text:
# How do runescribes activate the desired effect from ideographic runes with multiple, different meanings?

There exists a radical offshoot of the Elder Futhark runic alphabet called kig-anr (“stone-symbols”) that evolved into an ideographic form consisting of over fifty different characters. People who use kig-anr to perform magic get referred as “runescribes” who do so through two means with tools that need to have magical energy channelled into them during useage: inscribing runes onto a solid surface with a sharp metal stylus which requires a lot of time and steady hand or custom-made wands called vǫndr to quickly project runes onto tangible objects by pointing at them.

Once manifested, the kig-anr runes take instant effect, burning out after they activate unless they have a klik-kig (“stop-symbol”) following them. A klik-kig serves as a punctuation marker of sorts indicating the end of a runic chain while syphoning life-force from living organisms (plants, animals or microbes) within a one-hundred-meter radius which becomes converted into energy used to fuel the runes’ effects. If the klik-kig gets destroyed through physical means, the runes will cease functioning and become nothing more than simple carvings.

However, each rune has multiple meanings attached to them and I'm clueless as to how a runescribe would invoke a desired effect from their chosen runes without the stupid, impractical idea of calling them out loud (something that you'd expect to find in shitty Shonen animu).

How can such a problem get solved?

• The only realistic option I see is adding enough runes to give context. is there a problem with being more verbose with your rune inscriptions? Jan 25, 2021 at 19:52

“When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean — neither more nor less.’

’The question is,’ said Alice, ‘whether you can make words mean so many different things.’

’The question is,’ said Humpty Dumpty, ‘which is to be master — that’s all.”$$^1$$

Like in common language a single word can have multiple meanings but the context in which that word is placed usually eliminates the ambiguity, so for this particular usage the context where the rune is placed self clarify which effect is desired.

Example with English, used as safe language since it doesn't have magic effects attached:

Rune: LEFT

• Usage 1: Turn LEFT alone (the target turns left)

• Usage 2: Be LEFT alone (the target is separated from its group)

$$^1$$ Source

• Left "laevus" and left ("residuus", pple. of leave) are different words, with different histories, which just happened to sound the same in Modern English. (In Old English they were lyft and lǣfd. The process you are describing is called phonetic borrowing (where the symbol for one word is used to write a different, unrelated but similarly sounding word), and is a characteristic of logographic writing system, such as the one used by the Chinese. Jan 25, 2021 at 8:54
• @AlexP They are different words but the arrangement of glyphs is identical, which is the heart of the question.
– rek
Jan 26, 2021 at 5:53
• @rek: That is because we are writing English with an alphabet; that is, we represent the sounds of words, as opposed to a logographic system which represents words more akin to Platonic ideals. In a logographic system, there is no particular reason for words with the same pronunciation to have the same logogram, and they usually do not. In a logographic writing system, when two words with the same (or similar) pronunciation share the same logogram (or similar logograms) we call that a "phonetic borrowing". Jan 26, 2021 at 6:09
• @AlexP OP's Futhark is described as an alphabet, not logographic.
– rek
Jan 26, 2021 at 15:29
• Consider the antonyms 'oversight': does the spell reveal hidden details, or make something harder to notice? Now what if 'sight' and 'site' were written phonetically, does it make the location levitate? This is OP's problem as I understand it.
– rek
Jan 26, 2021 at 16:23

In the language of the divine, each symbol has one and only one meaning.

Humans like the play fast and loose with words. The symbol for fire might come to mean "bright" (because fires are bright) or "burn" (because fires inflict burns) or "that feeling in the middle of autumn when you're sitting at a bonfire remembering how you used to do this as a kid", and then you check back in two hundred years and the last meaning has mutated into "reminisce" and nobody knows why. But that's only the word in human languages.

But in the divine language - the language of magic, or the gods, or the spirits, or whatever natural force is behind this phenomenon - the word has one and only one meaning. It will always mean fire, whether humans mean that or not. (Or even whether humans remember that it can mean fire or not...)

This may mean that expressing yourself precisely in the language of the divine is difficult or counterintuitive; magic often is. That's why you have dedicated runescribes rather than everyone's writing coming to life willy-nilly. It's why runescribes have to [do something that furthers your plot] rather than [do something else that completely ruins your plot], because it is difficult or impossible to precisely describe the latter in runes. It's why discovering a new rune, or one forgotten to time, is a huge deal (if you wish it to be).

How many runes are there and how many meaning per rune? EIther group runes together, or have enough of them to fomr a protolanguage of some sort, where the context makes grammar and disambiguates the meaning.

If there are only few meanings per rune, you can use diacritics to disambiguate.

In modern Pǔtōnghuà, you could, if you really want to, kind of get away with 800 runes and words of mostly two runes stuck together. In her predecessor Wényánwén you can have one rune = one meaning, however each rune is composed of a pronunciation part and the "core meaning" part, with about 200 of the latter (and you have many thousands of such runes).

Or use a special (determinative) runes to hint the meaning (as in Sumerian).

Eventually, if there are just few runes and many meaning depending on their combination, you will reinvent the alphabet.


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Search for heavy resonances decaying to WW, WZ, or WH boson pairs in the lepton plus merged jet final state in proton-proton collisions at $\sqrt{s}$ = 13 TeV

The collaboration
CMS-B2G-19-002, 2021.

Abstract
A search for new heavy resonances decaying to pairs of bosons (WW, WZ, or WH) is presented. The analysis uses data from proton-proton collisions collected with the CMS detector at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 137 fb$^{-1}$. One of the bosons is required to be a W boson decaying to an electron or muon and a neutrino, while the other boson is required to be reconstructed as a single jet with mass and substructure compatible with a quark pair from a W, Z, or Higgs boson decay. The search is performed in the resonance mass range between 1.0 and 4.5 TeV and includes a specific search for resonances produced via vector boson fusion. The signal is extracted using a two-dimensional maximum likelihood fit to the jet mass and the diboson invariant mass distributions. No significant excess is observed above the estimated background. Model-independent upper limits on the production cross sections of spin-0, spin-1, and spin-2 heavy resonances are derived as functions of the resonance mass and are interpreted in the context of bulk radion, heavy vector triplet, and bulk graviton models. The reported bounds are the most stringent to date.

format: plain


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1. Algebraic Expressions Sin(arctan x/3)

I want to make sure that i am right

let U= arctan x/3

tan u=x/3

i think this is correct

2. Originally Posted by NoAsherelol
I want to make sure that i am right

let U= arctan x/3

tan u=x/3

i think this is correct
yes


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# One Way Anova in SAS

One way ANOVA is a parametric test. It compares the means of two or more independent groups to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups.

This article will provide a brief introduction to the one way ANOVA in SAS, including the test’s assumptions and when you should use this test.

## What is Anova and why do we need it?

In statistical analysis when the objective is to test the means of two groups, then T-test for one sample or two samples is used, but when the researcher deals with the samples where testing of more than two groups is required simultaneously, the analysis of variance is used.

T-testing may be used to obtain the findings for the same scenario. Still, running T-tests on paired samples is laborious and time-consuming. Another downside of performing T-tests for every pairwise sample is that the level of significance increases as the number of tests increases.

Ronald Fisher developed the analysis of variance approach in 1918 to obtain quicker findings.

The ANOVA, or analysis of variance, basically analyzes the different estimates by utilizing the F-distribution to evaluate if the population means are equal or not.

Analysis of variance is regarded as one of the most powerful techniques to perform when working with data that you can categorize into groups. You can use Excel, SAS, SPSS, Minitab, and R programming to perform variance analysis.

Analysis of variance is beneficial when there is a difference between the means of the group, indicating that it is required to consider which of the means is causing the difference. For this, numerous ad-hoc tests can be performed.

The one-way and two-way analyses of variance are generally implemented in real-world data sets.

The term “one-way Anova” refers to two levels of a single independent variable, whereas “two-way Anova” refers to more than two or more levels or groups of a two-variable.

To carry out the analysis of variance, it is essential to check for the following assumptions:

• Normality- The population should be normal or close to normal distribution.
• Independence- Samples selected from the population should be independent.
• Homogeneity of Variances- The variances deduced from the population must be the same.
• The effects are additive, which means that the Xij is the component with the observation of three components:

$X_ij = \mu + t_j + E_ij$

Where $\mu$ is the overall mean, $t_j$ is the treatment effect and $E_ij$ is the random error.

The hypothesis for the analysis of variance is:

H0: There is no significant difference between the scores of the students
Or
H1: At the smallest one, the means are not equal.

The hypothesis for the analysis of variance is:

H0: There is no significant difference between the scores of the students

Or $H_0 : \mu_1 = \mu_2 = \mu_3$

H1: At the smallest one, the means are not equal.

## ANOVA with Repeated Measure

ANOVA with the repeated measures is similar to a one-way analysis of variance in how the same groups are analyzed repeatedly under different conditions.

For example, a group of patients is being monitored for their diabetes level for medication during the previous six months. Diabetes is the dependent variable in this situation.

When the cost is low, and the same observations are being observed repeatedly, the analysis of variance with repeated measures is used.

Also, because the sample is not divided into groups, the sample size stays constant, and most significantly, when data from the same observations are gathered repeatedly over time, the odds of disparities diminish.

