rise}}{\text{run}}.\,$From our example, we have$\,m=\frac{1}{2},$which means that the rise is 1 and the run is 2. So starting from our y-intercept$\,\left(0,1\right),$we can rise 1 and then run 2, or run 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in (Figure). ### Graphical Interpretation of a Linear Function In the equation$\,f\left(x\right)=mx+b$ • $b\,$is the y-intercept of the graph and indicates the point$\,\left(0,b\right)\,$at which the graph crosses the y-axis. • $m\,$is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for the slope: $m=\frac{\text{change in output (rise)}}{\text{change in input (run)}}=\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ Do all linear functions have y-intercepts? Yes. All linear functions cross the y-axis and therefore have y-intercepts. (Note: A vertical line is parallel to the y-axis does not have a y-intercept, but it is not a function.) ### How To Given the equation for a linear function, graph the function using the y-intercept and slope. 1. Evaluate the function at an input value of zero to find the y-intercept. 2. Identify the slope as the rate of change of the input value. 3. Plot the point represented by the y-intercept. 4. Use$\,\frac{\text{rise}}{\text{run}}\,$to determine at least two more points on the line. 5. Sketch the line that passes through the points. ### Graphing by Using the y-intercept and Slope Graph$\,f\left(x\right)=-\frac{2}{3}x+5\,$using the y-intercept and slope. [reveal-answer q=”fs-id2077051″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2077051″] Evaluate the function at$\,x=0\,$to find the y-intercept. The output value when$\,x=0\,$is 5, so the graph will cross the y-axis at$\,\left(0,5\right).$ According to the equation for the function, the slope of the line is$\,-\frac{2}{3}.\,$This tells us that for each vertical decrease in the “rise” of$\,–2\,$units, the “run” increases by 3 units in the horizontal direction. We can now graph the function by first plotting the y-intercept on the graph in (Figure). From the initial value$\,\left(0,5\right)\,$we move down 2 units and to the right 3 units. We can extend the line to the left and right by repeating, and then drawing a line through the points. [/hidden-answer] #### Analysis The graph slants downward from left to right, which means it has a negative slope as expected. ### Try It Find a point on the graph we drew in (Figure) that has a negative x-value. [reveal-answer q=”fs-id2084719″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2084719″] Possible answers include$\,\left(-3,7\right),\,$ $\left(-6,9\right),\,$or$\,\left(-9,11\right).$ [/hidden-answer] #### Graphing a Function Using Transformations Another option for graphing is to use a transformation of the identity function$\,f\left(x\right)=x.\,$A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression. ##### Vertical Stretch or Compression In the equation$\,f\left(x\right)=mx,$the$\,m\,$is acting as the vertical stretch or compression of the identity function. When$\,m\,$is negative, there is also a vertical reflection of the graph. Notice in (Figure) that multiplying the equation of$\,f\left(x\right)=x\,$by$\,m\,$stretches the graph of$\,f\,$by a factor of$\,m\,$units if$\,m>\text{1}\,$and compresses the graph of$\,f\,$by a factor of$\,m\,$units if$\,0 ##### Vertical Shift In[latex]\,f\left(x\right)=mx+b,$the$\,b\,$acts as the vertical shift, moving the graph up and down without affecting the slope of the line. Notice in (Figure) that adding a value of$\,b\,$to the equation of$\,f\left(x\right)=x\,$shifts the graph of$\,f\,$a total of$\,b\,$units up if$\,b\,$is positive and$\,|b|\,$units down if$\,b\,$is negative. Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method. ### How To Given the equation of a linear function, use transformations to graph the linear function in the form$\,f\left(x\right)=mx+b.$ 1. Graph$\,f\left(x\right)=x.$ 2. Vertically stretch or compress the graph by a factor$\,m.$ 3. Shift the graph up or down$\,b\,$units. ### Graphing by Using Transformations Graph$\,f\left(x\right)=\frac{1}{2}x-3\,$using transformations. [reveal-answer q=”fs-id2396043″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2396043″] The equation for the function shows that$\,m=\frac{1}{2}\,$so the identity function is vertically compressed by$\,\frac{1}{2}.\,$The equation for the function also shows that$\,b=-3\,$so the identity function is vertically shifted down 3 units. First, graph the identity function, and show the vertical compression as in (Figure). Then show the vertical shift as in (Figure). [/hidden-answer] ### Try It Graph$\,f\left(x\right)=4+2x\,$using transformations. [reveal-answer q=”2238414″]Show Solution[/reveal-answer][hidden-answer a=”2238414″][/hidden-answer] In (Figure), could we have sketched the graph by reversing the order of the transformations? No. The order of the transformations follows the order of operations. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. This is why we performed the compression first. For example, following the order: Let the input be 2. $\begin{array}{ccc}\hfill f\left(2\right)& =& \frac{1}{2}\left(2\right)-3\hfill \\ & =& 1-3\hfill \\ & =& -2\hfill \end{array}$ ### Writing the Equation for a Function from the Graph of a Line Earlier, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at (Figure). We can see right away that the graph crosses the y-axis at the point$\,\left(0,\text{4}\right)\,$so this is the y-intercept. Then we can calculate the slope by finding the rise and run. We can choose any two points, but let’s look at the point$\,\left(–2,0\right).\,$To get from this point to the y-intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be $m=\frac{\text{rise}}{\text{run}}=\frac{4}{2}=2$ Substituting the slope and y-intercept into the slope-intercept form of a line gives $y=2x+4$ ### How To Given a graph of linear function, find the equation to describe the function. 1. Identify the y-intercept of an equation. 2. Choose two points to determine the slope. 3. Substitute the y-intercept and slope into the slope-intercept form of a line. ### Matching Linear Functions to Their Graphs Match each equation of the linear functions with one of the lines in (Figure). $\begin{array}{cccc}a\text{.}\hfill & \hfill \text{ }f\left(x\right)& =& 2x+3\hfill \\ b\text{.}\hfill & \hfill g\left(x\right)& =& 2x-3\hfill \\ c\text{.}\hfill & \hfill h\left(x\right)& =& -2x+3\hfill \\ d\text{. }\hfill & \hfill j\left(x\right)& =& \frac{1}{2}x+3\hfill \end{array}$ [reveal-answer q=”fs-id2280188″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2280188″] Analyze the information for each function. 1. This function has a slope of 2 and a y-intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function$\,g\,$has the same slope, but a different y-intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through$\,\left(0,\text{3}\right)\,$so$\,f\,$must be represented by line I. 2. This function also has a slope of 2, but a y-intercept of$\,-3.\,$It must pass through the point$\,\left(0,-3\right)\,$and slant upward from left to right. It must be represented by line III. 3. This function has a slope of –2 and a y-intercept of 3. This is the only function listed with a negative slope, so it must be represented by line IV because it slants downward from left to right. 4. This function has a slope of$\,\frac{1}{2}\,$and a y-intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. Lines I and II pass through$\,\left(0,\text{3}\right),$but the slope of$\,j\,$is less than the slope of$\,f\,$so the line for$\,j\,$must be flatter. This function is represented by Line II. Now we can re-label the lines as in (Figure). [/hidden-answer] #### Finding the x-intercept of a Line So far we have been finding the y-intercepts of a function: the point at which the graph of the function crosses the y-axis. Recall that a function may also have an x-intercept, which is the x-coordinate of the point where the graph of the function crosses the x-axis. In other words, it is the input value when the output value is zero. To find the x-intercept, set a function$\,f\left(x\right)\,$equal to zero and solve for the value of$\,x.\,$For example, consider the function shown. $f\left(x\right)=3x-6$ Set the function equal to 0 and solve for$\,x.$ $\begin{array}{ccc}\hfill 0& =& 3x-6\hfill \\ \hfill 6& =& 3x\hfill \\ \hfill 2& =& x\hfill \\ \hfill x& =& 2\hfill \end{array}$ The graph of the function crosses the x-axis at the point$\,\left(2,\text{0}\right).$ Do all linear functions have x-intercepts? No. However, linear functions of the form$\,y=c,$where$\,c\,$is a nonzero real number are the only examples of linear functions with no x-intercept. For example,$\,y=5\,$is a horizontal line 5 units above the x-axis. This function has no x-intercepts, as shown in (Figure). ### x-intercept The x-intercept of the function is value of$\,x\,$when$\,f\left(x\right)=0.\,$It can be solved by the equation$\,0=mx+b.$ ### Finding an x-intercept Find the x-intercept of$\,f\left(x\right)=\frac{1}{2}x-3.$ [reveal-answer q=”fs-id2631972″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2631972″] Set the function equal to zero to solve for$\,x.$ $\begin{array}{ccc}\hfill 0& =& \frac{1}{2}x-3\hfill \\ \hfill 3& =& \frac{1}{2}x\hfill \\ \hfill 6& =& x\hfill \\ \hfill x& =& 6\hfill \end{array}$ The graph crosses the x-axis at the point$\,\left(6,\text{0}\right).$ [/hidden-answer] #### Analysis A graph of the function is shown in (Figure). We can see that the x-intercept is$\,\left(6,\text{0}\right)\,$as we expected. ### Try It Find the x-intercept of$\,f\left(x\right)=\frac{1}{4}x-4.$ [reveal-answer q=”fs-id2203514″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2203514″] $\,\left(16,\text{ 0}\right)$ [/hidden-answer] #### Describing Horizontal and Vertical Lines There are two special cases of lines on a graph—horizontal and vertical lines. A horizontal line indicates a constant output, or y-value. In (Figure), we see that the output has a value of 2 for every input value. The change in outputs between any two points, therefore, is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use$\,m=0\,$in the equation$\,f\left(x\right)=mx+b,$the equation simplifies to$\,f\left(x\right)=b.\,$In other words, the value of the function is a constant. This graph represents the function$\,f\left(x\right)=2.$ A vertical line indicates a constant input, or x-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined. A vertical line, such as the one in (Figure), has an x-intercept, but no y-intercept unless it’s the line$\,x=0.\,$This graph represents the line$\,x=2.$ ### Horizontal and Vertical Lines Lines can be horizontal or vertical. A horizontal line is a line defined by an equation in the form$\,f\left(x\right)=b.$ A vertical line is a line defined by an equation in the form$\,x=a.$ ### Writing the Equation of a Horizontal Line Write the equation of the line graphed in (Figure). [reveal-answer q=”fs-id2394901″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2394901″] For any x-value, the y-value is$\,-4,\,$so the equation is$\,y=-4.$ [/hidden-answer] ### Writing the Equation of a Vertical Line Write the equation of the line graphed in (Figure). [reveal-answer q=”fs-id2382624″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2382624″] The constant x-value is$\,7,$so the equation is$\,x=7.$ ### Determining Whether Lines are Parallel or Perpendicular The two lines in (Figure) are parallel lines: they will never intersect. They have exactly the same steepness, which means their slopes are identical. The only difference between the two lines is the y-intercept. If we shifted one line vertically toward the other, they would become coincident. We can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the same and the y-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel. $\begin{array}{cc}\begin{array}{l}f\left(x\right)=-2x+6\hfill \\ f\left(x\right)=-2x-4\hfill \end{array}\bigg\}\text{ parallel}& \begin{array}{l}f\left(x\right)=3x+2\hfill \\ f\left(x\right)=2x+2\hfill \end{array}\bigg\}\text{ not parallel}\end{array}$ Unlike parallel lines, perpendicular lines do intersect. Their intersection forms a right, or 90-degree, angle. The two lines in (Figure) are perpendicular. Perpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a specific way. The slope of one line is the negative reciprocal of the slope of the other line. The product of a number and its reciprocal is$\,1.\,$So, if$\,{m}_{1}\text{ and }{m}_{2}\,$are negative reciprocals of one another, they can be multiplied together to yield$\,–1.$ ${m}_{1}{m}_{2}=-1$ To find the reciprocal of a number, divide 1 by the number. So the reciprocal of 8 is$\,\frac{1}{8},\,$and the reciprocal of$\,\frac{1}{8}\,$is 8. To find the negative reciprocal, first find the reciprocal and then change the sign. As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor vertical. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular. $\begin{array}{cccc}\hfill f\left(x\right)& =& \frac{1}{4}x+2\hfill & \phantom{\rule{2em}{0ex}}\text{negative reciprocal of}\,\frac{1}{4}\text{ is }-4\hfill \\ \hfill f\left(x\right)& =& -4x+3\hfill & \phantom{\rule{2em}{0ex}}\text{negative reciprocal of}\,-4\text{ is }\frac{1}{4}\hfill \end{array}$ The product of the slopes is –1. $-4\left(\frac{1}{4}\right)=-1$ ### Parallel and Perpendicular Lines Two lines are parallel lines if they do not intersect. The slopes of the lines are the same. $f\left(x\right)={m}_{1}x+{b}_{1}\,\text{and}\,g\left(x\right)={m}_{2}x+{b}_{2}\,\text{are parallel if and only if }{m}_{1}={m}_{2}$ If and only if$\,{b}_{1}={b}_{2}\,$and$\,{m}_{1}={m}_{2},\,$we say the lines coincide. Coincident lines are the same line. Two lines are perpendicular lines if they intersect to form a right angle. $f\left(x\right)={m}_{1}x+{b}_{1}\,\text{and}\,g\left(x\right)={m}_{2}x+{b}_{2}\,\text{are perpendicular if and only if}$ ${m}_{1}{m}_{2}=-1,\text{so}\,{m}_{2}=-\frac{1}{{m}_{1}}$ ### Identifying Parallel and Perpendicular Lines Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines. $\begin{array}{cccccc}\hfill f\left(x\right)& =& 2x+3\hfill & \hfill \phantom{\rule{2em}{0ex}}h\left(x\right)& =& -2x+2\hfill \\ \hfill g\left(x\right)& =& \frac{1}{2}x-4\hfill & \hfill \phantom{\rule{2em}{0ex}}j\left(x\right)& =& 2x-6\hfill \end{array}$ [reveal-answer q=”839246″]Show Solution[/reveal-answer] [hidden-answer a=”839246″]Parallel lines have the same slope. Because the functions$\,f\left(x\right)=2x+3\,$and$\,j\left(x\right)=2x-6\,$each have a slope of 2, they represent parallel lines. Perpendicular lines have negative reciprocal slopes. Because −2 and$\,\frac{1}{2}\,$are negative reciprocals, the functions$\,g\left(x\right)=\frac{1}{2}x-4\,$and$\,h\left(x\right)=-2x+2\,$represent perpendicular lines.[/hidden-answer] #### Analysis A graph of the lines is shown in (Figure). The graph shows that the lines$\,f\left(x\right)=2x+3\,$and$\,j\left(x\right)=2x–6\,$are parallel, and the lines$\,g\left(x\right)=\frac{1}{2}x–4\,$and$\,h\left(x\right)=-2x+2\,$are perpendicular. ### Writing the Equation of a Line Parallel or Perpendicular to a Given Line If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line. #### Writing Equations of Parallel Lines Suppose for example, we are given the equation shown. $f\left(x\right)=3x+1$ We know that the slope of the line formed by the function is 3. We also know that the y-intercept is$\,\left(0,1\right).\,$Any other line with a slope of 3 will be parallel to$\,f\left(x\right).\,$So the lines formed by all of the following functions will be parallel to$\,f\left(x\right).$ $\begin{array}{ccc}\hfill g\left(x\right)& =& 3x+6\hfill \\ \hfill h\left(x\right)& =& 3x+1\hfill \\ \hfill p\left(x\right)& =& 3x+\frac{2}{3}\hfill \end{array}$ Suppose then we want to write the equation of a line that is parallel to$\,f\,$and passes through the point$\,\left(1,\text{7}\right).\,$This type of problem is often described as a point-slope problem because we have a point and a slope. In our example, we know that the slope is 3. We need to determine which value of$\,b\,$will give the correct line. We can begin with the point-slope form of an equation for a line, and then rewrite it in the slope-intercept form. $\begin{array}{ccc}\hfill y-{y}_{1}& =& m\left(x-{x}_{1}\right)\hfill \\ \hfill y-7& =& 3\left(x-1\right)\hfill \\ \hfill y-7& =& 3x-3\hfill \\ \hfill y& =& 3x+4\hfill \end{array}$ So$\,g\left(x\right)=3x+4\,$is parallel to$\,f\left(x\right)=3x+1\,$and passes through the point$\,\left(1,\text{7}\right).$ ### How To Given the equation of a function and a point through which its graph passes, write the equation of a line parallel to the given line that passes through the given point. 1. Find the slope of the function. 2. Substitute the given values into either the general point-slope equation or the slope-intercept equation for a line. 3. Simplify. ### Finding a Line Parallel to a Given Line Find a line parallel to the graph of$\,f\left(x\right)=3x+6\,$that passes through the point$\,\left(3,\text{0}\right).$ [reveal-answer q=”fs-id2499832″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2499832″]The slope of the given line is 3. If we choose the slope-intercept form, we can substitute$\,m=3,x=3,$and$\,f\left(x\right)=0\,$into the slope-intercept form to find the y-intercept. $\begin{array}{ccc}\hfill g\left(x\right)& =& 3x+b\hfill \\ \hfill 0& =& 3\left(3\right)+b\hfill \\ \hfill b& =& –9\hfill \end{array}$ The line parallel to$\,f\left(x\right)\,$that passes through$\,\left(3,\text{0}\right)\,$is$\,g\left(x\right)=3x-9.$ [/hidden-answer] #### Analysis We can confirm that the two lines are parallel by graphing them. (Figure) shows that the two lines will never intersect. #### Writing Equations of Perpendicular Lines We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the function shown. $f\left(x\right)=2x+4$ The slope of the line is 2, and its negative reciprocal is$\,-\frac{1}{2}.\,$Any function with a slope of$\,-\frac{1}{2}\,$will be perpendicular to$\,f\left(x\right).\,$So the lines formed by all of the following functions will be perpendicular to$\,f\left(x\right).$ $\begin{array}{ccc}\hfill g\left(x\right)& =& -\frac{1}{2}x+4\hfill \\ \hfill h\left(x\right)& =& -\frac{1}{2}x+2\hfill \\ \hfill p\left(x\right)& =& -\frac{1}{2}x-\frac{1}{2}\hfill \end{array}$ As before, we can narrow down our choices for a particular perpendicular line if we know that it passes through a given point. Suppose then we want to write the equation of a line that is perpendicular to$\,f\left(x\right)\,$and passes through the point$\,\left(4,\text{0}\right).\,$We already know that the slope is$\,-\frac{1}{2}.\,$Now we can use the point to find the y-intercept by substituting the given values into the slope-intercept form of a line and solving for$\,b.$ $\begin{array}{ccc}\hfill g\left(x\right)& =& mx+b\hfill \\ \hfill 0& =& -\frac{1}{2}\left(4\right)+b\hfill \\ \hfill 0& =& -2+b\hfill \\ \hfill 2& =& b\hfill \\ \hfill b& =& 2\hfill \end{array}$ The equation for the function with a slope of$\,-\frac{1}{2}\,$and a y-intercept of 2 is $g\left(x\right)=-\frac{1}{2}x+2$ So$\,g\left(x\right)=-\frac{1}{2}x+2\,$is perpendicular to$\,f\left(x\right)=2x+4\,$and passes through the point$\,\left(4,\text{0}\right).\,$Be aware that perpendicular lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature. A horizontal line has a slope of zero and a vertical line has an undefined slope. These two lines are perpendicular, but the product of their slopes is not –1. Doesn’t this fact contradict the definition of perpendicular lines? No. For two perpendicular linear functions, the product of their slopes is –1. However, a vertical line is not a function so the definition is not contradicted. ### How To Given the equation of a function and a point through which its graph passes, write the equation of a line perpendicular to the given line. 1. Find the slope of the function. 2. Determine the negative reciprocal of the slope. 3. Substitute the new slope and the values for$\,x\,$and$\,y\,$from the coordinate pair provided into$\,g\left(x\right)=mx+b.$ 4. Solve for$\,b.$ 5. Write the equation of the line. ### Finding the Equation of a Perpendicular Line Find the equation of a line perpendicular to$\,f\left(x\right)=3x+3\,$that passes through the point$\,\left(3,\text{0}\right).$ [reveal-answer q=”fs-id2238491″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2238491″] The original line has slope$\,m=3,\,$so the slope of the perpendicular line will be its negative reciprocal, or$\,-\frac{1}{3}.\,$Using this slope and the given point, we can find the equation of the line. $\begin{array}{ccc}\hfill g\left(x\right)& =& –\frac{1}{3}x+b\hfill \\ \hfill 0& =& –\frac{1}{3}\left(3\right)+b\hfill \\ \hfill 1& =& b\hfill \\ \hfill b& =& 1\hfill \end{array}$ The line perpendicular to$\,f\left(x\right)\,$that passes through$\,\left(3,\text{0}\right)\,$is$\,g\left(x\right)=-\frac{1}{3}x+1.$ [/hidden-answer] #### Analysis A graph of the two lines is shown in (Figure). Note that that if we graph perpendicular lines on a graphing calculator using standard zoom, the lines may not appear to be perpendicular. Adjusting the window will make it possible to zoom in further to see the intersection more closely. ### Try It Given the function$\,h\left(x\right)=2x-4,$write an equation for the line passing through$\,\left(0,0\right)\,$that is 1. parallel to$\,h\left(x\right)$ 2. perpendicular to$\,h\left(x\right)$ [reveal-answer q=”fs-id2363423″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2363423″] a.$\,f\left(x\right)=2x;$b.$\,g\left(x\right)=-\frac{1}{2}x$ [/hidden-answer] ### How To Given two points on a line and a third point, write the equation of the perpendicular line that passes through the point. 1. Determine the slope of the line passing through the points. 2. Find the negative reciprocal of the slope. 3. Use the slope-intercept form or point-slope form to write the equation by substituting the known values. 4. Simplify. ### Finding the Equation of a Line Perpendicular to a Given Line Passing through a Point A line passes through the points$\,\left(-2,\text{6}\right)\,$and$\,\left(4,5\right).\,$Find the equation of a perpendicular line that passes through the point$\,\left(4,5\right).$ [reveal-answer q=”fs-id2632112″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2632112″] From the two points of the given line, we can calculate the slope of that line. $\begin{array}{ccc}\hfill {m}_{1}& =& \frac{5-6}{4-\left(-2\right)}\hfill \\ & =& \frac{-1}{6}\hfill \\ & =& -\frac{1}{6}\hfill \end{array}$ Find the negative reciprocal of the slope. $\begin{array}{ccc}\hfill {m}_{2}& =& \frac{-1}{-\frac{1}{6}}\hfill \\ & =& -1\left(-\frac{6}{1}\right)\hfill \\ & =& 6\hfill \end{array}$ We can then solve for the y-intercept of the line passing through the point$\,\left(4,5\right).$ $\begin{array}{ccc}\hfill g\left(x\right)& =& 6x+b\hfill \\ \hfill 5& =& 6\left(4\right)+b\hfill \\ \hfill 5& =& 24+b\hfill \\ \hfill -19& =& b\hfill \\ \hfill b& =& -19\hfill \end{array}$ The equation for the line that is perpendicular to the line passing through the two given points and also passes through point$\,\left(4,5\right)\,$is $y=6x-19$[/hidden-answer] ### Try It A line passes through the points,$\,\left(-2,\text{−15}\right)\,$and$\,\left(2,-3\right).\,$Find the equation of a perpendicular line that passes through the point,$\,\left(6,4\right).$ [reveal-answer q=”fs-id1798920″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1798920″] $\,y=–\frac{1}{3}x+6$ [/hidden-answer] Access this online resource for additional instruction and practice with linear functions. ### Key Concepts • Linear functions can be represented in words, function notation, tabular form, and graphical form. See (Figure). • An increasing linear function results in a graph that slants upward from left to right and has a positive slope. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line. See (Figure). • Slope is a rate of change. The slope of a linear function can be calculated by dividing the difference between y-values by the difference in corresponding x-values of any two points on the line. See (Figure) and (Figure). • An equation for a linear function can be written from a graph. See (Figure). • The equation for a linear function can be written if the slope$\,m\,$and initial value$\,b\,$are known. See (Figure) and (Figure). • A linear function can be used to solve real-world problems given information in different forms. See (Figure), (Figure), and (Figure). • Linear functions can be graphed by plotting points or by using the y-intercept and slope. See (Figure) and (Figure). • Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See (Figure). • The equation for a linear function can be written by interpreting the graph. See (Figure). • The x-intercept is the point at which the graph of a linear function crosses the x-axis. See (Figure). • Horizontal lines are written in the form,$\,f\left(x\right)=b.\,$See (Figure). • Vertical lines are written in the form,$\,x=b.\,$See (Figure). • Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See (Figure). • A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the x– and y-values of the given point into the equation,$\,f\left(x\right)=mx+b,\,$and using the$\,b\,$that results. Similarly, the point-slope form of an equation can also be used. See (Figure). • A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope. See (Figure) and (Figure). ### Section Exercises #### Verbal Terry is skiing down a steep hill. Terry’s elevation,$\,E\left(t\right),$in feet after$\,t\,$seconds is given by$\,E\left(t\right)=3000-70t.\,$Write a complete sentence describing Terry’s starting elevation and how it is changing over time. [reveal-answer q=”fs-id2216168″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2216168″] Terry starts at an elevation of 3000 feet and descends 70 feet per second. [/hidden-answer] Jessica is walking home from a friend’s house. After 2 minutes she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 miles from home. What is her rate in miles per hour? A boat is 100 miles away from the marina, sailing directly toward it at 10 miles per hour. Write an equation for the distance of the boat from the marina after t hours. [reveal-answer q=”fs-id2216193″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2216193″] $d\left(t\right)=100-10t$ [/hidden-answer] If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the y-intercepts. If a horizontal line has the equation$\,f\left(x\right)=a\,$and a vertical line has the equation$\,x=a,\,$what is the point of intersection? Explain why what you found is the point of intersection. [reveal-answer q=”fs-id2270466″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2270466″] The point of intersection is$\,\left(a,\text{ }a\right).\,$This is because for the horizontal line, all of the$\,y\,$coordinates are$\,a\,$and for the vertical line, all of the$\,x\,$coordinates are$\,a.\,$The point of intersection is on both lines and therefore will have these two characteristics. [/hidden-answer] #### Algebraic For the following exercises, determine whether the equation of the curve can be written as a linear function. $y=\frac{1}{4}x+6$ $y=3x-5$ [reveal-answer q=”fs-id1798718″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1798718″] Yes [/hidden-answer] $y=3{x}^{2}-2$ $3x+5y=15$ [reveal-answer q=”fs-id1798782″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1798782″] Yes [/hidden-answer] $3{x}^{2}+5y=15$ $3x+5{y}^{2}=15$ [reveal-answer q=”fs-id2295754″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2295754″] No [/hidden-answer] $-2{x}^{2}+3{y}^{2}=6$ $-\frac{x-3}{5}=2y$ [reveal-answer q=”fs-id2295838″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2295838″] Yes [/hidden-answer] For the following exercises, determine whether each function is increasing or decreasing. $f\left(x\right)=4x+3$ $g\left(x\right)=5x+6$ [reveal-answer q=”fs-id1855518″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1855518″] Increasing [/hidden-answer] $a\left(x\right)=5-2x$ $b\left(x\right)=8-3x$ [reveal-answer q=”fs-id1855588″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1855588″] Decreasing [/hidden-answer] $h\left(x\right)=-2x+4$ $k\left(x\right)=-4x+1$ [reveal-answer q=”fs-id2239403″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2239403″] Decreasing [/hidden-answer] $j\left(x\right)=\frac{1}{2}x-3$ $p\left(x\right)=\frac{1}{4}x-5$ [reveal-answer q=”fs-id2239489″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2239489″] Increasing [/hidden-answer] $n\left(x\right)=-\frac{1}{3}x-2$ $m\left(x\right)=-\frac{3}{8}x+3$ [reveal-answer q=”fs-id2083770″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2083770″] Decreasing [/hidden-answer] For the following exercises, find the slope of the line that passes through the two given points. $\left(2,4\right)\,$and$\,\left(4,\text{10}\right)$ $\left(1,\text{5}\right)\,$and$\,\left(4,\text{11}\right)$ [reveal-answer q=”fs-id2597468″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2597468″] 2 [/hidden-answer] $\left(–1,\text{4}\right)\,$and$\,\left(5,\text{2}\right)$ $\left(8,–2\right)\,$and$\,\left(4,6\right)$ [reveal-answer q=”fs-id2597592″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2597592″] –2 [/hidden-answer] $\left(6,11\right)\,$and$\,\left(–4,\text{3}\right)$ For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. $f\left(-5\right)=-4,\,$and$\,f\left(5\right)=2$ [reveal-answer q=”fs-id2498518″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2498518″] $y=\frac{3}{5}x-1$ [/hidden-answer] $f\left(-1\right)=4,\,$and$\,f\left(5\right)=1$ Passes through$\,\left(2,4\right)\,$and$\,\left(4,10\right)$ [reveal-answer q=”fs-id2653227″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2653227″] $y=3x-2$ [/hidden-answer] Passes through$\,\left(1,5\right)\,$and$\,\left(4,11\right)$ Passes through$\,\left(-1,\text{4}\right)\,$and$\,\left(5,\text{2}\right)$ [reveal-answer q=”fs-id2282286″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2282286″] $y=-\frac{1}{3}x+\frac{11}{3}$ [/hidden-answer] Passes through$\,\left(-2,\text{8}\right)\,$and$\,\left(4,\text{6}\right)$ x intercept at$\,\left(-2,\text{0}\right)\,$and y intercept at$\,\left(0,-3\right)$ [reveal-answer q=”fs-id2604294″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2604294″] $y=-1.5x-3$ [/hidden-answer] x intercept at$\,\left(-5,\text{0}\right)\,$and y intercept at$\,\left(0,\text{4}\right)$ For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither. $\begin{array}{l}4x-7y=10\hfill \\ 7x+4y=1\hfill \end{array}$ [reveal-answer q=”fs-id2252420″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2252420″] perpendicular [/hidden-answer] $\begin{array}{c}3y+x=12\\ -y=8x+1\end{array}$ $\begin{array}{c}3y+4x=12\\ -6y=8x+1\end{array}$ [reveal-answer q=”fs-id2261926″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2261926″] parallel [/hidden-answer] $\begin{array}{l}6x-9y=10\hfill \\ 3x+2y=1\hfill \end{array}$ For the following exercises, find the x– and y-intercepts of each equation. $f\left(x\right)=-x+2$ [reveal-answer q=”fs-id2445722″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2445722″] $\begin{array}{l}f\left(0\right)=-\left(0\right)+2\\ f\left(0\right)=2\\ y-\mathrm{int}:\left(0,2\right)\\ 0=-x+2\\ x-\mathrm{int}:\left(2,0\right)\end{array}$ [/hidden-answer] $g\left(x\right)=2x+4$ $h\left(x\right)=3x-5$ [reveal-answer q=”fs-id2250144″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2250144″] $\begin{array}{l}h\left(0\right)=3\left(0\right)-5\\ h\left(0\right)=-5\\ y-\mathrm{int}:\left(0,-5\right)\\ 0=3x-5\\ x-\mathrm{int}:\left(\frac{5}{3},0\right)\end{array}$ [/hidden-answer] $k\left(x\right)=-5x+1$ $-2x+5y=20$ [reveal-answer q=”fs-id2479739″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2479739″] $\begin{array}{l}-2x+5y=20\\ -2\left(0\right)+5y=20\\ 5y=20\\ y=4\\ y-\mathrm{int}:\left(0,4\right)\\ -2x+5\left(0\right)=20\\ x=-10\\ x-\mathrm{int}:\left(-10,0\right)\end{array}$ [/hidden-answer] $7x+2y=56$ For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither? Line 1: Passes through$\,\left(0,6\right)\,$and$\,\left(3,-24\right)$ Line 2: Passes through$\,\left(-1,19\right)\,$and$\,\left(8,-71\right)$ [reveal-answer q=”fs-id2239328″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2239328″] Line 1: m = –10 Line 2: m = –10 Parallel [/hidden-answer] Line 1: Passes through$\,\left(-8,-55\right)\,$and$\,\left(10,89\right)$ Line 2: Passes through$\,\left(9,-44\right)\,$and$\,\left(4,-14\right)$ Line 1: Passes through$\,\left(2,3\right)\,$and$\,\left(4,-1\right)$ Line 2: Passes through$\,\left(6,3\right)\,$and$\,\left(8,5\right)$ [reveal-answer q=”fs-id1797548″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1797548″] Line 1: m = –2 Line 2: m = 1 Neither [/hidden-answer] Line 1: Passes through$\,\left(1,7\right)\,$and$\,\left(5,5\right)$ Line 2: Passes through$\,\left(-1,-3\right)\,$and$\,\left(1,1\right)$ Line 1: Passes through$\,\left(2,5\right)\,$and$\,\left(5,-1\right)$ Line 2: Passes through$\,\left(-3,7\right)\,$and$\,\left(3,-5\right)$ [reveal-answer q=”fs-id2424452″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2424452″] $\text{Line 1}: m=–2 \text{Line 2}: m=–2 \text{Parallel}$ [/hidden-answer] For the following exercises, write an equation for the line described. Write an equation for a line parallel to$\,f\left(x\right)=-5x-3\,$and passing through the point$\,\left(2,\text{–}12\right).$ Write an equation for a line parallel to$\,g\left(x\right)=3x-1\,$and passing through the point$\,\left(4,9\right).$ [reveal-answer q=”fs-id2407048″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2407048″] $y=3x-3$ [/hidden-answer] Write an equation for a line perpendicular to$\,h\left(t\right)=-2t+4\,$and passing through the point$\,\left(-4,–1\right).$ Write an equation for a line perpendicular to$\,p\left(t\right)=3t+4\,$and passing through the point$\,\left(3,1\right).$ [reveal-answer q=”fs-id2261296″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2261296″] $y=-\frac{1}{3}t+2$ [/hidden-answer] #### Graphical For the following exercises, find the slope of the line graphed. [reveal-answer q=”fs-id2261379″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2261379″] 0 [/hidden-answer] For the following exercises, write an equation for the line graphed. [reveal-answer q=”fs-id2261423″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2261423″] $y=-\frac{5}{4}x+5$ [/hidden-answer] [reveal-answer q=”fs-id2605056″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2605056″] $y=3x-1$ [/hidden-answer] [reveal-answer q=”fs-id2202054″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2202054″] $y=-2.5$ [/hidden-answer] For the following exercises, match the given linear equation with its graph in (Figure). $f\left(x\right)=-x-1$ $f\left(x\right)=-2x-1$ [reveal-answer q=”fs-id2605232″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2605232″] F [/hidden-answer] $f\left(x\right)=-\frac{1}{2}x-1$ $f\left(x\right)=2$ [reveal-answer q=”fs-id2354833″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2354833″] C [/hidden-answer] $f\left(x\right)=2+x$ $f\left(x\right)=3x+2$ [reveal-answer q=”fs-id2354901″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2354901″] A [/hidden-answer] For the following exercises, sketch a line with the given features. An x-intercept of$\,\left(–4,\text{0}\right)\,$and y-intercept of$\,\left(0,\text{–2}\right)$ An x-intercept$\,\left(–2,\text{0}\right)\,$and y-intercept of$\,\left(0,\text{4}\right)$ [reveal-answer q=”2393328″]Show Solution[/reveal-answer][hidden-answer a=”2393328″][/hidden-answer] A y-intercept of$\,\left(0,\text{7}\right)\,$and slope$\,-\frac{3}{2}$ A y-intercept of$\,\left(0,\text{3}\right)\,$and slope$\,\frac{2}{5}$ [reveal-answer q=”2393459″]Show Solution[/reveal-answer][hidden-answer a=”2393459″][/hidden-answer] Passing through the points$\,\left(–6,\text{–2}\right)\,$and$\,\left(6,\text{–6}\right)$ Passing through the points$\,\left(–3,\text{–4}\right)\,$and$\,\left(3,\text{0}\right)$ [reveal-answer q=”2239680″]Show Solution[/reveal-answer][hidden-answer a=”2239680″][/hidden-answer] For the following exercises, sketch the graph of each equation. $f\left(x\right)=-2x-1$ $f\left(x\right)=-3x+2$ [reveal-answer q=”2239776″]Show Solution[/reveal-answer][hidden-answer a=”2239776″][/hidden-answer] $f\left(x\right)=\frac{1}{3}x+2$ $f\left(x\right)=\frac{2}{3}x-3$ [reveal-answer q=”2447371″]Show Solution[/reveal-answer][hidden-answer a=”2447371″][/hidden-answer] $f\left(t\right)=3+2t$ $p\left(t\right)=-2+3t$ [reveal-answer q=”2447460″]Show Solution[/reveal-answer][hidden-answer a=”2447460″][/hidden-answer] $x=3$ $x=-2$ [reveal-answer q=”2447519″]Show Solution[/reveal-answer][hidden-answer a=”2447519″][/hidden-answer] $r\left(x\right)=4$ For the following exercises, write the equation of the line shown in the graph. [reveal-answer q=”fs-id1979133″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1979133″] $y=\text{3}$ [/hidden-answer] [reveal-answer q=”fs-id2084650″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2084650″] $x=-3$ [/hidden-answer] #### Numeric For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data. $x$ 0 5 10 15 $g\left(x\right)$ 5 –10 –25 –40 [reveal-answer q=”fs-id2214586″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2214586″] Linear,$\,g\left(x\right)=-3x+5$ [/hidden-answer] $x$ 0 5 10 15 $h\left(x\right)$ 5 30 105 230 $x$ 0 5 10 15 $f\left(x\right)$ –5 20 45 70 [reveal-answer q=”fs-id1799056″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1799056″] Linear,$\,f\left(x\right)=5x-5$ [/hidden-answer] $x$ 5 10 20 25 $k\left(x\right)$ 13 28 58 73 $x$ 0 2 4 6 $g\left(x\right)$ 6 –19 –44 –69 [reveal-answer q=”fs-id1828786″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1828786″] Linear,$\,g\left(x\right)=-\frac{25}{2}x+6$ [/hidden-answer] $x$ 2 4 8 10 $h\left(x\right)$ 13 23 43 53 $x$ 2 4 6 8 $f\left(x\right)$ –4 16 36 56 [reveal-answer q=”fs-id2654439″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2654439″] Linear,$\,f\left(x\right)=10x-24$[/hidden-answer] $x$ 0 2 6 8 $k\left(x\right)$ 6 31 106 231 #### Technology For the following exercises, use a calculator or graphing technology to complete the task. If$\,f\,$is a linear function,$\,f\left(0.1\right)=11.5, \text{and} f\left(0.4\right)=–5.9,\,$find an equation for the function. [reveal-answer q=”fs-id2353198″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2353198″] $f\left(x\right)=-58x+17.3$ [/hidden-answer] Graph the function$\,f\,$on a domain of$\,\left[–10,10\right]:f\left(x\right)=0.02x-0.01.\,$Enter the function in a graphing utility. For the viewing window, set the minimum value of$\,x\,$to be$\,-10\,$and the maximum value of$\,x\,$to be$\,10.$ Graph the function$\,f\,$on a domain of$\,\left[–10,10\right]:fx)=2,500x+4,000$ [reveal-answer q=”2241329″]Show Solution[/reveal-answer][hidden-answer a=”2241329″][/hidden-answer] (Figure) shows the input,$\,w,$and output,$\,k,$for a linear function$\,k.\,$a. Fill in the missing values of the table. b. Write the linear function$\,k,$round to 3 decimal places. w –10 5.5 67.5 b k 30 –26 a –44 [reveal-answer q=”fs-id2241524″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2241524″] $y=3.613x-6.129\text{ }$ [/hidden-answer] (Figure) shows the input,$\,p,$and output,$\,q,$for a linear function$\,q.\,$a. Fill in the missing values of the table. b. Write the linear function$\,k.$ p 0.5 0.8 12 b q 400 700 a 1,000,000 Graph the linear function$\,f\,$on a domain of$\,\left[-10,10\right]\,$for the function whose slope is$\,\frac{1}{8}\,$and y-intercept is$\,\frac{31}{16}.\,$Label the points for the input values of$\,-10\,$and$\,10.$ [reveal-answer q=”2511952″]Show Solution[/reveal-answer][hidden-answer a=”2511952″][/hidden-answer] Graph the linear function$\,f\,$on a domain of$\,\left[-0.1,0.1\right]\,$for the function whose slope is 75 and y-intercept is$\,-22.5.\,$Label the points for the input values of$\,-0.1\,$and$\,0.1.$ Graph the linear function$\,f\,$where$\,f\left(x\right)=ax+b\,$on the same set of axes on a domain of$\,\left[-4,4\right]\,$for the following values of$\,a\,$and$\,b.$ 1. $a=2;b=3$ 2. $a=2;b=4$ 3. $a=2;b=–4$ 4. $a=2;b=–5$ [reveal-answer q=”2266906″]Show Solution[/reveal-answer][hidden-answer a=”2266906″][/hidden-answer] #### Extensions Find the value of$\,x\,$if a linear function goes through the following points and has the following slope:$\,\left(x,2\right),\left(-4,6\right),\,m=3$ Find the value of y if a linear function goes through the following points and has the following slope:$\,\left(10,y\right),\left(25,100\right),\,m=-5$ [reveal-answer q=”fs-id2404800″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2404800″] y = 175 [/hidden-answer] Find the equation of the line that passes through the following points: $\,\left(a,\text{ }b\right)\,$and$\,\left(a,\text{ }b+1\right)$ Find the equation of the line that passes through the following points: $\left(2a,b\right)\,$and$\,\left(a,b+1\right)$ [reveal-answer q=”fs-id2673199″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2673199″] $y=-\frac{1}{2}x+b+2$ [/hidden-answer] Find the equation of the line that passes through the following points: $\left(a,0\right)$and$\,\left(c,d\right)$ Find the equation of the line parallel to the line$\,g\left(x\right)=-0.\text{01}x\text{+2}\text{.01}\,$through the point$\,\left(1,\text{2}\right).$ [reveal-answer q=”fs-id2762709″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2762709″] y = –0.01x + 2.01 [/hidden-answer] Find the equation of the line perpendicular to the line$\,g\left(x\right)=-0.\text{01}x\text{+2}\text{.01}\,$through the point$\,\left(1,\text{2}\right).$ For the following exercises, use the functions$\,f\left(x\right)=-0.\text{1}x\text{+200 and }g\left(x\right)=20x+0.1.$ Find the point of intersection of the lines$\,f\,$and$\,g.$ [reveal-answer q=”fs-id2393588″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2393588″] $\text{ }\left(\frac{1999}{201},\frac{400,001}{2010}\right)$ [/hidden-answer] Where is$\,f\left(x\right)\,$greater than$\,g\left(x\right)?\,$Where is$\,g\left(x\right)\,$greater than$\,f\left(x\right)?$ #### Real-World Applications At noon, a barista notices that she has$20 in her tip jar. If she makes an average of $0.50 from each customer, how much will she have in her tip jar if she serves$\,n\,$more customers during her shift? [reveal-answer q=”fs-id2393782″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id2393782″] $20+0.5n$ [/hidden-answer] A gym membership with two personal training sessions costs$125, while gym membership with five personal training sessions costs $260. What is cost per session? A clothing business finds there is a linear relationship between the number of shirts,$\,n,$it can sell and the price,$\,p,$it can charge per shirt. In particular, historical data shows that 1,000 shirts can be sold at a price of$\,30,$while 3,000 shirts can be sold at a price of$22. Find a linear equation in the form$\,p\left(n\right)=mn+b\,$that gives the price$\,p\,$they can charge for$\,n\,$shirts.
$p\left(n\right)=-0.004n+34$
A phone company charges for service according to the formula:$\,C\left(n\right)=24+0.1n,$where$\,n\,$is the number of minutes talked, and$\,C\left(n\right)\,$is the monthly charge, in dollars. Find and interpret the rate of change and initial value.
A farmer finds there is a linear relationship between the number of bean stalks,$\,n,$she plants and the yield,$\,y,$each plant produces. When she plants 30 stalks, each plant yields 30 oz of beans. When she plants 34 stalks, each plant produces 28 oz of beans. Find a linear relationships in the form$\,y=mn+b\,$that gives the yield when$\,n\,$stalks are planted.
$y=-0.5n+45$
A city’s population in the year 1960 was 287,500. In 1989 the population was 275,900. Compute the rate of growth of the population and make a statement about the population rate of change in people per year.
A town’s population has been growing linearly. In 2003, the population was 45,000, and the population has been growing by 1,700 people each year. Write an equation,$\,P\left(t\right),$for the population$\,t\,$years after 2003.
$P\left(t\right)=1700t+45,000$
Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function:$\,I\left(x\right)=1054x+23,286,$where$\,x\,$is the number of years after 1990. Which of the following interprets the slope in the context of the problem?
1. As of 1990, average annual income was $23,286. 2. In the ten-year period from 1990–1999, average annual income increased by a total of$1,054.
3. Each year in the decade of the 1990s, average annual income increased by $1,054. 4. Average annual income rose to a level of$23,286 by the end of 1999.
When temperature is 0 degrees Celsius, the Fahrenheit temperature is 32. When the Celsius temperature is 100, the corresponding Fahrenheit temperature is 212. Express the Fahrenheit temperature as a linear function of$\,C,$the Celsius temperature,$\,F\left(C\right).$
1. Find the rate of change of Fahrenheit temperature for each unit change temperature of Celsius.
2. Find and interpret$\,F\left(28\right).$
3. Find and interpret$\,F\left(–40\right).$
1. $\text{Rate of change =}\frac{\Delta F}{\Delta C}=\frac{212-32}{100-0}=1.8\text{ degrees F for one degree change in C}$
2. $F\left(28\right)=1.8\left(28\right)+32=82.4\text{ degrees F is 28 degrees C}$
3. $F\left(-40\right)=1.8\left(-40\right)+32=-40\text{ degrees F is -40 degrees C}$
### Glossary
decreasing linear function
a function with a negative slope: If$\,f\left(x\right)=mx+b, \text{then} m<0.$
horizontal line
a line defined by$\,f\left(x\right)=b,$where$\,b\,$is a real number. The slope of a horizontal line is 0.
increasing linear function
a function with a positive slope: If$\,f\left(x\right)=mx+b, \text{then} m>0.$
linear function
a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line
parallel lines
two or more lines with the same slope
perpendicular lines
two lines that intersect at right angles and have slopes that are negative reciprocals of each other
point-slope form
the equation for a line that represents a linear function of the form$\,y-{y}_{1}=m\left(x-{x}_{1}\right)$
slope
the ratio of the change in output values to the change in input values; a measure of the steepness of a line
slope-intercept form
the equation for a line that represents a linear function in the form$\,f\left(x\right)=mx+b$
vertical line
a line defined by$\,x=a,$where$\,a\,$is a real number. The slope of a vertical line is undefined. | 20,435 | 69,061 | {"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} | 4.75 | 5 | CC-MAIN-2024-26 | latest | en | 0.9156 |
https://inches-to-meters.appspot.com/2/hr/53.6-palac-u-metar.html | 1,610,746,298,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703496947.2/warc/CC-MAIN-20210115194851-20210115224851-00611.warc.gz | 395,358,103 | 6,072 | Palac U Metar
# 53.6 in na m53.6 Palaca na Metara
in
=
m
## Kako to pretvoriti 53.6 palaca na metara?
53.6 in * 0.0254 m = 1.36144 m 1 in
## Pretvori 53.6 in na zajedničke duljine
Mjerna jedinicaJedinica za duljinu
Nanometar1361440000.0 nm
mikrometar1361440.0 µm
Milimetar1361.44 mm
Centimetar136.144 cm
Palac53.6 in
Stopa4.4666666667 ft
Jard1.4888888889 yd
Metar1.36144 m
Kilometar0.00136144 km
Milja0.0008459596 mi
Nautička milja0.0007351188 nmi
## Alternativni pravopis
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https://www.coursehero.com/file/6248391/62-KEY/ | 1,498,352,724,000,000,000 | text/html | crawl-data/CC-MAIN-2017-26/segments/1498128320368.57/warc/CC-MAIN-20170624235551-20170625015551-00442.warc.gz | 860,793,509 | 88,051 | # 6.2 KEY - repeated zeros 2 3 4 8 6 x x x x f = 9 3 3-= x x...