The assumptions that need to be followed while analyzing variances with repeated measures are:

• The Dependent variables should be of the categorical type, and the independent should consist of continuous.
• The variances of all the groups should be equal.

## One Way Anova test in SAS

To study an example of one way ANOVA in SAS, let’s consider the data set of three students of different majors with their scores. The hypothesis for this study is:

H0: There is no significant difference between the scores of the students

H1: At the smallest one, the means are not equal.

In the example below, Score is the target variable and students’ independent variable.

Level of significance=0.05

Test Statistics:

One-way ANOVA can be conducted by using the PROC ANOVA Procedure. The first step for conducting an analysis of variance is to load the data or enter the data.

Step 1: Create a Sample data

data student_scores;
input f1 f2 f3;
datalines;
17 12 13
23 23 31
25 34 25
25 29 21
26 25 20
25 24 43
18 31 23
15 32 45
21 18 12
34 15 13
42 19 27
32 12 32
;
run;

Step 2: Transform the data if required.

After creating the data set, we need to transform the data set from a single Observation to multiple observations.

data student_scores2;
set student_scores;
array f{3} f1-f3;
do ind=1 to 3;
scores=f{ind};
f_ind=cats("f", ind);
output;
end;
keep scores f_ind;
run;
proc sort data=student_scores2;
by f_ind;
run;

Step : 3  Check the Assumptions

Before carrying out the analysis, it is essential to get information about the nature of the data and the assumptions for the analysis, for ANOVA, we need to check the normality assumptions.

You can use proc Means Procedure to check the basic statistics of the dataset. The Normality can be checked by plotting a box blot

proc means data=student_scores2 mean median min max q1 q3 maxdec=2 nonobs;
class f_ind;
run;

proc sgplot data=student_scores2;
vbox scores /category=f_ind;
run;

According to the summary statistics, the variables are normal since the median is the middle value of the boxplot and the distance between the whiskers is equal. Therefore, since the most crucial assumption has been met, we can analyze variance in SAS.

The ANOVA testing for one independent variable with three groups is shown in step 4. The sum of square residuals is (SS=2618.833), while the degree of freedom is (dof= 33).

Step: 4 Conduct the ANOVA test.

SAS has a procedure called SAS PROC ANOVA, which allows us to perform an Analysis of Variance.

PROC ANOVA procedure has two statements, a CLASS statement to give a name of a categorical variable. And MODEL statement helps us to provide a structure of model or analysis.

proc ANOVA data=student_scores2;
title Example of one-way ANOVA;
class f_ind;
model scores=f_ind;
run;

The “Class Level Information” table is shown above. It lists the variables that appear in the CLASS statement, their levels, and the number of observations in the data set.

The next set of outputs displays the ANOVA table, followed by some simple statistics and tests of effects.

The PROC ANOVA output shows the F test for the analysis of variances and the P-value. According to the decision rule, if the P-value is less than the pre-assigned alpha value, we reject H0 and conclude that there are significant differences between the means of the groups.
The (P-value = 0.731) is greater than the pre-assigned alpha level (α=0.05). Hence, we accept H0 and conclude that there is no significant difference between the groups.

When ODSGraphics is enabled and you fit a one-way analysis of the variance model, the ANOVA procedure output includes a box plot of the dependent variable values within each classification level of the independent variable.

Conclusion

Analysis of variance is one of the powerful statistics techniques to test the multiple means simultaneously instead of testing each pair individually. The assumptions need to be fulfilled to carry out the analysis. It gives valid and faster results and is highly useful if the data is categorical. ANOVA provides the control over the type I error and is it the parametric results, hence providing valid results if the assumption of normality is fulfilled.

## Article Sources

1. Faraway, J. J. (2002). Practical Regression and ANOVA using R. University of Bath., (Vol. 168).
2. Girden, E. R. (1992). ANOVA: Repeated measures. Sage, No. 84).
3. Miller Jr, R. G. ( (1997). ). Beyond ANOVA: basics of applied statistics. CRC press.
4. St, L. &. (1989). Analysis of variance (ANOVA). Chemometrics and intelligent laboratory systems, 6(4), 259-272.

Every week we'll send you SAS tips and in-depth tutorials


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# Une étude des sommes fortes : isomorphismes et formes normales

Abstract : The goal of this thesis is to study the sum and the zero within two principal frameworks: type isomorphisms and the normalization of lambda-terms. Type isomorphisms have already been studied within the framework of the simply typed lambda-calculus with surjective pairing but without sums. To handle the case with sums and zero, I first restricted the study to the case of linear isomorphisms, within the framework of linear logic, which led to a remarkably simple characterization of these isomorphisms, obtained thanks to a syntactic method on proof-nets. The more general framework of intuitionistic logic corresponds to the open problem of characterizing isomorphisms in bi-cartesian closed categories. I contributed to this study by showing that there is no finite axiomatization of these isomorphisms. To achieve this, I used some results in number theory regarding Alfred Tarski's so-called high school algebra'' problem. The whole of this work brought about the problem of finding a canonical form to represent lambda-terms, with the aim of either denying the existence of an isomorphism by a case study on the form of the term, or checking their existence in the case of the very complex functions I was brought to handle. This analysis led us to give an extensional'' definition of normal form for the lambda-calculus with sums and zero, obtained by categorical methods using Grothendieck logical relations. Finally, I could obtain an intentional'' version of this result by using normalization by evaluation. By adapting the technique of type-directed partial evaluation, it is possible to produce a result in the new normal form, reducing considerably its size in the case of the type isomorphisms considered before.
Keywords :
Document type :
Theses
Domain :

Cited literature [80 references]

https://tel.archives-ouvertes.fr/tel-00007880
Contributor : Vincent Balat <>
Submitted on : Sunday, January 2, 2005 - 9:29:14 PM
Last modification on : Friday, March 27, 2020 - 2:58:55 AM
Long-term archiving on: : Friday, April 2, 2010 - 9:13:32 PM

### Identifiers

• HAL Id : tel-00007880, version 1

### Citation

Vincent Balat. Une étude des sommes fortes : isomorphismes et formes normales. Modélisation et simulation. Université Paris-Diderot - Paris VII, 2002. Français. ⟨tel-00007880⟩

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# directx – DeleteVertexShader dx8.1 to dx9 conversion

i am currently trying to convert a game to use dx9 instead of dx8. I would say that i’m quite close to completing it, but I have a few errors that I don’t exactly know how to deal with atm.

1. DeleteVertexShader and DeletePixelShader do not exist anymore in directx 9. What do I do with those? I could not find any equivalent to them in dx9 so far.
Old code example:

D3D_CHECKERROR(hr); hr = _pGfx->gl_pd3dDevice->DeletePixelShader(ulHandle);

1. Render state D3DRS_PATCHSEGMENTS does not exist anymore, it was used for the number of segments per edge when drawing patches. Do I need to replace it with something? I could not find any equivalent for this either.
Code example:

HRESULT hr = _pGfx->gl_pd3dDevice->SetRenderState( D3DRS_PATCHSEGMENTS, *((DWORD*)&fSegments));

These two issues are the ones I have the most struggles with atm, so any help would be appreciated.


text:
Matrix Scalar Product is Associative/Proof

Theorem

Let $\Bbb F$ denote one of the standard number systems.

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$.

For $\mathbf A \in \map \MM {m, n}$ and $\lambda$ \in $\Bbb F$, let $\lambda \mathbf A$ be defined as the matrix scalar product of $\lambda$ and $\mathbf A$.

The matrix scalar product is associative on $\map \MM {m, n}$, in the following sense:

For all $\mathbf A$ in $\map \MM {m, n}$ and $\lambda, \mu \in \Bbb F$:

$\lambda \paren {\mu \mathbf A} = \paren {\lambda \mu} \mathbf A$

Proof

 $\ds \lambda \paren {\mu \mathbf A}$ $=$ $\ds \lambda \paren {\mu \sqbrk a_{m n} }$ Definition of $\mathbf A$ $\ds$ $=$ $\ds \lambda \paren {\sqbrk {\mu a}_{m n} }$ Definition of Matrix Scalar Product $\ds$ $=$ $\ds \sqbrk {\lambda \paren {\mu a} }_{m n}$ Definition of Matrix Scalar Product $\ds$ $=$ $\ds \sqbrk {\paren {\lambda \mu} a}_{m n}$ Associative Law of Multiplication $\ds$ $=$ $\ds \paren {\lambda \mu} \sqbrk a_{m n}$ Definition of Matrix Scalar Product $\ds$ $=$ $\ds \paren {\lambda \mu} \mathbf A$ Definition of $\mathbf A$

$\blacksquare$


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Since I have spent a lot of time studying the flow inside the running clearances in pumps, it becomes attractive if the calculated values on a rotor surface could be drawn as contours. So I decided to write a piece of Matlab code to do the job for me. The simulation data is exported from CFX-Post.

In order to generate the contour, I utilize the patch function of Matlab. For patch to work you have to tell it the topological structure of the grid and geometry. Thus, you have to export geometry data from CFX-Post. The geometry data holds the coordinate of each node and the connections among nodes for each element face. You feed these information into patch together with the calculated values, such as pressure, on all the nodes. Then patch will give you the contour image.

A sample contour looks like this ↓:

In fact, another Matlab function, griddata, seems more suitable for this purpose. I will look into that later.

The following list is the Matlab code. And associated m-files are listed here:


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# Update:

The VGA Simulator is up and running. You can go simulate your own designs or just use the example file to see it working. I also wrote a blog post to help anyone get started using it for their own VHDL designs.

I am making a VGA simulator or virtual monitor because Xilinx takes too long to synthesize VHDL (on my pc, and in general for large projects). With this VGA simulator, I can just run my testbench which logs to a file while running in a normal simulator such as Isim and then use that log file in my VGA simulator to generate the frames.

The VGA simulator decodes hsync, vsync, and rgb (5 major signals) to generate the frames.

Is there a way to get the resolution or pixel clock rate from just the 5 major signals? Right now I just have some text fields that you have to put it in manually (see form). I need the pixel clock rate in order to keep track of what pixel the system is working on.

I am assuming that every monitor uses 640x480 at 60 Hz as a default setting for VGA because I don't have to fiddle with anything to get it working. I understand that there is some EDID pins and some newer serial interface that are spec'ed out but are they actually used to determine a different resolution or fps/Hz? On the Basys 2, they are just grounded.

My simulator is working great right now but doesn't account for the back porch so I have blank space at the top left and some cut off at the bottom right. This is because I just look for a rising edge in hsync and vsync and start capturing. This is not a huge deal and easy to fix but I am unsure of the correct value to put for the back porch in order to work with a variety of peoples log files.

My VHDL design(based off of this design) has back porch of 48 clock cycles and for example this document says it should be 45, and this one says 48. What is the correct tolerance for the back porch or the proper way to synchronize to avoid these discrepancies?

Of course a monitor is designed to take this all into account but what are the proper ways to deal with tolerance in not only the pixel clock but how many cycles each process takes (front porch, back porch, resolution, fps)?

An example of a discrepancy: My design just divides the 50 Mhz clock on the Basys 2 by two to get a 25 Mhz pixel clock to drive the VGA when it is spec'ed at 25.175hz everywhere but of course you have to put your pixel clock in the form so it is non-issue at the moment for my design.

Here is just a nice diagram of timing of hsync and vsync (just a good reference):

• It may help to recall that back in the day, video was produced by steering an electron beam. Sync pulses were roughly those intervals in which the beam could be snapped back to the left (HS) or top (VS). The 'porches' are nothing more than some margin to allow the beam to get back into the visible display area. Porch times basically adjust the centering of the image. Get the overall line time right, and you can divvy it up between the porches and the active pixels more or less however you like. – JustJeff Dec 7 '13 at 4:00
• @JustJeff I knew that a CRT worked with a electron beam and scanning it across the screen but it did not quite click that the signal I am sending to the screen explicitly controls the analog circuitry in the monitor instead of it being interpreted and beamed out (the beam just being the technology running the screen instead of a lcd). When you change the position option on the monitor you essentially modify the back porch values and I have added a couple of range inputs so people can align their signal. – MLM Dec 7 '13 at 6:17
• @JustJeff: Although VGA is usually output using separate wires for the for sync signals, it was hardly uncommon to feed it into monitors that expect sync to be superimposed on the "green" signal. Extracting sync from a composite signal requires that the signal be at a consistent stable voltage before the sync pulse; the front porch ensures that's the case. If VGA did not include a front porch, then VGA-to-3-wire adapters would have had to delay horizontal sync in order to add one. – supercat Dec 7 '13 at 23:24
• @JustJeff: The "analog" behavior of monitors makes it possible for monitors (especially monochrome ones) to show a variety of resolutions "perfectly", unlike LCD screens which have a particular fixed resolution and have to scale any other resolution up or down. – supercat Dec 7 '13 at 23:30
• @supercat - just trying to lend some perspective to the situation. Often in electronics we find seemingly strange interfaces that preserve odd bits of OBE technology. Thanks for expanding on this. – JustJeff Dec 8 '13 at 15:33


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# Which process is depicted in the given diagram?

This question was previously asked in
UPPCL JE EC 25 March 2021 Official Paper (Shift 1)
View all UPPCL JE Papers >
1. Schottky emission
2. Thermionic emission
3. Photo-electric emission
4. Photo-excited emission

Option 3 : Photo-electric emission
Free
Free Mock Test Uttar Pradesh (History)
78.9 K Users
10 Questions 10 Marks 7 Mins

## Detailed Solution

When the beam of the light enlightens the surface of the metals like sodium, potassium, caesium and rubidium then their electrons start emitting. The phenomenon of emission of electrons is known as the photoelectric emission. The photoelectric emission is the process through which the free electrons are liberated from the surface of metals when it absorbs light. The photoelectric emission depends on the frequency of light and not on the intensity of light.

In the given above figure we can see  the phenomenon of photoelectric emission. The emitter and the anode are enclosed in the evacuated glass tube . The anode is made of thin metal and is kept at the positive potential concerning the emitter. When the light incident on the surface of metal then the electrons start ejected from the emitter surface. The electrons moved towards the positive anode and established the current in the external circuit. The flow of electrons is maintained till the illumination is continue.

2. In photoelectric emission, electron from the surface of metal can emit only when sufficient amount of energy incident on metal surface i.e

hf ≥ e ϕ

$$f \ge \frac{{e\phi }}{h}$$

Where hf = energy of photons.

ϕ = work function of metal surface.


text:
Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sucidALTVD Unicode version

Theorem sucidALTVD 32449
Description: A set belongs to its successor. Alternate proof of sucid 4915. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sucidALT 32450 is sucidALTVD 32449 without virtual deductions and was automatically derived from sucidALTVD 32449. This proof illustrates that completeusersproof.cmd will generate a Metamath proof from any User's Proof which is "conventional" in the sense that no step is a virtual deduction, provided that all necessary unification theorems and transformation deductions are in set.mm. completeusersproof.cmd automatically converts such a conventional proof into a Virtual Deduction proof for which each step happens to be a 0-virtual hypothesis virtual deduction. The user does not need to search for reference theorem labels or deduction labels nor does he(she) need to use theorems and deductions which unify with reference theorems and deductions in set.mm. All that is necessary is that each theorem or deduction of the User's Proof unifies with some reference theorem or deduction in set.mm or is a semantic variation of some theorem or deduction which unifies with some reference theorem or deduction in set.mm. The definition of "semantic variation" has not been precisely defined. If it is obvious that a theorem or deduction has the same meaning as another theorem or deduction, then it is a semantic variation of the latter theorem or deduction. For example, step 4 of the User's Proof is a semantic variation of the definition (axiom) , which unifies with df-suc 4842, a reference definition (axiom) in set.mm. Also, a theorem or deduction is said to be a semantic variation of another theorem or deduction if it is obvious upon cursory inspection that it has the same meaning as a weaker form of the latter theorem or deduction. For example, the deduction infers is a semantic variation of the theorem (OrdA<->(TrA/\A.xe.A A.ye.A(xe.y\/x=y\/ye.x))), which unifies with the set.mm reference definition (axiom) dford2 7963.
 h1:: |-Ae. 2:1: |-Ae.{A} 3:2: |-Ae.({A}u.A) 4:: |-sucA=({A}u.A) qed:3,4: |-Ae.sucA
(Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sucidALTVD.1
Assertion
Ref Expression
sucidALTVD

Proof of Theorem sucidALTVD
StepHypRef Expression
1 sucidALTVD.1 . . . 4
21snid 4021 . . 3
3 elun1 3637 . . 3
42, 3e0a 32348 . 2
5 df-suc 4842 . . 3
65equncomi 3616 . 2
74, 6eleqtrri 2541 1
 Colors of variables: wff setvar class Syntax hints:  e.wcel 1758   cvv 3081  u.cun 3440  {csn 3993  succsuc 4838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3083  df-un 3447  df-in 3449  df-ss 3456  df-sn 3994  df-suc 4842
 Copyright terms: Public domain W3C validator


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# Conformally invariant operators

(Difference between revisions)
 Revision as of 03:05, 24 September 2013 (view source)Tianling (Talk | contribs)← Older edit Latest revision as of 05:04, 24 September 2013 (view source)Tianling (Talk | contribs) Line 29: Line 29: * GJMS operators : this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl (see also Fefferman-Graham ). The formula are more explicit when they are on the standard spheres. * GJMS operators : this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl (see also Fefferman-Graham ). The formula are more explicit when they are on the standard spheres. - *Scattering operators , or the conformally invariant fractional powers of the Laplacian : This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. The authors reconciled the way of defining $P_\sigma$ in and the localization method of Caffarelli-Silvestre  for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$. + *Scattering operators , or the conformally invariant fractional powers of the Laplacian : This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. Chang and González reconciled the way of defining $P_\sigma$ in Graham-Zworski and the localization method of Caffarelli-Silvestre  for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.

## Latest revision as of 05:04, 24 September 2013

On a general compact Riemannian manifold $M$ with metric $g$, a metrically defined operator $A$ is said to be conformally invariant if under the conformal change in the metric $g_w=e^{2w}g$, the pair of the corresponding operators $A_w$ and $A$ are related by $A_w(\varphi)=e^{-bw} A(e^{aw}\varphi)\quad\mbox{for all }\varphi \in C^{\infty}(M),$ where $a, b$ are constant.