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1 Section 6.2 Zeros of polynomial functions Obejctives s Find the zeros (and their multiplicity) of a polynomial function s Understand the relationship between zeros, roots, solutions, x -intercepts and factors of a polynomial What is the difference between a polynomial and a polynomial function? 12 2 + - x x 12 ) ( 2 + - = x x x f s Let’s talk about graphs of polynomial functions 1 linear 4 quartic 3 cubic 2 quadratic Example function Example graph Degree Find the zeros and graph ) 3 )( 1 )( 2 ( + + - = x x x y ) 3 )( 1 ( ) ( 2 + + = x x x x f In this case, the root 0 happens twice. What do you notice about the graph at this root? Multipliciy s A repeated zero has a multiplicity equal to the number of times the zero occurs. s The multiplicity of a zero affect the graphs behavior at that value.
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2 Find all the zeros of the function, then state the multiplicity of any
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Unformatted text preview: repeated zeros 2 3 4 8 6 ) ( x x x x f + + = ) 9 ( ) 3 ( 3 +-= x x y Write a polynomial s Write a polynomial function in standard form with zeros -2, 3, and 3. s HINT: Use the Zero Product Property in reverse. Write a polynomial function in standard form with zeros 0, 1, and -3. Summary! s The following 4 statements are equivalent s-4 is a solution of s-4 is an x-intercept of the graph of s-4 is a zero of s X + 4 is a factor of 4 3 2 =-+ x x 4 3 2-+ = x x y 4 3 2-+ = x x y 4 3 2-+ x x Summary Questions s-3 is a zero of s Tell me one of the factors of s True / False 3 is an x-intercept of the graph of 6 2-+ = x x y 4 3 2-+ x x 6 2-+ = x x y Assignment # ___ s Page 311 (17-31 odd) &amp; 35...
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## This note was uploaded on 05/13/2011 for the course MTH 98 taught by Professor Johnson during the Fall '09 term at Grand Valley State.
### Page1 / 2
6.2 KEY - repeated zeros 2 3 4 8 6 x x x x f = 9 3 3-= x x...
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Ask a homework question - tutors are online | 695 | 2,296 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.59375 | 5 | CC-MAIN-2017-26 | longest | en | 0.857588 |
http://mathmatique.com/articles/where-start-pure-mathematics-set-theory | 1,725,966,285,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651241.17/warc/CC-MAIN-20240910093422-20240910123422-00082.warc.gz | 19,704,796 | 9,798 | ## Where To Start In Pure Mathematics? Set Theory
So you'd like to learn pure mathematics. Where should you start? Real analysis? Topology? Number Theory? You can find introductory, undergraduate, preliminary, and even "gentle" textbooks written on any topic can think of, and it's not obvious that one subject "comes before" another the same way that algebra comes before pre-calculus. Likewise, plenty of syllabi from august universities claim that their courses are "students' first introduction to rigorous mathematics," only for others to puzzlingly claim "no prerequisites." What's the deal, then? Which subject should you really study first?
The answer is set theory. Oddly enough, it's likely that set theory is at most a single course at most universities, rarely intended as an introductory pure mathematics course, and typically hidden in the course catalog by the profusion of analysis, topology, and abstract algebra courses. So then why is set theory the appropriate starting point? Because it's secretly the first subject taught in each of those classes. In fact, every standard text in higher mathematics starts with or assumes knowledge of set theory.
Don't believe me? Let's take a look at the canon of math textbooks for "real" math:
• Real Analysis: Principles of Mathematical Analysis, Rudin: The first definition in Chapter 1 reads "If $A$ is a set (whose elements may be numbers or any other objects), we write $x \in A$ to indicate that $x$ is a member (or an element) of $A$." The next few pages are dedicated to the basics of sets. In fact, Chapter 2, "Basic Topology" (of $\mathbb{R}$) starts off with a section on "Finite, Infinite, and Uncountable Sets," and a number of the later topological proofs in the chapter bring in set theory directly, of which several are pure set theory and have nothing to do with metric topology.
• Topology: Topology, James Munkres: Chapter 1 is entitled "Set Theory and Logic" and is over 70 pages long.
• Abstract Algebra: Abstract Algebra, Dummit and Foote: Set theory appears before the first chapter in "Preliminaries." The first sentence in the affectionately numbered section 0.1 reads "The basics of set theory: sets, $\cap$, $\cup$, $\in$, etc. should be familiar to the reader." Nonetheless, the authors spend three and a half pages rehashing some basic definitions.
• Graph Theory: Graph Theory, Bondy and Murty: After a perfunctory introductory paragraph, Chapter 1.1 "Graphs and Their Representations" trots out its first definition in terms of sets: "A graph $G$ is an ordered pair $(V(G), E(G))$ consisting of a set $V(G)$ of vertices and a set $E(G)$, disjoint from $V(G)$, of edges..."
• Linear Algebra: Linear Algebra Done Right, Axler: The first sentence of Chapter 1 section A is "You should already be familiar with basic properties of the set R of real numbers." Immediately following that paragraph is the set-theoretic definition of the complex numbers in terms of the real numbers, complete with the curly braces and colon notation. Section 1.B provides a definition of vector spaces that begins "A vector space is a set $V$...". (Oddly enough, this book is easiest one in the list, yet it assumes more knowledge of set theory than the others!)
You get the idea. Even if set theory doesn't get pride of place in the course catalog, all the textbooks seem to agree that knowledge of sets is critical to the "actual" topic at hand.
So what the heck is set theory? Set theory is the study of (wait for it) sets. A set is just a collection of objects. The letters of the alphabet form a set, as do the set of chairs at your kitchen table, the set of books on the top shelf of your bookcase, and so on. Due to their simplicity, sets form the basis for the other "higher" branches of math. Algebraic objects such as groups, rings, and fields are sets on which functions with particular properties are defined. "Spaces" such as vector spaces, topological spaces, measure spaces, and probability spaces are defined in terms of several sets that also meet additional requirements. And graphs, as we saw above, are defined in terms of a set of vertices and a set of edges between them.
Set theory also covers many of the basic building blocks of math, such as relations, functions, orders, and cardinality (aka "size"). And no course in set theory would be complete without a construction of the natural numbers from sets. Studying set theory therefore gives you a firm grasp on all of the loose tidbits that are called upon at will in other higher order subjects.
Constructions aside, set theory also has very little to do with numbers, which makes it perfect for learning proofs. In contrast, subjects like real analysis are a terrible fit for the job, because all of the theorems it wants to prove have been taken as The Obvious Truth by students for their entire lives up until that point, which needlessly complicates the important task of learnings how to do proofs themselves. Learning how to write a proof is a herculean undertaking wholly independent of any actual subject, and it's the big kids equivalent of learning algebra. If you don't "get" algebra, your math career dies around ninth grade, and if you don't "get" proofs, your math career never leaves the collegiate frisbee field.
One final benefit of set theory is that it does not induce the latent fear of yet more fundamental math. If you start with real analysis, you'll look at the definitions and ask "Well shouldn't I study fields first? Or perhaps rational numbers?" Both questions lead to a death spiral of looking for The Beginning. Well, that's set theory. If you want to study math from the bottom up, set theory is square one. Philosophers will argue that logic is even more fundamental, but we can unironically place basics of first order logic as the chapter 0 to our own book on set theory.
So there you have it. Set theory is your pure math boot camp. Get proving! | 1,337 | 5,935 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.484375 | 3 | CC-MAIN-2024-38 | latest | en | 0.917133 |
https://www.math.tu-berlin.de/fachgebiete_ag_modnumdiff/angewandtefunktionalanalysis/v_menue/forschung/angewandte_harmonische_analysis/parameter/de/mobil/ | 1,561,004,577,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627999130.98/warc/CC-MAIN-20190620024754-20190620050754-00092.warc.gz | 795,850,644 | 5,880 | FG Angewandte FunktionalanalysisAngewandte Harmonische Analysis
# Angewandte Harmonische Analysis
Applied harmonic analysis has established itself as the main area in applied mathematics focused on the efficient representation, analysis, and encoding of data. The primary object of this discipline is the process of ‘breaking into pieces’ – from the Greek word analysis –, to gain insight into an object.
The Fourier basis is historically the first representation system, which significantly impacted various areas within mathematics but also in applications. The associated Fast Fourier Transform (FFT) is to date in fact one of the most often utilized algorithm in numerical computations. However, the Fourier basis has certain disadvantages such as non-locality and also does not sparsely approximates singularities, which are the most distinct and hence important features of functions/signals.
Historically, the introduction of wavelets about 20 years ago represents a milestone in the development of efficient encoding of piecewise regular signals. The major reason for the spectacular success of wavelets consists not only in their ability to provide optimally sparse approximations of a large class of frequently occurring signals and to represent singularities much more efficiently than traditional Fourier methods, but also in the existence of fast algorithmic implementations which precisely digitalize the continuum domain transforms. Wavelets are nowadays widely used both for more theoretical tasks such as for elliptic partial differential equations or for more practical tasks such as for the image compression standard JPEG2000.
Despite their success, wavelets are not very effective when dealing with multivariate data.The reason for this failure is that multivariate data is typically governed by anisotopic features such as edges in images, whereas wavelets are isotropic objects, hence not optimally adapted to those. The limitations of wavelets and traditional multiscale systems have stimulated a flurry of activity involving mathematicians, engineers, and applied scientists. Many systems were suggested such as ridgelets and curvelets.
In 2006 the system of shearlets was introduced by Kutyniok and Labate and is by now the first anisotropic system, which optimally sparsely approximates anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for which a compactly supported version is available for high spatial localization, and which admits a unified treatment of the continuum and digital world to ensure faithful implementations. As most of the other anisotropic representation systems, they do not form an orthonormal basis, but a frame. Shearlets are to date often used in combination with compressed sensing techniques for solving inverse problems or developing efficient numerical solvers for certain partial differential equations.
## Some of our Research Topics
• Development of more general frameworks including shearlets such as parabolic molecules and alpha-molecules.
• Introducing novel shearlet systems for dealing with, for instance, data on bounded domains.
• Development of the software package ShearLab.
• Development and analysis of approaches using compressed sensing in combination with shearlets for solving inverse problems such as the inverse scattering problem.
• Development and analysis of approaches to numerically solve certain partial differential equations such as transport equations using shearlets as a trial basis for sparsely approximating the solution.
• Application of shearlets to speed up the data acquisition in Magnetic Resoance Imaging.
• Application of shearlets to solve imaging problems such as inpainting or feature extraction, also in real-world problems such as electron microscopy.
• Parametrized shearlet systems and learning of their parameters.
## Survey Papers
• G. Kutyniok, W.-Q Lim, and G. Steidl.
Shearlets: Theory and Applications.
GAMM-Mitteilungen 37 (2014), 259-280.
• G. Kutyniok and D. Labate.
Introduction to Shearlets.
Shearlets: Multiscale Analysis for Multivariate Data, 1-38, Birkhäuser Boston, 2012.
• G. Kutyniok, J. Lemvig, and W.-Q Lim.
Shearlets and Optimally Sparse Approximations.
Shearlets: Multiscale Analysis for Multivariate Data, 145-198, Birkhäuser Boston, 2012. | 857 | 4,327 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2019-26 | longest | en | 0.941744 |
http://mathhelpforum.com/advanced-algebra/14390-using-eisenstein.html | 1,529,730,116,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267864940.31/warc/CC-MAIN-20180623035301-20180623055301-00124.warc.gz | 208,699,840 | 9,289 | # Thread: Using eisenstein
1. ## Using eisenstein
I know how to prove this for a general case using Eisenstein criterion but not for all of p and q
If p and q are prime numbers and p does not = q, then x^5-p^5*q contained in Z(x) is irreducible in Q(x).
2. Originally Posted by chadlyter
I know how to prove this for a general case using Eisenstein criterion but not for all of p and q
If p and q are prime numbers and p does not = q, then x^5-p^5*q contained in Z(x) is irreducible in Q(x).
It is necessary and sufficient to show that x^5-p^5*q is irreducible in Z[x] for that will imply that it is irreducible in Q[x].
Use Eisenstein Criterion with q, then q|p^5*q obviously.
But q^2 does not divide p^5*q since p!=q.
Q.E.D. | 212 | 732 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.296875 | 3 | CC-MAIN-2018-26 | latest | en | 0.910122 |
http://stats.stackexchange.com/questions/93843/manually-calculated-chi-squared-statistic-differs-from-that-given-by-chisq-test | 1,469,342,859,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257823963.50/warc/CC-MAIN-20160723071023-00082-ip-10-185-27-174.ec2.internal.warc.gz | 241,113,827 | 18,609 | # Manually calculated chi-squared statistic differs from that given by chisq.test in R
I'm using R to do a chi-squared test on two sets of data to check if they follow a distribution. In the first case the chi-squared statistic given by chisq.test matches the one calculated by hand, but in the other case it does not.
First case
> e1 = c(18, 38, 78, 107, 107, 72, 35, 15)
> o1 = c(10 ,30 ,80 ,180 ,60 ,40 ,40 ,30)
> sum((o1-e1)**2/e1)
[1] 105.6762
> chisq.test(o1,p=e1,rescale.p=T)$statistic X-squared 105.6762 Both methods give 105.67 as the statistic. But in the other case: > o2=c(401, 1235, 2989, 5682, 8489, 9966, 9196, 6668, 3800, 1701, 599, 165) > e2=c(385, 1204, 2945, 5643, 8472, 9966, 9186, 6635, 3754, 1665, 578, 157) > sum((o2-e2)**2/e2) [1] 5.111822 > chisq.test(o2,p=e2,rescale.p=T)$statistic
X-squared
3.301292
Now the statistics are different and I don't understand why. Am I doing something wrong or missing something?
-
Loosely related: What is wrong with this chi-squared calculation? – gung Apr 15 '14 at 14:56
This isn't really a software issue, as it hinges on a misapplication or misunderstanding of chi-square tests.
1. Your manual calculation in the second case takes no account of the fact that o2 and e2 have different totals, e2 50590, o2 50891.
2. R's result is what I get independently in different software if I scale the expected e2 to sum to the sum of the observed o2.
3. It is surprising in any case that the expected frequencies come as integers. This is unusual in practice. My guess is that this is a two-way problem being presented wrongly as a one-way problem. If so, a chi-square test yields 1.6304 with 11 d.f. and $P =$ 0.999 and the result rings alarm bells as almost too good to be true!
-
Different totals in e2 and o2 were the problem. Thanks! But I don't understand what you mean by this being a two-way problem. I'm comparing two normal distributions and the type of test shouldn't depend on whether the data are integers of floats (though I admit floats for expected frequencies make more sense). – numentar Apr 15 '14 at 12:31
I set aside the claim that you have two "normal" distributions. The main point is that, from what you say, your two vectors of frequencies have the same standing as being both vectors of observed counts. That being so, it is not that one is observed and the other is expected; otherwise put, why not reverse which is observed and which expected? You can separately test whether each is compatible with a normal distribution, or you can jointly test whether they are consistent with the same distribution. Expected means what it says: the expected frequencies are based on some hypothesis. – Nick Cox Apr 15 '14 at 12:42
Chi-square is a lousy way to test for normality, by the way. Use a normal probability plot or (if a $P$-value is something you feel you really need) a dedicated test such as Shapiro-Wilk or Doornik-Hansen. – Nick Cox Apr 15 '14 at 12:47
Thanks for the tips! I'm actually not testing for normality per se. I just generated the data for this post from two normal distributions. But why cannot the expected frequencies be integers? Say I have census data for age groups from an entire country. Now I do random selection 3 years later and want to see if the sample age distribution is different from the population age distribution 3 years ago. Both datasets would originally be counts. – numentar Apr 15 '14 at 13:07
In your last case, the expected frequencies still have to have the same total as the observed frequencies. If your hypothesis is "like this empirical distribution", scaling to the same total means that expected frequencies being integers is unlikely, but admittedly not impossible. You may be spoiled by software doing the scaling for you, but it is needed. – Nick Cox Apr 15 '14 at 13:15 | 1,024 | 3,813 | {"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} | 3.171875 | 3 | CC-MAIN-2016-30 | latest | en | 0.890113 |
http://www.gurufocus.com/term/Cash%20from%20Financing/HLF/Cash%2BFlow%2Bfrom%2BFinancing/Herbalife%252C%2BLtd | 1,480,751,446,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698540909.75/warc/CC-MAIN-20161202170900-00474-ip-10-31-129-80.ec2.internal.warc.gz | 490,945,217 | 32,667 | Switch to:
Herbalife Ltd (NYSE:HLF)
Cash from Financing
\$-273 Mil (TTM As of Sep. 2016)
Cash from financing is the cash generated/spent from financial activities such as share issuance (buy back), debt issuance (repayment), and dividends paid to preferred and common stockholders.
For the three months ended in Sep. 2016, Herbalife Ltd paid \$8 Mil more to buy back shares than it received from issuing new shares. It spent \$3 Mil paying down its debt. It paid \$0 Mil more to buy back preferred shares than it received from issuing preferred shares. It received \$0 Mil from paying cash dividends to shareholders. It received \$4 Mil on other financial activities. In all, Herbalife Ltd spent \$8 Mil on financial activities for the three months ended in Sep. 2016.
Definition
This is the cash generated/spent from financial activities such as share issuance (buy back), debt issuance (repayment), and dividends paid to preferred and common stockholders. In the calculation of free cash flow, cash from financing is not calculated because it is not related to operating activities.
Herbalife Ltd's Cash from Financing for the fiscal year that ended in Dec. 2015 is calculated as:
Cash from Financing (A: Dec. 2015 ) = Net Issuance of Stock + Net Issuance of Debt + Net Issuance of Preferred + Dividends + Other Financing = -16.6 + -227.6 + 0 + 0 + -5.8 = -250
Herbalife Ltd's Cash from Financing for the quarter that ended in Sep. 2016 is
Cash from Financing (Q: Sep. 2016 ) = Net Issuance of Stock + Net Issuance of Debt + Net Issuance of Preferred + Dividends + Other Financing = -8 + -3.3 + 0 + 0 + 3.5 = -8
Herbalife Ltd Cash Flow from Financing for the trailing twelve months (TTM) ended in Sep. 2016 was -32.1 (Dec. 2015 ) + -233.7 (Mar. 2016 ) + 0.2 (Jun. 2016 ) + -7.8 (Sep. 2016 ) = \$-273 Mil.
* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.
Explanation
Cash from financing contains five items:
1. Net Issuance of Stock:
A company may raise cash from issuing new shares. It can also use cash to buy back shares. If this number is positive, it means that the company has received more cash from issuing shares than it has paid to buy back shares. If this number is negative, it means that company has paid more cash to buy back shares than it has received for issuing shares.
Herbalife Ltd's net issuance of stock for the three months ended in Sep. 2016 was \$-8 Mil. Herbalife Ltd paid \$8 Mil more to buy back shares than it received from issuing new shares.
2. Net Issuance of Debt:
Net issuance of debt is the cash a company received or spent through debt related activities such as debt issuance or debt repayment. If a company pays down its debt during the period, this number will be negative. If a company issued more debt, it receives cash and this number is positive.
Herbalife Ltd's net issuance of debt for the three months ended in Sep. 2016 was \$-3 Mil. Herbalife Ltd spent \$3 Mil paying down its debt.
3. Net Issuance of Preferred:
A company may raise cash from issuing new preferred shares. It can also use cash to buy back preferred shares. If this number is positive, it means that the company has received more cash from issuing preferred shares than it has paid to buy back preferred shares. If this number is negative, it means that company has paid more cash to buy back preferred shares than it has received for issuing preferred shares.
Herbalife Ltd's net issuance of preferred for the three months ended in Sep. 2016 was \$0 Mil. Herbalife Ltd paid \$0 Mil more to buy back preferred shares than it received from issuing preferred shares.
4. Dividends:
Cash flow for dividends refers to the payment of cash to shareholders as dividends when the company generates income.
Herbalife Ltd's cash flow for dividends for the three months ended in Sep. 2016 was \$0 Mil. Herbalife Ltd received \$0 Mil from paying cash dividends to shareholders.
5. Other Financing:
Money spent or earned by company from other financial activities.
Herbalife Ltd's other financing for the three months ended in Sep. 2016 was \$4 Mil. Herbalife Ltd received \$4 Mil on other financial activities.
Related Terms
Historical Data
* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.
Herbalife Ltd Annual Data
Dec06 Dec07 Dec08 Dec09 Dec10 Dec11 Dec12 Dec13 Dec14 Dec15 Net Issuance of Stock 12 -366 -139 -75 -160 -322 -557 -307 -1,292 -17 Net Issuance of Debt 80 171 -50 -101 -72 25 284 444 1,069 -228 Net Issuance of Preferred 0 0 0 0 0 0 0 0 0 0 Dividends 0 -42 -51 -49 -54 -85 -135 -123 -30 0 Other Financing -147 33 34 11 24 43 37 17 -136 -6 Cash from Financing -55 -204 -205 -213 -262 -339 -371 31 -390 -250
Herbalife Ltd Quarterly Data
Jun14 Sep14 Dec14 Mar15 Jun15 Sep15 Dec15 Mar16 Jun16 Sep16 Net Issuance of Stock -583 -1 -14 -9 -0 -2 -6 -2 -2 -8 Net Issuance of Debt -19 -19 -25 -25 -139 -39 -25 -230 0 -3 Net Issuance of Preferred 0 0 0 0 0 0 0 0 0 0 Dividends 0 0 0 0 0 0 0 0 0 0 Other Financing 5 -2 7 3 -7 -1 -1 -2 2 4 Cash from Financing -597 -21 -32 -31 -146 -41 -32 -234 0 -8
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GuruFocus Affiliate Program: Earn up to \$400 per referral. ( Learn More) | 1,427 | 5,387 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2016-50 | longest | en | 0.935716 |
https://stats.stackexchange.com/questions/578425/is-the-exact-value-of-any-likelihood-meaningless?noredirect=1 | 1,708,927,139,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474650.85/warc/CC-MAIN-20240226030734-20240226060734-00117.warc.gz | 558,936,394 | 48,773 | Is the exact value of any likelihood meaningless?
While reading about likelihood, I have heard that "the exact value of any likelihood is meaningless" why?
So, because of that we may use the likelihood ratio.
So, my question is, why the exact value of likelihood is meaningless? And what is the benefit of likelihood-ratio over the likelihood?
• There is no scale so some people say something like "the likelihood of $\theta$ is proportional to ..." Jun 11, 2022 at 10:42
• I was intrigued where the OP might have heard that "the exact value of any likelihood is meaningless". Quotes usually imply there is a reference. Google pointed me to A Guide on Data Analysis. Jun 11, 2022 at 17:01
It's “meaningless” in the sense that it's very hard to interpret, it's just “the bigger, the better”. That is the case because the likelihood is not probability and it is calculated without normalizing the constant so the numerical value can be any non-negative number. You still can maximize the likelihood because lack of normalization doesn't matter for optimization. See other questions tagged as for more details like What is the reason that a likelihood function is not a pdf? or What is the difference between "likelihood" and "probability"?.
• The likelihood is a probability (or probability density) in the data space (not the model-parameter space). Specifically, it is the probability of the data given the model parameters. The integral of the likelihood over all possible data comes to 1. In MCMC, one often works with an un-normalized version of the likelihood, and that un-normalized version is, of course, not a proper probability. Jun 12, 2022 at 12:45
• @apdnu it's not, it's more complicated, see stats.stackexchange.com/q/2641/35989 Also you seem to assume a Bayesian perspective, where the term exists also outside it.
– Tim
Jun 12, 2022 at 13:20
• The likelihood is literally defined as the probability of the data, given a model. It's a probability in data-space, integrates to one in that space, is non-negative, etc. Jun 13, 2022 at 11:46
• It sounds like your objections are more philosophical (Frequentist vs. Bayesian) than practical. Likelihood functions exactly like a probability (or probability density) in the data-space, and that is a perfectly valid interpretation of what it is. Indeed, that is exactly how it is generally defined, and any other definition is equivalent to this definition under a shift of your philosophical outlook (e.g., from Frequentist to Bayesian). Jun 13, 2022 at 11:59
• @apdnu Likelihood functions have the parameter(s) of interest as the x-axis scale. How is that "in the data-space"? Jun 13, 2022 at 21:18
When we use likelihood then we are comparing probability (density) of the data given a certain hypothesis/theory.
The actual probability is not important. It can actually become extremely small. Imagine that you are testing whether a coin is a fair coin and you observe 1000 000 flips with 500 000 heads and 500 000 tails. If the coin is fair (equal head and tails probability), then the probability of this particular observation is 00.0007978844.
• Can you show the proof for this here? :) Dec 28, 2022 at 10:04
• @JEDHK proof of what? Dec 28, 2022 at 10:32
• I mean the mathematical derivation that 1 000 000 flips with 500 000 heads and tails being 0.0007978844 Dec 28, 2022 at 16:05
• @JEDHK I had this calculated by a computer. In R you would use dbinom(500000,1000000,0.5). Alternatively you can use the asymptotic approximation for the middle term and compute $\frac{\sqrt{2}}{\sqrt{n\pi}}$. I guess that I used the first method because the second gives me 0.0007978846 Dec 28, 2022 at 16:33
The likelihood function is usually taken to be the PDF viewed as as a function of parameters for known data.
For example, if I have a coin with Heads probability $$\theta$$ and toss it $$n = 10$$ times, getting $$x = 3$$ heads, then I can take the likelihood function to be
$${n\choose x}\theta^x(1-\theta)^{n-x},$$ considered as a function of $$\theta.$$
If I want the MLE $$\hat \theta$$ of $$\theta,$$ then I might write the likelihood function as $$f(\theta \mid x = 3)\propto \theta^3(1-\theta)^7,$$ where the symbol $$\propto$$ (read as "proportional to") indicates that the constant $${n\choose x} = {10\choose 3} = 120$$ is omitted. The maximum of the likelihood function is at $$\hat \theta = x/n = 0.3,$$ whether I use or ignore the constant $${10\choose 3}.$$
So the values of the likelihood function might be considered less important than its shape, which leads to the MLE $$\hat\theta.$$
• It is more than that: If you saw THTTTTTHTH then the probability of that exact result would have been $\theta^3(1-\theta)^7$ while the probability of $3$ heads and $7$ tails in any order would have been ${10 \choose 3}\theta^3(1-\theta)^7$. The likelihood of $\theta$ being say $0.3$ rather than some other value does not change between these two ways of finding the probability so it is reasonable to say each is only proportional to the likelihood. Jun 12, 2022 at 23:48
• @Henry. Thanks for edit fixing typo. Jun 13, 2022 at 0:16
[Context]
@Henry and @JonathanLew firmly pointed out errors in my original answer, which argued that the statement "the exact value of any likelihood is meaningless" is glib and that you can't prove a logical claim about all likelihoods by providing specific examples where it's safe to compute the likelihood up to a constant (which admittedly is often the case).
Since I first posted my answer I've learned that continuous likelihoods have (theoretical) units given by 1/(units of the data) from @apdnu's answer to Units for likelihoods and probabilities.
I've come across examples where likelihood function should be be computed exactly to get the correct answer. These examples teach me that I should be careful with my likelihood calculations and don't presume that I can safely ignore normalizing constants.
And I've discovered that (a version of) this question has been asked and answered before: What does "likelihood is only defined up to a multiplicative constant of proportionality" mean in practice?
Example #1: Comparing a model with normal errors to a model with Cauchy errors
This example is from Chapter 6, Y. Pawitan In All Likelihood: Statistical Modelling And Inference Using Likelihood (2013).
We want to model Y in terms of X; there are a few unusual values in the data (outliers). We propose two models with the same mean structure E(Y) = β0 + β1X but different error structure: in one model the errors are iid Normal(0, σ2), in the other model the errors are iid Cauchy(0, γ). We fit the models by maximizing the likelihoods and next we use AIC = -2$$\log$$L + 2k (where k is the number of model parameters) to choose the "better" model. Both the Normal density and the Cauchy density have constant terms that are usually save to ignore: (2𝜋)-1/2 for the Normal and 𝜋-1 for the Cauchy. These are not the same constant for both models, so no parts of the likelihood functions can be dropped.
Example #2: Mixture of Bernoullis for latent class analysis
This example is from Chapter 9 of C. M. Bishop. Pattern Recognition and Machine Learning (2006).
We want to model a dataset of binary observations as a mixture of $$K$$ Bernoulli components with parameters $$\{\mu_k\}$$ and mixing proportions $$\pi_k$$. The log likelihood is:
$$\ln p(\mathbf{X}|\mathbf{\mu},\mathbf{\pi}) = \sum_{n=1}^N\ln\left\{\sum_{k=1}^K\pi_kp(\mathbf{x}_n|\mathbb{\mu}_k)\right\}$$
Since there is a summation inside a logarithm, the math doesn't simplify but the maximum likelihood solution can be found with the EM algorithm.
Example #3: Bayesian $$t$$-test
This example is from Chapter 4 of K. P. Murphy. Machine Learning: A Probabilistic Perspective (2012).
We want to test the hypothesis $$\mu > \mu_0$$ for some known value of $$\mu_0$$. The p-value for an one-side t-test is an integral over the likelihood:
\begin{aligned} p(\mu>\mu_0|\text{data}) = \int_{\mu_0}^\infty p(\mu|\text{data})d\mu \end{aligned}
We can't omit any terms inside the integral or we won't compute the p-value correctly.
In summary, there are both theory and examples to illustrate that the exact value of the likelihood function can be meaningful.
The statement "the exact value of any likelihood is meaningless" is abstract and imprecise at the same time. So let's start with the definition of likelihood. In the spirit of this question, the definition isn't mathematically rigorous.
We take a probabilistic model f(x,θ) for data x with parameter θ.
• As a function of the data x, f(x,θ) is a probability density/mass function. [pdf if x is continuous; pmf if x is discrete.]
• As a function of the parameter θ, f(x,θ) is the likelihood.
It's true that the likelihood doesn't integrate to 1. Many functions don't, yet we don't conclude that their exact value is meaningless.
• x f(x,θ) dx = 1 [replace the integral with a summation if x is discrete]
• θ f(x,θ) dθ = constant that depends on the model f and the data x
A common theme running through the answers is that likelihood computations often simplify. The logical argument seems to go something like this: in many computations a term in the likelihood is constant or behaves like a constant so we can simplify the math by dropping that term; ergo the exact value of a likelihood function is meaningless.
However, the likelihood has more uses than maximizing it to find the MLE or performing a likelihood ratio test. And a likelihood term that can be ignored in one computation is important to keep track of in another.
• But clearly you can rescale both likelihoods by the same amount without affecting the Likelihood Ratio or Bayes Factor. Suppose you toss a biased coin three times and want to consider the models of the probability of heads as $\theta_0=\frac15$ or $\theta_1=\frac45$. If you see $HHT$ it does not matter whether you say the likelihood of $\theta$ is then $\theta^2(1-\theta)$ or say ${3\choose 2}\theta^2(1-\theta)$ as you will get a ratio of $4$ in both cases. Jun 11, 2022 at 21:44
• In your $\theta_0 = 0.1$ and $\theta_1 = 0.12$ case and my $HHT$ example, it still does not matter whether you say the likelihood is $\theta^2(1-\theta)$ or ${3\choose 2}\theta^2(1-\theta)$ as you would get a ratio of $1.408$ either way. Jun 12, 2022 at 0:17
• You seem to be suggesting that if one model thinks order matters and the other that order does not matter then there is an issue over the likelihood of the latter. I am saying that since the calculations you do for the likelihoods are only meaningful up to proportionality, you can scale them to be on a comparable basis Jun 12, 2022 at 0:17
• $\theta^2(1-\theta)$ is the probability of Heads then Heads then Tails in that order. ${3\choose 2}\theta^2(1-\theta)$ is the probability of $2$ Heads and $1$ Tails in any order. It would be peculiar if your assessment of the likelihood of a particular value of $\theta$ having observed $HHT$ depends on whether you use the full observation or a sufficient statistic but, since likelihood is relative, you do need to be consistent in your choice Jun 12, 2022 at 0:32
• Not at all, and I did not say that - it is obviously important in finding a binomial probability and is not meaningless. I am saying it can be used or not in the calculation of a likelihood (so long as this is done consistently) and in that sense does not affect the likelihood of a parameter taking a particular value. Hence my original comment Jun 12, 2022 at 1:08
An important concept that should be mentioned in the context of this discussion is that of the Likelihood principle (See also Berger & Wolpert's book). The likelihood principle, which is one of the foundations of Bayesian statistics, states that all the evidence relevant to a model parameter $$\theta$$ is contained in the likelihood function $$\mathcal L(\theta | x) = f_X(x|\theta)$$.
The precise statement of the likelihood principle is that if two experiments about the same parameters $$\theta$$ are proportional to each other, namely if
$$f_X(x|\theta) = cf_Y(y|\theta)$$
with $$c$$ some positive constant, then the evidence on $$\theta$$ from the experiments is identical. In this sense, the likelihood is only meaningful (as evidence) up to a normalizing constant.
Indeed in Bayesian statistics those two likelihoods will lead to the same posterior probability of $$\theta$$ (given the same prior), hence Bayesian statistics 'automatically' respects the likelihood principle.
Frequentists statistics in general violates the likelihood principle, so discussion of the meaning of likelihood from a frequentist perspective is somewhat meaningless by itself. But practically most if not all frequentist uses of the likelihood function are via quantities (such as likelihood ratios or the maximum likelihood estimator) that are also invariant under scaling, | 3,189 | 12,889 | {"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": 29, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5 | 4 | CC-MAIN-2024-10 | latest | en | 0.946072 |
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(-7 + 5)/2= -1
(-7 + -3)/2= -5
(-1,-5)
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## 30+ Measuring Instruments for Mechanical Engineer
Here are 30 Measuring instruments for mechanical engineer. Every Mechanical Engineering student or employee should know about this.
Join online course of Mechanical instruments step by step
## 01. Vernier Caliper
Vernier Caliper is a widely used linear measurement instrument with a least count of 0.02 mm. It is used to measure linear dimensions like length, diameter, depth.
It is Basic Instrument of measurement, consist of two types of scale
The main scale and the Vernier scale that can slide along the main scale. Two types of measurement we can do, the first one is through external jaw (measure external dimensions) and another one is internal jaw (measure internal dimensions).
## 02. Outside Micrometer
External Micrometer is also known as Outside Micrometer or External Micrometer.
It is used to check outside diameter of circle by the means of accuracy of 0.01 mm or up to 0.001 mm.
Vernier type Micrometer gives highest acceptable accuracy of 1 micron such gauge is vernier type micrometer.
## 03. Vernier Height Gauge
Vernier height gauge used to measure vertical dimension from reference ground. Vernier Height Gauge consist of a graduated scale or bar is held in a vertical position by a finely ground fixed base.
The graduated scale has least count of 0.02 mm like Vernier Caliper has. And the way of taking a reading of measurement in Vernier Height Gauge is same as in Vernier gauge.
## 04. Steel Scale
Steel scale is single piece linear measuring instrument.
Steel scale indicates two units that are cm and inches, cm division on one side and inches, on another side.
## 05. Vernier Depth Gauge
Vernier Depth Gauge as the name suggests is used to measure the depth from the surface of reference of an object.
Vernier caliper also has depth bar but this can not be used as the standard measurement.
## 06. Vernier Bevel Protractor
A simple protractor is a basic device used for measuring angles with a least count of 1° or ½°. Bevel Protractor is an angular measuring instrument capable of measuring angles with a least count of 5’.
The protractor dial is graduated in degrees with every tenth degree numbered. The sliding blade is fitted into this dial I.e. it may be extended to either direction and set at an angle to the base.
## 07. Plunger Dial Gauge
The dial indicator or the Plunger dial gauge is one of the simplest and the most widely used mechanical comparator.
First of all the use of plunger dial gauge used to compare work-pieces against a master
## 08. Lever Dial Gauge
Lever Dial Gauge also knew as Test Indicator. It used to measure sensitive contact.
Lever Dial gauge usually measure up to 0.80 mm. But some special type lever dial design for measurement up to 2 mm.
## 09. Engineering Square
A ruler can be used to draw straight lines but there is no quarranty that the line drawn is accurate and exactly straight,this is where the engineering square is brought into use.
An engineering square also known as machinist square is similar in size and construction to a try square.
It is a tool used for making straight lines and are used to measure angles.
# 10. V Block
In Industries where precise marking and firm holding of objects is required, V-Blocks play a major role and are extremely important metalworking jigs.
The construction had two clamps:screw clamp and a U shaped handle like clamp and a V-block .
Gauges are derived from the French word”jauge” which means the result of the measurement. We are all aware that gauges are used to measure the thickness, size or capacity of something.
Likewise, radius gauges are instruments that are used to measure the radius of the object.
The radius gauge is combined with another gauge known as fillet gauge which in mechanics means a rounding of the part design
## 12. Digital Vernier Caliper
Digital Vernier Caliper is upgrading version of Analog Vernier Caliper, which is a widely used linear measurement instrument with a least count of 0.01 mm, more accurate than Analog.
This Digital type of vernier is similar to Analog Vernier Caliper Instead Dimension output in Digital Manner that is more convinces than analog type.
It also consists of two types of scale –
The main scale and the Digital display that can slide along the main scale. Two types of measurement we can do, the first one is through external jaw (that measure external dimensions like shaft) and another one is internal jaw (measure internal dimensions like hole).
## 13. Digital Micrometer
Digital Micrometer is a very popular instrument these days because of its easy of compactness of taking an observation.
Digital Micrometer can observe measurement in mm or inch depend upon our need basis.
## 14. Inside Micrometer
nside Micrometer is used to measure the larger internal dimension. Inside Micrometer can measure internal Diameter of holes and registers.
## Types of Micrometer
There are various types of micrometers available in the industry each used for a specific purpose. One such micrometer is the “Inside Micrometer”.
The inside micrometer is used to measure the inner diameter of objects that are bonded by walls,cylindrical bore or hollow pipe.There are two types of Inside Micrometer
1. Analog type Inside Micrometer
2. Digital type Inside Micrometer
## 15. Depth Micrometer
We also measure depth through Vernier caliper, but Vernier caliper does not deliver as much as accuracy and precision, because extension rod of Vernier caliper has no standard, it used only for comparison purpose.
Depth micrometer used to measure the depth of an object with precision and accuracy with least count of 0.01 mm. The measuring span is 25 mm just like in micrometer, that can change by changing upsetting rod.
## 16. Measuring Tape
Measuring tape is used to measuring long distance.
## 17. Digital Height Gauge
The electronic Height gauge is very compact, simple, easy to use instrument widely used in industries, tool rooms, workshops etc.
Some think you should consider while handling Electronic Height Gauge, Because of the cost of Electronic Height gauge matter in any industries or organization.
## 18. Electronic Height Gauge
Electronic height gauge is advance form of height gauge. It has higher accuracy and multi function.
An Electronic height gauge can measure diameter,angle, parallelism, squareness along with length.
## 19. Digital Universal Caliper
Digital universal caliper is widely used for measure inner and outer length of work-piece
## 20. Sine Bar
Sine bar is a precision angle measuring instrument along with slip gauges. The name suggests that Sine bar work on sine principle. Slip gauge used to build up the height of sine bar.
The required angle is obtained when the difference in height between the two rollers is equal to the Sine of the angle multiplied by the distance between the centers of the rollers.
## 21. Digital Protractor
Digital Protractor measure an angle over 360 degree with a high accuracy of 0.1 degree. Digital Protractor is very simple and ease to measure angle by just putting between levels or surface.
## 22. Spirit level
To check level of any surface or table, spirit level is used for leveling.
## 23. Slip Gauge Set
Slip Gauge is set standard size rectangular shape block. Slip gauges are available in standard sets in both metric and inch units.
In metric units, sets of 31, 48, 56, and 103 pieces are available.
For example, the set of 103 pieces consists of the following:
1. One piece of 1.005 mm
2. 49 pieces ranging from 1.01 to 1.49 mm in steps of 0.01 mm
3. 49 pieces ranging from 0.5 to 24.5 mm in steps of 0.5 mm
4. Four pieces ranging from 25 to 100 mm in steps of 25 mm
## 24. Bore Dial Gauge
Bore Dial Gauge use to measure the inner diameter of the hole, it is gauge fitted with a plunger dial gauge. In this video, you are able to know how to measure diameter with a bore dial gauge.
Bore dial gauge used within the standard range that means bore dial gauge have a standard set of pins(anvils) with the variable range of up to 2 mm.
## 25. Feeler Gauge
A Feeler gauge is used to measure the clearance between two parallel flat faces for example piston and cylinder. As the name suggests, feeler gauge called as to measure neither tide nor freely.
Feeler gauges are used for measurement of clearances.
It is a tool that measures air or narrow gap widths between two surfaces in engines and machinery.
Feeler gauge available in no. of the blade like 10,13,20 and 28. With a step of 0.05 and 0.10 mm.
## 26. Thermocouple
Thermocouple consist of two dissimilar conductor wire, made from different metals. These wire are welded together at one end, form a junction used to measure temperature.
## 27. Thermister
Thermistors have a high coefficient of resistivity. Thermistors are made of semiconductor of solid type.
The suitable temperature measuring range for thermistor is -100 degree Celsius to 300 degree Celsius. Some special type thermistor can measure up to 600 degree of temperature.
The change in temperature is measured by changes in its resistance. So, the Wheatstone bridge circuit is used for it.
Also, thermistor can convert changes in temperature to corresponding changes in voltage as a current.
## 28. Barometer
Air gets pulled down by gravity and hence air exerts pressure on objects. One such example of measurement is known as a barometer.
Barometers are scientific instruments that are used for the measurement of atmospheric or air pressure. It is an important and essential tool used in the meteorological department for forecasting short term weather and altitude changes.
The barometers should be kept at the same level and cannot measure air pressure above the altitude of 5,000 feet. Initially, water-based barometers were used, later it was replaced by mercury and aneroid barometers which the two most commonly used barometers.
## 29. Snap gauge
Snap gauge is GO and NOGO type gauge. That means snap gauge consist 2 fixed measured distance or gap, one is known as GO and another one is known as NOGO.
## 30. Plain Plug gauge
Plug gauge is used for assessment of hole or diameter of an object. With specified tolerance of GO and NOGO side, plug gauge is also known as pin gauge.
Behalf of checking the diameter plug gauge also use in comparing, setting, calibrating of other gauges.
## 31.Galvanometer
In an electric circuit, even if the switch or loop is closed, it is difficult to know whether the current is flowing, hence for this purpose galvanometer was introduced in the circuit.
Read More In Full Details and Videos
Like this Blog? | 2,332 | 10,662 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.90625 | 3 | CC-MAIN-2020-24 | latest | en | 0.904529 |
https://socratic.org/questions/how-do-you-simplify-3sqrt-1-8 | 1,701,941,293,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100651.34/warc/CC-MAIN-20231207090036-20231207120036-00622.warc.gz | 581,797,761 | 5,570 | How do you simplify 3sqrt( -1/8)?
$= \frac{3}{4} \sqrt{2} i$
$3 \sqrt{- \frac{1}{8}} = 3 \sqrt{\left(- 1\right) \frac{2}{2} ^ 4} = \frac{3}{2} ^ 2 \sqrt{2} i = \frac{3}{4} \sqrt{2} i$ | 98 | 184 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2023-50 | latest | en | 0.255371 |
https://www.shaalaa.com/question-bank-solutions/1-2-tan-x-2-1-4-tan-x-4-1-2-n-tan-x-2-n-1-2-n-cot-x-2-n-cot-x-all-n-n-0-x-2-principle-mathematical-induction_53390 | 1,618,733,337,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618038469494.59/warc/CC-MAIN-20210418073623-20210418103623-00435.warc.gz | 1,118,231,331 | 10,323 | Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
# 1 2 Tan ( X 2 ) + 1 4 Tan ( X 4 ) + . . . + 1 2 N Tan ( X 2 N ) = 1 2 N Cot ( X 2 N ) − Cot X for All N ∈ N and 0 < X < π 2 - Mathematics
$\frac{1}{2}\tan\left( \frac{x}{2} \right) + \frac{1}{4}\tan\left( \frac{x}{4} \right) + . . . + \frac{1}{2^n}\tan\left( \frac{x}{2^n} \right) = \frac{1}{2^n}\cot\left( \frac{x}{2^n} \right) - \cot x$ for all n ∈ and $0 < x < \frac{\pi}{2}$
#### Solution
We need to prove $\frac{1}{2}\tan\left( \frac{x}{2} \right) + \frac{1}{4}\tan\left( \frac{x}{4} \right) + . . . + \frac{1}{2^n}\tan\left( \frac{x}{2^n} \right) = \frac{1}{2^n}\cot\left( \frac{x}{2^n} \right) - \cot x$ for all n ∈ and $0 < x < \frac{\pi}{2}$ using mathematical induction.
For n = 1,
LHS = $\frac{1}{2}\tan\frac{x}{2}$ and
$RHS = \frac{1}{2}\cot\frac{x}{2} - \cot x = \frac{1}{2\tan\frac{x}{2}} - \frac{1}{\tan x}$
$\Rightarrow RHS = \frac{1}{2\tan\frac{x}{2}} - \frac{1}{\frac{2\tan\frac{x}{2}}{1 - \tan^2 \frac{x}{2}}}$
$\Rightarrow RHS = \frac{1}{2\tan\frac{x}{2}} - \frac{1 - \tan^2 \frac{x}{2}}{2 \tan\frac{x}{2}} = \frac{1 - 1 + \tan^2 \frac{x}{2}}{2\tan\frac{x}{2}} = \frac{\tan\frac{x}{2}}{2}$
Therefore, the given relation is true for n = 1.
Now, let the given relation be true for n = k.
We need to prove that the given relation is true for n = k + 1.