Examples of conformally invariant operators include:

• The conformal Laplacian:

$L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g,$ where $n$ is the dimension of the manifold, $-\Delta_g$ is the Laplace–Beltrami operator of $g$, and $R_g$ is the scalar curvature of $g$. This is a second order differential operator. One can check that in this case, $a=\frac{n-2}{2}$ and $b=\frac{n+2}{2}$.

• The Paneitz operator [1] [2]:

$P=(-\Delta_g)^2-\mbox{div}_g (a_n R_g g+b_n Ric_g)d+\frac{n-4}{2}Q,$ where $\mbox{div}_g$ is the divergence operator, $d$ is the differential operator, $Ric_g$ is the Ricci tensor, $Q=c_n|Ric_g|^2+d_nR_g^2-\frac{1}{2(n-2)}\Delta_gR$ and $a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, b_n=-\frac{4}{n-2}, c_n=-\frac{2}{(n-2)^2}, d_n=\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}.$ This is a fourth order operator with leading term $(-\Delta_g)^2$.

• GJMS operators [3]: this is a family of conformally invariant differential operators with leading term $(-\Delta_g)^k$ for all integers $k$ if $n$ is odd, and for $k\in \{1,2,\cdots,\frac{n}{2}\}$ if $n$ is even. A nonexistence result can be found in [4] for $k>\frac n2$ and $n\ge 4$ even. An explicit formula and a recursive formula each for GJMS operators and Q-curvatures have been found by Juhl [5][6] (see also Fefferman-Graham[7] ). The formula are more explicit when they are on the standard spheres.
• Scattering operators [8], or the conformally invariant fractional powers of the Laplacian [9]: This is a family of conformally invariant pseudo-differential operators $P_\sigma$ defined on the conformal infinity of asymptotically hyperbolic manifolds with leading term $(-\Delta_g)^\sigma$ for all real numbers $\sigma\in (0,\frac n2)$ except at most finite values. Chang and González [9] reconciled the way of defining $P_\sigma$ in Graham-Zworski [8] and the localization method of Caffarelli-Silvestre [10] for the fractional Laplacian $(-\Delta)^\sigma$ in the Euclidean space $\mathbb{R}^n$.

Special cases:

• On the Euclidean space $\mathbb{R}^n$: the operators mentioned above are just the fractional Laplacians.
• On the standard sphere $(\mathbb{S}^n, g_{\mathbb{S}^n})$ (which is the conformal infinity of the standar Poincare disk): they are the following intertwining operator [11] of explicit formula:

$P_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\mathbb{S}^n}}+\left(\frac{n-1}{2}\right)^2},$ where $\Gamma$ is the Gamma function and $\Delta_{g_{\mathbb{S}^n}}$ is the Laplace-Beltrami operator on $(\mathbb{S}^n, g_{\mathbb{S}^n})$. Moreover, the operator $P_{\sigma}$

• is the pull back of $(-\Delta)^{\sigma}$ under stereographic projections,
• has the eigenfunctions of spherical harmonics, and
• is the inverse of a spherical Riesz potential.

## References

1. Paneitz, S (1983), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds", preprint
2. Paneitz, S (2008), "A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary)", SIGMA Symmetry Integrability Geom. Methods Appl. (4)
3. Graham, C Robin; Jenne, Ralph; Mason, Lionel J; Sparling, George AJ (1992), "Conformally invariant powers of the Laplacian, I: Existence", Journal of the London Mathematical Society 2: 557--565
4. Gover, A; Hirachi, Kengo (2004), "Conformally invariant powers of the Laplacian—a complete nonexistence theorem", Journal of the American Mathematical Society 17: 389--405
5. Juhl, Andreas, "On the recursive structure of Branson’s Q-curvature", arXiv preprint arXiv:1004.1784
6. Juhl, Andreas (2013), "Explicit formulas for GJMS-operators and Q-curvatures", Geometric and Functional Analysis 23: 1278--1370
7. Fefferman, Charles; Graham, C (2013), "Juhl’s formulae for GJMS operators and 𝑄-curvatures", Journal of the American Mathematical Society 26: 1191--1207
8. 8.0 8.1 Graham, C Robin; Zworski, Maciej (2003), "Scattering matrix in conformal geometry", Inventiones mathematicae 152: 89--118
9. 9.0 9.1 Chang, Sun-Yung Alice; González, Maria del Mar (2011), "Fractional Laplacian in conformal geometry", Advances in Mathematics 226: 1410--1432
10. Caffarelli, Luis; Silvestre, Luis (2007), "An extension problem related to the fractional Laplacian", Communications in Partial Differential Equations 32: 1245--1260
11. Branson, Thomas P (1987), "Group representations arising from Lorentz conformal geometry", Journal of functional analysis 74: 199--291


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# Anyone ever use these W14s and what did you use them for?4

## Anyone ever use these W14s and what did you use them for?

(OP)
Saw these at a steel supplier.

### RE: Anyone ever use these W14s and what did you use them for?

Not quite that big, but have done W14s in the 300-400 lb/ft range for chord members of long span transfer trusses.

Edit: Actually remember another job that had a W14x500, both as a chord of a transfer truss and then for some of the columns supporting the transfer.

### RE: Anyone ever use these W14s and what did you use them for?

Columns to support the boiler at electrical generating stations. The boiler hangs from the top of the structural steel so that thermal expansion causes it to "grow" downward (several inches) at normal operating temperatures. Since the boiler structure may be 250 feet high, or so, these columns have to support astounding loads.

### RE: Anyone ever use these W14s and what did you use them for?

2
Largest I've ever used was a W36x720.
Top chord of a 378 ft. long bowstring truss which used three, 4 inch diameter bridge strand for the bottom chord.

The flanges were 3.9 inches thick with a 2.165 inch web. Got them from Belgium I believe.

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### RE: Anyone ever use these W14s and what did you use them for?

I think the larger W14's tend to be generally used for columns.

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### RE: Anyone ever use these W14s and what did you use them for?

Guardrail stanchions.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

### RE: Anyone ever use these W14s and what did you use them for?

Columns in high rise buildings.

### RE: Anyone ever use these W14s and what did you use them for?

(OP)
The largest column reaction I've seen on plans was 3,000,000 lbs. What's the largest axial reaction you've seen?

### RE: Anyone ever use these W14s and what did you use them for?

I saw an interesting one at a power plant in New Mexico 40 years ago where A W14x320 (or thereabouts) was used as a core section and then had approximately 20 x 3" cover plates welded to both flanges. As noted above, this was the supporting steel for the boiler structure.

### RE: Anyone ever use these W14s and what did you use them for?

I've seen them used as columns, though not on any job that I've worked on.

### RE: Anyone ever use these W14s and what did you use them for?

I've seen some big ones as truss chords at the local NHL arena. W14x455 Top chords, W14x311 bottom chords. Span around 280 feet.

### RE: Anyone ever use these W14s and what did you use them for?

If you look at the AISC manual you will see that W14's come in a wide range of sizes, from x22 all the way up to x730. No other shape has nearly that wide a range of profiles. This was clearly done* for use as columns in high rise buildings, as Hokie66 pointed out.

Dlclarkii, were those salvaged from the World Trade Center buildings, per chance? I find it unlikely that a fabricator would "just so happen" to have such sizes lying around. In preparation for a new job or left over from an old job gone bust, yes, but not as part of his general inventory unless he's intent on going out of business.

*EDIT: perhaps a better way to have worded this would have been: It seems to me that this was likely done to meet the requirements of high rise buildings and other structures that require members capable of taking extremely large compression members. W14 lends itself to being able to be rolled to a very large cross sectional area while keeping ry as large as possible. It is an "efficient" shape in that regard.

### RE: Anyone ever use these W14s and what did you use them for?

(OP)
To my knowledge, they're drops, left over pieces from a previous order. A customer ordered a specific length but didn't want the drop. So the supplier has these large beams stored incase someone else may need a W14 of this size.

edit: Which is stupid, when you buy a beam you're paying for the stock length, not just the length you want. So someone paid for these full length pieces plus cutting charges but didn't want the drop. At 60¢/lb that's $300/ft x 35' is a$12,000 beam! Granted I probably wouldn't want these drops if I were a fabricator and have to find storage in my shop. Just moving them around would be a bear at 600lb/ft. Must be a government job with no budget in mind. Would hurt if one of these rolled on your foot, steel toes or not!

### RE: Anyone ever use these W14s and what did you use them for?

Used similar size for moment frame columns, at one point used a larger size. Project now has a more "reasonable" size, W14x455.

### RE: Anyone ever use these W14s and what did you use them for?

Ah, it didn't dawn on me that they could be left over pieces. I wonder what's involved in cutting something like that?

### RE: Anyone ever use these W14s and what did you use them for?

Those are not normal crops, if they are crops or drops. No fab shop or stl. supplier in their right mind would mill order something like those members, 35' too long, and then assume they could put them out in stock, to be used or sold later. There is just no ready market or likely future use for something like that, during your lifetime. There must have been a major change or redesign in the project which caused something like that, or their purchasing guy shot a real bull. Our purchasing people had a handle on std. sizes, what we used and how quickly it turned over; plus all of the lengths of pieces of stock out in the warehouse. They would order std. sizes in 60' lengths, or max. lengths for a given size, a mixed lot, by the railcar load. Then we would just restock reasonable length crops for future use. But, something like those W14's, or other unusual sizes where ordered so all we had to do was clean up the ends to final length, with min. waste. For large jobs they would mill order most of the stl. in large lots from various mills, and that material was pretty well kept separate for that particular job.