$\therefore \frac{1}{2}\tan\left( \frac{x}{2} \right) + \frac{1}{4}\tan\left( \frac{x}{4} \right) + . . . + \frac{1}{2^k}\tan\left( \frac{x}{2^k} \right) = \frac{1}{2^k}\cot\left( \frac{x}{2^k} \right) - \cot x$
Now, $\frac{1}{2}\tan\left( \frac{x}{2} \right) + \frac{1}{4}\tan\left( \frac{x}{4} \right) + . . . + \frac{1}{2^k}\tan\left( \frac{x}{2^k} \right) + \frac{1}{2^{k + 1}}\tan\left( \frac{x}{2^{k + 1}} \right) = \frac{1}{2^k}\cot\left( \frac{x}{2^k} \right) - \cot x + \frac{1}{2^{k + 1}}\tan\left( \frac{x}{2^{k + 1}} \right)$
Let: $L = \frac{1}{2^k}\cot\left( \frac{x}{2^k} \right) - \cot x + \frac{1}{2^{k + 1}}\tan\left( \frac{x}{2^{k + 1}} \right)$
$\Rightarrow L = \frac{1}{2^k}\cot\left( \frac{x}{2^k} \right) + \frac{1}{2^{k + 1}}\tan\left( \frac{x}{2^{k + 1}} \right) - \cot x$
$\Rightarrow L = \frac{1}{2^k \tan\left( \frac{x}{2^k} \right)} + \frac{1}{2^{k + 1}}\tan\left( \frac{x}{2^{k + 1}} \right) - \cot x$
$\Rightarrow L = \frac{1}{2^k \tan2\left( \frac{x}{2^{k + 1}} \right)} + \frac{1}{2^{k + 1}}\tan\left( \frac{x}{2^{k + 1}} \right) - \cot x$
$\Rightarrow L = \frac{1}{2^k \times \frac{2\tan\left( \frac{x}{2^{k + 1}} \right)}{1 - \tan^2 \left( \frac{x}{2^{k + 1}} \right)}} + \frac{1}{2^{k + 1}}\tan\left( \frac{x}{2^{k + 1}} \right) - \cot x$
$\Rightarrow L = \frac{1 - \tan^2 \left( \frac{x}{2^{k + 1}} \right)}{2^{k + 1} \tan\left( \frac{x}{2^{k + 1}} \right)} + \frac{1}{2^{k + 1}}\tan\left( \frac{x}{2^{k + 1}} \right) - \cot x$
$\Rightarrow L = \frac{1 - \tan^2 \left( \frac{x}{2^{k + 1}} \right) + \tan^2 \left( \frac{x}{2^{k + 1}} \right)}{2^{k + 1} \tan\left( \frac{x}{2^{k + 1}} \right)} - \cot x = \frac{1}{2^{k + 1}}\cot\left( \frac{x}{2^{k + 1}} \right) - \cot x$
Now,
$\frac{1}{2}\tan\left( \frac{x}{2} \right) + \frac{1}{4}\tan\left( \frac{x}{4} \right) + . . . + \frac{1}{2^k}\tan\left( \frac{x}{2^k} \right) + \frac{1}{2^{k + 1}}\tan\left( \frac{x}{2^{k + 1}} \right) = \frac{1}{2^{k + 1}}\cot\left( \frac{x}{2^{k + 1}} \right) - \cot x$
Thus,
$\frac{1}{2}\tan\left( \frac{x}{2} \right) + \frac{1}{4}\tan\left( \frac{x}{4} \right) + . . . + \frac{1}{2^n}\tan\left( \frac{x}{2^n} \right) = \frac{1}{2^n}\cot\left( \frac{x}{2^n} \right) - \cot x$ for all n ∈ and $0 < x < \frac{\pi}{2}$
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Class 11 Mathematics Textbook
Chapter 12 Mathematical Induction
Exercise 12.2 | Q 34 | Page 28 | 1,750 | 3,780 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.28125 | 4 | CC-MAIN-2021-17 | latest | en | 0.311018 |
https://www.unitconverters.net/radiation-activity/becquerel-to-kilobecquerel.htm | 1,723,128,003,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640728444.43/warc/CC-MAIN-20240808124926-20240808154926-00843.warc.gz | 801,207,667 | 2,840 | Home / Radiation-Activity Conversion / Convert Becquerel to Kilobecquerel
# Convert Becquerel to Kilobecquerel
Please provide values below to convert becquerel [Bq] to kilobecquerel [kBq], or vice versa.
From: becquerel To: kilobecquerel
### Becquerel to Kilobecquerel Conversion Table
Becquerel [Bq]Kilobecquerel [kBq]
0.01 Bq1.0E-5 kBq
0.1 Bq0.0001 kBq
1 Bq0.001 kBq
2 Bq0.002 kBq
3 Bq0.003 kBq
5 Bq0.005 kBq
10 Bq0.01 kBq
20 Bq0.02 kBq
50 Bq0.05 kBq
100 Bq0.1 kBq
1000 Bq1 kBq
### How to Convert Becquerel to Kilobecquerel
1 Bq = 0.001 kBq
1 kBq = 1000 Bq
Example: convert 15 Bq to kBq:
15 Bq = 15 × 0.001 kBq = 0.015 kBq | 267 | 633 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2024-33 | latest | en | 0.184764 |
https://www.sudokuwiki.org/Print_Cage_Unit_Overlap | 1,718,439,288,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861584.65/warc/CC-MAIN-20240615062230-20240615092230-00310.warc.gz | 904,870,399 | 4,631 | Main Page - Back From SudokuWiki.org, the puzzle solver's site
# Cage Unit Overlap
## Type 1
This is an important Killer Sudoku strategy which I have placed at the start of all the more complex strategies in the solver because it is so useful and very easy to spot. It is related to Intersection Removal. Whereas IR is the overlap of rows/columns with boxes, this is the overlap of 'cages' with rows, columns and boxes.
Each 'cage' is made up of one or more 'combinations' - sets of numbers that total the cage clue. If you can find a candidate number inside that cage that is not found elsewhere on the row, columns or box the cage is aligned on, then you know that number must appear in the cage. That part is self-evident since, in the solver at least, these numbers are displayed. Given that number is true, we can remove all combinations which omit that number. Often that means we can remove a bunch of numbers.
In the first example below, two such Cage/Unit Overlaps occur. The 2-cell cage with the red box has a clue of 14, which means the two combinations (visible if you hover over the cage on the solver) are {5,9} and {6,8}. The 6s in the cage are unique to that cage and the cage is entirely inside the box. So the only combination that fits is {6,8}. Hence we can remove 5s and 9s from the cage. In fact, since {6,8} is the only combination left both those numbers must fit in the cell and ALL other candidates can be removed, so the 4s and 3s can also go.
The blue ringed cage, a 3-cell with a clue of 19 gives us five different combinations. But the 9 in that cage is unique to both the cage and row H, so only the combinations with 9 in them are valid. {4,7,8} and {5,6,8} are not possible. Of the remaining candidates, the 5s can be removed.
Example 2 (Link) In this second example, five Cage/Unit Overlaps have been found. Taking just the centre one as an example, the red ringed 4-cell cage has a clue of 28 - the combinations being {4,7,8,9} and {5,6,8,9}. 7 is, however, unique to the cage and the centre box, so only the first combination can be valid. 5s, 6s and other numbers not in that combination (the 1s) can be removed. I'll leave it to you to show how the other four cages have similar eliminations. Overall, most Killer Sudoku puzzles will have at least one example of this strategy so they are well worth looking out for, and often you can reduce the puzzle with this method while looking into the cages with multiple combinations. Keep an eye out on the rows, columns and boxes you are studying.
## Type 2
If Type 1 above is related to Box Line Reduction, then you'd recognize that Type 2 will look like Pointing Pairs, it's opposite. Credit to Caleb Andrews for finding this example and stating the logic so elegantly.
In this puzzle (also used in the Innies/Outies example) we consider the 4-cage in a square CD67. Because the 4 is fixed in C7 we only need to consider the combinations that have 4, but
1489 <- no 1 left in the cage
2479 <- no 2 left in the cage
3469
3478
4567 <- no 5 left in the cage
so we are left with two combinations. Both contain 3 and 4. 4 is not helpful as we've already placed it, but look where 3 fits in the cage. It can only go in D6 and D7. That makes 3 a certainty in those cells. So it acts like a Pointing Pair aligned on the row. We can remove 3 from D1235.
There is another large example of this strategy in the same puzzle, on the big L cage starting on D3.
## Bi-Value attack on a cage
Another type of elimination I've chosen, a bit arbitrarily, to insert into the search at this strategy point is to compare a cage with a bi-value cell. This was a tip from Ivar Ling.
In this puzzle the 2-cage with a sum of 12 can only be {4/8} or {5/7} (with {3/9} already removed). Now a bi-value cell B1 can see all the cells in this cage and it contains a pair of candidates which matches one of those combinations. Since B1 must be either 4 or 8 it removes {4/8} combination from the cage. This will be reported as
Cage/Unit Overlap
CAGE vs BIVALUE: the cage starting on B6 has combinations which are ruled out by bi-value cell B1 which can see all cells in the cage, so
- we can remove 4/8 from B6
- we can remove 4/8 from B7
It is easy to imagine a Triple on two cells effecting a 3-cage as well, but the solver does not currently look for this.
Go back to Killer Cage Combination Example Continue to Innies and Outies
Article created on 22-March-2011. Views: 59097 | 1,155 | 4,450 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.625 | 4 | CC-MAIN-2024-26 | latest | en | 0.950565 |
https://www.jiskha.com/similar?question=What+type+of+waves+are+Earthquakes%3F | 1,560,808,728,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998580.10/warc/CC-MAIN-20190617203228-20190617225228-00070.warc.gz | 806,120,957 | 16,275 | # What type of waves are Earthquakes?
10,200 questions
1. ## physics
Earthquakes produce several types of shock waves. The most well-known are the P-waves (P for primary or pressure) and the S-waves (S for secondary or shear). In the earth's crust, the P-waves travel at around 6.5 km/s while the S-waves move at about 3.5
asked by mikeM on June 30, 2011
2. ## math
Earthquakes produce several types of shock waves. The most well-known are the P-waves (P for primary or pressure) and the S-waves (S for secondary or shear). In the earth's crust, the P-waves travel at around 6.5 km/s while the S-waves move at about 3.5
asked by mikeM on June 30, 2011
3. ## physics
Earthquakes are motions of the earth's crust. Essentially, they are big sound waves that travel through the earth. There are two types of waves. The first type is called a longitudinal or P wave, where the earth compresses in the same direction as the wave
asked by rohit on May 9, 2013
4. ## Geology
What type of waves are Earthquakes?
asked by Gary on December 6, 2007
5. ## physics
Earthquakes are essentially sound waves—called seismic waves—traveling through the earth. Because the earth is solid, it can support both longitudinal and transverse seismic waves. The speed of longitudinal waves, called P waves, is 7800m/s .
asked by Kate on April 7, 2014
6. ## Physics HELP!!
Earthquakes are essentially sound waves travelling through the earth. They are called seismic waves. Because the earth is solid, it can support both longitudinal and transverse sismic waves, which travel at different speeds. The speed of longitudinal
asked by sarah on November 21, 2012
7. ## Physics
Earthquakes are essentially sound waves travelling through the earth. They are called seismic waves. Because the earth is solid, it can support both longitudinal and transverse sismic waves, which travel at different speeds. The speed of longitudinal
asked by Patrick on November 23, 2010
8. ## Physics
Earthquakes are essentially sound waves travelling through the earth. They are called seismic waves. Because the earth is solid, it can support both longitudinal and transverse sismic waves, which travel at different speeds. The speed of longitudinal
asked by Mike on November 23, 2010
9. ## Physics
Earthquakes are essentially sound waves travelling through the earth. They are called seismic waves. Because the earth is solid, it can support both longitudinal and transverse sismic waves, which travel at different speeds. The speed of longitudinal
asked by Stella on November 22, 2010
10. ## physics
Earthquakes are essentially sound waves travelling through the earth. They are called seismic waves. Because the earth is solid, it can support both longitudinal and transverse sismic waves, which travel at different speeds. The speed of longitudinal
asked by Anonymous on November 21, 2008
11. ## Physics Waves
Earthquakes are essentially sound waves traveling through the earth. They are called seismic waves. Because the earth is solid, it can support both longitudinal and transverse seismic waves. These travel at different speeds. The speed of longitudinal
asked by Mike on April 21, 2010
12. ## Phsyics Waves
Earthquakes are essentially sound waves traveling through the earth. They are called seismic waves. Because the earth is solid, it can support both longitudinal and transverse seismic waves. These travel at different speeds. The speed of longitudinal
asked by Michael Moskvich on April 25, 2010
•What are earthquakes? •Where do they occur most often? Why? •How likely is it that your area will experience an earthquake? •What are the types of seismic waves? Describe/explain each of them. •What instrument do seismologists use to
asked by FluttershyK22 on April 28, 2016
14. ## Science
1. Describe how the magma is generated for volcanoes parallel to a subduction zone. 2. What two factors result in a volcano at a plate boundary? 3. Why are volcanoes found at mid-ocean ridges but not at ocean trenches? 4. What causes a hot spot in the
asked by Anjan on December 31, 2010
15. ## physics
An earthquake generates three kinds of waves: surface waves (L waves), which are the slowest and weakest; shear (S) waves, which are transverse waves and carry most of the energy; and pressure (P) waves, which are longitudinal waves and travel the fastest.
asked by maria on October 24, 2011
16. ## physics
An earthquake generates three kinds of waves: surface waves (L waves), which are the slowest and weakest; shear (S) waves, which are transverse waves and carry most of the energy; and pressure (P) waves, which are longitudinal waves and travel the fastest.
asked by fiko on October 24, 2011
17. ## physics
An earthquake generates three kinds of waves: surface waves (L waves), which are the slowest and weakest; shear (S) waves, which are transverse waves and carry most of the energy; and pressure (P) waves, which are longitudinal waves and travel the fastest.
asked by maria on October 24, 2011
18. ## Physical Science
Here are a few questions that I have answered; can someone please check if they are correct? Thanks! 1.In which type of waves do the particles in a medium move parallel to the direction that the waves move? Ans- longitudinal waves 2.Which type of wave
asked by mysterychicken on May 4, 2009
19. ## physics
An earthquake generates three kinds of waves: surface waves (L waves), which are the slowest and weakest; shear (S) waves, which are transverse waves and carry most of the energy; and pressure (P) waves, which are longitudinal waves and travel the fastest.
asked by maria on October 24, 2011
20. ## Science
I'm currently studying for a science test and I need help on one of the question. - What is a tsunami and how is it related to an earthquake? For the part " What is a tsunami, my answer would be Tsunamis are giant waves caused by earthquakes or volcanic
21. ## earth science
Which of the following statements is FALSE about the magnitude of earthquakes? A. Magnitudes of earthquakes are based on powers of ten B. An earthquake of magnitude 3 on the scale is 1.5 times more intense than a magnitude 2 C. Great earthquakes can have a
asked by alz on November 12, 2018
22. ## Science
Im sorry i am totally stumped What is the main difference between mechanical and electromagnetic waves A. Mechanical waves involve the transfer of energy; electromagnetic waves do not B. Mechanical waves require a medium to travel in; electromagnetic waves
asked by help me on April 27, 2016
23. ## physics
Which of the following conditions is not necessary to observe an interference pattern between two sets of waves? Choose one answer. a. The waves must have the same wavelength. b. The waves must have a constant phase difference. c. The waves must have the
asked by ashley -- on April 17, 2011
24. ## Science
Electromagnetic and mechanical waves have different characteristics that make them advantageous for certain applications. Select the reasons that mechanical waves, such as ultrasound, are used for soft tissue imaging and to identify fractures in metals.
asked by Maya on March 14, 2017
25. ## Science
What is the main difference between mechanical and electromagnet waves? a)Mechanical waves involve transfer of energy; electromagnet waves do not. b)Mechanical waves require a medium to travel in; electromagnet waves do not. c)Mechanical waves have
asked by need help on October 30, 2017
26. ## general idea
Hi what is the name of a large wave created by some earthquakes it is not tsunami off course Help plesaeee Since this is not my area of expertise, I searched Google under the key words "earthquake wave" to get these possible sources:
asked by nun on January 2, 2007
27. ## Science
What is the main difference between mechanical and electromagnetic waves? A. Mechanical weaves involved transfers energy; electromagnetic waves do not B. Mechanical waves require a medium to travel in; electromagnetic waves do not***** C. mechanical waves
asked by I need help on March 13, 2017
28. ## science
Electromagnetic and mechanical waves have different characteristics that make them advantageous for certain applications. Select the reasons that mechanical waves, such as ultrasound, are used for soft tissue imaging and to identify fractures in metals. I)
asked by jessica on April 9, 2018
29. ## Science
Earthquakes and Seismic Waves •If the statement is true right true if the statement is false change the underlined word or words(I’ll put them in parentheses) to make this statement true.• 1.______The shaking and trembling that results for movement
asked by Molly on February 8, 2018
30. ## science
"Tsunamis" are giant waves caused by earthquakes under the ocean floor. They ________ islands and coastlines? a.) provide drinking water for b.) form c.) damage d.) irrigate c
asked by pam on September 4, 2010
31. ## Earth Science REPOST
A seismograph station is located 2000 km from an earthquakes epicenter. Explain the order that the S and P waves will arrive at the station.
asked by Camron on October 17, 2007
32. ## Earth Science
A seismograph station is located 2000 km from an earthquakes epicenter. Explain the order that the S and P waves will arrive at the station.
asked by Camron on October 16, 2007
33. ## Physics
A wave travels at a speed of 34 cm/s. It’s wavelength is 18 cm. What is the period of the wave? 0.75 s 1.2 s 0.66 s 0.53 s Regions of compression and rarefaction help define _______. Electromagnetic waves Longitudinal waves, but not transverse waves
asked by GlowBaby on December 4, 2018
34. ## Jones
1. To what extent can earthquakes be predicted? 3. What can a trench dug across an active fault show about past fault movement? 4. What is a seismic gap, and what is its significance in determining future fault activity? 5. What information indicates the
asked by Joe on October 10, 2010
35. ## Physics
Which of the following is correct for the two coherent light waves to produce zero light? The light waves should travel in perpendicular directions. The light waves are in phase with each other. The light waves are out of phase with each other. The light
asked by Blah on October 19, 2012
36. ## Science
What type of signal is measured in a sonar system to determine the depth of the ocean? A. absorption of sound waves by underwater features B. reflection of sound waves by underwater features C. absorption of electromagnet waves by underwater features D.
asked by Dayana on March 22, 2012
37. ## Physics
The speed of longitudinal waves, called P waves, is 7274.0 m/s. Transverse waves, called S waves, travel at a slower 3803.0 m/s. A seismograph records the two waves from a distant earthquake. If the S wave arrives 4.47 min after the P wave, how far away
asked by Pete on November 23, 2010
38. ## Chemistry
The oscillation of electrons in an antenna is an example of which type of radiation/waves/rays? A. Radio waves B. Infrared rays C. Microwaves D. VLF radiation
asked by Morgan on December 16, 2014
39. ## Science
1. Which of the following statements about mechanical waves is true? a. mechanical waves require a medium to travel through b. mechanical waves do not have amplitude and wavelength c. mechanical waves do not have frequency d. mechanical waves can travel
asked by Thank you! on May 14, 2018
40. ## Geological Events
In 2005 and 2006, several major earthquakes occured in Pakistan and Iran, located to the north and west of India. A) Why does the theory of plate tectonics support the occurence of earthquakes in these countries? The plate tectonics theory states that the
asked by Anonymous on January 11, 2011
Sources and uses for the following: Radio waves Microwaves Infrared Waves Visible Light Waves Ultraviolet Waves X-Rays Gamma rays
asked by Unknown on October 30, 2013
42. ## Physics
A pond is 12 metres across. The crests of two successive waves are 60.0 cm apart and they each move across the pond in 15 seconds. a)What is the velocity of the waves? b)What is the frequency of the waves? c)What is the period of the waves?
asked by Sasha on November 23, 2014
43. ## Physics
A pond is 12 metres across. The crests of two successive waves are 60.0 cm apart and they each move across the pond in 15 seconds. a)What is the velocity of the waves? b)What is the frequency of the waves? c)What is the period of the waves?
asked by Mark on February 24, 2013
44. ## science
1. Why do earthquakes occur along the San Andreas Fault? a. Two plates meet at the San Andreas Fault and they slide next to each other causing earthquakes b. Two plates converge at the San Andreas Fault causing earthquakes*** c. Two plates pull apart at
asked by what on May 4, 2014
45. ## Quick Physics help
1. What is the medium for waves on a rope? 2. What is the medium for earthquake waves? 3. A characteristic of waves is that after the wave has passed the medium is (disturbed, undisturbed) is this disturbed? 4. A cork is floating in the water 20 m from the
asked by Anonymous on April 10, 2016
46. ## Science
1. Which of the following statements about mechanical waves is true? Mechanical waves require a medium to travel through. Mechanical waves do not have amplitude and wavelength.** Mechanical waves do not have frequency. Mechanical waves can travel through
asked by Tristan Seymour on April 25, 2019
47. ## chemistry
The oscillation of electrons in an antenna is an example of which type of radiation/waves/rays? A. Radio waves B. Infrared rays C. Microwaves D. VLF radiation I don't understand what is being asked so I don't have a clue on the answer.
asked by Morgan on December 16, 2014
48. ## science
How do the speeds of light waves and sound waves compare? Sound waves can travel faster depending on the medium. Sound waves always travel faster than light waves.**** Light waves travel at the same speed as sound waves. Light waves travel about 1 million
asked by noodles on April 10, 2019
49. ## Physics
Which of the following are not transverse waves? a) Wind (Airy) waves on the ocean. b) Water ripples in a small pool. c) Microwaves inside a microwave oven. d) Vibrations travelling on the string of a guitar. e) Radio waves travelling through space. f)
asked by Brandy on May 18, 2011
50. ## Physics
While watching ocean waves at the dock of the bay, Otis notices that 10 waves pass beneath him in 30 seconds. He also notices that the crests of successive waves exactly coincide with the posts that are 5 meters apart. What are the period, frequency,
asked by Shannon on March 14, 2011
51. ## physics
while watching ocean waves at the dock of the bay. otis notices that 10 waves pass beneath him in 30 seconds. he also notices that the crests of succesive waves exactly coincide with the posts that are 5 meters apart. what are the period frequency
asked by charlie on February 11, 2011
52. ## science
Certain high-frequency radiations are likely to cause harm to astronauts landing on Mars. A detector for which type of electromagnetic radiation is most suitable to study this harmful radiation on Mars? I think it is radio waves A. Infrared B. Radio waves
asked by ayeitsweston on May 10, 2018
53. ## Science: Earthquakes
In Science, we have to make pamphlet about Earthquakes by answering questions from a textbook. One question is. "What are the ffects associated with earthquakes?" I do not understand this question. Could I get some help what they are asking? Or give me
asked by Wenmar on October 23, 2007
54. ## Earth/Environmental Science
"Tsunamis" are giant waves caused bhy earthquakes under the ocean floor. They _______ islands and coastlines. What does tsunamis do to islands and coastlines?
asked by Anonymous on October 26, 2010
55. ## college physics
A student observes 30 waves traveling down a string in a lab experiment. It takes 10 seconds for the waves to all go by. What is the frequency of the waves (in Hz)?
asked by Taylor on July 25, 2016
56. ## Science
I needed help with this question. I’m not sure I got it quite right. Is it correct to say that in every case, without exception, any radio wave travels faster than any sound wave? ANSWER: Yes, it is correct to say because radio waves are electromagnetic
asked by Lizzie on January 25, 2010
57. ## Physics
Standing waves are produced in a string that is 4.0 m long. If the waves are travelling at 125 cm/s and the distance between the first and the fifth nodes is 80.0 cm, find the frequency of the waves.
asked by Lizzie on November 25, 2008
58. ## Science
Hi! I'm in 8th grade science 8-B in Connections academy. I have a question that I'm stuck on. Can anyone help me? Thx! 2. Where do most Earthquakes occur most often? Why I am taking a guess that areas such as California have more earthquakes than most. I
asked by # love mystery books! on March 30, 2017
59. ## science
If radio waves are not compressional waves,like sund waves, what is their role in enabling us to hear music?
asked by Juliann on December 4, 2007
60. ## science
Which of the following travels the fastest? sound waves in air, sonar, microwave radiation, waves produced by a sonic boom, Ultrasonic waves?