We had a circle saw with about a 6' dia. blade, and hundreds of machine tool teeth, which literally machined (took small chips, like a milling machine might) through stl. members like that. That saw had about a 5/16 or 3/8" kerf width. With that saw we didn’t have to mill the ends for bearing when we were done cutting them to length. We also had a band saw type cutting system which would have worked. Then we had and end milling machine to dress the ends square and flat for end bearing.

### RE: Anyone ever use these W14s and what did you use them for?

Cut them in to smaller lengths and use them with a glass top for a nifty coffee table.

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### RE: Anyone ever use these W14s and what did you use them for?

you'd have to cut them into one 6" length just to be able to move the sucker.

### RE: Anyone ever use these W14s and what did you use them for?

#### Quote (KootK)

Guardrail stanchions.

Would not want to hit that!

### RE: Anyone ever use these W14s and what did you use them for?

(OP)
I think the customer but the entire beam and only wanted a short piece and told the supplier to keep the drop. IDK for sure!

### RE: Anyone ever use these W14s and what did you use them for?

Texas toothpicks...

Mike McCann, PE, SE (WA)

### RE: Anyone ever use these W14s and what did you use them for?

What you really need to do is use W14x730s... and then cover plate them!

"We shape our buildings, thereafter they shape us." -WSC

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text:
Find time given velocity and distance

1. Feb 5, 2010

Calhoun295

1. The problem statement, all variables and given/known data

A certain vehicle requires a distance of 29 m to stop when it is traveling at 28 m/s.
(a) What magnitude of acceleration is produced by braking under these conditions?
(b) If a jaywalker is 27 m in front of the vehicle when the driver first applies the brakes, how much time does the pedestrian have to get out of the way?

2. Relevant equations

v=vo + at
x - xo = vot + $$\frac{1}{2}$$at2
v2 = vo2 + 2a(x - xo)

are the given equations for what we're learning.

3. The attempt at a solution

(a) 13.52m/s2
(b) is what I cannot find

2. Feb 5, 2010

Staff: Mentor

I'll rephrase the question: How long does it take the vehicle to cover 27 m?


text:
Saturday, 25 April 2015

Friday wrap-up: SHiP, portals...

Wherein I list some (mostly) recent happenings, ramble a bit, and provide links, in an order roughly determined by importance and relevance to particle physics. Views are my own. Content very definitely skewed by my own leanings and by papers getting coverage, and it may not even be correct. It is a blog after all...

• The SHiP fixed target experiment has uploaded its 200+ page physics case to the arXiv. Working at the intensity frontier, fixed target experiments collide very many (in this case) protons on a heavy target, with a detector placed some distance away after a significant amount of shielding. Because of the sheer number of protons on target, the setup is particularly sensitive to any ≲GeV scale extremely weakly interacting particles (large number × small number = detectable number!). The document is very comprehensive and speaks for itself; evidently the experiment has the capacity to explore some very interesting new physics scenarios, for example...

If one modestly extends the standard model with a vector or a scalar field, it is always possible to write down gauge-invariant operators$$\epsilon F^{\mu\nu}F'^{\mu\nu},\\ \xi\phi^\dagger\phi S^2,$$where $F'^{\mu\nu}$ is a dark field strength operator and $S$ is a real singlet scalar. These are known as portal operators, and in the limit of very small $\epsilon$ or $\xi$ (which restores an enhanced Poincare symmetry and is therefore technically natural) the new states (referred to often as the dark photon and dark Higgs) are very long-lived and very weakly coupled to standard model states, so that they could still have gone undetected even if their masses are sub-GeV. If dark matter couples directly to these new states then they provide a "portal" from the standard model to the dark sector.

So I was very interested to see the reach of the proposed experiment with respect to those portals; that reach is shown below as a function of mass for the case without dark matter, or with $m_{DM}>m_{A},m_S$ (the g* in the singlet case is proportional to the $\xi$ parameter above)...

It is evident that the experiment would explore a significant amount of unexplored (not grey) parameter space (and the results are even stronger for a pseudoscalar). For the dark scalar case, the reach of the experiment comes from the unprecedented (in a fixed target experiment) number of B mesons produced, which can then subsequently decay to the light scalar state at a rate of one in a million or so. The states then live long enough to travel through the (~70m of) shielding before decaying in the detector.

It is of note that unfortunately the widths and branching ratios of the scalar in the region $2m_\pi < m_S \lesssim 4$ GeV have large hadronic uncertainties, and the plot above must assume one theoretical prediction, so the story is not as clear-cut as it seems; luckily the experiment would be sensitive to many final states, and this goes some way to making the reach independent of this uncertainty. (These uncertainties do not exist for the dark photon thanks to measurements of our very own photon!). The most recent theoretical calculation for the dark scalar widths in this region is >20 years old. I wonder if lattice QCD could have something to say if it was applied to the problem?
• Scientific American has an article on self-interacting dark matter on the back of the Abell 3827 cluster "hint" from last week.
• And now that we have warmed up with Hubble here are some photos from the week...


text:
# Options

There are some options to control what exactly is going to be computed to obtain the MDDF. These options can be defined by the user and passed to the mddf function, using, for example:

options = Options(lastframe=1000)
results = mddf(trajectory,options)

### Most common options that the user might want to control are:

firstframe: Integer, first frame of the trajectory to be considered.

lastframe: Integer, last frame of the trajectory to be considered.

stride: Integer, consider every stride frames, that is, if stride=5 only one in five frames will be considered.

binstep: Real, length of the bin step of the histograms, default = 0.02 Angstroms.

dbulk: Real, distance from which the solution is to be considered as a bulk solution, that is, where the presence of the solute does not affect the structure of the solution anymore. This parameter is important in particular for systems with a single solute molecule (a protein, for example), where the density of the solvent in the box is not the bulk density of the solvent, which must be computed independently. Default: 10 Angstroms.

cutoff: Real, the maximum distance to be considered in the construction of histograms. Default: 10 Angstroms.

usecutoff: true/false: If true, the cutoff distance might be different from dbulk and the density of the solvent in bulk will be estimated from the density within dbulk and cutoff. If false, the density of the solvent is estimated from the density outside dbulk by exclusion. Default: false.

### Options that most users will probably never change:

irefatom: Integer, index of the reference atom in the solvent molecule used to compute the shell volumes and domain volumes in the Monte-Carlo volume estimates. The final rdf data is reported for this atom as well. By default, we choose the atom which is closer to the center of coordinates of the molecule, but any choice should be fine.

n_random_samples: Integer, how many samples of random molecules are generated for each solvent molecule to compute the shell volumes and random MDDF counts. Default: 10. Increase this only if you have short trajectory and want to obtain reproducible results for that short trajectory. For long trajectories (most desirable and common), this value can even be decreased to speed up the calculations.

seed: Seed for random number generator. If -1, the seed will be generated from the entropy of the system. If your results are dependent on the seed, is is probable that you do not have enough sampling. Mostly used for testing purposes. Two runs are only identical if ran with the same seed and in serial mode.

StableRNG (::Bool), defaults to false. Use a stable random number generator from the StableRNGs package, to produce identical runs on different architectures and Julia versions. Only used for testing.

nthreads: How many threads to use. By default, all available threads will be used (for multi-threading, one for managing, n-1 for computation).

lcell: Integer, the cell length of the linked-cell method (actually the cell length is cutoff/lcell). Default: 2.

sleep: Real, the time between checks between multiple spawns of calculations in parallel. Default 0.1 s.

GC: Bool, force garbage collection in parallel runs, to avoid memory overflow. Default: true. That this might be required is probably a result of something that can vastly improved in memory management. This may slow down parallel runs significantly if the GC runs too often.

GC_threshold: Float64, minimum fraction of the total memory of the system required to force a GC run. That is, if GC_threshold=0.1, which is the default, every time the free memory becomes less or equal to 10% of the total memory available, a GC run occurs.


text:
wspDesign {DiceDesign} R Documentation

## WSP algorithm

### Description

The WSP (Wooton, Sergent, Phan-Tan-Luu) algorithm is an iterative algorithm based on suppression of some experiments from an initial design in each step. WSP leads to a space filling design

### Usage

wspDesign(design, dmin, init = "center")

### Arguments

 design a matrix (or a data.frame) corresponding to the design of experiments dmin a minimum bound for mindist value of the final design init defines the initialization point (input coordinates) of the algorithm: "center" (default value) takes the central point of the domain "random" takes a random point inside the domain

### Details

WSP enables to create a design D which is such that mindist(D)>dmin. However, it cannot assess the number of experiments. Similarly to straussDesign function, WSP is a powerful algorithm to construct space filling designs in high dimension

### Value

A list containing:

 InitialDesign the starting design dmin minimum bound for mindist value of the final design design the matrix of the final design ResidualDesign the matrix of the residual design (points of InitialDesign not in design)

### Author(s)

G. Damblin & B. Iooss

### References

J. Santiago, M. Claeys-Bruno, M.Sergent (2012). Construction of space filling designs using WSP algorithm for high dimensional spaces, Chenometrics and Intelligent Laboratory Systems, 113:26-31.