asked by Anonymous on April 30, 2011
61. ## physiiics
Which of the following is correct for the two coherent light waves to produce zero light? The light waves should travel in perpendicular directions. The light waves are in phase with each other The light waves are out of phase with each other. The light
asked by ooommgggg on February 11, 2013
62. ## Physics
Which of the following is correct for the two coherent light waves to produce zero light? The light waves should travel in perpendicular directions. The light waves are in phase with each other. The light waves are out of phase with each other. The light
asked by Brian on May 10, 2015
63. ## Physics
What is stimulated emission? a. the process of distinguishing between sources of electromagnetic waves. b. the process of reducing mechanical waves' energy. c. the process of amplifying sound waves. d. producing move light waves which are coherent with an
asked by Anonymous on March 9, 2017
64. ## Physics
Last question i'm stuck on please help me through this. Five waves pass a dock in 2.0 s. If the waves are traveling at 10.0 m/s, then what is the wavelength of the waves?
asked by Me and I on April 14, 2014
65. ## physics
1)The resultant wave from the interference of two identical waves traveling in opposite directions is described by the wave function y(x, t) = (2.49 m)sin(0.0458x)cos(5.40t), where x and y are in meters and t is in seconds. a) What is the frequency of the
asked by joy on April 25, 2018
66. ## Science
2. Which statement is true of P-waves? a. They travel the fastest through rocks. b. They vibrate at 90° to the wave motion. c. They are last to reach the epicenter. d. They are the most damaging. 4. When rocks break, they cause _____ that move throughout
asked by Jman on November 6, 2012
67. ## Physics
On December 26, 2004, a great earthquake occurred off the coast of Sumatra and triggered immense waves that killed hundreds of thousands of people. Satellites observing these waves from space measured 800 km from one wave crest to the next and a period
asked by Anonymous on July 14, 2010
68. ## Science
6. Sonar is used to inspect pipelines and bridge foundations. In this scenario, sonar takes advantage of the fact that (1 point) A.sound waves can reflect.*** B.sound waves can diffract. C.sound waves can spread out. D.sound waves can refract.
asked by jeje on January 21, 2015
69. ## Science
Certain seismic waves travel as.. A) electromagnetic and transverse waves. B) visible light and longitudinal waves. C) longitudinal and transverse waves. D) compression and visible light waves. I have no idea I never learned this:(
asked by Anon on January 14, 2016
70. ## Science
Earthquakes can form at what type(s) of plate boundaries? Convergent Divergent Transform Fault* All of the above You can look up pictures of all of these boundaries in google.
asked by Little Bird on January 2, 2019
71. ## science
do earthquakes occur in areas without identified large faults? The key words you use are "identified" and "large." As far as I know, earthquakes occur at fault lines, whether or not they have been identified or are large. Since this is not my area of
asked by alejandra on October 12, 2006
72. ## Physics
When interference happens with two monochromatic light waves, which of the following is a characteristic of the amplitude of the resultant wave? a. it is zero. b. it is equal to the sum of the amplitudes of the component waves. c. less than the amplitude
asked by Anonymous on March 8, 2017
73. ## Physics
The velocity of the transverse waves produced by an earthquake is 5.08 km/s, while that of the longitudinal waves is 8.3312 km/s. A seismograph records the arrival of the transverse waves 55.9 s after that of the longitudinal waves. How far away was the
asked by Laura on May 8, 2011
74. ## Algebra 1 high school
In deep water, the speed s (in meters per second) of a series of waves and the wavelength L (in meters) of the waves are related by the equation 2(pi)s^2=9.8L. a. Find the speed the the nearest hundredth of a meter per second of a series of waves with the
asked by Help now on August 27, 2013
75. ## physical science
can mechanical waves travel through any type of medium?
asked by Jam on May 6, 2017
76. ## PHISIC help
water waves in a ripple tank are diffracted by a narrow aperture.if the wavelength of the water waves is decreased slightly whilst keeping the aperture size the same ,which one of the following will happen? the waves will spread out more the waves will
asked by alicya on December 20, 2012
77. ## Physics
Outline the role of the medium in the propagation of: a) Longitudinal waves b) Transverse waves Not sure if I'm supposed to refer to the movement of particles in a medium compared to the direction of the energy for these waves...
asked by Anonymous on February 28, 2019
78. ## PHYSICS - Waves
Can someone tell me how to get the answer to this question? Just a formula would be great. It's on my physics study guide for my unit test. Sound waves with a wavelength of 2.5 m are traveling 7.5 m/s. How many waves will pass a microphone in 10.0 s?
asked by Diana on April 9, 2013
79. ## physics
In a double-slit experiment, two beams of coherent light traveling different paths arrive on a screen some distance away. What is the path difference between the two waves corresponding to the third bright band out from the central bright band? The path
asked by Anonymous on March 22, 2014
80. ## Science
1.What is refraction? The bending of light as it passes through matter 2.What type of lens would you use to magnify your view of a butterfly? Concave 3.What is a sound wave? Sound waves are waves of energy moving through matter.Sound waves are vibrations
asked by Derrick on March 20, 2012
81. ## Physical Science
I need help with a few of these questions: 1.Which of the following waves are mechanical waves? A.light waves B.microwaves C.radio waves D.sound waves I think it's D? 2.Which gives the frequency of the wave in terms of its peaks? A.the height of the
asked by mysterychicken on May 8, 2009
82. ## Physics
The velocity of the transverse waves produced by an earthquake is 7 km/s, while that of the longitudinal waves is 11.41 km/s. A seismo- graph records the arrival of the transverse waves 51.1 s after that of the longitudinal waves. How far away was the
asked by Evelyn on September 26, 2012
83. ## chemistry
We encounter a large number of waves every day without realizing it. One such source is radio waves. A typical frequency for radio transmission is 680 kHz. How much energy does one mole of radio waves carry? X rays used by doctors to view bones are much
asked by Maria on January 22, 2011
84. ## Science
What is the difference between p-waves, s-waves, and surface waves? Which do the most damage? Please and thank you!
asked by Autumn on November 14, 2007
85. ## Physics
The velocity of the transverse waves produced by an earthquake is 7.2 km/s, while that of the longitudinal waves is 4.2 km/s. A seismograph records the arrival of the transverse waves 68 s before that of the longitudinal waves. How far away was the
asked by Einstein on December 14, 2007
86. ## science, ipc
Which best describes longitudinal waves? ________________________________________ A. Compressions and rarefactions make up longitudinal waves, which can only travel in matter. B. Longitudinal waves carry sound energy and are made up of crests and troughs.
asked by Superman on July 15, 2010
87. ## physics
wonder waves in shallow yellow dA.isc are 6.0 cm long. at one point, the water oscillates up and down at a rate of 4.8 of oscillations per second. A. what is the speed of the water waves? B. what is the period Of water waves?
asked by Jordan on March 12, 2013
Win 1010” AM radio broadcasts at 1010KHz. Determine the wavelength of its signal, knowing that all radio waves travel at 3.00 X 10^8 m/s. f=1010kHz v=3.00 X10^8 m/s v=F X lambda lambda= 2.9 X 10^5 For a particular type of wave in a particular medium why
asked by susan on February 17, 2007
89. ## Physics
Radio waves are electromagnetic waves that travel at a speed of 3.00 108 m/s, the speed of light. An AM radio station has an assigned frequency of 1350 kHz, which means that the radio waves broadcast by the station are at this frequency. Find the
asked by Alex on April 22, 2012
90. ## social studies
1. Use the drop-down menu to answer the question. The religions of Santería and Vodou are examples of A. The impact of tourism on the Caribbean B. the influence of Creole on the Caribbean languages C. the survival of american Indian religious practices in
asked by eskit on November 6, 2018
91. ## mathematics
seeds of type A and type B are sold in packets, each must contain 1) both type a and type b seeds 2) at least twice the amount of type B as there are type A 3) no more than 12 seeds state the minimum number in each paket of type A and of type B if here are
asked by keisha on February 25, 2008
92. ## Science
3. Flood water pounding against a canyon wall and wearing it down is an example of * A Erosion B Deposition*** C Flood plain D Weathering 4. Many Processes on earth occur at or near tectonic plate boundaries. Which geological events are most common along
asked by not that that random guy that has a name on December 11, 2018
93. ## Physics
Two waves, identical except for phase, are traveling in the same direction through the same medium. Each of the waves has amplitude A. The amplitude of the combined (net) wave is 1.14A. What is the absolute value for the smallest possible phase difference
asked by Ck on October 8, 2016
94. ## physics
Radio waves are able to diffract readily around buildings, as anybody with a portable radio receiver can verify. However, light waves, which are also electromagnetic waves, undergo no discernible diffraction around buildings. Why not?
asked by Anonymous on June 26, 2008
95. ## physics
Which of the following are not transverse waves? a) Water ripples in a small pool. b) Sound waves traveling through water. c) Radio waves traveling through space. d) Microwaves inside a microwave oven. e) The vibrations on the surface of a drum.
asked by hilary on January 15, 2010
96. ## Physics
You are standing on the beach and you notice 4 waves travel over your feet in 8 seconds. Your friend is 4 meteres from you in the water and you count 3 waves between you. How fast do the waves travel on the water?
asked by conner on May 1, 2011
97. ## Physics
The velocity of the transverse waves produced by an earthquake is 6.13 km/s, while that of the longitudinal waves is 9.9919km/s. A seismograph records the arrival of the transverse waves 61.5 seconds after that of the longitudinal waves. How far away was
asked by Marie on September 21, 2014
98. ## physics
Water waves in a shallow dish are 5.0 cm long. At one point, the water moves up and down at a rate of 5.2 oscillations per second. (a) What is the speed of the water waves? (b) What is the period of the water waves?
asked by phillup on April 22, 2008
99. ## Physics
Two waves, both of amplitude A, are traveling in opposite directions along a rope. What is the range of displacement y that the two waves may cause when they overlap? -2A
asked by GlowBaby on December 4, 2018
100. ## physics
While watching ocean waves at the dock of the bay, Otis notices that 13 waves pass beneath him in 32 seconds. He also notices that the crests of successive waves exactly coincide with the posts that are 7 meters apart. What are the period, frequency,
asked by anna on April 14, 2011 | 7,146 | 28,736 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.40625 | 3 | CC-MAIN-2019-26 | latest | en | 0.95747 |
http://www.sawaal.com/average-questions-and-answers/without-any-stoppage-a-person-travels-a-certain-distance-at-an-average-speed-of-42-km-h-and-with-sto_11050 | 1,529,852,161,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267866965.84/warc/CC-MAIN-20180624141349-20180624161349-00510.warc.gz | 470,000,036 | 16,719 | 5
Q:
# Without any stoppage, a person travels a certain distance at an average speed of 42 km/h, and with stoppages he covers the same distance at an average speed of 28 km/h. How many minutes per hour does he stop?
A) 15 minutes B) 20 minutes C) 18 minutes D) 22 minutes
Explanation:
Let the total distance to be covered is 84 kms.
Time taken to cover the distance without stoppage = 84/42 hrs = 2 hrs
Time taken to cover the distance with stoppage = 84/28 = 3 hrs.
Thus, he takes 60 minutes to cover the same distance with stoppage.
Therefore, in 1 hour he stops for 20 minutes.
Q:
Third proportion of 10 and 20 is
A) 30 B) 40 C) 25 D) 20
Explanation:
The third proportional of two numbers p and q is defined to be that number r such that
p : q = q : r.
Here, required third proportional of 10 & 20, and let it be 'a'
=> 10 : 20 = 20 : a
10a = 20 x 20
=> a = 40
Hence, third proportional of 10 & 20 is 40.
3 351
Q:
The average weight of 45 passengers in a bus is 52 kg. 5 of them whose average weight is 48 kg leave the bus and other 5 passengers whose average weight is 54 kg join the bus at the same stop. What is the new average weight of the bus?
A) 54.21 kgs B) 51.07 kgs C) 52.66 kgs D) 53.45 kgs
Explanation:
Given total number of passengers in the bus = 45
First average weight of 45 passengers = 52 kgs
Average weight of 5 passengers who leave bus = 48
Average weight of passengers who joined the bus = 54
Therefore, the net average weight of the bus is given by
17 1212
Q:
12 boys decided to constribute Rs. 750 each to an Orphange. Suddenly few of them boys dropped out and consequently the rest had to pay Rs. 150 more. Then the number of boys who dropped out?
A) 4 B) 6 C) 2 D) 3
Explanation:
Total money decided to contribute = 750 x 12 = 9000
Let 'b' boys dropped
The rest paid 150/- more
=> (12 - b) x 900 = 9000
=> b = 2
Hence, the number of boys who dropped out is 2.
14 1404
Q:
P, Q, R, S, T and U are six consecutive odd numbers and their average is 52. What is the product of P and U ?
A) 2212 B) 2154 C) 2349 D) 2679
Explanation:
Let x, x+2, x+4, x+6, x+8 and x+10 are six consecutive odd numbers.
Given that their average is 52
Then, x + x+2 + x+4 + x+6 + x+8 + x+10 = 52×6
6x + 30 = 312
x = 47
So Product = 47 × 57 = 2679
13 1429
Q:
19 friends went to a restaurant for a combined weekend dinner party. 13 of them spent Rs. 79 each on their dinner and the rest spent Rs. 4 more than the average expenditure of all the 19. What was the total money spent by them?
A) Rs. 1398.96 B) Rs. 1457.09 C) Rs. 1662.35 D) Rs. 1536.07
Explanation:
Let average expenditure was Rs. R
13 x 79 + 6x(R + 4) = 19R
=> R = Rs. 80.84
Total money = 19 x 80.84 = Rs. 1536.07.
19 1306
Q:
The average age of a class is 19 years. While the average age of girls is 18 and that of boys is 21. If the number of girls in the class is 16, Find the number of Boys in the class?
A) 12 B) 10 C) 8 D) 6
Explanation:
Let the number of boys = x
From the given data,
=> [21x + 16(18)]/(x+16) = 19
=> 21x - 19x = 19(16) - 16(18)
=> 2x = 16
=> x = 8
Therefore, the number of boys in the class = 8.
13 1490
Q:
The mean of 100 observations is 40. It is found that an observation 84 was misread as 48. Then the correct mean is ?
A) 40.36 B) 41.24 C) 41.92 D) 42.05
Explanation:
Given,
mean of 100 observations is 40
=> Total of 100 observations = 40 x 100 = 4000
=> Difference = 84 - 48 = 36
=> Now, new total of 100 observations = 4000 + 36 = 4036
Correct Mean = 4036/100 = 40.36
16 1838
Q:
The average marks of a student in 7 subjects is 85. His average marks in 6 subjects except English is 83. What is his marks in English ?
A) 97 B) 98 C) 92 D) 93 | 1,244 | 3,718 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.5625 | 5 | CC-MAIN-2018-26 | latest | en | 0.917054 |
http://www.topgradepapers.com/yuma-needs-a-singer-singer-a-is-offering-her-services-for-an-initial-35-in-addition-to-25-per-hour-singer-b-is-offering-his-services-for-an-initial-65-in-addition-to-10-per-hour-when-will-the-t/ | 1,582,018,912,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875143646.38/warc/CC-MAIN-20200218085715-20200218115715-00413.warc.gz | 249,517,063 | 10,264 | You start with 80 and ADD 10 each week
—-> 80 + 10x
Your brother starts with 275 and SUBTRACTS 15 each week
—> 275 – 15x
Set expressions equal to find out when you have same amount as your brother.
80 +10x = 275 -15x
Add 15x to both sides
80 +25x = 275
Subtract 80 from both sides
25x = 195
Divide by 25
x = 7.8 (Round up to 8)
This means in 8 weeks you will have caught up and if fact have more money than your brother.
Since you save 10 per week, in 8 weeks the Total saved = 10*8 = 80 | 167 | 499 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2020-10 | longest | en | 0.943583 |
https://www.instructables.com/Arduino-Totally-Derivative-Fake-TV/ | 1,716,276,296,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058385.38/warc/CC-MAIN-20240521060250-20240521090250-00154.warc.gz | 727,744,079 | 41,248 | ## Introduction: Arduino & Neopixel Totally Derivative Fake TV
A maker named Jonathan Bush (JonBush) created an AT Tiny 85
powered Fake TV - Burglar Deterrent. This used Neopixel RGB LEDs and programming on the AT Tiny 85 chip to create a light show that, when seen through drapes or blinds at night, creates a nice illusion of someone being home watching TV. Please see his Instructable here.
Since my 7-yo son is interested in both electronics and programming, this seemed like a good simple project we could build together. So we quickly breadboarded one out using an Arduino Trinket and a strip of 8 Neopixels, all sourced from AdaFruit.com. JonBush's code compiled and ran perfectly on the Trinket.
Then we actually used the Fake TV when we went camping, leaving the thing standing guard in a bedroom window. Although the flickering light of the Fake TV was very realistic, I thought a few things could be improved, or if not improved, at least made ridiculously over complicated. So I set some goals...
## Step 1: Goals
These were my goals for this project.
1. Brighten it up. The original Fake TV (hereafter known as the oFTV) had a potentiometer (pot) to adjust the brightness. I had omitted that and hardcoded the brightness to the maximum. Still, with only 6 Neopixels in my version, it seemed a little dim.
2. Add a timer function. When I ran the oFTV while on vacation, I put it on plugin timer. Seemed ridiculous given that the Arduino has a clock (as it turns out, it doesn't actually have a real-time clock, just a timer - learned something there).
3. Change the "cut length" (time between lighting changes) to something more realistic than a linear distribution. The random() function returns a linear distribution, which seemed "unnatural". As it turn out there is significant interest in various directors' cut lengths and what distributions they best fit - scholarly papers on the subject even. I had assumed a normal Gaussian distribution (the famous bell curve), but most film scholars seem to think it's more of a lognormal distribution (although that remains contentious). http://www.cinemetrics.lv/cutting_on_salt.php
4. Add an occasional fade-to-black, again to increase the realism.
5. Stick some more controls on it. Mostly just because, you know, complications are fun.
This Instructable is my resulting dFTV (derivative Fake TV).
## Step 2: Parts
The oFTV used an AT Tiny 85 controller, which is literally just a bare 8 pin chip. No little board, no LEDs, no buttons, no headers, just that itty bitty chip and a sparse few components – a beautifully minimalistic design. In my dFTV redesign I went through an Arduino Uno, then a Trinket, and finally a Pro Trinket 5 volt. Everything about my dFTV is both more expensive and more complex.
1. Trinket Pro 5v, \$9.95 from adafruit.com
2. Perma-Proto Quarter-sized Breadboard, \$8.50 for a 3-pack from adafruit.com.
3. 16 RGB LED Neopixel compatible ring (\$4.00). I did all my testing with a 12 LED Neopixel ring from adafruit.com, but my final version used this a 16 LED ring from China. Ordering from China takes weeks, but you cannot beat the price. You can order it from here.
4. 5.5 x 2.1 mm female panel mount power jack. I had this on-hand, but should cost about a buck.
5. 100-200 µF electrolytic capacitor. The exact value doesn’t matter, you just want a fat little capacitor there to provide some filtering on the power rails. Does need to be a higher voltage rating than your power supply. Had on-hand, <\$1.
6. 10kΩ resistor. This could actually be 20k, 30k, or higher. 10kΩ is traditional for a pull-down resistor. Had on-hand, <\$1.
7. 470Ω resistor. You could go as low as 100Ω here and be fine I think. Had on-hand, <\$1.
8. Panel mount red LED. Had on-hand, ~\$1.
9. Three 100K pots. 10K would work fine as well. I got 10 for \$4.12 from China here.
10. Small panel mount push button. Had on-hand, ~\$1.
11. 5 – 16v power supply w. 5.5mm barrel connector. I cut the B side end off of a junker printer USB cable and soldered a barrel connector onto it. That way I could just use any available 5 volt USB phone/tablet charger as my power supply.
12. Project box to hold it. I found a lovely plastic project box with a clear top, again from China, \$2.69 here.
13. Assorted hookup wire and some solder. Because I’m doing this with a 7-yo, I used lead-free solder. Got to protect those brain cells!
## Step 3: About the Circuit
The heart and brains of the dFTV is the Pro Trinket 5v (\$9.95) from adafruit.com. I feel that this board is cool enough that it deserves a few words. Initially I started out with the original Trinket (\$6.95) but I quickly got into trouble with it and the project became unstable, but when I upgraded to the Pro Trinket all that went away.
The Pro Trinket is almost an Arduino Uno (same chip), but on a tiny little finger-tip sized PCB. It has a micro-USB connector that can be used to both, power the board, and to program it. About the only things you lose with the Pro Trinket over the Uno is: serial output through the USB port, and pins #2 and #7 support.
One feature that the Pro Trinket has that I used in this project is a built-in 5v power regulator (150mA), so this project can run on anything from 5 – 16v. As it happens, I’m running it from a 5v USB wall-wart, but I could run it off of something else, including batteries, if I wanted to.
You do have to make some minor changes to the Arduino IDE to program the Pro Trinket. Excellent documentation and tutorials on the Pro Trinket are here on the AdaFruit website, those folks do an amazing job.
Above is the fritzing drawing of my final project. I did all my testing and debugging on an Arduino Uno with a standard breadboard. Once I had everything working like I wanted, I replaced the Uno with the Pro Trinket, moving the wires one-for-one, pin-for-pin.
Then I squished all the parts together until it fit on just ¼ of the breadboard. That let me use the AdaFruit Perma-Proto Quarter-sized PCB for the final version. I love these little PCBs (also available in ½ and full size) because they exactly replicate a standard breadboard. All you have to do is move your components over and solder them down just as they were on the breadboard.
The fritzing drawing shows the AdaFruit 12 LED Neopixel ring, and I did actually use that during prototyping and testing, but the final version uses a 16 LED Neopixel compatible ring from China. For \$4 you can’t beat the price, and since I was ordering a bunch of stuff from there anyway, why not?
The 100kΩ pots are wired together point to point off of the PCB. Basically I’m using wires to extend the power rails to the three pots. This lets me reduce the number of wires snaking back and forth. The pushbutton also catches the 5v rail this way. As I mentioned, 10kΩ pots would also work fine here. I used the 100k ones because they have less current leakage across the rails.
The center connections on the pots are the wipers, and are connected to analog input pins A1, A2 and A3 respectively. The pots are acting as variable voltage dividers. The analog input pins will see a varying voltage, going from 0 – 5v, as each pot is turned.
R1 on the diagram is a 10kΩ pull-down resister connected to digital input pin 4 on the Pro Trinket, with its other side connect to the ground rail. The pushbutton is also connected to pin 4, with its other side connected to the 5v rail. If the button is not pushed, pin 4 “sees” the ground rail through the pull-down resister and reads that as logic 0. If the button is pushed, pin 4 see the full 5v from the power rail (minus a tiny bit that leaks through the pull-down resister) and reads that as 1. If the pull-down resister were not there, pin 4 would be “floating” when the button wasn’t pushed and randomness would result.
Interestingly, Arduinos have built-in pull-up resistors that can be activated by the INPUT_PULLUP mode parameter to the pinMode() function. Using that would have eliminated the external pull-down resistor. In that case, the button would be wired to the ground rail instead of the 5v rail, and its state would be reversed (1 when not pushed, 0 when pushed).
Either way of handling the button would be correct. I chose to use a pull-down resister just because I was trying to learn and understand the whole pull-up/down thing.
R3 is a current limiting resistor for the red LED activity light. The LED will light whenever pin 13 goes high. The Pro Trinket and the Uno already have a built-in LED connected to pin 13, which will light too - making my external LED both redundant and optional. The reason I have it there it to bring the LED outside of the project case so it’s easily visible to the user.
The electrolytic capacitor is there to provide some filtration on the power rails. Presumably as the Neopixel LEDs are flashing away, their current draw is fluctuating the voltage on the rails. The capacitor is a reservoir of charge that can smooth those fluctuations out. I’ve actually run this project without that capacitor, and most of the Neopixel examples on the Adafruit website also omit it. Still, having it there is a good idea. Watch the polarity! Electrolytic capacitors tend to explode when you plug them in backwards.