### Examples

dimension <- 2
n <- 100
X <- matrix(runif(n*dimension), n, dimension)
m <- wspDesign(X, 0.1)
plot(m\$design)
xDRDN(m, letter = "T", dgts = 2, range = c(-10, 10))

[Package DiceDesign version 1.9 Index]


text:
Detecting Axion-Like Particles With Gamma Ray Telescopes

# Detecting Axion-Like Particles With Gamma Ray Telescopes

Dan Hooper and Pasquale D. Serpico Center for Particle Astrophysics, Fermi National Accelerator Laboratory, Batavia, IL 60510-0500 USA
July 15, 2019
###### Abstract

We propose that axion-like particles (ALPs) with a two-photon vertex, consistent with all astrophysical and laboratory bounds, may lead to a detectable signature in the spectra of high-energy gamma ray sources. This occurs as a result of gamma rays being converted into ALPs in the magnetic fields of efficient astrophysical accelerators according to the “Hillas criterion”, such as jets of active galactic nuclei or hot spots of radio galaxies. The discovery of such an effect is possible by GLAST in the 1-100 GeV range and by ground based gamma ray telescopes in the TeV range.

###### pacs:
98.70.Rz, 14.80.Mz
preprint: FERMILAB-PUB-07-190-A

Introduction— The Peccei-Quinn (PQ) mechanism Pec77 () remains perhaps the most compelling explanation of the CP problem of QCD. A new chiral symmetry that is spontaneously broken at some large energy scale, , would allow for the dynamical restoration of the CP symmetry in strong interactions. An inevitable consequence of this mechanism is the existence of axions, the Nambu-Goldstone bosons of  Weinberg77 (). One of the most important phenomenological properties of the hypothetical axion is its two-photon vertex which allows for axion-photon conversions in the presence of external electric or magnetic fields Dicus:1978fp () through an interaction term

 Laγ=−14gaγFμν~Fμνa=gaγE⋅Ba, (1)

where is the axion field, is the electromagnetic field-strength tensor, its dual, the electric field, and the magnetic field. The axion-photon coupling strength is quantified by

 gaγ=ξα2π1fa, (2)

where is the fine-structure constant and is a parameter of (1) depending on the details of the electromagnetic and color anomalies of the axial current associated with the axion field. In particular, this coupling is used by the ADMX experiment to search for axion dark matter Bradley:2003kg () and by the CAST experiment to search for solar axions Zioutas:2004hi (); Andriamonje:2007ew (). The Peccei-Quinn axion has the important feature that its mass and interaction strength are inversely related to each other and are connected to the measured properties of pions. One may, however, conceive of a more general class of particles whose coupling and mass are unrelated to each other. Such states are known as axion-like particles (ALPs). ALPs may manifest themselves in the propagation of photons in magnetic fields, either in laboratory or astrophysical environments, and may have potentially interesting astrophysical and cosmological consequences acconseq ().

In this letter, we propose another way to potentially detect ALPs, namely through their distortion of the energy spectra of high-energy gamma ray sources (we note however that a light scalar particle coupling to in Eq. (1) would lead to similar effects). This idea is somewhat similar to that discussed in the recent Ref. Mirizzi:2007hr (), but with some important differences. In that paper the authors considered the ALP parameters needed to fit PVLAS data Zavattini:2005tm () (as in other recently proposed gamma ray signatures of ALPs otherPVLAS ()), and assumed that the conversion of photons above TeV into ALPs takes place in the turbulent component of the galactic magnetic field. Here, in contrast, we discuss the case in which the photon-ALP conversion occurs near or within the gamma ray sources. Interestingly, we find that, if the gamma sources are (or are hosted in) efficient astrophysical accelerators according to the “Hillas criterion” Hil84 (), significant conversion can occur in ALP models which are fully consistent with all laboratory and astrophysical constraints. In fact, the mechanism discussed here may offer the most practical way to detect ALPs over a significant range of masses and couplings.

Photon-ALP conversion in gamma ray sources— As a consequence of the interaction of Eq. (1), ALPs and photons oscillate into each other in the presence of an external magnetic field. For a photon of energy , the probability of converting into an ALP can be written mixingAG ()

 Posc=sin2(2θ)sin2[gaγBs2√1+(EEγ)2], (3)

where is the size of the domain and is the magnetic field component along the polarization vector of the photon, which is assumed to be approximately constant within that domain. We have also defined an effective mixing angle and characteristic energy via

 sin2(2θ)=11+(E/Eγ)2,E≡m22gaγB, (4)

where the effective ALP mass squared is , is the plasma frequency, the electron mass, and the electron density. For the following considerations, it is useful to introduce the dimensionless quantities: , , , eV, GeV. Recent results from the CAST experiment Andriamonje:2007ew () provide a direct bound on the ALP-photon coupling of for eV, nominally below the long-standing globular cluster limit acconseq (). Note that

 ωpl=0.37×10−4μeV√ne/cm−3, (5)

which means that in the interstellar medium (ISM) of the Milky Way, where , the effective mass of the ALP will not be smaller than , independently of how small is. For ultra-light ALPs (eV), the absence of gamma rays from SN 1987A yields a stringent limit of  Brockway:1996yr () or even  Grifols:1996id (). But for eV, the CAST bound is the most general and stringent (bounds from ADMX, although stronger for some masses, assume that the axion is the Galactic dark matter).

General properties— From Eqs. (3,4) it follows that: (i) At energies below , the mixing is small and a fortiori the conversion probability is small. Above this critical energy, the mixing is large, and a significant depletion probability might arise. In suitable units,

 EGeV≡m2μeV0.4g11BG. (6)

As we shall argue, when plugging in the previous formula typical astrophysical and ALP parameters, this critical energy naturally falls in the gamma ray energy range. Thus, the physics of light and weakly coupled ALPs naturally points to -rays as the most promising tool for discovery. (ii) A significant conversion into axions also requires that the argument of the oscillatory function in Eq. (3) is not too small, i.e.

 gaγBs/2≳1,i.e.15g11BGspc>\kern-7.5pt\lower 4.3pt\hbox{∼}1. (7)

The condition in Eq. (7) depends on the product , which also determines the maximum energy to which sources can confine and thus accelerate ultra-high energy cosmic rays (UHECRs). This is known as the Hillas criterion Hil84 (), and for protons it writes

 Emax≃9.3×1020eVBGspc. (8)

This connection between cosmic ray acceleration and Eq. (7) is important. Since UHECRs with energies of a few times 10eV have been observed, environments where must exist in nature. This implies that couplings as small as might be probed, almost two orders of magnitude below present bounds.

In the following, we shall examine in greater detail what are the signatures expected in -rays due to the ALP-photon conversion mechanism, what are the most promising sources to look at, and the perspectives for current instruments to probe the ALP parameter space.

Spectral signatures— The qualitative signatures of the scenario considered here are remarkably robust, although the quantitative aspects are model dependent. The reason for this is twofold: (i) concerning particle physics, we ignore the fundamental mass and coupling parameters and ; (ii) the complicated (and unknown) 3-D field configurations typically present in astrophysical environments do not allow to apply naively Eq. (3) for detailed quantitative predictions of the magnitude of the depletion. Nonetheless, the feature one should look for can be robustly parameterized as modification of the undistorted spectrum into a modified spectrum

 Fa(Eγ)=[1−A1+(E/Eγ)2]F0(Eγ), (9)

where the constant can be obtained in an idealized case from the oscillatory sinus function in Eq. (3). As an estimate, an efficient conversion of an unpolarized photon source resulting in a complete depolarization of the photon-ALP system would cause an average depletion of 33% of the initial photon flux (i.e. ) above an energy determined roughly by Eq. (6). In Fig. 1 we plot the spectral distortion for for the two cases GeV and TeV, and for a representative high energy gamma ray source. In particular, we assumed , with and consistent around 100 GeV with the flux of the blazar object Mkn 421 as reported by the MAGIC collaboration Albert:2006jd (). This object is a powerful emitter also observed by EGRET Hartman:1999fc () at GeV energy. Note that to detect the signature expected, one needs a wide dynamical range and sufficient statistics to detect a normalization shift of the typical power-law spectra at the level of 10-20%. Overall systematic errors in the energy scale, in the aperture and in the exposure are irrelevant to such a detection. Also, in variable sources the ALP signature should be impressed over the variable spectrum at all times, since it depends on the propagation and not on the emission of the photons.) EGRET statistics and energy resolution are not sufficient to look for such a signature, and only recently ground-based gamma-ray telescopes have reached comparable performances for powerful emitters within reasonable exposure times. So, it would not be a surprise if such a signature had escaped detection so far, but would show up in the coming years thanks to the GLAST satellite detector and present and planned ground telescopes. In principle, if the astrophysical parameters were known, the amplitude of the depletion could be used to constrain , see Eqs. (3,7), while the energy at which the effect is observed could be used to infer , see Eq. (6). If only upper limits were available for , a lower limit on could be obtained, at least. Another prediction is that if a hint for an ALP would show up in a source at energy , then gamma emitters sitting in regions with similar values of should show a feature of similar amplitude at a characteristic energy related to only by the value of their field strength, see Eq. (6).