The 5.5mm power jack is wired to the BAT (battery) and the G (ground) pins on the Pro Trinket. This means that power goes through the built-in voltage regulator. Input voltage can be anything from 5 – 16v. The power rails on the Perma-Proto board get their regulated 5volts from the G and 5v pins. The output of the Pro Trinket is rated at 150mA. I don’t know what the Neopixel ring is drawing, but nothing seems to be getting hot on the Pro Trinket, no magic blue smoke, so I guess everything’s okay.
## Step 4: The Code
This code is pretty heavily modified from what JonBush initially published. Major changes are:
• Eliminated the use of the delay() function. In the original code the time between each cut (lighting change) was spent in a delay() state. This causes a problem reading the sensors (pots and button) since the Arduino won’t detect any events or changes until the delay() expires. It’s not such a problem for the pots, but for the button you might have to hold it down for 4 seconds before it got picked up. This code spends all of its time looping as fast as possible looking for changes in sensors, or time counters.
• Used the multiMAP() function (from rob.tillaart@removethisgmail.com) to convert the linear distribution generated by the random() function into something else. I tried several distributions: Gaussian, lognormal, and some I just made up. I left all of them there in the comments. In the end, I’m not sure any of them made much difference, linear was probably just fine.
• Added a timer to end the light show after a certain amount of time. Controlled by a pot.
• Added a cut speed multiplier, controlled by a pot.
• Added a lightshow abort, controlled by the pushbutton.
• Added a soft reboot function after 24 hours (86,400,000 ms). This causes all the counters and the light show to restart at the same time every day.
• Added an Activity LED on pin 13.
• Initialized all of the various program parameters to reasonable values so that any or all of the pots, and the switch, can be eliminated.
• The routine softReset() forces a program jump to address 0. This has the effect of resetting all the counters and restarting the code from the beginning. It’s a trick to simulate pushing the reset button. It wouldn’t compile for the Trinket, but works on the Pro Trinket and Uno. I don’t know what other Arduinos it might or might not work on.
• Fade-to-black. This I never got around to implementing. Fail.
I also commented the code pretty completely, so read through it if you have questions.
## Step 5: Code Source Listing
`//DIY Fake TV`
`//Keep burglars at bay while you are away`
```//Created by Jonathan Bush
//Last Updated 3/24/2015
//Runnig on ATTiny 85```
```//Modified 8/4/2015 by Mark Werley for Trinket Pro
//Rewritten to avoid the use of the delay() function
//Initialized MAXBRIGHT to 255, the maximum
//Added the use of the multiMAP function to convert a uniform distribution to something nonuniform
//Added an auto-off feature so the LEDs go dark after a certain amount of time
//Added a soft reboot feature after 24 hours, so the show starts up again every day at the same time
//Added a blink on the builtin LED, so user can tell program is still running when neopixels are off```
`#include <Adafruit_NeoPixel.h>`
```#define PIN 3 //PIN 3 "runs" the NeoPixels. Works on Uno or Trinket Pro
#define ledPin 13 //PIN 13 has built-in LED for Uno or Trinket Pro```
`#define buttonPin 4 //PIN 4 has a button attached through a 10k pullup resister`
```int ledState = LOW; //Keep track of the state of the built-in LED
int buttonState = 0; //Keep tract of the state of the button
int endShow = false; //True when the show is over```
```int POTPIN = A1; //1st analog pot pin, used for adjusting brightness
int POTPIN2 = A2; //2nd analog pot pin, used for adjusting light show cut speed
int POTPIN3 = A3; //3rd analog pot pin, used for adjust the runtime of the show```
```//Neopixel library provided by Adafruit, change 1st parameter to number of LEDs in your neopixels
Adafruit_NeoPixel strip = Adafruit_NeoPixel(16, PIN, NEO_GRB + NEO_KHZ800);```
```int BRIGHTNESS = 0;
int RED = 0;
int BLUE = 0;
int GREEN = 0;
int TIMEDEL = 0;
int mapTIMEDEL = 0;
int MAXBRIGHT = 255; //set MAXBRIGHT to 255, the max, if no brightness pot
int speedDivider = 1; //set speedDivider to 1, if no cut speed pot
int potval = 0;
int potval2 = 0;
int potval3 = 0;```
```unsigned long runTimeMillis = 0; //How long the Fake TV light show will run in milliseconds
int runTime = 120; //How long the Fake TV light show will run in minutes, 2 hours if no runTime pot
unsigned long startMillis = 0; //The sampled starting time of the program, ususally just 0 or 1 milliseconds
unsigned long previousMillis = 0; //Remember the number of milliseconds from the previous cut trigger
unsigned long rebootTimeMillis = 0; //How long the program will run before a soft reset/reboot happens (24 hrs = 86,400,000 ms)
unsigned long currentMillis = 0; //How long the program has run so far```
`int in[] = {200, 520, 840, 1160, 1480, 1800, 2120, 2440, 2760, 3080, 3400, 3720, 4040}; //This is just linear`
```// int out[] = {200,392,968,2120,3272,3848,4040,3848,3272,2120,968,392,200}; //normal distribution
// int out[] = {200,392,2120,3848,4040,3848,3560,3272,2696,2120,968,392,200}; //LnNormal-ish
// int out[] = {200,250,300,400,600,1200,4040,1200,600,400,300,250,200}; //made up
int out[] = {200, 250, 300, 350, 400, 500, 600, 700, 800, 1200, 2000, 3000, 4040}; //made up #2```
```void setup() //Initialize everything
{
//Initialize the NeoPix
strip.begin();
strip.show(); // Initialize all pixels to 'off'
pinMode(PIN, OUTPUT); //set the neopixel control pin to output```
` pinMode(ledPin, OUTPUT); //set the onboard LED pin to output`
` pinMode(buttonPin, INPUT); //set the button pin to input`
``` //Initialize the serial com, used for debugging on the Uno or Trinket Pro (w FTDI cable), comment out for production
Serial.begin(9600);
Serial.println("--- Start Serial Monitor SEND_RCVE ---");
Serial.println("Serial is active");
Serial.println();
rebootTimeMillis = 24ul * 60ul * 60ul * 1000ul; //hardcode reboot in 24 hours```
``` startMillis = millis(); //sample the startup time of the program
}```
```void loop() //Start the main loop
{
currentMillis = millis(); //sample milliseconds since startup```
``` if (currentMillis > rebootTimeMillis) softReset(); //When rebootTimeMillis is reached, reboot
// Let's read our sensors/controls```
``` potval = analogRead(POTPIN); //Reads analog value from brightness Potentiometer/voltage divider, comment out if not using
MAXBRIGHT = map(potval, 0, 1023, 11, 255); //Maps voltage divider reading to set max brightness btwn 11 and 255, comment out if not using```
``` potval2 = analogRead(POTPIN2); //Reads analog value from cut speed Potentiometer, comment out if not using
speedDivider = map(potval2, 0, 1020, 1, 8); //Maps the second pot reading to between 1 and 8, comment out if not using
potval3 = analogRead(POTPIN3); //Reads analog valuse from show lenth Potentiometer, comment out if not using
runTime = map(potval3, 0, 1020, 15, 480); //Maps the third pot to between 15 and 480 minutes (1/4 to 6 hours), comment out if not using
runTimeMillis= long(runTime) * 60ul * 1000ul;
Serial.print("potval3=");
Serial.print(potval3);
Serial.print(" runTime=");
Serial.print(runTime);
Serial.print(" runTimeMillis=");
Serial.println(runTimeMillis);
buttonState = digitalRead(buttonPin); //Sample the state of the button
if (buttonState == HIGH) endShow = true; //Button was pressed, time to end tonight's show```
``` if ((currentMillis - previousMillis) > long(mapTIMEDEL)) //Test to see if we're due for a cut (lighting change)
{
BRIGHTNESS = random (10, MAXBRIGHT); //Change display brightness from 20% to 100% randomly each cycle
RED = random (150 , 256); //Set the red component value from 150 to 255
BLUE = random (150, 256); //Set the blue component value from 150 to 255
GREEN = random (150, 256); //Set the green component value from 150 to 255
TIMEDEL = random (200, 4040); //Change the time interval randomly between 0.2 of a second to 4.04 seconds```
``` mapTIMEDEL = multiMap(TIMEDEL, in, out, 13); //use the multiMap function to remap the delay to something non-uniform
mapTIMEDEL = mapTIMEDEL / speedDivider; //Divide by speedDivider to set rapidity of cuts```
` if ((currentMillis - startMillis) > runTimeMillis) endShow = true; //runTimeMillis has expired, time to end tonight's show`
``` if (endShow) //Show's over for the night, aw...
strip.setBrightness(0);
else //The show is on!
strip.setBrightness(BRIGHTNESS);```
` colorWipe(strip.Color(RED, GREEN, BLUE), 0); //Instantly change entire strip to new randomly generated color`
``` if (ledState == HIGH) //toggle the ledState variable
ledState = LOW;
else
ledState = HIGH;```
` digitalWrite(ledPin, ledState); //Flip the state of (blink) the built in LED`
``` previousMillis = currentMillis; //update previousMillis and loop back around
}
}```
```// Fill the dots one after the other with a color
void colorWipe(uint32_t c, uint8_t wait)
{
for (uint16_t i = 0; i < strip.numPixels(); i++)
{
strip.setPixelColor(i, c);
strip.show();
}
}```
```// Force a jump to address 0 to restart sketch. Does not reset hardware or registers
void softReset()
{
asm volatile(" jmp 0");
}```
```//multiMap is used to map one distribution onto another using interpolation
// note: the _in array should have increasing values
//Code by rob.tillaart@removethisgmail.com
int multiMap(int val, int* _in, int* _out, uint8_t size)
{
// take care the value is within range
// val = constrain(val, _in[0], _in[size-1]);
if (val <= _in[0]) return _out[0];
if (val >= _in[size - 1]) return _out[size - 1];```
``` // search right interval
uint8_t pos = 1; // _in[0] allready tested
while (val > _in[pos]) pos++;```
``` // this will handle all exact "points" in the _in array
if (val == _in[pos]) return _out[pos];```
``` // interpolate in the right segment for the rest
return (val - _in[pos - 1]) * (_out[pos] - _out[pos - 1]) / (_in[pos] - _in[pos - 1]) + _out[pos - 1];
}```
## Step 6: Assembly
I used the Adafruit Perma-Proto for this project because it makes it so easy to move from breadboard to final version. Of course, you could also use any regular perf board, and that would be cheaper.
My project box came from China and has a transparent top, which is great for letting the Neopixel light out into the room. With an opaque box you would have to mount the Neopixels on the outside of the box.
Holes are drilled through the sides of the box for the: pots, pushbutton, activity LED, and power jack. These are connected back to the Perma-Proto board with hookup wire.
I just taped the Neopixel ring to the inside of the lid. I should think of something better than that.
## Step 7: Controls and Usage
Yeah, that’s a lot of knobs. My Grandpa’s shortwave receiver had fewer dials on it. But they all do something useful(ish), and, as I said, I like complications.
• Pot1: Adjusts brightness. Varies between 11 and 255. Is initialized to the max 255 if pot is omitted.
• Pot2: Adjusts cut speed. Varies between 1x (RomCom) and 8x (ActionAdventure). Is initialized to 1x if no pot.
• Pot3: Adjusts the length of the light show. Varies between 15 minutes and 6 hours. Is initialized to 2 hours if no pot.
• Pushbutton: Aborts the current light show, but doesn’t end the program or change any settings. Useful for when the wife says, “Honey, can you turn that thing off now?”, but you don’t want to unplug it.
• Activity LED, continues to blink after the light show ends, to let you know the program hasn’t crashed.
Once the code is loaded on your Arduino, usage is easy. Just plug it in to power and the light show will start. This establishes the start time. Turn the Brightness knob to adjust the average brightness of the show.
Turn the Cut Rate pot to change how often the Neopixels change. Set all the way up (8x), they get quite frantic; very Michael Bay.
As the dFTV is running, it’s counting down to the end of the light show. When the time for the end of the light show is reached, the Neopixels go dark, but the program continues to run and the activity LED continues to blink. Turn the Show Length pot to adjust when the show will end.
Let’s say you want the light show to run from 8PM to 10PM, just plug it in at 8 and that sets the start time. Then at 10PM you can turn back the Show Length pot until the Neopixels just go out, and that sets the end time. 24 hours after the dFTV was initially plugged in, the software resets and the light show starts up again.
## Step 8: Future Upgrades
One of the things I like about this project is that the pushbutton and pots are all defined by the software, as is the Neopixel display. Want to have a tint knob? Just change the code. Want to turn the whole thing into a disco light show? Change the code again.
Adding a backup battery to keep the program running if there’s a power flicker would be a good addition.
Normally an Arduino doesn’t have a real-time clock, which is why all of my code is based on counting down from the moment power is applied. Adding a RT clock module would make programming and setting the thing much more intuitive.
Each Neopixel is individually addressable. Here they all show the same thing at the same time, but it doesn’t have to be that way. You could, for example, implement a clock face on a Neopixel ring that counts down till when the light show starts again.
Implementing a fade-to-black, which I failed to do, would also be a nice thing. Please share your code if you do. | 5,981 | 23,012 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2024-22 | latest | en | 0.951498 |
http://www.nwchem-sw.org/index.php?title=P-4b2&printable=yes | 1,580,161,769,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579251728207.68/warc/CC-MAIN-20200127205148-20200127235148-00516.warc.gz | 256,907,533 | 4,832 | ##### Forum Menu
modified on 19 March 2011 at 17:58 ••• 2,300 views
# P-4b2
Jump to: navigation, search
```group number = 117
group name = P-4b2
crystal system = Tetragonal
setting number = 1
number of symmetry operators = 8
+x,+y,+z
-x,-y,+z
+y,-x,-z
-y,+x,-z
+x+1/2,-y+1/2,+z
-x+1/2,+y+1/2,+z
+y+1/2,+x+1/2,-z
-y+1/2,-x+1/2,-z
= operator 1 =
1.0 0.0 0.0 0.0
0.0 1.0 0.0 0.0
0.0 0.0 1.0 0.0
= operator 2 =
-1.0 0.0 0.0 0.0
0.0 -1.0 0.0 0.0
0.0 0.0 1.0 0.0
= operator 3 =
0.0 1.0 0.0 0.0
-1.0 0.0 0.0 0.0
0.0 0.0 -1.0 0.0
= operator 4 =
0.0 -1.0 0.0 0.0
1.0 0.0 0.0 0.0
0.0 0.0 -1.0 0.0
= operator 5 =
1.0 0.0 0.0 0.5
0.0 -1.0 0.0 0.5
0.0 0.0 1.0 0.0
= operator 6 =
-1.0 0.0 0.0 0.5
0.0 1.0 0.0 0.5
0.0 0.0 1.0 0.0
= operator 7 =
0.0 1.0 0.0 0.5
1.0 0.0 0.0 0.5
0.0 0.0 -1.0 0.0
= operator 8 =
0.0 -1.0 0.0 0.5
-1.0 0.0 0.0 0.5
0.0 0.0 -1.0 0.0
``` | 593 | 926 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2020-05 | latest | en | 0.690071 |
https://www.jiskha.com/questions/1774057/Each-of-3micro-C-are-placed-at-three-corners-of-a-square-whose-diagonal-is-6-m-long | 1,556,124,763,000,000,000 | text/html | crawl-data/CC-MAIN-2019-18/segments/1555578650225.76/warc/CC-MAIN-20190424154437-20190424180437-00335.warc.gz | 738,638,716 | 5,212 | # Physics
Each of +3micro C are placed at three corners of a square whose diagonal is 6 m long find feild intensity at the point if intersection of diagonal
1. 👍 0
2. 👎 0
3. 👁 40
1. you only need to find the field intensity fro the "middle" corner
... the two diagonally opposite corners will cancel each other
1. 👍 0
2. 👎 0
posted by scott
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asked by Jnr John on August 10, 2014
8. ### calculus
A 33 by 33 square piece of cardboard is to be made into a box by cutting out equal square corners from each side of the square. What size corners should be cut out so that the volume of the box is maximized?
asked by james on February 20, 2012
9. ### Math
we are working on a puzzle similar to a magic square. 9 boxes are arranged in an X with the center box being shared by both diagonals. the goal is to use each integer 1-9 one time so that each diagonal sums to 26 and the 4 corners
asked by Chuck on March 30, 2011
10. ### Physics
Two adjacent corners of a square have charges q1 = +1.5 x 10-9 C and q2 = +4.0 x 10-9 C. The length of a side of the square L = 0.25 m. Find the EPE of a charge q3= -6.0 x 10-9 C placed at the 2 remaining corners.
asked by K on July 10, 2008
More Similar Questions | 794 | 2,894 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.34375 | 3 | CC-MAIN-2019-18 | latest | en | 0.95784 |
https://web2.0calc.com/questions/help-please_15783 | 1,660,546,856,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882572161.46/warc/CC-MAIN-20220815054743-20220815084743-00396.warc.gz | 524,158,823 | 5,912 | +0
0
248
2
+55
In the SuperLottery, three balls are drawn (at random) from ten white balls numbered from 1 to 10, and one SuperBall is drawn (at random) from ten red balls numbered from 11 to 20. When you buy a ticket, you choose three numbers from 1 to 10 and one number from 11 to 20.
If the numbers on your ticket match at least two of the white balls or match the red SuperBall, then you win a super prize. What is the probability that you win a super prize?
I'm asking this again, because the original post did not have any correct answers. Link to original post: https://web2.0calc.com/questions/in-the-superlottery-three-balls-are-drawn-at-random
Jul 22, 2021
#1
+36431
+1
Here is my take on this one:
Matching two AND Superball
10 c 2 = 45
10 c 1 = 10 (for Superball) 45 * 10 = 450 possibles for matching TWO numbers correct and Super
Matching THREE and superball
10C3 = 120
10C1 = 10 (for Superball) 120*10 = 1200 possibles for matching THREE numbers and Super
Matching two
45
matching three
120
matching superball
10
YOU get ONE ticket 1 out of 450 + 1 out of 1200 + 1 out of 45 + 1 out of 120 + 1 out of 10
1/450 1/1200 1/45 1/120 1/10 = 481/3600
Different answer....I do NOT know if it is correct!
Jul 22, 2021
edited by ElectricPavlov Jul 22, 2021
edited by ElectricPavlov Jul 22, 2021
#2
+55
0
Thanks so much for replying! But this isnt correct
BigBoiChungus Jul 22, 2021 | 477 | 1,510 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.9375 | 4 | CC-MAIN-2022-33 | latest | en | 0.806095 |
https://solvedbyme.live/answer/237636/ | 1,695,613,840,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233506676.95/warc/CC-MAIN-20230925015430-20230925045430-00072.warc.gz | 596,063,582 | 43,566 | 24x^2 - 45x - 21 = 0
$$24x^2-45x-21=0\\ 3(8x^2-15x-7)=0\\ 8x^2-15x-7=0\\ \Delta=(-15)^2-4\cdot8\cdot(-7)=225+224=449\\ \sqrt{\Delta}=\sqrt{449}\\ x_1=\dfrac{-(-15)-\sqrt{449}}{2\cdot8}=\dfrac{15-\sqrt{449}}{16}\\ x_2=\dfrac{-(-15)+\sqrt{449}}{2\cdot8}=\dfrac{15+\sqrt{449}}{16}$$
$$24x^2-45x-21=0 3(8x^2-15x-7)=0 /:3 8x^2-15x-7=0 a=8... b=-15.... c=-7 delta= b^2-4ac=449 x_1= \frac{15- \sqrt{449} }{16} x_2= \frac{15+ \sqrt{449} }{16}$$√ΔΔΔΔΔΔΔΔΔ | 267 | 447 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.953125 | 4 | CC-MAIN-2023-40 | latest | en | 0.164595 |
http://mathhelpforum.com/calculus/152799-how-do-i-solve-integral.html | 1,529,634,815,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267864337.41/warc/CC-MAIN-20180622010629-20180622030629-00202.warc.gz | 201,995,682 | 9,423 | # Thread: How do I solve this integral?
1. ## How do I solve this integral?
Hi there. Well, the problem statement says: Calculate the arc length of the cardioid $\displaystyle \rho=a(1+\cos\theta)$.
So I used the formula for the arc length in the polar form:
$\displaystyle \int_a^b \! \sqrt{(\rho)^2+(\displaystyle\frac{d\rho}{d\theta} )^2} \, d\theta$
Then I get
$\displaystyle \int_0^{2\pi} \! \sqrt{(a+a\cos\theta)^2+(-a\sin\theta)^2} \, d\theta$
$\displaystyle =\int_0^{2\pi} \! \sqrt{a^2+2a^2\cos\theta+a^2\cos^2\theta+a^2\sin^2 \theta} \, d\theta$
$\displaystyle =\int_0^{2\pi} \! \sqrt{a^2(2\cos\theta+\cos^2\theta+\sin^2\theta)} \, d\theta$
$\displaystyle =\int_0^{2\pi} \! |a|\sqrt{2\cos\theta+\cos^2\theta+\sin^2\theta} \, d\theta=|a|\int_0^{2\pi} \!\sqrt{2\cos\theta+1} \, d\theta$
I can't solve this integral, I don't know how to. Any help?
2. Originally Posted by Ulysses
Hi there. Well, the problem statement says: Calculate the arc length of the cardioid $\displaystyle \rho=a(1+\cos\theta)$.
So I used the formula for the arc length in the polar form:
$\displaystyle \int_a^b \! \sqrt{(\rho)^2+(\displaystyle\frac{d\rho}{d\theta} )^2} \, d\theta$
Then I get
$\displaystyle \int_0^{2\pi} \! \sqrt{(a+a\cos\theta)^2+(-a\sin\theta)^2} \, d\theta$
$\displaystyle =\int_0^{2\pi} \! \sqrt{a^2+2a^2\cos\theta+a^2\cos^2\theta+a^2\sin^2 \theta} \, d\theta$
$\displaystyle =\int_0^{2\pi} \! \sqrt{a^2(1+2\cos\theta+\cos^2\theta+\sin^2\theta) } \, d\theta$ ... algebra error here (fixed)
...
3. Thanks skeeter. | 604 | 1,532 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.984375 | 4 | CC-MAIN-2018-26 | latest | en | 0.488056 |
https://usaonlineessays.com/fte-variance_calculation/ | 1,568,552,085,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514571360.41/warc/CC-MAIN-20190915114318-20190915140318-00174.warc.gz | 637,242,091 | 16,471 | # FTE VARIANCE_CALCULATION
Order Description
calculate full-time equivalents for a nursing unit and analyzing a variance scenario.
Answer the questions and complete the calculations required for the two sections of the assignment.
• Calculations of full-time equivalents (FTEs)
• Variance analysis
Submit your answers on a Word document, with the heading of Week 4 Assignment. For the questions requiring a written response, please adhere to proper grammar and syntax, and provide references.
For the questions requiring calculations, show all your work and follow the format that has been provided for the calculations in the lesson for Week 4.
PREPARING THE PAPER
Section One—Calculation of Full-Time Equivalents
Personnel Budget Case Study
Linda Watson has recently accepted a position as the assistant administrator for the department of nursing at St. Joseph Hospital. There have been both personnel and financial problems at St.
Joseph Hospital, and recently, the Catholic Healthcare System purchased the hospital. Like Linda, all of the management staff is new. Linda’s first priority is to assist the nurse managers in
preparing their cost centers’ personnel budgets for the coming fiscal year. She has decided to begin with the nurse manager on 1 West, a step-down nursing unit. Linda wants to do extensive data
collection and create a solid base for making projections.
1. Suggest all the information Linda needs to collect to prepare a budget for 1 West; be sure to include background data on the unit itself as it relates to previous budgets, nursing staff, and the
organization. In other words, summarize your approach to this Case Study—briefly state the organization’s goals (customer-service orientation) for the coming year, trends, any changes in trends
that you may anticipate—such as growth in the patient population and the types of data that need to be collected in order to make decisions about staffing levels and ratios.
2. Linda has collected the following information, in addition to what was asked in question number one:
Patient Data:
Average Daily Census 30
Unit Capacity 32
Average HPPD 8.8
Total Care Hours 96,360
Staff Data:
Productive hours/employee/year 1,780
Nonproductive hours/employee/year 300
Total Hours/employee/year 2,080
Skill Mix:
RNs 80%
LVNs 10%
Nurse aides 10%
Average Salary per Employee Category:
RN \$40.00/hour
LVN \$22.00/hour
Nurses Aide \$14.00/hour
Calculate the number of productive full-time equivalents (FTEs) that would be needed. Show your calculations.
3. How many RNs, LVNs, and nurses aides will be needed to staff this unit, assuming staff will be working 12-hour shifts? In other words, how many person-shifts will be required during a 24-hour
period?
Day Shift 50%
Night Shift 50%
4. Assign staff by type and by shift.
Day Shift 50%
Night Shift 50%
5. Convert staff positions to FTE positions.
Section Two—Variance Analysis
Variance Analysis Case Study
During the month of February, an outpatient surgery clinic has incurred a significant unfavorable variance. The director of the clinic is quite concerned, as this has never occurred before. The
director gathers information on total nursing care hours, average hourly rate of the employees, and total patient visits to determine what caused the variance. In addition, the director receives
the patient acuity levels for the month of February.
1. BPi \$40.00
2. BQi 5.0 hours per care per patient
3. BQo 340 visits
4. APi \$45.00
5. AQi 5.6 hours per care per patient
6. AQo 400 visits
1. Compare the original budget to the flexible budget and calculate the volume variance. Compare the flexible budget to the actual budget and determine the price and quantity variances. Work
through the entire analysis process as presented in your lesson for Week 4, and remember to show your calculations.
2. What factors caused the difference in the variances? Analyze all the factors that made the differences and why the variances occurred.
Category Points % Description
Background Information 15 15% Typically, the description of the maximum level of performance is provided here
Calculation of Full-Time Equivalents (FTEs) 10 10%
Shifts Required During a 24-Hour Period
10 10%
Staff Assigned by Type and Shift 10 10%
Staff Positions to FTEs 10 10%
Calculation of Volume Variance 10 10%
Calculations of Price and Quantity 10 10%
Calculation of Total Variance 10 10%
Analysis of Variances 15 15%
100 100
A quality assignment will meet or exceed all of the above requirements.
Assignment Criteria Exceptional
(100%)
Outstanding or highest level of performance Exceeds
(88%)
Very good or high level of performance Meets
(80%)
Competent or satisfactory level of performance Needs Improvement
(38%)
Poor or failing level of performance Developing
(0)
Unsatisfactory level of performance
Content
Possible Points = Points
Background Information 15 Points 13 Points 12 Points 6 Points 0 Points
Background information for budget analysis is thoroughly and clearly identified and stated. Background information for budget analysis is partially identified and stated, but could be more thorough
or clear. Background information for budget analysis lacks thorough and clear statements. Background information for budget analysis is minimally identified and stated. Background information for
budget analysis is not identified or stated.
Calculation of Full-Time Equivalents (FTEs) 10 Points 0 Points
Calculation of FTEs is correct. Calculation of FTEs is incorrect.
Shifts Required During a 24-Hour Period
10 Points 0 Points
Shifts required during a 24-hour period are correctly calculated. Shifts required during a 24-hour period are incorrectly calculated.
Staff Assigned by Type and Shift 10 Points 5 Points 0 Points
Staff assigned by type and shift are correctly calculated.
Staff assigned by type is correctly calculated; staff assigned by shift is incorrectly calculated.
Staff assigned by type is incorrectly calculated; staff assigned by shift is correctly calculated.
Suggest: One of the following is not correctly calculated: staff assigned by type or staff assigned by shift. Both staff assigned by type and by shift are not correctly calculated.
Staff Positions to FTEs 10 Points 0 Points
Calculation of staff positions to FTEs is correct. Calculation of staff positions to FTEs is incorrect.
Calculation of Volume Variance 10 Points 0 Points
Calculation of Volume Variance is correct. Calculation of Volume Variance is incorrect.
Calculations of Price and Quantity Variance 10 Points 0 Points
Calculation of Flexible Budget Variance is correct. Calculation of Flexible Budget Variance is incorrect.
Calculation of Total Variance 10 Points 0 Points
Calculation of Total Variance is correct. Calculation of Total Variance is incorrect.
Analysis of Variances 15 Points 13 Points 12 Points 6 Points 0 Points
Analysis of variances is thoroughly and clearly identified and stated. Analysis of variances is adequately identified and stated. Analysis of variances is partially identified and stated. Analysis
of variances is minimally identified and stated. Analysis of variances is not identified and stated.
Content Subtotal 100 of points
Total Points _____of points
LESSION
With the current economic climate of healthcare, the nurse executive needs to be proficient in the management of fiscal and human resources. A key component of the nurse-executive role, as
identified by the American Organization of Nurse Executives (AONE) Nurse Executive Competencies (AONE, 2005) is business skills. Business skills encompass both financial management and human-
resource management. The first step in either area is an assessment of the resources of the organization, the strategic plan for the organization, and current needs of the organization. Without
this assessment, the nurse executive cannot effectively develop and manage resources.
Budget
“What is a budget? In simple terms, a budget is a statement of a financial plan over a period of time. A budget provides guidance on how the resources of an institution will be used to meet its
strategic plan. The organization’s financial performance is measured against the budget. In some healthcare institutions, middle- and upper-level managers’ performance appraisals and incentives are
aligned with the fiscal performance of the institution.
Budgeting Process
The nurse executive will start the budgeting process by comparing the actual results of the prior fiscal year with the proposed budget of that year. This review also involves an evaluation of the
outcomes of the strategic plan against the budget. The next step in the process is to review the current strategic plan, determine the resources needed to meet the goals for patient care, and
establish the needs of the nursing department. To accomplish that step, the nurse executive must have, at minimum, the following data: patient days, patient acuity, length of stay, staffing model,
staff mix, number of open positions by level, number of anticipated vacancies by level, cost of agency staff by unit, projected salary increases and bonuses, cost of staff benefits, educational
needs of the staff in the upcoming fiscal year, and anticipated nonproductive staff time. Other information that will be crucial to the budgeting process will include any anticipated regulatory
changes, regulatory surveys anticipated for the fiscal year, status of equipment on nursing units, and anticipated changes in reimbursement. The nurse executive cannot develop the budget in
isolation. Input by nurse managers, unit directors, the finance department, and the human resource department is critical to a successful budget development cycle.
Calculating a Salary Budget
The salary budget is generally the largest element in the budget of the organization. The salary budget is established by determining the number of full-time-equivalent (FTE) staff needed for the
fiscal year. To prepare a salary budget, the first step is to figure out the volume of work for the coming year. The amount of work performed by a unit/department is referred to as the workload.
Workload can be measured in a variety of ways, such as the number of patient days, number of outpatient visits, and number of operations per day. Each cost center determines the measure that is
most appropriate for its unit of service. The next step of the budget process will be to predict the number of units of service that will be provided the coming year. This is usually based on the
past year’s number of units, plus a predicted incremental increase for the coming year, based on population increases and needs.
Budgeting is based on a patient-classification system, which is the amount of resources used related to different levels or categories of patient acuity. For example, let’s say that a certain
cardiac-observation unit is forecasting 12,500 patient-days for the coming year. Assume that this unit classifies these 12,500 patients into one of four categories: category one being fewest
resources used or lowest acuity; and category four being most resources used or highest acuity patients. Assume that, for category one, the average-patient-care-hours equals 1.5 hours of nursing
care provided to these patients during a 24-hour period; for category two, 5.0 care hours; for category three, 8.0 care hours; and, for category four, 12.0 hours of nursing care is provided to
these patients during a 24-hour period. Hours-per-patient-day (HPPD) is the number of hours of nursing care delivered per patient. To calculate HPPD, the total-care-hours required for the year is
divided by the number of patient days.
Calculating Hours Per-patient-day
1. Establish Hours-Per-Patient-Day (HPPD)
a. Calculate the total-care-hours required for the year:
# of Patient Days X Average Care Hours/24 Hours = Total Care Hours
1 1,000 1.5 = 1,500
2 2,500 5.0 = 12,500
3 4,000 8.0 = 32,000
4 5,000 12.0 = 60,000
12,500 106,000 total-care-hours needed
b. Calculate the hours-per-care-per-patient-day (HPPD):
Total-Care-Hours Required for the Year / Number of Patient Days
106,000 / 12,500 = 8.48 care hours
Thus, for this cardiac observation unit, an average of 8.48 nursing-care hours are provided to patients per a 24-hour period. Once the manager has the workload forecasted, the next step is to
determine the staff requirements for the unit. Staff mix varies from hospital to hospital and may consist of RNs, LPNs, and aides. The manager needs to determine how many full-time equivalents
(FTEs) will be required to provide patient care. One FTE is equivalent to 2,080 hours per year (40 hours per week times 52 weeks). A 0.9 FTE is equivalent to 36 hours per week; a 0.6 FTE is equal
to 24 hours per week. If the manager calculated FTE requirements by dividing total-care-hours by 2,080 to determine the number of FTEs for the unit, this would underestimate the needed FTEs. This
calculation would assume that all 2,080 paid hours were worked by the staff member. However, employees take vacation time, sick leave, and educational days—which are nonworked time, or
nonproductive hours. Instead, we must relate to productive hours rather than paid hours. Data on nonproductive hours are generally provided by the institution’s payroll department and equal between
15–20% of paid hours. In our example, let’s say that productive hours equal 85% of the employee time worked, which would equal 1,768 productive hours.
FTE Calculation
2. Apply HPPD to Census and Determine Full-Time Equivalents (FTEs)
a. Determine FTE requirements for number of expected patient -care hours:
Total Care Hours /1,768 productive hours per FTE
(1,768 = 85% of 2,080; 40 hours per week X 52 weeks per year = 2,080; less vacation, sick, education time)
106,000 / 1,768 = 59.95 FTEs = 60 FTEs
Next, the manager must determine how to allocate the number of FTEs per position type and shifts. To calculate how many staff members will be working every day, first determine the number of care
hours that are required per day.
Staff Requirements
3. Establish Positions by Type and Shift
a. Calculate number of staff required per 24-hour period:
Divide total care hours by 365 days per year
106,000 / 365 = 290 hours of care per day
Assuming employees work 12 hour shifts:
290 / 12 = 24.2 shifts per 24-hour period
Let’s say in our example, 80% of the shifts are allocated to RNs and 20% to aides. Assume that 50% of the RN/aide shifts are for day shift (7a–7p) and 50% for the night shift (7p–7a).
Staff Requirements Continued
3. b. Assign staff by type:
80% RN staff; 20% aides
.80 X 24.2 = 19.4 RN shifts ? 20 RN shifts
.20 X 24.2 = 4.8 aide shifts ? 4 aide shifts
24.2 24
c. Assign staff by shift:
Day Shift Night Shift Total
50% 50%
RNs 9.7 ? 10 9.7 ? 10 19.4 ? 20
Aides 2.4 ? 2 2.4 ? 2 4.8 ? 4
24.2?24
To establish staff positions, it is important to calculate the relationship between shifts per day and FTEs. This is the point where replacement is accounted for as well assuring that the unit is
covered 24/7—the number of staff who are needed to the number of FTEs to be employed.
Staff to FTE
4. Convert Staff Positions to FTE Positions
a. Calculate FTEs by type and shift:
Divide total FTEs required by the 24.2 daily staff shifts
59.95 / 24.2 = 2.48
Each person-shift calls for employing 2.47 FTEs. An adequate number of FTEs are needed to provide coverage for the other 3–4 days out of the week, vacation, sick time, and education time.
Multiply 2.47 by the staff type needed per shift.
RNs days = 2.48 X 10 = 24.8
RNs nights = 2.48 X 10 = 24.8 (50 RNs)
Aides days = 2.48 X 2 = 4.96
Aides nights = 2.48 X 2 = 4.96 (10 aides)
59.5 FTEs = 60FTEs
Variances
Variance analysis is the difference between the actual results and the planned results, or the amount by which the results vary from the budget. Variances are calculated for three reasons. One is
to aid in preparing the budget for the coming year and future planning. The second reason is to aid in controlling results throughout the current year. By understanding why variances are occurring,
actions can be taken to eliminate some of the unfavorable variances over the coming months. The third reason is to evaluate the performance of units or departments and their managers.
At the end of a given time period, the organization compares actual results with the budget. Most organizations do this on a monthly basis. The simplest approach is to compare the total costs for
the entire organization with the budgeted costs for the organization. For example, suppose a hospital incurred \$5 million in expenses and was budgeted for \$4 million in expenses for the year; the
organization spent \$1 million more than it had budgeted. The difference between the amount budgeted and the amount actually incurred is the total hospital variance. This variance is referred to as
an unfavorable variance because the organization spent more than was budgeted.
The volume variance is defined as the amount of variance caused simply by the fact that the patient volume has changed. For example, if the budget calls for 25,000 patient days, and there were
actually 30,000 patient days, it would be expected that it would be necessary to spend more to care for the additional 5,000 patients. The additional cost of resources needed to care for the
additional 5,000 patients constitutes the volume variance.
The price variance is the portion of the total variance caused by spending more or less per unit—for some resources—than had been anticipated. For example, if the average wage rate for nurses is
more per hour than was anticipated with the use of agency nurse or overtime, it would give rise to a price variance. Price variance could also be related to supplies. The purchasing department
predicts all prices used for supplies; if there is a variance, this may be related to paying higher prices than anticipated for patient supplies.
The third type of variance is the quantity variance. This portion of the overall variance for a particular line item that results from using more of a resource than was expected for a given work
load. For example, if more supplies were used per patient day than expected, that would give rise to a quantity variance, because the quantity, or number, of supplies used per patient day exceeded
expectations. Basically, this variance focuses on how much resources were used. If more nursing care resources were used per patient than was budgeted, this would translate into greater hours-per-
care per patient.
Flexible Budget Variance
Suppose that during the month of May, the 2nd-floor critical-care unit (CCU) at Wagner Hospital incurs a total variance of \$57,840U (unfavorable). The unit manager of the CCU is extremely concerned
and wants to find out what caused the variance. To do so, the manager gathers the following information for the month of May: patient days (actual and forecasted), average pay rate for nurses, and
the budgeted hours-per-patient-day.
BPi = budgeted price—\$33.00 per-hour budgeted nursing rate
BQi = budgeted quantity—12.0 hours- per- care per patient
BQo = 240 budgeted patient days
APi = \$42.00 per-hour actual nursing rate
AQi = 14.0 actual hours-per-care per patient
AQo = 260 actual patient days
Suppose that the CCU had the following line item in their variance report for the month of May:
Actual Budget Variance
Nursing Labor \$152,880 \$95,040 \$57,840 U
1. The first step for the manager is to calculate the original budget in terms of cost per patient day. In order to calculate cost per patient day, the manager multiplies the budgeted price by
budgeted quantity: \$33.00 X 12.0 = \$396 per patient day. For the month of May, 240 patient days are expected, and the budget is: \$396 X 240 = \$95,040.
Original Budget
BPi X BQi X BQo
\$33 X 12.0 X 240
\$95,040
2. The next step for the manager is to find the change in volume of the flexible budget. This is the amount the manager would have expected to spend if the actual number of patient days would have
been known. The only change at this point is from BQo of 240 patient days to the new AQo of 260 actual patient days. The manager calculates the flexible budget as follows.
Flexible Budget
BPi X BQi X AQo
\$33 X 12.0 X 260
\$102,960
Then, the manager compares the original budgeted amount to the flexible budgeted amount to determine the volume variance.
Flexible Budget Original Budget
BPi X BQi X AQo BPi X BQi X BQo
\$33 X 12.0 X 260 \$33 X 12.0 X 240
\$102,960 \$95,040
Volume Variance
\$7,920 U
The difference between the original budget and the flexible budget is caused by a difference in the number of patient days. The volume variance is 7,920 U. Since patient days are higher than
expected, cost will be higher than expected. However, the manager notes that \$49,920 of the variance still remains to be unaccounted for.
3. At this point, the manager compares the flexible budget to the actual budget in order to find the flexible-budget variance. The actual hours-per care, the actual average-price paid-per=nurse,
and the actual number- of-patient-days are used.
Actual Budget Flexible Budget
APi X AQi X AQo BPi X BQi X AQo
\$42 X 14.0 X 260 \$33 X 12.0 X 260
\$152,880 \$102,960
Flexible Budget
Variance
\$49,920 U
4. The manager is simply breaking down the total variance into parts. Next, the manager combines the volume variance with the flexible-budget variance to determine the total variance.
Actual Budget Flexible Budget Original Budget
APi X AQi X AQo BPi X BQi X AQo BPi X BQi X BQo
\$42 X 14.0 X 260 \$33 X 12.0 X 260 \$33 X 12.0 X 240
\$152,880 \$102,960 \$95,04
Flexible Budget Volume
Variance Variance
\$49,920 U \$7,920 U
Total Variance
\$57,840 U
At this point, the total variance has been broken down into a flexible-budget variance and volume variance. The flexible-budget variance is of greater concern to the nurse manager. The volume
variance is from changes in patient days and is usually outside of the manager’s control.
5. In order to find out what is causing the greatest variance in the flexible budget, the manager separates the budget into two pieces: the price variance (average nursing rate per hour) and the
quantity variance (hours per care). First, the manager determined if any part of the variance in the flexible budget was caused by differences in the quantity—or, nursing-care hours. In other
words, did the added patient days have an effect on the quantity of nursing time per patient? To do this, a new budget category is created, called the subcategory. The subcategory is compared to
the flexible budget—which equals the difference between actual patient-care hours compared to the patient-care hours that were budgeted. The AQo is the actual volume for both categories. The BQi is
the budgeted price-per-hour-of-nursing- time. The only difference between the two categories is the AQi, or the patient-care hours.
Subcategory Flexible Budget
BPi X AQi X AQo BPi X BQi X AQo
\$33 X 14.0 X 260 \$33 X 12.0 X 260
\$120,120 \$102,960
Quantity Variance
\$17,160
The quantity variance, or hours per care, was related to the patient population being more acuity ill, thus requiring more patient-care hours, and usage of agency personnel who were not as
efficient in caring for patients due to their unfamiliarity with the institution.
Second, the manager determines if some part of the flexible-budget variance was caused by differences in nursing-rate per hour. The Qi, or quantity (patient-care hours), is the actual value for
both. The Qo, or the volume, is the actual value for both. The price is the only difference. Thus, to calculate the price variance, compare the subcategory to the actual budget—this is the
difference between the average hourly rate paid to nurses versus the average hourly rate budgeted.
Actual Budget Subcategory
APi X AQi X AQo BPi X AQi X AQo
\$42 X 14.0 X 260 \$33 X 14.0 X 260
\$152,880 \$120,120
Price Variance
\$32,760
The price variance of \$32,760 results from the fact that, on average, \$42 was paid per hour for nursing time instead of the budgeted \$33. This was due to an excessive amount of overtime and use of
agency nurses due to a greater workload of patients and a nursing shortage (i.e., several senior nurses retired, several nurses changed units/hospitals/careers due to burn-out, and hospital has a
policy to hire only seasoned critical-care nurses to work in their CCU).
6. Finally, the manager incorporates the price and quantity variance together to note how they compromise the flexible-budget variance:
Actual Budget Subcategory Flexible Budget
APi X AQi X AQo BPi X AQi X AQo BPi X BQi X AQo
\$42 X 14.0 X 260 \$33 X 14.0 X 260 \$33 X 12.0 X 260
\$152,880 \$120,120 \$102,960
Price Variance Quantity Variance
\$32,760 \$17,160
Flexible Budget
Variance
\$49,920
In conclusion, it can be determined that the total variance was caused by only a slight increase in patient volume, yet a higher acuity of patients—causing a greater usage of overtime and agency
nurse utilization resulting in a higher average rate paid to nurses.
Actual Subcategory Flexible Budget Original Budget
APi X AQi X AQo BPi X AQi X AQo BPi X BQi X AQo BPi X BQi X BQo
\$42 X 14.0 X 260 \$33 X 14.0 X 260 \$33 X 12.0 X 260 \$33 X 12.0 X 240
\$152,880 \$120,120 \$102,960 \$95,040
Price Variance Quantity Variance
\$32,760 \$17,160
Flexible Budget Variance Volume Variance
\$49,920 \$7,920
Total Variance
\$57,840
FTE Practice
FTE Calculations | 6,017 | 25,405 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.703125 | 3 | CC-MAIN-2019-39 | longest | en | 0.926941 |
https://mail.haskell.org/pipermail/haskell-cafe/2015-February/118415.html | 1,669,868,326,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710789.95/warc/CC-MAIN-20221201021257-20221201051257-00366.warc.gz | 406,220,747 | 3,280 | # [Haskell-cafe] A Proposed Law for Foldable?
Edward Kmett ekmett at gmail.com
Sat Feb 28 17:57:25 UTC 2015
```The Foldable/Monoid constraints are unrelated in behavior, so not
necessarily.
Why?
At first blush you might think you'd get there via the free theorems, but
Monoid is only on *, so you can use the inclusion of the argument in the
Monoid instance to incur further constraints.
newtype Foo a = Foo a deriving (Monoid, Foldable)
now Foo has
instance Foldable Foo
instance Monoid m => Monoid (Foo m)
it lifts the Monoid to the element inside, but if you fold it you get the
single value contained inside of it, not mempty.
Now if you want to upgrade this approach all the way to Alternative, rather
than Monoid then the free theorem arguments start getting teeth.
foldMap f empty = mempty
should then (almost?) hold. I say almost because you *might* be able to
have empty be an infinitely large tree which never gets down to values
somehow, in which case that law would require deciding an infinite amount
of work. I haven't sat down to prove if the latter is impossible, so I
characterize it as at least plausible.
-Edward
On Fri, Feb 27, 2015 at 6:00 PM, Daniel Díaz <diaz.carrete at gmail.com> wrote:
> Hi,
>
> Sorry for the slight derail, but I wanted to ask the following doubt: if a
> Foldable type also happens to be a Monoid (say, like Set) does that
> automatically imply that toList mempty = [] ?
>
> On Friday, February 27, 2015 at 8:18:05 PM UTC+1, Gershom B wrote:
>>
>> On February 27, 2015 at 1:39:10 AM, David Feuer (david... at gmail.com)
>> wrote:
>> > I am still struggling to understand why you want this to be a law for
>> > Foldable. It seems an interesting property of some Foldable instances,
>> > but, unlike Edward Kmett's proposed monoid morphism law, it's not
>> > clear to me how you can use this low to prove useful properties of
>> > programs. Could you explain?
>>
>> I think there are a number of purposes for laws. Some can be thought of
>> as “suggested rewrite rules” — and the monoid morphism law is one such, as
>> are many related free approaches.
>>
>> Note that the monoid morphism law that Edward provides is _not_ a
>> “proposed” law — it is an “almost free theorem” — given a monoid morphism,
>> it follows for free for any Foldable. There is no possible foldable
>> instance that can violate this law, assuming you have an actual monoid
>> morphism.
>>
>> So Edward may have proposed adding it to the documentation (which makes
>> sense to me) — but it provides absolutely no guidance or constraints as to
>> what an “allowable” instance of Foldable is or is not.
>>
>> But there are other reasons for laws than just to provide rewrite rules,
>> even though it is often desirable to express laws in such terms. Consider
>> the first Functor law for example — fmap id === id. Now clearly we can use
>> it to go eliminate a bunch of “fmap id” calls in our program, should we
>> have had them. But when would that ever be the case? Instead, the law is
>> important because it _restricts_ the range of allowable instances — and so
>> if you know you have a data type, and you know it has a functor instance,
>> you then know what that functor instance must do, without looking at the
>> source code.
>>
>>
>>
> _______________________________________________
> Libraries mailing list | 850 | 3,346 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2022-49 | latest | en | 0.919896 |
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