Promising sources— To be consistent with existing bounds Andriamonje:2007ew (), an ALP should have a coupling , which implies that must hold at the source. In Fig. 2, the Hillas diagram is shown, reporting the typical and values for UHECR candidate sources. It is clear that virtually all the objects proposed as UHECR accelerators, from gamma-ray bursts to clusters of galaxies, appear suitable for the search of ALP signatures. This is fortunate, since many observed (e.g. blazars) or expected (e.g. galaxy clusters) gamma ray sources are hosted in or near putative UHECR accelerators. However, at least the compact sources on the Hillas plot are not likely the best candidates due to their higher densities. Even a density of g/cm, very low for terrestrial standards, would imply meV [see Eq. (5)] and thus the condition cannot be satisfied in the energy range probed by gamma ray astronomy. This is a general, although qualitative, argument disfavoring too compact (and presumably dense) sources as possible sites to observe photon-ALP mixing. Very promising sources are instead AGN jets and hot spots in radio galaxies such as Cygnus A and M87. For example, typical parameters for the hot spots of Cygnus A are ,  Wilson:2000zu (), and similar numbers apply to the hot spots of M87 Stawarz:2005wh (), which has been detected in the TeV range. In these environments, the quantity of Eq. (7) is near unity for , while for propagation in our galaxy the same coupling would fail to satisfy that condition by more than one order of magnitude.

A remark is in order. If it were proved that conservative estimates for the product of a detected -ray emitter satisfy Eq. (7) for , then gamma observations would turn into powerful probes of ALP physics. But vice versa is not necessarily true: indeed, although some fits assuming synchrotron-self-Compton models seem to indicate that many detected gamma ray sources (see e.g. Refs. Albert:2006jd (); othergamma ()) fall short of the requirement of Eq. (7) by one order of magnitude or more, it is important to remember that the ALP conversion feature depends on the properties of the environment crossed, not of the emitting region. In these cases, although a negative result can not be used to put significant bounds, a serendipitous discovery is by no means excluded.

Exploring the ALP parameter space— One can easily estimate the range of ALP parameters observationally accessible. As we argued earlier, from the highest energy UHECR observed the Hillas criterion suggests conservatively that sites where must exist in nature. Once plugged into Eq. (7) (assuming equality), this implies that at least couplings as small as may produce significant depletions in gamma-ray spectra. To deduce the range of masses which can be probed, we proceed as follows: (i) we neglect too compact objects on the Hillas plot and restrict our attention to the most promising range of astrophysical source sizes previously discussed, to , deducing the corresponding field strength (see Fig. 2); (ii) we plug these values in Eq.  (6,) thus obtaining Unless eV and the region is very compact (which is disfavored, as previously discussed), the transition energy is expected to fall in the gamma ray band, confirming again our initial, general remarks. Considering the energy range most interesting for GLAST (sub-GeV to a few tens of GeV), we deduce that ALPs in the mass range can be probed. For ground based gamma ray telescopes such as HESS, MAGIC and VERITAS, which are sensitive to gamma rays in the approximate range of to GeV, the photon-ALP conversion can also be significant for masses in the range of to . Globally, we estimate the approximate range of parameters which could lead to observable effects as the region schematically shown in Fig. 3, along with the range excluded by CAST and the band preferred by QCD axion models Andriamonje:2007ew (). The particularly interesting region which overlaps with the QCD models band corresponds to TeV transition energies with typical parameters of AGN cores.

Summary— Space and ground-based gamma ray telescopes may have a chance to observe the effects of ALP-like particles (ALPs) through the mechanism of photon-ALP mixing. This mechanism leads to the depletion of gamma rays at high energies, resulting in a peculiar signature in the spectra of gamma ray sources such as the jets of active galactic nuclei, the hot spots of radio galaxies, or clusters of galaxies. If the astrophysical parameters at the sources were known sufficiently well, the mass and coupling of the ALP could be reconstructed or, in case of negative outcome, excluded. Without a detailed knowledge of the field strength and geometry around the gamma source a detection is still possible, but a negative result cannot easily be translated into interesting exclusion plots in the ALP parameter space. We believe that the mechanism discussed here should be thought of as an example of the opportunities that the new generation of gamma ray telescopes will offer for studying fundamental physics. Note that gamma ray telescopes may observe the effects of ALPs with parameters which are exceedingly difficult to explore otherwise, as shown in Fig. 3. An appearance experiment such as CAST is actually sensitive to , and thus would require an improvement of five to six orders of magnitude in sensitivity to cover the entire range of parameters considered here. However, a detection of the kind described here could be confirmed, in part of the parameter space, by other astrophysical techniques, such as that suggested in Ref. Davoudiasl:2005nh ().

We would like to thank Tom Weiler for early discussions and comments and G. Raffelt for comments on the manuscript. This work was supported by the DOE and NASA grant NAG5-10842. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the US Department of Energy.

## References

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text:
# 2 Prime Numbers Problem

Number Theory Level 2

If $$p$$ and $$q$$ are prime numbers which satisfy the equation

$p+q = (p-q)^3,$

what is the value of $$p - q + 2pq$$?

×


text:
+86 0371 8654 9132

## description of kernel

##### What is Kernel? Definition from Techopedia

2 天前 · A kernel is the core component of an operating system. Using interprocess communication and system calls, it acts as a bridge between applications and the data processing performed at the hardware level. When an operating system is loaded into memory, the kernel loads first and remains in memory until the operating system is shut down again.

##### Geometric description of a kernel Physics Forums

2007-11-12 · For a general linear transformation, about the most you can say are that the kernel and range are both subspaces (if they are not trivial). For different the kernel might be {(0,0,0)}, a line through (0,0,0), a plane through (0,0,0), or all of R 3. Same for the range.

##### The Linux Kernel documentation — The Linux Kernel

2021-5-30 · The Linux Kernel documentation¶. This is the top level of the kernel’s documentation tree. Kernel documentation, like the kernel itself, is very much a work in progress; that is especially true as we work to integrate our many scattered documents into a coherent whole.

##### Geometric description of kernel Physics Forums

2008-3-6 · T is the projection onto the xy-coordinate plane: T(x,y,z)=(x,y,0) I have to give a geometric description of the kernel and range of T. my geometric description of the kernel: a line along the z-axis. Is this correct? whats the geometric description of the range of T?

##### linux Description of kernel.printk values Unix &

2021-4-7 · 6. Description of kernel.printk values. "0" → Emergency messages, system is about to crash or is unstable pr_emerg. "1" → Something bad happened and action must be taken immediately pr_alert. "2" → A critical condition occurred like a serious hardware/software failure pr_crit.

##### Kernel of a Linear Transformation

2017-4-3 · same field $$\mathbb{F}$$. The kernelof $$T$$, denoted by $$\ker(T)$$, is the set of vectors from $$V$$ that gets mapped to the zero vector in $$W$$; that is, $\ker(T) = \{ v \in V : T(v) = 0_W\}.$

##### An Explicit Description of the Reproducing Kernel

2006-9-25 · Abstract: Although Gaussian radial basis function (RBF) kernels are one of the most often used kernels in modern machine learning methods such as support vector machines (SVMs), little is known about the structure of their reproducing kernel Hilbert spaces (RKHSs). In this work, two distinct explicit descriptions of the RKHSs corresponding to Gaussian RBF kernels are given and some

##### description of linux kernel boot parameters 'real_root

2021-4-5 · The usual reason that happens is when the kernel doesn't have the modules it needs to mount the root file system. In that case a system called initrd is used. An initrd image is basically a small compressed file system with a few goodies like drive controller or network modules that the kernel is going to need to read the real root file system and continue booting.

##### Full Description of SAP Executable (Kernel

2015-2-4 · rsyn.bin For each kernel version of the R/3 System, there exists an appropriate file rsyn.bin which contains the ABAP/4 syntax description. It describes what to do when compiling an ABAP statement. sapccm4x CCMS Agent for Abap. sapccmsr CCMS Agent for Java

##### Linux内核文档：如何写符合 kernel-doc 规范的注释 广漠飘

2020-3-20 · 而这篇文档，就是介绍如何写符合 *kernel-doc* 格式要求的注释。. 简介 #. Linux内核使用 Sphinx 实现把 Documentation 目录下的 reStructuredText 文件转换为非常漂亮的文档。. 文档既可以通过 make htmldocs 转换成 HTML 格式，也可以通过 make pdfdocs 转换为 PDF 格式。. 转换生成的文档存放于 Documentation/output 目录下。. Linux内核强大的文档功能，除了直接转换 .rst 文档之外，还能从

##### Kernel Definition LINFO

2004-5-25 · The kernel is a program that constitutes the central core of a computer operating system. It has complete control over everything that occurs in the system. A kernel can be contrasted with a shell (such as bash, csh or ksh in Unix-like operating systems), which is the outermost part of an operating system and a program that interacts with user

##### Kernel Definition of Kernel at Dictionary

Kernel definition, the softer, usually edible part contained in the shell of a nut or the stone of a fruit. See more.

##### What is the Linux Kernel and What Does It Do?

2017-7-13 · A kernel is the lowest level of easily replaceable software that interfaces with the hardware in your computer. It is responsible for interfacing all of your applications that are running in “user mode” down to the physical hardware, and allowing processes, known as servers, to get information from each other using inter-process communication (IPC).

##### Geometric description of a kernel Physics Forums

2007-11-12 · For a general linear transformation, about the most you can say are that the kernel and range are both subspaces (if they are not trivial). For different the kernel might be {(0,0,0)}, a line through (0,0,0), a plane through (0,0,0), or all of R 3. Same for the range.

##### Geometric description of kernel Physics Forums

2008-3-6 · T is the projection onto the xy-coordinate plane: T(x,y,z)=(x,y,0) I have to give a geometric description of the kernel and range of T. my geometric description of the kernel: a line along the z-axis. Is this correct? whats the geometric description of the range of T?

##### The Linux Kernel documentation — The Linux Kernel

2021-5-30 · The kernel community is quite large, with thousands of developers contributing over the course of a year. As with any large community, knowing how things are done will make the process of getting your changes merged much easier.

##### Kernel of a Linear Transformation

2017-4-3 · The kernelof $$T$$, denoted by $$\ker(T)$$,is the set of vectors from $$V$$ that gets mapped to the zero vector in $$W$$;that is, $\ker(T) = \{ v \in V : T(v) = 0_W\}.$ Examples. Let $$T$$ be given by $$T(x) = Ax$$ for some $$A \in \mathbb{R}^{m \times n}$$.

##### description of linux kernel boot parameters 'real_root

2021-4-5 · The usual reason that happens is when the kernel doesn't have the modules it needs to mount the root file system. In that case a system called initrd is used. An initrd image is basically a small compressed file system with a few goodies like drive controller or network modules that the kernel is going to need to read the real root file system and continue booting.


text:
# Bar Graphs

Peter wanted to arrange the birthdays of his classmates according to their months.

He then wants to display the data in a way that he can visually understand them.

This is where he faces a problem.

He doesn't want to use a pictograph since it takes too much time also might be difficult.

He wants to do it in an easier method.

Can you suggest a better option for Peter?

Bar graphs can help him represent the data using rectangular bars, aiding better visualization.

In this chapter, we will learn about the types of bar graphs and double bar graphs. Check out the interactive simulation to know more about the concept of Bar Graph/ Bar Chart and try your hand at solving a few interesting practice questions at the end of the page.

## Lesson Plan

 1 What is Bar Graph? 2 Important Notes on Bar Graph 3 Solved Examples on Bar Graph 4 Tips and Tricks 5 Interactive Questions on Bar Graph

## What is Bar Graph?

Bar graph is a specific way of representing data using rectangular bars where the length of each bar is proportional to the value they represent.

Bar graphs are the pictorial representation of grouped data and are also referred to as bar charts.

Bar graphs are one of the examples of data handling.

Bar graphs are an excellent tool when representing data that are independent of one another and don’t need to be in any specific order while being represented.

The bars provide a visual display for comparing quantities in different categories.

The graphs can be plotted vertically (bars standing up) or horizontally (bars laying flat from left to right), but normally we use vertical bars.

Example:

Liza went to the market for buying different types of fruits. She then wants to display the data in a way that she can visually understand which type of fruits she buys the most.

We can start by counting each type of fruit and putting the data in a table.

Types of Fruits

Number of Fruits

Apples

5

Mangoes

3

Watermelons

2

Strawberries

3

Oranges

6

Let’s represent the data given below using a bar graph.

Now, the above bar graph shows the different types of fruits and the number of fruits bought by Liza.

From the above graph, we can easily say that orange is the type of fruit that Liza buys the most by looking.

This is the advantage of bar graphs.

### Properties of Bar Graph

• All bars must be of equal width.
• The bars can be drawn horizontally as well as vertically as per your needs.
• The height of the bar is proportional to the data they represent.
• All bars must share a common base.

## Types of Bar Graphs

To presenting the data visually, there are several different types of bar graphs we can consider.

There are different types of bar graphs and they are as follows:

1. Vertical bar graph
2. Horizontal bar graph

### 1. Vertical Graph

The most common type of bar graph is the vertical bar graph. It is very useful when representing a series of data over time.

"The bar graphs that represent the grouped data vertically are called vertical bar graphs."

The horizontal (x)-axis represents the categories and the vertical (y)-axis represents the data value for those categories.

This is how a vertical graph looks like:

### 2. Horizontal Graph

"The bar graphs that represent the grouped data horizontally are called horizontal bar graphs."

The data categories are shown on the vertical axis and the data values are shown on the horizontal axis.

This is how a horizontal graph looks like:

### Double bar graph

A double bar graph is just like a normal bar graph. The only difference is that it is used to compares 2 data groups.

In other words, a double bar graph is a graphical representation of information using two bars beside each other at various heights.

The bars can be arranged vertically or horizontally.

It is also referred to as double bar charts.

For example:

The sales of English and Science books in the years  2008, 2009, 2010, 2011 and 2012 are given below in the table:

Years

2015

2016

2017

2018

2019

English

30

35

20

25

40

Science

20

40

25

35

20

Let’s represent the data given above using a double bar graph.

Important Notes
1. If the frequency of data is very large, then bar graphs are always advisable since pictographs become time-consuming and very difficult to make.
2.  There is one disadvantage of vertical bar graphs is that they don't leave much space at the bottom of the chart if long labels are required to write.

## How to Draw a Bar Graph?

The following steps must be followed while drawing a bar graph:

Step 1: First, draw a horizontal and a vertical line( i.e. the x-axis and y-axis ).

Step 2: Sort the data collected according to the category.

Make a table to build a bar graph.

Class

Number of Girls

Class 5

30

Class 6

35

Class 7

20

Class 8

25

Class 9

40

Step 3: Choose a scale. The scale represents how much of the data is represented by a unit length of the bar.

And label the x-axis & y-axis.

Step 4: Now, draw the bar with heights corresponding to the value of the data.

### Uses of Bar Graph

A bar graph or bar chart is a pictorial representation of data. It is mostly used in maths and statistics.

Some of the uses are as follows:

• A bar graph makes comparisons between different variables easy and convenient.
• Bar graphs are also the easiest diagram to prepare. They do not require too much effort.
• Bar graphs are the most widely used method of data representation. They are used in various industries.
• Bar graphs are used to compare data sets that are independent of one another.
• They help in analyzing patterns over long periods of time.

## Make a Bar Graph

Test yourself by trying out the simulation given below:

## Solved Examples

 Example 1

The number of children in five different batches of an educational institute is given below.

i) Represent the data on a bar graph.

Batches

Number of Children

Batch 1

120

Batch 2

80

Batch 3

95

Batch 4

100

Batch 5

60

ii) Find the ratio of students of Batch 1 to the students of Batch 3

Solution

i) The data is represented by the bar graph is as follows:

ii)   Number of students in Batch 1 = 120

Number of students in Batch 3 = 95

\begin{align}\frac{\text{Number of students in Batch 1}}{\text{ Number of students in Batch 3}}&=\frac{120}{95}\\&=\frac{6}{5}\end{align}

 $$\therefore$$ The Ratio is = $${ 6: 5 }$$
 Example 2

Observe this bar graph which is showing the baking of cakes in a bakery from Monday to Saturday.

On which day were the maximum cakes were baked? How many cakes were baked on that day?

Solution

From the above graph, it is shown that

On Saturday the maximum cakes were baked.

60 cakes were baked on that day.

 $$\therefore$$ Maximum cakes on Saturday = $${ 60 \text{ cakes}}$$
 Example 3

The following table shows the number of apple trees planted by the Gardener of a school in different years.

Years

Number of apple trees

2005

150

2006

220

2007

350

2008

150

2009

300

2010

380

Draw the bar graph to represent the data.

Solution

The data is represented by the bar graph is as follows:

 Example 4

The number of cars manufactured by a factory for five consecutive weeks is shown below:

Week Number of Cars

First

600

Second

850

Third

700

Fourth

300

Fifth

900

I) Draw a bar graph to represent the data.

II) How many cars were manufactured on week third?

Solution

i) The data is represented by the bar graph is as follows:

ii)   From the above graph, it is shown that

Number of cars were manufactured on week third = 700

 $$\therefore$$ Number of cars on week third = $${ 700 \text{ cars}}$$
 Example 5

The data collected from a survey of a city is given below.

Draw a double bar graph, choosing the appropriate scale, and answer the following:

Sports

$$\text{Foot}$$ ball

$$\text{Base}$$ ball

$$\text{Basket}$$ ball

Tennis

Golf

Watching

1200

570

700

500

400

Playing

800

400

350

400

105

What is the average number of watching sports per person?

Solution

The data is represented by the double bar graph is as follows:

Average of watching sports

\begin{align}&= \frac{(1200 + 470 + 510 + 430 + 250)} {5}\\&= \frac{2860}{5}\\&= 572 \end{align}

 $$\therefore$$ Average number of watching sports$${ = 572 }$$
Tips and Tricks
1. To solve the questions on the bar graph, first, understand the data presented on the x-axis and y-axis and what's the relation between the two in terms of length bars.
2. There must be equal spacing between the bars.

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

We hope you enjoyed learning about the Bar Graph/ Bar Chart with the simulations and practice questions. Now you will be able to easily solve problems on types of bar graph, and double bar graph.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What are vertical and horizontal bar graphs?

The bar graphs that represent the grouped data vertically are called vertical bar graphs.

The data categories are shown on the horizontal axis and the data values are shown on the vertical axis.

The bar graphs that represent the grouped data horizontally are called horizontal bar graphs.

The data categories are shown on the vertical axis and the data values are shown on the horizontal axis.

## 2. What is a line graph?

A line graph is a type of chart or graph which is used to shows information when a series of data is connected by a line.

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